^{1}, G. F. Wang

^{1}and S. K. Lai

^{1,a)}

### Abstract

Treating the repulsive part of a pairwise potential by the hard-sphere form and its attractive part by the effective depletion potential form, we calculate using this model potential the colloidal domains of phase separation. Differing from the usual recipe of applying the thermodynamic conditions of equal pressure and equal chemical potential where the branches of coexisting phases are the ultimate target, we employ the free energy density minimization approach [G. F. Wang and S. K. Lai, Phys. Rev. E70, 051402 (2004)] to crosshatch the domains of equilibrium phases, which consist of the gas, liquid, and solid homogeneous phases as well as the coexistence of these phases. This numerical procedure is attractive since it yields naturally the colloidal volume of space occupied by each of the coexisting phases. In this work, we first examine the change in structures of the fluid and solid free energy density landscapes with the effective polymer concentration. We show by explicit illustration the link between the free energy density landscapes and the development of both the metastable and stable coexisting phases. Then, attention is paid to the spatial volumes predicted at the triple point. It is found here that the volumes of spaces of the three coexisting phases at the triple point vary one dimensionally, whereas for the two coexisting phases, they are uniquely determined.

This work was 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. The depletion potential

C. Helmholtz free energy densities of colloid-polymer mixtures: Fluid and solid phases

III. NUMERICAL RESULTS AND DISCUSSION

A. Phase-diagram domains and free energy density landscapes

IV. VOLUME PROPORTIONS AT TRIPLE POINT

V. CONCLUSION

### Key Topics

- Colloidal systems
- 45.0
- Free energy
- 31.0
- Polymers
- 14.0
- Spatial dimensions
- 10.0
- Liquid solid interfaces
- 9.0

## Figures

Theoretical phase-diagram domains of a colloid-polymer system calculated at a fixed size ratio . The reservoir polymer volume fraction is plotted against the colloidal volume fraction . FEDM was carried out for several initial colloidal volume fractions shown in the figure at , 0.35, 0.45, and 0.54. All vertical line-domains lead to the same phase boundaries of coexisting phases, which are displayed as (black) crosses for L1S2 and L2S2 and (green) pluses for L1L2. The horizontal lines indicate several selected , two above and two below the triple point . The latter FEDLs are depicted in Figs. 2(a)–2(e). The triple-point line is given by the full curve. Other notations used are as follows: solid circles, fluid; open circles, gas-liquid; open triangle up, liquid-solid; solid square, gas-solid; (blue) triangle left, continuation of Ll of L1S2 to spinodal decomposition curve (solid triangle down) at ; and (red) triangle right, continuation of L2 of L2S2 to spinodal decomposition curve (solid triangle down) at .

Theoretical phase-diagram domains of a colloid-polymer system calculated at a fixed size ratio . The reservoir polymer volume fraction is plotted against the colloidal volume fraction . FEDM was carried out for several initial colloidal volume fractions shown in the figure at , 0.35, 0.45, and 0.54. All vertical line-domains lead to the same phase boundaries of coexisting phases, which are displayed as (black) crosses for L1S2 and L2S2 and (green) pluses for L1L2. The horizontal lines indicate several selected , two above and two below the triple point . The latter FEDLs are depicted in Figs. 2(a)–2(e). The triple-point line is given by the full curve. Other notations used are as follows: solid circles, fluid; open circles, gas-liquid; open triangle up, liquid-solid; solid square, gas-solid; (blue) triangle left, continuation of Ll of L1S2 to spinodal decomposition curve (solid triangle down) at ; and (red) triangle right, continuation of L2 of L2S2 to spinodal decomposition curve (solid triangle down) at .

Free energy density vs colloidal volume fraction of colloid-polymer system calculated at a fixed size ratio . The dashed curve is the fluid free energy density , and the full curve is the solid free energy density . The reservoir polymer volume fractions are at (a) , (b) , (c) , (d) , and (e) . In all these figures, the (blue) dash-dotted line is L1L2, the (orange) dash-dot-dotted line is L1S2, and the (green) dash-dash-dotted line is L2S2. The triple points in (c) are denoted by solid circles.

Free energy density vs colloidal volume fraction of colloid-polymer system calculated at a fixed size ratio . The dashed curve is the fluid free energy density , and the full curve is the solid free energy density . The reservoir polymer volume fractions are at (a) , (b) , (c) , (d) , and (e) . In all these figures, the (blue) dash-dotted line is L1L2, the (orange) dash-dot-dotted line is L1S2, and the (green) dash-dash-dotted line is L2S2. The triple points in (c) are denoted by solid circles.

Schematic diagram depicting the geometric analysis of the spatial volumes of triple coexistence. The triangle shown as (orange) dashed lines shows the intersection of the equation of the plane on the axes , , and . The (red) solid line is the result of intersection of the (pink) plane with the (green) plane .

Schematic diagram depicting the geometric analysis of the spatial volumes of triple coexistence. The triangle shown as (orange) dashed lines shows the intersection of the equation of the plane on the axes , , and . The (red) solid line is the result of intersection of the (pink) plane with the (green) plane .

Schematic diagram of the Helmholtz free energy densities, (dashed line for fluid) and (solid line for solid), vs colloidal volume fraction . Note that two and three phases in coexistence are shown.

Schematic diagram of the Helmholtz free energy densities, (dashed line for fluid) and (solid line for solid), vs colloidal volume fraction . Note that two and three phases in coexistence are shown.

Spatial volumes of three coexisting phases for a colloid-polymer system at triple point (see Fig. 1). Straight lines on the triangular plane correspond to different initial volume fractions , and they are the results of the intersection of planes and . The (blue) thick solid line corresponds to , whereas the (red) solid circle and (red) open circle correspond to and , respectively. Other notations are as follows: open square, two coexisting phases L1S2; full square, liquid.

Spatial volumes of three coexisting phases for a colloid-polymer system at triple point (see Fig. 1). Straight lines on the triangular plane correspond to different initial volume fractions , and they are the results of the intersection of planes and . The (blue) thick solid line corresponds to , whereas the (red) solid circle and (red) open circle correspond to and , respectively. Other notations are as follows: open square, two coexisting phases L1S2; full square, liquid.

Pressure vs for the colloid-polymer system calculated at a fixed size ratio . Along L1S2, L2S2, and L1L2, two equilibrium phases coexist, and their respective and , where , are uniquely determined. The point where the L1S2, L2S2, and L1L2 meet is the triple point whose , , and take on innumerable values, all of which yield a same free energy density value.

Pressure vs for the colloid-polymer system calculated at a fixed size ratio . Along L1S2, L2S2, and L1L2, two equilibrium phases coexist, and their respective and , where , are uniquely determined. The point where the L1S2, L2S2, and L1L2 meet is the triple point whose , , and take on innumerable values, all of which yield a same free energy density value.

## Tables

Spatial volume parameter and composite free energy density [in units of ] (see text) for a colloid-polymer system characterized by an attractive depletion potential. The size ratio is fixed at , and the initial volume fraction is given by (blue thick line in Fig. 5).

Spatial volume parameter and composite free energy density [in units of ] (see text) for a colloid-polymer system characterized by an attractive depletion potential. The size ratio is fixed at , and the initial volume fraction is given by (blue thick line in Fig. 5).

Spatial volume parameter and composite free energy density [in units of ] (see text) for a colloid-polymer system characterized by an attractive depletion potential. The size ratio is fixed at , and the initial volume fraction is given by (red thin line in Fig. 5).

Spatial volume parameter and composite free energy density [in units of ] (see text) for a colloid-polymer system characterized by an attractive depletion potential. The size ratio is fixed at , and the initial volume fraction is given by (red thin line in Fig. 5).

Spatial volume parameter and composite free energy density [in units of ] (see text) for a colloid-polymer system characterized by an attractive depletion potential. The size ratio is fixed at , and the initial volume fraction is given by (orange thin line in Fig. 5).

Spatial volume parameter and composite free energy density [in units of ] (see text) for a colloid-polymer system characterized by an attractive depletion potential. The size ratio is fixed at , and the initial volume fraction is given by (orange thin line in Fig. 5).

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