Application of One-Dimensional Lattice Model to Mixed-Layer Muscovite-Montmorillonite

Abstract

The one-dimensional Ising (regular solution) model is a first-order statistical mechanical approximation to real muscovite-montmorillonite mixed layer clays. The model assumes a constant excess interaction energy, w, between the unlike layers; $$w = {w_{ab}} - {\textstyle{1 \over 2}}\left({w_{aa}} + {w_{bb}} \right)$$ Exact solution of the model, applicable to infinitely long chains, can be given by the quasi-chemical formula $$\overline{N_{aa}}\overline{N_{bb}}/\overline{N_{{ab}^2}} = \left( {{\textstyle{1 \over 4}}} \right)\,\exp \,\left( {2w/kT} \right)$$ where Nab is the equilibrium value of the number of a–b type of neighbors, etc. When w → + ∞, discrete crystals result; when w → − ∞ and Na = Nb, regular l:l mixed layer crystals result; when w = 0, random mixed layering results. For finite values of w, the mixed layering is irregular though non-random. Practically, however, either discrete or regularly mixed-layer crystals can obtain even for finite values of w calorimetrically too small to measure. Using the Ising model, the values of w/kT and µi/kT (where µi is the excess chemical potential of the ith type of layers) were calculated for three clays whose probability of layer succession, pij, had been evaluated by the MacEwan method. For two muscovite-montmorillonite mixed layer clays, w 0, as is expectable from crystallochemical considerations. For thin plates of equal numbers of a, b layers, a correction factor [(N − 2)/N]2 (where N = Na + Nb) must be applied even for ideal crystals. For such finite crystals, the partition function for non-ideal mixtures of specified Na and Nb can be evaluated directly, introducing a second correction to the quasi-chemical relation. Because of end effects, it is possible that Naa ≠ Nbb even for Na = Nb and w = 0, provided waa ≠ wbb. Application of the Ising model to real crystals depends on our ability to correlate X-ray diffraction patterns with run sequences in crystals. Computer calculations of expected diffraction patterns for thin crystals having various values of Na and Nb are being undertaken and should be useful towards this end.