The origin of the attraction between like charged hydrophobic and hydrophilic walls confining a near-critical binary aqueous mixture with ions

A near-critical binary mixture confined in a slit induces effective attraction or repulsion between the confining walls if the adsorption preferences of the two walls are the same or opposite respectively [1–3]. The range of this so called thermodynamic Casimir force is of the order of the bulk correlation length ξ. Parallel walls covered with electric charges of the same sign repel each other. One could thus expect stronger repulsion between like charged walls with opposite adsorption preferences confining the near-critical binary mixture. In striking contrast to the above expectation, in the recent experiments of [4] strong attraction was observed between a charged hydrophobic colloid particle and a charged hydrophilic substrate for some range of temperatures and concentrations of a hydrophilic salt added to the solution. The effective interactions between the colloid particles separated by distances much smaller than their radii are similar to the interactions between planar surfaces. The possibility of changing these interactions from attraction to repulsion through minute changes of temperature or salinity opens possibilities for designing and controlling reversible structural changes, in particular, aggregation or adsorption. It is thus important to understand the mutual influence of the critical adsorption and the distribution of ions that leads to the attraction between the walls instead of the expected repulsion. We address this issue in this communication.

We consider a mixture of water and organic liquid, containing hydrophilic ions in a slit with selective, charged walls of the area A→∞, separated by the distance L ≫ 1 (figure 1). We choose the average diameter of the molecules, a≡1, as the length unit and all the corresponding functions are dimensionless. The grand thermodynamic potential of the system can be written in terms of the local dimensionless densities ρ_i(r), where i = 1–4 for water, oil, and  +  and − ions respectively, in the form [5]

$$\begin{eqnarray} \displaystyle \fl \Omega =-p A L+A{\omega }_{\mathrm{ex}}&&\displaystyle \\ \displaystyle = \frac{1}{2}\int \nolimits _{V}\mathrm{d}\mathbf{r}\int \nolimits _{V}~\mathrm{d}{\mathbf{r}}^{\mathrm{^{\prime}}}{\rho }_{i}(\mathbf{r}){V}_{i j}(\mathbf{r}-{\mathbf{r}}^{\mathrm{^{\prime}}}){g}_{i j}(\mathbf{r}-{\mathbf{r}}^{\mathrm{^{\prime}}}){\rho }_{j}({\mathbf{r}}^{\mathrm{^{\prime}}})&&\displaystyle \\ \displaystyle +\int \nolimits _{V}~\mathrm{d}\mathbf{r}\hspace{0.167em} {\rho }_{i}(\mathbf{r})({V}_{i}^{s}(\mathbf{r})-{\mu }_{i})+{U}_{\mathrm{el}}-T S&&\displaystyle \end{eqnarray} \tag{ 1 }$$

where p is the bulk pressure, the integration is over the system volume V = AL, periodic boundary conditions are imposed in the directions parallel to the walls, and S, U_el, T and μ_i are the entropy, electrostatic energy, temperature and chemical potential of the ith species respectively. V_ij and g_ij are the van der Waals (vdW) interactions and the pair correlations between the corresponding components respectively, and the summation convention for repeated indices is assumed in the whole communication. ${V}_{i}^{s}(\mathbf{r})$ is the sum of the direct wall–fluid potentials acting on the component i. Finally,

$$\begin{eqnarray} &&{\omega }_{\mathrm{ex}}=({\gamma }_{0}+{\gamma }_{L})+\Psi (L) \end{eqnarray} \tag{ 2 }$$

is the excess grand potential per surface area, γ_n is the surface tension at the nth wall (n = 0,L), and the effective potential Ψ(L) is the subject of our study. Because of the translational symmetry in the parallel directions, the densities depend only on the distance from the left wall, z. We make the standard approximation

$$\begin{eqnarray} &&S/A=-{k}_{b}\sum _{i=1}^{4}\int _{0}^{\mathrm{L}}~\mathrm{d}z{\rho }_{i}(z)\ln {\rho }_{i}(z), \end{eqnarray} \tag{ 3 }$$

and the standard assumption [7]

$$\begin{eqnarray} \displaystyle \fl {U}_{\mathrm{el}}[\phi ]/A=\int \nolimits _{0}^{\mathrm{L}}\mathrm{d}z\left[\frac{-\epsilon }{8 \pi }\mathop{(\bigtriangledown \psi )}\nolimits ^{2}+~e \phi \psi \right]&&\displaystyle \\ \displaystyle +~e \sigma (0)\psi (0)+e \sigma (L)\psi (L),&&\displaystyle \end{eqnarray} \tag{ 4 }$$

where the electrostatic potential ψ satisfies the Poisson equation,

$$\begin{eqnarray} &&\frac{\epsilon }{4 \pi }\frac{{\mathrm{d}}^{2}\psi (z)}{\mathrm{d}{z}^{2}}+e \phi (z)=0, \end{eqnarray} \tag{ 5 }$$

e is the elementary charge, is the dielectric constant of the solvent, σ(n) is the dimensionless surface charge density at the nth wall, and

$$\begin{eqnarray} &&\phi (z)={\rho }_{3}(z)-{\rho }_{4}(z) \end{eqnarray} \tag{ 6 }$$

is the dimensionless charge density. Compressibility of the liquid can be neglected, and we assume that ${\mathop{\sum }\nolimits }_{i=1}^{4}{\rho }_{i}=1$. We choose ϕ, the solvent concentration s = ρ₁−ρ₂, and the density of ions ρ_c = ρ₃ + ρ₄ as the three independent variables. Bulk equilibrium densities for given T and μ_i correspond to the minimum of −pAL, and are denoted by $\bar {s}$ and ${\bar {\rho }}_{\mathrm{c}}$. In equilibrium, ϕ(z) and the deviations from the bulk values

$$\begin{eqnarray} &&\begin{array}{@{}r@{}} \displaystyle {\vartheta }_{1}(z)={\rho }_{1}(z)-{\rho }_{2}(z)-\bar {s}\\ \displaystyle {\vartheta }_{2}(z)={\rho }_{3}(z)+{\rho }_{4}(z)-{\bar {\rho }}_{\mathrm{c}} \end{array} \end{eqnarray} \tag{ 7 }$$

correspond to the minimum of ${\omega }_{\mathrm{ex}}[{\vartheta }_{1},{\vartheta }_{2},\phi ]=(\Omega [{\vartheta }_{1}+\bar {s},{\vartheta }_{2}+{\bar {\rho }}_{\mathrm{c}},\phi ]-\Omega [\bar {s},{\bar {\rho }}_{\mathrm{c}},0])/A$ with $\bar {s}$ and ${\bar {\rho }}_{\mathrm{c}}$ fixed. We choose for $\bar {s}$ and ${\bar {\rho }}_{\mathrm{c}}$ the values corresponding to the critical point.

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Common salts are soluble in water and insoluble in organic solvents. We thus assume that the difference in chemical nature of the anion and the cation is negligible, and postulate the same vdW interactions, V_i,3 = V_i,4. From equation (1) it easily follows that the vdW contribution to the internal energy expressed in terms of the new variables is independent of ϕ when V_i,3 = V_i,4 [6]. Because U_el is independent of ϑ_i (see equation (4)), in this approximation the vdW and the electrostatic contributions to the internal energy are decoupled. The coupling is present in the entropic part. The excess entropy, ${s}_{\mathrm{ex}}[{\vartheta }_{1},{\vartheta }_{2},\phi ]=(S[{\vartheta }_{1}+\bar {s},{\vartheta }_{2}+{\bar {\rho }}_{\mathrm{c}},\phi ]-S[\bar {s},{\bar {\rho }}_{\mathrm{c}},0])/A$, can be Taylor expanded in terms of ϑ_i(z) and ϕ(z). For a near-critical mixture with small amount of ions the expansion can be truncated, because ϑ_i(z) and ϕ(z) are small (except at microscopic distances from the surfaces). Using equation (3) one can verify that the excess entropy contains no terms proportional to ${\phi }^{n}{\vartheta }_{1}^{m}$, and the lowest-order mixed term is ϕ²ϑ₂. Thus, the excess grand potential can be split in two leading-order terms and the correction ΔL:

$$\begin{eqnarray} &&{\omega }_{\mathrm{ex}}[{\vartheta }_{1},{\vartheta }_{2},\phi ]\approx {\mathcal{L}}_{\mathrm{C}}[{\vartheta }_{1},{\vartheta }_{2}]+{\mathcal{L}}_{\mathrm{DH}}[\phi ]+\Delta \mathcal{L}[{\vartheta }_{2},\phi ]. \end{eqnarray} \tag{ 8 }$$

From the minimum condition for ω_ex it follows that the linear terms vanish, and the dominant terms in equation (8) are quadratic in the fields ϑ_i(z) and ϕ(z). The second term on the RHS of equation (8) has the form

$$\begin{eqnarray} &&{\mathcal{L}}_{\mathrm{DH}}[\phi ]={U}_{\mathrm{el}}[\phi ]/A+\frac{{k}_{\mathrm{B}}T}{2{\bar {\rho }}_{\mathrm{c}}}\int \nolimits _{0}^{L}\mathrm{d}z \hspace{0.167em} ({\phi }^{2}(z)+\mathrm{O}({\phi }^{4})). \end{eqnarray} \tag{ 9 }$$

Equations (9), (4) and (5) agree with the Debye–Huckel (DH) theory for the excess grand potential of ions in a homogeneous solvent confined in a slit with charged walls. The first term in equation (8) is equal to the excess grand potential per unit area for one kind of neutral solute in a two-component solvent, where the excess concentration of the solvent and the excess solute density are denoted by ϑ₁ and ϑ₂ respectively and the total density is fixed. This is because we assumed no difference between the vdW interactions of the anion and the cation—when uncharged, they represent the same species in this theory. Close to the critical temperature T_c the fields ϑ_i(z) vary on the length scale ξ∝|(T−T_c)/T_c|^−ν ≫ 1 with ν≈0.63, and the standard coarse-graining procedures leading to the Landau functional can be applied [8]. Our coarse-graining of the first term in equation (1) (expressed in terms of the new variables) is based on the Taylor expansion of ϑ_i(z') about z' = z. The excess grand potential is expressed in terms of the fields ϑ_i and their derivatives, and in terms of the appropriate moments of the vdW interaction potentials,

$$\begin{eqnarray} \displaystyle \fl {\mathcal{L}}_{\mathrm{C}}[{\vartheta }_{1},{\vartheta }_{2}]\approx \frac{1}{2}\int \nolimits _{0}^{L}~\mathrm{d}z\{ {\vartheta }_{i}(z){C}_{i j}^{0}{\vartheta }_{j}(z)&&\displaystyle \\ \displaystyle +~\nabla {\vartheta }_{i}(z){J}_{i j}\nabla {\vartheta }_{j}(z)+\mathrm{O}({\vartheta }_{1}^{4},{\vartheta }_{2}^{3},{\vartheta }_{1}^{2}{\vartheta }_{2})\} &&\displaystyle \\ \displaystyle +~\frac{{\vartheta }_{i}(0){J}_{i j}{\vartheta }_{j}(0)}{2}-{h}_{i}(0){\vartheta }_{i}(0)&&\displaystyle \\ \displaystyle +~\frac{{\vartheta }_{i}(L){J}_{i j}{\vartheta }_{j}(L)}{2}-{h}_{i}(L){\vartheta }_{i}(L),&&\displaystyle \end{eqnarray} \tag{ 10 }$$

where

$$\begin{eqnarray} &&{C}_{i j}^{0}=-{J}_{i j}^{0}-T\frac{{\partial }^{2}{s}_{\mathrm{ex}}[{\vartheta }_{1},{\vartheta }_{2},0]}{\partial {\vartheta }_{i}\partial {\vartheta }_{j}}, \end{eqnarray} \tag{ 11 }$$

${J}_{i j}^{0}=\int \mathrm{d}\mathbf{r}{J}_{i j}(r)$ and ${J}_{i j}=\frac{1}{6}\int ~\mathrm{d}\mathbf{r}{J}_{i j}(r){r}^{2}$. −J_ij(r) represents the vdW interactions for ϑ_i and ϑ_j, and can be obtained from the vdW contribution to equation (1) with the densities expressed in terms of the new variables. We shall assume that the interaction ranges ζ_ij defined by ${\zeta }_{i j}^{2}=6{J}_{i j}/{J}_{i j}^{0}$ are all ζ_ij≈1, and characterize the system with three interaction parameters, ${J}_{i j}={J}_{i j}^{0}/6$ (for the length unit a≡1). Finally, h_i(n) is the surface field describing interactions with the nth wall. The remaining surface terms result from the compensation for the interactions with the missing fluid neighbors at the wall. These interactions are included in the bulk term, but should be absent if the wall is present. In [6] the same functional was obtained from a lattice model for the four-component mixture. When the mixture phase separates, both the solvent concentration and the density of solute are different in the coexisting phases, because the solute is soluble only in water. Thus, L_C must depend on both ϑ₁ and ϑ₂. The critical order parameter is the eigenvector of ${C}_{i j}^{0}$ corresponding to the eigenvalue vanishing for T = T_c.

In equations (10) and (9) the terms proportional to k_BT represent the leading-order contributions to the excess entropy per surface area. The next-to-leading-order contribution has the form

$$\begin{eqnarray} \displaystyle \fl \Delta \mathcal{L}[{\vartheta }_{2},\phi ]=-{k}_{\mathrm{B}}T\int \nolimits _{0}^{L}~\mathrm{d}z \hspace{0.167em} \left[\frac{{\vartheta }_{2}(z){\phi }^{2}(z)}{2{\bar {\rho }}_{\mathrm{c}}^{2}}+\mathrm{O}({\vartheta }_{2}^{2}{\phi }^{2},{\vartheta }_{2}{\phi }^{4})\right]&&\displaystyle \\ \displaystyle &&\displaystyle \end{eqnarray} \tag{ 12 }$$

and results from the fact that there are more ways of introducing a local difference in the concentrations of the anions and the cations, ϕ(z), in the regions where there are more ions (ϑ₂(z)>0) than in the regions where there are fewer ions (ϑ₂(z)>0).

The Euler–Lagrange (EL) equations for the functional equations (8)–(12), with the higher-order terms defined in equations (10), (9) and (12) neglected, take the forms

$$\begin{eqnarray} \displaystyle \frac{{\mathrm{d}}^{2}{\vartheta }_{i}(z)}{\mathrm{d}{z}^{2}}={M}_{i j}{\vartheta }_{j}(z)+{\mathrm{d}}_{i}{\phi }^{2}(z)&&\displaystyle \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} \displaystyle \frac{{\mathrm{d}}^{2}\phi (z)}{\mathrm{d}{z}^{2}}={\kappa }^{2}\phi (z)+\frac{1}{{\bar {\rho }}_{\mathrm{c}}}\frac{{\mathrm{d}}^{2}(\phi (z){\vartheta }_{2}(z))}{\mathrm{d}{z}^{2}}.&&\displaystyle \end{eqnarray} \tag{ 14 }$$

The charge-neutrality condition, $\int \nolimits _{0}^{L}\mathrm{d}z \hspace{0.167em} \phi (z)+{\sigma }_{0}+{\sigma }_{L}=0$, is imposed on ϕ, and the boundary conditions for ϑ_i are

$$\begin{eqnarray} &&\begin{array}{@{}r@{}} \displaystyle {\left.\frac{\mathrm{d}{\vartheta }_{i}(z)}{\mathrm{d}z}\right\vert }_{z=0}-{\vartheta }_{i}(0)={H}_{i}(0) \\ \displaystyle {\left.-~\frac{\mathrm{d}{\vartheta }_{i}(z)}{\mathrm{d}z}\right\vert }_{z=L}-{\vartheta }_{i}(L)={H}_{i}(L). \end{array} \end{eqnarray} \tag{ 15 }$$

In the above, ${M}_{i j}=\mathop{({J}^{-1})}\nolimits _{i k}{C}_{k j}^{0}$, where (J⁻¹)_ik is the (i,k)th element of the matrix inverse to the matrix J_ij [6]. The remaining parameters are $({d}_{1},{d}_{2})=-\frac{{k}_{\mathrm{B}}T}{2{\bar {\rho }}_{\mathrm{c}}^{2}}(\mathop{({J}^{-1})}\nolimits _{1,2},\mathop{({J}^{-1})}\nolimits _{2,2})$, and H_i(n)=−(J⁻¹)_ijh_j(n). For a hydrophilic (hydrophobic) wall, H₁ < 0 (H₁ > 0).

When ΔL in equation (8) is neglected, the Casimir and the electrostatic potentials are independent contributions to ω_ex, and the EL equations are linear and decoupled (the second terms on the RHS of equations (13) and (14) are absent). In a semi-infinite system the solutions of the linearized EL equations (14) and (13) are

$$\begin{eqnarray} &&{\phi }^{(1)}(z)=-\kappa \sigma \exp (-\kappa z) \end{eqnarray} \tag{ 16 }$$

where ${\kappa }^{2}=\frac{4 \pi {e}^{2}{\bar {\rho }}_{\mathrm{c}}}{{k}_{\mathrm{B}}T\bar {\epsilon }}$ [7–9], and

$$\begin{eqnarray} &&{\vartheta }_{i}^{(1)}(z)={A}_{i}\exp (-z/\xi )+{C}_{i}\exp (-{\lambda }_{2}z), \end{eqnarray} \tag{ 17 }$$

where A_i and C_i depend linearly on H_i. The superscript (1) refers to the solutions of the linearized EL equations. In the critical region, ξ→∞ and λ₂ ≫ 1/ξ; therefore the second term on the RHS of equation (17) can be neglected. In the slit, the equilibrium fields ϕ⁽¹⁾(z) and ${\vartheta }_{i}^{(1)}(z)$ also contain terms ∝exp(−κ(L−z)) and ∝exp(−(L−z)/ξ) respectively (and the amplitudes are modified). The effective potential Ψ(L) is obtained by subtracting the L-independent part from ω_ex calculated for the equilibrium profiles. Neglecting ΔL in equation (8) we obtain Ψ(L)=Ψ_DH(L)+Ψ_C(L) with Ψ_DH(L)∝exp(−κL) and Ψ_C(L)∝exp(−L/ξ).

The nonlinear terms in the EL equations (13) and (14) can be neglected when their magnitudes are much smaller than the magnitudes of the linear terms at the relevant length scales. The nonlinear contributions to equations (13) and (14) can be estimated by examining ϕ⁽¹⁾²(z) and ${\phi }^{(1)}(z){\vartheta }_{2}^{(1)}(z)$ for z = ξ and z = κ⁻¹ respectively. This is because the linear terms in equations (13) and (14) decay on the length scales ξ and κ⁻¹ respectively. From equations (16) and (17) we obtain ϕ⁽¹⁾(ξ)∝exp(−ξκ) and ${\vartheta }_{2}^{(1)}({\kappa }^{-1})\propto \exp (-\mathop{(\xi \kappa )}\nolimits ^{-1})$. Thus, the magnitudes of the correction terms depend crucially on the ratio between the correlation and the screening length, ξκ. When ξκ→∞, then ϕ⁽¹⁾(ξ)→0 and ${\vartheta }_{2}^{(1)}({\kappa }^{-1})=\mathrm{O}(1)$; therefore we may consider the linearized equation (13), and treat equation (14) perturbatively. This case was considered in [6] for a semi-infinite system, and in [10] for a slit. On the other hand, for ξκ→0 we have ϕ⁽¹⁾(ξ)=O(1) and ${\vartheta }_{2}^{(1)}({\kappa }^{-1})\rightarrow 0$; therefore the linearized equation (14) can be considered. As a consequence, ϕ² in equation (13) can be approximated by ϕ⁽¹⁾². In this approximation equation (13) takes the form of a linear inhomogeneous equation. We assume that this approximation is reasonable as long as ξκ < 1, and the magnitudes of σ² and A_i are comparable.

In the experiments showing unusual dependence of the effective potential Ψ(L) on T, and consequently on ξκ, the relevant length ratio was ξκ < 1 [4]; therefore in this work we assume that ϕ = ϕ⁽¹⁾. Equation (13) with ϕ approximated by ϕ⁽¹⁾ can be easily solved analytically. The excess concentration of the solvent in the semi-infinite geometry takes the form

$$\begin{eqnarray} \displaystyle \fl {\vartheta }_{1}(z)=[{A}_{1}+{B}_{1}{\sigma }^{2}(f(\xi \kappa )+{f}_{1}(\kappa ,\kappa \xi ))]\exp (-z/\xi )&&\displaystyle \\ \displaystyle -{B}_{1}{\sigma }^{2}f(\xi \kappa )\exp (-2 \kappa z),&&\displaystyle \end{eqnarray} \tag{ 18 }$$

where ${B}_{1}=\frac{{B}^{\ast }}{2{\bar {\rho }}_{\mathrm{c}}}{k}_{\mathrm{B}}T$, B^* depends on the vdW interactions and

$$\begin{eqnarray} \displaystyle \fl f(y)=\frac{{y}^{2}}{\mathop{(2 y)}\nolimits ^{2}-1}{\mathop{\rightarrow }\nolimits }_{y\rightarrow \infty }\frac{1}{4},&&\displaystyle \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} \displaystyle \fl {f}_{1}(\kappa ,y)=\frac{\kappa }{y+\kappa }\left\{\vphantom {\frac {(J^{-1})_{1,2}}{B^{\ast }}} f(y)(2 y-1)-(1+2 \kappa )\kappa y\right.&&\displaystyle \\ \displaystyle \left.\times ~\left[\frac{\mathop{({J}^{-1})}\nolimits _{1,2}}{{B}^{\ast }}\left(\frac{{\kappa }^{2}}{f(y)}-1\right)+1\right]\right\}.&&\displaystyle \end{eqnarray} \tag{ 20 }$$

The excess solvent concentration at the distance z from the hydrophobic surface with weak and strong surface charge is shown in figure 2 for a few values of ξκ ≤ 1 (note that in the figure captions the length unit a is re-introduced). In all the cases we observe an excess of organic liquid close to the surface. In some cases, however, ϑ₁(z) is non-monotonic and changes sign for z₀∼ξ. An excess of water appears at the distances z > z₀ from the surface for all values of ξκ ≤ 1 in the case of strong surface charges. For weak surface charges an excess of water appears only for ξκ < y₀(σ); for ξκ > y₀(σ) a monotonic decay of |ϑ₁(z)| occurs, as in the uncharged system. Thus, the presence of the surface charge can change a (weakly) hydrophobic surface to an effectively hydrophilic one if we pay attention to the concentration of water at sufficiently large distances from the wall, z∼ξ. A change of the adsorption preferences caused by increased surface charge was observed experimentally [3, 11]. We emphasize that the change of the adsorption preference for small or moderate surface charges is present only sufficiently far from the critical point.

The above properties can be understood by examining equation (18) for the hydrophobic surface (A₁ < 0). For simplicity we neglect f₁(κ,ξ) (f₁(κ,κξ)→0 for ξ→∞ and κ→0, i.e. for T→T_c and ${\bar {\rho }}_{\mathrm{c}}\rightarrow 0$), and assume ξκ > 0.5. For ξκ > 0.5 the second term in equation (18) decays faster, and at the length scale ξ the excess of water is found when the prefactor of the first term is positive. From equations (18) and (19) we can conclude that for T→T_c (i.e. ξκ→∞) the excess of water occurs (i.e. ϑ₁(ξ)>0) when ${A}_{1}+\frac{{\sigma }^{2}}{4}{B}_{1}\gt 0$, which leads to the condition for the surface charge σ² > 4|A₁|/B₁. When σ² < 4|A₁|/B₁, an excess of organic liquid occurs (i.e. ϑ₁(ξ)<0) for T→T_c. Since σ²B₁f(ξκ) increases substantially when ξκ decreases (see equation (19)), the prefactor of the first term in equation (18), A₁ + σ²B₁f(ξκ), changes sign for ξκ = y₀. Thus, for σ² < 4|A₁|/B₁ a crossover from the excess of the organic liquid for ξκ > y₀ (close to T_c) to the excess of water for ξκ < y₀ (far from T_c) occurs for sufficiently large distances from the hydrophobic surface, z∼ξ.

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The physics behind such behavior is quite simple. The charged wall with no adsorption preference attracts ions. The ions insoluble in the organic liquid attract in turn water molecules to this wall. The excess number density of the hydrated ions (and thus the excess of water) appears in the layer of thickness ∼(2κ)⁻¹ [7, 9] and depends on the surface charge. The charge-neutral, hydrophobic surface attracts organic molecules. An excess of organic liquid is found in the layer of thickness ∼ξ, and depends on the hydrophobicity of the surface. Competition between the excess of organic liquid and the excess of water near the surface which is both hydrophobic and charged depends on ξκ, on the surface charge and on the hydrophobicity of the wall, and leads to the nontrivial concentration profiles.

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The Casimir potential between the walls results from the change of the concentration near the first wall caused by the presence of the second wall. Let us consider the vicinity of the hydrophilic wall when the weakly hydrophobic wall is present at the distance L∼ξ. The uncharged hydrophobic wall leads to depletion of water, but as discussed above and shown in figure 2, in the presence of the surface charge the hydrophilic ions can lead to the opposite effect. Thus, for the range of temperatures corresponding to the change of the adsorption preference of the weakly hydrophobic surface, the Casimir potential can be attractive. For a not too large surface charge it could overcome the electrostatic repulsion. We calculated Ψ(L) from equations (8)–(12) by inserting the solutions of equation (13) and linearized equation (14) with the boundary conditions given in equation (15). The result is shown in figure 3 for a particular model system. Indeed, the potential is repulsive far from the critical point because the electrostatic repulsion dominates, and becomes attractive and again repulsive when the critical temperature is approached.

The above theory is derived from the microscopic statistical mechanical description by a systematic coarse-graining procedure. We neglected any difference in chemical nature of the cation and the anion. The coupling between the excess concentration of the solvent, ϑ₁(z), and the charge density, ϕ(z), results first from the coupling between ϑ₁(z) and ϑ₂(z) in equation (10), originating from the large difference in solubility of the hydrophilic ions in the two components of the solvent, and next from the coupling between ϑ₂(z) and ϕ(z) of the entropic origin (see equation (12)). Very recently, similar behavior of Ψ(L) was obtained in [12]. The change of the adsorption preference in [12] results from different solubilities of the anion and the cation in water. Further studies are necessary to verify which mechanism plays the key role for the experimental results reported in [4].

Acknowledgment