Phase Transformation Contributions to Heat Capacity and Impact on Thermal Diffusivity, Thermal Conductivity, and Thermoelectric Performance

Abstract

Although direct measurements of κ are possible, [2] it is more frequent in measurement above room temperature to calculate thermal conductivity through the relation (Note S1, Supporting Information)Here, D [m 2 s −1 ] is the thermal diffusivity and ρc p = (∂H/∂T) p is the constant pressure heat capacity [J m −3 K −1 ] typically calculated from experimental bulk density, ρ [kg m −3 ] and the mass specific heat c p [J kg −1 K −1 ]. Equation (1) is utilized largely because measurements of thermal diffusivity, D, are accurate (within ≈3%) and easily accessible. [2] Thus, the uncertainty in calculating κ (often 10% or more) is frequently attributed to measurement of heat capacity. [2,3] Due to the uncertainty in the absolute magnitude of high temperature heat capacity measurements, it is often more accurate to use a model. Usually, the temperature independent Dulong-Petit heat capacity is a good approximation (within ≈5%) near room temperature. At higher temperatures, a model that incorporates a linear temperature dependence may be appropriate. [2,4,5] However, an incomplete understanding of heat capacity measurements and models can lead to inaccurate estimations of κ in some systems, especially those having substantial latent heats (e.g., during phase transitions). The recent debate surrounding the thermoelectric material, Cu 2 Se, is an excellent example. [6][7][8][9][10][11] In this material, and others, [12][13][14] the thermal diffusivity drops markedly as the material undergoes a phase transition. Depending on the heat capacity used to calculate κ, a maximum zT between 0.6 [7] and 2.3 [6] has been reported due to the superionic phase transition in Cu 2 Se. As exemplified in Figure 1, the choice of heat capacity can have a drastic impact on zT values.To properly characterize the behavior of thermal conductivity through a phase transition, it is paramount to understand the concurrent behavior of heat capacity. Recognizing that the total capacity of a material to absorb heat includes both the intrinsic heat capacity of the phases present and the enthalpy (heat) of transformation ΔH that is required to maintain equilibrium (characterized by the order parameter φ) as the temperature is changedThe accurate characterization of thermal conductivity κ, particularly at high temperature, is of paramount importance to many materials, thermoelectrics in particular. The ease and access of thermal diffusivity D measurements allows for the calculation of κ when the volumetric heat capacity, ρc p , of the material is known. However, in the relation κ = ρc p D, there is some confusion as to what value of c p should be used in materials undergoing phase transformations. Herein, it is demonstrated that the Dulong-Petit estimate of c p at high temperature is not appropriate for materials having phase transformations with kinetic timescales relevant to thermal transport. In these materials, there is an additional capacity to store heat in the material through the enthalpy of transformation ΔH. This can be described using a generalize...

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