Cha racterization of nanopa rticles b y continuous
contrast va riation in small-angle X-ra y scattering
vo rgelegt von
M.Sc.
Raul Ga rcia Diez
geb o ren in Ba rcelona, Spanien
V on der F akultät I I - Mathematik und Naturwissenschaften
der T echnischen Universität Berlin
zur Erlangung des ak ademischen Grades
Dokto r der Naturwissenschaften
Dr. rer. nat.
genehmigte Dissertation
Promotionsausschuss:
V o rsitzender: Prof. Dr. No rb ert Esser
Gutachter: Prof. Dr. Stefan Eisebitt
Gutachterin: Prof. Dr. Simone Raoux
Gutachter: Prof. Dr. Mathias Richter
T ag der wissenschaftlichen Aussp rache: 24. Mai 2017
Berlin 2017

Abstract
In the continuously gr owing field of nanomedicine, nanoparticles have a pr e-eminent
position. The particle morphology is a defining aspect of their functionality , yet most
curr ent characterization techniques possess certain limitations. This work pr oposes a novel
appr oach to contrast variation in small-angle X-ray scattering based on the constitution of a
solvent density gradient in a glass capillary in or der to choose in situ the most appropriate
contrast and to acquir e extensive datasets in a short time interval.
By examining the scattering curves measur ed at differ ent aqueous sucr ose concentra-
tions, information about the internal structur e of the nanoparticles as well as their size
distribution is obtained. Additionally , the particle density can be estimated fr om the
Guinier r egion of the scattering curve, as is shown for polymeric colloids acr oss a wide
spectrum of polymers. These r esults are successfully compar ed with imaging methods
and other techniques such as Dif ferential Centrifugal Sedimentation.
The continuous contrast variation technique is also employed to characterize the nano-
drug Caelyx, a PEGylated liposomal formulation of doxorubicin, using iodixanol as
contrast agent, an iso-osmolar suspending medium. The mean size of the nanocarrier is
obtained by a model-fr ee analysis of the scattering curves based on the position of the
so-called isoscattering point , while the traceable determination of the particle size highlights
the advantages in comparison to widespr ead characterization techniques as Dynamic
Light Scattering and T ransmission Electr on Microscopy .
Furthermor e, the response of the nanocarrier to incr easing solvent osmolality is evalu-
ated with sucr ose contrast variation and compared to the dif ferent r esponse of PEGylated
and plain liposomes to osmotic pr essure depending on their size. Ther efore, the os-
motic pr essure necessary for the liposomal shrinkage is quantitatively studied and the
morphological changes induced by this deformation ar e thoroughly examined.
The capabilities of the continuous contrast variation method as a sizing technique
ar e further investigated on relevant bio-materials like human lipopr oteins or polymeric
nanocarriers coated with antibodies. In addition, this technique is employed to determine
the density of the lipopr oteins, one of the most characteristic traits of these blood plasma
components.

Zusammenfassung
Im kontinuierlich wachsenden Ber eich der Nanomedizin haben Nanopartikel eine heraus-
ragende Stellung. Die funktionalen Eigenschaften der Nanopartikeln wer den durch ihr e
Morphologie beeinflusst, jedoch haben die meisten gegenwärtigen Charakterisierungs-
techniken gewisse Einschränkungen. Die vorliegende Arbeit schlägt einen neuartigen
Ansatz zur Kontrastvariation in Röntgen-Kleinwinkel-Str euung ( Small-Angle X-ray Scat-
tering , SAXS) auf der Grundlage des Aufbaus eines Lösungsmitteldichtegradienten in
einer Glaskapillar e vor , um in situ den geeignetsten Kontrast zu wählen und umfangr eiche
Datensätze inner halb eines kurzen Zeitraums zu sammeln.
Informationen über die inner e Struktur von Nanopartikeln sowie deren Größenve rtei-
lung können dur ch Untersuchung der Streukurven, die bei verschiedenen Konzentra-
tionen von Zucker in W asser gemessen wer den, erhalten wer den. Zusätzlich kann die
T eilchendichte bestimmt wer den, indem der Guinier-Ber eich der Streukurven analysiert
wir d, was für polymere Nanopartikel über ein br eites Spektrum von T eilchendichten
gezeigt wir d. Diese Ergebnisse wur den erfolgreich mit mikr oskopischen und ander en
T echniken wie Sedimentation in einem Dichtegradient ( Differ ential Centrifugal Sedimentati-
on , DCS) ver glichen.
Die T echnik der kontinuierlichen Kontrastvariation wur de mit dem iso-osmolaren
Kontrastmittel Iodixanol auch an dem Nano-Arzneimittel Caelyx dur chgeführt, einer
PEGylierten liposomalen Zuber eitung des Medikaments Doxorubicin. Die mittler e Größe
des Nanocarriers wir d durch eine modellfr eie Analyse der Streukurven basier end auf
der Position der sogenannten Isoscattering-Punkte er halten, während die rückführbar e
Bestimmung der Partikelgrößen die V orteile im V er gleich zu weit verbreiteten Charakteri-
sierungstechniken wie dynamischer Lichtstr euung ( Dynamic Light Scattering , DLS) und
T ransmissionselektr onenmikroskopie (TEM) unterstr eicht.
Zusätzlich wir d die Reaktion des Nanocarriers auf eine zunehmende Lösungsmittel-
Osmolalität mittels Zucker -Konzentrationsvariation untersucht und die unterschiedlichen
Reaktionen von PEGylierten und einfachen Liposomen auf den osmotischen Druck in
Abhängigkeit ihr er Größe verglichen. Dafür wir d der für die liposomale Schrumpfung
benötigte osmotische Druck quantitativ analysiert und die dur ch diese Deformation
induzierten morphologischen V eränderungen sor gfältig untersucht.
Die Möglichkeiten der kontinuierlichen Kontrastvariationmethode als T echnik zur Grös-
senbestimmung wer den weiter anhand von relevanten Biomaterialien untersucht, wie
menschlichen Lipopr oteinen oder polymeren Nanocarriern, die mit Antikörpern beschich-
tet sind. Außer dem wird diese T echnik verwendet, um die Dichte von Lipoproteinen zu
bestimmen, eine der Haupteigenschaften dieser Blutplasmakomponenten.

Contents
1 Intro duction 1
2 Theo retical background 7
2.1 Interaction of X-rays and matter . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.1 Beer -Lambert law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.1.2 Elastic scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Small-angle X-ray scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.1 Scattering by an ensemble of particles . . . . . . . . . . . . . . . . . 13
2.2.2 The scattering curve . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Modelling of the scattering intensity: form factors . . . . . . . . . . 14
2.3 Contrast variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Isoscattering point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Basic functions appr oach . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Instrumentation and exp erimental setup fo r SAXS measurements 21
3.1 Synchr otron radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.1 Insertion devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 The BESSY II electron storage ring . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 FCM beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 UHV X-ray r eflectometer . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 SAXS setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.1 X-ray area detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.4.2 HZB SAXS instrument and W AXS configuration . . . . . . . . . . . 28
3.5 Sample environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5.1 Round capillaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5.2 Rectangular capillaries . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5.3 Cell for low-ener gies . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 Data reduction: the scattering curve . . . . . . . . . . . . . . . . . . . . . . 32
4 Continuous contrast va riation in SAXS: the densit y gradient technique 35
4.1 Experimental procedur e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.1.1 Pr eparation of the density gradient capillaries . . . . . . . . . . . . 36
vii

Contents
4.1.2 Calibration of the solvent density: X-ray transmission . . . . . . . . 37
4.1.3 SAXS measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Pr oof of principle: application to the PS-COOH particles . . . . . . . . . . 40
4.3 Results and data evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.1 Cor e-shell form factor fit . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.3.2 Isoscattering point . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.3.3 Guinier region . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3.4 Consistency of the results . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4 Applicability and comparison with other contrast variation appr oaches . . 48
4.4.1 Other possible applications of the density gradient capillary . . . . 49
5 Simultaneous size and densit y determination of p olymeric colloids 51
5.1 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1.1 Polymeric particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.1.2 Dif ferential Centrifugal Sedimentation . . . . . . . . . . . . . . . . . 53
5.2 Determination of the particle size distribution . . . . . . . . . . . . . . . . 55
5.2.1 Inter -laboratory comparison of the mean particle diameter . . . . . 57
5.2.2 Particle size distribution of the PS-Plain particles . . . . . . . . . . . 59
5.3 Considerations about scattering data evaluation . . . . . . . . . . . . . . . 60
5.3.1 Shape scattering function formalism . . . . . . . . . . . . . . . . . . 60
5.3.2 Isoscattering point appr oach . . . . . . . . . . . . . . . . . . . . . . 61
5.4 Determination of the particle mass density . . . . . . . . . . . . . . . . . . 63
5.4.1 Mass density of the PS-Plain particles: validation with DCS . . . . 64
5.4.2 Density determination of heavier polymeric colloids . . . . . . . . 65
6 Continuous contrast va riation applied to relevant bio-materials 67
6.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2 T raceable size determination of a liposomal drug . . . . . . . . . . . . . . . 69
6.2.1 Isoscattering point appr oach . . . . . . . . . . . . . . . . . . . . . . 71
6.2.2 Shape scattering function calculation . . . . . . . . . . . . . . . . . . 71
6.2.3 A verage electron density . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.3 Osmotic effects in liposomes . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.3.1 Application to drug-stabilized liposomes . . . . . . . . . . . . . . . 74
6.3.2 Does PEGylation af fect the osmotic activity of liposomes? . . . . . 76
6.4 Sizing of blood plasma componenents . . . . . . . . . . . . . . . . . . . . . 82
6.5 Pr otein-coated low-density nanoparticles . . . . . . . . . . . . . . . . . . . 85
6.5.1 Har d protein cor ona characterization with contrast variation . . . . 86
7 Summa ry 89
Bibliography 93
viii

List of Figur es
1.1 Sizing techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Depiction of the Beer-Lambert law . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Contributions to the X-ray attenuation coef ficient of water . . . . . . . . . . 9
2.3 Schematics of a scattering pr ocess and graphical definition of q . . . . . . . 10
2.4 The scattering curve and its relevant regions. . . . . . . . . . . . . . . . . . 14
2.5 Solvent contrast variation experiment and contrast matching scheme. . . . 16
2.6 Isoscattering points and particle polydispersity . . . . . . . . . . . . . . . . . 17
3.1 Scheme of the electron storage ring BESSY II. . . . . . . . . . . . . . . . . . 23
3.2 Radiant power of BESSY II. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Diagram of the four -crystal monochromator beamline. . . . . . . . . . . . 25
3.4 Scheme of the four-crystal monochromator . . . . . . . . . . . . . . . . . . . 25
3.5 Photon flux of the FCM beamline. . . . . . . . . . . . . . . . . . . . . . . . . 26
3.6
Sample-to-detector distance calibration and scattering pattern of AgBehe at
lar ge distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.7 Homogeneity of the rectangular capillaries. . . . . . . . . . . . . . . . . . . 30
3.8 X-ray transmission of a r ectangular capillary half-filled with water . . . . . 31
3.9 Sample environments for SAXS experiments in vacuum. . . . . . . . . . . 32
4.1
Scheme of the contrast variation technique in SAXS with a density gradient
capillary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Calibration of the solvent electr on density by X-ray transmission. . . . . . 38
4.3 X-ray transmittance of the density gradient capillary at dif ferent ener gies. 39
4.4
Experimental scattering curves of the PS-COOH particles for differ ent sus-
pending medium electr on densities. . . . . . . . . . . . . . . . . . . . . . . 40
4.5 Backgr ound subtraction of the scattering curves of the PS-COOH particles. 41
4.6 Cor e-shell model fit to the PS-COOH particles experimental data. . . . . . 42
4.7 Isoscattering points of the PS-COOH particles. . . . . . . . . . . . . . . . . 44
4.8
Deviation fr om the
I (
0
)
used in the evaluation of the PS-COOH particles
experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.9 Radius of gyration of the PS-COOH particles. . . . . . . . . . . . . . . . . . 46
4.10 Zero-angle intensity of the PS-COOH particles. . . . . . . . . . . . . . . . . 47
4.11 Concentration gradient of 12 nm silica particles measured at 8000 eV . . . . 50
5.1 Scheme of the dif ferential centrifugal sedimentation setup. . . . . . . . . . 53
5.2 Scattering curve of the PS-Plain particles in buffer . . . . . . . . . . . . . . . 55
ix

List of Figures
5.3 Continuous contrast variation experimental data of the PS-Plain particles. 56
5.4 Experimental shape scattering function of the PS-Plain particles. . . . . . . 57
5.5
Comparison of the PS-Plain particles average diameter with dif ferent tech-
niques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.6
Simultaneous size and density determination of the PS-Plain particles with
a DCS combined approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.7 Number -weighted size distribution of the PS-Plain particles. . . . . . . . . 59
5.8
Diameter of the PS-Plain particles obtained fr om the shape scattering func-
tion as a function of the number of scattering curves. . . . . . . . . . . . . 60
5.9
Deviation of the size of the PS-Plain particles obtained with
q ?
1
fr om the
nominal value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.10 Zero-angle intensity of the PS-Plain particles. . . . . . . . . . . . . . . . . . 63
5.11 Mass densities of three polymeric colloids measur ed with SAXS and DCS. 64
6.1 Cryo-TEM micr ograph and schematic repr esentation of Caelyx. . . . . . . 68
6.2 Continuous contrast variation experimental data of Caelyx. . . . . . . . . . 70
6.3 Shape scattering function and zer o-angle intensity of Caelyx. . . . . . . . . 72
6.4
Relationship between the solvent electr on density and the solvent osmolal-
ity for an aqueous sucr ose solution. . . . . . . . . . . . . . . . . . . . . . . . 73
6.5 Osmotic ef fects of Caelyx in an aqueous sucrose density gradient. . . . . . 74
6.6 Osmotic ef fects in the intraliposomal doxorubicin-precipitate. . . . . . . . 75
6.7 Isoscattering point position of Caelyx with dif ferent solvents. . . . . . . . 76
6.8 Scattering curves of the liposomes measured in buf fer . . . . . . . . . . . . . 77
6.9 Schematic repr esentation of UL Vs and ML Vs. . . . . . . . . . . . . . . . . . 78
6.10
Scattering curves of the liposomes measur ed at differ ent solvent osmolalities.
79
6.11 Isoscattering point intensity of two differ ent liposomes. . . . . . . . . . . . 80
6.12 Osmotic effects in the phospholipid bilayer of the liposomes. . . . . . . . . 81
6.13 Continuous contrast variation experimental data of HDL and LDL. . . . . 82
6.14 Model free-appr oaches to the experimental data of HDL and LDL. . . . . 83
6.15 Squared radius of the HDL scattering data. . . . . . . . . . . . . . . . . . . 84
6.16 Scattering curves of the PS-COOH particles coated with IgG. . . . . . . . . 85
6.17 Isoscattering point position before and after attaching IgG. . . . . . . . . . 87
x

List of T ables
3.1 T wo dif ferent SAXS experimental setups and their accessible q -range. . . . 28
4.1 Uncertainty contributions associated to the core-shell fit. . . . . . . . . . . 43
4.2 Uncertainty contributions associated to the isoscattering point position. . 44
4.3
Comparison of the r esults obtained by differ ent evaluation approaches to
contrast variation SAXS data. . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.1 Parameters of the differ ent DCS setups. . . . . . . . . . . . . . . . . . . . . 54
5.2 Isoscattering points position and their corr esponding particle diameter . . . 56
5.3 Comparison of the diameters obtained by dif ferent evaluation appr oaches. 59
6.1 Diameter of Caelyx obtained by differ ent methods. . . . . . . . . . . . . . . 71
6.2
Concentration of IgG and IgG shell thickness ar ound the PS-COOH particles.
86
xi

Symbols
a Crystal lattice constant
A Atomic mass
B Magnetic field str ength
c Speed of light in vacuum
D Characteristic length of an object
d σ / d Ω Dif ferential scattering cr oss-section
d Σ / d Ω Differ ential scattering cross-section per volume
∆ η Scattering contrast
e Electr on charge
eV Electr onvolt
E Photon’s ener gy
E c Critical ener gy of the bending magnet
e 0 V acuum permittivity
η Dynamic viscosity of a fluid
f Scattering amplitude or form factor
f 0 Scattering amplitude at the limit q → 0
f 0 , f 0 0 Real and imaginary part of the anomalous scattering coef ficient
g Size distribution function
h Planck’s constant
¯ h Reduced Planck’s constant, defined as ¯ h = h / 2 π
I Scattering intensity
I s Shape scattering function or r esonant term
k = | k | Photons’s wavenumber
K Deflection parameter of the insertion device
K B Boltzmann constant
λ Photon’s wavelength
m e Electr on mass
µ Attenuation coef ficient
n Refractive index
N Number of particles
N A A vogadr o constant
p d Polydispersity degr ee
q Momentum transfer
q ? Isoscattering point
xii

r e Classical electr on radius or Thomson radius
R Radius of the particle
R g Radius of gyration
R Mean radius of the particle size distribution
ρ Mass density
ρ 0 A verage electr on density of the particle
ρ e Electr on density
ρ solv Electr on density of the suspending medium
σ Attenuation cr oss-section
σ R Standard deviation of the particle size distribution
T T emperatur e
2 θ Scattering angle
V V olume
˜
X Intensity-weighted average of the parameter X
Z Atomic number
xiii

Abbr eviations
AFM Atomic For ce Microscopy
AgBehe Silver behenate (CH 3 (CH 2 ) 20 COO · Ag)
BESSY Berliner Elektr onenspeicherring-Gesellschaft für Synchrotr onstrahlung
COOH Carboxyl gr oup
DCS Dif ferential Centrifugal Sedimentation
DLS Dynamic Light Scattering
DO X Doxorubicin
DSPE 1,2-Distear oyl-sn-glycero-3-phosphoethanolamine
F CM Four-crystal Monochr omator
HDL High Density Lipopr otein
HSPC Hydr ogenated soy phosphatidylcholine
HZB Helmholtz-Zentrum Berlin
IgG Immunoglobulin G
LDL Low Density Lipopr otein
MAA Methacrylic acid
ML V Multilamellar vesicles
MMA Methyl methacrylate
NP Nanoparticle
NPL National Physical Laboratory
OL V Oligolamellar vesicles
PEG Polyethylene glycol
PMMA Poly(methyl methacrylate)
PS Polystyr ene
PT A Particle T racking Analysis
PTB Physikalisch-T echnische Bundesanstalt
SANS Small-angle Neutr on Scattering
SAXS Small-angle X-ray Scattering
SEM Scanning Electr on Microscopy
SI International System of Units
SPT Sodium polytungstate (3Na 2 WO 4 · 9W0 3 · H 2 0)
SSL Sterically Stabilized Liposomes
TEM T ransmission Electron Micr oscopy
TSEM T ransmission Scanning Electron Micr oscopy
TXM T ransmission X-ray Microscopy
UHV Ultra-high V acuum
UL V Unilamellar vesicles
UV Ultraviolet light
xv

The most exciting phrase to hear in science,
the one that heralds new discoveries,
is not "Eur eka!" but "That’ s funny ..."
ISAAC ASIMOV

1
Intr oduction
In 1966, Richard Fleischer dir ected Fantastic V oyage , a film about the voyage of a mini-
aturized submarine used to cruise along human blood vessels and r epair the damage
caused to the scientist’s brain by a blood clot. The idea of tr eating damaged cells or organs
fr om the inside fuelled the imagination of the next generation scientists and shaped the
incipient field of nanomedicine. Less than 30 years later , science fiction became science
fact and Doxil was appr oved by the US Food and Drug Administration in 1995 as the
first nano-drug commer cially available (Barenholz, 2012). Although 20 years after this
milestone nano-submarines ar e still a long way off, nanomedicine is a well-established
r esearch field and dozens of pr oducts are under clinical trials or have been appr oved by
the r elevant health agencies (Etheridge et al. , 2013).
The origins of the nanomedicine br eakthrough can be found in the tr emendous progr ess
in nanoparticles r esearch observed in the 60s and 70s of the last century . Nanoparticles
(NPs) ar e objects with one or more external dimensions in the size range fr om 1 nm to 100 nm
(Eur opean Commission Recommendation for nanomaterial (2011/696/EU)) and have a
pr e-eminent position in the continuously growing world of nanotechnology , employed
as paints or cosmetic pr oducts (Guterres et al. , 2007). Besides, the application of NPs in
the emer ging field of nanomedicine opens up exciting prospects (Sahoo & Labhasetwar,
2003; W ickline & Lanza, 2003; Rosen & Abribat, 2005; Nie et al. , 2007; Zhou et al. , 2014),
especially considering their possibilities as platforms for drug-delivery (W ang et al. , 2012)
or encapsulating imaging agents (T ao et al. , 2011).
The development of NPs is curr ently focused towards tailoring nano-drug carriers with
flexible surface functionalizations and contr olled morphologies (Euliss et al. , 2006; Petros
& DeSimone, 2010; Nicolas et al. , 2013). The morphology of NPs is typically specified by
parameters like size, shape, density or chemical composition of the particle, which are
fundamental and defining aspects of the particle functions and determine their suitability
in r eal-world medical applications (V ittaz et al. , 1996; Canelas et al. , 2009). In this r egard,
the size of NPs is one of the most crucial physicochemical pr operties of nano-drugs,
because it determines whether they can intrude into the biological cells or the tar geted
tumor sites. An accurate and r eliable description of the morphological traits of the NPs is
ther efore of vital importance for their favourable translation into successful nanomaterials.
1

Chapter 1 INTRODUCTION
The term nanometr ology refers to the science of accurate and corr ect measurement of
r elevant properties at the nanometr e range. A central concept in metr ology is traceability ,
which r efers to the ability of relating the measur ed value i.e. measurand to a base unit
definition of the International System of Units (SI system) by an unbr oken chain of
comparisons with known uncertainties. This allows an objective comparison of the r esults
obtained by dif ferent methods based on a consistent uncertainty budget associated to the
measurand. The fundamental r esearch in the field of metr ology in Germany is addr essed
by its national metr ology institute, the Physikalisch-T echnische Bundesanstalt (PTB).
Founded in 1887, the PTB is devoted among other metrological activities to the new
definition of units based on natural constants or the technology transfer with the industry .
At the nanoscale level, PTB is involved in the development of the dimensional nanomet-
r ology field, which studies the measurement of the physical size or distances of a given
nanomaterial and traces it back to the unit metre . Ther e are several available techniques
which ar e suitable for the sizing of NPs, though not all pr ovide a traceable measurement. A
prime example is dynamic light scattering (DLS), the most widely used tool in nanomedi-
cine (Murphy, 1997; Hallett et al. , 1991; Egelhaaf et al. , 1996; T akahashi et al. , 2008; Jans
et al. , 2009; Hoo et al. , 2008). DLS is well-established and has indisputable advantages in
the size characterization of the NPs, e.g. easy-to-use instrumentation, fast and low-cost
operation, but it is not capable of a traceable size determination as there is no general
r elationship between the measured hydr odynamic diameter and the physical size of the
NPs (Meli et al. , 2012).
Other ensemble techniques extensively used ar e differ ential centrifugal sedimentation
(DCS) (Fielding et al. , 2012) and Particle T racking Analysis (PT A), both capable of meas-
uring the NPs in suspension. While DCS is based on the sedimentation of NPs thr ough
a density gradient, PT A is a single-particle counting method that relates the Br ownian
movement of the particles with the measur ed laser light scattering. The particle size
distribution obtained with DCS is calibrated with a r eference material of known size and
density and r equires of pr ecise information about the NPs density for the calculation,
r esulting in a measurement that is har dly traceable to SI units. Similarly to DLS, the PT A
measurand derives fr om the hydrodynamic pr operties of the NPs (V ar ga et al. , 2014 b ).
Micr oscopic tools are also fr equently used for structural investigations (Joensson et al. ,
1991; Silverstein et al. , 1989), and have pr oved to be useful techniques for solid NPs
due to their SI traceability achieved by coupling the measurement table with a laser
interfer ometer (Meli et al. , 2012). Nevertheless, techniques such as transmission electr on
micr oscopy (TEM), transmission scanning electron micr oscopy (TSEM), transmission
X-ray micr oscopy (TXM) or atomic force micr oscopy (AFM) are not ensemble averaged
and the statistical accuracy of non-ensemble methods is often not suf ficient. Besides, the
r emoval of the original suspending medium can be considered another drawback, as well
as the possible distortion of the particle morphology during the drying pr ocess, though it
can be partially over come by cryo-TEM (Li et al. , 1998). A schematic r epresentation of the
available measuring range of dif ferent sizing techniques is depicted in figur e 1.1.
The nanoparticles envisioned for medical use ar e typically in the soft matter regime and
thus the characterization tools must be care fully chosen considering the measurement
limitations. For example, liposomes and biodegradable NPs, e.g. polymeric colloids, ar e
finding many medical applications, especially as drug-carriers (Kattan et al. , 1992; V icent
& Duncan, 2006) and ar e starting to undergo clinical trials (Patel et al. , 2012; Beija et al. ,
2012; Cabral & Kataoka, 2014). However , the size determination of polymeric NPs with a
well-known technique like AFM is rather challenging due to their elastic pr operties (W u
2

1.0
1 nm 10 nm 100 nm 1 µ m 10 µ m 100 µ m
1 Å
SAXS
DLS
PT A
AFM
DCS
TSEM
TXM
Pr oteins
Nanoparticles
Cells
Hair
Crystal lattice
TEM
Figure 1.1 | Some available sizing techniques fo r nanoparticles and their measuring size range.
et al. , 2014) and suggests alternative appr oaches.
Liposomes ar e spherical vesicles composed of a closed phospholipid bilayer membrane
capable of encapsulating hydr ophilic compounds. The importance of lipid vesicles in
the pr ogress of nanomedicine is indisputable, as the first appr oved nano-drug is a lipo-
somal formulation of doxorubicin, Doxil
®
(Caelyx
®
in Eur ope). Nowadays, liposomes
continue to be a widespr ead instrument for drug delivery (Pér ez-Herrer o & Fernández-
Medar de, 2015), but their complicated internal structur e requir es typically more than a
single characterization tool (Khorasani et al. , 2014). Likewise, relevant biological str uctures
in nanomedicine possess heter ogeneous morphologies which are rather dif ficult to detect
with imaging techniques (Baumstark et al. , 1990; V arga et al. , 2010). For instance, electr on
micr oscopy is an effective tool for dir ect observation of the shape and size distribution of
nanoparticles, but it cannot conclusively elucidate their inner composition.
The use of an ensemble-averaged and non-destructive technique such as small-angle
X-ray scattering (SAXS) is r evealed as an appropriate alternative (Leonar d Jr et al. , 1952;
Motzkus, 1959). This technique can discern electron density dif ferences in the str ucture
of NPs and of fers advantages over other methods which requir e prior treatment of the
sample and ar e not averaging. SAXS is based on the elastic scattering of X-ray photons by
the electr on density distribution of an object and is traceable down to the SI unit m for the
size determination of suf ficiently monodisperse NPs (Meli et al. , 2012). The traceability of
SAXS arises fr om the precise determination of the oscillation period on the momentum
transfer axis, which is calibrated using SI traceable values of the X-ray wavelength and
the scattering angle (Krumr ey et al. , 2011).
The first SAXS phenomena wer e observed in the 1930s by P . Krishnamurti and B.E.
W arr en (Krishnamurti, 1930 a , b ; W arren, 1934) while investigating colloidal suspensions
and carbon black systems. The instrumental advances introduced by Kratky (1938) and
Guinier (1937) sparked inter est in the technique, while the seminal work of Guinier (1939)
3

Chapter 1 INTRODUCTION
paved the way for the development of a SAXS theor etical background by scientists like
Kratky , P . Debye or G. Por od (Kratky & Sekora, 1943; Debye & Bueche, 1949; Kratky &
Por od, 1949; Guinier, 1950; Guinier & Fournet, 1955). Stuhrmann’s new approach to the
understanding of the scatter ed intensity (Stuhrmann & Kirste, 1965) and the appearance
of dedicated synchr otron radiation sour ces stimulated the scientific community to employ
SAXS as a characterization tool. Since then, SAXS has been extensively employed in the
characterization of polymeric colloids (Dingenouts et al. , 1999; Chu & Hsiao, 2001; Ballauf f,
2011) and its use in liposome r esearch is also ubiquitous. For instance, it has been applied
to characterize the lamellarity , bilayer thickness, area per lipid ratio (Pabst et al. , 2010;
Bouwstra et al. , 1993; Brzustowicz & Brunger, 2005) and the thickness of the PEG-layer of
dif ferent liposomal samples (V arga et al. , 2010, 2012), as well as to describe the influence
of extrusion on the average number of bilayers (Jousma et al. , 1987) and to determine the
electr on density profile of liposomes (Bouwstra et al. , 1993; Brzustowicz & Brunger, 2005;
Hirai et al. , 2003) and biological vesicles (Castorph et al. , 2010).
Despite being a highly informative method for the accurate characterization of NPs, the
interpr etation of the scattering curves in the recipr ocal space, i.e. the uniqueness of the
solution of the model fitting, is frequently intricate for complex samples (Mykhaylyk, 2012)
and can af fect the traceability of SAXS to SI units or increase the uncertainty associated to
the r esult. This demands either the application of model-free appr oaches to the scattering
data analysis or the acquisition of complementary experimental information. The solvent
contrast variation appr oach is a noteworthy candidate due to the complementary data
that can be collected at each independent contrast and the availability of extended data
evaluation possibilities.
The contrast variation method in SAXS varies systematically the electr on density of the
suspending medium by adding a suitable contrast agent, e.g. sucr ose, in order to r esolve
the dif ferent contributions of the particle components to the scattering. By measuring
SAXS patterns as a function of the adjusted contrast, a more detailed insight into the
particle morphology can be obtained in comparison to single-contrast experiments (Bolze
et al. , 2004). For instance, the internal structur e can be modelled in terms of the radial
electr on density (Dingenouts et al. , 1994 b , 1999; Ballauff, 2011; Ballauf f et al. , 1996) and
the individual contribution of each component can be distinguished (Beyer et al. , 1990;
Grunder et al. , 1991, 1993; Ottewill et al. , 1995; Bolze et al. , 1997; Dingenouts et al. , 1994 c )
as well as its density (Mykhaylyk et al. , 2007). Additionally , model-fr ee approaches like
the isoscattering point position (Kawaguchi & Hamanaka, 1992) can be applied to the
evaluation of the contrast variation data sets.
This work was performed in the PTB laboratory at the electr on storage ring BESSY
II and pr oposes a novel approach to solvent contrast variation in SAXS, based on the
formation of a solvent density gradient within a capillary which enables the acquisition
of SAXS patterns at a continuous range of contrasts, and, as a r esult, collect an extensive
data set of complementary scattering curves in a relatively short timespan. This original
strategy averts the most pr oblematic issues of the classic solvent contrast variation tech-
nique, namely the discr ete range of available solvent electron densities and the pr olonged
time r equired for the pr eparation of the complementary samples and for obtaining the
experimental data. Besides, the possibility to choose during the experiment the most ap-
pr opriate contrast within the available range allows to tune in situ the performance of the
contrast variation technique in SAXS without any a priori knowledge of the investigated
nanoparticles.
The structur e of this thesis builds organically ar ound the main concept pr esented in
4

1.0
this work, i.e. the contrast variation technique in SAXS by means of a solvent density
gradient capillary . Following this intr oduction, chapter 2 is dedicated to describe the
theor etical framework requir ed to understand the contrast variation method in small-
angle X-ray scattering. The instrumentation employed to obtain the experimental r esults
pr esented in this thesis is thoroughly described in chapter 3. These two chapters serve as
the necessary building blocks for the development of the continuous contrast variation
method based on the idea of a density gradient column. The detailed r eview of its
performance is pr esented in chapter 4, where the technique is used to characterize low-
density nanoparticles. The metrological possibilities of the newly intr oduced method
ar e further evaluated in chapter 5, mainly focusing on its ability to determine the size
and density of polymeric NPs in a traceable way . Finally , the scope of the technique is
investigated in chapter 6 by using the continuous contrast variation method in a myriad
of r elevant nanomaterials related to nanomedicine or human biology . The final chapter 7
summarizes the r esults presented in this thesis while adding some conclusive r emarks.
Extensive parts of the work pr esented in chapters 4 to 6 have been published in peer-
r eviewed journals (Minelli et al. , 2014; Garcia-Diez et al. , 2015, 2016 a , b ).
5

2
Theor etical backgr ound
In this chapter , the basic physical principles underlying the operation of small-angle
X-ray scattering ar e presented, focusing principally on the interaction between X-rays and
matter and the elastic scattering of X-rays by an ensemble of electrons. The fundamental
theor etical background of SAXS is also intr oduced, jointly with the analytical expressions
of the form factors used in this work. An entir e section is devoted to the theoretical
framework used in contrast variation experiments in SAXS, wher e concepts such as the
isoscattering point and the basic functions appr oach are intr oduced.
2.1 Interaction of X-rays and matter
X-rays ar e electromagnetic waves which pr opagate in vacuum along the direction of the
wavevector
k
. The incident X-ray radiation can be described by the wave function of a
monochr omatic plane wave:
Ψ 0 ( r ) = A 0 e i kr (2.1)
wher e the wavenumber
k = | k |
is r elated to the X-ray wavelength
λ
by
k = 2 π / λ
.
Conventionally , X-ray wavelengths range between 0.01 and a few nanometr es, although
SAXS experiments ar e conducted normally at the hard X-ray range, e.g. at wavelenghts
between 0.02 and 0.8 nm. Due to the wave-particle duality of electromagnetic radiation,
X-rays possess a particle natur e as well, repr esented by the quantization of light into
an ensemble of photons with an ener gy
¯ h ω
. The photon ener gy is related to the X-ray
wavelength by (Als-Nielsen & McMorr ow, 2011)
λ = h c
E p h
(2.2)
wher e
h
is the Planck’s constant and
c
is the speed of light in vacuum. The photon
ener gies employed typically in SAXS experiments stretch between the silicon K-edge at
1.7 keV and some dozens of keV , including the classic copper K α emission line at 8 keV .
7

Chapter 2 THEORETICAL BA CK GROUND
Figure 2.1 |
The Beer-
Lamb ert la w is schematic-
ally depicted: The atten-
uation of X-ra ys through
a medium of thickness
d
and attenuation co effi-
cient
µ
b ehaves acco rd-
ingly to the exp ression
2.3.
I 0 I = I 0 e − µ d
d
x
2.1.1 Beer -Lambert law
The interaction of X-ray photons and matter produce an attenuation of the incident
radiation intensity
I 0
which is r elated to the properties and volume of the material. The
decr ease of the intensity through a medium is schematically depicted in figur e 2.1 and
described by the Beer -Lambert law (Als-Nielsen & McMorrow, 2011):
I ( x ) = I 0 e − µ x (2.3)
wher e
µ
is the linear attenuation coef ficient and
x
is the radiation path length. The
attenuation coef ficient is dependent on the material composition and the photon energy
and is dir ectly related to the extinction coef ficient
β
, e.g. the imaginary part of the r efraction
index n , by (Marr, 1987)
µ ( E ) = 4 π
h c E β ( E ) (2.4)
Considering that the r efractive index is expressed generally by
n =
1
− δ + i β
and
δ <
10
− 3
in the X-ray r egime (Henke et al. , 1993), refraction ef fects can be neglected in
scattering experiments because < ( n ) is very close but smaller than unity .
When the attenuating medium is composed of dif ferent atomic species,
µ
can be ex-
pr essed as the summation of each component attenuation coefficient µ i :
µ = ∑
i
µ i = ∑
i
ρ i
e σ i = N A ∑
i
Z i
A i ρ i σ i (2.5)
wher e
N A
is the A vogadr o constant,
σ
is the attenuation cr oss-section and
ρ e
is the
number density of absorbing centr es. The cross-section
σ
is defined as the ef fective area
in which photon-matter events occur . In the X-ray regime, photons interact principally
with the atomic electr ons, thus
ρ e
is the electr on density and is directly pr oportional to the
atomic number
Z
, the atomic mass number
A
and the mass density
ρ
of the component
i
.
In fact, the attenuation cross-section
σ
is dependent upon the several dif ferent mech-
anisms in which a X-ray photon interacts with the atomic electr ons. The 3 most relevant
ef fects are the photoelectr on absorption, the coherent scattering and the incoher ent scatter-
ing, which sum up to the total attenuation coef ficient:
µ = ρ e ( τ abs + σ scat, coh + σ scat, incoh ) (2.6)
8

Interaction of X-ra ys and matter 2.1
10 − 6
10 − 5
10 − 4
10 − 3
10 − 2
10 − 1
10 0
10 1
10 2
10 3
10 4
1 10 100 1000
W ater attenuation / cm − 1
Photon Ener gy / keV
Photoelectron Absorption
Coherent Scattering
Incoherent Scattering Figure 2.2 |
The differ-
ent contributions to the
attenuation of w ater at
ro om temp erature a re de-
picted as a function of the
photon energy (Henk e
et al. , 1993) and the total
attenuation is the sum-
mation of all the other
contributions. The pair
p ro duction in nuclea r and
electron field can b e neg-
lected at the displa y ed
photon energies.
When the X-ray photon is completely absorbed by the atom, the event is called photo-
electr on absorption because a photoelectron with the excess ener gy is expelled from an
inner atomic shell, leaving the atom ionized. The created cor e-hole is consequently filled
by an electr on from an outer shell either by a radiative pr ocess, i.e. fluorescence , or by a
non-radiative mechanism emitting a secondary electr on, i.e. Auger effect . The photoelectric
ef fect is the predominant contribution to the attenuation cr oss-section principally at low
X-ray ener gies and the ultraviolet regime, as shown in figur e 2.2.
The other r elevant contributions in the X-ray range are r elated to scattering pr ocesses. In
an inelastic scattering event, the ener gy of the incident photon is partially transfered to a
loosely bound electr on resulting in a scatter ed photon with a longer wavelength, accor ding
to the Compton r elation
∆ λ = h / m e c ( 1 − cos 2 θ )
(Als-Nielsen & McMorr ow, 2011), where
2
θ
is the scattering angle. The Compton scattering is incoherent and contributes generally
less than the elastic scattering at ener gies below 10 keV , as observed in figur e 2.2. Besides,
the coher ent scattering signal is the summation of the constructive interfer ences of the
electr omagnetic wave, which pr oduces a higher scattering intensity than the inelastic
scattering. In fact, the elastic scattering of X-rays, typically coherent, is the main pr ocess
used in material investigations and the physical principle behind SAXS.
2.1.2 Elastic scattering
When the wavelength of the scatter ed wave is the same than that of the incident one, the
pr occess is named elastic scattering or coherent scattering and the r esulting intensity is
the absolute squar e of the sum of the scattering amplitudes. In the following sections, the
elastic scattering theory will be pr esented for the classical case and for an ensemble of
electr ons.
9

Chapter 2 THEORETICAL BA CK GROUND
Figure 2.3 |
Scheme of
an scattering event b y an
object with a p otential
function
φ ( r 0 )
at a dis-
tance
| r | = r
. A geomet-
rical definition of the mo-
mentum transfer vecto r
q
is depicted on the right
hand side, where
k
and
k s
a re the incident and
scattered w avevecto r re-
sp ectively .
r
r 0
r − r 0
φ ( r 0 )
k s q
2 θ
2 θ
Detector
k
k s
k
Thomson scattering
Classically , the elastic scattering of a photon by a free electr on is described by the con-
servation of the photon ener gy , i.e. the wavenumber of the scatter ed wave is the same
than the incident one (
| k s | = | k |
). Consequently for unpolarized incident radiation, the
intensity of the scatter ed wave at a distance
r
and with a scattering angle 2
θ
is defined by
(W arr en, 1969):
I scat ( r , θ ) = I 0  r e
r  2 1 + cos 2 2 θ
2 ! (2.7)
wher e
r e = e 2 / 4 π e 0 m e c 2 =
2.82
·
10
− 15
m is the Thomson or classical electr on radius. A
r elevant quantity in scattering processes is the dif ferential scattering cr oss-section
d σ / d Ω
,
which is dir ectly proportional to the scattering intensity
I scat
. It is defined as the the
number of scatter ed photons per time and per solid angle over the incident intensity per
time and per ar ea (Als-Nielsen & McMorrow, 2011):
d σ
d Ω = I scat ·  r 2 ∆Ω 
I 0 ∆Ω = r 2
e 1 + cos 2 2 θ
2 ! (2.8)
wher e
r 2 ∆ Ω
is the detector surface in the plane of the impact parameter . The total
Thomson scattering cr oss-section is
σ = 8 π r 2
0 / 3 =
0.665
·
10
− 24
cm
2
and similarly to
d σ / d Ω
is
pr oportional to
r 2
e
and independent fr om the photon energy if the photon wavelength is
distant of an X-ray absorption edge.
Scattering by an ensemble of electrons
The scattering of a photon by an ensemble of weakly bound electr ons can be studied by
considering the interaction of particles with a thr ee-dimensional weak potential
V ( r ) =
V 0 · φ ( r )
, wher e
V 0
is the str ength of the potential and
φ ( r )
is the so-called potential function .
The r esulting wave can be expressed as a linear combination of the incident plane wave
(see equation 2.1) and the scatter ed spherical wave at the position r :
Ψ ( r ) = Ψ 0 ( r ) + Ψ scat ( r ) (2.9)
Inserting this expr ession at the time-independent Schrödinger equation and considering
the scattering wave as a perturbation pr oduced by the scattering potential function
φ ( r )
(Cowley, 1995), it can be derived that
10

Interaction of X-ra ys and matter 2.1
Ψ scat ( r ) = C Z e i k | r − r 0 |
| r − r 0 | φ ( r 0 ) Ψ  r 0  d r 0 3 (2.10)
wher e
C
is the so-called scattering length . If the detection position
r
is at distance
much lar ger than the scattering object size, as outlined in figur e 2.3, the Fraunhofer
appr oximation applies and   r − r 0   ≃ r (Feigin & Sver gun, 1987), resulting in
Ψ scat ( r ) = C e i kr
r Z e − i kr 0 φ ( r 0 ) Ψ  r 0  d r 0 3 (2.11)
Assuming that ther e are no multiple scattering events due to the low concentration of
scatter ers and that the interaction potential is weak, the first Born approximation can be
employed ( Ψ ( r ) ≃ Ψ 0 ( r ) ) (Cowley, 1995), leading to
Ψ scat ( r ) = C A 0 e i k r
r Z e i qr 0 φ ( r 0 ) d r 0 3 (2.12)
wher e
q = k s − k
is the momentum transfer vector and
k s
the scatter ed wavevector .
Analogously to equation 2.8, the dif ferential scattering cr oss-section is:
d σ
d Ω = | Ψ scat | 2 ·  r 2 ∆ Ω 
| Ψ 0 | 2 ∆ Ω = r 2
e   f ( q )  
2 = r 2
e I ( q ) (2.13)
wher e
f ( q ) = R e i qr 0 φ ( r 0 ) d r 0 3
is the scattering amplitude,
I ( q ) =   f ( q )  
2
is the scatter -
ing intensity and the scattering length is the classical electr on radius
r e
. The scattering
amplitude f ( q ) is simply the Fourier transform of the scattering potential function φ ( r ) .
This type of scattering mechanism is named Rayleigh-Gans-Debye when the refractive
index of the object
n obj
is close to unity and the condition
2 π / λ · D ·    n med − n obj    
1 is
fulfilled, being
D
the size of the object and
n med
the r efractive index of the suspending me-
dium. For X-ray photons with wavelenghts
λ
ar ound 0.1 nm and nanoscaled objects, this
appr oximation can be applied and it can be safely assumed that the same electromagnetic
wave impinges each part of the object (van de Hulst, 1957; Barber & W ang, 1978). In the
case of optical radiation scatter ed by colloids, the Mie scattering framework is used, while
the Rayleigh scattering corr esponds to light wavelengths much larger than the scattering
object.
Anomalous scattering
In X-ray scattering experiments, the scattering centr es are the electr ons of the atom and
the scattering potential function is the electron char ge density about the nucleous, so
φ ( r ) = ρ e ( r )
. The electron density is r elated to the atomic properties as intr oduced in
equation 2.5 and ther efore the scattering amplitude incr eases with the atomic number
Z
as can be shown by calculating equation 2.13 at the limit q → 0
f ( q → 0 ) = Z ρ e ( r 0 ) d r 0 3 = Z (2.14)
This is valid when the incident photon ener gy is much larger than the ener gy corres-
ponding to a r esonant excitation. When the X-ray energy is close to an absorption edge,
the anomalous dispersion becomes relevant and the scattering amplitude depends on
the ener gy of the X-ray by adding the anomalous corrections (Als-Nielsen & McMorr ow,
2011):
11

Chapter 2 THEORETICAL BA CK GROUND
f ( E ) = f 0 + f 0 ( E ) + i f 0 0 ( E ) (2.15)
wher e the imaginary part
f 0 0
is r elated to the attenuation coefficient
µ
by (Feigin &
Sver gun, 1987)
f 0 0 ( E ) = A ρ
2 N A r e h c E µ ( E ) (2.16)
wher e A is the atomic mass of the resonant atom and ρ its mass density . The term f 0 is
r elated to the imaginary anomalous coefficient by the Kramers-Kr onig relationship (de L.
Kr onig, 1926; Kramers, 1927):
f 0 ( E ) = 2
π Z ∞
0
E 0 f 0 0 ( E 0 ) d E 0
E 2 − E 0 2 (2.17)
The values of the anomalous scattering amplitude
f ( E )
ar e usually calculated using the
experimentally measur ed attenuation coefficient µ ( E ) .
2.2 Small-angle X-ray scattering
Small-angle X-ray scattering is a powerful technique that can elucidate the structural
featur es of particles with sizes ranging from a few nanometr es up to some hundreds of
nanometr es. By investigating the photons elastically scatter ed by the electron density dis-
tribution of the particle
ρ e ( r )
, the r esulting patterns can be analysed employing equation
2.13 to obtain information about the particle size, shape and composition. T wo funda-
mental quantities in a SAXS experiment ar e the scattering intensity
I ( q )
, pr oportional to
d σ / d Ω
, and the scattering amplitude or form factor
f ( q )
. The latter is expr essed for objects
with spherical symmetry wher e ρ e ( r ) = ρ e ( r ) by
f ( q ) = 4 π Z ∞
0 r 0 2 ρ e ( r 0 ) sin ( q r 0 )
qr 0 d r 0 (2.18)
wher e the modulus of the momentum transfer vector is defined by
q =   q   = | k s − k |
.
Considering that SAXS is an elastic scattering process (
| k s | = | k | = 2 π / λ
), the momentum
transfer is expr essed as
q = 4 π
λ sin θ = 4 π E
h c sin θ , (2.19)
wher e
θ
is half of the scattering angle as depicted in figur e 2.3,
h
is the Planck constant
and c is the speed of light.
The systems studied by SAXS in this work consist of particles suspended in a uniform
medium, e.g. water or buffer , with a differ ent electron density
ρ medium
than the studied
particle. In fact, the measur ed scattering amplitude is the addition of the medium and
the particle contributions. Ther efore, the scattering of the studied object is expr essed in
terms of the contrast ,
∆ η ( r ) = ρ e ( r ) − ρ medium
, the electron density dif ference between
the particle and the embedded matrix or surr ounding medium. This leads to a slight
modification of equation 2.18, wher e
ρ e ( r )
can be substituted by the contrast
∆ η ( r )
to
distinguish the contribution of the investigated particle fr om that of the medium.
12

Small-angle X-ra y scattering 2.2
2.2.1 Scattering by an ensemble of particles
For diluted systems with low particle concentration, the wave scattered by a particle does
not interfer e coherently with the neighboring particles, hence the scattering intensity can
be expr essed as a sum of the scattering of the individual particles, i.e. the structur e factor
contribution can be neglected because
S ( q ) =
1 (Feigin & Svergun, 1987). Assuming
this pr emise, the scattering intensity of an ensemble of randomly oriented spherically
symmetric nanoparticles in a diluted suspension can be expr essed as
I ( q ) = N Z ∞
0 g ( R )   f ( q , R )  
2 d R , (2.20)
wher e
N
is the number of scatter ers i.e particles,
g ( R )
is their size distribution function
and
f ( q
,
R )
is the particle form factor , which depends on the radial structure of the
particle as determined in equation 2.18. Generally , the particles in suspension are not
monodisperse and show a certain size distribution which is often r elated with their
chemical pr eparation. For systems of relatively low size polydispersity , a gaussian size
distribution is typically a good choice, which is expr essed by:
g Gauss ( R ) = 1
σ R √ 2 π e − ( R − R ) 2
2 σ 2
R (2.21)
wher e
R
is the mean radius of the particles and
σ R
is the standar d deviation of the
size distribution. For smaller particles or higher polydispersity degrees, a log-normal
distribution is pr eferred, defined as
g LN ( R ) = 1
R σ R √ 2 π e − ( ln ( R ) − ln ( R ) ) 2
2 σ 2
R (2.22)
whose mean radius is given by
R e σ 2
R
2
and the variance is
R 2 e σ 2
R ( e σ 2
R −
1
)
. Other ap-
pr oaches to the size distribution of particles in solution are based in numerical techniques,
like the Monte-Carlo appr oach to form-free particle size distributions (Pauw et al. , 2013).
A useful parameter for comparative purposes between samples is the polydispersity
degr ee
p d
, which is defined as the full width at half maximum (FWHM) of the number-
weighted particle size distribution divided by its average value. For a normal size distri-
bution, the FWHM is simply 2 √ 2 ln 2 times its standar d deviation σ R .
2.2.2 The scattering curve
The dif ferential scattering cr oss-section
d σ / d Ω
is the fundamental measurand in a SAXS
experiment, as described in section 2.1.2. Nevertheless, some comparability challenges
arise fr om this quantity as it depends on the sample volume
V
used in the experiment.
This can be solved by intr oducing the differ ential scattering cross-section per volume,
historically given in cm
− 1
. The expression of this quantity is derived fr om equations 2.13
and 2.20 and leads to:
d Σ
d Ω ( q ) = d σ / d Ω ( q )
V = r 2
e I ( q )
V = r 2
e · N
V · Z ∞
0 g ( R )   f ( q , R )  
2 d R (2.23)
wher e N / V is the concentration of scatterers, i.e. particles.
For isotr opically scattering samples, the scattering patterns consist of concentric rings, as
shown in figur e 2.4a. By azimuthally averaging the scattering pattern, the data is reduced
13

Chapter 2 THEORETICAL BA CK GROUND
(a) The scattering pattern
d Σ / d Ω / cm − 1
q / nm − 1
10 − 3
10 − 2
10 − 1
10 0
10 1
10 2
10 3
10 4
0.01 0.1 1
Guinier
region Por od
region
Fourier region
(b) The scattering curve and its relevant r egions
Figure 2.4 |
a) Radially symmetric scattering pattern of a nanopa rticle ensemble in susp ension with radius
50 nm and a p olydisp ersit y degree of 25 % . b) The scattering curve is the azimuthal integration of the 2D
image. The three different regions of the scattering curve discussed in the text are highlighted in the figure
as w ell.
fr om 2D images to 1D scattering curves. The scattering curve is the typical form to present
the experimental data, which displays the dif ferential scattering cr oss-section per volume
d Σ / d Ω
versus the momentum transfer
q
in a log–log graph as depicted in figure 2.4b for an
ensemble of spherical particles with radius 50 nm and polydispersity degr ee 25 %. Three
dif ferent r egions can be distinguished in a scattering curve (Schnablegger & Singh, 2006):
•
The
Guinier region
comprises the low-
q
r egion where
q D <
1.3 (Feigin & Sver gun,
1987), being
D
the characteristic length of the investigated object. This region
pr ovides principally information about the size of the particle.
•
The high-
q
r egion is called the
Porod region
, wher e information about the surface-
to-volume ratio of the particles can be derived. For a smooth particle surface, the
scattering intensity decays as
q − 4
, while for r ough or fractal surfaces the slope is a
function of q − b with 2 < b < 4 (Glatter & Kratky, 1982).
•
For suf ficiently monodisperse particle suspensions, the
Fourier region
or middle-
q
r egion of the scattering curve shows pronounced minima that characterize the
particle structur e, size and shape.
2.2.3 Modelling of the scattering intensity: form factors
Besides the information obtained about the size distribution of the particle ensemble, the
scattering intensity
I ( q )
pr ovides information about the shape and composition of the
particles, accessible by modelling the form factor . In the simple case of a solid sphere with
uniform density
ρ 0
, the radial electr on density profile is described by
ρ e ( r > R ) =
0 and
ρ e ( r < R ) = ρ 0
, whilst the integral of expr ession 2.18 is limited only to the radius of the
particle R . The form factor of a homogeneous solid spher e is
f sph ( q , R ) = 4
3 π R 3  ρ 0 − ρ medium  
 3 sin ( q R ) − q R cos ( q R )
 q R  3 
 = ∆ η · F sph ( q , R ) (2.24)
14

Small-angle X-ra y scattering 2.3
wher e
∆ η = ρ 0 − ρ medium
is the contrast and
F sph ( q
,
R )
is defined for convenience. When
the shape of the particle deviates fr om a sphere, the assumptions made in equation 2.18
ar e not applicable and the scattering intensity must be integrated over all available angles
numerically . For a homogeneous ellipsoid of r evolution with two equal semi-axes of
length
R
and a semi-principal axis of length
ν R
, the squar e of the form factor is expressed
as:
   f ellip ( q , R )   
2 = ∆ η 2 Z 1
0    
F sph  q , R q u 2  ν 2 − 1  + 1     
2
d u (2.25)
wher e
ν
is the ellipticity ,
u = cos α
and
α ∈ [ 0, π / 2 ]
. If
ν >
1, the expression defines a
pr olate spheroid, whilst ν < 1 defines an oblate spheroid.
Fr equently , nanoparticles show an internal heterogeneity , leading to an inner electr on
density distribution. If the components are radially distinguishable, the form factor
corr esponding to a morphology defined by sharp interfaces between the radial symmetric
components of the particle with radius R i is
f  q , R  = ∆ η F sph ( q , R ) +
n − 1
∑
i = 1
∆ ρ i  F sph ( q , R i + 1 ) − F sph ( q , R i )  , (2.26)
wher e
R
is the external radius of the particle and
n
is the number of concentric shells.
The excess of electr on density of each component is
∆ ρ i = ρ i − ρ core
and the contrast is
defined in this case as
∆ η = ρ core − ρ medium
in or der to isolate the electron density of the
surr ounding medium in one term.
The simplest case of expr ession 2.26 arises for cor e-shell particles in suspension. This
model r epresents a radially symmetric particle with a sharp interface between the outer
shell and the inner cor e. The form factor is described by
f CS ( q , R ) = ∆ η F sph ( q , R ) + ∆ ρ  F sph ( q , R ) − F sph ( q , R core )  , (2.27)
wher e
R
and
R core
ar e the outer shell and inner core radii r espectively , the excess of
electr on density is
∆ ρ = ρ shell − ρ core
and the contrast is expr essed as
∆ η = ρ core − ρ solv
,
wher e ρ solv is the electron density of the suspending medium.
Depending on the synthesis of the particles, the interface between the dif ferent phases
might show a linear electr on density gradient between the particle’s components. Analog-
ously to expr ession 2.26, the form factor of a multicomponent spherical particle with a
linear gradient interface is
f  q , R  =
n − 1
∑
i = 0  m i  F lin ( q , R i + 1 ) − F lin ( q , R i )  + b i  F sph ( q , R i + 1 ) − F sph ( q , R i )   (2.28)
wher e
m i =  ρ i + 1 − ρ i  / ( R i + 1 − R i )
and
b i =  ρ i − ρ solv  − R i m i
and the linear form factor is
defined by
F lin ( q , R ) = 4 π  2 q R sin ( q R ) + 2 cos ( q R ) − ( q R ) 2 cos ( q R ) 
q 4 (2.29)
The pr esented form factors are the models used in this work to analyse the experimental
SAXS data of nanoparticles in suspension which will be discussed in chapters 4, 5 and 6.
15

Chapter 2 THEORETICAL BA CK GROUND
ρ solv
Cor e
Shell
Electr on density
Solvent
∆ η cor e
∆ η shell
(a) Core-shell particle in solvent
ρ solv
Cor e
Solvent
Electr on density
∆ η cor e
(b) Contrast matching of the shell
Figure 2.5 |
V a riation of the solvent electron density rep resented by the electron densit y p rofile of a spherical
co re-shell pa rticle: a) The contrast of b oth co re and shell comp onents is high (
∆ η co re > ∆ η shell >
0 ), while
in figure b) the solvent electron densit y is increased to match the shell’s (
∆ η shell =
0 ). In this case, the
only contribution to the scattering intensit y will a rise from the co re.
2.3 Contrast variation
In the contrast variation method, the electr on density of the particle or the surrounding
medium is systematically alter ed in order to obtain independent scattering curves with
dif ferent contrasts
∆ η ( r )
. This technique is useful to characterize the dif ferent components
of heter ogeneous particles, due to the complementary data that can be collected at each
contrast. The work presented in this thesis is focused in the solvent contrast variation
method, wher e only the electron density of the suspending medium is varied.
By means of the solvent contrast variation appr oach, the electron density of a single
phase of the investigated particle can be matched (i.e. match point ), resulting in a incr eased
scattering amplitude of the other components of the object, as depicted in figures 2.5a
and 2.5b. This effect enables a much mor e detailed study of the dif ferent contributions of
the particle’s components to the scattering intensity , which can be isolated by choosing
the solvent electr on density appropriately . In the following paragraphs, the theor etical
framework r equired to interpr et a SAXS contrast variation experiment will be presented,
focusing mainly on the ef fects produced by the variation of the solvent electr on density
ρ solv .
2.3.1 Isoscattering point
One of the best known featur es appearing in a contrast variation experiment with hetero-
geneous nanoparticles is the existence of isoscattering points , first formulated by Kawaguchi
et al. (1983). At these specific
q
-values, the scattering intensity is independent of the adjus-
ted solvent contrast, i.e. all scattering curves intersect in the isoscattering points regar dless
of the contrast. The isoscattering points
q ?
ar e particularly interesting because they emer ge
for any spherical particle with an inner structur e and a suf ficiently narrow size distribu-
tion. From the contrast-depending part of equation
(2.26)
, a model-fr ee expression can be
derived which r elates the position of the isoscattering points
q ?
i
with the external radius of
the particle
R
, independent of its radial structur e (Kawaguchi et al. , 1983; Kawaguchi &
Hamanaka, 1992):
16

Contrast va riation 2.3
10 1
10 2
10 3
10 4
10 5
10 6
10 7
10 8
0.02 0.05 0.1 0.2 0.5
Scattering Intensity / a.u.
q / nm − 1
0.08 0.09 0.1
-100
-80
-60
-40
-20
0
20
40
60
∆ η / nm − 3
(a) Monodisperse nanoparticles
10 2
10 3
10 4
10 5
10 6
10 7
0.02 0.05 0.1 0.2 0.5
Scattering Intensity / a.u.
q / nm − 1
0.08 0.09
-100
-80
-60
-40
-20
0
20
40
60
∆ η / nm − 3
(b) Polydisperse ensemble: p d = 30 %
Figure 2.6 |
A isoscattering p oint is the
q
-value where all the scattering curves measured at different
contrasts
∆ η
intersect. a) In the mono disp erse case, the first isoscattering p oint is w ell-defined as depicted
in the inset, while the inset of b) sho ws how the high polydisp ersit y of the ensemble produces a diffuse
isoscattering p oint and the intersection p oint is smea red out.
tan ( q ?
i R ) = q ?
i R (2.30)
The solutions for this equation fulfill
q ? R =
4.493, 7.725, 10.904, ..., where the positions
of the isoscattering points corr espond to the minima positions of the scattering intensity
of a compact spherical particle with radius
R
. This expression r elates in a simple way the
position of
q ?
to the size of the particle inaccessible to the suspending medium and, thus,
a good method to determine the diameter of the colloid.
Although this expr ession is derived for the monodisperse case, it can still be applied up
to a moderate degr ee of polydispersity , if care is taken r egarding the shift of the minima
position due to polydispersity (van Beurten & V rij, 1981). For size distributions with
p d
lar ger than
≈
30 %, the isoscattering point is not well defined and the intersection point
of the curves is smear ed out, showing a dif fuseness in the isoscattering point position
(Kawaguchi & Hamanaka, 1992). The ef fect of polydispersity in the isoscattering point is
illustrated by simulating a 100 nm cor e-shell particle for the ideal case of a monodisperse
ensemble (figur e 2.6a) and with a degree of polydispersity of 30 % (figur e 2.6b). The shift
of the isoscatterig point position to smaller
q
-values and the dif fuseness of the intersection
point due to the high p d are clearly evident in the inset of figur e 2.6b.
Similarly , any deviation fr om the spherical shape produces a dif fuseness in the
q ?
position. Unfortunately , this ef fect cannot be distinguished from the smearing pr oduced
by the size polydispersity and the investigation of the particle shape needs to be performed
by other means.
17

Chapter 2 THEORETICAL BA CK GROUND
2.3.2 Basic functions appr oach
When analysing contrast variation data, a widespread theor etical appr oach is based on the
non-interacting model pr oposed by Stuhrmann & Kirste (1965; 1967) for monodisperse
particles. The so-called basic functions formulation differ entiates, independently of the
particle inner structur e, the contributions which depend on the varying solvent density or
contrast ( ∆ η ) and on the excess of electron density of each component of the particle.
Deriving fr om this approach, the scattering intensity can be expr essed as the combina-
tion of contributions corr esponding to differ ent features of the particles:
I ( q ) = I c ( q ) + ∆ η I s c ( q ) + ( ∆ η ) 2 I s ( q ) (2.31)
The
I c
function contains the contributions fr om the density fluctuations inside the particle,
the contribution
I s
is the so-called shape scattering function and
I sc
is the cr oss-term function.
Shape scattering function
The
I s ( q )
function corr esponds to the scattering contributions from particles with homo-
geneous density and a size equivalent to the volume inaccessible to the solvent, typically
the external size of the nanoparticle. By modelling the shape scattering function, the
shape and size distribution of the particles can be determined independently of their
inner structur e. The functions
I sc ( q )
and
I c ( q )
ar e more rar ely employed due to their
complex interpr etation.
I c ( q )
contains the electr on density deviations in the particle from
the average electr on density , while
I sc ( q )
includes cr ossed contributions from both
I c ( q )
and I s ( q ) .
In a system measur ed at
N
dif ferent solvent electr on densities i.e. contrasts, the shape
scattering function
I s
at each
q
-value can be calculated by solving the following matrix
equation:




I 1 ( q )
.
.
.
I N ( q )




= 



1 ∆ η 1  ∆ η 1  2
.
.
. .
.
. .
.
.
1 ∆ η N  ∆ η N  2



 


I c ( q )
I sc ( q )
I s ( q ) 

 (2.32)
wher e
I i ( q )
is the measur ed scattering intensity at each solvent electron density and
∆ η i
is the contrast corr esponding to each suspending medium density . A minimum of 3
independent scattering curves measur ed at differ ent contrasts are r equir ed to solve this
system of equations, while an accurate determination of the suspending medium electr on
density is also necessary for the calculation of the dif ferent ∆ η i .
Guinier approximation
The radius of gyration of a particle about its centre of mass
R g
is defined as the second
moment of the electr on density distribution and can be calculated by
R 2
g = R ρ e ( r ) r 2 dr
R ρ e ( r ) dr (2.33)
The radius of gyration is systematically employed in small-angle scattering as an evalu-
ation tool, due to its applicability to a lar ge diversity of samples, e.g. pr oteins, colloids,
suprastructur es (Mertens & Svergun, 2010; Sim et al. , 2012). If the object is spherical, the
gyration radius is dir ectly related with its external radius by R 2
g = 3 / 5 R 2 .
18

Contrast va riation 2.3
In SAXS,
R g
can be calculated using the Guinier appr oximation (Guinier, 1939; Guinier
& Fournet, 1955), which assumes that the scattering intensity behaves in the limit of small
q as
I ( q ) ≃ I ( 0 ) exp − R 2
g
3 q 2 ! , (2.34)
wher e
I (
0
)
is known as forwar d scattering or intensity at zero angle. Using the basic
functions appr oach, the radius of gyration of a monodisperse, heterogeneous particle can
be expr essed as a function of the solvent electron density
ρ solv
and the average electron
density of the particle ρ 0 (Feigin & Svergun, 1987)
R 2
g = R 2
g , c + α
ρ 0 − ρ solv − β
( ρ 0 − ρ solv ) 2 , (2.35)
wher e
R g , c
is the radius of gyration of the particle shape corr esponding to the volume
inaccessible for the solvent
V c
,
α
characterizes the distribution of dif ferent phases inside
the particle and
β >
0 considers the eccentricity of the dif ferent scattering contributions
(Stuhrmann, 2008). Particle aggregation influences the scattering curves esp ecially in the
Guinier r egion and must be explicitly avoided.
A vdeev (2007) pr oposed an extended version to equation
(2.35)
for the case of a poly-
disperse particle ensemble by intr oducing the effective values ˜
R 2
g , c , ˜
α and ˜
β , which ar e the
intensity-weighted averages of the corr esponding parameters over the polydispersity . The
observed average electr on density is not affected by the polydispersity (
˜
ρ 0 = ρ 0
) if the
volume ratio between the dif ferent particle components is constant for all particles in the
ensemble.
Assuming the pr emise of a constant average electron density for all the particles, the
intensity at zer o angle for a polydisperse system can be expressed as
I ( 0 ) ∝ N  ρ 0 − ρ solv  2 , (2.36)
with a minimum of the parabolic function at
ρ solv = ρ 0
. Ther efor e, by analysing the
Guinier r egion of the scattering curves in a contrast variation experiment, the average
electr on density of the particle can be obtained without assuming an a priori inner structure.
Using the models pr esented above, it is possible to obtain by independent means the
external radius and the average electr on density of the particles in suspension.
19

3
Instr umentation and experimental
setup for SAXS measur ements
Since the appearance of thir d generation synchrotr on radiation facilities devoted to dedic-
ated insertion devices and optimized for brightness, synchr otron radiation sour ces have
become of importance in Small-angle X-ray Scattering experiments due to their high
brilliance and collimation, favoring the application of SAXS in a wide variety of scientific
fields. The most relevant instr umentation requir ed in a small-angle X-ray scattering exper -
iment ar e the X-ray source, a sample envir onment and an area detector which collects the
elastically scatter ed photons.
The first part of this chapter (section 3.1) reviews the fundamentals of synchr otr on
radiation, while section 3.2 describes the synchrotr on radiation sour ce, the electron storage
ring BESSY II. After these intr oductory sections, the four-crystal monochr omator (FCM)
beamline operated in the PTB laboratory at BESSY II is intr oduced (section 3.3), wher e
all the r eported results wer e measured. Following this, the area detector mounted on
the HZB-SAXS instrument is r eviewed (section 3.4), highlighting the newly developed
in-vacuum Pilatus X-ray detector and the low uncertainty associated to the sample-detector
distance that can be achieved with this setup. Finally , section 3.5 presents a detailed insight
into the dif ferent sample envir onments needed for the nanoparticles in suspension studied
in this work. A brief overview of the data reduction is given in section 3.6, emphasizing
the a posteriori corr ections requir ed by the scattering curve.
21

Chapter 3
INSTRUMENT A TION AND EXPERIMENT AL SETUP F OR SAXS MEASUREMENTS
3.1 Synchr otr on radiation
Synchr otron radiation is the electr omagnetic dipole radiation which is emitted by ultra-
r elativistic charged particles when they ar e circularly accelerated by an external magnetic
field. The kinetic energy loss of the char ged particles (typically electrons) due to the
Br emsstrahlung process (Blumenthal & Gould, 1970) is tangentially radiated in form of a
light cone with high brilliance and a wide photon ener gy range.
The total radiant power emitted by a single ultra-relativistic electr on accelerated radially
( ~
a ⊥ ~
v ) by a homogeneous magnetic field is described by:
P sync = e 2 c
6 π e 0 R 2  E
m e c 2  4
∝ E 4 m − 4
e R − 2 (3.1)
wher e
m e
and
E
ar e the rest mass and ener gy of the electron, respectively .
R
is the
radius of the electr on trajectory in the circular storage ring and is r elated to the external
magnetic field str ength of the bending magnet (
B
) by
R = E
e c B
. The use of light particles
(electr ons or positrons) in storage rings such as BESSY II is explained by the pr oduction of
a radiative power
∼
10
13
times lar ger than heavier particles like protons due to the lar ge
pr oton-to-electron mass ratio ( ( m p / m e ) > 1800).
Synchr otron radiation sour ces generating X-rays photons have arisen as an important
tool in many scientific fields like physical chemistry , life science or physics. The br oad
spectral range and the high brilliance open new experimental possibilities in materials
science as well as in metr ology . For instance, the synchrotr on radiation can be employed
as a primary calibration standar d for electromagnetic radiation (Thornagel et al. , 2001) by
means of the Schwinger equation (Schwinger, 1949), which describes the radiant power
emitted by an electr on as a function of the photon energy , and the determination of the
number of electr ons, the electron ener gy and the magnetic field of the bending magnet.
The two most characteristic featur es of a synchrotr on radiation sour ce are the brilliance
and the critical ener gy or critical wavelength. The spectral brilliance is defined as the
number of photons per second, per electr on beam source cr oss section, per angular
diver gence and per 0.1 % bandwidth at a certain wavelength
λ
(Marr, 1987). The critical
ener gy E C is defined by (Schwinger, 1949):
E C = 3 h c
4 π R  E
m e c 2  3
(3.2)
The critical ener gy
E C
divides the spectral range into two parts with equal radiant power
(Marr, 1987).
22

Synchrotron radiation 3.1
0 10
Storage Ring
240 m
Bending Magnet
1.3 T
Booster
Synchtrotr on
1.72 GeV
Electron Beam
90 keV
Linac
500 MeV
rf-Cavity
500 MHz
Beamline
20 m
Insertion Device
Undulator
Figure 3.1 |
Scheme of the electron sto rage ring BESSY I I. The different comp onents involved in the
creation of X-ra ys a re depicted.
3.1.1 Insertion devices
The synchr otron radiation sour ces of the third generation ar e designed with the goal of
optimizing the insertion devices and, ther efore, enhance the spectral brightness (Robinson,
2015). The employment of insertion devices, such as wigglers or undulators, on the
straight sections of the storage ring impr oves the brilliance in comparison to the bending
magnets or pr oduces light polarizations differ ent from that pr oduced by bending magnets.
Both insertion devices consist on the same principle: a lar ge number (
N ∼
100) of
equally spaced alternately polarised dipole magnets stimulate the emission of synchr otron
radiation on the experiment dir ection, due to the coherent addition of the contributions
fr om the passage of a single electron. By this approach, the photon flux can be incr eased
in a factor
N
, the number of magnets separated with a spatial field period
λ 0
in the range
of cm. The distinguishable pr operty between wigglers and undulators is their deflection
parameter K , defined by (Marr, 1987):
K = e
m e 2 π c B 0 λ 0 (3.3)
wher e
B 0
is the magnetic field amplitude of the dipole magnet. Normally ,
K
can be
modified by varying the space between the dipole magnets ( gap ) and, thus, whether the
insertion device is called a wiggler or an undulator depends on its particular configuration.
The value of
K
is rather lar ge in the case of wigglers, emitting radiation in a br oad
spectral range and incr easing the
E C
of the storage ring. On the other hand, undulator
devices have
K ≤
1, emitting an almost monochr omatic and highly intense photon beam.
The sharp harmonic peaks observed in the undulator spectrum ar e produced by the
23

Chapter 3
INSTRUMENT A TION AND EXPERIMENT AL SETUP F OR SAXS MEASUREMENTS
10 − 5
10 − 4
1 10 100 1000 10000
Radiant Power / W
Ener gy / eV
E C
Figure 3.2 |
Radiant p o w er of the b ending magnets at BESSY I I under standa rd op eration (300 mA)
through an ap erture of 10 x 10 mm
2
situated at 30 m of the source calculated using the Schwinger equation.
The critical energy E C is 2.5 k e V and divides the sp ectrum into t w o pa rts with equal p o wer.
coher ent constructive interference of the radiation fr om the dif ferent dipole magnets.
3.2 The BESSY II electr on storage ring
The facility BESSY II situated in Berlin (Germany) is a synchrotr on X-ray and UV light
sour ce of the third generation. The electrons ar e accelerated to 1.72 GeV in a booster
synchr otron and injected into a storage ring with 240 m cir cumfer ence and an electron
curr ent of approximately 300 mA in the T opUp Mode (Couprie & Filhol, 2008). The
following paragraphs describe the cr eation of X-rays from the acceleration of the electr on
beam until the emission of synchr otron radiation on the bending magnets situated along
the storage ring (Bakker et al. , 1998; Bakker, 1999).
The cr eation of the free electr ons on the electron beam is the first step, as depicted
schematically in figur e 3.1. A standard DC grid cathode emits electr ons which are accel-
erated with a high voltage to the anode up to a 90 keV ener gy . These electr ons ar e the
sour ce of a 50 MeV Linac, which brings the electr on beam to r elativistic velocities. The
0.4 nC char ged electrons bunches ar e further transported to a 10 Hz booster synchrotr on
by a long Injection Line. The acceleration process in the rapid-cycling synchr otron takes
about 50 ms and is achieved by the disposition of a set of magnets and 500 MHz rf-cavities
coupled with the magnets in linear paths that ramp the electr on beam to its final operation
ener gy of 1.72 GeV .
At this point the electr ons are injected into the storage ring, wher e 32 bending magnets
with a magnetic field str ength of 1.3 T and a bending radius of 4.35 m (Klein et al. , 2014) are
equipped to maintain the cir cular trajectory of the electron beam at 1.25 MHz r evolution
fr equency . Figure 3.2 depicts the calculated radiant power of the bending magnets at
BESSY II as a function of the photon ener gy , where the critical ener gy
E C
of 2.5 keV is
shown.
24

F CM b eamline 3.3
Figure 3.3 |
A scheme
of the F CM b eamline in
the PTB lab o rato ry at
BESSY I I. The distance
of each comp onent to the
b ending magnet is sho wn
(Krumrey, 1998).
Θ
1st wheel 2nd wheel
Θ
Photon
beam
Crystal
Figure 3.4 |
Scheme of
the four-crystal mono-
chromato r: The rotation
angle of the wheel
Θ
defines the Bragg angle
on the crystal. The out-
going photon b eam is pa r-
allel to the incoming radi-
ation due to the geomet-
rical disp osition of the 4
crystals.
3.3 FCM beamline
The four -crystal monochromator bending magnet beamline operating in the PTB laborat-
ory at BESSY II (Krumr ey, 1998; Krumr ey & Ulm, 2001) pr ovides a monochromatic beam
in the 1.75 to 10 keV ener gy range at a fixed sample position with very high photon flux
r eproducibility and high ener gy resolving power . A schematic depiction of the beamline
and its components is shown in figur e 3.3.
At 14 m fr om the bending magnet, a Pt-coated tor oidal mirr or is located to focus the
beam in the horizontal dir ection and to collimate it in the vertical direction. The radiation
coming fr om the bending magnet is monochromatised further downstr eam by a set of 4
single crystals which r eflect the light according to the Bragg’s law for the (1 1 1) r eflection
as schematically depicted in figur e 3.4. As the 4 crystals are mounted on two wheels (one
on the r otation centre and one parallelly aligned), the r otation angle of the wheel
Θ
defines
the ener gy of the outgoing photon beam by
E = √ 3 h c
2 a sin Θ
, wher e
a
is the lattice constant of
the crystal.
T wo types of exchangeable crystal sets, Si (
a =
0.543 nm) and InSb (
a =
0.648 nm) (Kittel,
2004), ar e available to cover the energy range fr om 1.75 keV to 10 keV . The convolution
of the 4 Bragg r eflections provides a very high ener gy resolving power (
E / ∆ E =
10
4
)
thr ough the full energy range. Besides, the geometric disposition of the crystals fixes the
position of the outgoing radiation. The monochr omator is operated under a 10
− 8
mbar
vacuum.
The ener gy is traced back to the well-known lattice constant of Si (Kittel, 2004). The
back-r eflection of a silicon crystal at differ ent lattice planes is measured at distinct ener gies
and the ener gy is calibrated to the dips positions appearing when the Bragg condition is
fulfilled (Krumr ey & Ulm, 2001). This appr oach was employed at the sensitive surface
25

Chapter 3
INSTRUMENT A TION AND EXPERIMENT AL SETUP F OR SAXS MEASUREMENTS
Figure 3.5 |
Photon flux
of the F CM b eamline
using different crystals
(InSb(111) and Si(111))
and mirro r coatings
(MgF
2
and Pt) at
standa rd op eration (300
mA).
10 10
10 11
2 3 4 5 6 7 8 9 10
Photon flux / s − 1
Photon Ener gy / keV
InSb(111) / MgF 2
Si(111) / Pt
Si(111) / MgF 2
of the X-ray detector intr oduced in section 3.4.1 for an energy range between 4 keV and
10 keV (Gollwitzer & Krumr ey, 2014). T o check the accuracy of the energy calibration for
daily measur ements, a transmission scan around the K-edge of a copper foil (8980.5 eV) is
measur ed.
About 10 m befor e the sample chamber , a bendable plane mirror focuses the beam in
the vertical dir ection. The mirr or is coated with two differ ent materials in separated areas.
The Pt-coating is employed to maximize the r eflectivity at energies above 4 keV , while
the MgF
2
suppr esses the higher orders at ener gies below 4 keV . The photon flux achieved
with the dif ferent configurations available at the FCM beamline is shown in figur e 3.5,
although it can vary depending on the pr ecise disposition of the differ ent apertures along
the beam path.
The first slit behind the bending magnet is used to limit the acceptance angle of the
radiation into the monochr omator . T wo moveable slits more ar e employed downstream
to block the parasitic scattering. A germanium 520
µ
m cir cular pinhole (Scatex, Incoatec,
Geesthacht, Germany) situated befor e the sample chamber shapes the photon beam
into a cir cular spot on the sample and strongly r educes the parasitic radiation. A 8
µ
m
thick silicon photodiode diode is installed behind these components and can monitor
continuously the incoming photon flux for ener gies above 3 keV .
3.3.1 UHV X-ray r eflectometer
The sample chamber is situated 37 m away fr om the dipole i.e. bending magnet, right
behind the flux monitor diode. The UHV X-ray reflectometer disposes of a lar ge volume
(60 cm diameter and 70 cm length) which is fully evacuated to reach pr essur es of approx-
imately 10
− 7
mbar . High vacuum is needed to perform experiments at the low energies
accessible at the FCM beamline, as the attenuation length of air at ener gies below 2 keV is
less than 1 cm. A smaller lock chamber connected to the sample chamber by a 200 mm
diameter flange is used to exchange samples without br eaking the vacuum of the larger
UHV X-ray r eflectometer .
The motors of the sample holder can be moved linearly in thr ee mutually perpendicular
dir ections with very high precision and r eproducibility . The broad range of the
x
-motor
26

SAXS setup 3.4
perpendicular to the incoming beam (160 mm) permits the measur ement of differ ent
sample capillaries (ca. 20) at once without venting the chamber for exchanging the sample.
The lar ge volume of the reflectometer pr ovides enough space to allocate other compon-
ents close to the sample holder . For example, about 10 cm before the sample position,
a 1 mm cir cular guard pinhole (Incoatec, Geesthacht, Germany) is installed to remove
the parasitic scattering r esulting from the collimating system. Behind the sample, the
transmitted radiation is measur ed with a (10 x 10) mm
2
silicon photodiode. The thick
Can500C diode (Canberra, Meriden, USA) is capable of measuring through the entir e
beamline ener gy range, fr om 1.75 keV to 10 keV , and is calibrated against a cryogenic
electric substitution radiometer with a r elative uncertainty of 1 % (Krumr ey & Ulm, 2001).
3.4 SAXS setup
The intensity scatter ed by the sample is recor ded at a certain distance behind the sample
(sample-detector distance) with an ar ea X-ray detector mounted on the HZB SAXS instru-
ment and connected to the sample chamber . T ypically , long sample-detector distances ar e
r equired to access the small angles employed in SAXS experiments.
3.4.1 X-ray ar ea detector
The scatter ed X-ray photons are collected by a lar ge-area hybrid pixel detector . The Pilatus
1M (Dectris Ltd, Baden, Switzerland) has a sensitive surface of (179 x 169) mm
2
and
consists of a silicon pixel matrix with a pixel size of
d = (
172.1
±
0.2
) µ
m which operates
in single-photon counting mode, pr oviding very low darkcount rates, very good signal-to-
noise ratios and a high dynamic range. For instance, the detector quantum efficiency is
about 97 % at 8 keV using the ultra-high gain mode and almost 86 % at 4 keV (W ernecke
et al. , 2014).
Besides, the Pilatus 1M detector was modified to operate under vacuum to cover the full
ener getic range available at the beamline, down to 1.75 keV . The windowless detector is
dir ectly connected to the sample chamber with an evacuated bellow and cooled down at 5
to 10
◦
C. The narr ow point-spread function of the detector and the available low ener gies
incr ease the momentum transfer resolution and the accessible q -range.
A moveable beamstop mounted at thin wir es is installed just in front of the detector to
block the intense transmitted photon beam, avoiding saturation effects in the central pixels.
The beamstop is constructed within a funnel-like cavity (

5 mm) to r educe geometrically
the r eflections on the beamstop surface, which ar e damped by the cavity . Since April
2016, a silicon photodiode with a sensitive ar ea of (2.5 x 2.5) mm
2
(S10356-01, Hamamatsu,
Shizuoka, Japan) covers the beamstop to monitor the sample transmission during the
experiment, r evealing the possible radiation damage of the sample.
27

Chapter 3
INSTRUMENT A TION AND EXPERIMENT AL SETUP F OR SAXS MEASUREMENTS
T able 3.1 |
T w o differ-
ent exp erimental setups
which span the accessible
q
-range fo r almost 3 dec-
ades. The overall max-
imum and minimum ac-
cessible
q
-values a re high-
lighted in b old letters.
SAXS W AXS
Distance (mm) 4540 760
Ener gy (eV) 4000 10000
q min (nm − 1 ) 0.015 0.2
q max (nm − 1 ) 0.56 7
3.4.2 HZB SAXS instrument and W AXS configuration
The in-vacuum Pilatus 1M detector is mounted on the SAXS instrument of the Helmholtz-
Zentrum Berlin (HZB) (Gleber et al. , 2010), which is connected via a

100 mm flange to
the UHV X-ray Reflectometer . The HZB-SAXS apparatus is equipped with a large below
system and a motorized stage that can vary the sample-detector distance continuously
between 2.3 m and 4.6 m in vacuum (about 10 − 4 mbar) with an uncertainty of 20 µ m.
Complementary to the HZB-SAXS instrument, the sample-detector distance can be
r educed down to about 760 mm by attaching the X-ray detector dir ectly to the sample
chamber , increasing the scattering angles to ar ound 8
◦
. This short-distance setup, or
W ide-angle X-ray Scattering (W AXS) configuration, is used for the study of nanoparticles
with diameters below 10 nm, whose characteristic features appear beyond 1 nm
− 1
. The
accessible
q
-range of this setup is summarized in table 3.1 for the high-energy case, which
pr ovides the highest
q
-value available. Similarly , the table shows the limit
q
-values
achieved with the HZB-SAXS apparatus at low-ener gy .
Calibration of the sample-detector distance
In small-angle scattering experiments, it is crucial to know pr ecisely the distance between
the irradiated sample and the detector , in or der to calibrate the momentum transfer
q
. T ypically , a calibration standard material with a pr eviously measured crystal lattice
parameter is employed, which produces well-defined dif fraction rings in the low-angle
r egion. A material extensively used is dry rat-tail tendon collagen, with a
d
-spacing of 65
nm (Amenitsch et al. , 1997), corresponding to
q =
0.097 nm
− 1
. The degradation of this
material upon pr olonged radiation suggested the use of harder calibrants such as silver
behenate (CH 3 (CH 2 ) 20 COO · Ag) (Huang et al. , 1993).
AgBehe has a very narr ow diffraction ring at
q =
1.0763 nm
− 1
, arising from a long-
period spacing (
d 001
) value of 5.84 nm (Blanton et al. , 1995), although this value depends
slightly on the synthesis. A deviation of 0.5 % in the diffraction peak position could
be observed for dif ferent sample pr eparations. In order to incr ease the accuracy of the
calibration, the sample-detector distance was determined by the detection of the scattering
pattern of AgBehe at dif ferent positions of the HZB SAXS instrument, measur ed with the
built-in 3 m long Heidenhain optical encoder . By triangulating the radius of the diffraction
ring to the sour ce point, as depicted in figure 3.6a, the sample-detector distance is obtained
in a traceable way .
By measuring the AgBehe pattern along a distance range of 2200 mm with 100 mm steps
at 8000 eV , the relative uncertainty associated to the linear fitting is 0.03 %, corr esponding
to 1.5 mm. As observed in the residuals of the fitting in figur e 3.6a, the deviation incr eases
for long distances, due to the r elatively small
d
-spacing of AgBehe, disabling the use of
distances lar ger than
∼
3600 mm. In figure 3.6b, it is visible how the diffraction ring
surpasses the surface of the detector at a distance of 3638.2 mm and, thus, diminishes the
28

SAXS setup 3.5
300
400
500
600
700
Radius size / pixel
-3
-2
-1
0
1
2500 3000 3500 4000 4500
Residuals / pixel
Measur ed distance / mm
AgBehe
SBA
(a) Distance Calibration (b) AgBehe at large distance
Figure 3.6 |
Sample-to-detecto r distance calib ration: a) Radius of the diffraction ring of AgBehe and
SBA-15 as a function of the sample-detecto r distance. A linea r function is fitted to obtain the source
p oint distance. The residuals of the fitting are sho wn in the b ottom plot. b) Scattering pattern of AgBehe
measured at a distance of 3638.2 mm. At such long sample-to-detecto r distances, the diffraction rings
exceeds the detecto r a rea and the asso ciated uncertaint y increases.
accuracy of the peak determination.
By using a material with lower
q
-value, such as the templated mesoporous silica SBA-15
with
q =
0.681 nm
− 1
(Zhao et al. , 1998), this limitation can be mitigated as shown in
figur e 3.6a, where the r esiduals of SBA ar e minimal for the entire distance range. By
using SBA (kindly pr ovided by R. Schmack (T echnische Universität Berlin, Germany))
and incr easing the accessible distance range, the relative uncertainty of the fit decr eases
in a factor 5, r eaching an uncertainty of 0.004 % (0.2 mm) when measuring with 50 mm
steps. This improvement is also r elated with the narr ower diffraction peak of SBA-15
(FWHM/ q = 2.6 %) in comparison to AgBehe (5.5 %).
Although the fit uncertainty is smaller in the SBA case, the position and shape of
dif fraction peak depend strongly on the sample pr eparation (e.g. template pore size). For
the same polymer template, the
q
-value of the ring can vary until 1 % for dif ferent thermal
tr eatments and radiation damage effects ar e visible for short calcination times. On the
other side, prolonged beam exposur e of AgBehe can damage the sample as well and create
small silver nanoparticles, which incr ease the scattering background (Liu et al. , 2006).
The choice of the calibration standar d depends strictly on the needs of the experiment.
Besides, the lar gest contributions to the sample-detector distance uncertainty come from
the thickness of the sample (ca. 0.5 mm) and fr om the differ ence between the calibration
with AgBehe and SBA-15 (also 0.5 mm). Normally the r elative uncertainty associated with
the distance calibration is 10
− 4
, similar to the ener gy resolving power described in section
3.3.
29

Chapter 3
INSTRUMENT A TION AND EXPERIMENT AL SETUP F OR SAXS MEASUREMENTS
-2 -1 0 1 2
Horizontal Position / mm
5
6
7
8
9
V ertical Position / mm
-2
-1
0
1
Deviation /%
(a) Glass thickness
-2 -1 0 1 2
Horizontal Position / mm
-4
-2
0
2
4
V ertical Position / mm
0.99
1
1.01
Sample Thickness / mm
(b) Sample thickness
Figure 3.7 |
Homogeneit y of the rectangula r capillaries: a) Deviation of the empty capilla ry tranmission,
i.e. glass w all thickness, b) Sample thickness calculated from the water transmission of a filled capilla ry .
3.5 Sample envir onment
The sample consists normally of a few micr oliters of nanoparticles in solution which are
measur ed in a vacuum-proof container positioned inside the r eflectometer . The sample
envir onment must fulfill some requir ements:
•
The container ’s material should minimize the unnecessary absorption of the X-ray
photon flux by the sample envir onment.
•
The container volume should be small enough to enable the measur ement of valu-
able, limited samples.
•
The optimal sample thickness for a transmission dif fraction experiment is the inverse
of its attenuation coef ficient
µ ( E )
, which r educes the incoming intensity to
∼
37 %.
For example, the optimal thickness of water at 8000 eV is ar ound 1 mm.
T ypically , the samples ar e introduced in thin glass capillaries which maintain the tem-
peratur e and pressur e of the sample close to the ambient conditions. However , ther e are
dif ferent sample envir onments which can be used depending on the requir ements of the
experiment. In this work, only nanoparticles suspended in aqueous media have been
employed, allowing the use of a similar attenuation coef ficient for almost all experiments.
3.5.1 Round capillaries
For single-contrast SAXS measur ements, borosilicate glass r ound capillaries of 100 mm
length wer e used. They were pur chased at WJM Glass (Berlin, Germany) and had a
nominal inner diameter of 1 mm and a wall thickness of 10
µ
m. The sample is filled into
the capillary with a long syringe (Sterican
®
21 x 4 3/4", Braun, Melsungen, Germany),
avoiding the contact of the needle with the capillary walls. The top end of the capillary is
closed by welding.
The very narr ow glass walls (with a density of about 2.23 g cm
− 3
) absorb only 14 % of the
incoming flux at 8000 eV and pr oduce very low scattering background. Therefor e, these
capillaries ar e suitable for standard SAXS measur ements. Unfortunately , the capillaries
30

Sample environment 3.5
0.07
0.1
0.15
0.2
-4 -2 0 2 4 6 8 10
T ransmission
V ertical position / mm
T ransmission
Glass only
H 2 O
Figure 3.8 |
X-ra y trans-
mission of a rectangula r
capilla ry half-filled with
w ater along the main ver-
tical axis situated at
x =
− 0.15 mm.
sample thickness shows a significant deviation along the vertical axis and ar e inappropriate
for measur ements at differ ent capillary heights, as needed for the continuous contrast
variation technique.
3.5.2 Rectangular capillaries
The capillaries used for the contrast variation experiments ar e vacuum-proof bor osilicate
glass capillaries fr om Hilgenberg (Malsfeld, Germany) with a nominal r ectangular cross
section with outer dimensions of (4.2
±
0.2) x (1.25
±
0.05) mm
2
, a length of (80
±
0.5)
mm and a wall thickness of ca. 120
µ
m. The thicker glass walls reduce the transmitted
intensity to about 80 % at 8000 eV , but in contrast both the glass and sample thicknesses
ar e very homogeneous for the entire capillary .
The transmission of an empty capillary is mapped in figur e 3.7a, where it can be ob-
served that the deviation of the glass wall thickness is less than 2 % for an horizontal
range of 2.5 mm (of a total width of 4.2 mm). This range is at least 5 times larger than the
typical beam diameter , avoiding the convolution of differ ent thicknesses in the measur e-
ment. Similarly , figur e 3.7b depicts the sample thickness in the capillary , calculated from
a capillary filled with water using the Beer -Lambert law , the glass transmission and the
mass attenuation coef ficient of water at 8000 eV , 10.37 cm
2
g
− 1
(Hubbell et al. , 1996). The
thickness of the sample intr oduced in the capillary is homogeneous within 2 % for a width
range of ca. 2.5 mm.
Fr om these figures, it is clear that the homogeneity of the sample envir onment is even
better along the main vertical axis of the capillary . Figure 3.8 shows the measur ed X-ray
transmission of a half-filled capillary along its vertical axis (at the horizontal position -0.15
mm). For example, the glass transmission within a 6 mm vertical range is 20.1 %, with
an associated r elative uncertainty of
δ r T =
0.6 %. By calculating
δ r d = δ r T
l o g ( T )
wher e
T
is the glass transmission, the relative uncertainty of the glass thickness is
δ r d =
0.4 %.
Analogously , the uncertainty of the water transmission is 0.9 % and the sample thickness
has an uncertainty of 0.9 % along the vertical axis.
These r ectangular capillaries are a very suitable sample envir onment for measur ements
which r equire a high homogeneity along the vertical axis of the capillary . The thickness of
31

Chapter 3
INSTRUMENT A TION AND EXPERIMENT AL SETUP F OR SAXS MEASUREMENTS
Figure 3.9 |
Different
sample environments: On
the left side, the disas-
sembled lo w-energy cell
with the t w o silicon-
nitride windo ws, the p oly-
meric ring spacer and the
t w o parts of the metallic
holder. On the right side,
the round and rectangula r
capilla ries.
the wall varies only by 0.4 % and the sample thickness less than 0.9 %, although the thick
glass walls r educe considerably the transmitted intensity and produce lar ger background
scattering than the r ound capillaries.
3.5.3 Cell for low-ener gies
Samples with lar ger structures r equir e the measurement of scattering curves at lower
q
-values. T o extend the measurable
q
-range, one possibility is to reduce the photon ener gy ,
though this involves r educing the sample thickness, due to the short penetration length of
X-rays at low ener gies. Therefor e, a custom-made sample holder is used utilizing silicon-
nitride windows (NX7150E, Nor cada Inc., Edmonton, Canada). The 500 nm thickness
windows pr oduce very low scattering and have a negligible absorbance (
<
5 %) for
ener gies above 4000 eV .
A polymeric 100
µ
m ring cut with a micr otome is used as spacer between the 2 windows,
in or der to achieve the desired 120
µ
m sample thickness which optimizes the intensity
attenuation at 4000 eV . The access to smaller
q
-values using this cell is shown in V ar ga
et al. (2014 b ), wher e a value of
q =
0.015 nm
− 1
is achieved. The differ ent components of
the cell ar e shown in figure 3.9.
3.6 Data r eduction: the scattering curve
In the case of nanoparticles in suspension and other isotr opically scattering samples, the
scattering patterns collected in the ar ea detector consist of concentric rings whose centre
is the transmitted beam. The dimensionality of the data can be r educed by performing
a radial integration of the measur ed pattern, converting the 2D images into 1D scatter -
ing curves. This reduction step is based on the
q
-binning: the gr ouping of pixels with
similar scattering angle
q
irr espective of their azimuthal angle on the detector (Pauw,
2013). By averaging the scatter ed intensity of the pixels within the same
q
-bin (
I ( q )
), the
uncertainty of the data decr eases in the scattering curve. The size of the bins depends
on the r equirements of the data evaluation while the bins ar e typically spaced uniformly ,
32

Data reduction: the scattering curve 3.6
although logarithmic distributions ar e also extensively used. The differ ence in solid angle
for each pixel due to the spherical pr ojection of the scattering on a flat detector is also
consider ed in this step.
The uncertainty associated to the intensity
I ( q )
is calculated as the standar d deviation
between each pixel intensity in the
q
-bin, which gives a better estimate than the uncertainty
associated to the photon-counting Poisson statistics (Pauw, 2013). The pixels discar ded
(or masked out ) for the weighted average of the
n
th
q
-bin (
q n
) ar e those whose intensity is
not comprised within the range
"   I med  q n − 1  − I med  q n   
2 − 3 σ ,   I med  q n + 1  − I med  q n   
2 + 3 σ # , (3.4)
wher e
I med  q n 
is the median intensity of the pixels prior to this masking pr ocedure and
σ
is the standar d deviation. W ith a confidence level of 99.7 %, the pixels excluded of the
r eduction process ar e those pixels whose intensity lies clearly out of the radial average,
such as hot pixels , anisotropic scattering fr om the glass capillary or undesir ed reflexes
without radial simmetry .
The position of the centr e of the scattering pattern is of vital importance for the radial
integration step. A standard calibrant with very narr ow dif fraction rings such as AgBehe
can be used to locate the centr e with high precision. Nevertheless, the masking process
pr eviously described can be used as well to determine the centre position by minimizing
the number of masked pixels and the standar d deviation uncertainty of the
q
-bins. The
accuracy of the centr e determination is sub-pixel using both approaches, but the masking
pr ocedure does not r equire a calibration standar d material.
The scattering curve obtained by radial integration still requir es of some data corr ection.
For instance,
I meas ( q )
(photon counts) must be normalized to the exposur e time
∆ t
, the
solid angle
∆ Ω
, the incident photon flux
Φ 0
, measur ed by the flux monitor described in
section 3.3, and the measured transmittance of the sample
T
, which implicitly contains
information about the density and chemical composition of the sample. In or der to
pr esent the scattering cross section
d σ / d Ω
per volume
V
(
d Σ / d Ω
) in absolute units (cm
− 1
),
the measur ed intensity must be normalized to the sample thickness
t
and the quantum
ef ficiencies of the X-ray detector and the silicon diodes η Q E :
d Σ
d Ω  q  =
d σ
d Ω  q 
V = I meas  q 
Φ 0 · T · ∆ Ω · ∆ t · η Q E · t (3.5)
By using the monitor diode on the beamstop as described in section 3.4.1,
T
and
Φ 0
can be measur ed simultaneously during the experiment, without the necessity of the flux
monitor diode.
Alternatively a standar d material like lupolen (Kratky et al. , 1966; Shaffer & Hendricks,
1974) or glassy carbon (Perr et & Ruland, 1972) can be employed to scale the measured
scattering intensity to the known values of these materials.
The normalized scattering curve r equires an accurate backgr ound correction. The
scattering of the pur e suspending medium and the sample environment can af fect the eval-
uation of the data, specially for low-scatterers, and, ther efor e, the normalized scattering
backgr ound must be subtracted to obtain a usable scattering curve.
33

4
Continuous contrast variation in
SAXS: the density gradient technique
The contrast variation method in Small Angle X-ray Scattering (SAXS) experiments consists
in systematically varying the electr on density of the dispersing media to study the differ ent
contributions to the scattering intensity in gr eater detail as compared to measur ements at
a single contrast, as described in chapter 2. It emerges as an ideally suited technique to
elucidate the structur e of particles with a complicated inner composition and has been
r epeatedly employed to investigate the radial structur e of nanoparticles in suspension, e.g.
latex particles suspended in an aqueous medium (Dingenouts et al. , 1999; Ballauf f, 2011).
In Small Angle Neutr on Scattering (SANS) the contrast variation technique is widely
used by mixing water and deuterium oxide, but the use of deuterated chemicals and
the incoher ent contribution to the background as well as the limited access to neutr ons
r estrict the application of this technique. Other methods for structural investigation (e.g.
transmission electr on microscopy (Joensson et al. , 1991; Silverstein et al. , 1989)) r equire
prior tr eatment of the sample and are not ensemble averaged.
In SAXS, the solvent contrast variation technique is achieved by adding a suitable
contrast agent to the suspending medium (e.g. sucrose) and r ecor ding the scattering data
as a function of the adjusted solvent electr on density
ρ solv
(Ballauf f, 2001; Bolze et al. , 2003).
In or der to resolve small changes of the radial str ucture, the average electr on density of
the colloidal particles must be close to the dispersant’s, i.e., the match point should be
appr oached, where the average contrast of the particle vanishes. In the case of polymeric
latexes with electr on densities ranging from 335 to 390
nm − 3
, an aqueous sucr ose solution
is very well suited as the suspension medium, due to the easy r ealization of concentrated
solutions with electr on densities of up to 400
nm − 3
. Previous studies on globular solutes
(Kawaguchi & Hamanaka, 1992) and the influence of the sucrose on the size distribution
of vesicles (Kiselev et al. , 2001 a ) show the feasibility of this technique, while further studies
have investigated the ef fect of the penetration of the solvent into the particles (Kawaguchi,
1993).
The pr eparation of a number of differ ent sucrose solutions has been a major inconveni-
ence in solvent contrast variation experiments, due to the tedious, time-consuming process,
35

Chapter 4
CONTINUOUS CONTRAST V ARIA TION IN SAXS: THE DENSITY GRADIEN T
. . .
possible inaccuracy in the sucr ose concentration and the discrete range of available solvent
electr on densities. In this chapter , a novel approach using a density gradient column is
intr oduced, which allows the tuning of the solvent contrast within the provided density
range, r esulting in a virtually continuous solvent contrast variation. By filling the bottom
part of the capillary with a particle dispersion in a concentrated sucr ose solution and the
top part with an aqueous solution of the same particle concentration, a solvent density
gradient is initiated with a constant concentration of nanoparticles along the capillary .
Density gradient columns ar e extensively used in fields like marine biology (Coombs,
1981) or biochemistry together with centrifugation (Hinton & Dobrota, 1978), to cr eate a
continuously graded aqueous sucr ose solution by diffusion of the sucr ose molecules. By
measuring the density gradient column at dif ferent points in time during the dif fusion
pr ocess of the sucrose, it is possible to choose in situ the most appr opriate solvent densities
to perform measur ements close to the contrast match point. Combining this approach with
SAXS, a very extensive dataset with a virtually continuous variation in the suspending
medium density can be acquir ed in a short interval of time.
The experimental details of the pr oposed approach ar e shown in section 4.1, followed
by the example of the continuous contrast variation technique applied to polymeric
nanoparticles in section 4.2. The evaluation of the SAXS data using differ ent methods is
r eviewed in section 4.3, jointly with the discussion of the experimental measurements and
a summary of the obtained r esults. Finally in section 4.4 the applicability of the solvent
contrast variation technique in SAXS is discussed and compar ed to other contrast variation
techniques. Parts of this chapter have been adapted from an article published pr eviously
(Gar cia-Diez et al. , 2015).
4.1 Experimental pr ocedur e
4.1.1 Pr eparation of the density gradient capillaries
The solvent density gradient is pr epared in the r ectangular glass capillaries presented
in section 3.5, which are extraor dinary homogeneous and show very uniform sample
thickness within 0.9 % and glass thickness within 0.4 %. The bottom end of the capillary is
closed by welding and the lower section, up to a height of ca. 1 cm, is filled with Galden
®
PFPE SV90 fr om Solvay Plastics (Brussels, Belgium). This fluid has an exceptionally
high density of 1.69 g cm
− 3
, low viscosity and is immiscible with aqueous solutions.
Consequently , a uniform interface with the particle suspension is formed at the bottom,
which is employed as r eference position for the X-ray transmittance measur ements.
The studied nanoparticles in suspension ar e mixed with a high sucrose concentration
(Sigma-Aldrich, Missouri, USA) and diluted in an aqueous solution, creating two mixtur es
with dif ferent solvent densities but equal particle concentrations. Directly above the
Galden fluid, the denser of these two mixtur es is filled into the capillary using a syringe up
to a height of about 1 cm. The lighter aqueous dilution is then filled on top of the aqueous
sucr ose solution along ca. 1 cm. By the time the two components come into contact, the
density gradient is initiated and the sucr ose starts diffusing along the ca. 20 mm length of
the filled capillary .
The calculated dif fusion time constant of the solvent density gradient is ca. 10 minutes,
considering that the dif fusion coefficient of sucr ose in water at 25
◦
C is
D =
5.2
·
10
− 10 m 2 s − 1
(Uedaira & Uedaira, 1985; Ribeir o et al. , 2006) and assuming that convection
ef fects are negligible due to the small length-scale of the capillary (Berberan-Santos et al. ,
36

Exp erimental p ro cedure 4.1
q
Sample Motor
X-ray
beam x
y
Density gradient
capillary
2 θ
Galden
Figure 4.1 |
The rectan-
gula r densit y gradient ca-
pilla ry is placed in the
X-ra y b eam and can b e
moved b y sample moto rs
in b oth directions p erpen-
dicula r to the incoming
b eam.
1997). The time needed for the transfer of the sample into the UHV sample chamber
amounts to ca. 1 hour . W ithin this time duration, the simulations of the sucr ose diffusion
show that the deviation of the solvent density at both ends of the gradient fr om the initial
value can be estimated with an uncertainty below 0.5 %.
4.1.2 Calibration of the solvent density: X-ray transmission
The r ectangular capillary is placed in the sample holder inside the UHV reflectometer
described in section 3.3 which allows the movement with micr ometer precision in the
dir ections perpendicular to the incoming beam, as depicted in figure 4.1. In order to
determine the central vertical capillary axis, a horizontal X-ray transmission scan is
performed at two dif ferent vertical positions of the capillary spaced by 20 mm. The central
vertical axis can be drawn fr om the centres of both measur ements and the sample can
be moved along this axis by the simultaneous operation of the vertical and horizontal
motors.
The transmitted intensity thr ough the sample is recor ded at a photon energy of
E =
(
5500.0
±
0.5
)
eV for 10 seconds at each position. The measurement points ar e spaced
0.5 mm along the central vertical axis of the capillary , starting at the bottom refer ence
interface with Galden
®
PFPE SV90. The overall X-ray transmission measurement r equir es
appr oximately 5 minutes, which is within the calculated dif fusion timescale of the aqueous
sucr ose solution. This transmission measurement is performed both immediately befor e
and after r ecording the scattering patterns, which should not take much longer than the
sucr ose diffusion timescale (15 to 20 min). The transmittance values used for the density
calibration ar e then linearly interpolated between both data sets taking into account the
time-dependence.
These values can be converted to solvent electr on densities via the Beer-Lambert law
intr oduced in section 2.1.1, which relates the density of the solution with the transmitted
intensity:
ρ e ( z ) = A ln  I 0
I ( z )  (4.1)
Her e
ρ e
is the electr on density of the suspending medium,
I
and
I 0
ar e the transmitted
and incoming intensities r espectively and
A
is a factor determined by the refer ence values
of the solvent electr on density at the vertical limits of the capillary at the initial time. The
sucr ose concentration in solution expressed as the mass fraction
M
at these r eference points
37

Chapter 4
CONTINUOUS CONTRAST V ARIA TION IN SAXS: THE DENSITY GRADIEN T
. . .
335
340
345
350
355
360
0 2 4 6 8 10
0.027
0.028
0.0295
0.0305
Solvent electr on density / nm − 3
X-ray T ransmission / %
V ertical Position / mm
100
120
140
160
180
200
220
240
Diffusion T ime / min
Figure 4.2 |
Solvent densit y along the gradient capilla ry vertical axis at different diffusion times, calculated
from the transmission measurements at 5500 e V of an aqueous solution with a maximum sucrose mass
fraction of 23.5 % at the b ottom of the capilla ry . The corresponding X-ray transmission is sho wn on the
right axis, revealing the lo w transmittances of the filled capillary at lo w energies.
can be converted to electr on densities with the empirical formula
ρ e =
1.2681
M +
333.19
nm
− 3
, which relates the experimentally measur ed density of aqueous sucr ose with its
concentration (Haynes, 2012). The solvent electron density pr ofile within the density
gradient capillary derived fr om this measurement is depicted in figur e 4.2 at dif ferent
dif fusion times for an aqueous solution with a maximum sucrose mass fraction of 23.5 %
at the bottom of the capillary .
The focused X-ray beam has a vertical size at the capillary of ar ound 0.5 mm which
convolutes all the available sucr ose concentrations within the illuminated sample volume
and pr oduces a scattering curve with an averaged suspending medium electron density .
The lar gest averaging effect occurs at the interface between the two mixtur es, where the
steepest density variation is found. Although the uncertainty contribution of the beam size
has typically a maximum value of 1 nm
− 3
, the uncertainty associated to the suspending
medium electr on density depends on the experimental conditions, e.g. dif fusion time,
sucr ose concentration. In the results shown in section 4.2, a maximum uncertainty of 1.5
nm − 3 was estimated.
The X-ray transmission measur ements are performed at a low incident photon ener gy
of
E =
5500 eV to incr ease the transmittance differ ences for the less absorbing sucrose
solution. In figure 4.3a, the calculated transmittances of a 65 % concentrated sucr ose
mixtur e and water (0 %) are depicted, along with the ratio between both transmissions.
This ratio str ongly decreases for high ener gies, suppressing the transmission dif fer ences
between both components of the density gradient column. This fact is revealed in figur e
4.3b, wher e the X-ray transmittance of an aqueous sucrose density gradient measur ed at
5500 eV shows a better signal-to-noise ratio than the same measur ement at 10000 eV .
The calculated transmission of the empty r ectangular capillary is less than 1 % below
6000 eV as shown in figur e 4.3a and the filled capillary just transmits 0.03 % of the incoming
photon flux at 5500 eV , as observed in figur e 4.2. Therefor e, a compromise between the
absorbance ratio and the capillary transmittance was taken at a photon ener gy of 5500 eV .
38

Exp erimental p ro cedure 4.1
0.1
1
10
5000 6000 7000 8000 9000 10000
1.5
2
2.5
3
3.5
X-ray T ransmission / %
Ratio
Ener gy / eV
Capillary
0% sucrose
65% sucrose
T 0% /T 65%
(a) Calculated transmittance
0
10
20
30
40
50
60
70
0 2 4 6 8 10 12 14 16 18
Sucr ose Mass Fraction / %
V ertical Position / mm
5500 eV
10000 eV
(b) Calibrated sucrose concentration
Figure 4.3 |
X-ra y transmittance as a function of the photon energy . a) Calculated transmittances (Henke
et al. , 1993) of an empt y capilla ry , w ater and an aqueous sucrose mixture with 65 % mass fraction assuming
a 1 mm sample thickness and the nominal sp ecifications of the glass capilla ry . The ratio b et w een the
w ater and the sucrose mixture transmittances is sho wn in the right axis. b) Sucrose mass fraction derived
from an exp erimental transmittance measurement of a 65 % sucrose density gradient measured at t w o
different energies under simila r exp erimental conditions. The abso rbance differences a re smaller for the
higher energy .
4.1.3 SAXS measur ements
In or der to collect the scattering patterns, the sample is moved in steps of 0.5 mm along the
central vertical capillary axis and exposed at each position for about 1 minute. The acquis-
ition time depends notably on the experimental parameters (e.g. sample concentration,
scattering power of the material...), though it is strictly limited by the dif fusion time of the
contrast agent. At these positions, the solution transmittances were pr eviously measur ed
and the suspending medium electr on density calibrated, as described previously . Due to
a vertical beam size of about 0.5 mm, the measured scattering curve is an average over
a range of solvent electr on densities, specially r elevant at the height where the density
gradient is steeper .
As a consequence of the observations from figur e 4.3a, the incident photon energy
E = ( 8000.0 ± 0.8 )
eV was chosen to be higher than the photon energy employed for the
transmission measur ements to improve the r ecorded statistics, due to a ca. 200 higher
transmission (Henke et al. , 1993). On the other hand, the decr easing photon flux at the
FCM beamline for high ener gies as depicted in figure 3.5, suggest the utilization of photon
ener gies below 9 keV in scattering experiments.
The dimensions of the investigated particle defines the r equired
q
-range of the exper-
iment, which is delimited by the photon ener gy and the sample-detector distance, as
discussed in section 3.4. The photon ener gy is generally limited by the needs of the sample
envir onment, but the distance can be adjusted with the HZB-SAXS instrument to the
nanoparticle r equirements and can compensate the ener gy r estriction. For sizes typically
ranging fr om 10 to 200 nm, the sample-detector distance is fixed at 4500 mm and enables
q -values between 0.03 and 1.1 nm − 1 at 8000 eV .
Since the installation in April 2016 of the monitor diode on the beamstop pr esented in
section 3.4.1, the sample transmittance can be r ecorded simultaneously with the scattering
patterns. The longer integration times requir ed for the scattering experiments (around
60 s) incr ease by a factor 6 the statistics of the simultaneous X-ray transmission meas-
ur ement, improving the quality of the transmittance data. The possibility to collect the
39

Chapter 4
CONTINUOUS CONTRAST V ARIA TION IN SAXS: THE DENSITY GRADIEN T
. . .
Figure 4.4 |
Exp erimental
scattering curves of the
PS-COOH nanopa rticles
fo r different susp ending
medium electron densit-
ies measured b et w een 78
and 93 minutes after the
inception of the densit y
gradient. The gray line
sho ws the exp erimental
background, containing
scattering contributions
from the capilla ry and the
pure solvent. 1
10
100
1000
0.03 0.05 0.1 0.2 0.3 0.5
Scattering Intensity / a.u.
q / nm − 1
335
340
345
350
355
360
Solvent Electron Density / nm − 3
scattering data at the same photon ener gy that the solution transmittances improves the
normalization of the scattering curve and the calibration of the solvent electr on densities.
However , all the results pr esented in this work were r ecor ded before the commissioning
of the beamstop diode.
4.2 Pr oof of principle: application to the PS-COOH particles
In or der to demonstrate the proposed continuous contrast variation technique, carboxylated
polystyr ene nanoparticles with a nominal size of 105 nm suspended in water (Kisker ,
Steinfurt, Germany) wer e measured following the pr ocedure described pr eviously in
this chapter . The particles have a narr ow size distribution and consist of a spherical
polystyr ene (PS) core enclosed by a thin shell of a denser polymer , most likely poly(methyl
methacrylate) (PMMA). The synthesis by multi-addition emulsion polymerization is re-
sponsible of the cor e-shell structure found in these PS-COOH particles and suggests that
all the particles have the same average density independent on their size.
The density gradient capillary was built accor ding to the description in section 4.1.1
using two aqueous mixtur es with a particle concentration of 12.6 mg ml
− 1
accor ding to
the pr oducer ’s specification. The dense aqueous solution was prepar ed with 21.23 %
sucr ose mass fraction with a mass density of
ρ 1 =
1.088 g cm
− 3
, whereas a lighter one was
pr oduced without sucrose (
ρ 2 =
0.997 g cm
− 3
). In total, 40 scattering curves with differ ent
solvent electr on densities were measur ed at two differ ent times
t 1 =
78 min and
t 2 =
156
min after filling the capillaries.
The measur ed scattering curves of the PS-COOH particles are displayed in figur e 4.4.
In the r egion for
q
fr om 0.03 nm
− 1
to 0.5 nm
− 1
it is possible to observe the variation of
the curve featur es corresponding to the particle form factor thr ough the increase of the
solvent electr on density from 333.7 nm
− 3
at the top edge of the density gradient to 360.3
nm
− 3
at the maximum sucr ose concentration. In this r egion, the experimental background
is composed mainly by the contribution of the capillary scattering at the low
q
-r egion
and the uniform scattering of the suspending medium. The experimental background
scattering varies for dif ferent sucr ose concentrations, but their variations are small and the
40

Pro of of p rinciple: application to the PS-COOH particles 4.3
1
10
0.05 0.1 0.2 0.5 1
Scattering Intensity / a.u.
q / nm − 1
Original curve
W ater Background
Subtracted Curve Figure 4. 5 |
The thick
blue line sho ws the scat-
tering curve measured
at
ρ solv
= 345.4 nm
− 3
,
close to the match p oint,
and the black line displa ys
the exp erimental back-
ground. The red sym-
b ols with erro rba rs sho w
the background co rrected
scattering curve.
backgr ound remains one or der of magnitude below the sample scattering in the relevant
Fourier r egion.
Upon incr easing the solvent density , the position of the first minimum shifts fr om
0.07 nm
− 1
towar ds smaller
q
-values until it vanishes when the solvent electron density
matches the average electr on density of the measured particle. In the Fourier region of the
scattering curves, several minima are observed which shift towar ds smaller
q
-values when
incr easing the solvent electron density . Upon subtracting the experimental background
fr om the scattering curve, a decrease of the scattering intensity towar ds
q =
0 is observed
only for the solvent electr on density closest to the match point as depicted in figure
4.5. Ther efore, backgr ound corrections can be neglected for systems with r elatively high
scattering power like in this study . For low-scatterers, an accurate backgr ound corr ection
by measuring the pur e suspending medium at differ ent sucrose concentrations might be
r equired. The behaviour at low
q
-values will be further discussed in section 4.3.3 when
evaluating the zer o-angle intensity .
The pr esence of the clearly visible isoscattering point around
q =
0.09 nm
− 1
confirms
the existence of an inner structur e. This heter ogeneous composition was previously
r eported for the same colloids by Minelli et al. (2014), who observed methacrylic acid
(MAA) and methylmethacrylate (MMA) at the particle surface, both monomer pr ecursors
of PMMA polymerization. A more detailed insight into the radial morphology is pr esented
subsequently , using the theoretical framework intr oduced in chapter 2.
41

Chapter 4
CONTINUOUS CONTRAST V ARIA TION IN SAXS: THE DENSITY GRADIEN T
. . .
Figure 4.6 |
The sim-
ulated scattering curves
from the co re-shell mo del
fit at three selected
contrasts
ρ 0 − ρ solv
a re
sho wn as lines together
with the exp erimental
data p oints. In the in-
set, the electron density
p rofile co rresp onding to
the fitted co re-shell fo rm
facto r is displa yed. 1
10
100
1000
0.05 0.1 0.2 0.3 0.5
Scattering Intensity / a.u.
q / nm − 1
330
340
350
360
0 10 20 30 40 50
ρ e / nm − 3
R / nm
11.4 nm − 3
0.4 nm − 3
-11.2 nm − 3
4.3 Results and data evaluation
The scattering curves of the PS-COOH nanoparticles measur ed at several contrasts can
be analysed using dif ferent, complementary evaluation methods. In this section, both
a model-fr ee theoretical framework as well as a cor e-shell model fit are applied and, in
combination, deliver a detailed insight into the inner structur e of particles.
4.3.1 Cor e-shell form factor fit
A cor e-shell model fit to the scattering curves is displayed in figure 4.6 for thr ee rep-
r esentative contrasts, which employs the form factor described by expression 2.27. The
simultaneous fitting of the form factor to the 40 measured scattering curves was performed
by means of the method of least squar es in the Fourier region (Pedersen, 1997). The cal-
culated scatter ed intensity was modelled as the sum of the particle contributions and a
two-component backgr ound
I BG = C 0 + C 4 q − γ
. The parameters
ρ core
,
ρ shell
,
R
,
R core
and
γ
wer e fitted simultaneously for all curves, whilst
C 0
and
C 4
wer e adjusted independently
for each solvent density . A Gaussian size distribution was assumed. For the suspending
medium electr on density
ρ solv
appearing in the contrast
∆ η
, the value determined fr om
the transmission measur ement was used for each curve.
The obtained r esults are
R = ( 49.7 ± 2.8 )
nm,
R core = ( 44.2 ± 0.9 )
nm,
ρ core = ( 339.7 ± 0.1 )
nm
− 3
and
ρ shell = ( 361.9 ± 2.0 )
nm
− 3
, which repr esent the radial structur e of a dense, thin
shell surr ounding a lighter core, as seen in the inset of figur e 4.6. The r esulting average
electr on density of the particle is
ρ 0 = ( 345.9 ± 1.5 )
nm
− 3
and the polydispersity degr ee,
p d = ( 22.8 ± 6.0 )
%. The best fitting backgr ound corresponds to a value of
γ =
4.3
±
0.5,
close to the case
γ =
4 originating fr om large impurities or pr ecipitates (Pedersen, 1994).
The fit uncertainty was calculated with a confidence interval of one standar d deviation.
The dif ferent contributions to the uncertainty associated to the external radius of the
particle
R
ar e detailed in table 4.1, where the uncertainties given ar e standar d uncertain-
ties (
k =
1). Besides the fit uncertainty , the table summarizes the contributions fr om
the ener gy resolution of the photon beam (Krumr ey & Ulm, 2001), the accuracy of the
distance between the irradiated sample and the scattering detector , the detector pixel size
42

Results and data evaluation 4.3
Input quantity u I u r Contribution
Photon ener gy 0.9 eV 10 − 4 0.005 nm
Sample-detector distance 5 mm 10 − 3 0.05 nm
Pixel size 0.2 mm 10 − 3 0.05 nm
Centr e determination 2 pixels n.a. 0.5 nm
Cor e-shell fitting 2.8 nm 6 · 10 − 2 2.8 nm
Combined standard uncertainty 2.8 nm
T able 4.1 |
Uncertaint y
contributions asso ciated
to the PS-COOH radius
R
determined b y a a co re-
shell mo del fit, where
u I
and
u r
co rresp ond to the
input uncertaint y and rel-
ative uncertaint y resp ect-
ively .
(W ernecke et al. , 2014) and the determination of the scattering centr e. As in this case and
in the examples appearing in chapter 5, the uncertainty is typically dominated by the
contribution arising fr om the fitting procedur e.
Besides, it is noticeable that the calculated electron density of the cor e coincides exactly
with the theor etical polystyrene electr on density , although the electron density of the shell
is r emarkably lower than the theoretical value of 383.4 nm
− 3
for PMMA (Ballauf f, 2001).
This might arise fr om the lower density of the monomers used in the particle synthesis
(MAA and MMA), which could have mixed with the styrene monomers r esulting in a
less dense material than PMMA. This model might pr esent some differ ences with the
r eal colloid system, as a dif fusive interfacial layer could be expected between polymer
phases in colloids (Dingenouts et al. , 1994 a ), especially for incompatible polymers such as
PMMA and PS. On the other hand, the large quantity of scattering curves used for the
fitting pr ocess and, accordingly , the decreased uncertainty suggests that the chosen sharp
cor e-shell model has a great r esemblance to the real particle.
4.3.2 Isoscattering point
Although the first isoscattering point is clearly visible in figur e 4.4, a model-free appr oach
like the isoscattering point r equires of a mor e precise determination of the position and
a quantitative evaluation. For this purpose, the r elative standard deviation
σ r
of the 40
measur ed curves at each q is calculated according to
σ r ( q ) = 1
¯
I ( q ) s ∑ M
i = 1 ( I i ( q ) − ¯
I ( q ) ) 2
M − 1 , (4.2)
wher e
¯
I ( q )
is the mean value of the intensity at
q
and
M
is the number of scattering curves.
This value becomes minimal at an isoscattering point. In order to r educe the influence
of outliers, a truncated mean value was utilized, disr egarding the 10 % most dispersed
data points. In figure 4.7a, the r elative standard deviation is plotted as a function of the
momentum transfer
q
, which shows several distinguishable minima corresponding to
isoscattering points.
A pr ecise determination of the isoscattering point positions is performed by fitting
Lor entzian functions to the minima in the relative standar d deviation plot, which allows
the calculation of the model-fr ee external radius of the particle by means of equation
2.30. The results ar e presented in table 4.7b together with their associated uncertainties
calculated accor ding to the uncertainty budget presented in table 4.2. The sources con-
tributing to the uncertainty associated to the position of
q ?
ar e similar to those reviewed
in table 4.1 for the cor e-shell fit. In addition, the chosen
q
-bin size and the corr ection of
the backgr ound contributions from the solvent ar e also consider ed. The dif fuseness of the
isoscattering point is quantified by computing the width of the momentum transfer (
∆ q
)
43

Chapter 4
CONTINUOUS CONTRAST V ARIA TION IN SAXS: THE DENSITY GRADIEN T
. . .
0.1
0.2
0.5
1
0.07 0.1 0.2 0.5
Rel. Std. Deviation
q / nm − 1
q ?
1 q ?
2 q ?
3
q ?
4
q ?
5
0.1
0.2
0.085 0.09 0.095
∆ q
∆ q
(a) Relative standard deviation
* ∆ q at 2 σ rel ( q ? ) can not be computed.
q ? (nm − 1 ) R (nm) u c (nm)
q ?
1 0.090 ± 0.006 49.9 3.3
q ?
2 0.152 ± 0.013 51.0 4.4
q ?
3 0.23 ± 0.05 48.1 9.5
q ?
4 0.28 ± 0.07 49.9 12.4
q ?
5 0.34 50.3 *
(b) Isoscattering point positions
Figure 4.7 |
Isoscattering p oints of the PS-COOH pa rticles: a) Relative standard deviation of the scattering
curves as a function of the momentum transfer. The lab elled minima co rresp ond to the first five isoscattering
p oint p ositions calculated b y fitting a Lo rentzian function (black line). In the inset, the width
∆ q
of the
first minimum at a value of 2
σ rel ( q ? ) =
0.22 is depicted, which quantifies the diffuseness of
q ?
1
due to
p olydisp ersit y effects or deviations from the spherical shape. b) Exp erimentally determined p osition of the
first five isoscattering p oints and the co rresp onding external pa rticle radius
R
. The combined standa rd
uncertaint y
u c
asso ciated to the radius is calculated acco rding to table 4.2, where the diffuseness of
q ?
p rovides the la rger contribution.
at a r elative standard deviation value two times lar ger than the value at the minimum
(
σ rel ( q ? )
), as depicted in the inset of figur e 4.7a. The width
∆ q
gives an estimation of the
uncertainty associated to the dif fuseness of
q ?
intr oduced in section 2.3.1 related to the
non-ideality of the particles, i.e. their polydispersity or the deviation of the particle shape
fr om the spherical model. As observed in the uncertainty calculation associated to the
q ?
1
position in table 4.2, the dif fuseness of
q ?
is the lar gest contribution to the combined
standar d uncertainty .
The obtained particle radii displayed in table 4.7b vary in the range fr om 48.1 nm to 51.0
nm, although as pr edicted by Kawaguchi & Hamanaka (1992) for a polydisperse system,
the isoscattering points get smear ed out for larger
q
-values and the pr ecision decreases,
simultaneously with the incr ease of the solvent background at higher
q
-values. This can
be dir ectly observed in the quality of the experimental data, as the first two minima are
T able 4.2 |
Uncertaint y contributions asso ciated to the first isoscattering p oint
q ?
1
p osition. The main
contribution a rises from the diffuseness of
q ?
which is quantified b y calculating the width
∆ q
at a value of
2
σ rel ( q ? )
. The uncertainty associated to
R
is derived from the exp ression 2.30, which preserves the relative
uncertaint y of q ? and R .
Input quantity u I u r Contribution
Photon ener gy 0.9 eV 10 − 4 0.000009 nm − 1
Sample-detector distance 5 mm 10 − 3 0.00009 nm − 1
Pixel size 0.2 mm 10 − 3 0.00009 nm − 1
Centr e determination 2 pixels n.a. 0.0009 nm − 1
q -bin size 0.0017 nm − 1 2 · 10 − 2 0.0017 nm − 1
Solvent backgr ound 0.0015 nm − 1 2 · 10 − 2 0.0015 nm − 1
Dif fuseness of q ? ( ∆ q ) 0.006 nm − 1 7 · 10 − 2 0.006 nm − 1
Combined standard uncertainty of q ? 7 · 10 − 2 0.006 nm − 1
Combined standard uncertainty of R 7 · 10 − 2 3.3 nm
44

Results and data evaluation 4.3
-40
-20
0
20
40
60
335 340 345 350 355 360
Deviation fr om I ( 0 ) / %
Solvent electr on density / nm − 3
Guinier approximation
Lowest available q
Figure 4.8 |
Deviation
from the
I (
0
)
values used
in the data evaluation:
The Guinier app ro xima-
tion overestimates the ex-
p erimental values, while
the intensit y at
q =
0.03
nm
− 1
underestimates the
zero-angle intensit y .
clearly mor e pronounced and have smaller uncertainties than the subsequent minima,
which appear smear ed out. For instance, the isoscattering point
q ?
5
is alr eady too weak
for an accurate evaluation and the thir d minimum shows two remarkably close smaller
minima which might af fect the shape of the function. Ther efore,
q ?
1
and
q ?
2
yield the most
r eliable values for evaluating the external radius of the particles. The weighted average
value derived fr om the first two isoscattering points
R = ( 50.3 ± 2.8 )
nm dif fers by only
1.2 % fr om the radius calculated from the model fit in the pr evious section.
Due to the existance of the isoscattering point dif fuseness, a quantitative determination
of the polydispersity of the suspended nanoparticles by means of the Lor entzian profile is
rather challenging. Nevertheless, the narrow size distribution of the sample becomes clear
by comparing the r elative standard deviation values of the observed minima in figur e
4.7a with a simulation using the structural parameters obtained in section 4.3.1. The value
σ r ( q ?
1 ) =
0.11 corr esponds to a calculated ensemble polydispersity of 24 %. This value
serves as an upper
p d
limit due to the possible over estimation caused by the scattering
contribution of the suspending medium.
4.3.3 Guinier r egion
By analysing the low
q
-r egion of the scattering curves, the so-called Guinier region, two
important parameters can be obtained: the radius of gyration
R g
r elated to the size of
particle and the average electr on density
ρ 0
derived fr om the intensity at zero angle
I (
0
)
.
Accor ding to Feigin & Svergun (1987), the fit of equation 2.34 to the Guinier region is
mainly valid up to
q R g <
1.3. In this r estricted
q
-range, too few data points ar e available
for a r eliable data analysis. Ther efore, an extrapolation using the spherical form factor
F sph ( q
,
R )
over the range available befor e the first minimum has been employed instead
to obtain
R g
and
I (
0
)
. This arises as a good choice because the Guinier appr oximation
over estimates the values of the zero-angle intensity due to its limitation to monodisperse
systems (Feigin & Sver gun, 1987), as observed in figure 4.8. On the other hand, a more
primitive appr oach, e.g. the intensity of the lowest accessible
q
-value (
q min =
0.03 nm
− 1
),
under estimates the
I (
0
)
values, because it neglects the extrapolation to
q →
0, as shown
also in figur e 4.8.
45

Chapter 4
CONTINUOUS CONTRAST V ARIA TION IN SAXS: THE DENSITY GRADIEN T
. . .
Figure 4.9 |
Exp erimental
squa red radius of gyra-
tion as a function of the
solvent electron densit y .
Equation 2.35 is fitted
to the data and sho wn
as a thick line. The
vertical and ho rizontal
asymptotes co rresp ond to
ρ 0 and ˜
R 2
g , c resp ectively .
500
1000
1500
2000
2500
3000
330 335 340 345 350 355 360 365 370
R 2
g / nm 2
Solvent electr on density / nm − 3
ρ 0
˜
R 2
g , c
As described in section 2.3.2, the radius of gyration of a heterogeneous particle in a
contrast variation experiment should behave accor ding to equation 2.35. In figure 4.9,
the experimental squar ed radius of gyration is displayed as a function of the suspending
medium electr on density . The best fit to the measur ed data with values
ρ 0 = ( 343.7 ± 1.5 )
nm − 3 , ˜
R g , c = ( 39.0 ± 5.2 ) nm, ˜
α = 4470 nm − 1 and ˜
β = 0 nm − 4 is shown by the solid line.
The uncertainty associated to the average electr on density of the particle
ρ 0
originates
mainly fr om the beam size, as described in section 4.1.2. On the other hand, the uncer -
tainty r esulting from the fit of equation 2.34 is the dominant contribution to the radius
uncertainty .
The positive value of
˜
α
validates the hypothesis that a mor e dense polymer like PMMA
surr ounds a lighter core (PS) (Stuhrmann, 2008). The calculated average electron density
of the particle
ρ 0
suggests a very thin layer of PMMA shell ar ound the PS core, due to
the pr oximity of its value to the polystyrene electr on density (339.7 nm
− 3
). The value
of
˜
β =
0 pr oves a concentric model, wher e cor e and shell share the same centr e. Using
the same polydispersity value of 22.8 % obtained in the fitting pr ocess, the value for
the particle shape radius of gyration r esults in
R g , c = ( 36.9 ± 4.9 )
nm and the external
radius of the particle can be calculated assuming the particle as a spherical object. This
calculation gives
R = ( 47.6 ± 6.4 )
nm, which is only 2.1 nm smaller than the external
radius
R = ( 49.7 ± 2.8 )
nm calculated with the cor e-shell model fit, though it might be
under estimated due to the choice of a possibly inflated polydispersity .
A verage electron density
Using the same set of 40 scattering curves, the behaviour of the zero-angle intensity under
the contrast variation is also investigated by fitting equation 2.36 to the experimental
I (
0
)
,
as depicted in figur e 4.10. A minimum in the curve is observed at
ρ solv = ( 346.0 ± 1.5 )
nm
− 3
, which corresponds to the value of the average electr on density of the particle.
This value is in very good agr eement with the result obtained by fitting the cor e-shell
form factor of
ρ 0 = ( 345.9 ± 1.5 )
nm
− 3
. It is also noticeable that the minimum intensity
is appr oximately 0, which means that the ef fective average density of the ensemble
˜
ρ 0
is equal to the average density of the particle
ρ 0
(A vdeev, 2007). This r esult further
46

Results and data evaluation 4.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
335 340 345 350 355 360
I ( 0 ) / a.u.
Solvent electr on density / nm − 3
Figure 4.10 |
Exp eri-
mental zero-angle intens-
it y as a function of the
solvent electron densit y .
The function co rresp ond-
ing to equation 2.36 is fit-
ted to the data and sho wn
as a thick line. The min-
imum in the pa rab ola co r-
resp onds to ρ solv = ρ 0 .
legitimates the assumption made pr eviously in section 4.2 about the PS-COOH particles
that the ratio between the particle components’ volumes is constant independent of the
polydispersity and hence ˜
ρ 0 = ρ 0 , i.e. the average density of the particle is not altered by
the size polydispersity .
4.3.4 Consistency of the r esults
T able 4.3 summarizes the r esults of all three pr esented methods. Fr om the first two
isoscattering points, a value for the external radius of
( 50.3 ± 2.8 )
nm and an upper bound
to the polydispersity degr ee have been derived. Focusing on the Guinier r egion of the
scattering curves, a value for the average electron density of the particles
ρ 0
is found using
the radius of gyration (
( 343.7 ± 1.5 )
nm
− 3
) as well as the zer o-angle intensity (
( 346.0 ± 1.5 )
nm
− 3
), the values of which differ by 2.3 nm
− 3
and lie within their confidence intervals.
By fitting a cor e-shell model, an external radius of
R = ( 49.7 ± 2.8 )
nm and an average
electr on density
ρ 0 = ( 345.9 ± 1.5 )
nm
− 3
have been obtained, which ar e in considerable
agr eement with the previous r esults. In fact, the values of
R
and
ρ 0
determined by dif ferent
methods agr ee with each other within their stated confidence ranges.
Fr om the results pr esented in table 4.3, the radius of gyration interpr etation produces
the most deviant values and the lar gest uncertainties. This might be founded in the
complicated function fitted to the data and the r educed availability of
q
-range employed to
obtain
R g
. The r esulting polydispersity degree of the measur ed particles from the model
fit is in agr eement with the upper limit obtained with the radii of gyration. Nevertheless
the polydispersity is the parameter determined with the lar gest uncertainty in the fitting
pr ocess and therefor e this result must be consider ed with car e.
It can be concluded that the dif ferent appr oaches show consistent and complementary
r esults about the size distribution of nanoparticles with radial inner structur e, especially
for the external radius of the particle and its average electr on density . A precise value
for the polydispersity degr ee could not be obtained as explained previously , although a
cr edible upper limit to the polydispersity degree of 24 % could be given.
47

Chapter 4
CONTINUOUS CONTRAST V ARIA TION IN SAXS: THE DENSITY GRADIEN T
. . .
T able 4.3 |
Compa rison of the results obtained b y the different approaches p resented in section 4.3 to
evaluate contrast va riation SAXS data.
R (nm) ρ 0 (nm − 3 ) p d (%)
Cor e-shell fitting 49.7 ± 2.8 345.9 ± 1.5 22.8 ± 6.0
Isoscattering point 50.3 ± 2.8* - < 24
Radius of gyration 47.6 ± 6.4** 343.7 ± 1.5 -
Zer o-angle intensity - 346.0 ± 1.5 -
*W eighted average value of q ?
1 and q ?
2
**Using the polydispersity degr ee from the cor e-shell model fitting
4.4 Applicability and comparison with other contrast vari-
ation appr oaches
The accessible electr on density range defines the possible applications of the proposed
technique and is consequently the most decisive factor to choose the contrast agent. W ith
saccharides like sucr ose or fructose, high concentrated mixtures with low viscosity can
be achieved, r eaching electron densities up to 400 nm
− 3
. Sugars ar e suitable for contrast
variation experiments with bio-materials and polymeric nanoparticles, whose densities
typically range between 0.9 and 1.4 g cm
− 3
(fr om 300 to 450 nm
− 3
). On the other hand,
contrast agents like ethanol can r educe the electron density of the suspending medium
until 270 nm
− 3
and, besides, is perfectly miscible with water . A wide variety of biological
particles exist within the available density range achieved between ethanol and sugar .
Mor e dense solutions prepar ed with heavy salts (e.g. sodium polytungstate (SPT))
could be an alternative for heavier particles e.g. silica, similarly to the application in
sink-float analysis and density gradient centrifugation (Rhodes & Miles, 1991; Mitchell &
Heckert, 2010). Nevertheless, the salt can compromise the stability of the particles inducing
aggr egation and lead to more complicated handling of the sample due to a decr eased
dif fusion timescale. The chemical stability of the suspension is a crucial parameter that
depends specifically on the investigated sample, but in general neutral contrast agents
like sugars ar e preferr ed to salts.
Another r elevant characteristic of the contrast agents is its scattering contribution to the
backgr ound. Generally , the background scattering of the suspending medium is dir ectly
pr oportional to the contrast agent concentration and can affect notably the scattering
data at the Fourier r egion, as observed in this chapter . Besides, the size of the diff using
molecule r elates to the background intensity , where lar ger molecules like sucr ose (ca. 342
g mol
− 1
) have a higher scattering power than smaller ones like fructose (180 g mol
− 1
)
at the same mass fraction. Ther efore, a compr omise is requir ed between the size of the
contrast agent molecule, its solubility in an aqueous medium and the diffusion timescale
of the solute.
In addition to solvent contrast variation in SAXS, other possible methods that vary
the contrast of a single medium have alr eady been proposed. Contrast variation in
SANS is the most widespr ead technique (Ballauff, 2011, 2001), reaching high contrasts
between sample and medium thr ough the opportune substitution of hydrogen atoms by
deuterium atoms. T ypically , the scattering length density of the medium is changed by
the appr opriate mixture of water and deuterated water , although the scattering density
of polymeric particles can also be modified by substituting a polymeric species by its
48

Applicabilit y and compa rison with other contrast va riation app roaches 4.4
deuterated equivalent (Rosenfeldt et al. , 2002). The contrast range achieved with this
technique is much br oader than that possible with SAXS, but the intrinsic experimental
dif ficulties of neutron scattering experiments limit its usage to specific sample systems.
Other appr oaches to contrast variation in X-ray scattering are based on the anomalous
behaviour of the atomic scattering amplitude near an absorption edge of an element con-
tained in the sample or in the medium. Anomalous SAXS (ASAXS) has been a well-known
technique in material science since its intr oduction by Stuhrmann in 1985 (Stuhrmann,
1985) and has been applied to a variety of colloids and polyelectr olytes at the hard X-ray
r egion (Goerigk et al. , 2003; Stuhrmann, 2007; Lages et al. , 2013). The r ecently introduced
Resonant Soft X-ray Scattering (RSoXS) method aims for absorption edges at much lower
ener gies than ASAXS, like the so-called water window below 530 eV . By focusing the photon
beam into a micr ometric spot, the polymeric components of latex nanoparticles could
be characterized due to their dif ferent chemical bond sensitivity near the carbon K-edge
(ar ound 285 eV) (Mitchell et al. , 2006; Araki et al. , 2006). The application of these tech-
niques r equire of a sample system specially tailor ed for the experimental needs, where
the pr obed atomic element is found in high concentrations. Besides, technical dif ficulties
ar e also present, like the need for very thin sample thicknesses in RSoXS or the high
monochr omacy of the hard X-ray photon beam r equired in ASAXS.
Although the contrast variation appr oach presented in this work pr esents certain limita-
tions, it shows evident advantages with respect to the other existing contrast variation
techniques. For instance, solvent contrast variation is not element specific and the photon
ener gy can be selected more fr eely , within the r estrictions arising from the sample attenu-
ation described pr eviously . Moreover , the investigated particles can be used without any
chemical tr eatment, unlike deuteration in SANS or atomic labelling in ASAXS. On the
other side, the accessible density range of the contrast agent r educes the employment of
the technique to r elatively low density particles.
4.4.1 Other possible applications of the density gradient capillary
The dif fusion time of a particle depends mainly on its size, as described by the Stokes-
Einstein expr ession of the diffusion constant (Einstein, 1905):
D = K B T
6 π η
1
R (4.3)
wher e
K B
is the Boltzmann constant,
T
is the solvent temperatur e,
η
is the dynamic
viscosity and
R
is the radius of the particle. For example, the small size of ions (below
200 pm) decr eases the diffusion timescale in a factor 5 in comparison with a disaccharide
molecule like sucr ose. On the opposite side, colloids can be considered dif fusive agents
which multiply the dif fusion time up to 100 times.
In figur e 4.11, the calibrated transmittance of a density gradient capillary of aqueous 12
nm silica nanoparticles (Ludox HS40, Sigma-Aldrich, Missouri, USA) is depicted, where
the particle concentration is a function of the capillary height. The slower evolution of the
concentration gradient compar ed with sucrose in figur e 4.2 and the large density dif fer ence
between water and the high concentrated particle suspension (ca. 1.3 g cm
− 3
) can impr ove
the quality of the X-ray transmission data and pr ovide an alternative application of
the density gradient technique in SAXS, wher e the diffusive agent is the investigated
object. Moreover , table-top X-ray sour ces can be an alternative to high photon flux
synchr otron radiation sour ces due to the extended experimental timescale achieved when
49

Chapter 4
CONTINUOUS CONTRAST V ARIA TION IN SAXS: THE DENSITY GRADIEN T
. . .
0
5
10
15
20
25
30
35
40
0 2 4 6 8 10 12 14
1.2
2
4
7
Particle Mass Concentration / %
X-ray T ransmission / %
V ertical Position / mm
100
120
140
160
180
200
220
240
Diffusion T ime / min
Figure 4.11 |
Concentration gradient of 12 nm silica pa rticles measured at 8000 e V. The la rge size of the
colloids in compa rison to a saccha ride molecule provides a longer diffusion time than a t ypical contrast
agent lik e sucrose.
using colloids as dif fusing agent.
A colloidal concentration gradient as pr esented in figure 4.11 can be used to study the
ef fects of concentration on the diffusion constant of the particles or investigate the type
of inter -particle interactions as a function of the colloidal concentration. For example,
standar d dilution series can be performed in situ with this approach or examine the
crystallization of the particles under gravitational for ces (Hellsing et al. , 2012).
50

5
Simultaneous size and density
determination of polymeric colloids
The curr ent advances in nanomaterial development for medical applications are focused
towar ds tailoring polymeric nano-drug carriers with flexible surface functionalisation and
contr olled morphologies (Euliss et al. , 2006; Y ang et al. , 2005). Size and shape, combined
with the choice of polymer and the mechanical pr operties, are fundamental and defining
aspects of the particle functions, e.g. their in-vivo biodistribution (V ittaz et al. , 1996;
Mitragotri & Lahann, 2009; Doshi & Mitragotri, 2009) or their drug-delivery ef ficacy
(Powers et al. , 2006). Therefor e, a full and consistent characterization of all properties
of nanoparticles is of crucial importance and must be car efully adr essed, especially for
polymeric NPs due to their typical complicate internal structur e.
This chapter demonstrates the simultaneous size and density determination using
continuous contrast variation technique in SAXS with 3 polymeric particles of differ ent
sizes and polymeric species. By means of an aqueous sucr ose density gradient, the
measur ements were achieved along a lar ge range of suspending medium densities, fr om
water density to that of poly(methyl methacrylate)’s, highlighting the relevance of the
technique acr oss a wide spectrum of polymers.
The applicability of this method for the traceable size determination of these colloids
is discussed in this chapter , wher e a high-resolution size distribution of the particles is
pr esented. Focusing on a low-density colloid, differ ent evaluation appr oaches to SAXS
contrast variation experiments ar e discussed and the advantages and drawbacks of a
model-fr ee formulation like the isoscattering point position are discussed, together with
the accuracy of the shape scattering function. In addition, a form factor model is fitted to
the scattering curves to obtain decisive information about the internal morphology of the
particle, which is not dir ectly available by other techniques such as transmission scanning
electr on microscopy (TSEM), dif ferential centrifugal sedimentation (DCS) (Fielding et al. ,
2012) or atomic for ce microscopy (AFM).
Besides, the ability of this technique to determine the density of polymeric colloids in
suspension is also discussed. Normally , the density of the suspended particles can not
be compar ed to the bulk density of the dry material. Such a complex question has been
51

Chapter 5
SIMUL T ANEOUS SIZE AND DENSITY DETERMINA TION OF POL YMERIC
. . .
addr essed by differ ent methods, though with evident limitations. For example, the density
of polymeric beads has been measur ed previously with field-flow fractionation (FFF) with
high-accuracy but at the expense of a priori assumptions about the morphology of the
particle (Giddings et al. , 1981; Y ang et al. , 1983; Caldwell et al. , 1986). Another method
which r equires of pr evious knowledge about the size of the particle is isopycnic centri-
fugation, widely used in biology (V authier et al. , 1999). Assuming the Stokes’ diameter
as the actual size of the colloid, recent advances in analytical ultracentrifugation allow
the complementary characterization of the size, density and molecular weight of gold
nanoparticles (Carney et al. , 2011).
The density of the 3 polymeric colloids was also analysed by DCS and the r esults
compar ed and discussed with those obtained by SAXS. DCS uses the sedimentation of
particles thr ough a density gradient to measure high r esolution particle size distributions
(Minelli et al. , 2014). Its accuracy typically depends on the knowledge of the density of
the particles. When the size of the particle is known, DCS can alternatively be used to
measur e average particle’s density .
In this study , the size and density of low-density particles is independently determined
by performing DCS measur ements with two differ ent discs using the sedimentation
and flotation r espectively of the particles through a density gradient and solving the
r elative Stokes’ equations. A similar approach to DCS which combines the r esults of two
independent measur ements has been investigated previously . For example, Neumann
et al. (2013) used two sucr ose gradients resulting in dif ferent viscosities and densities,
wher e the altered settling velocity combined with linear r egression analysis was used for
the calculation of the size and density of silica nanoparticles and viruses. Bell et al. (2012)
adopted a two gradient method based on the variation of the sucr ose concentration to
determine the density of the Stöber silica and the calibration standar ds used in DCS. Parts
of this chapter have been adapted fr om an article published previously (Gar cia-Diez et al. ,
2016 b ).
5.1 Materials and methods
In this section, a detailed description of the polymeric nanoparticles employed in the
experiments is pr esented. The experimental procedur e of the continuous contrast variation
technique is thor oughly discussed already in chapter 4, thus the focus of the section lies
only on the DCS technique. Special interest is put on the description of the combined DCS
appr oach based on the floating-sedimentation principle.
5.1.1 Polymeric particles
The experiments wer e performed using 3 differ ent types of polymeric nanoparticles, whose
diameters range fr om 100 nm to around 187 nm. Carboxylated poly(methyl methacrylate)
colloids (PMMA-COOH) with a nominal diameter of 187 nm and plain polystyr ene
particles (PS-Plain) polymerized with
<
1 wt% of a surface-active co-monomer with a
nominal diameter of 147 nm wer e purchased fr om Microparticles (Berlin, Germany). The
PS-COOH particles ar e described in detail in chapter 4 and are composed of a PS cor e
surr ounded by a PMMA shell. The phyisical densities of the NPs range from that of PS
(1.05 g cm − 3 ) until PMMA ’s, which has a density of ca. 1.18 g cm − 3 .
For the pr eparation of the high density aqueous sucrose solutions employed in the
density gradient capillaries, the suspended colloids were mixed with a sucr ose mass
52

Materials and metho ds 5.1
Detector
light beam
Injection
point
Gradient
fluid
Rotating disc
(a) DCS setup at the initial time
Small
particles
Lar ge
particles
(b) Size fractionation after a time t
Figure 5.1 |
Scheme of the differential centrifugal sedimentation technique. a) A DCS setup consists of a
disc rotating with a sp eed
Ω
filled with a gradient liquid with average densit y
ρ f
. At a certain distance of
the p oint where the pa rticles a re injected, the attenuation of the light b eam is measured. b) After a time
t
,
the pa rticles a re separated due to the centrifugal fo rce dep ending on their size, where la rger pa rticles a re
detected ea rlier than smaller ones.
fraction of 21.2 %, 42.5 % and 13.4 % for the PS-COOH, PMMA-COOH and PS-Plain
particles r espectively .
5.1.2 Dif ferential Centrifugal Sedimentation
The Dif ferential Centrifugal Sedimentation (DCS) technique is based on the fractionation
of particles in suspension by centrifugal sedimentation within a r otating, optically clear
disc containing a liquid medium with a density gradient, as depicted in figure 5.1a. The
time needed by the particles to r each the detector light beam at the edge of the disc
depends on their pr operties (e.g. size and shape) and can be converted into a particle size
distribution, as schematically pr esented in figure 5.1b. Particles with densities lower or
similar to water ’s can be measured in a mor e dense liquid medium within the centrifuge
by focusing on their buoyancy and observing how they float towar d the fluid surface.
DCS measur ements were performed by the National Physical Laboratory (NPL, T ed-
dington, UK) with a CPS DC20000 instrument (CPS Instr uments, Prairieville, LA, USA)
upgraded to DC24000 for the PS-Plain particles measurements. The radial position of
the detector was measur ed by injecting 100
µ
L aliquots of water into the spinning disc
initially empty until the accumulation of water pr oduced a response in the detector . For
the density gradient formation, the disc was filled with 14.4 mL of a sucrose (Amr esco
LLC, OH, USA) solution topped with 0.5 mL of dodecane to prevent evaporation. The
detailed information of the gradients is summarised in table 5.1. Measur ements of the
PS-COOH and PMMA-COOH particles at 0.05 % w/v concentration wer e performed in
triplicate. The measurements of the PS-Plain particles wer e r epeated seven times for each
setup. Injection volumes wer e 100 µ L.
The measur ed attenuation at 405 nm was converted to the number of particles for each
measur ed diameter by treating the particles as spherical Mie scatter ers with no optical
absorbance at the incident wavelength. Three dif fer ent types of calibration particles were
used: poly(vinyl chloride) colloids in water with density of 1.385 g cm
− 3
and nominal size
53

Chapter 5
SIMUL T ANEOUS SIZE AND DENSITY DETERMINA TION OF POL YMERIC
. . .
T able 5.1 |
P a rameters of the different DCS setups: comp osition of the sucrose gradients, average densit y
of the gradients ρ f , rotation sp eed of the centrifuge Ω and type of calib rant.
Sucr ose concentration (w/w) ρ f (g cm − 3 ) Ω (rpm) Calibrant
PS-COOH from 2 % to 8 % in H 2 0 1.013 2.0 · 10 4 A
PMMA-COOH fr om 4 % to 12 % in H 2 0 1.025 2.0 · 10 4 B
PS-Plain fr om 2 % to 8 % in H 2 0 1.013 2.4 · 10 4 B
PS-Plain* fr om 4 % to 12 % in D 2 0 1.140 2.4 · 10 4 C
* Low density disc
of
(
223
±
5
)
nm (calibrant A) and
(
239
±
5
)
nm (calibrant B) and polybutadiene colloids in
16 % sucr ose mass fraction in heavy water with nominal size of
(
510
±
20
)
nm and density
of 0.91 g cm − 3 (calibrant C).
A standar d disc configuration where the particles sediment thr ough a lower density
gradient was used and additionally , a more r ecently developed setup which makes use
of a disc wher e colloids float through a higher density gradient was also used for the
PS-Plain colloids due to their low density (Fitzpatrick, 1998). T ypically , the DCS diameter
D p or density ρ p of a spherical particle is derived from the Stokes’ law:
D p = v
u
u
t
18 η ln R f / R i
 ρ p − ρ fluid  ω 2 t p
(5.1)
wher e
t p
is the sedimentation time between radii of r otation
R f
and
R i
of the particle,
η
and
ρ f
ar e the viscosity and the density of the fluid respectively and
ω
is the disc angular
fr equency . If a calibrant of known diameter
D c
and density
ρ c
is measur ed with the same
setup, the investigated particle diameter can be expr essed as:
D p = D c v
u
u
t  ρ c − ρ fluid  t c
 ρ p − ρ fluid  t p
(5.2)
By using the combination of DCS measur ements performed in two differ ent fluids,
one with density
ρ L
and one with higher density
ρ H
, the values of
D p
and
ρ p
can be
independently found by solving analytically the following system of equations:
D p = D c H s  ρ c H − ρ H  t c H
 ρ p − ρ H  t p H
= D c L s  ρ c L − ρ L  t c L
 ρ p − ρ L  t p L
(5.3)
wher e
c H
and
c L
denote the calibrants used with high and low density fluids r espect-
ively and
t p H
and
t p L
ar e the sedimentation times of the particles measured in the high and
low density fluids r espectively . The measurement uncertanties given in the text include
both statistical and systematic uncertainty pr opagated from Stokes’ equations.
54

Determination of the pa rticle size distribution 5.2
0.1
1
10
100
1000
10000
0.02 0.03 0.05 0.1 0.2 0.3 0.5
Scattering Intensity / a.u.
q / nm − 1
333
337
341
345
0 25 50 75
ρ e / nm − 3
R / nm
PS-Plain in buffer
Core-Shell Fit
Figure 5.2 |
Scattering
curve of the PS-Plain
pa rticles in buffer: A
co re–shell fit to the exp er-
imental scattering curve
is p resented. In the inset,
the electron densit y radial
p rofile of this fit is sho wn,
assuming the co re is p oly-
st yrene with a densit y of
339.7 nm − 3 .
5.2 Determination of the particle size distribution
In figur e 5.2, the SAXS curve of the PS-Plain particles in buffer at a single-contrast is
shown. The large number of minima observed in the curve is r emarkable and indicates
the high monodispersity of the sample, which allows a traceable size determination of
these colloids.
Upon trying dif ferent form factor fits detailed in section 2.2, a simple cor e-shell structur e
with a sharp interface (eq. 2.27) was found to be the most suitable, suggesting a heterogen-
eous structur e which is eluded by other characterization techniques, e.g. microscopy . The
obtained particle diameter was
(
147.0
±
4.7
)
nm, wher e the fit uncertainty was calculated
with a confidence level of one standar d deviation (
k =
1) by examining the change in
χ 2
when varying the diameter . The radial electron density pr ofile of the core-shell fit is
shown in the inset of figur e 5.2, where a thin shell with high density surr ounds a lighter
cor e. This structure is likely due to the non-r eacted monomers in the main matrix or the
highly hydr ophilic behaviour of the co-monomer , segregating polystyr ene to the cor e.
The fit of the form factor 2.26 with 7 shells with a linear electron density gradient is
in very good agr eement with the experimental data as well and pr esents a
χ 2
value 20
times lower than the compact spher es model fit. Although the calculated
χ 2
value is very
similar to that of the cor e-shell model and the radial electron density pr ofile coincides
qualitatively as well, the uniquess of the solution can be debated due to the lar ge number
of fit parameters (14). In case of coinciding results, the simpler cor e-shell model might
solve the overfitting pr oblem and appears as the best solution.
The morphology of the PS-Plain particles was further studied using the density gradient
contrast variation technique described in chapter 4 by varying the suspending medium
electr on density from 333.2 to 350.2 nm
− 3
. By increasing the solvent contrast, the changes
of the featur es in the scattering curves presented in figur e 5.3a and the appearance of
isoscattering points pr ove the multi-component composition of this colloid.
Fr om the 40 experimental scattering curves shown in figure 5.3a, a model-fr ee size
determination can be performed by locating the isoscattering points
I i
. This is achieved
by calculating the r elative standard deviation, as shown in figur e 5.3b, where the minima
55

Chapter 5
SIMUL T ANEOUS SIZE AND DENSITY DETERMINA TION OF POL YMERIC
. . .
1
10
100
1000
0.03 0.05 0.1 0.2 0.3 0.5
Scattering Intensity / a.u.
q / nm − 1
334
336
338
340
342
344
346
348
Solvent Electron Density / nm − 3
(a) Scattering curves
0.02
0.05
0.1
0.2
0.5
1
0.05 0.1 0.2
Rel. Std. Deviation
q / nm − 1
I 1 I 2 I 3
I 4 I 5
Backgr ound subtracted
Raw data
(b) Isoscattering point positions
Figure 5.3 |
Continuous contrast va riation on the PS-Plain pa rticles: a) SAXS curves of the PS-Plain
pa rticles obtained b y density gradient contrast va riation after solvent background subtraction. b) The
relative standa rd deviation of each
q
calculated across all the measured scattering curves, where the minima
co rresp ond to the isoscattering p oints
I i
. The background subtraction shifts the p osition of
I i
, esp ecially
fo r high q -values.
T able 5.2 |
Isoscattering p oints p osition and the co rresp onding pa rticle diameter fo r the scattering curves
b efo re and after background co rrection. The diameter deviation b et ween both values is also shown, with
la rger deviation fo r higher
q
-values. The uncertaint y asso ciated to the diameter is calculated as describ ed
in chapter 4.
Raw data Corr ected data Deviation
q ? (nm − 1 ) Diameter (nm) q ? (nm − 1 ) Diameter (nm) nm
q ?
1 0.063 ± 0.002 142.0 ± 5.4 0.063 ± 0.002 142.4 ± 5.6 0.4
q ?
2 0.109 ± 0.003 142.0 ± 4.2 0.108 ± 0.003 143.6 ± 4.1 1.6
q ?
3 0.154 ± 0.005 141.9 ± 4.6 0.151 ± 0.004 144.4 ± 3.7 2.5
q ?
4 0.206 ± 0.016 136.6 ± 10.6 0.195 ± 0.011 144.3 ± 7.9 7.7
corr espond to the fulfillment of the isoscattering condition expressed by equation 2.30.
T able 5.2 summarizes the particle diameters obtained fr om the first 4 isoscattering
points (
I 1
to
I 4
), which range between 142.4 and 144.4 nm after background corr ection. The
pr ecision of the isoscattering point determination decreases for incr easing
q
as described
by Kawaguchi & Hamanaka (1992) and it is exemplified by the br oadening of the minima
for higher
q
and the incr ease of the associated uncertainties, as discussed previously in
chapter 4. As observed in figure 5.3b, the effect of the solvent backgr ound is r elevant
principally at high
q
-values as well. These effects ar e studied in more detail in section
5.3.2.
The data can also be analysed by using the shape scattering function described in section
2.3.2. The shape scattering function describes the external shape of the particle inde-
pendently of its inner structur e and is an appr opriate approach for the PS-Plain colloid,
because it enables the size distribution determination of the particles avoiding any a priori
consideration about the particle composition.
The experimental shape scattering function is calculated fr om the measured scattering
curves pr esented in figure 5.3a. The result is depicted in figur e 5.4 together with the
spherical model fitted to the data, which employs a simple form factor that ignor es the
internal structur e (eq. 2.24) and a gaussian size distribution expr essed by equation 2.21.
Fr om this fit, a mean particle size of
(
146.8
±
1.3
)
nm was determined. The associated
56

Determination of the pa rticle size distribution 5.2
1
10
100
1000
10000
100000
0.03 0.05 0.1 0.2 0.3
Scattering Intensity / a.u.
q / nm − 1
Shape scattering function
Spherical model
Figure 5.4 |
Exp erimental
shap e scattering function
of the PS-Plain pa rticles
calculated from 40 scat-
tering curves and the
spherical fo rm facto r fit-
ted to the calculated
shap e scattering func-
tion.
uncertainty calculated with this appr oach is 3.5 times smaller than the one obtained with
the single-contrast SAXS experiment. By fitting the ellipsoid model given by expression
2.25 to the shape scattering function, a sphericity of 98 % was obtained.
5.2.1 Inter -laboratory comparison of the mean particle diameter
The impr ovement in the size accuracy with the shape scattering function approach is
summarized in figur e 5.5, wher e the diameter of the PS-Plain particles determined by
dif ferent techniques in an inter -laboratory study is also presented (Nicolet et al. , 2016).
The figur e compares the PS-Plain diameter measur ed by the ensemble techniques SAXS
and DCS and the imaging methods AFM and TSEM and presents the weighted mean
value of all the r esults as a grey line, which corr esponds to a diameter of 145.0 nm with
an associated expanded uncertainty (
k =
2) of 1.6 nm. The SAXS results tend to lar ger
values when modelling the scattering form factor , whilst the diameter obtained from the
isoscattering points positions
I i
pr esent values slightly smaller than the calculated mean
value. However , the maximum deviation fr om the weighted mean is less than 2 %.
The DCS r esult is obtained by a combined analysis of two complementary centrifuge
configurations as detailed in section 5.1.2, where figur e 5.6 depicts the dependency of
the measur ed particle diameter on the density values for the two setups. The two setups
measur e the same diameter and density at the crossing point of the data, which occurs
for a diameter of (138.8
±
5.8) nm and a density of (1.052
±
0.010) g cm
− 3
. The measur ed
diameter fits within its uncertainty in the confidence interval of one standar d deviation of
the inter -laboratory comparison.
All the techniques ar e in very good agreement, even considering that they ar e based
on dif ferent physical principles. The improvement in accuracy for the size determination
with SAXS by using the shape scattering function appr oach is further sustained by this
comparison.
This impr ovement was confirmed by employing the same approach with the PS-COOH
colloids. The diameter obtained from the cor e-shell model fit in chapter 4 is (99.4
±
5.6)
nm, while the value obtained fr om the shape scattering function calculation is (101.4
±
2.4)
57

Chapter 5
SIMUL T ANEOUS SIZE AND DENSITY DETERMINA TION OF POL YMERIC
. . .
130
135
140
145
150
155
AFM (SMD)
AFM (MET AS)
AFM (VSL)
TSEM
DCS
Cor e-shell
Shape function
I 1 I 2 I 3 I 4
Diameter / nm
Figure 5.5 |
Compa rison of the PS-Plain average diameter obtained with different techniques, where the
erro rba rs correspond to the expanded uncertainty (
k =
2 ). The circles correspond to results obtained with
SAXS in the F CM b eamline and the diamond to combined DCS measurements p erfo rmed b y NPL. The
gra y line defines the w eighted average value of all the results. The microscopy values a re obtained from
Belgian Service Métrologie-Metrologische Dienst (SMD), Swiss F ederal Institute of Metrology (MET AS)
and Dutch Metrology Institute (VSL).
100
150
200
250
300
1.02 1.04 1.06 1.08 1.1 1.12
PS-Plain diameter / nm
PS-Plain density / g cm − 3
Standard disc
Low density disc
Figure 5.6 |
Dep endence of the intensit y-based mo dal Stok es’ diameter on the pa rticle densit y for the
PS-Plain pa rticles analysed in H
2
O-sucrose (black) and D
2
O-sucrose (red) gradients. The a rrow indicates
the crossing p oint of the data, where the t w o setups measure the same diameter and densit y of the colloid.
This o ccurs fo r a diameter of ( 138.8 ± 5.8 ) nm and a density of ( 1.052 ± 0.010 ) g cm − 3
58

Determination of the pa rticle size distribution 5.2
PS-Plain PS-COOH
Cor e-shell fitting 147.0 ± 4.7 nm 99.4 ± 5.6 nm
Shape scattering function 146.8 ± 1.3 nm 101.4 ± 2.4 nm
First isoscattering point 142.4 ± 5.6 nm 99.8 ± 6.6 nm
Second isoscattering point 143.6 ± 4.1 nm 102.0 ± 8.8 nm
T able 5.3 |
Compa rison
of the diameters of the
PS-Plain and PS-COOH
pa rticles obtained b y the
different app roaches de-
scrib ed in section 5.2.
0
0.2
0.4
0.6
0.8
1
1.2
80 100 120 140 160 180
Fr equency / a.u.
Diameter / nm
TSEM
Shape Scat. Func. SAXS
Standard DCS
Low density DCS Figure 5.7 |
Numb er-
w eighted size distribution
of the PS-Plain pa rticles
measured b y DCS, TSEM
(Nicolet et al. , 2016) and
SAXS with the shap e
scattering function ap-
p roach.
nm. Again, the uncertainty associated to the size decreases by
∼
60 %, whilst it is still in
accor dance with the diameter obtained with the first two isoscattering points positions
of 100.6
±
5.6 nm. The diameters of the PS-Plain and PS-COOH particles obtained by the
dif ferent appr oaches to contrast variation data ar e compared in table 5.3. In both examples,
the smallest uncertainty is associated to the shape scattering function formalism, while
the cor e-shell model and the position of the first two isoscattering points produce lar ger
combined standar d uncertainties.
The thir d polymeric particles used in the study are the PMMA-COOH colloids intr o-
duced in section 5.1. Due to the low polydispersity of these particles, a spherical form
factor fit to the single-contrast scattering curve provides alr eady a very accurate diameter
of (186.5
±
2.3) nm. In this case, contrast variation experiments in SAXS show no advant-
ages because of the homogeneous composition of the particles. The application of the
shape scattering function formalism or the isoscattering point approach is only feasible if
the NPs possess an internal structur e.
5.2.2 Particle size distribution of the PS-Plain particles
An important attribute of polymeric colloids is their polydispersity , as the suitability for
specific applications depends on their spr ead in size. For example, colloids are known to
induce dif ferent inflammatory r esponses depending on their size (Kusaka et al. , 2014).
The SAXS r esults determine a polydispersity degree
p d
for the PS-Plain colloids of 6.1
%, which is an indicator of a very monodisperse distribution, as also suggested by the
r egular minima observed in figure 5.2. Particle polydispersities measured by DCS ar e also
low as observed in figur e 5.7, ranging from 7.8 % measur ed with the standard setup, to
59

Chapter 5
SIMUL T ANEOUS SIZE AND DENSITY DETERMINA TION OF POL YMERIC
. . .
Figure 5.8 |
Diameter of
the PS-Plain pa rticles as
a function of the num-
b er of scattering curves
used in the shap e scatter-
ing function calculation.
The ho rizontal line sho ws
a diameter of 146.8 nm.
140
142
144
146
148
150
0 5 10 15 20 25 30 35
Diameter / nm
Number of scattering curves
11.3 % measur ed with the low density disc setup. The standard setup appears ther efor e to
achieve a higher r esolution size distribution. The size distribution measur ed by TSEM
with a p d of 8.3 % shows good agreement with the ensemble techniques.
The measur ements obtained by AFM provide polydispersity degr ees larger than 10 %
(Nicolet et al. , 2016) and, ther efore, slightly br oader size distributions than those calculated
by SAXS, TSEM and standar d DCS. This can be in part attributed to the low statistics
that typically af fect imaging methods, along with artefacts associated with the posterior
analysis.
For instance, in the TSEM images (Nicolet et al. , 2016), smaller and lar ger populations
with dif ferent contrasts have been observed which could af fect the evaluation of the
density measur ed by ensemble techniques in the following section 5.4, as the particle
average density might vary . Indeed, when a bimodal distribution is used to analyse the
SAXS shape scattering function of the PS-Plain particles, a second size population is found
at 101 nm in agr eement with TSEM, while the main mode maintains a p d of ca. 5 %.
5.3 Considerations about scattering data evaluation
In the pr evious section, the mean diameter of polymeric nanoparticles was obtained using
two dif ferent model-fr ee approaches, i.e. the isoscattering point and the shape scattering
function. The method employed to analyse the scattering curves measured with the
continuous contrast variation technique in SAXS affects the size determination and its
accuracy , as suggested by the r esults. Following, a discussion about both approaches is
pr esented based on the scattering data shown in figure 5.3a.
5.3.1 Shape scattering function formalism
The shape scattering function obtained by density gradient contrast variation has been
demonstrated as a powerful technique which can pr ovide precise information about the
size distribution and shape of the colloid by fitting a simple form factor .
60

Considerations ab out scattering data evaluation 5.3
However , an accurate determination of the suspending medium density for each scat-
tering curve is r equired, due to the incr eased uncertainties (Lefebvre et al. , 2000) that can
arise fr om the resolution of the system of linear equations described in section 2.3.2.
Besides, a minimum of 3 scattering curves measur ed at differ ent contrasts is necessary
to obtain the r esonant term, although an increasing number impr oves the determination
of the size distribution. This issue has been addressed with the experimental data of
the PS-Plain colloids measur ed by the density gradient contrast variation. Fr om the 40
experimental curves, only a limited number
N
was randomly selected to compute the
shape scattering function, while this pr ocess was repeated 100 times. The mean diameter
obtained fr om this data set and its statistical standard deviation ar e plotted in figure 5.8 as
a function of N .
The ef fect of increasing the number of measur ed contrasts evidences that the result
tends asymptotically to the value of 146.8 nm discussed in section 5.2 and the standar d
deviation of the 100 iterations decr eases for large
N
, e.g. the associated uncertainty is
r educed. This outcome emphasizes further the advantages of the continuous contrast
variation technique due to the lar ge number of scattering curves at differ ent contrasts
which can be easily measur ed.
In summary , it has been demonstrated that the possibility to determine the particle size
distribution by the shape scattering function is a clear impr ovement to single-contrast
SAXS techniques r educing relevantly the uncertainty , although an accurate determination
of the contrast and a r elatively high number of scattering curves are r equired.
5.3.2 Isoscattering point appr oach
The theory defines the isoscattering point
q ?
as a morphological parameter independent
of the suspending medium density , which is a enormous practical advantage as it can be
located without the pr oper calibration of the contrast. In cases where the composition
of the buf fer is unknown or the density of the solvent cannot be properly calibrated,
the isoscattering point position can still be used to determine the size of the particles by
calculating the r elative standard deviation of all the measur ed scattering curves.
In or der to obtain reliable r esults, a pr oper subtraction of the solvent scattering must be
performed. It is clear in figure 5.3b that the corr ection of the solvent contribution to the
scattering intensity plays an important r ole in the determination of the
q ?
values as the
curve shifts to smaller
q
-values when subtracting the solvent backgr ound. Although this
ef fect is larger at high
q
-values pr oducing deviations up to 7.7 nm, the solvent background
influences the position of all the isoscattering points as summarized in table 5.2.
It has been discussed befor e in this work that the polydispersity of the latex and its
deviation fr om the spherical shape influence the position and diffuseness of
q ?
, principally
at high
q
-values. This can disturb the size determination for polymeric particles with
br oad size distributions and limit the applicability of this technique. In fact, the largest
contribution to the uncertainty associated to the position of the isoscattering points ori-
ginates fr om the diffuseness of
q ?
due to the deviation fr om ideality of the particle, as
r eviewed in chapter 4.
In or der to prove the isoscattering point dependency on the particle polydispersity ,
the diameter obtained fr om the first isoscattering point position is simulated for thr ee
cor e-shell particle with differ ent core-to-size ratios, as depicted in figur e 5.9a. The devi-
ation of the calculated size fr om the nominal size becomes larger for incr easing particle
polydispersities, r eaching size deviations up to 8 % at
p d =
30 %. Mor eover , the size
61

Chapter 5
SIMUL T ANEOUS SIZE AND DENSITY DETERMINA TION OF POL YMERIC
. . .
-7
-5
-3
-1
0
1
0 5 10 15 20 25 30
Isoscattering point deviation / %
Polydispersity degree / %
Ratio R cor e / R
69 %
83 %
94 %
(a) Polydispersity effects
-0.5
0
0.5
1
1.5
330 340 350 360 370 380 390 400
Isoscattering point deviation / %
ρ max / nm − 3
(b) Solvent electron density range
Figure 5.9 |
Deviation of the size of the PS-Plain pa rticles calculated using the
q ?
1
p osition from the
nominal value dep ending on a) the size p olydispersity of co re-shell particles with different co re-to-size ratios
o r b) the solvent electron densit y range employ ed in the exp eriment, where ρ e ∈ ( 330 nm − 3 , ρ max ) .
deviation behaves dif ferently depending on the internal str ucture of the particle, tending
to lar ger deviations for thicker shells and positive deviations for thinner ones.
This work demonstrates also that the
q ?
value determined with the pr eviously described
method depends on the range of solvent densities used in the contrast variation experi-
ment. For this purpose, a contrast variation experiment with 10 differ ent solvent densities
was simulated for a polymeric particle with the morphology and size distribution obtained
with the cor e-shell model in section 5.2. Using a lower bound to the contrast range close to
the electr on density of water (
ρ min =
330 nm
− 3
) and incr easing systematically the upper
limit, it is shown in figur e 5.9b that the calculated result deviates fr om the nominal value
up to 1.5 %.
In this example, the lar gest deviations occur when the average density of the latex i.e.
match point (depicted as a vertical line in figur e 5.9b) is excluded from the experimental
contrast range or when
ρ max
is close to this matching density . This observation conflicts
partly with the initial intuition that this technique is independent of the experimental
pr ocedure, although this pr oblem can be avoided by selecting the solvent electron density
range skillfully i.e. equidistantly distributed around the match point. This could be
one explanation behind the slight size dif ferences observed in figur e 5.5 between the
issocattering appr oach and the other SAXS results.
The isoscattering point appr oach to contrast variation SAXS data evaluation presents
certain assets which can not be ignor ed. For instance, the independence of
q ?
fr om the
sample contrast facilitates its easy application, although the solvent electron density range
must be chosen with car e and always around the average electr on density of the particle
to maximize its accuracy . On the other hand, the diffuseness of the isoscattering point
position due to the polydispersity and ellipticity of the sample arises as an indisputable
drawback and pr oduces larger associated uncertainties than the shape scattering function
appr oach.
62

Determination of the pa rticle mass densit y 5.4
0
2
4
6
8
10
334 336 338 340 342 344
I ( 0 ) / a.u.
Solvent Electr on density / nm − 3
Figure 5.10 |
Intensit y
at zero-angle of the PS-
Plain pa rticles as a func-
tion of the solvent elec-
tron densit y measured
with continuous contrast
va riation in SAXS. The
minimum defines the av-
erage electron densit y of
the pa rticle.
5.4 Determination of the particle mass density
In contrast variation SAXS, the solvent electr on density which matches the average electron
density of the particle
ρ 0
corr esponds to a minimum in the intensity of the scattering
curve accor ding to expression 2.36. In order to quantify the particle density , the scattering
intensity of the PS-Plain particles at zer o angle
I (
0
)
is examined along the contrast range of
the experiment as shown in figur e 5.10. The value of
I (
0
)
was determined by extrapolation
to
q →
0 using a spherical form factor function fitted to the available range befor e the
first minimum, as discussed in section 4.3.3. The parabolic fit to the data is plotted as a
black line in figur e 5.10 and results in
ρ 0 = ( 339.2 ± 1.0 )
nm
− 3
, which is consistent with
the tabulated value of dry bulk polystyr ene 339.7 nm − 3 (Dingenouts et al. , 1999).
The mass density of the particle can also be determined by this appr oach because the
electr on density is directly pr oportional to the mass density , as reviewed in chapter 2.
A PS-Plain density of (1.043
±
0.003) g cm
− 3
is obtained, although an assumption about
the polymer (or monomer) components and their atomic structur e is necessary for the
calculation. Ther efore, a typical value of
Z / A =
0.54 was adopted for this conversion,
wher e
Z
and
A
ar e the average atomic number and mass of the polymer respectively . This
value is characteristic of polymers (or monomers) such as PS, PMMA or MMA, and very
close to the Z / A ratio of MAA (0.53), polyvynil chloride (0.51) or polyethylene (0.57).
The density uncertainty is associated to the vertical size of the focused X-ray beam as
discussed in 4.1.2, which typically corresponds to an associated uncertainty of 1 nm
− 1
or a r elative uncertainty of around 3 %. Furthermor e, the result can be af fected by the
polymeric composition of the colloid, and ther efore, the assumption of
Z / A
, although an
upper limit of 5 % is expected fr om this contribution.
63

Chapter 5
SIMUL T ANEOUS SIZE AND DENSITY DETERMINA TION OF POL YMERIC
. . .
Figure 5.11 |
Compa rison
b et w een the mass dens-
ities of three p olymeric
colloids measured with
SAXS using the
I (
0
)
app roach (black) and
DCS (red): PS-Plain
(squa res), PS-COOH
(circles) and PMMA-
COOH (diamonds).
The nominal densities
of p olyst yrene (1.05 g
cm
− 3
) and PMMA (1.18
g cm
− 3
) a re also sho wn
in the plot as ho rizontal
lines (Dingenouts et al. ,
1999).
1.04
1.05
1.06
1.07
Normal Disc
Combined
Low Density
SAXS
DCS
SAXS
DCS
SAXS
Density / g cm − 3
1.17
1.18
1.19
PS-Plain PS-COOH
PMMA-COOH
5.4.1 Mass density of the PS-Plain particles: validation with DCS
In figur e 5.11, the value measured with the
I (
0
)
appr oach from the continuous contrast
variation experiment is compar ed to the average density of the PS-Plain colloid measured
with dif ferent DCS configurations. For the standard centrifuge setup and the low density
disc configuration, the size value used for the density calculation was 147 nm, as measur ed
by single-contrast SAXS, while combining the information fr om both setups allowed the
measur ement of the density independently of the particle diameter , as explained in
section 5.2.1.
The r esults agree with each other within their stated measur ement uncertainties, al-
though DCS measur ements exhibit slightly higher densities than SAXS. T ypical causes of
systematic uncertainties in DCS ar e the inaccuracy of the size and density of the calibration
standar d and the thermal variation in the centrifuge gradient during the measurements,
which af fect its viscosity and density (Kamiti et al. , 2012). A temperature variation within
the gradient of about 7
◦
C befor e and after measurements was detected and a period
of 30 min was consider ed appropriate to r each reliable thermal equilibrium. In the low
density disc configuration, the determination of the average density of the D
2
0 sucr ose
gradient becomes an important sour ce of uncertainty which might explain the larger
associated uncertainty in comparison to the standar d configuration. Besides, the normal
disc setup shows a higher r esolution size distribution as discussed in section 5.2.2 which
also translates in smaller uncertainties associated to the mass density .
64

Determination of the pa rticle mass densit y 5.4
5.4.2 Density determination of heavier polymeric colloids
The applicability of the continuous contrast variation techniques is further discussed
by comparing with DCS for higher -density polymeric colloids, as summarized in fig-
ur e 5.11. The density of the PS-COOH particles derived fr om the
I (
0
)
appr oach is in
excellent agr eement with that measured by DCS using a standar d configuration and
assuming a particle diameter of 99.4 nm, which was obtained by SAXS. Considering the
similar electr onic composition of these polymers and the average electron density of the
particle
ρ 0 = ( 346.0 ± 1.5 )
nm
− 3
obtained in chapter 4, an average mass density of the
particles of (1.068
±
0.005) g cm
− 3
can be calculated. These cor e-shell particles, more dense
than polystyr ene as detailed in section 4.3.1, illustrate the tendency during the emulsion
polymerization to segr egate polar and nonpolar components (Dingenouts et al. , 1994 c ).
Similarly , the density of the PMMA-COOH colloids was measured using the standar d
DCS setup and assuming a diameter of 186.5 nm, as measured by SAXS. This value is
compar ed to the density of (1.173
±
0.003) g cm
− 3
obtained by computing the intensity at
zer o-angle of a continuous contrast variation experiment with a minimum at
( 381.5 ± 1.0 )
nm
− 3
. Again, both techniques are in excellent agr eement and reveal a mass density slightly
lower than the expected PMMA density of 1.18 g cm − 3 (Dingenouts et al. , 1999).
This r esult highlights the fact that the density of polymeric colloids in suspension
may vary fr om that of bulk materials, for example dry particles. For instance, a volume
variation can be expected when going fr om the MMA monomer to the polymer PMMA
(Nichols & Flowers, 1950) which might r educe the colloid density .
65

6
Continuous contrast variation applied
to r elevant bio-materials
In the continuously gr owing world of nanotechnology , nanoscience provides understand-
ing for biological structur es at the nanometre length scale, such as lipopr otein biology ,
while the application of nanoparticles in medicine opens exciting new possibilities in this
field (Nie et al. , 2007; Sahoo & Labhasetwar, 2003; W ickline & Lanza, 2003; Zhou et al. ,
2014; Rosen & Abribat, 2005). For example, polymeric colloids and other biodegradable
nanocarriers ar e finding many medical applications (V icent & Duncan, 2006) and are
starting to under go clinical trials (Patel et al. , 2012; Beija et al. , 2012; Cabral & Kataoka,
2014).
In this sense, lipid vesicles, or liposomes, have an increasing importance in the emer ging
field of nanomedicine, due to their capacity to encapsulate hydrophilic compounds within
the closed phospholipid bilayer membrane. In fact, liposomal nanocarriers are nowadays
a widespr ead instrument for drug delivery (Pér ez-Herrer o & Fernández-Medarde, 2015),
like the liposomal formulation of doxorubicin coated with polyethylene glycol (PEG):
Caelyx® (Bar enholz, 2012).
Despite SAXS being a usual method of choice for the accurate characterization of nano-
materials, the interpr etation of the scattering curves, i.e. the model fitting, is frequently
intricate for complex samples. Liposomal drugs or loaded polymeric nanoparticles belong
to this class, as both the carrier and the incorporated biotar get contribute to the scattering
intensity . These heter ogenous samples requir e either a priori knowledge about their mor-
phology or the measur ement of complementary scattering curves obtained under differ ent
experimental conditions, like in solvent contrast variation in SAXS
In this chapter , the utilization of continuous contrast variation in SAXS is examined
for the nano-drug Caelyx and for typical nanocarriers like lipid vesicles or polymeric
colloids. In the latter case, the particle is coated with an antibody to resemble the biological
conditions found upon injection in the bloodstr eam. Other components of the blood
plasma like lipopr oteins are also investigated with this technique. Parts of this chapter
have been adapted fr om articles published previously (Minelli et al. , 2014; Gar cia-Diez
et al. , 2016 a ).
67

Chapter 6
CONTINUOUS CONTRAST V ARIA TION APPLIED TO RELEV ANT BIO-
. . .
(a) Cryo-TEM
PEG
chains
Phospholipid
bilayer
Doxorubicin
Aqueous
buf fer
(b) Scheme
Figure 6.1 |
a) Cry o-TEM micrograph of Caelyx
®
(Ba renholz, 2012) and b) schematic rep resentation of
the PEGylated lip osomal do xo rubicin mo rphology .
6.1 Materials
In this chapter , the continuous contrast variation method in SAXS presented in chapters 4
and 5 has been employed in a variety of samples r elated with nanomedicine. In this
section, the dif ferent samples characterized with this technique ar e described and the
mor e relevant aspects of the experiments ar e detailed.
The r esults obtained on the Caelyx nano-drug are described in detail in section 6.2 and
6.3.1, while the empty liposomes ar e investigated under osmotic pressur e in section 6.3.2.
The size measur ements on the lipoproteins ar e presented in section 6.4 and the use of the
pr otein-coated nanoparticles is detailed in section 6.5.
Caelyx: PEGylated liposomal doxorubicin
Caelyx
®
(SP Eur ope, Brussels, Belgium) was purchased fr om Hungar opharma Ltd and
consists of liposomes suspended in 10 mM histidine buf fered sucr ose solution (pH 6.5)
formed by fully hydr ogenated soy phosphatydilcholine (HSPC), cholesterol, and DSPE-
PEG 2000 (N-(carbonyl-methoxypolyethylene glycol 2000)-1,2-distear oyl-sn-glycero-3-
phosphoethanolamine). The latter yields a steric barrier at the liposomal surface due to
the PEG 2000 r esidues that extend the blood-circulation time, the so-called stealth function.
Doxorubicin is encapsulated in the PEGylated liposome via an active loading pr ocedure,
which r esults in a crystal-like doxorubicin pr ecipitate inside the liposomes, as observed in
the micr ograph 6.1a (Barenholz, 2012). A schematic depiction of the sample morphology
is shown in figur e 6.1b.
68

T raceable size determination of a lip osomal drug 6.2
Lipid vesicles: PEGylated and plain liposomes
The PEGylated liposomes wer e prepar ed by the Institute of Materials and Envir onmental
Chemistry (Hungarian Academy of Sciences, Budapest, Hungary) with the same lipid
composition as the commer cially available Caelyx for comparison purposes: the weight
ratios of HSPC:DSPE-PEG 2000:cholester ol were 3:1:1 (corr esponding to molar ratios of
0.565:0.053:0.382). The samples were extr uded through polycarbonate filters (Nucleopor e,
Whatman Inc., Little Chalfont, UK) of five differ ent por e sizes, fr om 50 to 400 nm. A
mor e detailed description of the preparation is found elsewher e (V arga et al. , 2014 a ).
The components of the plain liposomes ar e HSPC:cholesterol with a weight ratio of 3:1
(corr esponding to molar ratios of 0.6:0.4). The preparation is identical to the PEGylated
liposomes. All the liposome samples are suspended in a 10 mM phosphate buf fered saline
(PBS) pH 7.4 buf fer solution.
Human lipoproteins
Native lipopr oteins from human plasma wer e purchased fr om Mer ck Milipore (Darmstadt,
Germany) and suspended in 150 mM NaCl, 0.01 % EDT A buffer with pH 7.4. The High
Density Lipopr otein (HDL) has a protein concentration of 14.3 g L
− 1
, while the Low
Density Lipopr otein (LDL) has a protein concentration of 5.96 g L
− 1
, considering that the
weight ratio between lipids and pr oteins is approximately 4:1 in the LDL sample.
PS-COOH particles coated with IgG
The polystyr ene nanoparticles with carboxylated surfaces (PS-COOH) described in chapter
4 ar e coated with the protein Immunoglobulin G (IgG). A set of four IgG-coated poly-
styr ene nanoparticle samples was prepar ed by the Surface and Nanoanalysis group of
NPL (T eddington, UK) by incubating 0.05 % (w/w) particles with varying concentrations
of IgG fr om 0.5 to 4 g L
− 1
in 100 mM T ris buf fer at pH 8 under continuous shaking for
2 h. Any unbound IgG was then r emoved from the particle samples by thr ee cycles of
centrifugation and r edispersion in clean buffer .
In the continuous contrast variation experiment with sucr ose as contrast agent, a protein
concentration of 4 g L
− 1
IgG was physisorbed at the surface of the bar e PS-COOH particles.
The details of the density gradient capillary ar e discussed in section 6.5.1.
6.2 T raceable size determination of a liposomal drug
The first appr oved nano-drug, Caelyx, was rapidly followed by a few other pr oducts
(Y eh et al. , 2011; Bar enholz, 2012). Nowadays there ar e approximately 250 nanomedicine
pr oducts that are either appr oved by the relevant health agencies or ar e under clinical
trials (Etheridge et al. , 2013). On the other hand, ther e is a translational gap between
the experimental work devoted to the development of new nano-drug candidates and
the clinical r ealization of their use, which is also r eflected in the high number of studies
dealing with nanomedicine and the number of appr oved products on the market (V enditto
& Szoka Jr ., 2013). As highlighted in a recent r eview by Khorasani et al. (2014), one of
the main r easons for this translational gap is that the current characterization techniques
possess limitations and ther e is a need for standardization in this field.
Among many r elevant physicochemical properties of nano-dr ugs, one of the most
important to be accurately determined is the size of the nanocarriers, which dir ectly
69

Chapter 6
CONTINUOUS CONTRAST V ARIA TION APPLIED TO RELEV ANT BIO-
. . .
0.01
0.1
1
10
100
0.05 0.1 0.2 0.5 1
Scattering Intensity / cm − 1
q / nm − 1
345
350
355
360
365
Solvent Electron Density / nm − 3
(a) Contrast variation with density gradient
0.1
1
0.05 0.1 0.2 0.5 1
Rel. Std. Deviation
q / nm − 1
Backgr ound subtracted
Raw data
(b) Isoscattering point positions
Figure 6.2 |
a) Scattering curves at different susp ending medium electron densities obtained with a solvent
densit y gradient of Caelyx in aqueous io dixanol with constant buffer osmolalit y . Figure b) shows the p recise
p osition of the isoscattering p oints before and after the p rop er co rrection of the background.
r elates to the in vivo biodistribution of the drug. The ultimate goal in this regar d is to r each
a traceable size determination of the nanomaterial and ther efore the continuous contrast
variation technique in SAXS is a suitable method to assess the size of a complex liposomal
drug, such as the PEGylated liposomal formulation of doxor ubicin.
Osmolality is a measur e of the balance in an aqueous medium between water and
the solvated chemical components. It quantifies the osmotic pressur e being exerted by
the solute in the studied membrane and is typically given in osmoles (Osm) of solute
per kilogram of solvent. The need of an iso-osmolal suspending medium to mimic the
physiological conditions of plasma and avoid osmotic ef fects in the vesicle membrane
r equires the use of Optipr ep
TM
(Sigma-Aldrich, Missouri, USA) as contrast agent, an
aqueous solution of iodixanol, which has an osmolality of 290 to 310 mOsm kg
− 1
. By
employing Optipr ep, the suspending medium osmolality can be kept constant along the
density gradient capillary .
SAXS curves of the liposomal doxorubicin sample measur ed at differ ent suspending
medium electr on densities are shown in figur e 6.2a, where a maximum solvent electr on
density of 365.2 nm
− 3
was r eached with an Optiprep mass fraction of 35 %. In the
scattering curves, it is possible to observe the variation of the curve featur es through the
incr ease of the suspending medium density , which indicates the complexity of the internal
structur e of the nanocarrier . Besides, the appearance of an isoscattering point ar ound
q = 0.12 nm − 1 is a further indicator of the structural complexity of the drug-carrier .
The solvent backgr ound has been subtracted by measuring the scattering curves of a
density gradient of Optipr ep and buffer without nanocarriers. The low scattering power of
the PEGylated liposomal doxorubicin at high
q
values and the contribution of the Optipr ep
backgr ound result in a decr eased signal-to-noise ratio in the high-
q
range of the corr ected
scattering curves, although in the Fourier region below
q =
0.3 nm
− 1
the backgr ound
ef fect is much less dominant.
70

T raceable size determination of a lip osomal drug 6.2
T able 6.1 |
Diameter of Caelyx obtained b y t wo different SAXS app roaches, DLS and Cry o-TEM. The result
from DLS w as obtained b y the Institute of Materials and Environmental Chemistry (Hungarian A cademy of
Sciences, Budap est, Hunga ry), whilst the Cry o-TEM diameter w as extracted from Ba renholz (2001).
Diameter (nm)
Shape scattering function 65.5 ± 4.7
First isoscattering point 73 ± 9
DLS 86
Cryo-TEM 75
6.2.1 Isoscattering point appr oach
In the low
q
part of the scattering curve, an isoscattering point is clearly visible as high-
lighted in figur e 6.2a. The isoscattering point position relates dir ectly to the external radius
of the measur ed particle inaccessible to the solvent, as explained in section 2.3.1. Ther efore,
the PEG-chains attached to the liposome surface might not be quantified in this approach
due to the permeability of the polymer layer . The isoscattering point position is pr ecisely
determined by calculating the r elative standard deviation of all the scattering curves at
each
q
-value, as shown in figur e 6.2b. As discussed in chapter 4, the pr oper subtraction of
the solvent backgr ound is essential for the right interpretation of the data, specially for
intense scatter ers like Optiprep. A clear shift in the minima of the relative standar d devi-
ation curve is observed in figur e 6.2b after correcting the backgr ound effects. Hence, the
first isoscattering point
q ?
1
is located at
q ?
1 = ( 0.123 ± 0.016 )
nm
− 1
, which corr esponds to a
diameter of
( 73 ± 9 )
nm. A second isoscattering point at
q ?
2 = ( 0.25 ± 0.06 )
nm
− 1
is still
visible, although the large diffuseness of the isoscattering points at higher
q
values, r elated
with the polydispersity of the ensemble and the possible ellipticity of the doxorubicin
loaded liposomes, makes it less r eliable for the determination of the outer diameter .
6.2.2 Shape scattering function calculation
In or der to provide a complementary r esult to the diameter value obtained with the
isoscattering point appr oach, an alternative evaluation procedur e has been used, namely
the calculation of the shape scattering function intr oduced in section 2.3.2 which extracts
all contributions fr om the 30 measured scattering curves that change with the contrast at
dif ferent solvent densities. The shape scattering function of the Caelyx sample contains
essentially information only about the shape and size distribution of the space filled up
by the liposomes, i.e. the contributions of the phospholipid bilayer and the encapsulated
doxorubicin to the scattering intensity ar e cancelled. Thus, the complex interpretation
of the original SAXS curve of Caelyx is avoided and enables the size determination of
the liposomal carrier by fitting the analytical model for homogeneous spherical objects
expr essed by equation 2.24 . A model with a certain ellipticity was also attempted using
the expr ession 2.25, due to the slight liposomal eccentricity observed in TEM images
(Bar enholz, 2012) though the best fit was accomplished with a spherical model.
The shape scattering function calculated fr om the SAXS curves and the theoretical model
fitting ar e depicted in figure 6.3a. The diameter obtained from the spherical form factor fit
is (65.5
±
4.7) nm, smaller than the value calculated from the isoscattering point position
and with a smaller associated uncertainty . Both values are in good agr eement within
their combined measur ement uncertainties, considering that the uncertainty associated
to the dif fuseness of the isoscattering point arises principally from the polydispersity of
71

Chapter 6
CONTINUOUS CONTRAST V ARIA TION APPLIED TO RELEV ANT BIO-
. . .
1
10
100
1000
10000
0.05 0.1 0.2 0.3
Scattering Intensity / a.u.
q / nm − 1
Shape scattering function
Sphere fit
(a) Shape scattering function
0
20
40
60
80
100
120
345 350 355 360 365
I ( 0 ) / cm − 1
Solvent Electron Density / nm − 3
(b) Zero-angle intensity
Figure 6.3 |
Evaluation of the scattering curves of Caelyx. a) The exp erimental shap e scattering function
of the lip osomes is sho wn with symb ols whilst the mo del fit fo r homogeneous spherical pa rticles is depicted
with a thick line. b) The measured intensit y at zero-angle of Caelyx as a function of the electron density of
the aqueous io dixanol susp ending medium is sho wn with symb ols and the function fitted to the exp erimental
data is depicted in black: The average densit y is 346.4 nm − 3 and there is an offset in I ( 0 ) of 1.6 cm − 1 .
the sample. This fact is supported by the broad size distribution determined by the shape
scattering function fitting. When assuming a Gaussian size distribution, the polydispersity
degr ee of the nanocarrier is ca. 40%. Therefor e, the weighted average value of (67
±
5) nm
can be embraced as a r eliable external diameter for the liposomal drug-carrier . The results
of both appr oaches are summarized in table 6.1 together with the diameter of Caelyx
obtained with other techniques.
The average diameter obtained by contrast variation in SAXS is smaller than the r esult
obtained with DLS of ca. 86 nm, performed on a W130i apparatus (A vid Nano Ltd, High
W ycombe, UK) by the Institute of Materials and Envir onmental Chemistry (Hungarian
Academy of Sciences, Budapest, Hungary) similarly to the protocol described in V arga
et al. (2014 a ). This deviation between both results can be attributed to the fact that the DLS
measurand is the hydr odynamic size of the nanoparticles, while SAXS pr ovides the size
of the spherical volume inaccessible to the solvent. As the 2 kDa PEG-chains attached
to the surface of the liposomes contribute to the hydr odynamic radius but that layer is
permeable to the solvent and, ther efore, invisible to contrast variation SAXS, the ca. 20 nm
dif ference between the diameters determined by DLS and SAXS is justified.
6.2.3 A verage electro n density
At low
q
-values, the Guinier appr oximation can be used as explained in section 2.3.2. By
fitting the spherical form factor to the
q
-range just below the first minimum of the scattering
curves, an extrapolated value for the intensity at zero-angle
I (
0
)
could be obtained as
displayed in figur e 6.3b. The minimum of the parabola fitted to the experimental points
determines the average electr on density of the drug carrier system, accor ding to the
equation 2.36.
Fr om this calculation, a value of
ρ 0
= (346.2
±
1.2) nm
− 3
is obtained which corr esponds
to a combination of the electr on density of the liposomal nanocarrier and the precipitated
doxorubicin dr ug. The uncertainty of 1.2 nm
− 3
is associated with the vertical size of the
focused X-ray beam. The obtained density is slightly higher than the value of 338 nm
− 3
estimated for empty PEGylated liposomes (Ku ˇ cerka et al. , 2006) due to the pr esence of the
72

Osmotic effects in lip osomes 6.3
340
350
360
370
380
0 5 10 15 20 25 30 35 40 0
250
500
750
1000
1250
1500
1750
Electr on Density / nm − 3
Osmolality / mOsm kg − 1
Sucr ose Mass Fraction / %
Figure 6.4 |
Relation-
ship b et w een the solvent
electron densit y and the
solvent osmolalit y fo r an
aqueous sucrose solution.
doxorubicin-sulfate aggr egate in the intraliposomal volume.
6.3 Osmotic ef fects in liposomes
The rigidity of the nanocarriers is a r elevant property dir ectly related with its dr ug delivery
ef ficacy , the particle stability or the r elease rate of the encapsulated drug. In fact, some
of these characteristics might change upon injection into the blood vessels due to the
mechanical str ess applied to the nanocarriers in the process. In the case of lipid vesicles,
i.e. liposomes, the permeability of water through the phospholipid bilayer is a defining
aspect of their physicochemical behaviour . Although many aspects about the membrane
permeability have been studied (Nagle et al. , 2008; Mathai et al. , 2008; Olbrich et al. , 2000),
the evaluation of the liposomes rigidity and its osmotic activity is still challenging.
The osmotic behaviour of liposomes depends, basically , on their size and chemical
composition. For example, the incorporation of cholester ol can vary the fluidity of the
lipid bilayer . Larger liposomes tend to be osmotically active (de Gier, 1993) and behave
accor ding to the Laplace law: the osmotic pr essure needed to deform them decr eases
for incr easing sizes. In the case of liposomal nanocarriers, the intraliposomal osmolality
should be equal to the buf fer outside of the liposomes to enhance the particle stability .
Ther efore, it is an important question whether the incorporation of a drug into the
intraliposomal volume might modify its osmotic activity . For example, it is expected
that the small size of Caelyx and the doxorubicin-sulfate aggr egate in the intraliposomal
volume incr ease the resistance against the buf fer osmotic pressur e in comparison to an
empty liposomal particle. No osmotic pressur e ef fects were observed in the size or density
of the liposomal drug Caelyx in the pr evious section 6.2 due to the constant osmolality
of the suspending medium along the whole density gradient that was achieved using
Optipr ep as contrast agent.
However , this ef fect can be studied by increasing systematically the osmolality of the
suspending medium using aqueous sucr ose in the buffer . As shown in figure 6.4, the
sucr ose molecule acts simultaneously as a contrast agent and as an instrument to incr ease
the solvent osmolality . This enables the study of the osmotic effects in liposomes by the
density gradient technique in SAXS using aqueous sucr ose as suspending medium.
In this section, a thorough investigation of Caelyx under the ef fects of an increasing
73

Chapter 6
CONTINUOUS CONTRAST V ARIA TION APPLIED TO RELEV ANT BIO-
. . .
0.1
1
10
0.05 0.1 0.2 0.5 1
Scattering Intensity / cm − 1
q / nm − 1
200
400
600
800
1000
1200
1400
1600
1800
2000
Solvent Osmolality / mOsm kg − 1
(a) Caelyx scattering curves
Intensity at q = 0.123 nm − 1 / cm − 1
Solvent Osmolality / mOsm kg − 1
SAXS
W AXS
0.8
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
300 600 900 1200 1500 1800 2100
Osmotic
shrinkage
Constant
shape
and size
(b) Isoscattering point intensity
Figure 6.5 |
Osmotic effects of Caelyx in an aqueous sucrose densit y gradient. a) Scattering curves
measured at different solvent osmolalities b) Scattering intensit y at
q ?
1
as a function on the aqueous sucrose
solution osmolalit y . An osmotic threshold app ear s at 670 mOsm kg
− 1
. The exp eriment w as measured with
t w o different configurations with identical results.
solvent osmolality is performed, complementary to the study of the empty liposomal
nanocarrier under similar conditions. Besides, the consequences of PEGylation on the
liposomal structur e are also studied using this technique, focusing principally in its
osmotic activity .
6.3.1 Application to drug-stabilized liposomes
By means of the density gradient technique, scattering curves of the liposomal doxorubicin
wer e recor ded at differ ent sucr ose concentrations of the suspending medium, i.e. at
dif ferent buf fer osmolalities, as shown in figure 6.5a. A maximum osmolality preparation
was achieved with a 37.8 % sucr ose mass fraction, which corr esponds to an electr on
density of 381.1 nm
− 3
and a solvent osmolality of 1776 mOsm kg
− 1
, whereas a lighter
solution was pr oduced without sucrose by adding pur e water to get the same Caelyx
concentration. Considering the sucrose mass fraction of the Caelyx buf fer to be 10%, this
latter pr eparation has an electron density of 339.4 nm
− 3
and an osmolality of 151 mOsm
kg
− 1
. The X-ray scattering measur ements were performed at two dif ferent detector -to-
sample distances, in or der to study a broader
q
-range, spanning fr om 0.03 to 5.55 nm
− 1
.
Using the W AXS configuration described in section 3.4.2, the 1,0-dif fraction peak of the
doxorubicin fiber -like precipitate ar ound
q =
2.3 nm
− 1
(Li et al. , 1998) was observed, as
depicted in the figur e 6.6a after proper backgr ound corr ection. This Bragg diffraction arises
fr om the crystalline nature of the doxorubicin aggr egate in the intraliposomal volume.
As discussed in the pr evious section, by increasing the electr on density of the suspend-
ing medium, the scattering curves of the drug carrier change drastically due to contrast
variation. In the case of the aqueous sucrose gradient shown in figur e 6.5a, this effect is also
observed and str ongly resembles the curves measur ed with the Optiprep density gradient
depicted in figur e 6.2a. Nevertheless, upon a certain sucr ose concentration (corr esponding
to osmolalities ar ound 900 mOsm kg
− 1
in figur e 6.5a), the features of the scattering curves
change abruptly , because the suspending medium osmolality is so high that it induces
morphological changes in the liposomal structur e and, consequently , the scattering form
factor of the particles changes.
This ef fect can be quantified by examining the scattering intensity at the first isoscatter-
74

Osmotic effects in lip osomes 6.3
0
0.0002
0.0004
0.0006
0.0008
0.001
2 2.5 3
Scattering Intensity / cm − 1
q / nm − 1
200
400
600
800
1000
1200
1400
Solvent Osmolality / mOsm kg − 1
(a) DOX diffraction peak
-1
-0.5
0
0.5
1
250 500 750 1000 1250 1500
Dif fraction Peak Deviation / %
Solvent Osmolality / mOsm kg − 1
(b) Peak position deviation
Figure 6.6 |
Osmotic effects in the intralip osomal do xo rubicin-p recipitate by using sucrose as contrast agent:
a) (1,0) diffraction p eak of do xo rubicin after background subtraction fo r increasing solvent osmolalit y .
The mean FWHM of the p eak is 0.333 nm
− 1
. b) Deviation of the do xorubicin aggregate diffraction peak
p osition from the w eighted average q = 2.28 nm − 1 .
ing point position (
q ?
1 =
0.123 nm
− 1
) as a function of the suspending medium osmolality ,
as shown in figur e 6.5b. The intensity of the isoscattering points is independent of the
electr on density of the solvent as long as the size and the shape of the investigated particle
r emain constant. However , there is a clear osmolality thr eshold at 670 mOsm kg
− 1
in
figur e 6.5b when the intensity at
q ?
1
decays drastically . Above this threshold, the osmotic
pr essure at the liposomal bilayer is so high that the liposome starts shrinking and changes
its size, structure and, consequently , scattering form factor . The increased r esistance
against osmotic pr essure, mor e than double the blood plasma osmolality and much higher
than the osmolality needed to shrink empty PEGylated liposomes (V arga et al. , 2014 a ), is
explained by the encapsulation of crystal-like doxorubicin inside the liposome.
The lar ge osmotic pressur e produces a r eversible shrinkage of the liposome though it is
not capable of cracking it. This was proved in an additional experiment by incr easing the
osmolality of the buf fer to 1334 mOsm kg
− 1
with a sucr ose mass fraction of 31.4% and
then r educing it to 565 mOsm kg
− 1
by adding distilled water , wher e it was observed in
the scattering curves that the osmotic shrinkage pr ocess is reversible.
The behaviour of the nano-drug for an incr easing solvent osmolality can be further
studied by evaluating the crystal structur e of the doxorubicin aggr egate, r epresented by the
dif fraction peak displayed in the figure 6.6a. For this purpose, a W AXS configuration was
employed which extends the available
q
-range until 5.55 nm
− 1
by r educing the sample-
to-detector distance to
L = (
569
±
1
)
mm. The position of the peak in the recipr ocal
space depending on the suspending medium osmolality is depicted in figur e 6.6b and
shows that its position deviates less than 1 % fr om the weighted average
q =
2.28 nm
− 1
along the whole osmolality range. This proves that the fiber -like structur e of the drug
inside the liposome is also constant during the osmotic shrinkage of the liposomes. The
measur ed position of the (1,0) diffraction peak matches exactly the value measur ed from
doxorubicin-sulfate complexes in solution (Lasic et al. , 1992).
T o conclude this section, the diameter obtained from the isoscattering position in the
Optipr ep solution can be compared with what is measur ed in an aqueous sucrose suspend-
ing medium. In the latter , if only the scattering curves below this osmolality threshold
ar e considered, the relative standar d deviation for each
q
value r eveals a pronounced
minimum for the first isoscattering point as depicted in figur e 6.7. When comparing this
75

Chapter 6
CONTINUOUS CONTRAST V ARIA TION APPLIED TO RELEV ANT BIO-
. . .
Figure 6.7 |
Isoscatter-
ing p oint p osition quan-
tified b y the calculation
of the relative standa rd
deviation of the scatter-
ing curves fo r different
solvent densit y gradients.
In the case of the aqueous
sucrose solution (black
line), only the scattering
curves b elo w the osmo-
lalit y threshold w ere em-
plo y ed for the calculation.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.1 0.15 0.2 0.25
Relative Standar d Deviation
q / nm − 1
Optiprep
Aqueous sucrose
r esult with the relative standar d deviation curve obtained from the Optipr ep contrast
variation measur ements, both values for the size of the drug carrier agr ee remarkably well
within 2 %. This reflects the independence of the technique fr om the contrast agent added
to the suspending medium and shows the r epeatability of the results.
6.3.2 Does PEGylation af fect the osmotic activity of liposomes?
T ypically , unilamellar liposomes present a very narr ow size distribution and spherical
shape, whose diameter ranges fr om 50 nm to some hundreds of nanometr es. The covalent
attachment of biocompatible polymers can impr ove the liposome stability . For example,
PEG polymer chains show very low toxicity (Y amaoka et al. , 1994) and ar e widely used
as stabilizer (Sou et al. , 2000). PEGylated liposomal formulations, also called sterically
stabilized liposomes (SSL) or stealth liposomes, show longer blood circulation times in
vivo (Bar enholz, 2001) and exhibit a slow drug release rate. PEG-modified liposomes
have become of importance lately due to their incr eased drug pharmakinetics, decreased
plasma clearance and impr oved patient convenience (Gabizon & Martin, 1997; Harris
& Chess, 2003). Ther efore, the self-assembly of lipid structur es in the presence of PEG
moieties has been studied for dif ferent lipids (Lee & Pastor, 2011).
The incorporation of biocompatible polymers incr eases the phospholipid bilayer strength
and enhances the vesicle rigidity , which r elates to the increase of the bending modulus
(Liang et al. , 2005; Sou et al. , 2000). The higher membrane stiffness of SSLs has been
extensively characterized with methods such as AFM (Spyratou et al. , 2009) though other
techniques such as light scattering have found a higher osmotic activity in SSLs in compar-
ison to their non-PEGylated counterparts when incubated in serum (W olfram et al. , 2014).
Further investigations about the r elationship between PEGylation and the liposomal os-
motic behaviour in suspension ar e essential. In the following work, the differ ent r esponse
of SSLs and plain liposomes to osmotic pr essure is studied by SAXS.
For this purpose, five PEGylated and three plain liposomes wer e extruded with dif ferent
por e sizes, as explained in section 6.1. T o simplify the following discussion, the liposomes
ar e named after the hydrodynamic diameter measur ed by DLS. It is appar ent from these
measur ements that the size of the pore and the polydispersity degr ee of the liposome
76

Osmotic effects in lip osomes 6.3
1
10
100
1000
10000
100000
0.03 0.05 0.1 0.2 0.5 1
Scattering Intensity / a.u.
q / nm − 1
PEG 81 nm
PEG 87 nm
PEG 103 nm
PEG 179 nm
PEG 274 nm
plain 89 nm
plain 116 nm
plain 128 nm
(a) Scattering curves of the liposomes in buffer
1
10
100
0.5 1 2
Scattering Intensity / a.u.
q / nm − 1
(b) Bilayer scattering feature
Figure 6.8 |
a) Scattering curves of the different lip osomes in buffer. The curves are intensit y shifted fo r
cla rit y . The five SSLs a re presented in the lo wer pa rt of the plot. The diameters in the legend a re extracted
from DLS measurements. b) The phospholipid bila yer scattering feature of the liposomes in buffer: High
q
-region of the scattering curves of t w o plain lip osomes and the three la rgest SSLs in buffer. The colo r
co de of the scattering curves is sha red with figure a).
sample ar e directly r elated.
The SAXS measur ements of the eight liposomes are shown in figur e 6.8a, wher e the first
minimum
q
-value ranges fr om
∼
0.1 nm
− 1
in the 81 nm SSL to
∼
0.05 nm
− 1
for lar ger sizes.
For high polydispersities this scattering minimum gets smear ed out, as it can be observed
for the 274 nm SSL. It can be stated fr om these measurements and the DLS r esults that
the polydispersity degr ee rises for increasing liposomal sizes. Besides, non-PEGylated
liposomes show slightly br oader size distributions than SSLs.
Focusing on the high
q
-r egion of the single-contrast SAXS curves as displayed in figure
6.8b, the scattering featur e related to the phospholipid bilayer structur e is observed. For
Unilamellar V esicles (UL V), the feature shape is typically r ound with a maximum ar ound
q =
0.86 nm
− 1
(V ar ga et al. , 2012), related to a distance (
d = 2 π / q
) of 7.3 nm, as it can be
seen in the case of small PEGylated liposomes. For SSLs extruded with lar ger pores, the
bilayer shape shows incipient Bragg peaks which suggest the simultaneous pr esence of
Multilamellar V esicles (ML V) with a lamellar r epeat distance of 7.3 nm and unilamellar
SSLs. These quasi Bragg peaks arise from the periodic str ucture of the phospholipid bilayer
and the water layers, which interact thr ough a combination of the electrostatic potential,
the V an der W als attraction and other hydration terms. Nevertheless, the ML V population
cannot exceed the total number of unilamellar liposomes because the scattering contribu-
tion fr om UL V is still clearly dominant (Sakuragi et al. , 2011). The schematic repr esentation
of the dif ferent types of liposomes and the graphical definition of the lamellar r epeat
distance ar e depicted in figure 6.9.
The bilayer featur e of the plain liposomes differs completely fr om the round shape
visible in unilamellar vesicles. The diffraction peaks appearing at
q 1 =
0.88 and
q 2 =
1.9
≃
2
q 1
nm
− 1
corr espond to a slightly smaller lamellar repeat distance of 7.1 nm and
ar e related to a mor e pronounced pr esence of ML Vs, possibly of Oligolamellar V esicles
77

Chapter 6
CONTINUOUS CONTRAST V ARIA TION APPLIED TO RELEV ANT BIO-
. . .
Intraliposomal
volume
Hydrophilic
head
Hydrophobic
tail
(a) UL V (b) ML V
Aqueous
medium
Repeat
distance
Phospholipid
bilayer
(c) Bilayer in a stack
Figure 6.9 |
Schematic rep resentation of the different t yp es of lip osomes: a) Unilamella r vesicle (UL V) and
the different comp onents of a phospholipid bila y er. b) Multilamella r vesicle (ML V) comp osed of concentric
lipid bila y ers. While an oligolamella r vesicle (OL V) is a ML V with only a few lamellae, a bilamella r lip osome
consists of only t w o concentric phosp olipid bila yers. A stack of phospholipid b ila yers is highlighted with a
red b o x and depicted in mo re detail in c) with the graphical definition of the lamellar repeat distance.
(OL V) with only a few lamellae. This tendency is emphasized for larger plain vesicles, as
observed for the 128 nm plain liposome, where the r ound shape of the scattering featur e
practically disappears. This observation suggests that the plain liposomes distribution
consists of bi-, oligo- and multilamellar liposomes in a much higher ratio than the SSLs.
The ef fect of PEGylation induces a higher ratio of UL Vs due to the increased negative
char ge of the phospholipid bilayer in comparison to plain liposomes, which hinders the
cr eation of periodic lamellar structures. Nevertheless, small populations of OL Vs and
ML Vs coexisting with unilamellar liposomes can be observed for large extrusion por e sizes
in SSLs as well. In conclusion, the size and composition of the liposomes affect r emarkably
the formation of unilamellar vesicles and the shape of the phospholipid bilayer .
The behaviour of the dif ferent liposomal structur es to osmotic str ess can be examined
with a continuous contrast variation experiment using sucr ose as contrast agent, similarly
to the measur ements with the Caelyx sample in section 6.3.1. The scattering curves
measur ed for a PEGylated liposome with diameter 81 nm are displayed in figur e 6.10a,
wher e the solvent osmolality has been increased until 1409 mOsm kg
− 1
using a maximum
sucr ose mass fraction of 27.3 %. From the low
q
-r egion of these scattering curves some
facts can be extracted which r eveal preliminary the structural changes of the liposome
induced by the osmotic pr essure.
The curves do not intersect clearly in one point, even for low sucrose concentrations as
occur ed in the Caelyx case. The absence of an evident isoscattering point can be related
with the shape variation of the liposome alr eady at small osmotic pressur es. However , a
dif fuse intersection point, or pseudo isoscattering point (Kawaguchi, 2004), is visible at
q =
0.18 nm
− 1
. A very similar behaviour can be observed for the plain 89 nm liposome
in figur e 6.10b, where the suspending medium osmolality is incr eased until 1885 mOsm
kg
− 1
by a 35 % sucr ose mass fraction. In analogy to figure 6.5b, the intensity at the pseudo
q ?
as a function of the solvent osmolality is depicted in figure 6.11, as the deviation fr om
the original intensity for a plain and a PEGylated liposome of similar diameters.
The intensity at
q ?
starts diver ging from the original value alr eady at very low solvent
osmolalitites and r eflects the continuous change in shape or size of the liposome when
incr easing the osmotic pressur e. This behaviour occurs for both SSLs and plain liposomes
and suggest that a sharp osmotic thr eshold, like in the Caelyx case, does not exist. Thus,
78

Osmotic effects in lip osomes 6.3
1
10
100
1000
0.05 0.1 0.2 0.5 1
Scattering Intensity / a.u.
q / nm − 1
Pseudo
Isoscattering
Point
0
5
10
15
20
25
Sucr ose Mass Fraction / %
(a) SSL 81 nm
1
10
100
1000
0.05 0.1 0.2 0.5 1
Scattering Intensity / a.u.
q / nm − 1
Pseudo
Isoscattering
Point 0
5
10
15
20
25
30
35
Sucr ose Mass Fraction / %
(b) Plain 89 nm
Figure 6.10 |
Scattering curves of the 81 nm SSL and the 89 nm plain lip osomes measured at different
solvent osmolalities with an aqueous sucrose densit y gradient. The p ositions of the pseudo isoscattering
p oints at
q =
0.18 nm
− 1
and
q =
0.16 nm
− 1
a re ma rked fo r the PEGylated and plain lip osomes resp ectively .
the r esponse of liposomes to osmotic pressur e is steady and is already appar ent at low
osmolalities.
Besides, an evident variation of the scattering curves below
q ≤
0.3 nm
− 1
is observed in
figur e 6.10a when increasing the solvent osmolality . For example, the minimum originally
appearing at 0.1 nm
− 1
shifts slightly to lar ger
q
-values and disappears almost completely
for high sucr ose concentrations. This variation of the form factor can be caused by the
flattening of the liposomal shape observed with Fr eeze-fracture TEM (V arga et al. , 2014 a ).
Due to the incr eased osmotic activity , the original spherical liposome shrinks into an oblate
spher oid. This hypothesis can be further explored by focusing on the scattering featur e
r elated to the phospholipid bilayer at the high q -region.
For this purpose, the bilayer feature of the 179 nm PEGylated liposome is shown in
figur e 6.12a for increasing solvent osmolalities. As observed in figur e 6.10a for sucr ose
concentrations above 15 %, the bilayer scattering feature shifts abr uptly to smaller
q
-values.
This lar ge contrast effect occurs at solvent densities close to the average electr on density
of the phospholipid bilayer (ca. 348 nm
− 3
), which corr esponds to a sucrose mass fraction
of
∼
12 %. The convolution of the contrast-r elated effects with the variations induced
by the osmotic pr essure demands a mor e challenging evaluation, can prevent the right
interpr etation of the data and is, thus, unwanted. Therefor e, the scattering curves shown
her e were measur ed with sucrose concentrations ≤ 10 %.
The original double-peak structur e of the SSL at 0 % sucrose concentration observed in
figur e 6.12a transforms upon increasing the solvent osmolality and splits into thr ee peaks
of decr easing intensity at
q 1 =
0.48 nm
− 1
,
q 2 =
0.86 nm
− 1
and
q 3 =
1.28 nm
− 1
. These
Bragg peaks superimposed on the bilayer form factor r eveal a periodic structure which
can be r elated with a partial oligolamellar structure in the liposome system (Fernandez
et al. , 2008). The three mentioned dif fraction peaks translate into a lamellar r epeat distance
of ca. 13 nm, approximately doubling the thickness of the single phospholipid bilayer
(Kenworthy et al. , 1995) and suggesting the appearance of a bilamellar structur e (Demé
et al. , 2002).
The transition between a single bilayer phase and a bilamellar phase at 10 % sucr ose
concentration supports the hypothesis pr esented above that the liposome shrinks into
lens-shaped vesicles due to the osmotic pr essure. The bilamellar structure might arise
79

Chapter 6
CONTINUOUS CONTRAST V ARIA TION APPLIED TO RELEV ANT BIO-
. . .
Figure 6.11 |
Isoscat-
tering p oint intensit y of
the lip osomes: Deviation
from the initial intensit y
at
q ?
at different solvent
osmolalities measured fo r
a PEGylated and plain
lip osome of simila r dia-
meters. A clea r osmotic
threshold can not b e ob-
served. 0
1
2
3
4
5
0 5 10 15 20
Deviation fr om q ? intensity / %
Sucr ose Mass Fraction / %
plain - 89 nm
PEG - 87 nm
fr om the close bilayer contacts at the outest part of the elliptical liposomes, while the
single bilayer conformation still r emains dominant in the midsection of the liposomes.
A similar morphology has been observed after the osmotic shrinkage of DPPC/DSPE-
PEG
2000
vesicles (T err eno et al. , 2009). In fact, this behaviour was identical for all five
studied PEGylated liposomes, independent of their size.
Besides, the changes of the phospholipid bilayer form factor are smooth upon incr easing
the osmotic pr essure as shown in figur e 6.12a, where the bilayer scattering featur e starts
varying at very low sucr ose concentrations. This validates the observation from figur e
6.11 and confirms that the incr easing solvent osmolality affects continuously the str ucture
of the liposomes and not as abruptly as in the case of Caelyx.
Contrarily , the phospholipid bilayer of the plain liposomes r emains unchanged upon
incr easing the solvent osmolality until 1285 mOsm kg
− 1
, as displayed in figur e 6.12b. This
suggests that the ML V structur e of the non-PEGylated vesicles increase their r esilience
and the multiple phospholipid bilayers str engthen the elastic modulus of the liposome
membrane.
The fact that the incorporation of PEG moieties influences alr eady the preparation and
formation of the liposomes pr events a proper comparison of the osmotic ef fects between
SSLs and plain liposomes of similar diameters. The existence of ML Vs for non-PEGylated
liposomes acts as a limiting factor for the osmotic activity and contrasts with the osmotic
ef fects observed in unilamellar SSLs already at low sucr ose concentrations, which shrinks
the PEGylated liposomes into oblated ellipsoids.
The chemical ef fect of sucrose on the SSL membrane is a subject of discussion, because
it can be ar gued that the disaccharide molecule penetrates the lipid membrane or creates a
solvation shell ar ound the liposomes. However , previous studies in this subject (Kiselev
et al. , 2001 a , b , 2003), the lar ge size of the sucrose molecule and similar r esults with other
experiments performed with salt (V ar ga et al. , 2014 a ) suggest otherwise. Therefor e, it
can be concluded that the study of the osmotic activity of liposomes can be performed
successfully using aqueous sucr ose and shows very distinguishable effects for UL Vs
(PEGylated liposomes) and ML Vs (plain liposomes).
80

Osmotic effects in lip osomes 6.4
0.2
0.5
1
2
0.5 1 2
Scattering Intensity / a.u.
q / nm − 1
0
2
4
6
8
10
Sucr ose Concentration / %
(a) PEG 179 nm
1
10
100
0.5 1 2
Scattering Intensity / a.u.
q / nm − 1
0
5
10
15
20
25
Sucr ose Concentration / %
(b) Plain 128 nm
Figure 6.12 |
Osmotic effects in the phospholipid bila y er of the lip osomes: Scattering curves measured at
different solvent osmolalities fo r a 179 nm SSL and a 128 nm plain lip osome. The app ea rance of Bragg
p eaks in the SSL memb rane contrasts with the unaltered shap e of the bila y er in the plain lip osome.
81

Chapter 6
CONTINUOUS CONTRAST V ARIA TION APPLIED TO RELEV ANT BIO-
. . .
0.1
1
10
0.2 0.5 1 2
Scattering Intensity / a.u.
q / nm − 1
340
350
360
370
380
Solvent Electron Density / nm − 3
(a) HDL
1
10
100
0.2 0.5 1 2
Scattering Intensity / a.u.
q / nm − 1
340
350
360
370
380
Solvent Electron Density / nm − 3
(b) LDL
Figure 6.13 |
Scattering curves of HDL and LDL measured at different solvent densities b y using an aqueous
sucrose densit y gradient.
6.4 Sizing of blood plasma componenents
Fr om a nanoscience point of view , human blood can be seen as a suspension of particles
with dif ferent physiological r oles, where important components ar e in the nanorange.
Serum lipopr oteins are the colloidal particles involved in the transport and metabolism of
insoluble lipids and ar e among the most studied biological particles. The interest in their
activity is understandable due to their direct r elationship with very extended diseases
in the W estern world population, such as obesity or ather ogenesis, e.g. obturation of the
arterial walls. For example, the dysregulation of cholester ol in plasma, primarily carried
within lipopr oteins, is responsible of ather osclerosis (Munr o & Cotran, 1988). Besides,
they ar e a convenient model for lipid-protein interactions (Assmann & Br ewer, 1974) due
to their lipid cor e and the hydrated proteins isometrically situated on its surface.
Lipopr oteins are isolated fr om blood plasma by ultracentrifugation (Havel et al. , 1955)
and ar e normally classified by their density range, showing differ ent chemical composition,
size and pathological condition for each class (German et al. , 2006). Indeed, the size of
lipopr oteins is critically connected with disease risk (Gardner et al. , 1996) and Low Density
Lipopr oteins (LDL) are suggested to be mor e or less atherogenic depending on their size
(Dr eon et al. , 1994). The effect of diabetes on the lipopr otein size is also of great inter est,
especially the sex-dependency of the High Density Lipopr otein (HDL) size (Colhoun et al. ,
2002).
Ther efore, pr ecise sizing techniques are a cr ucial tool to understand the physiological
pr ocesses of lipoproteins (German et al. , 2006). The naturally narrow size distributions
of LDL and HDL suggest small-angle scattering as a well-suited method and their het-
er ogeneous morphology advises the use of a contrast variation approach. For instance,
the first characterization attempts date back to the late 1970s with neutr on scattering
(Stuhrmann et al. , 1975), using salt (T ar dieu et al. , 1976) and sucrose (Müller et al. , 1978) as
SAXS contrast agents or modifying the sample temperatur e (Laggner et al. , 1977; Luzzati
et al. , 1979).
The complicated inner structur e of the lipoproteins r evealed in more r ecent studies
(Baumstark et al. , 1990; Schnitzer & Lichtenber g, 1994) encourages the use of parameter-
independent and model-fr ee analysis of the scattering data. W ith this objective, LDL
and HDL samples wer e measured with continuous contrast variation in SAXS using 40
82

Sizing of blo o d plasma comp onenents 6.4
0.1
0.2
0.5
1
0.3 0.5 1 1.5
Rel. Std. Deviation
q / nm − 1
HDL
LDL
(a) Isoscattering point position
0
0.5
1
1.5
2
2.5
340 350 360 370 380
I ( 0 ) / a.u.
Solvent Electron Density / nm − 3
HDL
LDL
(b) A verage electron density
Figure 6.14 | Compa rison of the mo del free app roaches for HDL (red) and LDL (black)
% sucr ose mass fraction to increase the solvent electr on density until 384 nm
− 3
. The
scattering curves obtained for HDL and LDL ar e presented in figur es 6.13a and 6.13b
r espectively .
In the case of HDL in buf fer , the first minimum appears at
q ≈
0.5 nm
− 1
. By increasing
the solvent density , this minimum shifts to smaller
q
-values hinting the denser composition
of the pr otein shell in comparison to the lighter lipid and cholesterol cor e. A lighter core
morphology is also expected for LDL (Luzzati et al. , 1979) and it agr ees with the contrast
ef fect observed in the scattering curves displayed in figure 6.13b.
The lar ge number of observable minima indicates the narrow size distributions of both
samples, providing ideal conditions to use the isoscattering point
q ?
appr oach. The relative
standar d deviation as a function of
q
calculated for both lipopr oteins is shown in figure
6.14a, wher e the minima correspond to the position of
q ?
i
. The clear minimum for HDL
is located at
q ? = ( 0.83 ± 0.14 )
nm
− 1
, corr esponding to an impenetrable diameter for
the solvent of
( 11 ± 2 )
nm. The position of the first
q ?
in LDL is shifted to smaller
q
,
q ? = ( 0.42 ± 0.08 )
nm
− 1
, which translates into a solvent-excluded diameter of
( 21 ± 4 )
nm.
Considering that the lipopr oteins are quasi-spherical (Stuhrmann et al. , 1975), these
r esults can be compared to those extracted fr om literature. The differ ent cholester ol
tranport necessities r eflect into a large variety of HDL subclasses with a diameter range
between 7 and 13 nm (German et al. , 2006). For example, a diameter of 13 nm was observed
for the subclass type HDL3 (T ardieu et al. , 1976), which deviates only 15 % fr om the result
measur ed in our study . Difficulties to know the measur ed subclass of the commercially
pur chased HDL hinders a more thor ough comparison.
In the case of LDL, several studies pr ovide diameters between 21 and 28 nm (T ardieu
et al. , 1976; Colhoun et al. , 2002; German et al. , 2006), though the most r epeated values lay
ar ound 22 to 23 nm (Müller et al. , 1978; Luzzati et al. , 1979), less than 10 % deviation from
our r esult. Nevertheless, the possible solvent penetration into the outer layers of LDL
(Stuhrmann et al. , 1975; T ardieu et al. , 1976) calls for caution as the diameter obtained fr om
the q ? position considers an impenetrable particle.
The ef fects of permeability and protein hydration might be r elated to the density of the
lipopr otein, which is the most characteristic feature of each lipopr otein class. As described
pr eviously , the intensity at zero-angle is r elated to the average electr on density by the
expr ession 2.36 and can be measured. The experimental
I ( q =
0
)
values ar e depicted in
83

Chapter 6
CONTINUOUS CONTRAST V ARIA TION APPLIED TO RELEV ANT BIO-
. . .
Figure 6.15 |
Squa red ra-
dius of the HDL scatter-
ing data. The analytical
fit results in an average
densit y of
( 353.6 ± 1.5 )
nm
− 3
and an external dia-
meter of ( 12 ± 2 ) nm.
0
50
100
150
200
250
300
340 350 360 370 380
Radius 2 / nm 2
Solvent Electr on Density / nm − 3
figur e 6.14b, where the fits of the pr evious equation are shown as solid lines.
Accor ding to the analytical fit, the average density of HDL is
( 358.4 ± 1.5 )
nm
− 3
and
the density measur ed in the LDL case is
( 345 ± 2 )
nm
− 3
. In the latter , the low number
of points measur ed below the average density of LDL due to the limited range given by
the water electr on density (333 nm
− 3
) incr eases the uncertainty of the result, although
the value is still in pr etty good agreement with other SAXS studies (T ardieu et al. , 1976;
Luzzati et al. , 1979). The pr otein-rich (
∼
50 %) structur e of HDL explains its higher density
in comparison to LDL, composed mainly of lipids ( ∼ 80 %).
Another model-fr ee interpretation of the HDL scattering data is pr esented in figur e 6.15,
wher e the the squared radius of the Guinier r egion is presented as a function of the solvent
electr on density . As previously shown, the analytical expr ession 2.34 can be fitted to the
experimental data, r esulting in an average electron density
ρ 0 = ( 353.6 ± 1.5 )
nm
− 3
and
a particle shape radius of
R c = ( 6 ± 1 )
nm. The diameter obtained with this appr oach,
( 12 ± 2 )
nm, is consistent with the previous r esult. Probably due to the absence of r elevant
experimental points ar ound the match point, the average density dif fers by almost 5 nm
− 3
fr om the I ( 0 ) r esult.
The continuous contrast variation technique and the subsequent model-fr ee analysis are
easy and ef fective tools to measure the size and density of lipopr oteins, very important
attributes to understand the biological pr ocesses related to cholester ol and lipid transport.
A mor e detailed analysis and modelling of the scattering data could have addressed
some issues such as the hydration and distribution of the pr oteins on the surface, the
permeability of the steric and lipid cor e or the radial distribution of cholesterol and
triglycerides in the lipopr otein.
84

Protein-coated lo w-densit y nanopa rticles 6.5
1
10
100
1000
0.03 0.05 0.1 0.2 0.4
Scattering Intensity / a.u.
q / nm − 1
PS-COOH
0.5 mg/ml IgG
1 mg/ml IgG
2 mg/ml IgG
4 mg/ml IgG Figure 6.16 |
SAXS
curves at a single con-
trast of the PS-COOH
pa rticles coated with IgG
at different concentra-
tions.
6.5 Pr otein-coated low-density nanoparticles
The most r ecent efforts in nanomedicine aim for a high contr ol of the characteristics of the
nanocarrier surface, as the surface’s pr operties are a defining element of its ef ficiency as
drug carrier . Besides, nanoparticles interact with proteins when intr oduced into biological
media, leading to the formation of the so-called pr otein corona surr ounding the nanocarrier
(Cedervall et al. , 2007; Monopoli et al. , 2011; Casals et al. , 2010). The identity of the
biomolecule coating depends on the particle size, surface functionalization and charge
(Lundqvist et al. , 2008; T enzer et al. , 2013; Gessner et al. , 2003) and its detailed description
is challenging. Y et, the ability to quantitatively characterise this interface is important in
understanding particle behaviour in these complex envir onments and improving their
surface engineering for enhanced functionality .
IgG is the most common type of antibody found in human serum and, therefor e, a
logical candidate to coat the studied nanoparticles with. In this case, we used commer -
cially available PS-COOH particles, because polystyr ene NPs are commonly used in the
development of nanoparticle-based strategies for medicine, thanks to the low cost of their
material and the versatility of their surface functionalization. The carboxylated surface
pr events the agglomeration of the particles and also provides a chemical anchor for the
pr otein binding. The use of SAXS to obtain a quantitative description of the protein cor ona
is examined for dif ferent IgG concentrations, e.g. shell thicknesses, and compared with
DLS and DCS (Minelli et al. , 2014).
The bar e PS-COOH particles are highly char ged, translating into a high
ζ
-potential, i.e.
str ong repulsive electr ostatic potential on the particle surface. A
ζ
-potential of (
−
49
±
1)
mV was measur ed, which is drastically reduced to ar ound
−
10 mV following the binding
of the positively char ged IgG. The SAXS measurements of the IgG-coated particles with
dif ferent pr otein concentrations are shown in figur e 6.16, where a clear shift to smaller
q
-values is observed for incr easing concentration of IgG. This effect is clearly r elated with
the incr ease in size for higher IgG concentration, although a quantitative description is
complicated.
Due to the cor e-shell morphology of the polymeric bare particle observed in chapter
85

Chapter 6
CONTINUOUS CONTRAST V ARIA TION APPLIED TO RELEV ANT BIO-
. . .
T able 6.2 |
Concentration of IgG incubated with PS-COOH pa rticles and IgG shell thickness as measured
b y single-contrast SAXS, DCS and DLS (Minelli et al. , 2014). A double-shell mo del with sha rp interfaces
w as used fo r the SAXS results. The uncertainties a re the standa rd deviations of rep eated measurements.
ρ I g G / mg mL − 1 ζ -potential / mV T D L S / nm T D CS / nm T S A X S / nm
0.5 -10.8 ± 0.9 10 ± 1 3.7 ± 0.6 7.7 ± 1.4
1 -10.7 ± 0.6 11 ± 2 5.9 ± 0.5 8.4 ± 1.4
2 -9.6 ± 0.5 12 ± 2 7.6 ± 0.4 9.6 ± 1.5
4 -9.7 ± 0.5 15 ± 2 8.3 ± 0.4 9.6 ± 1.5
4, SAXS curves wer e analysed using a double-shell model based in the form factor 2.26,
considering a sharp interface between the dif ferent components and a constant thickness
and density of the IgG cor ona. In order to focus on the total diameter instead of the details
of the internal structur e, the limits of the inner and outer radii of the polymer shell ar e not
fixed and ar e treated as fitting parameters together with the outer radius and the contrast
dif ference of each shell with the polystyr ene core.
The IgG shell thickness obtained for IgG-coated particles with dif ferent pr otein con-
centrations is shown in table 6.2 and compar ed to the size measurements performed
with other techniques by the Surface and Nanoanalysis gr oup of NPL (T eddington, UK).
All techniques (DLS, DCS and SAXS) show an incr ease in the IgG-shell thickness with
incr easing concentration of the protein in solution during incubation. As expected, DLS
pr ovides higher values than the other techniques, as the measured thickness is r elated to
the hydr odynamic properties of the system.
Although all techniques show an incr ease of the IgG shell thickness with increasing
concentration of the pr otein, full consistency among them r equires the calculation of a com-
bined measur ement uncertainty and further refinements of the SAXS and DCS modelling.
For instance, the SAXS evaluation has neglected the possible spatial heter ogeneity and
hydration of the IgG cor ona and the model employed for the core particle over estimates
the diameter by almost 10 % (chapter 4 and Minelli et al. (2014)).
6.5.1 Har d protein cor ona characterization with contrast variation
The possible inaccuracies arising fr om the previous modelling appr oach might be pr even-
ted by using continuous contrast variation and a model-fr ee evaluation. For this purpose,
the pr otein-coated particle with 4 mg mL
− 1
IgG was intr oduced in a density gradient with
sucr ose as contrast agent, resulting in an incr ease of the solvent electron density until 350.8
nm
− 3
at the maximum sucr ose concentration of 14.7 %. The isoscattering point position is
quantified by calculating the r elative standard deviation of the 20 measur ed curves at each
q , as depicted in figur e 6.17. This value becomes minimal at q = ( 0.080 ± 0.011 ) nm − 1 .
By comparing in figur e 6.17 the relative standar d deviation curves of the bare PS-COOH
particle obtained in chapter 4 and the IgG-coated sample, it is noticeable that the position
of the minimum is shifted to smaller
q
-values after the adsorption of pr oteins to the
surface as a consequence of the increase in size. The diameter incr ease
t
is quantified by
inserting in equation 2.30 the isoscattering positions befor e and after the target attachment,
q ? = ( 0.090 ± 0.006 )
nm
− 1
and
q ?
I g G = ( 0.080 ± 0.011 )
nm
− 1
r espectively . Combining
both r esults, t is expressed by:
t = R IgG − R = κ
q ?
IgG − κ
q ? , (6.1)
86

Protein-coated lo w-densit y nanopa rticles 6.5
0.1
1
0.075 0.1 0.125 0.15
Relative Standar d Deviation
q / nm − 1
Plain PS-COOH
After attaching IgG
Figure 6.17 |
Isoscatter-
ing p oint p osition before
and after attaching IgG
(4 mg mL
− 1
) to the PS-
COOH pa rticles. A shift
of the first minimum to
lo w er
q
-values is observed
after attaching the biota r-
get to the nanopa rticle.
wher e
κ =
4.493,
t
is the IgG-shell thickness and
R
and
R I g G
ar e the particle radii
befor e and after IgG incubation. This results in a shell thickness of (7
±
8) nm, wher e the
uncertainty associated to the thickness ( δ t ) is derived from the expr ession 6.1 as:
δ t 2 = 



κ
 q ?
IgG  2 · δ q ?
IgG 



2
+ 

κ
 q ?  2 · δ q ? 

2
(6.2)
wher e
δ q ?
IgG =
0.011 nm
− 1
and
δ q ? =
0.006 nm
− 1
arise fr om the diffuseness of the
isoscattering point position. This large uncertainty is mainly explained by the low concen-
tration of coated particles in suspension due to the IgG-incubation pr ocess. The decr eased
scattering contribution of the particles in comparison to the medium limits the signal-to-
noise ratio and thus the accuracy of the isoscattering point determination. Besides, the use
of sucr ose in the solution might disturb the solvation shell around the particles and vary
the hydration pr operties of the protein-cor ona. Such an effect is dif ficult to detect though
it can af fect strongly the scattering curves when the electr on density of the medium and
the pr obed particle are similar .
Although the r elative uncertainty associated to the shell thickness is
>
100 %, it is
important to highlight that
t
corr esponds to the volume inaccessible for the solvent
and, thus, it can be identified with the hard pr otein cor ona surrounding the polymeric
nanoparticle, i.e. the impermeable part of the IgG shell. Nevertheless, the lar ge associated
uncertainty suggests that this technique is inappr opriate for the accurate determination of
the thickness of a har d protein cor ona.
87

7
Summary
This thesis demonstrates how continuous contrast variation in small-angle X-ray scattering
(SAXS) by means of a density gradient capillary emer ges as a powerful characterization
technique for low-density nanoparticles. The technique has proven ef ficient on a gr eat
variety of systems r elevant to nanomedicine such as polymeric nanocarriers, the PEGylated
liposomal nano-drug Caelyx, empty liposomal nanocarriers and human lipopr oteins. The
possibility to collect an extensive data set of scattering curves in a short timespan and
the ability to tune the contrast range during the experiment arise as clear advantages
of the method. The scattering data acquired with this newly intr oduced technique has
been analysed with complementary appr oaches to reveal a consistent insight into the
size distribution and the inner structur e of the suspended nanoparticles, resulting in the
determination of the size and density of the nanoparticles in a traceable way .
The application of the continuous contrast variation technique in SAXS to characterize
low-density polymeric nanoparticles has been thor oughly reviewed in chapter 4. Up
to thr ee differ ent evaluation approaches wer e employed to determine the size of the
PS-COOH nanoparticles. By using a model-free analysis of the experimental data based
on the isoscattering point theory , an average particle diameter of
( 100.6 ± 5.6 )
nm was
obtained, which was in very good agr eement with the value obtained from a cor e-shell
model fit of ( 99.4 ± 5.6 ) nm.
The scope of the continuous contrast variation method as a sizing technique was r e-
vealed in chapter 5 by the consistency of the r esults of the PS-Plain particles obtained
with dif ferent evaluation appr oaches and techniques, like atomic force micr oscopy (AFM),
dif ferential centrifugal sedimentation (DCS) and transmission scanning electr on micro-
scopy (TSEM). Furthermor e, differ ent evaluation approaches to contrast variation SAXS
data ar e examined in detail. The model-fr ee isoscattering point framework is found to
be of easy use and very appr opriate for the size determination of spherical and quite
monodisperse colloids. On the other hand, the calculation of the shape scattering function
arises as a pr ecise sizing technique which can additionally provide an insight into the
particle shape, although a high number of measurements with dif fer ent contrasts and an
accurate calibration of the system ar e requir ed.
Due to the high sensitivity of SAXS to small electr on density differ ences in the colloid
89

Chapter 7 SUMMARY
morphology , information about the heterogeneous composition of the particles can be
r etrieved. For instance, the analysis of the Guinier region of the scattering curves per -
formed in section 4.3.3 showed that the radial inner structur e of the PS-COOH particles
consisted of a thin, mor e dense layer coating the polystyrene cor e. Complementing these
r esults, the form factor fit presented in section 4.3.1 r evealed that the core component of
the particle had exactly the same electr on density expected for polystyrene and the shell
was composed of a compound with a density below that of PMMA. This observation
is of paramount importance in polymeric particle characterization because the dir ect
observation by imaging techniques is inadequate for this purpose. In fact, the detection
of cor e-shell structures in polymeric colloids appears as essential for understanding the
possible pr ocesses occurring during the formation of the particle, e.g. the consequences of
emulsion polymerization synthesis or the segr egation of components due to their differ ent
hydr ophobicity .
Besides, a high accuracy in the density information is achieved with the density gradient
technique and extends along a rather large density range of polymers as shown in chapter 5.
For instance, SAXS measur ements of the density of three dif ferent polymeric colloids ar e
in excellent agr eement with those performed by DCS, a technique extensively used in
nanoparticle characterization. As r eviewed in section 4.3, the determination of the average
electr on density of the particle by differ ent evaluation approaches pr oves the continuous
contrast variation technique as a useful tool and an alternative to other techniques like
analytical ultracentrifugation, isopycnic centrifugation or field-flow fractionation.
At this point, the performance of the continuous contrast variation in SAXS for the
simultaneous size and density determination of low-density polymeric nanoparticles
has been successfully pr oven. The technique has evident advantages in comparison to
other contrast variation techniques in small-angle scattering like deuterated small-angle
neutr on scattering (SANS) or anomalous SAXS (ASAXS), but certain limitations do also
arise, namely its restriction to low-density nanomaterials due to the r elatively low electr on
densities achievable with standar d contrast agents. Nevertheless, the importance of the
technique has been justified with its application to multiple nanomaterials relevant to
r esearch fields like medicine or biology in chapter 6.
In the case of the nano-drug Caelyx, a liposomal formulation of doxorubicin coated with
polyethylene glycol (PEG), the position of the isoscattering point was measured by means
of an iso-osmolal density gradient wher eby the size of the liposomal drug was determined
with this model-fr ee approach. Supplemented by the model fitting of the shape scattering
function of the liposomes, the size was also obtained from an independent evaluation
pr ocedure and an average diameter of (67
±
5) nm was determined. This size is smaller
than the value measur ed by dynamic light scattering (DLS), which can be attributed to the
fact that the contrast variation SAXS determines the size of the liposomes impermeable
to the contrast agent, i.e. the outer PEG layer of the liposomes is not pr obed. This
demonstrates that the combination of SAXS with DLS can r eveal the differ ence between
the hydr odynamic diameter and the "core" size of the nanocarrier , which is r elated to the
thickness of the PEG-layer in case of the stealth liposomes. Moreover it is shown that
by means of the shape scattering function fitting, complementary information about the
shape of the nanocarrier can be obtained. Additionally , it was found that the average
electr on density of the liposomal doxorubicin was higher than that of the empty PEGylated
liposomes.
Using an aqueous sucr ose density gradient, it was possible to study the behaviour
of the liposomal drug carrier under dif fer ent osmotic conditions. It was shown that an
90

7.0
incr easing osmolality of the buffer pr oduces an osmotic shrinkage of the liposomal struc-
tur e, although this structural deformation is reversible and does not af fect the crystalline
structur e of the intraliposomal doxorubicin. For comparison purposes with the liposomal
doxorubicin system, the osmotic activity of empty liposomes was also investigated using
aqueous sucr ose. The distinguishable osmotic effects observed in PEGylated and plain
liposomes arise fr om the differ ent formation of the liposomes, which is influenced by
the pr esence of PEG moieties in the pr eparation. The creation of multilamellar domains
in the phospholipid layer was evaluated and the r ole of the PEG moieties in the mem-
brane r esilience was also investigated. The multilamellar structur e of the plain liposomes
shows higher r esilience against osmotic pressur e than the unilamellar membrane of the
PEGylated vesicles. In the latter , the unilamellar vesicle shrinks due to the osmotic pr es-
sur e and deforms the liposomes into obloid ellipsoids, creating a bilamellar str ucture at
the outest part of the vesicles.
The continuous contrast variation technique was also used to determine the most
distinctive traits of human lipopr oteins: size and density , while the application of the
technique on nanoparticles incubated in dif ferent concentrations of Immunoglobulin G
(IgG) r evealed that the large uncertainty associated to the dif fuseness of the isoscattering
points makes the contrast variation appr oach inappropriate for the accurate and traceable
determination of the pr otein-shell thickness. Nevertheless, the use of complementary
techniques such as SAXS, DLS and DCS shows an incr ease of the protein-cor ona thickness
with incr easing concentration of the proteins during incubation as expected.
The work pr esented in this thesis proves that the r ecently developed continuous con-
trast variation technique in SAXS extends the possibilities of the classic solvent contrast
variation appr oach to unexpected new heights. The use of a density gradient capillary
r esults in a virtually continuous range of available solvent electron densities and opens up
new perspectives in the characterization of low-density nanoparticles in suspension.
91

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106

Acknowledgments
I would like to use this opportunity to thank all the people that have assisted me during
these years to r each my goals and have contributed to the conclusion of this thesis. Though
I tried to avoid any verbose language during the main text, it will be extr emely challenging
to r emain synthetic when acknowledging the contributions of the following people.
First of all, I would like to thank Dr . Michael Krumrey , the leader of the working group
Röntgenradiometrie of the Physikalisch-T echnische Bundesanstalt (PTB) and the person
who pr ovided me the proper human and scientific envir onment to perform successful
experiments and pursue my r esearch inter ests. Under his leadership, I could concentrate
in the r elevant aspects of my investigations and focus all my energy into my r esearch.
I am very grateful also to Pr of. Dr . Mathias Richter for giving me the opportunity to
participate on the activity of the PTB in BESSY II and encourage me to chase my scientific
goals. His motivation and constructive advices during these years have been really helpful
and ar e highly appreciated.
Pr of. Dr . Stefan Eisebitt and Pr of. Dr . Simone Raoux are also kindly acknowledged for the
pr ecious advice given to complete my resear ch work and for the concern to read and pr ove
this written thesis. Their many r esearch inter ests inspired me to find new alternatives to
old scientific pr oblems.
I am gr eatly indebted to my mentor Dr . Christian Gollwitzer for his supervision and honest
inter est throughout these last 4 years. The valuable scientific expertise he provided me
with cannot overshadow the gr eat moments we spent together in the laboratory . W ithout
his support and expert advise, the completion of this thesis would have been virtually
impossible.
And also my most sincer ely acknowledgement to the whole Arbeitsgruppe 7.11 of PTB,
whose individuals have contributed to my work both technically and personally . I am
especially thankful to all the engineers who have provided the technical support to
perform SAXS experiments in an outstanding way . During these years, the gr oup line-up
included Levent Cibik, Ulf Knoll, Stefanie Langner , Swenja Schr eiber , Layla Riemann and Peter
Müller .
I don’t want to for get the many graduate students and postdocs with whom I have
cr ossed paths in PTB and who have influenced and enhanced my resear ch like Dr . Jan
W ernecke, Analía Fernández Herrer o, Anton Haase, Mika Pflüger , Oleksey Mariasov and Dr .
V ictor Soltwisch .
I am glad to acknowledge also the excellent job that all the members of the Laboratory
of the PTB in BESSY II perform day after day as well as the Helmholtz-Zentrum Berlin
(HZB) scientists who operate the synchr otron facility . W ithout the continuous and stable
performance of BESSY II, most of the experimental data shown in this thesis could not
have been collected.
I am in debt both personally and scientifically with Dr . Zoltan V arga fr om the Institute of
Materials and Envir onmental Chemistry (Research Centr e for Natural Sciences, Budapest,
Hungary). His expertise in SAXS and liposomal structur es is unparalleled and some of the

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