Eur. Phys. J. E 22, 157–161 (2007)DOI: 10.1140/epje/e2007-00021-2 THE EUROPEAN

PHYSICAL JOURNAL E

Hydrodynamic effects in bicontinuous microemulsions measuredby inelastic neutron scattering

O. Holderer1,a, H. Frielinghaus1, M. Monkenbusch1, J. Allgaier1, D. Richter1, and B. Farago2

1 Institut fur Festkorperforschung, Forschungszentrum Julich GmbH, 52425 Julich, Germany2 Institut Laue Langevin, Grenoble, BP 156, 38042 Grenoble Cedex 9, France

Received 22 November 2006Published online: 14 March 2007 – c© EDP Sciences, Societa Italiana di Fisica, Springer-Verlag 2007

Abstract. The dynamical properties of bicontinuous microemulsions have been studied with neutron spinecho spectroscopy around length scales corresponding to the correlation peak q0. Comparison of sampleswith different contrasts for neutrons shed light on the two modes dominated either by variation of theoil/water difference or surfactant concentration in the hydrodynamic regime. The results have been com-pared to theoretical predictions of the relaxation rates of bicontinuous microemulsions by Nonomura andOhta [M. Nonomura, T. Ohta, J. Chem. Phys. 110, 7516 (1999)]. The influence of modification of thesurfactant layer bending constants in the microemulsion by addition of homopolymers (polyethylenepropy-lene: PEPX and polyethyleneoxide: PEOX , X = 5 kg/mol), dissolved in the oil phase and water, has beeninvestigated.

PACS. 61.12.Ex Neutron scattering (including small-angle scattering) – 68.05.Gh Interfacial propertiesof microemulsions – 61.20.Lc Time-dependent properties; relaxation

1 Introduction

Bicontinuous microemulsions consisting of water, oil and asurfactant, are thermodynamically stable systems with asponge-like structure, where water and oil domains are in-terpenetrating each other, separated by a monolayer ofamphiphilic surfactant. A well established approach indescribing theoretically the physical properties of bicon-tinuous microemulsions [1–3] is based on the elastic cur-vature energy as described by Helfrich [4]. Alternatively,Ginzburg-Landau models, where the microemulsion is de-scribed by two fields, Φ for the local surfactant concen-tration and Ψ for the concentration difference betweenoil and water, have been applied to interpret phase di-agrams and structural parameters as obtained e.g. withsmall angle neutron scattering (SANS) [5]. The charac-teristic distance d between oil-oil or water-water domainsand the correlation length ξ, a measure for the spatialrange of local order, can be obtained with SANS by fit-ting the data around q0 with the Teubner-Strey formula [5]I(q) ∝ 1/[q4−2(q2

0−ξ−2)q2 +(q20 +ξ−2)2] with q0 = 2π/d.

An overview of the theoretical descriptions of microemul-sions provides reference [6]. Different approaches havebeen made to describe the dynamics of bicontinuous mi-croemulsions [7–11]. For the high-q regime (q > 4q0, i.e.significantly larger than the “correlation peak” close to q0

in the SANS spectrum), Zilman and Granek [12] devel-

a e-mail: o.holderer@fz-juelich.de

oped a theory treating the microemulsion as an ensem-ble of randomly oriented membrane patches of size ∼ξsurrounded by a viscous medium. The balance betweenrestoring forces by the bending modulus and friction dueto the fluid flow induced by the undulations controls therelaxation rate of thermal fluctuations. The patch sizelimits the undulation spectrum. The scattering functionresulting from relaxation of such a membrane ensemblecan be approximated by a stretched exponential function,S(q, t) ∝ exp(−(Γt)β) where the relaxation rate Γ ∝ q3.However, only a model expression for the intermediatescattering function that avoids further simplifications thatlead to the approximative expression yields proper valuesfor the bending modulus. This model uses numerical inte-gration over the full undulation mode spectrum contain-ing all lengthscales from the surfactant molecule size tothe patch size. I.e. the calculation yields directly the barebending modulus κ of the membrane [13]. At very smallq-values, i.e. much smaller than q0, dynamic light scat-tering on bicontinuous microemulsions of this kind [17]yields intermediate scattering functions with a monoex-ponential behavior and a rate proportional to q2. This re-flects the diffusion of oil-water concentration fluctuationsof microemulsion domains much larger than the character-istic distance. The theoretical description of S(q, t) in theintermediate q-range is more complicated and no simpleexpression is available in this range. Different approacheshave been made to get an expression for the intermediate

158 The European Physical Journal E

scattering function for scattering vectors smaller and cor-responding to the inverse characteristic distance q0. Theyare all based on Ginzburg–Landau models with two or-der parameters, Ψ for the local concentration of oil andwater, and Φ for the deviation of surfactant concentra-tion from its average value. Hennes and Gompper [9,10]applied this theory to bicontinuous microemulsions, Nono-mura and Ohta [11] used the same approach with differentapproximations. In our experiment, the membrane prop-erties could be modified by adding co-surfactants to a bi-continuous microemulsion. For example, the addition ofdiblock copolymers of the type PEO-PEP leads to an in-creased surfactant efficiency, i.e. less surfactant is neededto emulsify a given amount of water and oil (“boostingeffect”) [14]. The corresponding homopolymers PEP andPEO as cosurfactants have the opposite effect of reduc-ing the surfactant efficiency. Microscopically, the bendingelasticity κ of the membrane is modified by the cosurfac-tants. Homopolymers, as used in this study, decrease κ,as described by the theory of Eisenriegler [15]. This effecthas been investigated experimentally e.g. by SANS [16]and NSE [13]. The correlation length decreases with in-creasing homopolymer concentration, while the character-istic distance between domains is mainly governed by thesurfactant concentration and is almost not modified bythe addition of homopolymers. Inelastic neutron scatter-ing proved to be a suitable tool for the investigation of thedynamics of ternary microemulsions [13,17–19]. Neutronspin echo (NSE) spectroscopy is a technique with a veryhigh energy resolution, able to probe fluctuation dynam-ics in soft matter samples. At large scattering vectors q,i.e. small distances in the sample, local membrane fluctua-tions are probed with NSE, as reported e.g. in [17]. At verysmall q, beyond the q-region accessible with NSE for thistype of samples, normal diffusion of concentration fluc-tuations in a virtually homogeneous microemulsion takesplace as seen by DLS [17]. After a q-range gap of abouthalf a decade NSE spectroscopy sets in. In this paper wereport on dynamical properties of microemulsions in thisintermediate to small q-regime (q � q0), investigated withNSE. Dynamics on lengthscales of the order of the charac-teristic distance between water-water and oil-oil domainson the margin of the validity of the description in terms ofthe densities Ψ and Φ. The experimental results from NSEare compared to the theoretical description of Nonomuraand Ohta [11,17].

2 Experiments

The experiments have been carried out at the IN15 atthe ILL in Grenoble, which currently is the NSE spec-trometer with the largest Fourier time range available.The latter enabled to analyze the slow dynamics of thefluctuations close to the correlation peak. Contrast varia-tion by the exchange of hydrogen with deuterium allowedto measure two different scattering functions using sam-ples of two different types: “film” contrast and “bulk”contrast. Microemulsions consisting of deuterated water(D2O), deuterated decane (d-decane) and the hydrogen

containing surfactant C10E4 have a contrast between sur-factant film and surrounding medium oil and water (“film”contrast). “Bulk” contrast (D2O, h-decane and C10E4,contrast between D2O and the other components) revealsinformation on the dynamics of the local volume fractionof the oil or water layer. Whereas the latter exposes themodes with variations of the order parameter Ψ , the for-mer show sensitivity on Φ.

All samples were prepared with a water/oil volume ra-tio of 1 and surfactant concentrations of 16% and 18.5%.For both surfactant concentrations, samples with addedhomopolymers (polyethylenepropylene (PEP) in the oilphase and polyethyleneoxide (PEO) in the water phase,each with a molecular weight of 5 kg/mol) have been pre-pared. The homopolymer concentrations were 0.25 and0.5 vol % for both the water and oil component. Ho-mopolymers increase the viscosity of water and oil [13](e.g. the average viscosity for the 16% film contrast sam-ples were 0.96, 1.00 and 1.04×10−3 kg/(m s) for 0%, 0.25%and 0.5% homopolymer contents). The characteristic dis-tance in the samples used here were d = 250 A andd = 206 A for 16% and 18.5% respectively. The correla-tion length ξ for 0%, 0.25%, 0.5% homopolymer contentswere 129, 122, 117 A for the lower surfactant concentrationand 115, 107, 101 A for the higher one. These parametershave been obtained by fitting SANS data around q0 withthe Teubner-Strey formula. Phase diagram measurementson such systems [14,16] and other experiments on water-alcane-CiEj type microemulsions, e.g. with Freeze Frac-ture Electron Microscopy (FFEM) [20], have shown thatthis type of samples is indeed bicontinuous and that theparameters obtained by the Teubner-Strey formula corre-spond to those of bicontinuous structures.

Film contrast samples are sensitive to modes, whichcreate local deviations from the average surfactant density,while in “bulk” contrast only those modes that move waterand oil out of phase will be observed. At high q-values(q � q0), the dynamical signal is dominated by the localinterface fluctuations and the observed relaxation rates aretherefore expected be the same in both contrasts, while forthe low-q regime, the water and oil density fluctuations ofthe bulk contrast samples lead to a faster relaxation thanthe surfactant concentration fluctuation seen in the filmcontrast samples.

3 Results

Figures 1 and 2 show NSE-spectra of the pure microemul-sion in “film” and “bulk” contrast respectively with 16%surfactant concentration. The experimental data havebeen fitted with a simple exponential function for the en-tire q-range, S(q, t)/S(q) = exp(−Γb,f t), to get the ratesΓb for bulk contrast and Γf for film contrast samples. Forthe large q-values only the initial slope can be describedin this way, since the decay obeys a stretched exponentiallaw [12]. Therefore, a cutoff for the maximum fouriertimeτ has been applied in the fitting procedure such that theχ2 test of the fit is <7. The fits displayed in Figures 1and 2 show the cutoffs for each q-value. In film contrast,

O. Holderer et al.: Hydrodynamic effects in bicontinuous microemulsions measured by inelastic neutron scattering 159

0 50 100 150τ / ns

0

0.2

0.4

0.6

0.8

1

S(q,

τ) /

S(q

,0)

0 50 100 150τ / ns

0

0.2

0.4

0.6

0.8

1

S(q,

τ) /

S(q

,0)

Fig. 1. NSE measurements for q = 0.011 (filled cir-cles), 0.018 (open squares), 0.026 (shaded diamonds),0.035, 0.051, 0.072, 0.095, 0.118 A−1 for a sample with 16% sur-factant in film contrast. Dashed lines are fits with an exponen-tial function, showing also the limits of the theory at higherq-values (see text). More q-values have been measured at highq (up to 0.134 A−1, not displayed for clarity reasons).

0 50 100 150τ / ns

0

0.2

0.4

0.6

0.8

1

S(q,

τ) /

S(q

,0)

Fig. 2. NSE measurements for q = 0.011 (filled cir-cles), 0.018 (open squares), 0.026 (shaded diamonds),0.035, 0.051, 0.072, 0.095, 0.118 A−1 for a sample with 16% sur-factant in bulk contrast. Dashed lines are fits with an exponen-tial function, showing also the limits of the theory at higherq-values (see text). More q-values have been measured at highq (up to 0.134 A−1, not displayed for clarity reasons).

the relaxation rate increases with increasing scattering an-gle throughout the measured q-range. In contrast, a slow-ing down of the relaxation is observed when approachingthe correlation peak at q � 0.025 A−1 for the “bulk” con-trast sample. A slowing down of the dynamics at lengthscales of the order of the typical length scale in the sam-ple is observed in many systems and is often referred toas “De Gennes narrowing” [21].

The different dynamical behavior is clearly visiblewhen looking at the ratio of the relaxation rates of thetwo types of samples, bulk and film contrast. The ratiox = Γb/Γf is presented in Figure 3. It has to be pointedout that in first instance, Γ is merely a measure of thedecay of the spectra, without using a particular theory forS(q, t). Taking simply the logarithm of the experimentalS(q, t)/S(q) at, say, 100 ns would be another way in ob-

0 0.04 0.08 0.12

q (A o -1

)

0.6

0.8

1

1.2

Γ b /

Γ f

0 0.04 0.08 0.12

q (A o -1

)

0.6

0.8

1

1.2

16 % surfactant 18.5 % surfactant

Fig. 3. Ratio of relaxation rates of bulk and film contrastsamples, left for 16%, right for 18.5% surfactant concentration.Filled circles: pure microemulsion, shaded squares: 0.25% ho-mopolymer contents, open diamonds: 0.5% homopolymer con-tents.

taining ratios of the decay rates or at least a good approxi-mation to it. All samples show the characteristic minimumof x. The position of the minimum shifts as expected from∼0.025 A−1 for the 16% samples to ∼0.030 A−1 for the18.5% samples, which is in excellent agreement with thevalues for q0 of 0.0251 A−1 and 0.0305 A−1 respectively,as obtained by SANS [16]. Adding homopolymers to themicroemulsion reduced the depth of the dip at q0 for bothsurfactant concentrations. The position of the dip in theratio of relaxation rates does not change, since also thestructural length only depends on the surfactant concen-tration, but the dip is less pronounced.

4 Discussion

The observed behavior of the relaxation rates can be ex-plained qualitatively with the expression developed byNonomura and Ohta [11] for the intermediate q-range. Theequations of motion of a ternary fluid are derived from avariational approach. The microemulsion is described interms of the concentration fields Ψ and Φ and an averagevelocity field v. Hydrodynamic interactions in the bicon-tinuous microemulsion are taken into account by an Oseentensor. The intermediate scattering functions S(q, t) forbulk and film contrast microemulsions are then derived asFourier transforms of the time-correlation functions

S11(r, t) = 〈δΨ(r1, t)δΨ(r2, 0)〉 (1)

andS22(r, t) = 〈δΦ(r1, t)δΦ(r2, 0)〉. (2)

For microemulsions with a water/oil volume ratio of 1and under the assumption that there is no difference inthe local friction between the surfactant layer and the oilon the one hand and water on the other hand, S(q, t) canbe written as

S11(q, τ) = [(1 − f)e−Γ11(q)τ + fe−Γ22(q)τ ] (3)

160 The European Physical Journal E

for bulk contrast and

S22(q, τ) = [fe−Γ11(q)τ + (1 − f)e−Γ22(q)τ ] (4)

for film contrast samples, where the decay rates of therelaxation are:

Γ11 = l11q2[χψ(q)]−1 + q2Dq Γ22 = l22q

2 (5)

with the static correlation function χψ(q). The constantsl11, l22 (diagonal parts of the Onsager coefficients, con-stant in the framework of the theory in Ref. [11]) pertainto the coupling of the flow of Ψ and Φ to the free en-ergy variation depending on Ψ and Φ. The q-dependenddiffusion coefficient is defined as

Dq = kBTk0

6πηN(q/k0, u/k0) (6)

with the variables k0 =√

q20 − ξ−2 and u =

√2q0/ξ and

the scaling function N(q/k0, u/k0):

N(x, y) =3√

28

(x2 − 1)2 + y4

y2

∫ π

0

dθ sin3 θ

×[[(x2 sin2 θ − 1)2 + y4]1/2 + 1 − x2 sin2 θ

(x2 sin2 θ − 1)2 + y4

]1/2

(7)

which shows a minimum at q = k0 (see Refs. [17,11] fordetails).

Hydrodynamic effects appear only in Γ11, the relax-ation rate of the bulk contrast microemulsion [11]. Forsymmetric microemulsions, the weighting factor f is �0,which simplifies the above expressions to

S11(q, τ) = e−Γ11(q)τ S22(q, τ) = e−Γ22(q)τ . (8)

Small deviations of f from zero (i.e. a deviation from thesimple exponential behavior) would be difficult to dis-criminate at low q in the present experiments. Accord-ing to the above equations, the ratio x = Γ11/Γ22 =l11/l22χψ(q)−1 + Dq/l22 of the relaxation rates for bulkand film contrast has a minimum at q0. This is in agree-ment with the observed slowing down of the relaxationrate of the bulk contrast samples at q = q0 in Figure 2.For large q, x should increase since χψ(q)−1 as well as Dq

increases with increasing q. This is not observed experi-mentally in Figure 3, where the ratio is decreasing slightlyat larger scattering vectors. It is confirmed by Figure 3,that Γ11 < Γ22 at q = q0. The relaxation rate obeys avery similar behavior for film and for bulk contrast if qis significantly larger than q0, as expected, since the lo-cal membrane dynamics dominates at these short length-scales. It has been shown in previous experiments, that therelaxation follows a stretched exponential function with aq3-dependence [17] for large q and a stretching exponentof 2/3, as also predicted by theory [12]. The simplified ex-pressions in equation (8) are not valid in this limit. Fittingthe data at high q with a stretched exponential functionleads to values of the ratio Γb/Γf of the bulk and filmcontrast samples (same sample, only different contrast),

0.02 0.03 0.04

q (A o -1

)

0.6

0.8

1

1.2

Γ 11 /

Γ 22

0 0.04 0.08 0.12

q (A o -1

)

0

2

4

6

816 % surfactant

18.5 % surfactant

16 % surfactant

Fig. 4. Calculated ratio of relaxation rates of bulk and filmcontrast samples with negligible l11. Left: 16% surfactant con-centration, different homopolymer contents (0%: Black line,0.25%: dotted line, 0.5%: dashed line). Right: 16% and 18.5%surfactant concentration. The box indicates the scale of thegraph on the left.

which are lower by about 10%. This reflects the limitationsof the theoretical approximations made here with respectto the large q-vectors. A better phenomenological repre-sentation of the data would be achieved by using stretchedexponentials with q-dependent stretching exponent β(q).However, the statistical accuracy of the obtained valuesis poor and the available theoretical framework does notcover this behaviour. Therefore, as mentioned earlier, fora uniform evaluation scheme and to keep in the frameworkof the theory applied here, a simple exponential functionhas been used throughout, for each q-value the fit rangein Fourier time has been limited to τmax(q) such that χ2

does not exceed 7. Upon homopolymer addition, the mem-brane bending modulus as well as the correlation length ξdecrease [16], which broadens the peak in χψ(q).

The value of the diffusion constant Dq from equa-tion (6) has been calculated with the experimental val-ues for q0, ξ and η to estimate the hydrodynamic con-tribution to the relaxation rate. For q ≤ q0, q2Dq � Γb,e.g. for q = 0.018 A−1 < q0, the experimental relaxationrate Γb = 0.0011 ± 0.00005 ns−1 compares to the calcu-lated value q2Dq = 0.0016 ns−1 for the pure 16% mi-croemulsion, i.e. the relaxation rate of the bulk contrastsample is dominated by the hydrodynamic contribution.Dq alone already overestimates the observed relaxationrate by 45%, i.e. the value of l11 has to be small and isset to zero in the calculations below. The reduction of thedepth of the dip at q0 by an increase of the effective vis-cosity of the microemulsion due to homopolymers is onlyreflected in the ratio x of the relaxation rates, if the “diffu-sion constant” for film contrast samples, l22, is also takento be viscosity dependent. With l22 = k0/(6πη), the effectof homopolymer addition and also the depth of the dipat q0 can be reproduced as shown in Figure 4. At largerq-values, the calculated curve increases and deviates fromthe experimental data since the approximations made toobtain equation (8) are not valid for q � q0.

O. Holderer et al.: Hydrodynamic effects in bicontinuous microemulsions measured by inelastic neutron scattering 161

5 Conclusion

The hydrodynamic effect on the dynamics of bicontinu-ous microemulsions has been observed directly by chang-ing the contrast of the samples. While “bulk” contrastsamples are sensitive to the dynamics of oil and waterdomains, “film” contrast samples show the fluctuations ofthe surfactant concentration. The slowing down of the dy-namics on length scales of the order of the domain size atq = q0 is observed only in “bulk” contrast and attributedto hydrodynamic effects. At higher q-values, only a slightdecrease of the ratio of the rates is visible through thescaling function N(x, y) (Eq. (7)). The Ginzburg-Landaudescription of ternary liquids including linear hydrody-namics by Nonomura and Ohta [11] yields relaxation ratesfor microemulsions in bulk and film contrast which agreewell with the experimental results for the intermediate q-range (around q0), if the influence of l11 is assumed to besmall. The approximations applied to the theory are notapplicable for q significantly larger than q0.

Addition of homopolymers in the water and oil phaseresult in a slightly higher viscosity of the embeddingmedium of the membrane and in a decrease of the cor-relation length ξ. This leads to the same overall behav-ior but the dip is less pronounced. While homopolymersare strongly affecting the phase diagram of microemul-sions and significantly modify the bending modulus κ ofthe membrane, the effect of homopolymer addition is rel-atively small at intermediate lengthscales in the hydro-dynamic regime but well in agreement with the expectedmagnitude estimated from theory.

We would like to thank G. Gompper for fruitful discussions.

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