Oct., 1950
INTERFACIAL ENERGIES IN SOLID-LIQUID-VAPOR SYSTEMS
1655
CONFLICTS BETWEEN GIBBSIAN THERMODYNAMICS AND RECENT TREATMENTS OF INTERFACIAL ENERGIES I N SOLID-LIQUID-VAPOR SYSTEMS BY RULON E. JOHNSON, JR. Contribution No. 984, Jackson Laboratory, Organic Chemicals Department, E. I . d u Pont de N e m u r s and Company, Inc. Wilmington, Delaware Received M a y 11, 1969
Many recent pa ers on the thermodynamics of solid-liquid-vapor systems are in conflict with J. Willard Gibhs’ treatment of the subject. &e nature of these differences is discussed. Using the techniques of Gibbs, Young’s equation is derived for a sesde drop on a solid. Adsorption and gravity are explicitly considered. It ie shown that Young’s equation is valid iii all situations studied, provided it is understood to be a relationship among surface tensions rather than surface free energies.
In recent years, Young’s equation (equation (31) or (32)) has generated considerable discussion and controversy. Bikerman,‘ for example, denies the validity of Young’s equation both on theoretical and experiment,al grounds. I n discussing this equation a t the Second International Congress of Surface Activity he states, “perhaps it is not too much to hope that this Congress will mark the downfall of this relation.”2 Pethica and Pethica3 deny the validity of the equation in a gravitational field. Others4J deny its validity for zero contact angle. Moreover it is widely assumed that the equation should be written in terms of surface free energies. Gibbs6 developed the thermodynamics of solidliquid-vapor systems very completely and elegantly. He derived Young’s equation for the nongravitational case and outlined its derivation for a system in a gravitational field. A derivation of the equation using the methods of Gibbs is given below. Adsorption and gravity are explicitly considered. It is shown that the equation is generally valid. Four reasoiis can be advanced for the confusion that exists concerning this subject. 1. It is assumed by many (e.g., reference 3) that the surface tension y of an interface defined by equation (1) is numerically equal to the specific surface free energy f defined by equation (2). The two in fact are related by equation (3). (dF/dQ)?’,V.ni Y (1) where F = Helmholtz free energy of the system = surface area of interface T = temperature V = volume ni = number of moles of component i f = ( F - Fa FP)/Q (2) where Fa = Helmholtz free energy of a unit of volume in the homogeneous part of CY multiplied by the volume of CY FB = Helmholtz free energy of 6 defined analogously to Fa
-
(1) J. J. Bikerman, ”Solid Surfaces,” in “Second International Congress of Surface Activity,” Vol. 111, Academic Press, New York, N. Y.,1957, p. 131. (2) Ibid., p. 101. (3) B. A. Pethica and T. J. P. Pethica, “The Contact Angle Equilibrium,” in “Second International Congress of Surface Activity,” Vol. 111, Academic Press, New York, N. Y., 1957, p. 131. (4) 9. Baxter and A. B. D. Cassie, J . Tezl. Ins!., 36, TG7 (1945). (5) J. L. Moilliet and B. Collie, “Surface Activity,’’ D. Van Nostrand Co., New York, N. Y., 1951. pp. 98-99. (6) J. Willard Gibbs, “The Collected Works of J. Willard Gibbs Volume I. Thermodynamics,” Yale University Press, New Haven, Conn., 1928, pp. 314-331.
(3) where
ri = surface excess of
component i per unit area (dF/bni)~,v,sa,njLe., the chemical potential of component i K = number of components in the system
pi =
Often the adsorption of a solvent can be considered to be zero so that for a one-component system y does equal f. For those systems in which adsorption phenomena are important, the two are definitely not equal. As shown below, Young’s equation is valid in terms of y not f. 2. It has been assumed (e.g., by Harkins’) that for systems containing surfaces the necessary and sufficient condition for equilibrium is that the free surface energy of the system be a minimum. Actually, the total free energy of the system, a t constant T, V , and mass must be a minimum. This misconception has allowed derivations to be made such that specific surface free energies appear for the terms in Young’s e q u a t i ~ n . ~ ~ ~ 3. A general lack of rigor in discussing surface energetics has contributed immeasurably to the confusion. For example, the quantities in Young’s equations have been variously identified as surface tensions, surface free energies, specific surface free energies and surface energies. While it is permissible (although sometimes confusing) for an author t o name physical quantities as he desires, the quantities in question should be unambiguously defined. Unfortunately, in most discussions of surface free energy, the name itself is the only definition given for the quantity. It is only when the quantities are relatively well defined, as in the paper of Pethica and Pethica, that errors in physical meaning become o b v i o u ~ . ~ 4. The physical interpretation of y has created (7) W. D. Harkins. “The Physical Chemistry of Surface Films,” Reinhold Publishing Corp., New York 36, N. Y., 1952. pp. 80-82. (8) C. G. Sumner in ”Symposium on Wetting and Detergency,” Chem. Puhl. Co., New York, N. Y., 1937, p. 15. (9) Examples of the type of confusion discussed here are given in the writings of Adam and Harkins. Adamlo rigorously defines surface free energy as a surface excess in the manner of Gibbs. R u t he also saya, “It is always possible mathematically, to replace a free energy per unit area of a surface by a tension acting parallel to the surface” (ibid. pp. 2-3 and p. 178). I n a later paper” he says he has simply used the name surface free energy for surface tension. Harkins’ BEYS t h a t he too defines the surface free energy as Gibbs, yet the equations he develops can only be reconciled with the definition for y given above. (IO) N. K. Adam. “The Physics and Chemistry of Surfaces.” 3rd ed., Oxford University Press, London, 1941, pp. 107-110 and pp. 404-407. (11) N. K. Adam, Disc. Faraday Soc., 3, 5 (19481.
RULON E. JOHNSON, JR.
1656
vapor ( a )
solld (s)
-
Vol. 63
(GF)T,V,nr = 0 (7) I n our notation the symbol 6 refers to a variation of the system in the sense of a virtual work variation. The symbol d refers to an element or differential of a quantity. Consider the system of Fig. 1. The solid is insoluble in the liquid. Its surface is homogeneous, continuous, and isotropic. The superscripts s, 1, and a refer to the solid, liquid, and vapor respectively. The superscripts (SI), (sa) and (la) refer to the solid-liquid, solid-vapor and liquid-vapor interfaces, respectively. Some or all of the components in the liquid may exist in the vapor. Any summation of components is understood to be a summation of the components in the specified phase or interface. Since we cannot consider the pressures and surface tensions to be uniform throughout the phases and surfaces of a system in a gravitational field, we must consider elements small enough that the pressure or surface tension is uniform in each. The intrinsic energy of an element in a is given by
- PdVa
d U a = Tdsa
+
Nidnp
(8)
i
and the total intrinsic energy of a is then
u‘
(2+ au + auz)
=
dzdydz =
The limits of integration are chosen to define the volume of a. Similar equations can be written for other phases and for surfaces. In order to prevent the equations from becoming too cumbersome the symbols and will indicate integrations over all the volumes and surfaces, respectively. The symbol 6dF1 indicates the variation of the intrinsic free energy in an element of volume in the liquid phase. Similar notation is used for the other quantities. The intrinsic free energy of an element is that free energy not explicitly dependent on gravity. The total free energy of the system is given by
sv
h
where g = acceleration of gravity d m v = the mass of an element d V dmn = the mass of an element d o z = the height of the element above a horizontal plane
6
lv
gzdmV
sV + In
+6
sQ
gzdmQ = 0 (11)
Because we are dealing with reversible variations 6F
=
6dFv
6dFa
+
Sv
g6zdmV
+
Now dFv = -PdV
and
+
pidni’ i
(13)
INTERFACIAL ENERGIES IN SOLID-LIQUID-VAPOR SYSTEMS
Oct., 1959 dFn = yda
+
pidnin
(14)
i
6 Jn(lal y(la)dWa)=
cos 0 6TdL
-+la)
h(la)[; + ;]
Therefore
7
SL
p a )
pisdnin
Note: ni = component i.
+
Sv
+ Jv gz6dmv + g6zdmn + jn gz6m = 0 g.3zdmV
sa
mi/Mi,
(15) where Mi is the molecular weight of
Because the variations of mass are independent of variations of V and D equation (15) can be separated into two independent equations
jag6zdmn = 0
(16)
and 6
LIal,
y(d) d
6diW = 6
JncBa,
W) =
$n,sa,
dmv
-
pdV
(18)
and where:
dma = Ada (19) density of the volume element in units of mass/ volume X = surface density of the surface element in unit,s of maselarea p =
Furthermore
SvI
P6dVI = 6
SVI SVI PdV' -
6PdV'
and
where 6N(Ia) = the normal component of motion of the element of surface into the vapor = the pressure on the element measured in the PI liquid
Equation (21) is valid since each side represents the virtual work done by the liquid phase in the variation. The integral is only over the liquidvapor interface since the solid surface is not considered to vary normal t,o itself. Similarly
sva
P6dVa = 6
6
SVs
Sva
PdVB =
sv.
(22)
Pa6N('")dn(la)
(23)
PdVn -
- [a(Ial
sPdVa
where: Pa = the pressure on the element of surface measured in the vapor
and
GN('a)6Q(la)
Y ( ~ ' GTdL )
(25)
(27)
_ '
h(aar
Gy(aa)dWB)
(28)
and 6 $*(Ea)
y(nd
-
dQ(sa) =
SL
6TdL
(29)
Substituting equations (18) to (29) into equation (16) and collecting terms gives
h(la, [(Pa
6y(la)]
?Cia)
(20)
SL
&(sa)
,,(sa)
gz.3ma = 0 (17)
Equation (17) contains the conditions for chemical equilibrium in a gravitational field and (16) the conditions for mechanical equilibrium. Now
+
where dL is the element of length of the line formed by the intersection of the three surfaces and 6T is the virtual motion of this line normal to dL along the solid and in a direction which increases D@). 0 is the contact angle and rl and r2 are the principal radii of curvature of dn. The final term of equation (25) comes from the change of area of the liquidvapor interface due to the change of shape of the drop in the variation. The geometrical significance of this term is discussed by J. Rice.I4
and
Sn
1657
COS
PIMN
+ (: + i) GN + gX(la) 6z +la)
+ SvI[6P+ gp6z]dV1+ jL + [gX(d) - 6y(sI)]dW) + 0 - y'")]lGTdL + diP8)
[y(sl)
snlm,
62
[gX(") Sz
- 6y(na)]dQ(sa) =0
(30;
Since the variations are arbitrary it is necessary and sufficient that each integrand be zero. The last two integrals are zero if the solid surface is level and flat. The coefficient of dV gives the distribution of pressure within the liquid. The coefficient of d W ) gives the effect of surface tension and curvature on the pressure within the iquid. The coefficient of dL yields Young's equation y(la) COS
8
E
,,(BB)
- y(d)
(31)
If we repeat the analysis using and [(e1) (defined by equation (4))then it is found that y(la) cos e = {(sa) - p I ) (32) For a solid in contact with a pure liquid and its vapor, the Gibbs adsorption equation yields (33)
where: PO = vapor pressure of the liquid. Combining with equation (32)
(14) J. Rice in "Commentary on the Scientific Writings of J. Willard Gibbs. Volume I, Thermodynamics," F. Q. Donnan and A. Haas, editors, Yale University Press, New Haven, Conn., 1936, p. 13.
RULONE.JOHNSON, JR.
1658
These last two equations have been discussed by Bangham and Razouk,lb and Boyd and Livingston. l6 When reading these authors and others, it should be understood clearly that the Gibbs adsorption equation (e.g., equation (33)) gives a change in surface tensions (equation (l)), not specific surface free energies (equation (2)).
DISCUSSION J. J. BIKERMAN (Massachusetts Institute of Technology). -1 shall not speak of semantics or nomenclature because in my opinion these questions are really not important for deciding whether Young’s equaticn is or isn’t correct. There are two t,ypes of proofs of Young’s equation-one using surface energy and the other using surface tension. Since Dr. Johnson agrees that the proofs of the first type are incorrect, we have a SOY0 agreement because I believe that also the proof based on tensions is unsatisfactory. As given by Gibbs it was fundamentally not different from the treatment given by Young who considered the result evident. It is clear that the force pushing the three-phase line of contact to the right must be equal to that pushing it to the left, if 7 2 3 = 718. equilibrium is present; thus, ylzcos e However, this condition is not sufficient for equilibrium as follows, for instance, from Gibbs’ discussion of the liquidliquid-gas system. For this system, i.e., for the case of a liquid drop floating on another liquid in a gas, Gibbs writes z y b l = 0. In this equation y represents the various surface and interfacial tensions, and a2 is the virtual displacement of the three-phase line. I n fluids, this equation can be satisfied. It is not essentially different from the rule of the paralldogram of forces. It also has been confirmed by experiments. However, this equation cannot be satisfied in solids, a t least as long as the solid surface is treated as a plane, Le., as long as the two tensions y13and ylilare supposed to act along one straight line. If we substitute 7 1 3 - yzsfor two separate tensions, we are left with two forces acting in two different directions. They cannot compensate each other. The resultant of these tensions is yI2sin e. As long as this force is not balanced, no equilibrium can exist. Presumably, the tension ylz sin 6 has a real existence and causes effects which I hope to measure. Miss Bailey has mentioned that she observed curling of a mica sheet on which a drop of water was deposited. I hope to show that the force ylz sin 0 raised a ridge and is compensated by stresses. We hope to detect it on gels. If we succeed, Young’sequation would be disproved, first of all because and yZ3 would have to act along different directions and, consequently, the difference 7 1 3 - Y23 would be meaningless. The main thing, however, is that if this ridge exists, another form of energy, namely, the strain energy of the solid, must be considered when writing down the equilibrium conditions. This would be true also when displacement of the 3-phase, ;.e., a dynamic process, is considered. When this line moves along the solid, not only the surface areas and, consequently, the surface energies vary but also the shape of the solid and tho work of deformation mu& be taken into account. I should like to stress the statement made by Dr. Johnson that equilibrium is determined by the free energy of the
+
(15) D. H. Bangham and R. I. Raaouk, T~ans.Faraday Soc., 33, 1459 (1937).
(IO) G. E. Boyd and H.IC. Livingston, J . Am. Chem. Snc., 64, 2383
(1942).
Vol. 63
whole system, not by the capillary art of this energy. If a dro on a solid deforms the solid, t f e strain energy is a part of t%e free energy of the system. Gibbs didn’t consider it because Gibbs restricted himself to “unchangeable” solids, but real solids are not “unchangeable.” R. E. JOHNSON.-It is not claimed, nor is it true, that Young’s equation is the only requirement for total mechanical equilibrium for solid-liquid-vapor systems. I n fact equation (30) will ’eld six separate equations which must be satisfied. What G b b s has shown is that no matter what other conditions must be fulfilled, the respective surface tensions must obey Young’s equation at equilibrium. If the component “y12 sin 8’’ mentioned by Dr. Bikerman is discussed in terms of equation (30) as a whole, the apparent anomaly of an unbalanced force will disappear. While I have indeed considered the solid to be rigid (but not necessarily plane) the proof would have to be modified only slightly for mobile surfaces (e.g., gelatin). Equation (30) would then contain an additional term depending on the curvature of the solid, but Young’s equation would still appear in the form given. Similarly, equations describing the internal stresses in a solid would appear separate from Young’s equation. This is physically reasonable since stresses in the interior of a solid should not influence the spreading of a liquid over the solid. W. A. ZIBMAN (U. S. Naval Research Laboratory).-I wish to congratulate Dr. Johnson for his illuminating and well presented paper. As I understand it, the basic problem of concern here goes back t o the recognition over a 100 years ago that a purely mechanistic explanation of surface tension and surface energy was out of the question until such time as a precise knowledge would be available of the field of force between neighboring molecules in liquids and solids. Since a detailed mechanism was not possible, and still is not known sufficiently precisely, refuge had to be sought in a thermod namic approach. The problem before us is hnw to a p p g thermodynamics correctly to surface behavior. Dr. Johnson’s paper does much to clear up the unfavorable situation now existing widely in the literature which has been caused by frequent and continued carelessness in the application of thermod amics to surface systems. Unquestionably, there is mucgonfusion in the literature because of the use of the Gibbs free energy instead of the Helmholtz free energy, or the work function as physicists often call it. Especially helpful is Dr. Johnson’s discussion of the incorrectness of identifying the surface tension of a solution with the free energy per unit area. I agree with the author’s treatment of the Young equation. It would be a backward step, indeed, to derive the Young equation by treating the problem as a Rtatic equilibrium of a group of vector tensions acting as a point. It is more logical, more general, and more teachable to discriminating students to show that the Young equation follows necessarily from thermodynamics. Johnson’s point is usually missed that at equilibrium an infinitesimal virtual displacement of the disposable parameters must cause a negligible change in the free energy of the entire system. Almost invariably the condition for equilibrium is expressed in terms of the vii*tual change in the free energy resident in the surfaces of the system. Although Johnson’s discussion can be derived from material to be found in Gibbs’ papers, it is a valuable contribution because it will do much to eliminate errors and confusion which have arisen because these aspects of Gihbs’ prepentation were much too condensed and sketchy.
