ADSORPTION ON CURVED SURF..ICES -4ND EMULSIFICATION HANS $1. CASSEL Colgate-Palmolive-Peet Company, Jersey City, .V. J . Received January 8, 1938
The increase of the vapor pressure (potential) of a onc-coniponent liquid, owing to an increase of the curvature of its surface, can be describrd, from the thermodynamical point of view, as a consequence of the capillary pressure exerted upon the internal phase on account of thc surface tension, From the molecular point of view, on the othrr liand, this effect is t o be understood as the reduction of the attractivc field, oning to the removal of the fluid masses contained in the interspacr b e t w r n tlic tangential plane and the spherical body of the liquid. Although the sanic considerations, undoubtedly, hold in the case of adsorbcd moleculcs bound by a spherical adsorbent, the change of the potential of the adsorbed substance as a function of the curvature of the adsorbing surface, apparently, has ]lot been treated thermodynamically. The following deductions concerning this problem nrc hascd entirely upon the Gibbsian theory of capillarity. THERMODYNAMIC T H E O R l
We consider a apheriral inass of radius, r, consihting of a solvrllt (1) and a capillary-activc substance ( 2 ) dissolved in it being in cqiiilibriuni n i t h the vapor (external) phase containing thc sanie two components. By p1, p~ we shall designate the potentials; by clin, c?,,, and clrx, czcl the concentrations (masses per unit volume) of thc components in the internal and external phases, respectively. Wc also h a w to take into consideration, of course, the interface between the homogeneous phases with the superficial densities l?l and r2. Let p , , and p,, = p,, OC br the total pressures in the external and internal phaws, \%herer denotes the interfacial tension and C = 2 / r the curvature, supposing that thc radius ii: wfficiently defined by the size of the iiiternal mass. The problem we have t o deal with concern3 the potential, p2, of a film of a given density, rz,as a function of the curvature, C; i.e., ~ v cwish t o
+
knoiv the value of the differential coefficient, 476
(s)~:as a function of the
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HANS M. C.4SSEL
interfacial tension, the curvature, and the mass distribution given by the Clint Czin and Clex, Czea. Generally, p~ is a function of I‘z, p l J and C, so that
On t,he other side, for a given curvature, I’z is a function of
p1
and pz:
For d r Z = 0 we therefore obtain:
and
The establishment of equilibrium, furthermore, requires the following equations (of state) t o be fulfilled simultaneously:
Hence we may derive the relation:
as an expression of the “displacement effect” (4). Consequently equation la, because of equation 2a, can be written in the form:
Now, for the case of plane surfaces, Gibbs (12) has shown that we can consider the dividing surface as being located so that the total quantity of the first component in the vicinity of the surface of discontinuity is the same as if the density of this component were uniform on each side quite up to the dividing surface. Such a location might be objectionable only with regard to a component which has very nearly the same density in the adjacent phases. Extending this procedure to the case of curved surfaces we could say that the radius of the dividing surface had to be determined so as t o make
.4DSORPTION ON CURVED SURFACES AND EMULSIFICATION
477
the quantity rl vanish. However, an even less restrictive demand is implied in the assumption that the value of rl shall be constant for a given value of the curvature. On this basis it is justified to consider arl/ar2 as very small in comparison to arz/apz and t o use
as a reasonable approximation. If, finally, we take into account the condit,ion of mechanical equilibrium: dpin
- dpe,
+ Cda
= UdC
(7)
we obtain the result:
(g)
r1
= -
+ Crz -
(8)
0.
~2in
~zex
In perfect analogy to the behavior of a one-component droplet (GibbsThomson formula), the potential of a film of given thickness (density), r2, is larger upon the smaller droplets than upon the largcr ones if Czin
+ C r > Czcx
(9)
whereas the opposite will be true if
> C2in
+
(10) In other words, for a given potential, p2, under the first condition (inequality 9) the films of greater curvature must be thinner; under the second condition (inequality lo), however, they must be thicker than those of smaller curvature. This might be formulated quantitatively: C2ex
RT- a log rzaC
Cr2
--U
+ Crz - czeX
~2in
In regard to the analogy with droplets in bulk the inequality 10 will be considered as the “stability criterion” for films of given curvatuve and interfacial density. Some consequences drawn from this definition will be briefly discussed. The decision as to whether or not condition 10 can be satisfied depends entirely on the individual shape of the adsorption isotherm represented in a rz-czeX diagram (see figure 1). Here it is assumed as a first approximation that the course of the isotherms does not vary with the curvature. The adsorption isotherms A and A‘ correspond to those of the normal type, the latter representing an increase of the adsorption as possibly due to a lowering of the temperature. On the other hand, the stability condition is represented in the r2-c2== diagram by the course of the discriminant line:
ra
= r/z (czex -
czin)
(12)
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HANS M. CASSEL
only those parts of the idotherms located below the diqcriminant coriesponding t o “stable states” of films with given curvature. Since the discriminants are approximately straight lines starting in the origin of the diagram, there may occur one or two intersections with the isotherms or none a t all If we had to deal with nionolayers only, the isotherms would reach -aturation and only one intersection of the discriminants could occur, but we know, principally from the work of Perrin (16), that, for instance on the surface of zoap solutions, a multiplicity of layers can exist This means that the isotheims surpass saturation, as could also be shown in the case of other organic molecules adsorbed from the vapor phase on iiiercury ( 7 , 8). Then, of course, a second intersection may take place. I n general, a ininimuin concentration, cPex,has to be reached before stability is possible, and, in case of the second intersection, a definite maximum concentration cannot be surpassed without annihilating the stability
FIG.1
FIQ.2
bIoreover: for any given isotherm, a minimum radius must be guaranteed in order to obtain stability (see the tangent discriminants in figure 1). If the course of the isotherm is shifted in the direction of smaller concentrations (as indicated by the isotherm A’), the range of stability on a given discriminant, Le., for a given curvature, shrinks more and more until the isotherm fails t o be cut. Stability, then, is possible only for discritninants of a steeper slope, i.e., for films of larger radii. It might, finally, be emphasized that the present deductions are not restricted by the special assumption? concerning the adsorption isotherms as drawn in figure 1. On the basis of equation 11 tlie influence of the curvature upon the course of the isotherms may be easily derived. I n figure 2 the essential Yeaturcs of this effect are shown for a series of adsorption isotherrns, A I , > r2 > 73. A,, A 3 oil surfaces n i t h presumably increasing curvatures ‘The iiiterscctions 2gI9 F2,Sawith the discriminants D,,D2,D3 are now spread or’er a ~,~.icler rangr instc:ici of defining a sharp limit.
ADSORPTION OK CURVED SURF.4CES A S D EYI‘LSIFICATION
479
COMPARISON WITH EXPERIMENTAL F.ICTS
No direct experiments seem to be available which mould allow of n verification of this theory. An indirect test, however, is possible in the field of emulsions (5, 6). By the word “emulsion” we will understand a dispersion of finely divided droplets or gas bubbles suspended in a liquid medium. Tolman (18) has shown that such systems are therniodyiianiically stable only if the interfacial tension is zero. Otherwise they collapse as time goes on and change to the stable state of two coexisting masses in bulk. However, a “pseudo-stabilization’’ may be obtained, at least to a certain degree, by the addition of a third component, the emulsifying agent, which iq soluble in the external phase and produces a film around the particles, thus preventing their coalescence (9). This implies the assumption that the growth of solvent droplets through condensation is negligible. One has t o be aware, however, that the collisioii of particles does not necessarily result in their coalescence, Le., in the formation of larger homogeneous units. In many cases, usually regarded as examples of reversible coagulation (the terms “agglutination” and “flocculation” are applied in different fields), the particle boundaries are preserved, being separated and sticking together by means of the interfacial layers of emulsifying agent, According to the present aspect the coagulation iq simply underbtood as due t o the capillary condensation of the emulsifier in between adjacent particles, which naturally must occur if the stability condition, defined above, is not fulfilled. Generally coagulation has to be considered as the pre-stage of final coalescence. In comparing the present theory with experimental facts we introduce the somewhat hypothetical assumption that the criterion, apz/aC < 0 , represents a necessary condition for the pseudo-stability of emulsions, but one has to keep in mind that it can not be a suficient stability conditiofa. For this condition might well be satisfied also when the films are not thick enough to exert the protective action. Moreover, the compensation of gravity is necessary, which may be partially ascribed t o thc Brownian movement, and partially to the electric forces of adsorbed ions. Since the potential of the external solvent present in the case of miulsions is almost independent of the variations in concentration of thr other components which always form sufficiently dilute solutions, the formulas derived above are applicable without alterations. (To destroy any doubts one might consider the same assumption to be the basis of the generally acknowledged pH scale!) THE IATERSION O F EMULSIOA7S
As a matter of fact, two types of emulsions may be formed with the same solvents (15). The decision as to which phase is to be the dispersing
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HANS M. CASSEL
agent depends greatly on the nature of the emulsifying substance. Alkali soaps, which are readily soluble in water and far less soluble in oil, give oilin-water emulsions, while soaps of other bases, whose solubility is greater in oil, yield the water-in-oil type (2). Generally the rule of Bancroft holds, that the better solvent for the emulsifier is the external phase. This statement is in full conformity with the stability condition for the films as defined above. A few exceptions stand in the way of acknowledging the generality of Bancroft’s rule (2). For instance, oleic acid is soluble in benzene but not markedly soluble in water. Contrary to expectation, however, benzene is actually emulsified in water. Other examples are those of ethyl ether and some organic solvents emulsified in water, supposedly by the action of iodine as emulsifier. From the standpoint of the present theory, it seems probable that in these cases a fourth component not taken into consideration has actually operated in producing the protective film. In this regard, the oxygen of the air is open to suspicion because, in the presence of air, unsaturated acids as well as ether easily undergo changes. Undoubtedly, the possibility of emulsification offers a very sensitive analytical test for certain conipounds, the type of emulsion obtained, furthermore, allowing a statement concerning their relative solubility. In the case of supposedly insoluble emulsifiers, as, for instance, soot or alumina, we remain in agreement with the theory by regarding the particles of these substances as the molecules dissolved in the adjacent phases, the solubilities being regulated by the interfacial energies of these “molecules” (14). This view of the mechanism of emulsification is practically identical with that first suggested by Ramsden (17). THE STABILITY RANGE O F EMULSIONS
The theory provides that an excess as well as the removal of the emulsifier causes the collapse of emulsions. This explains the fact that for the practice of preparing emulsions it is advisable to apply the emulsifier in small portions instead of as a whole. An obvious confirmation of the theory is the observation of Bartsch (3) and others that optimum concentrations of stabilizing agents exist for the durability of dispersed systems. Experiment, furthermore, proves the theoretical expectation that the size of the emulsified particles is limited by a minimum radius. For example, Kistler (13),applying a colloid mill for the emulsification of water in toluene with aluminum stearate as the emulsifier, could not detect ~ diameter, estimated microscopically. droplets smaller than 0 . 2 in The theory also explains the otherwise astonishing fact that (9) the stability of emulsions can be noticeably improved by “homogenization”,
ADSORPTION ON CURVED SURFACES AND EMULSIFICATION
481
i.e., by reducing the radii of the droplets and decreasing the concentration of the emulsifying agent through the developmeilt of new surfaces. Finally, the theory accounts for the “salting out of emulsions” without recurring to the simple idea of neutralizing the electric surface charges. The effect of electrolytes upon the emulsifying agents, if it j s not a chemical reaction, consists in changing their activity coefficients (10). Accordingly, in case of increased activity (2, ll), the adsorption isotherm must be shifted into the direction of lower concentrations. The effect upon the stability of an emulsion, under such conditions, corresponds to the effect of temperature changes, as described in connection with figure 1. But reliable data are lacking for decision as to whether or not the breaking of emulsions by heat treatment is in agreement with the theoretical deductions. Probably the thermal effects can be understood only if the conditions sufficient for emulsification are also taken into account. CONCLUSION
The theory, on the whole, is in satisfactory agreement with the facts known about emulsions. However, experiments are missing which would permit a quantitative test in detail. On the other hand, it seems to be allowable to extend the theory here developed to the inhibiting action of “protective colloids” and “peptizing agents” in general. SUMMARY
The thermodynamic theory of adsorption on curved surfaces leads to a relation analogous to the Gibbs-Thomson formula for the vapor pressure of droplets as a function of the curvature. In regard to the lack of direct experiments the theoretical expectations are checked by experience with emulsions. The basic idea is that emulsions can be stable only if the interfacial density of the emulsifying films is greater for surfaces of larger than for those of smaller curvature. There are no data available which would allow a quantitative comparison, but no facts seem to be known which are in contradiction with the theory. REFERENCES (1) BANCROFT, W. D. : Applied Colloid Chemistry, 2nd edition. McGraw-Hill Book Co., New York (1932). (2) BANCROFT, W. D. AND BARNETT,C. E . : Colloid Symposium Monograph VI, 76 (1928). (3) BARTICH, 0.: Kolloid-chem. Beihefte 20, 1 (1924). (4) CASSEL,H. M.: Ergeb. exakt. Naturw. 6, 108 (1927). (5) CASSEL,H. M. : Nature 137,405 (1936), letter to the Editor. (6) CASSEL,H. M.: Preliminary report, Acta Physicochim. U. R. S. S. 6, 289 (1937). (7) CASSEL,H. bI.,AND SALDITT, F.: 2.physik. Chem. A166, 321 (1931).
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(8) CASSEL,J. M, A N D GCREWITCH, B.: Trans. Faraday SOC. 28, 177 (1932). (9) CLAYTON,V. : Emulsions and Emulsification, 3rd edition. P. Blakiston and Son, Philadelphia (1935). (10) DEBYE,P., AXD PHI CLAY, F.: Physik. 2 . 26, 22 (1925). (11) FREUNDLICH, H., AND SCHNELL, A.: 2. physik. Chem. 133, 151 (1928). (12) GIBBS, H. W.:Collected Works, Vol. I, p. 234 (1928 edition). (13) KISTLER,S. S : J. Am. Chem. SOC.68, 904 (1936). (14) LANGMUIR, I.: Chem. Rev. 6, 451 (1929). (15) OSTWALD,Wo.: Ilolloid-2. 6, 103 (1910). (16) PERRIN, J.: Kolloid-Z. 61, 2 (1930). (17) RAMSDEN, IT.: Kature 112 (1923). (18) T O L v i N , R . C . : J. Ani. Chem. Soc. 36, 317 (1913).
