The development of the lattice Boltzmann methods of simulating flows in multiphase liquids has developed rapidly in recent years. The time has come to evaluate the critical phenomena that characterize basic thermodynamic and hydrodynamic properties of this type of model. We performed numerical experiments on a LB fluid model to determine the equilibrium critical properties that are most important for comparison with real fluids. The results of those simulations are represented in a reduced variable description that is largely independent of the particulars of the model facilitating comparisons with other models of fluid mixtures and with experiment. This type of representation should also be advantageous in expressing experimental measurements in a model independent form. Our observations indicate that the critical properties (coexistence curves, correlation length, interfacial profile, surface tension) of the LB fluid correspond to an ideal mean-field fluid over a broad range of temperatures. This makes comparisons of the model to experiment particularly appropriate to the high molecular weight polymer blends and other fluid mixtures (perhaps also including some ionic fluids and fluid mixtures containing dissolved salts [90,91]) that can be reasonably modeled by mean-field theory.
Now that we have established the equilibrium critical properties of LB fluid mixtures, we are in a position to study much more complicated problems involving fluid flow, phase separation and interacting complex boundaries. We illustrated this type of problem in the case of phase separation in critical and off-critical fluid mixtures with and without shear. We also considered the perturbing influence of boundaries on quiescent phase separation and the flow instabilities that can occur in late stages of phase separation.
Our bulk blend phase separation studies show that the morphology of the phase separation process in its early and intermediate stages depends on the fluid composition. The early stage of phase separation corresponds to the growth of the local composition to the value of the coexisting composition. Coarsening proceeds at a later stage. The phase separation morphology had a bicontinuous form under off-critical conditions and the minority phase had a droplet morphology in a far off-critical blend. Previous LB calculations have emphasized the kinetics of phase separation, which is not the emphasized phenomena here [11,98,99,100]. In future work, we plan to study the crossover between the droplet to bicontinuous phase separation morphology as a function of viscosity mismatch, composition, and quench depth.
Next, we considered the more challenging problem of phase separation under steady shear. Again we considered on and off-critical blends and found the morphologies to be qualitatively different. Shear had the effect of causing the phase separation morphology to "streak" into a string-like morphology near the boundary of the critical composition phase-separating blend, leading to a complex gradient structure within the blend. The "penetration"depth of the surface-induced strings seemed to grow with time in the course of phase separation. The ultimate configuration of the phase-separated blend, alignment parallel or perpendicular to the flow direction, depends on the polymer surface interaction. These observations of fluid heterogeneity on intermediate time scales clearly raise questions about the proper interpretation of light scattering and optical microscopy studies of blends under shear, since these methods often involve an averaging over the gradient structure or are limited to observations of the near-surface properties of the mixture, respectively. The off-critical sheared blend simulations revealed a tendency toward droplet distortion and tilted alignment with respect to the shear flow direction. At a latter stage we observed droplet alignment string and the droplets subsequently coalescenced into a string-like morphology. These strings seem to be very stable under shear, which we expect to arise from the strong hydrodynamic interactions between the strings and the shearing boundaries in these highly confined phase separating fluids (see Figure 17). A similar phenomenon in off-critical blends sheared at low rates in a confined geometry has recently been observed experimentally [110].
In our final illustrative example we considered the perturbing influence of solid and air-boundaries on the phase separation of a blend. The existence of a "free" deformable boundary (polymer-air interface) makes this a particularly instructive example of some of the advantages of the LB method. We observe the development of composition waves in the phase separating blend, as observed in many previous experimental and simulation studies with a preferential interaction between one of the blend components and the boundaries [110,112,113,114,115,116]. The simulation illustrates the process by which layers are lost in the course of phase coarsening. These film coarsening processes apparently lead to a destabilization of the layer structure in a late stage of phase separation. The fluctuations within the film associated with successive film rupture processes cause the layer structure to collapse like a disturbed "house of cards," leading to a polymer blend morphology superficially resembling a bicontinuous "spinodal" pattern. These observations emphasize the importance of time dependent studies of blend film morphologies in measurements on real blend films to properly interpret their origin.
Our illustrations of LB calculations of blend phase separation were purposely restricted to relatively simple geometries and flows that are under current study for their potential relevance to processing applications. It is also possible to incorporate many other fluid properties of interest (shear dependence of fluid viscosity) and other important effects (temperature gradients and time dependent temperatures variations, density mismatch of fluid components and segregation with gravitational and centrifugal fields, fluid wetting and dewetting on heterogeneous substrates, phase separation of blend films on patterned substrates, phase separation in electric fields, phase separation at high rates of flow where inertial effects are important, flow in complex geometries and with the presence of filler inclusions, etc.). There are many possibilities for further application. An important challenge for future theoretical work is the incorporation of fluctuation effects to better describe fluid properties near the critical point for phase separation.