Rigidity of randomly intercalated layered solids

We set up a harmonic spring model that describes both the layer rigidity and the size and stiffness of the intercalant species. In certain limiting cases, when (1) the layers are either perfectly floppy or perfectly rigid and (2) the stiffness of the two intercalant species are the same, the model can be solved exactly. We also give an effective-medium solution that reproduces all the known exact results, and agrees well with numerical simulations in other cases. These simulations are performed for both one- and two-dimensional systems. If the two intercalant species have the same spring constant, Vegards law is recovered. We compute the probability distribution of the various interlayer distances and apply the results to two-dimensional alloys of Li and vacancies in graphite, and to K and Rb in graphite.