# Phase Transitions in Soft Matter
Induced by Selective Solvation
^{1}^{1}1Accounts: Bull. Chem. Soc. Jpn. June (2011)

###### Abstract

We review our recent studies on selective solvation effects in phase separation in polar binary mixtures with a small amount of solutes. Such hydrophilic or hydrophobic particles are preferentially attracted to one of the solvent components. We discuss the role of antagonistic salt composed of hydrophilic and hydrophobic ions, which undergo microphase separation at water-oil interfaces leading to mesophases. We then discuss phase separation induced by a strong selective solvent above a critical solute density , which occurs far from the solvent coexistence curve. We also give theories of ionic surfactant systems and weakly ionized polyelectrolytes including solvation among charged particles and polar molecules. We point out that the Gibbs formula for the surface tension needs to include an electrostatic contribution in the presence of an electric double layer.

###### pacs:

## I Introduction

In soft matter physics, much attention has been paid to the consequences of the Coulombic interaction among charged objects, such as small ions, charged colloids, charged gels, and polyelectrolytes Levin (); Barrat (); Holm (); Rubinstein (); PG (); Safran (). However, not enough effort has been made on solvation effects among solutes (including hydrophobic particles) and polar solvent molecules Is (); Marcus (); Gut (); Chandler (). Solvation is also called hydration for water and for aqueous mixtures. In mixtures of a water-like fluid and a less polar fluid (including polymer solutions), the solvation is preferential or selective, depending on whether the solute is hydrophilic or hydrophobic. See Fig.1 for its illustration. The typical solvation free energy much exceeds the thermal energy per solute particle. Hence selective solvation should strongly influence phase behavior or even induce a new phase transition. In experiments on aqueous mixtures, it is well known that a small amount of salt drastically alters phase behavior polar1 (); polar3 (); Misawa (); Kumar (); cluster (); Taka (); Anisimov (). In biology, preferential interactions between water and cosolvents with proteins are of crutial importance Tima (); Tima1 (). Thus selective solvation is relevant in diverse fields, but its understanding from physics is still in its infancy.

Around 1980, Nabutovskii et al.R1 (); R2 () proposed a possibility of mesophases in electrolytes from a coupling between the composition and the charge density in the free energy. In aqueous mixtures, such a coupling originates from the selective solvationOnukiJCP04 (); OnukiPRE (). It is in many cases very strong, as suggested by data of the Gibbs transfer free energy in electrochemistry (see Sec.2). Recently, several theoretical groups have proposed Ginzburg-Landau theories on the solvation in mixture solvents for electrolytes OnukiJCP04 (); OnukiPRE (); OnukiJCP (); Tsori (); Ro1 (); Ro2 (); An1 (); Araki (); Nara (); Daan (); Okamoto (), polyelectrolytesOnuki-Okamoto (); Oka (), and ionic surfactants OnukiEPL (). In soft matter physics, such coarse-grained approaches have been used to understand cooperative effects on mesoscopic scales PG (); Safran (); Onukibook (), though they are inaccurate on the angstrom scale. They are even more usuful when selective solvation comes into play in the strong coupling limit. This review presents such examples found in our recent research.

1.1 Antagonistic salt. An antagonistic salt consists of hydrophilic and hydrophobic ions. An example is sodium tetraphenylborate NaBPh, which dissociates into hydrophilic Na and hydrophobic BPh. The latter ion consists of four phenyl rings bonded to an ionized boron. Such ion pairs in aqueous mixtures behave antagonistically in the presence of composition heterogeneity. (i) Around a water-oil interface, they undergo microphase separation on the scale of the Debye screening length , while homogeneity holds far from the interface to satisfy the charge neutrality (see the right bottom plate in Fig.2). This unique ion distribution produces a large electric double layer and a large Galvani potential difference OnukiPRE (); OnukiJCP (); Araki (); Nara (). We found that this ion distribution serves to much decrease the surface tension OnukiJCP (), in agreement with experiments Reid (); Luo (). From x-ray reflectivity measurements, Luo et al. Luo () determined such ion distributions around a water-nitrobenzene(NB) interface by adding BPh and two species of hydrophilic ions. (ii) In the vicinity of the solvent criticality, antagonistic ion pairs interact differently with water-rich and oil-rich composition fluctuations, leading to mesophases (charge density waves). In accord with this prediction, Sadakane et al. Sadakane (); Seto () added a small amount of NaBPh to a near-critical mixture of DO and 3-methylpyridine (3MP) to find a peak at an intermediate wave number (Å) in the intensity of small-angle neutron scattering. The peak height was much enhanced with formation of periodic structures. (iii) Moreover, Sadakane et al. observed multi-lamellar (onion) structures at small volume fractions of 3MP (in DO-rich solvent) far from the criticality SadakanePRL (), where BPh and solvating 3MP form charged lamellae. These findings demonstrate very strong hydrophobicity of BPh. (iv) Another interesting phenomenon is spontaneous emulsification (formation of small water droplets) at a water-NB interface Aoki (); Poland (). It was observed when a large pure water droplet was pushed into a cell containing NB and antagonistic salt (tetraalkylammonium chloride). This instability was caused by ion transport through the interface.

1.2 Precipitation due to selective solvation. Many experimental groups have detected large-scale, long-lived heterogeneities (aggregates or domains) emerging with addition of a hydrophilic salt or a hydrophobic solute in one-phase states of aqueous mixturesS1 (); S2 (); S3 (); S4 (); S5 (); S6 (); S7 (). Their small diffusion constants indicate that their typical size is of order at very small volume fractions. In two-phase states, they also observed a third phase visible as a thin solid-like plate at a liquid-liquid interface in two-phase states third (). In our recent theory Okamoto (), for sufficiently strong solvation preference, a selective solute can induce formation of domains rich in the selected component even very far from the solvent coexistence curve. This phenomenon occurs when the volume fraction of the selected component is relatively small. If it is a majority component, its aggregation is not needed. This precipitation phenomenon should be widely observable for various combinations of solutes and mixture solvents.

1.3 Selective hydrogen bonding. Hydrogen bonding is of primary importance in the phase behavior of soft matter. In particular, using statistical-mechanical theories, the origin of closed-loop coexistence curves was ascribed to the hydrogen bonding for liquid mixtures Wheeler (); Goldstein () and for polymer solutions hydrogen1 (); hydrogen2 (). Interestingly, water itself can be a selective solute triggering phase separation when the hydrogen bonding differs significantly between the two components, as observed in a mixture of methanol-cyclohexane Jacobs (); Be (). More drastically, even water absorbed from air changed the phase behavior in films of polystyrene(PS)- polyvinylmethylether(PVME) Hashimoto (). That is, a small amount of water induces precipitation of PVME-rich domains. For block copolymers, similar precipitation of micelles can well be expected when a small amount of water is added.

1.4 Ionic surfactant. Surfactant molecules are strongly trapped at an interface due to the amphiphilic interaction even if their bulk density is very low PG-Taupin (); Safran (). They can thus efficiently reduce the surface tension, giving rise to various mesoscopic structures. However, most theoretical studies have treated nonionic surfactants, while ionic surfactants are important in biology and technology. In this review, we also discuss selective solvation in systems of ionic surfactants, counterions, and added ions in water-oilOnukiEPL (). We shall see that the adsorption behavior strongly depends on the selective solvation.

1.5 Polyelectrolytes. Polyelectrolytes are already very complex because of the electrostatic interaction among charged particles (ionized monomers and mobile ions) Barrat (); Holm (); Rubinstein (). Furthermore, we should take into account two ingredientsOnuki-Okamoto (), which have not yet attracted enough attention. First, the dissociation (or ionization) on the chains should be treated as a chemical reaction in many polyelectrolytes containing weak acidic monomers Joanny (); Bu1 (); Bu2 (). Then the degree of ionization is a space-dependent annealed variable. Second, the solvation effects should come into play because the solvent molecules and the charged particles interact via ion-dipole interaction. Many polymers themselves are hydrophobic and become hydrophilic with progress of ionization in water. This is because the decrease of the free energy upon ionization is very large. It is also worth noting that the selective solvation effect can be dramatic in mixture solventsOka (). As an example, precipitation of DNA has been observed with addition of ethanol in water B1 (); B2 (); B3 (), where the ethanol added is excluded from condensed DNA, suggesting solvation-induced wetting of DNA by water. In polyelectrolyte solutions, macroscopic phase separation and mesophase formation can both take place, sensitively depending on many parameters.

The organization of this paper is as follows. In Sec.2, we will present the background of the solvation on the basis of some experiments. In Sec.3, we will explain a Ginzburg-Landau model for electrolytes accounting for selective solvation. In Sec.4, we will treat ionic surfactants by introducing the amphiphilic interaction together with the solvation interaction. In Sec.5, we will examine precipitation induced by a strong selective solute, where a simulation of the precipitation dynamics will also be presented. In Sec.6, we will give a Ginzburg-Landau model for weakly ionized polyelectrolytes accounting for the ionization fluctuations and the solvation interaction. In Appendix A, we will give a statistical theory of hydrophilic solvation at small water contents in oil.

## Ii Background of selective solvation of ions

2.1 Hydrophilic ions in aqueous mixtures. Several water molecules form a solvation shell surrounding a small ion via ion-dipole interaction Is (), as in Fig.1. Cluster structures produced by solvation have been observed by mass spectrometric analysis W1 (); W2 (). Here we mention an experiment by Osakai et al Osakai (), which demonstrated the presence of a solvation shell in a water-NB mixture in two-phase coexistence. They measured the amount of water extracted together with hydrophilic ions in a NB-rich region coexisting with a salted water-rich region. A water content of M (the water volume fraction being 0.003) was already present in the NB-rich phase without ions. They estimated the average number of solvating water molecules in the NB-rich phase to be 4 for Na, 6 for Li, and 15 for Ca per ion. Thus, when a hydrophilic ion moves from a water-rich region to an oil-rich region across an interface, a considerable fraction of water molecules solvating it remain attached to it. Furthermore, using proton NMR spectroscopy, Osakai et al. proton () studied successive formation of complex structures of anions X (such as Cl and Br) and water molecules by gradually increasing the water content in NB. This hydration reaction is schematically written as X(HO) + HO X(HO) (). For Br, these clusters are appreciable for water content larger than M or for water volume fraction exceeding .

In Appendix A, we will calculate the statistical distribution of clusters composed of ions and polar molecules. Let the free energy typically decrease by upon binding of a polar molecule to a hydrophilic ion of species . A well-defined solvation shell is formed for , where the water volume fraction needs to satisfy

(2.1) |

For there is almost no solvation. The crossover volume fraction is very small for strongly hydrophilic ions with . For Br in water-NB, we estimate from the experiment by Osakai et al. as discussed aboveproton ().

2.2 Hydrophobic particles. Hydrophobic objects are ubiquitous in nature, which repel water because of the strong attraction among hydrogen-bonded water molecules themselves Is (); Chandler (). Hydrophobic particles tend to form aggregates in water Pratt-hyd (); Wolde-Chandler () and are more soluble in oil than in water. They can be either neutral or charged. A widely used hydrophobic anion is BPh. In water, a large hydrophobic particle nm) is even in a cavity separating the particle surface from water Chandler (). In a water-oil mixture, on the other hand, hydrophobic particles should be in contact with oil molecules instead. This attraction can produce significant composition heterogeneities on mesoscopic scales around hydrophobic objects, which indeed takes place around protein surfaces Tima (); Tima1 ().

2.3 Solvation chemical potential. We introduce a solvation chemical potential in the dilute limit of solute species . It is the solvation part of the chemical potential of one particle (see Eq.(B1) in Appendix B). It is a statistical average over the thermal fluctuations of the molecular configurations. In mixture solvents, it depends on the ambient water volume fraction . For planar surfaces or large particles (such as proteins), we may consider the solvation free energy per unit area.

Born Born () calculated the polarization energy of a polar fluid around a hydrophilic ion with charge using continuum electrostatics to obtain the classic formula,

(2.2) |

The contribution without polarization () or in vacuum is subtracted and the dependence here arises from that of the dielectric constant (see Eq.(3.5) below). The lower cutoff is called the Born radius, which is on the order of for small metallic ions Is (); RMarcus (). The hydrophilic solvation is stronger for smaller ions, since it arises from the ion-dipole interaction. In this original formula, 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 between a hydrophilic ion and a polar molecule is estimated from the Born formula (2.1) as or OnukiJCP04 ()

(2.3) |

where and is the Bjerrum length ( for water at room temperatures). For , a well-defined shell appears for . The solvation chemical potential of hydrophilic ions in water-oil should largely decrease to negative values in the narrow range , as will be described in Appendix A. In the wide composition range (2.1), its composition dependence is still very strong such that holds (see the next subsection). The solubility of hydrophilic ions should also increase abruptly in the narrow range with addition of water to oil. Therefore, solubility measurements of hydrophilic ions would be informative at very small water contents in oil.

In sharp contrast, of a neutral hydrophobic particle increases with increasing the particle radius in water Chandler (). It is roughly proportional to the surface area for nm and is estimated to be about at nm. In water-oil solvents, on the other hand, it is not easy to estimate the dependence of . However, should strongly increase with increasing the water composition , since hydrophobic particles (including ions) effectively attract oil moleculesTima1 (); SadakanePRL ().

2.4 Gibbs transfer free energy. We consider a liquid-liquid interface between a polar (water-rich) phase and a less polar (oil-rich) phase with bulk compositions and with . The solvation chemical potential takes different values in the two phases due to its composition dependence. So we define

(2.4) |

In electrochemistry Hung (); Ham (); Koryta (); Osakai (); Sabela (), the difference of the solvation free energies between two coexisting phases is called the standard Gibbs transfer free energy denoted by for each ion species . Since it is usually measured in units of kJ per mole, its division by the Avogadro number gives our or . With being the water-rich phase, is positive for hydrophilic ions and is negative for hydrophobic ions from its definition (2.4). See Appendix B for relations between and other interface quantities.

For ions, most data of are at present limited on water-NB Hung (); Koryta (); Osakai () and water-1,2-dichloroethane(EDC) Sabela () at room temperatures, where the dielectric constant of NB () is larger than that of EDC (). They then yield the ratio . In the case of water-NB, it is for Na, for Ca, and for Br as examples of hydrophilic ions, while it is for hydrophobic BPh. In the case of water-EDC, it is for Na and for Br, while it is for BPh. The amplitude for hydrophilic ions is larger for EDC than for NB and is very large for multivalent ions. Interestingly, for H (more precisely hydronium ions HO) assumes positive values close to those for Na in these two mixtures.

In these experiments on ions, a hydration shell should have been formed around hydrophilic ions even in the water-poor phase (presumably not completelyproton ()). From Eq.(2.1) should be exceeding the crossover volume fraction . The data of the Gibbs transfer free energy quantitatively demonstrate very strong selective solvation even in the range (2.1) or after the shell formation.

On the other hand, neutral hydrophobic particles are less soluble in a water-rich phase than in an oil-rich phase . Their chemical potential is given by where represents the particle species and is the thermal de Broglie wavelength. From homogeneity of across an interface, we obtain the ratio of their equilibrium bulk densities as

(2.5) |

where from the definition (2.4).

## Iii Ginzburg-Landau theory of mixture electrolytes

3.1 Electrostatic and solvation interactions. We present a Ginzburg-Landau free energy for a polar binary mixture (water-oil) containing a small amount of a monovalent salt (, ). The multivalent case should be studied separately. The ions are dilute and their volume fractions are negligible, so we are allowed to neglect the formation of ion clusters Levin (); pair1 (); pair2 (). The variables , , and are coarse-grained ones varying smoothly on the molecular scale. For simplicity, we also neglect the image interaction Onsager (); Levin-Flores (). At a water-air interface the image interaction serves to push ions into the water region. However, hydrophilic ions are already strongly depleted from an interface due to their position-dependent hydration Levin-Flores (). See our previous analysis OnukiPRE () for relative importance between the image interaction and the solvation interactiion at a liquid-liquid interface.

The free energy is the space integral of the free energy density of the form,

(3.1) |

The first term depends on , , and as

(3.2) |

In this paper, the solvent molecular volumes of the two components are assumed to take a common value , though they are often very different in real binary mixtures. Then is of the Bragg-Williams form Safran (); Onukibook (),

(3.3) |

where is the interaction parameter dependent on . The critical value of is 2 without ions. The in Eq.(3.2) is the thermal de Broglie wavelength of the species with being the molecular mass. The and are the solvation coupling constants. In addition, the coefficient in the gradient part of Eq.(3.1) remains an arbitrary constant. To explain experiments, however, it is desirable to determine from the surface tension data or from the scattering data.

In the electrostatic part of Eq.(3.1), the electric field is written as . The electric potential satisfies the Poisson equation,

(3.4) |

The dielectric constant is assumed to depend on as

(3.5) |

where and are positive constants. Though there is no reliable theory of for a polar mixture, a linear composition dependence of was observed by Debye and Kleboth for a mixture of nitrobenzene-2,2,4-trimethylpentane Debye (). In addition, the form of the electrostatic part of the free energy density depends on the experimental method Tojo (). Our form in Eq.(3.1) follows if we insert the fluid between parallel plates and fix the charge densities on the two plate surfaces.

We explain the solvation terms in in more detail. They follow if () depend on linearly as

(3.6) |

Here the first term is a constant yielding a contribution linear with respect to in , so it is irrelevant at constant ion numbers. The second term gives rise to the solvation coupling in . In this approximation, for hydrophilic ions and for hydrophobic ions. The difference of the solvation chemical potentials in two-phase coexistence in Eq.(2.4) is given by

(3.7) |

where is the composition difference. From Eqs.(B4) and (B5), the Galvani potential difference is

(3.8) |

and the ion reduction factor is

(3.9) |

The discussion in subsection 2.3 indicates (23) for Na ions and (-14) for BPh in water-NB (water-EDC) at 300K. For multivalent ions can be very large ( for Ca in water-NB). The linear form (3.6) is adopted for the mathematical simplicity and is valid for after the solvation shell formation (see Eq.(2.1)). The results in Appendix A suggest a more complicated functional form of .

In equilibrium, we require the homogeneity of the chemical potentials and . Here,

(3.10) | |||

(3.11) |

where . The ion distributions are expressed in terms of and in the modified Poisson-Boltzmann relations OnukiPRE (),

(3.12) |

The coefficients are determined from the conservation of the ion numbers, where denotes the space average with being the cell volume. The average is a given constant density in the monovalent case.

It is worth noting that a similar Ginzburg-Landau free energy was proposed by Aerov et al. Aerov () for mixtures of ionic and nonionic liquids composed of anions, cations, and water-like molecules. In such mixtures, the interactions among neutral molecules and ions can be preferential, leading to mesophase formation, as has been predicted also by molecular dynamic simulations Ionic ().

3.2 Structure factors and effective ion-ion interaction in one-phase states. The simplest application of our model is to calculate the structure factors of the the composition and the ion densities in one-phase states. They can be measured by scattering experiments.

We superimpose small deviations and on the averages and . The monovalent case is treated. As thermal fluctuations, the statistical distributions of and are Gaussian in the mean field theory. We may neglect the composition-dependence of for such small deviations. We calculate the following,

(3.13) |

where , , and are the Fourier components of , , and the charge density with wave vector and denotes taking the thermal average. We introduce the Bjerrum length and the Debye wave number .

First, the inverse of is written asOnukiJCP04 (); OnukiPRE ()

(3.14) |

where with . The second term is large for large even for small average ion density , giving rise to a large shift of the spinodal curve. If , this factor is of order . In the previous experiments polar1 (); polar3 (); Misawa (); Kumar (); cluster (); Taka (); Anisimov (), the shift of the coexistence curve is typically a few Kelvins with addition of a 10 mole fraction of a hydrophilic salt like NaCl. The parameter in the third term represents asymmetry of the solvation of the two ion species and is defined by

(3.15) |

If the right hand side of Eq.(3.14) is expanded with respect to , the coefficient of is . Thus a Lifshitz point is realized at . For , has a peak at an intermediate wave number,

(3.16) |

The peak height and the long wavelength limit are related by

(3.17) |

A mesophase appears with decreasing or increasing , as observed by Sadakane et al. Sadakane (). In our mean-field theory, the criticality of a binary mixture disappears if a salt with is added however small its content is. Very close to the solvent criticality, Sadakane et al. Seto (). recently measured anomalous scattering from DO-3MP-NaBPh stronger than that from DO-3MP without NaBPh. There, the observed scattering amplitude is not well described by our in Eq.(3.14), requiring more improvement.

Second, retaining the fluctuations of the ion densities, we eliminate the composition fluctuations in to obtain the effective interactions among the ions mediated by the composition fluctuations. The resultant free energy of ions is written as OnukiPRE ()

(3.18) |

The effective interaction potentials are given by

(3.19) |

where and are in the monovalent case, , and is the correlation length. The second term in Eq.(3.19) arises from the selective solvation and is effective in the range and can be increasingly important on approaching the solvent criticality (for ). It is attractive among the ions of the same species dominating over the Coulomb repulsion for

(3.20) |

Under the above condition there should be a tendency of ion aggregation of the same species. In the antagonistic case (), the cations and anions can repel one another in the range for

(3.21) |

under which charge density waves are triggered near the solvent criticality.

The ionic structure factors can readily be calculated from Eqs.(3.18) and (3.19). Some calculations give Nara ()

(3.22) |

where and

(3.23) |

is the structure factor for the cations (or for the anions) divided by in the absence of solvation. The solvation parts in Eq.(3.22) are all proportional to , where is given by Eq.(3.14). The Coulomb interaction suppresses large-scale charge-density fluctuations, so tends to zero as .

It should be noted that de Gennes deGennes () derived the effective interaction among monomers on a chain () mediated by the composition fluctuations in a mixture solvent. He then predicted anomalous size behavior of a chain near the solvent criticality. However, as shown in Eq.(3.19), the effective interaction is much more amplified among charged particles than among neutral particles. This indicates importance of the selective solvation for a charged polymer in a mixture solvent, even leading to a prewetting transition around a chain Oka (). We also point out that an attractive interaction arises among charged colloid particles due to the selective solvation in a mixture solvent, on which we will report shortly.

3.3 Liquid-liquid interface profiles. The second application is to calculate a one-dimensional liquid-liquid interface at taking the axis in its normal direction, where in water-rich phase () and in oil-rich phase (). In Fig.2, we give numerical results of typical interface profiles, where we measure space in units of and set , , and . In these examples, the correlation length is shorter than the Debye lengths and in the two phases. However, near the solvent criticality, grows above and and we encounter another regime, which is not treated in this review.

The upper plates give , , , and for hydrophilic ion pairs with and at and . The ion reduction factor in Eq.(3.9) is . The potential varies mostly in the right side (phase ) on the scale of the Debye length (which is much longer than that in phase ). The Galvani potential difference is here. The surface tension here is and is slightly larger than that without ions (see the next subsection).

The lower plates display the same quantities for antagonistic ion pairs with and for and . The anions and the cations are undergoing microphase separation at the interface on the scale of the Debye lengths and , resulting in a large electric double layer and a large potential drop (). The surface tension here is and is about half of that without ions. This large decrease in is marked in view of small . A large decrease of the surface tension was observed for an antagonistic salt Reid (); Luo ()

3.4 Surface tension. There have been numerous measurements of the surface tension of an air-water interface with a salt in the water region. In this case, almost all salts lead to an increase in the surface tension Onsager (); Levin-Flores (), while acids tend to lower it because hydronium ions are trapped at an air-water interface.h1 (); Levinhyd ()

Here, we consider the surface tension of a liquid-liquid interface in our Ginzburg-Landau scheme, where ions can be present in the two sides of the interface. In equilibrium we minimize , where is the grand potential density,

(3.24) |

Using Eqs.(3.10) and (3.11) we find , where and . Thus,

(3.25) |

Since and tend to zero far from the interface, tends to a common constant as . The surface tension is then written as OnukiPRE (); OnukiJCP ()

(3.26) |

where we introduce the areal densities of the gradient free energy and the electrostatic energy as

(3.27) |

The expression is well-known in the Ginzburg-Landau theory without the electrostatic interaction Onukibook ().

In our previous work OnukiPRE (); OnukiJCP (), we obtained the following approximate expression for valid for small ion densities:

(3.28) |

where is the surface tension without ions and is the adsorption to the interface. In terms of the total ion density , it may be expressed as

(3.29) |

where (, ), , and the integrand tends to zero as . From Eqs.(3.26) and (3.28) is expressed at small ion densities as

(3.30) |

In the Gibbs formula () Safran (); Gibbs (), the electrostatic contribution is neglected. However, it is crucial for antagonistic saltOnukiJCP (); Araki () and for ionic surfactant OnukiEPL ().

In Fig.3, numerical results of , , , and the combination are plotted as functions of the bulk ion density for the antagonistic case , where and . The parameter in Eq.(3.15) exceeds unity (being equal to 1.89 for ). In this example, weakly depends on and is fairly in accord with Eq.(3.30), while steeply increases with increasing . As a result, increases up to , leading to vanishing of at .

We may understand the behavior of as a function of and by solving the nonlinear Poisson-Boltzmann equation OnukiJCP (), with an interface at . That is, away from the interface , the normalized potential obeys

(3.31) |

in the two phases (, ), where , , and with . In solving Eq.(3.31) we assume the continuity of the electric induction at (but this does not hold in the presence of interfacial orientation of molecular dipoles, as will be remarked in the summary section). The Poisson-Boltzmann approximation for is of the form,

(3.32) | |||||

We should have in the thin interface limit . In the first line, the coefficient is defined by

(3.33) |

and is the normalized potential difference calculated from Eq.(3.8). In the second line, is the Bjerrum length in phase . The second line indicates that the electrostatic contribution to the surface tension is negative and is of order as away from the solvent criticality, as first predicted by Nicols and Pratt for liquid-liquid interfaces Pratt (). Remarkably, the surface tension of air-water interfaces exhibited the same behavior at very small salt densities (known as the Jones-Ray effect) Jones (), though it has not yet been explained reliably OnukiJCP ().

In the asymptotic limit of antagonistic ion pairs, we assume , where the coefficient in the second line of eq.(3.32) grows as

(3.34) |

We may also examine the usual case of hydrophilic ion pairs in water-oil, where and are both considerably larger than unity. In this case becomes small as

(3.35) | |||||

In this case, the electrostatic contribution in could be detected only at extremely small salt densities.

Analogously, between ionic and nonionic liquids, Aerov el al. Aerov () calculated the surface tension. They showed that if the affinities of cations and anions to neutral molecules are very different, the surface tension becomes negative.

3.5 Mesophase formation with antagonistic salt. Adding an antagonistic salt with to water-oil, we have found instability of one-phase states with increasing below Eq.(3.15) and vanishing of the surface tension with increasing the ion content as in Fig.3. In such cases, a thermodynamic instability is induced with increasing at a fixed ion density , leading to a mesophase. To examine this phase ordering, we performed two-dimensional simulations Araki (); Nara () and presented an approximate phase diagram Nara (). We here present preliminary three-dimensional results. The patterns to follow resemble those in block copolymers and surfactant systems Onukibook (); Seul (). In our case, mesophases emerge due to the selective solvation and the Coulomb interaction without complex molecular structures. Solvation-induced mesophase formation can well be expected in polyelectrolytes and mixtures of ionic and polar liquids.

We are interested in slow composition evolution with antagonistic ion pairs, so we assume that the ion distributions are given by the modified Poisson-Boltzmann relations in Eq.(3.12). The water composition obeys Araki (); Nara ()

(3.36) |

where is the kinetic coefficient and is defined by Eq.(3.10). Neglecting the acceleration term, we determine the velocity field using the Stokes approximation,

(3.37) |

where is the shear viscosity and are defined by Eq.(3.11), We introduce to ensure the incompressibility condition . The right hand side of Eq.(3.37) is also written as , where is the stress tensor arising from the fluctuations of and . Here the total free energy in Eq.(3.1) satisfies with these equations (if the boundary effect arising from the surface free energy is neglected).

We integrated Eq.(3.36) using the relations (3.4), (3.12), and (3.37) on a lattice under the periodic boundary condition. The system was quenched to an unstable state with at . Space and time are measured in units of and , respectively. Without ions, the diffusion constant of the composition is given by in one-phase states in the long wavelength limit (see Eq.(3.14)). We set , , , , , and .

In Fig.4, we show the simulated domain patterns at , where we can see a bicontinuous structure for and a droplet structure for . There is almost no further time evolution from this stage. In Fig.5, the ion distributions are displayed for these two cases in the - plane at . For , the ion distributions are peaked at the interfaces forming electric double layers (as in the right bottom plate of Fig.2). For , the anions are broadly distributed in the percolated oil region, but we expect formation of electric double layers with increasing the domain size also in the off-critical condition. In Fig.6, the structure factor in steady states are plotted for , and 0.003. The peak position decreases with increasing in accord with Eq.(3.16). Sadakane et al. Sadakane (); SadakanePRL () observed the structure factor similar to those in Fig.6.

Finally, we remark that the thermal noise, which is absent in our simulation, should be crucial near the criticality of low-molecilar-weight solvents. It is needed to explain anomalously enhanced composition fluctuations induced by NaBPh near the solvent criticalitySeto ().

## Iv Ionic surfactant with amphiphilic and solvation interactions

4.1 Ginzburg-Landau theory. In this section, we will give a diffuse-interface model of ionic surfactantsOnukiEPL (), 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 thickness . In our model, the adsorption of ionic surfactant molecules and counterions to an oil-water interface strongly depends on the selective solvation parameters and and that the surface tension contains the electrostatic 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 densities of water, oil, surfactant, and counterion are , , , and , respectively. The volume fractions of the first three components are , , and , where is the common molecular volume of water and oil and is the surfactant molecular volume. The volume ratio can be large, so we do not neglect the surfactant volume fraction, while we neglect the counterion volume fraction supposing a small size of the counterions. We assume the space-filling condition,

(4.1) |

Let be the composition difference between water and oil; then,

(4.2) |

The total free energy is again expressed as in Eq.(3.1). Similarly to Eq.(3.2), the first part reads

(4.3) | |||||

The coefficients and are the solvation parameters of the ionic surfactant and the counterions, respectively. Though a surfactant molecule is amphiphilic, it can have preference to water or oil on the average. The last term represents the amphiphilic interaction between the surfactant and the composition. That is, is the partition function of a rod-like dipole with its center at the position . We assume that the surfactant molecules take a rod-like shape with a length considerably longer than . It is given by the following integral on the surface of a sphere with radius ,

(4.4) |

where is the unit vector along the rod direction and represents the integration over the angles of . The two ends of the rod are at and under the influence of the solvation potentials given by and . The parameter represents the strength of the amphiphilic interaction.

Adsorption is strong for large , where is the difference of between the two phases and . In the one-dimensional case, all the quantities vary along the axis and is rewritten as

(4.5) |

where . In the thin interface limit , we place the interface at to find for , while

(4.6) |

for . Furthermore, in the dilute limit and without the electrostatic interaction, we have for and for , where and are the bulk surfactant densities. The surfactant adsorption then grows as

(4.7) | |||||

However, the steric effect comes into play at the interface with increasing the surfactant volume fraction at the interface ().