Algorithms for thermal diffusivity measurement of heterogeneous 1D materials based on Infrared Microscopy Enhanced Angstrom Method

An infrared microscopy enhanced Angstrom method has been develpoed to measure the thermal diffusivity. Infrared microscopy technique can acquire temperatures of multiple points at one shot. Two algorithms for calculating thermal diffusivity were proposed and compared in practice. One is based on global temperature data and the other is based on local temperature data. The according calculated thermal diffusivities are denoted as αnG and αnL . Three 1D materials of different heterogeneity (Cu wire, Ni-Cu wire and PVA-CNT fiber) were measured on the experimental platform. The calculated αnG and αnL values show that for homogeneous material such as Cu, these two algorithms give similar results, while for heterogeneous ones (Ni-Cu and PVA-CNT), they come to be discrepant. The data fluctuation analysis of fnL zooms in the discrepancy and verifies that αnL is more sensitive to local property change and more competent in revealing heterogeneous properties.


Introduction
The Angstrom's method has been widely used for measuring the thermal diffusivity of materials [1][2][3]. Thermal diffusivities of materials such as copper, carbon fibers, carbon nanotubes and other materials have been measured by Angstrom method [4][5][6]. This method should lead to many local details missing for heterogeneous material. Other traditional methods such as laser flash method, T-type method [7][8] have the same trouble. T-type method read current and voltage signals. Laser flash method cannot measure temperature of multiple points due to the limitation of experimental setup. This work introduces a new measurement technique to measure thermal diffusivity based on infrared (IR) camera enhanced Angstrom method. The large area temperature data can be acquire by the IR camera at one shot. This method overcomes the averaging issue that bothering all the traditional methods mentioned above. Two algorithms for calculating thermal diffusivity have been developed on basis of IR Microscopy measurement technique. The data fluctuation analysis provide an evaluation method for the heterogeneity of 1D material.

Theory of thermal diffusivity measurement
Thermal diffusivity can be derived by Angstrom method. Considering 1D heat transfer diffusion with heat loss, the equation can be expressed as follow [1]. where is thermal diffusivity, T is the temperature The schematic diagrams of Angstrom method are shown in Figure 1. A sinusoidal heat wave is loaded to one end of the sample at S position. Then temperature at position X1 and also show the sinusoidal feature. By solving Eq. (1), the thermal diffusivity can be obtained as follow [2]: where L is the distance between position X1 and X2; A1 and A2 is the amplitude of the temperature waves at X1 and X2; dt is the phase delay between the waves at X1 and X2.

Algorithms of Angstrom method based on Infrared Microscopy
Infrared Microscopy enhanced Angstrom method can simultaneously read temperature of multiple points. Here we pick n segments (n+1 points) and employ two algorithms for thermal diffusivity In general, the thermal properties in any part of 1D homogenerous materials should be the same, while that of 1D heterogeneous materials may be discrepant. Hence and can be used as the basis for heterogeneity judging.

Thermal diffusivity of heterogeneous 1D materials
In the experiment, a Cu wire, a Ni-Cu wire and a PVA-CNT fibre were picked for the comparison measurement of α and α . Figure 4 shows the calculated α and α of three materials. In Figure 4a, the change trends of α and α for the Cu wire are the same. When n=2, there is a slight difference between them, which may be caused by experimental error. In Figure 4b, the change trends of α and α for the Ni-Cu wire and PVA-CNT fibre are quite different. The change of α is small, which makes the α almost a stable value, while on contrast, α shows evident fluctuations around α . is the thermal diffusivity of the entire sample, representing an average value. It is a manifestation of overall thermal performance.
is the location thermal diffusivity at certain point of the sample. For homogeneous materials, thermal properties are invariable at any position. Then and should be same. That is what we have observed in Figure 4a. But for heterogeneous materials such as composites, should be different with showing more dramatic fluctuation due to the uneven mixing of substances and inconsistent molecular ordering. This is what happens in Figure 4b. The difference between them can be used to evaluate the degree of heterogeneity of 1D material. The suggested quantification details are provided below.

Fluctuation analysis
To distinguish the fluctuation origins, whether it is from the experimental error or from the heterogeneity of materials, the data analysis should be designed. As mentioned before, the algorithm of has an On contrast, is the local thermal diffusivity of each segment. Its average value ( ) is the true average of the entire sample. It fluctuates more drastically than when the sample is heterogeneous, thus its relative fluctuation (Eq. (5)) renders more heterogeneity information. Figure 5 exhibits their comparison. For homogeneous material Cu, the two algorithms of relative fluctuation give no big difference. Thus its standard deviation of the fluactuation from the local algorithm ( ( )) can be used as a benchmark for experimental error, only the n=1 data should be crossed out due to obvious deviation, which may be caused by system error. For heterogeneous material Ni-Cu and PVA-CNT, the error origin and hetero-origin are mixed and both of them contributed to the apparent standard deviation ( ( )). Thus the heterogeneity (h) hidden in the drastic fluctuations of should be defined as ℎ = ( ) − ( ). And for Ni-Cu, h=0.0273, while for PVA-CNT, h=0.1148.

Conclusion
The use of Infrared Microscopy temperature measurement can directly measure and read temperature of multiple points on 1D sample under test. Two algorithms of 1D material thermal diffusivity measurement are developed based on the multiple points' temperature acquisition. One is based on global temperature data and the other is based on local temperature data. The according calculated thermal diffusivities are denoted as and . The measured thermal diffusivities of Cu, Ni-Cu and PVA-CNT and data fluctuations were discussed. and of Cu are almost the same, while there are obvious differences between and of Ni-Cu and PVA-CNT. In addition, the error-origin and the hetero-origin for the fluctuations were distinguished by designed data analysis and the heterogeneity h is defined accordingly.