*T*

_{ c }

^{1}and Yoav Tsori

^{1,a)}

### Abstract

We calculate the interaction potential between two charged colloids immersed in an aqueous mixture containing salt near or above the critical temperature. We find an attractive interaction far from the coexistence curve due to the combination of preferential solventadsorption at the colloids’ surface and preferential ion solvation. We show that the ion-specific interaction strongly depends on the amount of salt added as well as on the mixture composition. The calculations are in good agreement with recent experiments. For a highly antagonistic salt of hydrophilic anions and hydrophobic cations, a repulsive interaction at an intermediate inter-colloid distance is predicted even though both the electrostatic and adsorption forces alone are attractive.

We gratefully acknowledge numerous discussions with D. Andelman, C. Bechinger, M. Bier, J. Dietrich, L. Helden, O. Nellen, A. Onuki, H. Orland, P. Pincus, and R. Podgornik. This work was supported by the Israel Science Foundation under Grant No. 11/10 and the European Research Council “Starting Grant” No. 259205.

I. INTRODUCTION

II. MODEL

III. RESULTS

A. Colloidalinteraction in a mixture at a critical composition

B. Surfaceinteraction in an off-critical mixture

C. Interaction between hydrophilic and hydrophobiccolloids

D. Ion specific effects

IV. CONCLUSIONS

### Key Topics

- Hydrophilic interactions
- 36.0
- Solvents
- 31.0
- Double layers
- 28.0
- Surface charge
- 27.0
- Hydrophobic interactions
- 24.0

##### B01J13/00

## Figures

(a) The interaction potential *U*(*D*) between two colloids at a distance *D* at different temperatures τ ≡ *T*/*T* _{ c } − 1 > 0 immersed in a mixture at a critical composition (ϕ_{0} = ϕ_{ c }). *U*(*D*) becomes attractive as τ decreases. Here *n* _{0} = 10 mM and Δγ_{ R, L } = 0.1*T*/*a* ^{2}, corresponding to about 3.4 mN/m. For the solid curves, Δ*u* ^{±} = 4 and the surfaces have the same charge σ_{ L, R } = −σ_{ sat }. Dashed-dotted curve: the same as the solid curve for τ = 0.008 except that σ_{ L } = 3σ_{ R } = −1.5σ_{ sat }. Dashed curve: the same as for τ = 0.008 except that Δ*u* ^{−} = 8. (b) The corresponding excess surface adsorption Γ. In this and in other figures, as an approximation of a water–2,6-lutidine mixture we used *T* _{ c } = 307.2 K, *v* _{0} = 3.9 × 10^{−29}m^{3}, *C* = χ/*a*,^{27} ɛ_{2, 6-lutidine} = 6.9, and ɛ_{water} = 79.5, and the surface area is taken to be *S* = 0.01μm^{2}.

(a) The interaction potential *U*(*D*) between two colloids at a distance *D* at different temperatures τ ≡ *T*/*T* _{ c } − 1 > 0 immersed in a mixture at a critical composition (ϕ_{0} = ϕ_{ c }). *U*(*D*) becomes attractive as τ decreases. Here *n* _{0} = 10 mM and Δγ_{ R, L } = 0.1*T*/*a* ^{2}, corresponding to about 3.4 mN/m. For the solid curves, Δ*u* ^{±} = 4 and the surfaces have the same charge σ_{ L, R } = −σ_{ sat }. Dashed-dotted curve: the same as the solid curve for τ = 0.008 except that σ_{ L } = 3σ_{ R } = −1.5σ_{ sat }. Dashed curve: the same as for τ = 0.008 except that Δ*u* ^{−} = 8. (b) The corresponding excess surface adsorption Γ. In this and in other figures, as an approximation of a water–2,6-lutidine mixture we used *T* _{ c } = 307.2 K, *v* _{0} = 3.9 × 10^{−29}m^{3}, *C* = χ/*a*,^{27} ɛ_{2, 6-lutidine} = 6.9, and ɛ_{water} = 79.5, and the surface area is taken to be *S* = 0.01μm^{2}.

The effect of removing preferential solvation or short-range chemical interactions on the potential *U*(*D*) between the colloids at temperatures given by (a) τ = 0.009 and (b) τ = 0.005. The dashed and dashed-dotted curves show *U*(*D*) when either the surface chemical affinity or preferential solvation parameters are zero, respectively. In the solid curves both short-range chemical preference and solvation are included. These two interactions are clearly nonadditive as the solid curve is not the sum of the dashed and dashed-dotted lines.

The effect of removing preferential solvation or short-range chemical interactions on the potential *U*(*D*) between the colloids at temperatures given by (a) τ = 0.009 and (b) τ = 0.005. The dashed and dashed-dotted curves show *U*(*D*) when either the surface chemical affinity or preferential solvation parameters are zero, respectively. In the solid curves both short-range chemical preference and solvation are included. These two interactions are clearly nonadditive as the solid curve is not the sum of the dashed and dashed-dotted lines.

Interaction potentials at different temperatures τ, (a) for ϕ_{0} = 0.48 < ϕ_{ c } with Δγ_{ R, L } = 0.1*T*/*a* ^{2} and (b) for ϕ_{0} = 0.52 > ϕ_{ c } with Δγ_{ R, L } = 0.4*T*/*a* ^{2}. The onset temperature for attraction is higher for ϕ_{0} < ϕ_{ c } due to preferential solvation. In (a), at intermediate temperatures the potential has metastable states. Here we used Δ*u* ^{+} = 4, Δ*u* ^{−} = 8, and σ_{ L } = 3σ_{ R } = −1.5σ_{ sat }. We took the average ion density to be *n* _{0} = 10 mM leading to κ values of κ ≃ 2.69 nm.

Interaction potentials at different temperatures τ, (a) for ϕ_{0} = 0.48 < ϕ_{ c } with Δγ_{ R, L } = 0.1*T*/*a* ^{2} and (b) for ϕ_{0} = 0.52 > ϕ_{ c } with Δγ_{ R, L } = 0.4*T*/*a* ^{2}. The onset temperature for attraction is higher for ϕ_{0} < ϕ_{ c } due to preferential solvation. In (a), at intermediate temperatures the potential has metastable states. Here we used Δ*u* ^{+} = 4, Δ*u* ^{−} = 8, and σ_{ L } = 3σ_{ R } = −1.5σ_{ sat }. We took the average ion density to be *n* _{0} = 10 mM leading to κ values of κ ≃ 2.69 nm.

Solid curves show the dependence of *U*(*D*) on the mixture salt concentration *n* _{0} with Δ*u* ^{+} = 4 and Δ*u* ^{−} = 8; colloidal attraction increases with the addition of salt. The attraction is weak for a salt concentration of *n* _{0} = 0.01 M but with Δ*u* ^{±} = 0 (dashed curve). Here ϕ_{0} = ϕ_{ c }, τ = 0.008, Δγ_{ L, R } = 0.1*T*/*a* ^{2}, and σ_{ L } = 3σ_{ R } = −1.5σ_{ sat }.

Solid curves show the dependence of *U*(*D*) on the mixture salt concentration *n* _{0} with Δ*u* ^{+} = 4 and Δ*u* ^{−} = 8; colloidal attraction increases with the addition of salt. The attraction is weak for a salt concentration of *n* _{0} = 0.01 M but with Δ*u* ^{±} = 0 (dashed curve). Here ϕ_{0} = ϕ_{ c }, τ = 0.008, Δγ_{ L, R } = 0.1*T*/*a* ^{2}, and σ_{ L } = 3σ_{ R } = −1.5σ_{ sat }.

Inter colloid potentials U(D) for hydrophilic and hydrophobic colloids (antisymmetric boundary conditions). For the surface on the right we used Δγ_{ R } = 0.1*T*/*a* ^{2} and σ_{ R } = −σ_{ sat }. For the surface on the left we used Δγ_{ L } = −0.4*T*/*a* ^{2} and σ_{ L } = −0.01σ_{ sat } ≪ σ_{ sat }. (a) The interaction potential *U*(*D*) at different temperatures τ showing that *U* becomes attractive when τ decreases, but repulsive close to *T* _{ c }. Here we took for the ions Δ*u* ^{+} = 4 and Δ*u* ^{−} = 8. (b) *U*(*D*) at τ = 0.048 and different values of Δ*u* ^{±}. The interaction is purely repulsive for Δ*u* ^{ d } = Δ*u* ^{+} − Δ*u* ^{−} = 0 (dashed-dotted curve) and weakly attractive for Δ*u* ^{ d } = 2 (dashed curve). The attraction is much stronger in the solid curves, all having different values of Δ*u* ^{±} but the same difference Δ*u* ^{ d } = −4. Among these curves, the attraction is strongest for the antagonistic salt (Δ*u* ^{−} = −Δ*u* ^{+} = 2).

Inter colloid potentials U(D) for hydrophilic and hydrophobic colloids (antisymmetric boundary conditions). For the surface on the right we used Δγ_{ R } = 0.1*T*/*a* ^{2} and σ_{ R } = −σ_{ sat }. For the surface on the left we used Δγ_{ L } = −0.4*T*/*a* ^{2} and σ_{ L } = −0.01σ_{ sat } ≪ σ_{ sat }. (a) The interaction potential *U*(*D*) at different temperatures τ showing that *U* becomes attractive when τ decreases, but repulsive close to *T* _{ c }. Here we took for the ions Δ*u* ^{+} = 4 and Δ*u* ^{−} = 8. (b) *U*(*D*) at τ = 0.048 and different values of Δ*u* ^{±}. The interaction is purely repulsive for Δ*u* ^{ d } = Δ*u* ^{+} − Δ*u* ^{−} = 0 (dashed-dotted curve) and weakly attractive for Δ*u* ^{ d } = 2 (dashed curve). The attraction is much stronger in the solid curves, all having different values of Δ*u* ^{±} but the same difference Δ*u* ^{ d } = −4. Among these curves, the attraction is strongest for the antagonistic salt (Δ*u* ^{−} = −Δ*u* ^{+} = 2).

(a) The effect of the sign of the colloids’ charge on the inter-colloid potential *U*(*D*). The interaction is attractive for two positively charged surfaces and is repulsive for two negatively charged surfaces; compare the dashed-dotted and dashed curves. We used Δ*u* ^{+} = 2, Δ*u* ^{−} = 4, Δγ_{ R, L } = 0.1*T*/*a* ^{2}, and τ = 0.008. In the solid curve the surfaces are both hydrophilic (Δγ_{ R, L } = 0.1*T*/*a* ^{2}) but oppositely charged. For an antagonistic salt with Δ*u* ^{−} = −Δ*u* ^{+} = 6 there is a repulsive regime at an intermediate range. (b) The effect of the hydrophilicity or hydrophobicity of the colloid's surface. In the absence of preferential solvation (Δ*u* ^{±} = 0) two hydrophobic (and hydrophilic, not shown) surfaces weakly attract (dashed-dotted curve). For a hydrophilic salt (Δ*u* ^{±} = 4), hydrophobic surfaces repel (solid curve) whereas hydrophilic surfaces attract (dashed curve). We used τ = 0.003 and σ_{ L, R } = −σ_{ sat }.

(a) The effect of the sign of the colloids’ charge on the inter-colloid potential *U*(*D*). The interaction is attractive for two positively charged surfaces and is repulsive for two negatively charged surfaces; compare the dashed-dotted and dashed curves. We used Δ*u* ^{+} = 2, Δ*u* ^{−} = 4, Δγ_{ R, L } = 0.1*T*/*a* ^{2}, and τ = 0.008. In the solid curve the surfaces are both hydrophilic (Δγ_{ R, L } = 0.1*T*/*a* ^{2}) but oppositely charged. For an antagonistic salt with Δ*u* ^{−} = −Δ*u* ^{+} = 6 there is a repulsive regime at an intermediate range. (b) The effect of the hydrophilicity or hydrophobicity of the colloid's surface. In the absence of preferential solvation (Δ*u* ^{±} = 0) two hydrophobic (and hydrophilic, not shown) surfaces weakly attract (dashed-dotted curve). For a hydrophilic salt (Δ*u* ^{±} = 4), hydrophobic surfaces repel (solid curve) whereas hydrophilic surfaces attract (dashed curve). We used τ = 0.003 and σ_{ L, R } = −σ_{ sat }.

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