ELEMENTS OF DOUBLE-LAYER THEORY Mathematical description of the diffuse double layer

has been developed for the assumptions of both planar and spherical surfaces. The planar (one-dimensional) case is a reasonable assumption for platy clay particles.

The following idealizing assumptions are made:

1. Ions in the double layer are point charges, and there are no interactions between them.

2. Charge on the particle surface is uniformly dis-tributed.

3. The particle surface is a plate that is large relative to the thickness of the double layer (one-dimensional condition).

Figure 6.11 Variation of electrical potential with distance from a charged surface according to the idealized Gouy–

Chapman theory. Except in very unusual cases,_in soils is negative.

Figure 6.12 Ion concentrations in a potential field.

4. The permittivity⁴ of the medium adjacent to the particle surface is independent of position.

The concentration of ions (ions / m³) of type i, ni, in a force field at equilibrium is given by the Boltzmann equation:

E_i0⫺ Ei

nⁱ_⫽ n expⁱ⁰ 冉 冊^kT (6.9) The subscript 0 represents the reference state, taken to be at a large distance from the surface, E is the poten-tial energy, T is temperature (K), and k is the Boltz-mann constant (the gas constant per molecule) (1.38_⫻ 10^⫺23J K^⫺1).

The potential energy of an ion in an electric field is

Ei_{⫽ v}ie_ (6.10)

wherev_iis the ionic valence, e is the electronic charge (1.602 _⫻ 10^⫺19C), and_ is the electrical potential at the point.⁵ Potential varies with distance from a charged surface in the manner shown by Fig. 6.11. In clays, _ is negative because of the negative surface charge. The potential at the surface is designated as

0. As E_i0⫽0, because_{ ⫽}0 at a large distance from the surface,

E_i0⫺ Ei_{⫽ ⫺v}ie_

so the Boltzmann equation [Eq. (6.7)] becomes

⫺vie_

nⁱ_⫽ n expⁱ⁰ 冉 冊^kT (6.11) Equation (6.11) relates concentration to potential, as illustrated by Fig. 6.12. For negatively charged clay

4The permittivity is a measure of the ease with which molecules can be polarized and oriented in an electric field. Quantitatively, the per-mittivity is defined by in Coulomb’s equation for the force of elec-trostatic attraction F between two charges Q and Q separated by a distance d; that is,

F⫽QQ₂

d

The relative permittivity or dielectric constant D is given by ⫽ 0D in which0is the permittivity of vacuum. D is the ratio of the elec-trostatic capacity of condenser plates separated by the given material to that of the same condenser with vacuum between the plates. The dielectric constant of free water at 20C is about 80. The permittivity of vacuum0is 8.8542⫻ 10^⫺12C²J^⫺1m^⫺1.

5The electrical potential is defined as the work to bring a positive unit charge from a reference state to the specified point in the electric field.

particles, n^⫹i _⬎ n and ni0 ^⫺i _⬍ n ,i0, where _⫹ and _⫺ are for cations and anions, respectively.

The Poisson equation relates potential, charge, and distance. In one-dimension:

d2_{ }

⫽ ⫺ (6.12)

dx2 _

156 6 SOIL–WATER–CHEMICAL INTERACTIONS

in which x is distance from the surface (m),is charge density (C / m³), and _is the static permittivity of the medium (C² J^⫺1 m^⫺1 or F m^⫺1). The charge density in the diffuse layer is contributed by the ions so that

 ⫽e冘v^{i i}n (6.13) with n_iexpressed as ions per unit volume.

Substitution for n_i from Eq. (6.11) gives

⫺vie_

 ⫽e冘 v^{i i0}n exp冉 冊^kT (6.14) which, when substituted into Eq. (6.12), yields

d2_ e _⫺vie_

⫽ ⫺ 冘v^{i i0}n exp冉 冊 (6.15)

dx2 _ kT

Equation (6.15) is the differential equation for the elec-tric double layer adjacent to a planar surface. Solutions of this equation provide a basis for computation of electrical potential and ion concentrations as a function of distance from the surface.

For the case of a single cation and anion species of equal valence, that is, i _⫽ 2 and n^⫹_{0 ⫽} n^⫺_{0 ⫽} n₀ and

Some explicit solutions of Eq. (6.16) are available (Verwey and Overbeek, 1948; Bolt, 1955, 1956; Bab-cock, 1963; van Olphen, 1977; Mitchell, 1993).

Clay particles are characterized by a constant sur-face charge density determined by the amount of unbalanced isomorphous substitution in the clay structure. The surface charge density is proportional to the cation exchange capacity divided by the specific surface. Double layers in many colloidal systems are controlled by a constant surface potential, determined by the concentration of ‘‘potential-determining ions’’

in solution. Diffuse layers that form at the edges of clay particles are of this type because the ions in so-lution control the amount of dissociation of alumina in the octahedral sheets of clay minerals. The equations given here are for constant surface charge.

Single Diffuse Double Layer

Solutions of the differential equation of the double layer are usually given in terms of the dimensionless quantities

ve_ y _⫽

kT Potential functions (6.17) ve_0

冧

z _⫽ kT and

 ⫽Kx Distance function (6.18) where

2n e₀ 2 2v

K2_⫽ (6.19)

kT

The solution to Eq. (6.16) describes a roughly expo-nential decay of potential with distance from the sur-face. For surface potentials less than about 25 mV, the potential decreases purely exponentially with distance, and the center of gravity of the diffuse charge is at a distance x _⫽ 1 / K from the surface. This distance is a measure of the ‘‘thickness’’ of the double layer.

According to Eq. (6.19) the value of 1 /K depends only on the characteristics of the dissolved salts and the fluid phase. However, the actual values of concen-tration and potential at any distance from the surface also depend on the particle surface charge, surface po-tential, and specific surface and dissolved ion interac-tions, and these depend on the type of clay and conditions in the pore solution.

The double-layer charge is given by

 ⫽ ⫺冕0⬁^{ dx (6.20)}

the solution of which is

1 / 2 z

 ⫽(8n_0kT) sinh (6.21) 2

which for small values of_0 reduces to

 ⫽ k_0 (6.22)

A single diffuse double layer is not representative of the actual conditions in most clay systems because double layers of adjacent particles will overlap. None-theless, the above equations are useful for understand-ing some effects of changes in solution composition and concentration on diffuse layer thickness, which can then be related to the behavior of clay suspensions, as discussed later.

Figure 6.13 Potential and charge distributions for interact-ing double layers from parallel flat plates: (a) potential and (b) charge.

Interacting Double Layers

The electrical potential and charge distributions for the case of interacting double layers from parallel flat plates, separated at distance 2d are shown in Fig. 6.13.

The potential function at the midplane y_{⫽ v}e_c/ kT is denoted by u, and the integration boundary conditions for Eq. (6.20) are that for  ⫽ Kd, y _⫽ u, and dy / d

⫽ 0. Values of u for conditions of constant surface charge are tabulated by van Olphen (1977) for given values of surface potential, 2d, and K. For small inter-actions, that is, for large values of Kd, as would be the case for large plate separation, high n, highv, or small

0, the midplane potential is close to the sum of the double-layer potentials at distance d based on the

so-lutions for a single plate.⁶ Concentrations at the mid-plane can be obtained from Boltzmann’s equation:

⫺ ⫹

n _⫽ n exp(u)₀ n _⫽n exp(_{0 ⫺}u) (6.23) As the overlap of double layers of the same sign generates an interparticle repulsion, it is important to investigate whether double layers in typical fine-grained soils are sufficiently thick that interactions be-tween adjacent particles will actually occur. Using 1 /K, the distance from the surface to the center of gravity of the diffuse layer, as the thickness of double layer, values of 1 nm in a 0.1 M solution of cation, increasing to 10 nm in a 0.001 M solution are obtained.

For water distributed uniformly on surfaces of clay particles, the water layer thickness is equal to half the particle spacing, or d in Fig. 6.13. This thickness is given by the volumetric water content (cm³/ g) divided by the specific surface area (m²/ g). On this basis, for a water content of 50 percent (water weight / dry solid weight) and a specific surface in the range of 50 to 300 m²/ g, values of d from 1.7 to 10.0 nm are ob-tained. These spacings are well within the range where interactions can be important.

There is much higher concentration of divalent cat-ions than monovalent catcat-ions near the particle surface in a system that contains both monovalent and divalent cations, even if the concentration of monovalent cati-ons is much greater in the bulk solution. According to the DLVO theory, the ratio of concentrations of diva-lent cations to monovadiva-lent cations required to coagu-late colloidal suspensions of clay minerals is only 0.0156 (Sposito, 1989). This value is consistent with experimental observations of the concentrations re-quired to cause coagulation according to the Shultze–

Hardy rule, developed over 100 years ago. This rule states that the critical coagulation concentration of ions in suspension of opposite sign to the charge on the colloid is proportional to an inverse power of the val-ence of the ion, and the power according to DLVO is 6 (Sposito, 1989).

6.9 INFLUENCES OF SYSTEM VARIABLES ON