EFFECTS OF AN IMPOSED FLOW ON PHASE-SEPARATING BINARY MIXTURES

Transcription

1 Fractals, Vol. 11, Supplementary Issue (February 2003) c World Scientific Publishing Company EFFECTS OF AN IMPOSED FLOW ON PHASE-SEPARATING BINARY MIXTURES F. CORBERI Istituto Nazionale per la Fisica della Materia, Unità di Salerno Dipartimento di Fisica, Università di Salerno, Baronissi (Salerno), Italy G. GONNELLA Istituto Nazionale per la Fisica della Materia, Unità di Bari Dipartimento di Fisica, Università di Bari Istituto Nazionale di Fisica Nucleare, Sezione di Bari, via Amendola 173, Bari, Italy A. LAMURA Institut für Festkörperforschung, Forschungszentrum Jülich, Jülich, Germany Abstract We study the phase separation of a binary mixture in uniform shear flow in the framework of the continuum convection-diffusion equation based on a Ginzburg- Landau free energy. This equation is solved both numerically and in the context of large-n approximation. Our results show the existence of domains with two typical sizes, whose relative abundance changes in time. As a consequence log-time periodic oscillations 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

2 120 F. Corberi et al. with the power law R(t) t α t 1/3 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 finitesize 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 effects. 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 model 6 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 = γye x, 2 γ and e x 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 2 δ(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 2 ϕ ϕ4 + 1 } 2 ϕ 2 (3) in the ordered phase.

3 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 dkc(k, t) R x (t) = (5) dk kx C(k, t) and analogously for the other directions. Of experimental interest are also the rheological quantities. We consider the excess viscosity which can be defined as 2 η(t) = 1 dk γ (2π) d k xk y C(k, t) (6) where d denotes the number of spatial dimensions. Fig. 1 Configurations of a portion of sites of the whole lattice are shown at different values of the shear strain γt at T = 0. The x axis is in the horizontal direction.

4 122 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 conditions 7 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 increases, 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 k y 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 Kx Ky Kx Ky k y k x k y k x Fig. 2 Three-dimensional plot of the structure factor at γt = 11, 20. Only one foil of C(k, t) is shown (see the text for details). Kx

5 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 R x (t), R y (t). Their behavior is shown in Fig. 3. Due to the alternative dominance of the peaks of C(k, t), R x and R y increase with amplitudes modulated by an oscillation in time. Using a generalization of the renormalization group scheme for the unsheared case 9 we have shown that R x γt 4/3 and R y t 1/3. The simulations cannot give evidence of such a scaling regime because finitesize effects 8 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 = 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 γk x C(k, t) k y = k 2 [k 2 + S(t) 1]C(k, t) + k 2 T (7) where S(t) is obtained through the self-consistent prescription S(t) = and q is a phenomenological cutoff. k <q dk C(k, t) (8) (2π) d

6 124 F. Corberi et al. γt =0.05 γt =1 k y k y γt =6 k x γt =10 k x k y k y k x k x Fig. 4 The structure factor at consecutive times for γ = in the large-n approximation. The scales on the k x and k y axes have been enlarged differently for a better view. Using a scaling ansatz 14 we found that R x γt 5/4 (9) and R t 1/4 (10) in the direction of the flow and perpendicular to it, respectively. behaves as The excess viscosity η γ 2 t 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 R x and R y grow in time with a power law behavior, R x γt 5/4 and R y t 1/4, decorated by oscillations which are found to be periodic in the logarithm

7 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 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,

8 126 F. Corberi et al. Fig. 6 3/2. The excess viscosity as a function of the shear strain γt. The slope of the straight line is 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 alternatively 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.

9 REFERENCES Effects of an Imposed Flow on Phase-Separating Binary Mixtures See e.g., A. J. Bray, Adv. Phys. 43, 357 (1994). 2. A. Onuki, J. Phys. Condens. Matter 9, 6119 (1997). 3. T. Ohta, H. Nozaki and M. Doi, Phys. Lett. A145, 304 (1990); J. Chem. Phys. 93, 2664 (1990). 4. D. H. Rothman, Europhys. Lett. 14, 337 (1991). 5. Z. Shou and A. Chakrabarti, Phys. Rev. E61, R2200 (2000). 6. For simulations with hydrodynamics, see M. E. Cates, V. M. Kendon, P. Bladon and J. C. Desplat, Faraday Discuss. 112, 1 (1999); A. J. Wagner and J. M. Yeomans, Phys. Rev. E59, 4366 (1999); A. Lamura and G. Gonnella, e-print cond-mat/ , in press in Physica A. 7. A. W. Lees and S. F. Edwards, J. Phys. C5, 1921 (1972). 8. F. Corberi, G. Gonnella and A. Lamura, Phys. Rev. Lett. 83, 4057 (1999). 9. A. J. Bray, Phys. Rev. B41, 6724 (1990). 10. F. Corberi, G. Gonnella and A. Lamura, Phys. Rev. E62, 8064 (2000). 11. G. F. Mazenko and M. Zannetti, Phys. Rev. Lett. 53, 2106 (1984); Phys. Rev. B32, 4565 (1985). 12. G. Pätzold and K. Dawson, Phys. Rev. E54, 1669 (1996). 13. A. Coniglio, P. Ruggiero and M. Zannetti, Phys. Rev. E50, 1046 (1994). 14. F. Corberi, G. Gonnella and A. Lamura, Phys. Rev. Lett. 81, 3852 (1998). 15. N. P. Rapapa and A. J. Bray, Phys. Rev. Lett. 83, 3856 (1999). 16. D. Sornette, Phys. Rep. 297, 239 (1998). 17. F. Corberi, G. Gonnella and A. Lamura, Phys. Rev. E61, 6621 (2000). 18. K. Migler, C. Liu and D. J. Pine, Macromolecules 29, 1422 (1996).