Phase separation under a steady shearing flow

We next illustrate a non-trivial application of the LB method to a situation where fluid flow is crucially important. Fig. 14 shows the phase separation of a critical composition ( $\phi_A=\phi_B;M_A=M_B$_A = $\phi_A=\phi_B;M_A=M_B$_B , M_A = M_B ) blend for the same quench as shown in Fig. 11. The upper and lower boundaries in the figure are energetically "neutral" (neither fluid preferentially wets the surface.) A hydrodynamic "stick" boundary condition is imposed at the walls. The top and bottom walls move at velocities u_w and -u_w such that the dimensionless shear rate equals, t_ps = (2u_w / d)t_ps = 0.18, where d is the spacing between the walls. The boundary condition is periodic in the direction parallel to the translating planes. We observe in the top view that the phase separation pattern appears to have a "string-like" form at the boundary, but that the structure within the film is actually more complicated. The phase separation in the plane perpendicular to the flow is remarkably undisturbed by the flow and closely resembles a two-dimensional phase separation pattern in the absence of shear. As time proceeds, the "penetration depth" of the surface-induced "string-like" structures in Fig. 14 increases and the phase separation morphology ultimately coarsens to a state where the fluid interface lies perpendicular to the shearing planes and parallel to the flow direction. The surface-induced "structuring" of the blend morphology would be accentuated in the early stages of phase separation if one of the blend components has a preferential attraction to the shearing boundaries. This surface interaction evidently has a symmetry breaking effect, since the fluid interface was found to lie parallel to the interface of the shearing boundaries at long times, when the boundary interaction was modified in this fashion. This illustrative calculation shows that the surface interaction can have a large influence on the ultimate alignment of a phase separated fluid under shear. It is also apparent that the interpretation of optical and scattering data on sheared phase separated fluids is complicated by the existence of gradients in the composition and structure of the fluid. Clearly, these observations warrant a thorough investigation of the many parameters that seem relevant to the phenomena (quench depth, surface energy, viscosity, molecular weight mismatch, roughness of shearing surface, steady and oscillatory shear, etc.). We note that many experiments have recently reported "string-like" structures in sheared phase-separating fluids using both light scattering and optical microscopy techniques [48,103,104] and "string-like" structures have also been reported in two-dimensional LB phase separation simulations [105]. The case of two dimensions is somewhat special, however, since the Taylor-Tomotika [106] instability is suppressed in two dimensions [107].

  

Figure 14: Sheared critical composition blend. The "string" structures are observed along shearing planes where the strings are oriented in the direction of the fluid flow. The quench depth $\tau _G=0.537$_G = 0.537, $\bar{t}=8.5$ = 8.5 and t_p = 0.18.

An illustration of phase separation in an off-critical blend under steady shear also provides important insights into the kinds of phase separation morphologies that can be expected experimentally. Fig. 15 shows an off-critical blend (15-85) for a dimensionless shear rate,
t_ps = (2u_w / d)t_ps = 0.58. Initially, long narrow filaments formed and eventually they broke apart into droplets due to the well-know Taylor-Tomotika instability [106]. The droplets (Fig. 16) then became elongated and tilted at approximately 45 degrees relative to the shear plane, as predicted in the limit of a low concentration of dispersed droplets [108]. However, at a later stage of phase separation (Fig. 17), the droplets coalesce to form undulating string structures that seem to persist indefinitely in a "dynamic string state" (a video of our simulation can be found in reference [109]). The correlated motions of the strings suggest that the hydrodynamic interactions between the strings and/or between the strings and the boundary of the sheared fluid seem to be playing an important role in the stabilization and formation of the extended string structures. Subsequent experiments have indicated a similar string formation phenomenon in an off-critical blend sheared at low shear rates in a parallel plate geometry having a narrow gap relative to the droplet size [110].

  

Figure 15: Early stage off critical (15-85) phase separation under shear. For Figures 15-17, orange represents the regions of high localized phase fraction of fluid A ( $\phi _A=0.15$_A = 0.15). The green regions represent the transition to a high localized phase fraction of fluid B, $\phi _B=0.85$_B = 0.85. Note the incipient Taylor-Tomitaka instability in some of the fluid "strings". The quench depth $\tau _G=0.287$_G = 0.287, $\bar{t}=16.4$ = 16.4, and the dimensionless shear rate $\dot{\gamma}t_p=0.56$t_p = 0.56.

  

Figure 16: Intermediate stage off-critical (15-85) phase separation ( $\bar{t}=27.6$ = 27.6).
Droplets form after the fluid threads in Fig. 15 break up by the Taylor-Tomotika instability.

  

Figure 17: Late stage off-critical (15-85) phase separation ( $\bar{t}=56.8$ = 56.8).
Droplets shown in Fig. 15 reconnect into string-like structures that appear to persist and coarsen in cross-sectional dimension with time. The strings in Fig. 17 are contrasted with the "strings-like" structures near the mixture surface in Fig. 14, representative of a critical composition fluid mixture.