Quantum Corrections of Heat Capacity invalidate Nanoscale Heat Transfer by Molecular Dynamics

Nov. 6, 2012 - PRLog -- Background

MD is commonly used in computational heat transfer to derive the thermal response of nanostructures. MD stands for molecular dynamics. Finding basis in SM, MD relates the thermal energy of the atom to its momentum through the equipartition theorem. SM stands for statistical mechanics. Momenta of atoms in an ensemble are determined by solving Newton’s equations with inter-atomic forces derived from Lennard-Jones potentials. SM always assumes the atom has heat capacity as otherwise the momenta of the atoms cannot be related to their temperature. See e.g. Allen & Tildesley [1].

In heat transfer simulations of bulk materials, MD assumes the atoms have heat capacity. Under periodic boundary conditions, MD simulations of the bulk are valid by QM because atoms in the bulk do indeed have heat capacity. QM stands for quantum mechanics.

Problem

MD simulations of heat transfer in discrete nanostructures differ from those in the bulk. The problem is QM precludes atoms in nanostructures under TIR confinement from having the heat capacity necessary to conserve absorbed EM energy by an increase in temperature.  TIR stands for total internal reflection and EM for electromagnetic. Today, the problem is amplified as commercially available MD computer programs assume the atoms always have heat capacity. What this means is the uncountable number of heat transfer solutions derived by MD that abound the literature are invalid by QM.

Theory of QED Radiation

The theory of QED radiation avoids the invalidity of MD by QM. QED stands for quantum electrodynamics. QM based on the E-H relation [3] shows the heat capacity given by the thermal kT energy of the atom vanishes at the submicron TIR wavelengths of nanostructures. E-H stands for Einstein-Hopf. Lacking heat capacity, conservation of absorbed EM energy proceeds by the QED induced creation of non-thermal EM radiation inside the nanostructure at  its TIR frequency – the EM radiation having the necessary Planck energy to produce charge by the photoelectric effect, and if not, is emitted to the surroundings. In this regard, numerous papers have argued that MD heat transfer simulations of discrete nanostructures based on SM are invalid by QM. See “Validity of Molecular Dynamics in Heat Transfer”, “Unphysical Heat Transfer by Molecular Dynamics”, “MOLEC 2012”, and “NAP 2012” at http://www.nanoqed.org, 2012.

QC of Thermodynamic Variables

QCs of classical SM thermodynamic variables support QED radiation theory that relies on a vanishing heat capacity of the atoms in nanostructures.  QC stands for quantum correction. Indeed, Allen &Tildesley give procedures for performing QCs (See “Quantum Corrections” Sect. 2.9 of [1]). In nanostructures, the atoms are assumed to behave as a set of QM harmonic oscillators, the density of states given by the Fourier transform of the velocity autocorrelation function. The QC for the response of the thermodynamic variable is determined at each frequency of the density of states by the response of the QM oscillator, the total QC obtained by integrating over all frequencies. QCs for the water molecule [2] generalized for the energy E, constant volume heat capacity C_V, Helmholtz free energy A, and entropy S of any QM system are summarized in the thumbnail. For nanostructures, a simpler approach is to determine the QC of the thermodynamic variable at the TIR resonance.

In the thumbnail, the QC weighting function W = Q – C is the difference “delta” between the quantum Q and classical C values of the thermodynamic variable. Mouse over the the thumbnail to enlarge,or double click for fine resolution. The bottom abscissa is u = h * Nu / kT, where Nu = TIR frequency of the nanostructure. The top abscissa is the wave number equivalent to the parameter u at 300 K. All W go to zero for u < 1 consistent with the low frequency anharmonic region of classical SM where QCs are insignificant; whereas, W for u > 1corresponds to the harmonic approximation where QCs are significant.
 
Summary and Conclusions

1.  QED radiation is based on the energy E of the atom given by the E-H relation for the thermal kT energy of the QM oscillator. The quantum Q heat capacity C_V is the derivative dE / dT of the energy E of the E-H relation with respect to temperature T.

                            C_V =  u²exp(u) / [1-exp(u)]²      

The E-H relation does not include the ZPE in Planck’s derivation, but this is inconsequential because the derivative of the ZPE with respect to temperature vanishes. ZPE stands for zero point energy. For C_V, the W < 0 because Q < C and vanishes for u > 5. Depending on TIR confinement, C _V may vanish in nanostructures at ambient temperature, i.e., vanishing heat capacity need not require temperatures at absolute zero as in SM. What this means is heat capacity by QM is always less than classical and vanishes under TIR confinement in nanostructures at ambient temperature thereby supporting the theory of QED radiation.  

2.  The theory of QED radiation that relies on the vanishing heat capacity required by the E-H relation of QM is consistent with QCs. Moreover, the QED radiation created at the speed of light is far faster than phonons can respond at acoustic velocities, thereby negating heat flow by conduction in nanostructures. What this means is the EM energy absorbed by a nanostructure is almost totally conserved by QED radiation.

3.  Absent heat capacity, nanostructures do not change in temperature upon the absorption of EM energy, and so the Fourier equation has no meaning at the nanoscale. It is therefore not surprising that applying QCs to MD derivations of thermal conductivity based on phonons with or without the ZPE was found [4] to not agree with QCs.  What this means is the thermal conductivity of nanostructures remains at bulk, but heat flow by conduction is negated because temperature changes do not occur without heat capacity, i.e., MD derivations of thermal conductivity at the nanoscale based on electron scattering are unphysical.

4.  MD simulations of heat transfer in nanostructures that assume the atoms have heat capacity are unequivocally invalid by QM.

References

[1] M. P. Allen and  D. J. Tildesley, Computer Simulations of Liquids, (Oxford: Clarendon Press: 1987.
[2] P. H. Berens, et  al., “Thermodynamics and quantum corrections from molecular
dynamics for liquid water,” J. Chem. Phys., 79, 1983.
[3] A. Einstein and L. Hopf, “Statistische Untersuchung der Bewegung eines Resonators in einem Strahlungsfeld,” Ann. Physik, 33, 1105-1120, 1910.
[4] J. E. Turney, et al., “Assessing the applicability of quantum corrections to classical thermal conductivity predictions,” Phys. Rev. B, 79, 224305, 2009.