## Department of Physics Papers

#### Document Type

Journal Article

#### Date of this Version

10-1-1989

#### Publication Source

Physical Review B

#### Volume

40

#### Issue

10

#### Start Page

7256

#### Last Page

7264

#### DOI

10.1103/PhysRevB.40.7256

#### Abstract

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=c_{S}q, where *q* is the wave vector and c_{S}∼(K/ρ)^{1/2}, where *ρ* is the mass density. These results are generalized to obtain a scaling form for c_{S} and the density of states of the random model which is valid when the correlation length for compressional rigidity becomes large.

#### Recommended Citation

Wang, J.,
&
Harris, A.
(1989).
Central-Force Models Which Exhibit a Splay-Rigid Phase.
*Physical Review B,*
*40*
(10),
7256-7264.
http://dx.doi.org/10.1103/PhysRevB.40.7256

**Date Posted:** 12 August 2015

This document has been peer reviewed.