Surface Tension of a Charged and Polarized System Surface Tension of a Charged and Polarized System

Usually, the formula for the surface tension of a planar charged and polarized interface is obtained from that for a system involving only short-range forces, y = J??- dz [p - px(z)], by replacing the tangential pressure p, by p, + E2/8u. Problems with this include (a) p, is no longer explicitly defined, (b) the electrostatic stress term E2/8u is not correct in general but only if polarization is proportional to density of polarizable species, (c) the derivation of the formula in terms of p and p, involves calculating the work to expand a volume containing the interface, and this work cannot be written in terms of the pressure of the surroundings when there are long-range forces. To derive a formula free from these objections, we consider the spherical system contained between r = R, and r = R2 and containing charged and dipolar particles, the orientation of the latter giving rise to the electrical polarization. There is no electric field, electric polarization, or local charge density for r < R, or for r > R2. If this system is expanded keeping the ratios of all radii fixed, the work done by the surroundings is 4u@1R,2bRl -p2R&3R2), which is set equal to the change in free energy, calculated from the canonical partition function. The surface tension is defined as (R,,/2)(pl -p2), where R,, is the surface of tension. When R, becomes infinite (plane interface), the value of R,, becomes irrelevant. Both long-range and short-range terms in the surface tension are shown to behave properly for R, - m, the long-range terms being proportional to l d r [-p(r) V(r) + 3P(r) E(?)] (P = polarization). If only charged particles are present (no polarization), correlations and short-range forces are neglected, and the distribution of each charged species ni follows the Boltzmann law with energy qiV, it is shown that kmini - E2/8r is independent of z. Using this fact with our surface tension formula, we prove the Lippmann equation. If dipolar particles are present as well as charged particles, the former must be included in Cini. Then the quantity k E i n , - E2/8u - EP is shown independent of z, and our surface tension formula again leads to the Lippmann equation.


Introduction
An understanding of the interfacial tension of the electrochemical interface and how it varies with the compositions of the phases adjoining the interface and with the potential drop across the interface is central to modern theories of electrochemistry.I4 In the polarizable interface, with which we are concerned here, a change in U, the electrostatic potential drop across the interface, produces a new equilibrium state (no current flows), with the change in interfacial tension y given by the Gibbs-Lippmann equation, (dy/aU)T,wmp = -Q, where Q is the surface charge density on one side of the interface. Surface tension calculations for phases containing charged and polarizable particles may be based on density-functional theories or involve molecular distribution functions and intermolecular forces expli~itly;~ of course, the surface tension of the interface between two phases is not simply the sum of single-phase surface tensions. Although thermodynamic approaches have been fruitful in electrochemistry (indeed, the Gibbs-Lippmann equation is normally derived thermodynami~ally'*~), a statistical mechanical formula for the surface tension is needed to relate thermodynamic properties to molecular properties. Because the interface between two phases containing charged and polarizable particles is a region within which there exist large electric fields, local electric charge densities, and rapidly varying species densities, the usual derivation of the expression for the interfacial tension in terms of molecular distribution functions is not valid. A correctly derived formula for surface tension is needed for a true molecular theory of the interface. We give a new derivation of such a formula here, and isolate the electrostatic or long-range contribution. For several physical models, we will show that our surface tension formula verifies the Gibbs-Lippmann equation.
Commonly, the surface tension formula for a charged and polarized interface is obtained starting from a formula such as for a plane interface, perpendicular to the z axis, in which only short-range forces exist. Here, p is the pressure in either homogeneous phase and p,, sometimes called the tangential pressure, involves particle densities and interparticle forces in a direction parallel to the interface. For a system containing Coulombic long-range forces, it can be argued6 that a term proportional to the square of the electric field should be added top,, so that one writes instead of (1)

(2)
A term in Ez also appears in the surface tension formula in terms of correlation functions for a multicomponent charged fluid. If each partial direct correlation function is written as the surface tension becomes -(4u)-ISdz E2(z) plus a term in the The argument for the term -E2/8u given by Sanfeld6 is as follows: In the absence of electric fields, the condition of mechanical equilibrium for a spherically symmetric system is where r is distance from the center, and pn and pl are the pressures normal (radial) and tangential to the interface. If an electric field is present, a force pE + P(dE/dr) should be added to the force -(ap,/ar). Here p is the charge density, P the electric polarization, and E the field; P and E are in the radial direction, and V.(E + 4uP) = 41rp. Then

ap, ~( E z / s * ) a(Ep)
This simplifies to Good i s m a n which, combined with (4), suggests that one should replacep, and Pn by ptl= Pt + -Pn' = Pn --EP

( 5 )
E2 8* the added terms being pet and pen. Now integrating eq 3 from the interior to the exterior of the interface gives -2lCr-l(p,,p() dr = pc -pi Since, for a spherical interface, the surface tension y is defined by the Young-Laplace equation9 as 1/2r.,(pi -pc), where r., is the surface of tension: For an interface of large radius, approaching planarity, this becomes l@,,p() &, and mechanical equilibrium requires that p i be constant through the interface and equal to the pressure of either homogeneous phase bounding the interface. Therefore, y = J(pp;) dz, which is eq 2 (with p, = pJ. There are several problems with this derivation. Most important is the lack of a precise explicit definition of pn and pt in the presence of electric fields. If E2/8u is the contribution to the tangential pressure of the long-range forces, the remainder of p( should include the kinetic pressure k m n i and the contribution of the short-range forces. But there should also be a short-range contribution to the pressure involving the electrostatic interaction and short-range correlations, as identified in a previous treatment* of the Lippmann equation for the ideally polarizable electrode. Furthermore, Hurwitz and d'Alkaine'O derive, starting from the same electrostatic force as Sanfeld, pE + P(aE/ar) which differs from Sanfeld's result6 by the term -27rP2. In fact, Sanfeld concludes from his thermodynamic discussion of surface tension in a charged and polarized system that the definition of pn or p, is a matter of convention. If pn and p, are not well defined, one cannot say a formula for y has been derived. Only a statistical mechanical treatment can resolve the ambiguity.
Another problem is that the replacements of ( 5 ) are not correct in general. Both Sanfeld6 and Hurwitz and DAlkaine'O use the formulas for the stresses in a charged and polarized system to derive their formulas. The stress tensor u was derived by Landau and Lifschitz" by considering the work done in displacing unit area of surface within a charged and polarized medium. They found for a polarized one-component system of density n, where po is the local pressure that would exist in the absence of the field, but with the same matter (presumably including the same distributions and interparticle correlations) present. If the polarization P is proportional to n, then and 4UUik = 6,k[-4'lrp0 -'/2E2] + EiDk. Since = -Uxx and Pn = -uzz, one may recover eq 5 . Thus ( 5 ) is not valid in general but requires P to be proportional to n.
Looking further, one sees that for a system involving long-range forces the usual derivation12J3 of a basic surface tension formula such as (1) is not possible. Such a derivation defines the surface tension as the work required to change the surface a m of a system by unity. Let the system be defined by 0 I x I a, 0 I y I b, 0 I z I c, where the interface is near z = c / 2 . Then one can increase a by the infinitesimal amount 6a and b by bb and decrease c by (c/a)6u + (c/b)6b, so that the volume is constant to fmt order and the interfacial area increases by abb + b6a. Calculating the change in free energy leads to a formula such as eq 1 if there are only short-range forces, because the work involved can be written in terms of pressures in the surroundings. One need not consider the surroundings in detail since the importance of interactions in regions far from the boundaries of the system can be made as small as desired by increasing the size of the system. This is not so if there are electric fields for some value of z for all x and y . In this case, one must specify what deformation is carried out on the surroundings during the deformation of the system. The surroundings must, like the system, be charged and polarized, so that interaction of the system with faroff parts of the surroundings may not be neglected.
In the present article we give a new derivation of a formula for the surface tension of a system with long-range forces, free from the above objections. To avoid considering the surroundings explicitly, we consider a spherically symmetric system, with unpolarized surroundings, which is eventually allowed to approach planarity. Long-range terms in the surface tension formula are isolated, and an explicit unambiguous expression for the "preSSUTe" or short-range terms is obtained.

Basic Formula
Consider a spherically symmetric system, extending from r = R , to r = R2 (where r is the distance from the center), as shown in Figure 1. The regions r < Rl and r > R2 contain homogeneous and isotropic phases 1 and 2, so that the electric field, charge density, and polarization density vanish in these regions. Since the forces exerted by the surroundings are short-range only and can be expressed as pressure with the conventional definitions, we can conveniently consider the work done during a deformation of the system to derive our formula. We will eventually allow the radii R, and R2 to become large compared to their difference, which corresponds to a planar interface.
The system is at equilibrium at temperature T. Let p1 and p2 be the pressures of the homogeneous phases 1 and 2. Within the system, ri(s) is the position of particle number s of species i, so ri(s) is the distance of this particle from the origin. Each particle of species i carries a charge qi and a permanent dipole moment p i . The electric polarization arises from the orientation of these moments; electronic polarization is not included in the model.
We calculate the free energy change for a deformation of the system at constant temperature in which each ri(s) changes by the same fractional amount, Le., R2 -. R2 + 6R2, R , -R, + 6RI with 6RI = (R,/R2)bR2, and ri(s)ri(s) + 6r,(S) with 6rp) = (ri(s)/R2)6R2. The change in the free energy of the system is the work done on the system by the surroundings, i.e.
In terms of the partition function, 6A = -kT(d In QN/dR2)6R2, where the partition function QN is given by

QN = [nN,!]-l(nfi(
Here Ni is the number of particles of species i, and 9 is the interparticle interaction energy:

= f/zCCC~ij(ri(s),r~),Qi(s),Q~))
The interaction between particles depends on their species (4Jij): if dipolar species are involved, particle orientations as well as particle positions enter &p The integral over ri(s) extends over the volume of the system, and that over Qi(s), which gives the orientation of particle s of species i, extends over the angles defining the orientation, so that SdQi(s) = 4rC, where C = 1 for a cylindrically symmetric particle and 27r for others (because of the third Euler angle). We have The differentiation of QN is carried out using scaled coordinates. Let R,, be between R , and R2; although its precise value is later shown to be unimportant, it should correspond to the region in which electric field and polarization are nonzero. Equation 10 may be rewritten as so that the pressure term cancels off the others in the integrand of (1 1 ) when rl is near R , or R2.
We may now introduce the surface tension according to Laplace's eq~ation:~

( 1 3 )
Here, must be taken as the surface of tension,63l3 its value being defined in terms of the moments of the tangential forces. On a macroscopic level, the surface tension is meaningful only when Ro is large compared to the thickness of the interface, which is also the situation which permits the measurement of Ro. Combining eqs 1 1 and 13, we have The one-particle distribution ni(rlQl) may be integrated over Q, to give ni(rl). This follows because Sdr2 dQ2 g(rlQlr2Q3 approaches a finite limit, G(rl,Ql), independent of Ro (the effective volume of integration does not change as R , , R2, and Ro increase). The value of this limit, by the symmetry of our system, depends only on rl and a,.
(e = electrostatic, s = short-range) and13 where the correlation function hkl approaches 0 for large rI2. The integrand in the second term of (14) is a sum of a short-range term from +kJ and a term involving the long-range electrostatic potential which includes the true long-range terms Returning to (14), we write t #~~~ as 4kf + nk/(rlQlrZQZ) = nk(rlQ1) n/(r2Q2) [1 + hk/(rlQIr2QZ)1 (15) Goodisman and short-range terms from h, , which are nonzero only over a limited range of r2 around rl. Thus the terms involving 4$ in (14), as well as the terms involving hij and the terms depending only on rl (terms from the ni), make a contribution to the curly bracket of (14) which is proportional to RoZ. Their contribution to y is then independent of Ro as Ro --.
The true long-range terms will be considered separately in the next section. We emphasize here that the long-range (electrostatic) potentials make a contribution to the short-range terms involving hi,. If the short-range terms are to be called pressures, like p , and pt in (1)-(5), one should remember that the electrostatic forces contribute to the pres~ures.~J~ If the interaction potential 4i is wholly short range and isotropic, we can pass to the limit of a pianar interface and show that (14) is equivalent to a familiar surface tension expression. When R1, R2, and Ro become infinite, p1 and p 2 become equal to po, the pressure in the homogeneous phases, and (14) a well-known expression13 for the surface tension.

Long-Range Terms
To show that the long-range terms in (14) can in fact be written as 4rb2 multiplied by an integral independent of &, we rewrite them in terms of the electric field and electrostatic potential. We thus consider n,(rlQl) is the total density of particles of species i at rl and C,ni(r)qi is the charge density p(r). Similarly, the average dipole moment of molecules of species i located at rl is (18) and Cini(r)mi is the dipole density or electric polarization P(ri), which is in the radial direction. Then, after carrying out the differentiations with respect to rl and rz, the long-range terms become mi = JdQi ni(rzQi)~i/JdQi ni(riQl) We now introduce the electric potential and electrostatic field, calculated in terms of the charge density and dipole density of polarization.
The relation between the "microscopic fields" resulting from molecular charge densities and the "macroscopic fields" of the Maxwell equations have been discussed by many authors, including the classic treatments of Van VleckI6 and EMttcher." It is shown that one must average the microscopic fields, which result only from true charges, over small volumes which contain large numbers of molecules to get the macroscopic fields. This allows the use of a multipole expansion in r/R, where r is a displacement vector from an origin at the center of gravity of the molecular charge distribution to an element of the charge density (nuclei and electrons) assigned to the molecule, and R is the vector from the center of gravity to the position of the observer, so that r/R is small. Then a molecule is characterized by its total charge and dipole moment, the latter being an integral over the molecule of charge density multiplied by r. In calculating the field, one has thus to exclude contributions of a small volume about the observation point. This is responsible for the ambiguity in the definition of the field discussed by Sanfeld6 and by Hurwitz.Io It is shown by B6ttcher" that the electric field, calculated as the force on a unit charge, will obey the Maxwell equation V.D = 4up if one calculates the force due to all matter outside a cylinder of infinite length and infinitesimally small radius oriented in the field direction. De Groot and SuttorpI4 show that E satisfies the Maxwell equation when written as W l ) = -P j d r z b(r2) + P(r2)'vz1v11rl -r21-lj b s 2 n2.P(rz)Vllrl -r2I-l where the second integral is over the surface of a small volume around rl, equivalent to BBttcher's cylinder. They also show that one may write (20) The electrostatic potential at point rl due to distributions of W I ) = -V,jIp(rz) + P(r,).V,)Ir, -r21-l with convergent integrals. charges and dipoles is Surface Tension of a Charged and Polarized System V(rl) = S d r 2 p(rz)lrlrzl-l + S d r 2 P(r2).V21rl -r21-I = The1 field using (20)
The surface tension formula now becomes where p is the pressure in either homogeneous phase. The integrand of (25) is nonzero only in the region of inhomogeneity, so integrals over z may be extended to the range -to -. To ~m p a r e (25) with other formulas for the surface tension of a charged and polarized system, one requires a quantity, like p,,' of (9, which is conserved through the interface (independent of z ) and hence becomes equal to p in the homogeneous phases.

L i p p m~ Equation
The Lippmann equation, normally derived thermodynamically,194J8 is (&y/aU)p,T,mp = -Q where pressure, temperature, and composition of the homogeneous phases bounding the interface are kept constant. Here, U is the potential drop across the interface, V ( -) -V(--), and Q the surface charge density on the + side of the interface. The total surface charge density of the interface, which is electrically neutral, is zero. To define Q, each charged species is assigned to one side or the other of the interface, Le., to the homogeneous phase at z -+or that at z ---.
Then, denoting by Ci(+) and Xi(-) sums over species belonging to the + andsides (xi = xi(+) + x!-)), the surface charge densities for each side of the interface are The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6359 with Q-= -Q+ because of overall electrical neutrality.
The most commonly used model for the electrochemical in-terface1.4J8 is the Gouy-Chapman model, although it is usually applied to only one side of the interface, the other being taken as surroundings. In this model interparticle correlations and short-range forces are neglected, and each charged species obeys a Boltzmann relation: Each species is referred to the homogeneous phase at +or the phase at --.

z [pAV-C(+)qiniAV(+m) -C(-)qiniAV(--) -pAV
Introducing the surface charge densities Q+ and Q-, we have which is the Lippmann equation. Note that it was necessary to take a dielectric constant of unity here and assume no polarizable species were present, even though it is usual in the Gouy-Chapman model to represent the polarizable species (solvent) by introducing a dielectric constant different from unity.
In the presence of polarizable species, k m i n I -E2/8u is not independent of z, because the density of the polarizable species should be included in Cini, and P # 0. It has been suggested (see eq 5 ) that in this case k m i n i -(E2/8u) -EP is independent of z, with eE = D = E + 4uP. To verify this, we write X i = + E,(-) + C/O), where P = Cl(o)n&i (sum over polarizable species) and the dipole moment pi may depend on E. Then we write a Boltzmann expression for each polarizable species: i i Now,using (27) for the charged species equal -JfdE'pi(E'). In previous ~o r k ,~*~~1 ; . , which is the work required to introduce a dipolar particle into an electric field, was written in this way by analogy to (27), in which qi(V-V(*:m)) appears as the work required to introduce a charged particle.

ni(z) = ni(f-)e-f@)/k* (30)
According to (31), kmini -E2/87r -EP is independent of z, so that it may replacep in the surface tension formula (25). Then we get E y = X I d z (-" + p -kTCni 1 Good i s m a n we show that the Lippmann equation, eq 29, is verified. Commonly, the Gouy-Chapman model is used for ions in a solution of dielectric constant different from unity, but k m i n i ( z ) -E-( z )~/~x is then no longer conserved through the interface. This inconsistency with mechanical equilibrium can be remedied if polarizable species are included in Cini and

nj(z) = n j ( * m ) exp[ S, dE' ~j ( E ' l ]
for each polarizable species. Then k E i n i -(E2/8r) -EP is independent of z, and the Lippmann equation is satisfied. The variation of the density of polarizable species with position means that the dielectric constant is not a constant but a function of position.
It will be of interest to investigate other m o d e l~.~~~~ in the same way as we have investigated the Gouy-Chapman model, to see whether the Lippmann equation is satisfied. Similarly, our surface tension formula should be analyzed to see under what conditions the surface tension can be written as a sum of the contributions of the two phases (e.g., metal and electrolyte), and when one is justified in considering one phase only and treating the other as an impenetrable barrier and a source of electric field. Other commonly used approximations should also be investigated, now that we have a completely explicit formula for the surface tension.

Q-Av(-m) = -Q+A[v(m) -V(-m)l (32)
and the Lippmann equation is satisfied for this model. According to (30), there is an inconsistency in the Gouy-Chapman model when the polarizable species are considered only as providing a dielectric constant; the density of polarizable species must vary in space because of the variation in electric field (electrOStriCtiOn). This implies a Spatial Variation Of dielectric constant. Many models for the electrolyte part of the electrochemical double layer posit a pition-dependent dielectric constant. Almost always, one distinguishes between the compact and diffuse parts20 of the double layer, with very different dielectric constants, and some workers21 have introduced a dielectric constant which varies as a function of distance d from the electrode, approaching the bulk -OD* The present work suggests that the spatial variation of dielectric constant should depend on the state of charge of the electrode. (Note that the constant is generally manent dipoles; the electronic polarizability of solvent molecules has not been considered here.)

Conclusions
We have derived a formula for the surface tension of an interfacial system containing charged and polarizable species in terms of the interparticle potentials and the one-and two-particle distribution functions, eq 14. This formula is for a spherical interface and involves the surface of tension, &. The long-range t m " involving of products ofthe electrostatic interaction potentials and one-particle distributions, are extracted and rewritten in terms of the electrostatic potential, electric charge density, and electric polarization, eq 23. Then, on passing to the limit of a planar interface, the value of & becomes irrelevant, as short-range terms approach proper limits, leading to an explicit formula for the surface tension of a planar interface, eq 25.
Several simple models for an electrochemical interface are then considered in light of the derived formula. The commonly used Gouy-Chapman model neglects short-range forces and interparticle correlations, and the distribution of each charged species is Boltzmann-like. we show that the pressure in the direction normal to the interface is k E i n i ( z ) -E (~)~/ 8 7 7 , which is independent of z, if there are no polarizable species present. Then