Joint Applied Mathematics -  Physics Seminar

                  Rigidity in Random Networks               

                             by 

      	Michael F. Thorpe, Physics Dept., Michigan State

             INSCC 110, 3:30pm  Monday, May 11th, 1998


                          Abstract




The simple yet  powerful ideas of  percolation theory have  found
their way into many different areas of research.  In this talk we show
how RIGIDITY PERCOLATION can be  studied at a  similar  level of  
sophistication,  using a  powerful new program THE PEBBLE GAME [D.  J.
 Jacobs and M.  F.  Thorpe,  Phys. Rev. E 53,  3682 (1996)] 
that  uses an integer algorithm.  This program can analyse the
rigidity of two and three  dimensional networks containing more than
one million bars and joints.  We find the total number of floppy
modes, and find the critical behavior as the network  goes from floppy
to rigid as more bars are added.

We discuss the relevance of this work to network glasses,  and how it
relates to experiments that involve mechanical  properties like
hardness and elasticity of covalent  glassy  networks such as 
Ge_xAs_ySe_{1-x-y} and  dicuss recent experiments   that 
suggest that  the  rigidity  transition  may be  first order [Xingwei 
Feng,  W.   J.Bresser  and P.   Boolchand,  Phys.  Rev.  Lett
78, 4422 (1997)].

We have found that rigidity nucleates  most readily on small rings
[smaller than six bonds] and that if such nucleation sites are
supressed, the transition is more likely to be first order in the
absence of nucleating rings.  In addition a ring can only be
nucleating if it has three connections or more to the network
coming from it - so that fourfold rings in GeSe glasses, formed from
edge sharing tetrahedra, are not effective as nucleation centers. 
Thus the nature of the rigidity transtion, as the composition of the
glass is changed, can give us useful information about the ring
structure and hence medium range order.

This approach  is also useful  in  macromolecules and  proteins, 
where detailed information about the rigid domain structure can be
obtained.

This work is done jointly with Don. A. Jacobs.

Requests for preprints and reprints to: thorpe@pa.msu.edu      
This source can be found at http://www.math.utah.edu/applied-math/