Document [original]

J. Chem. Phys. 148, 104901 (2018); https://doi.org/10.1063/1.5009424 148, 104901

Dynamics of small unilamellar vesicles

Cite as: J. Chem. Phys. 148, 104901 (2018); https://doi.org/10.1063/1.5009424

Submitted: 16 October 2017 . Accepted: 23 February 2018 . Published Online: 12 March 2018

Ingo Hoffmann, Claudia Hoffmann, Bela Farago, Sylvain Prévost , and Michael Gradzielski

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THE JOURNAL OF CHEMICAL PHYSICS 148, 104901 (2018)

Dynamics of small unilamellar vesicles

Ingo Hoffmann,1,2,a) Claudia Hoffmann,1Bela Farago,2Sylvain Pr´

evost,1,2,3

and Michael Gradzielski1,b)

1Stranski-Laboratorium f¨

ur Physikalische und Theoretische Chemie, Institut f¨

ur Chemie, Technische Universit¨

Berlin, Straße des 17. Juni 124, Sekr. TC 7, D-10623 Berlin, Germany

2Institut Max von Laue-Paul Langevin (ILL), 71 Avenue des Martyrs, CS 20156, F-38042 Grenoble Cedex 9,

France

3Helmholtz-Zentrum Berlin, Hahn-Meitner-Platz 1, D-14109 Berlin, Germany

(Received 16 October 2017; accepted 23 February 2018; published online 12 March 2018)

In this paper, we investigate the dynamics of small unilamellar vesicles with the aid of neutron spin-

echo spectroscopy. The purpose of this investigation is twofold. On the one hand, we investigate the

influence of solubilised cosurfactant on the dynamics of the vesicle’s surfactant bilayer. On the other

hand, the small unilamellar vesicles used here have a size between larger vesicles, with dynamics

being well described by the Zilman-Granek model and smaller microemulsion droplets which can be

described by the Milner-Safran model. Therefore, we want to elucidate the question, which model

is more suitable for the description of the membrane dynamics of small vesicles, where the finite

curvature of the bilayer is felt by the contained amphiphilic molecules. This question is of substantial

relevance for our understanding of membranes and how their dynamics is affected by curvature, a

problem that is also of key importance in a number of biological questions. Our results indicate

the even down to vesicle radii of 20 nm the Zilman-Granek model appears to be the more suitable

one. Published by AIP Publishing. https://doi.org/10.1063/1.5009424

I. INTRODUCTION

Closed surfactant bilayers are called vesicles. They have

attracted quite some interest over the past decades, and as

opposed to manyother carrier systems, they can be used for the

delivery of both hydrophilic and hydrophobic molecules.1–9

There is also quite some interest in them, as they can be used

assimple modelsystemsfor cells,especially whentheyconsist

of lipids instead of synthetic surfactants, when such vesicles

are also called liposomes.2,4,7,10–12

As a significant part of their stability stems from the

undulation motion of their membrane,13 there is quite some

relevance to the study of their membrane dynamics and its

change when loading the membrane with another molecule.

In general, the membrane elasticity is a key property of such

bilayer systems and is quantitatively described by its bend-

ing moduli, the mean bending modulus κand the Gaussian

modulus ¯κ.14 Accordingly, there have been many studies to

determine the bending moduli of membranes often via optical

methods15,16 or light scattering,17 and these approaches have

been reviewed comprehensively,18–20 showing that in general

the bending rigidity depends strongly on the type of mem-

brane studied (and for instance a stiffening is observed upon

the introduction of cholesterol into phospholipid bilayers21).

An interesting but less easily accessible method to study mem-

brane dynamics is neutron spin-echo (NSE) spectroscopy22,23

that allows us to follow membrane fluctuations in the range

of 1-20 nm.24,25 NSE is complementary to dynamic light

a)Electronic mail: hof[email protected]

b)Electronic mail: [email protected]

scattering that probes length scales of several hundreds of

nm. This is relevant as the bending properties of membranes

and layers are known to depend on the length scale probed,26

becoming smaller with smaller probed length scale. NSE has

the advantage of probing rather directly the mean bending

modulus that can be derived from the intermediate scattering

function S(Q,t).

A particularly interesting parameter for the understanding

ofthebilayerelasticityis itsthickness,andthereare somestud-

ies investigating its influence27–31 on the membrane rigidity. A

study on bilayer vesicles assembled from amphiphilic diblock

copolymers showed that the mean bending rigidity κfor large

polymersomes scales with the hydrophobic membrane thick-

nesstothepowertwo.32 Mostotherworkhasbeendealingwith

surfactant bilayers which are loaded with rather large amounts

of other compounds, and therefore, the two monolayers are

decoupled and a decrease of the bending rigidity is observed.

However, initially an increase is observed,27 as the bilayer

behaves as a thicker membrane. Here, we want to focus on

that region and investigate the scaling of the bending rigidity

κwith the bilayer thickness when relatively small amounts of

other molecules are added. If theybehave as an isotropic mate-

rial, they should scale as κ∝d3,33 while as long as they behave

as a true bilayer κshould scale as κ∝d2.14,34 Szleifer et al.,35

on the other hand, predicted a strong reduction by more than

a factor 2 upon addition of relatively small amounts of shorter

chains to a bilayer based on their conformational entropy with

only relatively little impact on the membrane thickness.

Recently Mell et al.36 investigated whether the Zilman-

Granek model or the Milner-Safran model is more appropriate

to describe the dynamics of relatively large liposomes and

0021-9606/2018/148(10)/104901/8/$30.00 148, 104901-1 Published by AIP Publishing.

104901-2 Hoffmann et al. J. Chem. Phys. 148, 104901 (2018)

found better agreement with the Zilman-Granek model. How-

ever,whiletheZilman-Granekmodelhasbeenlargelysuccess-

ful in describing the dynamics in relatively large membrane

structures with sizes of 50 nm and more, small microemulsion

droplets with sizes on the order of 5 nm are well described by

the Milner-Safran model.37,38 It is reasonable to assume that

thereis asize below which theMilner-Safranmodelworksbet-

ter, as apparently it applies well to microemulsion droplets. In

order to elucidate this interesting point at which size vesicles

become small enough to be better treated with the Milner-

Safranmodel, we investigatedthe dynamics ofsmall unilamel-

lar vesicles, which are larger than microemulsion droplets but

smaller than the vesicles usually studied before.

For this purpose, we chose a well-studied system based

on sodium oleate or isostearate, where by addition of medium

chain alcohols such as octanol39,40 or decanol,41 spontaneous

formation of rather monodisperse unilamellar vesicles has

been observed. These systems contain an excess of alco-

hol compared with the surfactant (typically a molar ratio

of 2-3:1 of alcohol to oleate) and form at higher concen-

tration a well-ordered vesicle gel. However, upon dilution

with water, one ends up in an isotropic solution that con-

tains well-defined unilamellar vesicles. Such vesicles were

then chosen for our NSE experiments to determine the mean

bending modulus and to elaborate whether the Milner-Safran

model or the Zilman-Granek model is more suited to describe

them.

II. MATERIALS AND METHODS

A. Materials

Oleic acid (purum 98%, Fluka) and isostearic acid (Emer-

sol 874) were dissolved in D2O (99.9% Euriso-top) with

equimolar amounts of sodium hydroxide (98% puris, Fluka)

to yield 80 mM stock solutions of sodium oleate and sodium

isostearate. The surfactant stock solutions were added to

the appropriate amounts of octanol (per synthesis, Merck)

and vigorously stirred overnight to obtain solutions of small

vesicles.

B. Methods

Small angle neutron scattering42 (SANS) measurements

were performed on the instrument V443,44 at Helmholtz-

Zentrum Berlin (HZB) using sample to detector distances of

1, 4, and 15.85 m and a wavelength λ= 0.457 nm allowing

to cover a Qrange from 0.036 to 5.7 1/nm, where Qis the

magnitude of the scattering vector Q= 4π/λsin(θ/2) and θis

the scattering angle. The software package BerSANS45 was

used for data reduction.

The SANS curves were modeled as slightly polydisperse

(10%relativestandarddeviationinradius)hollowsphereswith

a single shell. The form factor for such a spherical shell reads

P(Q,Ri,∆SLDi)=*

i=0

F(Q,Ri,∆SLDi)+

, (1)

where R0is the radius of the aqueous core, R1is the outer

radius, and the thickness of the surfactant bilayer is given by

ds=R1



R0and the ∆SLDiare the differences in scatter-

ing length density (SLD) going from R>Rito R<Ri. The

amplitude Fis given by

F(Q,Ri,∆SLDi)=

4πR3

3∆SLDi3sin(QRi)−QRicos QRi

(QRi)3,

(2)

and the scattering intensity for a non-interacting, monodis-

persesystemisgivenbyI=1NP(Q,Ri,∆SLDi)+Iinc,where1N

is the particle number density and Iinc is the incoherent scatter-

ing contribution which has been subtracted. In a monodisperse

system,1Nis givenby1N=φ/V(R), withthevolumefraction φ

and the volume of the particles V(R). In a polydisperse system,

it is given by

1N=

∫∞

0f(R)V(R)dR, (3)

where f(R) is the size distribution of the vesicles. Here, we

used the log-normal size distribution

f(x,µ,σ)=

√2πσ ·xexp −(ln(x/µ))2

2σ2!, (4)

with arithmetic mean M=µexp(σ2/2) and its standard

deviation is given by SD =µp(exp(σ2)−1)exp(σ2/2).

For a system with interactions, it is necessary to intro-

duce a structure factor S(Q), and in the local monodisperse

approach,46 theintensity for a polydisperse, interacting system

now reads

I=1N∞

f(R,µ,σ)P(Q,Ri,∆SLDi)S(Q,R,φ)dR+Iinc, (5)

where the hard-sphere structure factor in the Percus-Yevick

approximation47 has been used. See the supplementary mate-

rial for details.

Neutron spin-echo22,23 (NSE) measurements were per-

formed on the instrument IN1548 at Institut Laue-Langevin

(ILL, Grenoble, France). Using wavelengths of 10 and 17 Å,

maximum Fourier times of 50 and 200 ns were reached

covering a Qrange from 0.13 to 1.3 1/nm.

NSE accesses the intermediate scattering function S(q,t).

To get a general idea about the system’s behaviour, data

can be described with an apparent diffusion coefficient

Dapp,

S(Q,t)=exp(−DappQ2t). (6)

III. THEORY

The general form of S(Q,t) for the description of

membrane dynamics in diffusing systems is given by

S(Q,t)=exp(−D(Q)Q2t)((1 −A(Q)) + A(Q)Sund(Q,t)), (7)

where D(Q) is the Q-dependent translational diffusion coeffi-

cient

D(Q)=D0/S(Q), (8)

with the translational diffusion coefficient at infinite dilution

D0, which is related to the size of the object via the Stokes-

Einstein equation

D0=

kBT

6πηRH

, (9)

104901-3 Hoffmann et al. J. Chem. Phys. 148, 104901 (2018)

with the hydrodynamic radius RH, solvent viscosity η, Boltz-

mann constant kB, and temperature T.Sund(Q,t) describes the

undulation motion of the membrane.

For the description of the membrane dynamics in

microemulsion droplets, the Milner-Safran model37,38,49–52

has proven successful. In the framework of this model mem-

brane motions are described as spherical harmonics and

Sund is given by a series of simple exponentials Sund,MS

=P∞

l=2alexp(−ΓMS,lt) with relaxation rates ΓMS,l

ΓMS,l=

ηR3

l(l+ 1) −6+4w−3¯κ

κ−3kBT

4πκ Φ(φ)

Z(l), (10)

where κisthe bendingrigidity, ¯κisthe saddlesplay modulus, φ

is defined as is accounting for the entropy of mixing and can be

approximated as Φ(φ)=1

φ(φln(φ) + (1 −φ) ln(1 −φ)),φis

the volume fraction of the dispersed microemulsion droplets,

Zisgiven by Z(l)=(2l+1)(2l2+2l−1)

l(l+1)(l+2)(l−1) , and w=c0R,where c0is the

spontaneous curvature. The sum in Eq. (10) starts at 2 as the

zero order term corresponds to size fluctuations, which are too

slow for NSE to capture and the first order term corresponds

to diffusion. The non-normalised amplitudes are given by

al=

kBT

κ(l+ 2)(l−1) l(l+ 1) −6+4w−3¯κ

κ−3kBTΦ(φ)

4πκ .

(11)

The amplitude of the dynamic contribution A(Q) is pre-

dicted to show a maximum around the form factor minimum,

and in fitting experimental data using the model usually only

a single exponential term is kept to account for membrane

dynamics,52 asboththeamplitudesquicklydecreasewithl(see

Fig. S1 of the supplementary material) and the ΓMS,lquickly

increase and therefore move out of the dynamic range. Apply-

ing the model to the membrane dynamics of larger objects

such as large unilamellar vesicles, it is usually assumed that

the relaxation at a given Qare dominated by undulations of

thecorresponding lengthscale resulting ina single exponential

accounting for the membrane dynamics with a relaxation rate

Γ=κ

4ηQ3.36,53,54 See the electronic supplementary material

[Eqs. (S3)–(S5)] for more details.

The Zilman-Granek model55,56 on the other hand pre-

dicts a stretched exponential decay of Sund(Q,t) with a stretch

exponent of 2/3,

Sund,ZG(Q,t)=exp(−(ΓZGQ3t)2/3), (12)

with ΓZG =0.025γqkBT

kBT

ηand for κ/kBT1, γ≈1. As it

was developed for large bilayer structures where translational

diffusion is negligible, no explicit prediction is made for A(Q).

The concept of both models is similar in that they both

start from a Helfrich bending Hamiltonian14 but the Milner-

Safran model only takes into account the smallest wave vector

undulation, which corresponds to the radius of the particle

[first fraction in Eq. (10)]. The Zilman-Granek model on the

other hand results from an integration over all undulation wave

vectors between the length scale of the particle and a lower

cut-off molecular length scale. Therefore, the Milner-Safran

model is suited for the description of relatively small droplets,

while the Zilman-Granek model is suited for the description

of larger membrane structures.

The most prominent difference between the 2 models is

the scaling of Γwith κ. While the Milner-Safran model pre-

dicts ΓMS ∝κ, the Zilman-Granek model predicts ΓZG ∝1/√κ,

and it was its benefit that it managed to give an explana-

tion for the previously observed scaling of the relaxation rate

with κ.17,24 The second difference is the pronounced increase

of the amplitude Aaround the form factor minimum in the

Milner-Safran model. While no explicit prediction is made

by the Zilman-Granek model, as it is the result of averaging

over a range of wave vectors, a more smooth behaviour is

expected and also predicted by the Milner-Safran model at

higher Qand when taking into account larger undulation wave

vectors.36

The third difference is the shape of the function. How-

ever, differentiating between a stretched exponential with a

stretch exponent of 2/3 and a simple exponential is not neces-

sarily straightforward especially, when combined with another

exponential as in Eq. (7).

While the scaling of Γwith κis correctly predicted, the

absolute values of κobtained from NSE measurements have

often shown to be wrong and a number of approaches exist to

correct them.25,57–63 For example, Seifert and Langer60 took

into account contributions from lateral flow in the membrane

and interbilayer friction.64 Watson and Brown62 adapted that

theory to the framework of the Zilman-Granek theory, and the

measured,effectivebending rigidityisgivenby κeff =κ+2d2k,

where kis the monolayer compressibility modulus and dis the

height of the monolayer neutral surface. Monkenbusch et al.61

explicitly evaluated the integrals which arise in the derivation

of Eq. (12) to account for the finite size of the film and the

molecules which comprise it. As here, we are mainly inter-

ested in relative changes in κ, we will limit ourselves to the

simplest correction, which consists of using an effective sol-

vent viscosity ηeff = 3η,57–59 which previously led to realistic

values of κ.

IV. RESULTS AND DISCUSSION

For our experiments, we choose well-defined, sponta-

neously forming vesicles that are not too concentrated, so

the individual vesicles are able to move and undulate freely.

This was accomplished by using a surfactant (Na oleate or

Na isostearate) concentration of 80 mM, which is well below

the gelation concentration of ∼100 mM but not too dilute as

then the vesicles stop being well-defined.1The octanol con-

centration was varied in the range of 150–300 mM, where

formationof unilamellar vesiclesshould occur, in order to vary

systematically the thickness and composition of the bilayer.

We performed SANS measurements to verify the presence of

vesicles in the samples and precisely determine their structure,

especially the thickness of the surfactant layer as a function

of surfactant to cosurfactant ratio. The SANS curves taken for

samples with 80 mM Na oleate and 150, 225, and 300 mM

1-octanol (corresponding to cosurfactant mass fractions of

2, 3, and 4 wt. % or surfactant to cosurfactant volume ratios

φs/φOctanol of 1.04, 0.68, and 0.52) and 80 mM Na isostearate

and 225 mM 1-octanol (corresponding to a cosurfactant mass

fraction of 3 wt.% or a surfactant to cosurfactant volume ratio

104901-4 Hoffmann et al. J. Chem. Phys. 148, 104901 (2018)

FIG. 1. Top: SANS curves of samples indicated in the graph with fits accord-

ing to Eq. (5); bottom: Kratky plot (IQ2vs. Q) of the same SANS curves.

Inset: Magnification of the region between 2 and 5 1/nm showing the form

factor minimum due to the membrane thickness. The slight shift in its position

shows that the membrane thickness changes slightly.

φs/φOctanol of 0.66) are given in Fig. 1and confirm the pres-

ence of small vesicles with radii between 20 and 40 nm (seen

by the form factor minimum around 0.1 1/nm), which can be

seen from the Q



2slope at intermediate Qand the resulting

plateau in the Kratky plot (see Fig. 1, bottom). The minimum

at high Q(around 3 1/nm) is slightly different between the

various samples (also best seen in the Kratky plot), which

means that the membrane thickness (dshell = 2π/Qmin) varies

with different ratios between the surfactant and octanol. In

Fig. 2, it can be seen that the layer thickness determined from

the position of the high-Qminimum decreases linearly with

the addition of the cosurfactant octanol, which has a signif-

icantly shorter chain than the surfactant itself. This means

FIG. 2. Thickness of the surfactant layer (left axis, circles) and vesicle radius

(right axis, triangles) as a function of surfactant to cosurfactant ratio. The sur-

factant layer thickness dsincreases linearly with the surfactant to cosurfactant

volume ratio. The overall size of the vesicles increases as well.

that the concentration of octanol is a suitable tool for con-

trolling the thickness of the vesicle bilayers. The values for

the measurement with sodium oleate and sodium isostearate

seem to fall on the same line for dshell, but both systems dif-

fer with respect to the radius. The smaller radius for sodium

isostearate indicates that it has a more flexible membrane that

tends to a smaller spontaneous curvature. The double peak at

low Q(see Fig. 1) has been observed previously for similar

systems1and is due to the overlap between form factor min-

imum and structure factor maximum. While the combination

of hard sphere structure factor and a hollow sphere manages

to qualitatively reproduce the double peak with a dip in the

middle at low Q, it fails to quantitatively describe the lowest

Qpart; however, this is beyond the scope of this paper and

of no relevance, as the NSE measurements do not reach such

small Qvalues. Anyway this Q-range probes already largely

the interactions between the vesicles and therefore would be

less relevant for determining the shape fluctuations of the

vesicles.

After having elucidated the structure of the vesicles, we

proceeded to perform NSE measurements in the Q-range of

0.13–1.3 1/nm to investigate their dynamics. The measured

NSE curves (see Fig. 3and Figs. S2–S4 of the supplementary

material) show non-exponential relaxation behaviour, as can

be seen from their curvature in the log-lin plot.

A preliminary analysis using Eq. (6) yields relatively con-

stantvaluesofDapp (seeFig.4).Additionallythereisaconstant

offset between the values obtained from 10 or 17 Å measure-

ments which indicates a constant non-exponentiality over the

whole Qrange. Only at the lowest Q,Dapp increases. At these

values, however S(Q,t) only decays from 1 to more than 0.9,

and this behaviour should not be mistaken to be due to the

increase in Dapp around the form factor minimum predicted by

the Milner-Safran model, especially as it is similarly observed

for the sample with sodium isostearate which forms smaller

vesicles and therefore has its form factor minimum at some-

what higher Q. A more likely source of this effect is de Gennes

narrowing65 [see Eq. (8)]. The absolute values obtained are

not in agreement with the values expected for simple transla-

tional diffusion of vesicles of the sizes found in SANS. For

FIG. 3. S(Q,t) of 80 mM sodium oleate with 3 wt.% 225 mM octanol; fits

using the Zilman-Granek model: Eq. (7) with Eqs. (8) and (12); fits are in

good agreement with the data. It can be seen that S(Q,t) is nonexponential

throughout the whole Q-range.

104901-5 Hoffmann et al. J. Chem. Phys. 148, 104901 (2018)

FIG. 4. Dapp obtained from fitting Eq. (6) to NSE data indicated in the graph.

The values are relatively constant over the whole Qrange, and there is a

constantoffsetbetween10(higher values,opensymbols) and17Å wavelength

data (lower values, filled symbols), indicating non-exponentiality.

example, for vesicles with radii between 20 and 40 nm Eq. (9)

would give diffusion coefficients between 1 and 0.5 Å2/ns,

which are substantially lower than the values obtained, and

obviously the membrane dynamics give a significant contribu-

tion to S(Q,t) at all investigated Qvalues. The fact that there is

no pronounced change with Qand that S(Q,t) has a similarly

non-exponential shape over the whole Q-range indicates that

the vesicles are already large enough for the Zilman-Granek

model to be the more appropriate description. Specifically,

the lack of an increase of Dapp toward smaller Qshows that

using the Milner-Safran model simply using a relaxation rate

Γ=κ

4ηQ3is inappropriate, here.

Therefore, the NSE data were fitted using Eq. (7) with

Eq. (12), and the contribution for translational diffusion was

described by Eq. (8) with Eq. (S1) using the radius obtained

from SANS. The fits are in good agreement with the data (see

Fig. 3), and it can also be seen that S(Q,t) is nonexponential

at all Q.

Nevertheless, the data were also fitted with the Milner-

Safran model using the same translational contribution but

replacing the stretched exponential of Eq. (12) with a simple

exponential with a relaxation rate which does not depend on

Qas predicted by Eq. (10), and significantly worse fit results

are obtained (see Fig. 5) with a χ2higher by roughly a factor

5 (see Fig. 6). Not neglecting higher order terms up to l= 5

in the Milner-Safran model only slightly improves the situa-

tion, and including even higher order terms have no significant

effect as their amplitude is extremely small. See Sec. II and

Fig. S1 of the supplementary material for details. The inset

in Fig. 6shows that the Zilman-Granek model also yields a

well-defined minimum for the relaxation rate ΓZG. In addi-

tion, the amplitude of the undulation contribution was left as a

free parameter and while it shows a slight increase with Qfor

the Zilman-Granek model, a strong increase is observed for

the Milner-Safran model, and it fails to produce the predicted

maximum around the form factor minimum (see Fig. S5 of the

supplementary material). Furthermore the values for κ[ignor-

ing the corrections in the second fraction of Eq. (10)] obtained

withthe Milner-Safran modeldo not showaclear trendin dshell

(see Fig. S6 of the supplementary material), which would not

really be expected.

FIG. 5. S(Q,t) of 80 mM sodium oleate with 3 wt.% 225 mM octanol; fits

using the Milner-Safran model: Eq. (7) with Eq. (8), Eq. (10), and a simple

exponential for Sund (full lines) and terms up to l= 5 according to Eq. (S3)

and Eq. (11) (dashed lines); fits are in significantly worse agreement with

the data than fits using the Zilman-Granek model, and the additional constant

relaxation rate fails to describe the data over the whole Qrange.

The values obtained for κwith the Zilman-Granek model,

using an effective solvent viscosity of ηeff = 3ηto account

for additional dissipation in the membrane,57–59 are shown in

Fig. 7. It can be seen that κincreases with dshell. The obtained

values for κare quite small and similar to values observed

in microemulsions.49,61,66–72 However, it should be noted that

the bilayers investigated here contain as majority component

octanol and are with 2.0-2.2 nm thickness thinner by about a

factor 2 compared with phospholipid bilayers. Accordingly,

one would expect about a factor 4 to 8 lower values for κ.

Another remarkable observation is the fact that (within the

admittedly small range of dshell)κscales strongly with the

shell thickness and both a scaling of κ∝d2

shell and κ∝d3

shell

(dash-dotted and dashed line in Fig. 7) is too weak and the

data rather support κ∝d8

shell (full line). An explanation here

would be that by adding octanol one does not only make the

membrane thinner, but the increased presence of the short

chain alcohol octanol also leads to an additional substantial

FIG. 6. χ2obtainedfromfittingtheZilman-GranekandMilner-Safran(using

only the l= 2 term and terms up to l= 5, ΓMS corresponds to the value for l= 2)

model to data from 80 mM sodium oleate with 225 mM octanol by varying

the respective relaxation rate (ΓZG or ΓMS ) by 20% around their optima and

optimising A(Q); inset: χ2for the Zilman-Granek model with linear axis.

104901-6 Hoffmann et al. J. Chem. Phys. 148, 104901 (2018)

FIG. 7. Bendingrigidity κof the bilayer asobtainedfrom NSE measurements

as a function of bilayer thickness dshell. The bilayer is quite soft, similar to

microemulsion systems, and κincreases strongly with dshell.

softening of the membrane. It might be noted here that, of

course, the changes in the relaxation are not entirely due to

the changing bending elasticity of the bilayers but also reflect

changes of the internal dissipation (internal friction) in the

membrane,asthe ratiobetween surfactantandoctanolchanges

here quite largely. However, we follow here the conventional

notion and attribute the changes in stiffness to a changing

bending rigidity, and our findings are also in principal agree-

ment with theoretical calculations that indicate a softening of

bilayers due to having a mixed bilayer of chains with largely

different chain length. It might be argued that hydrodynamics

[which are ignored in our analysis by using Eq. (8) and not

D(Q)=D0H(Q)

S(Q)with the hydrodynamic function H(Q)] influ-

ence the obtained results, as at higher concentrations, the short

timediffusioncoefficientshouldbeincreasinglyreducedasthe

high Qlimit of H(Q)<1.73–75 This would mean that the dif-

fusional component calculated based on Eqs. (8) and (9) with

the radius obtained from SANS would yield too high values

of Dwhich would lead to too low values of ΓZG (too high val-

ues of κ) with the effect being more pronounced for higher

concentrations. However, it is found experimentally that κ

decreases with concentration (dshell decreases with concentra-

tion) and thus the change of κwould be even more pronounced

if we were accounting for hydrodynamic interactions in our

analysis.

V. CONCLUSION

We have investigated the membrane dynamics of small

unilamellar vesicles using neutron spin-echo (NSE) spec-

troscopy. While the Milner-Safran model is well suited for

the description of the undulation motions of small droplet

microemulsions, the Zilman-Granek model can be applied to

the description of undulation motions in larger self assem-

bled structures, such as large vesicles or bilayer structures or

nanoemulsions. We found that even the relatively small unil-

amellar vesicles with radii between 20 and 40 nm used in

this study are much better described by the Zilman-Granek

model, while the Milner-Safran model leads to an unrealis-

tic interpretation of the elastic properties of the bilayers. The

bending rigidities found here for these small vesicles are in a

similar range as those found for microemulsion droplets.

Therefore, for such low bending rigidities, the size below

which the Milner-Safran model is better suited must be

somewhere between 5 and 20 nm.

It was also found that the bending rigidity κincreases

strongly with the thickness of the surfactant bilayer. While the

layer thickness dshell changes by only about 10%, the bending

rigidity changes by a factor 2 and consequently the relaxation

rate changes by a factor of 1/√2 as ΓZG ∝1/√κ. This very

pronounced reduction of κupon addition of octanol cannot

just be explained by the thinning of the membrane but must

be associated with a marked softening due to the presence

of this shorter chain molecules, which is in good qualitative

agreement with calculations35,76 of the bending rigidity based

on chain entropy.

SUPPLEMENTARY MATERIAL

See supplementary material for additional formulas con-

cerning the hard sphere structure factor and the Milner-Safran

model and additional figures.

ACKNOWLEDGMENTS

Financial support from the BMBF Project No. 05K13KT1

is gratefully acknowledged as well as allocation of beamtime

by ILL and HZB.

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