^{1}and Marisol Ripoll

^{1,a)}

### Abstract

The thermophoretic behavior of concentrated colloidalsuspensions can be understood as the sum of single particle and collective effects. Here, we present a simulation model to investigate the particularities of the collective thermodiffusive effects in concentrated uncharged solutions, where the influence of different colloid-colloid interactions is analyzed. The concentration dependence found in our simulations qualitatively agrees with experimental results. Colloids with repulsive interactions are found to accumulate more effectively than the solvent in the warm areas, such that the corresponding Soret coefficients are negative and decrease with increasing concentration. The accumulation of colloids in the cold regions is facilitated by attraction, such that colloids with attractive interactions have larger values of the Soret coefficient. A thermodynamic argument that explains our results from equilibrium quantities is discussed as well.

The authors want to thank helpful discussions with Simone Wiegand, Gerrit A. Vliegenthart, Mingcheng Yang, and Gerhard Gompper. We also want to acknowledge S. Wiegand for making directly available to us experimental data in Ref. 22 and to G. A. Vliegenthart for proofreading the paper.

I. INTRODUCTION

II. THEORY

III. EXPERIMENTAL RESULTS

IV. SIMULATION MODEL

A. MPC solvent

B. Colloid-solvent interactions

C. Colloid-colloid interactions

D. Temperature gradient establishment

E. Determination of the Soret coefficient

F. Condensation effects

V. SIMULATION RESULTS

A. Limiting cases

1. Limit of infinite dilution

2. Closed packed limit

B. Effect of repulsion and repulsion softness

C. Effect of attraction and attraction range

D. Effect of the attraction strength

VI. ANALYTICAL APPROACH

VII. DISCUSSION

VIII. SUMMARY AND CONCLUSIONS

### Key Topics

- Colloidal systems
- 114.0
- Solvents
- 37.0
- Suspensions
- 14.0
- Thermal diffusion
- 13.0
- Equations of state
- 7.0

##### B01D17/00

##### B01J13/00

## Figures

(a) Soret coefficient for octadecyl coated silica particles in toluene as a function of the volume fraction ϕ for different values of the average temperature. Solid lines are a fit to Eq. (5) for low ϕ, and dashed lines a fit to Eq. (7) for high ϕ. (b) Values of the Soret coefficient at the limit of zero volume fraction, and the single particle contribution as a function of the temperature, obtained from the fit of the data to Eq. (5). (c) Temperature dependence contribution of *S* _{ T } at high ϕ values are obtained from a fit of the data to Eq. (7). (a) Reprinted with permission from J. Chem. Phys. **125**, 204911 (2006). Copyright 2006 American Institute of Physics.

(a) Soret coefficient for octadecyl coated silica particles in toluene as a function of the volume fraction ϕ for different values of the average temperature. Solid lines are a fit to Eq. (5) for low ϕ, and dashed lines a fit to Eq. (7) for high ϕ. (b) Values of the Soret coefficient at the limit of zero volume fraction, and the single particle contribution as a function of the temperature, obtained from the fit of the data to Eq. (5). (c) Temperature dependence contribution of *S* _{ T } at high ϕ values are obtained from a fit of the data to Eq. (7). (a) Reprinted with permission from J. Chem. Phys. **125**, 204911 (2006). Copyright 2006 American Institute of Physics.

Displaced Soret coefficient as a function ϕ, corresponding to the collective contribution. Lines correspond to those in Fig. 1. The arrow indicates increasing colloid-colloid attraction. (Inset) Difference between the “zero concentration extrapolated Soret coefficients” at low concentration in Eq. (6) and high concentration in Eq. (7).

Displaced Soret coefficient as a function ϕ, corresponding to the collective contribution. Lines correspond to those in Fig. 1. The arrow indicates increasing colloid-colloid attraction. (Inset) Difference between the “zero concentration extrapolated Soret coefficients” at low concentration in Eq. (6) and high concentration in Eq. (7).

Profiles for concentrated colloids suspensions interacting with the LJ *n* = 6 potential. Crosses indicate the temperature profile with values in the right axis. The left axis quantifies the normalized relative density of colloids (down triangles), solvent particles (up triangles), and the normalized relative molar fraction of colloids (bullets). Lines correspond to the linear fits to determine the gradients of temperature and molar fraction. (a) ϕ_{ c } = 0.2 as example of a positive *S* _{ T } with colloids excess on the cold side. (b) ϕ_{ c } = 0.3 as example of a negative *S* _{ T }.

Profiles for concentrated colloids suspensions interacting with the LJ *n* = 6 potential. Crosses indicate the temperature profile with values in the right axis. The left axis quantifies the normalized relative density of colloids (down triangles), solvent particles (up triangles), and the normalized relative molar fraction of colloids (bullets). Lines correspond to the linear fits to determine the gradients of temperature and molar fraction. (a) ϕ_{ c } = 0.2 as example of a positive *S* _{ T } with colloids excess on the cold side. (b) ϕ_{ c } = 0.3 as example of a negative *S* _{ T }.

Temperature gradient induced phase separation of colloids in liquid-gas coexistence regime below critical point with LJ 12-6, ε = 1.0, ϕ_{ c } = 0.1, and *T* _{0} = 1.0. The volume fraction on the cold side of the system is ϕ_{ c } ≃ 0.37 corresponding to a liquid state. On the hot side, ϕ_{ c } ≃ 0.03 corresponding to a gas phase.

Temperature gradient induced phase separation of colloids in liquid-gas coexistence regime below critical point with LJ 12-6, ε = 1.0, ϕ_{ c } = 0.1, and *T* _{0} = 1.0. The volume fraction on the cold side of the system is ϕ_{ c } ≃ 0.37 corresponding to a liquid state. On the hot side, ϕ_{ c } ≃ 0.03 corresponding to a gas phase.

Soret coefficient *S* _{ T } for different mean volume fractions ϕ_{ c } for colloids with rLJ potentials in Eq. (10) with *n* = 6 bullets, *n* = 12 up-triangles, and *n* = 24 down-triangles. The inset displays the employed potentials.

Soret coefficient *S* _{ T } for different mean volume fractions ϕ_{ c } for colloids with rLJ potentials in Eq. (10) with *n* = 6 bullets, *n* = 12 up-triangles, and *n* = 24 down-triangles. The inset displays the employed potentials.

Soret coefficients *S* _{ T } as a function of the volume fraction ϕ_{ c } for LJ potentials in Eq. (10) with *n* = 6 squares, *n* = 12 up-triangles. The inset displays the employed potentials.

Soret coefficients *S* _{ T } as a function of the volume fraction ϕ_{ c } for LJ potentials in Eq. (10) with *n* = 6 squares, *n* = 12 up-triangles. The inset displays the employed potentials.

Soret coefficients *S* _{ T } as a function of the volume fraction ϕ_{ c } at various attraction ranges, squares *r* _{ c } = 1.2, up-triangles *r* _{ c } = 1.1, and bullets *r* _{ c } = 1.0365. Arrows indicate increasing attraction ranges. The attraction strength are (a) ε = 2.0, (b) ε = 1.0, and (c) ε = 0.5. The inset in (b) is an example of employed sticky potentials in Eq. (13).

Soret coefficients *S* _{ T } as a function of the volume fraction ϕ_{ c } at various attraction ranges, squares *r* _{ c } = 1.2, up-triangles *r* _{ c } = 1.1, and bullets *r* _{ c } = 1.0365. Arrows indicate increasing attraction ranges. The attraction strength are (a) ε = 2.0, (b) ε = 1.0, and (c) ε = 0.5. The inset in (b) is an example of employed sticky potentials in Eq. (13).

Soret coefficients *S* _{ T } as a function of the volume fraction ϕ_{ c } at various attraction ranges, squares ε = 2.0, up-triangles ε = 1.0, and bullets ε = 0.5. Arrows indicate increasing attraction strength. The attraction strength are (a) *r* _{ c } = 1.2, (b) *r* _{ c } = 1.1, and (c) *r* _{ c } = 1.0365. The inset in (b) is an example of employed sticky potentials in Eq. (13).

Soret coefficients *S* _{ T } as a function of the volume fraction ϕ_{ c } at various attraction ranges, squares ε = 2.0, up-triangles ε = 1.0, and bullets ε = 0.5. Arrows indicate increasing attraction strength. The attraction strength are (a) *r* _{ c } = 1.2, (b) *r* _{ c } = 1.1, and (c) *r* _{ c } = 1.0365. The inset in (b) is an example of employed sticky potentials in Eq. (13).

Soret coefficient *S* _{ T } as a function of the volume fraction ϕ_{ c }. Symbols are simulation results with rLJ *n* = 6 as displayed in Fig. 5. Dashed line corresponds to Eq. (19) with β_{ T, c } in Eq. (20). Solid line includes the additional contribution .

Soret coefficient *S* _{ T } as a function of the volume fraction ϕ_{ c }. Symbols are simulation results with rLJ *n* = 6 as displayed in Fig. 5. Dashed line corresponds to Eq. (19) with β_{ T, c } in Eq. (20). Solid line includes the additional contribution .

## Tables

Parameters employed for the sticky potentials in Eq. (13).

Parameters employed for the sticky potentials in Eq. (13).

Values of the Soret coefficient for the MPC binary mixture at various concentrations, and for two potential interactions at very low concentrations.

Values of the Soret coefficient for the MPC binary mixture at various concentrations, and for two potential interactions at very low concentrations.

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