A number of observations related to interfacial electrostatics of polar liquids question the traditional assumption of dielectric theories that bulk dielectric properties can be continuously extended to the dividing surface separating the solute from the solvent. The deficiency of this approximation can be remedied by introducing local interface susceptibilities and the interface dielectric constant. Asymmetries of ionic hydration thermodynamics and of the mobility between cations and anions can be related to different propensities of the water molecules to orient their dipole toward and outward from solutes of opposite charges. This electrostatic asymmetry is reflected in different interface dielectric constants for cations and anions. The interface of water with neutral solutes is spontaneously polarized due to preferential water orientations in the interface. This phenomenon is responsible for a nonzero cavity potential directly related to a nonzero surface charge. This connection predicts that particles allowing a nonzero cavity potential must show mobility in an external electric field even if the net charge of the particle is zero. The theory predicts that a positive cavity potential and a positive surface charge translate to an effectively negative solute charge reported by mobility measurements. Passing of the cavity potential through a minimum found in simulations might be the origin of the maximum of mobility vs the ionic size observed experimentally. Finally, mobility of proteins in the field gradient (dielectrophoresis) is many orders of magnitude greater than predicted by the traditionally used Clausius-Mossotti equation. Two reasons contribute to this disagreement: (i) a failure of Maxwell’s electrostatics to describe the cavity-field susceptibility and (ii) the neglect of the protein permanent dipole by the Clausius-Mossotti equation. An analytical relation between the dielectrophoretic susceptibility and dielectric spectroscopy of solutions provides direct access to this parameter, confirming the failure of the Clausius-Mossotti equation in application to protein dielectrophresis.
All materials satisfy the Coulomb law at the microscopic level. It states that for any two particles carrying charges and and separated by the distance , the potential energy is written as (Gaussian units1)
A certain level of coarse-graining is typically introduced in condensed-matter calculations and in atomistic numerical simulations by assigning charges to composite particles: atoms or molecular groups. Such a formulation is still referred to as the microscopic form of the Coulomb law since all atomic coordinates are followed in the simulation trajectory. We will follow this adopted practice and designate this very detailed level of description as the microscopic form of the Coulomb law.
Most practical calculations applied to complex systems, solutions, and interfaces depart from this detailed microscopic picture by applying some level of averaging. A powerful approach, leading to equations of electrodynamics of materials,2 replaces the complete information about the charges and positions of the atoms with continuous scalar and vector fields. The main challenge here is how to incorporate specific properties of interfaces into the field equations. The traditional formulation of electrodynamics of continuous media2 assumes that bulk material properties can be extended all the way to the dividing surfaces separating different components of the inhomogeneous material. If one component is a liquid, this mixed substance is known as a solution. The focus of this discussion is on electrostatics of solutions.
If one is concerned only with the interaction of charges and in a condensed medium, the only parameters relevant for such an observation are the distance between the charges and the thermodynamic state of the material. When one fixes only the thermodynamic parameters of the system (temperature, pressure, etc.) allowing all possible microscopic configurations, one arrives at the free energy of the solution characterizing its thermodynamic state. If, in addition, the microscopic distance is fixed while allowing all possible configurations consistent with this constraint, the result is the potential of mean force, which is a partial free energy depending on a single remaining microscopic coordinate . Application of polarization fields of classical electrostatics leads to the factor in Eq. (1),
which is often called the screened potential. The dependence of the dielectric constant on the thermodynamic state of the material is an indication that carries the meaning of the free energy, i.e., the reversible work required to separate two charges to the infinite distance in the liquid. Within the dielectric theories, the same function appears when measuring polarization of bulk dielectrics in dielectric experiments.3
A similar procedure of reducing the system’s manifold of degrees of freedom is used in the calculations of the solvation free energy, i.e., the chemical potential of a solute at infinite dilution. Here, the polarization field of the medium is integrated out in the external interaction potential produced by the solute. One naturally wonders if the polarization of the interface induced by the solute field is described by the same rules as dielectric screening at large distances in Eq. (2); i.e., whether the same dielectric constant can be used to construct the electrostatic response for both problems. The answer given to this question by electrostatic theories of dielectrics is affirmative. Here, we argue that there are a number of observations where departure from this answer is required and a separate susceptibility of the interface needs to be constructed. It is not reduced to either of bulk dielectric or to its algebraic manipulations.
The assumption of continuous electrodynamics that bulk properties apply to all components of the solution up to the dividing surfaces supposes that the spatial dimension of the interface is much smaller than the distance on which continuous electrostatic fields vary. It is this assumption that often fails in describing solutions of molecular solutes in molecular solvents, a situation typical for solution chemistry and biochemistry. In molecular systems, the field of solute charges and the response of the solvent in the interface often vary on the same length-scale, thus invalidating the continuum assumption. Alternative coarse-graining solutions, at least partially incorporating the microscopic structure of the solute-solvent interface into the equations for continuous fields, are required. We will discuss below how such extensions of the continuum picture can be realized for electrostatics of solutions and for the calculation of free energies and forces acting on particles dissolved in polar liquids. It turns out that the basic equations of electrostatics of continuous fields can be preserved at the expense of introducing local interface susceptibilities distinct from material bulk properties.
II. ELECTROSTATICS OF SOLUTIONS
In applications to electrostatics of solutions, the common practice is to separate the molecular charges into those of the solutes and those of the solvent. Since solutes are brought into the liquid solvent to create the solution, the charges of the solutes, in accord with the standard formulations of electrostatics,1 are viewed as external (free) charges with the density characterizing the distribution of partial atomic charges placed at positions within a given solute molecule. The charges of all solutes can be obtained from this charge distribution by translation and rotation transformations. Since we assume an ideal solution here, considering a single solute is sufficient for our purposes. For instance, a simple ion carrying the charge can be placed at the origin of the coordinate system producing and .
In addition to the solute charge density , there are atomic charges at the molecules of the solvent. Their distribution, regardless of the assignment of the charges to a specific molecule of the solvent, can be characterized by a scalar field of the instantaneous microscopic (subscript “m”) charge density representing a specific instantaneous configuration of the solvent. Thermal agitation alters making it a fluctuating field, but the Coulomb law applies to each configuration of the nuclei in the system. The instantaneous microscopic (subscript “m”) electrostatic potential is
where is the electrostatic potential created by the solute charges .
The conservation of the total charge of the solvent requires that the fluctuating scalar field of the solvent charge density can be replaced by the divergence of the vector field known as the polarization density,2,4
This equation indicates that there is a charge associated with any small volume in the liquid in which is nonuniform. Quoting from Feynman:4 “If there is a nonuniform polarization, its divergence gives the net density of charge appearing in the material. We emphasize that this is a perfectly real charge density: we call it ‘polarization charge’ only to remind ourselves how it got there.” The emphasis on “real” is a warning against a common misconception that is not a physical charge. Indeed, because of complexities of describing the distribution of molecular charges in condensed materials, classical electrostatic theories avoid calculating by formulating the problem in terms of free charges .1 Nevertheless, there is no fundamental distinction between and , and modern atomistic simulations treat both on equal footing.5
Equation (4), which is based on the conservation of charge, in principle applies to both bulk materials and interfaces. However, for interfacial problems, the formulation is significantly simplified by separating the electrostatics arising from the slowly varying in the bulk from fast changing liquid polarization in the interface. The microscopic potential then becomes a sum of the interface (surface), , and bulk, , contribution. As we show below, the bulk contribution is constant and thus vanishes in the electrostatic field .
Fast alteration of the polarization field in the interface can be accounted for by multiplying the bulk polarization field with a function changing sharply from unity to zero in the interface. In the limit when the size of the interfacial region is significantly smaller than the characteristic size of the solute, a step function can be applied. This is a Heaviside function, which is equal to zero inside the solute and is equal to unity inside the solvent. If is substituted instead of in Eq. (4), one obtains
Here, is the projection of the polarization density field on the unit vector normal to the dividing surface at the point and directed outward from the dielectric and is the delta-function. The instantaneous microscopic potential created by the solvent then becomes the sum of the potential of the bulk charges and of the surface charges ,
The potential components are given by integrals over the surface of the solute and over the volume occupied by the solvent,
As in Eq. (3), defines the microscopic (instantaneous) charge density in the bulk. The presence of the interface is responsible for the surface electrostatic potential , which fluctuates due to thermal fluctuations of . The next step is to specify the observable statistical averages of the electrostatic potentials and .
Statistically averaged projection of the polarization field on the outward normal direction is identified in electrostatic theories with the surface charge density4
where angular brackets denote an ensemble average. It is often stated that “the surface charge description accounts for the polarization of the entire solvent.”6 This result comes from the observation, following from the Helmholtz theorem,1 that the field produced by the bulk density can be identified with the longitudinal component of the polarization field,7
Here, the “L” subscript denotes the longitudinal [irrotational, in contrast to solenoidal (transverse);1 see Fig. 1] projection of the vector field . The main result here is that the field of the bulk can be identified with the local polarization field at the point where is calculated. Since there is no polarization in the volume excluded from the liquid, inside the solute. Therefore, the bulk of the liquid produces no field inside the solute, and the electrostatic potential must be a constant. Only the knowledge of the surface charge is required to calculate the electrostatic field in a void.
It is important to stress that this result is solely a consequence of the Coulomb law, combined with the requirement of expulsion of the liquid polarization from the solute volume. This conceptual framework puts emphasis on the interfacial liquid structure in formulating solution electrostatics. The bulk turns out to be irrelevant, and finding the electrostatic field in a void becomes an interfacial problem.8,9 We next explore this perspective with specific examples connecting the surface charge density with solvation and mobility of solutes in polar liquids.
A. Surface charge density and interface susceptibility
The discussion presented above has shown that the problem of finding the statistically averaged electric field in a void,
can be fully formulated in terms of the surface charge density given by Eq. (8) and the density of external charges . The microscopic origin of is not specified: it might originate from molecular multipoles (dipoles, quadrupoles, etc.), induced dipoles due to electronic molecular polarizability, and surface hydrogen bonds. All these microscopic origins of the surface charge are averaged over the microscopic configurations to produce . As we show below, involves two components: (i) static surface charge due to spontaneous polarization of the interface and (ii) surface charge induced by external charges. The second component can be expressed in terms of the interface susceptibility or, alternatively, in terms of the interface dielectric constant. It is instructive to start the discussion with dielectric theories, where only the second component of the surface charge is considered and is calculated in terms of the dielectric constant of the bulk.
We start with the simplest configuration of an ion centered in the cavity created by its repulsive core [Fig. 2(a)]. The cavity radius is and the charge is placed at its geometrical center. In Maxwell’s electrostatics,1 the ion carrying the charge polarizes the surrounding liquid to create the surface charge density
It carries a sign opposite to , thus screening the electric field of the external charge (“M” designates Maxwell’s electrostatics). When multiplied with the surface area , this uniformly distributed surface charge density yields the total charge of the interface (surface charge)
The electrostatic potential in the liquid is a sum of the solute potential and the surface potential , thus yielding the standard expression for the screened potential inside the liquid. In contrast to the decaying potential in the liquid, a constant potential is created by the uniform surface charge inside the cavity
The reversible work of bringing charge into the void is the electrostatic component of the free energy of ion solvation. Since , the reversible work of charging the cavity is quadratic in ,
This is the celebrated Born equation.11 The measurable free energy of bringing an ion from vacuum to solution includes, in addition to , a number of additional components12 (see below). However, as we discuss next, even the free energy of electrostatic solvation needs modification to account for the static electrostatic potential, independent of , within the solute cavity.
The quadratic scaling with the ion charge predicted by the Born equation has been confirmed with great accuracy by computer simulations of hydrated ions.10,13 Nevertheless, two deviations from the Born equation were identified: (i) a nonzero offset potential at zero ionic charge and (ii) different slopes of for anions and cations.10,13–15 According to simulations10,15–18 and from experimental hydration enthalpies,19 accounting for different slopes requires either or in Eq. (14) [Fig. 2(b)]. Here, the subscript is applied not to the charge of the ion, but instead to the surface charge in the interface: cations produce a negative screening charge (“” subscript) and anions create a positive screening charge (“” subscript). Therefore, hydration of anions leads to a more compact and more polar hydration shell compared to hydration of cations.20,21
If preferential molecular orientations are caused by specifics of the solute-solvent and solvent-solvent interactions, a static (spontaneous) interfacial polarization must follow. For interfaces with water, a nonuniform orientational distribution is usually related to two sources: (i) surface hydrogen bonds viewed as local interactions not reducible to the total charge carried by the solute and (ii) competition between water’s dipoles and quadrupoles to minimize their free energy in the interface, which is present even for the water-vapor interface.22,23 A nonuniform orientational distribution of interfacial multipoles leads to a nonzero normal polarization density component and to a uniform surface charge density integrating to a nonzero total surface charge . This uniform surface charge will produce a static cavity potential23–25 [Eq. (7) and Fig. 2(b)]
With the account for the static potential, the Born solvation free energy becomes only a part of the overall electrostatic free energy16,23,26 (also labeled as the “intrinsic” hydration free energy27)
The first term here is the energy of the charge in the static potential of the interface , while the second term is the free energy of polarizing the interface by charge , which is quadratic in the charge, [Eq. (14)].
Two issues related to the static cavity potential need to be clarified. First, one has to realize that the polarization field is a composite quantity including both the molecular dipoles (induced and permanent) and molecular quadrupoles (as well as higher multipoles),
is the traceless molecular quadrupole28 defined through partial atomic charges at the coordinates relative to the center of mass of the solvent molecule. The sum in Eq. (17) runs over the solvent molecules with coordinates , thus producing the vector field of the polarization density.1,28 Therefore, the static potential is caused by both dipolar and quadrupolar molecular orientations in the interface.
Direct calculations for SPC/E (extended simple point charge) water have shown that is dominated by quadrupolar polarization for small cavities.23 This contribution, decaying as , becomes insignificant29 for (assuming zero potential assigned to the bulk). The constant reported for large cavities is mostly the result of preferential dipolar orientations in the interface. Second, the cavity potential is only one part in the overall surface potential , which represents the potential drop between the vapor and the potential inside the liquid cavity. In addition to the potential drop at the liquid-solute interface considered here, it contains the potential drop at the vapor-liquid interface.23,25,30,31 The total potential drop , which comes as an incomplete compensation between and , needs to be considered for comparing calculated solvation free energies with experimental free energies for transferring ions from vacuum into the liquid. In addition to , the total is affected by the free energy of cavity formation and solute-solvent dispersion and induction interactions.12,31 The free energy of electrostatic solvation in Eq. (16) is only one component of the more complex .
The second term, , in Eq. (16) is the Born solvation free energy due to the surface charge induced by the field of the solute. If one assumes that the cavity radius is not affected by altering solute charge (large solutes),24 the asymmetry of the slope of the induced cavity potential between cations and anions [Fig. 2(b)] implies different propensities of water to orient its dipole inward or outward from the ion, depending on the direction of the external field. To account for this effect, one can assign different induced charge densities to the positive () and negative () surface charge and corresponding interface dielectric constants . Obviously, this change of perspective implies that is not a material property anymore but instead is an interfacial parameter.
Two specific interface dielectric constants are special cases of the interface dielectric constant characterizing the integral ability of the interface to be polarized by a probe charge placed inside a void.7 For a spherical cavity with the charge at its center, the interface dielectric constant defines the induced surface charge, which adds to the static charge to make the total surface charge
The corresponding cavity potential becomes
The cavity potential is the sum of the static (first term) and induced (second term) components. The difference between the microscopically derived charge in Eq. (19) and the dielectric charge in Eq. (12) is caused by two effects: and . To distinguish between the static and induced surface charge, one can define the interface susceptibility
The superscript “q” here marks the response of the interface to the placement of the charge to the solute, in contrast to the dipolar susceptibility of the interface considered below, which describes the response to a dipole placed into the cavity. The charge susceptibility is obviously related to the Born solvation energy by the following equation:
When ions of opposite charge are concerned, one can assign to for cations and to for anions by assuming constant slopes of for each ion charge [Fig. 2(b)]. The dielectric constants can, therefore, be estimated by analyzing the interfacial polarization induced in the surrounding polar liquid by cations and anions. This analysis has been recently performed for charged fullerenes in SPC/E water.32 The result is and for and , respectively. These values qualitatively agree with a steeper slope of for anions compared to cations [Fig. 2(b)]. One, therefore, generally expects for interfacial water,
Inequality (23) is illustrated in Fig. 3 where is calculated from Eq. (22) by using simulated16 electrostatic solvation free energies and the cavity potentials for a number of cations and anions in SPC (simple point charge) force-field water. The cavity radius was not separately established in that study, and the sum of the ionic, , and water, , Lennard-Jones (LJ) radii is used to estimate . The distinction between for cations and anions is qualitatively consistent with Eqs. (21) and (23). One, however, finds , which corresponds to a negative . This difficulty implies that taken as a sum of LJ radii is not a reliable approximation.
Solvation of small ions is affected not only by the interface susceptibility, but also by often a significant change of the structure of the hydration shell induced by the ion. This difficulty is dealt with by different formalisms to specify the effective size of the solute, which is often expressed in terms of the solute-solvent radial distribution function (RDF). The cavity size roughly coincides with the first peak of the solute-solvent RDF, but more complex expressions involving integrals of the RDF follow from perturbation theories.18 We do not consider these more special cases here focusing solely on the induced surface charge and the corresponding interface susceptibility.
Both dielectric constants, and , calculated for fullerenes in water32 are substantially below the dielectric constant of bulk SPC/E water (). This outcome, shared by a number of simulation studies,7,32–36 suggests suppression of the interfacial response in the direction normal to the interface. Dipoles in the interface, frustrated by the local fields and geometric constraints,37–40 do not develop the complete dielectric screening of the bulk material. One anticipates a general result,41–44
Microscopic interfaces will screen the ionic charge less effectively than prescribed by dielectric theories.
B. Electrostatics of cavities
We now remove the ion from the void and consider the simplest configuration related to mobility in a uniform external field: an empty cavity in a uniformly polarized liquid (Fig. 4). Polarization of the liquid leads to a surface charge at the void. According to electrostatics of dielectrics,1,2 the surface of the cavity gains the surface charge density corresponding to the first-order, , term in the expansion of the surface density in Legendre polynomials . This dipolar surface charge density creates a cavity dipole opposite in the direction to the external field with the magnitude , where is the cavity volume. If the uniform polarization is created in the medium, Maxwell’s electrostatics predicts1,2
The total charges of the negative, , and positive, , lobes of the surface charge density are equal in the magnitude and the total surface charge is zero.
From the discussion of ion solvation, it is easy to realize possible pitfalls of this picture when applied to liquids with asymmetric distribution of molecular charges, such as water.22,45 If the propensities of the solvent multipoles to orient toward and away from the uncharged solute are different, this should result in different surface charge densities for the positive and negative lobes of the surface charge. For hydration of solutes of the nanometer size, such as proteins, one can anticipate a scenario in which the solute surface provides sites with strong hydrogen bonds with the interfacial water molecules. In that case, orientations of water dipoles pointing toward the solute will be more probable, and the negative lobe in Fig. 4 will be strongly depopulated. One would expect for this scenario.
The phenomenology of ion solvation suggests a route to improve the standard picture in terms of charge-specific and . If is assigned to the positive lobe () and is assigned to the negative lobe (, Fig. 4), the total charge of the interface becomes
Correspondingly, the dipole moment of the void is oriented opposite to the external field and carries the magnitude
The standard case of Maxwell’s electrostatic in Eq. (26) is recovered when there is no preference for the surface dipoles to orient either inward or outward and .
Two scenarios can be anticipated for these interfacial multipoles. The static charge leads to the electrostatic force acting on the solute (see below), which is linear in the external field.46 On the contrary, the induced surface charge densities can differ as a consequence of different susceptibilities of the interface with respect to the external field pointing toward the solute and away from it, as is shown in Fig. 4. In this case, the induced surface charge
will be positive given inequality (23) and scale linearly with the applied field. Correspondingly, the resulting force will be quadratic in the electric field of external charges ,
For both static and induced surface charges, no gradient of the external field is required to cause the force, in contrast to the dielectrophoretic force discussed below. The result is that uncharged solutes will experience a dragging force in a uniform external field.
An important consequence of Eq. (15) is that the static surface charge and mobility of uncharged solutes can be evaluated from the static cavity potential often available from simulations.15,18,24,47 The energy is substantial: from for a hollow cavity24 with the radius up to to for an uncharged protein where all partial charges were set equal to zero.47 A number of simulations have shown that is nearly constant at Å. This observation translates into increasing approximately linearly with the cavity size [Eq. (15)]. Given that mobility in the field is inversely proportional to the particle radius [Eq. (39)], one gets a nearly constant contribution of the static surface charge to the mobility of the solute. This contribution is affected not only by the identity of the solvent, but also by the structure of the interface.
This is illustrated in Fig. 5, where calculated from Eq. (15) is plotted against the cavity radius identified with the position of the first maximum of the solute-solvent RDF.18 Two sets of points in Fig. 5, obtained from molecular dynamics simulations of nonpolar solutes of changing size, identify different strengths of LJ attraction between a nonpolar solute and TIP3P water.48 The surface charge density produced by these simulations is below the range of typically extracted from mobility measurements of air bubbles and hydrophobic oil drops.49,50 However, the laboratory data are affected by ionic adsorption,51,52 and corresponding modifications of the surface charge are not included in our analysis. Surface hydrogen bonds and release of dangling O–H bonds53,54 will additionally affect .
III. MOBILITY OF IONS
Consider a sample of polar liquid with solutes carrying charge each placed in a plane capacitor producing the vacuum field . If the field is along the -axis of the laboratory frame, the -component of the total force acting on the sample is
where the volume can be chosen in a way that there is no polarization field at the volume boundary. If the density of the bound charge is taken as above, , then the total bound charge is zero and
If the overall neutral electrolyte is considered instead, the total force on the sample is obviously zero.
Calculating the mobility of an ion in an external field requires a somewhat different procedure. Instead of calculating the force acting on the sample, one calculates the force acting on all charges within the spherical shear surface of radius , which then becomes hydrodynamic or the Stokes radius. The force acting on this volume is obviously
Here, is the cavity susceptibility correcting the external field to the cavity field inside the shear surface. By allowing two dividing surfaces for the polar liquid, at the cavity radius and at the Stokes radius , one obtains
Here, is the radial projection of the polarization field estimated at and . The radial projection at r = a is the negative of the surface charge density composed of the field-induced, , and static, , components [Eq. (19)]: . If one assumes that the field-induced components of and are specified by and , respectively, one obtains
Alternatively, one gets
where is the screening charge in Eq. (19).
The standard dielectric result follows at and . In this limit, there is no divergence of the radial polarization since and (Fig. 6). According to the standard rules of electrostatics discussed above, there is no bound charge density created by this polarization field, and there is no net bound charge between two spherical surfaces at and . A nonzero bound charge of the hydration shell comes from the static charge due to spontaneous polarization of the interface combined with polarization divergence caused by the solvent dipoles polarized differently between the cavity surface and the shear surface (). The radial polarization projection is typically a decaying oscillatory function in the interface.55
The stationary drift of an ion is established by balancing the electrostatic force with the force of hydrodynamic friction56,57 with the friction constant
Here, the stick boundary conditions are assumed and is the solvent viscosity. Equating the drag force in Eq. (34) with the friction force, one arrives at the Hückel equation for the product of ionic mobility and viscosity (the Walden product58),
where refers to the absolute mobility (mobility at zero ionic strength) and is given by Eq. (36).
Here, as above, the superscript “M” is used to specify Maxwell’s electrostatics. This form of , which should be used with care (see below), allows one to assume for in Eq. (39). When this approximation is adopted, becomes equal to , which corresponds to the slip boundary conditions in the standard models of mobility not accounting for the cavity-field corrections.58 Equation (39) can be changed to
The hydrodynamic radius is likely to deviate upward from the cavity radius , and one can anticipate that the induced surface charge density at the cavity radius should generally be different from the induced surface charge density at the shear surface. The extent of this deviation is hard to estimate even from microscopic considerations. For instance, the interface dielectric constant in Eq. (19) is in practice determined from simulation configurations by averaging the dipolar response over about three solvation shells.32,33 The cavity and Stokes radii fall within this coarse-graining range for most small ions, and induced charges at the cavity and shear surfaces are hard to distinguish.
An intriguing experimental observation much discussed in the past is the appearance of the mobility maximum as a function of the ionic radius found for both anions and cations.58 Theories of this effect have focused on modifying the friction component in the force balance. The overall friction experienced by an ion can be separated57 into the hydrodynamic component given by Eq. (38) (due to solute-solvent repulsion) and the friction arising from the solute-solvent soft interactions: . The assumed additivity between the repulsive hydrodynamic and soft friction components is not, however, supported by computer simulations.60 This observation questions the validity of all theoretical formalisms based on the additivity assumption.57,61
If the soft component is described by the electrostatic interaction with a continuum solvent, the result is the dielectric friction,21,57,60–63
Here, the numerical constant differs between Zwanzig62 and Hubbard-Onsager63 formulations and is the Debye relaxation time of the solvent. While these traditional theories considered dielectric friction originating from collective dielectric relaxation of the medium, the molecular model of ionic mobility by Bagchi and Biswas61 placed the emphasis on fast ballistic (single-particle) dynamics of water in the interface with the solute as the driving force for mobility of small ions. Calculations supporting this view were based on the mean-spherical approximation, which, similarly to Eq. (42), results in scaling for the friction component originating from soft solute-solvent interactions.
Tests of the predicted scaling by numerical simulations have not been entirely consistent. While no maximum in conductivity for uncharged solutes was found in early simulations,21 more recent studies using low-charge solutes, , have shown the existence of a maximum in mobility as a function of the solute radius.64 These recent results, confirming earlier simulations with neutral solutes,65 suggest that the mobility maximum is not related to dielectric friction but, instead, is a result of optimization, at a certain size, of solute’s migration between the voids in the liquid. Therefore, if a nonlinear dependence of mobility on the ionic size is to be related to soft friction, it is not driven by the electrostatic solute-solvent interactions. Furthermore, the maximum in the experimental conductivity (for among monovalent cations in water) turns out to coincide with the minimum of the activation enthalpy as a function of the ionic size.64 This experimental fact does not seem to allow an obvious explanation in terms of dielectric friction.
Along these lines, a connection between the mobility maximum, the residence time of water molecules in the hydration shell, and the entropy of solvation was also proposed.21 These simulations, therefore, suggest that conductivity reaches maximum for the most disordered hydration shell. This explanation might be just another reflection of the same physical reality of reaching optimum migration of ions between voids in the liquid when the hydration shells are most disordered and loose.64 Similarly, simulations at elevated pressure found the mobility to increase with increasing pressure,66 which was explained by increased disorder of hydration shells caused by breaking the network of hydrogen bonds induced by pressure. To summarize, all these studies have placed the focus on the structure of the hydration shell as the origin of the mobility maximum, in contrast to the dissipation mechanisms advocated by the theories focused on dielectric friction.
The present perspective is focused on microscopic screening in the interface and does not offer new insights into the mechanism of energy dissipation of the moving ion into the polar liquid. In contrast to previous theoretical studies concerned solely with the friction mechanisms, the focus here is on the electrostatic drag force acting on the ion. From this perspective, the theory predicts a linear dependence of the Walden product on the static cavity potential [Eq. (41)]. The static cavity potential was indeed found to pass through a minimum16 at the ionic size roughly consistent with the observed conductivity maximum (for Cs among monovalent cations58,61). This result implies that in Eq. (41) is expected to pass through a maximum, as observed for ionic conductivity.
The results of simulations16 for are collected in Fig. 7. The range of achieved in simulations is insufficient to explain the conductivity maximum for cations. Both and might need to be considered.7,67 In particular, found in simulations67,68 and discussed below in connection with protein dielectrophoresis might be the amplification factor converting a weak maximum of to a much more robust mobility maximum found for cations. Importantly, the term containing in Eq. (41) is independent of the ionic charge and, therefore, would explain the appearance of the mobility maximum in simulation of solutes with a low or zero charge.64,65 Physically, the minimum of should reflect the most orientationally disordered hydration shell, in line with previous qualitative explanations connecting the shell disorder with the mobility maximum.21,66
Explaining the conductivity maximum for anions will require additional studies. For anions, enters with the positive sign the right-hand side of Eq. (41), thus producing a minimum. Whether the combination of this minimum with the first term can produce a mobility maximum is hard to establish given that is likely to be a nonmonotonic function of the solute radius. Interfacial water molecules next to a negatively charged surface undergo a structural crossover releasing dangling O–H bonds pointing to the substrate with increasing the charge of the substrate.38,69 This transition is accompanied by a spike of , which decays on both sides of the crossover point.32
Equation (41) resolves an apparent contradiction between simulations of voids and LJ solutes producing a positive surface charge15,18,24,47 and a formally negative charge assigned to nonpolar, nonionic particles in water (oil drops, air bubbles, etc.) based on their mobility.49,70 Since enters the effective ionic charge interacting with the field with the negative sign, the static surface charge yields a negative contribution to . The negative charge of nonpolar solutes has been traditionally ascribed to adsorption of hydroxide ions to the interface. An alternative mechanism discussed here requires static interfacial orientation of the water molecules pointing their dipoles toward the nonpolar component [ in Eqs. (15) and (36)]. Whether hydroxide49 or bicarbonate51 ions alone, or their combination with , can explain the mobility of nonionic particles in water requires further studies. The perspective presented here allows direct access to this problem through atomistic simulations: one needs to calculate the cavity potential from simulations to access the static surface charge and combine this input with the average total charge within the shear surface from preferentially adsorbed ions.
IV. DIPOLE IN SOLUTION
A somewhat idealized configuration of the void in a polar liquid considered above can now be extended to the configuration relevant to a number of problems appearing when solutions are placed in an external electric field. Consider the solution placed in a spatially nonuniform electric field , which creates the nonuniform Maxwell field . For most practical problems, can be viewed as uniform on the length-scale of the solute. We, therefore, drop the spatial dependence and put . If the solute carries its own dipole moment , the external field aligns the dipole, thus creating an average dipole moment along the field,
Here, the solute dipolar susceptibility yields the average dipole induced by the field, which otherwise averages out to zero by free rotations in solution, . The angular brackets denote an ensemble average in the presence of an external field, in contrast to denoting the average taken when the field is switched off.
The averages at zero field can be expressed in terms of material liquid properties and interfacial susceptibilities. Since the external fields are always significantly weaker than the internal microscopic fields, perturbation expansion in terms of the interaction energy between the external field and the sample dipole moment is always a good approximation. The susceptibility follows directly from the first-order perturbation theory and is given by the following equation:
Here, and are deviations from average dipole moments, which can be assigned zero values in isotropic solutions; is the inverse temperature.
Despite a simple algebraic form given by Eq. (44), the susceptibility is hard to calculate. The main conceptual and technical difficulty comes from the cross correlation between the solute dipole and the solvent dipole entering . This difficulty is overcome in mean-field theories of dielectrics by introducing the concept of the cavity field,3 which is the field produced by the dielectric inside the solute. The cavity-field susceptibility68 is defined as the ratio of the field inside the solute cavity to the external field. A significant simplification of the problem is achieved71 by realizing that the same susceptibility is the ratio of in Eq. (44) and the self-correlation of the solute dipole ,
This relation incorporates all cross correlations between the solute dipole and the dipole moments of the solvent molecules into a single parameter, which can be independently calculated from either the statistical theories of solutions or from numerical simulations.71 These calculations turned out to be critical for understanding dielectrophoresis of proteins: fluctuations of the protein-water interface, related to protein’s elastic flexibility, produce substantial deviations from the solution of the standard dielectric boundary-value problem1 leading to Eq. (40). More specifically, the dielectric solution assumes that thermal agitation can only alter orientations of the solvent dipoles in the interface. In contrast, the protein-water interface is affected by elastic fluctuations of the protein coupled to polarized hydration water. These elastic fluctuations enhance the dipolar solute-solvent correlations in Eq. (45), leading to significantly exceeding the dielectric result.71,72
The total ensemble-averaged dipole moment at the solute induced by the external field is the sum of the ensemble-averaged solute dipole and the dipole moment of the interface considered above,
Here, in contrast to the case of a void shown in Fig. 4, we have added angular brackets to the interface dipole to stress that it is now determined by the combined effect of the external field polarizing the interface and the field of the solute. In analogy with Eq. (43), one can define the susceptibility of the interface as follows:
where the superscript “d” refers to the dipolar symmetry of the problem.
The great utility of the cavity susceptibility is that it not only helps to eliminate many-particle cross correlations from the dipolar response of the solute dipole, but can also be applied to fully characterize the interface susceptibility in Eq. (47). It is given by the following expression:71
In this equation, is the cavity susceptibility for a specific model of interfacial polarization known as the virtual Lorentz cavity.3 This is an imaginary cavity obtained by separating a closed volume of the liquid from the rest of the bulk. The field created in this separated volume by the bulk polarized by a uniform external field makes the Lorentz cavity field. Since there is no physical interface, there is no physical interfacial polarization and no surface charge density . The Lorentz cavity is, therefore, a useful physical limit corresponding to . The cavity-field susceptibility in the Lorentz limit is
It is clear that significantly exceeds in Eq. (40) when : .
Equation (48) shows that the appearance of the interface dipole is linked to the surface charge and is proportional to the deviation of the cavity susceptibility from the Lorentz limit representing . If one assumes that the rules of Maxwell’s electrostatics apply, one can use from Eq. (40) in Eq. (48) to obtain
V. DIELECTROPHORESIS OF PROTEINS
Dielectrophoresis represents the force acting on a particle in a solution polarized by a nonuniform electric field. The presence of the field gradient, which is required to produce the force acting on a dipole, distinguishes dielectrophoresis from ionic mobility considered so far. A nonzero average dipole is created by the external field considered to be uniform on the size of the solute. As described above, the same external field polarizes the solute-solvent interface inducing the interface dipole . Since both of these dipoles are proportional to the external field, the reversible work of orienting the solute dipole and of polarizing the interface is given by the free energy,1,2 which involves the factor of ,
Correspondingly, since , the dielectrophoretic force is proportional to the gradient of the squared field,73–75
Here, it is assumed that the external field is produced by a plane capacitor with the Maxwell field given as . Establishing the connection between and requires solving the dielectric boundary-value problem for more complex geometries of external field sources. Finally, in Eq. (52) is the dielectrophoretic susceptibility,
The parameters in Eq. (52) are chosen in such a way that becomes the Clausius-Mossotti factor when continuum electrostatics is used to calculate the induced dipole moments. When the solute dipole is neglected and , represents polarization of a void in a polar liquid. One obtains from Eqs. (50) and (53)
This result corresponds to a negative dielectrophoresis () when the particle is repelled from the region of a higher electric field.
The standard applications of the Clausius-Mossotti equation assume that the spherical solute carries its own dielectric constant . This limit is easy to obtain from the result for a void by noting that all results of electrostatics are sensitive only to the ratio of two dielectric constants, , at the dielectric dividing surface.2 Substituting in place of in Eq. (54), one arrives at the commonly used expression,73,74
The Clausius-Mossotti factor assumes that the material of the solute is polarizable but does not possess its own permanent dipole moment. In terms of properties of bulk materials, the assumption is that the solute is paraelectric. On the contrary, any material possessing a nonvanishing permanent dipole is characterized as a ferroelectric. When proteins are concerns, each protein molecule can be viewed as a ferroelectric domain, and neglecting in Eq. (53) is not justified. Dropping the permanent dipole of the solute applies only at sufficiently high frequencies when rotations of the dipole are dynamically frozen, and it does not have sufficient time to respond to an oscillating external field.76 As the solute size increases, one anticipates that the dipole moment scales linearly with the linear dimension of the solute, and scales inversely proportional to the linear dimension; e.g., changes as with the solute radius . Since the interface susceptibility in Eq. (53) is constant, it becomes the dominant factor in the dielectrophoretic response for sufficiently large solutes. This limit is, however, not reached for soluble globular proteins.
Proteins typically possess a large density of the surface charge produced by protonation/deprotonation of the water-exposed residues.77 The negative and positive charges mostly compensate each other78 to add to a typically negative overall protein charge at physiological conditions. The nonspherical shape of the molecule and an incomplete compensation between positive and negative charges lead to an overall large dipole moment of several hundreds of debye units when calculated relative to the protein center of mass.78–84
When both the permanent dipole and interface components are included in the dielectrophoretic susceptibility, one obtains71
Here, is the effective polarity76,85 of the protein as measured by the combination of the variance of its dipole moment,
and the polarity parameter quantifying the density of induced dipoles in the protein molecule; is the protein volume. Since at sufficiently low frequencies, is typically a very good approximation.
Large dipole moments of globular proteins78–84 are responsible for large values of the parameter in Eq. (56). For instance, molecular dynamics simulations of cytochrome c in TIP3P water72 have produced the protein dipole of D and at . Combined with from simulations, the susceptibility becomes equal to . This result points to a dramatic failure of the Clausius-Mossotti equation,86 which limits the dielectrophoretic susceptibility by the condition [Eq. (55)]. Given that for most proteins at sufficiently low frequencies of the external field, one can write a simplified equation for ,
Here, , is in debyes, and the protein volume is in .
Equation (58) gives the dielectrophoretic susceptibility at frequencies below the frequency of protein tumbling (), which determines rotational relaxation of protein’s permanent dipole moment (-relaxation in dielectric spectroscopy of proteins78). All functions entering Eq. (56) become dependent on the frequency when an oscillatory external field is applied. The difficulty encountered with the direct extension of the static equation (58) to the dynamic domain is that the frequency dependence of is mostly unknown. One has to turn to alternative means to access . This connection is provided by dielectric spectroscopy of solutions discussed next.
VI. DIELECTRIC SPECTROSCOPY OF SOLUTIONS
Equation (58) suggests that the product is required to determine in Eq. (52). The same product enters the dielectric increment of a low-concentration (ideal) solution compared to the dielectric constant of the solvent. For ideal solutions, the dielectric increment is a linear function of the volume fraction occupied by the solutes. One can derive the following equation connecting dielectric measurements to dielectrophoresis:72
Here, both the dielectrophoretic susceptibility and the dielectric function are written as functions of the circular frequency of the oscillating external field. Correspondingly, is the increment of the frequency-dependent dielectric function of the solution over that of the solvent, .
The static dielectric constant of the protein solution is typically higher than that of the solvent because of a large dipole moment of the protein (dielectric increment).78,87 According to Eq. (60), the range of frequencies with corresponds to positive dielectrophoresis, . However, the solution dielectric constant falls below the solvent value at the frequencies exceeding the frequency of overdamped rotational tumbling of the protein. A crossover to negative dielectrophoresis is, therefore, predicted by Eq. (60) for frequencies exceeding the crossover frequency at which and .
Note that the dielectrophoretic force is proportional to the gradient of in the whole range of frequencies, both below and above it. The crossover itself is merely a dynamic effect of dynamic freezing of protein dipole rotations at . A linear scaling with the field appears from the static polarization of the water dipoles in the interface producing . This force does not require a field gradient and is present even for a uniform external field. The overall force acting on the protein is the sum of the linear and quadratic terms in the field,
where is the effective charge of the solute given by Eq. (36).
Figure 8 shows the frequency-dependent dielectrophoretic susceptibility calculated from Eq. (60) by using dielectric functions reported for solutions of the ribonuclease A protein.83 The dielectric function for water was taken from Ref. 88. The low-frequency turns out to be in the same range of as calculated from molecular dynamics simulations of cytochrome c72 and a number of other proteins.71 The crossover to negative dielectrophoresis at is also clearly observed.
The parameter , where is the protein concentration in , is often used to determine the dipole moment of the protein from dielectric spectroscopy.73,78,84,87,89 Dielectric experiment provides access to the root mean square (rms) protein dipole,
Here, is the molar mass of the protein in kilodaltons and the protein dipole is in units of debyes.
If fluctuations of the dipole moment due to elastic motions of the protein are neglected, pure rotations of the permanent dipole produce and . The rms dipole can be identified with the permanent dipole moment in this case. This is the standard application of Oncley’s formula79 used to estimate protein’s dipole moment from dielectric spectroscopy of solutions.84,90
which turns out to be very close to – calculated from Eq. (45) applied to molecular dynamics trajectories for hydrated globular proteins.71,72 This value of the cavity susceptibility is substantially above the prediction of Maxwell’s electrostatics [Eq. (40)] and also exceeds the Lorenz result [Eq. (49)]. Consequently, both empirical and simulation evidence suggest that Maxwell’s cavity field is a poor approximation for this property when applied to hydrated proteins.68 Failure of Maxwell’s electrostatics was also documented for absorption of radiation by protein solutions, where the cavity susceptibility becomes a prominent parameter of the theory.76 The absorption of THz radiation follows the Lorentz cavity equation [Eq. (49)].
The Lorentz cavity field can be brought in a better agreement with reported by simulations if another recent calculation is taken into account. From the analysis of simulations, the dielectric constant of hydration shells of protein was found to be very low, .34 If in Eq. (49) is replaced with to account for polarization of the protein, one obtains
The simulation and empirical values of [Eq. (64)] are consistent with this equation at and . One still has to recognize that a large reported by simulations is likely related to phenomenology not accounted for by dielectric models. Elastic deformation of the protein coupled with hydration water is likely to be responsible for the cross correlations between the protein and water dipole moments in Eq. (45), leading to the reported values of .
VII. CONCLUDING REMARKS
The physical principle unifying different phenomena discussed here is that the local interfacial response of a polar liquid is distinct from the macroscopic dipolar response responsible for the long-distance dielectric screening and measured by the dielectric experiment. Studies of ion solvation have shown asymmetry of solvation thermodynamics between anions and cations and, in addition, the presence of spontaneous polarization of the interface around solutes carrying a zero charge. This observation translates to asymmetry in the local interfacial susceptibilities between anions and cations and to the presence of a nonzero surface charge, even in the case of uncharged solutes. Furthermore, mobility of ions is determined by the total effective charge within the shear surface. In the limit of Maxwell’s electrostatics, the total charge becomes the ion charge since there is no overall bound charge from the solvent within the shear surface. Alternatively, if the polarization response is nonuniform in the interface, an effective charge [Eq. (36)] appears in the electrostatic force acting on the ion. The difference between and is due to the preferential molecular orientation in the interface (static charge) and different local polar susceptibilities at the cavity and shear surfaces.
Protein dielectrophoresis is another problem where dielectric calculations have been traditionally applied. The dielectrophoretic susceptibility of a large biopolymer is typically calculated from the Clausius-Mossotti equation, which carries two deficiencies: (i) it is based on Maxwell’s cavity susceptibility and (ii) it neglects the permanent dipole of the solute. The first deficiency might be related to elastic flexibility of the solute-solvent interface characteristic of biomolecules, which leads to much stronger cross correlations between the solute and solvent dipoles than anticipated by dielectric theories [Eq. (45)]. The second deficiency is most important since the large dipole moment of the solute (protein) produces orders-of-magnitude higher dielectrophoretic susceptibility than predicted by the Clausius-Mossotti equation. Proteins are, therefore, poised to show “gigantic” dielectrophoresis compared to the standard predictions.
This research was supported by the National Science Foundation (NSF) (No. CHE-1800243). The author is grateful to Mark Hayes for many discussions and for comments on the manuscript.