Equation of state of polydisperse hard-disk mixtures in the high-density regime

A proposal to link the equation of state of a monocomponent hard-disk fluid to the equation of state of a polydisperse hard-disk mixture is presented. Event-driven molecular dynamics simulations are performed to obtain data for the compressibility factor of the monocomponent fluid and of 26 polydisperse mixtures with different size distributions. Those data are used to assess the proposal and to infer the values of the compressibility factor of the monocomponent hard-disk fluid in the metastable region from those of mixtures in the high-density region. The collapse of the curves for the different mixtures is excellent in the stable region. In the metastable regime, except for two mixtures in which crystallization is present, the outcome of the approach exhibits a rather good performance. The simulation results indicate that a (reduced) variance of the size distribution larger than about 0.01 is sufficient to avoid crystallization and explore the metastable fluid branch.

Figure 1

Schematical view of the polydisperse $\leftrightarrow$ monocomponent mapping represented by Eqs. (1, 2, 3, 4).

Figure 2

Scatter plot of ${\overline{B}}_3^{\text{app}}/b_3$ vs ${\overline{B}}_3/b_3$ for the 26 mixtures considered in Table 1. The straight dashed line represents the equality ${\overline{B}}_3^{\text{app}}/b_3={\overline{B}}_3/b_3$. The inset shows a scatter plot of $\lambda$ vs ${\overline{B}}_3/b_3$.

Figure 3

Plot of $Z\left(\varphi \right)$ as obtained from our simulations for the HD mixture B5. The curves correspond to $\mathcal{N}=2^{12}$ (red), $\mathcal{N}=2^{13}$ (blue), and $\mathcal{N}=2^{14}$ (black), with growth rates $\mathrm{\Gamma }=1.6\times {10}^{-3}$ (dotted), $\mathrm{\Gamma }=1.6\times {10}^{-4}$ (dash-dotted), and $\mathrm{\Gamma }=1.6\times {10}^{-5}$ (solid). Note that the case $\mathcal{N}=2^{14}$ with $\mathrm{\Gamma }=1.6\times {10}^{-3}$ has not been included.

Figure 4

(a) Plot of $Z\left(\varphi \right)$ as obtained from our simulations for the monocomponent HD fluid and for various polydisperse HD mixtures. (b) Plot of the inferred monocomponent compressibility factor $Z_s\left({\varphi }_{\text{eff}}\right)$ as obtained from Eqs. (2) and (14), complemented by Eqs. (11) and (15). The thick black solid line corresponds to the monocomponent system, while the thin red (B1–B8), blue (TH3–TH11), and green (IP1–IP7) solid lines correspond to polydisperse systems. The two thick blue dashed lines refer to mixtures TH1 and TH2. Note the tiny error bars in the curves B4–B8 of panel (a).

Figure 5

Plot of the inferred monocomponent compressibility factor $Z_s\left(\varphi \right)$ as obtained from Eq. (19). The meaning of the curves is the same as in Fig. 4.

Figure 6

Plot of the compressibility factor for the mixture B8 (black solid line) and for the inferred monocomponent compressibility factor as obtained from Eq. (14) (blue solid line) and from Eq. (19) (dotted red line). The circles joined by straight lines denote pairs $({\varphi }_{\text{eff}},\varphi )$ as given by Eq. (2).

Figure 7

Plot of the inferred monocomponent compressibility factor $Z_s\left(\varphi \right)$ as obtained from Eq. (21) with ${\overline{B}}_3$ given by Eq. (11). The meaning of the curves is the same as in Fig. 4.

Figure 8

Plot of our MD results for the monocomponent compressibility factor $Z_s^{\text{in}}\left(\varphi \right)$ (—$\circ$—, black) and the output functions $Z_s^{\text{out}}\left(\varphi \right)$ obtained from $Z_s^{\text{in}}\left(\varphi \right)$ by using Eq. (28) with $\lambda =1.5$ (–$·$–$·$–, dark yellow), $\lambda =2$ (– – –, red), and $\lambda =3$ ($\cdots$, blue). The (black) thin solid line represents the function $Z_s^{\text{SPT}}\left(\varphi \right)$. The inset shows the differences $Z_s\left(\varphi \right)-Z_s^{\text{SPT}}\left(\varphi \right)$.