Layer Rigidity and Collective Effects in Pillared Lamellar Solids

The $x$ dependence of the normalized basal spacing, $d_n\mathrm{}\left(x\right)\mathrm{}$ of pillared vermiculite (Vm) has been measured for the mixed-layer system ${\left[{\left(\mathrm{C}{\mathrm{H}}_3\right)}_4{\mathrm{N}}^{+}\right]}_x{\left[{\left(\mathrm{C}{\mathrm{H}}_3\right)}_3\mathrm{N}{\mathrm{H}}^{+}\right]}_{1-x}-\mathrm{Vm}$ and compared with that of ${\mathrm{Cs}}_x{\mathrm{Rb}}_{1-x}-\mathrm{Vm}$. Both systems exhibit a nonlinear $d_n\mathrm{}\left(x\right)\mathrm{}$ with approximate thresholds of $x\cong 0.2 \mathrm{and} 0.5$, respectively. A model which related $d_n\mathrm{}\left(x\right)\mathrm{}$ to layer rigidity and the binding energies of gallery and defect sites yields excellent fits to the basal spacing data and to monolayer simulations if collective effects are included. This model should be applicable to other types of lamellar solids.