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Simultaneous measurement of interdiffusion and intrinsic diffusion coefficients in liquid metals on the ground
International Journal of Heat and Mass Transfer 133 (2019) 531–541
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Simultaneous measurement of interdiffusion and intrinsic diffusion coefficients in liquid metals on the ground Hideto Fukuda a, Masato Shiinoki a,⇑, Yuki Nishimura a, Shinsuke Suzuki b a
Department of Applied Mechanics, Faculty of Science and Engineering, Waseda University, Okubo 3-4-1 Shinjuku-ku, 169-8555 Tokyo, Japan Kagami Memorial Research Institute of Materials Science and Technology, Department of Applied Mechanics and Aerospace Engineering, Faculty of Science and Engineering, Waseda University, Okubo 3-4-1 Shinjuku-ku, 169-8555 Tokyo, Japan b
a r t i c l e
i n f o
Article history: Received 7 August 2018 Received in revised form 26 October 2018 Accepted 6 December 2018
Keywords: Diffusion coefficient Liquid metal Interdiffusion Intrinsic diffusion Boltzmann-Matano method Darken’s equation
a b s t r a c t Simultaneous measurements of interdiffusion and intrinsic diffusion coefficients in liquid Sn-Pb were performed using the shear cell method and a stable density layering. The binary diffusion couples of Sn-Pb and Sn0.6Pb0.4-Sn0.4Pb0.6 were used in two experiments with the average molar fraction of diffusion couples NSn = 0.5. The concentration dependency of the interdiffusion coefficient was confirmed to exhibit a downward convexity through the comparison of the interdiffusion coefficients. However, the Boltzmann–Matano method reveals a diminished influence. The sample measurements of the intrinsic diffusion coefficients of a diffusion couple of pure metals demonstrate a dependency on concentration. On the other hand, the difference of diffusion coefficients was small in a diffusion couple of alloys. The difference between the values calculated by fitting the error function based on the superposition of fluxes and by the conventional analysis methods was small. We propose that a reasonable intrinsic diffusion coefficient can be calculated using the superposed equation of error function. The theoretical interdiffusion coefficients were calculated by substituting the measured intrinsic diffusion coefficients into the Darken’s equation and comparing the results with the measured values. As a result, the difference between the theoretical and measured interdiffusion coefficients is not very large. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Diffusion coefficient is one of the important physical properties in understanding diffusion phenomena. In particular, the interdiffusion coefficient in liquid metals is very important in the discus~ can be sion of diffusion in alloys. The interdiffusion coefficient D expressed theoretically by the Darken’s equation [1], expressed by the following equations:
D ¼ Da Nb þ Db Na U;
ð1Þ
D ¼ Da Nb þ Db Na Di ¼ Di U; i ¼ a or b ;
ð2Þ
where Di is the self-diffusion coefficient, Di is the intrinsic diffusion coefficient, Ni is the molar fraction, and U is the thermodynamic factor. However, the validity of the Darken’s equation in liquid metals has not been confirmed by experiments. A method to verify its ⇑ Corresponding author. E-mail addresses: [email protected] (H. Fukuda), stag7211@asagi. waseda.jp (M. Shiinoki), [email protected] (Y. Nishimura), suzuki-s@ waseda.jp (S. Suzuki). https://doi.org/10.1016/j.ijheatmasstransfer.2018.12.036 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
validity compares the measured value of the interdiffusion coefficient and the theoretical value calculated by Eq. (1). Another method simultaneously measures the interdiffusion and the intrinsic diffusion coefficient and compares them with Eq. (2) [2]. The isotope is used in the measurement of the intrinsic diffusion coefficient. There are at least two problems with measuring the diffusion coefficient in liquid metals on the ground. One problem is that it is unclear whether the error derived from the experimental method was appropriately corrected. Klassen et al. measured the interdiffusion coefficients of the Sn-Pb system and researched the concentration dependency of the interdiffusion coefficient [3]. The Sn-Pb system was selected for two reasons: first, the selfdiffusion coefficients of both Sn and Pb had already been investigated. Next, the molten Sn-Pb alloys could be prepared and handled with relative ease. They concluded that the diffusion coefficients may be measured with higher accuracy using the shear cell method rather than the capillary reservoir method, which they used. In addition, their results showed discontinuity in the diffusion coefficient with eutectic composition, although they did not mention it in the paper. The other problem is that natural convection might occur. It was found that the microgravity environment
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Nomenclature C Ch Cl C0 Cisotope Catom CDSIS CTLD ~ D
D D D207Pb D124Sn DD H N NSn
concentration (at%) initial higher concentration (at%) (variable parameter) initial lower concentration (at%) (variable parameter) initial concentration of intermediate sample (at%) (variable parameter) concentration of isotope (at%) concentration of element (at%) concentration that expressed diffusion between semiinfinite samples (at%) concentration that expressed thick layer diffusion (at%) interdiffusion coefficient (m2/s) self-diffusion coefficient (m2/s) intrinsic diffusion coefficient (m2/s) intrinsic diffusion coefficient of 207Pb (m2/s) intrinsic diffusion coefficient of 124Sn (m2/s) standard deviation of diffusion coefficient (m2/s) thickness of the intermediate sample (m) (fixed parameter) molar fraction molar fraction of Sn
is effective in suppressing natural convection [4]. However, diffusion experiments in microgravity have some problems, e.g., high cost, long preparation time, limitation of opportunities, and high risks during launch and recovery. Thus, the establishment of ground-based experimental methods for diffusion measurements is strongly required. The method of combining the shear cell method and the stable density layering, in which the sample is placed so that the density of the samples monotonically increases in the gravity direction, is valid for measuring the diffusion coefficient on the ground [5]. However, there have been few reports on interdiffusion experiments using the combined method. Therefore, it is necessary to perform interdiffusion experiments using this method. Suzuki et al. measured the interdiffusion coefficients of the Sn-Pb system for NSn = 0.05 and 0.95 using this method [6]. However, the diffusion coefficients at higher solute concentrations have not been reported. In addition, the resulting concentration profiles have large variations. Therefore, it is necessary to obtain concentration profiles with small variations for each diffusion experiment. This can be performed using the shear cell method with high solute concentration, e.g., NSn = 0.5. To reduce the variation in the concentration profiles, pure metals are used as a diffusion couple. Moreover, when the density of the samples monotonically varies with composition, such as in the Sn-Pb system [7], stable density layering is easily created by this technique because the couple has the maximum density difference in the system. On the other hand, the measured values are considered to be influenced by the concentration dependency of the diffusion coefficient. However, these influences have not been confirmed by experiments. Kraatz et al. simultaneously measured the interdiffusion and the intrinsic diffusion coefficients by taking the ratio of the measured atomic and isotopic concentrations in a sample [2]. However, the ratio is not applicable to a diffusion couple of two pure metals, which is expected to exhibit suppression of convection, as mentioned above. In this case, arithmetic calculation cannot be performed by this analysis method. As a result, the intrinsic diffusion coefficients cannot be calculated. The purpose of this study is to establish a simultaneous measurement method of interdiffusion and intrinsic diffusion coefficients in liquid metals. The validity of the experimental and analytical methods was investigated. Interdiffusion and intrinsic
NPb P R2 t T x x0 xDv E
molar fraction of Pb filling pressure of samples (kPa) coefficient of determination diffusion time (s) diffusion temperature (K) position (m) central position (m) (fixed parameter) vertex position (m)
X2
mean square diffusion depth (m2) E X 2measure mean square diffusion depth of measurement (m2) D E (variable parameter) X 2average additional mean square diffusion depth from concen2 D E tration averaging (m ) X 2shear additional mean square diffusion depth from shear2 D E induced convection (m ) X 2initial additional mean square diffusion depth from initial condition (m2) q density (kg/m3) U thermodynamic factor D
diffusion experiments were conducted using the Foton shear cell [5] and stable density layering on the ground. The validity of the experimental method was examined by the concentration error in the concentration profiles, regardless of the occurrence of natural convection, and the influence of the concentration dependency on the measured value. Furthermore, the validity of a new analytical method based on superposition of fluxes for diffusion coefficient calculation was investigated. Finally, the validity of the Darken’s equation was examined by comparing measured values and theoretical values of the Darken’s equation at NS = 0.50.
2. Experimental procedure 2.1. Sample preparation In this study, the molar fraction of Sn, NSn, representing the diffusion couple was defined as the average composition of each sample constituting the diffusion couple. For NSn = 0.5, two experiments were performed in which the compositions of the diffusion couples were changed. In the first experiment, the diffusion couple was constituted by pure Sn, Sn0.5Pb0.5, and pure Pb (this is called ‘‘a diffusion couple of pure metals”). In the second experiment, the diffusion couple was constituted by Sn0.6Pb0.4, Sn0.5Pb0.5, and Sn0.4Pb0.6 (this is called ‘‘a diffusion couple of alloys”). The composition of the intermediate sample was determined to be the average composition of the samples on both sides so that the central position of interdiffusion corresponds to one of intrinsic diffusion in the concentration profile. Pure Sn (99.999 wt%) and pure Pb (99.998 wt%) produced by Kojundo Chemical Laboratory Co., Ltd. were used to prepare samples. Pure Sn, isotope 124Sn (99.90 at%), and isotope 207Pb (99.10 at%) produced by ISOFLEX USA, Inc., as well as common lead isotopic standard SRM 981 (concentration as shown in Table 1) produced by National Institute of Standards and Technology, were used for the preparation of the intermediate samples. The concentration of 207Pb and 124Sn were adjusted to 35 at% and 20 at%, respectively. The adjusted samples were set in a crucible and held at 1073 K in vacuum atmosphere for 900 s before cooling. The samples were constantly vibrated during cooling so that segregation was suppressed. The solidified samples were cut and shaped into a cylindrical shape through drawing dies.
H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541 Table 1 Certified isotopic concentration of SRM981. Type of isotope
Certified isotopic concentration (at%)
204
1.4255 0.0012 24.1442 0.0057 22.0833 0.0027 52.3470 0.0086
Pb Pb Pb 208 Pb 206 207
2.2. Diffusion experiment and concentration analysis Fig. 1 shows a schematic illustration of the diffusion experiment using the Foton shear cell [5] and a stable density layering. The shear cell unit consists of 20 graphite disks of 3 mm thickness and reservoirs. Each disk had four capillaries of 1.5 mm in diameter. The capillaries, named A, B, C and D, were located 5 mm away from the center of the axis. The reservoirs were equipped on both ends of the capillary so that these filled the capillary with the samples under pressure P = 20 kPa. The diffusion couple of pure metals consisted of pure Pb with a length of 27 mm, an intermediate sample Sn0.5Pb0.5 with length H, and pure Sn with a length of 30 mm. The diffusion couple of alloys consisted of Sn0.4Pb0.6 with a length of 27 mm, an intermediate sample Sn0.5Pb0.5 with length H, and Sn0.6Pb0.4 with a height of 30 mm. The mass of the samples was measured before the diffusion experiments. In each experiment, intermediate samples with enriched isotopes were used in only one of the four capillaries. The compositions of the intermediate sample in the remaining three capillaries were the isotope natural abundance. To suppress natural convection, each sample was set in a stable density layering (cf. Fig. 1a). As shown in Fig. 1b, the couples were assembled with the intermediate sample separated from the diffusion axis. To prevent oxidation of the samples, the air in the vacuum chamber was replaced three times with Ar gas after evacuation. The air continued the evacuation after third replacement. The diffusion couples of alloys were set vertical to the gravity direction,
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heated to 773 K, and then homogenized for 1.5 h. After that, the diffusion couples of alloys were reset parallel to the direction of gravity and homogenized for 3 h. Since it was necessary to use the diffusion couples of pure metals, the diffusion couples were set parallel to the direction of gravity from the beginning, heated to 773 K, and homogenized for 3 h. Tanimoto et al. also set diffusion couples vertically in the direction of gravity during homogenization and then reset them in parallel. They conducted this technique to fill the samples in the capillary [8]. However, the aim of this technique was not only to fill the sample in the capillary but also to homogenize the alloy sample in this study. In both experiments, intermediate samples were inserted into the diffusion axis by a rotation with a motor at a speed of 0.5 mm/s (0.1 rad/s) after homogenization, and as a result, diffusion couples were formed (cf. Fig. 1c). The diffusion temperature T and diffusion time t were 773 K and 3600 s, respectively (cf. Fig. 1d). The atmosphere was 1–2 Pa during both experiments. The temperature was measured by thermocouples at three points along the axis of capillaries, that is, close to both ends of the capillary and the middle position. At that time, the temperature gradient was 40 K/m. At the end of the diffusion period, diffusion couples were divided into 20 pieces by the rotation with the motor again and cooling (cf. Fig. 1e). By the shear cell method, errors due to mass transport and volume change were eliminated during heating and cooling. The mass of each piece was also measured after the diffusion experiments. The pieces were dissolved in a mixture composed of nitric acid, hydrochloric acid, and tartaric acid. The concentrations of Pb and Sn in each piece were analyzed by inductively coupled plasma - optical emission spectrometry (ICP-OES, Agilent 5100). The isotope concentrations of 204Pb, 206Pb, 207Pb, 208Pb, and 124Sn were analyzed by inductively coupled plasma - mass spectroscopy (ICP-MS, Agilent 7700). The measured concentration was normalized by Eq. (3) [6] under the assumption that the sum of the measured Pb and Sn concentrations should be 100 at%
Fig. 1. Schematic illustration of the diffusion experiment (a) set-up, (b) heating and homogenization, (c) diffusion couple formation, (d) diffusion (the symbol ‘‘ the gravity direction.), and (e) separation and cooling.
ɡ” indicates
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Ca
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nor
¼
Ca ; Ca þ Cb
ð3Þ
here, Ci is the concentration of atom i (i = a, b) and Ca_nor is the normalized concentration of atom a. 3. Results Fig. 2a and b show the normalized Pb concentration profiles with the diffusion couples of pure metals and alloys, respectively. Fig. 2 shows the profiles of the four parallel experiments. The results of the four diffusion couples simultaneously overlap in Fig. 2a. On the other hand, the results in Fig. 2b do not overlap because there was a difference in the initial concentration, which is an inevitable error in the experiments. However, the resulting concentration was approximately equal to the target concentration. The concentrations on both ends of the five plots have small dispersion in Fig. 2a and b. Therefore, the concentration of each plot is considered not to change. Furthermore, the concentration profiles were smooth. Fig. 3a and b show the isotopic concentration profiles of 207Pb and 124Sn with diffusion couples of pure metals or alloys, respectively. The concentration change from both ends in the five plots is not confirmed in Fig. 3a or b. The concentration of the enriched
Fig. 3. Concentration profiles of intrinsic diffusion. The symbol ‘‘ ɡ” indicates the gravity direction. The square and triangle show the measured concentrations of 207Pb and 124Sn, respectively. The dark gray line and light gray solid line are the fitting curves: (a) fitting curves of 207Pb and 124Sn were obtained by fitting the superposed equation of the error function with the values shown above; (b) fitting curves of 207Pb and 124Sn were obtained by fitting the equation with the values shown above. The dark gray dashed line and light gray dashed line show the initial concentrations of 207Pb and 124Sn, respectively.
isotopes is high near the intermediate sample. Also, the concentration peak of 207Pb with the diffusion couple of pure metals slightly increases from the natural abundance of the isotope. 4. Discussion In this section, we explain the principles of the methods of analysis of interdiffusion and intrinsic diffusion, and discuss the validity of these analysis methods, the confirmation of suppression of natural convection, and the validity of the Darken’s equation in liquid metals. 4.1. Analysis methods
Fig. 2. Concentration profiles of interdiffusion. The symbol ‘‘ ɡ” indicates the gravity direction. The profiles of the four parallel experiments are shown. At the initial concentration, the sides of the capillaries were filled with the Pb-rich samples up to x = 0.027 m by the reservoir mechanism. (a) The fitting curves of capillaries A, B, C, and D were obtained by fitting the superposed equation of the error function with the values shown above. (b) The fitting curves of capillaries A, B, C, and D were obtained by fitting the superposed equation of the error function with the values shown above.
4.1.1. Analysis methods of interdiffusion To investigate the interdiffusion coefficient, we used two analysis methods. The first method involved fitting an error function. For diffusion between semi-infinite samples (DSIS), the analytical solution of Fick’s second law of diffusion was applied. Derivations of the following equations are shown in Appendix A.
0 C DSISðxÞ
1
Ch þ Cl Ch Cl B x x0 C ¼ erf @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; 2 2 2X 2measure
ð4Þ
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here, Ch and Cl are the initial highest and lowest concentrations of the targeted atom, respectively, and x0 is the central position of diffusion. The parameter X 2measure is the mean square diffusion depth. On the other hand, the concentration profile of thick layer diffusion (TLD) is described as
C TLDðxÞ
8 0 1 0 19 > > < = C0 Cl B x x0 C Bx ðx0 þ HÞC ¼ erf @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA-erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A : > > 2 : 2X 2measure 2X 2measure ;
ð5Þ
C ðxÞ ¼ C DSISðxÞ þ C TLDðxÞ 0
1
ð6Þ
In other words, Eq. (6) describes the diffusion of the intermediate sample with thickness H during diffusion between double semi-infinite samples. In this study, x0 was 0.027 m, and H was determined using the filling ratio of the intermediate cell, calculated by the mass of the samples measured before and after the diffusion experiment. The interdiffusion coefficient obtained by fitting the error function was calculated by the compensation provided by the average effect X 2average and shear convection X 2shear following Eq. (7) [9].
E D E D E 1 D 2 X measure X 2average X 2shear 2t E 1 D 2 X measure 1:99 106 : ¼ 2t
C¼C i
ð8Þ
here, to determine the Matano plane on the order of 104 m, 20 concentration profiles were subdivided into 571 plots by cubic spline interpolation curves. The Matano plane was determined by Eq. (9),
R
xM ¼
C ðxÞ dx : Ch Cl
ð10Þ Eq. (6) describes the diffusion with the initial concentration in less than two steps, as shown in Fig. 3 (see Appendix A). The second method involves fitting the error function to profiles by taking the ratio of the measured atomic and isotopic concentrations in a sample [2]. Eq. (5) was used for fitting. Substituting the square of the mean diffusion depth obtained by both the first and second methods into Eq. (7), the intrinsic diffusion coefficient ~ can be applied to D. was calculated. Then, D The third method involved fitting a Gaussian function. TLD is described by Eq. (11):
C TLDðxÞ
0 1 ðC 0 C l ÞH ðx xv Þ2 A @ E : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E exp D 2 2 X measure 2p X 2measure
ð11Þ
Eq. (11) approximately expresses the diffusion of a sample with thickness H. On the other hand, Eq. (5) describes the diffusion of a sample with H. As with the first analysis method, we assumed that the isotopic concentration profiles consisted of the sum of the diffusion between semi-infinite samples (DSIS) and TLD. Then, Eq. (12) was calculated.
0
ð7Þ
This analysis method does not determine the concentration dependency of the diffusion coefficient. However, the interdiffusion coefficient generally depends on the concentration of the solute. The second method was the Boltzmann–Matano method [10]. The slope of the tangent of the concentration profile at C = Ci and the area are used to obtain the mean square diffusion depth by this method (Eq. (8)).
Z 0 dx ¼ xdC; dc C¼C i C i
C B Ch þ Cl Ch Cl x x0 C rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ erf B D E A @ 2 2 2 X 2measure 8 0 1 19 0 > > > > < C B B x ðx0 þ HÞ C= C0 Ch x x0 C C B B þ -erf @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : erf @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E D E A A > > 2 > > : ; 2 X 2measure 2 X 2measure
C ðxÞ C DSISðxÞ þ C TLDðxÞ
D¼
D E X2
1
0 C ðxÞ
This equation was calculated from Fick’s second law to describe the diffusion of an intermediate sample from both sides into semiinfinite samples, for a thickness H and initial concentration C0. Considering the superposition of fluxes, Eq. (6) was obtained from the sum of Eq. (4) and (5),
Ch þ Cl Ch Cl B x x0 C ¼ erf @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 2X 2measure 8 0 1 0 19 > > < = C0 Cl B x x0 C Bx ðx0 þ HÞC erf @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA-erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A : þ > 2 > : 2X 2measure 2X 2measure ;
isotopic concentration profile of 207Pb. For the analysis of the isotopic concentration profile of 124Sn, the following equation was applied:
ð9Þ
~ obtained by the Boltzmann– The interdiffusion coefficient D Matano method was calculated using Eq. (7). 4.1.2. Analysis methods of intrinsic diffusion To discuss the intrinsic diffusion coefficient, we used three analysis methods. One method was to fit the error function through the method of interdiffusion. Eq. (6) was applied to the analysis of the
¼
1
C B Ch þ Cl Ch Cl x x0 C erf B D EA @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 X measure 0 1 ðC 0 C l ÞH ðx xv Þ2 A @ E þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E exp D 2 2 X measure 2p X 2measure
ð12Þ
Rewriting Eq. (12) gives
C ðxÞ C DSISðxÞ
0 1 ðC 0 C l ÞH ðx xv Þ2 A @ E : ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E exp D 2 2 X measure 2p X 2measure
ð13Þ
Taking the natural logarithm of both sides, the following equation is obtained from
1 ðC 0 C l ÞH E ðx xv Þ2 þ ln rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln C ðxÞ C DSIS ¼ D D E: 2 2 X measure 2p X 2
ð14Þ
measure
In this analysis method, xv is the vertex position of the spline curve of concentration obtained by subtracting CDSIS from the measured concentration. Eq. (14) reveals the linear relationship between ln(CðxÞ -C DSIS ) and ðx xv Þ2 . The intrinsic diffusion coefficient D obtained from fitting the Gaussian function was calculated by Eq. (15)
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Table 2 Interdiffusion coefficients for a molar fraction NSn = 0.5, obtained by fitting error function and Boltzmann–Matano method. determination.
DD ~ : D
relative standard deviation; R2: coefficient of
Analysis method
Type of diffusion couple
~ (109 m2/s) D
DD (109 m2/s)
DD ~ D
Error function
Pure metals Alloys Pure metals Alloys
2.66 2.29 2.57 2.46
0.06 0.14 0.06 0.14
2.07 6.30 2.24 5.72
Boltzmann–Matano
E D E D E D E 1 D 2 X measure X 2average X 2shear X 2initial 2t ! 1 H2 6 2 ; hX i 1:99 10 ¼ 2t 6
D¼
ð15Þ
here, X 2initial is the mean square diffusion depth from the start profile [11]. The reason for using Eq. (15) only in the third analysis method is that Eq. (11) was approximated as Eq. (5). 4.2. Validity of obtained interdiffusion coefficient Table 2 and Fig. 4 show the obtained interdiffusion coefficients. In Fig. 4, the results of the Boltzmann–Matano method show only the range with a significant concentration change.
Fig. 4. Interdiffusion coefficients of the Sn-Pb system. (a) Total system of Sn-Pb. (b) Enlarged view of the range 0.48 < NSn < 0.52. The red circle and blue circle are the calculated interdiffusion coefficients by the superposed equation of the error function. Error bars show the reproducibility of the simultaneous diffusion of the four diffusion couples. The lines are the values calculated by the Boltzmann– Matano method. Pure metals: a diffusion couple of pure metals, Alloys: a diffusion couple of alloys, SC: shear cell technique, LC: long capillary technique, CR: capillary reservoir technique, 1 ɡ: on the ground, lɡ: microgravity.
(%)
R2 0.99972 0.99965 – –
The results calculated by fitting the error function in Table 2 show that fitting curves can be considered to be in good agreement with the concentration profiles since the coefficient of determination R2 is larger than 0.999 for both experiments. R2 is the value of the variance in the dependent variable that is predictable from the independent variable. R2 ranges from 0 to 1. In general, larger R2 indicates a better fit to the data. The results calculated by the Boltzmann–Matano method in Fig. 4a show that the measured interdiffusion coefficients differ from each other despite the same molar fraction from 0.6 to 0.9. The reason is due to the errors in the concentration measurement and analysis. However, the relative standard deviation (RSD) is 5.72% at NSn = 0.5, which is much smaller than the 15% of the previous interdiffusion coefficient measurement [12]. In addition, Fig. 4a shows that the Sn-Pb system has a downward-convex concentration dependency. The tendency of the concentration dependency is consistent with the tendency of the reference data. Furthermore, the results of Boltzmann–Matano method show a tendency to converge with the reference data of the self-diffusion coefficients of Sn and Pb. Therefore, the results from the Boltzmann–Matano method are reasonable. The interdiffusion coefficients obtained by the Boltzmann–Matano method show that the difference between the results of pure metals or alloys is small, as they are within the reproducibility of each experiment (cf. Fig. 4b). As demonstrated above, the use of the error function to measure the interdiffusion coefficient is influenced by the concentration dependency of the diffusion coefficient because the measured interdiffusion coefficient of the diffusion couple of pure metals is larger than the coefficient of the diffusion couple of alloys. This tendency is reasonable, considering the downward
Fig. 5. Profiles of the ratio of the measured atomic and isotopic concentrations C isotope =C atom in the diffusion couple of alloys. The square is the ratio of the measured Pb and 207Pb concentrations in a sample. The triangle is the ratio of the measured Sn and 124Sn concentrations in a sample. The solid lines are fitting curves: fitting curves of 207Pb and 124Sn were obtained by fitting the error function with D207Pb = 3.09 109 m2/s and 3.65 109 m2/s, respectively. The symbol ‘‘ ɡ” indicates the direction of gravity.
H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541
convexity of the concentration dependency revealed by the Boltzmann–Matano method. In short, we think that the interdiffusion coefficient obtained by the Boltzmann–Matano method is the most appropriate value because the influence of the concentration dependency is small.
4.3. Validity of obtained intrinsic diffusion coefficient The intrinsic diffusion coefficients are calculated by fitting the error function. Fig. 3 shows that the fitting agrees well with the concentration profiles since the coefficient of determination R2 is larger than 0.99 in each experiment. Fig. 5 shows the results of using the second analysis method. The fitting curve agrees well with the concentration profiles. Fig. 6 shows the results obtained from the third analysis method. The concentration profiles turn over at the vertex position of the concentration, xv , as shown in Fig. 6. Fig. 6b and d show that the plots are aligned. Table 3 shows the intrinsic diffusion coefficients obtained by the three analysis methods, in which the intrinsic diffusion coefficient calculated by
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fitting the error function differs slightly from the values calculated by the second method using the ratio C isotope =C atom . Table 3 shows that the diffusion coefficient calculated by the Gaussian function in the experiment with the diffusion couple of pure metals differs on the left and right sides with the vertex as the boundary. In other words, there is a concentration dependency of the intrinsic diffusion coefficient in the diffusion couple. On the other hand, there was no significant difference in the diffusion coefficient between the left and right sides of the concentration profile in the experiment with the diffusion couple of alloys. Therefore, we suggest that the measured intrinsic diffusion coefficient with the diffusion couple of alloys should be carefully treated because the influence of the concentration dependency of the intrinsic diffusion coefficient is small in the case of the diffusion couple of alloys. The difference among the intrinsic diffusion coefficients with the diffusion couple of alloys calculated by the three analysis methods is small, as shown in Table 3. Therefore, a reasonable value at NSn = 0.5 is obtained by the method of fitting the error function. For a more accurate measurement of the intrinsic
Fig. 6. Results of intrinsic diffusion by fitting the Gaussian function. The symbol ‘‘ ɡ” indicates the direction of gravity. The dark and light gray plots show the results of pure metals and alloys, respectively. (a) Profiles of 207Pb obtained by subtracting the concentration of diffusion between semi-infinite samples (DSIS) from the measured concentration. (b) Profiles of 207Pb obtained by the natural logarithm of the subtracted value. (c) Profiles of 124Sn obtained by subtracting the concentration of diffusion between semi-infinite samples (DSIS) from the measured concentration. (d) Profiles of 124Sn obtained by the natural logarithm of the subtracted value. The dash-dotted lines show vertex position xv of the spline curve of the measured value. Parts (c) and (d) show only the range where the concentration change is confirmed.
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Table 3 Intrinsic diffusion coefficients at molar fraction NSn = 0.5 obtained by three analysis methods. Analysis method Error function
Type of diffusion couple
Type of isotope
D (109 m2/s)
Coefficient of determination R2
Pure metals
207
2.68 2.43 3.04 3.44 * a * a 3.09 3.65 1.98 4.66 2.68 3.51 2.93 2.76 3.54 3.67
0.99882 0.99494 0.99926 0.99097 * a * a 0.97237 0.9948 0.99646 0.99992 0.99694 0.99669 0.9996 0.99712 0.99883 0.99609
Pb Sn Pb 124 Sn 207 Pb 124 Sn 207 Pb 124 Sn 207 Pb 124 207
Alloys Fitting to the ratio C isotope =C atom
Pure metals Alloys
Gaussian function
Pure metals
x < xv x > xv x < xv x > xv x < xv x > xv x < xv x > xv
124
Sn
207
Alloys
Pb
124
Sn
*
a: analysis is not possible.
diffusion coefficient, an analysis method considering the concentration dependency of the intrinsic diffusion coefficient should be considered. 4.4. Confirmation of suppression of natural convection Fig. 6b and d reveal that the influence of natural convection was small, given that the concentration of 207Pb and 124Sn was in good agreement with Eq. (14), in the form of a linear relationship. The linearity is a prerequisite for suppressing natural convection. As shown in Table 3, the intrinsic diffusion coefficient calculated by the Gaussian function was larger in the region x > xv than in x < xv for pure metals. In other words, the results show that Pb atoms were transferred in a direction opposite to the direction of gravity. As Pb has a larger density than Sn, the results indicate that there is no gravity-driven mass transport, that is, convection in pure metals. On the other hand, the correlation of alloys was inverted with that of pure metals in Table 3. The reasons for the correlation could not be affirmed. For example, measurement error or the influence of natural convection can be considered. However, in this research, a stable density layering, which is effective in the suppression of natural convection, was set. Therefore, natural convection is also considered to be suppressed in the diffusion experiment of alloys. 4.5. Validity of the Darken’s equation in liquid metals We verified the Darken’s equation at NSn = 0.50. Table 4 shows the theoretical values of the interdiffusion coefficient calculated by the Darken’s equation using the measured intrinsic diffusion coefficient and the value calculated by the Boltzmann–Matano method. The measured values with the diffusion couple of pure metals show good agreement with theoretical values calculated by Eq. (1). However, considering the concentration dependency of the
intrinsic diffusion coefficient, we surmise that the results with the diffusion couple of alloys should be used for calculation and comparison. The relative error was 24.1%, as shown in Table 4. The differences are not very large between the theoretical value calculated by Eq. (2) and the measured values with the diffusion couple of pure metals and alloys. In addition, the concentration dependency of the diffusion coefficient of theoretical values shows a downward-convex tendency. The tendency corresponds with the results calculated by the Boltzmann–Matano method. The reason why the measured and calculated values do not correspond should be the error of substituted values. As error sources, D (or D⁄), U; ~ can be considered. However, D ~ is eliminated from the error and D ~ calculated by the Boltzsource because the correspondence of D mann–Matano method with diffusion couples of pure metals or alloys was confirmed at NSn = 0.5. Therefore, the error sources would be D, D⁄, and/or U. 4.6. Summary of experimental and analysis methods Table 5 shows the features of each experimental method. It is evident that using stable density layering with a diffusion couple of pure metals would be preferable to using it with a diffusion couple of alloys.
Table 5 Features of each experimental method.
Interdiffusion Intrinsic diffusion
Type of diffusion couple
Possibility of stable density layering
Pure metals Alloys Pure metals Alloys
A A (limit) A A (limit)
A: available, limit: only under the condition of stable density layering during diffusion time.
Table 4 Comparison between theoretical interdiffusion coefficient calculated by the Darken’s equation and the value calculated by the Boltzmann–Matano method at a molar fraction NSn = 0.50. The intrinsic diffusion coefficients are the measured values in this study. The self-diffusion coefficients are reference data [13,14]. The thermodynamic factor was calculated by a quasi-chemical model [15]. Type of diffusion couple
Theoretical value (109 m2/s) ~ eq:ð1Þ D
~ eq:ð2Þ D
Measured value (109 m2/s) ~ meas D
Relative error (%) Dmeasure D eq:ð1Þ
Dmeasure D eq:ð2Þ
Deq:ð1Þ
Pure metals Alloys
2.56 3.24
2.95
2.57 2.46
0.391 24.1
Deq:ð2Þ
12.9 16.6
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H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541 Table 6 Features of each analysis method.
Interdiffusion Intrinsic diffusion
Analysis method
Type of diffusion couple
Possibility of analysis
Consideration of concentration dependence
Simple calculation process
Validity of calculated value
Superposed equation of error function Boltzmann–Matano Superposed equation of error function Fitting to the ratio C isotope =C atom
Pure metals Pure metals Pure metals Pure metals Alloys Pure metals
A A A N/A A A
N/A A N/A ─ N/A A (limit)
A N/A A ─ N/A N/A
A (limit) A A ─ A A
Gauss function
or Alloys or Alloys or Alloys
or Alloys
A: available, N/A: not available, limit: only under the condition of stable density layering during diffusion.
Table 6 shows the features of each analysis method. It is possible to calculate the interdiffusion coefficients by two methods. The analysis method using the Boltzmann–Matano method is better than the analysis method using the superposed equation of the error function because the latter is influenced by the concentration dependency of the diffusion coefficient. It is possible to calculate the intrinsic diffusion coefficients by the three methods. However, the analysis method of fitting to the ratio has applicability limits. On the other hand, the analysis method with the superposed equation of error function is convenient due to the simplicity of the calculation and no limits.
Aerospace Exploration Agency (JAXA) ‘‘Diffusion Phenomena in Melts” Working Group. We thank Kimura Foundry Co., Ltd. for financial support. Appendix A A.1. Derivation of superposed equation of error function The concentration profile of the diffusion between semi-infinite samples (DSIS) is described by the following Eq. (A1), as shown in Fig. A1.
0
5. Conclusions
C DSISðxÞ We performed experiments and analyses and clarified the following: A simultaneous measuring method of interdiffusion and intrinsic diffusion coefficients in liquid metals was established. Simultaneous measurements in the Sn-Pb system were conducted on the ground. The dispersion in concentration was suppressed by the method of making the concentration of the sample uniform before starting diffusion time. The calculated value with the method of fitting the error function is influenced by the concentration dependency of the diffusion coefficient rather than the average concentration. On the other hand, at a certain concentration, the interdiffusion coefficients can be calculated for different concentrations while eliminating other influences by the Boltzmann–Matano method. The intrinsic diffusion coefficients were calculated by fitting the superposed equation of error function. Then, the experimental method using the diffusion couple of alloys and the homogenization technique is more suitable. The applicability of the Darken’s equation was examined using the measured diffusion coefficient. The results showed that the difference between the theoretical and measured interdiffusion coefficients is not very large. Therefore, the possibility of adapting the Darken’s equation in liquid metals was confirmed in this experiment. However, further experiments are necessary to determine the detailed conditions for the adaptation of the Darken’s equation. The error factor would be intrinsic diffusion or self-diffusion coefficients and/or the thermodynamic factor.
1
B x x0 C Ch þ Cl Ch Cl C ¼ erf B D EA @rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 X
ðA1Þ
The concentration profile of thick layer diffusion (TLD) is described by the following Eq. (A2).
C TLDðxÞ
8 9 Z b < ðx dÞ2 = C0 Cl ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp D E dd D Effi : 2 X2 ; a 2p X 2 8 0 1 0 19 > > > > < B xa C B x b C= C0 Cl B C B C r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi erf erf @rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ D E D E A @ A>: 2 > > > : ; 2 X2 2 X2
ðA2Þ
Here, if a = x0 and b = x0 + H, rewriting Eq. (A2) gives
Conflict of interest The authors declared that there is no conflict of interest. Acknowledgments We sincerely would like to express our all gratitude to staff members of Environmental Safety Center, Waseda University for sample analyses. This study was supported by grant-in-aid from Mitsubishi Materials Corporation in fiscal year 2017 and the Japan
Fig. A1. Concentration profile of diffusion between semi-infinite samples.
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H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541
C TLDðxÞ
8 0 1 19 0 > > > > < B x x0 C Bx ðx0 þ HÞC= C0 Cl C C B B erf @ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ erf @rffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E D E A A > 2 > > > : ; 2 X2 2 X2
ðA3Þ
This equation is shown in Fig. A2. Considering the superposition of fluxes, Eq. (A4) was obtained by the sum of Eqs. (A1) and (A3).
0 C ðxÞ ¼
1
B x x0 C Ch þ Cl Ch Cl C erf B D EA @rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 X2 8 0 1 19 0 > > > > < B x x0 C Bx ðx0 þ HÞC= C0 Cl C C B B r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi erf þ erf @rffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E D E A A> @ 2 > > > : ; 2 X2 2 X2
ðA4Þ
The number of steps changes according to C0, as shown in Fig. A3. Appendix B. Supplementary material Fig. A2. Concentration profile of thick layer diffusion.
Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2018.12.036.
Fig. A3. Schematic concentration profiles. All profiles have two or less steps. (a) C0 = Cl, (b) Cl < C0 < Ch, (c) C0 = Ch, (d) Ch < C0.
H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541
References [1] L.S. Darken, Diffusion mobility and their interrelation through free energy in binary metallic systems, Trans. AIME 175 (1948) 184–201. [2] K.-H. Kraatz, G. Frohberg, S. Suzuki, Final Report of ‘‘Refurbishment der AGAT Hardware für die Mission FOTON M2. Präparation und Durchführung von 6 Scherzellenexperimenten in metallischen Schmelzen (50WM0348). Deutsches Zentrum für Luft- und Raumfahrt, Germany, pp. 1–18. [3] M. Klassen, J.R. Cahoon, Interdiffusion of Sn and Pb in Liquid Pb-Sn alloys, Metall. Mater. Trans. A 31A (2000) 1343–1352. [4] G. Frohberg, K.H. Kraatz, H. Wever, Selfdiffusion of Sn112 and Sn124 in Liquid Tin, ESA-SP-222, 1984, pp. 201–205. [5] S. Suzuki, K.-H. Kraatz, G. Frohberg, Ann. New York Acad. Sci. 1027 (2004) 169– 181. [6] K. Suzuki, N. Yamada, A. Tanaka, S. Suzuki, Improvement of a measurement method of interdiffusion coefficient of liquid alloys by using the Foton shear cell and verification of Darken’s equation in liquid alloys, Trans. Japan Soc. Aeronaut. Space Sci., Aerospace Technol. Japan 12(ists29) (2014) Ph19– Ph24. [7] W. Gasior, Z. Moser, J. Pstrus, Density and surface tension of the Pb-Sn liquid alloys, J. Phase Equilibria 22 (1) (2001) 20–25.
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[8] K. Tanimoto, Y. Ando, K. Suzuki, Y. Yokoyama, S. Suzuki, Measurement of interdiffusion coefficients in liquid AuGe alloy, Microgravity Sci. Technol. 27 (5) (2015) 345–351. [9] S. Suzuki, K.-H. Kraatz, G. Frohberg, The effect of shear convection on diffusion measurements in liquid metals using the Foton shear cell, microgravity science and technology, XVIII-3/4 (2006) 155–159. [10] C. Matano, On the relation between the diffusion-coefficients and concentrations of solid metals (the nickel-copper system), J. Phys. Soc. Japan 8 (3) (1933) 109–113. [11] S. Suzuki, K.-H. Kraatz, G. Frohberg, Reduction of convection in diffusion measurement using the shear cell by stabilization of density layering on the ground, Japan Soc. Microgravity Appl. 28 (2) (2011) s100–s104. [12] J. Cahoon, Y. Jiao, K. Tandon, M. Chaturvedi, Interdiffusion in liquid tin, J. Basic Appl. Res. 27 (4) (2006) 325–332. [13] G. Mathiak, A. Griesche, K.H. Kraatz, G. Frohberg, Diffusion in liquid metals, J. Non-Crystalline Solids 205–207 (1996) 412–416. [14] M. Shiinoki, N. Hashimoto, H. Fukuda, Y. Ando, S. Suzuki, Self-diffusion measurements of liquid Sn using the shear cell technique and stable density layering, Metall. Mater. Trans. B 49 (6) (2018) 3357–3366. [15] R.N. Singh, Short-range order and concentration fluctuations in binary molten alloys, Canadian J. Phys. 65 (3) (1987) 309–325.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Simultaneous measurement of interdiffusion and intrinsic diffusion coefficients in liquid metals on the ground Hideto Fukuda a, Masato Shiinoki a,⇑, Yuki Nishimura a, Shinsuke Suzuki b a
Department of Applied Mechanics, Faculty of Science and Engineering, Waseda University, Okubo 3-4-1 Shinjuku-ku, 169-8555 Tokyo, Japan Kagami Memorial Research Institute of Materials Science and Technology, Department of Applied Mechanics and Aerospace Engineering, Faculty of Science and Engineering, Waseda University, Okubo 3-4-1 Shinjuku-ku, 169-8555 Tokyo, Japan b
a r t i c l e
i n f o
Article history: Received 7 August 2018 Received in revised form 26 October 2018 Accepted 6 December 2018
Keywords: Diffusion coefficient Liquid metal Interdiffusion Intrinsic diffusion Boltzmann-Matano method Darken’s equation
a b s t r a c t Simultaneous measurements of interdiffusion and intrinsic diffusion coefficients in liquid Sn-Pb were performed using the shear cell method and a stable density layering. The binary diffusion couples of Sn-Pb and Sn0.6Pb0.4-Sn0.4Pb0.6 were used in two experiments with the average molar fraction of diffusion couples NSn = 0.5. The concentration dependency of the interdiffusion coefficient was confirmed to exhibit a downward convexity through the comparison of the interdiffusion coefficients. However, the Boltzmann–Matano method reveals a diminished influence. The sample measurements of the intrinsic diffusion coefficients of a diffusion couple of pure metals demonstrate a dependency on concentration. On the other hand, the difference of diffusion coefficients was small in a diffusion couple of alloys. The difference between the values calculated by fitting the error function based on the superposition of fluxes and by the conventional analysis methods was small. We propose that a reasonable intrinsic diffusion coefficient can be calculated using the superposed equation of error function. The theoretical interdiffusion coefficients were calculated by substituting the measured intrinsic diffusion coefficients into the Darken’s equation and comparing the results with the measured values. As a result, the difference between the theoretical and measured interdiffusion coefficients is not very large. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Diffusion coefficient is one of the important physical properties in understanding diffusion phenomena. In particular, the interdiffusion coefficient in liquid metals is very important in the discus~ can be sion of diffusion in alloys. The interdiffusion coefficient D expressed theoretically by the Darken’s equation [1], expressed by the following equations:
D ¼ Da Nb þ Db Na U;
ð1Þ
D ¼ Da Nb þ Db Na Di ¼ Di U; i ¼ a or b ;
ð2Þ
where Di is the self-diffusion coefficient, Di is the intrinsic diffusion coefficient, Ni is the molar fraction, and U is the thermodynamic factor. However, the validity of the Darken’s equation in liquid metals has not been confirmed by experiments. A method to verify its ⇑ Corresponding author. E-mail addresses: [email protected] (H. Fukuda), stag7211@asagi. waseda.jp (M. Shiinoki), [email protected] (Y. Nishimura), suzuki-s@ waseda.jp (S. Suzuki). https://doi.org/10.1016/j.ijheatmasstransfer.2018.12.036 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
validity compares the measured value of the interdiffusion coefficient and the theoretical value calculated by Eq. (1). Another method simultaneously measures the interdiffusion and the intrinsic diffusion coefficient and compares them with Eq. (2) [2]. The isotope is used in the measurement of the intrinsic diffusion coefficient. There are at least two problems with measuring the diffusion coefficient in liquid metals on the ground. One problem is that it is unclear whether the error derived from the experimental method was appropriately corrected. Klassen et al. measured the interdiffusion coefficients of the Sn-Pb system and researched the concentration dependency of the interdiffusion coefficient [3]. The Sn-Pb system was selected for two reasons: first, the selfdiffusion coefficients of both Sn and Pb had already been investigated. Next, the molten Sn-Pb alloys could be prepared and handled with relative ease. They concluded that the diffusion coefficients may be measured with higher accuracy using the shear cell method rather than the capillary reservoir method, which they used. In addition, their results showed discontinuity in the diffusion coefficient with eutectic composition, although they did not mention it in the paper. The other problem is that natural convection might occur. It was found that the microgravity environment
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H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541
Nomenclature C Ch Cl C0 Cisotope Catom CDSIS CTLD ~ D
D D D207Pb D124Sn DD H N NSn
concentration (at%) initial higher concentration (at%) (variable parameter) initial lower concentration (at%) (variable parameter) initial concentration of intermediate sample (at%) (variable parameter) concentration of isotope (at%) concentration of element (at%) concentration that expressed diffusion between semiinfinite samples (at%) concentration that expressed thick layer diffusion (at%) interdiffusion coefficient (m2/s) self-diffusion coefficient (m2/s) intrinsic diffusion coefficient (m2/s) intrinsic diffusion coefficient of 207Pb (m2/s) intrinsic diffusion coefficient of 124Sn (m2/s) standard deviation of diffusion coefficient (m2/s) thickness of the intermediate sample (m) (fixed parameter) molar fraction molar fraction of Sn
is effective in suppressing natural convection [4]. However, diffusion experiments in microgravity have some problems, e.g., high cost, long preparation time, limitation of opportunities, and high risks during launch and recovery. Thus, the establishment of ground-based experimental methods for diffusion measurements is strongly required. The method of combining the shear cell method and the stable density layering, in which the sample is placed so that the density of the samples monotonically increases in the gravity direction, is valid for measuring the diffusion coefficient on the ground [5]. However, there have been few reports on interdiffusion experiments using the combined method. Therefore, it is necessary to perform interdiffusion experiments using this method. Suzuki et al. measured the interdiffusion coefficients of the Sn-Pb system for NSn = 0.05 and 0.95 using this method [6]. However, the diffusion coefficients at higher solute concentrations have not been reported. In addition, the resulting concentration profiles have large variations. Therefore, it is necessary to obtain concentration profiles with small variations for each diffusion experiment. This can be performed using the shear cell method with high solute concentration, e.g., NSn = 0.5. To reduce the variation in the concentration profiles, pure metals are used as a diffusion couple. Moreover, when the density of the samples monotonically varies with composition, such as in the Sn-Pb system [7], stable density layering is easily created by this technique because the couple has the maximum density difference in the system. On the other hand, the measured values are considered to be influenced by the concentration dependency of the diffusion coefficient. However, these influences have not been confirmed by experiments. Kraatz et al. simultaneously measured the interdiffusion and the intrinsic diffusion coefficients by taking the ratio of the measured atomic and isotopic concentrations in a sample [2]. However, the ratio is not applicable to a diffusion couple of two pure metals, which is expected to exhibit suppression of convection, as mentioned above. In this case, arithmetic calculation cannot be performed by this analysis method. As a result, the intrinsic diffusion coefficients cannot be calculated. The purpose of this study is to establish a simultaneous measurement method of interdiffusion and intrinsic diffusion coefficients in liquid metals. The validity of the experimental and analytical methods was investigated. Interdiffusion and intrinsic
NPb P R2 t T x x0 xDv E
molar fraction of Pb filling pressure of samples (kPa) coefficient of determination diffusion time (s) diffusion temperature (K) position (m) central position (m) (fixed parameter) vertex position (m)
X2
mean square diffusion depth (m2) E X 2measure mean square diffusion depth of measurement (m2) D E (variable parameter) X 2average additional mean square diffusion depth from concen2 D E tration averaging (m ) X 2shear additional mean square diffusion depth from shear2 D E induced convection (m ) X 2initial additional mean square diffusion depth from initial condition (m2) q density (kg/m3) U thermodynamic factor D
diffusion experiments were conducted using the Foton shear cell [5] and stable density layering on the ground. The validity of the experimental method was examined by the concentration error in the concentration profiles, regardless of the occurrence of natural convection, and the influence of the concentration dependency on the measured value. Furthermore, the validity of a new analytical method based on superposition of fluxes for diffusion coefficient calculation was investigated. Finally, the validity of the Darken’s equation was examined by comparing measured values and theoretical values of the Darken’s equation at NS = 0.50.
2. Experimental procedure 2.1. Sample preparation In this study, the molar fraction of Sn, NSn, representing the diffusion couple was defined as the average composition of each sample constituting the diffusion couple. For NSn = 0.5, two experiments were performed in which the compositions of the diffusion couples were changed. In the first experiment, the diffusion couple was constituted by pure Sn, Sn0.5Pb0.5, and pure Pb (this is called ‘‘a diffusion couple of pure metals”). In the second experiment, the diffusion couple was constituted by Sn0.6Pb0.4, Sn0.5Pb0.5, and Sn0.4Pb0.6 (this is called ‘‘a diffusion couple of alloys”). The composition of the intermediate sample was determined to be the average composition of the samples on both sides so that the central position of interdiffusion corresponds to one of intrinsic diffusion in the concentration profile. Pure Sn (99.999 wt%) and pure Pb (99.998 wt%) produced by Kojundo Chemical Laboratory Co., Ltd. were used to prepare samples. Pure Sn, isotope 124Sn (99.90 at%), and isotope 207Pb (99.10 at%) produced by ISOFLEX USA, Inc., as well as common lead isotopic standard SRM 981 (concentration as shown in Table 1) produced by National Institute of Standards and Technology, were used for the preparation of the intermediate samples. The concentration of 207Pb and 124Sn were adjusted to 35 at% and 20 at%, respectively. The adjusted samples were set in a crucible and held at 1073 K in vacuum atmosphere for 900 s before cooling. The samples were constantly vibrated during cooling so that segregation was suppressed. The solidified samples were cut and shaped into a cylindrical shape through drawing dies.
H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541 Table 1 Certified isotopic concentration of SRM981. Type of isotope
Certified isotopic concentration (at%)
204
1.4255 0.0012 24.1442 0.0057 22.0833 0.0027 52.3470 0.0086
Pb Pb Pb 208 Pb 206 207
2.2. Diffusion experiment and concentration analysis Fig. 1 shows a schematic illustration of the diffusion experiment using the Foton shear cell [5] and a stable density layering. The shear cell unit consists of 20 graphite disks of 3 mm thickness and reservoirs. Each disk had four capillaries of 1.5 mm in diameter. The capillaries, named A, B, C and D, were located 5 mm away from the center of the axis. The reservoirs were equipped on both ends of the capillary so that these filled the capillary with the samples under pressure P = 20 kPa. The diffusion couple of pure metals consisted of pure Pb with a length of 27 mm, an intermediate sample Sn0.5Pb0.5 with length H, and pure Sn with a length of 30 mm. The diffusion couple of alloys consisted of Sn0.4Pb0.6 with a length of 27 mm, an intermediate sample Sn0.5Pb0.5 with length H, and Sn0.6Pb0.4 with a height of 30 mm. The mass of the samples was measured before the diffusion experiments. In each experiment, intermediate samples with enriched isotopes were used in only one of the four capillaries. The compositions of the intermediate sample in the remaining three capillaries were the isotope natural abundance. To suppress natural convection, each sample was set in a stable density layering (cf. Fig. 1a). As shown in Fig. 1b, the couples were assembled with the intermediate sample separated from the diffusion axis. To prevent oxidation of the samples, the air in the vacuum chamber was replaced three times with Ar gas after evacuation. The air continued the evacuation after third replacement. The diffusion couples of alloys were set vertical to the gravity direction,
533
heated to 773 K, and then homogenized for 1.5 h. After that, the diffusion couples of alloys were reset parallel to the direction of gravity and homogenized for 3 h. Since it was necessary to use the diffusion couples of pure metals, the diffusion couples were set parallel to the direction of gravity from the beginning, heated to 773 K, and homogenized for 3 h. Tanimoto et al. also set diffusion couples vertically in the direction of gravity during homogenization and then reset them in parallel. They conducted this technique to fill the samples in the capillary [8]. However, the aim of this technique was not only to fill the sample in the capillary but also to homogenize the alloy sample in this study. In both experiments, intermediate samples were inserted into the diffusion axis by a rotation with a motor at a speed of 0.5 mm/s (0.1 rad/s) after homogenization, and as a result, diffusion couples were formed (cf. Fig. 1c). The diffusion temperature T and diffusion time t were 773 K and 3600 s, respectively (cf. Fig. 1d). The atmosphere was 1–2 Pa during both experiments. The temperature was measured by thermocouples at three points along the axis of capillaries, that is, close to both ends of the capillary and the middle position. At that time, the temperature gradient was 40 K/m. At the end of the diffusion period, diffusion couples were divided into 20 pieces by the rotation with the motor again and cooling (cf. Fig. 1e). By the shear cell method, errors due to mass transport and volume change were eliminated during heating and cooling. The mass of each piece was also measured after the diffusion experiments. The pieces were dissolved in a mixture composed of nitric acid, hydrochloric acid, and tartaric acid. The concentrations of Pb and Sn in each piece were analyzed by inductively coupled plasma - optical emission spectrometry (ICP-OES, Agilent 5100). The isotope concentrations of 204Pb, 206Pb, 207Pb, 208Pb, and 124Sn were analyzed by inductively coupled plasma - mass spectroscopy (ICP-MS, Agilent 7700). The measured concentration was normalized by Eq. (3) [6] under the assumption that the sum of the measured Pb and Sn concentrations should be 100 at%
Fig. 1. Schematic illustration of the diffusion experiment (a) set-up, (b) heating and homogenization, (c) diffusion couple formation, (d) diffusion (the symbol ‘‘ the gravity direction.), and (e) separation and cooling.
ɡ” indicates
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Ca
H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541
nor
¼
Ca ; Ca þ Cb
ð3Þ
here, Ci is the concentration of atom i (i = a, b) and Ca_nor is the normalized concentration of atom a. 3. Results Fig. 2a and b show the normalized Pb concentration profiles with the diffusion couples of pure metals and alloys, respectively. Fig. 2 shows the profiles of the four parallel experiments. The results of the four diffusion couples simultaneously overlap in Fig. 2a. On the other hand, the results in Fig. 2b do not overlap because there was a difference in the initial concentration, which is an inevitable error in the experiments. However, the resulting concentration was approximately equal to the target concentration. The concentrations on both ends of the five plots have small dispersion in Fig. 2a and b. Therefore, the concentration of each plot is considered not to change. Furthermore, the concentration profiles were smooth. Fig. 3a and b show the isotopic concentration profiles of 207Pb and 124Sn with diffusion couples of pure metals or alloys, respectively. The concentration change from both ends in the five plots is not confirmed in Fig. 3a or b. The concentration of the enriched
Fig. 3. Concentration profiles of intrinsic diffusion. The symbol ‘‘ ɡ” indicates the gravity direction. The square and triangle show the measured concentrations of 207Pb and 124Sn, respectively. The dark gray line and light gray solid line are the fitting curves: (a) fitting curves of 207Pb and 124Sn were obtained by fitting the superposed equation of the error function with the values shown above; (b) fitting curves of 207Pb and 124Sn were obtained by fitting the equation with the values shown above. The dark gray dashed line and light gray dashed line show the initial concentrations of 207Pb and 124Sn, respectively.
isotopes is high near the intermediate sample. Also, the concentration peak of 207Pb with the diffusion couple of pure metals slightly increases from the natural abundance of the isotope. 4. Discussion In this section, we explain the principles of the methods of analysis of interdiffusion and intrinsic diffusion, and discuss the validity of these analysis methods, the confirmation of suppression of natural convection, and the validity of the Darken’s equation in liquid metals. 4.1. Analysis methods
Fig. 2. Concentration profiles of interdiffusion. The symbol ‘‘ ɡ” indicates the gravity direction. The profiles of the four parallel experiments are shown. At the initial concentration, the sides of the capillaries were filled with the Pb-rich samples up to x = 0.027 m by the reservoir mechanism. (a) The fitting curves of capillaries A, B, C, and D were obtained by fitting the superposed equation of the error function with the values shown above. (b) The fitting curves of capillaries A, B, C, and D were obtained by fitting the superposed equation of the error function with the values shown above.
4.1.1. Analysis methods of interdiffusion To investigate the interdiffusion coefficient, we used two analysis methods. The first method involved fitting an error function. For diffusion between semi-infinite samples (DSIS), the analytical solution of Fick’s second law of diffusion was applied. Derivations of the following equations are shown in Appendix A.
0 C DSISðxÞ
1
Ch þ Cl Ch Cl B x x0 C ¼ erf @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA; 2 2 2X 2measure
ð4Þ
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H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541
here, Ch and Cl are the initial highest and lowest concentrations of the targeted atom, respectively, and x0 is the central position of diffusion. The parameter X 2measure is the mean square diffusion depth. On the other hand, the concentration profile of thick layer diffusion (TLD) is described as
C TLDðxÞ
8 0 1 0 19 > > < = C0 Cl B x x0 C Bx ðx0 þ HÞC ¼ erf @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA-erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A : > > 2 : 2X 2measure 2X 2measure ;
ð5Þ
C ðxÞ ¼ C DSISðxÞ þ C TLDðxÞ 0
1
ð6Þ
In other words, Eq. (6) describes the diffusion of the intermediate sample with thickness H during diffusion between double semi-infinite samples. In this study, x0 was 0.027 m, and H was determined using the filling ratio of the intermediate cell, calculated by the mass of the samples measured before and after the diffusion experiment. The interdiffusion coefficient obtained by fitting the error function was calculated by the compensation provided by the average effect X 2average and shear convection X 2shear following Eq. (7) [9].
E D E D E 1 D 2 X measure X 2average X 2shear 2t E 1 D 2 X measure 1:99 106 : ¼ 2t
C¼C i
ð8Þ
here, to determine the Matano plane on the order of 104 m, 20 concentration profiles were subdivided into 571 plots by cubic spline interpolation curves. The Matano plane was determined by Eq. (9),
R
xM ¼
C ðxÞ dx : Ch Cl
ð10Þ Eq. (6) describes the diffusion with the initial concentration in less than two steps, as shown in Fig. 3 (see Appendix A). The second method involves fitting the error function to profiles by taking the ratio of the measured atomic and isotopic concentrations in a sample [2]. Eq. (5) was used for fitting. Substituting the square of the mean diffusion depth obtained by both the first and second methods into Eq. (7), the intrinsic diffusion coefficient ~ can be applied to D. was calculated. Then, D The third method involved fitting a Gaussian function. TLD is described by Eq. (11):
C TLDðxÞ
0 1 ðC 0 C l ÞH ðx xv Þ2 A @ E : rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E exp D 2 2 X measure 2p X 2measure
ð11Þ
Eq. (11) approximately expresses the diffusion of a sample with thickness H. On the other hand, Eq. (5) describes the diffusion of a sample with H. As with the first analysis method, we assumed that the isotopic concentration profiles consisted of the sum of the diffusion between semi-infinite samples (DSIS) and TLD. Then, Eq. (12) was calculated.
0
ð7Þ
This analysis method does not determine the concentration dependency of the diffusion coefficient. However, the interdiffusion coefficient generally depends on the concentration of the solute. The second method was the Boltzmann–Matano method [10]. The slope of the tangent of the concentration profile at C = Ci and the area are used to obtain the mean square diffusion depth by this method (Eq. (8)).
Z 0 dx ¼ xdC; dc C¼C i C i
C B Ch þ Cl Ch Cl x x0 C rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ þ erf B D E A @ 2 2 2 X 2measure 8 0 1 19 0 > > > > < C B B x ðx0 þ HÞ C= C0 Ch x x0 C C B B þ -erf @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : erf @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E D E A A > > 2 > > : ; 2 X 2measure 2 X 2measure
C ðxÞ C DSISðxÞ þ C TLDðxÞ
D¼
D E X2
1
0 C ðxÞ
This equation was calculated from Fick’s second law to describe the diffusion of an intermediate sample from both sides into semiinfinite samples, for a thickness H and initial concentration C0. Considering the superposition of fluxes, Eq. (6) was obtained from the sum of Eq. (4) and (5),
Ch þ Cl Ch Cl B x x0 C ¼ erf @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 2 2 2X 2measure 8 0 1 0 19 > > < = C0 Cl B x x0 C Bx ðx0 þ HÞC erf @qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA-erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A : þ > 2 > : 2X 2measure 2X 2measure ;
isotopic concentration profile of 207Pb. For the analysis of the isotopic concentration profile of 124Sn, the following equation was applied:
ð9Þ
~ obtained by the Boltzmann– The interdiffusion coefficient D Matano method was calculated using Eq. (7). 4.1.2. Analysis methods of intrinsic diffusion To discuss the intrinsic diffusion coefficient, we used three analysis methods. One method was to fit the error function through the method of interdiffusion. Eq. (6) was applied to the analysis of the
¼
1
C B Ch þ Cl Ch Cl x x0 C erf B D EA @rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 X measure 0 1 ðC 0 C l ÞH ðx xv Þ2 A @ E þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E exp D 2 2 X measure 2p X 2measure
ð12Þ
Rewriting Eq. (12) gives
C ðxÞ C DSISðxÞ
0 1 ðC 0 C l ÞH ðx xv Þ2 A @ E : ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E exp D 2 2 X measure 2p X 2measure
ð13Þ
Taking the natural logarithm of both sides, the following equation is obtained from
1 ðC 0 C l ÞH E ðx xv Þ2 þ ln rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln C ðxÞ C DSIS ¼ D D E: 2 2 X measure 2p X 2
ð14Þ
measure
In this analysis method, xv is the vertex position of the spline curve of concentration obtained by subtracting CDSIS from the measured concentration. Eq. (14) reveals the linear relationship between ln(CðxÞ -C DSIS ) and ðx xv Þ2 . The intrinsic diffusion coefficient D obtained from fitting the Gaussian function was calculated by Eq. (15)
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Table 2 Interdiffusion coefficients for a molar fraction NSn = 0.5, obtained by fitting error function and Boltzmann–Matano method. determination.
DD ~ : D
relative standard deviation; R2: coefficient of
Analysis method
Type of diffusion couple
~ (109 m2/s) D
DD (109 m2/s)
DD ~ D
Error function
Pure metals Alloys Pure metals Alloys
2.66 2.29 2.57 2.46
0.06 0.14 0.06 0.14
2.07 6.30 2.24 5.72
Boltzmann–Matano
E D E D E D E 1 D 2 X measure X 2average X 2shear X 2initial 2t ! 1 H2 6 2 ; hX i 1:99 10 ¼ 2t 6
D¼
ð15Þ
here, X 2initial is the mean square diffusion depth from the start profile [11]. The reason for using Eq. (15) only in the third analysis method is that Eq. (11) was approximated as Eq. (5). 4.2. Validity of obtained interdiffusion coefficient Table 2 and Fig. 4 show the obtained interdiffusion coefficients. In Fig. 4, the results of the Boltzmann–Matano method show only the range with a significant concentration change.
Fig. 4. Interdiffusion coefficients of the Sn-Pb system. (a) Total system of Sn-Pb. (b) Enlarged view of the range 0.48 < NSn < 0.52. The red circle and blue circle are the calculated interdiffusion coefficients by the superposed equation of the error function. Error bars show the reproducibility of the simultaneous diffusion of the four diffusion couples. The lines are the values calculated by the Boltzmann– Matano method. Pure metals: a diffusion couple of pure metals, Alloys: a diffusion couple of alloys, SC: shear cell technique, LC: long capillary technique, CR: capillary reservoir technique, 1 ɡ: on the ground, lɡ: microgravity.
(%)
R2 0.99972 0.99965 – –
The results calculated by fitting the error function in Table 2 show that fitting curves can be considered to be in good agreement with the concentration profiles since the coefficient of determination R2 is larger than 0.999 for both experiments. R2 is the value of the variance in the dependent variable that is predictable from the independent variable. R2 ranges from 0 to 1. In general, larger R2 indicates a better fit to the data. The results calculated by the Boltzmann–Matano method in Fig. 4a show that the measured interdiffusion coefficients differ from each other despite the same molar fraction from 0.6 to 0.9. The reason is due to the errors in the concentration measurement and analysis. However, the relative standard deviation (RSD) is 5.72% at NSn = 0.5, which is much smaller than the 15% of the previous interdiffusion coefficient measurement [12]. In addition, Fig. 4a shows that the Sn-Pb system has a downward-convex concentration dependency. The tendency of the concentration dependency is consistent with the tendency of the reference data. Furthermore, the results of Boltzmann–Matano method show a tendency to converge with the reference data of the self-diffusion coefficients of Sn and Pb. Therefore, the results from the Boltzmann–Matano method are reasonable. The interdiffusion coefficients obtained by the Boltzmann–Matano method show that the difference between the results of pure metals or alloys is small, as they are within the reproducibility of each experiment (cf. Fig. 4b). As demonstrated above, the use of the error function to measure the interdiffusion coefficient is influenced by the concentration dependency of the diffusion coefficient because the measured interdiffusion coefficient of the diffusion couple of pure metals is larger than the coefficient of the diffusion couple of alloys. This tendency is reasonable, considering the downward
Fig. 5. Profiles of the ratio of the measured atomic and isotopic concentrations C isotope =C atom in the diffusion couple of alloys. The square is the ratio of the measured Pb and 207Pb concentrations in a sample. The triangle is the ratio of the measured Sn and 124Sn concentrations in a sample. The solid lines are fitting curves: fitting curves of 207Pb and 124Sn were obtained by fitting the error function with D207Pb = 3.09 109 m2/s and 3.65 109 m2/s, respectively. The symbol ‘‘ ɡ” indicates the direction of gravity.
H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541
convexity of the concentration dependency revealed by the Boltzmann–Matano method. In short, we think that the interdiffusion coefficient obtained by the Boltzmann–Matano method is the most appropriate value because the influence of the concentration dependency is small.
4.3. Validity of obtained intrinsic diffusion coefficient The intrinsic diffusion coefficients are calculated by fitting the error function. Fig. 3 shows that the fitting agrees well with the concentration profiles since the coefficient of determination R2 is larger than 0.99 in each experiment. Fig. 5 shows the results of using the second analysis method. The fitting curve agrees well with the concentration profiles. Fig. 6 shows the results obtained from the third analysis method. The concentration profiles turn over at the vertex position of the concentration, xv , as shown in Fig. 6. Fig. 6b and d show that the plots are aligned. Table 3 shows the intrinsic diffusion coefficients obtained by the three analysis methods, in which the intrinsic diffusion coefficient calculated by
537
fitting the error function differs slightly from the values calculated by the second method using the ratio C isotope =C atom . Table 3 shows that the diffusion coefficient calculated by the Gaussian function in the experiment with the diffusion couple of pure metals differs on the left and right sides with the vertex as the boundary. In other words, there is a concentration dependency of the intrinsic diffusion coefficient in the diffusion couple. On the other hand, there was no significant difference in the diffusion coefficient between the left and right sides of the concentration profile in the experiment with the diffusion couple of alloys. Therefore, we suggest that the measured intrinsic diffusion coefficient with the diffusion couple of alloys should be carefully treated because the influence of the concentration dependency of the intrinsic diffusion coefficient is small in the case of the diffusion couple of alloys. The difference among the intrinsic diffusion coefficients with the diffusion couple of alloys calculated by the three analysis methods is small, as shown in Table 3. Therefore, a reasonable value at NSn = 0.5 is obtained by the method of fitting the error function. For a more accurate measurement of the intrinsic
Fig. 6. Results of intrinsic diffusion by fitting the Gaussian function. The symbol ‘‘ ɡ” indicates the direction of gravity. The dark and light gray plots show the results of pure metals and alloys, respectively. (a) Profiles of 207Pb obtained by subtracting the concentration of diffusion between semi-infinite samples (DSIS) from the measured concentration. (b) Profiles of 207Pb obtained by the natural logarithm of the subtracted value. (c) Profiles of 124Sn obtained by subtracting the concentration of diffusion between semi-infinite samples (DSIS) from the measured concentration. (d) Profiles of 124Sn obtained by the natural logarithm of the subtracted value. The dash-dotted lines show vertex position xv of the spline curve of the measured value. Parts (c) and (d) show only the range where the concentration change is confirmed.
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Table 3 Intrinsic diffusion coefficients at molar fraction NSn = 0.5 obtained by three analysis methods. Analysis method Error function
Type of diffusion couple
Type of isotope
D (109 m2/s)
Coefficient of determination R2
Pure metals
207
2.68 2.43 3.04 3.44 * a * a 3.09 3.65 1.98 4.66 2.68 3.51 2.93 2.76 3.54 3.67
0.99882 0.99494 0.99926 0.99097 * a * a 0.97237 0.9948 0.99646 0.99992 0.99694 0.99669 0.9996 0.99712 0.99883 0.99609
Pb Sn Pb 124 Sn 207 Pb 124 Sn 207 Pb 124 Sn 207 Pb 124 207
Alloys Fitting to the ratio C isotope =C atom
Pure metals Alloys
Gaussian function
Pure metals
x < xv x > xv x < xv x > xv x < xv x > xv x < xv x > xv
124
Sn
207
Alloys
Pb
124
Sn
*
a: analysis is not possible.
diffusion coefficient, an analysis method considering the concentration dependency of the intrinsic diffusion coefficient should be considered. 4.4. Confirmation of suppression of natural convection Fig. 6b and d reveal that the influence of natural convection was small, given that the concentration of 207Pb and 124Sn was in good agreement with Eq. (14), in the form of a linear relationship. The linearity is a prerequisite for suppressing natural convection. As shown in Table 3, the intrinsic diffusion coefficient calculated by the Gaussian function was larger in the region x > xv than in x < xv for pure metals. In other words, the results show that Pb atoms were transferred in a direction opposite to the direction of gravity. As Pb has a larger density than Sn, the results indicate that there is no gravity-driven mass transport, that is, convection in pure metals. On the other hand, the correlation of alloys was inverted with that of pure metals in Table 3. The reasons for the correlation could not be affirmed. For example, measurement error or the influence of natural convection can be considered. However, in this research, a stable density layering, which is effective in the suppression of natural convection, was set. Therefore, natural convection is also considered to be suppressed in the diffusion experiment of alloys. 4.5. Validity of the Darken’s equation in liquid metals We verified the Darken’s equation at NSn = 0.50. Table 4 shows the theoretical values of the interdiffusion coefficient calculated by the Darken’s equation using the measured intrinsic diffusion coefficient and the value calculated by the Boltzmann–Matano method. The measured values with the diffusion couple of pure metals show good agreement with theoretical values calculated by Eq. (1). However, considering the concentration dependency of the
intrinsic diffusion coefficient, we surmise that the results with the diffusion couple of alloys should be used for calculation and comparison. The relative error was 24.1%, as shown in Table 4. The differences are not very large between the theoretical value calculated by Eq. (2) and the measured values with the diffusion couple of pure metals and alloys. In addition, the concentration dependency of the diffusion coefficient of theoretical values shows a downward-convex tendency. The tendency corresponds with the results calculated by the Boltzmann–Matano method. The reason why the measured and calculated values do not correspond should be the error of substituted values. As error sources, D (or D⁄), U; ~ can be considered. However, D ~ is eliminated from the error and D ~ calculated by the Boltzsource because the correspondence of D mann–Matano method with diffusion couples of pure metals or alloys was confirmed at NSn = 0.5. Therefore, the error sources would be D, D⁄, and/or U. 4.6. Summary of experimental and analysis methods Table 5 shows the features of each experimental method. It is evident that using stable density layering with a diffusion couple of pure metals would be preferable to using it with a diffusion couple of alloys.
Table 5 Features of each experimental method.
Interdiffusion Intrinsic diffusion
Type of diffusion couple
Possibility of stable density layering
Pure metals Alloys Pure metals Alloys
A A (limit) A A (limit)
A: available, limit: only under the condition of stable density layering during diffusion time.
Table 4 Comparison between theoretical interdiffusion coefficient calculated by the Darken’s equation and the value calculated by the Boltzmann–Matano method at a molar fraction NSn = 0.50. The intrinsic diffusion coefficients are the measured values in this study. The self-diffusion coefficients are reference data [13,14]. The thermodynamic factor was calculated by a quasi-chemical model [15]. Type of diffusion couple
Theoretical value (109 m2/s) ~ eq:ð1Þ D
~ eq:ð2Þ D
Measured value (109 m2/s) ~ meas D
Relative error (%) Dmeasure D eq:ð1Þ
Dmeasure D eq:ð2Þ
Deq:ð1Þ
Pure metals Alloys
2.56 3.24
2.95
2.57 2.46
0.391 24.1
Deq:ð2Þ
12.9 16.6
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H. Fukuda et al. / International Journal of Heat and Mass Transfer 133 (2019) 531–541 Table 6 Features of each analysis method.
Interdiffusion Intrinsic diffusion
Analysis method
Type of diffusion couple
Possibility of analysis
Consideration of concentration dependence
Simple calculation process
Validity of calculated value
Superposed equation of error function Boltzmann–Matano Superposed equation of error function Fitting to the ratio C isotope =C atom
Pure metals Pure metals Pure metals Pure metals Alloys Pure metals
A A A N/A A A
N/A A N/A ─ N/A A (limit)
A N/A A ─ N/A N/A
A (limit) A A ─ A A
Gauss function
or Alloys or Alloys or Alloys
or Alloys
A: available, N/A: not available, limit: only under the condition of stable density layering during diffusion.
Table 6 shows the features of each analysis method. It is possible to calculate the interdiffusion coefficients by two methods. The analysis method using the Boltzmann–Matano method is better than the analysis method using the superposed equation of the error function because the latter is influenced by the concentration dependency of the diffusion coefficient. It is possible to calculate the intrinsic diffusion coefficients by the three methods. However, the analysis method of fitting to the ratio has applicability limits. On the other hand, the analysis method with the superposed equation of error function is convenient due to the simplicity of the calculation and no limits.
Aerospace Exploration Agency (JAXA) ‘‘Diffusion Phenomena in Melts” Working Group. We thank Kimura Foundry Co., Ltd. for financial support. Appendix A A.1. Derivation of superposed equation of error function The concentration profile of the diffusion between semi-infinite samples (DSIS) is described by the following Eq. (A1), as shown in Fig. A1.
0
5. Conclusions
C DSISðxÞ We performed experiments and analyses and clarified the following: A simultaneous measuring method of interdiffusion and intrinsic diffusion coefficients in liquid metals was established. Simultaneous measurements in the Sn-Pb system were conducted on the ground. The dispersion in concentration was suppressed by the method of making the concentration of the sample uniform before starting diffusion time. The calculated value with the method of fitting the error function is influenced by the concentration dependency of the diffusion coefficient rather than the average concentration. On the other hand, at a certain concentration, the interdiffusion coefficients can be calculated for different concentrations while eliminating other influences by the Boltzmann–Matano method. The intrinsic diffusion coefficients were calculated by fitting the superposed equation of error function. Then, the experimental method using the diffusion couple of alloys and the homogenization technique is more suitable. The applicability of the Darken’s equation was examined using the measured diffusion coefficient. The results showed that the difference between the theoretical and measured interdiffusion coefficients is not very large. Therefore, the possibility of adapting the Darken’s equation in liquid metals was confirmed in this experiment. However, further experiments are necessary to determine the detailed conditions for the adaptation of the Darken’s equation. The error factor would be intrinsic diffusion or self-diffusion coefficients and/or the thermodynamic factor.
1
B x x0 C Ch þ Cl Ch Cl C ¼ erf B D EA @rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 X
ðA1Þ
The concentration profile of thick layer diffusion (TLD) is described by the following Eq. (A2).
C TLDðxÞ
8 9 Z b < ðx dÞ2 = C0 Cl ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp D E dd D Effi : 2 X2 ; a 2p X 2 8 0 1 0 19 > > > > < B xa C B x b C= C0 Cl B C B C r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi erf erf @rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ D E D E A @ A>: 2 > > > : ; 2 X2 2 X2
ðA2Þ
Here, if a = x0 and b = x0 + H, rewriting Eq. (A2) gives
Conflict of interest The authors declared that there is no conflict of interest. Acknowledgments We sincerely would like to express our all gratitude to staff members of Environmental Safety Center, Waseda University for sample analyses. This study was supported by grant-in-aid from Mitsubishi Materials Corporation in fiscal year 2017 and the Japan
Fig. A1. Concentration profile of diffusion between semi-infinite samples.
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C TLDðxÞ
8 0 1 19 0 > > > > < B x x0 C Bx ðx0 þ HÞC= C0 Cl C C B B erf @ rffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ erf @rffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E D E A A > 2 > > > : ; 2 X2 2 X2
ðA3Þ
This equation is shown in Fig. A2. Considering the superposition of fluxes, Eq. (A4) was obtained by the sum of Eqs. (A1) and (A3).
0 C ðxÞ ¼
1
B x x0 C Ch þ Cl Ch Cl C erf B D EA @rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 X2 8 0 1 19 0 > > > > < B x x0 C Bx ðx0 þ HÞC= C0 Cl C C B B r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi erf þ erf @rffiffiffiffiffiffiffiffiffiffiffiffiffiffi D E D E A A> @ 2 > > > : ; 2 X2 2 X2
ðA4Þ
The number of steps changes according to C0, as shown in Fig. A3. Appendix B. Supplementary material Fig. A2. Concentration profile of thick layer diffusion.
Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2018.12.036.
Fig. A3. Schematic concentration profiles. All profiles have two or less steps. (a) C0 = Cl, (b) Cl < C0 < Ch, (c) C0 = Ch, (d) Ch < C0.
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