9022 J. Phys. Chem. 1994, 98, 9022-9032 Double Layer Forces between Heterogeneous Charged Surfaces S. J. Miklavic,'*t D. Y. C. Chan,# L. R. White,# and T. W. Healyg Department of Mathematics and Department of Chemistry, University of Melbourne, Parkville, 3052 Victoria, Australia, and Divisions of Physical Chemistry and Food Technology, Chemical Center, University of Lund, S-221 00 Lund, Sweden Received: March 25, 1994" In this paper we study the double layer interaction between two heterogeneous surfaces of either constant charge or constant potential. The surface heterogeneities are assumed to be distributed on periodic lattices of arbitrary structure. General expressions for the 3-D electrostatic potential distribution and the interaction free energy between the two surfaces are given. Asymptotic forms and numerical examples for the interaction potential are provided for the symmetric lattice problem. In general, the interaction potential or osmotic pressure decays exponentially at large separations. When a nonuniform, but net neutral, surface interacts with a uniform surface (charged or uncharged), the interaction can be either attractive or repulsive depending on whether the surfaces are constant potential or constant charge. For two nonuniform net neutral surfaces, the interaction can be either attractive or repulsive depending on whether the surfaces are constrained in configurations in which regions of unlike or like charge are in opposition. For this case, a statistical mechanical average over all relative lateral displacements shows that asymptotically the interaction will always be attractive. The magnitude of the attraction is comparable to or can exceed the van der Waals interaction. The results given here would warrant inclusion in any interpretation of surface force measurements in systems involving adsorbing, neutralizing surfactants. I. Introduction Kuin6 recently considered surfaces held at constant but nonuni- form surface potential. However, Kuin's results were obtained A knowledge of the interaction between particles or between with the superposition approximation which restricted their surfaces in general is fundamental to our understanding of many validity to large separations. properties of colloid science. In a large class of colloidal systems, We recognize that it is difficult to consider a completely the surface forces are dominated by long-range electrostatics, arbitrary charge distribution in a quantitative calculation. the familiar double layer force.' For many applications under Consequently, we shall consider sources of charge or potential colloidal conditions the traditional approach to double layer which are periodically distributed on the surface. In the next two interactions, the mean-field model based on the Poisson-Boltz- sections we present a general formalism, outlining the process of mann equation, has proven to be very useful for experimental solution of the electrostatic potential problem, and the evaluation analysis. In recent years, considerable effort has gone into of the interaction free energy invoking only the assumption that improving the mean-field description of the electrolyte behavior on each surface there is some periodic but otherwise unspecified under more extreme conditions (see, for example, ref 2). In arbitrary two-dimensional lattice distribution of charge or comparison, considerably less effort has been spent in studying potential. We consider the two cases of constant charge and departures from the classical assumptions which arise from a constant potential surfaces when thesurfaces approach each other. more detailed prescription of surface structure. In section IV we present numerical and asymptotic results for the Techniques for scanning surface irregularities with atomic interaction between two surfaces with the same periodic lattice accuracy, such as atomic force microscopy (AFM), now show structure. The calculations are based on distributions of inho- quite convincingly that many systems comprise surfaces which mogeneities which are arguably more relevant than the examples cannot be treated as uniform. In fact, there are many situations given by either Richmond or Kuin. The main effect of periodic wherein nonuniformities can be envisaged to arise. In some cases charge heterogeneities is that the additional interaction that arises this feature has been confirmed by AFM. For example, a host can be repulsive or attractive depending on the relative transverse of these involve solutions of highly surface active species, that is, displacement of the surfaces. In section V, we discuss the species which adsorb strongly. These species may range from procedure of performing the proper statistical mechanical average polyelectrolytes, or bulky multivalent ions, of charge opposite to over possible transverse displacements of the patchy heterogeneous that of the surface to ionic amphiphiles. Our focus is on systems surfaces. The general conclusion is that when averaged over which possess isolated surface regions which are occupied by a transverse displacements, the extra contribution to the interaction concentration of charge opposite to that of the surface as for between from charge heterogeneities is an additional attraction instance in the case of negatively charged mica surfaces with between the surfaces. The paper closes with a summary of the adsorbed or deposited cationic surfactants.3 major results. Richmond4and Nelson and McQuarrieS examined the effects of discrete surface charge distributions on the double layer 11. Electrostatic Double Layer Potential interaction between surfaces. From these studies one already sees that interesting and potentially important behavior can be Consider two infinite planar half-spaces with dielectric per- found. While Richmond studied surfaces held at constant charge, mittivities, c1 and t3. These are separated by a third dielectric continuum of permittivity e2 and of width h; see Figure 1. This * To whom correspondence should be addressed. central medium contains a solution of simple electrolyte of species f University of Lund. valence, in chemical equilibrium with a bulk solution of molar t Department of Mathematics, University of Melbourne. zi, 8 Department of Chemistry, University of Melbourne. concentration cm, or number density, nm, at a temperature T.We Abstract published in Aduance ACS Abstracts. July 15, 1994. are interested in determining the mean electrostatic potential, 0022-3654194 /2098-9022%04.5O JO 0 1994 American Chemical Society Double Layer Forces between Heterogeneous Surfaces The Journal of Physical Chemistry. Vol. 98. No. 36, 1994 9023 For an arbitrary electrolyte the electrostatic potential, $(r), satisfies the following equations V2$(r) = 0 for z < 0, z > h (34 47r V2J.(r) = --p(r) for 0 < z < h (3h) e2 with volume charge density, p(r) at r given by (,9 = 1 JksT) p(r) = xz,qoexp(-z,ep$(r)) (4) i Since the surface source distributions are no longer uniform, the scalar mean potential retains its dependence on the three- dimensional vector position, r, and the equations to be solved are oartial differential eauations. A simolifvine assumotion that can bften be made is to &me that $6)ii small everywhere, Le., &(r)/ksT< I; wemaythen linearize thedependenceoftheionic densities on the mean potential to give Figure 1. A schematic representation of the system under study. Two surfaces with a periodic distribution of sources of charge separated by an electrolyte solution of thickness h. p(r) = -Cz,’e2ndJ.(r) (5) i $(I)across this intervening region, when the interfaces between Note that this approximation is better for symmetric electrolytes medium 1 and 2, at = 0, and between medium 2 and 3, at z z as the O(Y) term in theexpansionvanishes identically. Basically, = h, carry specified surface charge or surface potential distribu- the linearization assumption is needed to achieve any analytical tions. In this paper we concentrate on either of the two limiting progress. However, what qualitative features appear in the conditions where the surface charge or surface potential distribu- linearized model are certain to remain in the more complex tions are independent of separation. Weshall refer to thesecases nonlinear calculation. The approximation we adopt therefore as constant charge or constant potential. The intermediatecases does not detract from our conclusions. Given the form of the of self-consistent regulating surfaces and the asymmetric system source functions the boundaries, together with the linearity of one surface held at constant potential and the other held at on of the problem, the solution for the electrostatic potential $(I) constant charge are made the subject of a separate report. can be expressed as a Fourier expansion in terms of the transverse We consider nonuniform distributions of sources +(s) and coordinates s = (x,y) yR(s)on the left and right surfaces, respectively. We use the subscripts and superscripts Lor R to refer quantities to the left (z = 0) or right (z = h) interfaces, respectively. The variable y represents either the surface potential, 9,or surface charge, g, as the case may be, and s = (x,y) represents the transverse vector where the Fourier coefficients, &), depend on the wave vector position. k and the normal coordinate z. Without lossof generality, wecanassume that for eachsurface, Substitution of (6) into (3) gives a set of four ordinary y contain a uniform part, yo,on which is superimposed a periodic differential equations for each k distribution of sources, yp. Since yp.and