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TEMPEL, J. LUCASSEN, AND E. H. LUCASSEN-REYNDERS
Application of Surface Thermodynamics to Gibbs Elasticity
by M. van den Tempel, J. Lucassen, and E. H. Lucassen-Reynders Unilever Research Laboratory, Vlaardingen, The Netherlands
(Received April 13, 1964)
Rupture of a thin liquid film drawn from a surfactant solution can be opposed by the surface tension gradient arising from local extension of the film surfaces. A measure for this effect, which is thought to be responsible for the large influence of certain minor impurities on foam stability, is the Gibbs elasticity, relating the excess surface tension in the extended region to the relative increment of the surface area. For a quantitative evaluation of the Gibbs elasticity of a,film, the relation between the surface tension and the composition of the film liquid must be known. Rigorous surface thermodynamics have been developed to derive this relation. Simplified versions of the general equations are used to calculate the Gibbs elasticity, especially for films of mixed surfactant solutions. Compounds of suitable surface activity, even if present in minor amounts, appear to have large effects on the Gibbs elasticity. Their influence is already considerable a t rather large film thicknesses, assuming the most simple, ideal behavior of both the surface and the bulk liquid in the film. The effect of deviations from ideality is discussed.
Introduction One of the mechanisms that are generally accepted to contribute to the stability of thin films obtained from surfactant solutions is the Plateau-Marangoni-Gibbs effect.l Rupture of the film is supposed to be introduced by local thinning, during which the surface area in part of the film is enlarged. The amount of surfactant in the extended region of the film is insufficient to maintain the original values of both the adsorption and the bulk concentration; in general, the surface tension in the extended region is, therefore, higher than in the neighboring parts of the film. The Plateau-Marangoni-Gibbs effect accounts for this surface tension gradient and for the resulting elasticity of the surface, which tries to restore the original shape of the film. Equilibrium between the enlarged surface and the underlying film liquid is reestablished immediately (or a t least much more rapidly than the equilibrium between neighboring parts of the film) in film elements of dimensions comparable with the film thickness; for such elements the elasticity is called Gibbs elasticity. It is defined2 as the ratio of the increase in the film tension (2da) resulting from an infinitesimal increase in area and the relative increment of the area (d In A )
E = - 2d u d In A The Journal of Physical Chemistry
(1)
Evaluation of the Gibbs elasticity as influenced by the composition of the surfactant solution would be useful for understanding the behavior of the thin films produced therefrom. Such a calculation of E requires the variations in the surface tension B and in the surface area A to be expressed in measurable parameters of the a m : its thickness h, the concentrations c4 of the surfactants contained in it and their properties. In the following, the influence of properties and amounts of surfactants will be considered. Gibbs Elasticity. The variation of u in an extending film element is related to the surfactant adsorptions I?( and the concentrations of the surfactants (i = 2 , 3 . . . n) by Gibbs adsorption law -du = RT
is.2
r4d In y4c4
(2)
where y c represents the activity coefficient of i. The variation in A of the element can be related to the surface and bulk concentrations by considering the volume of the element and the total amount of each independent component in it to be constant (1) B. V. Derjaguin and A. S. Titijevskaya, Proc. Intern. Congr. Surface Adivity, 8nd, London, I, 211 (1957); L.E.Scriven and C. V. Sternling, Nature, 187, 186 (1960). (2) J. W. Gibbs, “Collected Works,” Vol. I, Dover Publishing Co., Inc., New York, N.Y., 1961,p. 301.
APPLICATIONOF SURFACE THERMODYNAMICS TO GIBBSELASTICITY
-dlnA = dlnh =
hdc%3. 2drt ( i = 1 , 2 ... n) 2rt (3)
Consequently, the general relation for the Gibbs elasticity reads dln
E
=
4RTC i=2
~i
rt21 + - d In ct Ct
hf2-
dri dct
(4)
giving the influence of film thickness and surfadant concentrations. The values of ri and drt/dct occurring moreover in eq. 4 follow from the surface equation of state for the system considered. It should be noted that the latter quantity is not, equal to the partial differential quotient or to the slope of the adsorption isotherm of i, since during extension the concentrations of the other components also vary
ar,_ar,+c_x;lF; br, dc, dci
dci
+j
dcj
(5)
where the variations in cj with respect to ci follow from the conservation of matter, as expressed in eq. 3. Further quantitative treatment requires the surface equation of state to be given, the thermodynamics of which will be presented in the following section. Even qualitatively, however, eq. 4 shows two remarkable properties of the Gibbs elasticity. First, the Gibbs elasticity of films made from approximately ideal dilute solutions of one surfactant has a maximum value at an intermediate surfactant concentration. At very low concentrations E increases since the adsorption in this region is proportional to the concentration, whereas at higher concentrations the adsorption no longer varies measurably with concentration, causing E to decrease asymptotically to zero. Secondly, eq. 4 shows that the value of E' is very sensitive to the presence of coniponents with high adsorption values at very low concentrations. Knowledge of the surface equation of state for mixed surfactant solutions will permit a quantitative evaluation of these effects. Surface Thermodynamics. Thermodynamics of the equilibrium distribution of matter between parts of a system-e.g., between surface and bulk solution-has to start from the uniformity of the thermodynamic potential pt for each component throughout the system. The relation between pi and the concentration or mole fraction of i in any part of the system, then, gives the desired relation between the concentrations of i in these different parts of the system. The relation between pt and the concentration ct of that component
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can, however, only be given in terms of an activity coefficient, yi. This coefficient is defined by this very relationship, e.g., in the case of the bulk solution a pta =
cia+ RT In ytaci
(6)
Tta being a constant characterizing the standard state of i. (In the following, the symbol will be considered to denote infinite dilution as a standard state; i.e., the value of lt is adjusted so as to make y i -t 1 for cf --t 0.) Consequently, a general surface equation of state contains two activity coefficients of each component, one for the solution, y;, and one for the surface, y?. As is discussed in detail el~ewhere,~ the relation between the thermodynamic potential of i in the surface and the surface concentration of i, can be written as
Here l? is a constant characterizing the standard state of i in the surface. For solutions containing one surfactant, r" is the saturation adsorption, defined as the surfactant adsorption corresponding with the limiting slope of the surface tension-log surfactant activity plot at high surfactant activities. The derivation of eq. 7, which gives all information about the surface equation of state, can only briefly be indicated here. It was obtained by considering the surface phase, s, to be a geometrical dividing surface according to Gibbs between the two bulk phases. The position of this surface, however, was not fixed as suggested by Gibbs, i.e., not so as to make the adsorption of the solvent equal to zero. For deriving a surface equation of state, the distribution of all components, including the solvent, between bulk and surface is considered; therefore a convention which defines the solvent to be not adsorbed is not suitable. A new convention has been chosen which fixes the position of the dividing surface so as to make the sum of solvent and surfactant adsorptions equal to the constant amount of r" moles cm.-2; this enables the derivation of eq. 7. The ratio of and r- thus gives the mole fraction of i in the surface, in analogy with ci in eq. 6 measuring the mole fraction of i in the bulk solution. In principle this treatment, which considers the solvent adsorption, rl, not to be equal to zero, would necessitate the introduction of a term for i = 1 into Gibbs adsorption law as given in eq. 2. For aqueous surfactant solutions, however, where the mole fraction of water is always higher than 0.999, this term (3) E. H. Lucassen-Reynders and M. van den Tempel, paper submitted for the IVth International Congress on Surface Active Substances, Brussels, 1964.
volume 69, NUmbET 6
June 1966
M.
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can be neglected due to the extremely small variations in water activity. Equating the expressions given for p f in eq. 6 and 7 , the surface equation of state can be obtained in three equivalent formulations, from which either r for ct or u has been eliminated, all the expressions containing three parameters : (1) the surface activity coefficients y t , determining the shape of the adsorption isotherm; (2) the saturation adsorption Fm,determining the magnitude of the adsorption, i.e., the scale of the adsorption isotherm in the vertical direction; (3) a constant to be denoted by at, determining the magnitude of the surfactant concentration at which a given adsorption value is reached, i.e., the scale of the adsorption isotherm in the horizontal direction. This constant equals the ratio of surface and bulk mole fractions of the surfactant a t infinite dilution and, therefore, depends on the standard state parameters f t and on the surface tension UO at infinite dilution
RT In ai
UO
=
p1" - lis - r-
=
-A
to
tants. Therefore, the concentration at half saturation adsorption is a sensitive measure for the surface activity of a compound, low values of it corresponding with high surface activity. These general considerations following from eq. 6 and 7 can equally be applied to solutions of several surfactants. Surfactants each forming ideal surface solutions with water are assumed to behave ideally in each other's presence; the following treatment will be limited to ideal surface solutions. The parameter I" for solutions of an arbitrary number of surfactants is generalized to be a total saturation adsorption corresponding to given ratios k , between the surfactant activities. Its value is found from the limiting slope of the d o g yiact plot for any of the surfactants at such constant ratios
(8)
The parameter Afou thus defined is the molar free enthalpy of adsorption (from phase a) of i at infinite dilut i ~ n . For ~ ideal surface solutions, with 7; = 1 a t all surface compositions, at is the surfactant activity at which the surfactant adsorption has reached half of its saturation value. It is emphasized that the general equations giving the surface behavior of a surfactant in terms of these three parameters were derived without any model for the surface layer; in particular, the model of a monomolecular surface layer is not involved, nor have any assumptions been made as to the areas occupied by the molecules in this layer. As to the values of the surface activity coefficients ,:y it has been shown3that from single surfactant solutions ideal surface solutions are often formed, in which the surface activity coefficients of solvent and surfactant equal unity at all surface compositions. The shape of the adsorption isotherm for such ideal surface behavior is given by the well-known equation of Langmuir. This was shown to occur even in cases where the bulk activity coefficients of solvent and surfactant are by no means equal to unity, e.g., in the case of micelleforming compounds. The value of the second parameter, the saturation adsorption, is not very sensitive to variations in the nature of the surfactant; at the air-water interface its value for surfactants with one single aliphatic chain varies between 2.5 X and 5.5 X mole cm.-2. The value of the third parameter, a, however, can differ by a factor as high as lo5 for two different surfacThe Journal of Physical Chemistry
TEMPEL, J. LUCASSEN, AND E. H. LUCASSEN-REYNDERS
Surface tension and adsorptions as a function of surfactant concentrations, then, are given by generalized Szyszkowski-Langmuir equations uo - u =
RTI" In
[ et+ 11 is2
r$= r-
at
7t"ctlat
c 71ac,la5 + 1
(10) (11)
j-2
Occurrence of Szyszkowski-Langmuir adsorption has been generally supposed to be limited to adsorption from ideal bulk solutions of a single nonionic nonvolatile component; it has been considered essential that equal cross-sectional areas could be predicted from the geometry of solvent and solute. Equations 10 and 11 are generalized to include also nonideal solutions of an arbitrary number of surfactants, the geometry of which is considered irrelevant. In principle, the value of rdepends on the concentration ratios of the various surfactants present; this effect is considered negligible in comparison with the influence of the difference in a values for the surfactants. In the following section, the Gibbs elasticity for films containing one or two Surfactants will be evaluated by combining eq. 4 and 11. Gibbs Elasticity for Ideal Surface Behavior. Usually the bulk activity coefficients of surfactants (not that of the solvent) are nearly equal to unity below the critical (4) A. Vignes, J. chim. phys., 57, 966 (1960).
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APPLICATION OF SURFACE THERMODYNAMICS TO GIBBSELASTICITY
micellar concentrations: yza = yap = 1. For such solutions it is convenient to express surfactant concentrations in units at cr X( = ut
and likewise to express the total amount of surfactant present in unit volume of the film, gf, in units a,
I" = 2.6 X 10-1O mole cm.-2; u2 =
0.44 X
mole cm.-*
The only known elasticity measurements have been performed in this system6; they yielded values between 8 and 15 dynes cm.-l at a concentration around the c.m.c., in agreement with the calculated values just below the c.m.c. in Figure l. Gibbr dlasticilv
and, finally, to d e h e another dimensionless parameter, yt, measuring the amount of i present in the film liquid (xt) relative t o the amount adsorbed at the surfaces (st
-
st>
In terms of these dimensionless parameters, the Gibbs elasticity of single surfactant solutions is found to be given by
E
=
2RTP
x2
1
+ YZ(1 +
=
2RTF"
Xd2
xz(g2 g2
- 22)
+ xz2
Figure 1. Gibbs elasticity of films with thickness 10-4 cm. containing sodium dodecyl sulfate.
(15) At a given film thickness, eq. 15 predicts a linear increase of the elasticity with increasing surfactant concentration in the Traube region (XZ
