^{1,a)}, Alexander J. Williamson

^{1,a)}, Jonathan P. K. Doye

^{1,b)}, Jesús Carrete

^{2}, Luis M. Varela

^{2}and Ard A. Louis

^{3}

### Abstract

In patchy particle systems where there is a competition between the self-assembly of finite clusters and liquid–vapor phase separation, re-entrant phase behavior can be observed, with the system passing from a monomeric vapor phase to a region of liquid–vapor phase coexistence and then to a vapor phase of clusters as the temperature is decreased at constant density. Here, we present a classical statistical mechanical approach to the determination of the complete phase diagram of such a system. We model the system as a van der Waals fluid, but one where the monomers can assemble into monodisperse clusters that have no attractive interactions with any of the other species. The resulting phase diagrams show a clear region of re-entrance. However, for the most physically reasonable parameter values of the model, this behavior is restricted to a certain range of density, with phase separation still persisting at high densities.

We would like to thank the Engineering and Physical Sciences Research Council, the Royal Society, the Spanish Ministry of Education and Science, and the European Regional Development Fund (Grant No. FIS2007-66823-C02-02) for financial support. J.C. wishes to thank the Spanish Ministry of Education and Science for an FPU grant.

I. INTRODUCTION

II. MONOMER-CLUSTER EQUILIBRIUM

A. Partition functions

B. Estimation of free volume coefficients

C. Clusteringtransition

III. SELF-ASSEMBLING VAN DER WAALS FLUID

A. Partition functions

B. Coexistence curve

C. Phase diagrams

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Polymers
- 78.0
- Phase diagrams
- 48.0
- Self assembly
- 25.0
- Phase separation
- 21.0
- Cluster analysis
- 15.0

## Figures

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 × 10^{8} Monte Carlo steps at a constant number density of ρσ^{3} = 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.

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 × 10^{8} Monte Carlo steps at a constant number density of ρσ^{3} = 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*/σ^{3} = 1, and dotted lines show , the maximum density at which particles can have an equal probability of being monomeric or in a cluster. .

for the transition from monomers to icosahedral clusters for different ratios of the free volume coefficients, *B*/*b*. For the nonideal systems, *b*/σ^{3} = 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 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*/εσ^{3} = 1, *b*/σ^{3} = 1, .

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*/εσ^{3} = 1, *b*/σ^{3} = 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*/εσ^{3} = 1, *b*/σ^{3} = 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*/εσ^{3} = 1, *b*/σ^{3} = 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*/εσ^{3} = 1, *b*/σ^{3} = 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*/εσ^{3} = 1, *b*/σ^{3} = 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*/σ^{3} = 1, *b*/*B* = 24, .

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*/σ^{3} = 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*/εσ^{3} = 0.6, *B*/*b* = 11, *b*/σ^{3} = 1, .

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*/εσ^{3} = 0.6, *B*/*b* = 11, *b*/σ^{3} = 1, .

A schematic *T*–ρ phase diagram to illustrate the potential effects of crystallization on the phase diagram in Fig. 4.

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*/εσ^{3} = 1, *b*/σ^{3} = 1, , *B*/*b* = 24.

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*/εσ^{3} = 1, *b*/σ^{3} = 1, , *B*/*b* = 24.

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