Nonlinear Molecular Dynamics and Monte Carlo Simulations Of Crystals at Constant Temperature and Tensorial Pressure
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NONLINEAR MOLECULAR DYNAMICS AND MONTE CARLO SIMULATIONS OF CRYSTALS AT CONSTANT TEMPE]RATURE AND TENSORIAL_PR1ESSUR1E
1. V. IWII Naval lResearch Laboratory Complex Systems Theory B•ranch, Code 6692 Washington, DC 20375 - 5345 Abstract: Complimentary molecular dynamics and Metropolis Monte Carlo algorithms for the atomistic simulation of crystals at constant temperature and homogeneous tensorial pressure are summarized. The novel aspect of computational physics which unites these methods is the extension of the virial theorem to nonlinear elastic media. This guarantees the dynamical balance between the internal pressure, as determined by the interatomic potential,* and effective external pressure, as determined by the applied laboratory pressure, and includes the elastic response of the material. Numerical examples are presented.
§11.
Introduction: This article will present a qualitative discussion of several features of new molecular dynamics (MD) and Monte Carlo (MC) methods for the atomistic simulation of crystals in the Gibbsian isothermal-isostress ensemble. This ensemble is the crystalline analog of the isothermalisobaric ensemble for fluids, but the simulation of finite homogeneous deformations of an elastic crystal is considerably more complicated The complications arise when combining elements of the continuum theory of nonlinear elasticity with the statistical mechanics of atomistic simulations. A preliminary report of this work has recently appearedl, and a lengthy paper including derivations and additional computational results is in preparation 2. The principal complication is the nonlinearity in the problem due to the material response of the elastic crystal to the applied stress. That is, as the crystal is deformed the differential of the work required to perform a further deformation changes. This implies the equilibrium pressure within the material changes as a result of the deformation. Thus when considering finite deformations of the crystal, the nonlinear expression for the elastic energy must be employed. The principal theoretical result of this analysis is a formulation of the virial theorem that includes nonlinear elastic effects. The new MD equations of motion guarantee that the internal and external pressures balance, even when a structural phase transition has occurred. The new Metropolis MC algorithm works in the same physical phase space and generates stochastic trajectories comparable to the deterministic
Mat. Res. Soc. Symp. Proc. Vol. 291. 4ý1993 Materials Research Society
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trajectories of the MD algorithm, and independently verifies the interpretation of the nonlinear virial theorem. In §2, matrix notation for the geometrical description of finite homogeneous deformations of a crystal and the associated elastic energy is summarized. The molecular dynamics algorithm is presented in §3, and the Monte Carlo algorithm in §4. Some numerical results are described in §5. A brief discussion of some limitations of the present MD and MC methods follows in §6. §2.
Finite Deformations
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