Two-Scale Competition in Phase Separation with Shear

Abstract

The behavior of a phase separating binary mixture in uniform shear flow is investigated by numerical simulations and in a renormalization group (RG) approach. Results show the simultaneous existence of domains of two characteristic scales. Stretching and cooperative ruptures of the network produce a rich interplay where the recurrent prevalence of thick and thin domains determines log-time periodic oscillations. A power law growth R(t) ∼ t α of the average domain size, with α = 4/3 and α = 1/3 in the flow and shear direction respectively, is shown to be obeyed. numbers: 47.20Hw; 05.70Ln; 83.50Ax The application of a shear flow to a disordered binary mixture quenched into a coexistence region greatly affects the phase-separation process [1]. A large anisotropy is observed in typical patterns of domains which appear greatly elongated in the direction of the flow [2]. This behavior has consequences for the rheological properties of the mixture. A rapid strain-induced thickening followed by a gradual thinning regime is observed [3]. Various numerical simulations confirm these observations [4][5][6]. PACSIn a recent paper [7] we have studied the phaseseparation kinetics in the context of a self-consistent approximation, also known as large-N limit. Within this approach the existence of a scaling regime characterized by an anisotropic power-law growth of the average size of domains was established. The segregation process, however, cannot be fully described by this technique because interfaces are absent for large-N [8].The aim of this letter is to investigate the ordering process by extensive numerical simulations. Our main result concerns the simultaneous existence of two length scales which characterize the thickness of the growing domains. The competition between these scales produces a rich dynamical pattern with an oscillatory behavior due to the cyclical prevalence of one of the two lengths. A power law growth R(t) ∼ t α of the average domain size, with α = 4/3 and α = 1/3 in the flow and shear direction respectively, is shown to be obeyed by a renormalization group (RG) analysis.The kinetic behaviour of the binary mixture is described by the Langevin equationwhere the scalar field ϕ represents the concentration difference between the two components of the mixture [1]. The equilibrium free-energy can be chosen as usual to bewhere b, κ > 0 and a < 0 in the ordered phase. v is an external velocity field describing plane shear flow with average profile given bywhere γ is the shear rate and e x is the unit vector in the flow direction. η is a gaussian white noise, representing thermal fluctuations, with mean zero and correlation η( r, t)η( r ′ , t ′ ) = −2T Γ∇ 2 δ( r − r ′ )δ(t − t ′ ), where Γ is a mobility coefficient, T is the temperature of the fluid, and the symbol ... denotes the ensemble average. The Langevin equation (1) has been simulated in d = 2 by first-order Euler discretization scheme. Periodic boundary conditions have been used in the flow direction while in the y direction the point at (x, y) is i...

Spinodal decomposition of asymmetric binary fluids in a micro-Couette geometry simulated with molecular dynamics

“…An acceleration of the phase separation under shear has also been reported in previous experimental and simulation studies. 27,28 The breakup of domains under shear is also expected to introduce a maximum attainable domain size, but this limit is apparently not reached under the current simulation conditions.…”

“…25,26 In combination with the interfacial driving force, these give rise to anisotropic growth of the domains, with two linear growth processes in the flow-vorticity plane and a possibly supralinear growth, ␣ Ͼ 1, in the sheargradient direction. 25,27,28 The domains cannot grow indefinitely in the shear-gradient direction but can attain a ratedependent nonequilibrium steady state; it is not clear whether the growth in the two perpendicular directions also saturates. 25,29 Shear may also be applied to induce phase separation in dynamically asymmetric mixtures.…”

Structure and rheology of binary mixtures in shear flow

“…A numerical analysis of the exact TDGL model has been performed recently in [14] where it is shown that the global picture of the one-loop approximation is adequate. In particular the oscillatory pattern is recovered.…”

“…The existence of a scaling symmetry and the determination of the related exponents, however, has not been clearly established numerically mainly due to finite size effects limitations. The actual value of the growth exponents can be inferred by scaling [13] or renormalization group [14] arguments to be α ⊥ = 1/3, as in the case without shear (we stress the fact that hydrodynamic effects are neglected in this model), and α x = 4/3.…”

Equilibrium and dynamical behavior in the Vicsek model for self-propelled particles under shear

“…Even in passive materials an imposed flow may lead to flow-sustained non-equilibrium steady states. For instance, wormlike micelles and liquid crystals form bands when sheared [12][13][14], whereas a sheared binary fluid arrests spinodal decomposition, leaving domains of a well-defined size, provided that hydrodynamic coupling between the order parameter and the velocity field is retained [15][16][17][18][19][20]. The self-driving characteristics of SPP suspensions is likely to add even more richness to this already fascinating physics.…”