^{1,a)}, Domenico Gazzillo

^{1,b)}, Achille Giacometti

^{1,c)}and Peter Sollich

^{2,d)}

### Abstract

We study the effects of size polydispersity on the gas-liquid phase behavior of mixtures of sticky hard spheres. To achieve this, the system of coupled quadratic equations for the contact values of the partial cavity functions of the Percus-Yevick solution [R. J. Baxter, J. Chem. Phys.49, 2770 (1968)] is solved within a perturbation expansion in the polydispersity, i.e., the normalized width of the size distribution. This allows us to make predictions for various thermodynamic quantities which can be tested against numerical simulations and experiments. In particular, we determine the leading order effects of size polydispersity on the cloud curve delimiting the region of two-phase coexistence and on the associated shadow curve; we also study the extent of size fractionation between the coexisting phases. Different choices for the size dependence of the adhesion strengths are examined carefully; the Asakura-Oosawa model [J. Chem. Phys.22, 1255 (1954)] of a mixture of polydisperse colloids and small polymers is studied as a specific example.

I. INTRODUCTION

II. THE SHS MODEL

III. THE STICKINESS COEFFICIENTS

A. General arguments

B. Stickiness coefficients for the Asakura-Oosawa model

IV. PERTURBATION THEORY FOR THE POLYDISPERSE PY CLOSURE

A. Evans’ perturbative expansion

B. Perturbative analysis of the PY closure

C. Excess free energy

V. PHASE BEHAVIOR

A. PY closure

B. Other approximation schemes

VI. CONCLUSIONS

### Key Topics

- Polymers
- 35.0
- Free energy
- 31.0
- Colloidal systems
- 24.0
- Critical point phenomena
- 16.0
- Adhesion
- 13.0

## Figures

Equation of state, from the energy route, for a one-component fluid of SHS. From left to right and top to bottom the four panels refer to a reduced temperature of , 0.50, 0.20, and 0.15, respectively. The continuous line corresponds to the MSA approximation, the dotted line to the mMSA approximation, the short dashed line to the C1 approximation, the long dashed line to the PY approximation, the dot-dashed line to the WCA first order perturbation theory, squares to the WCA second order perturbation theory (with error bars indicating the range where the true value should lie with probability of 99.7%), and triangles to the MC simulations of Miller and Frenkel (Ref. 20). In all cases the HS component of the pressure was chosen to be the one obtained from the compressibility route of the PY approximation (Ref. 39).

Equation of state, from the energy route, for a one-component fluid of SHS. From left to right and top to bottom the four panels refer to a reduced temperature of , 0.50, 0.20, and 0.15, respectively. The continuous line corresponds to the MSA approximation, the dotted line to the mMSA approximation, the short dashed line to the C1 approximation, the long dashed line to the PY approximation, the dot-dashed line to the WCA first order perturbation theory, squares to the WCA second order perturbation theory (with error bars indicating the range where the true value should lie with probability of 99.7%), and triangles to the MC simulations of Miller and Frenkel (Ref. 20). In all cases the HS component of the pressure was chosen to be the one obtained from the compressibility route of the PY approximation (Ref. 39).

The overlap volume of the two exclusion zones around colloid particles of diameter and which cannot be accessed by polymers of diameter .

The overlap volume of the two exclusion zones around colloid particles of diameter and which cannot be accessed by polymers of diameter .

Phase diagram of the monodisperse SHS fluid obtained with the PY closure and the energy route to thermodynamics. Shown are the binodal and spinodal curves and the region where the PY equation has no solution [see Eq. (25)].

Phase diagram of the monodisperse SHS fluid obtained with the PY closure and the energy route to thermodynamics. Shown are the binodal and spinodal curves and the region where the PY equation has no solution [see Eq. (25)].

Pressure from the energy route of the PY approximation for a single (parent) phase with case IV stickiness coefficients, plotted against volume fraction. Results are shown for several small values of the polydispersity (see legend) and well above, just above, and below (from left to right) the critical point of the monodisperse system. The pressure was determined using Eq. 9 of Ref. 22.

Pressure from the energy route of the PY approximation for a single (parent) phase with case IV stickiness coefficients, plotted against volume fraction. Results are shown for several small values of the polydispersity (see legend) and well above, just above, and below (from left to right) the critical point of the monodisperse system. The pressure was determined using Eq. 9 of Ref. 22.

Cloud and shadow curves for SHS mixtures with polydispersity , as obtained within the PY approximation and the energy route to thermodynamics, for coefficients chosen according to cases II and IV from Eq. (5). The shifts from the binodal of the monodisperse system (labeled “mono”) were calculated using Eq. (15) and give the leading corrections in a perturbative treatment of polydispersity. Note the collapse of the cloud and shadow curves, as expected from this order of the perturbation theory for purely size-polydisperse models (Refs. 32 and 33), and the divergence of the perturbation theory at the monodisperse critical point.

Cloud and shadow curves for SHS mixtures with polydispersity , as obtained within the PY approximation and the energy route to thermodynamics, for coefficients chosen according to cases II and IV from Eq. (5). The shifts from the binodal of the monodisperse system (labeled “mono”) were calculated using Eq. (15) and give the leading corrections in a perturbative treatment of polydispersity. Note the collapse of the cloud and shadow curves, as expected from this order of the perturbation theory for purely size-polydisperse models (Refs. 32 and 33), and the divergence of the perturbation theory at the monodisperse critical point.

Cloud and shadow curves for the SHS model with polydispersity and case V stickiness coefficients. The binodal of the monodisperse system is shown for comparison.

Cloud and shadow curves for the SHS model with polydispersity and case V stickiness coefficients. The binodal of the monodisperse system is shown for comparison.

Cloud and shadow curves for the SHS model with polydispersity and case I stickiness coefficients. The binodal of the monodisperse system is shown for comparison.

Cloud and shadow curves for the SHS model with polydispersity and case I stickiness coefficients. The binodal of the monodisperse system is shown for comparison.

Cloud and shadow curves for the AO model with polymer-to-colloid size ratio and (colloid) polydispersity . The binodal of the monodisperse system is shown for comparison.

Cloud and shadow curves for the AO model with polymer-to-colloid size ratio and (colloid) polydispersity . The binodal of the monodisperse system is shown for comparison.

Fractionation in SHS mixtures with stickiness coefficients chosen according to cases II and I, at and for polydispersities as in the corresponding Figs. 5 and 7. Shown are the cloud (parent) size distribution , taken to be of the Schulz form, and the size distributions in the liquid shadow and gas shadow phases that form when coexistence is approached from low densities (gas cloud phase) and high densities (liquid cloud phase), respectively. For case II (main graph) the larger particles tend to accumulate in the liquid phase, while for case I (inset) the opposite is true.

Fractionation in SHS mixtures with stickiness coefficients chosen according to cases II and I, at and for polydispersities as in the corresponding Figs. 5 and 7. Shown are the cloud (parent) size distribution , taken to be of the Schulz form, and the size distributions in the liquid shadow and gas shadow phases that form when coexistence is approached from low densities (gas cloud phase) and high densities (liquid cloud phase), respectively. For case II (main graph) the larger particles tend to accumulate in the liquid phase, while for case I (inset) the opposite is true.

Decomposition of the difference in between gas and liquid phases. The two contributions and are plotted separately against ; the latter quantity is graphed on the vertical rather than the horizontal axis for ease of comparison with Figs. 5–8. Inset: ratio of .

Decomposition of the difference in between gas and liquid phases. The two contributions and are plotted separately against ; the latter quantity is graphed on the vertical rather than the horizontal axis for ease of comparison with Figs. 5–8. Inset: ratio of .

Cloud and shadow curves for case II stickiness coefficients and with polydispersity , calculated using the BCMSL-type free energy [Eq. (29)] rather than the PY approximation, as in Fig. 5. The binodal of the monodisperse system, which differs from the PY result, is shown for comparison. Main graph: region around the critical point. Inset: global view of the results on the same scale as in Fig. 5.

Cloud and shadow curves for case II stickiness coefficients and with polydispersity , calculated using the BCMSL-type free energy [Eq. (29)] rather than the PY approximation, as in Fig. 5. The binodal of the monodisperse system, which differs from the PY result, is shown for comparison. Main graph: region around the critical point. Inset: global view of the results on the same scale as in Fig. 5.

Comparison of predictions for the AO model with polymer-to-colloid size ratio . Left: results of SHS mapping analyzed within the PY approximation; as in Fig. 8 cloud and shadow curves are shown for colloid polydispersity , along with the monodisperse binodal for comparison. The vertical axis now shows the polymer volume fraction rather than the reduced temperature . Right: analogous results obtained from free volume theory. Inset, right: fractionation coefficient for the two approximation schemes.

Comparison of predictions for the AO model with polymer-to-colloid size ratio . Left: results of SHS mapping analyzed within the PY approximation; as in Fig. 8 cloud and shadow curves are shown for colloid polydispersity , along with the monodisperse binodal for comparison. The vertical axis now shows the polymer volume fraction rather than the reduced temperature . Right: analogous results obtained from free volume theory. Inset, right: fractionation coefficient for the two approximation schemes.

## Tables

Coefficient of the perturbative expansion (6) of the adhesion parameters for the four cases listed in Eq. (5).

Coefficient of the perturbative expansion (6) of the adhesion parameters for the four cases listed in Eq. (5).

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