Investigations of vesicle gels by pulsed and modulated gradient NMR diffusion
techniques
Samo Lasi
c,*
ab
Ingrid
Aslund,
a
Claudia Oppel,
c
Daniel Topgaard,
a
Olle S€
oderman
a
and Michael Gradzielski
c
Received 8th November 2010, Accepted 28th January 2011
DOI: 10.1039/c0sm01278e
Vesicle gels are surfactant systems that form stiff gels with rather low amounts of surfactant. So far
their structures have mostly been investigated using scattering techniques, which are generally
appropriate for the study of structures on the nm-length-scale. Here we examine these gels using two
complementary diffusion NMR techniques, which are both sensitive to structures on the mm-scale. The
presented results imply structural features on the mm-scale, indicating a more complex structure than
just that of densely packed small vesicles, as previously found for these systems. It is demonstrated that
a combination of the diffusion NMR methods, described here, can provide useful insights, when
morphological features extend over a wide range of length scales.
1 Introduction
A multitude of different structures can be obtained from self-
assembled surfactant and block copolymer systems, including
micelles and liquid crystals.
1–4
An important class of structures
formed are constituted by bilayers and vesicles, i.e. spherically
closed bilayers. Such vesicles may occur as unilamellar or mul-
tilamellar vesicles.
5
They can be used in diverse applications, such
as formulations for softeners, nanoreactors, material templates
and cell membrane models. They are also interesting for
a fundamental understanding of molecular and colloidal inter-
actions and for being model systems for membranes.
For sufficiently high concentrations of surfactants, vesicles will
be densely packed and this will lead to a substantial increase in
viscosity and even to the formation of vesicle gels, i.e. systems
that posses a yield stress. Such behaviour has for instance been
observed for densely packed multilamellar vesicles.
5–7
However,
it has also been observed that unilamellar vesicles composed of
anionic surfactant, cosurfactant, and water can form such
a vesicle gel spontaneously. These systems are viscous and
according to previous experimental investigations they consist of
monodisperse unilamellar small vesicles with a radius of about
15–25 nm.
8–11
The fact that they are unilamellar and relatively
small, is rather uncommon for spontaneously forming structures,
i.e. their formation must be driven by the spontaneous curvature
of the bilayer due to an asymmetry induced by the mixture of
surfactant and cosurfactant.
Earlier works have generally focused on scattering techniques
along with electron microscopy and conductivity measure-
ments.
5,8–10
In this work the focus is on nuclear magnetic reso-
nance (NMR). Here two diffusion NMR techniques are used to
examine the structure, viz. the pulsed-field-gradient spin-echo
(PGSE) and the modulated gradient spin-echo (MGSE)
techniques.
12
Diffusion NMR techniques can detect different features of
the diffusion process.
12,13
Since diffusion is affected by the
sample morphology, structural information can be obtained.
With the available experimental equipment to date, such tech-
niques can probe structures typically down to the micrometer
length scale.
In the PGSE approach a pair of short magnetic field gradient
pulses is used to label the start and the end positions of molecular
diffusive paths during the interval between the two pulses to yield
the molecular mean square displacement. In fact, this is only true
if molecular displacements during the application of the pulses
are shorter then the characteristic size of the restrictions present
in the sample.
14
This condition is generally known as the short-
gradient-pulse (SGP) approximation.
K€
arger and Heink
15
realized that the PGSE experiment can be
used to probe the Fourier representation of the real space
average displacement propagator. Callaghan et al.
16–18
used the
q-space as a structure analysis concept in PGSE experiments,
when the SGP approximation is fulfilled. The parameter qis the
wave vector along the applied magnetic field gradient and is
determined by the strength and duration of the gradient pulses.
Since the average displacement propagator is related to the
position dependent spin density, which reflects geometrical
features of materials, q-space imaging
16
is similar to scattering
techniques.
12
Diffraction-like effects in the echo attenuations
result when q-space imaging is applied to locally ordered
structures.
18
a
Division of Physical Chemistry, Center of Chemistry and Chemical
Engineering, Lund University, Lund, Sweden
b
Colloidal Resource AB, Kemicentrum, Box 124, 221 00 Lund, Sweden.
c
Stranski-Laboratorium f€
ur Physikalische Chemie und Theoretische
Chemie, Institut f€
ur Chemie, Technische Universit€
at Berlin, Berlin,
Germany
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Due to instrumental limitations, it is often difficult to fulfil the
SGP condition.
19,20
The effects of the finite duration of the
gradient pulses have been experimentally investigated
21,22
and
turned into an advantage to probe the intracellular fraction
22
and
diffusion coefficient of water in cell suspensions.
23
The variable
pulse duration can be used to determine the minimum length-
scale at which the diffusion appears Gaussian. For this purpose,
the concept of the homogeneous length scale, l
hom
, has been
introduced in PGSE experiments and applied to determine the
characteristic domain size of a micro-heterogeneous lamellar
liquid crystal system.
24
For distances shorter than l
hom
the
sample appears inhomogeneous, which could be caused by either
anisotropic domains
24
or restricting confinements.
25
For
distances longer than l
hom
the structural features are blurred out
and the structural information is lost. The PGSE protocol
24
for
determining the l
hom
was used in this study.
The characterization of diffusion by the PGSE experiment can
be complemented and extended by the MGSE approach.
26–29
By
generating a periodically oscillating effective magnetic field
gradient, the MGSE experiment provides information about
diffusion in the frequency domain. The result of the experiment is
the signal attenuation, which is proportional to the spectrum of
the velocity auto-correlation function (VCF) of the spin-bearing
particles. The effect of spin de-phasing, due to diffusion, is accu-
mulated over many gradient modulation cycles, which results in
stronger diffusive attenuation on a shorter time scale compared to
the conventional PGSE method. The VCF spectrum, also known
as the diffusion spectrum, D(u), probed by the MGSE approach,
is related to the mean square displacement which contains infor-
mation about the sample morphology.
30
The zero frequency
component of the diffusion spectrum corresponds to long range
diffusion, while higher frequency components approach the bulk
diffusion value. The MGSE approach is able to separate the
effects of restricted diffusion from molecular migration across the
interconnected porous structure.
31–33
This fact allows one to
determine tortuosity as well as the size of restrictions experienced
by the diffusing molecules. Different gradient modulation
schemes have been employed in MGSE experiments to study
water diffusion in porous media
32
and in emulsions.
34
Here we apply a MGSE method which uses sinusoidal gradient
pulses separated by 180radio frequency (RF) pulses.
33
Such an
experiment can be used to determine polydispersity in highly
concentrated emulsions on length-scales below mm. Further-
more, it has been shown that MGSE can yield complementary
morphological information, as compared to the results of the
PGSE experiments applied to the same kind of system.
21
The above described methods were applied to a system
composed of densely packed vesicles. The vesicles are formed in
the sodium oleate/octanol/water system, with concentrations
of z0.2 M sodium oleate and z0.6 M octanol.
9
A sample
containing tetramethylammonium as a counterion was also
examined in order to obtain results from the (proton containing)
counterions. The vesicles are densely packed with a volume
fraction of about 0.5.
The aim of this study was two-fold. Firstly, we wish to
demonstrate how the novel diffusion NMR methods described
above can be used to obtain morphological information for
complex soft matter on the micrometer length scale. Secondly, we
seek to further characterize the structure of the vesicle gels, in
particular on the micrometer length-scale; a length scale which is
difficult to investigate by alternative methods. By applying these
novel NMR methods we are able to deduce larger scale structural
information about the investigated systems.
2 Theory
Diffusion is characterized through the attenuation of signal
intensities measured in NMR spin-echo experiments. If there is
no net flow and the relaxation is factored out, the normalized
echo intensity Ecan be expressed by the Stejskal–Tanner equa-
tion.
35
The expression for the echo intensity can be extended to
include contributions from spins with different diffusion
properties k
E¼X
k
Ekebk;(1)
where P
k
E
k
¼1.
The attenuation b
k
is proportional to the measured diffusion
coefficient D
k
b
k
¼bD
k
, (2)
where the diffusion weighting factor bdepends on the effective
magnetic field gradient waveform. When diffusion is measured in
a porous medium, the D
k
values do not necessarily correspond to
the bulk diffusion value. If the structure hindrance or confine-
ments affect the spin trajectory during the measurement time, the
D
k
values will be smaller then the bulk value.
36
In case of
microscopically heterogeneous samples, the echo attenuation
might be multi-exponential (see eqn (1)) even when spins have
a uniform bulk diffusion coefficient. This is due to an occurrence
of different sub-ensembles of spins experiencing different
morphological features. One example is when morphologically
different parts of a sample are separated by distances larger then
the spins can diffuse during the experiment. However, a situation
characteristic for both the PGSE and the MGSE approaches
described here, is the fact that the echo attenuation is given by
contributions from different sub-ensembles of spins with
different diffusion properties even when the spins can migrate
over all the characteristic restrictions during the experimental
time. The weighting of the contributions depends on the exper-
imental conditions. Note that due to the low gradient strengths,
the MGSE echo decays are mono-exponential, yet given by
a weighted average over several sub-ensemble contributions
(see section 2.2).
In the limits of very long and very short diffusion times,
diffusion can be considered a Gaussian process,
37
while at
intermediate times, correlations in the spin trajectory due to
restrictions are apparent and diffusion is a non-Gaussian
process. At longer diffusion times D
k
yields the so-called long
range diffusion coefficient D
N
, while at shorter times it
approaches the free diffusion value in bulk D
0
. The ratio D
0
/D
N
is known as the tortuosity, which is related to the connectivity in
the porous sample.
2.1 Pulsed-field-gradient experiments
In the PGSE experiment used in this study, long diffusion times
were used to allow the spins migrating over all characteristic
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length scales present in the sample.
24
The magnetic field gradient
strength gand the pulse duration dwere varied. These two
parameters determine the sensitivity to diffusion and define the
wave number qas
q¼gdg
2p;(3)
where gis the gyromagnetic ratio. The parameter qsets the
inverse length scale at which the spin displacements are detected.
At low q,i.e. low diffusion sensitivity, only large displacements
can be detected, while at large qthe increased sensitivity allows
the detection of smaller displacements.
For the PGSE experiment the parameter bis calculated
according to
b¼(2pq)
2
t
d
, (4)
where t
d
is the effective diffusion time, i.e. the time between the
start of the two gradient pulses Dreduced by a correction term,
which depends on the shape of the gradient pulses.
38,39
By combining the eqn (3) and (4), it is evident that in the PGSE
experiment the echo attenuation in eqn (2) is proportional to the
product D
k
t
d
, which is related to the mean square displacement
(MSD) along the applied gradient
hZ
k
2
i¼2D
k
t
d
. (5)
When diffusion is unrestricted or at diffusion times so short that
the probability for spins to meet the confinement is negligible, the
D
k
values do not depend on time and the echo attenuation is
proportional to t
d
. At intermediate times, when the confinements
significantly affect the diffusion, a decrease of D
k
values is
observed at increasing t
d
. At long diffusion times, when all the
spins are affected by the confinement, the echo attenuation
becomes independent of t
d
, if the pores are closed. In such a case,
a time independent MSD value is reached, which depends on the
confinement shape and size as well as on the gradient pulse
duration.
40
The maximum MSD is observed in the SGP limit.
Since the spins are labelled for a position given by the center-of-
mass average of their path during the application of the gradient
pulse, increasing dresults in position averaging and consequently
in a decrease of the MSD and of the echo attenuation.
21,23
When the PGSE echo decay is probed for a semi-permeable
structure and long diffusion times are used, the information
extracted from the echo decay depends on the q-value range
(see Fig. 3 and Fig. 4). In the range of low qvalues, where only
large displacements contribute to the echo decay, a Gaussian
character is retrieved. In this regime, the influence of the struc-
tural boundaries has a negligible effect on the position labelled
during the application of the pulse and consequently the echo
decay at a given qdoes not depend on d. Since the structural
features cannot be detected at low qvalues, the sample appears
homogeneous. At increasing qvalues smaller displacements
contribute to the echo decay, and the influence of the boundaries
on the labelled spin position cause the echo decay to depend on
d.
21
At large qvalues, corresponding to short length scales, the
sample appears inhomogeneous. The q-value at which the echo
decay goes from being independent to dependent of ddefines the
inverse of l
hom
.
When performing a l
hom
analysis, it is important to make sure
that t
d
is sufficiently long so that the spins have passed through
enough of the sample during the diffusion time to probe all of its
representative parts. To verify this, one needs to examine the
initial slope, i.e. the low b(or q) values, of the echo attenuations.
This part of the data contains information about D
N
, which
should be independent of t
d
for long times. In one dimension, the
corresponding square-root of the long range MSD is given by
hZirms¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2DNtd:
p(6)
For restricted diffusion, a previous study shows that l
hom
is
directly proportional to the length of the periodicity of a micro-
heterogeneous medium. Based on the simulations by
Aslund
et al.
25
and assuming spherical restrictions, the characteristic
radius of restrictions in given as l
hom
/6.
We stress that the l
hom
analysis is a model free way of esti-
mating the range for the characteristic sizes in the system. The
value obtained should be considered as a length scale at which
there is a cross-over from locally anisotropic to globally isotropic
diffusion.
24,25
2.2 Modulated gradient spin-echo
As in the homogeneous length scale experiment described above,
the MGSE experiment also presumes that all the characteristic
diffusive modes are observed during the relevant experimental
time t
e
. The MGSE approach is complementary to the PGSE
experiment in several ways. While the PGSE protocol for
determining the l
hom
monitors diffusion on different length
scales, the MGSE approach detects diffusion modes at different
frequencies. While the PGSE experiment can accurately deter-
mine the long-range diffusion, the MGSE approach probes
diffusion on much shorter time scales and is therefore more
sensitive to structural features on smaller length scales. In
summary, the MGSE operates in the range of small qvalues.
Here, the signal attenuation can be approximated by the second
order term in the cumulant expansion, known as the Gaussian
approximation.
41
The signal attenuation for the k-th species is
given by
bk¼1
pðN
0jFðuÞj2DkðuÞdu:(7)
where F(u) is the spectrum of the phase factor
FðtÞ¼gðt
0
Gðt0Þdt0, which is determined by the effective gradient
waveform G(t), and D
k
(u) is the spectrum of the VCF or the
diffusion spectrum in the direction along the applied gradient.
26–30
While the PGSE method uses a single pair of spin labelling and
de-labelling gradient pulses, the spin-labelling and de-labelling is
repeated several times within the MGSE experiment. Thus,
adequate signal attenuation can be achieved by using much lower
gradient field strength compared to the PGSE approach. This is
an essential feature of the MGSE experiment. It means that the
spin phase grating, which is inversely proportional to the
amplitude of the phase factor, can always be made larger than
the size of the confinements under investigation. Thus, the signal
intensity of the MGSE experiment can be analyzed within the
Gaussian approximation of the cumulant expansion.
42
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The method relies on the rate of the gradient modulation rather
than on its magnitude.
If the spectrum of the phase factor F(u) has a narrow
frequency peak at a non-zero frequency u
m
, the component of
the diffusion spectrum at u
m
can be measured. This is achieved
by an appropriate choice of the gradient modulation waveform.
The pulse sequence used here (presented in Fig. 1) consists of the
Carr–Purcell–Meiboom–Gill (CPMG) RF train with 2Nrefo-
cusing 180pulses and interleaved gradient pulses with sinu-
soidal shape. The time of echo acquisition is given by t
e
¼NT
m
,
where Nis the number of modulation periods and T
m
is the
modulation period. The refocusing pulses are separated by T
m
/2.
The shape of the gradient pulses is defined by gsin(u
m
t), where
u
m
¼2p/T
m
is the modulation frequency. Note that the first and
the last pulse in the sequence oscillate twice as fast as the rest of
the pulses. The corresponding effective gradient G(t) has an
apodized cosine shape and the phase factor oscillates periodically
around zero with the modulation period T
m
. The spectrum of the
phase factor F(u) has a peak centered around u
m
with a width
inversely proportional to t
e
. When several modulation periods
are applied, the echo attenuation is proportional to the spectral
component of the diffusion spectrum at the modulation
frequency. It can be expressed in a form similar to eqn (2), as
b
k
zbD
k
(u
m
), (8)
where bis the attenuation factor. For the pulse train in Fig. 1, bis
given by
b¼gg
8p2
t3
e
8N1
N3:(9)
In order to factor out relaxation effects, while probing the
diffusion spectrum, the number of modulation periods Nis
adjusted along with the period T
m
providing a constant
t
e
¼NT
m
. The lowest value of u
m
at which the diffusion spec-
trum can be probed is limited by the longest t
e
value that the T
2
relaxation rate allows, while the upper limit is set by hardware
restrictions.
The model for the restricted diffusion spectrum D(u) can be
analytically derived in cases of simple geometries (planar, cylin-
drical and spherical).
30
Complex systems often consist of inter-
connected restricted environments or permeable pores, which
define the long-range diffusivity D
N
. This is reflected in the
diffusion spectrum as a finite component at zero frequency.
Assuming that the long-range diffusion mode is probed during t
e
,
the expression for D(u) can be extended by an additional
tortuosity term as
DðuÞ¼aD0þð1aÞD0X
N
k¼1
akBku2
a2
kD2
0þu2;(10)
where a¼D
N
/D
0
is the inverse tortuosity. The coefficients a
k
and B
k
depend on the geometry under consideration.
30,33
As
a rough model of our system we assume restrictions of spherical
geometry. For a pore radius R, the coefficients a
k
and B
k
are
given by
ak¼zk
R2
(11a)
Bk¼2ðR=zkÞ2
z2
k2;(11b)
where z
k
are kernels of
mJ
1/2
(z)–2J
3/2
(z)¼0 (12)
and J
n
denotes n-th order Bessel function of the first kind.
For the case of polydispersity in restriction sizes, the echo
decay would generally be multi-exponential (see eqn (1)).
However, if the molecules can migrate between regions with
different morphology, so that they experience all characteristic
restrictions in the sample during the MGSE experiment, the echo
decay is mono-exponential and determined by the average
diffusion spectrum
D(u
m
,a), which is given by the volume
weighted sum over sub-ensembles of spins originating in regions
of the sample with different restrictions
33
Dðum;aÞ¼X
R
PðRÞDðR;um;aÞ:(13)
2.3 Diffusion in vesicles
In vesicle systems, three regions with different diffusive proper-
ties can be identified. The first region is the interior of the vesicle.
The second region is the membrane or shell of the vesicle. The
effect the membrane has on the diffusion depends on both the
solubility of the molecules in the membrane and on the diffusion
of the molecules inside the membrane. The third region is the
continuous surrounding media. It should be noted that the dense
packing of the vesicles imposes restrictions on the diffusion also
in this region.
In these types of systems with different regions the measured
diffusion coefficient depends on the time-scale of the experiment
and the time-scale of the molecular exchange between the
different regions. If the exchange time is long compared to the
experimental time-scale, the measured signal will be a superpo-
sition of the signal from the different regions and eqn (1) can be
used. If, on the other hand, the exchange time is shorter than the
measuring time-scale the signal attenuation is given by the
Fig. 1 Schematic timing of the RF pulse sequence with sinusoidal gradient lobes. An even number of the refocusing pulses is used, separated by T
m
/2.
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time-averaged signal of diffusivity in the different regions. Such
is the case in the MGSE experiment (see eqn (13)).
The exchange time sfor migration of water molecules between
the internal vesicular space and the surrounding can be estimated
with appropriate assumptions. For a spherical vesicle, the rela-
tion between the membrane permeability P, the vesicle radius
rand the exchange rate k¼1/sis
43
P¼kr
3
0s¼r
3P:(14)
For the vesicles used in this work a radius of z20 nm has been
deduced using neutron scattering data.
5
The permeability of
bilayers is generally within 2–100 mms
1
for water.
1
This would
suggest an exchange time of 0.07–3 ms for the present system.
In the diffusion-NMR experiments the time scale of the
measurement is typically between 1 ms and a few 1 s, depending
on sample, technique and instrument. Thus, if only small vesicles
are present any structural information related to the vesicles
should be averaged out and not be measurable with the NMR
techniques.
The long-range diffusion coefficient D
N
can be estimated using
the cell model approach for a spherical region 1 being sur-
rounded by a region 2:
44
DN¼D2
1
11C1
C2F
1qF
1þqF
2
(15)
where
q¼D2C2D1C1
D2C2þD1C1
2
:(16)
D
i
is the diffusion coefficient for the molecules in region i,C
i
is
the concentration in region iand Fis the volume fraction of the
enclosed region 1. Note that this model is only valid if the
diffusion of vesicles during the experiment can be considered
negligible.
The diffusion properties in the vesicle gels can be obtained by
using the model in two steps. First, one calculates D
N
for the
vesicle, with region 1 being the inner core and region 2 being
the surrounding bilayer. In the next step one calculates D
N
for
the entire system by using the obtained D
N
from the first step and
a calculated mean concentration for the vesicle (including the
interior and the enclosing bilayer) as region 1 and the
surrounding medium (the continuous water region) as region 2.
For reasonable values for the different parameters using water
45
(including an obstruction factor for the diffusion coefficient of
the surrounding water) a value of D
N
of z110
9
m
2
s
1
is
obtained, not much different than the bulk diffusion coefficient
of water (2.3 10
9
m
2
s
1
).
3 Experimental
3.1 Sample
The two samples used in this study are identical except for the
type of surfactant counterion. They are made from a stock
solution of 200 mM cation-oleate in a mixture of H
2
O and D
2
O
(20 : 80), in order to adjust the concentration of H
2
O so that in
the NMR experiments one can follow the diffusion of water,
surfactant, and counterion independently. The cation is either
tetrametylammonium (TMA
+
) or sodium (Na
+
). The stock
solution has then been mixed with 1-octanol to a final concen-
tration of 7.5 wt-%.
3.2 NMR experiments
The experiments were performed at 25 C on a Bruker AVII-200
spectrometer at 200.13 MHz proton resonance frequency.
Gradients where generated by a Bruker DIFF-25 probe
controlled by a Bruker GREAT 1/40 gradient amplifier with
a maximum gradient strength of 9.6 T/m in the z-direction.
For the PGSE-experiments a stimulated echo sequence was
used.
46
During the experiments both dand t
d
were varied. Four
different values for t
d
were used (logarithmically spaced between
30.1 and 200 ms), while dwas logarithmically spaced between
2 and 20 ms in 4 steps. The set of q-values was kept constant by
adjusting the gradient strength as dwas changed. For the
smallest value of d, the gradient strength was logarithmically
spaced between 1 and 100% of the maximum value possible.
The MGSE experiment was performed with constant
t
e
¼100 ms. Diffusion was probed at 8 different modulation
frequencies. The numbers of modulation periods Nwere between
10 and 45 incremented by 5. This corresponds to 8 modulation
periods T
m
between 2.2 ms and 10 ms. The gradient amplitude
was adjusted so that the same 16 linearly stepped bvalues were
applied at each modulation period. The maximum gradient
amplitude used at the shortest value of T
m
was 3.51 T m
1
. The
standard 8 steps CPMG phase cycling was applied.
3.3 SANS experiments
The SANS measurements were performed at the Laboratoire
L
eon Brillouin, Saclay (France), on the instrument PAXE. A
wavelength of 0.5 nm (FWHM 10%) was chosen, and a fixed
sample to detector distance of 5 m was used, with the detector
off-center. All samples were measured in 1 mm thick rectangular
quartz cells. The data were recorded on a 64 64 cm
2
two-
dimensional detector. The data reduction was done by the
program BerSANS.
47
Data were corrected for detector back-
ground, the scattering of the empty cell, radially averaged and
converted into absolute units using the direct beam flux.
Sample volumes of about 1 mL were prepared by weighing in
first stock solutions of Na- and/or TMA-oleate in D
2
O and then
an appropriate amount of 1-octanol. Homogenization was done
by vigorous shaking with a Vortex for about 15 s. Afterwards the
mixed components were immediately transferred into quartz
cuvettes (Hellma), while still being relatively low viscous.
Samples were aged in cuvettes about one and a half day before
being measured.
4 Results and discussion
This section is organized as follows. We start by giving some
introductory remarks, followed by the short account on the
SANS data. We then proceed to describe and discuss the results
from the PGSE and MGSE experiments. A short summarizing
discussion is given in the end of this section.
As an introductory remark, we note that in both the PGSE and
the MGSE experiments the surfactant/cosurfactant peaks show
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such a small attenuation and low signal-to-noise ratio, caused by
both low concentration and relaxation effects, that no quanti-
tative conclusions can be made. However, a slight diffusive
attenuation of these peaks can be observed (not shown here).
This indicates a very slow diffusion of the surfactant and/or co-
surfactant molecules. These peaks will not be discussed further
and throughout the text concerning NMR below focus will be on
the water and TMA peaks.
4.1 SANS measurements
To investigate whether the type of counterion influences the
morphology of the vesicle gels SANS measurements were carried
out. Two otherwise identical vesicle gel systems with Na
+
or
TMA
+
as counterions were examined. Scattering curves of these
systems are shown in Fig. 2. The total composition of the
investigated samples correspond with the ones used in the NMR
studies. In the SANS experiments the surfactant was varied
systematically from pure Na oleate over the corresponding
mixtures to pure TMA oleate in D
2
O. The scattering curves of
the vesicle gels are similar for both the pure and the mixed
counterion samples. Accordingly, the structural ordering, that is
manifested in the q-range from 0.2 to 0.5 nm
1
, is very similar. In
addition, the large q-behaviour is almost identical for all the
samples, which means that their local bilayer structure is similar,
with a bilayer thickness of around 2.2 nm. In summary, the
vesicle gels possess a similar structural organisation for the
different counterions. Thus the comparison of the NMR results
for these samples is reliable.
4.2 l
hom
Measurements
From the D
N
values presented in Table 1 it can be concluded that
the long-range diffusion regime was reached for water. On the
other hand, one should take care when interpreting the results for
TMA, which can not be confirmed to reach the long-range limit.
It is not surprising that the different compounds reach the long-
range regime at different diffusion times. In fact, they are
expected to reach the long-time regime at a the same Z
rms
. One
reason is the difference in the bulk diffusion coefficient (see the
MGSE results) and another is the difference in solubility in the
bilayer (the ions can be expected to have a very low solubility in
the non-polar bilayer while the water molecules have a slight
solubility).
21,33
The values of D
N
presented in Table 1 are lower
by roughly a factor of 2 than the estimation given in section 2.3.
We will return to this fact below.
The shape of the decays are presented in Fig. 3 and 4, clearly
showing more than one exponential component, indicating the
presence of larger confinements hindering the diffusion. In these
figures the echo attenuations for two of the different t
d
and for all
four different dvalues are presented. The data for the TMA-
oleate sample are presented in Fig. 3, while the data for the
Na-oleate sample can be found in Fig. 4.
The l
hom
is obtained as an inverse of the qvalue at which the
relative difference between the signal intensities for the longest
and the shortest dat a constant t
d
reach a certain threshold
value.
24
Because of the concentration and attenuation differ-
ences, the different species were assigned different threshold
values to define a deviation (10% for the water peak and 2% for
the TMA-peak). The l
hom
is only well defined for long-range
diffusion and thus the data at the longest t
d
is used to determine
it. The threshold is reached at a qvalue of about 4 10
4
m
1
,
which is similar for both species (water and ion) and both
samples, as indicated by the vertical lines in Fig. 3 and 4, along
with the values given in Table 2. This indicates a l
hom
of about
25 mm, and thus a restriction radius of about 4 mm
(see section 2.1).
The water peak for the Na-oleate system appears to have
a more distinct biexponential decay. This would indicate a longer
exchange time between the compartments, as discussed in the
theory. The increase in exchange time could be caused by, for
example, increase in restriction size and/or decrease in
permeability (see eqn (14)).
4.3 Diffusion spectrum measurements
The intensities of the water and TMA peaks were analyzed in
terms of Stejskal-Tanner plots, which reveal mono-exponential
decays (not shown here). Exponential fits on the graphs of echo
intensities E vs. b at different T
m
give the diffusion spectra results,
shown in Fig. 5 and 6.
The mono-exponential decays observed by the MGSE exper-
iments indicate that different morphological regions are con-
nected, so that the molecules can migrate between them during
the experimental time t
e
(see the Theory section). This is
consistent with the large long-range diffusion values obtained
from the PGSE experiments and confirmed by the MGSE
experiments (see Table 1). Therefore the analysis in terms of
eqn (13) can be used to describe the measured D(u) data.
The single-size model cannot be fitted to the diffusion spectra
shown in Fig. 5 and 6. Instead eqn (13) is fitted to the data by
tentatively assuming a log-normal volume-weighted size distri-
bution of restrictions with spherical geometry. Such a choice of
the size distribution is justified as it is commonly used to describe
Fig. 2 SANS curves of the vesicle gel samples with gradual exchange of
TMA-oleate for Na-oleate at 25 C, taken at about one and a half days
after preparation. The different TMA/Na-ratios are shown with different
symbols: 0/100 (circles), 20.3/79. 7 (squares), 40.6/59.4 (triangles up),
59.7/40.3 (diamonds), 79.5/20.5 (triangles down) and 100/0 (crosses). For
clarity the intensities starting from the second curve in the plots are
multiplied by a consecutive factor of 2.
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size polydispersity.
48–50
The probability density of a log-normal
distribution is given by
PðRÞ¼ 1
ffiffiffiffiffiffi
2p
psR exp"ðln Rln R0Þ2
2s2#;(17)
where R
0
is the median and sis the standard deviation of lnR.
The mean value is provided by hRi¼R
0
e
s
2
/2
.
The bulk diffusion values D
0
used in the model (10) are
assumed to be 2.0 10
9
m
2
s
1
for water (or rather a mixture of
H
2
O/D
2
O of 20/80) and 1 10
9
m
2
s
1
for TMA.
33
The size
distribution is represented by 100 linearly spaced bins between
1 nm and 50 mm. In the case of TMA-oleate both water and TMA
data are well described by a common size distribution of spher-
ical restrictions (Fig. 5). It should be noted that a fit of reasonable
quality is obtained only if the two inverse tortuosities aare
allowed to be different for the water and TMA data. This is likely
due to different membrane permeabilities for water and salt. A
global fit yields volume-weighted distribution parameters
R
0
¼0.4 mm and s¼1.9 and inverse tortuosities of a¼20% for
water and 7.3% for TMA.
A large value of smeans that the size distribution has a long
tail. Even though the general restriction size is in the nm-range
there are a few but much larger restrictions (in the range of
10 mm) present in the sample. Both the PGSE and MGSE
diffusion experiments are sensitive to these larger restrictions,
because of volume-weighting effects.
In the case of the TMA-oleate system the size distribution
parameters correspond to a mean restriction size of approxi-
mately hRi¼2.4 mm. The inverse tortuosity values give the long-
range diffusion coefficients (see eqn (10)) which are in good
agreement with the values observed by the PGSE method. These
can be estimated from the attenuation of the echo intensities in
the limit of low bvalues (see Table 1). The small differences of
D
N
between the PGSE and MGSE experiments are most likely
due to the fact that the long-time limit has not been reached,
particularly in the case of TMA, as discussed above.
In the case of the Na-oleate system, the water data (Fig. 6) was
fitted by the two size distribution parameters, while awas fixed to
the value of 19.6%, which was obtained from the PGSE results
(see Table 1). The resulting distribution parameters are
R
0
¼0.2 mm and s¼2 corresponding to the mean size of
approximately 1.5 mm. The sizes obtained from the PGSE and
the MGSE methods respectively are summarized in the Table 2.
In summary, both diffusion methods reveal confinements,
which are polydisperse in size and on the mm length scale. The
results from the MGSE experiments suggest that most of the
confinements are below mm in size. However, a small fraction of
confinements in the range of 10 mm-scale suffices to account for
the restricted diffusion effects observed by both methods. We
stress that these results do not exclude the occurrence of
Table 1 The obtained D
N
and Z
rms
using the initial slope of the echo
attenuations and eqn (1) and 6 for the PGSE experiments for different
values of t
d
. For each t
d
value, an average is calculated for the different
dvalues used. For comparison the values obtained from the MGSE
experiments by extrapolation to zero frequency are given
t
d
(ms) 30.1 56.6 106 200 MGSE
H
2
Oin
TMA-oleate
D
N
(10
10
m
2
s
1
) 4.7 4.7 4.5 4.2 4.6
Z
rms
(mm) 5.3 7.3 9.8 13
TMA
+
in
TMA-oleate
D
N
(10
10
m
2
s
1
) 1.2 1.1 0.96 0.79 0.73
Z
rms
(mm) 2.7 3.6 4.5 5.6
H
2
Oin
Na-oleate
D
N
(10
10
m
2
s
1
) 4.0 4.4 4.5 4.5 4.5
Z
rms
(mm) 4.9 7.0 9.8 13
Fig. 3 The normalized echo attenuations for the water peak (A) and the
TMA peak (B) for the TMA-oleate system. For clarity only two of the
used t
d
values are shown. The different line types (which are included to
guide the eye and for identification) indicate the value of t
d
used
(56.7 ms ¼solid and 200 ms ¼dashed). The markers indicate different
dvalues; 2.0 ms (circles), 4.3 ms (squares), 9.8 ms (triangles) and 20 ms
(diamonds). The vertical dashed line indicates the inverse of l
hom
for the
longest t
d
. Note the different scales of the ordinate axis for the two graphs.
Fig. 4 The echo attenuations for the water peak in the Na-oleate vesicle
gel. For clearness only two of the used t
d
values are shown. The different
line types indicate different values of t
d
used (56.7 ms ¼solid and
200 ms ¼dashed). The markers indicate different dvalues; 2.0 ms
(circles), 4.3 ms (squares), 9.8 ms (triangles) and 20 ms (diamonds). The
vertical dashed line indicates the inverse of l
hom
for the longest t
d
.
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confinements in the nm-range but they do indicate the presence
of polydisperse confinements with sizes in the mm-range.
As noted above, the long-time diffusion values obtained for
water (see Table 1) are lower by a factor of 2 then the ones
predicted in the theory section for a system of densely packed
vesicles. However, it should be noted that the vesicles in this
systems are densely packed, i.e. the hydrated bilayers are in
contact and along these contact regions of the vesicles basically
no free water is present. The additional barriers on the mm
length-scale, as indicated by the l
hom
and the D(u) measure-
ments, might be responsible for the difference between the
measured and the predicted long-range diffusion values.
Combining results presented here with prior results,
5,8–10
suggests a complex structure where both small unilamellar vesi-
cles and large confinements are present. An inhomogeneous
sample with spatial variation in the volume fraction of vesicles
seems unreasonable on account of the electrostatic interactions
that are present between the vesicles. The small vesicles being
confined in either bigger vesicles or a bilayer-network is a more
reasonable suggestion.
5 Conclusions
The presented NMR diffusion results indicate the presence of
confinements on the mm-range in a vesicle gel, composed of small
unilamellar vesicles as building blocks. This length-scale is
considerably larger than the small unilamellar vesicles which has
been shown to be present before.
5
The new results do not
contradict former experimental findings, since the NMR
measuring techniques are sensitive to different length scales than
those previously probed. This is evidenced by the different
q-ranges used in the scattering and NMR techniques. Both the
scattering and the electron microscopy techniques monitor small
structures, down to the nm-range and would not be able to
identify the structures on the mm-scale to which the diffusion
NMR techniques are sensitive.
A combination of the two NMR techniques presented here
(PGSE and MGSE) provides a robust evidence of a complex
morphology of the vesicle gels. The results of the two techniques
are complementary, consistent, and therefore give strong support
in favour of the validity of the structural results obtained. The
PGSE experiment offers a fast way of identifying the length scale
on which structural features are present in the samples with
complex morphology. The MGSE experiment provides addi-
tional information on size polydispersity of the confinements.
The ‘‘super-structure’’ in the mm-range of highly ordered
vesicle gel systems, is a novel and important finding with respect
to a better understanding of these systems. These results are also
interesting for a variety of applications of vesicle gels. Such
a ‘‘super-structure’’ might be important for a comprehensive
understanding not only of their structure but also of the release
properties in the case of encapsulated active agents.
6 Acknowledgements
We would like to thank Dr L. Noirez and Dr S. Prevost for help
with the SANS experiments.
This work is financially supported by the Swedish Foundation
for Strategic Research (SSF) and the Swedish Research Council
Table 2 Summary of the sizes obtained from the PGSE (l
hom
, the
characteristic radius of restriction l
hom
/6, see text for details) and the
MGSE (R
0
,s,hRi) experiments for the systems with TMA-oleate and Na-
oleate. The meaning of the symbols is described in the text
l
hom
(mm) l
hom
/6 (mm) R
0
(mm) shRi(mm)
H
2
Oin
TMA-oleate
25 4.2 0.4 1.9 2.4
TMA
+
in
TMA-oleate
26 4.3 0.4 1.9 2.4
H
2
Oin
Na-oleate
25 4.2 0.2 2 1.5
Fig. 5 Diffusion spectrum for water (A) and TMA (B) in the TMA-
oleate sample. Theoretical model (solid lines, according to eqn (14)) is
fitted to the data (circles). Error bars indicate confidence intervals of 95%.
Fig. 6 Diffusion spectrum water in the Na-oleate sample. Theoretical
model (solid line) is fitted to the data (circles). Error bars indicate
confidence intervals of 95%.
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(VR) through the Linnaeus Center of Excellence on Organizing
Molecular Matter (OMM) along with the Crafoord Foundation.
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