Cottrill Research Paper

Ultra-high thermal effusivity materials

Mathematically, the thermal effusivity, e, is related to the product of the thermal conductivity, k, and volumetric heat capacity, C:1

where ρ is density and Cp is specific heat.

If a PCM is utilized in a dynamic environment in proximity to its phase transition temperature, an alternate, effective thermal effusivity, eeff, should be used instead:

where h is the latent heat per unit mass (Supplementary Note 1, Supplementary Fig. 1). The incorporation of PCMs in proximity to their phase transition temperatures results in an enhancement of approximately an order of magnitude in energy storage per unit volume in comparison to sensible heat storage7.

We have prepared two ultra-high thermal effusivity materials through the incorporation of conformal, chemical-vapor-deposited (CVD) multi-layer graphene (G) into PCM composites, which consist of the PCM octadecane (OD) and a copper (Cu) or nickel (Ni) foam (Methods). We label these materials Cu/G/OD and Ni/G/OD, respectively. The metal foam serves as a highly thermally conducting and porous matrix, while the OD bolsters the thermal capacitance through the latent heat of its phase change. OD is chosen as the PCM because its phase transition temperature (27.7 °C) makes it amenable to ambient energy harvesting6. The incorporation of carbon nanomaterials, such as graphene, can enhance the thermal conductivity, while negligibly affecting the thermal capacitance because of the low volume fractions needed to affect thermal conduction13,14.

To fabricate Cu/G/OD and Ni/G/OD materials, we grow multi-layer graphene on Cu32 and Ni33 foams, which are then vacuum impregnated with molten OD (Methods). Scanning electron micrographs (SEMs) for the pristine Ni foam, OD-impregnated Ni foam (Ni/OD), and Ni/G/OD are shown in Fig. 1a. One can clearly identify the grain boundaries of the Ni lattice for the original Ni foam, especially in the magnified micrograph on the bottom. Once infused with OD, one can identify the filled pores, with the detailed Ni grain boundaries remaining unchanged. The multi-layer graphene coating shown in the third column, however, appears to have completely covered the Ni surface, bridging the grain boundaries, as evidenced by the layered fingerprint-type structure shown in the micrograph. This connectivity can improve heat transfer across Ni grain boundaries, as well as augment interfacial heat transfer between the OD and the Ni foam. SEMs for the pristine Cu foam, OD impregnated Cu foam, and Cu/G/OD are also provided in Fig. 1b. Similar interpretations and conclusions can be developed for the Cu foam PCM composites.

To further characterize the Ni/G/OD and Cu/G/OD materials, we perform Raman spectroscopy, latent heat, and thermal conductivity measurements. Raman spectra for the metal foams before and after conformal, CVD graphene growth indicate the presence of multi-layer graphene (Supplementary Fig. 2, Supplementary Methods), as evidenced by the intensities of the 2D peaks relative to the G peaks (ratio < 1)32. Furthermore, the results of the thermal conductivity and latent heat measurements are used to calculate thermal effusivity, according to Eq. (2) (Methods, Supplementary Notes 2, 3, Supplementary Figs. 3–7). We observe a significant enhancement in the thermal conductivity and effective thermal effusivity of our Ni/G/OD ultra-high thermal effusivity material as compared with Ni/OD (Fig. 1c). Similar enhancements in thermal conductivity and effective thermal effusivity are observed when comparing Cu/G/OD to Cu/OD. Our values of effective thermal effusivity and latent heat agree well with predictions from Hashin–Shtrikman (HS) theory (Supplementary Fig. 8, Supplementary Note 4) and Supplementary Eq. (10), respectively. The latter agreement supports complete OD impregnation of the metal foams. To compare our synthesized materials with the existing literature, we generated a master plot of effective thermal effusivity vs. phase transition temperature, T* for PCMs (Fig. 2)6,34,35,36,37. The PCMs incorporated into the master plot are ambient (T* between 20 and 40 °C) and isotropic (Supplementary Table 1). As can be seen from the master plot, our ultra-high thermal effusivity materials exceed all values in the literature to date.

The enhancements for conductivity and effusivity are not as drastic for the Cu composites as compared to the Ni composites. We attribute this to a couple of reasons. First, the thermal conductivity of pristine Cu (385 W m−1 K−1) is larger than pristine Ni (90 W m−1 K−1), which would reduce the impact of the graphene on the Cu/G/OD composite (Supplementary Note 4). Also, the Cu/G composite shows a larger amount of defects in the graphene compared with the Ni/G composite, as evidenced by the magnitude of the D peaks in the Raman spectra (Supplementary Fig. 2). A comparison of our thermal conductivity measurements with the H–S model38,39,40 supports that a reduction in interfacial thermal resistance between the metal foam and the OD, as imparted by the graphene, is likely the main factor for the thermal conductivity enhancement of the composites (Supplementary Note 4).

Application in thermal resonance devices

A general schematic of a thermal resonance device is shown in Fig. 3a. The generic device consists of a heat engine encased between two thermal masses, which convert temperature fluctuations to a spatial temperature difference for power extraction via the heat engine. Consider the thermal circuit depicted in Fig. 3b, where the ambient thermal environment is taken to be a time varying source term:

where is the ambient temperature, T0 is the median ambient temperature, TA is the amplitude for ambient thermal fluctuations, ω is the angular frequency of temperature oscillations, and t is time.

This source is connected to a generic heat engine capable of the conversion of a temperature difference, , to usable work. As a specific example, consider a thermoelectric device, where a temperature difference approximately yields the power output:41

where P is power, is the Seebeck coefficient, and Reff is an effective resistance that accounts for thermal and electrical resistances in the system41. Note that in the absence of additional thermal circuit elements, the power generated is zero for all time. However, as in the case of the circuit in Fig. 3b, the heat engine is encased between two thermal masses (1 and 2), having constant thermal conductivities () and thermal capacitances (). These resistive and capacitive elements are differential in size and form a finite transmission line42. The linear thermal masses can be specifically paired to maximize the spatial and temporal temperature gradient existing across the heat engine, such that persistent power can be harvested according to Eq. (4) from temperature fluctuations.

To optimize the power harvesting capabilities of thermal resonance devices, we postulate for thermal mass 1 (dominant thermal mass) to be tuned to the temperature oscillation frequency, while thermal mass 2 should have negligible thermal resistance (extrinsic). We also postulate that maximizing the thermal effusivity (intrinsic) of thermal mass 1 will optimize the power output. As we show later, these design criteria fall into the optimal performance regime. Furthermore, because our thermal resonators operate in a dynamic temperature environment, we can select a PCM with a transition temperature in proximity to the median temperature of oscillations and evoke Eq. (2) for an enhanced thermal effusivity. Figure 3c, d illustrates the thermal resonator’s operation under such constraints. Thermal mass 2 rapidly responds to the temperature of the ambient environment and adopts its temperature, whereas the dominant thermal mass does not rapidly respond to the environment and exists mainly at the phase transition temperature of the high thermal effusivity PCM. Throughout this paper, we will optimize thermal mass 1, while maintaining an invariant thermal mass 2, which is characterized by low thermal resistance.

The beneficial effects of high thermal effusivity materials toward thermal energy harvesting via thermal resonators, in terms of power output, are demonstrated experimentally in Fig. 3e–g. The oscillatory temperature inputs, which mimic ambient temperature fluctuations, are generated using a programmable, temperature-controlled stage (Methods, Supplementary Fig. 9). The power output profiles for various resonators—incorporating thermal masses ranging from standard masses (e.g., Styrofoam, PVC, Teflon, etc.) to our high thermal effusivity materials—are measured, and the time-averaged power outputs (proportional to the extracted energy for a given time frame) are plotted with respect to the thermal effusivities of the dominant thermal masses (Fig. 3g, Supplementary Notes 5, 6, Supplementary Fig. 10). Typical power output profiles and input temperature oscillations for OD and Cu/G/OD thermal resonators are shown in Fig. 3e, f, respectively. Note the temperature oscillates around the phase transition temperature of our high effusivity materials, which enable activation of the enhanced, effective thermal effusivity definition in Eq. (2). As a control, we also investigate the performance of the device in the absence of thermal masses 1 and 2 (Supplementary Fig. 9), and the results show insignificant power generation compared to our tuned devices. Furthermore, the average power output of each device is closely related to the thermal effusivity (Fig. 3g, mathematical details in the following section). Comparisons of the average power vs. thermal conductivity and thermal capacity (Supplementary Fig. 11) do not show significant trends. The maximum average power that we observe is 786 μW for the Cu/G/OD resonator. This value can be compared with previous efforts to harvest from temperature fluctuations using non-PCMs23,24,25,26,27 and unmodified PCMs30,31 with thermoelectrics by considering the power output of the non-PCMs and pure OD in Fig. 3g. For these simulated temperature oscillations, we also calculated the anticipated average power output of similarly sized, common (250 μW) and state-of-the-art (490 μW) pyroelectrics for comparison to our thermal resonators (Supplementary Note 8).

Theoretical design of a tuned thermal resonator

To model the thermal circuit shown in Fig. 3b, we start with the continuum energy conservation equation for a linear, 1-D system:

where Tj is the spatial and temporal temperature of thermal mass j, αj refers to the thermal diffusivity of thermal mass j, and xj refers to the spatial dimension of thermal mass j.

Equation (5) completely describes the diffusion of heat for the non-PCMs in our thermal resonator. However, in modeling the PCMs in our devices, the following assumptions are made in order to apply Eq. (5): thermal fluctuations exist around the phase transition temperature and within the melting range of the PCM43 and thermal diffusion occurs due to the movement of the phase change boundary (Supplementary Note 1). The constant thermal diffusivity that we assume in this work appears to adequately describe the frequency response of our high effusivity materials.

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March 11th, 2018

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