Differential geometric theory of capillarity

COLLOIDS AND ELSEVIER

Colloids and Surfaces A: Physicochemicaland Engineering Aspects 114 (1996) 1 22

A

SURFACES

Differential geometric theory of capillarity J. Gaydos Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, Ont. KIS-5B6 Canada Received 10 January 1996; accepted 20 February 1996

Abstract

Interfacial physics is a rich area of study with many practical manifestations and significant complexity inherent in the underlying two-dimensional behaviour. For example, structures formed from aggregates of self-assembled amphiphiles may show a variety of forms and properties ranging from ordered arrays of micelles to disordered, bicontinuous microemulsions. Any theoretical study of this behaviour must begin with a characterization of both the shape and energetic state of the interface. Often one has terms in the energy that depend on both the area (e.g. surface tension) and the curvature. We present a review of the various "curvature measures" that have historically been employed to evaluate the degree of surface bending, and their relationship to both the generalized theory of capillarity and the form of the corresponding equilibrium conditions (e.g. the Young-Laplace equation of capillarity). Keywords: Capillarity theory; Curvature measures; Differential geometry; Line tension; Young-Laplace equation

1. Introduction

As developments in the characterization of microemulsions, micellar solutions and mesomorphic phases advanced there was a corresponding need to explain the variations in properties and phase behaviour that accompanied these systems. An integral part of the approach to explain various effects was a correct formulation of both the fundamental equation for the surface free energy and the form of the Young-Laplace equation of capillarity. Even in situations where one is concerned with substructure models of multilayer surfactant or biological films, an appropriate choice of "curvature measure" is important if the transition zone between two adjacent bulk phases is to be modelled properly as either one or m a n y two-dimensional surfaces. In this review article we (i) survey the historical insight that lead to the original explanation of the 0927-7757/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved PH S0927-7757 (96)03591-1

connection between interfacial bending and the pressure j u m p across a surface; (ii) provide a detailed account of the surface-excess concept and its role in the proper definition of a dividing surface fundamental equation; (iii) formulate the basic variational problem that characterizes all capillary systems; (iv) discuss the important role of the free energy formulation in capillary variational problems; (v) survey alternative curvature measures; (vi) contrast the form of alternative curvature measures with the attempts by Gibbs and other workers to develop a non-moderately curved capillary theory with a corresponding generalization of the classical Young-Laplace equation (see Table 1); (vii) provide a generalization of the classical Young-Laplace equation that is completely second-order in the principal curvatures (see Eq. (78)); and (viii) provide a numerical example of the type of solution one would obtain for the shape of an axisymmetric sessile drop influenced

2

J. Gaydos/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1 22

by only first-order bending energy effects (see Fig. 1).

2. The historical Young-Laplace equation The study of phenomena that are, in some manner, influenced by the presence of a liquidfluid interface is as old as the first recorded observations of water rising in a small capillary tube by Leonardo da Vinci (1490). (It seems only appropriate for a workshop held in the region of Tuscany, Italy that we acknowledge the contributions of da Vinci to yet another field of human endeavour.) Much later, Honoratus Fabry (1676) found that the height of water rise in a glass capillary tube was inversely proportional to the tube's radius. The first measurements of capillary rise were by Francis Hawksbee (1709) and a physician by the

name of James Jurin (1719). They attributed the water rise to an attraction between the glass and the water. The important concept of interfacial or surface tension was introduced by J.A. Von Segnar (1751) who ascribed the surface tension to attractive forces of extremely short range between different, but adjacent, portions of the liquid. John Leslie (1802) demonstrated that the attractive force between the glass wall of the capillary tube and the thin layer of liquid in contact with it could be both normal to the tube wall and responsible for the capillary rise. However, it was not until the significant investigations of T. Young and P.S. Laplace that a proper formulation of the phenomena was established. Young (1805) proposed a theory whereby two forces, one attractive and the other repulsive, acting between fluid bodies in a surface were responsible for the existence of interfacial tension. Subsequently, he concluded that

0.00 J/m -0.05 ~

'

k "~

Ill -0.I0

~0~ I ~

/ j J

-0,20 0.00

= lO-7Jlm

CI = 10-12JIm

,

i 0.10

,

I 0.20

0.,.30

X Fig. l. Influence of the bending moment Ca > 0 on the profile of a (dimensionless) sessile drop. The (X, E) origin corresponds to the apex or top of the drop. The complete drop surface for any profile curve is obtained by revolving the curve in question about the vertical E axis. The volume Y~"J equals the region enclosed by the drop surface and a horizontal plane which intersects the contact line (i.e. end-point of the profile curve) of the drop. All curves are obtained by commencing the integration at the origin and progressing until they enclose the same volume, i.e. Y') =0.0041473434. Consequently, each profile curve terminates at a different end-point and at a different contact angle.

3". Gaydos/Colloids Surfaces .4: Physicochem. Eng. Aspects 114 (1996) 1-22

on a curved surface, a net force, proportional to the surface's mean curvature, must act on a superficial body to force it towards the centre of curvature of the surface. Laplace (1806) obtained essentially the same result as Young, but with the important difference that he expressed the result (for a spherical surface) via the mathematical relationship 27

P=Pm+~

(1)

where the pressure P at a point in the interior of a liquid is given by the sum of a constant "molecular pressure" P,, and a term that includes both the surface tension 7 and the radius of curvature 1/R of the spherical surface. Even though Laplace's initial assumption about the density being uniform within the transition zone was wrong, Poisson (1831) was able to demonstrate that the form of Eq. (1) remains unaltered [1,2]. J.W. Gibbs (1876/8) created a "pure statics of the effects of temperature and heat" [3]. His application placed the static equilibrium behaviour of the transition zone on a sound conceptual basis while demonstrating that Eq. (1) applies for a "moderately curved dividing surface" [4] representation of the interfacial zone. Many subsequent descriptions of capillary phenomena have relied upon the form of the Young-Laplace equation of capillarity, i.e. AP = ?J (where AP is the pressure difference across an interface that separates adjacent bulk phases and J is the mean curvature), to properly characterize both the static balance of forces across an interface and the interfacial linear momentum. (We shall define the mean curvature as the sum of principal curvatures, i.e. J = cl + c2. This definition is found in many of the older texts, such as Ref. [5].) In this article, we shall review the various "curvature measures" that have been employed to evaluate the degree of surface bending and their relationship to both the generalized theory of capillarity and the form of the corresponding equilibrium Young-Laplace condition. We shall not examine the appropriateness of employing Gibbs' "surface-excess" approach to the transition zone from a molecular point of view [ 6 - 1 8 ] nor explain

3

in any detail sub-structure models of multi-layer surfactant or biological films [19-45], nor be concerned with the dynamic behaviour of the transition zone [46-80]. However, for most of these situations, an appropriate choice of the "curvature measure" is still important if the transition zone is to be modelled as either one or many two-dimensional surfaces. The references listed in this paragraph have been included to provide a sense of the scope of activity rather than a complete, historically accurate listing of relevant publications.

3. The excess concept and the planar surface fundamental equation To develop a generalized thermodynamic formalism which takes into account the interfacial regions (i.e. the confluent zones between two fluid phases), it is convenient to start by considering the smoothed volume densities of the internal energy, entropy and mass of the ith component, i.e. u (v), s (v) and p!V), throughout the whole fluid system. In any equilibrium state, the actual densities u (v), s t~) and p~) will be functions of position. They will vary slowly through each bulk phase because of the influence of body forces like gravity but may vary quite rapidly across each interface. Using the methodology developed by Gibbs, we may represent each interface by a single mathematical surface or dividing surface. To avoid the "empty" spaces between the bulk fluid phases which result from this reduction of the interface to a strictly twodimensional boundary, it is necessary to extrapolate the bulk properties from the interior of each fluid phase right up to the dividing surface. The extrapolations are performed such that the densities u (~), s (v) and p~) on either side of the dividing surface conform with the bulk fundamental equations and with the influence of gravity; however, they are uninfluenced by the proximity of the other bulk phase. As a consequence, one may define the excess quantities u(4v), s(a~) and p~i) by the relationships

u]~)(r) = u(V)(r)- u~)(r)

(2)

S~v) (r) ---- s (v) (r) -- s ~ ' ( r )

(3 )

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J. Gaydos/Colloids Surfaces A." Physicochem. Eng. Aspects 114 (1996) 1-22

and p~)i (r) = plV~(r) - p ~ ! (r)

(4)

where r represents the position vector and the subscript infinity symbol indicates an idealized bulk phase density based on an infinite bulk phase without surfaces. In other words, the actual densities u(v), s ~) and p~V) will, in general, be different from the ideal densities u~ ), s~ ) and p ~ that one would have for an exclusively bulk system that is uninfluenced by external boundaries or forces. These excess volume densities (i.e. the actual volume density at location r in excess of the extrapolated one) are zero outside or sufficiently far away from the interface. Integration of these excess quantities u(v~, s(~~) and p~,-)along a path which is directly across the interface yields the complete or total excess amount for the interface at that location. These total excess quantities, which are attributed to the dividing surface, are denoted by u ~"), s ("~ and p!~) and are commonly called the surface excess densities of the internal energy, entropy and mass of the ith component. It should, however, be realized that to a certain extent, the surface densities u (~), s ~) and p}a) depend on the integration path across the interfacial zone and they must be evaluated at an interface location which is sufficiently far away from any contact line. The dividing surface, which is initially constructed as a geometrical surface of bulk separation, may be transformed into a thermodynamic, autonomous system governed by a suitable fundamental equation for the interface which is dependent only on excess or surface quantities. To quote Delay and Prigogine [81]: "This fundamental difference between bulk phases and surface phases is taken account of by expressing the properties of bulk phases in terms of variables relating solely to these phases, while the properties of surface phases are presumed to depend not only on the variables describing the surface, but also on the variables which define the state of neighbouring bulk phases. Bulk phases are said to be a u t o n o m o u s , while surface phases are n o n - a u t o n o m o u s . This distinction loses its importance for equilibrium states, since then the intensive variables characterizing one phase fix, through the equilibrium conditions, the intensive variables characterizing all the coexisting phases."

Ultimately, this means that we assume that it is physically meaningful to be able to discuss surface densities defined at a point on the dividing surface and that it is reasonable to treat the surface phase as a mathematical surface of zero thickness amenable to differential geometry. Thus, surface densities defined at a point in the dividing surface will be considered in exactly the same manner as volume densities defined at a point in the bulk fluid. As far as the geometric variables are concerned, the flmdamental equation for bulk phases is complete since a volume region has no extensive geometric variables (besides its volume), and hence no geometric point variables upon which the volume densities u (v) could be assumed to depend. Likewise, we require that a fundamental equation for surfaces be complete as far as the geometric variables are concerned. We do not seek any additional variables (besides u ~a), s (a) and p~) where i = 1, 2, ..., r) other than geometric ones since the corresponding properties would have to be considered also in the fundamental equation for bulk phases and the resulting theory would be more general (e.g. electrocapillarity) than presently desired. A specific density form of a fundamental equation for surfaces can now be set up by analogy with the corresponding bulk phase expression. For a planar dividing surface, we can see that a surface domain in two-dimensional space (analogous to a volume region in three-dimensional space) has no extensive geometric properties other than its surface area. Therefore, the complete fundamental equation for planar surfaces is identical to the one suggested by Gibbs more than a century ago, namely

u~"~= u~")[s~"~,PT, pT ..... p?)]

(5)

The corresponding intensive parameters, are given by

T=

l

L as~"~J~!o,~

(6)

F°u"'l

I,, = L ap?--~j,,°,,~os,a, ~ and the specific free energy T = u ~) -- Ts ~") -- ~ i=I

#ipl ")

(8)

J. Gaydos/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1-22

where ~, represents the surface tension for a planar interface. In definition (7), the subscript {P}~)i} denotes that all surface densities except the density p~a) are held constant during the differentiation. The corresponding extensive or total quantities are defined in analogy with the bulk phase definitions; however, all integrations are carried out over the surface instead of the volume. Fundamental relationships may also be developed for both linear phases and point phases. We shall denote these functions by the expressions u °) and U (°) respectively. In the remainder of this paper we shall be concerned with alternative forms for the specific surface fundamental equation and corresponding Young-Laplace equations.

4. Equilibrium conditions Any particular configuration of the total system in which the thermodynamic parameters are distributed in compliance with the fundamental equations and also in compliance with the constraints on and within the system is called a possible state of the system. In our case, this means that we maintain the total entropy and the total mass of each component in the system as a constant. Therefore, the fundamental equations determine and describe the thermodynamic states in all parts of the fluid system, while the minimum principle is a necessary condition which allows determination of the equilibrium states from the multitude of thermodynamic states allowed by the governing fundamental equations. Mathematically, the thermal, chemical and mechanical equilibrium conditions are obtained through application of the calculus of variations. Gibbs applied the criterion necessary for equilibrium of a volume region to the internal portion of a fluid system with the condition of isolation imposed by enclosing the internal portion of the composite system with an imaginary envelope or bounding wall [-82]. Following this approach, we may write the necessary condition for equilibrium of a composite system with volume, area, line and point phases as ~(Et)s~V,,v,{M~V}} = 0

(9)

5

where E t = U t -q-g2¢ represents the total internal energy and external field energies of the composite system. The expression for Ut is given by Eq. (12) below while the corresponding expression for g2, is given by Eq. (12) if one replaces u (v) with p~V)~, u {a) with p(")¢, U (1) with p(1)¢ and U I°) with M(°)¢ where ¢(r) represents the potential energy associated with the external field. The three subsidiary conditions denoted by the subscripted quantities above are necessary if one requires that the variational problem remains equivalent to the problem stated by Gibbs for an isolated composite system. In other words, an isolated system does not permit the transfer of either heat, mass or work across its outer boundary. If these restrictions are imposed on our system and on the formulation of the variational problem which accompanies the system, then we must force all dissipation processes to vanish, restrict the total mass of each species in the system to remain fixed and require that all outer boundary variations that would perform work be zero. We impose the first condition that all dissipation processes vanish in the composite system by requiring that the total entropy remains fixed. Imposition of the second condition simply requires that the mass of each species remains constant. The final boundary condition, which requires that no virtual work be possible on the outer wall, requires that 6rl{aw,A -- 0

(10)

and 6t~l(Lw.kI = 0

(11)

where {Awj} denotes the union of all internal surfaces that would intersect the bounding wall during a variation and {Lw.k} denotes the union of all internal contact lines that would intersect the bounding wall during a variation. The first condition fixes the "imaginary" bounding wall by imposing the condition that all internal surfaces remain unvaried along the bounding wall while the second condition fixes the unit normals to the dividing surfaces along all contact lines which contact the bounding wall. The outer wall may have arbitrary shape; however, to insure that the total internal energy Ut is unambiguously determined, it is necessary to place

6

J. Gaydos/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1-22

certain geometric constraints on the manner in which internal surfaces, lines and points contact the outer wall. Specifically, it shall be required that no portion of a dividing surface, with the exception of its boundary lines or points (i.e. no amount of its area), lie on the outer wall. In addition, it shall also be required that no segment of a dividing line, with the exception of its end points (i.e. no amount of its length), may lie on the outer wall. Finally, it is necessary to require that a dividing point may not be an outer wall point. If any of these conditions are violated, then one would obtain a constrained variation, or the mechanical equilibrium conditions for the dividing surfaces, lines or points would be connected to the geometric shape of the imaginary bounding surface of the composite fluid system. The total energy is divided into parts which belong to the bulk, surface, line and point regions of the composite system. If the total number of bulk phases, dividing surfaces, dividing lines and dividing points inside the composite system are denoted by the symbols Vkl, Ak2, Lk3 and Pk4, respectively, then it is possible to write the total internal energy of the system as

Vklf;;

Ut = ~ k=l

Vk

Ak2;;

u (~)dV+ ~ k=l

u~ d A

Ak

Lk3 I Pk4 + ~ u(1) d L + ~ [U(°)]p k k=l k=l Lk

(12)

where Vk denotes a particular volume region with a particular specific internal energy out of a total Vkl volume regions which contribute to the composite system. Likewise, A k, Lk and Pk denote particular dividing surfaces, lines and points. The k subscripts on the symbols Vkl, Ak2, Lk3 and Pk4 acquire values, in general, such that kl # k2 # k3 # k4. However, these seemingly unrelated quantities are in fact connected by a topological or combinatorial quantity )~ which is called the Euler characteristic [83-87]. (For any compact surface in three-dimensional space, the Euler characteristic Z is related to the geometric genus of the surface gs by the relation Z = 2 ( 1 - g ~ ) . Furthermore, if the surface can be segmented and

represented by a large number of regions or patches, then the number of vertices Ps4, edges Ls3 and patches A~z are related to the Euler characteristic by the expression X = A s 2 - L~3 + P~4. A surface which is representable in this fashion is known as a differential geometric surface.) Upon solution of the variational problem Eq. (9) one finds that the condition of thermal equilibrium in isolation is T=T

13)

Physically, this states that the equilibrium temperature T is the same in all bulk phases, dividing surfaces, and linear regions. Similarly, considering the chemical components to be independent (with no chemical reactions permitted), one finds that the conditions of chemical equilibrium for each component are

ill + ~b=/~i

for i = 1, 2..... r

(14)

throughout the system, where #i are the equilibrium chemical potentials of the chemical constituents of the system at the reference surface, ~(r) = 0. In addition to the thermal and chemical equilibrium conditions that are given by Eqs. (13) and (14) there are mechanical equilibrium conditions that also arise for liquid-fluid interfaces (i.e. dividing surfaces), and for three-phase dividing lines. When the surface fundamental equation has the functional dependence indicated in Eq. (5), then the condition of mechanical equilibrium for each dividing surface is given by the classical form of the Young-Laplace equation [4]. Alternative expressions or models for the transition zone yield different forms of the Young-Laplace equation.

5. The free energy representation A thermodynamic investigation into the equilibrium of any system begins with the selection of a suitable thermodynamic fundamental equation and the appropriate equilibrium principle or condition. The fundamental equation describes the thermodynamic states in all parts of the fluid system, while the minimum principle determines only the equilibrium states possible from the multitude of thermodynamic states permitted by the fundamen-

.L Gaydos/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1-22

tal equation. Various forms or representations of the minimum principle and the fundamental equation are possible. The connection between the various expressions of the fundamental equation (i.e. the thermodynamic potentials) is performed by means of a mathematical technique known as a Legendre transformation [88-92]. Using this technique, parameters defining the fundamental equation may be replaced by their corresponding intensive quantities. Therefore, in essence, it becomes possible to design the thermodynamic formalism so that parameters like the entropy, volume, or interfacial area, which are not easily manipulated experimentally, may be replaced by quantities like the temperature, pressure, and surface tension which are much easier to control. We shall consider some of the alternative Legendre transformed versions of the fundamental equations for capillary systems. As noted by Callen [88] the energy formulation (internal energy plus gravitational) is not really suited for capillary systems because the representation does not take advantage of the thermal equilibrium present in the system (i.e. the temperature is constant throughout and is known). The next thermodynamic potential to consider is the Helmholtz function. In this representation, the entropy as an independent variable is replaced by the temperature, which is kept constant throughout the system. The Helmholtz function is "admirably" [93] suited to assure thermal equilibrium since the search for configurations that are at complete equilibrium is reduced to the identification of configurations that already are at thermal equilibrium. However, the equilibrium principle for the Helmholtz function still requires fixed component masses inside a fixed system volume which eliminates the possibility of considering open systems. If the Helmholtz function is used, the desired constant pressure within each phase and the composition of the phase can only be obtained indirectly. The next thermodynamic potential, the Gibbs function, is rejected immediately for capillary systems because it requires that each pressure be controlled by a pressure reservoir [94]. This is impossible for a small bubble or drop phase surrounded by another larger fluid phase since it is obvious that the smaller phase does not have a

7

pressure reservoir. At this point, the well-known thermodynamic potentials have been exhausted. Thus, to no surprise, it is the Helmholtz function that is usually selected when treating capillary systems. Conceptually, the relevant Legendre transformations have not really been exhausted because neither the Helmholtz nor the Gibbs potential considers the possibility of changes in mass or mole numbers, and hence the possibility of chemical equilibrium with one or more components, expressed by the equality of the chemical potentials. Thus, the thermodynamic potential in which the independent variables "entropy" and "mass" of the individual chemical constituents are replaced respectively by the temperature and the chemical potentials is a suitable fundamental equation for investigating capillary systems. This thermodynamic potential, often called the grand canonical potential and denoted by ~, does not seem to have been used much in the field of thermodynamics (Gibbs refers to it once without a name), although it is well known in statistical mechanics [95]. When it comes to capillary systems, there are many instances of either the Helmholtz or the Gibbs functions being used in applications where the free energy or grand canonical potential would have been far more suitable and appropriate. (However, for closed, isothermal surface systems such as red blood cells the Helmholtz function is employed with the side constraint or condition that the surface mass remain fixed, usually stated as the requirement that the surface area remain constrained [35,38].) Consequently, since the conditions of thermal and chemical equilibrium are the same throughout the system, this presents the possibility of using the conditions of thermal and chemical equilibrium beforehand to reduce the minimum (internal energy) problem described above in Eq.(12). Evidently, in the reduced minimum problem, the state of complete equilibrium is sought only among those thermodynamic states that already are in thermal and chemical equilibrium. Thus, using the equilibrium conditions which exist between the temperature and the chemical potentials throughout the system (i.e. Eqs. (13) and (14)), we may write the grand canonical potential density for the

8

J. Gaydos/Colloids Surfaces A: Physicoehem. Eng. Aspects 114 (1996) 1-22

bulk phase as ~o~) = u ~v) -- Ts lv) -- ~ plp~ v)

(15)

i

where all the quantities are to be evaluated at the equilibrium temperature T = T and chemical potentials #i=/~i-~b ( i = 1, 2 ..... r). In essence, Eq.(15) defines a Legendre transformation from the specific volume internal energy u (v~ to the specific grand canonical potential co(~)= c°(v)(T, ~q, #2 ..... #r)

(16)

which is the specific free energy representation of the fundamental equation for bulk phases which are known to be in thermal and chemical equilibrium. Expression (16) simultaneously replaces the entropy density by the temperature and the mass densities by the chemical potentials as the independent parameters in the fundamental equation. The differential form of the fundamental equation is obtained by taking a total differential of Eq. (15) and using the expression for du ~) to obtain de) (~) = - s (~) d T - ~ p~V)d/~i

(17)

i

A comparison with the Euler relationship P = Ts (~1 + ~ #ip~ ~) -- u (~

(18)

i and Eq, (15) yields oj(v) = _ p

through the given external potential ~b(r). However, to evaluate g2(v) one still needs to know the exact functional relation for the fundamental equation, The reduction of the dividing surface part of g2t can be carried out in complete analogy with that of the bulk phase. The conditions of thermal and chemical equilibrium permit one to use Eq. (5) to write that o) (") = u (") - Ts (a} - ~ l~ip~a)

which introduces the specific free energy representation of the fundamental equation for surfaces as (D(a) = o)(a)( T , /'21, ~ 2 . . . . .

~r)

(22)

The differential form of Eq. (22) is given by dcola)= --S (a) d T - - ~ p!~)d/l i

(23)

i

and, from Eqs. (23) and (8), the surface version of the Euler relationship is given by cot~ = ~

(24)

which defines the specific free energy of a dividing surface. Thus, only in the restrictive case of a flat interface, will the surface free energy o)(~ be equal to the surface tension ?. The contribution of the dividing surfaces to the total free energy function f2t becomes

( 19 ) o(A'=

which shows that the negative of the pressure in a bulk phase is the expression for the specific free energy. Alternatively, the quantity oJ(v) dV = -PdV may be interpreted as representing the work done on the bulk system when there is an associated volume change d V. The contribution of the bulk phases to the total free energy (2t is then written as

V

(21)

i

(20)

This expression is considerably reduced in the sense that the independent functions of co(~) which remain in the integrand of Eq. (20) are known so that oJ(v) becomes a known function of position

; C co(a)(T, f~ - - ¢ , # 2 - - ¢ . . . . . fir-~b) dA A

(25)

where the integrand becomes a known function of ~b(r) on each dividing surface in the system. Once again, the functional expression for co(a)(r) remains unknown. Reduction of the total free energy (2t into its separate geometric contributions when the system also contains linear and point phases follows directly and in an analogous manner to that of the bulk and surface phases discussed above. After a suitable reduction, the total free energy f2t remains a thermodynamic potential with the same extremum properties (yielding the same solution) as any other suitable thermodynamic potential. Mathematically, the difference between the total

J. Gaydos/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1-22

energy and the total free energy extremum formulations is that the constraints in the first definition (namely that the total entropy and masses remain constant) are replaced by the subsidiary conditions T = constant and /~ + ~b-- constant in the second definition such that both problems yield the same solution. The transformations between such conjugate extremum problems are known as involutory transformations [96]. Finally, the advantage of employing the free energy O t is that there is a direct connection between the variation ( ~ t and the virtual work. The modified free (free of constraints) energy integral is given by the expression -('2t = U t + ~'24~-- )cS t - - ~

(26)

};iMti

i=1

where 2 and 2i are the Lagrange multipliers for the total entropy constraint and the ith component total mass constraint. Any variation of the total free energy £2t, together with the boundary conditions (10) and (11), is handled as an unconstrained problem. The Lagrange multipliers can be evaluated from the boundary conditions. The final equilibrium conditions are obtained by eliminating the Lagrange multipliers using the constraint conditions that the total entropy and the total mass of each component must remain fixed. Accordingly, the variation of the total free energy can be written as Vkl

Ak2

k=l

k=l Lk3 Pk4 -1- 2 ~'2(1)'4- 2 (~¢~(k0)= 0 k=l k=l

(27)

where

(28)

Vk

where ~5(v)= u (v) + m~vl - 2s (v~- ~

2ipl~)

Ak

where ~ta) + t/(a) + o)~a) __ /~s(a) __ ~ ~ip!a) i=1

f~(k1)= f 6)In dL ,]

L~

(29)

(30)

where o5°~ + u (l~+ co~)-- 2s m -- ~ 2ipl l~ i=l

and n(o) _-- u(o~ + ~ o ) -- As(o) _ ~ ~iJVli - .-(o) i=1

(31)

The solution of the variational problem posed by Eq. (27) will depend critically upon the choice of parameterization and upon the generality of the functional expressions which are adopted for the free energies co(v), co(a), co(l) and ~(o).

6. Free energy and alternative curvature measures

The earliest attempts at solving problem (27) (i.e. determining the mechanical equilibrium conditions that would render the integrals stationary) usually considered a capillary system as a composite system of at most three bulk phases with three surface phases and one contact line of mutual intersection. Any mobile interface that existed between adjacent deformable bulk phases was considered to possess an energy that was proportional to the surface area of the interface. In virtually all cases, this proportionality factor was treated as a constant or uniform tension on the surface. The only real exception to this state of affairs, until the studies of Buff and Saltsburg [97-102] and Hill [103], was the impressive fundamental capillarity work of Gibbs [ 104]. The mechanical equilibrium condition for the surface that arises from the solution to the variational problem simplifies approximately to a problem which renders the area of the interface a minimum, or

6ffdA=O

i=1

9

(32)

When solved, this problem yields a minimal surface of negligible thickness and mass whose mean cur-

10

J. Gaydos/ColloidsSurfacesA: Physieochem.Eng.Aspects114 (1996) 1-22

vature J = c1 + c2 vanishes. If the surface bounds a phase of fixed volume, a constraint must be added to the variational problem; that is

off dA-fff ,dV=O

(33)

where AP represents the Lagrange multiplier for the fixed volume constraint. A formulation based on Eq. (33) leads to a surface of constant, but not vanishing, mean curvature and a Young-Laplace equation of the form AP = ~J. The unique properties of these surfaces with either fixed or zero mean curvature soon captivated the imagination and interest of many mathematicians. In both cases, the problem was restricted by fixing the position of the boundary so that no boundary conditions occur and by excluding the constraint of fixed volume. In addition, alternative surface integral expressions such as

bffj2dA=O

(34)

designed by Poisson in the nineteenth century to characterize the potential energy of a membrane started to appear [105]. Another example, was provided by Casorati (1889) [ 106]

6ff(j2-2K)dA=O

[35)

where K = clc2 is the Gaussian curvature. It might be argued, as was done by Nitsche [107] that a more appropriate surface integral to investigate would be

6ff''dA=O

(36)

where co~) denotes a positive, symmetric but not necessarily homogeneous function of the curvatures J and K, that is, o~~) = co(~ (J, K ). Polynomial examples are o~~1 = a + bJ 2 - cK, with both b and c much less than a [107] or ~o(a)= b ( J - Jo)2+ cK [108]. If ~ol~)=~P(J)-cK, then the EulerLagrange equation, which is a necessary condition for the variation of the surface integral to vanish,

is given by [107]

Abel+

(SZ--2K)~-STt

=0

(37)

where Ab denotes the Beltrami-Laplace operator [ 109]. For the special case co~a~= j2 the differential equation (37) reduces to J

AbJ + ~ ( j 2 - - 4 K ) = O

(38)

and was derived by Schadow, referred to in Ref. [ 110]. Regardless of what particular expression is adopted for the surface energy, the EulerLagrange equation for the variational problem

fiff~,.¢"'(J,K)dA=O

(39)

is lengthy and involves the fourth-order derivatives of the position vector for the surface. Recent mathematical investigations have centred on the expression (34) and its higher dimensional extensions. The case of surfaces with non-fixed or free boundaries "requires the discussion of appropriate boundary conditions and has not attracted much attention so far" [ 107,111 ]. Recent extensions and elucidations of Gibbs' and Buff's efforts, which consider non-fixed boundary conditions with volume constraints, by Murphy [112], Melrose [113-115] Hoffman and Cahn [116], Cahn and Hoffman [117], Helfrich [108] Boruvka and Neumann [118], Scriven [119], Benner et al. [120], Rowlinson and Widom [121,122], Alexander and Johnson [123,124], Shanahan and de Gennes [125], Shanahan [126-128], Neogi et al. [129], Neogi and Friberg [130], Markin et al. [131] and Kozlov and Markin [132], Povstenko [133], Kralchevsky and co-workers [ 134-139], Eriksson and Ljunggren [ 140,141] and Ljunggren et al. [142], have been primarily directed at the determination of the appropriate mechanical equilibrium conditions across a surface (i.e. the Young-Laplace equation) and at a contact line boundary (i.e. either Young's equation or Neumann's triangle relationship) for quite general differential geometric surfaces. However, a certain amount of contention among these investigators

J. Gaydos/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1-22

has occurred over the particular functional expression that one might expect for the free energies.

7. The non-moderately curved surface of Gibbs

Part of the difficulty with the selection of a suitable curvature measure to describe surface bending stems from the original suggestion of Gibbs [104] that one considers an area Aa to be regarded as sufficiently small to be uniform throughout with respect to its curvature and in respect to the state of the surrounding matter so that the expression for the variation of the surface energy will be determined not only by the variables in Eq. (22), but also by the variations of its principal (orthogonal) curvatures 6c~ and 6c2, such that 612 (A) = 75A + CI (~C1 + C2(~c 2

~

yfA +

l ( C 1 -4- C2)(~(c 1 -]- c2)

-~ ½(C 1 - C2)t~(c I - c2)

(41)

The principal curvatures are related to the principal radii of curvature by the relationships Rx = 1/Cl and R2 = 1/c2, respectively. They are assumed to be uniform on the surface piece Aa. The variables C1 and C2 represent the energy inherent in the bent, non-planar surface. Gibbs never provided an expression for his form of the dividing surface fundamental equation, but proceeded immediately to show that it is possible to select a position for the dividing surface where higher-order bending effects are insignificant. However, if we proceed towards a fundamental equation and write down the form of Gibbs' expression, we would obtain, in the free energy notation, the relationship I2 (a) = g2(a)(T, A, #i, cl, C2)

(42)

The other equilibrium expressions implied by Eq. (42) are d,,Q (A) =

where Eq. (43) is integrated at constant principal curvatures to yield Eq. (44) since both principal curvatures are intensive. Gibbs did not expand upon the physical meaning of the quantities Cx and C2 but proceeded immediately to eliminate any consideration of these terms by shifting the dividing surface to the surface of tension position defined by the condition Cx + C2 = 0 [143]. It is not surprising that he eliminated these dependences on the curvature almost immediately since he was primarily interested in investigating the effects of capillarity for systems which are "composed of parts which are approximately plane" [ 144] or for those common situations in which "our measurements are practically confined to cases in which the difference of the pressures in the homogeneous masses is small" [145]. Furthermore, from the partial derivative

(40)

or

6Q (a) =

'?' dA + C1 dci + C2 dc2

(43)

and g2(A) = yA

11

(44)

Jr,.~,,~ - L

aA

Jr,,.~,.~ =

it may be seen that this quantity represents the average specific free energy density of the interface (sometimes referred to as the average specific grand canonical potential) or 09ta) and not the surface tension ~ unless the surface is planer. In addition, it was made quite clear by Gibbs that [ 146] "The value of ~ is therefore independent of the position of the dividing surface, when this surface is plane. But when we call this quantity the superficial tension, we must remember that it will not have its characteristic properties as a tension with reference to any arbitrary surface. Considered as a tension, its position is in the surface which we have called the surface of tension, and, strictly speaking, nowhere else." In the current vernacular, this means that o9~") = only at the surface of tension position where ), is a pure tension. At any other position, the equality between cot~ and ~ will not hold since the specific free energy of the surface will also contain energetic curvature contributions. Throughout his analysis, Gibbs was very much aware of the constraints which he imposed on his formalism. Thus, when he considered, for instance, the surface tension y, he was very careful to distinguish between its value at the surface of tension and its value at any other

12

J. GaydoslColloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1-22

dividing surface location. It would seem quite apparent that Gibbs had no intention of generalizing his analysis beyond capillary systems with moderate curvatures. (It would seem reasonable to conclude that Gibbs was aware of invariant curvature measures, namely the sum of curvatures c1 + cq and the product of curvatures ci, c2 based on his text [ 1471.) The specific point is that one should employ a definition of the “superficial tension” or specific surface free energy that yields a quantity which is a pure tension at the “surface of tension” dividing surface location. At any other position of the dividing surface, the quantity y remains as a pure tension so that there is a distinction between the surface tension y and the specific surface free energy w(~). In the discussion which follows on non-uniformly curved interfaces, we shall express the specific surface free energy as a simple summation of Gibbs’ planar surface tension y and a symmetric function of the curvatures J and K. This approach will simplify the formal thermodynamic description enormously. One primary advantage of writing cJa) as W@)= y + w&J, K)

(46)

is that the surface tension portion, i.e. y, of the expression remains unambiguously in its definition while remaining identical to the commonly measured experimental quantity (for planar or nearly planar surfaces).

8. Developments

after Gibbs

The next significant developments after Gibbs were those of Buff [97-99,102], Buff and Saltsburg [ lOO,lOl], Hill [ 1031, and somewhat later Murphy [ 1121 and Melrose [ 113-l 151. Collectively, they provided a fundamental extension to Gibbs’ original work and a potential reinterpretation of his surface free energy expression to dSZcA’= y dA + A((?, dc, + CZ dc2)

(47)

where the bending moments C, and C, were assumed constant, In the general case, one could imagine situations where the bending moments C1 and C, could vary from point to point on the surface. Perhaps, a

variation could arise from inherent inhomogeneities in the properties of the system or it could result from modifications induced by the effects of the contact lines on the surface. Regardless of the particular cause, the adjustment to Gibbs’ original work was to assume that Ci dA = C, A

C2 dA = C,A

and

.i‘i

ss (48)

In this form, three distinct consequences arise: (i) It becomes impossible a priori to consider systems which may possess gradients in the bending moments, i.e. VZC1 and VZC2 are both zero by definition along the surface. (ii) For a given temperature T and chemical potential pi state, the bending moments are related by (49) which implies bending moment symmetry in the principal directions. (iii) The Young-Laplace equation acquires the form [97] Al’ = y(q + c2) - c, cf - c,c;

(50)

where we have introduced the symbol 7 to indicate that this quantity is not equivalent to the planar or nearly planar surface tension that is commonly measured (see Table 1 for a comparison of definitions). For a spherical surface (i.e. c1 = c,), a rather unique situation arises because the bending moments are equal; i.e. C, = C, = C by Eq. (49) and the Young-Laplace equation simplifies to AP = y,(cl + cJ - c(c: + c;)

(51)

or dP=~J-C(JZ-2K)

(52)

In this situation, some authors have suggested that “this is the special case of a plane interface” [ 1481 suggested by Gibbs [ 1493 because the “unrigid dividing surface (C = 0) coincides with the surface of tension” [ 1481. Subsequently, it was suggested that Eq. (52) was “not accurate for highly curved

J. Gaydos/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1-22

regions" [ 150] and that for a transversely uniform interface the appropriate surface excess energy should be given by [ 151 ] d O (A) =

y dA + A(Cj dJ + tJr dK)

(53)

where the two intrinsic and invariant (unlike the difference c , - c2 expression in Eq. (41)) surface curvature measures are the mean curvature J = c1+c2 and the Gaussian curvature K = c l c 2. Using our notation, the bending modulus Cj and the torsional modulus Cx are defined by the integral expressions

ffC, dA=CjA

(54)

f CK dA

and without self-intersection (e.g. Klein bottles are excluded) the statement in Eq.(57) applies. Unusual but acceptable surfaces include both Schwarz's and Neovius' periodic minimal surfaces which partition space into two equal, infinitely connected, interpenetrating sub-volumes [6,119]. Mathematically, the condition stated in Eq. (57) permits one to conclude that the three variational expressions

6ff(cl+c)dA=O

(58)

a f f +c)dA=O

(59)

and

and

f

13

CKA

(55)

which yields the corresponding generalization of the Young-Laplace equation in the form [ 112] AP = ~J - Cs(J z - 2K) - C ~ J K

(56)

In this form, two distinct consequences arise: (i) as in the case above, it remains impossible to consider systems which may possess gradients in the bending moments; and (ii) it was discovered many years ago by Poisson [152,153] that the coefficient Cr should be absent from the final form of the EulerLagrange equation that leads to the YoungLaplace equation and that this conclusion applies whether or not the surface system has a boundary or it is closed. Furthermore, by the Gauss-Bonnet theorem [84-86], it is possible to demonstrate that if the surface is an orientable, compact surface of sufficient "smoothness" or continuity then the surface integral is a topological invariant, that is ffKdA=2~z=aconstant

6ff(c,--c2)dA=O

(60)

are equivalent (when the boundary is fixed) because they all provide an integrand which depends upon the square of the mean curvature and a linear term for the Gaussian curvature which does not change provided the Euler characteristic does not change. In their sequence of papers [118,154-156] Boruvka et al. and Rotenburg et al. [ 157] supposed that both Buff and Melrose had mixed up extensive and intensive thermodynamic quantities. To avoid any ambiguity, they explicitly defined the total mean and Gaussian curvatures as extensive curvature terms using the local mean J and Gaussian K curvatures to obtain

J=ffJdA

(61)

and

(57)

where Z is the Euler characteristic (see section 4). The important point is that the value of the integral will remain fixed provided the surface system does not change its genus. Genus is the topological property of "holeiness" such that the genus of a sphere is zero, of a torus one, etc. Thus, for surfaces that are continuous, orientable, of positive genus

If one employs these definitions, it is no longer necessary to restrict one's consideration to surface systems which have zero bending moment gradients as was necessary with both expressions used in Eqs. (47) and (53). However, in order to be able to compare the Boruvka et al. formulation to previous efforts we shall restrict our interest to the

J. Gaydos/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1 ~ 2

14

case of a homogeneous dividing surface, i.e. a dividing surface where both V2C1 and V 2 C 2 a r e zero along the surface. With this restriction, the corresponding differential and integrated forms are dr2 (a) = dU ~A~- T dS ( a ) -

E #1dM! A~ i

= ~ dA + Cs d J + CK d ~

(63)

and g'2¢a) = U~A) -- TS¢AJ - E #iM~A) i

(64)

= 7A + C s J + C K ~

and the Young-Laplace equation is given by (65)

A P = ~J + 2 K C s

If one had not assumed that the surface mechanical potentials 7, Cs and CK were constant along the dividing surface, then the condition of mechanical equilibrium across each dividing surface would be [118] AP=~J + 2KCs-VZzCs-KV

* "(V2CK)

(66)

where V2z and V* are surface differential operators [5,118]. The corresponding definition for the specific or density form of the surface free energy, based on the fundamental equation o9(~)= ~o(a)(T, #1, #2 ..... #,, J, K )

(67)

is straightforward and given by [-154 156] ~o~) = u ~ - Ts (") - ~ #~pl ~) = ? + C j J + CKK i

(68) and, in differential form, by d¢o(a) = - s (~ d T -

~ p~a) d# i + Cj dJ + CK d K i

(69) where all quantities are defined locally on the dividing surface. When the surface is planar or nearly planar, the expression for ~o~al simplifies to that given in Eq. (24). For a non-uniformly bent surface, Eq. (68) represents the energy required to bend a planar surface using an approximation that includes the first two differential invariants of the surface. The form of the expression in Eq. (68) is not unlike the energy density expressions obtained

per unit area of the middle surface of a plate or shell [158]. In both cases, an integral across the middle surface yields two terms, one proportional to the mean curvature J and another proportional to the Gaussian curvature K. One may also consider the bending energy ~o~a I = C J J + C K K to represent the energy required per unit area to bend a surface away from a planar reference surface/configuration. Thus, the surface tension y is a measure of the change in free energy with change in area at constant mean and Gaussian curvature. The bending moment Cs is a measure of the change in free energy with change in mean curvature at constant area and Gaussian curvature and will be of importance when the excess pressure distribution is an odd function about the reference dividing surface [156]. This might be imagined to occur for surface-active long-chained molecules which are non-symmetric about their midpoint when present at an interface. The second bending moment CK is, from the mechanical point of view, the second moment of the excess pressure distribution about the reference dividing surface [156] and represents a change in free energy with change in Gaussian curvature at constant area and mean curvature. Under these conditions, a straight forward calculation shows that changes in the Gaussian curvature 6 K are equal to either - 1 / 4 ( 6 D 2) where D is the deviatoric curvature (defined below) or - 1 / 4 1 6 ( H 2 - K ) ] where H = J/2. Alternatively, if one had selected the expression o~,a) oc j 2 _ 4K or o9~a) oc H z -- K initially, one would have found that this quantity is zero for all spheres and spherical caps. As a consequence, the free energy co~a~would represent the energy required per unit area to bend a surface away from a spherical reference configuration (e.g. spherical microemulsions or symmetric bilayer membranes) and the systems would be characterized by either their surface area or enclosed volume [ 159]. As with all the previous relationships, i.e. Eqs. (50), (52) and (56), relationship (66) expresses the balance which exists in equilibrium between the internal surface forces and the forces external to the dividing surface (namely, the pressure difference) when gravitational effects are not present. However, unlike previous relationships, Eq. (66) is not restricted to uniformly curved systems because

15

J. Gavdos/ Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1-22

the curvature potentials Cj and CK are local variables rather than global averages; compare the average definitions in Eqs. (48), (54) and (55). In 1990, the first of a long series of papers [134-142] based on the work of Kralchevsky [135] commenced. The work is a blend of the original curvature expression of Gibbs and the explicit extensive total curvature definitions of Boruvka et al. In particular, Kralchevsky and co-workers opted to define the surface mechanical work per unit area, our specific free energy e)~a~,as y d~ + B d H + 0 dD

d¢o ~a) =

the more complicated relationships [ 141] (73)

B = Cs + 2CKH

and (74)

0 = --2CKD

have been derived which seem to combine the definitions of the bending moment or curvature potential with the local curvatures. This coupling of curvature and bending moment definitions is not necessary if the obvious definition 2B = Cj is used in combination with the definition.

(70)

(75)

= ~ + BH + OO

where

Eq. (13) of Ref. [141] should read

d(zJa) Aa

(H 2 + D 2 ) B + 2 H D O = Cj(2H 2 -- K ) + 2 C K H K

is the relative dilation of the area element Aa of the dividing surface, H = 1/2(cl +c2), B is the associated bending moment, D = 1/2(cl -- c2) is the deviatoric curvature and O is the associated bending moment. Their expressions are defined locally so that their bending moments, i.e., B and O, are not necessarily uniform across the surface. Under similar "homogeneous dividing surface" assumptions to those used to simplify Eqs. (66) to (65), their form of the Young-Laplace equation is given by [135,141] A P = 2 ~ H - B ( H 2 + D 2) -- 2 0 H D

(71)

or, after using the expression K = H 2 - D 2, the slightly modified expression A P = 2 ~ H -- B ( 2 H z - K ) -

20H~/(H

z -

K)

(72)

which would seem to imply the presence of both the Gaussian curvature and a bending moment related to K (see Eq.(74) below) in the final expression, contrary to Poisson's earlier discoveries [152]. Despite the apparent similarity in form between Eqs. (69) and (70), especially in the fourth term of Eq. (69) and second term of Eq. (70), it has been claimed that the bending moments are not related by simple expressions such as 2B = C s . Instead,

with a plus sign in front of the final term Substitution of this expression for ~ into Eq. (71) causes the term with O to drop out directly so that when 2B is set equal to Cj, one recovers the earlier expression (65) for the Young-Laplace equation. Thus, rather than the overtly cumbersome approach using definition Eqs. (73) and (74) it is possible to represent ~ as a Legendre transformation of ~. In Table 1 we present a brief synopsis of the relationships that exist between the various definitions, i.e. 37 in Eq. (50), 37c in Eq. (52), 37 in Eq. (56), ~ in Eq. (65) and ~ in Eq. (71), and the surface tension ? that Gibbs defined for planar or nearly planar surfaces. Substitution of either Murphy's 37 expression from Table 1 into Eq. (56) or Kralchevsky's } expression from Table 1 into Eq. (71) simplifies both Young-Laplace relationships to the form given in Eq. (65). Eq. (65) has two primary advantages: (i) it is the most compact representation; and (ii) the surface tension 7 definition of Gibbs for planar or nearly planar surfaces Table 1 Synopsis of the various superficialtension definitions Author

Equation

Expression

Buff Sphere Murphy Boruvka et al. Kralchevsky et al.

(50) (52) (56) (65) (71)

?7= ), + Clcl + C2c2 ffc = )' -}- C(Cl "~ c2) ~ = y + CaJ + CKK ~= ~ = ), + B H + OD

16

J. Gaydos/Colloids Surfaces A: Phvsicochem. Eng. Aspects 114 (1996) 1-22

occurs explicitly. However, it is not the most general expression that may be derived for a homogeneous dividing surface.

9. A general second-order Young-Laplace equation A significant body of research on the behaviour of vesicles, membranes and microemulsions has been performed considering an energetic contribution to the free energy that is proportional to ( J - Jo) 2 [19-45]. As this term is second-order in the principal curvatures, it would seem arbitrary to exclude this term from consideration while including the Gaussian curvature. Thus, we modify the expression for ~o~a) in Eq. (46) to the form o)t") = 7 + C j J + CH(J -- Jo) 2 + C K K

(761

where the bending moments (i.e. curvature potentials) C j and CK are defined in an analogous manner to their definitions in Eq.(69), and we shall denote the factor CH as the Helfrich curvature potential (Helfrich used the symbols kc for 2CH and kc for C;~) [108]. The final quantity J0 represents the spontaneous curvature. Several comments need to be made about this choice of free energy: (i) the contribution to the free energy has been limited to energetic terms up to second order in the curvature with the understanding that the inclusion of higher order terms would involve quantities such as j3, H 3, J K , H K , etc. [119]; (ii) from stability considerations one has 7/>0; (iii) based on the order of the curvature, the magnitude of the various bending moments should be related by IC~l > ICHI /> ICK[/> 0; (iv) inclusion of the third term, involving the coefficient CH, provides an energetic mechanism for describing and obtaining the shape of surface systems such as vesicles, bilayer membranes, microemulsions, etc.; and (v) for a symmetric, bilayer membrane with two identical sides Jo = 0. If one slightly modifies the variational problem in Eq. (33) to include the possibility that the surface area A could remain constant, one obtains the variational problem

, ff o' 'dA-ff, dA--fff ,dV=O (77)

where Ae denotes the surface's or membrane's lateral tension and A P represents the pressure difference across the interface. Employing the expression for cota) from Eq.(76) permits us to determine the appropriate Young-Laplace equation for a bilayer vesicle with surface area A and enclosed volume V. A tedious manipulation yields the Young-Laplace equation, at the same level of generality as Eqs. (56), (65) or (71), as [160] A P = (7 + A e ) J - CH(J - Jo)(J 2 - 4K + JJo) -- CHAbJ + 2 K C j

(78)

where Ah denotes the Beltrami-Laplace operator [109] and it should be anticipated that 5' will be approximately zero for cases where Ae is non-zero. At present, most numerical schemes for solving Eq. (78) have been limited to axisymmetric geometries under the side conditions (appropriate for vesicles) that 7 ~ 0 and C j = 0 [161-168]. Analytical investigations are likewise limited to the shapes of spheres, cylinders, a Clifford torus and its conformal transformations [169,170]. In the next section we propose a scheme for evaluating the shape of a sessile drop when one assumes that the dominant correction to the surface free energy is, to first-order, just C j J . This represents an approach that is consistent with the assumption adopted by Buff [97-99], and Buff and Saltsburg [100,101]. The pendant drop configuration may be solved in a similar fashion.

10. Numerical integration of the generalized Young-Laplace equation When the specific surface free energy ~o~") is a constant, i.e. 7, the Young-Laplace equation of capillarity is given by the relatively simple expression [171] 27 7J = Roo + A p g [ ~ ( r 4: 0) - ~(r = 0)]

(79)

which may be numerically integrated when the natural boundary condition Ro is given as input [172]. In Eq.(79), Ap represents the density difference across the interface, z = ~(r) is an axisym-

J. Gaydos/ Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1-22

metric function which defines the position of the surface, r is an independent coordinate that measures the radial distance from the axis of symmetry to a point on the surface such that ¢(r # 0) - ~(r = 0) represents an elevation difference, and 1/R o is the radius of curvature of the surface at the axis of symmetry location r = 0. However, for the case considered above where ~ ) is restricted to the particular choice coC~)= ~ + CsJ + CKK

(80)

the Young-Laplace equation, given by Eq. (65), becomes ?J + 2 C j K = Roo ~

Ro

Ro ,1

+ Apg[~(r#O)--~(r=O)]

(81)

In this particular form, the Young-Laplace equation may be numerically integrated once suitable values for the physical parameters have been selected to show the effect of the non-classical bending moments Cj and CK upon the surface shape. To avoid difficulties with vertical gradients and infinite derivatives which may arise in functions that are given by explicit equations (e.g. in the form z = ~(r)), the Young-Laplace equation was not directly integrated, as given in Eq. (81), but was converted to a dimensionless, parameterdependent expression. The relationship is derived elsewhere [173] and is given by 2(

dO

1

~

Ck)

B

N

sinO

+S--2--

(Sx0)

dY-

1 + 2d~ - -

(82)

where all lengths, including the arc-length s, are made dimensionless using the capillary constant c, defined by c-

Apg

(83)

The dimensionless lengths are given by Y

= s c 1/2

B = Ro cl/2

X = re 1/2 ~,

(84) = Z C 1/2

17

while the dimensionless bending moment constants dj and Ck are defined by ~cc C so (i = - -

~ --

CCKo

(85)

Extensive numerical calculations show that a non-zero value for the bending moment Ca causes the surface profile to flatten and the contact angle to decrease for the case of a sessile drop on a flat surface. Fig. 1 provides a typical illustration of the influence of the bending moment Cj on the profile of a (dimensionless) sessile drop (with CK set to zero). Each curve in this figure represents a distinct sessile drop profile. The complete drop surface for any profile curve is obtained by revolving the curve in question about the vertical ~ axis (the axis of symmetry). The dimensionless volume y~v) (which equals Vc3/2 where V is the volume) equals the region enclosed by the drop surface and a horizontal plane which intersects the contact line (i.e. endpoint of the profile curve) of the drop. Each profile curve terminates at a different endpoint since they each enclose the same volume, i.e. y~v~= 0.0041473434. It is also apparent from this figure that a non-zero, positive value for Cj causes the surface to resist bending and to flatten itself out subject to the constraint of fixed volume and the tendency of the surface tension ~ to pull the system into a spherical shape. Furthermore, the flattening tendency of a surface with Ca > 0 also manifests itself by reducing the contact angle which the sessile drop would form on its solid support. For the purposes of illustration, a liquidvapour sessile drop system with the following physical parameters: Ap~'V~= 103kgm-3; ~v~= 0.072 J m -E, g -----9.81 m s-2; and B~v) = 0.1 was selected for presentation. Other choices are possible. This system corresponds to water near room temperature and has a dimension (i.e. R~v) ~ 0.271 mm) that approximates the characteristic size of the sessile drops that are encountered in many laboratory situations. Integration was performed in each case until the contact angle 01 = 180 ° . Comparison of the same system with different values of Cs requires that one selects an arbitrary dimensionless volume Y~ and then finds the same volume, possibly by interpolation, for

18

J. Gaydos/Colloids Surfaces A: Physicochem. Eng. Aspects 114 (1996) 1-22

other systems with different values of Ca. In this way, it is possible to compare the effect of nonzero Cs values on the shape or profile of a sessile drop whose volume is the same in all cases (i.e. the incompressible assumption). In Fig. 1 are plotted four constant volume curves of differing Ca determined in this fashion. In each case, the volume ytv) enclosed by the curve is 0.0041473434 and the other curves are plotted from the position (0, 0) to that value of (X, ~ which enclosed the designated volume ytv). For example, the C j = 1 0 - 1 2 j m 1 curve travels from the origin to a point where the contact angle 01 = 175 °. The classical case of Cj = 0 corresponds to a sessile drop with a contact angle of 180 ° and dimensionless volume y~v~ of 0.0041474. From the sessile drop profile curves it is possible to tabulate the influence of the non-zero bending m o m e n t on the magnitude of the contact angle. For example, if the dimensionless volume y~v~ is selected as 0.0041473434, then the contact angle 01 corresponding to this volume would be 175 ° when Cs = 10-12 j m-1. At other positive values for Cj the sessile drop profile flattens and the contact angle decreases according to the results presented in Table 2. It should be realized that one is not restricted to a volume of y~v)= 0.0041473434 but could just as easily have considered a value less than this for comparison. For example, if one had selected the volume Y~V)= 0.0012919995 instead of Y~}= 0.0041473434, then the results would be given as in Table 3. Other choices of Y~V)are possible as are other sets of curves from other choices for B. As a consequence of these results, it is possible to appreciate situations in which both the line tension and the bending moment Cs may influence Table 2. Influence of Cj upon the contact angle of a sessile drop whose dimensionless volume y tv~ =0.0041473434

Cj (J m - 1)

01 (Deg)

10-12 10- 7 10 -s 1.715 × 10 -5

175.00 142.59 46.62 16.81

Table 3 Influence of Cs upon the contact angle of a sessile drop whose dimensionless volume ytv~ =0.0012919995 Cj (J m 1)

0l (Deg)

10 1-" 10 7 10 -5 1.715 × 10 -5

75.00 74.20 33.89 12.47

the measured contact angle. If there are geometric arrangements in which the radius of curvature or the system's properties are such as to permit Cj to have a perceptible influence, then an attempt to measure the line tension from a measurement of the contact angle would be ambiguous. However, in virtually all cases in which the interface is open with respect to mass transport from the adjacent bulk phases, the magnitude of Cj is expected to be quite small so that any effect would be imperceptibly small. The situation is somewhat different for interacting condensed membranes. These structures, usually formed from mixtures of diacyl-chain lipids and other amphiphilic constituents, exhibit surface cohesion with restricted surface compressibility. As a result, when the specific free energy of a membrane surface is changed (e.g. by adding electric charges or by screening charges by electrolytes) what occurs is a slight contraction or expansion in surface density until a new equilibrium configuration is achieved. The relatively closed, with respect to mass transfer, nature of these membrane structures means that if the membrane is constrained it will exhibit both resistance to area dilation and to deformations which cause bending. According to Evans and Skalak [8], "this bending rigidity is dominated by elastic expansion of one layer of the bilayer relative to compression of the adjacent layer when the membrane is curved". Furthermore, "the differential tension between layers produces a membrane torque or stress couple about contour lines in the surface" [174]. But, even for these kinds of systems Evans [174] estimates that the resistance to bending is extremely small (i.e. his estimate yields a value of Cj= 10 -ix ~tN) and that it "offers little visible resistance to deformation for vesicles with diameters greater than 1 0 - 6 m '' [175]. However, it

J. Gaydos/Colloids Surfaces A." Physicochem. Eng. Aspects 114 (1996) 1-22 should be realized that there is still a great deal of u n c e r t a i n t y a b o u t the range of m a g n i t u d e s that is possible for the b e n d i n g m o m e n t . Very recent d y n a m i c m e a s u r e m e n t s [ 1 7 6 ] at a frequency of 5 G H z indicate that this b e n d i n g (or rigidity m o d u lus) might be 103 times larger t h a n previously believed for surfactant layers in swollen lyotropic lamellar liquid-crystal phases. Even if this value for Cs h a d been used to calculate a profile curve for Fig. 1, the difference between the C s - - 0 a n d Cj = 10 -8 ~tN profiles w o u l d have been m u c h less t h a n the thickness of the lines plotted in Fig. 1. Therefore, it seems r e a s o n a b l e to assume that for relatively large, pure liquid sessile drops that the b e n d i n g m o m e n t Cs does n o t yield a perceptible effect o n either the surface profile or the c o n t a c t angle. However, for surfactant systems one m a y need to be m o r e cautious a b o u t dismissing the i m p o r t a n c e of the b e n d i n g m o m e n t .

Acknowledgements This investigation was s u p p o r t e d by the N a t u r a l Science a n d E n g i n e e r i n g Research C o u n c i l of C a n a d a ( N S E R C ) t h r o u g h g r a n t O G P 0155053 a n d by C a r l e t o n U n i v e r s i t y t h r o u g h a G R - 5 G r a n t . The a u t h o r w o u l d also like to t h a n k S.S. Chetty, who compiled Table 1, for m a n y informative discussions. I n addition, the a u t h o r has benefited from discussions with P. Chen, D. K w o k , M. M o r i a n d S. Treppo.

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