The mean size of clusters formed as a function of the reduced temperature and the patch width for patchy particles designed to form 12-particle icosahedral clusters. These Monte Carlo simulations involved 120 particles simulated for 1.2 × 108 Monte Carlo steps at a constant number density of ρσ3 = 0.15 and used the virtual move Monte Carlo algorithm (Ref. 41). Results were averaged over five independent simulations. Decreasing the temperature at a patch width of ∼0.5 radians involves two phase transitions; from a monomer gas through a liquid–gas coexistence region to a cluster gas.
for the transition from monomers to icosahedral clusters for different ratios of the free volume coefficients, B/b. For the nonideal systems, b/σ3 = 1, and dotted lines show , the maximum density at which particles can have an equal probability of being monomeric or in a cluster. .
The relative proportion of monomers and clusters in the equilibrium mixture is shown as a function of the density at kT/ε = 0.2 for B/b = 24, . The function takes the value −1 when only monomers are present, +1 when only clusters are in the system, and 0 at the center of the clustering transition (i.e., on the line). Vertical dotted lines indicate the maximum particle density associated with clusters , the maximum density at which the monomers and clusters can have an equal probability and the maximum monomer density , as labeled. The solid line is calculated from the partition functions, while the dashed line is the extrapolation in the regime where the available volume is held at zero.
The T–ρ phase diagram for the self-assembling van der Waals fluid for B/b = 24. The thick lines give the binodals for the system. Also represented are lines which correspond to points A, B, D, and E in the p–v plots in Fig. 10, where the derivative of the pressure with respect to the volume is zero or ±∞, where they are true limits of stability. The binodals and spinodals for the standard, pure van der Waals fluids are plotted for comparison (dotted lines). Finally, the clustering temperature () is also plotted (thick dotted line). a/εσ3 = 1, b/σ3 = 1, .
The p–T phase diagram for the self-assembling van der Waals fluid at different values of B/b. The critical point is labeled “CP”. The clustering temperature () is also plotted for B/b = 24 (thick dotted line) in the monomer-cluster region only. a/εσ3 = 1, b/σ3 = 1, .
Binodal lines for (a) B/b = 12.5 and (b) B/b = 12 are shown in violet, compared with the dotted red line representing the pure van der Waals binodal. The labels “c”, “m”, “l”, and “f” refer to cluster gas, monomer gas, liquid and fluid states, respectively. a/εσ3 = 1, b/σ3 = 1, .
A T–ρ phase diagram for the self-assembling van der Waals fluid with a selection of values for the van der Waals attractive parameter a. b/σ3 = 1, b/B = 24, .
A T–ρ phase diagram for the self-assembling van der Waals fluid forming tetrahedral clusters. The solid line is the binodal curve, while the thin dotted line represents the pure van der Waals system binodal. The thick dotted line is the clustering temperature, , for this system. The labels “c”, “m”, “l”, and “f” refer to cluster gas, monomer gas, liquid and fluid states, respectively. a/εσ3 = 0.6, B/b = 11, b/σ3 = 1, .
A schematic T–ρ phase diagram to illustrate the potential effects of crystallization on the phase diagram in Fig. 4.
The p–v [(a), (c), and (e)] and p–μ [(b), (d), and (f)] curves for the self-assembling van der Waals fluid at reduced temperatures of (a) and (b) kT/ε = 0.21, (c) and (d) kT/ε = 0.17, and (e) and (f) kT/ε = 0.02. Regions of mechanical stability are shown in blue, those of mechanical instability in red, and the mechanically stable, but compositionally unstable, region of back-bending is shown in green. Binodal points are marked by violet asterisks connected by a tie line. Spinodal points, where the derivative of the pressure with respect to the volume is either zero or infinity, are also shown with the labeling and color-coding matching that used in Fig. 4. v = V/N, a/εσ3 = 1, b/σ3 = 1, , B/b = 24.
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