### Abstract

We study bi- and polydisperse mixtures of hard sphere fluids with extreme size ratios up to 100. Simulation results are compared with previously found analytical equations of state by looking at the compressibility factor, *Z*, and agreement is found with much better than 1% deviation in the fluid regime. A slightly improved empirical correction to *Z* is proposed. When the density is further increased, excluded volume becomes important, but there is still a close relationship between many-component mixtures and their binary, two-component equivalents (which are defined on basis of the first three moments of the size distribution). Furthermore, we determine the size ratios for which the liquid-solid transition exhibits crystalline, amorphous or mixed systemstructure. Near the jamming density, *Z* is independent of the size distribution and follows a −1 power law as function of the difference from the jamming density (*Z* → ∞). In this limit, *Z* depends only on one free parameter, the jamming density itself, as reported for several different size distributions with a wide range of widths.

The authors would like to thank Andrés Santos, Martin van der Hoef, Nicolás Rivas, Sebastián González, Fatih Göncü, and two anonymous reviewers for helpful suggestions, and Monica Skoge and Aleksandar Donev for consultations about the event-driven code. This research is supported by the Dutch Technology Foundation STW, which is the applied science division of NWO, and the Technology Programme of the Ministry of Economic Affairs, project Nr. STW-MUST 10120.

I. INTRODUCTION

II. THEORY

A. Bidisperse systems

B. Polydisperse systems

C. Discussion

III. COMPARISON WITH SIMULATIONS

A. Particle size distributions

B. Equation of state in the fluid regime

C. How much disorder is necessary to avoid order?

D. Bidisperse versus polydisperse

E. Towards the jamming density

F. Different growth rates

G. Super dense limit

IV. SUMMARY AND CONCLUSIONS

### Key Topics

- Crystallization
- 31.0
- Equations of state
- 30.0
- Colloidal systems
- 4.0
- Crystal structure
- 4.0
- Cumulative distribution functions
- 4.0

## Figures

The quality factor, i.e., the numerical *Z* scaled by the theoretical predictions for different densities and for different size distributions (a) monodisperse, and polydisperse with (b) uniform size, ω = 5, (c) uniform size, ω = 100, and (d) uniform volume, ω = 4. The growth rate for all data is Γ = 16 × 10^{−6}. The error bars indicate the standard deviation of the quality factor within an averaging bin. The error bars are shown only for the BMCSL EOS since in the other cases they have the same trend and magnitude.

The quality factor, i.e., the numerical *Z* scaled by the theoretical predictions for different densities and for different size distributions (a) monodisperse, and polydisperse with (b) uniform size, ω = 5, (c) uniform size, ω = 100, and (d) uniform volume, ω = 4. The growth rate for all data is Γ = 16 × 10^{−6}. The error bars indicate the standard deviation of the quality factor within an averaging bin. The error bars are shown only for the BMCSL EOS since in the other cases they have the same trend and magnitude.

The estimated jamming density ϕ_{ J } as a function of volume fraction ν for systems with uniform size distribution. Shown are systems of 4096 spheres with various size ratios ω. In (a) ω, given in the inset, corresponds to decreasing ϕ_{ J } (top-to-bottom) and in (b) increasing ω corresponds to bottom-to-top. Also plotted are the fluid theory (BMCSL EOS) and the approximation for the crystal phase.^{67} The used growth rate here is Γ = 8 × 10^{−6}. For ω = 1.2 data for two different runs are shown, marked with 1.2 and 1.2*.

The estimated jamming density ϕ_{ J } as a function of volume fraction ν for systems with uniform size distribution. Shown are systems of 4096 spheres with various size ratios ω. In (a) ω, given in the inset, corresponds to decreasing ϕ_{ J } (top-to-bottom) and in (b) increasing ω corresponds to bottom-to-top. Also plotted are the fluid theory (BMCSL EOS) and the approximation for the crystal phase.^{67} The used growth rate here is Γ = 8 × 10^{−6}. For ω = 1.2 data for two different runs are shown, marked with 1.2 and 1.2*.

The estimated jamming density ϕ_{ J } as a function of volume fraction ν for bidisperse systems corresponding to uniform size distribution. Increasing ω corresponds to bottom-to-top, with growth rate Γ = 16 × 10^{−6}. The (metastable) freezing density is decreasing with increasing size ratio. The BMCSL EOS are shown by dashed-dotted lines.

The estimated jamming density ϕ_{ J } as a function of volume fraction ν for bidisperse systems corresponding to uniform size distribution. Increasing ω corresponds to bottom-to-top, with growth rate Γ = 16 × 10^{−6}. The (metastable) freezing density is decreasing with increasing size ratio. The BMCSL EOS are shown by dashed-dotted lines.

Particle systems with uniform size distribution with different size ratios at densities very close to jamming. Size ratios are ω = 1 (a), ω = 1.12 (b), ω = 1.18 (c), and ω = 1.22 (d). The order-disorder transition can be clearly seen as the size ratio increases. Color is by relative size, i.e., yellow (light) corresponds to small particles and blue (dark) corresponds to big ones. The used growth rate to reach these configurations was Γ = 8 × 10^{−6}.

Particle systems with uniform size distribution with different size ratios at densities very close to jamming. Size ratios are ω = 1 (a), ω = 1.12 (b), ω = 1.18 (c), and ω = 1.22 (d). The order-disorder transition can be clearly seen as the size ratio increases. Color is by relative size, i.e., yellow (light) corresponds to small particles and blue (dark) corresponds to big ones. The used growth rate to reach these configurations was Γ = 8 × 10^{−6}.

Particle systems with bidisperse size distribution corresponding to a uniform size distribution, i.e., *n* _{1} = 1/2, with different size ratios at densities very close to jamming. Size ratios are *R* ^{bi} ≈ 1.100 (ω = 1.18) (a), *R* ^{bi} ≈ 1.111 (ω = 1.2) (b), *R* ^{bi} ≈ 2.060 (ω = 4) (c), and *R* ^{bi} ≈ 2.404 (ω = 6) (d). The order-disorder transition can be clearly seen. Note that in (d) also some signs of segregation/clusterization of small particles can be seen, though investigation of this is beyond the scope of this paper.

Particle systems with bidisperse size distribution corresponding to a uniform size distribution, i.e., *n* _{1} = 1/2, with different size ratios at densities very close to jamming. Size ratios are *R* ^{bi} ≈ 1.100 (ω = 1.18) (a), *R* ^{bi} ≈ 1.111 (ω = 1.2) (b), *R* ^{bi} ≈ 2.060 (ω = 4) (c), and *R* ^{bi} ≈ 2.404 (ω = 6) (d). The order-disorder transition can be clearly seen. Note that in (d) also some signs of segregation/clusterization of small particles can be seen, though investigation of this is beyond the scope of this paper.

The estimated jamming density ϕ_{ J } as a function of volume fraction ν for systems with (a) uniform size distribution and their bidisperse equivalents, using Γ = 8 × 10^{−6}, and for systems with (b) uniform volume distribution and their bidisperse equivalents, using Γ = 16 × 10^{−6}. Size ratios ω and *R* ^{bi} are displayed in the inset, where the latter is given in brackets. Also plotted are the fluid theory (BMCSL EOS) and the approximation for the crystal phase.^{67} Data for polydisperse systems in (b) are shown for ω ⩽ 4. In the inset of (b) the zoomed data for ω = 8 are shown.

The estimated jamming density ϕ_{ J } as a function of volume fraction ν for systems with (a) uniform size distribution and their bidisperse equivalents, using Γ = 8 × 10^{−6}, and for systems with (b) uniform volume distribution and their bidisperse equivalents, using Γ = 16 × 10^{−6}. Size ratios ω and *R* ^{bi} are displayed in the inset, where the latter is given in brackets. Also plotted are the fluid theory (BMCSL EOS) and the approximation for the crystal phase.^{67} Data for polydisperse systems in (b) are shown for ω ⩽ 4. In the inset of (b) the zoomed data for ω = 8 are shown.

The maximum density ν_{max} as a function of the inverse size ratio ω^{−1} for different size distributions and for different compression rates, in the inset subscript 1 corresponds to the reference growth rate Γ = 8 × 10^{−6} and subscript 2 corresponds to Γ = 16 × 10^{−6}, i.e., two times faster. Size distributions are (a) uniform size distribution (US) and their bidisperse equivalents (BUS) or (b) uniform volume (UV) and their bidisperse equivalents (BUV). The size ratio ω corresponds to the polydisperse systems, while bidisperse ones are constructed using C. Results for BUV systems with *N* = 8192, using Γ = 16 × 10^{−5}, are shown as pluses in (b), which are fitted for ω ⩾ 10 by Eq. (16) with swapped values of ϕ_{RCP} and (dash-dotted line). The left most point (+, ω = 30) is higher than expected because of partial crystallization, setting in also here at very large ω. Results from Ref. 83 obtained by compression of soft frictionless particles with a uniform size distribution are shown as crosses in (a). In the inset of (a), ν_{max} is plotted for different growth rates Γ for the uniform size distribution (ω = 2) and their bidisperse equivalent (*R* ^{bi} ≈ 1.48). In the inset of (b), the deviation of data from the fits is shown with corresponding symbols.

The maximum density ν_{max} as a function of the inverse size ratio ω^{−1} for different size distributions and for different compression rates, in the inset subscript 1 corresponds to the reference growth rate Γ = 8 × 10^{−6} and subscript 2 corresponds to Γ = 16 × 10^{−6}, i.e., two times faster. Size distributions are (a) uniform size distribution (US) and their bidisperse equivalents (BUS) or (b) uniform volume (UV) and their bidisperse equivalents (BUV). The size ratio ω corresponds to the polydisperse systems, while bidisperse ones are constructed using C. Results for BUV systems with *N* = 8192, using Γ = 16 × 10^{−5}, are shown as pluses in (b), which are fitted for ω ⩾ 10 by Eq. (16) with swapped values of ϕ_{RCP} and (dash-dotted line). The left most point (+, ω = 30) is higher than expected because of partial crystallization, setting in also here at very large ω. Results from Ref. 83 obtained by compression of soft frictionless particles with a uniform size distribution are shown as crosses in (a). In the inset of (a), ν_{max} is plotted for different growth rates Γ for the uniform size distribution (ω = 2) and their bidisperse equivalent (*R* ^{bi} ≈ 1.48). In the inset of (b), the deviation of data from the fits is shown with corresponding symbols.

The compressibility factor scaled by the free volume equation of state (14) in the limit of diverging pressure for systems with uniform size distribution for different size ratios, increasing ω corresponds to top-to-bottom, with growth rate Γ = 16 × 10^{−6}. When partial crystallization happens we see some fine structures as shown in the inset for the monodisperse system (ω = 1).

The compressibility factor scaled by the free volume equation of state (14) in the limit of diverging pressure for systems with uniform size distribution for different size ratios, increasing ω corresponds to top-to-bottom, with growth rate Γ = 16 × 10^{−6}. When partial crystallization happens we see some fine structures as shown in the inset for the monodisperse system (ω = 1).

## Tables

Given are size ratios ω and dimensionless moments *O* _{1} and *O* _{2} for a few polydisperse systems with uniform size (US) and uniform volume (UV) radii distributions and *R* ^{bi} with for their bidisperse equivalents.

Given are size ratios ω and dimensionless moments *O* _{1} and *O* _{2} for a few polydisperse systems with uniform size (US) and uniform volume (UV) radii distributions and *R* ^{bi} with for their bidisperse equivalents.

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