Heat transfer between nanoparticles: Thermal conductance for near-field interactions

We analyze the heat transfer between two nanoparticles separated by a distance lying in the near-field domain in which energy interchange is due to Coulomb interactions. The thermal conductance is computed by assuming that the particles have charge distributions characterized by fluctuating multipole moments in equilibrium with heat baths at two different temperatures. This quantity follows from the fluctuation-dissipation theorem (FDT) for the fluctuations of the multipolar moments. We compare the behavior of the conductance as a function of the distance between the particles with the result obtained by means of molecular dynamics simulations. The formalism proposed enables us to provide a comprehensive explanation of the marked growth of the conductance when decreasing the distance between the nanoparticles.


I. INTRODUCTION
The study of energy transfer mechanisms at the nanoscale 1,2 has aroused increasing interest due to the emergence of the interdisciplinary field of nanoscience where such wide-ranging fields as for example solid state physics 3 , nanothermodynamics 4,5,6 or electrical engineering 7 coexist. One of the basic problems in this field is to determine the energy exchange between two nanoparticles (NPs) at different temperatures. The way in which this energy is transfered depends crucially on the distance between the particles. For sufficiently large distances, heat exchange proceeds via thermal radiation, through emission or absorption of photons whereas at smaller distances recent molecular dynamics simulations have shown that Coulomb interaction (near-field radiation) is the dominant mechanism 8 . For near-field interactions, the thermal conductance was calculated under the assumption that both NPs behave as effective dipoles at different temperatures 8 . Hence, since these dipoles undergo thermal fluctuations, the fluctuation-dissipation theorem (FDT) 9,10,11,12 provides the energy which dissipates into heat in each NP. It was found that the heat and therefore the conductance varies according to d −6 a very different behavior from the one observed in the case of thermal radiation: d −2 . Molecular dynamics simulations agree with the dipoledipole model when the two NPs are separated by a distance on the order of a few nanometers. However, near contact the conductance deviates dramatically from the prediction of the dipole model, as the simulations show. This behavior is a consequence of the fact that when particles become very close the position of the atoms are highly correlated, consequently the charge distributions become nonsymmetric and cannot be described merely as two interacting dipoles. To account for this distortion of the distribution of charges a more general formalism which focuses more convoluted interactions involving higher order multipoles aside from the dipoles is required.
Our purpose in this paper is to provide this general formalism enabling us to analyze the behavior of the conductance beyond the dipolar approximation. We will use the linear response theory to derive an expression of the FDT for the fluctuations of the higher order multipoles. In particular, we will focus on the quadrupolar contributions to the conductance which are able to reproduce the behavior observed in the simulations for some sizes of the NPs.
The paper is organized as follows. In Section 2, we present the multipolar expansion of the Coulomb forces 13 between both NPs and derive a general expression of the FDT valid for multipoles of any order which leads to the heat transfer between the NPs. In Section 3, we analyze the particular case of quadrupolar contributions and derive the expresion of the conductance. We compare our result with the molecular dynamics simulations 8 . Finally, in Section 4, we emphasize our main conclusions.

II. HEAT TRANSFER BETWEEN TWO NANOPARTICLES
In this section, we will study the near-filed radiative heat transfer flux between two NPs which interact through Coulomb forces.

A. Multipolar expansion
To analyze the Coulomb interaction between two NPs (see Fig. 1) it is necessary to know the charge distribution inside each of them. This can be performed by specifying their multipole moments so that the multipole moment of order n of the NPi,M  where e r is the charge at the position r inside the NP and the X (n) α (r) are symmetric irreducible tensors (see the Appendix I for more details) where α = (α 1 , . . . , α n ) and α j = 1, 2, 3 for j = 1, ..., n. Thus, the case n = 0 corresponds to the monopoleM (0) (i);α (r) = r e r , n = 1 is related to the dipole momentM (1) (i);α (r) = r e r r α1 , and n = 2 for the quadrupole momentM (2) (i);α (r) = 1/2 r e r (3r α1α2 − δ α1α2 ). Hence, in terms of the spherical surface tensors Y The above mentioned interaction between these NPs modifies their respective Hamiltonians. The interaction between NPi with NPj introduces a time-dependent per-turbationĤ (ij) in its Hamiltonian which can be written as a multipolar expansion 13 : with c m = 1/(2m − 1)!! and ⊙ stands for the full contraction of indexes,M (m) withV (i,j) (d) being the interaction potential between both NPs and d the separation between their centers. In terms of the first contributions, the perturbation can be expressed aŝ where −V (i,j) is the electric field induced in the NPi, −V (2) (i,j) is the gradient of this induced field, andM (2) (i) is the conjugated quadrupolar moment.
Likewise, the electrostatic potential admits a multipolar expansion as well which expresses the fact that the potential acting on the NPi depends on the charge distribution in the NPj. Here, M (n) (j) are the multipolar moments of the NPj and is the Green propagator, with d the vector connection the centers of the particles andd the corresponding unit vector. Thus, from Eq. (6) where B. Heat transfer from the fluctuation-dissipation theorem In the linear response regime, the multipolar moments can be expressed as where P (n,m) (ω) are the multipolar polarizabilities which may in general depend on frequency. The energy transferred between the particles and converted into heat can be obtained from the linear response theory 11,12 . One obtains (see the Appendix II) where the symbol * stands for the complex conjugated, and the brakets express thermal average. According to Eq. (8), the term in Eq. (11) containing the thermal average can be transformed as where we have defined Moreover, from Eqs. (A-3), (7) and (9) one can prove that the S (l,k) (j) are symmetric tensors. Therefore, making use of Eq. (13), Eq. (11) becomes The dependence of the energy transferred on the distance d resides in S (l,k) (j) , as follows from Eq. (13) and the expression of the propagators given through Eqs. (7) and (9). The multipole-multipole correlation can be obtained by using the FDT 9,11,12 where Θ(ω, T i ) = ω {1/2 + 1/ exp( ω/kT i − 1)} is the mean energy of an oscillator. As an illustration, for the dipolar case 8 , we obtain where P is the dipole-dipole polarizability which we assume to be given through and △ (1) For the quadrupolar case one has where P is the quadrupole-quadrupole polarizability given through with β (i) (ω) = β (i) (ω) + iβ (i) (ω) and is the isotropic tetradric. Thus, with Eqs. (20) and (21), Eq. (19) is written as Up to the quadrupolar order one has to take into account also the cross correlation dipole-quadrupole where P (1,2) (i) is the dipole-quadrupole polarizability, given through with γ (i) (ω) = γ (i) (ω) + iγ (i) (ω) and It must be emphasize that the FDT, Eq. (15), applies whenever the charge distribution of each particle in the presence of mutual interactions has reached equilibrium with the heat bath. When this is not the case, in the nonaged regime 12 , one can still use a similar expression of the FDT in terms of an effective temperature. This can be done through a generalized Langevin equation 10 , which takes into account the heat exchange between the NPi and its thermal bath. Relating the momentum variance of the NPi with its temperature by the equipartition theorem, one can obtain the effective temperature (T (l,k) ef f (i) ) through the response of the system due to fluctuations of the multipolar moments 15 . Thus, where T 0 stands for the initial temperature of the NPi and T b is the bath temperature. This expression shows that when the multipole moments of the particles are uncorrelated, i.e. when both particles equilibrate independently at two different temperatures, the effective temperature coincides with that of the bath. This is the situation addressed in this paper.
The effective temperature, defined as that for which the system would equilibrate, is a parameter measuring the distance to the stationary state in which both particles reach two different temperatures. It can be also calculated using a relaxation model 16,17 .

III. THE THERMAL CONDUCTANCE
In this section, we will calculate the thermal conductance between the two NPs in the presence of quadrupolar contributions. To this end, we start by writing Eq. (14) as follows where from Eqs. (7), (9) and (13) S (l,k) with Hence, by substituting Eq. (30) into Eq. (29) and the resulting equation into Eq. (28) we obtain where C n,m,l,k Therefore, from Eq. (31) we obtain the net heat flux between both NPs In view of Eq. (25) and the symmetric character of the spherical surface tensors given through Eq. (A-3) one can prove from Eq. (30) that Moreover, it can be shown that when m + n = 2p + 1 (p > n), P (n,m) (j) is proportional to an isotropic skewsymmetric tensor (p) of order 2p + 1 which satisfies 14 Therefore, by symmetry reasons only coefficients C n,m,l,k for which n + m = 2q and l + k = 2s, with q and s two positive integers, contribute to the heat flux. Hence, up to quadrupolar order we can write from Eq. (31) and consequently When NPs are at the same temperature T , Eq. (38) reduces to whence since the system is in thermal equilibrium In the general case, i.e. out of equilibrium, we can linearize Eq. (38) with respect to the temperature diference ∆T = T 1 − T 2 in order to obtain the conductance given through G 12 (T 0 ) = ∂Q 12 /∂ ∆T | T1=T2=T0 . We obtain where T 0 = (T 1 +T 2 )/2 is the average temperature, which corresponds to the final equilibrium temperature that two bodies would reach when brought into contact and a heat flow established between them 18 . In the expression we have obtained for the conductance, we can identify the following contributions: (i) Dipolar which coincides with the expression obtained in Ref. 8 .
In order to verify our results, in Fig. 2 we reexhibit a graph obtained by Domingues et al. 8 extending the logarithmic scale for conductance in the more usual form. This graph displays the thermal conductance as a function of distance between the NPs, both with radius R, in three significant situations: in mechanical contact (d = 2R), in the intermediate region shortly before contact (2R < d < 4R), and in the most distant region (d 4R) where the near-field interaction is still valid. In this situation, the results corresponding to the grey dotted lines show the behavior d −6 which was obtained in Ref. 8 . Our results are in broad agreement for this region where the dipolar domain is present. When the particles are close togheter, their charge distributions becomes very disorderly and higher orders than dipolar interactions come into play in the calculation of the thermal conductance. In this case, as predicted by Domingues et al. 8 the thermal conductance is about 4 orders of magnitude larger than that of the dipole model given in Eq. (42). In more extreme conditions when the particles come into contact to each other, the same authors also predicted that the conductance would be 2 to 3 orders of magnitude lower than the conductance just before contact. These numerical predictions are covered by the result we give in Eq. (41) where one can see that the dominant contribution (d −10 ) is 4 orders of magnitude lower than the dipolar case (d −6 ) while an intermediate case would give a value d −8 .
It must be stressed that we have obtained the conductance up to quadrupolar order, nonetheless through our formalism it is possible to obtain the conductance for any order of multipolar interaction.

IV. CONCLUSIONS
In this paper, we have presented a theory to explain the exchange of energy between two NPs at different temperature. Our theory provides a general formalism based on the multipolar expansion of the electrostatic field in order to study heat transfer between two NPs for arbitrary small distances provided that the FDT be satisfied. However, out of the FDT regime and when the system possesses fast and slow degrees of freedom it is possible to formulate a FDT in terms of a non stationary effective temperature which depends on the slow degrees of freedom 16,17 .
We have found that our analysis of the heat interchanged between two NPs separated by a few submicrons agrees with the explains the rapid growth of the conductance observed in the simulation 8 , even when the NPs are in contact. Hence, we are able to provide a comprehensive explanation of the numerical results reported in Ref. 8 .
The formalism presented could also be applied to other situations such as the radiative heat transfer between a small dielectric particle and a surface 19 and the study of the optical forces due the radiation of a thermal source 20 , enabling us to go beyond the dipolar approximation.
Since the total heat must be a real quantity Therefore, the heat at the frequency ω is given through which after performing the thermal average leads to the equation equivalent to Eq. (11) .