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Phase Separation without Shear

In Fig. 11 we illustrate the case of a critical composition ( $\phi_A=1/2;m_A=m_B$A = ½ mA = mB). The value of the "quench depth" equals $\tau _G=0.537$G = 0.537 and the reduced time, = 3.6. Periodic boundary conditions are employed in this LB calculation to minimize wall effects. Figure 12 shows separation after a later time,
$\bar{t}=17.8$ = 17.8. The pattern is similar in geometric form to Fig. 11, but the characteristic scale of the pattern is larger after longer times, geometrically illustrating the notion of dynamic similarity in the phase separation coarsening process. A study of the time dependence of the growth of the phase separation shows that the pattern scale grows slowly in the early stage of the phase separation process, as the local composition builds up to its coexisting composition value (one of the coexisting phases, $\phi_B$B, has been rendered transparent in Fig. 11). At a later stage of phase separation, the pattern scale growth is roughly linear in time [96,97]. Other LB studies have recently focused on modeling the kinetics of phase separation so we do not dwell on this well known phenomena in the present paper [11,97,98,99,100]. We next illustrate the qualitative change in the phase separation morphology that occurs under off-critical conditions. Fig. 13 shows the phase separation morphology for a quench depth, $\tau _G=0.133$G = 0.133. The rendered composition $\phi _A$A is taken to have the off-critical value, $\phi_A=0.1$A = 0.1. The phase separation then occurs through droplet formation rather then the formation of a bicontinuous "spinodal" phase separation pattern. At a later time, we observe the droplets to coarsen by coalescence in a normal manner for off-critical fluids [101]. In future work we plan to explore the conditions (e.g. quench depth, viscosity mismatch, etc.) that determine the crossover between the droplet and bicontinuous phase separation patterns observed in the early stages of phase separation. Even this basic aspect of fluid phase separation remains poorly understood so that materials scientists must rely on engineering correlations [102].


  
\begin{figure} \begin{center} \special{psfile=fig11.ps angle=-0 hoffset=83 voffset=-375 vscale=55 hscale=55} \end{center}\vspace{9.75 cm}\end{figure}

Figure 11: "Spinodal" phase separation morphology in critical composition (50-50 relative composition) fluid mixture.
The quench depth equals $\tau _G=0.537$G = 0.537 and $\bar{t}=3.6$ = 3.6.


  
\begin{figure} \begin{center} \special{psfile=fig12.ps angle=-0 hoffset=83 voffset=-325 vscale=55 hscale=55} \end{center}\vspace{8.0 cm}\end{figure}

Figure 12: Spinodal phase separation in critical composition fluid mixture. Note similarity of the structure in Fig. 12 to Fig. 11, apart from scale.
This observation reflects the existence of dynamic scaling in the mixture coarsening. The quench depth equals $\tau _G=0.537$G and $\bar{t}=17.8$ = 17.8.


  
\begin{figure} \begin{center} \special{psfile=fig13.ps angle=-0 hoffset=83 voffset=-375 vscale=50 hscale=50} \end{center}\vspace{9. cm}\end{figure}

Figure 13: Off-critical (10-90 relative composition) phase separation showing droplet formation and coarsening.
The quench depth $\tau _G=0.133$G = 0.133 and $\bar{t}=8.4$ = 8.4.

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