Free energy function ΔF(η c )/k B T plotted vs. η c for a quasi-two-dimensional system, choosing linear dimensions L x = 20, L y = 60 and D = 1.5, for three choices of as indicated. In all these cases, μ was adjusted to μcoex (which depends on , of course 19,20 ). The curves for the two smaller values of where moved upward by arbitrary amounts, for better visibility. Note that the choice L y = 3L x has advantages for the estimation of the interfacial tension, because then the two interfaces (of area A = L x D) are well separated along the y-axis, and hence do not interact with each other.
Phase diagram of the CAO model, comparing the system in the bulk (taken from Ref. 12 shown by crosses) and in the thin film geometry, confined by two planar walls a distance D apart, for D = 5 and D = 10 (as indicated in the figure). Several linear dimensions of L x = L y = 6.735, L x = L y = 10 and L x = 8, L y = 20 were used to exclude noticeable finite-size effects. Symbols indicate the colloid packing fraction in the polymer-rich phase (left branches) and in the colloid-rich phase (right branches).
Phase diagram of the confined CAO model in the plane of variables polymer packing fraction η p (ordinate) versus colloid packing fraction η c (abscissa), for D = 5 (a) and D = 10 (b). Crosses show the corresponding bulk data (taken from Ref. 12 ). The broken straight lines indicate the tie lines connecting coexisting phases. Arrows indicate the quenches (performed via reduction of the volume at constant particle numbers N c and N p of colloids and polymers, respectively, for different ratios N c /N p ) that will be discussed in Sec. III .
Shear viscosity η of the pure solvent as a function of the mean free path λ, estimated from a study of Poiseuille flow between walls (open circles) or inside of a cylinder (open diamonds), implementing stick boundary conditions at the walls. 77 The curve shows the corresponding prediction from kinetic theory 63–66 for this case. 77 The full symbols highlight the choice made for the present simulations of phase separation kinetics, including also the estimates of the viscosities in the pure coexisting phases.
Snapshot of the colloids and polymers during the demixing process visualizing the three different length scales: the point-like solvent particles (shown in blue), the colloids and polymers (shown in yellow and black respectively) as well as the length scale of the colloid-rich and polymer-rich domains. The snapshot corresponds to an ultrathin film of thickness D = 1.5 at . For a clearer view on the large domains, a layer containing many solvent particles was removed from the top. Solvent particles inside the colloids and polymers are also not visible.
Radial pair distribution function g(r, t) of the colloids in lateral direction for D = 5 at a quench to the state with , and , Three times after the quench are shown, as indicated. Arrows indicate the choice of r c2(t), see text. These data refer to the choice of stick boundary conditions.
Snapshots of the demixing process, showing times t = 12 (a), 1200 (b), 12 000 (c) and 29 000 (d) MD time units after the quench. Only xy-coordinates of the polymer positions of a 256 × 256 × 5 system are shown as black dots (the colloids are shown in yellow). The data are for the “critical quench” as characterized in Fig. 6 , and refer to slip boundary conditions (left), stick boundary conditions (middle) and switched-off hydrodynamics (right).
Domain size ℓ d (t) vs. time t, for the “critical quench” of Figs. 6 and 7 in the film of thickness D = 5, and the three choices of dynamics: slip boundary conditions, stick boundary conditions, and switched-off hydrodynamics. Straight lines illustrate growth exponents of 1/3 and 2/3, respectively, on this log-log plot.
Two-dimensional velocity autocorrelation functions plotted versus collision time steps τ = 0.008 on a log-log plot, for various wall distances D and boundary conditions, as indicated. Full symbols mark negative values, i.e., an anticorrelation of the velocity over time.
Early stages of the demixing process for quench 1 in the system with wall distance D = 10 and slip boundary condition. (a) Side view on a center slice of width 8 for several times: (from top to bottom) 228, 540, 936, 1224, 1512, 6306. Here, colloids are represented as black dots, while the polymers are not shown. (b) Top view of the system at time t = 1512 where at the left only the colloids are shown as black dots and at the right only the polymers are shown as black dots.
Reduced average domains size ℓ′(t) = ℓ d (t) − ℓ s for the system with wall distance D = 5 for all three b.c. (compare with Fig. 9 ).
(a) Domain size ℓ d (t) vs. time t, for two off-critical quenches with relative concentration 75:25 (L) and 25:75 (R) of the polymer-rich and colloid-rich phases, studied with slip boundary conditions for wall distance D = 10. For comparison, the system with 50:50 concentrations (but studied with stick rather than slip boundary conditions) is included, to show that all these systems evolve with a growth exponent of about 1/3 (indicated by the straight line). (b) Corresponding snapshots for the two quenches (L,R), as described in part (a) and indicated also in Fig. 3(a) , for time t = 4000 and t = 16 000.
(a) Snapshots at time t = 1500 (upper row) and at time t = 6700 (lower row) for the three quenches 1, 2, 3 shown in Fig. 3(b) . Only the polymers are shown, as black dots. (b) Average Euler characteristic of colloids χ c (t), upper part, and of polymers, χ p (t), lower part, plotted vs. time, for the three quenches 1, 2, 3 shown in Figs. 3(b) and 13(a) . All runs have been made with slip boundary conditions. (c) Domain size ℓ d (t) on a log-log plot versus time, for the three quenches of Figs. 3(b) and 13(a) . Straight lines illustrate the growth law exponents 2/3 and 1/3, as indicated. System sizes were 256 × 256 × 10 throughout.
Comparison of the time evolution of the domain patterns for (upper row) and (lower row), for the 512 × 512 × 1.5 system, and slip boundary conditions.
Average domain size ℓ d (t) vs. t for the 512 × 512 × 1.5 system and the choice (a) and (b). In each case, slip boundary conditions lead to the fast growth with a growth exponent 1/2 (a) or 2/3 (b), while for both the stick boundary condition and in the case of switched-off hydrodynamics, the effective growth exponent exceeds 1/4 only slightly, as indicated by the straight lines.
Reduced average domain size ℓ′ = l d − ℓ s with ℓ s = 2.5 for a system with D = 1.5 and (a) and (b). The insets show the effective exponent α i (see Eq. (11) ) as a function of inverse reduced domain size ℓ′(t). Horizontal dotted lines are located at 1/2 and 1/3 (a) and 2/3 and 1/3 (b) serving as a guide to the eye.
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