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Validity of Molecular Dynamics in Heat Transfer
Molecular dynamics simulations of heat transfer in bulk materials consistent with statistical mechanics and valid under periodic boundary conditions are invalid by quantum mechanics for discrete nanostructures lacking periodic boundaries
By: Thomas Prevenslik
MD in computational heat transfer [1-3] finds theoretical basis in SM. MD stands for molecular dynamics and SM for statistical mechanics. SM relates the thermal energy of the atom to its momentum by the equipartition theorem, the momenta of atoms in a submicron MD computation box for an ensemble of atoms determined by solving Newton’s equations of motion for inter-atomic forces derived from Lennard-Jones potentials.
Following SM, MD assumes the ensemble of atoms in the computation box always have thermal energy, or equivalently the capacity to absorb heat. Otherwise, the temperature of the atoms cannot be related by the equipartition theorem to their momenta. For bulk materials, MD heat transfer is performed for the MD computation box under periodic boundary conditions. MD simulations of the bulk are therefore valid because atoms having heat capacity in discrete submicron boxes under periodic boundary conditions are equivalent to those in the bulk that do indeed have heat capacity
Unlike SM that always allows the atom to have heat capacity, QM restricts the heat capacity of the atom depending on its EM confinement. QM stands for quantum mechanics and EM for electromagnetic. The heat capacity of the atom by QM and SM in terms of the EM confinement wavelength is shown in the thumbnail. In macrostructures under EM confinement at long wavelengths, QM allows the atom to have the same heat capacity as in SM, and therefore MD simulations of the bulk performed in submicron computation boxes under periodic boundary conditions are valid by QM. In discrete nanostructures with short EM wavelengths, however, QM precludes atoms from having heat capacity.
Discrete nanostructures are unambiguously not periodic, and therefore periodic boundary conditions cannot be imposed as in MD simulations of the bulk. Nevertheless, the literature is replete with MD simulations of discrete nanostructures with atoms having heat capacity. Although consistent with SM, MD of discrete nanostructures is invalid by QM. What this means is QM forbids atoms in discrete nanostructures from conserving absorbed heat by an increase in temperature, and therefore the classical modes of heat transfer – convection, radiation, and conduction that depend on changes in temperature have no meaning. Indeed, the time dependent Fourier equation of conduction heat transfer is not valid for nanostructures.
Problem and Question
Given that MD simulations of heat transfer in discrete nanostructures are a priori invalid by QM, the relevant question is: How can MD heat transfer simulations of discrete nanostructures be performed that are at least consistent with QM?
By QM, conservation of EM energy in nanostructures proceeds by the creation of non-thermal QED induced EM radiation that is emitted to the surroundings, or charges the discrete nanostructures by Einstein’s photoelectric effect. QED stands for quantum electrodynamics. QED radiation may be understood by considering the Einstein-Hopf relation for the QM harmonic oscillator shown in the thumbnail. QM gives the dispersion of the average Planck energy of photons within a macro or nanostructure in terms of the wavelength of their EM confinement. By QED theory, the Planck energy of the QM oscillator based on photons is taken as the measure of the capacity of the ensemble of atoms in a structure to absorb EM energy by an increase in temperature. Photons and not phonons are therefore the heat carriers in nanostructures simply because photons respond far faster than phonons to conserve absorbed EM energy. In macrostructures, phonons may carry thermal energy, but in nanostructures, photons trump phonons. Upon absorption of EM energy in nanostructures, the QED induced photons are spontaneously created well before the phonons can respond, thereby effectively negating phonons as heat carriers at the nanoscale. MD simulations of heat transfer in nanostructures based on phonons are therefore not only invalid by QM, but also meaningless, as there is no need for heat conduction to conserve absorbed EM energy that already has already been conserved by the prompt creation of QED photons.
Recommendations for how invalid MD heat transfer of discrete nanostructures may be made consistent with QM include holding the temperature of the nanostructure constant as required by QM, say by using the Nose-Hoover thermostat [1-3] during the MD solution run, whereby the QED radiation created by the nanostructure is the accumulated thermostatic heat. For interacting nanostructures, MD heat transfer simulations consistent with QM are computationally intractable, and therefore finite element simulations are proposed using as input simple estimates of QED radiation for the nanostructures in programs such as ANSYS and COMSOL. Details are given in “Validity of Molecular Dynamics in Heat Transfer,” at http://www.nanoqed.org, 2012.
1. MD heat transfer based on SM is valid by QM only for bulk simulations.
2. MD heat transfer assuming atoms in discrete nanostructures have heat capacity is invalid by QM.
3. Heat transfer of nanostructures consistent with QM is intractable by MD.
4. Consistency of heat transfer of nanostructures with QM may be found with finite element simulations that treat naostructures as point sources of QED radiation with the thermal response of the macroscopic surroundings determined by classical heat transfer.
 C. G. Gray, K. E. Gubbins, Theory of Molecular Fluids, Oxford, Clarendon Press, 1985.
 J-P Hansen, I. R. McDonald, Theory of Simple Liquids, London, Academic Press, 1986.
 M. P. Allen, D. J. Tildesley, Computer Simulations of Liquids, Oxford, Clarendon Press, 1987.
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Classically, absorbed heat from EM sources at the macroscale is conserved by an increase in temperature. But at the nanoscale, temperature increases are forbidden by quantum mechanics. QED radiation explains how heat is conserved by the emission of non-thermal EM radiation to the surroundings.