Detecting Thermal Cloaks via Transient Effects

Recent research on the development of a thermal cloak has concentrated on engineering an inhomogeneous thermal conductivity and an approximate, homogeneous volumetric heat capacity. While the perfect cloak of inhomogeneous κ and inhomogeneous ρcp is known to be exact (no signals scattering and only mean values penetrating to the cloak’s interior), the sensitivity of diffusive cloaks to defects and approximations has not been analyzed. We analytically demonstrate that these approximate cloaks are detectable. Although they work as perfect cloaks in the steady-state, their transient (time-dependent) response is imperfect and a small amount of heat is scattered. This is sufficient to determine the presence of a cloak and any heat source it contains, but the material composition hidden within the cloak is not detectable in practice. To demonstrate the feasibility of this technique, we constructed a cloak with similar approximation and directly detected its presence using these transient temperature deviations outside the cloak. Due to limitations in the range of experimentally accessible volumetric specific heats, our detection scheme should allow us to find any realizable cloak, assuming a sufficiently large temperature difference.


SCATTERING SOLUTION TO THE HEAT EQUATION
Given the heat equation with homogeneous materials ρc p ∂ t T = ∇ · (κ∇T ) (1) in polar coordinates we take the Fourier transform of time and use a separable solution T (r, θ, t) = R(r)e ilθ e iωt giving This is the differential equation for a modified Bessel function (I l (z) or K l (z)) of z = we can make the coordinate transformation to reduce the solution in the primed coordinates to the homogeneous case. * e-mail: Baowen.Li@Colorado.EDU For a steady-state cloak (κ as for the perfect cloak, ρc p = ρ 0 c p0 (b/(b − a))η, i.e. evaluating ρc p at r = b when η = 1) no transformation will reproduce a homogeneous solution. Using x = iωρ 0 c p0 ηb/κ 0 (b − a)(r − a) and separation of variables we find . This can be solved by the method of Frobenius R l (x) = Σb ± nl x n±l with recurrence relation This relation is exact, but additional insight can be gained by expanding the solution by powers of Ka. For even terms in the series this is which is the same as series expansion for I l and K l respectively. On the other hand, for odd terms it becomes 2n,l the odd terms are therefore a function of the modified Bessel functions. Ergo, we term these components F[R l (x)]. A similar derivation can be carried out for a spherical cloak where l becomes half-integer instead of integer.

SIMULATIONS OF THE SSC Simulation Details
We model a rectangular domain of dimensions L =70 mm by L ⊥ =50 mm centered around a cloak of dimension a =13 mm, b =20 mm. The background medium is κ 0 = 71.4W/m · K, ρ 0 = 2100kg/m 3 , and c p0 = 1000J/kg · K. This gives a diffusivity of D = κ 0 /ρ 0 c p0 = 3.4 · 10 −5 m 2 /s and diffusion timescale τ D = L 2 /D = 144.12s. The initial temperature was 293.15K with thermal baths at 300K, and T 0 =293.15K giving a ∆T of 6.85K. After confirming that the simulations were invariant under a change of scale we use the natural units of x/L, y/L, t/τ D , (T − T 0 )/∆T . Space Dependence of the Deviation of the SSC In Fig S1 we take several slices of δT along y =constant for t = 2.08τ D /100, 2.08τ D /10, and 2.08τ D (or 3s, 30s, and 300s) (blue, green, and red respectively) to observe the spatial dependence more precisely. Slices are centered, offset, and outside the cloak. Initially the perturbation is well confined to the portion of the cloak that has been reached by the applied heat current. As time passes and heat has spread relatively far into the domain the δT grows and spreads throughout the domain. As the system approaches steady state, δT falls. The linear dependence inside the cloak for steady state implies that T (SSC) inside this domain is essentially constant. Outside the cloak δT is effectively a sine curve. This is clearest for the slice outside the cloak (after the initial curve, which contains higher that decay faster than the fundamental mode), but even for the other two their linear drop-off away from the surface of the cloak corresponds to the linear section of a sine curve.
where ∂r are the boundaries of the domain and the boundary conditions are stationary. In this case, there exists a steady state profile ∂ t T (SS) = 0 that uniquely satisfies the boundary conditions. By linearity, Assuming that the materials are everywhere homogeneous for some coordinate system we can apply a spatial Fourier transform (∇ 2 T (tr) ≡ −k 2 T (tr) ) and therefore where D = κ 0 /ρ 0 c p0 is the thermal diffusivity. For two systems that differ only in ρc p the difference between the two Note the time dependence is a sum of the difference of exponentials. This implies that for short times δT is approximately linear while for long times it decays exponentially. In the case that only a single Fourier mode is excited δT is separable. This is also approximately true if a small number of well separated Fourier modes dominate T (tr) ( k, 0).

SENSITIVITY OF A CLOAK TO THE INNER BOUNDARY
Following [1] we consider a PC that has lost a section of the inner boundary of thickness δ. Defining the domains I, II, III to be external to the cloak, the cloak, and the interior the boundary conditions (continuity of T andn · κ∇T ) are a (I) which, given an arbitrary b implies that a which can be expanded in the limit δ → 0. For l = 0 this gives which vanish at δ = 0 For l = 0 this gives which also vanishes at δ = 0 but converges more slowly than the previous case. For ω = 0 repeating the same procedure gives = 0 for l = 0. Thus for a PC (δ → 0) the temperature inside is a constant and the scattering field vanishes. This confirms that a PC is truly perfect, as expected.

SIMULATIONS AND EXPERIMENTAL STUDY OF THE BC
We follow [2] to model the BC as rectangular domain of dimensions L =45 mm by L ⊥ =45 mm centered around a cloak with hidden region of size a =6 mm, first layer of r 2 =9.5 mm, and second layer of b =12 mm. The background medium is κ 0 = 2.3W/m · K, ρ 0 = 2000kg/m 3 , and c p0 = 1500J/kg · K, the outer layer's medium is κ 1 = 9.8W/m · K, ρ 1 = 8440kg/m 3 , and c p1 = 400J/kg · K, the inner layer's medium is κ 2 = 0.03W/m · K, ρ 2 = 50kg/m 3 , and c p2 = 1300J/kg · K, and the interior medium is κ 3 = 205W/m · K, ρ 3 = 2700kg/m 3 , and c p3 = 900J/kg · K. This gives a diffusivity of D 0 = κ 0 /ρ 0 c p0 = 7.67 · 10 −7 m 2 /s and diffusion timescale τ D0 = L 2 /D = 2641.3s. The initial temperature was 273.15K with thermal baths at 333.15K, and T 0 =273.15K giving a ∆T of 60K. For plotting we use the natural units of x/L, y/L, t/τ D0 , (T − T 0 )/∆T . The results are shown in Fig. S2, confirming that the cloak is visible in the transient response We also test the time-dependence of δT using Fig. S3. As expected, the time-dependence is a sum of exponential terms like those predicted in eq. 15. Because of the additional boundaries in this system we see that there are more Fourier modes excited. What's more, the addition of these Fourier modes implies that the solution is not fully separable. This is clear from the separation of δT at the nearest point around the cloak to the heat source. This can also be seen with the initial plot of δT in Fig. S2 where there is initially relative cooling outside the cloak that is not found elsewhere along it's surface.
To verify our simulations we follow the procedure of Fig. S2 with an experimental realization of the BC and its homogeneous background. Since the temperatures at each boundary are not perfectly fixed, we normalize the data using the infimum of T 0 =285.08K and supremum of T =325.96K giving ∆T =40.88K. Using the normalization (T − T 0 )/∆T shows good agreement with the theoretical result. Results are plotted in Fig. S4. There is a slight discrepancy in the temperature deviation between the simulations and experiment. This is due to a slight difference in temperature gradients applied to the BC and homogeneous cases. Hence, this effect is strongest at the boundaries of the system and more negligible near the cloak itself.