# Environmetal Soil Properties and Behaviour

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\u3c1 is the space charge density and \u3b5 is the dielectric constant. \u2022 The ions in solution are considered to be point-like in nature, that is, zero-volume condition. \u2022 The space charge density \u3c1 of the ions that contribute to the interactions can be described by the Boltzmann distribution: \u3c1 = \u2211nizie exp (\u2013zie\u3c8 / \u43aT). With these conditions, the Poisson\u2013Boltzmann relationship for \u3c8 can be obtained as follows: d dx n z e z e Ti i i i 2 2 4\u3c8 pi \u3b5 \u3c8 \u3ba = \u2212 \u2212 \uf8eb \uf8ed\uf8ec \uf8f6 \uf8f8\uf8f7\u2211 exp (3.2) The relationship for \u3c8 can be obtained (e.g., Kruyt, 1952; van Olphen, 1977; Yong, 2001) as follows: \u3c8 \u3ba pi \u3b5\u3ba = \u2212 \uf8eb \uf8ed\uf8ec\uf8ec \uf8f6 \uf8f8\uf8f7\uf8f7 2 2 8 2 2T e x e z n T i iln coth (3.3) The relationship between the surface charge \u3c3s and surface potential \u3c8s for constant surface charge minerals is \u3c3 \u3b5\u3ba pi \u3ba s i i s n T z e T = \uf8eb \uf8ed\uf8ec \uf8f6 \uf8f8\uf8f7 2 2 1 2 sinh \u3c8 (3.4) For constant surface charge minerals, the surface charge density is constant. Accordingly, if ni and zi are increased, the electric potential \u3c8s at the surface of the clay particle will be reduced, and the thickness of the diffuse double- layer will be correspondingly reduced. For pH-dependent surface charge clay particles, we need to account for the dependence of the surface potential \u3c8s on the potential determining ions (pdis). The Nernst equation for the sur- face potential \u3c8s is given as \u3c8s = 2.303 \u43aT/\u454 (pHo \u2013 pH), where pHo is the pH at which the surface potential \u3c8s = 0. Accordingly, the relationship between the surface charge density \u3c3s and surface potential \u3c8s for pH-dependent surface charge clay particles is \u3c3 \u3b5\u3ba pi \u3b5s i i o n T z e pH pH= \uf8eb \uf8ed\uf8ec \uf8f6 \uf8f8\uf8f7 \u2212( )2 1 15 1 2 sinh . (3.5) 93Soil\u2013Water Systems 3.3.2 Interaction Energies 3.3.2.1 Energies of Interaction in the Stern Layer The energy of interaction in the inner Helmholtz plane (IHP), Eihp , is obtained directly from a knowledge of the interaction between the ions in solution and forces emanating from the surface of a clay particle. From a consideration of Coulomb forces, we obtain (Yong, 2001): E E z z e rihp c i j = = 2 4pi\u3b5 (3.6) where Ec is the Coulombic interaction energy, zi and zj are the valences of the i and j species of ion in the region under consideration, and r is the distance between the centre of the ith ion and jth ion. The interaction energy in the outer Helmholtz plane (OHP), Eohp, includes interaction with the Coulomb forces and also the short-range forces due to (a) ion\u2013dipole interaction, (b) dipole\u2013dipole interaction, and (c) dipole\u2013site interaction. The Coulomb force is the strongest, and its effect is felt at long distances, quite often exceeding some chemical bonding forces. Taking r as the distance between the centre of a dipole and the correspond- ing ion, the ion\u2013dipole interaction Eid is given as follows: \u2022 Ion\u2013dipole interaction Eid given as Eid = \u2013 \u3bcze cos\u3b8 / 4\u3c0\u3b5r2 \u2022 Dipole\u2013dipole interaction Edd is given as follows: E rdd = \u2212 µ µ pi\u3b5 1 1 34 D where \u3bc is the dipole moment, D is a function of angles of the dipoles, and the subscripts 1 and 2 refer to the respective dipoles. \u2022 Dipole\u2013site interaction energy Eds is similar in almost all respects to the ion\u2013dipole interaction energy, with the exception that the dis- tance r is now taken to be the distance between the centre of a dipole and the charge site on the particle. 3.3.2.2 London\u2013van der Waals Energy and Total Intermolecular Pair Potential The London\u2013van der Waals force, which is the force between ions and between molecules, is sometimes called London dispersion forces or simply van der Waals forces. This force, which operates between all molecules regardless of the electrostatic conditions of molecules, exceeds the dipole-dependent forces. The van der Waals force is operative from 0.2 nm to more than 10 nm in distance and is a repulsive or attractive force depending on the distance between molecules. The van der Waals interaction energy Evdw is given by Evdw = \u2013 cicj / r6, where r is the distance between molecules, and ci, cj are the London dispersion force constants of molecules i and j. These consist mainly of the ionization potential, dielectric constant, and polarizability. 94 Environmental Soil Properties and Behaviour The potential due to the repulsive force between a pair of molecules can be determined or expressed in two different ways: (a) using an empirical form of hard-sphere potential or power-law potential and (b) using the exponential potential established on the basis of quantum mechanics. The power law potential Erp is described by Erp = A/rn, where r is the distance between the molecules, A is a coefficient, and a commonly adopted value is n = 12. The exponential potential is given as follows: Erp = b exp(\u2013cr), where b and c are coefficients. The total intermolecular pair potential Etip used for uncharged colloidal particles, which is called the Lennard\u2013Jones form or 12-6 potential, is com- monly described as follows: E A r c c rtip i j = \u221212 6 (3.7) In clay\u2013water systems with charged ions and charged clay minerals, the total intermolecular pair potential is given as E z z e r c c r b crtip i j i j = \u2212 + \u2212 2 64pi\u3b5 exp( ) (3.8) This intermolecular pair potential will describe the mode of ion desorption on clay minerals, and in addition, the properties of soil water such as density, vis- cosity, and self-diffusion coefficient of water molecules, with proper constants. 3.3.2.3 DLVO Model and Particle Interaction Energies The DLVO (Derjaguin, Landau, Verwey, and Overbeek) particle interaction energy model, to which the diffuse double-layer (DDL) concept is applied, is useful for calculations of interparticle or interaggregate interaction. In this instance, calculations account for the nature of the charged clay particle surfaces, chemical composition of the clay-water system, and particle con- figuration. Figure 3.7 shows the three principal configurations for particle interaction used in most calculations. The energy interaction calculations consider van der Waals\u2019 attractive force and the DDL forces as the primary forces responsible for the energies of interaction between the particles. The particle interaction models reported by Flegmann et al. (1969) form the basis for calculation of the maximum energies of interparticle action for similar charged surfaces in face-to-face and edge-to-edge particle arrangement, and the relationship reported by Hogg et al. (1966) is used for dissimilar charged surfaces (edge-to-face). In this instance, the surface potentials are defined by the zeta potential at the edges and at the faces of the particles. The zeta potential at the edge of the particle is assumed to be one-fifth that on the face of the particle, consistent with the observation reported by Ferris and Jepson (1975). 95Soil\u2013Water Systems Assuming constant surface potential and potential-determining ion influ- ence on surface potentials (i.e., pH dependent), the energy of repulsion between interacting parallel-faced particles, Erff, is obtained as follows: E n T yz D x D D xr ff i i H H H = \u2212 \uf8eb \uf8ed\uf8ec \uf8f6 \uf8f8\uf8f7 + \u2212 4 2 1 2\u3ba ( ) exp exp( )\uf8ee\uf8ee\uf8f0 \uf8f9\uf8fb (3.9) where y = \u2212(e\u3c8oh /\u3baT) is a dimensionless potential; e, \u3c8oh, \u3ba, and T have been previously defined as electronic charge, potential at the ohp (see Figure 3.5), Boltzmann constant, and temperature, respectively; zi and ni are valence and bulk concentration of counterions, respectively; x is the interparticle distance; and DH is the Debye\u2013Hueckle reciprocal length. The distance x can be taken as the distance between particle surfaces minus the distance of closest approach determined by the size