Charged colloids in an aqueous mixture with a salt

Ryuichi Okamoto and Akira Onuki

Phys. Rev. E 84, 051401 – Published 2 November 2011

Abstract

We calculate the ion and composition distributions around colloid particles in an aqueous mixture, accounting for preferential adsorption, electrostatic interaction, selective solvation among ions and polar molecules, and composition-dependent ionization. On the colloid surface, we predict a precipitation transition induced by a strong preference of hydrophilic ions to water and a prewetting transition between weak and strong adsorption and ionization. These transition lines extend far from the solvent coexistence curve in the plane of the interaction parameter $χ$ (or the temperature) and the average solvent composition. The colloid interaction is drastically altered by these phase transitions on the surface. In particular, the interaction is much amplified on bridging of wetting layers formed above the precipitation line. Such wetting layers can either completely or partially cover the colloid surface depending on the average solvent composition.

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Received 26 August 2011

DOI:https://doi.org/10.1103/PhysRevE.84.051401

Authors & Affiliations

Ryuichi Okamoto and Akira Onuki

Department of Physics, Kyoto University, Kyoto 606-8502, Japan

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Vol. 84, Iss. 5 — November 2011

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Figure 1
Real and imaginary parts of $q_{1} / κ$ and $q_{2} / κ$ in the plane of $γ_{p}$ and $M = 1 / κ \bar{ξ}$ in the linear theory. Here $q_{1}$ and $q_{2}$ are positive for $γ_{p} < | M - 1 |$ [yellow (light gray)] and are complex conjugates for $| M - 1 | < γ_{p} < M + 1$ [green (dark gray)], while the system is unstable for $γ_{p} > M + 1$ (in white). These three regions are depicted in the right bottom panel, where the profiles and the interaction free energy will be shown at four $\times$ points (A), (B), (C), and (D) in Fig. 2 and those at three $•$ points (E), (F), and (G) in Fig. 3 in the linear theory.Reuse & Permissions
Figure 2
Results of the linear theory with $a = 600 a_{0}$ for hydrophilic ions with $γ_{p} = 0.23$ , $g_{a} = 2$ , $\bar{σ} = 0.001 a_{0}^{- 2}$ , and $κ = 0.1 a_{0}^{- 1}$ , where $M$ is (A) 0.4, (B) 0.5, (C) 1, and (D) 3 (see Fig. 1 for their locations in the $M$ - $γ_{p}$ plane). (Top) Normalized potential $U_{0} (r)$ (left) and composition deviation $ψ_{0} (r)$ (right) in Eq. (3.25) vs $r / a_{0}$ around a colloid. (Left bottom) Interaction free energy $F_{int} (d)$ between two colloids in Eq. (3.30) vs $d / a_{0}$ . (Right bottom) Minimum of $F_{int}$ , denoted by $F_{int}^{\min}$ , in the $M$ - $\bar{γ} / \bar{σ}$ plane with the common $\bar{σ}$ , $κ$ , and $a$ . It is zero (attained at infinity) in the right (for $M ≳ 1.3)$ and is negative in the left (for smaller $M$ ).Reuse & Permissions
Figure 3
Results of the linear theory with $a = 600 a_{0}$ for antagonistic ions with $γ_{p} = 1.727$ , $g_{a} = 15$ , $\bar{γ} / \bar{σ} = - 1$ , and $κ = 0.1 a_{0}^{- 1}$ , where $M$ is (E) 0.769, (F) 0.833, and (G) 0.909 (see Fig. 1 for their locations). (Left) Normalized potential $U_{0} (r) / a^{2} \bar{σ}$ vs $r / a_{0}$ around a colloid. (Right) Normalized interaction free energy $F_{int} (d) / a^{4} {\bar{σ}}^{2}$ between two colloids from Eq. (3.30) vs $d / a_{0}$ .Reuse & Permissions
Figure 4
Minimum interaction parameter $χ_{p}$ vs $\bar{φ}$ at $n_{0} = 3 \times 10^{- 4} v_{0}^{- 1}$ in the range $0 < \bar{φ} < 0.47$ for $g = 7, 8, 8.5, 9$ , 9.5, and 10 from above (left) and in the range $0.4 < \bar{φ} < 0.75$ for $g = 10$ (right). Two-phase coexistence region (in yellow) without solute is in the upper part. Spinodal curve $τ = τ_{c}$ is also written by dotted line (right).Reuse & Permissions
Figure 5
$ψ (r) = φ (r) - \bar{φ}$ and $U (r)$ vs $r / a_{0}$ around a colloid from the linear and nonlinear theories below the precipitation line at $v_{0} n_{0} = 3 \times 10^{- 4}$ . (Upper plates) Those for a hydrophobic surface with $γ = 0.08 a_{0}^{- 2}$ at $σ_{0} = 0.018 a_{0}^{- 2}$ , where $| ψ |$ at the surface is smaller than $1 / g_{i}$ and $| U |$ is at most of order unity so the linear results fairly agree with the nonlinear results. (Lower plates) Those for a hydrophilic surface with $γ = 0.2 a_{0}^{- 2}$ at small $σ_{0} = 0.003 a_{0}^{- 2}$ . From $ψ \sim 0.3$ as $r \to a$ , the cations are considerably accumulated near the surface and $U$ becomes positive in the nonlinear theory, while $U$ remains negative in the linear theory.Reuse & Permissions
Figure 6
Prewetting and precipitation transitions on a hydrophilic colloid surface with $a = 15 a_{0}$ and $γ = - 0.2 a_{0}^{- 2}$ . (Left) Colloidal precipitation line (red broken line) and prewetting lines for $a_{0}^{2} σ_{0} = 0$ , 0.003, and 0.005, around which $α ≅ 1$ here. (Right) Normalized adsorption $Γ / (4 π a^{3} / 3)$ vs $χ$ at $a_{0}^{2} σ_{0} = 0.003$ for (a) $\bar{φ} = 0.42$ , (b) 0.4176, (c) 0.415, (d) 0.412, and (e) 0.41. No phase transition appears on path (a), the prewetting critical point is on path (b), and the transition is discontinuous for paths (c), (d), and (e).Reuse & Permissions
Figure 7
$φ (r)$ (left) and $v_{0} [n_{1} (r) + n_{2} (r)]$ (right) at four points (A)–(D) marked in Fig. 6 for a hydrophilic colloid surface with $γ = - 0.2 a_{0}^{- 2}$ and $\bar{φ} = 0.412$ . There is a weakly discontinuous prewetting transition between (A) and (B) and a strongly discontinuous precipitation transition between (C) and (D).Reuse & Permissions
Figure 8
Phase diagram of prewetting and precipitation transitions on a hydrophobic colloid with $a_{0}^{2} γ = 0.08$ . Three prewetting lines, corresponding to $a_{0}^{2} σ_{0} = 0.015$ , 0.018, and 0.021, are more extended than in the hydrophilic case in Fig. 6. Precipitation on the colloid surface occurs above the broken red line. See Fig. 9 for $α$ and $Γ$ on vertical paths (a)–(g).Reuse & Permissions
Figure 9
Degree of ionization $α$ (left) and normalized adsorption $Γ / (4 π a^{3} / 3)$ (right) vs $χ$ on a hydrophobic colloid with $a_{0}^{2} γ = 0.08$ and $a_{0}^{2} σ_{0} = 0.018$ along paths (a) $\bar{φ} = 0.46$ , (b) 0.455, (c) 0.45, (d) 0.44, (e) 0.425, (f) $0.415$ , and (g) 0.41 (see their locations in Fig. 8). (Top) Behavior around the prewetting line at $a_{0}^{2} σ_{0} = 0.018$ (middle line in Fig. 8) along paths (a)–(d). (Bottom) Large jumps across the precipitation line along the paths (e)–(g) (dotted lines). On path (e), there is a prewetting transition between (A) and (B) and a precipitation transition between (C) and (D). Path (f) also passes the two transitions. Path (g) passes only the precipitation line.Reuse & Permissions
Figure 10
$φ (r)$ (left top), $U (r)$ (right top), $v_{0} n_{1} (r)$ on a semilogarithmic scale (left bottom), and $v_{0} [n_{1} (r) - n_{2} (r)]$ (right bottom) on a hydrophobic colloid at four points (A), (B), (C), and (D) on path (e) in Fig. 9, where $a_{0}^{2} γ = 0.08$ , $a_{0}^{2} σ_{0} = 0.018$ , and $\bar{φ} = 0.425$ . A prewetting transition occurs between (A) and (B), while a precipitation transition between (C) and (D). At (B), (C), and (D), the ions are enriched in the layer. In the inset in the right bottom plate, a small electric double layer is shown in the interface region around $r ≅ 40 a_{0}$ at (D).Reuse & Permissions
Figure 11
Normalized interaction free energy $F_{int} (d) / T$ vs normalized separation distance $d / a_{0}$ (left) and $φ (r, z)$ at $d = 44 a_{0} = 2.93 a$ (right) for $\bar{φ} = 0.425$ , where $a_{0}^{2} γ = 0.08$ and $a_{0}^{2} σ_{0} = 0.018$ . The system is below a prewetting line at $χ = 1.88$ and above it at $χ = 1.92$ , where $F_{int}$ changes its sign from positive to negative. The surface is hydrophobic with $φ^{'} = ν \cdot \nabla φ > 0$ [see Eq. (2.24)] at $χ = 1.88$ but is hydrophilic with $φ^{'} < 0$ due to an increase in $α$ at $χ = 1.92$ . The color represents $φ$ according to the color bar attached.Reuse & Permissions
Figure 12
Normalized interaction free energy $F_{int} (d) / k_{B} T$ vs normalized separation distance $d / a_{0}$ (left) and $φ (r, z)$ at $d = 44 a_{0} = 2.93 a$ (right) at the critical composition $\bar{φ} = 0.5$ , where there is no prewetting transition. (Top) Those for (originally) weakly hydrophobic surface with $a_{0}^{2} γ = 0.08$ and $a_{0}^{2} σ_{0} = 0.018$ at $χ = 1.2$ and $χ = 1.965$ . Here $F_{int}$ changes its sign on approaching the solvent criticality. The surface becomes hydrophilic ( $φ^{'} < 0$ ) weakly for $χ = 1.2$ and strongly for $χ = 1.965$ due to ionization. (Bottom) Those for strongly hydrophobic surface with $a_{0}^{2} γ = 0.2$ and $a_{0}^{2} σ_{0} = 0.01$ at $χ = 1.2$ and $χ = 1.965$ . Here the surface remains hydrophobic even for full ionization.Reuse & Permissions
Figure 13
$φ (r, z)$ (left) and $U (r, z)$ (right) around two hydrophobic colloids above a precipitation line, where $a_{0}^{2} γ = 0.08$ , $a_{0}^{2} σ_{0} = 0.018$ , and $\bar{φ} = 0.425$ . Wetting layers of two colloids are bridged at $d = 5.33 a = 80 a_{0}$ (top) and are disconnected for $6.8 a = 102 a_{0}$ (bottom). The surface has become hydrophilic with $φ^{'} < 0$ .Reuse & Permissions
Figure 14
Normalized interaction free energy $F_{int} (d) / T$ as a function of normalized distance $d / a_{0}$ above a precipitation line, where $χ = 1.94$ , $1.95$ , and $1.96$ for three curves (from above). Wetting layers of two colloids are bridged for small $d$ but are disconnected for large $d$ . There is hysteretic behavior between these two states (see Fig. 15).Reuse & Permissions
Figure 15
(Left) Normalized interaction free energy $F_{int} (d) / T$ , exhibiting hysteresis between bridged and disconnected states of wetting layers for $χ = 1.95$ in the box region in Fig. 14. There can be a metastable state in an interval of $d$ (left). Composition $φ (r, z)$ in wetting layers at the two states marked by the arrows in the left.Reuse & Permissions
Figure 16
Partially or completely wetting profiles of $φ (r, z)$ around two hydrophobic colloids for $\bar{φ} = 0.3$ with $d = 44 a_{0} = 2.93 a$ , where $χ = 1.25$ , 1.55, 1.85, and 2 above the bulk precipitation value $χ_{p} = 1.12$ , The other parameters are common to those in Figs. 11 and 13, 14, 15.Reuse & Permissions
Figure 17
Two partially wetted, hydrophobic colloids in a bridged or disconnected state for $\bar{φ} = 0.3$ at $χ = 1.55$ . The other parameters are common to those in Fig. 16. (Left) $F_{int} (d) / k_{B} T$ vs $d / a_{0}$ with hysteresis between the two states. Arrows indicate $d / a_{0} = 44$ , 60, 72. (Right) $φ (r, z)$ at $d / a_{0} = 60$ (bridged) and 72 (disconnected). Profile at $d / a_{0} = 44$ is in the top right panel in Fig. 16.Reuse & Permissions
Figure 18
$φ (r)$ and $U (r)$ around a hydrophobic colloid for antagonistic salt with $g_{1} = - g_{2} = 13$ , where $a_{0}^{2} γ = 0.2$ , $a_{0}^{2} σ_{0} = 0.016$ , and $\bar{φ} = 0.5$ . Three curves correspond to $χ = 1.961$ (green), $1.965$ (blue), and $1.966$ (red). Parameter $M$ in Eq. (3.21) in the linear theory is equal to 0.8808, 0.8344, 0.8224, respectively. Oscillatory behavior is amplified with increasing $χ$ . The system tends to be homogeneous far from the colloid for the first two curves, while a mesophase is realized for the third curve.Reuse & Permissions
Figure 19
(Left) Normalized interaction free energy $F_{int} / k_{B} T$ as a function of $d$ for two colloids with antagonistic ions, where $a_{0}^{2} γ = 0.2 > 0$ , $a_{0}^{2} σ_{0} = 0.016$ , and $\bar{φ} = 0.5$ . Three curves correspond to $χ = 1.950$ , $1.961$ , and $1.965$ from below. Instability occurs at $χ > 1.9651$ , so homogeneity is attained far from the colloids. (Right) $φ (r, z)$ around two hydrophobic colloids at $d = 49 a_{0} = 3.27 a$ for $χ = 1.965$ .Reuse & Permissions

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