Langmuir 1985,1, 673-678 673 A Quantitative Theory of Negative Adsorption of Nonelectrolytes Caused by Repulsive van der Waals Forces Manoj K. Chaudhury* and Robert J. Good Department of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 14260 Received June 6, 1984. In Final Form: April 1, 1985 It is known that sucrose raises the surface tension of water; it is negatively adsorbed at the airlwater interface. We have analyzed this phenomenon from the point of view that the solute molecules are repelled from the interface because of the existence of a repulsive van der Waals force. The theoretical treatment employs the Lifshitz theory of forces, together with the thermodynamic theory of interfacial distances. Our calculation has yielded a quantitative prediction of the increase in surface tension of water as a function of solute concentration, which agrees well with experimental observations. It also predicts the concentration gradient of solute molecules in the region below the surface. Introduction where p is the dipole moment of the solute molecule, em It is a familiar fact of surface science that electrolyte is the dielectric constant of water, z is the distance of the solutes in water cause an increase in surface tension above dipole from the airlwater interface, and 4’ is a dimen- that of the pure solvent. It is not so well-known that sionless correction factor, which ranges from 1 to 2.23. certain nonelectrolytes also cause an increase in surface Since pz/3kT has the dimensions of the static dipolar tension. For both types of solutes, the change in surface polarizability, we can rearrange eq 1 in a convenient form, tension is certainly due to negative adsorption. such as Wagner’ and Onsager and Samaras2 have developed the theory of the effect of electrostatic forces on ions at and w(2) = near an airlwater interface. The electrostatic “image force” z(:) causes a net repulsion to act on the ions, so that they are excluded from the interfacial region. Onsager and Samaras where k is Boltzmann constant, Tis absolute temperature, were able to make predictions of the phenomenon which and a is the dipole polarizability of the solute molecule in were in a fair agreement with experimental observations. a vacuum. The negative adsorption of nonelectrolytes such as su- We note, however, that the electrostatic behavior of a crose at the airlwater interface suggests that a repulsive solute molecule that is due to the presence of the phase force should be inferred, acting on the solute molecules. boundary (liquid/vapor or liquidlair) cannot be accounted Initial attempts to treat the case of nonelectrolyte so- for merely by the polarizability of the solute. The solute lutions were made by Buff and G~el.~They applied their molecule replaces an equal volume of solvent, which (in calculations to the case of aqueous amino acids. The the absence of solute) had a self-energy due to the sur- surface excess quantity was distributed between two pa- rounding solvent and to the absence of solvent on the other rameters, a and 8, where the former designated the surface side of the phase boundary. Thus, it is more correct to activity of the fatty acid portion of the amino acids. The employ an “excess” polarizability, a*,rather than a which numerical magnitude of a was derived utilizing the em- is given by p2/3kT. pirical relation between the surface tension decrease and So, in the limit of a continuum approximation, one the chain length of the lower fatty acids, known as would expect that w(z) should be zero when the solute has Traube’s rule. (The surface tension of aqueous fatty acid the same dipole moment and dielectric constant as the solutions is a linear function of the solute concentration solvent. Also, w(z) should be negative if the dipolar po- in the Henry’s law region. Traube’s rule4 states that the larizability of the solute is less than that of the solvent. proportionality factor between the decrease in surface Unfortunately, these two expectations cannot be derived tension and solute concentration is, in turn, a linear from eq 1 and 2. In order to elucidate the limits of validity function of the chain length of the fatty acids.) The second of eq 2, we will now focus our attention on an improved parameter was assigned to the effect arising from an in- expression for the self-energy of the solute molecule, as verse cubic interaction law. This law was derived from an derived by Imura and OkanoS6 Israelachvili,’ following analytical solution of the electrostatic problem of multi- the procedure of previous authors, has shown that the poles embedded in a spherical cavity. Although Buff and electrodynamic self-energy of a randomly orienting solute Goel took into account the effects due to multipole in- molecule, at a distance z from the airlwater interface, can teractions, they showed that the dipolar contribution be expressed as constituted the dominant term in the net cavity image potential. Later, Clay, Goel, and Buff,5 in a more detailed treatment, considered the effects of finite solute size, an- isotropy, and also the diffuse nature of the interface. The approximate mathematical expression for the image potential of a randomly oriented dipole, obtained by Buff and Goel, was (1) Wagner, C. Phys. 2. 1924, 25, 474. (2) Onsager, L.; Samaras, N. N. T. J. Chem. Phys. 1934,2,528. P241 (3)Buff, F. P.; Goel, N. S. J. Chem. Phys. 1972,56,2405. w(2) = - (4)Adamson, A. W. “Physical Chemistry of Surfaces”, 4th ed.; Wiley 12tmz3 Interscience: New York, 1982. (5)Clay, J. R.; Goel, N. S.; Buff, F. P. J. Chem. Phys. 1972,56,4245. (6)Imura, H.;Okano, K. J. Chem. Phys. 1973,58,2763. *Present address: Dow Corning Corp., Midland, MI 48640-0994. (7) Israelachvili, J. N. Q.Reu. Biophys. 1974, 6, 341. 0743-7463/85/2401-0673$01.50/0 0 1985 American Chemical Society 674 Langmuir, Vol. 1, No. 6, 1985 Chaudhury and Good where a*,(iw,) is the excess polarizability of the solute a sharp dichotomy of short-range (SR) interactions vs. molecule and tj(iw,) and t,(iw,) are respectively the di- long-range interactions. The latter can be treated by the electric permeability of air and of water; these functions methods of London15J6and Lif~hitz;~we refer to them, are evaluated along the complex frequency axis, The elsewhere,17J8as “LW”. prime in the sum indicates that the zero frequency term In principle, if the enthalpy of hydration of a solute is is given half-weight in the summation. The zero frequency known, the contribution of the H-bond interaction to the term in eq 3 originates from the orientation and the in- net surface excess quantity can be estimated by using a duction contributions to the multipole image intera~tion.~ step-function Boltzmann equation for the concentration Since the static dielectric constant of air is unity and that profile. Fowkeslghas put forth the idea that the H-bond of water is about 80, E,,, >> tj, and we can obtain a sim- interaction is a subset of acid/base interaction, and pro- plified, approximate expression for g(z),=, from eq (3), as posed the use of Drago’s empirical acid/ base parameters20 follows: in obtaining a quantitative estimate of the H-bond in- teraction. Drago’s table of these parameters is, unfortu- nately, far from complete, and hence the Drago-Fowkes (4) approach cannot be followed at the present time. If we encounter a situation where the solute is strongly where CY*,is now the excess static polarizability of the hydrated, so that the energy of hydration is much larger solute, the dominant term of which is due to the dipolar than the average kinetic energy, then we can simplify the contribution, for a polar solute. picture by assuming that at least the first monolayer of It is worthwhile to note the similarity of form, between water at the interface will be devoid of solute. This as- eq 4 and 2. Equation 4 is, however, significantly different sumption points directly to a negative adsorption of the from eq 2, since, in the derivation of eq 3, and hence of solute. But it does not, in general, account for the mag- eq 4, the concept of excess polarizability was invoked. nitude of the negative adsorption. The profile of solute Equation 4 potentially can predict that g(z),=, can be zero, concentration as a function of distance from the surface negative or positive, depending upon the sign of CY*,. is needed. For example, it would be possible to envision An approximate expression for CY*,has been given by a case where the profile of concentration vs. distance could McLachlan,8 as follows: have a maximum just below the surface layer, even though the very last layer was solute-depleted. Such a distribution would be rare, because it would require the combining of low polarizability of the solute with a strong tendency toward hydrogen bonding to water. In principle, we can where, t, is the dielectric constant of the solute and r is calculate the contribution due to electrodynamic forces by the radius of the solute, in the spherical approximation. using the expression for the self-energy of the solute, From eq 4 and 5, it is clear that if t, E,, a*, will be following Israelachvili’s appr~ach.~There is, however, an negative, and it will be positive when t, > E,. The values immediate difficulty: it is not very clear as to what should of the static dielectric constants for most water-soluble be the best mathematical form for a*,(iw,) as a function nonelectrolytes are generally smaller than that of water, of dielectric permeabilities of the solute and the solvent.