40
Langmuir 1985, I , 40-44
Free Energy Formulation of the Theory of Capillarity L. Boruvka, Y. Rotenberg, and A. W. Neumann* Department of Mechanical Engineering, University of Toronto, Toronto, Ontario, Canada M5S IA4 Received July 9, 1984. I n Final Form: October 8, 1984 On the basis of a review of the basic fundamental equation for a capillary system, it is shown that the appropriate free energy potential is the grand canonical potential. The equilibrium conditions which correspond to the minimum in the free energy are set up and a simplified derivation of the generalized Laplace equation is presented.
Introd uction This paper contributes to the development of the general thermodynamic theory of capillarity which was initiated by Buff et al.1-3 and later reformulated by the present
author^.^ The latter work was recently criticized by Rowlinson5 on the grounds that, in integrating over curvature terms to obtain the internal energy, we have mixed intensive and extensive parameters. Rowlinson states that we have ignored the fact that the energy is not a homogeneous first-order function of curvature, which makes such an integration improper. The reader of our paper4 will easily convince himself that this criticism by Rowlinson is inappropriate. Indeed, at the outset of our paper, we remove the well-known difficulty invoked by Rowlinson by introducing appropriate extensive curvature terms, so that the internal energy becomes indeed a homogeneous firstorder function of curvatures. The general thermodynamic formalism so far available4 consists of the fundamental equations in the energy representation, the energy minimum principle, and the resulting conditions of thermal, chemical, and mechanical equilibrium, including general forms of the Laplace equation of capillarity and of Young’s equation. The formulation of the fundamental equations with the associated equilibrium principle is the basis of the theory, and the equilibrium conditions are a consequence. However, the theoretical formalism is far from complete as there are other consequences and aspects which have yet to be developed. One major development, already started by Buff, is the task of linking or unifying the general thermodynamic theory with the nonthermodynamic hydrostatic approach to capillarity. Another major area, the problem of stability, presently remains untouched; it has not even been fully treated at the classical level. Besides the theoretical aspects there are many questions of an applied nature which should be addressed within the context of the general theory, for example, the interrelated questions of (a) selection of the dividing surface, (b) determination of fundamental equations for surfaces from the available surface tension data, and (c) curvature dependence of surface tension. A remarkable contribution to this latter point is the Ph.D. Thesis of C. L. Murphy,6 who formulated a thermodynamic description of systems of high curvatures and low interfacial tensions with the aim of describing the behavior of emulsions. Thus, this thesis of 1966 addresses (1) Buff, F. P. J. Chem. Phys. 1956,25, 146. (2) Buff, F. P.; Saltsburg, H. J. Chem. Phys. 1957, 26, 23. (3) Buff, F. P.; Saltsburg, H. J. Chem. Phys. 1957, 26, 1526. (4) Boruvka, L.; Neumann, A. W. J . Chem. Phys. 1977, 66, 5464. (5) Rowlinson, J. S.J. Chem. SOC.,Faraday Trans. 2 1983, 79, 77-90. (6) Murphy, C. L. Ph.D. Thesis, University of Minnesota, Minneapolis, MN, 1966.
0743-7463/85/2401-0040$01.50/0
issues that are currently of great interest. Unfortunately, the description of Murphy does indeed mix extensive and intensive parameters (see,e.g., his equation B 3.49), so that his actual results are open to question. On the other hand, his strategies for dealing with practical matters such as emulsification seem very promising and could, after corrections of the thermodynamics, provide much needed insight. When attempting to approach the above problems it has been found that an intermediate step is needed: the free energy representation and its consequences. We will show that the proper free energy is the “grand canonical potential”, 52 (Callen7). The free energy does not seem to have been used much in the field of thermodynamics (Gibbs8 refers to it once without a name), although it is well-known in statistical mechanics. While not needed in general, when it comes to capillarity the introduction of the free energy is long overdue. There are many instances of Helmholtz or Gibbs functions being used in applications where the free energy would have been far more suitable and appropriate. As an example, consider a class of two-phase capillary systems in which one phase contains a small bubble (or droplet) of the second phase either attached to a phase boundary or free (gravity neglected in the latter case). The larger phase is kept at constant temperature, pressure, and composition either directly, through contacts with appropriate reservoirs, or indirectly, by using a rigid adiabatic enclosure that is sufficiently large such that the appearance and growth of the second phase will not affect the first phase appreciably. Note that a rather large class of capillary systems is thus being considered (homogeneousand heterogeneous nucleation systems, capillary condensation systems, some of the model systems for investigation of contact angle hysteresis, etc.). A thermodynamic investigation into the stability and/or the equilibrium of such systems begins with the selection of a suitable thermodynamic potential and the equilibrium principle that goes with it. As noted by Callen (ref 7, p 107), the energy (internal plus gravitational) is not very suitable because the energy representation does not take advantage of the thermal equilibrium (temperature is constant throughout and known). The next thermodynamic potential to consider is the Helmholtz function. In this representation the entropy as an independent variable is replaced by temperature, which is kept constant throughout the system. The Helmholtz function is “admirably” (Callen) suited to assure thermal equilibrium (the search for configurations that are at complete equilibrium is now reduced to configurations that already are (7) Callen, H. B. ‘Thermodynamics”; Wiley: New York, 1960. (8) Gibbs, J. W. “The Scientific Papers”; Dover: New York, 1961; Vol 1, pp 55-371.
(C 1985 American Chemical Society
Langmuir, Vol. 1, No. I, 1985 41
Formulation of the Theory of Capillarity at thermal equilibrium). However, the equilibrium principle for the Helmholtz function still requires fixed component masses inside the fixed system volume. 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, has to be rejected because it requires that each pressure be controlled by a pressure reservoir (Callen, p 115) but here the smaller phase has none. At this point the suitable and well-known thermodynamic potentials have been exhausted. So, to no surprise, it is the Helmholtz function that is usually selected when treating the capillary systems under consideration. However, conceptually, the relevant Legendre transformations have not really been exhausted, because neither the Helmholtz nor the Gibbs function considers the possibilities of changes in masses or mole numbers, and hence the possibility of chemical equilibrium, expressed by the equality of chemical potentials, ab initio. Thus, the thermodynamic potential in which the independent variables "entropy" and "masses" of the individual chemical constituents are replaced respectively by the temperature and the chemical potentials is a suitable fundamental equation in the investigation of the above capillary systems. This thermodynamic potential, often called the grand canonical potential, we call the free energy. In the following sections we will reformulate the fundamental equation of capillarity in the free energy representation. We will define a corresponding minimum principle and will obtain the conditions of equilibrium for a two-phase capillary system.
Fundamental Equations and Minimum Principles The descriptive formalism of the classical equilibrium thermodynamics for fluid systems is based on the fundamental equations and a minimum principle. The fundamental equations determine and describe the thermodynamic states in all parts of the fluid system. The minimum principle then determines the stable equilibrium states from the multitude of thermodynamic states allowed by the fundamental equations. Various forms or representations of the minimum principle and the fundamental equations are possible. The energy minimum principle and some of the fundamental equations in the energy representation4 are reviewed here. Fundamental Equation for Bulk Phases. The relation u(" = u("(s(",p1(",p2'",
...)
(1)
is a fundamental equation. The specific (per unit volume) internal energy, u(v,is the thermodynamic potential. The specific entropy, s(", and the mass densities, P ~ ( " ) , P ~ ( ~ , . . . , are the natural variables of u ( v . The differential form of the fundamental equation is du'" = T dd"
+ Z p i dpi" 6)
(2)
which shows the meaning of the partial derivatives of u(" (T is temperature and p1,p2,...are the chemical potentials). The Euler relation in this representation is u(" = Ts'" xpipi(" - P (3)
+
(i)
which brings in the last important property, the pressure, P. Fundamental Equation for Surfaces. In Gibbsian thermodynamics, an interface is represented by a dividing surface which carries the excess interfacial properties and is governed by a fundamental equation. The fundamental
equation in the specific (per unit area, superscript A ) energy representation is the relation4
u ( ~=) U ( ~ ) ( S ( ~ ), ,PP~ ~( (~..~) ,,J,K) ).
(4)
The differential form and the Euler relation are dU(A)= T ds'A' + Cpi dpi'A)+ C1 d J + C2 dK (5) (i)
u ( ~=)TdA)+
+ C I J + CzK + y
(6)
(i)
The quantities J and K are the first (mean) and the second (Gaussian) curvatures of the surface ( J = c1 + c2 and K = clcz, where c1 and c2 are the two principal curvatures); C1 and Cz are the corresponding potentials (bending moments). The rest of the variables follow in analogy with the bulk phases. See ref 4 for the reasoning that leads to the above form of the fundamental equation for surfaces. Fundamental equations for the dividing lines and dividing points are not needed here. Energy Minimum Principle. The general energy minimum principle, as used by Gibbs,8 is as follows: An isolated system is in stable equilibrium if and only if its total energy (UT) is at a minimum relative to other thermodynamic states of the system with the same entropy (S), (UT)s = min (7) The total energy of the system, UT, consists of the internal energy and the potential energy in the external field. It should be noted that, for any thermodynamic state, the value of UT is based on (and calculated from) the fundamental equations for the individual parts of the system, i.e., bulk phases and surfaces. No extra internal constraints, which would reduce the number of admissible (possible) thermodynamic states, are considered here. For systems with independent components (no chemical reactions), the energy minimum principle reduces to (uT)S,Ml,M2,... =
min
(8)
subject only to the condition of mechanical isolation (displacements that would transfer work through the boundary are disallowed). The quantities Mlf12,... are the total masses of the individual chemical components inside the system. An equivalent unconstrained minimum problem is QT = UT - TS - CpiMi = min (9) (i)
where T,p1,p2,... are constant Lagrange multipliers. This minimum problem is free of constraints; the mechanical isolation remains as the boundary condition. Just as in eq 7 , it should be noted that for any thermodynamic state, the value of ilT is calculated from the corresponding equations for the individual parts of the system, i.e., QT
= QT("
+ QTcA)
A contribution to
QT, as defined by eq 9, from the bulk phases inside the system can (quite generally) be expressed as
where
(VI
and C$ = 4(?) is the external potential. The integration in eq 10 extends through the volume (V)of the whole system, using the appropriate fundamental equation for u ( v in each bulk part. By complete analogy, the contribution to
42 Langmuir, Vol. 1, No. 1, 1985 QT
Boruvka, Rotenberg, and Neumann
from the dividing surfaces is (A)
where wT(A)
=
U(A)
+ &,(A)+
- Ts(A)
- Cplp,(A)
(1)
(13)
(1)
The integration extends over all dividing surfaces in the system; the appropriate fundamental equation is used on each surface. Equilibrium Conditions. The condition of thermal equilibrium is
T=T
(14)
The equilibrium temperature, T, is the same throughout all bulk phases, dividing surfaces, etc. The conditions of chemical equilibrium are
uT(v)
=
(i)
which introduces the specific free energy representation of the fundamental equation for surfaces:
u(v)
-
Ts("
-
P,Pl(v)
The differential form of eq 22 is dw(A)= -dA)dT - Cpi(A) dpi + C1 d J + Cz dK
u(v)=
W(~(T,/.L,1,/L~,...)
(23)
(i)
and from eq 21 and 6 w(A)
+ C1J + C2K
=
(24)
Equation 24 defines the specific free energy of a dividing surface. In the special case of a flat interface, both J and K are identically equal to zero. Hence, for this specific case is equal to the surface tension, only, the free energy, dA), y. Using eq 22 to 24, one can derive additional relations for dividing surfaces: The mechanical potentials of a dividing surface (surface tension and bending moments) in the free energy representation y = Y(T,~17~Z,***,J,K)
C1 = Ci(T,pi,CiZ,...,J,K)
cz = CZ(T,PI,CLZ,...,J,K)
(25)
must satisfy the following integrability conditions:
(16)
acl
-ay + dJ
(1)
where all quantities are to be evaluated at the equilibrium temperature T = T and chemical potentials p l = p, - 4, i = 1,2,... With the understanding that p, = p I - 4, i = 1,2,..., we eliminate the subscript (T) from the expressions for o and Q and understand these expressions now as free energy terms. It is noted that eq 16 in fact defines a Legendre transformation from u(v)to
= w(~)(T,~~,cL~,...,J,K) (22)
w(A)
p, - 4
i = 1,2,..., (15) throughout the system, where p, are the equilibrium chemical potentials of the chemical constituents of the system at the reference surface, 4 = 0. These conditions, eq 14 and 15, together with the conditions of mechanical equilibrium were derived4from the appropriate variational extremum principle. Free Energy Representation. Unlike the conditions of mechanical equilibrium, the conditions of thermal and chemical equilibrium are simple and the same throughout the system. This presents the possibility of using the conditions of thermal and chemical equilibrium beforehand to reduce the minimum problem. 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. The contribution to QT, eq 9, from bulk phases will be reduced first. Using the equilibrium conditions (14) and (15) in eq 11 yields p1 =
becomes a known function of position through the given external potential, 4(?). The reduction of the dividing surface part of Q can be carried out in complete analogy with that of the bulk part. Again, the conditions of thermal and chemical equilibrium reduce eq 13 to u ( A ) = u(A)- T s ( A ) - C p ,. ( A ) (21) ,PI
J-+KdJ
acz= O aJ
ay + J -acl + K-acz = 0 dK dK dK
-
These conditions follow from the definitions of y, C1, and
c,:
=
u(A)
-
C1J - CZK
(29)
(17)
which is the specific free energy representation of the fundamental equation for bulk phases. The differential form of the fundamental equation is obtained by taking a total differential of eq 16 and using eq 2 for du'"): do(")= -s(W dT - CP," dpl
(18)
(1)
Also, by comparing the Euler relation, eq 3, and eq 16, (19) Thus, the negative of the pressure in a bulk phase is the expression for the specific free energy. The contribution of the bulk phases to the functional R can now be written as ,(v)
= -p
Compared to the full expression of Q("'J in eq 10 and 11, this expression is considerably reduced. There are no unknown functions left in the integrand of eq 20; dv)
by differentiation with respect to J and K and comparing. The relations 25 are equivalent to the fundamental equation, eq 22, provided they satisfy the integrability conditions, eq 26 to 28. By taking a total differential of eq 24 and comparing to eq 23, one obtains the differential identity (Gibbs-Duhem type): d r + J dC1 + K dCz + dA)d T + C P ~dpi ' ~ =' 0 (32) (i)
A t constant temperature and chemical potentials, eq 32 reduces to d r + J dC, + K dCz = 0 d T dpi = 0 (33) which is equivalent to eq 27 and 28 combined.
Langmuir, Vol. 1, No. 1, 1985 43
Formulation of the Theory of Capillarity Now the contribution of dividing surfaces to the reduced function Q becomes Q(A)
=
ss
~ ( ~ ) ( ~ , i i ~ - I $,..., , pJ~, -KI )$dA
(34)
(A)
where the integrand becomes a known function of 7,J , and K on each dividing surface. To complete the transition to the free energy representation, the appropriate minimum principle, the free energy minimum principle, is formulated as follows: A mechanically isolated system in contact with a source of constant temperature and chemical potentials at some point (a reservoir of the thermodynamic state) will be in stable equilibrium if and only if its free energy is at a minimum, =
Q(V)
+
= min
(35)
where Q(nand QcA) are given in eq 10 and 34. The free energy is thus a thermodynamic potential, the natural variables of which are the temperature, the chemical potentials, p1,p2,..., and the extensive geometric parameters which describe the system. To be more precise, for a single homogeneous bulk system, Q is a function of T, pcLi)s and V (v)
For a system of two homogeneous bulk phases with an interface of uniform curvatures, R is still a function of T, p[s, and the extensive geometric parameters which describe the system: Q = -P1V1- P2V2 y A Cl6 +C2X = Q ( T,PI,PZ,.-Vi, Vz,A,63) (37) where A, 6, and 3c are the extensive geometric parameters of the i n t e r f a ~ e .Specifically, ~ we have d = J A and X = KA. But for nonhomogeneous systems (due to gravity) or systems with a nonuniformly curved interface, R is not a function but a functional (the same goes for any other thermodynamic potential). In any case, Q is a thermodynamic potential with the same extremum properties (and yielding the same solution) as any other thermodynamic potential. Mathematically, the difference between the total energy and the free energy extremum formulations is that the constraints in the first problem (S = constant, M i= constant) are replaced by subsidiary conditions (T = constant, pj I$ = constant) in the second problem such that both problems yield the same solution. The transformations between such conjugate extremum problems are known as involutory transformation~.~ This thermodynamic potential is also called’ the grand canonical potential. However, we suggest that the name free energy potential, 3,be retained since this name conveys the physical meaning of this potential. For a reversible work process, the work that is delivered by a system in contact with temperature and chemical potential reservoirs is equal to the change in the free energy, Q . Conditions of Mechanical Equilibrium. The complete conditions of mechanical equilibrium can be obtained by minimizing the free energy for an arbitrary capillary system. A fully general development in the free energy representation would be very similar to that using the energy representation. So, with a rigorous derivation a~ailable,~ this section takes the opportunity to enlighten the topic of the generalized Laplace equation by presenting
+
+
+
(9) Courant, R.; Hilbert, D. “Methods of Mathematical Physics”; Interscience: New York, 1970; Val. 1.
a simple derivation. The derivation of the secondary mechanical equilibrium conditions, Le., the hydrostatic balance equations, will also be presented below. Generalized Laplace Equation. Consider a spherical or cylindrical (between parallel plates) bubble, or a drop, inside a larger fluid volume. The free energy, R, can be written as (A)
(v)
where A and V are bubble area and volume, dA) is given by eq 24, and AP is the pressure difference (inside minus outside). Since gravity is excluded, pressures, curvatures, etc. are all uniform and eq 38 becomes simply 0 = y A + C1d + C2X - APV (39) According to the free energy minimum principle, the equilibrium radius of the bubble is determined by -dQdR-O with T,pl,p2, kept constant. Differentiating Q in eq 39 yields, in view of the Gibbs-Duhem identity, eq 32.
...
dA + CI-d 6 d R d R dR
dQ = y
+ C 2d-dR bX
AP-dV dR
= 0 (40)
Equation 40 will now be applied separately to a cylindrical and a spherical bubble. (i) Cylindrical Bubble. The relevant geometric parameters of a section of a cylindrical bubble of radius R and length L are V = aR2L (volume) A = 2aRL (area) 6 = J A = 2aL (1st total curvature) X = KA = 0 (2nd total curvature) By using these relations in eq 40, one obtains dQ _ - y2aL - AP2aRL = 0
dR
which means that the equilibrium radius of a cylindrical bubble has to satisfy the equation Y ( ~ / R=) fl (41)
(ii) Spherical Bubble. The relevant geometric parameters of a spherical bubble of radius R are V = Y3aR3 (volume) A = 4aR2 (area) d = 8aR (1st total curvature) X = 4a (2nd total curvature) Again, eq 40 yields
and the equilibrium radius of a spherical bubble will satisfy the equation r)
yv
+ cl-
r)
= AP
R R2 Equations 41 and 42 are specialized forms of the same equation, the generalized Laplace equation of capillarity, which may now be deduced by realizing that it should contain the invariant curvatures J and K (rather than R). Since for a cylinder J = 1JR and K = 0 and for a sphere J = 2/R and K = 1/R2the simplest common way of writing
44 Langmuir, Vol. 1, No. 1, 1985
Boruuka, Rotenberg, and Neumann
eq 41 and 42 is quite obviously y J + 2C1K =
AP
(43)
This is the generalized Laplace equation. A comparison with eq 73 in ref 4 confirms that the above simplistic derivation has led to the correct result. Equation 43 is not the most general form of the Laplace equation; it has been obtained (eq 59 in ref 4) by neglecting gravity agd neglecting the surface gradients of C1 and C2 (p(*)iiV4 = 0, a,c, = a2c2= 0). Hydrostatic Balance Equations. When reviewing the complete set of conditions for thermodynamic equilibrium (thermal, chemical, and mechanical), the conditions of mechanical equilibrium may appear incomplete as there are none for the bulk phases. Also, the Laplace equation for surfaces, eq 43, may appear incomplete since evidently it only balances forces perpendicular to the surface. In this section, it is shown that the “missing”hydrostatic balance equations are secondary conditions which follow from the thermal and chemical equilibrium conditions and from the fundamental equations. From eq 18 and 19, the pressure differential in a bulk phase is
dP = do d T + x p k ( Odp, (1)
?P = s(VdT + Cp,(V?p, (1)
4t therm$
and chemical equilibriu_m,see eq 14 and 15, V T = 0, V p l = -V4 and therefore V P reduces to
where
p(V
+ &2C1 + I&zC2
= - S ( ~ ) ? ~- TC ~ i ( ~ ) ? , p i (i)
At thermal and chemical equilibrium ?,T = 0, and the above equation reduces to a,y
+ &,cl +
= p(A’ijz4
= -v24
(45)
where p(A) is the total surface density of mass of the dividing surface. Equation 45 is the hydrostatic equation for a dividing surface which balances the external and internal forces at any point on the surfaces in the two directions tangential to the surface. Equation 45 is a surface analog of eq 44. Forces in the third direction (normal to the surface) are balanced by the Laplace equation. It may be noted that the use of the surface gradient rather than the space gradient in eq 45 is necessary because the quantities y, C1, and Czare defined only on the surface. For a quantity that is defined on the surface as well as away from it, the surface gradient is obtained from the space gradient by subtracting the normal component, e.g.: e24= 94 - ri(?i.a$)
(a2)
where 6 is the unit normal of the surface.
and therefore the pressure gradient is
? p = -p‘V$q)
a bulk phase (in all three directions), From eq 32, the surface gradients (V,) of the properties of a dividing surface have to satisfy the equation
(44)
is the total mass density in the bulk phase: p(V = =yPl(V (1)
Equation 44 is the well-known hydrostatic equation which balances the external and internal forces at any point in
Summary In the preceding sections, we have formally presented the free energy representation of a capillary system. This reformulation of the fundamental equations is particularly convenient for treatment of various phenomena of capillarity. The corresponding free energy minimum principle was stated, and the conditions of mechanical equilibrium were developed by implementing this principle. The complete set of the mechanical equilibrium conditions, along all three directions in space, was derived. For a dividing surface these conditions are the generalized Laplace equation and the hydrostatic balance equations.
