Electrostatic and Entropic Interactions between Parallel InterfacesSeparated by a Glassy Film

Karen Johnston and Michael W. Finnis

Atomistic Simulation Group, Department of Pure and Applied Physics, Queen’s University Belfast,Belfast BT7 1NN, United Kingdom

A simple classical density functional model is set up to describethe electrostatic and entropic interactions between two parallelplanar charged interfaces separated by a thin film of a phase(the glass) containing a distribution of charged ions. The totalcharge in the system is zero. Three cases are treated: (1) thetwo interfaces carry a fixed surface charge; (2) the firstinterface carries a fixed surface charge, simulating a ceramic,while the second is held at zero potential, simulating a metal;and (3) both interfaces are held at zero potential. A discretizedform of the nonlinear Poisson–Boltzmann equation is derivedand solved by a Newton–Raphson method. The continuumapproximation is compared with a model in which the ions areonly allowed to occupy discrete planes. The effect of correla-tion among the ions is included within the local densityapproximation. Inserting parameters appropriate to the cop-per–alumina interface, we find that the attractive image forcebetween the ceramic and metal dominates the entropic(DLVO) repulsive force in the 1–2 nm range.

I. Introduction

THE observation of thin (1–2 nm) and remarkably uniformglassy films at grain boundaries in ceramics1,2 has puzzled

materials scientists for nearly three decades.Two extremely different models have been applied in an

attempt to understand this phenomenon. The model proposed byClarke3,4 is of a continuum, statistical nature. An attractive forcebetween two grains separated by a film is attributed to the van derWaals interaction, which is opposed by two repulsive effects. Thefirst is loosely described as “steric” in origin. What is understoodby this term is the effect of short-range order, imposed by thecrystalline surfaces on the fluid film as it solidifies, but this has notbeen quantified. The second is the DLVO5,6 or entropic force dueto a distribution of positive counterions within the film, which tendto have a higher concentration closer to the crystalline surfaces inorder to screen an excess of negatively charged surface ions.DLVO stands for Derjaguin, Landau, Verwey, and Overbeek, whoderived the theory of repulsive forces due to overlapping chargedclouds of counterions. This second effect is well known as theforce which maintains the separation of colloidal particles, and itsmagnitude has been thoroughly studied. It can be estimated easilyfor low concentrations of ions, when the Poisson–Boltzmannequations can be linearized, giving the Debye–Huckel model.Following this approach, Clarke was able to estimate a realisticvalue for the thickness of the films by balancing repulsive and

attractive forces. His model accounts for the observation7 that anincrease in the concentration of counterions leads to thicker films.

The contrasting kind of model which has been studied isentirely atomistic. Empirical interatomic potentials have beendeveloped and used for molecular dynamics simulation of a glassyfilm between ceramic crystals. In the work of Blonski andGarofalini,8 films of SiO2 of various thickness containing 0–30%CaO were simulated between alumina surfaces. Interestingly, atendency of Ca to segregate was found, with also a layer of orderedoxygen atoms at the film/alumina interface. However, no indica-tion of an equilibrium thickness was found. This is perhaps notsurprising if one believes that van der Waals forces versus DLVOforces between the crystals are the deciding effects; the former arenot included in the interatomic force model, and it may have beendifficult to see DLVO forces, which require statistical equilibrationof all concentrations, being generated on the time scales (a fewpicoseconds) of the molecular dynamics simulation. A more recentpaper employing similar techniques9 nevertheless reports finding aminimum in energy versus thickness at around 1 nm. We believethe issue is far from settled.

Recently a thin, uniform glassy silicate phase was observed at aCu–Al2O3 interface by Scheu et al.10 They found that the thicknessof the glass film depended on the orientation of the crystalstructure of the alumina. Presumably whatever physical mecha-nisms are at work in the grain boundary cases also operate at thesemetal–alumina interfaces. An additional source of attraction to beconsidered in this case might be the image interaction of anyexcess charge on the ceramic surface, e.g., due to oxygen ions,with the metal surface. This, one can speculate, would also causea variation in film thickness with orientation of the alumina crystal.

Our aim here is to make a more rigorous study of the repulsiveforce acting between the interfaces across the glassy film byextending the original model of Clarke et al.3,4 The originaltreatment is extended in three ways:

(1) A Poisson–Boltzmann-type of equation is derived from asimple classical density functional and solved numerically by aNewton–Raphson method. This gets us beyond the linear screen-ing approximation.

(2) The charged counterions in the glass can occupy discretesites in the glass, which have a known density, up to a maximumoccupancy of one ion per site. This puts a realistic upper limit onthe counterion density and introduces a simple regular solutionmodel for the configuration entropy of the counterions.

(3) The effect of confining the ions to discrete planes isinvestigated. The idea is that since the glass film that we aresimulating may consist of only three or four layers of molecules,there are likely to be significant deviations from the continuumapproximation. Our simulation strategy will give an idea of theimportance of such deviations.

(4) The boundary conditions are specified to simulate the glassfilm separating an insulator from either a metal, two insulators, ortwo metals.

The method that we use involves several novel features, and wedescribe it in some detail below in Section II. After discussinggeneral features of the results for several test cases, we considerthe parameters appropriate to copper–alumina. In this case we

W.-Y. Ching—contributing editor

Manuscript No. 187561. Received July 25, 2001; approved February 6, 2002.Support for M. W. Finnis was provided by Technion University, Haifa, during a

visit that stimulated this work. Support for K. Johnston was provided by David Clarkeand the University of California at Santa Barbara during her study visit.

J. Am. Ceram. Soc., 85 [10] 2562–68 (2002)

2562

journal

make the assumption that the surface of the ceramic is carrying anegative charge, due to being terminated by a layer of oxygen ions.There are two types of attractive force between the interfaces toconsider in this case: the Hamaker (van der Waals) interaction andthe image interaction due to the charged ceramic surface. TheHamaker forces typically contribute less than 10�2 J�m�2 to theinteraction energy of interfaces separated by 1 nm. A detaileddiscussion of Hamaker forces is in the recent review article byFrench.11 We find that the image interaction dominates the van derWaals force by more than 2 orders of magnitude over the physicalrange of film thicknesses (1–2 nm).

II. The Model

(1) DefinitionsWe assume that the interfaces are located at x � 0 and x � l.

The glass between the interfaces contains a distribution of coun-terions over available sites. The charge on a counterion is q, thedensity of available sites is 1/�, and the concentration or occu-pancy of sites is given by the mean value c(x). The system is at atemperature T and k is Boltzmann’s constant. In the discreteversion of the model that we describe first, the concentration willbe specified by its mean values ci at positions xi.

The counterions are supposed to lie on N planes between x � 0and x � l, equally spaced by d � l/(N � 1). We shall sometimesrefer to these planes as nodes. The spacing of counterions withineach plane is given by dp where

ddp2 � � (1)

Each plane carries a charge �i per unit area, where the subscriptlabels the plane, 0 � i � N � 1, and

�i �ciq

dp2 (2)

For the interface planes i � 0 and i � N � 1 Eq. (2) is a definitionof the quantities c0 and cN�1, respectively. The concentrationssatisfy the condition for overall charge neutrality:

�i�0

N�1

ci � 0 (3)

In deriving the equations to be solved we need to consider thefollowing three cases:

Case 1—c0 and cN�1 are held fixed; this is the model of twocharged ceramic surfaces.

Case 2—c0, . . . , cN are variables and cN�1 is held fixed; this isthe model of a metal surface at i � 0, treated as a classicalconductor, and a ceramic surface at i � N � 1. In this case c0

represents an image charge.Case 3—All ci are variables; this is the model of two metal

surfaces, where c0 and cN�1 both represent image charges.

(2) Density Functional TheoryThe approach that we take is a classical density functional

theory, and is similar to the way Lowen et al.12 treated interactingcolloid particles. We introduce a functional of the concentrationdistribution, representing the free energy per unit area of theboundary, which is a minimum at the equilibrium concentrations:

��c� �1

dp2 kT �

1

N

�ci ln ci � 1 � ci ln 1 � ci�

� EH�c� � Ecorr�c� � � �i�1

Nci

dp2 � � �

i�0

N�1ci

dp2 (4)

The first term in Eq. (4) represents the entropy of the noninteract-ing system of counterions distributed over planes 1 to N. The

second term is the electrostatic, or Hartree, energy of the system ofcharges within the mean field approximation. The third termcontains all of the correlation energy missing from the previousterms. No exact formula is available for Ecorr, and we shall makea simple local density approximation to examine the importance ofthis term. The chemical potential � multiplies the total number ofcations per unit area, and the Lagrange multiplier � multiplies anexpression proportional to the total charge per unit area whichmust vanish at equilibrium.

The task is to minimize Eq. (4) with respect to the ci, for whichwe require explicit expressions for the Hartree and correlationenergies.

(A) Poisson Equation, Hartree Energy and Potential: Wederive here an expression for the Hartree or electrostatic energy,given a distribution of charge corresponding to particular siteoccupancies, supposing that the sites in each plane are smeareduniformly over their plane. First we need the Hartree potential vi

H,which is the electrostatic potential at plane i. The difficulty to bedealt with is that, because our system extends to infinity in the (y,z) plane, unless the total charge per unit area vanishes, theelectrostatic energy per unit area will be infinite. We shall buildthis constraint in from the start by requiring that the electric fieldvanishes outside the range (0, l ).

The Hartree potential is obtained as usual by solving thePoisson equation, which in our discrete model with the aboveboundary conditions takes the form

� qd

0dp2�c0 � v0

H � v1H (5a)

� qd

0dp2�cN�1 � vN�1

H � vNH (5b)

� qd

0dp2�ci � �vi�1

H � 2viH � vi�1

H �1 � ci � N� (5c)

It is most conveniently solved by integrating the electric field fromx � 0 to plane i, and making use of the relationship (2) between themean planar charge density �i and site occupancy ci. We find

viH � v0

H � �j�0

i�1

i � j� qd

0dp2�cj (6)

which we can write in the convenient form

viH � v0

H � �j�0

N�1

Lijcj (7)

where L is a lower triangular (N � 2) � (N � 2) matrix. To makethings symmetrical, we can also obtain vi

H by integrating the fieldback from x � l, which gives

viH � vN�1

H � �j�0

N�1

LijTcj (8)

where we have introduced the transpose of the L matrix. Thisenables us to write the potential in terms of the symmetrizedmatrix LS � 1⁄2(L � LT):

viH � �

j�0

N�1

LijScj �

1

2v0

H � vN�1H (9)

v0H and vN�1

H differ whenever the slab carries a net dipole moment.We make the further assumption that the Hartree potential

vanishes everywhere when the concentrations {ci, i � 0, . . . , N �1} are all zero. This is no restriction, since the addition of aconstant electrostatic potential to the system will not change theequilibrium distribution or total energy provided that the chemicalpotential is measured with respect to the same potential. If we now

October 2002 Interactions between Parallel Interfaces Separated by a Glassy Film 2563

take the concentrations from 0 to their final values {ci} by meansof a scaling parameter �, the total differential of the Hartree energywith respect to � is

dEH � �i�0

N�1

�viH�ci� �

ciq

dp2 d� (10)

By integrating with respect to � from 0 to 1, and applying thecharge neutrality condition (3) we obtain

EH �q

2dp2 �

i, j�0

N�1

ciLijScj (11)

Notice that because we are restricting variables to the spacedefined by charge neutrality, both the Hartree energy and thepotential are invariant to the addition of any term proportional to¥i�0

N�1 ci. The intuitive result

�EH

�ci� � q

2dp2�vi

H (12)

does not hold in general, but it can be enforced by adding to EH asuitable term proportional to ¥i�0

N�1 ci, as follows:

EH �q

2dp2 �

i, j�0

N�1

ciLijScj �

q

2dp2v0

H � vN�1H �

i�0

N�1

ci (13)

(B) Correlation Energy and Potential: Lowen et al. used aform for Ecorr which is asymptotically valid at high temperatureand/or low concentration. Their expression also makes the localdensity approximation and is in the form of a series expansion interms of the small parameter:

� �q2

0kT �4�c

3� � 1/3

(14)

In our case � is by no means likely to be less than unity so we musttake a different approach. This should be as simple as possiblesince it would be pointless to introduce a high level of sophisti-cation at this point when we are for example neglecting theinteratomic interactions within the solvent glass and ignoring allbut Coulomb interactions. A suitable approximation is motivatedby the observation that we are dealing with point charges whichare restricted to discrete sites. In an array of point charges thecorrelation energy is equivalent to the Madelung energy. Hence weintroduce a local density approximation to the correlation energyper ion of the Madelung form:

ecorr � ��corr

2 �� q2

4� 0�� c

��1/3

(15)

where �corr is a constant of order unity. Within this local densityapproximation

dp2Ecorr � �

i�1

N

cieicorr (16)

As we do not know the best form of ecorr we shall explore theeffect of including this term with a reasonable range of values of�corr.

We define the correlation potential vcorr such that qvcorr is thefunctional derivative of the correlation energy:

qvicorr � dp

2��Ecorr�c�

�ci� (17)

This local density approximation gives the relation

qvcorr � �4

3�ecorr (18)

(3) Boltzmann EquationsThe minimization of the free energy in Eq. (4) with respect to

the ci gives Boltzmann equations for 1 � i � N:

ci

1 � ci� exp�� � �

kT � exp��q

kTvi

H � vicorr� (19)

These must be solved together with the Poisson equations (5). Thetreatment of the multipliers � and � is critical to this process, andwe now consider the three cases in turn. For case 1, with both sidesceramic, c0 and cN�1 are predetermined. So for case 1 the Nequations (19) together with charge neutrality (3) are sufficient todetermine the N unknowns ci and the single remaining parameter� � �. There is no need for separate values of � and �. For case2, if the left-hand side is a metal, c0 is an additional variable, forwhich the derivative of Eq. (4) gives the additional constraint

qv0H � � � 0 (20)

This specifies � once we have set v0H, and we can arbitrarily choose

v0H � 0. There is still an extra variable c0 compared to case 1,

which means that we have the freedom to choose a value for �,which can be thought of in terms of a reference concentration.Rather than specify � explicitly, it is convenient to specify areference concentration c�0 which we define as the concentration atvanishing Hartree potential. Finally, in case 3, if both sides aremetal, cN�1 and c0 are both variables. The derivative of Eq. (4)with respect to cN�1 gives the further constraint

qvN�1H � � � 0 (21)

In case 3, Eqs. (20) and (21) imply that v0H � vN�1

H , as one wouldintuitively expect. As in case 2, we are free to choose �.

(4) Continuum LimitThe continuum limit of the above equations is obtained by

letting d tend to zero while the product ddp2 � � is fixed. We give

the resulting formulas here and report some solutions below forcomparison with the discrete model.

The energy functional becomes

��c� � � 1

��kT �0

l

�c x ln cx

� 1 � cx ln 1 � cx� dx

�q

2� �0

l

cxvHx dx � Ecorr�c�

�� � �

� �0

l

cx dx ��

q�0 � �N�1 (22)

The Poisson equation is

�2vH

�x2 � � � q

0��c (23)

and the Hartree potential is:

vHx � v0H �

q

0� �0

x

x � x�cx� dx� (24)

The solution Eq. (19) remains valid.One can think of the discrete equations as a finite difference

approximation to the continuum equations, which can be used to

2564 Journal of the American Ceramic Society—Johnston and Finnis Vol. 85, No. 10

solve them for a suitably large choice of N. If the purpose were tosolve the continuum equations economically, a more rapidlyconvergent discrete representation would make the distance be-tween zero and the first node d/2, and similarly between nodes Nand N � 1, so as to properly weight the endpoints.

We also see from the continuum equations that a natural lengthscale for the problem is

� � � 0�kT

q2 (25)

III. Method of Solution

The Poisson–Boltzmann equations were solved iteratively by anefficient Newton–Raphson procedure, which we briefly describehere. The first step is to guess starting values of ci, from which theHartree potentials vi

H(P) are calculated from the solution of Eq. (9)to the Poisson equations. A second estimate of the Hartreepotential vi

H(B) is obtained from the Boltzmann equation, which isinverted so that vi

H(B) is a function of ci. We base our iterations onthe differences ri between these quantities:

ri � viHB � vi

HP (26)

In the equilibrium state, ri � 0. The corrections to {ci} are foundby solving:

��viHB

�ci��ci � �vi

HP � ri �1 � i � N� (27)

By using the Poisson equation (5) we can express this in terms ofthe projected changes in vi

H(P):

�j

��viHB

�ci�� �ci

�vjHP��vj

HP � �viHP � ri (28)

The first factor in Eq. (28) is obtained by differentiating Eq. (19)and the second by differentiating Eqs. (5), yielding a tridiagonalmatrix which we invert to get the changes {�vi

H(P)}. The required{�ci} are obtained directly from the {�vi

H(P)} by inserting the latterinto the differentials of Eqs. (5).

Notice that whereas Eq. (27) is a set of N equations with Nunknown �ci, the transformation to Eq. (28) introduces extravariables �v0

H(P) and �vN�1H (P). Before Eq. (28) can be solved, it

is necessary to eliminate �v0H(P) and �vN�1

H (P), and at this point wemake the distinction between the three cases. In case 1, the electricfields to the surface planes are fixed, so we have

�vN�1H P � �vN

HP (29a)

�v0HP � �v1

HP (29b)

In case 2 we have

�vN�1H P � �vN

HP (30a)

�v0HP � 0 (30b)

and in case 3

�vN�1H P � 0 (31a)

�v0HP � 0 (31b)

IV. Results and Discussion—The Metal–Ceramic System

We describe some results obtained with our model for case 2,the metal–ceramic system, which is the unsymmetrical case andone that to our knowledge has not previously been analyzed by anymethods. First we discuss some general features of the model, andthen we consider the particular case of the alumina–copperboundary.

(1) Convergence of the Newton–Raphson MethodStarting with a uniform concentration profile, the method

quickly converged to high accuracy. The energy decreases to aminimum and the solution, correct to nine decimal places, isreached after just 10 iterations. The number of iterations requiredto reach a prescribed accuracy depends only slightly on the numberof nodes, so the scaling is just that required to solve a sparse N �N matrix.

(2) Effects of the Parameters(A) Width of the Film: Figure 1 shows the variation of the

equilibrium concentration profiles with number of nodes for N �5, 10, and 15. We regard the separation between the nodes as aconstant reflecting the spacing of counterion sites in the glass. Asexpected, we see a buildup of concentration at the ceramic surfacesince the positive counterions are attracted to the negative surfacecharge. As the distance between the surfaces is increased, thescreening profile or shielding length stays approximately the same.The surface charge on the metal decreases with increasing sepa-ration since the total number of counterions is increasing, to thepoint that for N � 15 the metal has a negative surface charge, andwe see a slight heaping up of counterion density at the metalsurface. In the limit of large separation, the counterion density isscreening both its induced image charge on the metal surface to theleft and the surface charge on the ceramic to the right.

The behavior at the metal surface with N � 15 is reproduced ifwe consider the symmetric case 3, with metal at both boundaries,with 10 nodes. Figure 2 shows the concentration c(x) and potentialvH(x) for this case.

(B) Surface Charge on the Ceramic: Concentration profilesfor three different surface charge densities on the ceramic are

Fig. 1. Variation of the concentration profile as the separation of themetal and ceramic surfaces is increased at a fixed node spacing. Cases for5, 10, and 15 nodes are shown. The length scale � is defined in Eq. (25).The minimum and maximum values of x correspond to the interface planesx � 0 and x � l.

October 2002 Interactions between Parallel Interfaces Separated by a Glassy Film 2565

shown in Fig. 3. As the surface charge on the ceramic becomesmore negative, more counterions are naturally attracted to it andthe concentration near the ceramic surface increases. For largevalues of surface charge the concentration becomes saturated at theceramic surface and this causes a considerable increase in thescreening length. This illustrates that site saturation is a nonlinearfeature of our model which can change the length scale of theinteraction of the screened ceramic surface.

(C) Reference Concentration: The variation of the concen-tration profile for different values of the reference concentration c�0

is shown in Fig. 4. The values illustrated are c�0 � 0.1, c�0 � 0.2,and c�0 � 0.5. Recall that c�0 would be the equilibrium concentra-tion at zero potential, which is the boundary condition imposed atthe metal surface. This is why c(x) extrapolates to c�0 at the metalsurface. As c�0 is increased, there is a slight increase in screeninglength at the ceramic surface, again illustrating that we are in avery nonlinear regime; in the dilute limit we would expect Debye

screening to operate, implying a screening length which wouldvary as 1/�c�0.

(D) Correlation Energy: The concentration profiles areshown in Fig. 5 for a range of values of �corr. Correlation energyin principle allows a higher density of counterions to screen theceramic surface. We might expect this to lead to a reduction in thescreening length, as the ions can bunch together and enhance themean density, while correlation allows them to keep out of eachother’s way. However, the width of the screening charge athalf-maximum actually increases very slightly, because even withno correlation energy we are already close to saturating theconcentration adjacent to the ceramic surface.

(E) Energy versus Film Width: The variation of the energywith the width of the glass film is shown for four cases in Fig.6. Taking A as a reference, it corresponds to the scenarios ofwhich three are shown in Fig. 1, for all values of N up to 15.The attractive force at less than about five nodes is due to theinteraction of the negative surface charge on the ceramic withits positive image in the metal surface. This is analogous to theforce between the plates of a capacitor. At higher separationsthis is screened by the counterions, and as mentioned above, thecharge on the metal surface changes sign as the influence of theceramic becomes negligible. The decrease in energy withincreasing separation length represents a repulsive force be-tween the metal and ceramic and arises from the mutualrepulsion between the positively charged counterions in theglass.

The effect of changing the surface charge on the ceramic can beseen by comparing case A with case B, in which the surface chargewas changed from �3.2 to �2.5 (dimensionless units). A decreasein surface charge results in a decrease in energy and a weakerattractive force.

The effect of changing the reference concentration can beseen by comparing cases A and C, in which the referenceconcentration was doubled from 0.1 to 0.2. An increase in c�0

results in a decrease in energy, the curve becoming moresharply peaked and the maximum shifting to a lower value ofseparation length. This is the effect of the slightly longer rangeof influence of the charged surface, indicated already by theconcentration profiles.

Fig. 4. Variation of the concentration profile with the reference concen-tration c�0, which would be the equilibrium concentration at zero potential.

Fig. 2. Concentration of the counterions and electrostatic potentialprofiles for the case of two metals separated by a glassy film.

Fig. 3. Dependence of the concentration profile on the surface charge ofthe ceramic.

2566 Journal of the American Ceramic Society—Johnston and Finnis Vol. 85, No. 10

The effect of adding the correlation energy is seen by compar-ing curves A and D (�corr � 40). Apart from lowering slightly theoverall energy, we do not see any significant difference.

(3) Physical Parameters for a Copper/Alumina InterfaceIn the study of Ref. 10 the samples were prepared at a

temperature of approximately 1600 K. Although the samples wereobserved at room temperature, it was assumed that the systemreached its equilibrium state at 1600 K and solidified on cooling sothat the equilibrium thickness was frozen in. The relative permit-tivity of pure SiO2 glass, 3.81, was used for . The distancebetween sites was calculated using the molar volume of glass,giving d � 3.566 � 10�10 m. The counterions were assumed to bemainly Ca2�, so the charge on a counterion was taken to be q �2e, where e is the elementary charge. An upper limit to the

magnitude of the surface charge was calculated by assuming thatthe alumina crystal presented an (0001) surface which was oxygenterminated. Each oxygen atom was assumed to carry an excesscharge of �e. This gave the value of the surface charge to be

�N�1 � �6e

�3a2 C�m�2 (32)

where a � 4.759 nm is the hexagonal lattice parameter ofcorundum. We take for �corr multiples of the Madelung constant1.79, having seen that the results will not be sensitive to thisparameter. The value of the reference concentration depends on theprecise experimental conditions and is not precisely known; it wasestimated to be 0.1.

The concentration and potential as functions of x are shown inFig. 7 and the energy versus thickness is shown in Fig. 8. This casediffers from the test cases reported above in that the concentrationfalls to practically zero at distances more than one node from theceramic surface. It was necessary in this case to impose a strictconvergence criterion, better than 10�12 in c, in order to achievethe agreement shown between Boltzmann and Poisson potentials.The reason for the almost complete absence of counterions awayfrom the surfaces is the very short screening length in this case.Consider the length scale � in Eq. (25), which is equivalent to theBjerrum length in colloid science. With the given parameters thisis 0.018 nm, much less than the node spacing. In other words, theCoulomb interaction between counterions dominates kT. As aconsequence we are in a rather simple regime of almost completescreening.

However, in spite of the small screening length, the interactionbetween the ceramic surface and the metal cannot be neglected.From the slope of energy versus thickness, at l � 1.4 nm theattractive force was estimated to be F � 7.17 � 107 N�m�2, whichis considerable, as we now show by comparison with the van derWaals or Hamaker force.

We estimate for comparison the conventional Hamaker forcebetween the surfaces. The standard expression for the Hamakerenergy is

EHamaker ��H

12�l2 (33)

where H is the Hamaker constant, which depends on the ceramicsystem, and l is the width of the SiO2 layer. The Hamaker constantfor a Cu–SiO2–Al2O3 system was not available, so instead theHamaker constant 22 � 10�21 J for a Al2O3–SiO2–Al2O3 systemwas used, taken from Ref. 3. To account for the differences

Fig. 6. Energy per unit area as a function of glass film thickness fordifferent parameters.

Fig. 5. Concentration profiles for different values of the correlationconstant �corr.

Fig. 7. Concentration of the counterions and electrostatic potentialcalculated for the parameters appropriate to copper/alumina.

October 2002 Interactions between Parallel Interfaces Separated by a Glassy Film 2567

between the systems it was assumed that the counterions in theSiO2 “saw” their images in the copper and therefore the effectivedistance would be twice that of the Al2O3–SiO2–Al2O3 system.We therefore replace the standard Hamaker energy by

EHamaker ��H

48�l2 (34)

Differentiating this with respect to l gave the Hamaker force

FHamaker �H

24�l3 (35)

This force is valid for distances down to about 1 nm. For l � 1.4nm, FHamaker � 1.06 � 105 N�m�2. The force between the copperand alumina using the simple classical density functional is muchgreater than the Hamaker force for the same separation.

V. Conclusion

A classical density functional model has been developed tocalculate the classical electrostatic and entropic forces betweentwo parallel planar charged interfaces separated by a glass phasecontaining a distribution of counterions. The formalism permits theinterfaces to be with a metal, characterized by the boundarycondition of zero potential and variable surface charge, or with aceramic whose surface carries a fixed charge. The positive ionswithin the glass are restricted to discrete sites with a maximumoccupancy of one per site. The approximation is made that there isfull translational symmetry parallel to the interfaces. The effect of

correlation between the ions is included within a local densityapproximation.

A nonlinear Poisson–Boltzmann equation for the counterionconcentration was derived from the density functional. The Pois-son–Boltzmann equation was solved numerically, using a New-ton–Raphson method, and a number of test cases were studied.

The energy and force were calculated for physical parametersappropriate to a copper–alumina interface separated by a glassyfilm. We have assumed that the concentration of counterions in thesystem is 0.1 of saturation. This value is large enough for nonlineareffects to be significant. For the parameters of this system, thescreening length is much smaller than the separation of counterionsites, and the interfaces are already strongly screened at distancesof one atomic spacing. Nevertheless the residual interactionbetween the interfaces is not negligible, and it was found that theelectrostatic and entropic forces dominated the Hamaker force atdistances of �1–2 nm. The film thickness in these conditionscollapses to the value set by the steric repulsion. This behavior isquantitatively different for films between two ceramic surfaces inwhich the electrostatic force is absent and the Hamaker force isimportant. It will be of interest to see if electronic structurecalculations would support this picture.

Acknowledgments

We are grateful for discussions with David Clarke, Wayne Kaplan, ChristinaScheu, and Gerhard Deym.

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Fig. 8. Energy per unit area as a function of glass film thicknesscalculated for the parameters appropriate to copper/alumina.

2568 Journal of the American Ceramic Society—Johnston and Finnis Vol. 85, No. 10