^{1}, Deepti Ballal

^{1}and Walter G. Chapman

^{1,a)}

### Abstract

We apply Wertheim's theory to develop an equation of state for one site patchy colloids where the patch can bond multiple times. We allow for the possibility of ring formation without the introduction of empirical parameters and show that for moderate patch coverage the infinite series of chain graphs is well represented by the first two terms. The theory is found to be in excellent agreement with new NVT and NPT Monte Carlo simulations. The approach described here can easily be converted to the form of a density functional theory to describe inhomogeneous patchy colloid systems.

The financial support for this work was provided by the Robert A. Welch Foundation (Grant No. C-1241) and by the National Science Foundation (CBET – 0756166).

I. INTRODUCTION

II. THEORY

III. SIMULATIONS

IV. RESULTS

V. CONCLUSIONS

### Key Topics

- Colloidal systems
- 50.0
- Perturbation theory
- 16.0
- Density functional theory
- 11.0
- Correlation functions
- 5.0
- Equations of state
- 5.0

##### B01J13/00

## Figures

Diagram of interacting colloids.

Diagram of interacting colloids.

Diagram of three interacting colloids where colloids 1 and 3 are both in contact with colloid 2.

Diagram of three interacting colloids where colloids 1 and 3 are both in contact with colloid 2.

Comparison of theoretical (current theory – solid lines, theory of Kalyuzhnyi et al. ^{13} – dashed lines), and simulation (symbols) predictions for the monomer fractions X o (blue), fraction of colloids bonded once (red) and fraction bonded twice (green) at packing fractions of η = 0.2 top and η = 0.4 bottom; each with α c = 35°.

Comparison of theoretical (current theory – solid lines, theory of Kalyuzhnyi et al. ^{13} – dashed lines), and simulation (symbols) predictions for the monomer fractions X o (blue), fraction of colloids bonded once (red) and fraction bonded twice (green) at packing fractions of η = 0.2 top and η = 0.4 bottom; each with α c = 35°.

Fractions bonded twice in a ring X ring (solid lines – theory, filled symbols – simulation) and twice in a chain X 2c (dashed line – theory, open symbols – simulation) at packing fractions of η = 0.2 (red) and η = 0.4 (black); for each case α c = 35°.

Fractions bonded twice in a ring X ring (solid lines – theory, filled symbols – simulation) and twice in a chain X 2c (dashed line – theory, open symbols – simulation) at packing fractions of η = 0.2 (red) and η = 0.4 (black); for each case α c = 35°.

Energy per colloid E* = E/Nk B T at a critical angle α c = 35° and packing fractions η = 0.2 (top) and η = 0.4 (bottom). Symbols are simulation results, solid line is from the current theory Eq. (38) , long dashed line is predictions from the theory of Kalyuzhnyi et al., ^{13} and short dashed line is TPT1 predictions.

Energy per colloid E* = E/Nk B T at a critical angle α c = 35° and packing fractions η = 0.2 (top) and η = 0.4 (bottom). Symbols are simulation results, solid line is from the current theory Eq. (38) , long dashed line is predictions from the theory of Kalyuzhnyi et al., ^{13} and short dashed line is TPT1 predictions.

Energy per colloid E* = E/Nk B T as a function of critical angle α c . The packing fraction is fixed at η = 0.3, symbols give simulation results, solid curves give current theory predictions Eq. (38) , and dashed curves are TPT1 predictions. Color code: red represents ɛ* = 5 and black represents ɛ* = 3. Inset gives the simulated fraction of colloids bonded 3 times X 3.

Energy per colloid E* = E/Nk B T as a function of critical angle α c . The packing fraction is fixed at η = 0.3, symbols give simulation results, solid curves give current theory predictions Eq. (38) , and dashed curves are TPT1 predictions. Color code: red represents ɛ* = 5 and black represents ɛ* = 3. Inset gives the simulated fraction of colloids bonded 3 times X 3.

Change in pressure due to association ΔP* for α c = 35° (left) and α c = 45° (right). Symbols give NPT simulation results, solid curves give theoretical predictions from the current work and dashed curves give theoretical predictions using the theory of Ref. 13 .

Change in pressure due to association ΔP* for α c = 35° (left) and α c = 45° (right). Symbols give NPT simulation results, solid curves give theoretical predictions from the current work and dashed curves give theoretical predictions using the theory of Ref. 13 .

## Tables

Numerical results for geometric integrals Γ and Ψ and angles and .

Numerical results for geometric integrals Γ and Ψ and angles and .

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