ORIGINAL RESEARCH article
Coupling Parameters for Modeling the Near-Field Heat Transfer Between Molecules
- Energy Transport Laboratory, Department of Energy Science and Engineering, Indian Institute of Technology Bombay, Mumbai, India
The behavior of near-field heat transfer between molecules at gaps which are small compared to wavelength of light is greatly influenced by non-radiative dipole-dipole interactions between the molecules. Here we derive the coupling parameters and estimate the near-field heat transfer between two molecules using coupled Drude oscillators. The predictions from this model are verified with results from standard fluctuational electrodynamics principles. The effect of orientation factor of the dipole moments in the molecules traditionally taken into consideration for analysis of resonance energy transfer between molecules but hitherto overlooked for near-field heat transfer is also discussed.
Fluctuating charge distribution of one molecule can interact with the fluctuating charge distribution of another proximate molecule so as to give rise to more energetically advantageous fluctuations. This interaction is responsible for fundamentally important phenomena such as London’s dispersion forces, and that relevant for this work–near-field heat transfer (NFHT). In order to model the dominant dipolar terms of these fluctuating charge interactions, and thus the ensuing phenomena, it is possible to use a Drude oscillator model where the electric charge distribution in the atom or molecule undergoes an oscillatory displacement in response to an oscillatory electric field. This has been successfully employed to model among others the London dispersion forces between molecules , van der Waals interaction between atom and a surface  and between two surfaces .
For this study we analyze NFHT between two closely spaced molecules. The study of NFHT has gained prominence in recent years due to theoretical predictions [4–6] and subsequent experimental confirmations [7–14] of heat transfer between vacuum separated objects exceeding Planck’s blackbody predictions by several orders of magnitude. This has led to development of multitude of novel applications such as near-field thermophotovoltaics [15, 16], thermal diodes , transistors [18, 19], rectifiers  and modulators . A more exhaustive list of applications can be found in any recent review on this topic such as in Refs. [22–24]. Experimental advancements has lead to measurement of NFHT with conductances of the order of 200 pWK−1 for configurations such as that between STM tip over substrate [14, 25]. Recently it has been shown that thermal conductance in a single molecule junction between two heaters maintained at different temperatures can be measured with resolution as low as 2 pWK−1 . A similar configuration can be adopted to measure thermal conductance between two molecules (or two nanoparticles in general) but now with vacuum gap between them. As the molecules are brought closer, nonradiative dipole coupling will result in a gap-dependent heat flux which we attempt to quantify in this theoretical study.
In our study we adopt coupled Drude oscillators to model the interaction between the molecules and estimate NFHT between them. The advantage of using this model stems not only from the fact that this picture offers mathematical simplicity and hence the physics is transparent but also from that it establishes a common theoretical framework to analyze the two phenomena of dispersion forces (for which the theory has been well developed) and NFHT both of which have common origin in the interaction between fluctuating charge distributions. This study also enables us to predict the effect of orientation factor of two dipoles on near-field heat transfer which has so far not been taken into account in the fluctuational electrodynamics framework traditionally used to analyze near-field heat transfer. Another added advantage is that the results from Drude oscillator model can be easily extended to estimate interaction between molecules by including either the tabulated oscillator strengths  for the respective molecules, or by calculating them from their relations with experimental measurements of fluorescence and absorption spectra . Recently , have proposed an alternative method to compute the heat transfer between molecules which requires computation of Green’s function for an approximate geometry of the molecule via boundary element or finite-difference methods, and the molecular susceptibilities obtained from density functional theory. In contrast, the procedure outlined in this manuscript proposes an alternative approach: to make use of experimentally determined fluorescent and emission spectra of the molecules (which inherently depend on the shape, size and composition of the molecule) to arrive at an expression for the near-field heat transfer.
The procedure for detailing the near-field heat transfer between two molecules is as follows: we first derive the eigenmodes for the case of two interacting dipoles in Dipole-Dipole Coupling for Near-Field Heat Transfer and by comparing these with the corresponding forms in the coupled harmonic oscillator model shown in Coupled Harmonic Oscillator Model we get the expressions for the parameters that can be substituted in the expression for heat transfer mediated by a coupled harmonic oscillator system between two heat baths maintained at two different temperatures. We show that results obtained from such a trivial substitution conforms to the predictions from fluctuational electrodynamics (FE) theory. The NFHT between two dipoles is then extended to predict that between two molecules by incorporating their oscillator strengths. This procedure has been previously applied to estimate the NHFT between two nanoparticles  and two planar surfaces [31, 32]. The model has the added advantage that it can be extended to cases where the local thermal equilibrium condition considered in FE is not valid - such as estimating dynamic heat transfer between objects [30, 33]. The procedure detailed in Dipole-Dipole Coupling for Near-Field Heat Transfer and Coupled Harmonic Oscillator Model is similar to that detailed in Ref.  and has been included for the purpose of completion and providing additional details in the relevant calculations. In addition the results shown in Ref.  differed from FE predictions by a constant factor. In this work we account for this discrepancy by taking into account degeneracy in eigenmodes and in NFHT Between Molecules we discuss the effect of orientation factor between dipoles on NFHT between molecules.
2 Dipole-Dipole Coupling for Near-Field Heat Transfer
Consider two transient dipoles
where, if we consider without loss of generality that the two dipoles are separated along z-axis,
where, the superscripts L and T, standing for longitudinal and transverse modes, have been included to differentiate between the two possible solutions with different degeneracies. Physically these two solutions denote the possibility of the dipoles being in either “head-to-tail” configuration or “side-by-side” configuration respectively , also termed “J-aggregate” and “H-aggregate”, respectively . The Lorentz form of polarizability has been assumed in Eq. 5 since it allows us to readily extend the analysis to estimate the NFHT between two molecules by including the appropriate spectral oscillator strength as shown in NFHT Between Molecules. By replacing this expression of polarizability with other forms applicable for larger objects, such as nanoparticles, one can trivially extend this analysis to estimate heat transfer between such objects. It should also be possible to generalize this method when there are several interacting dipolar structures such as that seen in hybrid graphene nanostructures .
3 Coupled Harmonic Oscillator Model
The equations of motion of two coupled harmonic oscillators of unit mass, with same natural frequency
where, the forcing function
FIGURE 1. Schematic showing the two systems which are being related: two dipoles
The equations of motion in frequency space reduce to:
where it is assumed that
where expressions for the natural frequencies
The net heat transferred in this coupled harmonic oscillator (CHO) system can be found by observing that in steady state the rate of heat transferred to the second oscillator via coupling with the first oscillator is equal to the decay in the second oscillator via damping:
We now ask: what are the equivalent expressions for the parameters
Substituting the coupling parameters from above we get:
We note here in passing that for
Comparison With Results From Fluctuation Electrodynamics
The heat transfer rate between two dipoles predicted from Eq. 18 needs to be compared with that predicted from fluctuational electrodynamics, the expression for which in the classical limit
Substituting the expression of
which, on evaluating the integral and taking the limit of small damping
4 NFHT Between Molecules
Here we use the expression of heat flux between two dipoles derived in Eq. 14 to predict the heat transfer between two molecules for the general case when resonant frequencies of the two molecules are different. For the case of two oscillators with different resonant frequencies
which, for cases where
Thus, as in Eq. 15, the equivalent expression for heat flux between molecules with natural frequencies
Here, c is the velocity of light in vacuum,
5 The Effect of Orientation Factor
We now consider the case of heat transfer between molecules when the relative orientation of dipole moments is fixed (possibly due to the molecules not being free to rotate) and indicated by the orientation factor κ. This situation is frequently encountered in the analysis of Förster resonance energy transfer (FRET) between two adjacently placed molecules. In fact, Förster in his original treatize on FRET  analyzed the interaction between molecules using two drude oscillators (an english version of the derivation can be found in Ref. ). We now consider the effect of this orientation factor on NFHT between the molecules.
Since only the component of electric field of one dipole along the other dipole is responsible for work done and hence the heat flow, we consider only the component of the electric field given in Eq. 2 along the driven dipole [43–45]:
For two similar molecules of polarizability
Comparison with Eq. 10 gives
Using this in the expression for heat flux between two dipoles in Eq. 18 gives:
The equivalent expression for heat flux between two molecules taking into account the dipole orientation factor κ in Eq. 23 will then be:
where expressions for
For demonstration, we now calculate the thermal conductance, defined as
We have shown here the framework that can be used for analyzing the NFHT between two molecules. This method employs the expression of power flow between two coupled Drude oscillators, and the necessary coupling parameters required to model the near-field dipole-dipole interactions. The advantage of adopting this approach is that the expression for power flow between the two Drude oscillators can be linked trivially to the NFHT between the two molecules via the experimentally determinable absorption and fluorescent spectra of the molecules as shown in NFHT Between Molecules. The effect of orientation factor on the NFHT between molecules is also considered. We have confirmed that this model in the dipole limit tallies with the results of NFHT between two dipoles as predicted from fluctuational electrodynamics principles and the discrepancy of a constant factor which was reported in an earlier work  has been accounted for by considering degeneracy of eigenmodes. It is expected that the expression of NFHT between molecules derived in this work will be used for comparison with experimental measurements. Further, by modifying the coupling term in Eq. 15 this method can be extended to include non-local effects arising from nuclear motion , an exercise which will be carried out in a follow-up article.
Data Availability Statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
KS conceived the idea for this work, derived the results, and wrote the manuscript.
The authors acknowledge funding from IITB seed grant RD/0518-IRCCSH0-004.
Conflict of Interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
We acknowledge useful conversations with Svend Age Biehs.
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Keywords: near-field heat transfer, fluctuational electrodynamics, coupled drude oscillators, dipole-dipole interaction, molecular interaciton
Citation: Sasihithlu K (2021) Coupling Parameters for Modeling the Near-Field Heat Transfer Between Molecules. Front. Phys. 9:682939. doi: 10.3389/fphy.2021.682939
Received: 19 March 2021; Accepted: 24 June 2021;
Published: 09 July 2021.
Edited by:Venu Gopal Achanta, Tata Institute of Fundamental Research, India
Reviewed by:Alejandro Gil-Villegas, University of Guanajuato, Mexico
Alok Ghanekar, University of Southern California, United States
Copyright © 2021 Sasihithlu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Karthik Sasihithlu, email@example.com