Central-Force Models Which Exhibit a Splay-Rigid Phase

Two models, one random the other periodic, are described which exhibit splay rigidity but are not rigid with respect to compression. The random model is based on a periodic lattice of rhombuses whose sides consist of central-force springs, which is perturbed in the following way: rhombuses can have diagonal central force struts with probability y or they can have one of the horizontal springs removed with probability x. For x,y≪1 we are led to consider a long-ranged anisotropic percolation process which is solved exactly on a Cayley tree. We show that for y/x near 2 the compressional rigidity of this system is zero but the Frank elastic constant, K, describing splay rigidity is nonzero. This is the first example of a percolation model for which this phenomenon, suggested earlier, is conclusively established. For y/x≳2 √2 the system has nonzero bulk and shear moduli. We also study the excitation spectrum for a periodic model which possesses only splay rigidity and obtain a libron dispersion relation ω=cSq, where q is the wave vector and cS∼(K/ρ)1/2, where ρ is the mass density. These results are generalized to obtain a scaling form for cS and the density of states of the random model which is valid when the correlation length for compressional rigidity becomes large.