Langmuir 1988,4,485-486
485
Notes A Simple Graphical Method To Evaluate Surface Elasticities and Surface Concentrations from Rectilinear Isothermal Plots
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Sydney Ross Rensselaer Polytechnic Institute, Troy, New York 12180
Ian D. Morrison* Xerox Corp., Webster, New York 14580 Received September 28, 1987. In Final Form: November 28, 1987
While preparing an appeal to fund a Maxwell museum at his birthplace in Edinburgh by reviewing his collected works, we came across his description of a simple graphical method to evaluate the modulus of bulk elasticity.’ An advantage of his method is that relative values of elasticity can be recognized at a glance. (The method is applicable whenever the derivative with respect to the logarithm is needed.) Elasticity under any given conditions is defined as the ratio of a small increase in the pressure to the fractional change of size: E = -ds/d In Q (1) where s is the pressure and Q is the generalized size, which for a three-dimensional body has the units of volume and for a surface film has the units of area. The modulus of elasticity is usually evaluated by plotting s versus the logarithm of Q and measuring the slope. Simple as this, Maxwell’s method is even simpler. The elasticity a t the point P on the rectilinear plot of s versus Q is obtained by extending the tangent a t the point P to intersect the s-axis a t the point G (see Figure 1). Let F be the value of s at P, then E = -Q ds/dQ = FP(FG/FP) = FG (2) Thus, for example, the elasticity of a monomolecular film can be obtained directly from the a-A curve at any point by drawing PG, a tangent to the curve at P, and PF, a horizontal line. The portion FG of the r-axis represents, in the same units as a, the Gibbs elasticity of the film at P. The relative values of elasticity can be recognized easily by mentally completing the operation at any given point on the *-A isotherm. For example, in Figure 2, the elasticity of the two-dimensional condensed phase at PI is clearly larger than that of the two-dimensional gaseous phase at P,; the elasticity at P2in the two-phase region is zero, which a moment’s reflection will confirm. The same method of recognition is useful to relate the Gibbs, or equilibrium, elasticity and the Marangoni, or dynamic, elasticity of a surface film.Marangoni elasticity is measured some time before equilibrium is reached, the time interval depending on the particular method of measurement. The Marangoni effect is dynamic and nonequilibrium, hence temporary. The first result of extending the surface is to increase the surface tension by reducing the surface concentration; but the initial increase (1) Maxwell, J. C. The Theory o f H e a t , 2nd ed.; Longmans, Green & Co.: London, 1872; p 106.
STRESS
Figure 1. Generalized stress-strain diagram showing the graphical evolution of elasticity by Maxwell’s method.
PI
SQUARE ANGSTROM UNITS PER MOLECULE Figure 2. Mid FA isotherm of a monomolecular f h ,showing a two-dimensional phase change between a gaseous and a condensed phase.
AREMMOLECULE
Figure 3. How to obtain elasticity of a surface film from a *-A curve. The solid line represents the equilibrium isotherm; the dotted line represents a nonequilibrium curve.
of tension immediately begins to relax as diffusion of the adsorbate, either laterally in the case of an insoluble monolayer or from the bulk solution in the case of a soluble layer, tends to bring the system to its new equilibrium. Gibbs elasticity is m k u r e d after the surface concentration has settled to its ultimate value, which, after extension, is less than it had been before. The dynamic effect always produces a greater 6r for a given 6A than is observed at equilibrium. The slope of the dynamic PA curve at any point is therefore greater than that of the equilibrium r - A curve (see Figure 3). The application of Maxwell’s method shows that Marangoni
0743-7463/88/24Q4-Q485$01.50/00 1988 American Chemical Society
Langmuir 1988,4,486-487
486
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ACTIVITY, mM Figure 4. Graphical evaluation of Gibbs excess surface concentration of sodium dodecyl sulfate, at 2.80 mM, from the data of M y ~ e l s . ~
elasticity is therefore always greater than Gibbs elasticity. The same graphical method can be used to evaluate the Gibbs excess surface concentration of a solute from a rectilinear plot of surface tension versus activity, by means of the Gibbs adsorption theorem: rz= -da/(vaRT d In a,) (3) where F2 is the Gibbs excess surface concentration of so-
lute, v is the number of ions per molecule of solute, a is the degree of dissociation, and a2 is the activity of the solute. A plot of surface tension versus activity (or concentration c2 if the solution is ideal) can be treated in the same way to obtain r2as the T-A curve was treated to obtain E. This application of Maxwell’s method has already been suggested by Couper.2 We take the recently published data of Mysels3 for surface tensions of dilute aqueous solutions of sodium dodecyl sulfate to illustrate Maxwell’s graphical method. In Figure 4 the value of da/d In a2 at a2 = 2.80 mM is measured by FG in mN/m. For a2 = 2.80 mM, da/d In a2 = 14.2 mN/m. The remainder of the calculation to evaluate P2 (a = 1, v = 2) is r,(in mol/cm2) = 2.02
X
10-”FG
(4)
where FG is in mN/m, which gives r2 = 2.86 X mol/cm2 at a, = 2.80 mM. By plotting cr/(vaRT) versus the activity of the solute, the distance FG then gives the value of rz directly. Maxwell’s method is equally as accurate and more illuminating than the usual procedures. (2)Couper, A. in IUPAC Commission 1.6 Physical Chemistry: Enriching Topics from Colloid and Surface Science: van Olphen, H.; Mysels, K. J., Eds.; Theorex: La Jolla, CA, 1975;pp 131-132. (3)Mysels, K. J. Langmuir 1986,2,423-428.
Comments On “An Unusual Gel without a Gelling Agent“
The enthusiasm for the peculiar rheological properties and potential applications of “polyaphron” gels, expressed in a recent letter in this journal,’ is understandable. It is shared by many others, including myself, who have worked with, or have been exposed to, these intriguing materials. It must be pointed out, however, that the paper lacks novelty, except for the name that the authors have attached to what more prosaic workers have referred to simply as “high-internal-phase-ratio emulsions”‘ (HIPREs) or “highly concentrated emulsions” (e.g., ref 2-14). These
(1)Bergeron, V.;Sebba, F. Langmuir 1987,3,857,858. (2)Lissant, K. J. J. Colloid Interface Sci. 1966,22,462;1973,42,201; 1974,47,416. (3) Lissant, K. J. SOC. Cosmet. Chem. 1970,21,141. (4)Nixon, J.; Beerbower, A. Prepr. Am. Chem. SOC.,Diu. Pet. Chem. 1969,14(1), 49,62. (5)Mannheimer, R.J. J. Colloid Interface Sci. 1972,40,370. (6)Princen, H.M. J. Colloid Interface Sci. 1979,71, 55. (7)Princen, H.M.;Aronson, M. P.; Moser, J. C. J . Colloid Interface Sci. 1980,75,246. (8) Prudhomme, R. K., 53rd Annual Society of Rheology Meeting, Louisville, KY, 1981. (9)Princen, H. M. J. Colloid Interface Sci. 1983,91, 160; 1986,105, 150. (10)Princen, H.M.;Kiss, A. D. J. Colloid Interface Sci. 1986,112,427. (11)Princen, H. M. Langmuir 1986,2,519; 1988,4,164-169. (12)Princen, H.M.;Kiss, A. D. Langmuir 1987,3,36. (13)Schwartz, L. W.;Princen, H. M. J . Colloid Interface Sci. 1987, 118, 201. (14)Princen, H. M.; Kiss, A. D. J . Colloid Interface Sci., submitted.
are emulsions (oil-in-water or water-in-oil) in which the volume fraction of the dispersed phase, 4, approaches or exceeds that of the close-packed-sphere packing, 4o= 0.74. As the volume fraction approaches unity, the droplets assume an increasingly pronounced polyhedral shape, while remaining separated (and protected against coalescence) by thin films of continuous phase. With the right kind and concentration of surfactant in the continuous phase, emulsions with 4 as high as 0.99 can be readily prepared. The presence of a second surfactant in the dispersed phase . may or may not be helpful. The structural and rheological properties of such emulsions are described by essentially the same laws as those of gas-liquid foams of 4 > 40. Because of the crowding, the deformable dispersed units (drops or bubbles) cannot move freely past each other when a small stress or strain is applied to the system. Therefore, up to , systems behave as viscoelastic the yield stress, T ~ these solids (“gels”),characterized by a shear modulus, G. Above the yield stress, they are shear-thinning fluids. This behavior has nothing to do with the presence of a “web of thin and, therefore, strong water which is ice-like and which has to be broken before the gel can flow”.’ The behavior is indeed determined by the three-dimensional network of films of continuous phase, but in a different way that is well understood from two-dimensional modeling.8~9J3J6-19~21-23 These models, and careful measure(15)Khan, s. A., Ph.D. Thesis, MIT, 1985. (16)Khan, S.A.; Armstrong, R. C. J . Non-Newtonian Fluid Mech. 1986,22, 1; 1987,25,61. (17)Khan, S. A. Rheologica Acta 1987,26,78. (18)Kraynik, A. M.;Hansen, M. G. J . Rheology (N.Y.)1986,30,409.
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