Mechanics of Solids (about journal) Mechanics of Solids
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IssuesArchive of Issues2003-4pp.76-88

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E. A. Ivanova, A. M. Krivtsov, N. F. Morozov, and A. D. Firsova, "Description of crystal packing of particles with torque interaction," Mech. Solids. 38 (4), 76-88 (2003)
Year 2003 Volume 38 Number 4 Pages 76-88
Title Description of crystal packing of particles with torque interaction
Author(s) E. A. Ivanova (St. Petersburg)
A. M. Krivtsov (St. Petersburg)
N. F. Morozov (St. Petersburg)
A. D. Firsova (St. Petersburg)
Abstract Pairwise interaction potentials, for example, Lennard-Jones, Mie, or Morse potentials, are widely utilized for modeling molecular systems. These potential have a clear physical meaning and, although rather simple, enable one to describe the properties of a great number of substances on the qualitative level. It is well known, however, that these potentials have essential drawbacks that restrict their application area. One of the major drawbacks is that the pairwise potentials can provide stability only for crystal lattices with dense packing. This is a triangular lattice in the 2D case and a face-centered cubic lattice (or one similar to that) in the 3D case. 3D and 2D lattices with lower packing densities (e.g., a simple cubic lattice) are unstable for most of the pairwise potentials. This is for that reason that the distance between the second neighbors (along the diagonals of the cube) lies on the unstable segment of the force-distance diagram, i.e., an increase in this distance leads to a destabilizing, rather than restoring, force. This leads, for example, to the fact that the deformation of a square lattice into a rhombic one turns out to be energetically profitable. The structures characteristic of covalent crystals-hexagonal lattice on plane (graphite structure) and diamond structure in space-are also unstable. The traditional solution of this problem implies the utilization of many-body interaction potentials [1, 2]. These potentials depend on the angles between the bonds, which enables structures with low packing density to be stabilized. On the other hand, the expressions for these potentials are rather complex and constants occurring in these expressions do not have a clear physical meaning. These constants are adjusted so as to provide the best agreement of the calculated characteristics of substances to be modeled with those known from experiments. However, when proceeding from a crystal structure to another structure (for, example, from graphite to diamond), one has to completely change the interaction potential.

An alternative approach involves taking into account the torque component of the interatomic interaction. Covalent bonds are known to be directional, which enables these to transmit torque interactions. The necessity to take into account the torque interaction has been confirmed for polymers. An adequate description of the structural properties of polymer molecules is impossible without taking into account the torque interaction (corresponding to rotation about the bond direction). In the present paper, we will show that by taking into account pairwise torque interaction (in addition to the pairwise force interaction) one can provide the stability of crystal structures with low packing density. The torque interaction has a clear physical meaning and, for this reason, should be preferred to the many-body interaction.

However, the torque interaction is also rather complex in the general case. Therefore, a mechanical model of the torque interaction is needed which should, on the one hand, be fairly simple to admit a clear analytical formulation and, on the other hand, provide stability for crystal lattices under consideration. Such a model can be represented by groups of particles rigidly connected to one another and possessing a symmetry characteristic of the crystal structure. Each particle is a point mass interacting with particles of another group by means of pairwise force interaction. However, the total interaction between the groups will contain both the force and torque components. The resulting law of interaction can be applied to describe the interatomic interaction in a crystal with low packing density. This approach needs rather few parameters to be involved in the model. These are the parameters of the pairwise interaction and the geometrical parameters that characterize the groups of rigidly connected particles.

We anticipate that such an approach will enable one to construct models for most of the existing crystal structures and, moreover, will make it possible to describe the transition between the structures without changing the interaction law.

Equations of continuous theory of elasticity of media with rotational degrees of freedom are well known; see, e.g., [3]. In the present paper, we develop a model of a crystal packing of particles which yields the linear theory of couple stress elasticity in the long-wave approximation [4]. To that end, we apply the approach of [5, 6] that has been suggested for the couple-free case. The most difficult point in the description of the torque interaction between particles at the nodes of the crystal lattice is the definition of a strain measure and the derivation of constitutive relations. For the force interaction, the force acting between a pair of particles is central, with the magnitude depending on the distance between the particles. The allowance for the torque interaction substantially complicates the situation. In this case, the force and torque vectors may have arbitrary directions, both of these vectors depending on the mutual position and mutual orientation of the particles. In what follows, to define the strain vector and to derive the constitutive relations for two interacting particles we utilize an approach similar to the technique employed in continuum mechanics to construct couple stress theories [7-10]. By means of this approach we show that the dynamics of the crystal lattice in the long-wave approximation is governed by the equations of macroscopic couple stress elasticity, irrespective of the structure of the particles. We obtain the expressions for the stiffness tensors of rank 4, which characterize the elastic properties of the crystal lattice, in terms of stiffness tensors of rank 2, which characterize the stiffness of the interatomic bonds, and vectors specifying the geometry of the lattice. To illustrate the proposed technique we investigate the stability of a square crystal lattice, with the torque interaction being taken into account. The torque interaction provides the stabilization of the crystal.
1.  J. Tersoff, "New empirical approach for the structure and energy of covalent systems," Phys. Rev. B, Vol. 37, pp. 6991-7000, 1988.
2.  D. W. Brenner, "Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films," Phys. Rev. B, Vol. 42, pp. 9458-9471, 1990.
3.  E. L. Aero and E. V. Kuvshinskii, "Basic equations of elasticity for media with rotational interactions between the particles," Fizika Tverdogo Tela, Vol. 2, No. 9, pp. 1399-1409, 1960.
4.  M. Born and K. Huang, Dynamical Theory of Crystal Lattices [Russian translation], Nauka, Moscow, 1958.
5.  A. M. Krivtsov, "To the theory of media with microstructure," in Transactions of St. Petersburg State Technical University [in Russian], No. 443, pp. 9-17, SPbGTU, St. Petersburg, 1992.
6.  A. M. Krivtsov, "Constitutive equations of the nonlinear crystal lattice," ZAMM, Vol. 79, No. S2, pp. 419-420.
7.  P. A. Zhilin, "Basic equations of the non-classical theory of shells," in Transactions of Leningrad Polytechnic Institute [in Russian], No. 386, pp. 29-46, LPI, Leningrad, 1982.
8.  Kh. Al'tenbakh and P. A. Zhilin, "General theory of elastic simple shells," Uspekhi Mekhaniki [Advances in Mechanics], No. 4, pp. 107-148, 1988.
9.  P. A. Zhilin, A. D. Sergeev, and T. P. Tovstik, "Nonlinear theory of rods: Statics, dynamics, and stability," in Transactions of All-Union School "Analysis and Synthesis of Nonlinear Mechanical Oscillating Systems" [in Russian], pp. 313-337, St. Petersburg, 1997.
10.  E. Grekova and P. Zhilin, "Basic equations of Kelvin's medium and analogy with ferromagnetics," J. Elasticity, Vol. 64, pp. 29-70, 2001.
11.  W. Nowacki, Theory of Elasticity [Russian translation], Mir, Moscow, 1975.
Received 24 April 2003
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