Bulk phase diagrams of a binary liquid mixture with added salt of constant chemical potential (per k B T) μ I = μ+ + μ− within the bilinear coupling approximation (BCA) [(a) and (b)] and within the local density approximation (LDA, see Eq. (3)) V ±(ϕ) = −ln (1 − ϕ(1 − exp ( − f ±))) [(c) and (d)] in terms of the Flory-Huggins parameter χ and the composition ϕ [(a) and (c)] or the chemical potential (per k B T) μϕ = μ A − μ B conjugate to the composition ϕ of the binary solvent [(b) and (d)]. The thick solid lines correspond to the binodals, which delimit the two-phase coexistence regions in the ϕ-χ diagrams [(a) and (c)] from below. The dashed lines are the spinodals and the thin horizontal line in panel (a) is the tie-line corresponding to the triple point (▲) found within BCA. Representative values for the solubility contrasts per k B T, (f +, f −) = (3, 26), have been chosen. The chemical potential μ I k B T of the salt corresponds to an ionic strength mM at the critical point with composition . The weak influence of the salt on the phase diagram within LDA leads to curves in panels (c) and (d) which are, on the present scale, almost (but not quite) symmetric with respect to and μϕ = 0, respectively. Whereas the LDA [(c) and (d)], in agreement with the experimental evidence, exhibits a single critical point (•, ), which slightly shifts upon changing the ionic strength (see Fig. 3), the standard BCA [(a) and (b)], in contrast to the available experimental observations, leads to a second critical point (•, ϕ c, 2 ≈ 0.1, χ c, 2 ≈ 2.1) as well as to a triple point (▲).
Comparison of the solvent-induced ion potential V ±(ϕ) [(a)] and its derivative [(b)] within LDA and BCA for ion solubility contrast f ± (see the main text). For small values of f ± (see the case f ± = 1) the differences between LDA and BCA are small. For large values of f ± (see the case f ± = 10) V ±(ϕ) and become large at solvent compositions ϕ ≈ 0.5 within BCA whereas they remain small within LDA. Within LDA and , while within BCA .
Variation of the critical volume fraction ϕ c [(a)] and the critical Flory-Huggins parameter χ c [(b)] as function of the ionic strength at the critical point for two representative sets of solubility contrasts: (f +, f −) = (3, 26) and (0, 20). These results show that both and χ c − 2 depend linearly on I c and that there are no quantitatively significant shifts of the critical point upon varying the ionic strength within experimentally reasonable ranges. On this scale, the phase diagrams for (f +, f −) = (3, 26) (see Figs. 1(c) and 1(d)) and for (f +, f −) = (0, 20) are almost indistinguishable. Note that and .
Poles k 1, …, k 4 of the Fourier transform of the two-point correlation functions G ij (r) in the complex plane , which correspond to the roots of the denominator L(k) (see Eq. (7)). According to the analytic structure of L(k) (see the main text) only the three distinct situations shown in panels (a)–(c) can occur. Purely imaginary poles [(a)] correspond to a monotonic decay of G ij (r → ∞) whereas a pole structure as in panel (b) (|k ν| all equal) corresponds to an oscillatory decay of G ij (r → ∞). Purely real poles [(c)] indicate an unstable bulk state, which does not occur in the one-phase region of the phase diagram in Fig. 1(c). The merging of two poles on the imaginary axis [(d)] corresponds to a Kirkwood crossover point.
Phase diagrams as in Fig. 1 with Kirkwood crossover lines (dotted lines) within the bilinear coupling approximation (BCA) [(a)] and the local density approximation (LDA) [(b)–(d)]. The parameters correspond to the binary liquid mixture water+3-methylpyridine (ɛ A = 10, ɛ B = 80, ); for simplicity the temperature dependence of the Bjerrum length ℓ B is ignored. The chemical potential μ I k B T of the salt is fixed such that at the (slightly shifted) critical point with composition there is an ionic strength . Outside the grey regions bounded by the dotted lines the two-point correlation functions exhibit asymptotically a monotonic decay, whereas inside these regions damped oscillatory decays occur. Within BCA a large portion of the phase diagram corresponds to oscillatory decay, whereas within LDA this occurs only in a narrow band within which the value of the bulk correlation length is close to that of the Debye screening length. Note the differences in scales for the axes in (a) and in (b)–(d).
Within LDA real and imaginary parts of the poles of the Fourier transform of the two-point correlation functions G ij (r) as functions of the deviation χ c − χ from the critical point at the critical composition for the parameters corresponding to Fig. 5(b). The four poles k 1, …, k 4 can be expressed in terms of (see Eqs. (13)–(17) and Fig. 4). If G ij (r → ∞) decays monotonically, the poles of are purely imaginary (), giving rise to two branches and of positive imaginary parts (see Fig. 4(a)). If G ij (r → ∞) decays oscillatorily, there is only one pole of with positive real and imaginary parts () (see Fig. 4(b)). The merging of the two branches and for monotonic asymptotic decay takes place at the Kirkwood crossover points (•) (see Fig. 4(d)). Upon varying ϕ these points form the Kirkwood crossover lines (dotted lines in Fig. 5). For comparison the inverse Debye length κ (dashed line) as well as the inverse Ornstein-Zernike length (dotted line, see Eq. (12)) are displayed. Within the range of values of χ leading to an oscillatory decay, depicted by the grey regions in Fig. 5, one has κ ≈ 1/ξ(OZ). Within the range of monotonic decay the decay rate of the leading contribution to G ij (r → ∞) is given by , whereas that of the subdominant contribution is . For χ c − χ ⩽ 1.3 × 10−3 the decay rates are and , whereas for χ c − χ ⩾ 4.3 × 10−3 the decay rates are and .
Profiles of the volume fraction ϕ of solvent component A [(a)], the electrostatic potential (with ) [(b)], the cation number density [(c)], and the anion number density [(d)] in a semi-infinite system bounded by a wall at with surface charge density and surface field strength h. These results correspond to Gibbs free energies of transfer f + = 0, f − = 20, the bulk volume fraction ϕ b = 0.5 of solvent component A, and the bulk ionic strength . The Flory-Huggins parameter χ(T) is chosen to correspond to that temperature, for which the bulk correlation length ξ is half of the Debye length 1/κ, which is taken to be temperature independent (see Fig. 6). For the specified surface fields h the solid lines are the numeric solutions obtained from the density functional model in Eq. (1) within LDA. For reasons of clarity in (b) and (c) the full lines for h = ±0.01 are not designated; they can be nonetheless identified in an obvious way. The dashed lines correspond to the approximate profiles , , and introduced in Eqs. (18)–(20), respectively. Note that and, due to the choice f + = 0, are independent of the magnitude |h|; therefore, both in (b) and (c) there is only one dashed line. For z > 2 the approximate profiles differ only slightly from the ones obtained by a full numerical minimization. Density oscillations close to the wall, which are expected in actual fluids, do not occur, because packing effects are not captured by the present square-gradient approach.
Composition profiles within BCA for strong ion-solvent coupling (f + = 30, f − = 0, solid line) and in the absence of ion-solvent coupling (f + = f − = 0, dashed line) taken from Fig. 3(a) in Ref. 19. Here a solvent with ɛ A = 80 and ɛ B = 20 at bulk composition ϕ b = 0.09 is considered. The bulk ionic strength is and the surface charge density is . At distances from the wall the two curves differ strongly from each other, whereas the differences are small at large distances.
Comparison of the numerically calculated excess adsorption Γ(ξ) obtained within the full model in Eq. (1) (•) with the predictions of Eq. (24) for the parameters used in Fig. 7 with h = 1. The Debye length κ−1 (marked by an arrow) corresponds to a bulk ionic strength . The term Γ0(ξ) (dotted line, see Eq. (25)), which contains the leading contribution to the excess adsorption and which corresponds to a vanishing surface charge density (σ = 0), exhibits visible deviations from the numerical results (•); nonetheless Γ1(ξ)/Γ0(ξ) → 0 for ξ → ∞. Taking into account, in addition, the term Γ1(ξ) (see Eq. (26)), which exhibits a dependence on the surface charge σ, quantitative agreement is found between Γ0(ξ) + Γ1(ξ) (solid line) and the numerical results (•) in the limit ξ → ∞. This finding also implies that those terms of Eq. (1), which have been left out upon deriving Eq. (21), do not contribute detectably.
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