Thermal homeostasis using microstructured phase-change materials

Shao-Hua Wu; Mingkun Chen; Michael T. Barako; Vladan Jankovic; Philip W.C. Hon; Luke A. Sweatlock; Michelle L. Povinelli

doi:10.1364/OPTICA.4.001390

1. INTRODUCTION

Thermal control schemes for space have focused on emission control since the absence of convection makes radiative emission the sole cooling mechanism. Radiators that emit significantly more when heated than cooled can be designed to dampen temperature fluctuations that arise from changes in solar illumination and from on-board heat generation [1]. Solid-state approaches to emission control [2,3] offer lightweight alternatives to approaches based on mechanically moving parts [4–7] or fluid-filled heat pipes [8,9]. The majority of these schemes, however, require electrical power [2,4–7], limiting their application space. Here, we present a novel, passive scheme for thermal self-regulation. Our design uses micropatterned phase-change materials to achieve a $10 \times$ difference in emissivity between high- and low-temperature states, resulting in an $\sim 20 \times$ reduction in temperature variation relative to ordinary materials.

Micropatterning has been a subject of intense research for applications in radiative cooling [10,11]. Recent work has shown that micropatterned materials can be designed to achieve near-unity infrared (IR) emissivity and steady-state radiative cooling [11]. To provide passive temperature regulation, however, temperature-switchable emissivity is required. Phase-change materials such as vanadium dioxide ( ${VO}_{2}$ ) show a dramatic change in optical properties near their phase-change temperature, $T_{c}$ [12–15]. ${VO}_{2}$ has been used previously to achieve switchable reflectivity and transmissivity in the IR [16,17] and visible ranges [13,18]; IR emissivity tuning has also been demonstrated with this unique material [19,20]. None of these works, however, considered a passive, switchable IR emitter for thermal self-regulation. Previous work on bulk perovskite manganese oxide used a metal–insulator phase transition to provide switchable emission [3,21,22]; however, the maximum difference in emissivity between high- and low-temperature states was only $\sim 0.5$ , and the width of temperature range for the phase transition was as large as $\sim 200 K$ [22]. As we will see below, the width of the transition limits temperature regulation. Chalcogenide phase-change materials such as Ge–Sb–Te (GST) can also provide switchable optical properties [23–25]. However, their non-volatile nature—in which the phase transition must be triggered by an energetic pulse—makes them nonideal for passive thermal homeostasis applications.

In this paper, we design a ${VO}_{2}$ -coated silicon microcone structure with a large emissivity difference of 0.8 between low and high-temperature states. We show that our structure’s sharp change in emissivity at the phase-change temperature (330 K) provides excellent thermal regulation capability due in part to the narrow width of the ${VO}_{2}$ insulator-to-metal phase transition, which can be as small as 4 K for high material quality [18]. In particular, we solve the time-dependent heat equation using a lumped capacitor approach to obtain the transient temperature in response to a time-varying heat load. Our results show an $\sim 20 \times$ reduction in temperature variation relative to an uncoated silicon film.

A. Design of Structures for Thermal Homeostasis

The concept of thermal homeostasis is illustrated in Fig. 1. The ideal surface for thermal homeostasis would have near-zero thermal emissivity below the design temperature set point, $T_{c}$ [Fig. 1(a)], and close-to-unity thermal emissivity above it [Fig. 1(b)]. In this case, fluctuations in temperature will be mitigated by changes in emissivity. When the object gets too cold, heat loss to the environment is minimized [Fig. 1(a)]; when the object gets too hot, heat loss is enhanced [Fig. 1(b)].

Fig. 1. Illustrations of thermal homeostasis in optics. A surface that radiates much more at a higher temperature will help maintain the object at the target temperature, $T_{c}$ .

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We have designed a structure with temperature-dependent emissivity needed for thermal homeostasis. Our design is shown in Fig. 2(a). A square array of silicon microcones is covered by a conformal layer of ${VO}_{2}$ . Cone arrays are known to show strong antireflection and to be relatively insensitive to the angle of incidence, making them well suited for absorber and emitter applications [26–29]. In the calculations below, we will take the height of silicon microcones to be 40 μm, the period to be 20 μm, and the thickness of the coating to be 200 nm. These dimensions were optimized by running a particle swarm optimization [30] to maximize the broadband emissivity difference between the insulating and metallic states. The lower and upper bounds on period, cone height, and ${VO}_{2}$ thickness were set to at 5–40 μm, 5–40 μm, and 0.2–1.0 μm, respectively. For reference, we will also consider a flat, ${VO}_{2}$ -coated silicon film [Fig. 2(b)] and an uncoated silicon film [Fig. 2(c)], and calculate thermal emissivity for all three structures.

Fig. 2. Design of structure for thermal homeostasis. (a) A square array of silicon microcones with a conformal ${VO}_{2}$ coating, residing on a silicon film. Note that layer thicknesses are not drawn to scale. (b) A flat, ${VO}_{2}$ -coated silicon film. (c) An uncoated silicon film.

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2. RESULTS

A. Spectral Emissivity

We first calculated the IR spectral emissivity for the ${VO}_{2}$ -coated Si microcones (see Methods). The results are shown in Fig. 3(a). For $T < T_{c}$ , ${VO}_{2}$ is in the insulating phase and the emissivity of the ${VO}_{2}$ -coated microcones is low (blue curve). For $T > T_{c}$ , the ${VO}_{2}$ layer is metallic and the emissivity is high (red curve). The ${VO}_{2}$ -coated microcones thus act as a switchable thermal emitter, with a nearly $10 \times$ difference in emission between the insulating and metallic states.

Fig. 3. Emissivity spectra. (a) ${VO}_{2}$ -coated silicon microcones, (b) a ${VO}_{2}$ -coated flat silicon film, and (c) an uncoated silicon film. Results are for normal incidence, averaged over polarization.

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The difference in emission can be understood as follows: consider IR light incident on the structure. The metallic state has a much larger imaginary part of permittivity than the insulating state, yielding strong attenuation in the thin ${VO}_{2}$ layer. From Kirchoff’s law, the increased attenuation (absorption) corresponds to an increase in emission. We note that the oscillatory features in Fig. 3(a) are due to reflection from the back surface of the sample, resulting from the negligibly small absorption in Si.

The emission from a microcone structure is far more switchable than that from a planar film. For the planar film, the difference in emissivity between metallic and insulating states is smaller [Fig. 3(b)]. Moreover, the emissivity for the metallic state [red curve; Fig. 3(b)] is much lower than for the microcones [red curve; Fig. 3(a)]. To understand this effect, we again consider incident IR light. In the metallic state, the planar structure is highly reflective and little light is absorbed in the ${VO}_{2}$ layer. In contrast, the microcones act as impedance-matching tapers and effectively serve as an antireflection coating, allowing light to be better absorbed in the ${VO}_{2}$ . The emissivity for the insulator state of the planar film [blue curve; Fig. 3(b)] is largely dominated by the properties of the silicon; above 10 μm, the spectrum of the ${VO}_{2}$ -coated film is nearly identical to that of the uncoated Si film [Fig. 3(c)]. We note that the sharp cut-off seen in Fig. 3(c) at 10 μm is due to the transparent nature of Si in this wavelength range (1–10 μm) [31].

We note that the calculations shown in Fig. 3 are obtained from coherent absorptivity at normal incidence. Experiments may not resolve the fine-scale wavelength features seen in the plots, and we have thus added smoothed lines as a guide to the eye.

B. Radiated Thermal Power

We next calculate the total radiated power in the insulating and metallic phases from the angle-averaged emissivity of each structure (see Methods). The radiated powers at the transition temperature, $P_{rad}$ ( $T_{c}$ ), are shown by symbols in Fig. 4. The microcones have a large difference in radiated power between the insulator (filled green circle) and metallic (unfilled green circle) states. The ${VO}_{2}$ -coated flat film (magenta diamonds) has a smaller difference, as expected from the smaller difference in thermal emissivity.

Fig. 4. Radiated thermal power. (a) The arrows indicate the direction of heating or cooling processes. The symbols represent the calculated values of thermal radiation for metallic (hollow symbols) ${VO}_{2}$ or insulating (filled symbols) ${VO}_{2}$ structures at 330 K. The solid curves represent the temperature-dependent model for radiated power assuming a phase transition width of 10 K. (b) Boundary conditions used to solve the heat equation.

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To model the temperature dependence of the radiated power, we assume a model that takes into account the hysteresis of the phase transition and temperature dependence of the blackbody spectrum (see Methods). The full model of $P_{rad}$ ( $T$ ) is shown by curves in Fig. 4(a). The directions of heating and cooling processes are indicated by arrows. For the microcone heating curve, the radiated power increases sharply with temperature through the ${VO}_{2}$ phase transition (green curve, upward arrow). This sharp increase is consistent with its function as a switchable emitter. When the temperature is decreased, the radiated power also drops sharply due to a change in emissivity across the phase transition. These trends are much more pronounced than for the planar film. The radiated power for the uncoated silicon film is shown for reference and increases slowly across the entire range.

C. Thermal Homeostasis

The large, sharp increase in radiated power across the phase transition helps regulate the temperature of the microcones. Given a fluctuating heat input, the temperature variation for the microcones is much smaller than for a Si film. To see this effect, consider the time-varying heat input shown in the top panel in Fig. 5(a). The value of $P_{in}$ oscillates between 150 and $550 W / m^{2}$ . Such an input could result, for example, from a time-varying solar illumination or internal heat load. We demonstrate the thermal dynamics of the system by solving the time-dependent heat equation for an isothermal mass (i.e., a “lumped capacitor”) [32] with an initial temperature of 330 K. We plot the system temperature as a function of rescaled time $t^{'} = t / ρ C L_{C}$ , where $ρ$ is the material density, $C$ is the thermal capacitance of the structure, and $L_{C}$ is the characteristic length scale (i.e., height) of the structure.

Fig. 5. Thermal homeostasis. (a) Temperature variation for different structures with a time-varying heat input flux. (b) Radiated power in an extended temperature range. The dotted–gray lines indicate the heat input range ( $150 - 550 W / m^{2}$ ) and the corresponding steady-state temperature values for each structure.

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For the bare silicon film, the temperature of the device oscillates strongly in response to the input, as shown by the dotted–dashed black curve in Fig. 5(a). The amplitude of the variation is 219.3 K. The ${VO}_{2}$ -coated flat film reduces these fluctuations to 147.3 K (dotted magenta curve). However, the microcone structure has a nearly constant temperature response: the fluctuation amplitude is reduced by nearly $20 \times$ relative to the silicon film, to 11.9 K (solid green curve). We refer to this behavior as thermal homeostasis; by proper design, the material can passively regulate its temperature far better than a bare silicon film. Moreover, the material also regulates temperature better than a blackbody emitter. Calculations show that the fluctuation amplitude of the microcone structure is approximately $8 \times$ smaller than for a perfect blackbody (see Supplement 1).

The origin of thermal homeostasis can be understood from power balance formalism. We assume that the input power varies slowly enough for the device to reach steady state at each step (increase or decrease in $P_{in}$ ). The steady-state temperature is determined by a balance between input and radiated power, shown schematically in Fig. 4(b): $P_{in} (T) = P_{rad} (T)$ . For convenience, we replot the radiated power curves from Fig. 4(a) over an extended temperature range in Fig. 5(b).

Starting with the uncoated Si (black curve), we determine that the temperature value corresponding to $P_{in} = P_{rad} = 550 W / m^{2}$ is 749 K. For the lower power $P_{in} = P_{rad} = 150 W / m^{2}$ , $T = 530 K$ . The temperature fluctuation is indicated by black arrows in Fig. 5(b). For the flat ${VO}_{2}$ -coated structure, a similar procedure gives a narrower temperature range, indicated by magenta arrows in Fig. 5(b). For the microcone structure, however, the range of temperature corresponding to powers between 150 and $550 W / m^{2}$ is much smaller. To find this range, we take into account the hysteresis in the curve. When $P_{in}$ is increased to $550 W / m^{2}$ , the heating curve (right side of hysteresis loop) gives a steady-state temperature of 337 K. When $P_{in}$ is decreased to $150 W / m^{2}$ , the cooling curve (left side of hysteresis loop) gives a temperature of 325 K. The overall temperature fluctuation (green arrows) is thus much smaller than for the other two structures. In summary, the design of the microcone structure, which yields a steep $d P_{rad} / d T$ at the phase transition, provides strong thermal regulation behavior with much smaller oscillation in temperature than a Si film.

D. Homeostatic Operating Range

The ability of the ${VO}_{2}$ -coated microcone device to maintain thermal homeostasis is limited by the width and height of the hysteresis loop associated with the insulator-to-metal phase around $T_{c}$ . In Fig. 5(b), the height of the loop (green curve) extends from 40 to $550 W / m^{2}$ . The width of the loop is approximately 10 K. Figure 6 shows the temperature variation of the microcones for three different input power oscillations. In Fig. 6(a), the values of $P_{in}$ fall well within the range of the loop (shaded yellow). The resulting thermal variation is 11.6 K, as in Fig. 5(a) (note the change in $y$ -axis scale). However, when the range of $P_{in}$ is lowered below [Fig. 6(b)] or above [Fig. 6(c)] the range of the hysteresis loop, the temperature variations over the cycle are larger. For optimal performance, the variations in input power should therefore fall within the range of the hysteresis loop. However, we note that for variations in input power larger than the range of the hysteresis loop, the microcone structure still outperforms the flat film or uncoated Si film.

Fig. 6. Homeostatic operating range. [(a)–(c)] Temperature variation of the silicon microcone structure for different heat inputs. (d) Reduction in temperature variation for microcones with a narrower hysteresis width. The yellow shade illustrates the range of the hysteresis loop (homeostatic operating range).

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The width of the loop will determine the size of the temperature fluctuations. As the width is reduced to 0, the fluctuations decrease as well, as shown in Fig. 6(d). Experimentally, the width of the hysteresis loop can be reduced via improvements in material quality [12,14,18,33], with some works showing hysteresis widths as low as 4 K. [33] The best thermal regulation performance will thus be obtained by using material with minimal hysteresis.

E. Dependence on ${VO}_{2}$ Thickness and Fabrication Feasibility

In our calculations, the ${VO}_{2}$ coating thickness was set to 200 nm for ease of computation, but better performance may be possible using a thinner coating. While a full optimization for the microcone structure is computationally prohibitive, we can easily calculate the emissivity of the flat, ${VO}_{2}$ -coated silicon film as a function of ${VO}_{2}$ thickness. For the best performance, the radiated power in the metallic and insulating states should be as different as possible at $T_{c}$ . In Fig. 7, we plot $P_{rad}$ ( $T_{c}$ ), normalized by the radiated power at $T_{c}$ for a perfect blackbody. Figure 7 shows that for a flat, ${VO}_{2}$ -coated film, the largest difference between metallic and insulating states occurs for a thickness of 0.03 μm, or 30 nm. For insulating ${VO}_{2}$ (blue circles), $P_{rad}$ ( $T_{c}$ ) increases with increasing ${VO}_{2}$ thickness. Insulating ${VO}_{2}$ is optically absorptive in the IR range. As the amount of ${VO}_{2}$ increases, the emissivity is increased at wavelengths where Si is transparent. The results suggest that a very thin layer of ${VO}_{2}$ is sufficient to provide maximum emissivity difference between metallic and insulating states.

Fig. 7. Dependence on coating thickness. Radiated power at phase transition for a flat, ${VO}_{2}$ -coated silicon film.

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Our design is amenable to standard microfabrication techniques. Si cone arrays with similar aspect ratios to our design have been fabricated by cryogenic, inductively coupled plasma reactive-ion etching [26,28,34,35]. Thin ${VO}_{2}$ conformal coatings can be achieved by using gas-phase reactions and deposition, such as sputtering deposition [12,13,15,33,36], pulsed-laser deposition [14,37], and atomic layer deposition [38–42]. Deposition of conformal coatings on microscale structures is an ongoing area of research [42]. In this work, we have assumed a perfect conformal coating for simplicity. However, further calculations show that deviations from perfect conformality do not change the qualitative difference in emissivity between metal and insulator states. Future work will design and test the concept of thermal homeostasis in experiment.

In the calculations above, we have considered a microcone structure surrounded by vacuum on both sides for simplicity. We note that the addition of an opaque material as the bottom boundary, e.g., a gold coating on the back surface of the Si substrate, has a minimal effect on the emissivity spectrum (see Supplement 1).

F. General Considerations

In the discussion and calculations above, we have analyzed specific structures based on ${VO}_{2}$ -coated microcones. We can abstract from our results to speculate on the ideal conditions for thermal homeostasis.

First, the temperature at which homeostasis is obtained corresponds to the phase-change temperature of the material. For ${VO}_{2}$ , this temperature can be tuned between 310 and 360 K [13,18] by adjusting the processing method [13,18,33,37,40], doping [13,36], or strain [12,43]. For applications at other temperatures, one could hope to identify a different phase-change material with a transition temperature in the target range.

When evaluating alternative materials, several considerations should be kept in mind. First, the time scale for the phase change should be shorter than both the thermal response time of the structure and the time scale for fluctuations in input power. For ${VO}_{2}$ , experimental measurements of phase transition time are in the picosecond range [44]. Second, materials with large changes in permittivity across the phase transition will generally make it easier to design a microstructure geometry that provides the desired change in emissivity. Emissivity should be as close as possible to 0 below the transition, and as close as possible to the blackbody above. The microcone structure presented here is optimized for ${VO}_{2}$ ; other materials will likely require different microstructures and/or metamaterial designs. Third, the width of the phase transition should be as small as possible. As discussed above, the residual temperature fluctuations for our material will be reduced as the width of the hysteresis loop shrinks [Fig. 6(d)].

3. CONCLUSION

In conclusion, we have proposed a route to thermal homeostasis using passive microstructures. We have presented a specific design that uses a thin film of ${VO}_{2}$ conformally coated on Si microcone structures to yield switchable thermal radiation. The design concept is based on a temperature-switchable thermal emitter: below the target temperature, emission is minimized, whereas emission is maximized above the target temperature. This sharp change in emission helps to lower or dampen the temperature variation of the structure due to a time-varying heat load. The proposed thermal homeostasis structure has a $10 \times$ difference in emissivity between the metallic and insulating states of ${VO}_{2}$ , resulting in a nearly $20 \times$ reduction in temperature variation relative to a Si film, and $8 \times$ reduction relative to a perfect blackbody. These numbers are obtained within a one-dimensional (1D) heat-transfer system in which radiation is the sole heat dissipation mechanism. Our results provide a light-weight, completely solid-state thermal control mechanism particularly well suited for space applications. The use of mechanically static structures, free of any moving parts, provides a complementary alternative to existing microelectromechanical systems (MEMS)-based approaches for thermal emission control [45].

A. Methods

1. Thermal Emissivity

Thermal emissivity $ϵ (λ, Ω)$ in the IR range is calculated via electromagnetic simulation, using the ISU-TMM package [46,47], an implementation of the plane-wave-based transfer matrix method. The simulation calculates absorptivity, where absorptivity is equal to emissivity, by Kirchoff’s law. The values shown in Fig. 3 are for normal incidence. The wavelength range shown is chosen to be 2.5–30 μm; outside this range, the blackbody radiance at room temperature is negligible. The calculated spectral resolution is 10 nm. The optical constants for ${VO}_{2}$ and Si are obtained from semi-empirical fitted experimental [15] data and experiments [31], respectively. We note that the measured optical constants for silicon in the IR range are obtained from intrinsic samples, and so the free-carrier contribution is minimal.

2. Radiated Power

The radiated thermal power can be written as

P_{rad} (T) = \int d Ω \cos θ \int_{2.5 μm}^{30 μm} d λ \cdot I_{B B} (λ, T) \cdot ϵ (λ, Ω),

where

I_{B B} (λ, T)

is the spectral blackbody radiance and

ϵ (λ, Ω)

is the computed emissivity at 330 K. The angle resolution was 5 deg. The calculated values of

P_{rad}

(

T_{c} = 330 K

) for Si cone structures coated with metallic and insulating

{VO}_{2}

are

550 W / m^{2}

and

40 W / m^{2}

(as shown in Fig. 4; unfilled and filled green circles), respectively.

Given the calculated values of $P_{rad}$ at 330 K (symbols in Fig. 4), we assume a model for $P_{rad}$ ( $T$ ) that takes into account the hysteresis of the phase transition and the temperature dependence of the blackbody spectrum (solid curves). Experimentally, the phase transition of ${VO}_{2}$ shows a hysteresis loop with a width of 4–15 K [13–15,18]. The hysteretic width can be reduced by annealing or depositing ${VO}_{2}$ onto a lattice-matched substrate to improve the quality of ${VO}_{2}$ [14,18]. We assume a smooth function that matches the calculated values at 330 K and a hysteretic width of 10 K:

P_{rad} (T) = \int d Ω \cos θ \int_{2.5 μm}^{30 μm} d λ \cdot I_{B B} (λ, T) \cdot {ϵ_{M} (λ, Ω) \cdot \frac{1}{2} [1 + \erf (T - T_{c} \pm \frac{1}{2} Δ T_{c})] + ϵ_{I} (λ, Ω) \cdot \frac{1}{2} [1 - \erf (T - T_{c} \pm \frac{1}{2} Δ T_{c})]},

where

I_{B B} (λ, T)

is the spectral blackbody radiance,

ϵ_{M} (λ, Ω)

is the emissivity for metallic

{VO}_{2}

,

ϵ_{I} (λ, Ω)

is the emissivity for insulating

{VO}_{2}

,

Δ T_{c}

is the hysteretic width of 10 K, and erf is the error function. The error function is used for convenience in interpolation; substituting another smooth function will slightly change the shape of the temperature response curves in Fig. 5, but will not alter their qualitative behavior. The + and − branches of Eq. (5) give the cooling and heating branches of the

P_{rad} (T)

loop, respectively.

3. Thermal Modeling Approach

We solve the time-dependent heat equation to obtain the transient temperature resulting from a time-varying heat input. When the resistance to heat spreading in the structure is small, the volume can be approximated as being isothermal, and we can neglect the spatial distribution of the temperature. The entire layered structure can thus be treated as a boundary, as shown in Fig. 4(b). $P_{in}$ is the heat source in the system and $P_{rad}$ is the radiated thermal power. Such a lumped capacitor approach is valid for Biot numbers $B i = L_{C} h_{rad} / k \leq 0.1$ , where $h_{rad}$ is a radiative heat transfer coefficient, $L_{C}$ is the characteristic length scale (e.g., the height of the structure in a 1D heat flow), and $k$ is the thermal conductivity [32]. The radiative heat transfer coefficient can be written as $h_{rad} \sim 4 ϵ σ T^{3}$ , where $σ$ is the Stefan–Boltzmann constant and $ϵ$ is the effective emissivity ( $P_{rad}$ normalized by radiated power for a perfect blackbody). Assuming a perfect blackbody with $T = 800 K$ and $L_{c} = 300 μm$ , the Biot number can be calculated as $B i = 8.34 \times 10^{- 4} ≪ 0.1$ . Since the blackbody value is an upper bound for our structure, the Biot number is smaller than this value.

The time-dependent heat equation can then be written as

ρ C L_{C} \frac{d T (t)}{d t} = P_{in} (t) - P_{rad} (T),

where

ρ

is the material density (

kg / m^{3}

),

C

is the heat capacitance (

J / K - kg

),

L_{C}

is the characteristic length scale (

m

),

P_{in} (t)

is the time-dependent heat source (

W / m^{2}

), and

P_{rad} (T)

is the radiated thermal power (

W / m^{2}

). By a change of variables, the rescaled time (

t^{'}

) can be written as

t^{'} = \frac{t}{ρ C L_{C}} .

Equation (3) can be further simplified to

\frac{d T (t^{'})}{d t^{'}} = P_{in} (t^{'}) - P_{rad} (T) .

Here we assume for simplicity that $C$ is the thermal capacitance of the structure itself. If the structure is in thermal contact with an additional object, the time-dependent response will depend on the lumped thermal capacitance [32] of the entire system.

Funding

National Science Foundation (NSF) (ECCS-1711268); NG Next Northrop Grumman Corporation.

Acknowledgment

Computation was supported by the University of Southern California Center for High Performance Computing and Communication. The authors thank Luqi Wang for performing calculations of absorption in the visible range. The authors also thank Dr. Virginia Wheeler for insightful discussion on ${VO}_{2}$ ALD methods.

M. L. P. laid out the concept and supervised the project. All authors conceived and designed the simulations. Simulations and data analysis were carried out by S.-H. W. and M. C. Interpretation of the data and writing of the manuscript were performed by S.-H. W., M. C., and M. L. P. All authors commented on the data and on the final version of the manuscript.

The authors declare no competing financial interests.

See Supplement 1 for supporting content.

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Thermal homeostasis using microstructured phase-change materials

Abstract

Corrections

1. INTRODUCTION

A. Design of Structures for Thermal Homeostasis