Progr Colloid Polym Sci (1997) 106:75 78
9 SteinkopffVerlag 1997

S. K o m u r a
H. K o d a m a

Received: 13 October 1996
Accepted: 7 March 1997
Dr. S. Komura (F:~)
Department of Mechanical System
Engineering
Kyushu Institute of Technology
Iizuka 820, Japan
Dr. H. Kodama
Graduate School of Science and Technology
Kobe University
Kobe 657, Japan

Microemulsions under steadyshear flow

Abstract Dynamic response ofmicroemulsions to shear deformationon the basis of two-order-parametertime dependent G i n z b u r ~ L a n d a u
model is investigated by means of celldynamical system approach. Timeevolution of anisotropic factor andexcess shear stress under steady shear
flow is studied by changing shear rate
and total amount of surfactant. As thesurfactant concentration is increased,

Microemulsions being mixture of oil, water and surfactant
are known to exhibit various interesting mesoscopic structures depending on the temperature or the composition[1]. For relatively low concentration of surfactant and if
the volumes of oil and water are not very different, microemulsions form a bicontinuous structure where a multiply
connected randomly oriented monolayer of surfactants
separate oil-rich and water-rich subvolumes with a mesoscopic length scale (10 ~ 100 nm).
When one applies an external flow to the microemulsion system, its mechanical response is deeply affected by
its internal structure. Using a single-order-parameter timedependent Ginzburg-Landau (TDGL) model, Mundy
et al. have investigated rheological properties of microemulsions theoretically [2]. In their model, the order parameter represents the concentration difference between oiland water, and the presence of surfactants is taken intoaccount through the surface tension parameter. Theirwork has been extended by P/itzold and Dawson, and it
was shown that the microemulsions behave in an essentially non-Newtonian manner [3].

overshoot peak height of theanisotropic factor increases whereasthat of the excess shear stress is almostunchanged.
Key words Microemulsions rheology - shear flow Ginzburg-Landau model

In this proceeding, we investigate the rheological properties of microemulsions using a two-order-parameter
T D G L model. The two-order-parameter Ginzburg-Landau
free energy for microemulsions has already been proposedby Laradji and his coworkers to study their microphaseseparation dynamics [4]. However, the proposed free energy is not applicable to the rheological study, since thepattern does not deform even in the presence of a flow [5].
This is related to the fact that their free energy is unbounded from below for the configurations with large
surfactant concentration at the oil/water interfaces [1].
Thus, we use a different free energy which has been proposed in our previous paper as an improved model [5].
Let O(r) describe the local concentration difference
between oil and water, and p(r) the local surfactant concentration. What we have assumed in our model is that (i)the profiles of 0 and p at oil/water interfaces do notdepend on the average values of ~ and p (denoted hereafteras ~ and fS, respectively) and that (ii) the coarse-graining
dynamics of 0 based on the free energy becomes slow
when the amplitude of p at the interface takes a certain

76

S. Komura and H. Kodama
Microernulsions under steady shear flow

saturated value. One of the m i n i m u m models which reproduce the properties of microemulsions is [5]equations are given by
04,a~ + V.(v~) -- M o v- 2 ~fiF + ~,(r, O,F = .tdr [w(V2O) 2 + d(VO) 2 - aO z + u~, 4+ ep~(p-p~)~ -sp(VO)q,(1)where w, d, a, u, e, Ps and s are positive constants. The firstterm w(V2~,) 2 with positive w prevents the model frombecoming unbounded, whereas the d o u b l e - m i n i m u mpotential e p 2 ( p - p,)2 guarantees that p locally takes
either 0 or p~. The last term - s p ( V ~ ) 2 favors the surfactants to sit at the oil/water interfaces [4].
F o r the evolution of ~b(r, t) and p(r, t), we assume thestandard T D G L equations with a macroscopic flow v.
Since both ~ and p are conserved quantities, T D G L

8p
c~F& + V.(vp) = M,,v~ ~op + ,7o(r, t).

(2)Here My, and Mp are transport coefficients, ~I~ and ~lt, represent the thermal noise which satisfy the fluctuationdissipation theorem 01o(p)(r, t)rl~,(p~(r', t')) = - 2kBTMo(p) x
V2c~(r - r')fi(t - t'), where kB is the Boltzmann constantand T the temperature. As regards the macroscopic flow in
Eq. (2), we consider a simple shear flow v~(r)= >;y,vy = v~ = 0, where the shear rate '2 is t h e time derivative of
the strain y. In our work, we have entirely ignored t h eFig. 1 Time evolution of ~ (left) and p (right) for (a) t5 = 0.1, ") = 2 x 10 ~ (9% = 0.1) and (b) fi = 0.1, 9 = 2 x 10 -3 (9% = 1)

Progr Colloid Polym Sci (1997) 106:75-78
9 Steinkopff Verlag 1997

hydrodynamic interactions which might play an important role in microemulsions.
We solved the above time-evolution equations numerically by using the "cell dynamical system" approach proposed by Oono et al. [6]. The details of the numericalmethod have been mentioned in our previous paper [5].
The simulation conducted here is essentially equivalent tosolving Eq. (2) with the parameter values fixed as w = 0.1,
d = 0.25, a = 0.3, u = 0.18, e = 0.125, p~ = l, s = 0.25 and
M 0 = Mp = 0.05. The noise terms are treated as a random
noise with an amplitude 0.02. We fixed ~ as ~ = 0, whereas/5 has been changed as/5 = 0.1, 0.2, 0.3 and 0.4. Here, we
restricted ourselves to a two-dimensional system. The system size is chosen as 128 x 128, and the sheared-periodic
boundary condition is imposed both for ~ and p. We firststarted from random uniform distributions of ~ and p in
the range [ ~ - 0.01, ~ + 0.01] and [/5 - 0.01,/5 + 0.01],
respectively. We relaxed them according to Eq. (2) without
any shear flow up to t = 5 • 105, and then applied theshear flow with a constant shear rate ~ = 2 x 10 -4,
5 x 10 4, 1 x 10 -3 and 2 x 10 -3. Since we have not included any hydrodynamic interactions, our model lacks
a bare viscous time scale against which we can compare
these shear rates. Nevertheless, we can choose the model
intrinsic time scale ro as the inverse of the initial growthrate of the most unstable mode, which is estimated asro ~ 5.0x 102 for the above parameter values. Theinvestigated shear ranges from weak to medium shear(~o = 0.1 ~ 1).
Typical time evolutions of ~ and p are shown in Fig. 1for (a) /5 = 0.1, o~= 2 x 10 -'~ ($Zo = 0.1) and (b) /5 = 0.1,
= 2 x 10-3 (gZo = 1). By changing/5 and ~, we found thefollowing general behaviors. When the shear rate is small(Fig. la), surfactants move under the flow keeping themselves attached to the oil/water interfaces. The total amountof the interface does not seem to change appreciably during the deformation. On the other hand, when the shear
rate is large (Fig. lb), some surfactants are blown off theinterfaces by the shear. The coagulation and break-up
processes take place as has been observed in the spinoidal
decomposition under steady shear flow [7], and the total
amount of the interface increases.
Given the evolving patterns, we have evaluated theanisotropic factor defined by [2, 3]

Q~,

=_

2d t ~ x ~ y t _ 4w

02~' ~ 2 ,-/\,(3)

where ~ ... ) denotes the average over the total volume.
Although this quantity essentially represents the xy-component of the macroscopic excess stress tensor in the caseof one-order-parameter model, this is not the case in thepresent model since there should be a contribution to the7735[-25<y,; /"'%":,iii, , .....

I24687Fig. 2 The anisotropic factor Q~, (upper 4 curves, see Eq. (3)) and theshear stress %y (lower 4 curves, see Eq. (4)) as a function of the shear
strain 7 = )t for fixed 9 = 2x 10.4 (~z0 = 0.1) and/5 = 0.1, 0.2, 0.3,0.4

stress due to the non-local coupling term in Eq. (1). Nevertheless, the problem of stress division between ~ and p istheoretically not yet clear. Instead, we propose here thefollowing quantity:•xr=-2 (d-sP)~x~y/-4w

~/,

(4)

which is assumed to express the excess shear stress. Figure
2 shows the plots ofand (rxy as a function of the shear
strain 7 for several values of/5. Here the shear rate is fixedas ~ = 2 x 10 -4 (~ro = 0.1). In the time region where
Qxy and axy increase the domains are elongated, whereas
these quantities start to decrease once the burst and therecombination take place. We observed that the strain
giving the peak position of Qxr and a~y is almost constantthroughout the present simulation (7 ~ 2). On the other
hand, the peak height of Q~y is larger than that of ~rxy as
a whole. /5 dependencies of the peak height of Qxy andaxy are also different; the peak height of Qxy increases
linearly with/5, while that of axy is almost independent of ft.
However, a clear shear rate dependence of the peak heightof axy could be observed as in ref. [7].
Details of the present work have been published elsewhere [8].

QxyThis work is supported by the Ministry of Education,
Science and Culture, Japan (Grant-in-Aid for Scientific
Research Nos. 08226233, 08740324 and 0657).

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S. Komura and H. Kodama
Microemulsions under steady shear flow

References
1. Gompper G, Schick M (1994) Self-Assembling Amphiphilic Systems. AcademicPress, New York
2. Mundy CJ, Levin Y, Dawson K (1992)
J Chem Phys 97:7695-7698
3. Pfitzold G, Dawson K (1996) J Chem
Phys 104:5932-5941

4. Laradji M, Guo H, Grant M, Zuckermann MJ (1991) J Phys A 24:L629 L635;(1992) J Phys Condens Matter 4:6715 6728
5. Komura S, Kodama H (1997) Phys Rev
E 55:1722-1727

6. Oono Y, Puri S (1987) Phys Rev Lett 58:836-839; (1988) Phys Rev A 38:434-453
7. Ohta T, Nozaki H, Doi M (1990) Phys
Lett A 145:304; (1990) J Chem Phys
93:2664 2675
8. Kodama H; Komura S (1997) J Phys IIFrance 7:7-14