Updated: Feb 15, 2003
Molecular Dynamic Simulation of an
Elastic Solid with Hexagonal Symmetry
A molecular dynamics simulation is developed for an elastic solid with general hexagonal symmetry. This allows the vibrational frequencies of solid objects made of a single crystal with hexagonal symmetry to be determined. The method is tested for the isotropic special case for exactly known frequencies for an isotropic sphere.

   A C++ computer program hc2mod.cpp is available to accurately compute the vibrational frequencies of an elastic sphere to within the order of 1%. A separate computer program hex2.cpp is used to determined the five force constants starting from the five elastic constants of the material. The lattice used is hexagonal with lattice constants a = 1 and c = 0.93. The ratio c/a must be chosen so as to avoid having some negative force constants. (For the case when Poisson ratio = 0.45, c was changed to 0.80 to avoid instability.) However, this c/a need not equal the c/a ratio of the actual crystal.
   Five different force constants are required in order to represent the general elasticity of a material with hexagonal symmetry. Such a material has five independent elastic constants, C11, C12, C13, C33 and C44. Other elastic constants are dependent on these through: C11=C22=C33, C12=C21, C13=C31=C23=C32, C44=C55, C66=(C11-C12)/2. Finally, constants Cij and Cji with i=1,2,3 and j=4,5,6 are all 0.
   In selecting force rules for the lattice, it is desirable that the connections be as short as possible in order to quickly achieve convergence to the continuum limit as the object increases in number of particles. It is not possible to represent general elastic properties in terms of two-particle force laws alone.
   The first three force constants are simple two-particle force laws: A spring with force constant ksp1 connects particles at distance a (assuming a is unequal to c). Each particle has six such neighbours. A spring with force constant ksp3 connects particles at distance c (assuming a is unequal to c). Each particle has two such neighbours. A spring with force constant ksp4 connects particles at distance sqrt(a2+c2). Each particle has twelve such neighbours.
   There is a three point force term involving a force constant k3pt. This force is calculated not for pairs of atoms, but for equilateral triangular groups of three atoms. Apart from atoms on the surface, each atom is involved in six of these clusters. Consider one such group with atoms numbered j = 0, 1, 2. Let p = 0, 1, 2 label the x, y and z axes. The coordinates of the atoms are rjp. The center of mass of the cluster is at RCM where

RCMp = (1/3) Σj rjp

Define the quantity D by

D = Σj,p (RCMp - rjp)2

The p-component of the force on atom j is given by

Fjp = k3pt (RCMp - rjp) (D - a2)

   Since, for an undistorted lattice, each cluster is a triangle of side a, the distance from each atom to the center of mass is a/sqrt(3). D, the sum of the squared distances to the center of mass, is nominally a2. So there is no force when the lattice is undistorted. From its construction, this force respects the translational and rotational symmetry of space just as the two-body forces do. It also satisfies the requirement that the net force exerted by a cluster on itself is zero. Furthermore, since the forces emanate directly outward from the center of mass, the net torque exerted on the cluster by itself is also zero. This 3-point force can equivalently be defined in terms of the potential energy,

U( r0, r1, r2) = (??coef??) k3pt (D - a2)2

   There is a six point force term involving a force constant k6pt. This force is calculated for groups of six atoms. Apart from atoms on the surface, each atom is involved in twelve of these clusters. Consider one such group with atoms numbered j = 0, 1, 2, ... 5. Let p = 0, 1, 2 label the x, y and z axes. The coordinates of the atoms are rjp. The center of mass of the cluster is at RCM where

RCMp = (1/6) Σj rjp

Define the quantity D by

D = Σj,p (RCMp - rjp)2

The p-component of the force on atom j is given by

Fjp = k6pt (RCMp - rjp) (D - 6(a2/3+c2/4))

   For an undistorted lattice, the distance from each atom to the center of mass of the cluster is (a2/3+c2/4). D, the sum of the squared distances to the center of mass, is nominally 6 (a2/3+c2/4). So there is no force when the lattice is undistorted. From its construction, this force respects the translational and rotational symmetry of space just as the two-body forces do. It also satisfies the requirement that the net force exerted by a cluster on itself is zero. Furthermore, since the forces emanate directly outward from the center of mass, the net torque exerted on the cluster by itself is also zero. This six-point force can equivalently be defined in terms of the potential energy,

U( r0, r1, r2, r3, r4, r5) = (??coef??) k6pt (D - 6 (a2/3+c2/4))2


Table I. Force constants for objects with
isotropic elasticity
(Using C++ program hex2.cpp.)
constant units ν=0.15 ν=0.25 ν=0.35 ν=0.45
a (metres) 1 1 1 1
c (metres) 0.93 0.93 0.93 0.8
C11 (GPa) 90 90 90 90
C44 (GPa) 37.06 30 - - 8.182
k6pt (???) -3.287e9 0 4.298e9 1.181e10
ksp1 (N/m) 5.659e10 4.581e10 3.171e10 9.211e9
k3pt (???) -3.909e9 0 5.112e9 -1.890e9
ksp3 (N/m) 4.970e10 3.548e10 1.690e10 1.807e10
ksp4 (N/m) 2.145e10 1.737e10 1.202e10 4.842e9

Figure 1. Fourier transforms of lattice simulations of isotropic elastic spheres with ν=0.35 are shown. The initial disturbance of the lattice is as shown at lower left. For each plot, the horizontal axis is η = ω R/CS. The vertical axis is 1/R, with infinite R at the top. R at the bottom is 3. The maximum radius and number of atoms, N, used is shown at the lower right. (hc2mod.cpp hcp35sp0.gif hcp35to2.gif...)

Figure 2. Fourier transforms of lattice simulations of isotropic elastic spheres with ν=0.45 are shown. The initial disturbance of the lattice is as shown at lower left. For each plot, the horizontal axis is η = ω R/CS. The vertical axis is 1/R, with infinite R at the top. R at the bottom is 3. The maximum radius and number of atoms, N, used is shown at the lower right. (hc2mod.cpp hcp45to2.gif...hcp45sp0.gif )

Figure 3. Fourier transforms of lattice simulations of isotropic elastic spheres with ν=0.15 are shown. The initial disturbance of the lattice is as shown at lower left. For each plot, the horizontal axis is η = ω R/CS. The vertical axis is 1/R, with infinite R at the top. R at the bottom is 3. The maximum radius and number of atoms, N, used is shown at the lower right. (hc2mod.cpp hcp15to2.gif...hcp15sp0.gif )

Table II: Comparison for ν = 0.35
Typel η
(simul.)		 η
(exact)		 % diff.
sph 0 5.80 5.790 +0.2%
tor 2 2.50 2.502 -0.1%
sph 2 2.65 2.652 -0.1%
sph 1 3.65 3.628 +0.6%
tor 3 3.95 3.865 +2.2%
sph 3 3.96 3.957 +0.1%

Table III: Comparison for ν = 0.45
Typel η
(simul.)		 η
(exact)		 % diff.
tor 2 2.52 2.502 +0.7%
sph 2 2.66 2.662 -0.1%
sph 1 no peak 3.798 N/A
tor 3 3.95 3.865 +2.2%
sph 3 3.98 3.989 -0.2%
sph 0 10.05 10.006 +0.4%

Table IV: Comparison for ν = 0.15
Typel η
(simul.)		 η
(exact)		 % diff.
tor 2 2.50 2.502 -0.1%
sph 2 2.65 2.625 +1.0%
sph 1 3.18 3.195 -0.5%
sph 0 3.68 3.680 0%
tor 3 3.93 3.865 +1.7%
sph 3 3.93 3.867 +1.6%

   The agreement between the molecular dynamics simulations and exactly known frequencies for spheres with isotropic elasticity is within 2%, and usually within 1%. Only in the l = 1 spheroidal case at ν = 0.45 did a peak fail to appear. The rate of convergence is variable, the general pattern being that convergence is best for the lowest frequency modes. The estimated frequencies are more often higher than the exact frequencies.
References:

D. B. Murray "Eight Point Force Molecular Dynamical Estimates of Vibrational Frequencies of an Isotropic Elastic Sphere" 2002 link to article

D. B. Murray "Breathing Mode Vibrations of an Isotropic Elastic Sphere Surrounded by a Fluid Medium" 2002 link to article

D. B. Murray "Vibrational Frequency of Silicon Nanoparticles" 2002 link to article

http://www.chem.vt.edu/chem-dept/tissue/nano/
"While the elastic modes of spherical nanoparticles can be calculated analytically, this is not the case for nonspherical nanoparticles." "Visscher et al. ...mode structure for ellipsoidal nanoparticles..." ::

W. M. Visscher, A. Migliori, T. M. Bell and R. A. Reinhart, "The normal modes of free vibration of inhomogenous and anisotropic elastic objects" J. Acoust. Soc. Am. (Journal of the Acoustic Society of America) Vol. 90 (1991) 2154-2161.

A. Stekel, J. L. Sarrao, T. M. Bell, M. Lei, R. G. Leisure, W. M. Visscher, A. Migliori, "Method of Identification of the Vibrational Modes of a Rectangular Parallelipiped" J. Acoust. Soc. Am. vol. 92 no. 2 pt. 1 (August 1992) pp. 663-668.

A. Migliori, J. L. Sarrao, W. M. Visscher, T. M. Bell, M. Lei, Z. Fisk and R. G. Leisure, "Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids" Physica B 183 1-24 (1993)

M. K�gl, L. Gaul A 3-D Boundary Element Method for Dynamic Analysis of Anisotropic Elastic Solids Computer Modeling in Engineering & Sciences, 1(4) (2000), pp. 27-43 abstract

Daniel Murray
Associate Professor
Math, Stats & Physics Unit
University of British Columbia - Okanagan
Kelowna, BC, Canada
daniel "dot" murray "at" ubc "dot" ca

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