Tunable analog thermal material

Naturally-occurring thermal materials usually possess specific thermal conductivity (κ), forming a digital set of κ values. Emerging thermal metamaterials have been deployed to realize effective thermal conductivities unattainable in natural materials. However, the effective thermal conductivities of such mixing-based thermal metamaterials are still in digital fashion, i.e., the effective conductivity remains discrete and static. Here, we report an analog thermal material whose effective conductivity can be in-situ tuned from near-zero to near-infinity κ. The proof-of-concept scheme consists of a spinning core made of uncured polydimethylsiloxane (PDMS) and fixed bilayer rings made of silicone grease and steel. Thanks to the spinning PDMS and its induced convective effects, we can mold the heat flow robustly with continuously changing and anisotropic κ. Our work enables a single functional thermal material to meet the challenging demands of flexible thermal manipulation. It also provides platforms to investigate heat transfer in systems with moving components.


Supplementary Note 1: Derivations of effective solid configuration in a bilayer structure
Considering the rotations of the actual fluid domain under finite angular velocity, the field deviation in the azimuthal direction should be considered at the solid-fluid interface. Hence, the entire system can be regarded as a solid bilayer system with an anisotropic center. To analyze the system, the rotated fluid center is considered as an effective solid plate with anisotropic conductivities of , ( ) and , ( ) along the principle axes (r and θ). To create expected behaviors in a solid system without external field distortion, a rigorous analysis based on steady-state heat conduction equation ( In equations (1) ~ (4), A ~ F are scaled components of the solutions, which can be derived from the boundary conditions. Due to the anisotropic and nonhomogeneous conductivities of the center, the temperature of region IV cannot be directly presented with the standard form of Laplace equation. Hence, the components of implicit functions G1(r) and G2(θ) are employed to describe the temperature distribution in the center. It's noted that B should be 0 under the above assumption. r denotes the specific locations. Under certain temperature gradient ∇ along radial direction generated at the system boundary, the field distribution in each region can be obtained with the temperature and flux consistencies at adjacent boundaries. Here, the boundary conditions at the regional interfaces can be written as: At the interface of r = R2: At the interface of r = R1: Considering the assumption of finite temperature of the center, the general solution of equation (7) can be expressed as: ( ) = • . Hence, the thermal field at the interface of r = R0 can be expressed as: Solving equation (9), the relations among the constants of E, F, and G can be achieved. Considering the temperature gradient ∇ along radial direction and conductivities of the system under the assumption of B = 0, the constants A ~ F and the component functions G1(r) and G2(θ) can be achieved. Furthermore, the following principle of matching adjacent field and avoiding external distortions can be derived by combining the scaled components.
In equation (11), and denote the effective temperatures at the interface between regions III and IV.

Boundary conditions of the actual fluid center
For matching the effective solid domain and rotated fluid, the same principle system is considered here. Then, the actual temperature distributions of the fluid domain (r < R0) at steady state can be also achieved with the following convective equation: In equation (12), κ0 is the initial conductivity of the employed fluid. As [2] indicated, the general solution of the above equation can be obtained with the first order Kelvin's function. Hence, the actual temperature distribution inside the fluid domain can be expressed as: In equation (13), ϕ(ω, r) is an argument in the solution to illustrate the rotational effect, and it is a continuous real function of ω and r, which can be defined in the general solution and calculated in the convective process.
According to the properties of Bessel function, the general temperature gradients at radial directions can be written as follows: As indicated in [2], θ-ϕ(ω)+π/4 should approach θ based on minimum entropy production, when ω is large enough. In this paper, the general case with modulated angular velocity is considered. Hence, the specific condition used at high speed should be motivated to a general one. It indicates that a larger temperature gradient can be obtained at a low angular velocity of fluid flow. That is, a finite thermal field deflection under the prerequisite of minimum entropy production is observed. Hence, the initial relation of ( , ) = = arctan 1 / 1 should be employed here 3 . According to the asymptotic expansions under the condition of → ∞, the value of approaches π/4.

Boundary coupling of the effective solid domain and actual fluid system
The next step is to couple the effective and actual values at the interface (same location). Since the effectively anisotropic solid domain is employed to make an equivalence of the actual fluid, its locally effective values at the boundary r = R0, including the temperature and radial heat flux, should be in accordance with the actual fluid on each point. Hence, the following relations at the boundary (r = R0) should be obeyed for a general case: Whether the actual fluid domain or the effective solid plate, the same interface of r = R0 is shared. Hence, both of the actual and effective temperatures on each point along the interface should be same. However, this condition cannot be achieved under a spinning system, due to the azimuthal field deformation. Hence, the differences between the actual and effective values on each point should be minimum.
Considering the diffusive system, the first order of the solution G2(θ) can be employed here to simplify equation (16), thus approximately leading to the following relation: The minimized temperature deviation can be achieved under specific angular velocity, once the following relation is matched.
Taking equation (18) into the boundary conduction of equation (15), the locally effective radial conductivity at the solid-fluid interface can be obtained.
To  should be involved to describe the deflected temperature fields and multifarious functions (not limited to cloaking).

Effect of fluid viscidity
The fluid viscidity would affect the practical operations of tunable analog thermal materials. Here, we respectively employ the uncured PDMS and water in the center to make a fair comparison of its effects at a moderate velocity. As shown in Supplementary Figure 4, the surface pressure of the uncured PDMS is much lower than that of water at the same moderate-velocity, which reveals fewer outflows along radial direction with large

Effective static solid center
Upon the effective conductivity along the internal boundary of the actual fluid domain, we indicate that the entire fluid center can be further considered as an anisotropic static solid domain with an axis rotation to the principle system (r, θ). Under the consideration of spinning effects of actual fluid, the system rotation to the principle system (r, θ) caused by varied rotation rates should be also involved in the effective solid center. As mentioned above, approaches π/4 once → ∞, while it changes continuously with the increasing rotation rates. Besides, the temperature gradient would deflect round and round with the increasing rotation rates, and become stable once the rotation rate is large enough in the rotated fluid domain. During this process, the deflection would reach π/4 (final stable azimuth) several times with the increasing rotation rates. That is, the system rotation in the effective static solid system can be considered as the superposition time of reaching π/4 at varied rotation rates of the actual fluid domain. Hence, the effective static solid system (r', θ') can be expressed as 4 : = , = + . Here, ω1 denote the rotation rate when the system rotation first reaches π/4. In the current system, the value of ω1 is 0.003 rad/s. The Considering the heat transfer between inhomogeneous solid media 5 , the heat flux deflection at the interface can be achieved with the heat flux components along x and y directions. The effective conductivity at the solidfluid interface in Cartesian coordinate can be written as: noteworthy that the behavior of sensitive cloaking can be observed in the cases with a wide range of spinning velocities, since the effectively large conductivity contributes to the rapid homogenization of thermal profiles with few temperature gradients.
Supplementary Figure 5. Temperature distributions and heat flux deflections at the regional boundary of the case at 1000 rad/min. a illustrates the temperature distributions; b denotes its heat flux deflections.
abovementioned one (equation (27)) without regional rotation of enhanced transparency case.

Supplementary Note 6: Differences between the tunable analog thermal material and thermal ground plane
We are aware that the technique of thermal ground plane (TGP) can be also used to achieve the effectively extreme-large conductivity and rapidly passive heat spreading. Though there exists a similarity of the effectively extreme-large conductivity, the TGP and tunable analog thermal materials can be regarded as two independent techniques respectively based on heat pipes and dynamic thermal metamaterials.
For a general TGP, the effectively extreme-large conductivity and rapidly passive heat spreading are obtained through cyclic two-phase fluid motions pumping by the continual condensation and evaporation of the working fluid 6,7 . Hence, the wicking structure for vapor chambers 6,7 (or a serpentine-arranged tube for oscillating heat pipes 6 ) is indispensable to the motions of working fluid, while additional heat sink for generating condensate and thermal source for generating vapor are also required to assist the entire cyclic phase changes. Due to the spreading effect, TGP should possess high effective conductivity and exhibit a uniform thermal profile without temperature gradients. However, the tunable range of conductivity is passive and quite dependent on the combined effects of designed parameters, including the external heat input, working fluid, filling ratio, and the actual structural design.
Thus, it is hard to achieve the active and continuous control in the full range of conductive demands without changing the external heat inputs or practically environmental parameters.
Compared with TGP, the tunable analog thermal material is obtained by modulating a single fluid without phase changes inside a bilayer structure, while its functionality is not limited to the effectively extreme-large conductivity and uniform thermal profiles at extreme velocity. By adjusting the fluid at arbitrary velocities, a full range of effective conductivity from near-zero to near-infinity can be actively and continuously observed without changing any structural and environmental parameters. Meanwhile, the fluid continuity and mobility further contribute to the inhomogeneous distributions of the effective conductivities at a specific velocity, which paves the most essential conditions for functional thermal meta-devices with different demands of temperature gradients.
Hence, various manipulative behaviors, including enhanced transparency, field contortion, field inversion, and sensitive cloaking can be also significantly realized and switched in-situ by simply adjusting the velocities.
In general, tunable analog thermal material possesses some unique properties, including the single-phase working fluid, minimalist structure, actively and robustly tunable conductivity, and various behaviors of field distributions. It is believed that these aspects would be complementary to conventional techniques, and further motivate the thermal managements with multifarious behaviors in-situ.