# Near-field radiative energy transfer between sphere-sphere and sphere-flat surfaces

This section is about radiative heat transfer between different geometries, in particular between two spheres and between a sphere and a half-space, when they are held at different temperatures and separated in vacuum.

#### Near-field radiative heat transfer between two spheres

Near-field thermal radiation between two spheres was first reported by Volokitin and Persson  by assuming them as dipoles. When there are two spherical nanoparticles whose dielectric constants are ε1 and ε2 , the spectral power dissipated in particle 2 by the electromagnetic field induced by particle 1 can be written using the dipolar approximation as

where x2 is the position of the particle 2 and α2 = 4πR32 − 1) / (ε2 + 2) is the polarizability of a sphere of radius R having the relative permittivity of ε2. The electric field incident on the particle 2, Einc(x2,ω), is created by the thermal fluctuating dipole of particle 1 at Temperature T1 :

where $\rm \overline {\overline G}_e (x_2,x_1,\omega)$ is the electric dyadic Green’s function between two dipoles in vacuum and expressed as 

where d = |x2-x1|, $\rm \overline {\overline I}$ is the identity tensor, and $\rm {{\hat u} _r}{{\hat u} _r}$ the dyadic notation of unit vectors. Because of thermal fluctuations, particle 1 has a random electric dipole that yields the correlation function of the dipole:

where < > represents ensemble averaging, and * denotes the complex conjugate. Equation (4) is in fact the modified form of the ensemble average derivation provided by fluctuation-dissipation theorem. By combining the above equations, we finally obtain the heat exchange between two dipoles at temperatures T1 and T2, written as

The conductance between two dipoles due to the near-field radiative heat transfer can thus be expresses as 

It should be noted that radiative heat transfer between two spheres has the 1/d6 spatial dependence, which is typical of the dipole-dipole interactions. The thermally fluctuating dipole at one nanoparticle induces electromagnetic field on the other nanoparticle to cause the second dipole to fluctuate. Equation (6) suggests that near-field heat transfer between two nanoparticles have a similar resonant behavior at the surface polariton resonance when the dielectric constant approaches –2. In such case, the polarizability α has a resonance. Provided that the surface polariton resonance occurs when the dielectric constant approaches –1 in case the material is interfaced with the vacuum , this resonant behavior is not directly because of the surface polariton resonance: instead, is named as the localized surface polariton resonance – grouped oscillations of the charge density confined to nanostructures . The localized surface polariton resonance appears in the visible range for metals and in the infrared for polar materials.

While the dipole approximation elucidates the d-6 dependence of the near-field radiative heat transfer between two spheres, this dependence is valid only when R << λT and d >> R1 + R2, where λT is the characteristic wavelength defined from Wien’s displacement law. For cases when the dipole approximation is not valid, calculation of near-field thermal radiation between two spheres becomes computationally challenging, mainly due to the difficulty in determining the dyadic Green’s function. Domingues et al.  attempted to address this challenge by using the molecular dynamics (MD) scheme. After computing all the atomic positions and velocities as function of time using the second Newton’s law, $\sum _{j} f_{ij} = m_i \ddot x _i$, where mi and $\ddot x_i$ are the atomic mass and acceleration and fij is the interatomic force exerted by atom j on atom i, the power exchange between two nanoparticles (NP1 and NP2) is computed as the net work done by a particle on the ions of the other particle:

The interatomic force fij is derived from the van Beest, Karmer, and van Santen (BKS) interaction potential , in which a Coulomb and a Buckingham potentials are included. On the other hand, the dyadic Green’s function for the two-sphere configurations was directly determined by using vector spherical wave functions and the recurrence relations . They investigated the scattering between two spheres by expanding the electromagnetic field in terms of the vector spherical waves at each sphere and re-expanding the vector spherical waves of one sphere with the vector spherical waves of the second sphere to satisfy the boundary conditions. Reoccurrence relations for vector spherical waves are also required to reduce the computational demands in determining translation coefficients of each spherical wave function term. Due to complexities in the formulations, we do not include equations of the dyadic Green’s function for two spheres: see  for more details.

Near-field radiative heat transfer between two spheres computed in both Refs.  and  agree well with the dipole approximation for the interparticle distance larger than the radii of spheres. However, when the distance is smaller than the particle radius, two studies show different trends: Ref.  shows a higher gap dependence than d-6 in the thermal conductance, but Ref.  has the d-1 gap dependence as the gap distance further decreases. In Ref. , a deviation between the MD and the dipole approximation appears when d < 8R , reaching four orders of magnitude higher than the dipole approximation. This enhanced heat transfer appears to be due to the contribution of multipolar Coulomb interactions. However, the thermal conductance in Ref.  asymptotically approaches a d-1 slope when the interparticle gap distance is much smaller than the particle radius. In fact, this slope change is consistent with the result of the proximity force approximation or the Derjaguin approximation , in which the flux between two curved surfaces can be locally described as a flux between two parallel plates separated by a distance d(r). For the near-field radiative heat transfer between two flat surfaces, the corresponding thermal conductance per unit area, or the radiative heat transfer coefficient hr, varies with the d-2 dependence in the short distance regime [9-11]. When the spheres are separated by a minimum gap distance d0 much smaller than the sphere radii, the thermal conductance between two spheres can be approximated as

where $d(r)=d_0 + R - \sqrt {R^2 - r^2}$ is the local distance between two spheres. This equation is further simplified to G12 = πd0Rhr(d0,T), providing the d-1 gap dependence for the near-field radiative thermal conductance between two spheres for small gap distances. These contradictory predictions suggest that further theoretical and experimental investigations on near-field thermal radiations between two spheres are required.

#### Near-field radiative heat transfer between a sphere and a half space

In this part, we discuss near-field thermal radiation between a small spherical particle and a semi-infinite medium, which can be considered as a simplified model of the scanning tunneling microscopy or scanning thermal microscopy . Similarly to the previous chapter, the small particle can be approximated as a dipole of radius R with dielectric constant εP(ω) and temperature Tp. The semi-infinite surface is maintained at temperature TB and has the dielectric constant εB(ω). The center of the particle is at a distance d above the interface. Then, the spectral mean power radiated by the half space and absorbed by the particle can be written as $P_{B\to P} (\omega )=\frac{2\omega ^{4} }{\pi c^{4} } Im[\varepsilon _{B} (\omega )]Im[\alpha (\omega )]\Theta (\omega ,T_{B} )\sum _{n,m=x,y,z}\int _{B}\left|G_{nm} (r_{p} ,r',\omega )\right|^{2} d^{3} r'$

(9)

When the fluctuating currents inside the particles radiates into the half space and dissipates, the power locally dissipated per unit volume at a point r inside the space can be written as $P_{P\to B} (r,\omega )=\frac{2\omega ^{4} }{\pi c^{4} } Im[\varepsilon _{B} (\omega )]Im[\alpha (\omega )]\Theta (\omega ,T_{B} )\sum _{n,m=x,y,z}\left|G_{nm} (r,r_{p} ,\omega )\right|^{2}$

(10)

where $\alpha (\omega )=4\pi R^{3} [\varepsilon _{P} (\omega )-1]/[\varepsilon _{P} (\omega )+2]$ is the polarizability of the dipole and Gnm(r,r',ω) is (n, m) component of the dyadic Green’s function at r due to a point source at r' in a system constituted by two semi-infinite media whose dielectric constants are either 1 if z ≥ 0 or εB(ω) if z < 0. It should be noted that the dipole polarizability needs to be corrected to take into account the interaction between the dipole and the interface when d is comparable to R.

When $P_{B\to P} (\omega )$ is calculated for a SiC spherical particle at temperature TP = 300 K of radius R = 5 nm at different distances above the SiC surface , two peaks are observed at frequency ω1≈1.756×1014 rad/s and ω2≈1.787×1014 rad/s, each of which corresponds to the localized surface polariton resonance at the SiC dipole (i.e., Re[εB(ω)]=-2 ) and the surface polariton resonance on the SiC surface (i.e., Re[εB(ω)]=-1 ), respectively. Since the electromagnetic waves associated with theses resonance behaviors are evanescent, strong coupling between the field and the particle enhances the radiative heat transfer while the particle lies in the evanescent region. Moreover, the near-field radiative heat transfer increases as d-3 , enhanced by several orders of magnitude than the far-field thermal radiation. Another example is shown in Fig. 1, which plots PP→B(ω), i.e., near-field thermal radiation emitted from a SiC particle of radius R = 25 nm heated at TP= 1000 K to a gold surface right below the particle. There is a sharp peak at ~ 930 cm-1, corresponding to 1.756×1014 rad/s or 10.6 μm when the tip is near the surface, suggesting that the heated SiC particle emits a quasi-monochromatic thermal radiation at around 10.6 μm in the near-field regime.

It should be noted again that the dipole approximation for a sphere is valid when R << λT and R < d. In the other extreme case, i.e., d << R~λT, the dipole approximation may not be valid, and instead we should use the proximity force approximation (or Derjaguin approximation). Using the same scheme as in near-field radiative heat transfer between two spheres, it can be shown that the near-field radiative heat transfer has the d-1 dependence, instead of d-3, when the proximity force approximation is used. This result will be further discussed in the consecutive section. Figure 1. Mean spectral power radiated from the SiC particle (of radius R = 25 nm and held at TP= 1000 K) to the gold surface right below the particle.

## References

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