APS March Meeting 2010
Volume 55, Number 2
Monday–Friday, March 15–19, 2010;
Portland, Oregon
Session Q2: Jamming
11:15 AM–2:15 PM,
Wednesday, March 17, 2010
Room: Oregon Ballroom 202
Sponsoring
Units:
DCMP GSNP
Chair: Wouter Ellenbroek, University of Pennsylvania
Abstract ID: BAPS.2010.MAR.Q2.5
Abstract: Q2.00005 : Elasticity and response in nearly isostatic periodic lattices
1:39 PM–2:15 PM
Preview Abstract
Abstract
Author:
Xiaoming Mao
(University of Pennsylvania)
The emergence of rigidity, especially when complicated by
disorder, is a subtle phenomenon that occurs in a wide variety of
systems. Isostatic lattices, such as the $d$-dimensional
hypercubic lattice and the $2d$ kagome lattice with nearest
neighbor springs, in which the number of contacts $z$ per
particle in $d$-dimensions is equal to $z_c=2d$, provide simple
models, which inform us about systems like jammed solids,
glasses, colloidal suspensions, and foams, for the analytic study
of general features of the onset of rigidity. Isostatic lattices
are marginally stable and may exhibit a non-extensive number of
zero modes that can be removed by the addition of bond-bending
forces, negative pressure, or additional springs. We use the
coherent potential approximation (CPA) to study the onset of
rigidity induced by randomly adding next-nearest-neighbor ($NNN$)
bonds to the square and kagome lattices, and we relate the
results to the random packings of frictionless spheres at point
J. We identify a characteristic frequency scale $\omega^*$ and
length scale $l^*$ and show that within the CPA they scale,
respectively, as $\Delta z$ and $\Delta z^{-1}$ where $\Delta z =
z - z_c$. This result, which replicates results near jamming, is
a result of strongly nonaffine elastic response at small $\Delta
z$. We find that the frequency-dependent effective $NNN$ spring
constant $\kappa$ obeys a scaling relation $\kappa ( \omega)/\kappa
(0) = f(\omega/\omega*)$, where $\kappa ( 0 ) \sim (\Delta z)^2$.
In the square lattice the shear modulus $G(\omega)$ is equal to
$\kappa ( \omega)$, whereas in the kagome lattice, the shear
modulus $G_0$ at $\Delta z = 0$ is finite and proportional to the
spring constant of nearest-neighbor bonds, and $G(\omega) -G_0
\sim \kappa ( \omega)$, Finally, we show that the CPA exhibits
strong phonon scattering for $\omega >\omega^*$ indicating a
Ioffe-Regel limit for heat transport. This work was supported by
NSF DMR 0804900.
To cite this abstract, use the following reference: http://meetings.aps.org/link/BAPS.2010.MAR.Q2.5