Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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(6.3), the terms Φ 1 , Φ 2 , Φ 3 , . . . are, respectively, contributions due to external fields (e.g., gravitational or the container wall), pair, triple and higher order interatomic interactions. In traditional MD, rigor is often sacrificed for the sake of efficiency, and interatomic inter actions among all atoms have been replaced by less computer-intensive approximations in which each individual atom interacts only with a certain number of nearest neighbors (so-called, first-nearest, second-nearest, etc.). Further, by neglecting the three-body inter actions (involving interatomic angles) and higher, the potential energy of the system can be approximated by the sum of isolated empirical biatomic potentials ( pairwise additivity assumption) Φ = 1 2 ∑ i ∑ j φ ( r i j ) . (6.4) It is obvious that the concept of pairwise additivity (6.4) represents a huge simplifica tion with far-reaching consequences. For some crystal lattices, the pairwise interaction is not able to take into account a good portion of the cohesive interaction [19]. Furthermore, the interaction in ionic crystals may be a consequence of the polarization effects, attributed to the action of the electric fields of the surrounding ions, which cannot be described by simple pair potentials. However, the main interactions in the ionic and Van der Waals crystal lattices are believed to be essentially pairwise [19]. Many more complex forms of potential can be used as needed at the cost of increasing the duration of the simulation. The consequence of choosing the central-force potential (6.4) is that the total energy of the system is conserved. In the language of statistical physics, classical MD generates a microcanonical ensemble ( N , E , V ). The pair potentials are the simplest potentials since the force of interaction of two atoms is completely determined by their mutual distance. There is an extensive literature on the ways in which these potentials are experimentally determined or theoretically modeled (e.g., [20]). Strictly speaking, they realistically describe only noble gases. The simplest potential of this type is the discontinuous potential of a "rigid sphere" which implies that the value of the interatomic force is equal to either: (i) zero, if the interatomic distance is greater than the prescribed value; or (ii) infinity, if the interatomic distance is equal to or less than the prescribed value. A more realistic interatomic interactions are obtained under the assumption that the interaction force gradually varies from strongly repulsive (at small interatomic distances) to attractive (at medium distances) until it finally converge asymptotically to zero (with further increase in distance) (Figure 6.5). The best known potential of this type, which has been widely used in the past when the focus of research was on the study of qualitative trends (essential physics) rather than narrowly specific issues, is the Lennard-Jones 6-12 potential φ i j = − ε LJ " 2 1 ¯ r i j 6 − 1 ¯ r i j 12 # (6.5) originally developed for noble gases from van der Waals cohesion [21]. In expression (6.5), ε LJ represents the depth of the potential well, and ¯ r i j = ( r / r 0 ) i j ratio of current and equilibrium distance between atoms i and j (Figure 6.5). These model parameters are chosen with the aim of optimally reproducing the most desirable physical and mechanical