Adsorption Kinetics of Ionic Surfactants after a Large Initial Perturbation. Effect of Surface Elasticity K. D. Danov, † V. L. Kolev, † P. A. Kralchevsky,* ,† G. Broze, ‡ and A. Mehreteab § Laboratory of Thermodynamics and Physicochemical Hydrodynamics, Faculty of Chemistry, University of Sofia, 1164 Sofia, Bulgaria, Colgate-Palmolive Research and Development, Inc., Avenue Du Parc Industriel, B-4041 Milmort (Herstal), Belgium, and Colgate-Palmolive Co., Technology Center, 909 River Road, Piscataway, New Jersey 08854-5596 Received August 2, 1999. In Final Form: November 19, 1999 This theoretical study is devoted to the relaxation of surface tension of an ionic surfactant solution for submicellar concentrations. The effects of added nonamphiphilic electrolyte and counterion binding are taken into account. We consider a large initial deviation from equilibrium, which is defined as the formation of a new interface: there is no adsorbed surfactant and electric double layer at the initial moment. Next, the surfactant solution and its interface are allowed to relax without any subsequent perturbation. The electrodiffusion equations, which describe the adsorption kinetics, are nonlinear, and it is impossible to find a general analytical solution, especially in the case of large initial deviations. Nevertheless, the problem can be linearized in the asymptotic case of long times. The derived theoretical expressions show that the relaxation times in the cases of large and small initial perturbations are numerically close to each other. For that reason the relaxation time can be considered as a general kinetic property of the adsorption monolayer. The theory predicts also the slope of the experimental plot of dynamic surface tension vs inverse square root of time; this makes the theory useful for interpretation of experimental data. The theoretical expressions involve the surface (Gibbs) elasticity, whose definition for adsorption monolayers of soluble ionic surfactants is discussed in detail. The Gibbs elasticity of such monolayers is found to increase strongly with the rise of salt concentration. The derived asymptotic expressions are verified against an exact computer solution of the electrodiffusion problem, and excellent agreement is found. 1. Introduction The characteristic time of surfactant adsorption at a fluid interface is an important parameter for surfactant- stabilized dynamic systems, like foams and emulsions. For example, it is expected that surfactant solutions with faster adsorption kinetics do exhibit a greater foaminess. 1,2 Sutherland, 3 derived an expression describing the relax- ation of a small dilatation of an initially equilibrium adsorption monolayer from a soluble nonionic surfactant where t is time is characteristic relaxation time, σ is surface tension, c 1 , Γ 1 , and D 1 are surfactant concentration, adsorption, and diffusivity; here and hereafter the superscript “(e)” denotes the equilibrium value of the respective parameter; erfc(x) is the complementary error function. 4-6 Using the asymptotics of the latter function for x . 1, one obtains Equation 1.3 is often used as a test to verify whether the adsorption process is under diffusion control: data for the dynamic surface tension σ(t) are plotted vs 1/t 1/2 and it is checked if the plot complies with a straight line; the extrapolation of this line to 1/t 1/2 f 0 is used to determine the equilibrium surface tension, σ (e) . 1,7 In the experiment one often deals with large initial deviations from equilibrium. In such a case there is no general analytical expression for the dynamic surface tension σ(t) since the adsorption isotherms (except that of Henry) are nonlinear. In this case one can use either a computer solution 8,9 or apply the von Karman’s ap- proximate approach. 10,11 Moreover, analytical asymptotic expressions for long times (t f ∞) can be obtained. Hansen 12 derived the long-time asymptotics of the sub- surface concentration, c 1s (t), of a nonionic surfactant: * Corresponding author. E-mail: pk@ltph.bol.bg. † University of Sofia. ‡ Colgate-Palmolive Research and Development, Inc. § Colgate-Palmolive Co., Technology Center. (1) Dukhin, S. S.; Kretzschmar, G.; Miller, R. Dynamics of Adsorption at Liquid Interfaces; Elsevier: Amsterdam, 1995. (2) Fainerman, V. B.; Khodos, S. R.; Pomazova, L. N. Kolloidn. Zh. (Russia) 1991, 53, 702. (3) Sutherland, K. Aust. J. Sci. Res., Ser. A 1952, 5, 683. (4) Janke, E.; Emde, F.; Lo¨sch, F. Tables of Higher Functions; McGraw-Hill: New York, 1960. (5) Abramowitz, M.; Stegun, I. A. Handbook of Mathematical Functions; Dover: New York, 1965. (6) Korn, G. A.; Korn, T. M. Mathematical Handbook; McGraw-Hill: New York, 1968. (7) Loglio, G.; Rillaerts, E.; Joos, P. Colloid Polym. Sci. 1981, 259, 1221. (8) Miller, R. Colloid Polym. Sci. 1981, 259, 375. (9) Rakita, Y. M.; Fainerman, V. B.; Zadara, V. M. Zh. Fiz. Khim. 1986, 60, 376. (10) Kralchevsky, P. A.; Radkov, Y. S.; Denkov, N. D. J. Colloid Interface Sci. 1993, 161, 361. (11) Danov, K. D.; Vlahovska, P. M.; Horozov, T. S.; Dushkin, C. D.; Kralchevsky, P. A.; Mehreteab, A.; Broze, G. J. Colloid Interface Sci. 1996, 183, 223. (12) Hansen, R. S. J. Chem. Phys. 1960, 64, 637. σ(t) - σ (e) σ(0) - σ (e) ) Γ 1 (t) - Γ 1 (e) Γ 1 (0) - Γ 1 (e) ) exp ( t τ 1 29 erfc [( t τ 1 29 1/2 ] (1.1) τ 1 ≡ 1 D 1 ( ∂Γ 1 ∂c 1 29 2 (1.2) σ(t) - σ (e) σ(0) - σ (e) ) Γ 1 (t) - Γ 1 (e) Γ 1 (0) - Γ 1 (e) ) ( τ 1 πt 29 1/2 + O(t -3/2 ) (1.3) t . τ 1 2942 Langmuir 2000, 16, 2942-2956 10.1021/la9910428 CCC: $19.00 © 2000 American Chemical Society Published on Web 01/22/2000