Fractals, Vol. 11, Supplementary Issue (February 2003) 119–127fc World Scientific Publishing Company

EFFECTS OF AN IMPOSED FLOW

ON PHASE-SEPARATING

BINARY MIXTURES

F. CORBERIIstituto Nazionale per la Fisica della Materia, Unità di Salerno

Dipartimento di Fisica, Università di Salerno,84081 Baronissi (Salerno), Italy

G. GONNELLAIstituto Nazionale per la Fisica della Materia, Unità di Bari

Dipartimento di Fisica, Università di BariIstituto Nazionale di Fisica Nucleare, Sezione di Bari,

via Amendola 173, 70126 Bari, ItalyA. LAMURA

Institut für Festkörperforschung, Forschungszentrum Jülich,52425 Jülich, Germany

Abstract

We study the phase separation of a binary mixture in uniform shear flow in theframework of the continuum convection-diffusion equation based on a Ginzburg-Landau free energy. This equation is solved both numerically and in the context oflarge-N approximation. Our results show the existence of domains with two typicalsizes, whose relative abundance changes in time. As a consequence log-time periodicoscillations are observed in the behavior of most thermodynamic observables.

1. INTRODUCTION

When a binary mixture is suddenly quenched from a disordered initial state to a coexistence

region below the critical temperature, the two components segregate and form domains

which grow with time. In the unsheared case a single length scale R(t), which measures the

average size of domains, characterizes the kinetics of phase separation. This length grows

119

120 F. Corberi et al.

with the power law R(t) ∼ tα ∼ t1/3 in a purely diffusive regime.1 The application of a

shear flow greatly affects the phase separation process. A large anisotropy is observed in

typical patterns of domains which appear elongated in the direction of the flow.2 Previous

numerical studies confirm these observations.3–5 These studies, however, were carried out

on rather small systems so that an accurate resolution of the spatial properties, which is

usually inferred from the knowledge of the structure factor, was not yet available.

In this article we investigate the segregation process both by numerical simulations of

large scale systems in order to compute at a fine level of resolution the structure factor and in

the context of a self-consistent approximation which allows to overcome any possible finite-

size effect of the simulations. We show that two typical lengths in each spatial direction

characterize the phase separation process. Domains are stretched and broken cyclically

producing an alternate prevalence of thin and thick domains. As a consequence of this

mechanism all the physical observables are modulated by oscillations which are periodic on

a logarithmic time scale.

This paper is organized as follows. In Sec. 2 we introduce the model. In Sec. 3 we

present the results of the numerical simulations. Section 4 is devoted to the analysis of the

behavior of the model in the large-N approximation. Finally we summarize and draw our

conclusions.

2. THE MODEL

In the following we describe a binary mixture by means of a model with a coupling between

a diffusive field ϕ, representing the concentration difference between the two components

of the mixture, and an applied velocity field. This approach neglects hydrodynamic ef-

fects. For weakly sheared polymer blends with large polymerization index and similar

mechanical properties of the two species the present model is expected to be satisfactory in

a preasymptotic time domain when velocity fluctuations are small.1 When hydrodynamic

effects become important, instead, the full dynamical model6 where the velocity is governed

by the Navier-Stokes equation must be considered.

The mixture, which we assume to be symmetric in the two components, is described by

the Langevin equation

∂ϕ

∂t+ ∇ · (ϕv) = ∇2

δF

δϕ+ η (1)

where v is the external velocity field describing plane shear flow with profile v = γyex,2

γ and ex being, respectively, the shear rate and the unit vector in the x direction. η is a

Gaussian white noise, representing the effects of thermal fluctuations with zero mean and

correlations that, according to the fluctuation-dissipation theorem, are given by

〈η(r, t)η(r′, t′)〉 = −2T∇2δ(r − r′)δ(t − t′) (2)

where T is the temperature of the mixture, and 〈· · · 〉 denotes the ensemble average. The

term ∇ · (ϕv) describes the advection of the field ϕ by the velocity and ∇2(δF/δϕ) takes

into account the diffusive transport of ϕ. The equilibrium free-energy can be chosen as

F{ϕ} =

∫

dr

{

−1

2ϕ2 +

1

4ϕ4 +

1

2|∇ϕ|2

}

(3)

in the ordered phase.

Effects of an Imposed Flow on Phase-Separating Binary Mixtures 121

The main observable for the study of the growth kinetics is the structure factor defined

as

C(k, t) = 〈ϕ(k, t)ϕ(−k, t)〉 (4)

where ϕ(k, t) are the Fourier components of ϕ. From the knowledge of the structure factor

the average sizes of domains in different directions can be computed as

Rx(t) =

∫

dkC(k, t)∫

dk|kx|C(k, t)(5)

and analogously for the other directions. Of experimental interest are also the rheological

quantities. We consider the excess viscosity which can be defined as2

∆η(t) = −1

γ

∫

dk

(2π)dkxkyC(k, t) (6)

where d denotes the number of spatial dimensions.

Fig. 1 Configurations of a portion of 512 × 512 sites of the whole lattice are shown at differentvalues of the shear strain γt at T = 0. The x axis is in the horizontal direction.

122 F. Corberi et al.

3. NUMERICAL SIMULATIONS

The most straightforward way of studying Eq. (1) is to simulate it numerically. We have

used a first-order Euler discretization scheme. Periodic boundary conditions have been

adopted in the x and z directions; Lees-Edwards boundary conditions7 were used in the

y direction. These boundary conditions require the identification of a point at (x, 0, z)

with one located at (x + γL∆t, L, z), where L is the size of the lattice and ∆t is the time

discretization interval. Simulations were run using lattices of size L = 4096 in d = 2 and

L = 256 in d = 3. Results are presented here for the two-dimensional case at temperature

T = 0.8

A sequence of configurations at different values of the strain γt is shown in Fig. 1. After

an early time, when well defined interfaces are forming, the pattern of domains starts to

be distorted for γt > 1. The growth is faster in the flow direction and domains assume the

typical striplike shape aligned with the flow direction. As the elongation of the domains in-

creases, non-uniformities appear in the system: Regions with domains of different thickness

can be clearly observed at γt = 11 and γt = 20.

An analysis of length scales present in the system can be done by studying the behavior

of the structure factor. At the beginning C(k, t) exhibits an almost circular shape. Then

shear-induced anisotropy deforms C(k, t) into an elliptical pattern. The profile of the

structure factor changes with time until C(k, t) is separated in two distinct foils, each of

them characterized by two peaks. Since the property C(k, t) = C(−k, t) holds, the two

foils are symmetric with respect to the origin of the k-space and it is sufficient to consider

only the two peaks of one foil. The position of each peak identifies a couple of typical

lengths, one in the flow and the other in the shear (y) direction. This corresponds to the

observation of domains with two characteristic thicknesses observed in Fig. 1. The relative

height of the peaks of C(k, t) is shown in Fig. 2 where the two maxima in each foil are

observed to dominate alternatively at the times γt = 11 and γt = 20.

The competition between two kinds of domains is a cooperative phenomenon. In a

situation like that at γt = 11, the peak with the larger ky dominates, describing a prevalence

of stretched thin domains. When the strain becomes larger, a cascade of ruptures occurs

in those regions of the network where the stress is higher and elastic energy is released. At

this point the thick domains, which have not yet been broken, prevale and the other peak

of C(k, t) dominates, as at γt = 20.

γt = 11 γt = 20

ky kx ky kxKy Kx KyKxKx

Fig. 2 Three-dimensional plot of the structure factor at γt = 11, 20. Only one foil of C(k, t) isshown (see the text for details).

Effects of an Imposed Flow on Phase-Separating Binary Mixtures 123

Fig. 3 Evolution of the average domains sizes in the shear (lower curve) and flow (upper curve)directions. The straight lines have slopes 1/3 and 4/3.

This dynamics affects the behavior of the average size of domains Rx(t), Ry(t). Their

behavior is shown in Fig. 3. Due to the alternative dominance of the peaks of C(k, t), Rxand Ry increase with amplitudes modulated by an oscillation in time. Using a generalization

of the renormalization group scheme for the unsheared case9 we have shown that Rx ∼ γt4/3

and Ry ∼ t1/3. The simulations cannot give evidence of such a scaling regime because finite-

size effects8 prevent to follow the alternate predominance of the two peaks on sufficiently

long times. The present result indicates that the growth in the shear direction is not affected

by the flow, while in the x direction the convective term in Eq. (1) increases the growth

exponent of 1 with respect to the unsheared case. Similar results have been obtained for

temperatures in the range 0 ≤ T ≤ 5 and in d = 3 at T = 0.10

4. A SELF-CONSISTENT APPROXIMATION

Another possible way of studying the present model is in the context of the large-N

approximation.11,12 In this context the model is generalized to a vectorial order parameter

with an arbitrary number N of components and the limit N → ∞ is taken. This special

limit is known to provide a mean-field picture of the phase-separation process which can

give an insight of the phenomenon at a semi-quantitative level.13 The governing equation

for the structure factor in this limit reads

∂C(k, t)

∂t− γkx

∂C(k, t)

∂ky= −k2[k2 + S(t) − 1]C(k, t) + k2T (7)

where S(t) is obtained through the self-consistent prescription

S(t) =

∫

|k|

124 F. Corberi et al.

γt = 0.05

kx

ky

γt = 6

kx

ky

γt = 1

kx

ky

γt = 10

kx

ky

Fig. 4 The structure factor at consecutive times for γ = 0.001 in the large-N approximation. Thescales on the kx and ky axes have been enlarged differently for a better view.

Using a scaling ansatz14 we found that

Rx ∼ γt5/4 (9)

and

R⊥ ∼ t1/4 (10)

in the direction of the flow and perpendicular to it, respectively. The excess viscosity

behaves as

∆η ∼ γ−2t−3/2. (11)

The self-consistency condition (8) has been worked out explicitly in the long-time domain.15

It is found that the model has a multiscaling symmetry and that the asymptotic behaviors

(9)–(11) have logarithmic corrections.

In Fig. 4 the solution of Eq. (7) in d = 2 at T = 0 is plotted for different values of the shear

strain γt. The picture resembles the one outlined in the previous section but the numerical

integration of Eqs. (7) and (8) is much less demanding than Eq. (1) and the evolution of

the structure factor can be neatly obtained over much longer timescales. The evolution

of C(k, t) shows that an alternate prevalence of the two peaks on each foil continues in

time. As a consequence Rx and Ry grow in time with a power law behavior, Rx ∼ γt5/4

and Ry ∼ t1/4, decorated by oscillations which are found to be periodic in the logarithm

Effects of an Imposed Flow on Phase-Separating Binary Mixtures 125

Fig. 5 Evolution of the average domains size in the shear (lower curve) and flow (upper curve)directions. The straight lines have slopes 1/4 and 5/4.

of time, as shown in Fig. 5. The growth exponent 1/4 is expected for systems with a

continuous symmetry (N > 1) without shear and corresponds to the diffusive exponent 1/3

characteristic of the scalar model. Again a difference 1 between the two growth exponents

is found.

Stretching of domains requires work against surface tension and burst of domains dissipate

energy resulting in an increase ∆η of the viscosity.2 The time behavior of ∆η is shown in

Fig. 6. Starting from zero, ∆η grows up to a global maximum, then it relaxes to smaller

values oscillating. The decay is consistent with the expected power-law behavior (11). The

oscillations are related to the dynamics of the peaks of C(k, t). Relative maxima of ∆η

occur when the domains are maximally stretched (see Figs. (4) and (5) at γt = 6). Later,

when the domains are deformed to such an extent that they start to break up, ∆η decreases

and more isotropic domains are formed. Then a minimum of ∆η is observed. The process of

storing elastic energy with consequent dissipation through cascades of ruptures reproduces

periodically in time with a characteristic frequency. Similar log-time periodic oscillations

have been observed in solid materials subject to an external strain.16 In three dimensions

we proved that the growth of domains is not affected by the external shear in the directions

perpendicular to the flow and the growth exponent 5/4 is still found to characterize the

growth in the x direction.17

5. CONCLUSIONS

In the theoretical framework above outlined we have shown and accounted for many of the

most significative features of the phase separation of sheared binary mixtures. Moreover,

126 F. Corberi et al.

Fig. 6 The excess viscosity as a function of the shear strain γt. The slope of the straight line is−3/2.

additional predictions are allowed. We could connect the alternating dominance of the

peaks of the structure factor and the overshoots in the excess viscosity. A four-peaked

structure factor was observed in shallow quenching of polymer solutions,18 together with a

double overshoot of the excess viscosity. The results of our study strongly suggest that the

interplay between the four peaks of the structure factor, with their alternate prevalence,

gives rise to an oscillatory phenomenon which reflects itself on the main observables. On

these bases our conjecture is that the experimental double overshoot can be interpreted

as the first part of an oscillatory pattern superimposed on the global trend of the excess

viscosity.

Our simulations indicate that domains of two characteristic sizes in each direction alter-

natively prevail, because the thicker are thinned by the strain and the thinner are thickened

after multiple ruptures in the network. This is reflected on the properties of the structure

factor. These results have been confirmed, at a semiquantitative level, in the context of

the large-N approximation, where the observed oscillations are periodic in the logarithm

of time. The logarithmic periodicity of these overshoots in time is one of the relevant

predictions of the theory to be tested in future experiments on longer time scales.

ACKNOWLEDGMENTS

F. C. and G. G. acknowledge support by the TMR network contract ERBFMRXCT980183,

by INFM PRA-HOP 99 and by MURST PRIN-2000.

Effects of an Imposed Flow on Phase-Separating Binary Mixtures 127

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