Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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The simple potentials also provide a direct way to estimate the modulus of elasticity and the theoretical strength using expressions

1 r 0

d 2 φ d r 2

0

d φ d r r = r D

1 r 2

, σ m =

E =

(6.6)

r = r 0

which follows from its physical nature [23]. In Equation (6.6) 2 , r D stands for the so-called separation distance defined by the maximum value of the interatomic force d f ( r ) / d r = 0, that is, the inflection point of the interparticle potential d 2 φ ( r ) / d r 2 = 0 (Figure 6.5). The basic purpose of any interatomic potential is to correctly reproduce the most prominent characteristics of atomic bonds. Therefore, it should always be borne in mind that the pair of potentials were “originally developed to describe atomic interaction in systems in which these forms of potentials are physically justified” [24] and resist the temptation to use them injudiciously for their simplicity. The pair potentials cannot accurately describe interatomic interactions in more complex systems [16] such as, for example, strongly covalent systems (e.g., SiC), most ceramics characterized by fully populated orbitals, metals characterized by delocalized “sea of electrons”, or semiconductors. However, in MD simulations from the 1950s to the 1980s, a couple of these simplified potentials were used almost exclusively. Significant progress was made during the 1980s with the development of many-body potentials for metals based on the concept of atomic density (e.g., [25]). The main observation that needed to be modeled was that interatomic bonds become weaker when nested in a "dense" local environment. Accordingly, the force acting on an atom depends not only on the distance separating its nucleus from nuclei of its neighbors, but also on the local atomic density. In other words, the forces between ions are characteristically dependent on many bodies ("many-body in character"), instead of simply being pairwise additive. The focus is usually on the attractive part of the potential [16]. Accordingly, the so-called "glue model" potentials have been developed (e.g., [26] and references cited therein). Among these potentials, the best known is the so-called embedded atom method ( EAM )

i "

φ ( r i j )+ ψ ( ¯ ρ i ) # ,

1 2 ∑ i

1 2 ∑ j̸ = i

∑ j = i

φ ( r i j )+ Ψ = ∑

Φ ( r i j , ρ i ) =

(6.7)

¯ ρ i = ∑ j̸ = i

ρ ( r i j ) .

developed to approximate the interaction between ions in metals. The various forms of (6.7) differ from each other only in the forms of functions: φ (pairwise term depending entirely on the interatomic distance), ψ (density-dependent contribution - the embedding energy necessary to insert the i -th atom into the background of the electron density ¯ ρ i ), and ρ i j (atomic density function) [26]. Differences between pair potentials and many-body potentials have been discussed in detail in the literature (e.g., [27], [28]). EAM potential has been used extensively to model ductile metals. Finally, empirical potentials in organic chemistry and molecular biology are often called force fields. It should be noted that for complex macro-molecular chains (e.g.,