Polarization energy of two charged dielectric spheres in close contact

Huada Lian a and Jian Qin *b
aDepartment of Materials Science and Engineering, Stanford University, Stanford, California 94305, USA
bDepartment of Chemical Engineering, Stanford University, Stanford, California 94305, USA. E-mail: jianq@stanford.edu

Received 18th September 2017 , Accepted 27th October 2017

First published on 27th October 2017

Evaluating the interaction energy between charged dielectric spheres in close contact is difficult because interfacial charge polarization becomes increasingly stronger as separation decreases. The limit of high dielectric permittivity is amenable to theoretical treatment because the particles become equipotential upon close contact. For particles with low permittivity, the constitutive equation for the induced surface charges demanded by the boundary conditions is derived in this work, based on which the contact energy of symmetric cases is evaluated. The known result for conducting spheres is recovered as a limiting case.

Design, System, Application

Successful design of the structural stability of molecular systems needs to be based on a thorough understanding of the inter-component interactions. This work focuses on the strong, nearly singular polarization interaction between charged dielectric spheres in close contact. The polarization interaction is caused by the surface charges accumulated at the interfaces between the particles and medium, which exhibit distinct dielectric permittivities. The interaction was known to be strong and singular for conducting spheres. We consider dielectric spheres and develop an analytical formalism that reveals a similar type of singularity and allows for calculation of polarization energy at small separations. For two spheres of equal size and charges, we evaluate the cohesive energy and show that the result includes the conducting case as a special example.

1 Introduction

Bringing two conducting spheres of identical size into contact produces a cohesive energy
image file: c7me00105c-t1.tif(1)
where εout is the medium dielectric permittivity, Q1 and Q2 are the charges on the spheres, a is the sphere radius, and (2[thin space (1/6-em)]ln[thin space (1/6-em)]2)a is the total capacitance of the two spheres in contact. The subtracted term (Q12 + Q22)/(2a) is the self-polarization energy of two distant spheres. Eqn (1) was known to Maxwell,1 and was re-considered later.2,3 The main conclusion was that when the charges carried by the two spheres produce a difference in electrostatic potential upon close contact, the induced polarization charges near the contact increase in amplitude according to image file: c7me00105c-t2.tif, where d is the gap distance. As a result, the majority of the potential difference is established when the two spheres are close to each other. This weak singularity as d → 0 leads to a strong separation dependence in energy, and makes accurate calculation of cohesive energy between two conducting spheres difficult. The same type of contact singularity is present in clusters of multiple conducting spheres, for which a generalized approach that combines the Derjaguin approximation with numerical extrapolation has been employed to evaluate the capacitance coefficients and the cohesive energy.4

Despite extensive efforts for dielectric particles,5–7 no analogous results on contact energy have been reported. When dielectric spheres have large separation, the polarization energy can be calculated by approximating the induced polarization charges as dipoles or, as separation decreases, as a sum of multipoles. A systematic approach based on multiple-scattering formalism has been developed to calculate the polarization energy of many polarizable spheres, which carry monopolar, dipolar or multipolar charges.8–10 This approach expresses the total electrostatic energy of dielectric spheres carrying charges of ideal multipoles as a sum of two-body, three-body and other interactions. The two-body terms are normal Coulomb interactions. The three-body terms are interactions between two spheres mediated by one polarizable surface. The four-body terms involve two polarizable surfaces, and the higher order terms are constructed analogously. Each of these terms can be efficiently evaluated by using image lines.8,10 At a scaling level, the two-body terms are proportional to the inverse of average separation R = ρ−1/3 between spheres, where ρ is the number density of spheres. The three-body terms are smaller by a factor (a/R)−3. Each higher order term is smaller by an additional factor (a/R)−6.

Therefore, it is expected that the three-body terms are sufficient for dilute systems or for spheres at large separation. As average separation increases, however, higher order terms become increasingly more relevant. Upon close contact, essentially all the higher order polarization interactions are needed, which is the origin of the singular behavior identified for contact energy between conducting spheres. For dielectric spheres, since the Derjaguin approximation4 cannot be applied to obtain the contact charge—the dielectrics are not equipotential—the contact singularity cannot be readily identified. Preliminary numerical calculation based on multiple-scattering formalism has revealed the difficulty of resolving this contact energy: in one case, the number of higher order polarization terms increases with reduced separation;9 in another, greater spatial resolution is needed.5

This work aims at calculating the contact energy between two charged dielectric spheres. Instead of relying on the perturbative multiple-scattering approach for electrostatic potential, the boundary conditions demanded by the induced surface charges are derived using a bispherical coordinate system. Then, the potential and energy are calculated from the induced charges for varying inter-particle separations. The expressions for elastance coefficients are derived. The contact energy for spheres with arbitrary dielectric permittivity εin is evaluated. It is demonstrated that, in the conducting limit, i.e., εin → ∞, the contact energy reduces to that of Maxwell's result (eqn (1)). The dependence on permittivity allows one to estimate the cohesive energy for dielectric dimers carrying symmetric charges.

2 Formalism

We consider two dielectric spheres with radius a1 and a2, each carrying a fixed charge distribution inside the sphere. An arbitrary charge distribution can be expanded in terms of multipoles. The multipoles can be projected onto surface charges. In order to calculate the total energy of a system of polarizable particles, it suffices to consider particles with embedded multipoles and apply the superposition principle. The formalism developed below can be applied to particles with embedded multipoles. For simplicity, we focus on monopolar particles, i.e., each sphere carries a net charge, Q1 and Q2, which is evenly distributed on the surface. The electrostatic potential outside the sphere generated by such surface charges is identical to that generated by a point source placed at the center of the sphere.

The particle configuration is sketched in Fig. 1. The dielectric permittivity inside and outside the two spheres are εin and εout, respectively. We are interested in the variation of polarization energy with gap distance d. At a large distance, the energy is the total of self-polarization energy of two isolated spheres, image file: c7me00105c-t3.tif, which is caused by the interaction between the charges Q1 and Q2 and the induced surface charges on their own surfaces. At a reduced distance, the normal Coulomb energy, E = Q1Q2/(4πεoutR12), is needed. In the limit of close contact, d → 0, the mutual polarization interaction becomes relevant.8 Evaluating the energy and electrostatic potential requires solving the boundary value problem for Poisson's equation explicitly.11

image file: c7me00105c-f1.tif
Fig. 1 Two approaching dielectric spheres with radius a1 (on sphere ζ) and a2 (on sphere η) and with embedded net charges Q1 and Q2. The inter-particle distance is labeled d. Two foci are located at ±b.

The boundary condition is conveniently formulated using the bispherical coordinate system,12 parameterized by (σ, τ, ϕ). Here, 0 ≤ σ ≤ π is the angle spanned between two foci located at (0, 0, −b) and (0, 0, +b). The term − < τ ≡ ln(d/d+) < is defined based on the ratio of distance d to foci (0, 0, −b) and distance d+ to foci (0, 0, +b), so that τ > 0 in half space z > 0, τ < 0 in half space z < 0, and τ = 0 in the xy plane. The azimuthal angle is defined by 0 ≤ ϕ < 2π.

The surfaces of the two spheres in Fig. 1 have a constant τ value: τ = ζ > 0 and τ = −η < 0. The coordinates of points on the surface satisfy x2 + y2 + (zb coth τ)2 = (b csch τ)2. The two spheres approach each other when ζ and η both approach zero. The center of sphere τ is (0, 0, b coth τ) in Cartesian and (0, 2τ, 0) in bispherical coordinate systems. The scale factors for coordinates σ, τ and ϕ are

image file: c7me00105c-t4.tif(2)
which define the surface area element and the volume element, and are needed by the gradient operator. The Cartesian coordinates are obtained using (x, y, z) = hσ (sin[thin space (1/6-em)]σ[thin space (1/6-em)]cos[thin space (1/6-em)]ϕ, sin[thin space (1/6-em)]σ[thin space (1/6-em)]sin[thin space (1/6-em)]ϕ, sinh[thin space (1/6-em)]τ). The distance between two points (σ, τ, ϕ) and (σ1, τ1, ϕ1) is
[2hσhσ1(cosh(ττ1) − cos[thin space (1/6-em)]γ)]1/2,(3)
where cos[thin space (1/6-em)]γ = cos[thin space (1/6-em)]σ[thin space (1/6-em)]cos[thin space (1/6-em)]σ1 + sin[thin space (1/6-em)]σ[thin space (1/6-em)]sin[thin space (1/6-em)]σ1[thin space (1/6-em)]cos(ϕϕ1).

By introducing the surface charge densities ρζ and ρη into the two spheres, the electrostatic potential ϕ(r) at position r can be written as a sum:

image file: c7me00105c-t5.tif(4)

Here, ϕ0(r) is the potential of source charges,

image file: c7me00105c-t6.tif(5)
where r1 and r2 are centers of spheres ζ and η. For clarity, Qζ and represent Q1 and Q2, respectively. Ref. 8 formulated the boundary condition equivalently by using the full electrostatic potential, which reads
image file: c7me00105c-t7.tif(6)

Here, εrεin/εout is the ratio of dielectric permittivities inside and outside the particle. It is clear that ρη can be interpreted as (1 − εr)n·∇ϕ(r(Sη)) and ρζ as (1 − εr)n·∇ϕ(r(Sζ)).

Eqn (4) is a formal solution for Poisson's equation valid at all separations. We are particularly interested in the limit when separation d = R12a1a2 is smaller compared to both radii, a1 and a2. In the bispherical coordinate system, this can be achieved by setting b → 0. At the same time, to keep a1 = b/sinh[thin space (1/6-em)]ζ and a2 = b/sinh[thin space (1/6-em)]η constant, b needs to be of the same order as sinh[thin space (1/6-em)]ζζ or sinh[thin space (1/6-em)]ηη. In the limit of close contact, we have ζ = b/a1 and η = b/a2. The distance between the two surfaces can be written as image file: c7me00105c-t8.tif, in which image file: c7me00105c-t9.tif. Using a as the unit of length, we find that b/a is asymptotically small and that b/a ≃ (d/a)1/2.

2.1 Boundary conditions for surface charge

The formal solution (eqn (4)) relates four quantities ϕ(r), ϕ0(r), ρζ and ρη to each other. The source potential ϕ0(r) is given. The surface charge densities ρζ and ρη produce the discontinuity of the electric field at dielectric boundaries, i.e.,
εinEς(−) = εoutEς(+), εinEη(−) = εoutEη(+).(7)
In the above equation, Eς(±) and Eη(±) are the normal components of the electric fields inside (−) and outside (+) the interfaces τ = ζ and τ = −η. The field Eτ relates to the potential ϕ(r) by image file: c7me00105c-t10.tif. For positive or negative τ, the above expression gives the component Eτ pointing towards the inward or outward directions.

The electrical field at the boundary of sphere ζ includes contributions E(0)ςς from the source charge in the ζ particle, E(0)ςη from the source charge in the η particle, E(s)ςς from the induced charge in the ζ particle, and E(s)ςη from the induced charge in the η particle. The electrical field at the boundary of sphere η can be decomposed similarly. Thus, we have Eς = E(0)ςς + E(0)ςη + E(s)ςς + E(s)ςη and Eη = E(0)ης + E(0)ηη + E(s)ης + E(s)ςς. The self-polarization term E(0)ςς equals image file: c7me00105c-t11.tif inside sphere ζ (at a1 − 0) and image file: c7me00105c-t12.tif outside sphere ζ (at a1 + 0). The self-polarization term E(0)ηη equals image file: c7me00105c-t13.tif inside sphere η and image file: c7me00105c-t14.tif outside sphere η. They both satisfy the required boundary conditions in eqn (7) alone. On the other hand, E(0)ςη and E(s)ςη are continuous across the ζ surface, and E(s)ςς is discontinuous; similar arguments hold for the η surface. So the boundary conditions can also be written as:

image file: c7me00105c-t15.tif(8)

The subscripts “out” and “in” remind us from which side the field is evaluated.

Eqn (8) serves as the basis for solving the surface charges. The source terms E(0)ςη and E(0)ης can be trivially evaluated. The remaining induced terms can all be expressed as integrals over the surface charges. For instance, the potential at an arbitrary point r = (σ, τ, ϕ) generated by the surface charge ρζ can be written as:

image file: c7me00105c-t16.tif(9)

Here, an auxiliary charge density, [small rho, Greek, tilde]ζρζ/(cosh[thin space (1/6-em)]ζ − cos[thin space (1/6-em)]σ′)3/2, has been introduced in order to simplify the notation. The factor cos[thin space (1/6-em)]γ ≡ cos[thin space (1/6-em)]σ[thin space (1/6-em)]cos[thin space (1/6-em)]σ′ + sin[thin space (1/6-em)]σ[thin space (1/6-em)]sin[thin space (1/6-em)]σ[thin space (1/6-em)]cos(ϕϕ′) is the angle cosine between unit vectors oriented along (σ, ϕ) and (σ′, ϕ′).

The straightforward differentiation image file: c7me00105c-t17.tif applied to the potential yields the electric field along the direction −[small tau, Greek, circumflex]

image file: c7me00105c-t18.tif(10)
Here, image file: c7me00105c-t19.tif. The above expression is applicable for any value of τ, and is continuous except at τ = ζ. To resolve the magnitude of the discontinuity, we focus on the cases τ = ζ ± 0. Using the fact that image file: c7me00105c-t20.tif as τζ ± 0, we get
image file: c7me00105c-t21.tif
The term in the bracket is continuous across the normal direction of the surface since the logarithmic divergence arising from the weak singularity at the denominator does not depend on whether surface ζ is approached from the inside or the outside. So we denote it as E(cont.)ζζ. The second term −(±ρζ/2) shows that the discontinuity is proportional to the charge density. By our convention, ζ is positive, so the term with the positive sign, for ζ + 0, is the field inside the particle, whereas the term with the negative sign, for ζ − 0, is the field outside the particle. Using these conventions, the first line of the boundary condition, eqn (5), can be written as
image file: c7me00105c-t22.tif
It is convenient to introduce the relative dielectric discontinuity, image file: c7me00105c-t23.tif, and divide both sides by (cosh[thin space (1/6-em)]ζ − cos[thin space (1/6-em)]σ)3/2, to convert ρ to [small rho, Greek, tilde] and, analogously, E to image file: c7me00105c-t24.tif. Then, we arrive at
image file: c7me00105c-t25.tif(11)
The terms on the right hand side that depend on the induced charge densities are given from differentiation by
image file: c7me00105c-t26.tif
Apart from the dependence on free charges, the equation above is identical to eqn (31) of ref. 12.

The analogous result for particle η is obtained by exchanging η and ζ:

image file: c7me00105c-t27.tif(12)

Eqn (11) and (12) are the boundary conditions that will be solved to obtain the induced charge densities.

2.2 Mode expansion

Given the symmetry of particle boundaries, it is convenient to expand the charge densities and the normal components of the electric fields using spherical harmonics, as follows:
image file: c7me00105c-t28.tif

The expansion coefficients are computed by projections, e.g.,

image file: c7me00105c-t29.tif
in which Ω = (σ, ϕ) and dΩ = dσdϕsin[thin space (1/6-em)]σ. The convention for spherical harmonic functions Y[small script l]m is that they are normalized, i.e., image file: c7me00105c-t30.tif, where Pm[small script l] is the associated Legendre polynomial. For configurations with uniaxial symmetry, such as the case with monopolar charges, only isotropic components with m = 0 survive, and the coefficients are simply written as A[small script l], B[small script l], U[small script l], and V[small script l], which are evaluated using the projection on the Legendre polynomial of the form, for instance, image file: c7me00105c-t31.tif Two points are made before proceeding further. First, the expansion is applied to the quantities with ‘∼’, i.e., those normalized by (cosh[thin space (1/6-em)]τ − cos[thin space (1/6-em)]σ)−3/2. Second, the field image file: c7me00105c-t32.tif in the most general situations includes contributions from all the sources other than those on particle ζ; here, the only external source is particle η, so η is explicitly labeled.

By substituting the above mode expansion coefficients to eqn (11) and (12), and using the expansions image file: c7me00105c-t33.tif and image file: c7me00105c-t34.tif, we obtain the following algebraic relation for the coefficients from the boundary conditions on sphere ζ,

image file: c7me00105c-t35.tif(13)

Here, the coefficients are defined by g[small script l] ≡ e−([small script l]+1/2)(ζ+η), image file: c7me00105c-t36.tif, and image file: c7me00105c-t37.tif An analogous expression can be found from the boundary conditions on sphere η. For the uniaxial case of interest here, only the m = 0 terms survive. So we drop the explicit reference to index m and adopt the convention that all the coefficients are evaluated at m = 0. Then, the constitutive equation can be written as:

image file: c7me00105c-t38.tif(14)

The coefficients C[small script l][small script l] and I[small script l][small script l] are reduced to image file: c7me00105c-t39.tif and image file: c7me00105c-t40.tif. The corresponding constitutive equation for particle η is obtained by exchanging A and B and by replacing U with V:

image file: c7me00105c-t41.tif(15)

In the conducting limit, ε → 1, the coefficient relation can be further simplified to:image file: c7me00105c-t42.tif

In the far field regime, ζ → ∞ and η → ∞, whereas at the same time ζ/sinh[thin space (1/6-em)]ζa1 and η/sinh[thin space (1/6-em)]ηa2. The cross terms all vanish because the coupling g[small script l] decays to zero.

In the near field regime, ζ → 0, η → 0, b/ζa1 and b/ηa2; the small parameters ζ, η, and b are of the same order of magnitude. In this limit, the coefficient I[small script l][small script l] is of the order image file: c7me00105c-t43.tif. To solve the unknown A[small script l] and B[small script l], the mode coefficients U[small script l] and V[small script l] from the fields produced by free charges are needed. The numerical results for the conducting limit obtained from such an approach are illustrated in Fig. 2 and compared to the prediction of ref. 3.

image file: c7me00105c-f2.tif
Fig. 2 Comparison of surface charge densities on two spheres with equal radius Qζ = 2Qη and separation distance R/a = 3 with known results for conducting spheres.3 The thick curves are the total densities and the dashed lines denote the free charge density, both of which have units of Qζ/4πa2. The bispherical coordinate σ is related to θ by image file: c7me00105c-t44.tif. The upper sign corresponds to the right half space and the lower sign corresponds to the left half space.

2.3 Fields U[small script l] and V[small script l] from point sources

To solve the induced surface charges, we first consider the potential ϕζ(r) at an arbitrary point r(σ, ζ, ϕ) on the surface of sphere ζ, produced by the charge Qη on sphere η, which is centered at rη = (0, −2η, 0). In the bispherical coordinate system, ϕζ(r) can be written as:
image file: c7me00105c-t45.tif(16)
Note that for the monopolar point source, the angle cosine cos[thin space (1/6-em)]γ reduces to cos[thin space (1/6-em)]σ. The field E(0)ζ along the outward-pointing normal direction can be evaluated by applying image file: c7me00105c-t46.tif to ϕζ(r), which gives rise to the following expression for the mode coefficient:
image file: c7me00105c-t47.tif(17)
where the auxiliary factor h[small script l](η, ζ) is defined by:
image file: c7me00105c-t48.tif
The coefficients V[small script l] on sphere η are found by exchanging ζ and η. Completely analogous results can be obtained for V[small script l] by exchanging ζ and η. It then follows that A[small script l] and B[small script l] can be calculated from eqn (14) and (15) by inverting the coefficient array.

2.4 Total energy

The total energy is the sum of self-energy, direct interaction between source charges, image file: c7me00105c-t49.tif, and the interaction between induced charges and source charges, Epolarization. We first consider the electrostatic potential at point (0, τ0, 0), where τ = 2ζ or −2η produced by the surface charge on sphere ζ:
image file: c7me00105c-t50.tif
Setting ζ = η and A[small script l] = B[small script l] gives the potential generated by surface charges on sphere η. The overall potential for τ = 2ζ and τ = −2η can be written in the matrix form as
image file: c7me00105c-t51.tif(18)

Once the coefficients A[small script l] and B[small script l] have been solved, the polarization energy can be calculated by using:

image file: c7me00105c-t52.tif(19)

By writing coefficients A[small script l] and B[small script l] explicitly in terms of charges Qζ and Qη, the elastance coefficients can be obtained from the summation of entries in array T.

3 Contact energy for symmetric charges

To demonstrate the application of eqn (14), (15), (18) and (19), we first compare the charge densities obtained from the above formalism to the known results for conducting spheres.3 The expressions in ref. 3 were derived by employing the facts that conducting spheres have a constant potential, which is inapplicable for dielectric spheres. However, within our formalism, the behavior of conducting spheres can be recovered by setting the dielectric permittivity εout = ∞ or ε = 1. Setting ε = 1 first then inverting the coefficient array (ICI)[small script l][small script l] leads to a singular behavior. In practice, it is found that inverting the array for ε values approaching 1 results in converged charge densities. Fig. 2 shows that for separation R/a = 3, the values ε = 0.9 and ε = 0.99 both produce charge densities that are indistinguishable from those predicted using the expressions in ref. 3.

Then, the contact energy for dielectric spheres with symmetric charges can be calculated as follows. First, the interaction energy for a given dielectric permittivity, which includes the pairwise Coulomb energy and the mutual polarization energy, with the self-polarization energy omitted, is calculated at several separations, close to the contact value 2, which is the last term given by eqn (18) and (19). Then, the separation dependence is fitted and extrapolated to the close-contact value. This procedure is demonstrated in Fig. 3, for two spheres with identical radii and charges Q1 = Q2 = 1. Two families of data points are calculated from the cases with strong polarization, i.e., ε → ±1. In both cases, the fitting is based on quadratic polynomials, and a smooth extrapolation towards R/a = 2 can be made.

image file: c7me00105c-f3.tif
Fig. 3 The variation of polarization energy E(R/a; ε) with separation and with dielectric permittivity, for identical spheres carrying symmetric charges. The dashed line is the analytical result for contact energy of a conducting dimer, 1/ln[thin space (1/6-em)]2 − 1, which is recovered using the extrapolated value at ε = 0.99 and at R/a = 2.

The extrapolated values for the contact energy are plotted against the values of ε in Fig. 4. Three limits are worth noting. (1) At ε = 0, the dielectric interface and polarization contribution vanishes. So the total energy is dominated by the normal Coulomb interaction, which is 0.5 on this normalized scale. (2) At ε = 1, the behavior approaches that of conducting spheres, as demonstrated in Fig. 2. The value for the contact energy obtained from the Maxwell expression (eqn (1)), which is 1/ln(2) − 1 ≃ 0.44, is recovered. (3) At ε = −1, the behavior of weak dielectrics, with εinεout, is recovered. This is the typical case for an aqueous solution of colloidal particles. In this limit, the polarization energy becomes independent of the internal dielectric permittivity, and is determined by the medium permittivity. From the known result13 for the electrostatic potential ϕw produced by a point charge outside dielectric particles with a low value of permittivity, the polarization energy can be estimated to be (1/2)(Q1ϕ1 + Q2ϕ2), in which ϕ1 and ϕ2 are the electrostatic potential ϕw produced by charges Q1 and Q2, and evaluated at the location of Q1 and Q2, which is a correction to the self-polarization energy. Taking the limit R/a = 2, it is straightforward to show that Q1ϕ1 = Q2ϕ2 = 1/3 + ln(3/4 = 0.05) on the normalized scale. Adding this value to the Coulomb energy gives an estimate of 0.55, a value very close to that obtained from the numerical approach.

image file: c7me00105c-f4.tif
Fig. 4 The contact energy of two charged dielectric spheres and its variation with dielectric permittivity. The energy approaches 1/ln[thin space (1/6-em)]2 − 1 ≈ 0.44 in the conducting limit (ε → 1), approaches approximately 5/6 + ln(3/4) ≈ 0.55 in the weak dielectric limit (ε → −1), and approaches 0.5 in the absence of the dielectric interface (ε = 0.5).

4 Summary

The main result is shown in Fig. 4, displaying the contact energy for symmetric dielectric spheres at all dielectric permittivity values, which includes the weak dielectric and conducting spheres as the special cases. Physically, this contact energy is the work needed for bringing the two dielectric spheres into contact. We found that the work is mainly dominated by the normal Coulomb interactions image file: c7me00105c-t53.tif, 0.5 in the reduced unit. The polarization correction is small for all possibilities, leading to at most ±10% of variation, which suggests that for most practical cases, the polarization correction for particles carrying the same amount of charges can be safely neglected.

The formalism developed, although only applied to spherical geometries, can be readily modified to treat the cases involving flat interfaces, by setting ζ = 0 or η = 0, or to charged cylinders, by modifying the metric hϕ related to the azimuthal degree of freedom. To adapt the formalism to arbitrary source charge distribution, the same type of multipole expansion can be applied, and eqn (13) is the most general form of constitutive equation. This formalism, however, only works for continuum media and breaks down when the inter-particle separation becomes comparable to the average separation between discrete charges on the surface. The dielectric permittivity of particles need not be identical, as is clearly shown by the expression for the boundary conditions. Further, the surface polarization can be induced by externally applied and oscillating fields, as demonstrated in ref. 12 in the treatment of van der Waals interactions.

The numerical data was only provided for the symmetric case. The contact energy for the asymmetric case such as Q1 = 1 and Q2 = −1 cannot be obtained by applying the extrapolation in Fig. 3, because the contact charge accumulates rapidly at small separations, which leads to a nearly singular behavior for the separation dependence of polarization energy, as demonstrated in ref. 4. For the conducting case, this leads to a image file: c7me00105c-t54.tif dependence. For the dielectric cases, this dependence is weakened; no analogous expression based on the Derjaguin approximation can be readily obtained, since the dielectrics are not equipotential. Yet, in the absence of an analytical expression for the strength of this singularity, any numerical extrapolation will fail at sufficiently small separation. We thus leave the clarification of this issue to future work.

Conflicts of interest

There are no conflicts to declare.


J. Q. acknowledges support from the Terman Faculty Fund, the 3M Non-Tenured Faculty Award, and the Hellman Scholar Award. This research has been supported by the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of Vehicle Technologies of the U.S. Department of Energy through the Advanced Battery Materials Research (BMR) Program (Battery500 Consortium).


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