An approach towards a perfect thermal diffuser

A method for the most efficient removal of heat, through an anisotropic composite, is proposed. It is shown that a rational placement of constituent materials, in the radial and the azimuthal directions, at a given point in the composite yields a uniform temperature distribution in spherical diffusers. Such arrangement is accompanied by a very significant reduction of the source temperature, in principle, to infinitesimally above the ambient temperature and forms the basis for the design of a perfect thermal diffuser with maximal heat dissipation. Orders of magnitude enhanced performance, compared to that obtained through the use of a diffuser constituted from a single material with isotropic thermal conductivity has been observed and the analytical principles underlying the design were validated through extensive computational simulations.


I. Obtaining a uniform temperature distribution (rectangular space) and isotropic heat flow (spherical space), through mutual mapping
The mapping transformation involves the mapping(1) of a rectangular block, formed from an isotropic material of thermal conductivity: κ, and of length: L Figure S1 (a), onto a cylindrical sector: Figure S1 (b), The correspondent mapping between a space spanned in rectangular geometry, i.e., U = (x, y, z) to that of a space spanned by cylindrical geometry, i.e., V = (X, Y, Z), accomplishes the incorporation of features relevant to a uniform temperature distribution coupled with isotropic diffusion.

Figure S1
Schematic mapping of a (a) rectangular block of length L, width 2W and thickness t mapped onto (b) a cylindrical sector with central angle 2θ, inner radius L/θ -W, outer radius L/θ +W and thickness t. Two representative correspondent points (indicated, for example, by the w and ★) are indicated. Note that, as depicted, θ = π/2, The mapping of the (x, y, z) space to (X, Y, Z) space, indicated in Eqn. S1, below, is conformal and preserves the parallelism of the isotherms in the two spaces. The mutual geometric relationships are represented through: The Jacobian (J), for the transformation, is given by(2-4) : (S2) An alternate representation related to Eqn. (S2) involves the use of a cylindrical coordinate system (from Eqn. S1b), and with reference to the rectangular coordinate system at the top of Figure S1(b), and uses: The modification of the thermal conductivity, due to the transformation given by Eqn. (S1), is then obtained through: The equivalent thermal conductivity in a rectangular (X, Y, Z) coordinate system is done through a rotation by an angle θ about the Z-axis, as follows: where, Such a form has been indicated as Eqn (2) of the main text and the schematic variation of conductivity ( , , and , has been plotted in Fig. 3(A), in the main text.

II. Constructing the spatial anisotropy represented in Eqn. (2)/(S5) using isotropic materials
The spatial anisotropy represented through the relation in Eqn.
(2) of the main text or (S5) in Section I of the Supplementary information, can be effectively reduced to a two-dimensional form, in the (r, θ ) system, as r R c Z c r = = κ κ . Also, it is noted from (S5) and Figure 3 A composite thermal conductivity satisfying the above conditions can be obtained through the arrangement of two layers, with isotropic thermal conductivity values of κ 1 and κ 2 . In our previous work (2, 7), it was shown that the effective thermal conductivity of such layers arranged in parallel, i.e., κ p was larger than that of layers arranged in series, parallel, i.e., κ s .

Figure S2
The alternate stacking of two layers, with isotropic thermal conductivity values of κ 1 and κ 2 , respectively, can yield an effective composite thermal conductivity measured in series, i.e., κ s , and another larger value for parallel measurement, i.e., κ p.
For example, with reference to Figure S2, it may be derived that: Consequently, for r < R, the orientation of layered elements would need to be parallel to the heat flux, with a correspondence of the (/ ) to κ p and with being related to the κ s . However, for r > R, the orientation of layered elements would need to be arranged in series with the heat flux, with a correspondence of the (/ ) to κ s and related to κ p . This would then imply a spatially varying κ 1 (r) and κ 2 (r).

III. Temperature variation in thermal diffuser employing spatial conductivity modulation
We consider cylindrical diffusers, of the type shown in Fig. 2(A) -in the main text, or Figure S1 (b), with an input heat flux at r = R i (= R-W) placed in an ambient (at a temperature = T amb ). A given convective heat transfer coefficient h between the outer surface at r = R o (= R+W) and the ambient is assumed (5, 6). The resulting temperature, at the outer surface, for the cylindrical diffusers, with (a) homogeneous and isotropic material ( T R O iso ), and (b) with anisotropic and spatial modulation of the constituents ( T R O aniso ), are then derived.
From thermal energy conservation,

a) Temperature T r iso in an isotropic cylindrical diffuser
From heat flux continuity, for an isotropic cylindrical diffuserκ , Integrating Eqn. (S7) between r =r (at a temperature, T = T r iso ) and r = R o (at T = T R O iso ) and using the relation in (S6) for the boundary conditions, we get, This is Eqn. (3a) in the main text.

b) Temperature T r aniso in an anisotropic cylindrical diffuser
From heat flux continuity, for an anisotropic cylindrical diffuser, with spatially varying thermal conductivity, given by κ c (r,θ,Z ) = κ r c = R r κ , the heat conservation equation yields: Integrating Eqn. (S12) between r =r (at T = T r aniso ) and r = R o (at T = T R O aniso ) and using the boundary conditions represented in (S8), we get, Then, This is Eqn. (3b) in the main text.

IV. The arrangement of materials used for simulation in Figure 5
We indicate a representative material arrangement underlying the computational simulations needed to validate our analytical derivations and the principles underlying the perfect thermal diffuser design. Following Fig. 2(B), we use representative R i = 0.5 cm, R o = 4.5 cm (implying R = 2.5 cm and W = 2 cm), and L = 2.5 π cm, with κ = 60 W/mK .
From Eqn.
(2)/ Eqn. S5, we obtain: At any given r, the thermal conductivity mimicking the spatial anisotropy may be determined using the above relation. We approximately sub-divide the diffuser into four regions of radial thickness 1 cm. The (/ ) and is subsequently computed from (S15), at any given r (taken for example, to be at the center of the individual regions, e.g., for the first region spanning r = 0.5 cm and r = 1.5 cm, the effective r = 1 cm. Following the rationale presented in Section III of the Supplementary Materials, for r < R, the (/ ) is corresponded to a κ p and the is related to the κ s . Consequently, the appropriate κ 1 and κ 2 is obtained through using Eqn. (S7b). Such details for all the four sections of the anisotropic diffuser are depicted in Table I, on the next page.  Table S1 The corresponding arrangement of materials in Fig. 5(A). It is discussed in the paper as to how the arrangement of isotropic materials in an anisotropic manner could yield uniform temperature distribution along with a lower source temperature. Following Fig. 2(B), R i = 0.5 cm, R o = 4.5 cm, and L = 2.5 π cm, with a representative κ = 60 W/mK. The 4 cm (= R o -R i ) wide strip is divided into four 1 cm wide regions (Column 1), with an average r (Column 2), and the use of Eqn. (S15) to compute the (Column 3) and (Column 4). Consequently, the relations in Eqn. S7b, were used to obtain κ 1 (Column 5) and κ 2 (Column 6). For a point away from the maximum, i.e., at R = 5 cm and W = 2 cm For a diffuser needing to dissipate a q in , of 2x10 6 W/m 2 , the corresponding anisotropic thermal conductivity was derived assuming a nominal isotropic κ = 60W/mK and T amb = 293K.
A comparison of the analytically predicted temperature variation with the computational simulation results is illustrated in Figure S3 (b), and found to be linear, in excellent accord with predictions ( Figure 4). A reduction in the source temperature by ~ 50 K in the engineered composite was observed, corresponding to temperatures at the center (r= R i ) of 1200 K and 1248 K, for the anisotropic and the isotropic cases, respectively.  Table S2 below.
(B) The resulting temperature variation, from the simulations, was in excellent accord with the analytical relation derived in Eqn. 3(b).

Table S2
The corresponding arrangement of materials in Fig. S3(a).