Electrostatic fields at fluid interfaces and surface tension

Journal o[ Electrostatics, 26 (1991) 275-289

275

Elsevier

Electrostatic fields at fluid interfaces and surface tension R. Cade Department ofMathematics and Research Centre, CatholicUniversity,Santiago,Dominican Republic (Received February 6, 1990; accepted in revised form August 1, 1991 )

Summary Electromechanical considerations at interfaces, in which an electrostatic field is provided by an electric double layer, lead to an identification of surface tension in terms of field quantities. In the present note, defects are pointed out in this theory; they are removed, and in the process, a deeper insight is gained into the electrical and mechanical relationship.

1. Introduction

We consider a liquid A and second fluid (liquid or gas) B which are separated by a common surface S, and take x i (i=0,1,2) as curvilinear coordinates such that the x°-direction is orthogonal to the other two, the surface x ° •0 is S, and, when x°ffi 0, the x I and x2-directions are tangential to S. We shall specify these coordinates more closely in due course. We suppose that there is no applied electrostatic field, but that, in the interfacial region, there is an electrostatic field due to the presence of an electric double layer, this being a general property of matter. Thus we contemplate a charge density p and polarization P' which are sources of an electric intensity E i (writing the latter two quantities as contravariant vectors with respect to the coordinates x ~). By assuming that there is no fluid motion and adopting the continuum electromechanical approach of Brown [ 1], we can write down an equation of hydrostatic equilibrium which takes into account the electrical forces. With the use of C.G.S. Electrostatic units, this equation, in our general tensor notation, is

pEi +PYEij- g l ~ +,~,j =0,

(1)

where/~ is the mass density,g the acceleration due to gravity,y a vertically upward Cartesian coordinate of the point x i,and ~r~ the (mixed) mechanical

0304-3886/91/$03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.

276

stress tensor, a comma before an index meaning that this index corresponds to covariant differentiation. It is found consistent with (1) to make the usual assumption that Z~ is the ordinary hydrostatic stress tensor -pJ~, where J~ is the fundamental mixed tensor (1 if i - j , 0 otherwise) and p is the pressure (a scalar). Then we can solve (I) for p. Denoting by CAthe value of x ° at a point adjacent to, and at the A-side of the interfacial region, and using the suffix A to mean the value of a function at this point, we find that, approximately,

1 fcA~°/~mEoDOd~O

P=PA +~-~

(2)

where D / is the electric displacement and/~m is the mean curvature of the surface ~o=constant. (The sign of ~'m is determined by the convention that a principal radius of normal curvature of the surface ~o= k (constant) is nonnegative if the arc to which it pertains is locally convex when the region for ~o~
"

¢cA

EoD °

dx ° + R,

(3)

where R is a relatively small residue. Comparing this with the classical Laplace formula, by which we relate the left-hand pressure difference to the surface tension T according to PA -PB ffi2TKm,

(4)

we are led to the obvious identification,

1 ~clJ Eo D ° dx°.

(5)

This theory has, however, some unsatisfactory features which we shall describe in the next section. Our object is to rectify the dubious points and, in so doing, to obtain a deeper understanding of the significance of (5). Before embarking upon our task, we must emphasize two points. In the first place, it is customary to study surface tension in terms of molecular theory, with a tendency to the view that macroscopic physics has no role in the understanding of the phenomenon. We discount this point of view, for clearly, under merely the statistically reasonable assumption that the interfacial region can

277

be treated as a continuum, the application of the laws of macroscopic physics in this thin region is something to which microscopic physics should conform. The fact that in molecular physics, electrochemistry in particular, one contemplates forces in the region which may not exist in bulk (such as ones concerned with chemical interaction ), is no contrary argument. For all forces which are not of an ordinary electrostatic nature are included in the stress tensor X~. The appearance on the right of eqn. (5) of only electrostatic quantities is what evolves from the equation of equilibrium (1) in which, in the anal:~sis, X~ is effectively determined by these electrostatic quantities, ones pertaining to the electric double layer. There is in fact nothing new about macroscopic theory of surface tension, which appears to have originated in the work of Bakker [5 ] and has accumulated a considerable literature; a number of references are given by Defay and Sanfeld [6 ]. Previous work has been based on the assumption of a mechanical stress tensor Xj different from the usual hydrostatic one, -pJ~, and a significant feature of the electrohydrostatic theory is that, in the purely statical context, no such departure from classical concepts is required. Our second point is that, while we speak of an "electrohydrostatic theory of surface tension", we make no attempt to "explain" surface tension. We consider that this is a difficult, perhaps not even meaningful, question. For the existence of surface tension and of the surface itself are inseparably linked (a gas, for example, has neither), and if one says that properties of a surface give rise to forces which we perceive as surface tension, the argument can be inverted to say that the occurrence of these forces gives rise to the surface. What we claim is simply an identification of surface tension in terms of other physical quantities, something which may or may not be a useful contribution to our practical knowledge.

2. Critique of the eleetrohydrostatie theory An essential feature of the theory leading to eqn. (5) is that what we have called the interfacial region, is a thin layer of thickness a, what we shall in future call the boundary layer, and that different physical quantities have different orders of magnitude in terms of a. For example, thinking of any ordinary electrostatic situation, without inclusion of a double layer, the charge density p must be a function that would tend to infinity if we took the limit a-*0, in order that in this limit, the surface density of charge a should have a non-zero value. Thus we say that p is 0 (a - ~), so that by its integration across the boundary layer, awill be O(a°). All 'ordinary' quantities, such as a, should come out O (a °), and all do, except for T given by (5). For on account of the double layer, Eo and D o in the boundary layer are (as we shall make clear in due course) 0 (a - ~), so that the integrand in (3) is O (a - 2), and the integral itself, O (a - ~), while the residue R is

278

O(a °). It can be argued that the surface tension is indeed quantitatively large by the general standard or surface entities, but the fact remains that (3) equates an O(a °) left member to an O(a -1 ) right member, and the acceptance of this makes nonsense of the scheme of classification on which the whole theory is based. Now while the indefinite x°ointegral of an O(a -2) function is an O(a -1) function, it can happen that the definite integral on [CA,CB]is O(a°), so that the possibility exists of the double layer charge and polarization being so distributed that the integral in (3) turns out so. But if this happens, the difficulty is replaced by another and worse one. For since now both terms on the right of (3) are O(a°), no identification of Tby (5) is possible at all. The purpose of this note is to address these difficulties, and we shall see that they have a satisfactory resolution. We shall prove that the residue R in (3) is in fact O (a), and so absolutely (not just relatively) negligible. After this, it is shown to be clearly indicated, and confirmed by physical arguments, that the integral in (3) is in fact O(a°), and with this information, the identification of T by (5) is acceptable. 3. Description of the boundary tayer We shall give a means of classifying boundary-layer functions in terms of orders O (a '' ) which we believe is both simpler and superior to that given originally [2]. But first let us describe precisely the coordinates x i in terms of which these functions are expressed and the integrals in (2)- (5), valid. We take x ° as signed distance normal to S, positive in the A - . B direction, and x 1, x 2 as variables by which 8 is smoothly parametrized. Any surface x" =constant is one parallel to S according to the definition of differential geometry [ 7 ], whereby the x ~are known as parallel-surface coordinates. They fulfil the condition of the x°.direction being orthogonal to the other two, but in general, are not as a whole an orthogonal system. The quantities cA and c~ introduced in Section 1, we take as constant values of x °, choosing these values so as to define the boundary layer as the region for which cA ( < 0) < x ° < c~ ( > 0). We then define the thickness a in a precise way as c~-ca, as we may since x ° is pure (signed) distance. Another consequence of this last fact, along with the fact that the x°-direction is orthogonal to the others, is that the metric tensors 7ij and yii have the properties ~'o: (ffiyio)ffi)'°i (ffi~:°)ffi0,

i=1,2,

~ooffi~°°ffil.

(6)

It follows that the respective 0-components of a covariant vector fi and its conjugate contravariant one, f ~, are equal (for f °ffi~°if~fy°°foffifo). In fact, they both coincide with the outward-normal Cartesian component (which is how Eo and D o ( ffiDo) in Section I can be interpreted). By 7 we shall mean as usual, the determinant of F~i.

279

Any function ~(x°,xl,x 2) defined for x°e [CA,CB], we call a boundary-layer function. Such functions will include ones of x ~and x 2 only, these being simply constant with respect to x °. Important boundary-layer functions for us are the charge density p (a scalar), and the electric intensity Ei, electric displacement D i and dielectric polarization Pi (vectors). An example of a boundary-layer function which is of x 1 and x 2 only is the surface density of charge ~, defined by

ffc .'B

O'(xl,x2 ) --

p(X°,Xl,X 2 ) dx °

(7)

A

Other examples are, given ¢~(x°,xl,x2),the functions (~(CA,X~,X'2) and (Cs,XI,X2),for we regard them as distinctboundary-layer functions. We now develop the following scheme for classifying boundary-layer functions. (i) We assume that any boundary-layer function immediately relevant to physical description outside the boundary layer, is of x I and x 2 alone. W e assume as a partial converse to (i) that, given a boundary-layer (ii) function ~, ~A--~(CA,Xl,X2) and ~s-~(cv,x~,x 2) (distinctboundarylayer functions, as we have said), are immediately relevant to description outside the boundary layer. (iii) W e say that a boundary-layer function ~ is of ordinary magnitude ifa~ for every x ° in [CA,CB], is negligible. (iv) W e assume that any boundary-layer function immediately relevant to description outside the boundary layer, is of ordinary magnitude. W e say that ~ is once-largeif its integral on [cA,x° ] is a function of (v) ordinary magnitude, and hence, defining inductively, that it is n-fold largeifits integral on [CA,X° ] is (n-- I )-fold large. Thus, the condition of a being or ordinary magnitude will be satisfied ifp is once-large.

(vi) (vii)

(viii)

W e assume that 0~/0x I,0~,0x2 and, if~ is a boundary-layer function of ordinary magnitude, ~ , are negligible,of ordinary magnitude, or, for specified n, f-foldlarge,accordingly as ~ is. The properties of O~/Ox° are not, however, at our disposal.If ~ is n-fold large, we find (since ~A is of ordinary magnitude ) and O0/Ox ° is ( n + 1 )fold large. If both ~ and ~ - ~ a are of ordinary magnitude, O0/Ox° is once-large, and in this case we say that 0 is quickly-varying. If ~ is of ordinary magnitude and ¢--~A is negligible, O0/Ox° is of ordinary magnitude; in this case we say that ~ is slowly-varying. Geometrical functions, such as the metric tensors ?ij and 7 ij, while defined independently of the boundary layer, are boundary-layer functions nevertheless when x ° is taken in [CA,CB]. We assume (as can be

280 proved under sufficient smoothness of S) that they and such of their partial derivatives that enter our discussion, are slowly-varying. As a code for facilitating the concise application of the above scheme, we shall say in future that functions which are negligible, of ordinary magnitude and n-fold large, are, respectively, O(a), O(a °) and O(a-n). Here we make contact with the previous mode of classification [ 2 ], but the present is better, assuming no specific mathematical properties but based only on one or two simple physical notions, principally that of the boundary layer being so thin that certain entities which would not otherwise be so, are negligible. Of course, the symbolism O(a), O(a°), O(a -n) carries with it the connotation of functions which tend, respectively, to zero, to a finite limit and to infinity in the limit a-~0, but while this may be a convenient way of thinking, we reject it in any literal sense, as the limit is not physically meaningful. We mentioned the fact that a sufficient condition for ~, given by (7), to be O(a °) (of ordinary magnitude), is that p be O(a -~) (once-large). But the condition is not a necessary one; p can be 0 (a - 2) (twice.large) if we require that the integral in (7) be O(a°), and this prompts the general mathematical definition of the electric double layer. Precisely, we include the polarization pi and define the double layer by the holding of the following conditions: (i) p--O(a -2) and the integral in (7) is O(a°); (ii) P ° f O ( a - ~ ) , pi ( i f 1,2) ffiO (a°). These conditions include the trivial possibilities ofp and po being of higher order in a, and if both are so, then we have no double layer. The strength ~ of the double layer (moment per unit area) is defined by

r(x t,x2) _ f,'lJ (px° + P ° ) dx °,

(8)

¢ CA

and is O(a°). Thus physically, our concept of the double layer is of a layer partly of high normally-directed polarization and partly of high 'equal and opposite' free charges. If po is only 0 (a°), of ordinary magnitude, we would speak of a charge double layer, and if p is only O (a - ~), as in an ordinary electrostatic situation, of a polarization double layer. In general, where both components are present, we expect the polarization component to be in a sense fixed and the charge component to have a certain mobility, considerations which will be important to us later on. Microscopically, we imagine the former to consist of polarized molecules (molecular dipoles ) with zero net charge, and the latter of free electrons and ions. Let us, however, take this opportunity to emphasize that our considerations belong exclusively to macroscopic continuum physics, with which, as we said earlier, we expect microscopic physics to comply. We say that a double layer is uniform if either condition p f O ( a - 2 ) , po = 0 (a - t ), is accompanied by the qualification that the function concerned does not differ more than at higher order in a from one constant with respect to x t and x 2. Then we find at once from (8) that z is constant but for a negli-

281

gible, that is,O (a), error, while as we shall see shortly, the potential V has a comparable property. W e say that the double layer is torque-free if EoPi-EiP°--O(a°), a condition that has an electromechanical significance (torque balance on a volume-element) that we need not go into here, but important in the general theory. W e shall assume that the double layer is both

uniform and torque-free. 4. The boundary-layer electrostatic field

W e have in the boundary layer the following general electrostaticequations:

the relationship between electric intensity and potential,

0V Ei

(9)

- - OX i ,

=

the definition of electric displacement,

(10)

Di = Ei + 4~Pi, and the tensorial equivalent of div D - 4 ~ p (see [7], p. 113), 1

~-~

0

(11)

(J?Di)ffi4~P •

The relation (10) is the only one between Di and El; we cannot, in contrast with matter in bulk, assume a proportionality relationship by a constant scalar permittivity. By (9), Eo in the boundary layer is of one order lower in a than E: (if 1,2 ), but by (10), (11) and the definition of the double layer, cannot non-trivially be of order lower than 0 (a - i). Hence, by the said definition,keeping in mind

(cf. (6)) that EoffiE°, DoffiD °, while E ~(i- l,2 ) -TiJEj-TiIEl +7i2E2 is ofthe same order as E:, we can reduce (11) to

1 07

oo + o

(aO)

(12)

=4 p.

According to the theory of parallel surfaces ( [71, §48),

(13)

1 07 ffi4/~m Oxo

(a formula correct under the sign convention of/~m described after (2)). Let us define

I fxO ~'

1

2(x°,xl,x 2) =exp \ 2 c AKm(tt,xl,x 2) du ,

(14)

282 a geometrical function, O(a °) and slowly-varying. Then with (13), (12) becomes 1 0

20x ° (2D°)=4up+O(a°)'

(15)

and by integration,

Do(x°,xl,x 2) --~-D o ( C A , X I , x

2)

1 ~o 4"/t(X°,XI,X 2) .4

+O(aO)} d~O.

{47tP(~°'XI'X2)~'(~O'xI'x2) (16)

The integral in (16) being O(a -1 ) {since p is O(a -2) ), while Do(cA,xl,x 2) is O(a°), it follows that Do(x°,xl,x 2) is O ( a - l ) . Hence, by (10) and the fact that Po is O (a - ~), Eo is O (a - ~). It follows from (9) that the potential V is O(a °) and quickly-varying. The potential difference V s - Va ~s therefore also O(a°), and so generally non-negligible; it is the 'contact potential', or 'phaseboundary potential', one of the best known of double-layer properties. The above argument is quite general, holding independently of our assumption that the double layer is uniform. To derive special consequence of this property ( additional to that of the constant strength r), we invoke the equation

4n(pxO+Po) dxOffidV+O(a °) dx °,

(17)

where d V is the elementary potential 'jump' on crossing (in the A-~B direction ) the element of thickness dx ° of the boundary layer. This equation follows by the theory of ideal surface distributions of dipole, appropriate to t,he uniformity condition (see [8 ], pp. 66-69). Account is taken in the derivation of ~he fact that px°dx °, unlike P°dx°, is not an ideal surface density of dipole moment, but the moment of surface charge densities pek °, -pdx °, separated by the distance x °, and the error is correctly, as one may verify, allowed for by the terra O(a°)ek°). The integration of (17) gives, through the uniformity condition, V(x°,xl,x2)- V(c~,xl,x'2)- W(x°) +O(a), (18) so that,just likethe strength v,the contact potential Vs- VA is constant with respect to x ~ and x 2,but for an O(a) quantity. One of the basic physical prescriptionsof this paper, that there be no applied electrostaticfield,an essential requirement for (1) to lead to (2), goes some way to assure that,although there isa fieldin the boundary layer,the situation is not 'electrostatic'in the conventional sense. To achieve this completely, we need to know that that the sources in the boundary layer,giving the fieldthere, give no appreciablefieldoutside.N o w by the mere factthat the only fieldsource isthe double layer,the considerations [8 ] providing (17 ) and (18), show that VA in (18) is constant but for an O (a) function, whence it follows from this equation and (9) that, although Eo--O(a-~), E~ ( i - l , 2 ) - O ( a ) . But this is

283

not enough. For there is no guarantee from (18) that EoA--~(--OV/Ox°)A and EoB'= (--OV/Ox°)Bare O(a) rather than O(a°). W e can only accomplish this by further refining our specificationof the boundary-layer field.W e shall assume that

+f(X°,Xl,X2), Do = ~ ( x °) + h(x°,xl,x2), ~,~=O(a-1), f,h=O(a), ~(CA)=~(Cs)=O, ~(CA)-~ ~(CB)-~O.

E o = C(X °)

(19)

We easily find by (10) and (11) that this implies that the double layer is uniform, but, as we have just said, a uniform double layer does not necessarily imply the property included in (19), that EoA--O(a), Eon-O(a)*. In fact, (19) is not necessary for the main considerations of the paper; nevertheless, it does have importance, and moreover, as we shall see in Section 6, we are almost obliged to consider it true. Finally in this section, we study an aspect of the boundary-layer electrostatics in which (19) has a role. If we take a closed surface Z formed by normals to S and segments of the parallel surfaces for x °-cA,x ° =x~, we find by (19) and Gauss's flux theorem that the total charge inside Z is O(a), whereas under general electrostatic conditions it would be O(a°). Our finding is consistent with our 'non-electrostatic' requirement. Let us, however, return to ( 16 ). By ( 14 ), A(~°,Xl,X2)/A(X°,Xl,X~)=exp

--2

0

Km(U,Xl,X2) du ,

(20)

so that, taking into account that/~m is slowly-varying and approximating by rejecting what will give only an O (a) contribution to ( 16 ), A(~°,xl,x2)/A(x°,xl,x 2) ,,, 1 - 2Kin(x°- ~°),

(21)

Km being, as we said before (3), the mean curvature of S. Hence and by the fact, from (19), that Do(CA,Xl,X 2) is O(a), (16) becomes

Do(xOx~,x2)ffi4~

A

{1-2Km(X°-?~°)}pd~°+O(a).

(22)

But when x ° ~e~, Do(c~,x~,x 2 ) also, by (19)i is O(a), so that (22) in this case becomes *If EOA=Eos-0, zero field throughout space outside the boundary layer follows in a wide class of circumstances by the uniqueness theorem for the Neumann problem [ 8 ]. When EOA,Eosare 0 (a), the corresponding approximation to this fact requires a stability theorem for the Neumann problem which appears not to be known, although one does not doubt intuitively that it is true.

284

4x

ffc~B{1-2Km(cs-x°)}pdx°=O(a),

(23)

,4

whence by (7), a=2Km

fcA"~B(cs-x°)pdx°+O(a).

(24)

This result apparently contradicts the previous conclusion, that under (19), net charge in the boundary layer is O (a). The paradox lies in the fact that, through the 'bending' of the boundary layer, net charge is not quite given by a surface integral of a defned by (7), if the charge density p is of the low order O(a-2). Thus (24) represents the bending effect, appropriately zero if S is plane, when Km =0. Another quantity of the nature of (24) will appear in a later stage of our work. In the next two sections we shall discuss the integral in (2), which for the purposes of this paper we regard as given. Its origin is in the appropriate extension to the boundary layer of the fundamental work of Brown [ 1] on electromechanical stresses, an extension whose first version [2 ] contained various imperfections, and an improved version of which has been given recently [9 ]. The general theory is couched in hydrodynamical terms, and a static situation has to be understood as 'quasi-static' in the sense that correctly, (1) should contain an O(a) residue arising from velocity terms of this order. This O(a) residue contributes to an O (a) residue which would appear on the right of (2) for this equation to hold exactly. The particular proof in the next section is also given in the new presentation of the general theory [9 ], but is repeated here in view of its central importance to the surface tension theory. 5. The reduction of the stress integral

With Km the mean curvature of S and ~m that of the parallel surface for arbitrary x ° in [CA,CB],we shall denote OK~/Ox° at a point between 0 and x ° by ~ m / 0 x °, and at x°ffi0, by OKJOx °. By the mean-value theorem and the fact that the derivative is slowly-varying, +x

° - -

Ox°

=K,. + x o OK. +x°O(a). Oxo

(25)

Taking the upper limit x ° of the integral in (2) as cB, and substituting for/~m the right member of (25), we obtain

t

t'B

I1 (' A

[fm EoDo dx ° -If,.

Eo Do dx ° + ~ t'A

+O(a),

285

A x°EoDo dx°,

J=

(26)

lowering, as we may, the upper suffix of D o, and using the fact that, although EoD o is O(a-2), the integral of its product with x°O(a) is O(a). We thus identify the residue R in (3) as 0Kin

R = Ox-----6- J + O ( a ) ,

(27)

and we see that, as indicated in Section 2, the integrand of J being O (a - ~), R must be assumed to be O(a °) unless we can prove a more special property. By (9), integration by parts, and the fact that x°VDo is O(a) when x ° is CA or cB,

J~ -

'fa OX 0 V°x o-~ Do

1

~

A

p(

d x °--"

yc~a V ~-~-(x°Do) dx°+O(a)

0Do ~ dx°

D o + x ° 8x o ]

0x °

(2s)

+O(a),

whereupon, with the use of (12) and the fact that Do is O (a - ~),

J-

V(Do+4nx°p) dx°+O(a).

(29)

A

Hence, using (10), Jffi feb Jca VEo dx ° + 4n fc/~ ~c~ V(px°+Po)

dx°+O(a).

(30)

Now, the double layer being uniform, we have (17), in whose right member we can take dV-(0V/Sx°)ck°ffi-Eodx °. Hence, substituting for px°+ Po in the second integralof (30),we find that this integral,but for an O (a) quantity, exactly cancels the first.It follows that J is O (a), and hence by (27), that R is so too. 6. The order of the stress integral

Equation (3) is now reduced to PA --PB = 2n

A

E°D° dx° + o ( a ) '

(31)

and the difficulty that remains is the fact that, while the left member is 0 (a ° ), the right member will be 0 (a - ~) for arbitrary Eo and Do, since both of these are O(a-1). Now it is legitimate at this stage to argue that the consistency of the theory

286

requires that the integral in (31) be O(a°), so that by this equation and (4), the surface tension will be given by (5). However, since we intuitively expect some arbitrariness to be possible in double-layer structures, while, as we have just said, under arbitrary functions Eo and Do, the integral will be non-trivially O ( a - l ) , we should like to find corroborative electromechanical ground,.~ for the inference. We first show that it is non-trivially possible for the integral to be O (aO). ~n fact, there is not even a problem in the case of a polarization double layer, for in this case, since p is O(a-~), Do is by (12), O(a°), so that the integrand is O(a-1). But we cannot assume that a double layer is necessarily a polarization one. What we might assume (cf. Section 3) is that, in the general case, the charge component will have a certain ability to adjust. Let us suppose that

OPo ). P=-~xO +C~(x°,xl,x ), ¢~=O(a -l

(32)

Then it follows at once by (10) and (12), that Eo-O(a°), so that again, the integral in (31) is O(a°). The two possibilities we have are extreme. In the former, Do, and in the latter, Eo, is O (a ° ). But we could have something intermediate, in which both Eo and Do are O (a - 1). For example, if for some subset F of [CA,Cs] (perhaps a single subinterval), we have

~)Po

P=°~° + O( x°,x l,x2), x°eF, p f c/( x',x l,x~ ) ,

x% [cA ,cl, ] \ F,

0ffiO(a-l), t ~ffiO(a~),

(33)

we shall have Eo coinciding with an O(a') function on F, and D, coinciding with such a function on the complement of F, so that again, the integral concerned is O(a °). The present possibility corresponds to a general double layer in which the effective (i.e. non-trivially 0 (a - '~) ) charge component occupies only part of the boundary layer. We prefer it to (32), for if EoffiO(a°), then V is slowly-varying, so that there is effectively no contact potential, the presence of which seems normally to be a fact. But in any case, the extreme possibilities pffiO(a -t ) and (32) simply correspond, respectively, to the special cases of (33) in which F = 0 (empty set) and Fffi [ca,cB], and from now on we shall regard them as such. Two equal and opposite layers of charge giving a density p--O(a-2), constituting a charge double layer, would tend to coalesce under their mutual attraction, represented by the normal electric intensity Eo ffiO (a - ~). Even if the charges were just the charge component of a general double layer with polarization component also present, Eo would still generally by O(a-~) and the tendency would still be present. But if the charge arranges itself in such a

287

manner that the net effect of the charge and polarization on the charge component is an Eo of relatively negligible magnitude O (a o ), it is a condition under which we would expect the possibility of equilibrium. It is realized by (33). This somewhat intuitive physical argument appears to serve the purpose intended, of reinforcing our confidence that the integral in ( 31 ) is indeed O (a ° ), and thereby, the justification of the identification of the surface tension by

(5). But we are in a position to deduce more. Let us use (9) to substitute for Eo in the integrand of (31), integrate by parts, and then substitute for ODo/OX° from (12):

yfS EoDo dxO = - f~'B OV D° d'r° = Ld (.~ V ODO 'A A ox° Oxo dxO + ( VDo ) A _ ( VDo ) B =4~

A

pV d x ° - 2

.~ ~, O-fix ° VD°

dx°+ ( VDo).4

-- (VDo)a + O ( a ) .

(34)

Hence by (13), the slow-variationof Kin, and the fact that V D o is O(a-'),

~

"~ E o D o dx° ffi4 n L L - 2Kin + ( V D o )A - ( V D o )~ + O (a ),

(35)

'A

L ffi

p V dx o,

VDo dx o.

M=

"A

¢ CA

The general O (a - i) nature of the left-hand integralisrepresented in the righthand integral L, for the other terms are O(a°), 2 K m M in particular,being a 'curvature effect'of the kind we encountered in (24). Thus, given that the left integral is actually O(a°), so must be L. At this stage we bring in the condition (19), which we have not until now had occasion to use, while using the fieldproperties implied by (33) to treat the integralM. The resultisthat the terms on the right of (35) except the first, are in fact O (a), whence, differentiatingthe reduced equation with respect to X i,

0

f,~'BEoDodxO_.47t~xi+O(a). 0L

(36)

We shall take the volume integral ofpV on (or on a segment of) the boundary layer. With the fact that the volume element dz, in terms of the surface element dS of the surface ~ parallel to S, is dzffidScix ° = x/~dx°dx 'dx ~, this integral is

288

LpVd~=~Qdx' dx=' £.7 x/TpVdx°'

(37)

where Q is the region of integration of x' and x 2. We apply to x/~ the same process as we used in (25) for/~m, SOthat (39) becomes

L

pV dz=

dx I dx 2

y(O,xl,x 2)

¢ CA

+2

k~7OX°/xO=o+x°O(a)

}p

V

,

(38)

whence by (13 ),

f,,v

CB

1,x '2 )

dx l dx 2

p V dx

0

"A

+'If,,

£:2'

x°pVdx°+O(a).

(39)

By (9), (12), integration by parts, and the fact that x°pVwhen x ° is Ca or cn,

is O(a),

I

'"' I f'" 0D. dx o x"p V dx" = x" Ox" V •,,'a 4K ¢,,a = -~

I

fell

4 ~ ,,'a

Do V dx"

(40)

I f""xODoEodx"+O(a). The first integral on the right is M, while the second is J of Section 5, both proven to be O (a). Hence (39) reduces to

i

pVdt=

IL x/P(0,xl,x'a)dx Idx ~ fc

pVdx°+O(a).

(41)

¢ ¢'A

But the left integral here is just the electrostatic energy of the free charge in the total field. We therefore interpret the quantity L as this energy per unit area of the surface S. Returning to (36), for whose derivation we have already used the condition (19), this condition shows also, as we see immediately, that the left member

289

of the equation is O (a), which in turn shows that the quantity L is, but for an O (a) quantity, constant over S. This is to say that there is a uniform distribution of the electrostatic energy, for as we seen by (41), segments of S of equal area support segments of the boundary layer for which the electrostatic energies are (but for O (a) quantities) equal. Now, going back to our intuitive discussion, before (34), we expect the equilibrium situation to one in which, on any segment, the potential energy of the charge free to adjust, is a minimum, itself implying precisely the uniformity we have described. This expectation is fulfilled on account of (19), which is our reason for anticipating that we are virtually obliged to assume this condition as a refinement to the basic specification for the surface tension theory. On account of (19), (5) becomes, when we ignore 0 (a) quantities, T--

1 fc~ d~(x°)~ (x°) dx°' •, c ~

(42)

indicating that T is constant over S, which is of course completely in accord with physical experience. References 1 W.F. Brown, Jr., American J. Phys., 19 (1951) 290. 2 Ft. Cade, Proc. Phys. Soc. London B, 67 (1954) 689. 3 R. Cade, Proc. Phys. Soc. London, 82 (1963) 216. 4 R.Cade, J. Phys. A, 11 (1978) 791. 5 G. Bakker, K. Ned. Akad. Wet., 8 (1899) 308. 6 R. Delay and A. Sanfeld, Electrochim. Acta, 12 (1967) 913. 7 L.P. Eisenhart, Differential Geometry, Princeton U.P., 1947. 8 0 . D . Kellogg, Potential Theory, Dover Publications, New York. 1953. 9 R. Cade, Q.J. Mech. Appl. Math., 44 ( 1991 ) 209.