Radiative heat transfer in fractal structures

The radiative properties of most structures are intimately connected to the way in which their constituents are ordered on the nanoscale. We have proposed a new representation for radiative heat transfer formalism in many-body systems. In this representation, we explain why collective effects depend on the morphology of structures, and how the arrangement of nanoparticles and their material affects the thermal properties in many-body systems. We investigated the radiative heat transfer problem in fractal (i.e., scale invariant) structures. In order to show the effect of the structure morphology on the collective properties, the radiative heat transfer and radiative cooling are studied and the results are compared for fractal and nonfractal structures. It is shown that fractal arranged nanoparticles display complex radiative behavior related to their scaling properties. We showed that, in contrast to nonfractal structures, heat flux in fractals is not of large-range character. By using the fractal dimension as a means to describe the structure morphology, we present a universal scaling behavior that quantitatively links the structure radiative cooling to the structure gyration radius.

Figure 1

Schematic illustration of Vicsek fractals (composed of the same nanoparticles, say, $N$ sphere of the same radius $R=5$ nm) of functionality $F$ (i.e., number of nearest neighbors of the branching sites): (a) $F=2$ (VF2); (b) $F=4$ (VF4); (c) two-dimensional Vicsek fractal with $F=6$ (VF6); and (d) three-dimensional Vicsek fractal with $F=6$ (3D-VF6). The growing fractals formed by repeated addition of copies of the initial configuration of nanoparticles. Moreover, $d=3R$ is the separation between nearest neighbors and $r$ is the typical separation between each pair of particles.

Figure 2

Double logarithmic plot of the radius of gyration for the Vicsek fractals of $N$ particles with functionality: $F=2$ (VF2), 4 (VF4), and 6 (VF6 for both 2D-VF6 and 3D-VF6). The dashed lines represent the asymptotic behavior of curves for large fractal sizes ($N>100$) and the scaling exponents. The fractal dimensions are $D_f=1,1.4649,1.77$ for $F=2,4,6$, respectively.

Figure 3

Eigenvalue histogram (a) and small piece of the histogram (b) of the interaction matrices $\mathbb{W}$ for Vicsek fractals (VF2, VF4, and VF6) and random gas of particles (RGP) consisting $N=1000$ nanoparticles with $d=3R$. Blue vertical lines denoting eigenvalues and red curves represent the eigenvalue density distribution for each configuration.

Figure 4

Transmission coefficient between two Ag nanoparticles with $R=5$ nm in (a) a two-body system, (b) inside a random gas of particles, as a function of separation distance $r$ and spectral variable $X$.

Figure 5

Color map of average transmission coefficient between a pair of particles in a Vicsek fractal as a function of pair separation distance $r$ and spectral variable $X$ for (a) VF2, (b) VF4, (c) 2D-VF6, and (d) 3D-VF6.

Figure 6

Average mutual conductance in collection of Ag nanoparticles at temperature $T=300$ K as a function intraparticle distance $r$. The result at each distance is normalized to the conductance between two isolate nanoparticles at same distance. Each configuration consists $N=1000$ spherical nanoparticle with $R=5$ nm.

Figure 7

(a) Average conductance of particles in a structure as a function of structure size $N$ for Vicsek fractal of functionality $F=2,4,6$ and periodic arrangement of nanoparticles in two (2D-P) and three (3D-P) dimensions. (b) The structure self-conductance as a function of normalized gyration radius for both fractal and nonfractal structures. (c) Structure conductance for VF4 fractals of polar materials (SiC and ${\mathrm{SiO}}_2$) at temperature $T=300$ and 350 K as a function of normalized gyration radius.