The Effect of Unequal Ionic Size on the Swelling Pressure in Clays

-In this paper, we use the unequal radius modified Gouy-Chapman theory to evaluate the effect of the ionic size of the electrolyte on the swelling pressures (H) in different clay systems immersed in electrolytic solutions. First the model is applied to a l:l electrolyte to show that the coion size is only important at surface charge densities much lower than those found in typical clay systems. The swelling pressure is calculated and the results are compared with experimental data. Literature ionic radii values are used to show the dependence of the swelling pressure on the specific counterions present. Next the model is applied to a 1:1 and 2:1 electrolyte mixture with unequal-sized counterions to show the swelling pressure is highly dependent on both counterion sizes. The unequal and same-sized cases are compared. Key Words--Clay, Swelling pressure, Unequal ion size. I N T R O D U C T I O N Clay swelling is important in many applications including petroleum engineering, soil permeability, waste disposal liner design and the development of many commercially available consumer products. Swelling is thought to be caused primarily by hydration of the clay surfaces at small separations and electrical double layer overlap at large separations. Most of the electrolyte ions in the electrical double layer are found in a narrow region near the charged surface. Finite ion size effects are expected to be most significant in this region. Finite size is usually accounted for by modifying the Poisson-Boltzmann formalism to allow for a region the width of a hydrated ion near the surface in which no charge is found. Given that ionic species are of different size, we will investigate the effect of the unequal size of the electrolyte within the Poisson-Boltzmann theory in the study of clay swelling. Some effort to understand the effect of unequal ionic sizes of the electrolyte on the electrostatic properties of the electrical double layer has been made in the last ten years. A non-zero potential at the electrode in the absence of a surface charge (potential-of-zero-charge) was found for a one-wall system by Valleau and Torrie (1982). That phenomenon was usually attributed to specific adsorption at the electrode, but it seems that it can be accounted for, at least in part, by introducing unequal ionic diameters. Bhuiyan et al. (1983) also reported the effect of considering different sizes in the electrolyte in a one-wall system using the nonlinear Poisson-Boltzmann equation and found results in agreement with Valleau and Torrie (1982). However the corresponding problem of the interaction between two planar surfaces has just recently received attention. McBroom and McQuarrie (1987) reported the effect of the unequal size of the electrolyte on the electrostatic force between two planar surfaces Copyright 9 1992, The Clay Minerals Society in a 1:1 electrolyte using the nonlinear Poisson-Boltzmann equation. The prediction of a non-zero force between two uncharged walls was an interesting result. We will use the unequal-radius-modified-GouyChapman (URMGC) approach to the interaction of two charged surfaces, introduced by McBroom and McQuarrie (1987), to describe the electrostatic interaction of two clay surfaces that are immersed in an electrolytic solution. First we consider a 1:1 electrolyte with an unequal-sized cation and anion. We will show that the anion size is not important in clay swelling. Secondly, due to the importance of clay stability problems, we apply the unequal radius model to a 1"1 and 2:1 electrolyte mixture such as NaC1 and CaC12 to show the dependence of the swelling pressure on both counteflon sizes. T H E O R Y Unequal radius model for a 1:1 electrolyte We model the clay surfaces as a two-wall system with a uniform surface charge density ao (ao < 0) immersed in an aqueous electrolyte solution. The solution is modelled by the unrestricted primitive model, where the ions are considered as hard spheres with the cation of radius R+ with a charge z+e and the anion of radius R_ with a charge z_e, where e (e > 0) is the protonic charge. In the first case we will consider a symmetrical, univalent salt with z+ = z = 1. The Poisson-Boltzmann equation for this system is d2~b zi -~ ~exp(-zi th) (1) i where ~ = Kx, with ~ the distance from one of the walls and K 2 = 2e2c0/eo&T the usual Debye-Hfickel parameter, co is the bulk number density of electrolyte, 4~ = eff/kT is a reduced electrostatic potential, e is the dielectric constant, k is the Boltzmann constant, and T 491 492 Huerta, Curry, and McQuarrie Clays and Clay Minerals