Phase Transitions in Soft Matter Induced by Selective Solvation
PPhase Transitions in Soft Matter Induced by Selective Solvation ∗ Akira Onuki, Ryuichi Okamoto, and Takeaki Araki
Department of Physics, Kyoto University, Kyoto 606-8502, Japan (Dated: November 13, 2018)We review our recent studies on selective solvation effects in phase separation in polar binarymixtures with a small amount of solutes. Such hydrophilic or hydrophobic particles are preferentiallyattracted to one of the solvent components. We discuss the role of antagonistic salt composedof hydrophilic and hydrophobic ions, which undergo microphase separation at water-oil interfacesleading to mesophases. We then discuss phase separation induced by a strong selective solventabove a critical solute density n p , which occurs far from the solvent coexistence curve. We also givetheories of ionic surfactant systems and weakly ionized polyelectrolytes including solvation amongcharged particles and polar molecules. We point out that the Gibbs formula for the surface tensionneeds to include an electrostatic contribution in the presence of an electric double layer. I. INTRODUCTION
In soft matter physics, much attention has been paidto the consequences of the Coulombic interaction amongcharged objects, such as small ions, charged colloids,charged gels, and polyelectrolytes . However, notenough effort has been made on solvation effects amongsolutes (including hydrophobic particles) and polar sol-vent molecules . Solvation is also called hydrationfor water and for aqueous mixtures. In mixtures of awater-like fluid and a less polar fluid (including poly-mer solutions), the solvation is preferential or selective,depending on whether the solute is hydrophilic or hy-drophobic. See Fig.1 for its illustration. The typicalsolvation free energy much exceeds the thermal energy k B T per solute particle. Hence selective solvation shouldstrongly influence phase behavior or even induce a newphase transition. In experiments on aqueous mixtures,it is well known that a small amount of salt drasticallyalters phase behavior . In biology, preferential inter-actions between water and cosolvents with proteins are ofcrutial importance . Thus selective solvation is rele-vant in diverse fields, but its understanding from physicsis still in its infancy.Around 1980, Nabutovskii et al. proposed a pos-sibility of mesophases in electrolytes from a couplingbetween the composition and the charge density inthe free energy. In aqueous mixtures, such a cou-pling originates from the selective solvation . It isin many cases very strong, as suggested by data ofthe Gibbs transfer free energy in electrochemistry (seeSec.2). Recently, several theoretical groups have pro-posed Ginzburg-Landau theories on the solvation in mix-ture solvents for electrolytes , polyelectrolytes ,and ionic surfactants . In soft matter physics, suchcoarse-grained approaches have been used to understandcooperative effects on mesoscopic scales , thoughthey are inaccurate on the angstrom scale. They are evenmore usuful when selective solvation comes into play in ∗ Accounts: Bull. Chem. Soc. Jpn. June (2011)
FIG. 1: Illustration of hydration of Na + surrounded by ashell composed of water molecules in (a) pure water and (b)water-nitrobenzene. The solvation chemical potential of Na + is higher for (b) than for (a). the strong coupling limit. This review presents such ex-amples found in our recent research. . An antagonistic salt con-sists of hydrophilic and hydrophobic ions. An exampleis sodium tetraphenylborate NaBPh , which dissociatesinto hydrophilic Na + and hydrophobic BPh − . The lat-ter ion consists of four phenyl rings bonded to an ion-ized boron. Such ion pairs in aqueous mixtures behaveantagonistically in the presence of composition hetero-geneity. (i) Around a water-oil interface, they undergomicrophase separation on the scale of the Debye screen-ing length κ − , while homogeneity holds far from theinterface to satisfy the charge neutrality (see the rightbottom plate in Fig.2). This unique ion distribution pro-duces a large electric double layer and a large Galvanipotential difference . We found that this ion dis-tribution serves to much decrease the surface tension ,in agreement with experiments . From x-ray re-flectivity measurements, Luo et al. determined suchion distributions around a water-nitrobenzene(NB) in-terface by adding BPh − and two species of hydrophilicions. (ii) In the vicinity of the solvent criticality, antag-onistic ion pairs interact differently with water-rich and a r X i v : . [ c ond - m a t . s o f t ] M a y oil-rich composition fluctuations, leading to mesophases(charge density waves). In accord with this prediction,Sadakane et al. added a small amount of NaBPh to a near-critical mixture of D O and 3-methylpyridine(3MP) to find a peak at an intermediate wave number q m ( ∼ . − ∼ κ ) in the intensity of small-angle neu-tron scattering. The peak height was much enhancedwith formation of periodic structures. (iii) Moreover,Sadakane et al. observed multi-lamellar (onion) struc-tures at small volume fractions of 3MP (in D O-rich sol-vent) far from the criticality , where BPh − and solvat-ing 3MP form charged lamellae. These findings demon-strate very strong hydrophobicity of BPh − . (iv) An-other interesting phenomenon is spontaneous emulsifica-tion (formation of small water droplets) at a water-NBinterface . It was observed when a large pure waterdroplet was pushed into a cell containing NB and antag-onistic salt (tetraalkylammonium chloride). This insta-bility was caused by ion transport through the interface. .Many experimental groups have detected large-scale,long-lived heterogeneities (aggregates or domains) emerg-ing with addition of a hydrophilic salt or a hydropho-bic solute in one-phase states of aqueous mixtures .Their small diffusion constants indicate that their typicalsize is of order 10 ˚A at very small volume fractions. Intwo-phase states, they also observed a third phase visi-ble as a thin solid-like plate at a liquid-liquid interfacein two-phase states . In our recent theory , for suffi-ciently strong solvation preference, a selective solute caninduce formation of domains rich in the selected com-ponent even very far from the solvent coexistence curve.This phenomenon occurs when the volume fraction of theselected component is relatively small. If it is a majoritycomponent, its aggregation is not needed. This precipita-tion phenomenon should be widely observable for variouscombinations of solutes and mixture solvents. . Hydrogen bond-ing is of primary importance in the phase behavior ofsoft matter. In particular, using statistical-mechanicaltheories, the origin of closed-loop coexistence curves wasascribed to the hydrogen bonding for liquid mixtures and for polymer solutions . Interestingly, water it-self can be a selective solute triggering phase separa-tion when the hydrogen bonding differs significantly be-tween the two components, as observed in a mixture ofmethanol-cyclohexane . More drastically, even waterabsorbed from air changed the phase behavior in films ofpolystyrene(PS)- polyvinylmethylether(PVME) . Thatis, a small amount of water induces precipitation ofPVME-rich domains. For block copolymers, similar pre-cipitation of micelles can well be expected when a smallamount of water is added. . Surfactant molecules arestrongly trapped at an interface due to the amphiphilicinteraction even if their bulk density is very low . Theycan thus efficiently reduce the surface tension, giving riseto various mesoscopic structures. However, most theo- retical studies have treated nonionic surfactants, whileionic surfactants are important in biology and technol-ogy. In this review, we also discuss selective solvation insystems of ionic surfactants, counterions, and added ionsin water-oil . We shall see that the adsorption behaviorstrongly depends on the selective solvation. . Polyelectrolytes are alreadyvery complex because of the electrostatic interactionamong charged particles (ionized monomers and mobileions) . Furthermore, we should take into account twoingredients , which have not yet attracted enough atten-tion. First, the dissociation (or ionization) on the chainsshould be treated as a chemical reaction in many poly-electrolytes containing weak acidic monomers . Thenthe degree of ionization is a space-dependent annealedvariable. Second, the solvation effects should come intoplay because the solvent molecules and the charged par-ticles interact via ion-dipole interaction. Many polymersthemselves are hydrophobic and become hydrophilic withprogress of ionization in water. This is because the de-crease of the free energy upon ionization is very large.It is also worth noting that the selective solvation ef-fect can be dramatic in mixture solvents . As an exam-ple, precipitation of DNA has been observed with addi-tion of ethanol in water , where the ethanol addedis excluded from condensed DNA, suggesting solvation-induced wetting of DNA by water. In polyelectrolytesolutions, macroscopic phase separation and mesophaseformation can both take place, sensitively depending onmany parameters.The organization of this paper is as follows. In Sec.2,we will present the background of the solvation on thebasis of some experiments. In Sec.3, we will explain aGinzburg-Landau model for electrolytes accounting forselective solvation. In Sec.4, we will treat ionic surfac-tants by introducing the amphiphilic interaction togetherwith the solvation interaction. In Sec.5, we will examineprecipitation induced by a strong selective solute, where asimulation of the precipitation dynamics will also be pre-sented. In Sec.6, we will give a Ginzburg-Landau modelfor weakly ionized polyelectrolytes accounting for the ion-ization fluctuations and the solvation interaction. In Ap-pendix A, we will give a statistical theory of hydrophilicsolvation at small water contents in oil. II. BACKGROUND OF SELECTIVESOLVATION OF IONS . Sev-eral water molecules form a solvation shell surroundinga small ion via ion-dipole interaction , as in Fig.1. Clus-ter structures produced by solvation have been observedby mass spectrometric analysis . Here we mention anexperiment by Osakai et al , which demonstrated thepresence of a solvation shell in a water-NB mixture intwo-phase coexistence. They measured the amount ofwater extracted together with hydrophilic ions in a NB-rich region coexisting with a salted water-rich region. Awater content of 0 . φ being 0.003) was already present in the NB-rich phasewithout ions. They estimated the average number of sol-vating water molecules in the NB-rich phase to be 4 forNa + , 6 for Li + , and 15 for Ca per ion. Thus, whena hydrophilic ion moves from a water-rich region to anoil-rich region across an interface, a considerable frac-tion of water molecules solvating it remain attached toit. Furthermore, using proton NMR spectroscopy, Osakai et al. studied successive formation of complex struc-tures of anions X − (such as Cl − and Br − ) and watermolecules by gradually increasing the water content inNB. This hydration reaction is schematically written asX − (H O) m − + H O ←→ X − (H O) m ( m = 1 , , , ... ).For Br − , these clusters are appreciable for water contentlarger than 0 .
1M or for water volume fraction φ exceeding0 . (cid:15) b i upon bind-ing of a polar molecule to a hydrophilic ion of species i .A well-defined solvation shell is formed for (cid:15) b i (cid:29) k B T ,where the water volume fraction φ needs to satisfy φ > φ i sol ∼ exp( − (cid:15) bi /k B T ) . (2.1)For φ (cid:28) φ i sol there is almost no solvation. The crossovervolume fraction φ i sol is very small for strongly hydrophilicions with (cid:15) bi (cid:29) k B T . For Br − in water-NB, we estimate φ i sol ∼ .
002 from the experiment by Osakai et al. asdiscussed above . . Hydrophobic objectsare ubiquitous in nature, which repel water becauseof the strong attraction among hydrogen-bonded watermolecules themselves . Hydrophobic particles tend toform aggregates in water and are more soluble in oilthan in water. They can be either neutral or charged. Awidely used hydrophobic anion is BPh − . In water, a largehydrophobic particle ( (cid:38) . In a water-oilmixture, on the other hand, hydrophobic particles shouldbe in contact with oil molecules instead. This attractioncan produce significant composition heterogeneities onmesoscopic scales around hydrophobic objects, which in-deed takes place around protein surfaces . . We introduce asolvation chemical potential µ i sol ( φ ) in the dilute limit ofsolute species i . It is the solvation part of the chemicalpotential of one particle (see Eq.(B1) in Appendix B).It is a statistical average over the thermal fluctuationsof the molecular configurations. In mixture solvents, itdepends on the ambient water volume fraction φ . Forplanar surfaces or large particles (such as proteins), wemay consider the solvation free energy per unit area.Born calculated the polarization energy of a polarfluid around a hydrophilic ion with charge Z i e using con-tinuum electrostatics to obtain the classic formula,( µ i sol ) Born = − ( Z i e / R i )(1 − /ε ) . (2.2) The contribution without polarization ( ε = 1) or in vac-uum is subtracted and the φ dependence here arises fromthat of the dielectric constant ε = ε ( φ ) (see Eq.(3.5) be-low). The lower cutoff R i is called the Born radius, whichis on the order of 1˚A for small metallic ions . The hy-drophilic solvation is stronger for smaller ions, since itarises from the ion-dipole interaction. In this originalformula, neglected are the formation of a solvation shell,the density and composition changes (electrostriction),and the nonlinear dielectric effect.For mixture solvents, the binding free energy betweena hydrophilic ion and a polar molecule is estimated fromthe Born formula (2.1) as (cid:15) bi ∼ − ∂ [( µ i sol ) Born ] /∂φ or (cid:15) bi ∼ Z i e ε /R i ε = k B T Z i (cid:96) B ε /R i ε, (2.3)where ε = ∂ε/∂φ and (cid:96) B = e /k B T ε is the Bjerrumlength ( ∼ ε ∼ ε , a well-defined shell appears for Z i (cid:96) B (cid:29) R i .The solvation chemical potential µ i sol ( φ ) of hydrophilicions in water-oil should largely decrease to negative val-ues in the narrow range 0 < φ < φ i sol , as will be de-scribed in Appendix A. In the wide composition range(2.1), its composition dependence is still very strong suchthat | ∂µ i sol /∂φ | (cid:29) k B T holds (see the next subsection).The solubility of hydrophilic ions should also increaseabruptly in the narrow range φ < φ i sol with addition ofwater to oil. Therefore, solubility measurements of hy-drophilic ions would be informative at very small watercontents in oil.In sharp contrast, µ i sol of a neutral hydrophobic par-ticle increases with increasing the particle radius R inwater . It is roughly proportional to the surface area4 πR for R (cid:38) k B T at R ∼ φ dependence of µ i sol . How-ever, µ i sol should strongly increase with increasing the wa-ter composition φ , since hydrophobic particles (includingions) effectively attract oil molecules . . We consider aliquid-liquid interface between a polar (water-rich) phase α and a less polar (oil-rich) phase β with bulk composi-tions φ α and φ β with φ α > φ β . The solvation chemicalpotential µ i sol ( φ ) takes different values in the two phasesdue to its composition dependence. So we define∆ µ iαβ = µ i sol ( φ β ) − µ i sol ( φ α ) . (2.4)In electrochemistry , the difference of the solvationfree energies between two coexisting phases is called thestandard Gibbs transfer free energy denoted by ∆ G iαβ foreach ion species i . Since it is usually measured in unitsof kJ per mole, its division by the Avogadro number N A gives our ∆ µ iαβ or ∆ µ iαβ = ∆ G iαβ /N A . With α being thewater-rich phase, ∆ µ iαβ is positive for hydrophilic ionsand is negative for hydrophobic ions from its definition(2.4). See Appendix B for relations between ∆ µ iαβ andother interface quantities.For ions, most data of ∆ G iαβ are at present limited onwater-NB and water-1,2-dichloroethane(EDC) at room temperatures, where the dielectric constant ofNB ( ∼
35) is larger than that of EDC ( ∼ µ iαβ /k B T . In the case of water-NB, it is13 . + , 27 . , and 11 . − as exam-ples of hydrophilic ions, while it is − . − . In the case of water-EDC, it is 22 , + and17 . − , while it is − . − . The ampli-tude | ∆ µ iαβ | /k B T for hydrophilic ions is larger for EDCthan for NB and is very large for multivalent ions. In-terestingly, ∆ µ iαβ for H + (more precisely hydronium ionsH O + ) assumes positive values close to those for Na + inthese two mixtures.In these experiments on ions, a hydration shell shouldhave been formed around hydrophilic ions even inthe water-poor phase β (presumably not completely ).From Eq.(2.1) φ β should be exceeding the crossover vol-ume fraction φ i sol . The data of the Gibbs transfer freeenergy quantitatively demonstrate very strong selectivesolvation even in the range (2.1) or after the shell forma-tion.On the other hand, neutral hydrophobic particles areless soluble in a water-rich phase α than in an oil-richphase β . Their chemical potential is given by µ i = k B T ln( n i λ i ) + µ i sol ( φ ) , where i represents the particlespecies and λ i is the thermal de Broglie wavelength.From homogeneity of µ i across an interface, we obtainthe ratio of their equilibrium bulk densities as n iβ /n iα = exp[ − ∆ µ iαβ /k B T ] , (2.5)where ∆ µ iαβ < III. GINZBURG-LANDAU THEORY OFMIXTURE ELECTROLYTES . Wepresent a Ginzburg-Landau free energy F for a polar bi-nary mixture (water-oil) containing a small amount ofa monovalent salt ( Z = 1, Z = − . The variables φ , n , and n are coarse-grained ones varying smoothlyon the molecular scale. For simplicity, we also neglectthe image interaction . At a water-air interface theimage interaction serves to push ions into the water re-gion. However, hydrophilic ions are already strongly de-pleted from an interface due to their position-dependenthydration . See our previous analysis for relative im-portance between the image interaction and the solvationinteractiion at a liquid-liquid interface.The free energy F is the space integral of the free en-ergy density of the form, F = (cid:90) d r (cid:20) f tot + 12 C |∇ φ | + ε π E (cid:21) . (3.1) The first term f tot depends on φ , n , and n as f tot = f ( φ ) + k B T (cid:88) i n i (cid:20) ln( n i λ i ) − − g i φ (cid:21) . (3.2)In this paper, the solvent molecular volumes of the twocomponents are assumed to take a common value v ,though they are often very different in real binary mix-tures. Then f is of the Bragg-Williams form , v fk B T = φ ln φ + (1 − φ ) ln(1 − φ ) + χφ (1 − φ ) , (3.3)where χ is the interaction parameter dependent on T .The critical value of χ is 2 without ions. The λ i = (cid:126) (2 π/m i k B T ) / in Eq.(3.2) is the thermal de Brogliewavelength of the species i with m i being the molecularmass. The g and g are the solvation coupling con-stants. In addition, the coefficient C in the gradient partof Eq.(3.1) remains an arbitrary constant. To explain ex-periments, however, it is desirable to determine C fromthe surface tension data or from the scattering data.In the electrostatic part of Eq.(3.1), the electric field iswritten as E = −∇ Φ. The electric potential Φ satisfiesthe Poisson equation, − ∇ · ε ∇ Φ = 4 πρ. (3.4)The dielectric constant ε is assumed to depend on φ as ε ( φ ) = ε + ε φ. (3.5)where ε and ε are positive constants. Though thereis no reliable theory of ε ( φ ) for a polar mixture, a lin-ear composition dependence of ε ( φ ) was observed by De-bye and Kleboth for a mixture of nitrobenzene-2,2,4-trimethylpentane . In addition, the form of the elec-trostatic part of the free energy density depends on theexperimental method . Our form in Eq.(3.1) follows ifwe insert the fluid between parallel plates and fix thecharge densities on the two plate surfaces.We explain the solvation terms in f tot in more detail.They follow if µ i sol ( φ ) ( i = 1 ,
2) depend on φ linearly as µ i sol ( φ ) = A i − k B T g i φ. (3.6)Here the first term A i is a constant yielding a contribu-tion linear with respect to n i in f tot , so it is irrelevantat constant ion numbers. The second term gives rise tothe solvation coupling in f tot . In this approximation, g i > g i < µ i sol = k B T g i ∆ φ, (3.7)where ∆ φ = φ α − φ β is the composition difference. FromEqs.(B4) and (B5), the Galvani potential difference is∆Φ = k B T ( g − g )∆ φ/ e, (3.8)and the ion reduction factor is n β /n α = n β /n α = exp[ − ( g + g )∆ φ/ . (3.9)The discussion in subsection 2.3 indicates g i ∼
14 (23)for Na + ions and g i ∼ −
14 (-14) for BPh − in water-NB(water-EDC) at 300K. For multivalent ions g i can be verylarge ( g i ∼
27 for Ca in water-NB). The linear form(3.6) is adopted for the mathematical simplicity and isvalid for φ > φ i sol after the solvation shell formation (seeEq.(2.1)). The results in Appendix A suggest a morecomplicated functional form of µ i sol ( φ ).In equilibrium, we require the homogeneity of thechemical potentials h = δF/δφ and µ i = δF/δn i . Here, h = f (cid:48) − C ∇ φ − ε π E − k B T (cid:88) i g i n i , (3.10) µ i = k B T [ln( n i λ i ) − g i φ ] + Z i e Φ , (3.11)where f (cid:48) = ∂f /∂φ . The ion distributions are expressedin terms of φ and Φ in the modified Poisson-Boltzmannrelations , n i = n i exp[ g i φ − Z i e Φ /k B T ] . (3.12)The coefficients n i are determined from the conservationof the ion numbers, (cid:104) n i (cid:105) = V − (cid:82) d r n i ( r ) = n , where (cid:104)· · ·(cid:105) = V − (cid:82) d r ( · · · ) denotes the space average with V being the cell volume. The average n = (cid:104) n (cid:105) = (cid:104) n (cid:105) isa given constant density in the monovalent case.It is worth noting that a similar Ginzburg-Landau freeenergy was proposed by Aerov et al . for mixtures ofionic and nonionic liquids composed of anions, cations,and water-like molecules. In such mixtures, the inter-actions among neutral molecules and ions can be pref-erential, leading to mesophase formation, as has beenpredicted also by molecular dynamic simulations . . The simplest applicationof our model is to calculate the structure factors of thethe composition and the ion densities in one-phase states.They can be measured by scattering experiments.We superimpose small deviations δφ ( r ) = φ ( r ) − (cid:104) φ (cid:105) and δn i ( r ) = n i ( r ) − n on the averages (cid:104) φ (cid:105) and n .The monovalent case ( Z = 1 , Z = −
1) is treated. Asthermal fluctuations, the statistical distributions of δφ ( r )and δn i are Gaussian in the mean field theory. We mayneglect the composition-dependence of ε for such smalldeviations. We calculate the following, S ( q ) = (cid:104)| φ q | (cid:105) e , G ij ( q ) = (cid:104) n i q n ∗ j q (cid:105) e /n , C ( q ) = (cid:104)| ρ q | (cid:105) e /e n , (3.13)where φ q , n i q , and ρ q are the Fourier components of δφ , δn i , and the charge density ρ = e ( n − n ) with wavevector q and (cid:104)· · ·(cid:105) e denotes taking the thermal average.We introduce the Bjerrum length (cid:96) B = e /εk B T and theDebye wave number κ = (8 π(cid:96) B n ) / . First, the inverse of S ( q ) is written as S ( q ) = ¯ r − ( g + g ) n Cq k B T (cid:20) − γ κ q + κ (cid:21) , (3.14)where ¯ r = f (cid:48)(cid:48) /k B T with f (cid:48)(cid:48) = ∂ f /∂φ . The secondterm is large for large ( g + g ) even for small averageion density n , giving rise to a large shift of the spin-odal curve. If g ∼ g ∼
15, this factor is of order 10 .In the previous experiments , the shift of the coexis-tence curve is typically a few Kelvins with addition of a10 − mole fraction of a hydrophilic salt like NaCl. Theparameter γ p in the third term represents asymmetry ofthe solvation of the two ion species and is defined by γ p = ( k B T / πC(cid:96) B ) / | g − g | . (3.15)If the right hand side of Eq.(3.14) is expanded with re-spect to q , the coefficient of q is C (1 − γ ) /k B T . Thusa Lifshitz point is realized at γ p = 1. For γ p > S ( q )has a peak at an intermediate wave number, q m = ( γ p − / κ. (3.16)The peak height S ( q m ) and the long wavelength limit S (0) are related by1 /S ( q m ) = 1 /S (0) − C ( γ p − κ /k B T. (3.17)A mesophase appears with decreasing ¯ r or increasing χ ,as observed by Sadakane et al. . In our mean-field the-ory, the criticality of a binary mixture disappears if asalt with γ p > etal. . recently measured anomalous scattering from D O-3MP-NaBPh stronger than that from D O-3MP with-out NaBPh . There, the observed scattering amplitudeis not well described by our S ( q ) in Eq.(3.14), requiringmore improvement.Second, retaining the fluctuations of the ion densities,we eliminate the composition fluctuations in F to obtainthe effective interactions among the ions mediated by thecomposition fluctuations. The resultant free energy ofions is written as F ion = (cid:90) d r (cid:88) i k B T n i ln( n i λ i )+ 12 (cid:90) d r (cid:90) d r (cid:48) (cid:88) i,j V ij ( | r − r (cid:48) | ) δn i ( r ) δn j ( r (cid:48) ) . (3.18)The effective interaction potentials V ij ( r ) are given by V ij ( r ) = Z i Z j e εr − g i g j A r e − r/ξ , (3.19)where Z and Z are ± A =4 πC/ ( k B T ) , and ξ = ( C/ ¯ r ) / is the correlation length.The second term in Eq.(3.19) arises from the selectivesolvation and is effective in the range a (cid:46) r (cid:46) ξ andcan be increasingly important on approaching the solventcriticality (for ξ (cid:29) a ). It is attractive among the ions ofthe same species ( i = j ) dominating over the Coulombrepulsion for g i > πC(cid:96) B /k B T. (3.20)Under the above condition there should be a tendency ofion aggregation of the same species. In the antagonisticcase ( g g < a (cid:46) r (cid:46) ξ for | g g | > πC(cid:96) B /k B T, (3.21)under which charge density waves are triggered near thesolvent criticality.The ionic structure factors can readily be calculatedfrom Eqs.(3.18) and (3.19). Some calculations give G ii ( q ) G ( q ) = 1 + n S ( q ) (cid:20) g + g − ( g − g ) u + 1) − g i u u + 1 (cid:21) ,G ( q ) = 12 + n S ( q )( g + g ) − C ( q ) ,C ( q ) = 2 uu + 1 + n S ( q ) ( g − g ) ( u + 1) u , (3.22)where u = q /κ and G ( q ) = u + 1 / u + 1 = q + κ / q + κ (3.23)is the structure factor for the cations (or for the anions)divided by n in the absence of solvation. The solva-tion parts in Eq.(3.22) are all proportional to n S ( q ),where S ( q ) is given by Eq.(3.14). The Coulomb inter-action suppresses large-scale charge-density fluctuations,so C ( q ) tends to zero as q → derived theeffective interaction among monomers on a chain ( ∝− e − r/ξ /r ) mediated by the composition fluctuations ina mixture solvent. He then predicted anomalous size be-havior of a chain near the solvent criticality. However,as shown in Eq.(3.19), the effective interaction is muchmore amplified among charged particles than among neu-tral particles. This indicates importance of the selectivesolvation for a charged polymer in a mixture solvent, evenleading to a prewetting transition around a chain . Wealso point out that an attractive interaction arises amongcharged colloid particles due to the selective solvation ina mixture solvent, on which we will report shortly. . The secondapplication is to calculate a one-dimensional liquid-liquidinterface at z = z taking the z axis in its normal direc-tion, where φ → φ α in water-rich phase α ( z − z → −∞ )and φ → φ β in oil-rich phase β ( z − z → ∞ ). InFig.2, we give numerical results of typical interface pro-files, where we measure space in units of a ≡ v / and set C = 2 . k B T /a , e /ε k B T = 3 a , and ε = ε . In theseexamples, the correlation length ξ is shorter than the Debye lengths κ − α and κ − β in the two phases. However,near the solvent criticality, ξ grows above κ − α and κ − β and we encounter another regime, which is not treatedin this review. FIG. 2: Top: normalized potential Φ( z ) / ∆Φ (left), compo-sition φ ( z ) (left), and normalized ion densities v n ( z ) and v n ( z ) (right on a semilogarithmic scale) for hydrophilic ionpairs, where χ = 2 . , g = 7 , g = 13, v n α = v n β =4 × − , and e ∆Φ = − . k B T . Bottom: those for an-tagonistic ion pairs, where χ = 2 . , g = 15 , g = − v n α = v n β = 1 . × − , and e ∆Φ = 10 . k B T . Inthese cases e /ε k B T = 3 a and ε = ε . The upper plates give φ ( z ), Φ( z ), n ( z ), and n ( z )for hydrophilic ion pairs with g = 7 and g = 13 at χ = 2 . n α = n α = 4 × − v − . The ion reduc-tion factor in Eq.(3.9) is 0 . β ) on the scale of the De-bye length κ − β = 67 . a (which is much longer than that κ − α = 6 . a in phase α ). The Galvani potential differ-ence ∆Φ is 0 . k B T /e here. The surface tension here is σ = 6 . × − k B T /a and is slightly larger than that σ = 6 . k B T /a without ions (see the next subsection).The lower plates display the same quantities for antag-onistic ion pairs with g = 15 and g = −
15 for χ = 2 . n α = n α = 1 . × − v − . The anions and thecations are undergoing microphase separation at the in-terface on the scale of the Debye lengths κ − α = 12 . a and κ − β = 9 . a , resulting in a large electric double layer anda large potential drop ( ∼ k B T /e ). The surface ten-sion here is σ = 0 . k B T /a and is about half of that σ = 0 . k B T /a without ions. This large decrease in σ is marked in view of small n α . A large decrease of thesurface tension was observed for an antagonistic salt . There have been numerousmeasurements of the surface tension of an air-water inter-face with a salt in the water region. In this case, almostall salts lead to an increase in the surface tension ,while acids tend to lower it because hydronium ions aretrapped at an air-water interface. Here, we consider the surface tension of a liquid-liquidinterface in our Ginzburg-Landau scheme, where ions canbe present in the two sides of the interface. In equilibriumwe minimize Ω = (cid:82) dzω , where ω is the grand potentialdensity, ω = f tot + 12 C |∇ φ | + ε π E − hφ − (cid:88) i µ i n i . (3.24)Using Eqs.(3.10) and (3.11) we find d ( ω + ρ Φ) /dz =2 Cφ (cid:48) φ (cid:48)(cid:48) , where φ (cid:48) = dφ/dz and φ (cid:48)(cid:48) = d φ/dz . Thus, ω = Cφ (cid:48) − ρ Φ + ω ∞ , (3.25)Since φ (cid:48) and ρ tend to zero far from the interface, ω ( z )tends to a common constant ω ∞ as z → ±∞ . The surfacetension σ = (cid:82) dz [ ω ( z ) − ω ∞ ] is then written as σ = (cid:90) dz (cid:20) Cφ (cid:48) − ε π E (cid:21) = 2 σ g − σ e , (3.26)where we introduce the areal densities of the gradient freeenergy and the electrostatic energy as σ g = (cid:90) dz Cφ (cid:48) / , σ e = (cid:90) dz ε E / π. (3.27)The expression σ = 2 σ g is well-known in the Ginzburg-Landau theory without the electrostatic interaction . FIG. 3: Surface quantities σ , 2 σ e , 2 σ g , and σ − k B T Γ + σ e inunits of k B T a − vs v n α for a monovalent, antagonistic saltwith g = − g = 15 at χ = 2 .
4. Here 2 σ g and σ − k B T Γ + σ e are close, supporting Eq.(3.30). Growth of 2 σ e gives rise tovanishing of the surface tension σ at v n α = 8 × − . In our previous work , we obtained the followingapproximate expression for σ valid for small ion densities: σ ∼ = σ − k B T Γ − σ e , (3.28) where σ is the surface tension without ions and Γ isthe adsorption to the interface. In terms of the total iondensity n = n + n , it may be expressed asΓ = (cid:90) dz (cid:20) n − n α − ∆ n ∆ φ ( φ − φ α ) (cid:21) , (3.29)where n K = n K + n K ( K = α , β ), ∆ n = n α − n β , andthe integrand tends to zero as z → ±∞ . From Eqs.(3.26)and (3.28) σ g is expressed at small ion densities as2 σ g ∼ = σ − k B T Γ + σ e . (3.30)In the Gibbs formula ( σ ∼ = σ − k B T Γ) , the electro-static contribution − σ e is neglected. However, it is cru-cial for antagonistic salt and for ionic surfactant .In Fig.3, numerical results of 2 σ e , 2 σ g , σ , and the com-bination σ − k B T Γ + σ e are plotted as functions of thebulk ion density n α = n α + n α for the antagonisticcase g = − g = 15, where χ = 2 . C = 2 . k B T /a .The parameter γ p in Eq.(3.15) exceeds unity (being equalto 1.89 for ε = 1 . ε ). In this example, σ g weaklydepends on n α and is fairly in accord with Eq.(3.30),while σ e steeply increases with increasing n α . As a re-sult, σ e increases up to σ g , leading to vanishing of σ at n α ∼ = 8 × − v − .We may understand the behavior of σ e as a functionof g and g by solving the nonlinear Poisson-Boltzmannequation , with an interface at z = 0. That is, awayfrom the interface | z | > ξ , the normalized potential U ( z ) ≡ e Φ( z ) /k B T obeys d dz U = κ K sinh( U − U K ) (3.31)in the two phases ( K = α , β ), where U α = U ( −∞ ), U β = U ( ∞ ), and κ K = (4 πn K e /ε K k B T ) / with ε K = ε + ε φ K . In solving Eq.(3.31) we assume the continuityof the electric induction − εd Φ /dz at z = 0 (but thisdoes not hold in the presence of interfacial orientation ofmolecular dipoles, as will be remarked in the summarysection). The Poisson-Boltzmann approximation for σ e is of the form, σ PBe k B T = 2 n α κ α (cid:20)(cid:112) b + 2 b cosh(∆ U / − b − (cid:21) = A s ( n α /(cid:96) B α ) / . (3.32)We should have σ e ∼ = σ PBe in the thin interface limit ξ (cid:28) κ − K . In the first line, the coefficient b is defined by b = ( ε β /ε α ) / exp[ − ( g + g )∆ φ/ , (3.33)and ∆ U = U α − U β = ( g − g )∆ φ/ (cid:96) B α = e /ε α k B T is the Bjerrum length inphase α . The second line indicates that the electrostaticcontribution to the surface tension is negative and is oforder n α / as n α → . Remarkably, the surface tension of air-waterinterfaces exhibited the same behavior at very small saltdensities (known as the Jones-Ray effect) , though ithas not yet been explained reliably .In the asymptotic limit of antagonistic ion pairs, weassume g ≥ − g (cid:29)
1, where the coefficient A s in thesecond line of eq.(3.32) grows as A s ∼ = π − / ( ε β /ε α ) / exp( | g | ∆ φ/
4) (3.34)We may also examine the usual case of hydrophilic ionpairs in water-oil, where g and g are both considerablylarger than unity. In this case A s becomes small as A s ∼ = ( ε β /πε α ) / [cosh(∆ U/ − × exp[ − ( g + g )∆ φ/ . (3.35)In this case, the electrostatic contribution − σ e ( ∝ n / α ) in σ could be detected only at extremely small salt densities.Analogously, between ionic and nonionic liquids, Aerov el al. calculated the surface tension. They showed thatif the affinities of cations and anions to neutral moleculesare very different, the surface tension becomes negative. .Adding an antagonistic salt with γ p > χ below Eq.(3.15) and vanishing of the surface ten-sion σ with increasing the ion content as in Fig.3. Insuch cases, a thermodynamic instability is induced withincreasing χ at a fixed ion density n = (cid:104) n (cid:105) = (cid:104) n (cid:105) ,leading to a mesophase. To examine this phase order-ing, we performed two-dimensional simulations andpresented an approximate phase diagram . We herepresent preliminary three-dimensional results. The pat-terns to follow resemble those in block copolymers andsurfactant systems . In our case, mesophases emergedue to the selective solvation and the Coulomb interac-tion without complex molecular structures. Solvation-induced mesophase formation can well be expected inpolyelectrolytes and mixtures of ionic and polar liquids.We are interested in slow composition evolution withantagonistic ion pairs, so we assume that the ion distri-butions are given by the modified Poisson-Boltzmann re-lations in Eq.(3.12). The water composition φ obeys ∂φ∂t + ∇ · ( φ v ) = L k B T ∇ h, (3.36)where L is the kinetic coefficient and h is defined byEq.(3.10). Neglecting the acceleration term, we deter-mine the velocity field v using the Stokes approximation, η ∇ v = ∇ p + φ ∇ h + (cid:88) i n i ∇ µ i , (3.37)where η is the shear viscosity and µ i are defined byEq.(3.11), We introduce p to ensure the incompressibil-ity condition ∇ · v = 0. The right hand side of Eq.(3.37)is also written as ∇ · Π ↔ , where Π ↔ is the stress tensor aris-ing from the fluctuations of φ and n i . Here the total free FIG. 4: Composition patterns at t = 5000 t for χ = 2 . v ¯ n = 3 × − with average composition (cid:104) φ (cid:105) being 0 . . g = − g = 10.These patterns are nearly pinned. Yellow surfaces are orientedto the regions of φ > . φ < . n (left) and anion n (right) in the x - y plane at z = 0 for antagonistic salt usingdata in Fig. 4. The domains are bicontinuous for (cid:104) φ (cid:105) = 0 . (cid:104) φ (cid:105) = 0 . energy F in Eq.(3.1) satisfies dF/dt ≤ × ×
64 lattice under theperiodic boundary condition. The system was quenchedto an unstable state with χ = 2 . t = 0. Space andtime are measured in units of a = v / and t = a /L ,respectively. Without ions, the diffusion constant of thecomposition is given by L f (cid:48)(cid:48) /k B T in one-phase statesin the long wavelength limit (see Eq.(3.14)). We set g = − g = 10, n = 3 × − v − , C = k B T / a , e /ε k B T = 3 a , ε = 0, and η L /k B T = 0 . a .In Fig.4, we show the simulated domain patterns at t = 5000 t , where we can see a bicontinuous structure for (cid:104) φ (cid:105) = 0 . (cid:104) φ (cid:105) = 0 .
4. There
FIG. 6: Normalized structure factor S ( q ) /v for antagonisticsalt from domain structures in pinned states at t = 5000 t ,where the patterns are bicontinuous at for (cid:104) φ (cid:105) = 0 .
5. A peakheight S ( q m ) /v of each curve much exceed unity. The aver-age ion density is v n = 0 . , . q < q m ). is almost no further time evolution from this stage. InFig.5, the ion distributions are displayed for these twocases in the x - y plane at z = 0. For (cid:104) φ (cid:105) = 0 .
5, the iondistributions are peaked at the interfaces forming elec-tric double layers (as in the right bottom plate of Fig.2).For (cid:104) φ (cid:105) = 0 .
4, the anions are broadly distributed in thepercolated oil region, but we expect formation of electricdouble layers with increasing the domain size also in theoff-critical condition. In Fig.6, the structure factor S ( q )in steady states are plotted for v n = 0 . , . n inaccord with Eq.(3.16). Sadakane et al. observed thestructure factor similar to those in Fig.6.Finally, we remark that the thermal noise, which isabsent in our simulation, should be crucial near the crit-icality of low-molecilar-weight solvents. It is needed toexplain anomalously enhanced composition fluctuationsinduced by NaBPh near the solvent criticality . IV. IONIC SURFACTANT WITHAMPHIPHILIC AND SOLVATIONINTERACTIONS . In this section, wewill give a diffuse-interface model of ionic surfactants ,where surfactant molecules are treated as ionized rods.Their two ends can stay in very different environments(water and oil) if they are longer than the interface thick-ness ξ . In our model, the adsorption of ionic surfac-tant molecules and counterions to an oil-water interfacestrongly depends on the selective solvation parameters g and g and that the surface tension contains the electro-static contribution as in Eqs.(3.26) and (3.28).We add a small amount of cationic surfactant, anionic counterions in water-oil in the monovalent case. The den-sities of water, oil, surfactant, and counterion are n A , n B , n , and n , respectively. The volume fractions ofthe first three components are φ A = v n A , φ B = v n B ,and v n , where v is the common molecular volume ofwater and oil and v is the surfactant molecular volume.The volume ratio N = v /v can be large, so we do notneglect the surfactant volume fraction, while we neglectthe counterion volume fraction supposing a small size ofthe counterions. We assume the space-filling condition, φ A + φ B + v n = 1 . (4.1)Let 2 ψ = φ A − φ B be the composition difference betweenwater and oil; then, φ A = (1 − v n ) / ψ,φ B = (1 − v n ) / − ψ. (4.2)The total free energy F is again expressed as inEq.(3.1). Similarly to Eq.(3.2), the first part reads f tot k B T = 1 v [ φ A ln φ A + φ B ln φ B + χφ A φ B ]+ (cid:88) i n i [ln( n i λ i ) − g i n i ψ ] − n ln Z a , (4.3)The coefficients g and g are the solvation parametersof the ionic surfactant and the counterions, respectively.Though a surfactant molecule is amphiphilic, it can havepreference to water or oil on the average. The last termrepresents the amphiphilic interaction between the sur-factant and the composition. That is, Z a is the partitionfunction of a rod-like dipole with its center at the posi-tion r . We assume that the surfactant molecules take arod-like shape with a length 2 (cid:96) considerably longer than a = v / . It is given by the following integral on thesurface of a sphere with radius (cid:96) , Z a ( r ) = (cid:90) d Ω4 π exp (cid:20) w a ψ ( r − (cid:96) u ) − w a ψ ( r + (cid:96) u ) (cid:21) , (4.4)where u is the unit vector along the rod direction and (cid:82) d Ω represents the integration over the angles of u . Thetwo ends of the rod are at r + (cid:96) u and r − (cid:96) u under theinfluence of the solvation potentials given by k B T w a ψ ( r + (cid:96) u ) and − k B T w a ψ ( r − (cid:96) u ). The parameter w a representsthe strength of the amphiphilic interaction.Adsorption is strong for large w a ∆ ψ (cid:29)
1, where ∆ ψ = ψ α − ψ β ( ∼ = φ Aα − φ Aβ ) is the difference of ψ between thetwo phases α and β . In the one-dimensional case, all thequantities vary along the z axis and Z a is rewritten as Z a ( z ) = (cid:90) (cid:96) − (cid:96) dζ (cid:96) exp[ w a ψ ( z − ζ ) − w a ψ ( z + ζ )] . (4.5)where ζ = (cid:96)u z . In the thin interface limit ξ (cid:28) (cid:96) , we placethe interface at z = 0 to find Z a = 1 for | z | > (cid:96) , while Z a ( z ) ∼ = 1 + (1 − | z | /(cid:96) )[cosh( w a ∆ ψ ) − , (4.6)0for | z | < (cid:96) . Furthermore, in the dilute limit v n (cid:28) n ( z ) = n α Z a ( z ) for z < n ( z ) = n β Z a ( z ) for z > n α and n β = e − g ∆ ψ n α are the bulk surfactantdensities. The surfactant adsorption then grows asΓ = (cid:90) −∞ dz [ n ( z ) − n α ] + (cid:90) ∞ dz [ n ( z ) − n β ]= ( n α + n β ) (cid:96) [cosh( w a ∆ ψ ) − / . (4.7)However, the steric effect comes into play at the interfacewith increasing the surfactant volume fraction at the in-terface ( ∼ Γ v /(cid:96) ). FIG. 7: Profiles for mixtures with cationic surfactant andanionic counterions with v = 5 v and w a = 12. Top: v n (bold line), φ A , and φ B . Middle: v n and v n . Bottom: e Φ /k B T exhibiting a maximum at the interface. Here g = 4, g = 10, and v n α = 10 − (left), while g = − g = 8 and v n α = 3 . × − (right). The counterion distribution hasa peak in the phase α (left) or β (right) depending on g .[From: A. Onuki, Europhys. Lett. , 58002 (2008)]. . We give typical one-dimensionalinterface profiles varying along the z -axis in Fig.7. We set v = 5 v , C = 3 k B T /a , χ = 3, and e /aε c k B T = 16 /π .The dielectric constant is assumed to be of the form ε = ε c (1 + 0 . ψ ), where ε c is the critical value. Then FIG. 8: σa /k B T , ( σ + σ e ) a /k B T , and Γ a as functionsof v n α with v = 5 v , χ = 3, and w a = 12. The curveschange on a scale of 10 − for hydrophilic ion pair g = 4 and g = 10 (top) and on a scale of 10 − for antagonistic ion pair g = − g = 8 (bottom). [From: A. Onuki, Europhys. Lett. , 58002 (2008)]. ε α ∼ = 2 ε β at χ = 3. This figure was produced in thepresence of the image interaction in our previous work (though it is not essential here).In Fig.7, we show the volume fractions φ A , φ B , and v n = 1 − φ A − φ B (top), the ion densities n and n (middle), and the potential e Φ /k B T with Φ α = 0(bottom). In the left, the counterions are more hy-drophilic than the cationic surfactant, where g = 4 and g = 10 leading to Γ = 0 . a − and σ = 0 . k B T a − at n α = 10 − v − . In the right plates, the surfac-tant cations are hydrophilic and the counterions are hy-drophobic, where g = − g = 8 leading to Γ = 0 . a − and σ = 0 . k B T a − at n α = 3 . × − v − . Thedistribution of the surfactant n is narrower than thatof the counterions n . This gives rise to a peak of Φ at z = z p , at which E ( z p ) ∝ (cid:82) z p −∞ dz ( n ( z ) − n ( z )) = 0.The adsorption strongly depends on the solvation pa-rameters g and g . It is much more enhanced for antag-onistic ion pairs than for hydrophilic ion pairs. . The grand potential density ω is again given by Eq.(3.24) and tends to a commonconstant ω ∞ as z → ±∞ , though its form is more com-plicated. The surface tension σ = (cid:82) dz [ ω ( z ) − ω ∞ ] isrewritten as Eq.(3.26) and is approximated as Eq.(3.28)for small n α . The areal electrostatic-energy density σ e in Eq.(3.27) is again important in the present case.1In Fig.8, we show σ , σ + σ e , and Γ as functions of v n α at w a = 12, where Γ is defined as in Eq.(3.28) for n = n + n . In the upper plate, the two species of ions areboth hydrophilic ( g = 4 and g = 10), while in the lowerplate the surfactant and the counterions are antagonistic( g = 8 and g = − σ even at very small n α . In thepresent case the Gibbs term k B T Γ is a few times largerthan σ e . Note that the approximate formula (3.28) maybe derived also in this case, but it is valid only for verysmall n α in Fig.8. V. PHASE SEPARATION DUE TO STRONGSELECTIVE SOLVATION .With addition of a strongly selective solute in a binarymixture in one-phase states, we predict precipitation ofdomains composed of the preferred component enrichedwith the solute . These precipitation phenomena occurboth for a hydrophilic salt (such as NaCl) and a neutralhydrophobic solute . In our scheme, a very large sizeof the selective solvation parameter g i is essential. InSecs.2 and 3, we have shown that | g i | can well exceed 10both for hydrophilic and hydrophobic solutes.With hydrophilic cations and anions, a charge den-sity appears only near the interfaces, shifting the surfacetension slightly. Thus, in the static aspect of precipita-tion, the electrostatic interaction is not essential, whilefusion of precipitated domains should be suppressed bythe presence of the electric double layers. We will firsttreat a hydrophilic neutral solute as a third component,but the following results are applicable also to a neutralhydrophobic solute if water and oil are exchanged. In ad-dition, in a numerical example in Fig.11, we will includethe electrostatic interaction among hydrophilic ions. . Addinga small amount of a highly selective solute in water-oil,we assume the following free energy density, f tot ( φ, n ) = f ( φ ) + k B T n [ln( nλ ) − − gφ ] . (5.1)This is a general model for a dilute solute. For monova-lent electrolytes, this form follows from Eq.(3.11) if thereis no charge density or if we set n = n = n/ , g = ( g + g ) / , (5.2)The first term f ( φ ) is assumed to be of the Bragg-Williams form (3.3). The λ is the thermal de Broglielength. The solvation term ( ∝ g ) arises from the so-lute preference of water over oil (or oil over water). Thestrength g is assumed to much exceed unity . We fix theamounts of the constituent components in the cell witha volume V . Then the averages ¯ φ = (cid:104) φ (cid:105) = (cid:82) d r φ/V and¯ n = (cid:104) n (cid:105) = (cid:82) d r n/V are given control parameters as wellas χ . In two phase coexistence in equilibrium, let the com-position and the solute density be ( φ α , n α ) in phase α and ( φ β , n β ) in phase β , where φ α > ¯ φ > φ β and n α > ¯ n > n β . We introduce the chemical potentials h = ∂f tot /∂φ and µ = ∂f tot /∂n . Equation (5.1) yields h = f (cid:48) ( φ ) − k B T gn, (5.3) µ/k B T = ln( nλ ) − gφ, (5.4)where f (cid:48) = ∂f /φ . The system is linearly stable for ∂h/∂φ − ( ∂h/∂n ) /∂µ/∂n > f (cid:48)(cid:48) ( φ ) − k B T g n > , (5.5)where f (cid:48)(cid:48) = ∂ f /∂φ . Spinodal decomposition occurs ifthe left hand side of Eq.(5.5) is negative.The homogeneity of µ yields n = ¯ ne gφ / (cid:104) e gφ (cid:105) , (5.6)where (cid:104)· · ·(cid:105) denotes taking the space average. The bulksolute densities are n K = ¯ ne gφ K / (cid:104) e gφ (cid:105) for K = α, β intwo-phase coexistence. In our approximation Eq.(5.6)holds even in the interface regions. We write the volumefraction of phase α as γ α . We then have (cid:104) e gφ (cid:105) = γ α e gφ α +(1 − γ α ) e gφ β in Eq.(5.6). In terms of ¯ φ and ¯ n , γ α isexpressed as γ α = ( ¯ φ − φ β ) / ∆ φ = (¯ n − n β ) / ∆ n. (5.7)where ∆ φ = φ α − φ β > n = n α − n β >
0. Inthese expressions we neglect the volume of the interfaceregions. Since n α /n β = e g ∆ φ (cid:29) g ∆ φ (cid:29)
1, thesolute is much more enriched in phase α than in phase β .Eliminating n using Eq.(5.6), we may express the averagefree energy density as (cid:104) f tot (cid:105) = (cid:104) f (cid:105) − k B T ¯ n ln[ (cid:104) e gφ (cid:105) ] + A , (5.8)where A = k B T ¯ n [ln(¯ nλ ) −
1] is a constant at fixed ¯ n .In terms of φ α , φ β , and γ α , Eq.(5.8) is rewritten as (cid:104) f tot (cid:105) = [ γ α f ( φ α ) + (1 − γ α ) f ( φ β )] − k B T ¯ n ln[ γ α e gφ α + (1 − γ α ) e gφ β ] + A . (5.9)The second term ( ∝ ¯ n ) is relevant for large g (even forsmall ¯ n ). Now we should minimize (cid:104) f tot (cid:105) − h [ γ α φ α +(1 − γ α ) φ β − ¯ φ ] with respect to φ α , φ β , and γ α at fixed ¯ φ ,where h appears as the Lagrange multiplier. Then we ob-tain the equilibrium conditions of two-phase coexistence, h = f (cid:48) ( φ α ) − k B T gn α = f (cid:48) ( φ β ) − k B T gn β , (5.10) f ( φ α ) − f ( φ β ) − k B T ∆ n = h ∆ φ. (5.11)These static relations hold even for ion pairs underEq.(5.2). .In Fig.9, we give numerical results on the phase behaviorof φ α and φ β in the left and n α and γ α in the rightas functions of χ . We set g = 11, ¯ φ = 0 .
35 and ¯ n =2 FIG. 9: Left: compositions φ α and φ β vs χ . Right: semi-logarithmic plots of volume fraction γ α and normalized solutedensity n α / ¯ n of the water-rich phase α vs χ . Here g = 11,¯ φ = 0 .
35, and ¯ n = 6 × − v − (top), while g = 10 .
5, ¯ φ =0 .
58, and ¯ n = 4 × − v − (bottom). Plotted in the leftalso are the coexisting region without solute (in right green)and the spinodal curve with solute (on which the left handside of Eq.(5.5) vanishes). See Fig.11 for simulation of phaseseparation on point (A) in the left bottom panel. [Upperplates are from: R. Okamoto and A. Onuki, Phys. Rev. E , 051501 (2010).] × − v − in the top plates and g = 10 .
5, ¯ φ = 0 . n = 4 × − v − in the bottom plates. The solutedensity is much larger in the latter case. Remarkably, aprecipitation branch appears in the range, χ p < χ < . (5.12)The volume fraction γ α decreases to zero as χ approachesthe lower bound χ p = χ p ( ¯ φ, ¯ n ). Without solute, the mix-ture would be in one-phase states for χ <
2. The pre-cipitated domains are solute-rich with φ α ∼ = 1, while φ β is slightly larger than ¯ φ . In the left upper plate φ α in-creases continuously with decreasing χ , while in the leftlower plate φ α jumps at χ = 1 .
937 and hysteresis appearsin the region 1 . < χ < . f (cid:48)(cid:48) ( ¯ φ ) − k B T g ¯ n = 0 , following from Eq.(5.5).Outside this curve, homogeneous states are metastableand precipitation can proceed via homogeneous nucle-ation in the bulk or via heterogeneous nucleation on hy-drophilic surfaces of boundary plates or colloids . Insidethis curve, the system is linearly unstable and precipita-tion occurs via spinodal decomposition. This unstableregion is expanded for ¯ n = 4 × − v − in the lowerplate. FIG. 10: Left: χ p ( ¯ φ, ¯ n ) vs ¯ φ for three values of ¯ n at g = 11,which nearly coincide with the asymptotic formula (5.22)(dotted line) for ¯ φ < .
35 and converge to the coexistencecurve for larger ¯ φ . Coexistence region without ions is in theupper region (in right green). Right: v n p ( ¯ φ, χ ) vs ¯ φ on asemi-logarithmic scale at g = 11, It coincides with the asymp-totic formula (5.23) (dotted line) for χ < , 051501(2010).] g .We present a theory of the precipitation branch in thelimit g (cid:29) χ p and n p . Assuming thebranch (5.12) at the starting point, we confirm its ex-istence self-consistently.We first neglect the term − k B T gn β in Eq.(5.10) from gv n β (cid:28) f ( φ α ) in Eq.(5.11) from φ α ∼ = 1.In fact gv n β (cid:28) h ∼ = f (cid:48) ( φ β ) ∼ = − [ f ( φ β ) + k B T n α ] / (1 − φ β ) . (5.13)This determines the solute density n α in phase α as afunction of φ β in the form, n α ∼ = G ( φ β ) /k B T, (5.14)where G ( φ ) is a function of φ defined as G ( φ ) = − f ( φ ) − (1 − φ ) f (cid:48) ( φ )= − ( k B T /v )[ln φ + χ (1 − φ ) ] . (5.15)From dG/dφ = − (1 − φ ) f (cid:48)(cid:48) ( φ ) < G (1) = 0, wehave G ( φ ) > n α > f (cid:48) ( φ α ) to obtain v − k B T [ − ln(1 − φ α ) − χ − gv n α ] ∼ = f (cid:48) ( φ β ) , (5.16)where the logarithmic term ( ∝ ln(1 − φ )) balances withthe solvation term ( ∝ gn α ). Use of Eq.(5.15) gives1 − φ α ∼ = A β exp[ − gv G ( φ β ) /k B T ] , (5.17)where the coefficient A β is given by A β = exp[ − χ − v f (cid:48) ( φ β ) /k B T ] , (5.18)3so A β is of order unity. The factor exp[ − gv G ( φ β ) /k B T ]in Eq.(5.17) is very small for g (cid:29)
1, leading to φ α ∼ = 1.Furthermore, from Eqs.(5.6) and (5.7), the volumefraction γ α of phase α is approximated as γ α ∼ = ¯ n/n α − e − g ∆ φ ∼ = k B T ¯ n/G ( φ β ) − e − g ∆ φ . (5.19)The above relation is rewritten as G ( φ β ) ∼ = k B T ¯ n/ ( γ α + e − g ∆ φ ) ∼ = k B T ¯ n (1 − φ β )¯ φ − φ β + (1 − φ β ) exp[ − g (1 − φ β )] . (5.20)From the first to second line, we have used Eq.(5.7) andreplaced ∆ φ by 1 − φ β . This equation determines φ β and γ α ∼ = ( ¯ φ − φ β ) / (1 − φ β ). We recognize that G ( φ β )increases up to T ¯ ne g ∆ φ ∼ = T ¯ ne g (1 − ¯ φ ) as γ α → φ β → ¯ φ . In this limit it follows the marginal relation, G ( ¯ φ ) ∼ = k B T ¯ ne g (1 − ¯ φ ) ( γ α → . (5.21)If ¯ n is fixed, this relation holds at χ = χ p so that χ p ∼ = [ − ln ¯ φ − v ¯ ne g (1 − ¯ φ ) ] / (1 − ¯ φ ) , (5.22)where we use the second line of Eq.(5.15). Here ¯ n appearsin the combination ¯ ne g (1 − ¯ φ ) ( (cid:29) ¯ n ). On the other hand, if χ is fixed, Eq.(5.21) holds at ¯ n = n p . Thus the minimumsolute density n p is estimated as n p ∼ = e − g (1 − ¯ φ ) G ( ¯ φ ) /k B T, (5.23)which is much decreased by the small factor e − g (1 − ¯ φ ) .In Fig.10, the curves of χ p and n p nearly coincide withthe asymptotic formulas (5.22) and (5.23) in the range¯ φ < .
35 for χ p and in the range χ < n p . Theyexhibit a minimum at ¯ φ ∼ e − g /v ¯ ng for χ p and at ¯ φ ∼ g − for n p . For larger ¯ φ > . χ p nearly coincide withthe coexistence curve, indicating disappearance of theprecipitation branch. Notice that n p decreases to zeroas ¯ φ approaches the coexistence composition φ cx = 0 . χ = 2 . χ = 2 . . In our theory we investigatedsolute-induced nucleation starting with homogeneousmetastable states outside the spinodal curve (dottedline) in the left panels of Fig.9. Here, we show two-dimensional numerical results of spinodal decompositionfor hydrophilic ions with g = 12, g = 9, e /ε k B T =3 a , and ε = ε . At t = 0, we started with point (A)inside the spinodal curve in the left lower panel in Fig.9using the common values of the static parameters givenby χ = 1 .
92, ¯ φ = 0 .
58, and ¯ n = (cid:104) n + n (cid:105) = 4 × − /v .In addition, we set C = 2 k B T a /v . On a 256 × φ ( r , t ) with the velocity field v ( r , t ) being determined by the Stokes ap-proximation in Eq.(3.37). The cations and anions obey ∂n i ∂t + ∇ · ( n i v ) = Dk B T ∇ · n ∇ µ i = D ∇ · (cid:20) ∇ n i − g i n i ∇ φ − Z i en i E (cid:21) , (5.24)where i = 1 ,
2. The chemical potentials µ i are definedin Eq.(3.11) and the ion diffusion constants are com-monly given by D . The space mesh size is a = v /d with d = 2. We measure time in units of t = a v /L and set D = a /t and η = 0 . k B T t /v , where L isthe kinetic coefficient for the composition and η is theshear viscosity.In Fig.11, we show the time evolution of the dropletvolume fraction, which is the fraction of the region φ > .
6. In the early stage, the droplet number decreasesin time with the evaporation and condensation mecha-nism. In the late stage, it changes very slowly tendingto a constant. In our simulation without random noise,the droplets do not undergo Brownian motion and thedroplet collision is suppressed. We also performed a sim-ulation for a neutral solute with g = 10 . FIG. 11: Time evolution of droplet volume fraction (that ofthe region φ > .
6) and composition snapshots of precipitateddroplets at three times. They are induced by hydrophilic ionswith g = 12 and g = 9. The initial state was at point (A)in Fig.9 with χ = 1 .
92, ¯ φ = 0 .
58, and ¯ n = 4 × − v − . Thedroplet volume fraction nearly tends to a constant. VI. THEORY OF POLYELECTROLYTES . In this sec-tion, we consider weakly charged polymers in a theta4or poor, one-component water-like solvent in the semidi-lute case φ > N − / . Following the literature of poly-mer physics , we use φ and N to represent the poly-mer volume fraction and the polymerization index. Herecharged particles interact differently between unchargedmonomers and solvent molecules. The selective solvationshould become more complicated for mixture solvents, asdiscuused in Sec.1.To ensure flexibility of the chains, we assume that thefraction of charged monomers on the chains, denoted by f ion , is small or f ion (cid:28)
1. From the scaling theory , thepolymers consist of blobs with monomer number g b = φ − with length ξ b = ag / b = aφ − . The electrostaticenergy within a blob is estimated as (cid:15) b = k B T ( f ion g ) (cid:96) B /ξ b = k B T f (cid:96) B /φ a. (6.1)where (cid:96) B is the Bjerrum length. The blobs are not muchdeformed under the weak charge condition (cid:15) b < k B T ,which is rewritten as φ > f / ( (cid:96) B /a ) / . (6.2) . The number of theionizable monomers (with charge − e ) on a chain is writ-ten as ν M N with ν M <
1. Then the degree of ionization(or dissociation) is ζ = f ion /ν M and the number densityof the ionized monomers is n p = v − f ion φ = v − ν M ζφ, (6.3)The charge density is expressed as ρ = e (cid:88) i = c, , Z i n i − en p . (6.4)Here i = c represents the counterions, i = 1 the addedcations, and i = 2 the added anions. The required rela-tion f ion (cid:28) ν M ζ (cid:28) ζ if ν M (cid:28) F accounting for the molec-ular interactions and the ionization equilibrium . Then F assumes the standard form (3.1), where the coefficientof the gradient free energy is written as C ( φ ) = k B T / aφ (1 − φ ) , (6.5)in terms of the molecular length a = v / and φ . The f tot consists of four parts as f tot = f ( φ ) + k B T (cid:88) i = c, , n i [ln( n i λ i ) − g i φ ]+ k B T (∆ + g p φ ) n p + f dis . (6.6)The first term f is of the Flory-Huggins form , v fk B T = φN ln φ + (1 − φ ) ln(1 − φ ) + χφ (1 − φ ) . (6.7)The coupling terms ( ∝ g i , g p ) arise from the molecularinteractions among the charged particles (the ions and the charged monomers) and the uncharged particles (thesolvent particles and the uncharged monomers), while k B T ∆ is the dissociation free energy in the dilute limitof polymers ( φ → f tot arises fromthe dissociation entropy on chains , v k B T f dis = ν M φ (cid:20) ζ ln ζ + (1 − ζ ) ln(1 − ζ ) (cid:21) . (6.8) . If F is minimizedwith respect to ζ , it follows the equation of ionizationequilibrium or the mass action law, n c ζ/ (1 − ζ ) = K ( φ ) , (6.9)where n c is the counter ion density and K ( φ ) is the dis-sociation constant of the form, K ( φ ) = v − exp[ − ∆ − ( g p + g c ) φ ] . (6.10)We may interpret k B T [∆ + ( g p + g c ) φ ] as thecomposition-dependent dissociation free energy. With in-creasing the polymer volume fraction φ , the dissociationdecreases for positive g p + g c and increases for negative g p + g c . If g p + g c (cid:29) K ( φ ) much decreases even for asmall increase of φ . Here K ( φ ) has the meaning of thecrossover counterion density since ζ is expressed as ζ = 1 / [1 + n c /K ( φ )] , (6.11)which decreases appreciably for n c > K ( φ ).In particular, if there is no charge density and no salt( n p = n c and n = n = 0), n c satisfies the quadraticequation n c ( n c + K ) = v − ν M φK, which is solved to give ζ = v n c /ν M φ = 2 / ( (cid:112) Q ( φ ) + 1 + 1) . (6.12)Here it is convenient to introduce Q ( φ ) = 4 ν M φ/v K ( φ ) . (6.13)We find ζ (cid:28) n c ∼ = ( ν M φK/v ) / for Q (cid:29)
1, while ζ → Q (cid:28)
1. The relation (6.12) holds approxi-mately for small charge densities without salt. . As in Sec.3, it is straightfor-ward to calculate the structure factor S ( q ) for the fluc-tuations of φ on the basis of f tot in Eq.(6.6). As a func-tion of the wave number q , it takes the same functionalform as in Eq.(3.14), while the coefficients in the poly-electrolyte case are much more complicated than those inthe electrolyte case. That is, the shift − ( g + g ) n / r dependent on n i and n p (for which see our paper ). In the followingexpressions (Eqs.(6.14)-(6.17)), n i , n p , and φ representthe average quantities. The Debye wave number of poly-electrolytes is given by κ = 4 π(cid:96) B (cid:20) (1 − ζ ) n p + (cid:88) i Z i n i (cid:21) , (6.14)5which contains the contribution from the (monovalent)ionized monomers ( ∝ n p ). The asymmetry parameter γ p in Eq.(3.14) is of the form, γ p = (4 π(cid:96) B k B T /C ) / A/κ , (6.15)where C is given by Eq.(6.5) and A = n p /φ − (1 − ζ ) g p n p + (cid:88) i Z i g i n i . (6.16)In this definition, γ p can be negative depending on theterms in A . Mesophase formation can appear for | γ p | > χ .The parameter γ p is determined by the ratios amongthe charge densities and is nonvanishing even in the dilutelimit of the charge densities. In particular, if n c = n p and n = n in the monovalent case, γ p is simplified as γ p = φ − − (1 − ζ ) g p + g c + ( g − g ) R (4 π(cid:96) B C/k B T ) / (2 − ζ + 2 R ) , (6.17)where the counterions and the added cations are differ-ent. The R ≡ n /n p is the ratio between the salt den-sity and the that of ionized monomers and Eq.(6.5) gives(4 π(cid:96) B C/k B T ) / = [ π(cid:96) B / aφ (1 − φ )] / .Some consequences follow from Eq.(6.17). (i) Withenriching a salt we eventually have R (cid:29) | g i | ; then, theabove formula tends to Eq.(3.15), which is applicable forneutral polymer solutions (and low-molecular-weight bi-nary mixtures for N = 1) with salt. (ii) Without thesolvation or for g i = 0, the above S ( q ) tends to the previ-ous expressions for polyelectrolytes , where γ p de-creases with increasing R . In accord with this, Braunet al. observed a mesophase at low salt contents andmacrophase separation at high salt contents. (iii) In ourtheory, neutral polymers in a polar solvent can exhibit amesophase for large | g − g | .Hakim et al . found a broad peak at an intermedi-ate wave number in the scattering amplitude in (neutral)polyethylene-oxide (PEO) in methanol and in acetoni-trile by adding a small amount of salt KI. They ascribedthe origin of the peak to binding of K + to PEO chains.Here more experiments are informative. An experimentby Sadakane et al. suggests that use of an antagonisticsalt would yield mesophases more easily. . We supposecoexistence of two salt-free phases ( n = n = 0), sepa-rated by a planar interface. Even without salt, the inter-face profiles are extremely varied, sensitively dependingon the molecular interaction parameters, ∆ , g p , and g c .If a salt is added, they furthermore depend on g , g , andthe salt amount. The quantities with the subscript α ( β )denote the bulk values in the polymer-rich (solvent-rich)phase attained as z → −∞ (as z → ∞ ). The ratio of thebulk counterion densities is given by n cα n cβ = φ α ζ α φ β ζ β = exp (cid:20) − g c ∆ φ − e ∆Φ k B T (cid:21) . (6.18) The Galvani potential difference ∆Φ = Φ α − Φ β is ex-pressed in terms of Q ( φ ) in Eq.(6.13) as e ∆Φ k B T = g p ∆ φ + ln (cid:20) (cid:112) Q ( φ β ) + 1 − (cid:112) Q ( φ α ) + 1 − (cid:21) , (6.19)If Q ( φ α ) (cid:29) Q ( φ β ) (cid:29) ζ α (cid:28) ζ β (cid:28) e ∆Φ /k B T ∼ = ( g p − g c )∆ φ/ φ β /φ α ). FIG. 12: Interface profiles in the salt-free case for (a) ∆ = 5, g p = 1, and g c = 4 (top) and for (b) ∆ = 8, g p = 2, and g c = − φ ( z ), normalizedpotential Φ( z ) / Φ M , and degree of ionization ζ ( z ) (left), andnormalized charge densities v n c ( z ) and v n p ( z ) (right). Theother parameters are common as χ = 1, N = 20, ν M = 0 . ε = − . ε , and (cid:96) B = 8 a/π . Here Φ( z ) is measured fromits minimum, and Φ M (= 2 . k B T /e in (a) and 6 . k B T /e in(b)) is the difference of its maximum and minimum. [From:A. Onuki and R. Okamoto, J. Phys. Chem. B, , 3988(2009).]
We give numerical results of one-dimensional profiles inequilibrium, where we set χ = 1, N = 20, ε = − . ε ,and (cid:96) B = e /ε T = 8 a/π . The dielectric constant ofthe solvent is 10 times larger than that of the polymer.The space will be measured in units of the molecular size a = v / . In Fig.12, we show salt-free interface profilesfor (a) ∆ = 5, g p = 1, and g c = 4 and (b) ∆ = 8, g p = 2, and g c = −
6. In the α and β regions, the degreeof ionization ζ is 0 .
071 and 0 .
65 in (a) and is 0 .
24 and6
FIG. 13: Periodic profiles in a salt-free mesophase with ν M =0 .
5, ∆ = 5, g p = 1, and g c = 4. Top: φ ( z ). Φ( z ) / Φ M with Φ M = 0 . T /e , and ζ ( z ). Bottom: v n c ( z ) and v n p ( z ).[From: A. Onuki and R. Okamoto, J. Phys. Chem. B, ,3988 (2009).] .
51 in (b), respectively. The normalized potential drop e (Φ α − Φ β ) /k B T is − .
67 in (a) and 0 .
099 in (b). In-terestingly, in (b), Φ( z ) exhibits a deep minimum at theinterface position. We can see appearance of the chargedensity n c − n p around the interface, resulting in an elec-tric double layer. The counterion density is shifted tothe β region in (a) because of positive g c and to the α region in (b) because of negative g c . The parameter γ p in Eq.(6.16) is 0 .
75 in (a) and 0 .
20 in (b) in the α region,ensuring the stability of the α region.The surface tension σ is again expressed as in Eq.(3.26)with the negative electrostatic contribution. It is calcu-lated as σ = 0 . k B T /a in (a) and as 0 . k B T /a in (b), while we obtain σ = 0 . k B T /a without ions atthe same χ = 1. In (a) σ is largely decreased because theelectrostatic term σ e in Eq.(3.27) is increased due to theformation of a large electric double layer. In (b), on thecontrary, it is increased by 10% due to depletion of thecharged particles from the interface .We mention calculations of the interface profiles inweakly charged polyelectrolytes in a poor solvent usingself-consistent field theory . In these papers, however,the solvation interaction was neglected. . With varyingthe temperature (or χ ), the average composition (cid:104) φ (cid:105) , theamount of salt, there can emerge a number of mesophases sensitively depending on the various molecular parame-ters ( g i , ∆ , and ν M ). In Fig.13, we show an exampleof a one-dimensional periodic state without salt. Here ν M is set equal to 0 . VII. SUMMARY AND REMARKS
In this review, we have tried to demonstrate the crucialrole of the selective solvation of a solute in phase tran-sitions of various soft materials. We have used coarse-grained approaches to investigate mesoscopic solvationeffects. Selective solvation should be relevant in under-standing a wide range of mysterious phenomena in wa-ter. Particularly remarkable in polar binary mixtures aremesophase formation induced by an antagonistic salt andprecipitation induced by a one-sided solute (a salt com-posed of hydrophilic cations and anions and a neutral hy-drophobic solute). Regarding the first problem, our the-ory is still insufficient and cannot well explain the compli-cated phase behavior disclosed by the experiments .To treat the second problem, we have started with thefree energy density f tot in Eq.(5.1), which looks ratherobvious but yields highly nontrivial results for large g .Systematic experiments are now possible. In particu-lar, this precipitation takes place on colloid surfaces as aprewetting phase transition near the precipitation curve χ = χ p as in Fig.10 .Though still preliminary, we have also treated an ionicsurfactant system, where added in water-oil are cationicsurfactant, anionic counterions, and ions from a salt. Inthis case, we have introduced the amphiphilic interactionas well as the solvation interaction to study the interfaceadsorption. For ionic surfactants, the Gibbs formula for the surface tension is insufficient, because it neglectsthe electrostatic interaction.In polyelectrolytes, the charge distributions are ex-tremely complex around interfaces and in mesophases,sensitively depending on the molecular interaction andthe dissociation process. Our continuum theory takesinto account these effects in the simplest manner, thoughour results are still fragmentary. Salt effects in polyelec-trolytes should also be further studied, on which somediscussions can be found in our previous paper . In thefuture, we should examine phase separation processes inpolyelectrolytes, where the composition, the ion densi-ties, and the degree of ionization are highly inhomoge-neous. In experiments, large scale heterogeneities havebeen observed to be pinned in space and time , giv-ing rise to enhanced scattering at small wave numbers.As discussed in Sec.1, there can be phase separationinduced by selective hydrogen bonding. In particular,the effect of moisture uptake is dramatic in PS-PVME ,where scattering experiments controlling the water con-tent are desirable. To investigate such polymer blends7theoretically, we may use the form in Eq.(5.1) with n be-ing the water density and f ( φ ) being the Flory-Hugginsfree energy for polymer blends . Similar problems shouldalso be encountered in block polymer systems containingions or water. It is also known that blends of block poly-mer and homopolymer exhibit complicated phase behav-ior for different interaction parameters χ ij .We mention two interesting effects not discussed in thisreview. First, there can be an intriguing interplay be-tween the solvation and the hydrogen bonding in phaseseparation. For example, in some aqueous mixtures, evenif they are miscible at all T at atmosphere pressure with-out salt, addition of a small amount of a hydrophilic saltgives rise to reentrant phase separation behavior .On the other hand, Sadakane et al. observed a shrinkageof a closed-loop coexistence curve by adding an atago-nistic salt or an ionic surfactant . Second, molecularpolarization of polar molecules or ions can give rise toa surface potential difference on the molecular scale atan interface. See such an example for water-hexane .This effect is particularly noteworthy for hydronium ionsin acid solutions .We will report on the wetting transition on chargedwalls, rods, and colloids and the solvation-induced col-loid interaction. These effects are much influenced bythe ion-induced precipitation discussed in Sec.5. In theseproblems, first-order prewetting transitions occur fromweak-to strong ionization and adsorption, as discussed inour paper on charged rods . We will also report that asmall amount of a hydrophobic solute can produce smallbubbles in water even outside the coexistence curve, onwhich there have been a large number of experiments. Acknowledgments
This work was supported by KAKENHI (Grant-in-Aidfor Scientific Research) on Priority Area Soft MatterPhysics from the Ministry of Education, Culture, Sports,Science and Technology of Japan. Thanks are due to in-formative discussions with M. Anisimov, K. Sadakane, H.Seto, T.Kanaya, K. Nishida, T. Osakai, T. Hashimoto,and F. Tanaka.
Appendix A: Statistical theory of selectivesolvation at small water composition
We present a simple statistical theory of binding ofpolar molecules to hydrophilic ions due to the ion-dipoleinteraction in a water-oil mixture when the water vol-ume fraction φ is small. We assume no macroscopic inho-mogeneity and do not treat the large-scale electrostaticinteraction. Similar arguments were given for hydrogenboning between water and polymer .Our system has a volume V and contains N w watermolecules. Using the water molecular volume v , we have φ = N w v /V = N w /N , (A1)where N = V /v . We fix N w or φ in the following. Thetotal ion numbers are denoted by N I i = V n i , where i = 1for the cations and i = 2 for the anions with n i being the average densities. The ionic volumes are assumed to besmall and their volume fractions are neglected. Then theoil volume fraction is given by 1 − φ . Each solvation shellconsists of ν water molecules with ν = 1 , · · · , Z i , where Z i is the maximum water number in a shell.Let the number of the ν -clusters composed of ν watermolecules around an ion be γ iν N w . The total number ofthe solvated ions is then γ i N I i with γ i = (cid:88) ν γ iν < , (A2)where 1 ≤ ν ≤ Z i . The number of the bound watermolecules in the ν -clusters is νγ iν N I i . The fraction ofthe unbound water molecules φ f satisfies φ f + v n i (cid:88) i,ν νγ iν = φ. (A3)We construct the free energy of the total system F tot for each given set of γ iν . In terms of the oil density n oil = v − (1 − φ ), the unbound water density n wf = v − φ f , theunbound ion densities n i f = n i (1 − γ i ), and the clusterdensities n iν = n i γ iν , we obtain F tot V k B T = n oil [ln( n oil λ ) −
1] + n wf [ln( n wf λ ) − (cid:88) i [ n if ln( n i f λ i ] − n i ] + (cid:88) i,ν n iν [ln( n iν λ iν ) − w iν ]+ χv n oil n wf + (cid:88) i,ν χ iν v n iν n oil , (A4)where λ oil , λ w , λ i , and λ iν are the thermal de Brogliewavelengths, k B T w iν are the ”bare” binding free ener-gies, and χ is the interaction parameter between the un-bound water and the oil. We asssume short-range inter-actions among the clusters and the oil characterized bythe interaction parameters χ iν to obtain the last term.At small φ , the interactions among the clusters and theunbound water are neglected. That is, we neglect thecontributions of order φ . We then calculate the solva-tion contribution F sol ≡ F tot − F , where F is the freeenergy without binding ( γ iν = 0). Some calculations give F sol N k B T = φ f (ln φ f −
1) + (cid:88) i n i (1 − γ i ) ln(1 − γ i )+ (cid:88) i,ν n i γ iν (ln γ iν − w iν ) − φ (ln φ − , (A5)where k B T w iν are the ”renormalized” binding free ener-gies written as w iν = w iν + 3 ln( λ i λ ν w /λ iν ) + νχ − χ iν . (A6)The fractions γ iν are determined by minimization of F sol with respect to γ iν under Eqs.(A2) and (A3) as γ iν = (1 − γ i ) φ ν f e w iν , (A7) γ i = 1 − / (1 + (cid:88) ν φ ν f e w iν ) . (A8)8Substitution of Eqs.(A7) and (A8) into Eq.(A5) yields F sol k B T N = φ ln φ f φ + φ − φ f + (cid:88) i v n i ln(1 − γ i ) . (A9)First, we assume the dilute limit of ions N I (cid:28) N w ,where we have φ − φ f (cid:28) φ . In the right hand sideof Eq.(A9), the sum of the first three terms becomes − ( φ − φ f ) / φ and is negligible. We write F sol as thesum V (cid:80) i n i µ i sol ( φ ) and use Eq.(A8) to obtain the solva-tion chemical potentials of the form, µ i sol ( φ ) = − k B T ln(1 + (cid:88) ν φ ν e w iν ) . (A10)Let the maximum of k B T w iν /ν (per molecule for various ν ) be (cid:15) b i for each i (see Eq.(2.3)). Then we obtain theexpression (2.2) for the crossover volume fraction φ i sol .For φ > φ i sol , γ i approaches unity.Second, we consider the dilute limit of water, φ (cid:28) φ i sol ( i = 1 , ν i are smalland the dimers with ν = 1 are dominant as indicated inthe experiment . Neglecting the contributions from theclusters with ν ≥
2, we obtain γ i ∼ = φ f e w i , φ f ∼ = φ/ (1 + S ) , (A11)where S is the parameter defined as S = v ( n e w + n e w ) . (A12)The solvation free energy behaves as F sol /N k B T ∼ = − φ ln(1 + S ) . (A13)For S (cid:28)
1, we find µ i sol ( φ ) ∼ = − k B T φe w i . However,if S (cid:38)
1, the solvation chemical potential are not welldefined.
Appendix B: Ions at liquid-liquid interface
In electrochemistry, attention has been paid to the iondistribution and the electric potential difference acrossa liquid-liquid interface . (In the vicinity of an air-water interface, virtually no ions are present in the bulkair region .) Let us suppose two species of ions ( i =1 ,
2) with charges Z e and Z e ( Z > Z < µ i ina mixture solvent are expressed as µ i = k B T ln( n i λ i ) + Z i e Φ + µ i sol ( φ ) , (B1) where λ i is the thermal de Broglie length (but is an ir-relevant constant in the isothermal condition) and Φ isthe local electric potential. This quantity is a constant inequilibrium. For neutral hydrophobic particles the elec-trostatic term is nonexistent, so we have Eq.(2.5).We consider a liquid-liquid interface between a polar(water-rich) phase α and a less polar (oil-rich) phase β with bulk compositions φ α and φ β with φ α > φ β . Thebulk ion densities far from the interface are written as n iα in phase α and n iβ in phase β . From the chargeneutrality condition in the bulk regions, we require Z n α + Z n α = 0 , Z n β + Z n β = 0 . (B2)The potential Φ tends to constants Φ α and Φ β in thebulk two phases, yielding a Galvani potential difference,∆Φ = Φ α − Φ β . Here Φ approaches its limits on thescale of the Debye screening lengths, κ − α and κ − β , awayfrom the interface, so we assume that the system extendslonger than κ − α in phase α and κ − β in phase β . Here weneglect molecular polarization of solvent molecules andsurfactant molecules at an interface (see comments in thesummary section).The solvation chemical potentials µ i sol ( φ ) also take dif-ferent values in the two phases due to their compositiondependence. So we define the differences ∆ µ iαβ as inEq.(2.4). The continuity of µ i across the interface gives k B T ln( n iα /n iβ ) + Z i e ∆Φ − ∆ µ iαβ = 0 , (B3)where i = 1 ,
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