A micromechanical derivation of the differential equations of interfacial statics. III. Line tension

A Micromechanical Derivation of the Differential Equations of Interfacial Statics III. Line Tension MICHI~LE VIGNES-ADLER*

AND H O W A R D

BRENNERt

*Laboratoire d'A#rothermique, 4", Route des Gardes, 92190-Meudon, France and tDepartment of Chernical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received March 7, 1984; accepted June 12, 1984 A continuum-mechanical theory of line-excess tension and adsorption is developed, quantitatively relating phenomena directly to the density-gradient intermolecular forces existing in the neighborhood of the three-phase contact line. This purely mechanical (i.e., nonthermodynamic) theory is rigorously derived via singular perturbation methods, utilizing a small parameter expansion involving the ratio of the microscopic length scale over which the density-gradient forces act to an appropriate macroscale associated with the macroscopic attributes (depth, curvature, lens radius, etc.) of the bulk phases. "Slender-body" theory serves as the geometrical basis of the perturbation scheme, with the "body" axis oriented along the contact line. In the sense of the theory of matched asymptotic expansions, the scheme automatically derives the proper pair of macroscopic "boundary conditions" at the common intersection of the phases, along the contact line. Specifically, these boundary conditions link together the "outer" or macroscopic stress fields existing within the three bulk phases, and along the three interfaces separating these bulk phases. Of the two boundary conditions thereby derived by the matching of inner and outer expansions, the condition normal to the contact line is found to be identical to the so-called "generalized Neumann force balance" of Buff and Saltsburg, involving a line tension. The second condition, tangent to the contact line, and arising from line-tension gradients, appears to be new. In nonequilibrium circumstances (where this line-tension gradient is not balanced by a line-excess external stress) it represents a potential source of lineal Marangoni flow phenomena, analogous to comparable areal phenomena existing at interfaces. Developments and techniques reported herein constitute the line-excess counterparts of comparable micromechanical surface-excess phenomena derived in the preceding papers in this series. © 1985AcademicPress,Inc. 1. INTRODUCTION

c o n d i t i o n s was derived by e m p l o y i n g the general m e t h o d s o f m a t c h e d a s y m p t o t i c exp a n s i o n s to the interfacial region. H e r e i n we seek to derive c o m p a r a b l e " b o u n d a r y c o n d i t i o n s " governing the equil i b r i u m statics o f the c o n t a c t line between three coexisting a n d i m m i s c i b l e phases, a n d c o n c o m i t a n t l y to p r o v i d e a m i c r o m e c h a n i c a l definition o f the so-called line tension. Again, a m a t h e m a t i c a l t h e o r y describing one-dim e n s i o n a l line p h e n o m e n a m u s t derive f r o m the physical, t h r e e - d i m e n s i o n a l case b y an a p p r o p r i a t e l i m i t i n g process. E v i d e n t a d v a n t a g e s o f this a p p r o a c h lie in

P r i o r c o n t r i b u t i o n s (1, 2) in this series have furnished a m i c r o m e c h a n i c a l d e r i v a t i o n o f the Laplace a n d Levich e q u a t i o n s o f equil i b r i u m interfacial statics, n o r m a l to a n d along the interface, respectively. R e c o g n i z i n g that all m a t t e r necessarily exists in three d i m e n s i o n a l space, a n d hence that a n y theory a t t e m p t i n g to describe t w o - d i m e n s i o n a l interfacial or surface p h e n o m e n a m u s t derive t h e r e f r o m b y an a p p r o p r i a t e limiting process, a m a t h e m a t i c a l l y rigorous m i c r o m e c h a n i c a l t h e o r y o f the s t a n d a r d interfacial b o u n d a r y

11 0021-9797/85 $3.00 Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

Copyright © 1985 by Academic Press, Inc. All rights of reproduction in any form reserved.

12

VIGNES-ADLER A N D B R E N N E R

the mathematical rigor of the resulting concepts, as well as in avoidance of the a priori introduction of an anisotropic local stress tensor approximate to the interface--as was originally done by Kirkwood and Buff (3) for interfacial tension and later by Buff and Saltsburg (4, 5) for line tension. Although the two alternative definitions of the microscopic stress tensor have been shown (2) to be formally equivalent (at least to zero-order terms), we believe that our isotropic viewpoint represents a more traditional, conservative and less ad hoc physical interpretation of the facts--one which should be abandoned only if it conflicts with existing experimental or theoretical facts. We are not aware of any such facts. Referring to Fig. la, consider a small quantity of immiscible fluid (II) placed at the interface between two immiscible fluids (I and III), where it has a negative spreading coefficient. As in Fig. 2a, let x' denote the position vector of an arbitrary point P (not necessarily lying on the line) with respect to an origin fixed at O. The whole system, assumed to exist in a state of thermodynamic equilibrium, may be regarded from the vantage point of some molecular length scale b (the "microscale") (Fig. lb) as being a

continuum--possessing continuously varying properties throughout the entire space occupied. At each point x' of the three-dimensional domain, the existence is assumed of both a pressure field p'(x') and density field c'(x'), e.g., mass density, molar density, or molecular number density. Viewed from a coarser length scale B [the "macroscopic" scale B >> b, say a characteristic linear dimension of the lens straddling the interface (Fig. la)], the previous properties may no longer appear continuous. Rather, at this level of description, singular surfaces will appear to exist, across which such surfaces discontinuities may occur. These surfaces constitute the interfaces between the bulk phases, while the generally skew singular curve lying at the common intersection of these three interfaces constitutes the so-called contact line, -£. An external force (mean) field F'e(x') (per molecule) will be assumed to act upon those molecules centered about the point x' (Fig. lb). The origin of this force is twofold: (i) from intrinsic intermolecular interactions arising from an inhomogeneous microscale spatial distribution of molecules (species number-density gradients) existing in the vicinity of the interfacial regions. (Indeed, the

b

Contact l i n e ~ F, e r~q\ \ \/ --, ~

interface B

i

~

/"~J[-----~'". b - - [ ~ ,(

-/

~

--'/.-" \

t-----

interface

Y ~-- Transition region / between]Z and 1]]: []

Transitionregion

± ~L

Transition region between TIT and I

"Cross section" of contact line transition region to ordinary interfacial transition regions

FIG. 1. (a) Macroscale view of the contact line L existing at the triple junction between three immiscible phases. As a mnemonic, the phases are labeled I, II, III in order of decreasing density. The thickness of the lens defines a characteristic macroscopic dimension B. (b) Microscale view of the contact line .£ in a plane ~0 lying normal to -£; b denotes a molecular length scale (the "microscale"). Depicted are the three transition regions between the bulk phases I, II, and III as viewed from this length scale, as well as the special transition region around the contact line. F'e(x') denotes the m e a n external force per molecule acting upon those molecules situated in proximity to the point x'. Journal of Colloid and Interface Science, Vol. 103,No. 1, January 1985

LINE TENSION

a

13

Contac~line,,~

b

Contact line

nrrE ~

D:E'rrr

pNiiornm a~ ~ " ~ _ ~ N ~

nl1II

\z I

FIG. 2. (a) Outer coordinates. -£ denotes the contact line; 1' denotes the physical distance to a point P lying on the contact line from an arbitrary point A lying along it; t = Ox'p/Ol'is the unit tangent vector along the contact line; I° denotes that normal plane to 1: at P. The position vector x' of an arbitrary point relative to an origin O can be parameterized by the Cartesian coordinates x' --- (x', y', z'). For the special case of a rectilinear contact line these coordinates may be conveniently chosen such that z' --- l' lies along .£. In that case the plane 10 normal to the contact line corresponds to the surface z' = const. (b) Outer coordinates in a plane 10 normal to -£. Conventions: Point P lies along the contact line. A general point lying on the interface formed by the common boundary of the c~ and fl phases will be represented as We. For convenience the three symbols (a,/3, 30, in that order, will be taken to be a cyclic permutation of (I, II, Ill). On those occasions where a neutral generic symbol is required to represent any one or all of these three phases (I, II, III) one may, for example, use the symbol v. The symbol n~e ~- - n e~ denotes a unit normal vector to the c~-flinterface, drawn in such a manner as to point from the a phase toward the fl phase. By convention, the direction of the unit tangent vector t to the contact line will then be drawn in such a sense that a right-handed screw turned in the direction of the vector n l'n o r nII'IHor n m'~ will advance in the t direction. Thus, in the above figure, t is directed out of the plane of the paper at P, toward the eye of the reader.

interface itself is the physical m a n i f e s t a t i o n o f the e x i s t e n c e o f s u c h steep gradients.) (ii) From an externally-imposed constraint which creates a n i n h o m o g e n e o u s macroscale distribution. F '~ m a y be r e g a r d e d as possessing a c o m p o n e n t F~~ = t . F ~ parallel to the ( u n i t t a n g e n t v e c t o r t to the) c o n t a c t l i n e .£ (Fig. 2a), a n d two c o m p o n e n t s ( F ~ , F i e) lying i n the n o r m a l p l a n e ~°(x', y') to .£(l'), w h e r e l' is t h e d i s t a n c e m e a s u r e d a l o n g t h e c o n t a c t line. It will suffice for p u r p o s e s o f the p r e s e n t analysis to restrict a t t e n t i o n to t h e q u i n t e s s e n t i a l case w h e r e F~e arises essentially f r o m n u m b e r density gradients varying on the macroscale

B, w h e r e a s ( F ~ , F ~e) arise exclusively f r o m number-density gradients varying on the microscale b. T h i s physical a s s i g n a t i o n o f o r d e r s o f m a g n i t u d e will h a v e i m p o r t a n t c o n s e q u e n c e s , in clarifying the s u b s e q u e n t exposition. W h e r e necessary, these very different types o f e x t e r n a l forces c a n be respectively distinguished i n textual references as extrinsic a n d intrinsic e x t e r n a l forces, t h o u g h we will e m p l o y o t h e r phrases too. T h e c o n t e x t surr o u n d i n g their use will always p r o v i d e t h e necessary d i s t i n c t i o n . As a c o n s e q u e n c e o f the a s s u m p t i o n t h a t the s y s t e m exists i n a state o f e q u i l i b r i u m , the hydrostatic equation, - V ' p + ~"e = 0,

~Gravity forces will be systematically neglected throughout as being of inconsequential magnitude in this context.

[1.1]

necessarily prevails at each p o i n t x'. H e r e , ~"~ = c ' F 'e, with c'(x') the local m o l e c u l a r n u m b e r Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

14

VIGNES-ADLER

density, is the mean external force per unit volume exerted by the surroundings on the molecules centered around x'. As in prior (1, 2) analyses, Eq. [ 1.1] implicitly assumes the local microscale stress tensor to be representable as an isotropic pressure. Equation [1.1] is to be regarded as a physically exact description of the mechanical state of the system (lacking only constitutive equations for the microscale pressure and intrinsic and extrinsic volumetric external force densities to be complete), at the continuum microscale level. Matched asymptotic expansion methods will be employed in the subsequent integration and interpretation of Eq. [1.1]. It is important to bear in mind that, in a strict sense, such asymptotic methods are purely mathematical--and hence introduce no new physics into the problem, beyond what is already explicitly and implicitly contained in [1.1], including the implicit scaling of the intrinsic and extrinsic forces to be discussed in Section 2. Adopting this purist stance vis~t-vis what is physical and what is mathematical, and insisting upon the maintenance of this distinction, does much to clarify the nature of subsequent operations, as in Ref. (2). Equally, one must distinguish between which subsequent equations are microscopic in scope, in the sense of [ 1.1 ], and which are macroscopic. For example, in our approach, concepts such as interfacial and line tensions, as well as solute interfacial- and line-excess adsorption phenomena, are regarded as strictly m a c r o s c o p i c concepts, that have no existence at the microscale. These macroscopic, singular, surface and line concepts arise naturally in the guise of matching (boundary) conditions at the (macroscopically) discontinuous interfacial and lineal regions separating the different regions of validity of the various asymptotic expansions. 2. A S Y M P T O T I C

ANALYSIS

Previous installments (1, 2) in this sequence have outlined in detail a scheme whereby all of the various mathematical (and "physicalJournal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

AND

BRENNER

appearing") entities occurring therein may be interpreted within the context of singular perturbation theory. Here, those arguments will be adapted to the present geometric configuration, without explicitly repeating existing details of the general procedure. Consider the contact region lying near the common intersection of the three immiscible bulk phases. This region may be envisioned as a long slender domain whose length is of order B and whose characteristic cross-sectional radius is of order b. "Slender-body" theory (6) furnishes a convenient theoretical framework for the subsequent asymptotic mathematical analysis of contact-line phenomena. In this domain a center line Z can be discerned. Distances along ./3, measured from an arbitrary point A (Fig. 2a), will be denoted by l', or its nondimensionalization, l = I'/B.

Other dimensionless quantities, similarly denoted by suppressing the prime superscript, will (unless otherwise stated) be based upon the length B, a reference pressure P~x and number density c~x. Note that .£ will normally be curvilinear rather than rectilinear. It will, however, be implicitly assumed throughout that its radius of curvature is of order B at all points. In nondimensional form the hydrostatic equation [ 1.1 ] adopts the form - V p + F'e = O.

[2.11

Define the dimensionless parameter E = b/B

,~ 1,

[2.2]

assumed small compared with unity at every point along .£. In terms of this parameter one may define an "outer" expansion in ~, with x as the independent variable and with "e p~, F~ (and, later, the number density coo, the chemical potential, #~, etc.) as dependent variables, as subsequently defined. At each point P (x -- Xp in Fig. 2a) of .£ one may also construct an "inner" expansion in e, in which the stretched position vector,

LINE TENSION

15

b

p]Im~/@iirff (b) pT~I

i

m

III

//

/

b)

OIIIZ . . . .

O(b)+~----L==~---~~~+O(b) °) " i ~ Contact line,~_,

O(b

t

FIG. 3. (a) Inner coordinates. Outer domains in the inner representation. At this microscale level of description, the interfaces are not singular surfaces, but appear rather as "diffuse" transition regions (whose thickness is of order b) between the three pairs of interfaces, I-II, II-III, and III-I. (b) Inner coordinates. Representation in the plane ~o normal to t; 8(a) is the cross section of the largest circular cylinder centered at P (radius = a) wherein the molecules are influenced simultaneously by all three of the bulk phases; the circle ag(a) bounds this cross section. For convenience the (planar) interfaces may be described by the "meridian planes" 0 = const = O"~, say, for the a-fl interface (0 ~< 0 ~< 27r). For simplicity, the plane 0 = 0 (and hence, 0 = 27r) may be conveniently chosen so as to coincide with one of the interfaces, say, 0TM for definiteness. With this choice the I-II interface may be assigned either of the values 0LH = 0 or 27r, according to circumstances. The a phase then corresponds to the set of 0 values, min 0~° < 0 ~
= (x -

Xp)/E

[2.3]

is used as the i n d e p e n d e n t variable, a n d / ~ , ~e (and, later, the n u m b e r density ?, the c h e m i c a l p o t e n t i a l /~, e t c . ) - - d e f i n e d subseq u e n t l y - a s d e p e n d e n t variables. In the o u t e r expansion, as e ~ 0 (with B fixed) the three-phase i n t e r a c t i o n region a r o u n d a[ a p p e a r s to c o n t r a c t to a line singularity (since b --- 0). M o r e o v e r , the three pairs o f two-phase i n t e r a c t i o n regions, sufficiently d i s t a n t f r o m the c o n t a c t line to be regarded as "fully d e v e l o p e d , " a n d h e n c e i n d e p e n d e n t o f one another, b e c o m e surface (i.e., "interfacial") singularities, possessing radii o f c u r v a t u r e o f o r d e r e. In the i n n e r expansion, as ~ --, 0 (with b fixed) the three-phase i n t e r a c t i o n region appears to u n d e r g o an essentially infinite m a g nification (since B ~ oo) with a E u c l i d e a n g e o m e t r y (no curvature). Actually, an infinite n u m b e r o f i n n e r e x p a n s i o n s exist, one for each p o i n t P o f the line singularity a[. H o w -

ever, these are all equivalent since the location o f p o i n t P along .£ has been chosen arbitrarily. A t P, erect a set o f local rectangular Cartesian axes (~,)7, z') with z lying along the c o n t a c t line .£, a n d t = OXp/Ol a unit t a n g e n t vector along z. O r t h o g o n a l axes ~, 37 lie in the p l a n e t° n o r m a l to a[, b u t otherwise possess a r b i t r a r y o r i e n t a t i o n s in this plane. Relative to these axes define a local cylindrical p o l a r system o f c o o r d i n a t e s (F, 0, i ) (with unit vectors i,, i0, iz) by m e a n s o f the relations ~? = ? cos O,

]7 = ? sin 0.

[2.4]

This same system o f axes m a y be e m p l o y e d in c o n n e c t i o n with the o u t e r p r o b l e m . As in Fig. 3b, d e n o t e b y 0"~ (0 ~< 0~ ~< 2~r) the angles o f the t a n g e n t planes to the three surface singularities at P; m o r e o v e r , 2 r=~?,

z=£

[2.5]

2 If curvature effects are negligible, dz = d£ .~ dl, in addition to t ~- iz. Journal of Colloid and Interface Science, Vol. 103, No. 1, J a n u a r y 1985

16

VIGNES-ADLER

Earlier statements regarding the macroscale variation of the extrinsic external force component F7 and the microscale variations of the intrinsic external force components (F], Fy,), may be explicitly quantified by the requirements that P~ --- P~(z)

[2.6a]

and ^e __ Fx = P~(E-1r, 0),

^e ~ /6~(e-lr, 0). [2.6b, c] Fy

(See also Eqs. [2.12b, c].) Furthermore, the exact pressure field obeying [2.1], will be supposed constant within each "bulk" (i.e., macroscopic) phase for z fixed, except in the inner region, which region spans the range r = O(0.

[2.7]

This leads to the proposition that the outer expansion of p is piecewise continuous (cf. [2.111),

p(r, O, z; ~) = p'(o)(Z) +

~p~I)(Z)

+ ~p~)(z) + . . . .

(a = I, II, III),

p~,

[2.8]

[2.9]

lying within the a phase. While this outer expansion is defined throughout the whole space (Fig. 3a), it is only asymptotically valid in the outer region, r = O(1).

BRENNER

Inner dependent variables may be defined via the relations def.

`6(F, O, z; e) = Fx,y(r, O, z;

[2.10]

There, for ~ ~ 1, the outer pressure fields p% and exact microscale pressure field p are asymptotically equal. In the outer region, [2. l] becomes

Opt/Or = O,

[2.11a]

op~/Oo = 0,

[2.11b]

Op~o/Oz = P~t(z),

[2.1 lc]

since, by hypothesis, F~ "e" does not possess a component in the plane ~(r, 0) normal to .£(z). Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

p(r = eF, O, z; e),

[2.12a]

ef'~z,y(r = eF, O, z; ~) [2.12b]

and FT(F, 0, z; ~) de__f.PT(r = eF, 0, z; E).

[2.12c]

The inner expansion assumed for ,6 is

`6(~, 0, z; ,) = `6~0)(7, 0 z) + ~`6,)(~, 0, z) + e2`6(2)(7,0, z) + • • . , [2.131 which is valid in the inner region, F = O(1).

[2.14]

In this inner region, [2.1 ] becomes _~`6 + ~,e = 0.

[2.151

Matching conditions between the inner and outer pressure field expansions require that, for fixed values of E ~ 1 and z, and for those points (min 0~ ~< 0 ~< max 0~) lying within phase a, lim `6(F, 0, z; e) = lim p%(r, 0, z; E) ~oo

for those points (Fig. 3b) rain 0~ ~< 0 ~< max 0"~

AND

[2.16]

r~O

for a = I, II, III. Similar relations and expansions hold for the number density c. In their experiments, Scheludko et al. (7, 8) measured the line tension existing in a soap film at the line of intersection between the film and its bulk solution. Though originally structured in a somewhat different context, our asymptotic analysis nevertheless remains completely valid for describing such phenomena. It provides a rigorous basis for the classical model (9) of a soap film as consisting of a plateau border joined to a thin parallel film. However, the thickness of the film must be of the same order of magnitude as the microscale dimension b in order to carry the asymptotic theory over to soap films. Thicker films, necessitating introduction of the so-called disjoining pressure, require a modified asymptotic procedure. This will be addressed in a future companion

LINE TENSION

17 m

s t u d y (10). F o r t h e p r e s e n t study, as originally c o n c e i v e d , t h e c o n t a c t line will be r e g a r d e d as b e i n g t h e a c t u a l t h r e e - p h a s e j u n c t i o n , as d e p i c t e d in Fig. l a. 3. DEFINITION OF LINE TENSION As in Fig. 4, c o n s i d e r t h e i n f i n i t e s i m a l c i r c u l a r c y l i n d r i c a l d o m a i n 63/" o f l e n g t h 6l a n d r a d i u s a, c e n t e r e d a b o u t t h e c o n t a c t line. T h e d i m e n s i o n a will b e t a k e n to be o f m a c r o s c o p i c size, w h e n c e the p e r i p h e r y 6 g ( a ) o f t h e circle will be a s s u m e d to lie in t h e o u t e r region. A d d i t i o n a l l y , t h e surfaces o f d i s c o n t i n u i t y s e p a r a t i n g t h e p h a s e s will be s u p p o s e d p l a n a r , at least to w i t h i n t e r m s o f d o m i n a n t o r d e r in t h e c u r v a t u r e . P r o j e c t i o n o f [2.1] u p o n t yields Op/Ol = P~.

[3. I I

S i m i l a r l y , f r o m [2. I I c ] , ~/. O p ~ l O l = F"~

[3.2]

F o r t h e I c o m p o n e n t , 6F~, o f t h e " t r u e " (i.e., m i c r o s c a l e ) e x t e r n a l force 6F e e x e r t e d u p o n t h e c o n t e n t s o f t h e i n f i n i t e s i m a l d o m a i n 83;, t h e f o r m e r o f t h e s e gives

fff

ff

~'V

[3.3]

¢(a)

where g(a) denotes the cross-sectional dom a i n o f the c i r c u l a r cylinder. T h e c o m p a r a b l e "macroscopic external force" derived from [3.2] is

8~V

[3.41 ~(a)

w h e r e i n , at this level o f d e s c r i p t i o n , w e m a y w r i t e 6~Y = 63f"1 ~ 6~VH ~ 6c~f Ill a n d 8 ( a ) = gI(a) ~ gH(a) @ 8hi(a). The macroscopic surface-excess external force e x e r t e d o n a n interfacial surface 6..4 "~

~5(a) 6 (a) I

FIG. 4. Infinitesimal circular cylindrical volume 6~V of radius a containing the contact line; g(a) with surface element dS is its circular cross section, girdled by the circle 0g(a); 6/, lying along the contact line, is the cylinder thickness; g~(3' = I, II, III) represents the pieshaped region formed by that portion of 3' phase lying inside the circle 0g; thus g ~ g~ (9 gn ~ gin; similarly, 5~V - 63; l @ 6~VH @ 631"HI. Areal domains &4 °~, with surface elements dA% are the appropriate interfacial areas contained within the interior of 63;. Line segment .£"~, with line element dr% is the trace resulting from the intersection of 6o4~ with the normal plane ~o to .£ (i.e., .£"a lies in the plane 0 = 0~ = const., and is of length a); n "e is a unit normal vector to the surface 6o4~B, as in Fig. 2b. Distance normal to the a-~ interface is denoted by y~e = (x - x"~).n ~ with x - x ~ the position vector x relative to any point x ~e lying on the a-/3 interface. Differential distance normal to the planar interface is thus dy °~ = dx. n ~a, where dx may now be measured from any origin; ds denotes a directed line element on Og(a), having the direction of the outer normal to this circle, and lying in the plane ~o. In terms of the system of circular cylindrical coordinates (r, 0, z) defined in the text, the various geometrical parameters characterizing the metrical properties of the infinitesimal domain ~ " = 6l ® g(a) may be explicitly expressed as 6l-= 6z; ~(a) ~ {0 ~< r < a, 0 ~< 0 ~< 2zr}; dS = rdrdO; Og(a) ~- {r = a}; dr ~ =- dr (on the plane 0 = 0"~); dA ~ =-dr~rl; n ~ --- i0 (on the plane 0=0~); t=i~; ds = ~adO, with v ~- L a unit outer normal vector on Og(a). As in Eq. [2.4], we have that y°e = r "~ sin(0- 0"~). Thus, for 0 - 0"~ --- 0, we obtain y ~ ~- r"~(O - 0~). Consequently, dy ~ - r "~ dO, with dy ~ reckoned as positive when proceeding in a direction pointing from the c~phase toward the/3 phase. In this same asymptotic sense, an alternative expression for the surface element dS on g(a) is now dS ~- dr"~dy ~. The volume element 63/" is the differential counterpart of thefinite volume depicted in Fig. 6. In effect, the differential line element 6l along the contact line .£ in Fig. 4 corresponds to the line segment P~P2 in Fig. 6. Since notation for these two cases has been so chosen as to display internal consistency, the geometrical domains referred to in the present figure may also be further comprehended by reference to Fig. 6. Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

18

VIGNES-ADLER AND BRENNER

(a ~ fl; a, fl = I, II, III) is, by definition (1,

2)

ff w

,

[3.51

with T} = - V s a ~

III

[3.6]

the surface-excess external force areal vector density (1, 2). Here, on the a-fl interface, 6,,4~, Vs--- V~~ = I~~ ' v

III

~ff"~--- E ~ w ~ ¢ "~ a,fl

[3.11]

a=l fl=l

for any function ¢~, in which w a~ is the interracial weighting function w ~ = 1 for (a,fl) = (I, II), (II, III), (III, I),

is the surface gradient operator, with I } e --- I -

(cf. external force vector 6Fse = ~,~,~ t6re'~afl ~ .,sj [3.5] and [3.11]) exerted on the combined totality of all three interfaces. In [3.10], the interfacial summation operator ~ , ~ denotes the special double sum

= 0 for (aft) = (II, I), (III, II), (I, III).

n ~ e n ~fl

[3.121 the surface idemfactor, in which (Fig. 2b) n ~fl = - n ed is a unit normal vector on 6-4 ~, pointing from the a phase into the/3 phase. In addition, a '~ =

(p~ - p)dy ~

[3.7]

oo

is the interfacial tension (1, 2). (A notationally more precise form of this formula is given in Eq. [8.9].) The l component of [3.5]-[3.6] is (6Fest)~ = - 6 l

=- - 6 l

L fo"

This counting operator insures that the same interfacial entities will not inadvertantly be counted twice. Closely related to the force defined by [3.9] is the line-excess external force lineal density, defined as T~l = 6F~t/6l. [3.13] Use of [3.3], [3.4], and [3.9] thereby yields

,p

Vst~r~dr ~

(Oc#~/Ol)dr ~ ,

[3.81

since no curvature effects are considered in this section. The line-excess external force that must be assigned to the contact line to rectify the disparity between the true (microscale) and (macroscopic) bulk plus surface combined external forces is here defined as ~F~t = lim (6F7 - 6 F % t - 6 F ~ t a~ct3

e 6Fst).

[3.15]

T eL = - - V L X ,

with VL the line-gradient operator, and

efff

X =

(p~ - p)dS - ~

£

~edr ~

[3.9]

[3.16]

[3.101

3 T h e algebraic sign in [3.15] is chosen so that a positive value of x corresponds to a tension, in the s a m e sense that in [6.4] a positive value of ~r corresponds to a tension, whereas in the c o m p a r a b l e t h r e e - d i m e n s i o n a l relation [2.1] a positive value o f p corresponds to a compression.

In this expression 6F~st = ~ (6F~sz)'~

for this excess density, wherein ~o denotes the whole plane normal to the contact line, and .£~e is the line of intersection of the a/3 interface with ~o. The component-free generalization of this relation is3

a,fl

is the l component of the total surface-excess Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

19

LINE TENSION

the so-called "line (or line-excess) tension." Explicitly, VL = IL. V, with IL -= tt the line idemfactor. This makes VL = tO/Ol. Equation [3.15] represents a fundamental dynamical relation, which generalizes to lines the comparable areal relation [3.6], representing the fundamental hydrostatic equation for a (planar) interface. Note that here and throughout the subsequent analysis, the symbol e refers either to "external" or "excess," according to context. In some cases it may even perform double duty, as for example in the capacity of a line- or surface-excess external force density. Definition [3.16] merits two essential clarifying remarks, discussed sequentially in each of the following two paragraphs. (i) Only the immediate neighborhood of the contact line contributes to the value of x. Indeed, denote by g(a, a~) an annular region (Fig. 5) of ~o centered at the point P, with very large radii a, a~ (a~o >> a >> 1), and consider the expression Ax(a,a~) =

_]'.]" (p~ - p)dS #(a,a~)

i

~dr ~

[3.17]

0¢3 d.g_~(a,a~)

for the incremental contribution to the value of the line tension made by this annular area. A fundamental aspect of the formulation [3.16] requiring emphasis is that, at O(e), the quantity p~ - p vanishes everywhere in f~, except near the lines .£~. Since (a, a~) are both very large, and since curvature effects are neglected, the surface element dS apppearing in [3.17] may be conveniently approximated by the asymptotic relation

dS ~- dr~dy~e,

~(a,a~)~

FIG. 5. Annular area in the plane ~ normal to the contact line and centered about the contact line.

where y ~ is the distance measured normal to .£"~. Consequently, with use of [3.7],

f j"

(p~ - p ) d S

#(a,a~j)

~- E

dr ~

ot,~

(p~ - p)dy ~ cx3

=- E

dr"~"e,

or,13

whence the right-hand side of [3.17] vanishes asymptotically. Q.E.D. (ii) In the definition [3.16] the integration domains ~o and £"~ were supposed infinite in extent. As a consequence of the preceding remark, it suffices for them merely to be of macroscopic size. This apparent vagueness in the precise location of the boundary delineating the transition region between the inner and outer expansions accords with the general ideas of matched asymptotic expansion methods--namely the existence of an extended overlap region, where both expansions are asymptotically valid. 4. ALTERNATIVE DEFINITION OF LINE TENSION

Another formulation for the line tension directly in terms of the intrinsic external force field/%# is easily derived. Translational symmetry along the contact line continues to be assumed as in Section 3. With r = irr designating a cylindrical polar position vector in the plane ~o measured from the contact line, the identity V.(rII) = r. V I I + I I V . r

[4.1]

obtains, in which II = Poo - p is generally discontinuous across the interfacial lines 2 Y . In this manner [3.16] becomes X = a-~lim{(1/2)~If [V. ( r I I ) - r. VII]dS g(a)

-- ~

a,fl

fz ~(a) ~dr~e}

[4.2]

Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

20

VIGNES-ADLER AND BRENNER

where g(a), .£~e(a) are defined in the caption to Fig. 4. Application of the two-dimensional divergence theorem gives

ff

V. (rH)dS

g(a)

Appearances to the contrary notwithstanding, the preceding expression for X is independent of a [cf. remark (i) of Section 3.] Finally, since tr"~ can itself be expressed directly in terms of the y component F~ of the external force density ~e as (2)

tr~ =

ao

o~ %

,~

y Fydy ,

oo

~=I g~(a)

=

f~

fo ~(a) ds" rH

f~ °'(~) naÈ"rIIdr~È' a,fl

+ ~

[4.3] where n ~ is the unit normal vector to .£~e, as in Fig. 2b. Since n ~e has a component only along i0 on Ore, whereas r has a component only along ir, then n ~e. r = 0. Hence, the second integral appearing on the righthand side of the last line of [4.3] vanishes identically. As in remark (i) of Section 3, the first integral can readily be transformed. Indeed, only those points of O~(a) lying close to . £ ~ contribute sensibly to the integral, and there (since r = Lr and ds = LrdO),

r. ds ~- ady ~ for each (a, r) pair, wherein dy ~ ,,, ado in the region around 0 = 0~e. Consequently, ds" rII ~- a ~, g(a)

et,~

f5

Hdy ~

co

it is clear that [4.4] relates the line tension directly to the volumetric external force density ~e. (A notationally more precise form of this formula for a ~ is given in Section 8; cf. Eq. [8.7]). In particular, it relates X to the intrinsic portion of ~e. AS such, it justifies the introduction of X as an intrinsic force, which is a basic molecularly derived property of the system. This will be discussed in Section 6. A more canonical form than [4.4] for X is discussed in Section 8. For the rectilinear configuration of Fig. 4, it can easily be shown that the expression [4.4] is actually independent of the precise location of the choice of origin for measurement of the polar position vector r within the largest (microscopic radius) cylinder for which the molecules are still simultaneously influenced by all three of the bulk phases. Indeed, for planar interfaces, Laplace's equation (1, 2) requires that there exist no discontinuity in the macroscopic pressure, whence p/

= a E ffa~ oq~

as a consequence of [3.7]. Moreover, r . Vp~ -- r . F~ = 0 since we have assumed in the Introduction that F~ is normal to ~o, and thus possesses a component only along the contact line. Since the interfacial tensions a "~, have been (for simplicity, implicitly) supposed constant along £"~, [4.2] may therefore be straightforwardly transformed into the form

ff r. eaS--½a a° . g(a)

[4.4]

a,B

Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

= pll

[4.5]

=pllI.

Imagine in [4.4] that the origin is displaced radially, such that r= r'+C,

with C a constant vector of microscopic size: This makes X= 5

( r ' + C ) . F e d S - ½a ~ o-~, g(a)

o~,B

in which we have utilized the facts that a "~ has already been shown to be independent of the choice of origin (2), and that since a is a macroscopic quantity it remains invariant

LINE TENSION

r=fff eav--fffv, a,.

under a microscopic displacement. Now, successively,

~V

ffc. °as=c.ff eds #(a)

21

Equivalently, since p is everywhere continuous,

g(a)

O'V

#(a)

However, the two-dimensional divergence theorem gives

f f ,~V p d S = f o

g(a)

pds~-fo g(a) poods =

where 0fit represents the outer surface of fit, and dS is the directed surface element. The macroscopic or bulk external force is I11

0.

~---

g(a)

Fo o , -r=l

For on O~(a), which is of macroscopic size, p and po~ are asymptotically equal, and

with

"Vv

fo ds=0 over any closed curve 0g bounding an arbitrary area ~. Of course, for the planar interfaces under consideration we have used the fact that poo is independent of a and hence is constant over O¢(a). Q.E.D. 5. HYDROSTATICS OF A C U R V E D

CONTACT LINE

Relations [3.15]-[3.16] for a rectilinear configuration may be generalized so as to apply to a curvilinear contact line as well. However, what follows is only valid if its radius of curvature remains macroscopic, of order B, at all points. Subsequent arguments underlying this transition from rectilinearity to curvilinearity at terms of dominant order in the curvature can be made rigorous by singular perturbation methods. Consider a cylindrical domain fit of finite extent surrounding the contact line, as shown in Fig. 6a. Its radius a is once again assumed to be of macroscopic size. The three macroscopic surface singularities are regarded as dividing fit into fitl ~ fitn ~ fitm. By definition, the true external force exerted upon the contents of fit is

~'r

(3' = I, II, III). Inasmuch as PL is continuous within fitv, but not generally so across its interphase boundaries ~0v and ~ " (Fig. 6b), application of the divergence theorem to the above must allow the possibility of discontinuities across these singular surfaces. Since, as in Fig. 6b, fit~ is bounded externally by the closed surface Ofity • ~4~ • ~4 v~ (wherein Off/"~ --- ~]~ ~ g~ • ~ ) , we have for 3' = I, II, III that

Oq/v

~y~

+

ff n'~Op~dAva,

(a 4: fl 4=3').

,A'rfl

Inasmuch as n "a = - n ~, we obtain

Journal of Colloid and Interface Science. Vol. 103, No. 1, January 1985

22

VIGNES-ADLER A N D BRENNER

b

Contactline

/"~'~ .~,~

,4

cr

/

/

line ~

P2~'

.A_~~ 1"~

F""'~Pz~'#"

FIG. 6. (a) Volume ~" with circular cross section g~ and ~2 of radii a capping its two ends, and centered about the contact line . L The open cylindrical surface (lateral surface) of radius a is designed as Z. Thereby, the total external surface O~V bounding ~V is O~V = g~ • g2 ~ ~. Insofar as labeling is concerned, sides 1 and 2 are distinguished by the fact that the positive sense of the contact line with respect to the unit tangent vector t is from P~ toward Pz, as in (c). The domain ~V m a y be decomposed into three "wedge-shaped" regions, ~" -= ~Vj @ ~Vn @ ~Vm, each ~ * (~, = I, II, III) corresponding to one of those portions of bulk phases I, II, or lII, contained within the interior of ~'. One of these three wedge-shaped volumes is shown in greater detail in (b). (b) Wedge-shaped, curvilinear prismatic volume ~ v containing that portion of the 3' phase lying within the cylindrical volume ~" of (a). The total volume ~V is made up of three such volumes, ~V --- ~'~ • ~'" • ~V~n. Domain fir ~ is bounded externally by a closed surface, consisting of the five open surfaces ~]~ @ g~ @ X~ • ,.4 ~v @ ..4 *". Each of these open surfaces is individually defined as follows: ~ is the pie-shaped curvilinear triangle P~P~P]~"P~ capping side 1; g~ is the analogous curvilinear triangle lying on side 2; ,14B~ is the curvilinear rectangle P2P~vP~vP~P2 lying on the fl-'r interface; .-4TM is the analogous curvilinear rectangle P2P~"P~"P~P2 lying on the "r-a interface; ~ represents the "cylindrical rectangle" ~t~VO~D~"~"D~V2 --~ --~ --Z --2 lying on the cylinder surface 2~. The relation of this wedge-shaped region to the complete cylinder (see (a)) is such that the surfaces ~.4~ and ..-4*~ appearing in the present figure are, in a sense, "internal" surfaces. Conversely, the "external" surface 0~¢ v partially bounding ~V~ is thus O~V~ --- g]' @ g~ @ Xv. Of course, in relation to the complete figure (a), ~ --= ~ @ E, @ E~H represents the total lateral surface of the open cylinder; ~ ~ ~ • gl I ~9 ~ " (i = 1, 2) is an end cap; and O~" --- 3 ~ ~ • o~flI ~) ocVll] is the total external surface area of the closed-ended cylindrical region bounding ~V. Line elements: The circular arc 6°~, formed by the intersection of ~ , with side 1 corresponds to the curve , , , , . Arc ~ is similarly constituted on side 2. The closed contour 3..4 "~ bounding the curvilinear rectangular area ..4 ~ is depicted in (c). Surface elements: dA~ (see (c)) is the scalar element of surface area lying on ~a~. A corresponding directed surface element on this generally curved c~-fl interface is aS = n ~ d A ~, where--as in the analogous planar case (Fig. 2 b ) - - n ~ lies normal to ~"~ and points from the fl phase toward the 7 phase (see (c)). Additionally, the various directed surface elements aS on the "external" surfaces c~V~ are dS = i~dS.~ (dSz = a~dOdl) on ~ or ~; a s = - t ~ d S on ~]~ or 8~ and dS = t a d S on 8~ or ~2, respectively, where, in (r, ~) coordinates, d S = rdrdO; t~ and t2 are shown explicitly in (c). (c) Rectangular curvilinear area ~ " ~ lying on the a-fl interface. The closed contour 3..4 "/~ bounding the area .-4"~ (---P~P2PI~P'{~P~) is composed of the four curves: ~ --- Pt Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

23

LINE TENSION

where we have written

3'=1 0W'~

Since (Fig. 6a) 0 ~ = 81 $ g2 ~ E, the first integral of [5.5] may be decomposed into the sum OW

since 0~V = 0~V~ • 0 ~ II • 0W ~". As in Refs. (1, 2) (e.g., Eqs. [5.8] to [5.14] of Ref. (1)), the surface-excess external force assigned to the curved a-fl interface is S] (F e'°°=

-ff tvs

(I~% ~)

..Aa3

+ n°e(pg - p~)]dA '~e. [5.3a] (This represents the generalization of [3.5][3.6] for curved interfaces of finite extent.) Hence, for the current triple interface situation, the total surface-excess external force is def.

F~ = Z (V9 °e.

[5.361

Similar to [3.9] the line-excess external force to be assigned to the contact line (to rectify the discrepancy between the true or microscopic external force and the macroscopic bulk plus surface combined external forces) is defined as F ~ = F ~ - (F~ + F}).

ff

0°2

E

[5.6] The second integral of [5.5] may be evaluated by using the general identity (GreenGauss-Ostrogadraski theorem)

f f Vs.(Iso)aA =

[5.7]

..-4

in which 004 is the closed contour bounding the area o4, and d L = vdL, say, is a directed line element on 004, having the direction of the outer normal to 04 and lying in the tangent plane to 04 (v = unit outer normal; dE = scalar line element). In conjunction with Figs. 6b and c, Eqs. [5.5] to [5.7] combine to yield

r>=-tl[ff (p-pojS+ &

[5.4]

Thereby, from [5.1] to [5.3] one obtains

y. f

a,~ d-£?~

l

+ t2[ff (p-poo)dS+= fc a~adr"e]

F ~ = .l'.l" d S ( p - P o o )

+ Y,

+f f

Vs" ( I ~ ) d A ~e. [5.51

f2

+

'~'8 ,..4~

t x E n~%~dl +

aFt,

[5.81

~,fl

P2; -£~e -= P2 ~ P~; A~ --- P~a ~ P~'e; .£~,e _= p ~ ~ p~. Thus, 0,,4 ~e -= a5 @ aS~~ @ A~ @ ~?e. From the convention adopted in (a), the unit tangent vector t is regarded as oriented from side 1 toward side 2. Directed line elements lying on 0o4 "a may be expressed generically as dL ~e = v~adL ~, with d L ~e = [dL~1 the scalar line element and v~ a unit outer normal vector to ,A °~ on 0o4 ~e, lying in the tangent plane to the a-fl interface. On each of the four different curves composing the closed contour 0o4 ~a, the various v "e and dL ~*~values and previously assigned special symbols are, respectively, Line

v~

dL ~

-~ -~ A.~

t X n~ t2 vae

dl dr °~ dLOe

"£~

-tl

dr ~

The scalar line elements dl, dr ~, and d L ~ are all taken to be positive.

Journal of Colloid and Interface Science, Vol.

103, No. 1, January 1985

24

VIGNES-ADLER AND BRENNER

in which a is the vector

in which

tr = nI'% IJl + nll'llttr I1'111 -1- nlll'lo-III'I.

AF~ = f f dS(p-p~) y,

+~ f

v~%"~dL% [5.9]

a,3 d A ~

Since ~ is the lateral surface of cV, which is necessarily of macroscopic radius, upon employing expression [3.7] for the interracial tension it is found that the incremental force [5.9] vanishes at O(0 [cf. remark (i) of Sect. 3]. 4 Accordingly, with use of [3.16], Eq. [5.8] adopts the form

F~ =

f

tlX1 - t2x2 +

t × ~ na3traadl.

[5.101 This may be written alternatively by utilizing the identity t2x2 -- tlX1 =

f

V L " (ttx)dl,

Now, the invariant vector form of [3.13], for the line-excess external force density vector T ec, applied to the finite length contact line is ~dl.

[5.111

Inasmuch as the respective locations of points 1 and 2 along the contact line may be arbitrarily chosen, this leads to the equality T~+VL.(ttx)-tXa=0,

6. INTERPRETATION OF THE GENERALIZED N E U M A N N EQUATION

VL" (ttx) = VLX + NrX

VL = tO/Ol.

f

Equation [5.12] applies locally, at every point along the contact line. Equation [5.12] represents the fundamental equation of hydrostatic equilibrium appropriate to a curvilinear contact line. It corresponds to the line analog of the standard equation of hydrostatic equilibrium for a curved interface (1, 2), from which Laplace's interfacial tension-curvature law and the socalled Levich equation [3.6] (13) were already deduced (1, 2) by employing matching arguments similar to those utilized here. We can readily transpose the results obtained previously for the surface configuration to the present line configuration.

With N the principal unit normal vector to .£, and r the curvature of .£, the identity5

whose validity is an immediate consequence of the representation

F~ =

[5.13]

[5.121

4 The only contribution to the first integral in [5.9] arises from those regions near A "~, where ~ intersects o4 "~, since p - p~ is effectively nonzero only in that neighborhood. But in those regions, dS -~ v"dL"dy ~. Thus, [5.9] is asymptotically of the form f f z dS(p - p~) = ~.~ fA~ v~dL~ f-~ (P - P~)dY ~. With use of [3.7] it therefore follows that f fz dS(p - p~) ~_ - ~ , ~ fAoo v"e~edL ~. Substitution into [5.9] shows that A ~ --~ 0, i.e., it vanishes to at least terms of O(1) in e. Journal of Colloid and Interface Science, VoL 103, No. 1, January 1985

follows as an immediate consequence of Frenet's relations (11). Its use in [5.12] thereby yields the alternative formulation T~ + VLX + Nr× - t × ~r = 0.

[6.1]

5 Frenet's relation for space curves may be written more suggestively in the form Vt'(ILx) = VLx + (VL" IL)X, wherein IL = tt is the line idemfactor and VL =- IL" V =- tO/Ol is the line gradient operator. In the last term of the above identity, we have that ( l l , 12) V t . It = rN. The curvature K at a point of the curve L is either positive or negative (at ordinary points) according as N points for the convex to the concave side of .£ or conversely. Its magnitude is [Kt = lOt~Oil =- [d2x/O12[, which is the inverse of the radius R of curvature of the osculating circle. The first displayed equation is the lineal analog, in both spirit and function, of the comparable areal identity, Vs. (Is~) = Vscr + (Vs. ls)tr. Herein, Is = I - nn is the surface idemfactor, Vs = I s - V is the surface gradient, and (11, 12) Vs.Is = 2Hn, with IH[ = (1/2)(RT ~ + R~-~) the magnitude of the mean curvature and R~, R2 the principal radii of curvature. By convention, n "¢ points from the a phase into the 3 phase, so that the mean curvature H is positive at ordinary points if the interface is concave toward 3 and negative if it is concave toward a.

25

LINE TENSION

Since t is normal to the vectors N and t × n "e, and since the line-excess external stress vector ~ lies parallel to t, Eq. [6.1] splits componentwise into the pair of relations T~ + VLX = 0

[6.2]

and NKX -- t × (nI'IIo"I'lI q'- nlI'llIo "II'III -]- nlII'lo "llI'l) = 0 .

[6.3]

Equation [6.3] may be recognized as Neumann's formula (4, 5) augmented by a linetension term, while [6.2] (cf. also [3.15]) is the lineal analog of the areal Levich equation (1, 2) (cf. [3.6]) ~ffs + Vs~ = 0

[6.4]

ber-density gradients 6 giving rise to unbalanced intermolecular interactions, as discussed in Ref. (2). Thus, it actually represents an "internal (i.e., intrinsic)-external" force 7 field--"external" in the sense evident from [2.1] and "internal" in the sense that its presence could have been implicitly included by the introduction of a physically equipollent (equilibrium) anisotropic stress tensor P* in place of the assumed isotropic stress tensor, P = - I p . Indeed, if we write P* = - I p + r,

with ~" the deviatoric stress arising from the anisotropy, and if the external force density field ~ is such that ~e = V. r,

for the surface-excess external stress. Equation [6.2] predicts the possibility of a "line Marangoni effect" in circumstances where X varies along the contact line and no line-excess external stress ~ exists to counterbalance the resulting "internal (i.e., Brownian gradient) stress" VLX. According to [6.2] such situations create a state of disequilibrium, and hence pose the possibility of accompanying lineal Marangoni-type effects. The physical meaning of the line-excess external force density vector ~ deserves comment. The initial volumetric force field density vector Fe, from which such excess dynamical entities derive, originates from two very different--and independent-sources: (i) It may arise from the existence of a "true" (i.e., extrinsic) external force field, such as gravity, due to an agency lying physically outside of the system. Except insofar as gravity governs the geometrical configuration of the equilibrium system (e.g., the denser fluid lies at the bottom of the container), its influence is completely negligible in present circumstances. Electrical and magnetic forces offer other external force possibilities in this extrinsic category. (ii) It arises from mutual interactions between one part of the system and the remainder, as a result of the existence of num-

[6.5]

[6.6]

then [ 1.1 ] becomes V. P* = 0.

[6.7]

The latter is equivalent to the initial formulation of the well-known Kirkwood and Buff (3) theory. This solenoidal anisotropic stress formulation is discussed at length in Ref. (2), where it is shown to be formally equivalent to our nonsolenoidal isotropic stress formulation, P = -Ip. This same philosophy can be adapted to the line-excess, external lineal force density by introducing an anisotropic line-stress tensor X*,8 defined by the relation

6 It can be foreseen that velocity gradients may also generate such force fields for dynamical, nonequilibrium systems. 7 The term "mutual force" is another possible appellation, but this description appears to be traditionally reserved for n-body interaction forces. Perhaps more appropriate is the name "resultant mutual force," in the sense employed by Truesdell and Toupin (14). As indicated in the text, however, our preferred terminology is "intrinsic external force," for its origin is to be found simultaneously in the domain external to the volume element under consideration, but nevertheless internal to the physicochemical system as a whole. 8 The general concept of a lineal tension stress dyadic x* may be defined as follows. Consider a curve .£. At some point P along -£ imagine the curve divided into two sides, respectively labeled the positive and negative Journal of Colloid and Inte~CaceScience, Vol. 103, No. 1, January 1985

26

VIGNES-ADLER A N D BRENNER

×* = ILX + rL,

[6.8]

with IL --- tt the line idemfactor and ~'L the lineal deviatoric stress dyadic, derived from T~ via the relation T~ = -VL" ~'L.

[6.9]

With this choice, [6.2] requires that VL" X* = 0.

[6.10]

The similarity of [6.10] with [6.7] is evident (although, of course, it must be recalled that [6.7] is a microscale relation, whereas [6.10] is a macroscale relation). A definition of the line-tension equivalent of [3.16] could have been based upon X*, which would have yielded a line-tension definition somewhat similar to that of Buff and Saltsburg (4, 5). Indeed, they succeeded in obtaining [5.12] sans the ~ term, 9 but with a totally negligible gravity force term (multiplied by a lineexcess mass density) in its stead. Moreover, since (at least for the floating lens configuration) the direction of the gravity force is normal to the contact line, there exists no possibility of reconciling these two alternative generalized Neumann formulas. In this same context, by failing to discover [6.2] they obviously could not foresee the possibility of a line Marangoni effect arising from possible line-tension gradients.

in origin. Having achieved this goal, however, the search for completeness leads naturally into the thermodynamic realm--initially in the guise of seeking an equation of state for X: more properly, the search for a line-excess adsorption isotherm (i.e., constitutive equation) for use in connection with [6.2]. Suppose that Fig. 4 pertains to a heterogeneous N-component system existing in a state of equilibrium. Upon denoting by FT(x) the external force per unit volume exerted upon those molecules of species i situated in proximity to x, Eq. [2.1 ] may be written as N

- V p + ~ b'7 = 0.

With ci the molecular number density of species i, the vector F7 = FT/ci represents the (mean) force acting upon a molecule of i. In circumstances where this force is conservative it may be derived from a species-specific (mean) potential Vi(x): F~ = - V V i .

Until now, our analysis has been totally free of thermodynamic considerations. This deliberate choice was made to demonstrate unequivocally that line tension may be regarded as being a purely mechanical concept sides. Let F denote the force exerted by the material lying on the positive side of the line upon the negative. Construct a unit tangent vector t at P pointing from the negative to the positive side. The line-tension dyadic X* at P is then defined as F = t . x*. For the isotropic case, X* = ILX, this reduces to F = t×. 9 In comparing our result with theirs, it should be noted that their "edge length parameter" .£* is opposite in algebraic sign to our line tension × (cf. the footnote pertaining to Eq. [3.15]). Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

[7.2]

To be consistent with the requirements stipulated for F7 in Section 1, this potential energy function must be additively separable in the coordinates normal (N) to, and along (L), the contact line. Explicitly (cf. Eq. [2.6]), Vi(x, y, z) = ViN(X, y) + ViL(Z).

7. T H E R M O D Y N A M I C S OF T H E C O N T A C T LINE

[7.1]

i=1

[7.3a]

Consistency with [2.12b, c] requires that the elements V i N and ViL of V~ be, respectively, micro- and macroscale variables. For a specified z, ViL is taken to possess the same value in all three bulk phases, a = I, II, III. (Thus, physicochemical distinctions between the bulk phases are embodied in the phase-dependent constant Viuo~ defined in the next paragraph, rather than ViL.) Define the limiting value (cf. Fig. 3b), def.

V~vo~ =

lim

ViN(C1r, 0),

[7.3b]

e~0 r=O(l) min 0 ~ < 0 ~ m a x 0 ~

of the function V/N in the outer region. In general, this constant will vary discontin-

27

LINE TENSION

uously with 0 as we pass from one bulk phase into the adjacent one. However, within any one phase a it is a material constant; hence the superscript notation. Of course, the potential energy is determined only to within an arbitrary additive constant. Thus, in [7.3a, b], only the numerical value of the constant difference VTN~ -- V~iu~ is physically relevant, rather than V~v~ and V~iuo~ individually. In the subsequent analysis it will prove convenient to introduce the function V i ~ ~. VTo~ def. = Vi%(O , z) = ViN~(O ) 3ff ViL(Z) '

[7.3C]

with ViNcothe discontinuous constant defined in [7.3b]. This two-term expansion represents the outer potential energy field for species i. As required by [2.1 2b, c], when considered in conjunction with [7.2] and the two-dimensional gradient operator (O/Ox, O/Oy) = ~-1(0/ 0Y, 0/0~), the potential energy function Viu(-f, 37) is of O(1) in the inner region, (Y, 37) = O(1), and tends to the "constant," phasedependent value [7.3b] in the outer region. On the other hand, ViL(Z) is of O(1) everywhere, since it is explicitly and implicitly independent of ~ and 37, and is thus by definition a macroscale variable. As in Ref. (1), with Fzithe mean molecular chemical potential (free-energy increase per molecule of i added to the system) of species i at x, the total potential is constant throughout the system. Hence, #i -~- Vi :

const = K,,

[7.4]

say. This expression represents an exact microscale relation. The comparable macroscale relation, obtained by passage to the limit ~ 0, with (x, y) fixed, is #i~ + Vi~ = Ki,

[7.5]

in which Ki has the same position-independent constant value as in [7.4], 1° and J0 The analogs of [7.2] and [7.1] are, respectively, ~ = - V V i ~ and - V p ~ + ~g_~ ~'7~ = 0, with ~ = cio~FT~. Furthermore, as in earlier references (1, 2), it

thereby--being a microscale constant~possesses the same value in each of the bulk phases. Specifically, in [7.5], K~ = KI I = KI II = Ki. In combination, equations [7.3] to [7.5] readily show that Oui/Oz = dl~iJdz,

[7.6]

since OVi/Oz =- dViL/dz, which from Eq. [7.3c] is also the value of the function dVio,/ dz. As in Ref. (1), Eqs. [7.1], [7.2], and [7.4] combine to show that N

V p = Z CiVUi • i=l

[7.7]

Since dp = dx. Vp, Eq. [7.7] is equivalent to the expression N

dp = ~, cidlai

[7.8]

i=1

for the pressure difference between the neighboring points x and x + dx. Again, [7.8] is a microscale relation, whose macroscopic counterpart is the Gibbs-Duhem equation for an isothermal system, namely N

dp~ = Z ci~d#io~.

[7.9]

i=1

is to be understood that in equations such as [7.5] and [7.9] the "phase label" superscript a (a = I, II, Ill) is to be regarded as implicitly attached to the macroscale variables, ~i~, V~, ci~, F~o, p ~ , etc., at least where necessary to avoid ambiguity. However, in this context three facts should be noted: (i) Ki possesses the same value for all a, as already observed; (ii) the symbol V~ previously employed in Ref. (1) corresponds not to the function defined in [7.3c], but rather to the function V~N~defined in [7.3b]. (This notational change was made to secure the "symmetrical" relations ~ = - V V i ~ as the bulk analog of [7.2] and #i~ + V,~ = Ki as the bulk analog of [7.4].); (iii) whereas one must strictly write [7.5] as u% + VT~ = Ki to avoid ambiguity, the same is not true of the corresponding differential relation d~i~ + dVi~ = 0, since from its finite predecessor in [7.3c], dVi~ ~ dViL(z) is independent of a. Hence, the same is necessarily true of d~i~ too. Therefore, while one generally needs to maintain the a distinction for thefinite function #i~ and Vi~, this need not be done for their differential forms, dt~i~ and dV,~.

Journal of Colloid and InterfaceScience, Vol. 103, No. l, January 1985

28

VIGNES-ADLER

Similarly, the comparable interfacial (macroscale) relation to the Gibbs adsorption isotherm,

AND

BRENNER

in which

fz

FT~dr~

,p

N

dtr~ = - ~ P~dtzi~,

[7.1 O]

[7.16]

i=1

is the line-excess number density of species i (molecules per unit length) at a point z along the contact line. Equivalently,

with r7 ~ =

f

oo

( c ~ - c~o~)dy ~

oo

the surface-excess areal number density of species i assigned to the a-fl interface (1, 2). (A notationally more precise form of the latter formula is given in [8.22].) Apply [7.8] to [7.10] specifically to changes along the t direction, and use [7.6] to derive

Oz

i=l ci dz '

0ffco

-OZ

N

[7.11]

d~i~

= ~ c~-i=l

[7.12]

dz

and 0ffai3

Oz

N

E r7 ~ d~t~ dz " i=1

[7.13]

Integrate [7.11] and [7.12] throughout the whole plane ~ normal to the contact line, and [7.13] along the interfacial lines .£"a. Subtraction of [7.12] and [7.13] from [7.11] thereby gives

ffoOz (p - p~)dS + ~_, f -~z ooodr"a i=1

,p

- Z

r7 ~

a,~ i= 1

dr ~ .

[7.14]

~

Recall that d#io~/dz is a function only of z. As such, it may be removed from beneath the integral signs in [7.14]. Employ the definition [3.16] of the line tension to then reduce [7.14] to the form

dx ----=dz

~ i=1 C i

dt.*ioo -~z

'

[7.15]

Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

N

dx = - ~ , Cidl~i~.

[7.17]

~=1 This relation represents the one-dimensional analog of the two-dimensional Gibbs adsorption isotherm [7.10] as well as of the threedimensional Gibbs-Duhem equation [7.9], since C~ is easily shown to represent the lineexcess concentration of species i. (cf. [8.21] et seq. for confirmation of this physical interpretation.) Equation [7.17] represents a macroscale relation, since all the quantities appearing therein possess physical meaning only at that level of description. In this sense it constitutes a state equation for the line tension. In this context it should be recalled that classical thermodynamics relates the macroscopic chemical potentials /~o to the macroscopic activities, ai~, say, for example. The paragraphs following provide an alternative derivation of [7.17], which parallels the classical thermodynamic derivation of the Gibbs-Duhem equation. Simultaneously, the derivation establishes a correspondence between the preceding mechanical approach and a more traditional thermodynamic one. This thermodynamic-dynamic equivalence can be established by first calculating the work done by displacing the boundaries O~V of a continuous closed volumetric system %r. With dS a directed surface element at a point x on O~V, let dF denote the force exerted by the surroundings on the contents of ~V, and acting across dS. Thus, the work done by the system on the surroundings when the point x is displaced by an infinitesimal amount 6x is, by definition, - r x - d F . Hence, for the whole surface,

29

LINE TENSION

t'£

~W = - | ]

6x. dF

cYV

represents the (infinitesimal) work. Now, in general, dF = d S . P, wherein P is the microscale stress tensor. Since this stress has been assumed to be of the isotropic form P = - I p , this yields

in which dV = 6x. dS is a differential volume element near the surface, and the volumetric domain ~fir is the infinitesimal volume formed upon displacing each point x of the surface Off; by bx. [In particular, the infinitesimal physical volume ~ V of this domain is given by the expression ~V = f f f ~ dV.] Specifically, consider the heterogeneous equilibrium system fir portrayed in Fig. 7, consisting of the fluid contained within a circular cylinder of length l and radius R (R

~R

>> a) centered about the contact line. Equation [7.18] then represents the work done by this system upon its surroundings under the infinitesimal reversible volume change fir ---, fir 6fir--which will be regarded as arising from separate increases 6l in length and 6R in radius. Interest in calculating this work term stems from the fact that its negative gives the increase 6F in the Helmholtz free energy of the system for an isothermal reversible expansion between the same initial and final stages, fir and fir • 6fir, respectively. In turn, this establishes a thermodynamic link with the purely mechanical considerations entering into the evaluation of [7.18] in subsequent paragraphs. Addition and subtraction of the macroscale pressure Po~ to and from the integrand p of [7.18] gives

[7.19] Since Po~ is generally discontinuous across the interfaces, the first integral above may be written alternatively as

S2:R

ot=l

Contoct line ~L

/

~ ~ 2" - - - ~ / t

FIG. 7. Infinitesimal increase 6fir in the volume ~V of a circular cylindrical container (radius R, length 1) surrounding the contact line .£, and resulting from the displacement 6x = (6/, 6R) of its surface 0~V. The original cylindrical domain fir undergoes an axial displacement 6l of its e n d ¢ t ( 0 ~ < r ~ < R , 0 ~ < 0 ~ < 2 ~ r , z = l) a n d a radial displacement 6R of its lateral surface ~R (r = R, 0 ~< 0 < 2~r, 0 ~< z ~< l), such that its original volume V = 7rR21 increases by an a m o u n t ~ V = 6 V(61) + 6 V(rR), in which 6 V(6I) = 7rR26l and 6 V ( r R ) = 27rRlrR. At the microscale level, neither the interfaces nor contact line exist. However, they do "exist" at the macroscale. Thus, at this level of description, changing the volume via the displacement (6/, 6R) has the simultaneous effect of stretching the contact line by an a m o u n t 6l and increasing the area of each of the three interfaces by an a m o u n t 6(RI) = R r l + lrR, as in Fig. 8.

with 8fir~ the region occupied by the a phase contained within 6fir. The integrand p~ - p of the remaining integral in [7.19] is nonzero only in the immediate neighborhood of each of the three interfaces, including their c o m m o n intersection along the contact line. The total volumetric change 6fir may be decomposed into the " s u m " 6fir --- 6fir(6l) @ 6fir(rR), these being the respective contributions arising from M and 6R. Explicitly, 6fir(M) -= ~z ® 6l (gt = circular cross section of cylinder of radius R at its end, z = l), and 6fir(SR) --- ~ ® 6R (~R = surface of the circular cylinder of radius R and length /, not including its two end surfaces, ~l and go). This decomposition of 6fir into the basic "orthogonal" Journal of Colloid and Interface Science, Vol. 103,No. 1, January 1985

30

VIGNES-ADLER AND BRENNER

pair of independent elements permits us to write

8~V

6~V(6l)

for any integrand ( . . . ) dependent upon position within the a-/3 interface. Here, 6,,4~(8l) =- R ® 6l is the infinitesimal rectangular area of the a-/3 interface shown in Fig. 8, resulting from "stretching" of the interface by the amount 6A (6l) =

~'(~R)

for the last term in [7.19]. In terms of the circular cylindrical coordinate system (r, 0, z), the first term on the RHS of the above may be written as

dA "~ =- R61.

[7.26]

The last term on the RHS of [7.21] may be written as

~(~R) 6~(6l)

=

=

l

dz

f2~rfR ,~0=0 ,Jr=0

( p ~ - p)rdrdO,

[7.22]

d z f (z) def. = fl+Tl d z f ( z ) - f ( l ) 6 l dl

[7.23]

f: f? =0

dz

def. fR+~R

R 1

dr

=0

( p ~ - p)RdO

[7.271

since r = R on the surface ZR. Here, f6

in which

R

drF(r) =

dR

drF(r) =- F ( R ) T R [7.28]

for any function F. As in remark (i) of Section 3 and the footnote pertaining to the

for any function f However, according to [3.16], for any R ~> a lying in the outer

regions, fo2" fr R=o (p°~ ~- X + ~ a,j3

~ ~edr~e.

Consequently,

6~(61)

I

a,# 6..4~(6l)

with dA ~ = d z d r ~ an element of area in the a-fl interface, and

[7.25] Journalof Colloidand InterfaceScience,Vol. 103, No. 1, January 1985

NG. 8. "Stretching" of the a-B interface, 0 = 0"~, contained within the circular cylinder of Fig. 7, via the displacement (~/, 6R). In circular cylindrical coordinates the initial interface o~~ is defined by the set of points (0 < r < R, 0 = 0~, 0 < z < l) and possesses the physical area A ~ = l × R. Stretching in the l direction by the amount 8l creates the infinitesimal rectangular domain 8.~4~(8l) = R ® 6l of physical area 8A"a(61) = RTl. Similarly, stretching in the R direction by the amount 6R produces the infinitesimal rectangular domain 8 ~ ( 6 R ) = l ® 6R of physical area 8A~(SR) = lTR. The new domain produced by the total displacement (61, 6R) is thus 8..4 ~ = 8~4"~(8l) ~ 8~4~(8R), whose physical area is 8A~ = 8A"~(6l) + 6A~(SR) = R 6 l + lSR =- 8(RI). An element of surface area on the a-/3 interface is denoted by dA ~ = dzdr ~ d z d r ~.

31

LINE TENSION

evaluation of Eq. [5.9], the only contribution to the 0 integration in [7.27] arises from those 0 values lying near to 0~. However, in those neighborhoods, one m a y set RdO ~- dy ~. Consequently,

process. Finally, substitute [7.32] and [7.20] into [7.19] to obtain"

6=I

L2~ (Poo -- p )RdO a'3 ?J.A~

~- E

~,~

oo

(p~

-P)dY °~

= E :~, ~,~

as in [3.7]. Replacement of dr by dr ~ in [7.27] thereby leads to the expression

l

In the elementary case where each of the macroscopic volumetric, surface and line "phases" is separately homogeneous, p%, a ~, and X appearing in [7.33] m a y each be taken as independent of position, whereupon [7.33] becomes IlI

~(~R)

6 W ~- ~ p%bV ~ - ~, a~rA "~ - xrl. [7.34]

'~ ~ = ~ ( ~ R )

a=l

[7.291

..o:fff,.o

wherein

ff

....

~.A-a(~R)

)"f.f f R

z=0

for any integrand ( . . . ) dependent upon position within the a-fl interface. Here, &.'4"~(rR) =- l ® fiR is the infinitesimal rectangular area of the a-/3 interface (Fig. 8), resulting from "stretching" of the interface by the a m o u n t

ff

6q/~

dA==(. . . )

[7.30]

t~A=t~(rR) def'=

a,3

In the preceding,

dA~e=--lrR. [7.31]

Introduce [7.29] and [7.24] into [7.21] to obtain

is the volume increase of the a phase, bA~ is the area increase of the a-fl interface (given by the sum of [7.26] and [7.31]), and 6l is the increase in length of the contact line. For these separately homogeneous subsystems we may, without ambiguity, replace the 6 symbols by more traditional d (thermodynamic) symbols, to obtain, in place of [7.34], '2

" If, by analogy to [7.18], we define the macroscopic work ~W~ accompanying the volume change 6~" as 6w~ = f f L~ p j v , then the difference,

,w-

= f f f (p- p ,av, 6"V

= 6Ws + ~wL,

= f l xdz + ,,~ ! ~ f a~dA ~,

[7.32]

in which (Fig. 8) 6o4 ~e = 6o4~(6l) ~ 6~4~e(6R) is the total increase in area of the a-fl interface arising from the combined (6l, 6R) stretching

say, with ~Ws de=f. __~a,~ ff~,Aa0 °'afld-Aa/3 and 6WL ~f -fnt xdz, might aptly be termed the (combined) surface-excessand line-excesswork terms, 6Ws and 6WL, respectively, to be assigned to the interface; that is 6W ~- 6W~ + 6Ws + 6WL.The latter representsan asymptotic decomposition of the true microscale work term into three distinct macroscopic work contributions--corresponding, respectively, to changes in volume, area, and length. ,2 In the calculations leading to [7.5] each of the three interfaces was stretched by the same amount, dA~a = Rdl Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

32

VIGNES-ADLER AND BRENNER III

d W ~- ~ p ~ d V ~ - ~ a~dA ~ a=l

a,fl

- xdl = - d E .

[7.35]

The asymptotic equality sign appearing in the above equation serves to emphasize its inexact character, in contrast with the rigorous (microscale) work term [7.18] from which it was derived. Equation [7.35] for the reversible work is a strictly mechanical result. It is clearly identical in form and substance to the comparable formula 13 dF = - ~

P " d V ~ + ~, a"~dA "~ + xdl [7.36]

that derives by applying Gibbsian thermodynamics to a discontinuous, equilibrium, isothermal system composed of separately homogeneous phases. Comparison of the last expressions confirms the (heretofore implied) + ldR, as a consequence of our use of cylindrical symmetry to simplify the analysis. Moreover, for this same reason, the volume changes dV~ appearing in [7.35] are not independent of one another. This apparent restriction upon the applicability of [7.35] is artificial, and is not demanded by the physics. A partial remedy of this "defect" in the proof could be achieved, for example, by not requiring that the center of R lie along the contact line. Rather, its eccentric location could be chosen such that the homogeneous circular displacement fR now produces areal changes fA"~ in the three interfaces in any desired proportions, fAUI:fAn'mlfA m']. However, a less ad hoc general proof can be achieved by introducing instead a local Cartesian system (x ~, y~a, z) associated with each separate interface (y~ = const.) in place of the single cylindrical coordinate system (r, O, z), and subsequently employing the cartesian volume element dV = dx"~dy"~dz and surface elements dA~" = dx~dz and dSz = dxdy to effect the Cartesian integrations of [7.21]. Similarly, the apparent restriction of [7.35] to planar interfaces and rectilinear contact lines is easily removed by recognizing that planarity and rectilinearity are strictly required only of the infinitesimals fl and fiR themselves (or their Cartesian counterparts), but not of the entire original l × R area, ..4~. And, being infinitesimals, these may always be regarded as being devoid of curvature to terms of leading (i.e., zero-) order. is The classical derivation of the Neumann relation [6.3] entails minimizing [7.36] while keeping the total volume constant. Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

equivalence between the macroscale (p~) and thermodynamic (P") pressures of the homogeneous a phase. These formulas may be extended to include the case of open systems by supposing that a total number d N of molecules of the N different species N

( d N = ~ dNi)

are added reversibly to the system through its lateral surface ~, jointly with the other volumetric, areal and lineal changes. Since #ioo is the (mean) chemical potential of a molecule of i on this boundary (which lies in the outer region), one now finds in place of [7.35] that d F = - ~ p ~ d V '~ + ~ a"~dA '~t~ ot

ot,[~ N

+ xdl + ~ #to~dNi.

[7.37]

Deriving from the latter is the GibbsDuhem relation -Z

V"dp~ + Z A~adtr"a ot

ce,fl N

+ ld× + ~ Nidu~oo = 0. ~=1

[7.38]

The total number of molecules of species i contained within the system fit is

.=fffc.,v

[7.39]

~V

To evaluate this integral, multiply [7.16] by dz and integrate from z = 0 to l so as to obtain

fffc., = fffc ,vo ff fo' o~=I

+ ~

rT~dA ~ +

C~dz.

[7.40]

a,# ,A-~

(This method of evaluation is formally equivalent to adding and subtracting ci~ to and

LINE TENSION

33

from the integrand of [7.39] and then pro- upon using the integral expression for a ~ ceeding as in the comparable evaluation of given in the displayed equation following [7.19].) Of particular thermodynamic interest [4.4]. In place of the now ad hoc symbol r ~ is the "homogeneous" case, where cTo~, F7~, we have employed x "~, thereby introducing and C~ are each position-independent con- a trio of local Cartesian coordinate systems stants. In that case a straightforward integra- (x ~, y ~ , z)--one for each interfacial pair, tion of [7.40] gives a-/3. In [8.1 ], 111 Uy.(3~,~ ) def. lim Fy(e ^e -1x, ~ - l y , ~ ) [8.2] " = N i = ~ c ~ V '~ + ~ a=I

FT~A~

+ Cil.

[7.41]

e~0

a,fl

x=O(I) ly~=O(l)

Substitute the latter into [7.38] to obtain N

denotes the y-component of the asymptotic, "fully developed" intrinsic external force a i-I a J3 density prevailing in the outer region E-Ix N N >> 1 of the a-13 interface, far from the contact + Z rT~d~i~) + / ( d × + Z Cid~i~) = O . line. i=1 i=l The lack of dependence of F~, upon dis[7.42] tance x from the contact line rests upon Each of these three parenthetical expressions intuitive, a priori information regarding the involves only intensive (macroscale) variables, functional form of the constitutive equation whereas the coefficients multiplying them are (2) relating Fx,y(x, y) (cf. [2.6] and [2.12]) to "e extensive (macroscale) variables, each of the number density gradients Vc;. That is, which may be varied independently of the we have assumed that at sufficient distances others. Thus, in view of [7.9] and [7.10] it from the contact line the x~-independent follows that limit [8.2] is achieved. In this same sense, N symmetry demands that dx + ~, Cidlz~ = 0. [7.43] " e def. i=1 Fx* = lim j ~ X ( ~ - - I x , e - l y ~ ) = 0. [8.3] ~0 This accords with the line adsorption isox=O(l) ~-ly~=O(l) therm [7.17], obtained by using purely mechanical arguments. In words, the transverse isotropy of fully developed binary interfaces cannot sustain 8. AN INFORMAL PHYSICAL an intrinsic, molecularly based force along INTERPRETATION OF LINE TENSION the interface owing to the absence of molecEquation [4.4] for the line tension X can ular concentration gradients in that direction. Equation [8.1] may be written as be cast into a canonical form, more revealing

- Z V"(dP% - Z cT~d#i~) + ~ A'~(dcr~

of its physical origin in relation to the molecularly based intrinsic external force density F~,y. To derive this expression, observe that the implicit lack of dependence of ~r~e upon the distance r ~ from the contact line enables [4.4] to be written alternatively as

fL =N ot,fl

a~dr~e y~P~*(y~)dx~dy ~, [8.11

~

aB(a)

a F, ~ ~,~

~

-,B

f;1x co

~=0

r. F~dx~dy ~,

[8.4]

since r - t = 0. Here, F'$ --- n~e/~.(y "~) is the fully developed intrinsic external force density vector. As in the region of the three "arms" depicted in Fig. 3b, F~ is sensible only in those neighborhoods, 10 - 0~1 ~ 0 [i.e., y ~ = O(E)], immediately on either side of each interface. This being the case, in [8.4] the summation symbols may be suppressed, the Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

34

VIGNES-ADLER

local Cartesian systems (x ~e, y~e) replaced by the single global Cartesian system (x, y), and the domain of integration extended over all of g(a). Explicitly,

a Z ,,*" = f f r. dxdy. a,3

[8.51

g(a)

Finally, put dxdy = dS in the above and introduce the resultant formula into [4.4] to obtain X = ½.l'.l" r'(~'e - F~)dS.

[8.6]

#(a)

Since r. t = 0, only the intrinsic external force components P:~,y in [2.6] contribute to the value of the above integral, but not the extrinsic component, FT(z). The aesthetically satisfying formula [8.6] shows that the physical origin of line tension resides in the very special nature of the molcularly derived density-gradient forces ar~,,e existing in the immediate neighborhood of the contact line, where the molecules are influenced by all three phases simultaneously (Fig. 3b). Such "three-phase" forces contrast with the fully developed "two-phase" forces Fy. prevailing far from the contact line (£ = ~-~x >> 1). These fully developed intrinsic forces are sensible only in the regions 37 = y/ E = O(1) surrounding each interface, where the molecules are influenced solely by the two bulk phases lying on either side. And they alone govern the ("fully developed") interfacial tensions a~a. Obviously, the absence of special threephase interaction forces in proximity to the contact line would require that F~ = F~,, leading via [8.6] to the value X = 0. This conclusion seems intuitively correct, just as does the analogous conclusion that the condition F~, = 0 on a-3 would lead to the vanishing interfacial tension a~a = 0. Equation [8.6] could of course, be extended over the entire plane ~o, rather than only over ~(a). For F~ - ~'~ presumably goes to zero sufficiently rapidly as Irl ~ ~ to assure convergence of the integral. The fundamental lournal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

AND

BRENNER

equation [8.6] which properly ascribes to X the dimensions of a force, expresses the line tension as the first polar moment (rdS) of the "residual" intrinsic force density F'~ -F~ over the plane normal to the contact line. Equation [8.6] is the line-tension counterpart of the invariant, coordinate-free formulation a" =

r. F~,n~ . dr

[8.71

oo

of the expression (2) for the interfacial tension a " given in the displayed equation following [4.4]. As an alternative to [8.6], one may express the line tension X in terms of the original pressure field p and the fully developed interfacial form thereof, namely def.

p . ( y ' ~ , z) =

lim

p(E-lx,

E-ly '~, z),

~0 x=O(I) ~-Iy~/3=O(l)

[8.8] prevailing in immediate proximity to the at3 interface, far from the contact line. This alternative form may be derived by utilizing the precursor (1, 2),

tr'~ =

(p~ - p , ) d y '~t~,

[8.9]

oo

of Eq. [8.7] (cf. [3.7]). This expression permits the last term of [3.16] to be written as

fz

f ,pf

[8.10]

Consequently, [3.16] adopts the form

x = ff (p, -

[8.11]

This expression may be regarded as the forerunner of [8.6]. Since p - p , is essentially zero at points lying outside of the cross section ~(a), there exists no essential difference between the integrations over g(a) and ~o in these two expressions. The foregoing analysis suggests the exis-

LINE TENSION

tence of a more explicit and easily understood approach to the present class of problems than considered heretofore. This consists essentially of expanding the exact microscale pressure field into the triple sum p=(p-p.)+(p.-p~)+po~

re= ffds'P,

[8.13]

8(a)

exerted on a portion 8(a) of the plane lying normal to the contact line, as in Fig. 9. Specifically, the force is that exerted by the material lying on that side of 8(a) into which dS is directed upon the material on the opposite side. Set P = - I p and introduce the decomposition [8.12] into [8.13] to obtain the expression

$(o)eP

~t

8"~)

F e = F~ + F~ + F~, ~

l

FIG. 9. Planar surface ~ containing the circular surface 8(a) of radius a lying normal to the contact line. The unit tangent vector t to the contact line is simultaneously the unit normal to 8. The side into which t is directed is designated as lying on the "positive" side of 8, the opposite side being designated as its "negative" side. At a point on 8 the directed surface element dS, pointing from - to +, is given by dS = tdS, with d S = rdrd8 a scalar element of surface area. The circular domain 8(a) may be decomposed into the three pie-shaped domains ¢"(a) • g~(a) • g~(a), representing those portions of ¢(a) lying within the (a, /3, 3') phases, respectively. The scalar element of arc length dr "~ lies along the line .E"~ in the a-fl interface, normal to the contact line.

[8.14]

in which

,p

[8.12]

by appropriately adding and subtracting both the fully developed interfacial field p . and bulk pressure field po~. These three terms, respectively, give rise to individual, easily identifiable, lineal, areal, and volumetric contributions. To observe explicitly how this occurs, consider the force,

35

V =tff(p -p,)dS ,p

[8.16]

= -t f f podS.

[8.17]

and #(a)

As a consequence of [8.11], Eq. [8.15] is equivalent to F~ = tx. [8.18] This obviously represents the line-excess contribution to the total force F c, as suggested by the subscript notation. From [8.9], Eq. [8.16] yields F se =

t'~e~aB t l'SI ,

[8.19a]

¢, J.c a ~ d L ~ '

[8.19b]

~ ot,~

in which (~e~e

with dL~e = tdr ~ a directed line element on the line .E~, formed by the intersection of the interfacial plane ~A~ with ~, and lying on ..4~. By definition, adL gives the surface tension force dF~ exerted across dL by that side of the interface ..4 into which dL is directed upon the material lying on the opposite side. Hence, Eq. [8.19a] obviously represents the surface-excess contributions of all three interfaces to the total force F c. Last, [8.17] may be written as Ill

F~ = ~ Fo~e~,

[8.20a]

a=]

in which

g"(a)

with ~ ( a ) that portion of ~(a) contained within the a phase. Equations [8.20] clearly represent the bulk phase surface contributions of all three phases to the total force F c. Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

36

VIGNES-ADLER AND BRENNER

These line, surface and bulk considerations confirm the categorical assignations implied by the choice of subscript notation in [8.14]. The elementary manner in which the pressure decomposition [8.12] gives rise to the comparable force decomposition [8.14] reflects the utility of this informal approach toward clarifying the nature of the asymptotic procedure. Indeed, this same general scheme might have been employed in the thermodynamic work computation, proceeding from [7.18] to [7.33]; that is, the t e r n a r y decomposition [8.12] is more rational to employ than is the b i n a r y decomposition, p = (p p~) + p~, actually used in the evaluation. Finally, in a similar context, we note that line- and surface-excess adsorption phenomena (as well as related density-gradient phenomena) may be equally easily analyzed by invoking the comparable ternary concentration decomposition,

in which it appears is sensibly nonzero only within the interior of the cylindrical region of radius a centered at the hypothetical contact line. Similarly, the contributions of p , p~ to the surface and line integrals in which it appears are sensible only within each of the three "slab-shaped" regions of thickness b bounding each of the hypothetical singular interfaces on both of its sides. In performing the informal, nonrigorous analysis of this section we have, in effect, implicitly assumed that a ~> b with b the thickness of each of the three distinct singular interfacial regionsl4--corresponding to the length scale of the gradient-induced external molecular force Fx,y(X, -e y). This inequality has permitted us to regard the three interfaces as already existing as singular surfaces when viewing the phenomenon from the viewpoint of the a scale. It is this fact that has permitted us to introduce the interfacial tension g ~ (cf. [3.6]) as already existing as a macroscopic ci = ( c i - ci,) + (ci, - ci~) + cio~. [8.21] entity, rather than having to proceed through For example, use of the latter in [7.39], the intermediate step of first relating these considered in conjunction with the defini- macroscopic "surface" forces to the compations, rable microscale pressure field p, as in [3.7] or, more precisely, in [8.7]. In turn, the clearr~ ~ = (ci, - c i ~ ) d y ~ [8.221 cut asymptotic limiting processes associated c~ with the three different length scales B >> a and >> b, which eventually result in respective [8.23] bulk, surface, and line matching "boundary ~p conditions," permits an unequivocal interpretation to be assigned to each of the three of the surface- and line-excess densities, per- terms in Eqs. [8.12] and [8.21]. mits a virtually immediate derivation of the It is important to recognize the informality asymptotic decomposition [7.39]-[7.40] of of the scheme outlined in the present section. Ni into distinct bulk, surface, and line con- In particular, the present ternary decompotributions. sitions, though suggestive, do not by themDecompositions [8.12] and [8.21] of the selves constitute a systematic asymptotic microscale pressure and species-density fields scheme. Specifically, each of the three quaninto three distinct contributions constitute i n f o r m a l representations of the several over14Our argumentscouldobviouslybe extendedwithout lapping asymptotic matching regimes required for the singular perturbation analysis. difficultyto the more generalcase, for whichthe common The significance of these regions and their b value could be taken as possessingdifferentvalues b"~ for each differentinterface.So long as b"~is of the same matching requirements are easily understood order for each a-B pair, no essential modificationsare in relation to Fig. 3b. Thus, the contribution required to the preceding analysis in order to achieve o f p - p , to the volume and surface integrals this generalization. -

y_°

c,= f f (c,-ci,)dS,

Journal of Colloid and Interface Science, VoL 103, No. 1, January 1985

-

37

LINE TENSION

tities p, p , , and poo (or their concentration counterparts) is defined on a different length scale. As such, adding them together is little more than symbolic, though the sum seems to possess at least some merit as a composite expansion. The analysis of the current section needs therefore to be regarded as informal, devoid of rigor. Only singular perturbation methods, such as those outlined in preceding sections, can provide the requisite mathematical structure necessary for a rigorous derivation of macroscale boundary conditions from microscale equations.

veloped interface limit and the bulk state, in combination with [9.2], therefore gives

ci, = Ciooexp(Ei~ - El,) =-- Cioo exp(EiNoo -- E i u * )

cf. [7.3a] and [7.3c]), with

Adsorption at the Contact Line Some tensioactive materials may preferentially accumulate along the contact line. This phenomenon would appear especially likely to occur if such solutes possessed a tripolar character--that is, a head, a tail, and a side branch--each of which extremity had a distinct physicochemical proclivity for residing in only one of the three phases. (By analogy to surfactants, which tend to accumulate preferentially at an interfacial surface as a result of their bipolar character, such special solutes might aptly be termed "linactants.") The quantitative extent to which a chemical species may be adsorbed at a contact line can be calculated from [8.23] by the following informal arguments. Application of [7.4] to the fully developed interface limit

denoting a normalized potential energy function. A similar analysis applied to the combination of [7.4] and [7.5] yields Ci = Cioo exp(EiNoo -- EiN)

Ci

I

I

~II

[9.1]

In combination with [7.5] this yields Vi,.

[9.2]

However, for an ideal solution,

d~ti = kTd In ci,

[9.3]

along with similar-appearing equations for the fully developed interface pair (#i,, ci,) and bulk pair (#ioo, ci~). Integration of [9.3] between the fully de-

11

LIII~III

= kiLCioo -1- KiLCioo "k- t~iLt~ioo,

[9.6]

[9.7]

wherein, for a = I, II, and III,

k';L-- f f [exp(E';No~-- EiN) -- exp(ETNoo -- Ei,)]dS

[9.8]

is the individual (i.e., a-phase-specific) lineexcess adsorption coefficient of species i. The integral [9.8] is, of course, assumed convergent--a fact that has permitted us to replace the complete region 1°~ by its subdomain #~. The three individual bulk solute concentrations c7~ (a = I, II, III) appearing in [9.7] are not independent of one another. Rather, they are interrelated through the "partitioning law" expression c';~ / c ~

gives

I.ti, --].l, ioo = Vioo -

[9.5]

Ei = Vi/kT

as the analog of [9.4]. Introduce [9.6] and [9.4] into [8.23] to obtain

9. DISCUSSION

#i, + Vi, = Ki.

[9.4]

= Ki~, ~

[9.9]

derived from the bulk form of [9.3] in conjunction with [7.5]. Here, Ki~ def.exp(E~u~o -- Ei~voo)

[9.10]

is the bulk thermodynamic interphase equilibrium partition coefficient, giving the distribution of species i between phases a and/3. Upon using [9.9], Eq. [9.7] can be expressed in terms of only a single one, say c]~, of the bulk concentrations. With this choice, Journal of Colloid and Interface Science. Vol. 103, No. 1, January 1985

38

VIGNES-ADLER AND BRENNER

[9.1 11 adsorption at the contact line. A derivation of this useful and simple formula is provided in which in the following paragraph. k~I [9.12] Apply [9.3] t o t h e bulk pair (gi~, cioo) and g], =/cl~ + ~ iL + 1.111/.."III,l ~i~,-i~ eliminate d#i~ between the resulting equation is the overall line-excess adsorption coefficient and [7.17] to obtain Ci = K]Lc~o~,

of i at the contact line, based upon phase I. ~5 d x = - k T ~, C i d l n Cioo. [9.14] (The terminology "individual" and "overall" i represents standard chemical engineering (Observe that whereas the finite entity In c;~ parlance.) requires a superscript ot in order to avoid ambiguity, the same is not true of its differLine-Tension Diminution Accompanying ential form d In cio~. For as is clear from Linactant Adsorption [9.9], in conjunction with the fact that the constant g i ~ is the independent of z--cf. Inadvertant solute adsorption at contact [9.10], we therefore have that d In c7~ = d lines may cause experimental error in linetension measurements, known (15) to be X In c ~ .) Without loss in generality it will error prone due to the presence of impurities. be convenient to affix the superscript I to d This phenomenon is easily explicable in terms X In c;~ in [9.14]. However, since the constant K~L defined in [9.12] is independent of of the (ideal solution) formula 16 z, we have by identity that d In ( g i L~C i ~ ) = d Xo - X = kTC, [9.13a] X In c~oo. With use of [9.11], Eq. [9.14] thereby adopts the form C = ~ Ci, [9.13b] i d× = - k T ~ Cid In C i i

giving the reduction in line tension (below the solute-free value ×o) arising from solute

-~ - k T ~ dCi. i

~SSymmetrical and asymmetrical equations [9.7] and [9.11], respectively, are the line-excess adsorption analogs of the comparable surface-excess adsorption equations (2) F7e = k ~ a l 3 ] c T ~ + keis[al3]cei~ and I'~~ = KTs[a~]cT~. Here, with y° measured from the a-/3 interface into the a phase, kTs[a~] = f o [1 - exp(ETuo~ - E i u . ) ] d y ~ (with a similar expression for the comparable t3 coefficient), is the i n d i v i d u a l (i.e., a-phase-specific) surface-excess adsorption coefficient for species i at the a-13 interface, and KTs[a~] = kTs[a~] + KT~k~s[aB] is the corresponding overall coefficient. These interface-bulk adsorption isotherms are readily derived (2) from [8.22], with the latter expressed in the symmetrical form PTa = f o (ci. - cT~)dy ~ + f o (c,. - c~i~)dy ~.

16 This is the counterpart of the well-known (2) ideal solution formula a~" - cr~ = k T F ~ ,

F ~ = ~ F7~, i

giving the interfacial tension reduction (below the solute-

free value a~~) accompanyingthe adsorptionof solute(s) at the a-~ interface. Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

Alternatively, in terms of the total solute concentration C defined in [9.13b], dx = -kTdC.

[9.151

The latter may be integrated subject to the solute-free value Xo at C = 0 to obtain [9.13a]. Q.E.D. "Dynamic" Line Tension

One advantage that accrues to our mechanical theory of line tension is that it automatically carries with it the concept of a "dynamic" line tension for nonequilibrium situations. Such generalizations entail replacing the microscale hydrostatic equation [2.1 ] by a comparable hydrodynamic equation, typically pDv/Dt = V . P + ~e

wherein P = - I p + r,

LINE TENSION supplemented (where appropriate) by auxiliary transport equations, involving, for example, the transport of m o m e n t of momentum, mass, species, energy, and electric charge. In this capacity for generalization our approach transcends the "traditional" thermodynamic one, which limits itself to equilibrium states. As our dynamic line tension arguments embody an obvious extension of those already outlined for dynamic interfacial tension in Ref. (2), no further comments will be offered here. In the present context of dynamics vs thermodynamic arguments for the analysis of line tension, attention should be directed to the recent thermodynamic analyses of the latter by Kerins and Widom (16) and Rowlinson (17). Their work was published after our own analysis was completed in the summer of 1982.

Point Tension As a natural extension of our sequence of papers in this series, one could imagine a point tension--possibly existing at the common intersection of four bulk phases--defining it by an appropriate sequence of limiting processes beginning with the physical, strictly three-dimensional phenomena. The procedure used in Section 3 could easily be generalized: A point-excess force would constitute the difference between the true force exerted on a small spherical volume element, and an appropriate sum of the macroscopic external forces--bulk, surface, and line, each of which would then be considered as an external force. Such a point would, of course, be highly singular. For this reason the notion of point tension might facilitate understanding of point singularities in related disciplines. Our analysis provides a clear-cut link between the strictly continuous three-dimensional physical phenomena (albeit on a rather minute length scale) and the discontinuous (i.e., singular) mathematical representations thereof.

39

Higher-Dimensional Spaces Finally, we would encourage extension of our ideas to higher-dimensional, "nonphysical" spaces. The history of physics is replete with examples illustrating the significant physical insights furnished by imaginative mathematical abstraction of physical phenomena. This despite Aris' ((12), p. 226) discouraging comment (in his introduction to the pedagogy of surface phenomena) that " . . . a bulk fluid can never be the interface of two four-dimensional fluids." APPENDIX:

a

a~

A A "~, (3A"~ dA "~

.~'~, 6.,-4"~

0o4 "~ b

NOMENCLATURE

j7

"macroscopic" radius of circular cylindrical cross section centered at the contact line, lying in the region of validity of the outer expansion (Fig. 3b) "macroscopic" radius of a very large circle centered at the contact line (Fig. 5) arbitrary point on the contact line (Fig. 2a) area on the a-/3 interface (Figs. 7, 8) scalar element of surface area on the c~-13 interface (Figs. 4, 6c, 8); or increase in interfacial area (Eq. [7.35]) curvilinear rectangular areal domain on the a-t3 interface, contained within the volumetric domain ~V, 8~V (Figs. 4, 6b-c, 7, 8) curvilinear rectangular contour bounding the area .A "~ (Fig. 6e) molecular ("microscopic") length scale corresponding to the range of intermolecular forces (Fig. lb); thickness of interfacial or contact line regions

~7Becauseof the geometricaland notationalcomplexity of the symbolsemployedin the text, terms in parentheses at the conclusion of each definition refer to the section, equation, figure, or footnote wherein the symbol is first defined, portrayedor otherwiseilluminated in context. Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

40 b,~

B

C

£i Ci~

C

G C

~, ~ Ei dF dF F~

Fe

£

Fe (F~, F~)

F~

6F?

VIGNES-ADLER AND BRENNER

thickness of transition region for the a-fl interface (footnote 14) macroscopic length scale (Fig. la); thickness of lens or length of contact line total molecular number density (Sect. 1) species-specific number density of i (Sect. 7) macroscale number density of species i (Sect. 7) total line-excess number density (Eq. [9.13b]) line-excess number density of species i (Eq. [7.16]) constant vector representing a radial displacement of the origin (Sect. 4) circular arcs lying in the 3" phase on sides 1 and 2 of the contact line (Fig. 6b) normalized potential energy function (Eq. [9.5]) Helmholtz free-energy change (Eq. [7.35]) force exerted on element of surface area aS (Sect. 7) mean external force per molecule in Sect. 1, or total external force exerted on the contents of fit in Eq. [5.1] mean external force per unit volume (Sect. 1) "fully developed" external force density, far from contact line (Eq. [8.41) inner external-force volumetric density (Sect. 2) scalar components of the mean external force per molecule F ~ normal to the contact line (Sect. 1) scalar component along the (tangent to the) contact line of the mean external force F ~ per molecule (Sect. 1) component along the contact line

Journal of Colloid and Interface Science. Vol. 103, No. 1, January 1985

of the external force 6Fe exerted on the contents ofrfit (Eq. [3.3]) F~ mean external force exerted on a molecule of species i (Sect. 7) F eL~ 6F~ line-excess external force assigned to that portion of the contact line contained within fit, ~fit (Eqs. [5.4], [3.9]) 6F~t scalar component along the contact line .£ of the line-excess external force 8F~ assigned to the length 6l of contact line (Eq. [3.9]) F~, 6F~ total surface-excess external force assigned collectively to the three interfaces contained within fit, ~fit (Eq. [5.3b], Sect. 3) (F~) ~, surface-excess external force as(~F~) ~ signed to the a-fl interface contained within fit, 6fit ([5.3a], Eq. [3.5]) 6Fesl component of 6F~ along the contact line (Eq. [3.10]) (6F~l) ~ component of (~F~) "~ along the contact line (Sect. 3) or, equivalently, contribution to ~F~t of the a-fl interface (Eq. [3.8]) F~ macroscopic external force exerted on the entire contents of the cylindrical domain fit (Sect. 5) 6F~l macroscopic counterpart of true force component 6F~(Eq. [3.4]) F~ macroscopic external force exerted on the 3, phase volume fit~ contained within the cylindrical volume fit (Sect. 5) F~'e macroscopic external-force volumetric density (Sect. 3) "e F~t scalar component of F~ along the (tangent to the) contact line (Sect. 3) F~ macroscopic external-force volumetric density at a point in the 3' phase (Sect. 5) i chemical species (Sect. 7) ir, i0, iz unit vectors in circular cylindrical coordinates (r, 0, z) (Sect. 2)

LINE TENSION

I IL

Is, I ~

k

k~ k~s[a~]

K~

KIL

KTsIa~]

dl

6l

L .£ .£'~(a)

dyadic idemfactor in three-dimensional space (Sect. 6) dyadic projection idemfactor on a dL ~, one-dimensional curved line dL ~ (Sect. 3) n~ dyadic projection idemfactor on a two-dimensional curved (a-r) N interface (Sect. 3) Boltzmann constant individual, a-phase-specific, lineexcess adsorption coefficient for N species i (Eq. [9.8]) individual, a-phase-specific, sur- Ni face-excess adsorption coeffi- O cient for species i at the a-~ interface (footnote 15) O(~) constant total potential for species p i (Eq. [7.4]) /~ overall, line-excess adsorption Po~ coefficient for species i at the p ~ contact line, based upon phase ! (Eq. [9.12]) P overall, surface-excess adsorption coefficient for species i at the a- ~o interface, based upon phase a (footnote 15) P~ bulk thermodynamic interphase equilibrium partition coefficient P for species i distribution be- P* tween a and fl phases (Eq. [9.10]) r component along the contact line (Sect. 1), or length of the contact dr line (Sect. 7) differential element of length mea- dr ~ sured along contact line (Sect. 2) infinitesimal length of contact line r defining infinitesimal cylinder 6~V (Fig. 4), or increase in length of contact line (Fig. 7) R line-excess quantity (Eqs. [3.9] and [5.4]) 6R contact line (Sect. 1) line or curve of intersection between a-fl interface and the

ds

41

normal plane ~o to the contact line (Figs. 4 and 6) scalar and directed line elements in the a-fl interface (Fig. 6c) unit normal vector to the a-fl interface (Figs. 2b and 6c) number of species in a multicomponent system (Sect. 7), or total number of molecules in a system principal unit normal vector to contact line (Sect. 6) molecules of species i in system arbitrary origin of a coordinate system (Fig. 2a) gage symbol (Sect. 2) microscale pressure field (Sect. 1) inner pressure field (Eq. [2.13]) macroscale pressure field (Sect. 2) macroscopic pressure in the a phase (Sect. 2) point in space, especially along contact line (Fig. 2a) plane (x, y) or (r, 0) normal to the contact line (Fig. 2a) "thermodynamic" pressure of the a phase (Eq. [7.36]) stress tensor (Sect. 7) anisotropic microscale stress tensor (Eq. [6.5]) circular cylindrical distance or coordinate (Sect. 2) differential displacement along r (Sect. 2) differential displacement dr measured along line . £ ~ (Figs. 4 and 6c) cylindrical coordinate radial vector in plane ~o normal to contact line (Sect. 4) radius of a circular cylinder (Fig. 7) increase in radius R of circular cylinder (Fig. 7) directed element of arc length on the circle O~(a) (Fig. 4)

Journal of Colloid and Interface Science, Vol. 103, No. t, January 1985

42

VIGNES-ADLER

AND

BRENNER

dS

scalar element of surface area on 8 (Figs. 4 and 6b) dS directed element of surface area (Fig. 6b) g, 8(a) cross section of a circle of radius a (Fig. 3b) 08, periphery of a circle of radius a 08(a) (Fig. 4) two ends of a circular cylinder 81, 82 (Fig. 6a) circular sectors containing the 3" phase at the two ends of a circular cylinder (Fig. 6b) two ends of a circular cylinder of 81, 80 length l (Sect. 7) t unit tangent vector to the contact line (Figs. 23, 6c) unit tangent vectors to the contact tl, t2 line at its two "ends" (Fig. 6c); also unit outer normals on (g~, 8]r) and (82, 8~), respectively (Fig. 6b) T absolute temperature T~ line-excess external lineal force density vector (Sect. 3 and Eq. [5.11]) TeLl component along the contact line of the line-excess external force density vector T~ (Eq. [3.13]) surface-excess external area force T~s density vector (Eq. [3.5]) microscale potential energy per Vi molecule of species i (Eq. [7.2]) v~o~ macroscale bulk phase potential energy of species i (Eq. [7.3c]) v, 6v volume (Fig. 7) dV differential volume element (Eq. [3.3]) V ~, 6 V ~ volume of c~ phase (Sect. 7) d V ,~ differential volume element of c~ phase (Eq. [7.35]) fit, 6fit volumetric domain inside a circular cylinder, finite or infinitesimal (Figs. 6 and 4) 0~ closed surface bounding the volume fit (Fig. 6a) ~ , 6~ ~ wedge-shaped volumetric domain Journalof Colloidand InterfaceScience,Vol. 103, No.

1, January 1985

0fit ~ w~ 6 W, d W

x, £

x 6x

x~

x~ y, )5

y~ y~

z, ~

of a phase within a circular cylinder (Figs. 6b and 4) partially closed, "external" surface of volume fit~ (Fig. 6b) interfacial weighting function (Eq. [3.12]) thermodynamic work (Eqs. [7.18], [7.35]) Cartesian coordinate in plane ~o normal to the contact line (Sect. 1, Eq. [2.4]) three-dimensional position vector of a point (Figs. lb, 2a) displacement of a point at a surface stretched position vector relative to a point P on the contact line (Eq. [2.31) distance from contact line along a-/3 interface (footnote to Eq. [7.35]) position vector of a point lying on the a-/3 interface (Fig. 4) Cartesian coordinate in a plane t° normal to the contact line (Sect. l, Eq. [2.4]) distance from interface into a phase (Sect. 9) distance from line . £ ~ or surface O~ in a plane ~o normal to the contact line (Fig. 4) Cartesian coordinate along contact line (Eq. [2.5])

Greek Letters

a, 3, 3' F~

I'7~

generic symbols for immiscible fluid phases (Fig. 2b) total surface-excess areal number density of all species at a-fl interface (footnote 16) surface-excess areal number density of species i at a-C/interface (Eq. [7.10]) quantity of infinitesimal extent, as in 6fit (Fig. 4) micro- to macroscale length ratio (Eq. [2.21)

LINE TENSION circular cylindrical polar angle (Eq. [2.4]) meridian plane defining the a-fl interface and line .£"~ (Fig. 3b) curvature of the contact line (Sect.

0 0~ K

6)

# i , #ioo

II

0"~ O"~3

O"

~, ~(a)

~R T

1/

line or curve formed by intersection of ~-3 phase with circular cylinder ~ (Fig. 6c) micro- and macroscale chemical potentials of species i (Eqs. [7.4], [7.5]) macro-/microscale "excess" pressure difference at a point in the a phase (Eq. [4.1 ]) interfacial tension of the a-3 interface (Eq. [3.7]) "composite" interfacial tension vector (Eq. [5.13]) open-ended lateral circular cylindrical surface (Fig. 6a) portion of ~ subtended by o~ phase (Fig. 6b) circular cylinder of radius R (Fig. 7) deviatoric stress dyadic (Eqs. [6.5], [6.8]) unit outer normal to circle O~(a) in the plane containing the circle (Fig. 4) unit outer normal to curve 6°"~ lying in a-3 interface (Fig. 6c) line (or line-excess) tension (Eq.

x

[3.16]) Xl,

X*

X2

line tensions at points 1 and 2 along the contact line (Fig. 6c) anisotropic line-stress dyadic (Eq. [6.8])

43

Subscripts i l

chemical species component of a vector along the contact line line or line-excess L normal to the contact line N circular cylinder of radius R R surface or surface-excess S g circular cross section x, y components in x and y directions, normal to contact line lateral open surface of a circular cylinder 0 solute-free value 1, 2 capped ends of a circular cylinder (0), (1), (2) . . . . inner perturbation fields 00 bulk-phase or macroscopic value * "fully developed" interfacial region, far from the contact line A reference value Superscripts e a,/3, 3' I, II, III '

external or excess generic symbols for immiscible fluid phases (Fig. 2) immiscible fluid phases (Fig. 1) dimensional physical quantity

Marks over Symbols inner variable or expansion volumetric density ACKNOWLEDGMENTS M. Adler was on leave from the Centre National de la Recherche Scientifique(CNRS) during the period in which this research was performed. She would like to thank both the University of Rochester and the Massachusetts Institute of Technology for their hospitality during her stays there.

Special Symbols V Vs VL

three-dimensional gradient operator two-dimensional surface gradient operator (defined following [3.6]) one-dimensional line gradient operator (defined following [3.16])

REFERENCES 1. Brenner,H., J. ColloidInterface Sci. 68, 422 (1979). 2. Brenner,H., and Vignes-Adler,M., J. Colloid Interface Sci. (submitted, 1985). 3. Kirkwood, J. G., and Buff, F. P., J. Chem. Phys. 17, 338 (1949). Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

44

VIGNES-ADLER AND BRENNER

4. Buff, F. P., and Saltsburg, H., £ Chem. Phys. 26, 23, 1526 (1957). 5. Buff, F. P., in "Handbook der Physik" (S. FliiggeMarburg, Ed.), Vol. X, p. 281. Springer-Vedag, Berlin, 1960. 6. Cox, R. G., J. FluidMech. 44, 4, 791 (1970). 7. Platikanov, D., Nedyalkov, M., and Scheludko, A., J. Colloid Interface Sci. 75, 2, 612 (1980). 8. Scheludko, A., Chakarov, V., and Toshev, B., J. Colloid Interface Sci. 82, 1, 83 (1981). 9. De Feijter, J. A., and Vrij, A., ElectroanaL Chem. Interfacial Electrochem. 37, 8 (1972). 10. Vignes-Adler, M., and Brenner, H., J. Colloid Interface Sci. (submitted, 1985). 11. Weatherburn, C. E., "Differential Geometry of Three

Journal of Colloid and Interface Science, Vol. 103, No. 1, January 1985

12.

13.

14.

15. 16. 17.

Dimensions," Vol. I. Cambridge Univ. Press, Cambridge, England, 1930. Aris, R., "Vectors, Tensors, and the Basic Equations of Fluid Mechanics." Prentice-Hall, Englewood Cliffs, N. J., 1962. Levich, V. G., "Physicochemical Hydrodynamics," p. 390. Prentice-Hall, Englewood Cliffs, N. J., 1962. Truesdell, C., and Toupin, R., in "Handbuch der Physik" (S. Fliigge, Ed.), Vol. III/1, p. 533. Springer-Verlag, Berlin, 1960. Pethica, B. A., J. Colloid Interface Sci. 62, 3 (1977). Kerins, J., and Widom, B., J. Chem. Phys. 77, 2061 (1982). Rowlinson, J. S., J. Chem. Soc., Faraday Trans. 79, 77 (1983).