Introduction

Electrical resistivity of materials is a basic property that is measured and used in a large variety of studies. In one type of study, electrical resistivity data are used to calculate thermal conductivity and interior heat flow of terrestrial-type bodies. The electrical resistivity of a sample is often obtained using the four-wire method as it reduces the contribution of the electrodes that is present in the two-wire method. The four-wire method is used in the cooling of industrial devices [1], soldering processes [2], electronic systems [3], temperature determination at depth [4], and simulations of physical properties of terrestrial-type bodies at high pressures and high temperatures. High pressure and high temperature measurements of electrical resistivity (ρ) in high pressure multi-anvil presses are used to calculate thermal conductivity (k) and the heat flow in the interiors of terrestrial-type bodies. Ezenwa and Secco [5] developed a four-wire method to solve the voltage bias, caused by passing a test current in a single direction, which previously resulted in erroneous analysis of ρ measurements due to parasitic voltages caused by the Seebeck effect or other sources. This polarity switching method allowed successful measurements of ρ of Zn [5], Nb [6], Ni [7], Co [8], Cu [9], Fe [10, 11], Ag [12], W [13], Re [13], Au [14], Fe-Si alloys [15, 16, 17], and Fe-S alloys [18]. Three other papers from different research groups report using similar data collection practices, while several other papers use the four-wire method without a polarity switch [19]. In these measurements, the sample is located in the middle of a high-pressure cell, between two metal discs which are each contacted by a thermocouple. The metal discs, ensuring contact between the thermocouples and the sample, are typically composed of W. The thermocouples are typically Type-C thermocouples (one leg is made of 95%W/5%Re and the other leg is made of 74%W/26%Re) where voltage (i.e., thermoelectric emf) is measured and then converted to temperature after the experiment. The voltage drop across the sample for resistivity measurement is made between a pair of wires on opposite sides of the sample (one wire leg from each thermocouple), and therefore includes the voltage contribution of the discs. Thus, the ρ of the sample is obtained by subtracting the disc contribution from the total measurement.

The four-wire method incorporates a polarity switch and a mode switch, and results in a pattern of alternating values of temperature (emf) and voltage in a single column when acquired by a single voltmeter. A complete measurement produces a pattern similar to the following: temperature emf before voltage measurement (T_b), positive voltage drop with polarity in one direction (V⁺), negative voltage drop with polarity in opposite direction (V), temperature emf after voltage measurement (T_a). The order of V⁺ and V^– may be alternated depending on the position of the polarity switch. At the University of Western Ontario, in the High-Pressure High-Temperature Laboratory, a programmable Keysight B2961 power supply is used to provide a constant direct test current of 0.2 A while data are acquired by a programmable Keysight 34470A meter operating at 20 Hz and 1 µV resolution. The selection of temperature and voltage values is typically done manually, and, to our knowledge, no software available in the literature is capable of automatically processing the temperature and voltage signals.

Implementation and architecture

The application Rho calculates the average of 10 data points for each temperature section and all available data points for each voltage section. It is expected that temperature remains stable during these 10 data points. At high temperatures, when the frequency of measurements is increased and the quantity of data points in each section is low (due to the rapid switching between each section), Rho selects all the available data for each section. The values in each selection are averaged. Then the V⁺ and V^– selections are averaged to give the voltage drop (V), while T_b and T_a selections are averaged to give the temperature (T). The voltage drop and current (I) are then used to calculate the electrical resistance (R) of the sample using Ohm’s law:

The electrical resistivity is then calculated using Pouillet’s law:

where A is the sample’s cross-sectional area and l is the sample’s length which are both measured on the post-experiment recovered sample. Values of ρ are then used to calculate the electronic component of thermal conductivity (k) via the empirical Wiedemann-Franz law:

where T is temperature and L_o is the theoretical Sommerfeld value (L_o = 2.44∙10^–8 W∙Ω∙K^–2) of the Lorenz number. The error in temperature measurement corresponds to the standard deviation of the data points of each temperature section. The error in ρ is obtained by error propagation using the uncertainties in sample geometry and standard deviation of the voltage measurements. Similarly, the error in k corresponds to the propagation of the error in ρ.

The main page of Rho is displayed in Figure 1a. The application requires data to be imported in .xlsx, .csv, or .xls format, where the T_b, V⁺, V^–, and T_a data alternate within the first column of the file (see Figure 1b). The first figure (see Figure 1a) that is generated when the file is loaded is of the raw data. Noise, or abrupt fluctuation in data, that are not representative of the measurements can be removed by selecting the starting and ending indices (x-axis) of the section. The application attributes NaN values to each data point of this section, which removes them from the future analysis. When ‘Plot Data’ is selected, Rho identifies the temperature and voltage sections based on deviations from the main increasing trend, which corresponds to the temperature increase with time.

First, the intervals of negative values are labelled as voltage measurements (V_total). This temporary selection is displayed in Figure 1 of the application (see Figure 2a below). In the first scenario, the pattern T_b V⁺ V^– T_a is considered and T_b and T_a are temporarily identified. To satisfy this scenario, the difference between T_b and V_total must be smaller than the difference between T_a and V_total considering that the part of V_total closer to T_b would be V⁺ and that closer to T_a would be V^– (and V^– < V⁺). Rho uses a loop to evaluate this scenario at each deviation from the main increasing trend. Whenever the scenario is observed, V⁺ is temporarily defined as the first trend in V_total. In the second scenario, the pattern T_b V^– V⁺ T_a is considered, and a similar loop is applied to identify the intervals where the difference between T_b and V_total is larger than that of T_a and V_total. Here, the part of V_total closer to T_b would be V^– and that closer to T_a would be V⁺. Another analysis of the first scenario identifies the stable intervals on the main increasing trend right before V⁺ as T_b, and that right after V^– as T_a. In other words, fluctuations, or noise, between temperature and voltage measurements are automatically ignored. A similar analysis of the second scenario identifies the stable intervals on the main increasing trend right before V^– as T_b and that right after V⁺ as T_a. The corresponding indexes (x-axis) and data (y-axis) for each selection are combined into variables for T_b, V^–, V⁺, and T_a. The ‘Outliers Degrees of Freedom’ parameter evaluates the differences between the trends of V^– and V⁺ and identifies data points outside of these trends as outliers. A larger degree of freedom will include more data points in the final variables for V^– and V⁺, while a smaller degree of freedom will ignore more data points. The value 20 seems to be the best fit for this parameter, although it can be adjusted by the user. In other words, fluctuations of 20 times and more are defined as outliers. The final selection is displayed in Figure 2 of the application (see Figure 2a and 2d). The contribution of the discs to the measured ρ is subtracted by fitting the ρ of the selected material. The ‘Discs’ dropdown menu contains options of W from 2–5 GPa [13], Re from 2–5 GPa [13] and Pt at 1 atm, 10 and 20 GPa [20]. Supplementary options may be added to future updates of the application as data become available in the literature for different metals used for the discs. Temperature is automatically converted from emf to (K) using the selected thermocouple calibration equation. The ‘Thermocouple’ dropdown menu contains options for Type-C [21], Type-S at 1 atm [22] and Type-S from 1–5 GPa thermocouples [23]. The input parameters are then used in equation (1), (2) and (3) to plot ρ in Figure 3 of the application and output values of T (K), ρ (µΩ·cm), error in ρ (µΩ·cm), k (Wm^–1K^–1) and error in k (Wm^–1K^–1) (see Figure 2b and 2c). The values are displayed in the ‘Output’ tab and can be copied and pasted on another platform for further analysis.

The script architecture is summarized in Figure 3 below.

Quality control

Examples of running the software are displayed in Figure 2. After loaded the raw data file, Figure 1 of the application should automatically display the raw data. If the raw data file is in the incorrect format, an error will be set (cell2mat error on MatLab), and Figure 1 will not display the raw data. The users should confirm the temperature and voltage selections since similarity in these values might result in the incorrect identification of some data points. The computed ρ can be compared with that of Fe at 1 atm [24]. Rho can successfully compute ρ of any material as long as the input file is in the correct format. Users may use example files of Au at 5 GPa [14], Fe-Ni-Si at 20 GPa [25], and Fe-Si at 9 GPa [19] available on the GitHub repository to get familiar with Rho and the format of the input file.

Operating system

Users may choose to download Rho.exe file from FigShare or Rho.m from GitHub.

Rho.exe – Any system capable of running a .exe file.

Rho.m – Any system capable of running Matlab R2021a or higher.

Rho.mlapp – Any system capable of running AppDesigner R2021a of higher.

Programming language

Tested in Matlab R2021a, likely to work in earlier versions also.

Additional system requirements

Rho.exe – 0.82 MB

Rho.m – 56 KB

Dependencies

N/A

List of contributors

N/A

Software location

Archive (e.g. institutional repository, general repository) (required – please see instructions on journal website for depositing archive copy of software in a suitable repository)

Name: FigShare

Persistent identifier: 10.6084/m9.figshare.19175432

Licence: GNU General Public License v3.0

Publisher: Meryem Berrada

Version published: 1.0

Date published: 15/02/22

Code repository (e.g. SourceForge, GitHub etc.) (required)

Name: GitHub

Identifier: https://github.com/meryemberradauwo/Rho.git

Licence: GNU General Public License v3.0

Date published: 02/02/22

Language

English