# Environmetal Soil Properties and Behaviour

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of the adsorbed hydrated ion. For hydrated Na ions for example, this would be 1.58 nm. Edge to face interaction Edge to edge interaction Clay particle Face to face interaction 1 µ FIguRE 3.7 Sketches show the principal modes of interaction between particles used for interparticle energy calculations. Transmission electron microscopic (TEM) picture shows an example of the three modes of interparticle and interaggregate interactions for a cleavage surface in a sheared kaolinite sample. 96 Environmental Soil Properties and Behaviour The energy of attraction between interacting parallel-faced particles, Eaff utilizes the London\u2013van der Waals attraction energy for two similar flat plates, and is obtained as follows: E A xa ff = 12 2pi (3.10) where A is the Hamaker constant. The total interaction energy ETff for the face-to-face configuration of particles is the sum of the Erff and Eaff energies. The relationship for this total energy as affected by NaCl concentration is shown in Figure 3.8, together with the results for the face-to-edge and edge- to-edge total energies. To calculate the face-to-edge repulsion and attraction energies, two interacting spheres are adopted as the model, with one sphere having a very large radius af with potential \u3c8f in comparison to the other with a much smaller radius ae with its corresponding potential \u3c8e. The interaction of the very large radius af sphere in comparison to the other very small radius sphere is meant to portray a flat particle interaction with a particle edge. The energies of face-to-edge repulsion Erfe and attraction Eafe are given as follows E a a a ar fe f e f e f e f e f e = +( ) + + \uf8ee \uf8f0 \uf8ef\uf8ef \u3b5 \u3c8 \u3c8 \u3c8 \u3c8 \u3c8 \u3c8 2 2 2 24 2 ( ) \uf8f9\uf8f9 \uf8fb \uf8fa\uf8fa + \u2212 \u2212 \u2212 + \u2212 \u2212ln exp( ) exp( ) ln exp( 1 1 1 2 D x D x D xH H H ))\uf8ee\uf8f0 \uf8f9\uf8fb (3.11) E A r x x r x r x x r r x a fe a a a a a a a a a a a = \u2212 + + + + + + 12 2 2 2 2 ln ++ + + + + \uf8eb \uf8ed\uf8ec \uf8f6 \uf8f8\uf8f7 x r x x x r x r a a a a a a a a 2 (3.12) where xa = x/(2af ), and ra = ae/af. The calculation of the edge-to-edge repulsion and attraction energies assumes small interacting spheres with identical radii of ae. The energies of repulsion Eree and attraction Eaee are obtained as follows E a D xr ee e e H= + \u2212\uf8ee\uf8f0 \uf8f9\uf8fb\u3b5 \u3c8 2 2 1ln exp( ) (3.13) E A x x x x x x xa ee a a a a a a a = \u2212 + + + + + + +12 1 2 1 2 1 2 2 2 2 2 2ln 22 1xa + \uf8eb \uf8ed\uf8ec \uf8f6 \uf8f8\uf8f7 (3.14) The calculated results shown in Figure 3.8 (Yong et al., 1979) used actual zeta potential measurements from experiments using a zetametre. The assump- tion of the edge zeta potential being one-fifth of that for the surface of the kaolinite particle needs to be verified or determined for other types of clay minerals, since the magnitude of the edge potential affects the magnitude 97Soil\u2013Water Systems of the net energies of interaction. For more precise calculations using the DLVO model relationships, different values of Hamakaer\u2019s constant for the faces and edges of particles should be used. Calculations based on the DLVO model show good accord for ideal systems where particle separation dis- tance is above 3 nm. At lower particle separation distances, the repulsive energy is overwhelmed by the van der Waals forces, as will be seen in the later discussion relating to swelling pressures of clays. 3.3.2.4 Energies in Particle Interaction and in Interlayer Space When two clay particles are less than 1.5 nm apart, the exchangeable ions are uniformly distributed in the interparticle space and do not separate into two diffuse double layers: one associated with each particle. Under these condi- tions, there is a net attraction between particles. For clay minerals, one should be mindful of interactions in interlayer space, that is, the space between repeating layers of clay minerals, in addition to interactions between clay particles. When interparticle and interlayer distances exceed 1.5 nm, diffuse double layers are formed, and net repulsion results. The concentration of ions is higher in the plane midway between the particles and repeating layers. Figure 3.9 provides a schematic illustration of the interactions in interlayer 1 2 3 1 2 3 0 \u201325 0 25 \u201350 Lo g x , n m E T \u3baT × 10 12 E T \u3baT × 10 0 NaCl Concentration, mEq/L 0.01 0.1 1.0 10 Face-to-edge Face-to-face Edge-to-edge FIguRE 3.8 Maximum interaction energies (ET/\u3baT) and their corresponding interparticle distances (x) at various NaCl concentrations for the three different particle configurations. The total interac- tion energy ET is the sum of the repulsive and attractive energies for the respective particle configurations. 98 Environmental Soil Properties and Behaviour space and development of the electronic potential, using the sketch of the montmorillonite mineral shown in Figure 2.5 in Chapter 2. The Poisson\u2013Boltzmann relationship between two charged particle sur- faces (see Equation 3.2) is rewritten as d y dx K y2 2 2 2 = exp( ) (3.15) where y = \u2013 zie\u3c8/\u3baT is a dimensionless potential, and K e z n Ti i= 8 2 2pi \u3b5\u3ba/ . To determine the midplane potential \u3c8m which tells one about the energy of interaction between the two particles or interlayers (Figure 3.9), the solution given by Langmuir (1938) is used. This solution, which specifies (a) the boundary condition that y = ym at the midplane between the particles for the first integration, and (b) the boundary condition that y \u2192 \u221e at x = 0 for the second integration for Equation (3.15), is given as y Kxm = 2 ln pi (3.16) The solution for the concentration of ions at the plane midway between the particles is given as Aluminium dioctahedral smectite (Montmorillonite) Repeating 2:1 unit layer thickness, 1 nm Potential \u3c8 Resultant interaction potential \u3c8ip Midplane potential \u3c8m FIguRE 3.9 Basic repeating 2:1 unit layers for dioctahedral smectite (montmorillonite) showing expanded view of repeating layers and the interaction of their electric potential \u3c8 and development of the resultant interaction potential \u3c8ip. 99Soil\u2013Water Systems c c K xc o = pi 2 2 2 (3.17) The Langmuir solution neglects anions, a condition that is acceptable at small distances between particles; that is, at low water contents, the anions are mostly excluded from the space between clay particles. 3.3.2.5 Calculated Swelling Pressures from Interaction Models Calculations of swelling pressures in clays are useful in the assessment of potential behaviour of swelling clays used for engineered barriers and buf- fers. In most instances, the theoretical diffuse double-layer (DDL) models used for these calculations assume interactions between parallel particles or unit layers. This simplifying assumption provides a basis for comparison with experimental swelling pressure information where samples are prepared with parallel particle orientation for swelling pressure tests. There have been tests conducted on clay samples without control on initial particle orientation, that is, on samples with indeterminate soil fabric and soil structure. Whilst swell- ing clay samples can be prepared without particle orientation control, it is dif- ficult for comparisons of measured swelling pressure to be made with results obtained from theoretical models, since these models will have to assign effec- tive particle spacings to various particle orientation configurations other than parallel. Whilst one could argue that the DLVO model might be used for such instances, one needs to remember that whilst the DLVO model permits one to make calculations