^{1,a)}and Xuhui Xiao

^{1}

### Abstract

The semigrand ensemble theory [H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey, A. Stroobants, and P. B. Warren, Europhys. Lett.20, 559 (1992)] in conjunction with the fundamental measure density functional theory [V. B. Warshavsky and X. Song, Phys. Rev. E69, 061113 (2004)] are used to construct the Helmholtz free energy densities of a mixture of uncharged colloidal hard spheres and colloidal rods in its solid and liquid phases. Given these free energy density functions, we apply the free energy density minimization method [G. F. Wang and S. K. Lai, Phys. Rev. E70, 051402 (2004)] to crosshatch the system’s regions of phases in coexistence. The calculated results show that the triangular area bounded by gas-liquid, gas-solid, and liquid-solid coexisting two phases which has been called the coexistence region of gas-liquid-solid corresponds in fact to sets of two phases in coexistence. The phase boundaries which define our calculated coexistence domains compare very well with previous theoretical calculations. The relevance of the phase-diagram domains to three phases in coexistence will be discussed.

This work is supported by the National Science Council, Taiwan (Grant No. NSC96-2112-M-008-018-MY3).

I. INTRODUCTION

II. THEORY

A. Free energy density minimization

B. Free energy densities for fluid and solid phases

C. Hard-sphere fluid and solid phases: fundamental measure density functional theory

III. RESULTS AND DISCUSSIONS

IV. CONCLUSIONS

### Key Topics

- Colloidal systems
- 82.0
- Free energy
- 29.0
- Liquid solid interfaces
- 17.0
- Chemical potential
- 16.0
- Helmholtz free energy
- 16.0

## Figures

Domains of reservoir volume fraction scaled by multiplication with (i.e., ) of colloidal rods vs colloidal hard spheres for the colloid-rod system. From top to bottom corresponds to (a) ; (b) ; (c) ; (d) ; (e) , respectively. Notations used are: gas-solid, red; fluid, black; liquid-solid, green; solid, blue; gas-liquid, yellow; kinetic phase separation region, gray. The solid triangle sits on the boundary between gas-liquid (yellow) and fluid (black). The asterisk symbol in the kinetic phase transition region is the initial volume fractions and of colloidal hard spheres and colloidal rods, respectively. In (e), the free energy density minimization yields two coexisting phases, one in the gas-liquid coexistence domain (left open circle) and another in the gas-solid coexistence domain (right open circle). The former is further phase-separated into gas-liquid (solid squares) and the latter into gas-solid (full triangles).

Domains of reservoir volume fraction scaled by multiplication with (i.e., ) of colloidal rods vs colloidal hard spheres for the colloid-rod system. From top to bottom corresponds to (a) ; (b) ; (c) ; (d) ; (e) , respectively. Notations used are: gas-solid, red; fluid, black; liquid-solid, green; solid, blue; gas-liquid, yellow; kinetic phase separation region, gray. The solid triangle sits on the boundary between gas-liquid (yellow) and fluid (black). The asterisk symbol in the kinetic phase transition region is the initial volume fractions and of colloidal hard spheres and colloidal rods, respectively. In (e), the free energy density minimization yields two coexisting phases, one in the gas-liquid coexistence domain (left open circle) and another in the gas-solid coexistence domain (right open circle). The former is further phase-separated into gas-liquid (solid squares) and the latter into gas-solid (full triangles).

Same as Figs. 1(a)–1(e), but for system’s volume fraction also scaled by multiplication with . Note the triangular gas-liquid-solid area. Our free energy density minimization shows that initial concentrations of colloidal hard spheres and colloidal rods phase-separated into two sets of two phases in coexistence. Figures (a)–(e) are one-to-one correspondence to Figs. 1(a)–1(e) and should therefore be read in parallel. The orange crosses inside the triangle are the initial volume fractions used in the minimization of in Eq. (19). All of the values of yield the same vertices of the triangle (numerical details are given in Table III) but with different , (see text). The inset in (e) enlarges the region containing phase transition points.

Same as Figs. 1(a)–1(e), but for system’s volume fraction also scaled by multiplication with . Note the triangular gas-liquid-solid area. Our free energy density minimization shows that initial concentrations of colloidal hard spheres and colloidal rods phase-separated into two sets of two phases in coexistence. Figures (a)–(e) are one-to-one correspondence to Figs. 1(a)–1(e) and should therefore be read in parallel. The orange crosses inside the triangle are the initial volume fractions used in the minimization of in Eq. (19). All of the values of yield the same vertices of the triangle (numerical details are given in Table III) but with different , (see text). The inset in (e) enlarges the region containing phase transition points.

(a) Coexistence phase boundaries of Fig. 1(e) reproduced for the convenience of discussion. Two initial reservoir volume fractions, and (9.01, 0.2) are selected. (b) Same as (a) but for the system’s volume fraction . The two cases (solid triangle) in (a) in this plot indicate the tie lines where phase separations occur. Along and for any inside the triangular area, there is an ambiguity if one were to use tie line to recover corresponding boundary points in Fig. 3(a). Our minimization of free energy density at and for any inside the triangular area yields domain points that lie in the gray region in Fig. 1(e).

(a) Coexistence phase boundaries of Fig. 1(e) reproduced for the convenience of discussion. Two initial reservoir volume fractions, and (9.01, 0.2) are selected. (b) Same as (a) but for the system’s volume fraction . The two cases (solid triangle) in (a) in this plot indicate the tie lines where phase separations occur. Along and for any inside the triangular area, there is an ambiguity if one were to use tie line to recover corresponding boundary points in Fig. 3(a). Our minimization of free energy density at and for any inside the triangular area yields domain points that lie in the gray region in Fig. 1(e).

## Tables

Minimized free energy density calculated at initial volume fractions [solid triangle in Fig. 1(d)] of colloidal rods and colloidal hard spheres . The , are volume fractions of phase-separated colloids, and are their corresponding Helmholtz free energy densities and ratio of spatial volumes to total volume, respectively.

Minimized free energy density calculated at initial volume fractions [solid triangle in Fig. 1(d)] of colloidal rods and colloidal hard spheres . The , are volume fractions of phase-separated colloids, and are their corresponding Helmholtz free energy densities and ratio of spatial volumes to total volume, respectively.

Initial volume fractions of reservoir colloidal rods at and colloidal hard spheres at . The free energy density minimization yields first the metastable liquid at coexisting with a solid at . The former subsequently phase-separates into equilibrium a gas phase at and a liquid phase at , whereas the latter phase-separates instead into equilibrium a gas phase at and a solid phase at . The variable is the ratio of the spatial volume of phase to the total volume and is the free energy density of phase.

Initial volume fractions of reservoir colloidal rods at and colloidal hard spheres at . The free energy density minimization yields first the metastable liquid at coexisting with a solid at . The former subsequently phase-separates into equilibrium a gas phase at and a liquid phase at , whereas the latter phase-separates instead into equilibrium a gas phase at and a solid phase at . The variable is the ratio of the spatial volume of phase to the total volume and is the free energy density of phase.

Triangular region of three-phase coexistence bounded by gas-liquid, gas-solid, and liquid-solid for a mixture of colloidal hard spheres and colloidal hard rods. The vertices of the triangle are: gas at , liquid at , and solid at . The spatial volume is the ratio of of phase to the total volume . Minimization of [Eq. (19) in text] subject to the conservation constraints given by Eqs. (20a)–(20c) for each of the initial volume fractions yields the same vertices of the triangle, whereas for initial volumes , (0.32, 4.2089), and (0.51, 0.5854) on the sides of the triangles, we obtain the gas-liquid with , gas-solid with , and liquid-solid with , respectively.

Triangular region of three-phase coexistence bounded by gas-liquid, gas-solid, and liquid-solid for a mixture of colloidal hard spheres and colloidal hard rods. The vertices of the triangle are: gas at , liquid at , and solid at . The spatial volume is the ratio of of phase to the total volume . Minimization of [Eq. (19) in text] subject to the conservation constraints given by Eqs. (20a)–(20c) for each of the initial volume fractions yields the same vertices of the triangle, whereas for initial volumes , (0.32, 4.2089), and (0.51, 0.5854) on the sides of the triangles, we obtain the gas-liquid with , gas-solid with , and liquid-solid with , respectively.

Comparison of the predicted rod reservoir volume fraction for an initially fixed coordinates and . Two cases are considered: one for [Fig. 1(a)] and another [Fig. 1(d)]. Both cases are set at .

Comparison of the predicted rod reservoir volume fraction for an initially fixed coordinates and . Two cases are considered: one for [Fig. 1(a)] and another [Fig. 1(d)]. Both cases are set at .

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