Effects of Dielectric Inhomogeneity and Electrostatic Correlation on the

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Effects of Dielectric Inhomogeneity and Electrostatic Correlation on the Solvation Energy of Ions in Liquids Issei Nakamura J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.8b01465 • Publication Date (Web): 03 May 2018 Downloaded from http://pubs.acs.org on May 5, 2018

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The Journal of Physical Chemistry

Effects of Dielectric Inhomogeneity and Electrostatic Correlation on the Solvation Energy of Ions in Liquids Issei Nakamura∗ Department of Physics, Michigan Technological University, Houghton, Michigan 49931, United States E-mail: [email protected]

Abstract

largely remains unknown because ions are typically strongly influenced by the spatially varying dielectric response of the species (the dielectric inhomogeneity) and electrostatic correlations. Freely controlling the electrochemical nature of ion-containing liquids via these effects is of emerging importance in electrochemistry. Among various challenges in this area, the question becomes considerably difficult when liquid mixtures, polymers, and different time and length scales are involved. One major gap in the current understanding is what effect the dielectric inhomogeneity and strong electrostatic correlation at the molecular scale has on the macroscopic physical properties. The considerable attention received by this subject over the past several decades has generated continuum electrostatic models that adequately account for the saturation of the orientational degree of electric dipoles in response to the strong electrostatic field near ions, 2,7–9 specifically coupled with hydrogen bonding and/or ion-ion correlation. Since Debye posed this question, 10 the dielectric nature has been carefully examined, for example, by Rashin and Honig, 11 Still et al. in their generalized Born model, 12 and Gong and Freed through a continuum Langevin-Debye model. 13 Moreover, coarse-grained mean-field theory also suggested that various experimental data for both pure liquids and liquid mixtures can be rationalized by consideration of the saturated dipoles. 14 Ref. 14 also suggests that the standard Born solvation energy is largely disparate from the experimental re-

Electrolytes often involve the spatially varying dielectric response of liquids and electrostatic correlation. Nevertheless, the complexity of their synergistic effects complicates our understanding of ion solvation and often limits theoretical approaches. Thus, we develop a Ginzburg-Landau-like (GL) theory that simultaneously considers these two features. We derive the modified Born solvation energy of ions, which accounts for the effect of saturated dipoles near the ions on the solvation energy, which is in good agreement with experimental data for different ionic charges and even for some selected liquid mixtures. Moreover, we consider the phase diagram of a mixture of polyelectrolyte and uncharged polymer and that of a mixture of ionic liquid and uncharged polymer. The GL theory encompasses the results of the previous mean-field theories, accounting for fluctuations of the electrostatic potentials, and hence serves as a simple alternative approach to dielectrically inhomogeneous media.

1 Introduction Dissolving ions in liquids is common in a broad range of electrochemical systems, 1–3 such as energy storage 4 and electrospray of ions. 5,6 Nevertheless, the complete mechanism of ion solvation * To

whom correspondence should be addressed

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consistent field theory (SCFT) and the integralequation theory. 17–20 Note that we perform the Fourier transform of the free energy functional, instead of invoking the modified Born solvation energy in Sec. 2, primarily because the boundary condition Er (r → ∞) → 0 for a single ion (see the Appendix) is inappropriate when ion concentration is substantially high. Finally, in Sec. 4, we consider the spinodal curve of a mixture of ionic liquid and uncharged polymer. The GL theory encompasses the results calculated by the Gaussian approximation (in field-theoretical jargon, the oneloop expansion) in SCFT. In both Sec. 3 and Sec. 4, the effects of the ion-ion correlation and dielectric inhomogeneity can be synergistically correlated. We summarize our conclusions in Sec. 5.

sults. Importantly, the macroscopic-phase behaviors of ion-containing liquids can be significantly altered by ion-ion correlation 15–20 and dielectric inhomogeneity. 14,21 Note that the dielectric inhomogeneity arises primarily from key factors such as dielectric saturation, electrostriction, 2 and the spatial heterogeneity of liquid mixtures. In any case, how the ion-ion correlation and the dielectric inhomogeneity affect each other remains unclear because the theoretical description of the multi-body interaction between the species at the microscopic level is usually challenging. Moreover, the GL theory suggests that ionic liquids (ILs) confined between charged surfaces may involve strong ionion correlation. 22 Thus, we must also develop a dielectric continuum theory that simultaneously accounts for various aspects of electrolytes that span different length scales and then examine its efficacy. Importantly, such a theory is expected to be theoretically tractable in anticipation of diverse applications, such as extension to dynamics and combination with simulation techniques. However, existing theories cannot adequately capture the diverse features of electrolytes simultaneously. For example, without significantly adjusting the intrinsic dipole moment, 23 the dielectric constant of liquids with hydrogen bonds is disparate from the experimental value, but ionic liquids often involve hydrogen bonds. In this study, we develop the GL functional for the total free energy of electrolytes suggested by Bazant, Storey, and Kornyshev 22 (hereafter, the BSK free energy functional). In Sec. 2, we first modify the standard Born solvation energy of ions using the BSK free energy functional. Our theory agrees with the experimental data for monovalent, divalent, and trivalent ions immersed in various types of pure liquids and liquid mixtures. The resultant modified Born solvation energy involves a model parameter that accounts for the effect of the dielectric saturation, but it scales as a linear function of the dielectric constant of various liquids. The consistency of the data fitting is remarkable. In Sec. 3, we consider the spinodal curve of a mixture of polyelectrolyte and uncharged polymer. The BSK free energy accounts for the strong electrostatic correlation and therefore compares favorably with the prediction from a hybrid of the self-

2 Modified Born solvation energy of ions We consider the BSK free energy functional that involves strong cor electrostatic R relations, G[ψ ] = d~r ρψ − ε2r |∇ψ |2 + n−1 ∇n ψ 2 . 22 Here, ρ (~r) is the α l ∑∞ n−2 c n=2 charge density of the ions, ψ (~r) is the electrostatic potential, and εr is the dielectric constant in units of the vacuum permittivity ε0 . The parameters αn are dimensionless constants, and the coefficient lc scales the lengths at which the non-local ionion correlation decays. By extremizing G with respect to the electrostatic potential ψ , the Poisson equation for ion-containing liquids is given by 2(n−1) (−)n ∇2n ψ (~r) = −εr ∇2 ψ (~r) + εr ∑∞ n=2 αn−2 lc ρ (~r). Importantly, the solution of this equation showed solid agreement with the results of a molecular dynamics simulation for ionic liquids between charged surfaces, when n = 2 was assumed. the GL functional becomes Then 2 R E(~r) . G[ψ ] ∼ d~r ρψ − ε2r ~E 2 (~r) + α0 lc ∇ · ~ Thus, the contribution from n = 2 indicates the lowest-order correction regarding O(E 2 ). Here, the correlation length lc was set to the size of an ionic liquid because the systems should involve strong electrostatic correlation. Most likely, this fact can also be more intuitively understood by invoking the case where the Debye screening length decreases with increasing ion concentration; as the


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Å at room temperature. Incidentally, this choice of b with lB = 7 Å for water at room temperature compares favorably with the length of the saturation layer of water molecules near the ions, which was calculated by Monte Carlo simulation in our previous study. 26 In general, the saturation parameter C depends on q and εr . However, it may also depend on the ionic radius a and solvent size, primarily because the compressibility of the solvents (or electrostriction) near the ions may be significant and Er (a) is expected to be equal or close to (e/4πε0a2 ). We next calculate the solvation energy upon transferring an ion from vacuum to a dielectric medium and obtain the modified Born energy as follows: Z 1 2 2 0 ∆G = d~r εr ~E (~r) − ~E0 (~r) = VBorn 2kT C[4qγ 2 l0 exp (γ a) + 2εrC + εrCaγ ] + , (3) [aγ 4 l0 exp (2γ a)]

ion concentration continues to increase, the correlation length may approach the hard-core diameter of the ion. 24 Our present theory to modify the Born solvation energy of ions immersed in liquids draws upon the theory of the BSK free energy functional. We can thus write our modified Poisson equation for an ion with ionic charge q located at~r = 0 as follows: −εr ∇2 ψ (~r) + εr lc2 ∇4 ψ (~r) = qδ (~r).

(1)

The solution of the inhomogeneous differential equation in Eq. (1) is given by Er (r) =

q C exp (−γ r)(γ r + 1) + , (2) 2 4πεr r (γ r)2

where Er (r) is the electrostatic field − ∂ ψ∂ (r) r at a distance of r from the origin, C is an arbitrary constant, and γ ≡ lc−1 . The derivation is discussed in the Appendix. The first term in Eq. (2) indicates Coulomb’s law for a single point charge in the implicit solvent with the dielectric constant εr . We note that microscopically, the orientational order of solvent dipoles is saturated near ions in response to the strong electrostatic field, which is widely known as dielectric saturation. 10 Thus, the dielectric saturation significantly alters the electrostatic fields in the vicinity of relatively small ions. This effect yields the nonlinear dielectric response near the ions and thus the Poisson equation for a dielectric continuum should be modified by the fielddependent term. Accordingly, we identify the second term to describe the effect of the saturated dipoles, which serves as a correction to the electrostatic field. Thus, the model parameter C is referred to as the saturation parameter, while γ provides the length scale of the dielectric saturation. However, to fit this parameter, we must consider the solvent dipoles explicitly. In our current√study based p on implicit solvation, we use γ = 1/ lB b = εr /(ql0) calculated by a coarse-grained mean-field theory that accounts for ion-dipole and dipole-dipole interactions, 25 where the order of the magnitude of the dipolar length b is 1 Å. Here, we used the Bjerrum length lB = ql0 /εr for ionic charge q and vacuum Bjerrum length l0 = e2 /(4πε0kT ). The value of l0 is 560

0 where VBorn = q2 l0 /(2a)(1/εr − 1) and a are the standard Born solvation energy and ionic radius, respectively. The subscript 0 denotes the corresponding variables in the vacuum. We performed the integral for the radial coordinate from a to infinity. Note that when the solvent has a very low per 2 mittivity (e.g., εr ≈ 1), the third term lc ∇ · ~E(~r) in the GL functional can be negligible. Therefore, in Eq. (3), we used an ansatz that the solvation energy of an ion primarily arises from the first and second terms ρψ − ε2r ~E 2 (~r) in the GL functional. That is, we used the modified the electrostatic field [Eq. (2)] to capture the effect of the dielectric saturation within linear-dielectric theory. As discussed in the subsequent paragraphs, we found that this treatment circumvents the multipleparameter fitting regarding C with quantitatively reasonable agreement between the theory and experiment. We first performed a non-linear regression of the experimental data for various monovalent ions and liquids with Eq. (3) via the saturation parameter C. Remarkably, Fig. 1 shows that the linearity of C is persistent regardless of the dielectric constant εr of the liquids. Thus, C is not an adjustable parameter but scales as C ≈ 0.66εr + 1.0.


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Figure 2 (a) shows the solvation energies of various liquids, given that the saturation parameter C is best fitted to the data for monovalent ions. We then used the values of C to predict the solvation energy of divalent (q = 2) and trivalent (q = 3) ions, as shown in the lower parts of Fig. 2 (a). Excellent agreement between theory and experiment is observed for various types of ions and liquids, as well as different ionic charges, despite the fact that the solvation energy is sensitive to ionic charges, as it scales with q2 . Thus, we further applied our theory to various liquid mixtures. For the mixing rule of the dielectric constant of liquid mixtures, we consider Lichtenecker’s mixture formulae, 27,28 which can account for the dielectric response of multiphase components. This method is specifically known to be successful when liquids gain serial or parallel alignment against an electrostatic field. Our rationale to adopt this rule draws upon the fact that the solvation energy of the ions may be significantly affected by local enrichment of a higher-dielectric component near the ions, 29,30 which may locally form a layer of the different dielectric species. Thus, to effectively capture such feature, we adopted Lichtenecker’s equation for the dielectric constant of liquid mixtures, εr = [εAν φA + εBν φB ]1/ν . Incidentally, when ν = 1, this equation is reduced to the simple volume-fraction weight average of the dielectric constant, εr = εA φA + εB φB . Note that C also varies with respect to changes in φA and φB , subject to C ≈ 0.66εr + 1.0. We then performed a nonlinear regression of the experimental data for selected liquid mixtures with respect to ν . Eq. 3 is quite non-linear, and thus, the fitting is nontrivial. Table 1 shows the results of the regression for monovalent and divalent ions immersed in liquid mixtures. The agreement between theory and experiment is quite reasonable [Fig. 2 (b)]. Moreover, the fitting values of ν for the monovalent ions fall in the range of 0.025 to 0.030; 32 thus, we regard ν as an almost unique constant, particularly in consideration of the experimental uncertainty. Such consistency in the parameter ν for multivalent ions requires further computational and experimental validation.

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Figure 1: Saturation parameter C vs. dielectric constant εr of certain liquids. 31 Linear-square fitting: C = 0.6614εr + 0.9909. The calculated solvation energy is presented in Fig. 2 (a).

3 Spinodal decomposition of a blend of polyelectrolyte and uncharged polymer A recent theoretical study based on a hybrid of SCFT and integral-equation theory revealed that the effects of strong correlation between ions and between an ion and a charged polymer may considerably alter various thermodynamic properties. 39–41 Indeed, we show that the BSK free energy functional also captures the effect of the strong electrostatic correlation between ions dissolved in a mixture of polymer A and polymer B. Polymer A is a polyelectrolyte with cationic groups in the backbone, whereas polymer B is an uncharged homopolymer. Thus, the counterions are anionic. The chain lengths and monomer volumes of polymers A and B are set to N and v0 , respectively. Here, we consider the incompressibility of the polymers; thus, φ and 1 − φ denotes the volume fractions of polymers A and B, respectively. The cation and anion concentrations are given by c0 . Note that we applied the stoichiometric constraint c0 = rφ /v0 , where r is the degree of ionization of polymer A. We write εr for the dielectric constant of the blend. Accordingly, we write the free energy of the polymer blend as


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Figure 2: (a) Our theory (black lines) vs. experimental data (red dots) 33–37 for eight different liquids. Ionic radius a [Å] (x-axis) and the solvation energy of an ion ∆G [kcal/mol] (y-axis). Ions include Li+ , + + − + 2+ 2+ 2+ 2+ 2+ 2+ Na+ , K+ , Rb+ , Cs+ , Br− , I− , ClO− 4 , Me4 N , Et4 N , Cl , Ag , Mg , Mn , Ca , Sr , Ba , Zn , Cd2+ , Pb2+ , Cu2+ , Ni2+ , Co2+ , Fe2+ , Cr2+ , V2+ , Mn2+ , Al3+ , Sc3+ , Y3+ , La3+ , Ga3+ , In3+ , Tl3+ , Yti3+ , Cr3+ , and Fe3+ . (b) Our theory (solid lines) vs. experimental data (colored dots) 38 for liquid mixtures. Volume fraction of added liquid (x-axis) and ∆G [kcal/mol] (y-axis) for selected ions and liquid mixtures (Table 1). Table 1: Parameter ν of Lichtenecker’s mixture formula for the liquid mixtures in Fig. 2 (b) Ag+ /MeCN/MeOH (red) ClO− 4 /MeOH/Water (purple) 0.03000 0.02625 + 2+ Ag /MeCN/Acetone (blue) Mn /MeCN/Water (black) 0.02519 0.006858 formed by shifting εr to εr [1 + (lck)2 ] according to the Fourier transform of the modified Poisson equation [Eq. (1)]. We also introduced the ultraviR olet cutoff Λ = 4a for the ~k-integral, casting d~k R 2π /Λ into 0 dk so that the conventional Born energy 0 VBorn of the ions is obtained when the system is a dilute solution consisting of ions dissolved in the blend and does not involve the dielectric saturation. Using Eq. (4), we can calculate spinodal curves by solving ∂ 2 F1 /∂ φ 2 = 0. The analytical expression for the spinodal curves is quite long and cumbersome. Thus, we suggest that ∂ 2 F1 /∂ φ 2 = 0 be solved using numerical software. For simplicity, we use ν = 1 for the dielectric constant of the blend, and hence, εr = εA φ + εB (1 − φ ). The ultraviolet cutoff Λ may also be fixed so that the modified Born solvation energy [Eq. (3)] is reproduced when the system is a dilute solution. However, it should also be noted that the dielec-

follows: v0 F1 kB TV

φ (1 − φ ) ln φ + ln (1 − φ ) + rφ ln rφ N N − rφ + χφ (1 − φ ) Z 2c 8 π l q v0 0 0 0 + d~k ln 1 + 2(2π )3 εr [1 + (lck)2 ]k2 (4) =

Here, the first and second terms account for the translational entropy of the polymers. The translational entropy of the counterions is given by rφ ln rφ − rφ , assuming charge neutrality. The Flory-Huggins interaction is given by χφ (1 − φ ). The last term accounts for the electrostatic energy of the ions in the blend and is obtained by, for example, the standard field-theory calculation. Interested readers can refer to, for example, Refs. 15 and 42 to derive the expression. A similar calculation for the electrostatic part can be per-


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the hybrid theory invokes a closure relation for the Ornstein-Zernike relation, which can account for the compressibility. 43 Thus, our theory indicates that the drastic change in the spinodal curves is caused primarily by the strong electrostatic correlation. Thus, we suggest that the BSK free energy functional is an alternative approach that yields a reasonable correction to the phase boundary of ion-containing polymers, which arises from the strong electrostatic correlation. Moreover, the GL theory also predicts enhanced miscibility between the polymers, as demonstrated in the present case with lc > 2a. Of particular interest is the correlation between the effects of the dielectric inhomogeneity and strong electrostatic correlation. Indeed, their synergistic effect largely remains unclear; however, this feature has become more important in the softmatter sciences. 21 In Fig. 3 (b), we showed that the two effects can be substantially correlated with respect to the correlation length lc . The critical point also shifts from a polymer A-poor to a polymer Arich region as lc is increased. The miscibility is enhanced over a broad range of lc values because the ions preferentially solvate the higher-dielectric polymer B and the phase separation gives rise to an entropic cost attributed to the effect of charge neutrality. 44,45 Thus, the effects of the dielectric inhomogeneity and strong electrostatic correlation may be equally significant to account for the phase behavior of ion-containing polymers.

Figure 3: Spinodal curves of a mixture of polyelectrolyte and uncharged polymer as a function of the volume fraction of the polyelectrolyte. For comparison, we show the spinodal curves of uncharged polymer blends (dashed line). The colors of the lines correspond to lc = a (black), lc = 2a (blue), lc = 4a (red), and lc = l0 /εr (purple). The degree of ionization of the polyelectrolyte is r = 0.1. (a) εA = εB = 8 and (b) εA = 2.6 and εB = 8. tric constant εr tends to decrease when ion concentration is increased. 26 Thus, such a parameter adjustment, with the fixed dielectric constant, is not necessarily ideal for highly concentrated ions. Thus, we attribute the complexity of the electrostatic interactions and other molecular features to the correlation parameter lc instead of C. Note that the modified Born solvation energy in Eq. (3) for a single ion is obtained using the boundary condition Er (r → ∞) → 0. However, this condition is inappropriate to account for the electrostatic field for highly concentrated ions. Thus, we used the Fourier transform of the free energy functional in Eq. (4) instead of invoking Eq. (3). Take for example the scaled Bjerrum length Γ = l0 /(2aεr ) = 17.1 with the ionic radius a = 2.5 Å. 19,20 Figure 3 shows the spinodal curves for both dielectrically homogeneous [Fig. 3 (a)] and inhomogeneous [Fig. 3 (b)] blends. Note that in Fig. 3 (a), the effect of the strong electrostatic correlation may substantially change the phase boundary. Importantly, the present GL theory predicts enhanced immiscibility when the correlation parameter is lc ≤ 2a. This trend of the phase boundary is consistent with that calculated by a hybrid of SCFT and integral-equation theory, 17–20 which accounts for the higher-order electrostatic correlation. In our theory, the ultraviolet cutoff Λ accounts for the excluded volume of the ions, but does not account for the compressible nature of the ions. However,

4 Spinodal decomposition of a blend of ionic liquid and uncharged polymer Finally, we consider a mixture of ionic liquid and uncharged polymer. Technological improvements to dissolve low-dielectric biodegradable polymers are being driven by the urgent demand for sustainable energy. Indeed, ionic liquids are considered promising substances for this purpose; 46,47 however, phase diagrams of these mixtures and their theoretical study remain significantly limited. Examples of biorenewable materials and energy storages include but are not limited to PEO (εPEO = 5 − 12 48–51 ), polystyrene (εPS = 2.6 − 4 52 ), and


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cellulose (εcell = 2 − 8 53–55 ). Thus, the dielectric constants of biodegradable polymers typically fall in the rage of 2 to 12. The dielectric constants of ionic liquids are typically larger than 6 and are often approximately 15, 56 but these values can be above 30 when ionic liquids are polymerized. 57 We write the free energy of the ionic liquidpolymer mixture along the same line as Eq. 4 as follows:

lation length of ionic liquids is on the order of the size of an ion. Note that Bazant et al. demonstrated that this parameter set reproduces the results of the molecular dynamics simulation for ionic liquids between charged surfaces. 22 Theoretically, this choice for the parameter is also reasonable when the concentration of ionic liquids increases as the charge screening effect becomes larger. Here, note that the screening length continues to become small enough to be compared with the size of an ion, i.e., until the hard-core nature of an ion is evident. Thus, the highly non-monotonic behavior of the phase diagram may need to be corrected in consideration of the dependence of lc on the ion concentration. As an analogue of the Debye screening length, we anticipate that lc may decrease as the ion concentration increases (i.e., the volume fraction of the polymer φ decreases). Nevertheless, the previous SCFT 42 does not account for the hard-core nature of ions, and accordingly, we do not set a lower limit for lc in the present study. Thus, we also show in Fig. 4 (b) that a better fitting between the results of the GL theory and SCFT can be obtained using lc = 0.5a. Thus, the GL theory can serve as an alternative to the SCFT within the one-loop level. The technical simplicity of the GL theory is another advantage over the higher-order calculations of SCFT and other classes of statistical mechanics theories. The dielectric contrast between the ionic liquid and polymer remains largely ignored in the main literature. However, Fig. 4 (c) shows that such an effect on the phase boundary may also be comparable to that of the strong electrostatic correlation in the case of εpol < εIL . Specifically, in this case, the increase in the dielectric contrast |εIL − εpol | enhances the miscibility between the ionic liquid and polymer, as predicted by the Gaussian approximation of SCFT. Note that the effects of the dielectric inhomogeneity and strong electrostatic correlation cannot be decoupled unless the ion concentration c0 is quite small. However, the previous mean-field theory showed that if the electrostatic correlation can be independently suppressed, the effect of the dielectric inhomogeneity may cause an unconventional shift in the critical point towards the polymer-rich region. 42 This feature may arise when ionic liquids are hydrogenbonded enough to inhibit the correlation effect, as

(1 − φ ) v0 F2 φ = ln φ + (1 − φ ) ln kB TV N 2 +χφ (1 − φ ) Z 8π l0 q20 c0 v0 ~ + . d k ln 1 + 2(2π )3 εr [1 + (lc k)2 ]k2 (5) The ionic charge of the ionic liquids is q0 = 1. For simplicity, the molecular volumes of the cation, anion, and monomer are set to be the same. We ignore the dielectric contrast between the cation and anion. The volume fractions of the polymer and cation (or anion) are given by φ and (1 − φ )/2, subject to the incompressibility of the mixtures. The first and second terms arise from the translational entropies of the polymer and cation (or anion). Here, we do not consider the specific interaction that allows ionic liquids to bind to polymers. Thus, both the cation and anion can move freely in the mixture. This situation distinguishes the entropic terms for the ions in F2 from those in F1 . The dielectric constant of the mixture is εr = εpol φ + εIL (1 − φ ), where εpol and εIL denote the dielectric constants of the uncharged polymer and ionic liquid, respectively. Again, the last term for the electrostatic interaction reduces to the Born solvation energy when the cation (or anion) concentration c0 is small. We first consider the case with no dielectric contrast between the species [Fig. 4 (a)]. The GL theory predicts that highly non-monotonic phase boundaries are caused primarily by the strong electrostatic correlation when the dielectric constant of the species is relatively small. Note that the GL theory also compares favorably with the trend of the phase behaviors calculated by the Gaussian approximation (or in field-theoretical jargon, the one-loop expansion) of SCFT. 42 In Fig. 4 (a), we used lc = 2a, where the corre-


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tion, and the dielectric inhomogeneity caused by the difference in the dielectric response between the species. Our modified Born solvation energy [Eq. (3)] accounts for the contribution of saturated dipoles near the ions and agrees with the solvation energies of various types of ions (Fig. 2). Notably, these cases include divalent and trivalent ions, as well as liquid mixtures, when Lichtenecker’s mixture formulae for the dielectric constant are invoked. Eq. (3) includes the saturation parameter C, which accounts for the degree of dielectric saturation. However, this parameter C can be fitted uniquely by a linear function C ≈ 0.66εr + 1.0 using the dielectric constant εr for various types of liquids (Fig. 1). Furthermore, we used the GL theory to calculate the spinodal curves of a mixture of polyelectrolyte and uncharged polymer (Fig. 3) and a mixture of ionic liquid and uncharged polymer (Fig. 4). Importantly, we did not use our modified Born solvation energy in these cases, primarily because the boundary condition Er (r → ∞) → 0 for a single ion cannot be used when ions are highly concentrated. The GL theory accounts for fluctuations of the local electrostatic potential, and accordingly, the phase boundaries can be unconventionally deformed. However, phase boundary trends compare favorably with those from a hybrid of SCFT and integral-equation theory. 17–20 Similarly, we showed that the phase diagrams calculated by the GL theory and SCFT are consistent within the Gaussian approximation (or so-called one-loop expansion). Our findings indicate that the results of the two theories can be reasonably matched by the unique choice of the correlation parameter lc . Given that the GL theory also compares favorably with the result of the MD simulation for ionic liquids between charged surfaces, 22 we suggest that the GL theory can be invoked to consider a broad class of (polymer) electrolytes. Acknowledgments This work was supported by the start-up funds of Michigan Technological University.

Figure 4: Spinodal curves of a mixture of ionic liquid and uncharged polymer as a function of the volume fraction of the polymer. The dashed lines indicate the spinodal curves calculated by the Gaussian approximation (one-loop expansion) of self-consistent field theory. 42 (a) and (b) εpol = εIL = 2.6, (c) and (d) εpol = 2 and εIL = 20 (black) and εpol = 2 and εIL = 6.5 (red). The inset of (d) indicates the case of εpol = 12 and εIL = 6.5 as an example of εpol > εIL . in some experimental cases. 58 Thus, the present GL theory is unlikely to independently elucidate the observed shift in the critical point towards the polymer-rich region. As shown in Fig. 4 (b), the result of the GL theory compares more favorably with that of SCFT when we use lc = 0.5a [Fig. 4 (d)]. Indeed, this choice for the parameter also yields excellent correspondence between the two theories when εpol > εIL [inset of Fig. 4 (d)]. Thus, we suggest that the phase boundaries calculated by the GL theory and SCFT can be more consistently matched by a unique lc value in the case of a broad range of dielectric values.

5 Conclusion Appendix

In summary, we showed that the GL theory and the resultant BSK free energy functional simultaneously account for various features related to ion-containing liquids, such as the dielectric saturation near ions, the strong electrostatic correla-

Herein, we derive Eq. (2) from Eq. (1). Us ∂ 2 ∂ ∂ 2 2 ∂ ing ∇4 = r2∂∂ r −2 r ∂ r r ∂ r + ∂ r2 r ∂ r , we perform the integral of ∇4 ψ (~r) in spherical coordi-


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R nates as follows: 0x d~r ∇4 ψ (~r) = 4π 2r ∂∂r r2 Er − x x 2 ∂2 r2 Er 0 = 4π 2Er − 2r ∂∂Err − r2 ∂∂ rE2r 0 . The ∂ r2 integral of Eq. (1) in spherical coordinates leads to the following inhomogeneous differential equation: Z x 2 εr ∂ 2 q = dr4π r 2 r Er r ∂r 0 2 x ∂ Er 2 ∂ Er 2 −r + 4πεr α0 lc 2Er − 2r ∂r ∂ r2 0 ∂ Er (x) = 4πεr x2 Er + 4πεr α0 lc2 2Er (x) − 2x ∂x ∂ 2 Er (x) ∂ Er (x) − x2 − 2Er (0) + lim 2x 2 x→0 ∂x ∂x 2 ∂ Er (x) + lim x2 . (6) x→0 ∂ x2

(2) Marcus, Y. Electrostriction, Ion Solvation, and Solvent Release on Ion Pairing. J. Phys. Chem. B 2005, 109, 18541–18549. (3) Kalidas, C.; Hefter, G.; Marcus, Y. Gibbs Energies of Transfer of Cations from Water to Mixed Aqueous Organic Solvents. Chem. Rev. 2000, 100, 819–852. (4) Lodge, T. P. Materials Science - A Unique Platform for Materials Design. Science 2008, 321, 50–51. (5) Dole, M.; Mack, L. L.; Hines, R. L. Molecular Beams of Macroions. J. Chem. Phys. 1968, 49, 2240–2249. (6) Jackson, G. P.; Duckworth, D. C. Electrospray Mass Spectrometry of Undiluted Ionic Liquids. Chem. Commun. 2004, 522–523.

Note that the general solution to an inhomogeneous differential equation is given by the general solution to the corresponding homogeneous equation and a particular solution to the inhomogeneous equation. Here, we rather heuristically find the particular solution of Eq. (6); Er (x → 0) = q/(4πεr x2 ). We write the homogeneous equation as follows:

(7) Papazyan, A.; Warshel, A. Continuum and Dipole-Lattice Models of Solvation. J. Phys. Chem. B 1997, 101, 11254–11264. (8) Papazyan, A.; Warshel, A. Effect of Solvent Discreteness on Solvation. J. Phys. Chem. B 1998, 102, 5348–5357.

∂ 2 ErH (x) ∂ ErH (x) + 2x − 2ErH (x) ∂ x2 ∂x (7) − γ 2 x2 ErH (x),

(9) Warshel, A.; Papazyan, A. Electrostatic Effects in Macromolecules: Fundamental Concepts and Practical Modeling. Curr. Opin. Struct. Biol. 1998, 8, 211–217.

0 = x2

where γ 2 ≡ (α0 lc2 )−1 . For convenience, we transform Eq. (7) into the dimensionless form, H 2 H η 2 ∂ E∂ rη 2(η ) + 2η ∂ E∂r η(η ) − (η 2 + 2)ErH (η ) = 0, where η ≡ γ x. Note that this equation is the Modified spherical Bessel (MSB) differential equation in the case of the integer index n = 1. Given ErH (η → ∞) = 0, the solution is given by the MSB function of the second kind, ErH (η ) = S exp (−η )(η +1)/η 2 , where S is constant. Moreover, without loss of generality, α0 is absorbed into lc (or alternatively, α0 is set to 1, as in Ref. 22.)

(10) Debye, P. J. W. Polar Molecules; Chemical Catalog Co.: New York, 1929; p 172 pages. (11) Rashin, A. A.; Honig, B. Reevaluation of the Born Model of Ion Hydration. J. Phys. Chem. 1985, 89, 5588–5593. (12) Still, W. C.; Tempczyk, A.; Hawley, R. C.; Hendrickson, T. Semianalytical Treatment of Solvation for Molecular Mechanics and Dynamics. J. Am. Chem. Soc. 1990, 112, 6127– 6129.

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where a is an ionic radius and λD is the Debye screening length. Here, the boundary condition is − d ψdr(r) r=a = 4πεe a2 . r The ion concentration around the tagged ion is c± (r) = c0 exp (∓ ekTψ ) = g± (r)c0 , where g± (r) is the pair correlation function for the cation (+) and anion (−) and c0 is the cation (or anion) concentration. Using exp (± ekTψ ) ≈ 1 ± ekTψ , the pair correlation function is expressed as l exp [−(r−a)/λ ] g± (r) = 1 ∓ 0 εr r(1+a/λD ) D . Thus, given that λD ≈ 2a for ionic liquids, the correlation length for ionic liquids may be estimated at λD .

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of the solvents. Agreement between the theory and experiment can be as excellent as that shown in Fig. 2 (b) with ν = 0.0185. Thus, the GL theory can be consistent with the field-theoretical calculation for dipolar solvents.

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Figure 5: TOC Graphic


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Effects of Dielectric Inhomogeneity and Electrostatic Correlation on the

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