Electrical Resistivity Measurement of Carbon Anodes Using the Van Der Pauw Method

metals

Article Electrical Resistivity Measurement of Carbon Anodes Using the Van der Pauw Method

Geoffroy Rouget 1,2, Hicham Chaouki 2, Donald Picard 2, Donald Ziegler 3 and Houshang Alamdari 1,2,*

1 Department of Mining, Metallurgical and Materials Engineering, Laval University, Quebec, QC G1V0A6, Canada; [email protected] 2 NSERC/Alcoa Industrial Research Chair MACE and Aluminum Research Center, Laval University, Quebec, QC G1V0A6, Canada; [email protected] (H.C.); [email protected] (D.P.) 3 Alcoa Primary Metals, Alcoa Technical Centre, 859 White Cloud Road, New Kensington, PA 15068, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-418-656-7666

Received: 27 July 2017; Accepted: 8 September 2017; Published: 13 September 2017

Abstract: The electrical resistivity of carbon anodes is an important parameter in the overall efficiency of the aluminum smelting process. The aim of this work is to explore the Van der Pauw (VdP) method as an alternative technique to the standard method, which is commonly used in the aluminum industry, in order to characterize the electrical resistivity of carbon anodes and to assess the accuracy of the method. For this purpose, a cylindrical core is extracted from the top of the anodes. The electrical resistivity of the core samples is measured according to the ISO 11713 standard method. This method consists of applying a 1 A current along the revolution axis of the sample, and then measuring the voltage drop on its side, along the same direction. Theoretically, this technique appears to be satisfying, but cracks in the sample that are generated either during the anode production or while coring the sample may induce high variations in the measured signal. The VdP method, as presented in 1958 by L.J. Van der Pauw, enables the electrical resistivity of any plain sample with an arbitrary shape and low thickness to be measured, even in the presence of cracks. In this work, measurements were performed using both the standard method and the Van der Pauw method, on both flawless and cracked samples. Results provided by the VdP method appeared to be more reliable and repeatable. Furthermore, numerical simulations using the finite element method (FEM) were performed in order to assess the effect of the presence of cracks and their thicknesses on the accuracy of the VdP method.

Keywords: carbon anodes; aluminum smelters; electrical resistivity; Van der Pauw

1. Introduction The Hall–Héroult process [1,2] is used to perform the electrolysis of alumina. To this end, an electrical current is applied through the electrolytic bath between carbon anodes and cathodes. The carbon anodes are used to provide carbon for the chemical reaction, as well as the electrical current necessary for the electrolysis. To ensure the proper functioning of the electrolysis cells, carbon anodes are subject to quality controls such as chemical reactivity [3,4] or mechanical properties such as strength [5] or density [6] during their production. The quality control is performed on core samples, and usually extracted from the top of the anodes beside the stub holes. This location may lead to core samples having some structural ﬂaws, especially at their bottom [7–9]. Core sampling itself may also induce cracks in the samples [10]. The electrical resistivity characterization of anode cores is achieved using the ISO 11713 standard method. This method is merely an adaptation of ASTM B193-02, which is used as a test method for the resistivity of electrical conductor materials, although this latter method requires a sample with neither cracks nor visible defects. Cracks located in the transverse axis (placed

Metals 2017, 7, 369; doi:10.3390/met7090369 www.mdpi.com/journal/metals Metals 2017, 7, 369 2 of 17 normal to the revolution axis of the sample) may most probably induce overestimated values on the measured electrical resistivity, which would not necessarily be representative of the electrical resistivity of the whole anodic block. To overcome this drawback, the Van der Pauw (VdP) method, developed to measure the electrical resistivity of samples with various shapes, represents a potential solution [11,12]. Koon studied the effect of size and place of contacts using the VdP method [13]. De Vries and Wieck targeted the current distribution in certain shapes of samples [14]. Rietveld et al. investigated the accuracy of the VdP method on several shapes of samples [15]. Kasl C. and Hoch M.J.R. [16] proposed a study using the VdP method on circular and cylindrical samples. This study shows that samples with a thickness smaller than their diameter give an accurate value of electrical resistivity. In addition, as exposed by Kasl and Hoch contacts placed along the edge of the sample are preferred to those placed on the top of the sample. Using the VdP method enables the use of samples much smaller than those used with ISO 11713 Standard. Therefore, by reducing the size of the sample, the risk of the presence of flaws reduces. In this study, the reliability of the VdP method for the electrical characterization of carbon anodes was investigated and compared with the standard method, which is currently used in the aluminum industry. Finally, defects were intentionally introduced to the anode cores, and their electrical resistivity was measured using both methods in order to assess their sensitivity to the presence of defects. The first part of this work consists of the comparison of both standard and VdP methods to confirm the reliability of the use of the VdP method for carbon anodes. In the second part, the electrical resistivity of industrial core samples is measured, and the results are compared with the values obtained by the industrial practice. For the third part of this study, another set of experiments is performed on the same batch of anode cores, as used in the preliminary experiment, in order to measure the effect of defect orientation on the measured electrical resistivity. Finally, finite element simulations were performed in order to verify the accuracy of the previously performed experiments, and to validate the use of the VdP method for cracked anode cores.

2. Experimental For both methods performed in the laboratory, the 1 ampere current is provided by a DC Power Supply GW-INSTEK, GPS-3030D, (GW-INSTEK, Taipei, Taiwan). Current and voltage are measured with an Agilent 34461A 61/2 Digit Multimeter (Agilent Technologies, Santa Clara, CA, USA).

2.1. Standard Method In the case of the standard method, the sample was maintained between two steel plates, both placed on springs to apply 3 MPa pressure, which is required by the standard. Voltage drop was measured with two thin pins mounted on spring, in order to apply the same pressure on each spot. These pins are located on a handle, keeping the same distance apart. The experimental setup is presented schematically in Figure1. Measurements according to the industrial practice were performed using the R&D Carbon RDC 150 for Specific Electrical Resistance apparatus (R&D Carbon, Sierre, Switzerland). To calculate the electrical resistivity of anode core using the ISO 11713, the following equation is used: V S ρ = (1) I L where ρ is the electrical resistivity of the material, V is the voltage drop measured, I is the intensity of the current applied, S is the cross-section of the sample, and L is the length between the voltage probes, as shown in the Figure1. Metals 2017, 7, 369 3 of 17 Metals 2017, 7, 369 3 of 17

Figure 1. Schematic view of the setting used for the measurement of electrical resistivity of carbon Figure 1. Schematic view of the setting used for the measurement of electrical resistivity of carbon Figureanodes 1. Schematic according view to the of ISO the 11713 setting standard. used for the measurement of electrical resistivity of carbon anodes according to the ISO 11713 standard. anodes according to the ISO 11713 standard. 2.2. Van der Pauw Method 2.2. Van der Pauw Method 2.2. Van derFor Pauw the VanMethod der Pauw method, since it was decided to take advantage of the cylindrical Forsymmetry, thethe VanVan the der dersample Pauw Pauw method,holder method, was since made since it was of ita decided wasself-centering dec toided take chuck,to advantage take which advantage of is the designed cylindrical of thefor lathes.cylindrical symmetry, Current suppliers and voltage probes were made of copper bars, with a V-shape facing the sample symmetry,the sample holderthe sample was made holder of awas self-centering made of a chuck,self-centering which is chuck, designed which for lathes. is designed Current for suppliers lathes. edge, in order to minimize the contact area, as represented in Figure 2. Currentand voltage suppliers probes and were voltage made probes of copper were bars, made with of copper a V-shape bars, facing with thea V-shape sample facing edge, the in order sample to edge,minimize in order the contactto minimize area, asthe represented contact area, in as Figure represented2. in Figure 2.

Figure 2. Picture of the Van der Pauw setting for laboratory measurement. The sample is placed at the center, between the four V-shaped copper probes.

Figure 2. Picture of the Van der Pauw setting for laboratory measurement. The sample is placed at Figure 2. Picture of the Van der Pauw setting for laboratory measurement. The sample is placed at the the center, between the four V-shaped copper probes. center, between the four V-shaped copper probes.

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Each copper bar is mounted with springs on plastic jaws that slide on the chuck’s body (Figure2). The plastic jaws are used to ensure a good electrical insulation between each copper bar and the chuck. The scroll ring turns to adjust the opening of the probes. All of the probes are related to move at the same time and with the same distance, in order to keep the sample always centered. As presented by Van der Pauw method [11,12], the electrical resistivity, the resistance, and the thickness of the sample between two contiguous sets of points of measurement on its edge must be related by the following equation: − · dL R − · L R e π ρ AB,DC + e π ρ BC,AD = 1 (2) where L is the thickness of the sample (m); RAB,DC is the measured electrical resistance when the current is injected between probes A and B, and the voltage drop is measured between D and C (µΩ); RBC,AD is the resistance measured when the current is injected between probes B and C, and the voltage drop is measured between A and D (µΩ); and ρ is the electrical resistivity (µΩ·m). For the standard method performed in the laboratory, four measurements were performed around each sample. The average value and the standard deviation were calculated. For the VdP method, eight measurements were performed: four in a ﬁrst position, and four after rotating the sample by 45◦. Considering the top view of the setting, which is presented in Figure2 and the Equation (2), it can be seen that a pair of two contiguous measurements are required to obtain a single value of electrical resistivity. Due to this requirement, four values of electrical resistivity would be obtained after the eight measurements. The average electrical resistivity and standard deviation were obtained after the four electrical resistivities were calculated. To avoid positioning errors, the sample remained in the sample holder the entire time for each set of four measurements, and only the wires from the copper bars were changed.

2.3. Samples Preparation A ﬁrst series of experiments allowed comparing the ISO 11713 standard method and the VdP method. To this end, eight samples, named 1 to 8, and each with a diameter of 5 cm and a length of 10 cm, were cored from the bottom of a single anode block. The electrical resistivity of each sample was measured using both techniques. This part was done to compare the reproducibility of the measurement, and assumed that the samples would have very close resistivity, as they were taken from the same anode block. To emphasize the comparison between the two techniques, a second series of experiments was carried out. Six industrial samples were tested, using the VdP method and the ISO 11713 standard method. All of the samples had their electrical resistivity measured previously using routine industrial practice (R&D Carbon method). Three samples were intact (in appearance), named Samples A, B and C, and the three other presented noticeable defects (broken into two pieces), named D, E and F. The intact samples were tested using the standard method at the university laboratory in order to compare the results with those obtained by the industrial laboratory. After the measurement using the standard method, three slices were cut out of the samples between the locations of the voltage probes. The electrical resistivity of each slice was measured using the VdP method. The position of the slices in an anode core is presented in Figure3. The three samples containing defects were directly cut in three slices, as their condition would not allow measuring their resistivity using the standard method. Figure4 shows the condition of the broken anode core after sampling for measurement by the VdP method. Metals 2017, 7, 369 5 of 17 Metals 2017, 7, 369 5 of 17 Metals 2017, 7, 369 5 of 17

Figure 3. Anode core shapes for measuring electrical resistivity using the standard method (A) and Figure 3. Anode corecore shapesshapesfor for measuring measuring electrical electrical resistivity resistivity using using the the standard standard method method (A) ( andA) and the the Van der Pauw method (B). Vanthe Van der Pauwder Pauw method method (B). (B).

Figure 4. Broken sample prepared for measurement using the Van der Pauw method. The failure was Figure 4. Broken sample prepared for measurement using the Van der Pauw method. The failure was circledFigure at 4. theBroken right sample of the sample. prepared for measurement using the Van der Pauw method. The failure was circled at the right of the sample. The third and last series of experiments was done in order to identify the effect of noticeable The third and last series of experiments was done in order to identify the effect of noticeable defectsThe in thirdthe sample and last on seriesthe measured of experiments electrical was resi donestivity. in The order defects to identify were intentionally the effect of created noticeable in defects in the sample on the measured electrical resistivity. The defects were intentionally created in thedefects sample. in the Samples sample were on themachined measured to create electrical cracks resistivity. placed in The radial defects and in were transversal intentionally positions, created as the sample. Samples were machined to create cracks placed in radial and in transversal positions, as shownin the sample.in the Figure Samples 5(A-2,B-2). were machined The samples to create used in cracks this experiment placed in radial were and taken in from transversal the same positions, anode shown in the Figure 5(A-2,B-2). The samples used in this experiment were taken from the same anode asas those shown used in thefor Figurethe first5(A-2,B-2). set of measurements. The samples used in this experiment were taken from the same as those used for the first set of measurements. anodeThe as measurements those used for thewere ﬁrst performed set of measurements. on samples containing defects before cutting, using the The measurements were performed on samples containing defects before cutting, using the standardThe method, measurements and then were after performed cutting, using on samplesthe VdP method. containing Finally, defects numerical before cutting,simulations using were the standard method, and then after cutting, using the VdP method. Finally, numerical simulations were carriedstandard out method, to verify and the thenpredictive after cutting, capabilities using of the the VdP Van method. der Pauw Finally, method. numerical To this end, simulations an electrical were carried out to verify the predictive capabilities of the Van der Pauw method. To this end, an electrical modelcarried was out todeveloped verify the in predictive the Abaqus capabilities software of the[17] Van. In der these Pauw simulations, method. To the this anode end, an electrical electrical model was developed in the Abaqus software [17]. In these simulations, the anode electrical resistivitymodel was is developed prescribed. in The the Abaquselectrical software resistances [17]. of In Equation these simulations, (2) are estimated. the anode Then, electrical the resistivityelectrical resistivity is prescribed. The electrical resistances of Equation (2) are estimated. Then, the electrical resistivityis prescribed. corresponding The electrical to the resistances VdP model of Equation is calculated, (2) are by estimated. solving Equation Then, the (2), electrical and comparing resistivity it resistivity corresponding to the VdP model is calculated, by solving Equation (2), and comparing it withcorresponding the prescribed to the value. VdP Moreover, model is calculated, by considerin byg solving samples Equation with different (2), and crack comparing configurations, it with the it with the prescribed value. Moreover, by considering samples with different crack configurations, it isprescribed possible to value. evaluate Moreover, the effect by consideringof these cracks samples on the with accuracy different of the crack VdP conﬁgurations, method. it is possible is possible to evaluate the effect of these cracks on the accuracy of the VdP method. to evaluate the effect of these cracks on the accuracy of the VdP method.

Metals 2017, 7, 369 6 of 17 Metals 2017, 7, 369 6 of 17

Figure 5. Schematic view of the cuts performed on sliced samples, along the revolution axis ( z-axis) (A--1,,A--2)) andand normalnormal toto thethe revolutionrevolution axisaxis ((BB--11,,BB--22).).

3. Results 3. Results All of the values presented in the experimental part are the resistivity values relative to the All of the values presented in the experimental part are the resistivity values relative to the average of the measurements performed on the first eight samples using the ISO 11713 standard average of the measurements performed on the ﬁrst eight samples using the ISO 11713 standard method, in which the average resistivity measured was of 52.5 μΩ∙m. The results presented are thus method, in which the average resistivity measured was of 52.5 µΩ·m. The results presented are thus a relative electrical resistivity, and a negative value refers to a measured electrical resistivity smaller a relative electrical resistivity, and a negative value refers to a measured electrical resistivity smaller than the average value obtained via standard method. than the average value obtained via standard method.

3.1. Comparison of Standard and Van de Pauw Methods The graph presented in Figure6 6 shows shows thethe valuesvalues ofof thethe relativerelative electricalelectrical resistivityresistivity ofof thesethese samples. The baseline of this graph is the average value of eight measurements obtained by the ISO 11713 standardstandard method.method. The The standard standard deviation deviation of of the the measurements measurements is presentedis presented as percentageas percentage of the of absolutethe absolute measured measured value, value, and and indicated indicated on eachon each value. value. The The data data points points marked marked as “average” as “average” on theon graphthe graph represent represent the averagethe average value value of all of eight all eight measurements measurements performed performed by twoby two techniques, techniques, thus thus the relativethe relative value value for the for ISO the 11713 ISO standard11713 standard method method is zero. Theis zero. results The show results that show the standard that the deviations standard ofdeviations the VdP methodof the VdP are muchmethod smaller are much than smaller those obtained than those by the obtained standard by method.the standard It can method. be noted It from can thebe noted ninth from set of the columns, ninth set named of columns, “Average”, named that “Ave the averagerage”, that value the obtained average value using obtained the VdP methodusing the is veryVdP closemethod to the is onevery obtained close to using the one the standardobtained method,using the while standard the standard method, deviation while the of the standard former isdeviation half that of of the the former standard is half method. that of This the ﬁrst standard set of measurementsmethod. This first showed set of the measurements high accuracy showed of the VdPthe high method accuracy for characterizing of the VdP method the electrical for characterizing resistivity of the the electrical anode core. resistivity of the anode core.

Metals 2017, 7, 369 7 of 17 Metals 2017, 7, 369 7 of 17

5 +/- 1.56 % Std-UL VdP 4 m)

μΩ∙ 3 +/- 2.26 %

2 +/- 1.13 % +/- 0.15 % +/- 0.89 % 1 +/- 1.93 % +/- 0.14 % +/- 2.30 % Relative resistivity ( 0 +/- 7.72 % +/- 4.86 % +/- 4.72 % -1 +/- 0.55 % +/- 0.76 % +/- 2.47 % +/- 0.07 % -2 +/- 0.11 % +/- 4.72 % -3

-4 +/- 0.87 %

-5 12345678Average

Sample Number FigureFigure 6. Comparison 6. Comparison of electrical of electrical resistivity resistivity using usin theg the standard standard and and Van Van der der Pauw Pauw methods methods for for carbon anodecarbon cores. anode cores.

3.2. Validation of Van der Pauw Method for Intact Samples 3.2. Validation of Van der Pauw Method for Intact Samples In the second set of measurements, the three intact industrial samples (A, B and C) were tested. InAll the the secondsamples setoriginated of measurements, from the same the anode three plant, intact Plant industrial 1. The results samples obtained (A, B in and the C) industrial were tested. All thelaboratory samples are originated labeled as from “Indus the”. sameThen, anodethose obtained plant, Plant using 1. the The four-point results obtained measurements in the of industrial the laboratorystandard are method labeled are as labeled “Indus as”. STD-UL. Then, those The three obtained slices usingcut out the of four-pointeach anode measurementscore, between the of the standardlocations method of the are voltage labeled sensors, as STD-UL. in order Theto use three the VdP slices method cut out were of eachlabeled anode VDP-1, core, VDP-2 between and the locationsVDP-3, of respectively. the voltage Finally, sensors, the in mean order electrical to use resistivity the VdP methodof each sample were labeledwas calculated VDP-1, using VDP-2 the and VDP-3,VdP respectively. method, and Finally,labeled as the VDP-AVG. mean electrical The results resistivity of the measurements of each sample are waspresented calculated in Figure using 7. the It can be seen that the electrical resistivity measured at the industrial laboratory and in our laboratory VdP method, and labeled as VDP-AVG. The results of the measurements are presented in Figure7. are very close when using the standard method. We notice that standard deviation for tests carried It canout be in seen the thatindustrial the electrical laboratory resistivity were not measuredprovided. at the industrial laboratory and in our laboratory are very closeThe samples when using A, B and the C standard were cored method. from different We notice anode that blocks standard that were deviation provided for by tests the carried same out in theanode industrial plant. laboratoryThis means werethey share not provided. the same raw materials and anode-forming process. However, Thethey samplesshow a strong A, B anddifference C were in coredelectrical from resist differentivity measured anode blocksby the standard that were method, provided both by at the the same anodeindustrial plant.This laboratory means and they at the share university the same (Indus raw and materials STD-UL and respectively). anode-forming On the other process. hand, However, the theyresults show aobtained strong through difference the inVdP electrical method resistivityshow that the measured electrical byresistivity the standard values for method, the three both at the industrialsamples are laboratory very close to and each at other. the university In addition, (Indus the standard and STD-UL deviations respectively). were smaller compared On the other with those provided by the standard method, which suggested that the VdP method could be a more hand, the results obtained through the VdP method show that the electrical resistivity values for accurate technique of measurement. The drastic change in results measured using the standard the three samples are very close to each other. In addition, the standard deviations were smaller method, both by the industrial partner and at the university, and the VdP method can be explained. comparedWhen withthe standard those provided method is by used, the the standard current method,flows through which the suggested sample along that the the axis VdP of revolution method could be a(Figure more accurate 5(A-1)), but technique the voltage of measurement.drop is measured The only drastic on the changeedge of the in resultssample. measuredDuring the usingcore the standardsampling method, of the both anode, by themicrocracks industrial and partner coarse and particles at the containing university, their and theown VdPdefects method can be can be explained.produced, When which the standardmay strongly method influence is used, the the curren currentt flow flows at the through specific the area sample of measurement. along the axis of revolutionHowever, (Figure these5 (A-1)),defects probably but the voltage cannot be drop seen is wi measuredth the naked only eye. on On the the edge other of hand, the sample. when using During the corethe VdP sampling method, of the the current anode, flows microcracks normally andto the coarse axis ofparticles revolution containing (Figure 5(B-1,B-2)), their own parallel defects to can be produced,the cross-section which of may the sample. strongly This influence implies that the the current defects flow generated at the during specific the areamachining of measurement. on any of the edges of the sample, may not have a strong influence on the measurements performed. It is However, these defects probably cannot be seen with the naked eye. On the other hand, when using suggested by the author that the measurement using the standard method occurs at the surface, while the VdP method, the current flows normally to the axis of revolution (Figure5(B-1,B-2)), parallel to the when using the VdP method, the measurement is over the volume. cross-section of the sample. This implies that the defects generated during the machining on any of the edges of the sample, may not have a strong influence on the measurements performed. It is suggested by the author that the measurement using the standard method occurs at the surface, while when using the VdP method, the measurement is over the volume. Metals 2017, 7, 369 8 of 17 Metals 2017, 7, 369 8 of 17

Sample A Sample B Sample C

+/- 0.43 % +/- 0.09 % +/- 0.76 % +/- 0.18 % +/- 0.70 % m) 0 μΩ∙ -0.5 N/C

-1

-1.5

-2

-2.5 +/- 0.06 % N/C -3 +/- 8.68 % Relative electrical resistivity ( resistivity electrical Relative -3.5 +/- 1.99 % -4 +/- 0.42 % +/- 0.50 % +/- 0.43 % N/C -4.5 +/- 0.25 % +/- 0.56 % +/- 0.07 % -5 +/- 5.71 % Indus STD-UL VDP-1 VDP-2 VDP-3 VDP AVG

Figure 7. Comparison of standard and Van der Pauw (VdP) methods for intact samples cored from Figure 7. Comparison of standard and Van der Pauw (VdP) methods for intact samples cored from the industrial laboratory. “Indus” and “STD-UL” are the measurements provided respectively by the the industrial laboratory. “Indus” and “STD-UL” are the measurements provided respectively by the industrial laboratory and those achieved using standard method in our laboratory. “VDP (1-3)” are industrial laboratory and those achieved using standard method in our laboratory. “VDP (1-3)” are measurements performed on slices 1, 2 and 3 of each sample (VdP method); VDP-AVG is the average measurements performed on slices 1, 2 and 3 of each sample (VdP method); VDP-AVG is the average ofof thethe threethree combined combined measurements. measurements.

3.3. Validation of Van der Pauw Method for Broken Samples 3.3. Validation of Van der Pauw Method for Broken Samples Another set of tests was performed on three samples provided by the industrial partner, and Another set of tests was performed on three samples provided by the industrial partner, containing noticeable flaws, as shown in Figure 4. Samples D and E originated from the same anode and containing noticeable ﬂaws, as shown in Figure4. Samples D and E originated from the same plant (Plant 2), while sample F came from a different anode plant (Plant 3). These samples were anode plant (Plant 2), while sample F came from a different anode plant (Plant 3). These samples already broken upon reception. This condition prevented the performance of any measurement using were already broken upon reception. This condition prevented the performance of any measurement the standard method. In this case, only the measurements provided by the industrial partner using the standard method. In this case, only the measurements provided by the industrial partner (standard) and VdP methods are compared. The anode core pieces were long enough to cut three (standard) and VdP methods are compared. The anode core pieces were long enough to cut three slices out of each sample. Figure 8 presents the results provided by the industrial laboratory and those slices out of each sample. Figure8 presents the results provided by the industrial laboratory and those obtained using the VdP method. Similar to the previous set of samples, the industrial laboratory did obtained using the VdP method. Similar to the previous set of samples, the industrial laboratory did not provide the standard deviation for the measurements. not provide the standard deviation for the measurements. It can be seen that there are strong differences between samples D and E when the standard It can be seen that there are strong differences between samples D and E when the standard method was used in the industrial laboratory. However, the VdP method showed that they exhibit method was used in the industrial laboratory. However, the VdP method showed that they exhibit very close relative electrical resistivities (−4.57 μΩ∙m for D and −4.91 μΩ∙m for E). The two samples very close relative electrical resistivities (−4.57 µΩ·m for D and −4.91 µΩ·m for E). The two samples D D and E showed a similar electrical resistivity using the VdP method, which might be because the and E showed a similar electrical resistivity using the VdP method, which might be because the same same raw materials and anode-forming process were used in the anode plant. For sample F, the raw materials and anode-forming process were used in the anode plant. For sample F, the standard standard and VdP methods lead to the same results. and VdP methods lead to the same results.

Metals 2017, 7, 369 9 of 17 Metals 2017, 7, 369 9 of 17

VDP Indus 12 N/C Metals 2017, 7, 369 9 of 17 m) 10 μΩ∙ 8 VDP Indus 12 N/C

m) 6 10 μΩ∙ 4 8

2 6 N/C 0 4

2 +/- 0.59 % -2 N/C N/C 0 -4 +/- 0.59 % Relative electrical resistivity ( electrical resistivity Relative -2+/- 2.20 %N/C -6 +/- 1.67 % -4 DEF Relative electrical resistivity ( electrical resistivity Relative +/- 2.20 % +/- 1.67 % Figure 8. Comparison-6 of the standard and Van der Pauw methods for flawed samples from the Figure 8. Comparison ofDEF the standard and Van der Pauw methods for ﬂawed samples from the industrial laboratory. Figure 8. Comparison of the standard and Van der Pauw methods for flawed samples from the 3.4. Validation of Van der Pauw Method for Intentionally-Generated Cracked Samples 3.4. Validationindustrial of Van laboratory. der Pauw Method for Intentionally-Generated Cracked Samples Finally, measurements were conducted on the samples containing noticeable intentionally- Finally,3.4. Validation measurements of Van der Pauw were conductedMethod for Intentionally-Genera on the samples containingted Cracked noticeable Samples intentionally-generated generated cracks. For all of the following experiments, the samples for the VdP method were sliced cracks. For all of the following experiments, the samples for the VdP method were sliced to 1 cm thick to 1 cm thickFinally, and measurements 5 cm in diameter. were conducted To simplify on thethe samplesproblem, containing cracks were noticeable made intentionally- either along the and 5generated cm in diameter. cracks. For To all simplify of the following the problem, experiments, cracks the were samples made for either the VdP along method the were revolution sliced axis revolution axis (radial crack), or normal to it (transverse crack). (radialto crack),1 cm thick or normal and 5 cm to itin (transversediameter. To crack). simplify the problem, cracks were made either along the A radial crack was generated on a sample before slicing. This allowed performing electrical Arevolution radial crack axis (radial was generatedcrack), or normal on a sampleto it (transverse before crack). slicing. This allowed performing electrical measurements by the standard method, both before and after performing the slice cut. Then, three measurementsA radial by crack the standardwas generated method, on a both sample before before and slicing. after performingThis allowed theperforming slice cut. electrical Then, three slices were cut out of the sample to measure the electrical resistivity using the VdP method. In such slicesmeasurements were cut out by of the standard sample tomethod, measure both the before electrical and after resistivity performing using the the slice VdP cut. method. Then, three In such a situation,slices were two cut positions out of the were sample used to formeasure the crack the el towardectrical resistivity the probes. using In theone VdP position, method. the In crack such was a situation, two positions were used for the crack toward the probes. In one position, the crack was placeda situation, at an equal two distance positions betweenwere used two for probes.the crack In toward the other the probes. position, In one the position, crack was the placed crack was close to placed at an equal distance between two probes. In the other position, the crack was placed close to one ofplaced the probes at an equal (Figure distance 9). Those between options two probes. were chosen In the other to assess position, the effectthe crack of position was placed of theclose crack to on one of the probes (Figure9). Those options were chosen to assess the effect of position of the crack on the results.one of the probes (Figure 9). Those options were chosen to assess the effect of position of the crack on the results.the results.

FigureFigure 9. Position 9. Position of the ofcrack the crack toward toward theprobes: the probes: (1) crack (1) crack placed placed between between two two probes probes at equal at equal distance; (2) crack placed close to one of the probes. Figuredistance; 9. Position (2) crack of placedthe crack close towardto one of the the probes.probes: (1) crack placed between two probes at equal distance; (2) crack placed close to one of the probes.

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The tests were also performed on samples that had transversal cracks (normal to the revolution axis). In order to compare the standard and VdP methods, cracks were created before slicing. A side view of the cracked sample is shown on Figure5(B-2). The obtained results were summarized in Table1. As the surface and length of samples were taken into account for the electrical resistivity calculation using the standard method, the void corresponding to the crack generation had to be removed from the total sample volume. We recall that the presented values are related to the average value of the electrical resistivity measured in the ﬁrst experiments, while the standard deviation is related to the real obtained values.

Table 1. Electrical resistivity measurement of samples containing defects using both the standard and the Van der Pauw methods.

Sample and Conﬁguration Relative Electrical Resistivity (µΩ·m) Measurements Using Standard Method Sample without defect −2.4 ± 12% Radial Crack +5 ± 12% void: 1.7% Transversal crack +12.9 ± 21% void: 0.18% Measurements using Van der Pauw method Sample without defect +2.5 ± 0.4% Radial Crack +3.2 ± 2.5% Crack placed between two probes Radial Crack +3.3 ± 2% Crack placed near the probe Transverse crack +4.5 ± 8%

The results presented in Table1 clearly show that when using the standard method, a noticeable difference is measured between the sample without any cracks, and the sample containing a crack. The electrical resistivity is overestimated in the presence of cracks. Moreover, it seems that the crack orientation substantially affects the measurements provided by the standard method. The results obtained using the VdP method show that this technique is considerably less sensitive to the cracks’ size and orientation than the standard method. In fact, small differences between relative electrical resistivity measurements for different samples were obtained with the VdP method. The bigger standard deviation is obtained for the transverse crack. The thickness of the crack behaves as a layer of insulating material parallel to the sample itself. Using the VdP technique enables the electrical resistivity to be retrieved when the crack is radial, as it does not interfere with any parameter of calculation. When the crack is placed transversally, it has an inﬂuence on the thickness of the sample (parameter “d” in the Equation (2)). This causes an overestimation of the electrical resistivity.

4. Finite Element Modelling of VdP Method In order to assess the robustness of the VdP method, numerical simulations were carried out using the commercial ﬁnite element software Abaqus [17]. The purpose of these numerical simulations was to simulate the experimental set up developed for the VdP method (Figures2 and9). Therefore, for a given sample, with or without crack, and an assigned electrical resistivity, all electrical potentials and resistances needed for the VdP method were evaluated numerically. This enabled the estimation of the electrical resistivity of a sample through the VdP method. To this end, the Newton–Raphson scheme for nonlinear equations was used to solve Equation (2). Furthermore, using this approach, some speciﬁc cases, such as very thin cracks, which are not easily achievable in laboratory, can be simulated. In order to reproduce conditions used in the laboratory, a cylindrical sample with 1 cm thickness and a 5 cm diameter was considered. An electrical current of 1 ampere was applied on the supply connector bars. The electrical resistivity of an anode is usually around 50 µΩ·m (a conductivity of Metals 2017, 7, 369 11 of 17 Metals 2017, 7, 369 11 of 17

connector bars. The electrical resistivity of an anode is usually around 50 μΩ∙m (a conductivity of 20,00020,000 S/m)S/m) [18]. [18]. Figures Figures 10 10 and and 11 11 show show the the dev developedeloped CAD CAD (Computer-Aaided (Computer-Aaided Design) Design) model, model, and andthe mesh the mesh used used for the for simulations the simulations in the incases the of cases the radial of the crack radial placed crack placedbetween between two probes two (Figures probes (Figures10A and 10 11A),A and and 11 A),the andtransv theerse transverse crack (Figure crack (Figure11A,B). 11A,B).

FigureFigure 10.10.CAD CAD (Computer-Aaided(Computer-AaidedDesign) Design)model modelused usedfor for thethe ﬁnitefiniteelement elementmethod method(FEM) (FEM)and andthe the VdPVdP method, method, ( A(A)) radial radial crack, crack, placed placed between between the the probes; probes; ( B(B)) transverse transverse crack. crack.

SinceSince AbaqusAbaqus softwaresoftware requiresrequires anan electricalelectrical conductivityconductivity asas thethe input,input, wewe decideddecided toto useuse thethe valuevalue of of 20,000 20,000 S/m S/m for simulations. Using thethe parametersparameters mentionedmentioned above,above, teststests werewere runrun withwith eacheach crackcrack conﬁguration configuration presented presented previously. previously. The electricalThe electrical potential potential at the voltageat the probesvoltage was probes calculated was andcalculated used in and the used VdP equationin the VdP in equation order to estimatein order theto estimate sample electricalthe sample resistivity. electrical Forresistivity. radial and For transversalradial and transversal defects, the defects, crack was the designed crack was as designe 2 mm thickd as 2 and mm 1.7 thick cm and deep. 1.7 In cm the deep. case In of the case radial of crack,the radial simulations crack, simulations were performed were performed with cracks with placed cracks between placed between the probes the and probes close and to close one ofto theone probes,of the probes, similar similar to that into thethat experimental in the experimental measurements. measurements.

Metals 2017, 7, 369 12 of 17 Metals 2017, 7, 369 12 of 17

Figure 11. Mesh model used for FEM and VdP methods; (A) radial crack, placed between the probes; Figure 11. Mesh model used for FEM and VdP methods; (A) radial crack, placed between the probes; (B) transverse cracks. (B) transverse cracks.

A first simulation was run on a simple circular sample. This simulation allowed us to retrieve an electricalA first simulation resistivity wasof 49.8 run μΩ∙ onm a simpleusing the circular VdP sample.method. ThisThis simulationresult shows allowed that the us error to retrieve between an µ electricalthe assigned resistivity electrical of 49.8resistivityΩ·m usingand the the calculated VdP method. one (using This result the VdP shows Method) that the is errorvery betweensmall. After the assignedthe validity electrical of the model resistivity was andconfirmed, the calculated tests were one run (using on other the VdP configurations. Method) is very The obtained small. After results the validityare stored of thein Tables model 2 wasand confirmed,3. tests were run on other configurations. The obtained results are storedTable in Tables 2 shows2 and the3. simulation results for samples containing a crack parallel to the axis of rotation (FigureTable 5(A-2)).2 shows The the crack simulation was placed results as for follow: samples (i) containingbetween two a crack sensors parallel at equal to the distance, axis of rotation and (ii) (Figureclose to5 (A-2)).one of Thethe cracksensors. was It placed is expected as follow: that, (i) fo betweenr a defect two parallel sensors to at the equal axis distance, of revolution and (ii) closebeing tonormal one of to the the sensors. electric It current is expected line, that, the forvolume a defect of parallelthe crack to does the axis not of substantially revolution being affect normal the results. to the electricThis assumption current line, can the be volumeexplained of thein that crack the does VdP not equation, substantially Equation affect (2), the is not results. related This to assumption the surface canarea be of explained the sample, in thatbut rather the VdP only equation, to its height. Equation Figure (2), 5(A-2) is not relateddepicts to the the position surface of area the of crack, the sample, which butin the rather case only of this to itsorientation, height. Figure does5 not(A-2) interfere depicts in the the position calculation. of the To crack, justify which the inlatter the statement, case of this a orientation,simulation was does performed not interfere on a insample the calculation. containing an To infinitesimally justify the latter thin statement, crack (0.1 amm), simulation which does was performednot represent on any a sample noticeable containing variation an infinitesimallyin the volume of thin the crack sample. (0.1 It mm), is important which does to note not representthat such anya thin noticeable crack may variation not be invisible the volume to the naked of the eye. sample. The Itcrack is important was placed to note in the that same such direction, a thin crack close may to notone beof visiblethe sensors. to the nakedIt appears eye. that The crackthe obtained was placed results in the seem same to direction,remain stable close whether to one of the sensors.crack is Itplaced appears close that to thea probe, obtained or centered results seem between to remain two, stableor even whether if its size the is crack highly is placedreduced. close Furthermore, to a probe, orthe centered relative betweenerror between two, or the even assigned if its size electrical is highly conductivity reduced. Furthermore, and the one theretrieved relative using error the between finite theelement assigned method electrical (FEM) conductivity and VdP methods and the remains one retrieved very low. using the finite element method (FEM) and VdP methods remains very low. Table 2. Electrical resistivity estimation, using the FEM and VdP methods, for a sample containing a Tablecrack 2.parallelElectrical to the resistivity axis of revolution. estimation, using the FEM and VdP methods, for a sample containing a crack parallel to the axis of revolution. Original Electrical Conductivity Electrical Conductivity Estimated Using CrackCrack TypeType and and Position Position Original ConductivityConductivity (S/m) Estimated Using FEM and ErrorError FEM and VdP Methods (S/m) 2-mm wide crack, parallel to the axis of (S/m) VdP Methods (S/m) 20,000 20,316 +1.58% 2-mmrevolution, wide equal crack, distance parallel to sensors to the axis 2-mm wide crack, parallel to the axis of of revolution, equal distance to 20,00020,000 20,26020,316 +1.30%+1.58% revolution, close to one sensor sensors0.1-mm thin crack, parallel to the axis of 20,000 20,253 +1.37% 2-mmrevolution, wide close crack, to one sensorparallel to the axis 20,000 20,260 +1.30% of revolution, close to one sensor 0.1-mmIn a secondthin crack, approach, parallel the to crack the axis was placed on a plane normal to the axis of rotation (Figure5(B-2)). 20,000 20,253 +1.37% Inof this revolution, case, the voidclose interferesto one sensor with the real height of the sample and most probably affects the results. To justify this assumption, two cases were investigated using the FEM and VdP methods. The crack was initiallyIn a second 2 mm approach, wide in the a 10 crack mm was thick placed sample, on a which plane creatednormal to a non-negligiblethe axis of rotation void (Figure within 5(B- the 2)). In this case, the void interferes with the real height of the sample and most probably affects the results. To justify this assumption, two cases were investigated using the FEM and VdP methods. The

Metals 2017, 7, 369 13 of 17 Metals 2017, 7, 369 13 of 17 crack was initially 2 mm wide in a 10 mm thick sample, which created a non-negligible void within the material. Referring to Figure 5(B-2) and Equation (2), one can easily state that the void created by thismaterial. crack would Referring most to probably Figure5(B-2) affect and the Equationcalculation (2), of one electrical can easily resistivity. state that After the the void 2 mm created crack, by a 0.1this mm crack crack would was mostplaced probably in the same affect spot the to calculation compare the of electrical results. Table resistivity. 3 shows After the theresults 2 mm of crack,both simulations.a 0.1 mm crack As wasexpected placed by in the the authors, same spot the toerror compare obtained the when results. the Table crack3 shows is macroscopic the results is ofbigger both thansimulations. when the As crack expected in infinitesimally by the authors, thin. the In errorthe ca obtainedse of the whenthin crack the crack placed is macroscopicnormal to the is axis bigger of revolution,than when the the error crack calculated in infinitesimally tended to thin. be even In the sma caseller of than the that thin in crack the case placed of the normal thin crack to the placed axis of parallelrevolution, to the the same error calculatedaxis. The re tendedsults presented to be even after smaller both than simula that intions the corroborate case of the thin the crackhypothesis placed initiallyparallel made to the by same the authors; axis. The i.e., results when presented the crack afteris placed both normal simulations to the corroborateaxis of revolution, the hypothesis namely parallelinitially to made the bycurrent the authors; lines, the i.e., thickness when the of crack the iscrack placed will normal have toan the influence axis of revolution,on the measured namely electricalparallel to resistivity. the current Moreover, lines, the th thicknesse thickness of the of the crack crack will will have induce an influence a non-negligible on the measured volume electrical of void withresistivity. a higher Moreover, electrical theresistivity, thickness after of thetaking crack into will account induce the a non-negligiblecalculation of the volume electrical of void resistivity with a ofhigher the sample. electrical This resistivity, leads to an after increase taking in into the account calculated the electrical calculation resistivity of the electrical or, as seen resistivity in Table of 3, the a decreasesample. Thisin electrical leads to anconductivity. increase in theThe calculated bigger the electrical crack, the resistivity stronger or, the as seen error in will Table be3, aobserved. decrease Figurein electrical 12 shows conductivity. the current The lines bigger flowing the crack, through the stronger the sample the errorin the will case be of observed. the radial Figure crack. 12 Figure shows 12Athe currentrepresents lines the flowing current throughflowing thewhen sample the current in the probes case of are the placed radial on crack. the same Figure side 12 Aof representsthe crack; meanwhile,the current Figure flowing 12B when shows the the current current probes lines are when placed the current on the sameprobes side are of placed the crack; on both meanwhile, sides of theFigure crack. 12 BIn showsboth cases, the current it can be lines noticed when that the th currente electrical probes current are placed flows in on a bothplan sidesperpendicular of the crack. to theIn bothrevolution cases, it axis. can beThese noticed results that theare electricalin agreemen currentt with flows authors’ in a plan hypothes perpendicularis about to the revolutionelectrical currentaxis. These flow. results are in agreement with authors’ hypothesis about the electrical current flow.

Figure 12. Cont.

Metals 2017, 7, 369 14 of 17 Metals 2017, 7, 369 14 of 17

Figure 12. Representation of the current line on a sample containing radial crack; (A) probes placed Figure 12. Representation of the current line on a sample containing radial crack; (A) probes placed on on the same side of the crack; (B) probes placed on both sides of the crack. the same side of the crack; (B) probes placed on both sides of the crack.

Table 3. Electrical resistivity of a sample containing a crack, normal to the axis of rotation, calculated Table 3. Electrical resistivity of a sample containing a crack, normal to the axis of rotation, calculated using the FEM and VdP methods. using the FEM and VdP methods. Electrical Conductivity Estimated Original Electrical Conductivity Estimated CrackCrack Type Type and and Position Original Conductivity (S/m)Using FEM and VdP Methods Error Error Conductivity (S/m) Using FEM and VdP Methods (S/m) 2-mm wide crack, normal to the axis of revolution 20,000(S/m) 19,362 3.19% 0.1-mm2-mm thin crack,wide normal crack, to the normal axis of revolution 20,000 20,215 +1.07% 20,000 19,362 3.19% to the axis of revolution 5. Detection0.1-mm ofthin Defects crack, Usingnormal VdP Method 20,000 20,215 +1.07% Into the the previous axis of revolution section, it was shown that the electrical resistivity could be easily retrieved using the VdP method, even if the sample contains structural flaws. Referring to Equation (2), the VdP method5. Detection requires of the Defects measurement Using VdP of twoMethod contiguous resistances, RAB,DC and RBC,AD, in order to calculateIn the the electrical previous resistivity section, it ofwas the shown material. that Thethe elec samplestrical haveresistivity a cylindrical could be geometry, easily retrieved and are using supposedthe VdP to method, be homogeneous even if the in sample composition. contains In stru a crack-freectural flaws. sample, Referring the resistances to EquationRAB,DC (2), theand VdP R present very close values. In the case of samples containing cracks along the axis of revolution, BC,ADmethod requires the measurement of two contiguous resistances, RAB,DC and RBC,AD, in order to thecalculate cylindrical the symmetry electrical re issistivity lost, and of onlythe material. the axis The of symmetry samples have remains a cylindrical along the geometry, crack. Thus, and are the resistances R and R show a significant difference. In the case of cracks normal to the axis supposed toAB,DC be homogeneousBC,AD in composition. In a crack-free sample, the resistances RAB,DC and RBC,AD of revolution,present very such close a difference values. In between the caseR AB,DCof samplesand R BC,ADcontainingis not cracks observed. along This the can axis be of explained revolution, by the the orientationcylindrical ofsymmetry the crack is toward lost, and the currentonly the flow. axis According of symmetry to the remains geometry along of the the system, crack. one Thus, can the state that the current flows mostly parallel to the faces of the sample; as a result, defects oriented in the resistances RAB,DC and RBC,AD show a significant difference. In the case of cracks normal to the axis of same direction as the current flows will have a lower effect than those placed in a different direction. revolution, such a difference between RAB,DC and RBC,AD is not observed. This can be explained by the For simplicity,orientationR ofAB,DC the andcrackR BC,ADtowardare the respectively current flow. expressed According as R1 toand theR 2,geometryhereafter. of Table the system,4 shows one the can resultsstate obtained that the after current the simulations.flows mostly It parallel reveals to that the for faces a crack of the placed sample; in a radialas a result, direction, defects parallel oriented to in the axisthe same of revolution, direction a as strong the current difference flows between will have the twoa lower contiguous effect than resistances those placed can be in observed, a different regardless of the thickness or the position of the crack toward the sensors. direction. For simplicity, RAB,DC and RBC,AD are respectively expressed as R1 and R2, hereafter. Table 4 shows the results obtained after the simulations. It reveals that for a crack placed in a radial direction, parallel to the axis of revolution, a strong difference between the two contiguous resistances can be observed, regardless of the thickness or the position of the crack toward the sensors.

Metals 2017, 7, 369 15 of 17

Table 4. Difference for two contiguous resistances for a sample containing a crack parallel to the axis of rotation, calculated using the FEM method.

Electrical Resistance Measured Crack Type and Position Original Conductivity (S/m) Difference Using Finite Element Method 2-mm wide crack, parallel to the axis of R = 0.7 mΩ 20,000 1 59.8% revolution, equal distance to sensors R2 = 1.74 mΩ 2-mm wide crack, parallel to the axis of R = 0.52 mΩ 20,000 1 73.9% revolution, close to one sensor R2 = 1.99 mΩ 0.1-mm thin crack, parallel to the axis of R = 0.553 mΩ 20,000 1 71.5% revolution, close to one sensor R2 = 1.91 mΩ

On the other hand, Table5 presents the results obtained in the case of a crack placed normally to the axis of revolution of the sample. This means that the crack shares the direction of the electrical current lines. In this case, referring to Equation (2) and considering the Figure5(B-2), the parameter “d” is obtained by subtracting the thickness of the crack from the thickness of the sample. Therefore, the thickness of the crack implicitly influences the measured values when the thickness of the crack is not negligible compared with that of the sample. In this configuration, the effect is limited to a diminution of the actual volume of matter through which the current flows. Results presented in Table5 show that the difference between contiguous probes is very low when compared with the case of a crack normal to the electrical field lines. The difference between the results in the two sections of Table5 is probably because when a crack is of a non-negligible size, the lack of material to conduct the current, which acts as a much more resistive part, will lead to higher resistance values and therefore a lower electrical conductivity. Consequently, the resistances measured for a 2 mm thick crack are slightly higher than that measured for a 0.1 mm thin crack. Those results are in a good adequacy with the above-mentioned hypothesis.

Table 5. Difference for two contiguous resistances for a sample containing a crack normal to the axis of rotation, calculated using the FEM method.

Electrical Resistance Measured Crack Type and Position Original Conductivity (S/m) Difference Using Finite Element Method R = 1.1 mΩ 2 mm wide crack, normal to the axis of revolution 20,000 1 6.8% R2 = 1.18 mΩ R = 1.086 mΩ 0.1 mm thin crack, normal to the axis of revolution 20,000 1 0.99% R2 = 1.0969 mΩ

The difference of values between R1 and R2 can indicate very precisely the presence of cracks, speciﬁcally if they are placed parallel to the axis of revolution. According to the obtained results, it seems that the Van der Pauw method, when coupled with FEM, helps detect the presence of defects in an anode core. Depending on the type, size, and orientation of the defects, the detection can be either good or totally inadequate. This method is complementary to the standard method that is usually used for the characterization of the anode core, and could provide better defect detection.

6. Conclusions In this work, the electrical resistivity of baked carbon anodes used in the aluminum industry was studied using both the ISO 11713 standard method and the Van der Pauw method. Comparison side by side was achieved by experimental analysis; then, numerical simulation was used to emphasize the previously obtained results. The results obtained experimentally show that on industrially produced anode cores, the Van der Pauw method appears to be more accurate and less sensitive to macro and micro structural defects in the material. As the cracks in anode cores are not necessarily representative of the microstructure of the whole anode, the knowledge of the electrical resistivity of the material without the effect of crack, which can be considered as intrinsic electrical resistivity of the material, gives better information on the Metals 2017, 7, 369 16 of 17 electrical properties of the anode block, i.e., inhomogeneity, paste formulation effect, etc. This method shows a good repeatability in the measurement of the electrical resistivity of samples. Numerical simulations using the ﬁnite element method showed that this method is perfectly suitable for the measurement of the electrical resistivity of samples containing defects or ﬂaws such as macroscopic cracks, in different orientations. It has to be noted that depending on the size and orientation of the defect toward the sensors, the defect may induce an error in the calculation of electrical resistivity. Also, the Van der Pauw method, complementary with the standard methods ISO 11713, can reveal the presence of defects in the material that cannot be seen by the naked eye. The Van der Pauw method is very simple and easy to use, and would be easy to automate for industrial purposes.

Acknowledgments: The authors gratefully acknowledge the financial support provided by Alcoa Inc., the Natural Sciences and Engineering Research Council of Canada, and the Centre Québécois de Recherche et de Développement de l’Aluminium. A part of the research presented in this article was financed by the Fonds de recherche du Québec-Nature et Technologies by the intermediary of the Aluminium Research Centre–REGAL. Author Contributions: Geoffroy Rouget and Houshang Alamdari conceived and designed the experiments; Geoffroy Rouget and Hicham Chaouki performed the experiments and numerical simulations; Donald Picard and Donald Ziegler contributed in data analysis; Geoffroy Rouget wrote the paper and all co-authors commented/corrected it. Conflicts of Interest: The authors declare no conflict of interest.

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