- Email: [email protected]
Measurements of aluminum diffusion in molten gallium and indium
Journal of Crystal Growth 186 (1998) 520—527
Measurements of aluminum diffusion in molten gallium and indium P. Bra¨uer, G. Mu¨ller-Vogt* Kristall- und Materiallabor der Fakulta( t fu( r Physik, Universita( t Karlsruhe, Kaiserstra}e 12, D-76128 Karlsruhe, Germany Received 8 August 1997; accepted 16 October 1997
Abstract In this work diffusion of Al in liquid Ga and In was investigated using a shear-cell method. The diffusion coefficient D of Al in Ga was determined with a reproducibility between $13% and $5% over the temperature range 130—800°C. The diffusion data of Al in Ga are consistent with viscosity data of pure Ga, when compared via a Stokes—Sutherland—Einstein relation. The same relationship between diffusion data of the solute and the viscosity of the solvent holds for Al in In. ( 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The shear-cell technique has proved to be a lowcost method for measuring diffusive transport in melts of semiconducting compounds. We gained extensive experimental experience on diffusion in multicomponent melts of group II—VI elements [1]. The accuracy of the extracted parameters was as high as $30%, much better than that of previous results (see, e.g. references in Ref. [1]). To increase the reliability of our data the influence of disturbing mechanisms such as forced convection due to the shearing process or buoyancy-driven convection was investigated and evaluated in a model system: a melt mainly consisting of Ga containing a low concentration of Al (max. 2.5 at%). This system
* Corresponding author. Fax: #49 721 697123.
provided a wide temperature region for diffusivity measurements.
2. Experimental procedure 2.1. Procedure We used shear cells designed by Ko¨{ler [2] as described in a previous paper [3]. The materials used for the experiments were Ga (m 6N), In (m 6N) and Al (m 5N). At the beginning of the experiment the shear cell was placed into a vacuum chamber which could be heated by a surrounding furnace. The samples within the supplying chambers of the shear cell were melted and homogenized at temperatures up to 1000°C for at least 15 h, then the capillaries were filled and the melt remained inside the capillaries for about 6—8 h. During this period the temperature needed for the measurement was
0022-0248/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 0 2 4 8 ( 9 7 ) 0 0 8 2 2 - 1
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
adjusted. Then the device was brought into the diffusion configuration. The period between the first contact of the two half capillaries and the complete alignment lasted about 10 s. Shearing of the whole stack of slices at the end of the diffusion experiment took less than 5 s. Compared to the usual diffusion times of 2—24 h this introduces an uncertainty of less than 10~6 in diffusion time. The shear cells used in this work contained two pairs of capillaries. So each run provided two parallel experiments which were carried out with identical concentration, time and temperature. During a diffusion run, the temperature was measured with an accuracy of 0.2% with two Pt—PtRh thermocouples located within the centring rod at the level of slice 1 (top) and slice 30 (bottom). To maintain reproducible temperature distributions within the shear cell for all diffusion experiments, we constructed a quickly removable measuring inset, brought into the apparatus in a monthly cycle. This inset was equal to an ordinary shear cell with regard to materials used, dimensions, set-up and position within the furnace. It contained 12 additional thermocouples. We chose five measuring points at the vertical positions of !5.8, !3, 0, #3 and #5.8 cm at the same radial distance from the axis as the diffusion capillaries. They were located at one side of the shear cell. A second group was arranged symmetrically to them at the other side (turned 180°). The two remaining measuring points were identical with those fixed within the shear cell at the centring rod during a normal diffusion run. 2.2. Influence of thermal conditions on the hydrodynamic stability of the melts Hydrodynamic stability demands that the axes of the diffusion capillaries are parallel to the gvector. Using Hart’s hydrodynamic analysis [4], Praizey [5,6] showed that for this configuration any radial gradients in temperature and/or concentration cause convection. To some extent, such convective flows can be suppressed by solutal stabilization (i.e., with the higher density melt situated below that of lower density) whereas the influence of axial temperature gradients on the stability seems to be weak.
521
Axial temperature gradients can also lead to first-order Marangoni convection [7], because the melts used are not wetting the graphite. Therefore, free surfaces may occur at pores in the graphite surface or at small kinks, which result from not so exactly aligned neighbouring slices. We measured the temperature distribution within the shear cell for two different temperature gradients set in the surrounding furnace: 1. Stabilizing temperature gradients along the axis (i.e., ¹ increases with height) up to 1.3 K/mm could be reached and were always accompanied by distinct gradients in radial direction caused by heat flows due to the construction of our experimental equipment. Our measurements revealed radial gradients of upto 5 K/mm under these conditions [8]. 2. Isothermal distributions (“zero gradient”) along the capillaries showed within the accuracy of the temperature measurements maximum gradients of about 0.05 K/mm in axial direction and of about 0.2 K/mm in radial direction. In order to minimize the influence of uncontrolled convection due to radial gradients and to avoid Marangoni convection at free surfaces, we carried out all experiments in the Al—Ga system under above-zero gradient conditions. In these experiments, the upper-half of the capillary always contained the melt of lower density (2.5 mol% Al in Ga). 2.3. Analysis of concentration data The content of each capillary segment was dissolved in HCl : HNO : H O"3 : 1 : 8. The abso3 2 lute masses of Al and Ga within the solution were measured by atomic absorption spectrometry (AAS) to determine the mean relative concentration (in at%) of Al within each slice. All samples containing more than about 0.3 at% of Al could be measured within the standard range of Flame-AAS, with a relative error dC/C between $1% (best case, precision limit of the AAS for Al) and $3% (worst case, additional scattering due to sample preparation). Lower concentrations of Al had to be measured using amplified signals. Therefore the resulting values were of lower precision (dC/C!$10%). The concentration range between 0 and 0.05 at% Al remained undetermined.
522
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
The concentration profiles C(x, t) which developed from an initial concentration step of [C , C ] at x"0 can be described by the equation 0 1 (given in Ref. [3], but with a modified local coordinate)
A
B
x!x C #C C #C 0 , 1" 0 1erf C(x, t)" 0 2 2 2JDt
(1)
where C and C are the contents of the solute Al 0 1 within the lower- and the upper-half of the capillary, erf(m) is the Gaussian error function, t the diffusion time, D the diffusion coefficient, x the local coordinate and x a shift of its origin, which is 0 supposed to lie in the middle of the shear cell at time t"0. To obtain the value of the diffusion coefficient, we plotted the mean concentration of Al within each slice against its position together with a weighed least-squares fit to Eq. (1). The fit considered the uncertainty of each measured concentration value and calculated the fit parameters (C , C , D and x ) and their standard deviation. 0 1 0 Fig. 1 gives an example of measured concentration data of Al in Ga and the fit curve. There were three reasons why we chose C as 0 0 and C as about 2.5 at%: 1 1. We got rid of the fit parameter C (there was 0 definitely no Al within the lower-half capillary). The remaining free parameters were C , x and 1 0 D. 2. Only in diluted solutions will diffusion experiments provide small statistical errors of D, due to the uncertainty of the concentration values. Concentration analysis by AAS leads to a constant relative error dC/C. This means an increasing absolute error dC with increasing concentration values. This absolute error must be compared to the concentration step (C !C ). Hence, the relative error dC/ 1 0 (C !C ) increases if the concentrations 1 0 C and C both shift to higher absolute values 0 1 and if the concentration step decreases. Increasing the initial concentration step (C !C ) 1 0 would reduce the relative error, but then the diffusion coefficient might be a function of concentration within the chosen concentration interval. 3. For concentrations of Al below 2.5 at%, all samples could be diluted in 5 ml acidic aqueous
Fig. 1. Diffusion profile which developed at 590°C from an initial concentration step of 0—2.5 at% Al in Ga after a diffusion time of 24 060 s. The open circles represent the measured concentration data, the solid line is the least-squares fit to Eq. (1), the dashed lines mark the diffusion zone d . 80
solution to fit the standard range of the FlameAAS directly, which meant a minimum of preparation steps and therefore fewer sources of error. The precision dC/C of the measured concentration values ($1% best case, $3% worst case) led to a statistical error dD/D between $3 and 9%.
3. Results 3.1. Diffusion of Al in molten Ga The diffusion of Al in Ga was investigated at five temperatures: 130, 330, 590, 690 and 800°C. All experimental concentration profiles were analyzed by a least-squares fit to Eq. (1). Within the series of measurements at identical temperatures and similar diffusion times the reproducibility of the values of D varied from about $10% to about $30%, which was clearly above the statistical error of one single experiment. No systematic deviations from the ideal shape of an error function could be detected, but often a recognizable shift x of the profiles 0 was found. We have estimated whether this shift could be due to drift movements caused by gravitational sedimentation or thermodiffusive separation of the components (see the appendix for an outline of the
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
considerations underlying these estimates). However, the resulting estimated values of x possibly 0 resulting from thermodiffusion and sedimentation, respectively, are at least one and four orders of magnitude smaller than those observed experimentally. Material losses can also be excluded, since Ga has a very low vapour pressure at all temperatures used. A comparison of identical experiments showed that for most cases the sign of x was correlated 0 with the deviation of D from the average value DM of each series. Therefore, we had to assume that the shifts were caused by an inhomogeneous distribution of Al within the upper-half capillary, which means that the initial boundary conditions were not fulfilled in every case. Hence, we simulated concentration profiles that would result from two cases of nonuniform initial concentration distributions. With either higher initial Al concentrations near the middle or near the upper end of the capillary, we obtained concentration profiles that clearly showed the correlation between the shift x and the 0 deviation of D [8] observed experimentally. As a consequence, we rejected all experiments, where the values of x exceeded the 90% probability re0 gion of the statistical error. This reduced the scattering of the residual diffusion coefficients between $13 and $5% within the series for all temperatures. We also wanted to get information about the contribution of forced initial transport, caused by the first shearing which brings in contact the lower capillary with the upper one. This problem has already been addressed in a previous paper [3]. One method to detect such transport contributions is to carry out experiments at different diffusion times. Hence, we performed experiments of different durations at temperatures of 300 and 800°C, respectively. For the lower temperature, no correction of the values of D was necessary but for 800°C this procedure led to corrected values of D of about 8% lower than the initial long-time value. The greater influence of initial shearing motion at the higher temperature is possibly due to the decreased viscosity of the melt. Table 1 summarizes the measured diffusivity data. Note that the error limit *D/D is not the statistical error of one single experiment but the
523
Table 1 Diffusion coefficients of 0—2.5 at% Al in Ga at different temperatures. *D/D represents the scatter within the number of experiments at the same temperature ¹ (°C)
D (10~4 cm2 s~1)
*D (10~4 cm2 s~1)
*D/D (%)
Number of experiments
130 330 590 690 800
0.267 0.552 0.922 1.17 1.36
$0.013 $0.055 $0.10 $0.060 $0.17
$5 $10 $10 $5 $13
2 4 2 3 5
reproducibility of data within a series of measurements at the same temperature. 3.2. Diffusion of Al in molten In We also investigated the diffusion of Al in In. The initial concentration step was C "0 and 0 C +5 at% Al. Within the standard range of the 1 Flame-AAS, the accuracy of the concentration determinations was about $3%. The resulting statistical error dD/D was between $5 and $10% (short-time experiment). Two experiments were carried out at 620°C and two at 680°C. The results were D"(8.4$1.5)]10~5 cm2 s~1 (at 620°C) and D"(9.3$2.5)]10~5 cm2 s~1 (at 680°C), respectively, where the first value in the parantheses represents the mean value of both experiments and the second one the difference between the single experiments.
4. Discussion 4.1. Temperature dependence In Fig. 2 we have presented the diffusivities of Al in Ga in the form of an Arrhenius plot. One sees that within the error bars it is possible to connect the data by a straight line (dashed) according to D(¹)"D e~*E@RT, (2) 0 with *E"(8.6$0.6) kJ mol~1 and D "(3.3$ 0 0.4)]10~4 cm2 s~1. However, the plot strongly suggests systematic deviations from this relation.
524
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
Fig. 2. Arrhenius plot of the diffusion coefficients given in Table 1 (open squares and error bars).
An alternative relation due to Swalin [9] is D(¹)"A¹2.
(3)
This relation was confirmed by Frohberg et al. in space experiments for tracer diffusion of 124Sn—112Sn [10] and interdiffusion of Sn and In [11]. A fit of our data to Eq. (3) leads to a value of A"(1.1$0.04)]10~10 cm2 K~2 s~1, but has an offset at ¹"0 K of (1.2$0.3)]10~5 cm2 s~1, which does not agree with theory. Thus, our data suggest a less than quadratic temperature dependence of D. A fit to the empirical relation D(¹)"B¹n
determined over rather limited temperature ranges [13—15]. These data are shown together with our results in Fig. 3. The thick line represents Ga selfdiffusivities obtained by Petit and Nachtrieb [13] between 30 and 100°C in capillaries with diameters of 0.5, 0.76 and 1 mm. Other experiments covered broader temperature ranges: about 260 K, using capillaries of 0.7 mm diameter (Larsson et al. [14], dashed line) and about 370 K, using capillaries of 1.6 mm diameter (Broome and Walls [15], medium-thickness solid line). The full squares represent our Al—Ga data (capillary diameter: 1.0 mm) with the thin-line fit to Eq. (4). Our extrapolated curve coincides with the results of Petit and Nachtrieb, but lies significantly lower than the values of the other two groups. No correlation between capillary diameters and diffusion coefficients can be found. Therefore, wall effects do not seem to be important. If we treat Al—Ga as an ideal system, we can apply the Darken equation [16] to our data. This results in a self-diffusion coefficient of Al lower than that of Ga. However, there is some reason to suppose that Al atoms have a higher mobility and therefore a higher self-diffusion coefficient than Ga. The most significant difference between Al and Ga lies in the atomic mass, Ga being about 2.5 times heavier than Al. (Comparisons of atomic volumes at the melting points [17] as well as diffraction data [18] clearly show that
(4)
yields B"(1.52$0.73)]10~9 cm2 s~1 K~n and n"(1.62$0.07), which is shown by the solid curve in Fig. 2. For a more definitive discussion of the temperature dependence of D — with regard to different diffusion models of liquids — the precision of our data is still too low. 4.2. Comparison with data for pure melts of Ga and In A comparison of our data with self-diffusion data using a modified Darken equation [12] is not possible, since neither the self-diffusion coefficients of Al nor the activity of Al in Ga are available. Only the self-diffusivities of Ga have been
Fig. 3. Compilation of self-diffusion coefficients of liquid Ga [13—15] and comparison to the diffusion data of Al in Ga (this work).
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
525
nearest-neighbour distances are nearly the same for liquid Ga and Al.) That means, Al—Ga is not an ideal system and the diffusion of Al is mainly determined by the properties of the Ga matrix, such as its viscosity. Iida and Guthrie [19] showed that the modified Stokes—Einstein equation k¹ D" , 4pkr
(5)
with k¹ the thermal energy and r the radius of the diffusing particle, can be used to correlate D and the viscosity k. Note that the factor of 4 in the denominator is based on a hydrodynamic calculation by Sutherland [20] for the case where the sizes of the diffusing and matrix particles are equal. According to the results of Ashcroft and Lekner [21], the values of r can be deduced from the atomic volume » at the melting point, assuming hard spheres and . a packing fraction of 0.45 using the relation
A B A
B
1@3 » 1@3 4p . " "2.1r, r30.45~1 N 3 A
Fig. 4. Compilation of diffusion coefficients of Al in Ga (this work) together with self-diffusivities deduced from viscosity data for pure Ga [22,23] using Eq. (5).
(6)
where N is the Avogadro’s number. Hard-sphere A radii of Al, Ga and In, calculated from Eq. (6) are between the onset and the maximum of the first peak of the pair-distribution functions deduced from diffraction data of Waseda [18]. The viscosity of liquid Ga was investigated by Spells [22] and more recently by Iida et al. [23]. Using Eq. (5) and r "2.67 A_ , we transformed G! these viscosity data to corresponding values of diffusion coefficients. The result is shown in Fig. 4, by open circles for Ref. [23] and by dots for Ref. [22]. Both are in very good agreement with our data (full squares and thin line) and obviously do not match the data of Refs. [14,15]. A compilation of data for In melts is plotted in Fig. 5. Self-diffusion data for pure In were given by Lodding [24] (solid line) and Careri et al. [25] (dashed line). Using the viscosity data of Culpin [26] and Iida et al. [23], and r "1.43 A_ , we I/ calculated diffusivities from Eq. (5). The results, presented in Fig. 5 by the dots and open circles, respectively, agree well with our experimental data (full squares).
Fig. 5. Compilation of diffusion coefficients of Al in In (this work) and self-diffusivities of pure liquid In [24,25], together with self-diffusivities deduced from viscosity data [23,26] using Eq. (5).
Although viscosities of binary mixtures of Al—Ga and Al—In are not known to the authors (even the published viscosity data of pure liquid Al scattered over half an order of magnitude), we assume that the viscosity of the Ga or In melt will not change significantly on addition of low concentrations of Al. Thus, our results appear to be representative for the behaviour of the pure melts. As a consequence, crystal growers should consider using our data when self-diffusion coefficients of liquid Ga are needed for transport estimates.
526
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
5. Conclusions and outlook The shear-cell technique has proved to be a successful method to measure diffusion coefficients in multicomponent melts on earth. We have demonstrated with the binary system Al—Ga that the adequate choice of the concentration interval, improvements of the thermal conditions as well as various methods for the detection of disturbing parameters provide values of diffusion coefficients with a scattering between $13 and $5%. The measurements were done at temperatures ranging from 130 to 800°C. Despite this large temperature interval, the values of the diffusion coefficients of Al in Ga change only by half an order of magnitude. Therefore, the limits of error are still too high to definitively decide on the applicability of different temperature dependences predicted by the various liquid diffusion theories. Good agreement was obtained between the diffusion data for Al (maximum addition 2.5 at%) in Ga and the viscosity data of pure liquid Ga on correlation with a Stokes—Sutherland—Einstein relation. Therefore, it is most likely that the diffusion of Al is determined by the properties of the Ga melt. This is in contrast with Ga self-diffusivities obtained by other authors whose values are significantly higher than our Al—Ga data. This may be due to weaker convective transport during our experiments. Because of the different atomic masses of the components, Al—Ga and Al—In melts are suitable systems to investigate thermodiffusive separation. If these measurements are performed with shear cells on the ground, our experience shows that unavoidable radial temperature gradients within the capillary will pose a serious problem. Using different capillary diameters on the ground and comparing the results to kg-data would help to quantify the influence of these radial gradients. Therefore experiments in microgravity will be essential.
Acknowledgements The authors gratefully acknowledge the assistance of T. Stra{er and D. Buck who programmed
the calculation routines, K. Hugle who carried out the AAS measurements and H. Mu¨ller who performed the thermodiffusion experiments. This work was financially supported by DFG. Appendix A The thermodiffusive drift velocity v was esti5) mated from the thermodiffusive coefficient D@ in the system Al—In (determined in our group as D@(Al)"!3.5]10~8 cm2 s~1 K~1) using ¹(x) v "!D@ , 5) x and assuming that (a) the thermodiffusive coefficient of Al—Ga does not exceed that of Al—In, (b) D@ is concentration independent, (c) the temperature gradient in the capillary is constant (which was experimentally determined) and (d) the solute (Al) concentration is low (it always was less than 2.5 at%). The gravitational sedimentation velocity was estimated from the relation v "uF, ' where u is the mobility of the diffusing particles (evaluated from the experimental values of D using the Einstein relation u"D/k¹) and F is the drift force resulting from the different density of the components (evaluated as F"g*m, with *m the difference of the atomic masses of Al and Ga, assuming equal atomic volume and a low concentration of the solute). References [1] G. Mu¨ller-Vogt, P. Bra¨uer, R. Ko¨{ler, Z. Peranic, J. Schlegelmilch, T. Stra{er, W. Trillsam, in: L. Ratke (Ed.), Immiscible Alloys and Organics, DGM-Informationsgesellschaft, Oberursel, 1992, p. 125. [2] R. Ko¨{ler, Thesis, Karlsruhe, 1990. [3] G. Mu¨ller-Vogt, R. Ko¨{ler, J. Crystal Growth 186 (1998) 511. [4] J.E. Hart, J. Fluid Mech. 49 (2) (1971) 279. [5] J.P. Praizey, Int. J. Heat Mass Transfer 32 (1989) 2385. [6] Y. Malmejac, J.P. Praizey, in: T.D. Guyenne, J. Hunt, (Eds.), Proc. 5th European Symp. on Material Sciences Under Microgravity, ESA Sp-222, ESTEC Reproduction Services, Noordwijk, The Netherlands, 1984, p. 147.
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527 [7] Y. Malmejac, G. Frohberg, in: H.U. Walter (Ed.), Fluid Sciences and Material Science in Space, Springer, Berlin, 1987, p. 159. [8] P. Bra¨uer, Thesis, Karlsruhe, 1995. [9] R.A. Swalin, Acta Metall. 7 (1959) 736. [10] G. Frohberg, K.H. Kraatz, H. Wever, ESA-SP-222, 1984, p. 201. [11] G. Frohberg, K.H. Kraatz, H. Wever, D1-Symp, Norderney. [12] P.G. Shewmon, Diffusion in Solids, McGraw-Hill, New York, 1963, p. 126. [13] J. Petit, N.H. Nachtrieb, J. Chem. Phys. 24 (1956) 1027. [14] S. Larsson, L. Broman, C. Roxbergh, A. Lodding, Z. Naturf. A 25 (1970) 1472. [15] E.F. Broome, H.A. Walls, Trans. Metall. Soc. AIME 245 (1969) 739. [16] L. Darken, Trans. AIME 174 (1948) 184.
527
[17] D.J. Steinberg, Metall. Trans. 5 (1974) 1341. [18] Y. Waseda, The Structure of Non-Crystalline Materials, Appendices 8 and 9, McGraw-Hill, New York, 1980. [19] T. Iida, R.I.L. Guthrie, The Physical Properties of Liquid Metals, Clarendon Press, Oxford, 1993, p. 214. [20] W. Sutherland, Philos. Mag. 9 (1905) 781. [21] N.W. Ashcroft, J. Lekner, Phys. Rev. 145 (1966) 83. [22] K.E. Spells, Proc. Phys. Soc. (London) 48 (1936) 299. [23] T. Iida, Z. Morita, S. Takeuchi, J. Jpn. Inst. Metals 39 (1975) 1169. [24] A. Lodding, Z. Naturf. A 11 (1956) 200. [25] G. Careri, A. Paoletti, M. Vicentini, Nuovo Cimento 10 (1958) 1088. [26] M.F. Culpin, Proc. Phys. Soc. (London) B 70 (1957) 1069.
Measurements of aluminum diffusion in molten gallium and indium P. Bra¨uer, G. Mu¨ller-Vogt* Kristall- und Materiallabor der Fakulta( t fu( r Physik, Universita( t Karlsruhe, Kaiserstra}e 12, D-76128 Karlsruhe, Germany Received 8 August 1997; accepted 16 October 1997
Abstract In this work diffusion of Al in liquid Ga and In was investigated using a shear-cell method. The diffusion coefficient D of Al in Ga was determined with a reproducibility between $13% and $5% over the temperature range 130—800°C. The diffusion data of Al in Ga are consistent with viscosity data of pure Ga, when compared via a Stokes—Sutherland—Einstein relation. The same relationship between diffusion data of the solute and the viscosity of the solvent holds for Al in In. ( 1998 Elsevier Science B.V. All rights reserved.
1. Introduction The shear-cell technique has proved to be a lowcost method for measuring diffusive transport in melts of semiconducting compounds. We gained extensive experimental experience on diffusion in multicomponent melts of group II—VI elements [1]. The accuracy of the extracted parameters was as high as $30%, much better than that of previous results (see, e.g. references in Ref. [1]). To increase the reliability of our data the influence of disturbing mechanisms such as forced convection due to the shearing process or buoyancy-driven convection was investigated and evaluated in a model system: a melt mainly consisting of Ga containing a low concentration of Al (max. 2.5 at%). This system
* Corresponding author. Fax: #49 721 697123.
provided a wide temperature region for diffusivity measurements.
2. Experimental procedure 2.1. Procedure We used shear cells designed by Ko¨{ler [2] as described in a previous paper [3]. The materials used for the experiments were Ga (m 6N), In (m 6N) and Al (m 5N). At the beginning of the experiment the shear cell was placed into a vacuum chamber which could be heated by a surrounding furnace. The samples within the supplying chambers of the shear cell were melted and homogenized at temperatures up to 1000°C for at least 15 h, then the capillaries were filled and the melt remained inside the capillaries for about 6—8 h. During this period the temperature needed for the measurement was
0022-0248/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 2 2 - 0 2 4 8 ( 9 7 ) 0 0 8 2 2 - 1
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
adjusted. Then the device was brought into the diffusion configuration. The period between the first contact of the two half capillaries and the complete alignment lasted about 10 s. Shearing of the whole stack of slices at the end of the diffusion experiment took less than 5 s. Compared to the usual diffusion times of 2—24 h this introduces an uncertainty of less than 10~6 in diffusion time. The shear cells used in this work contained two pairs of capillaries. So each run provided two parallel experiments which were carried out with identical concentration, time and temperature. During a diffusion run, the temperature was measured with an accuracy of 0.2% with two Pt—PtRh thermocouples located within the centring rod at the level of slice 1 (top) and slice 30 (bottom). To maintain reproducible temperature distributions within the shear cell for all diffusion experiments, we constructed a quickly removable measuring inset, brought into the apparatus in a monthly cycle. This inset was equal to an ordinary shear cell with regard to materials used, dimensions, set-up and position within the furnace. It contained 12 additional thermocouples. We chose five measuring points at the vertical positions of !5.8, !3, 0, #3 and #5.8 cm at the same radial distance from the axis as the diffusion capillaries. They were located at one side of the shear cell. A second group was arranged symmetrically to them at the other side (turned 180°). The two remaining measuring points were identical with those fixed within the shear cell at the centring rod during a normal diffusion run. 2.2. Influence of thermal conditions on the hydrodynamic stability of the melts Hydrodynamic stability demands that the axes of the diffusion capillaries are parallel to the gvector. Using Hart’s hydrodynamic analysis [4], Praizey [5,6] showed that for this configuration any radial gradients in temperature and/or concentration cause convection. To some extent, such convective flows can be suppressed by solutal stabilization (i.e., with the higher density melt situated below that of lower density) whereas the influence of axial temperature gradients on the stability seems to be weak.
521
Axial temperature gradients can also lead to first-order Marangoni convection [7], because the melts used are not wetting the graphite. Therefore, free surfaces may occur at pores in the graphite surface or at small kinks, which result from not so exactly aligned neighbouring slices. We measured the temperature distribution within the shear cell for two different temperature gradients set in the surrounding furnace: 1. Stabilizing temperature gradients along the axis (i.e., ¹ increases with height) up to 1.3 K/mm could be reached and were always accompanied by distinct gradients in radial direction caused by heat flows due to the construction of our experimental equipment. Our measurements revealed radial gradients of upto 5 K/mm under these conditions [8]. 2. Isothermal distributions (“zero gradient”) along the capillaries showed within the accuracy of the temperature measurements maximum gradients of about 0.05 K/mm in axial direction and of about 0.2 K/mm in radial direction. In order to minimize the influence of uncontrolled convection due to radial gradients and to avoid Marangoni convection at free surfaces, we carried out all experiments in the Al—Ga system under above-zero gradient conditions. In these experiments, the upper-half of the capillary always contained the melt of lower density (2.5 mol% Al in Ga). 2.3. Analysis of concentration data The content of each capillary segment was dissolved in HCl : HNO : H O"3 : 1 : 8. The abso3 2 lute masses of Al and Ga within the solution were measured by atomic absorption spectrometry (AAS) to determine the mean relative concentration (in at%) of Al within each slice. All samples containing more than about 0.3 at% of Al could be measured within the standard range of Flame-AAS, with a relative error dC/C between $1% (best case, precision limit of the AAS for Al) and $3% (worst case, additional scattering due to sample preparation). Lower concentrations of Al had to be measured using amplified signals. Therefore the resulting values were of lower precision (dC/C!$10%). The concentration range between 0 and 0.05 at% Al remained undetermined.
522
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
The concentration profiles C(x, t) which developed from an initial concentration step of [C , C ] at x"0 can be described by the equation 0 1 (given in Ref. [3], but with a modified local coordinate)
A
B
x!x C #C C #C 0 , 1" 0 1erf C(x, t)" 0 2 2 2JDt
(1)
where C and C are the contents of the solute Al 0 1 within the lower- and the upper-half of the capillary, erf(m) is the Gaussian error function, t the diffusion time, D the diffusion coefficient, x the local coordinate and x a shift of its origin, which is 0 supposed to lie in the middle of the shear cell at time t"0. To obtain the value of the diffusion coefficient, we plotted the mean concentration of Al within each slice against its position together with a weighed least-squares fit to Eq. (1). The fit considered the uncertainty of each measured concentration value and calculated the fit parameters (C , C , D and x ) and their standard deviation. 0 1 0 Fig. 1 gives an example of measured concentration data of Al in Ga and the fit curve. There were three reasons why we chose C as 0 0 and C as about 2.5 at%: 1 1. We got rid of the fit parameter C (there was 0 definitely no Al within the lower-half capillary). The remaining free parameters were C , x and 1 0 D. 2. Only in diluted solutions will diffusion experiments provide small statistical errors of D, due to the uncertainty of the concentration values. Concentration analysis by AAS leads to a constant relative error dC/C. This means an increasing absolute error dC with increasing concentration values. This absolute error must be compared to the concentration step (C !C ). Hence, the relative error dC/ 1 0 (C !C ) increases if the concentrations 1 0 C and C both shift to higher absolute values 0 1 and if the concentration step decreases. Increasing the initial concentration step (C !C ) 1 0 would reduce the relative error, but then the diffusion coefficient might be a function of concentration within the chosen concentration interval. 3. For concentrations of Al below 2.5 at%, all samples could be diluted in 5 ml acidic aqueous
Fig. 1. Diffusion profile which developed at 590°C from an initial concentration step of 0—2.5 at% Al in Ga after a diffusion time of 24 060 s. The open circles represent the measured concentration data, the solid line is the least-squares fit to Eq. (1), the dashed lines mark the diffusion zone d . 80
solution to fit the standard range of the FlameAAS directly, which meant a minimum of preparation steps and therefore fewer sources of error. The precision dC/C of the measured concentration values ($1% best case, $3% worst case) led to a statistical error dD/D between $3 and 9%.
3. Results 3.1. Diffusion of Al in molten Ga The diffusion of Al in Ga was investigated at five temperatures: 130, 330, 590, 690 and 800°C. All experimental concentration profiles were analyzed by a least-squares fit to Eq. (1). Within the series of measurements at identical temperatures and similar diffusion times the reproducibility of the values of D varied from about $10% to about $30%, which was clearly above the statistical error of one single experiment. No systematic deviations from the ideal shape of an error function could be detected, but often a recognizable shift x of the profiles 0 was found. We have estimated whether this shift could be due to drift movements caused by gravitational sedimentation or thermodiffusive separation of the components (see the appendix for an outline of the
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
considerations underlying these estimates). However, the resulting estimated values of x possibly 0 resulting from thermodiffusion and sedimentation, respectively, are at least one and four orders of magnitude smaller than those observed experimentally. Material losses can also be excluded, since Ga has a very low vapour pressure at all temperatures used. A comparison of identical experiments showed that for most cases the sign of x was correlated 0 with the deviation of D from the average value DM of each series. Therefore, we had to assume that the shifts were caused by an inhomogeneous distribution of Al within the upper-half capillary, which means that the initial boundary conditions were not fulfilled in every case. Hence, we simulated concentration profiles that would result from two cases of nonuniform initial concentration distributions. With either higher initial Al concentrations near the middle or near the upper end of the capillary, we obtained concentration profiles that clearly showed the correlation between the shift x and the 0 deviation of D [8] observed experimentally. As a consequence, we rejected all experiments, where the values of x exceeded the 90% probability re0 gion of the statistical error. This reduced the scattering of the residual diffusion coefficients between $13 and $5% within the series for all temperatures. We also wanted to get information about the contribution of forced initial transport, caused by the first shearing which brings in contact the lower capillary with the upper one. This problem has already been addressed in a previous paper [3]. One method to detect such transport contributions is to carry out experiments at different diffusion times. Hence, we performed experiments of different durations at temperatures of 300 and 800°C, respectively. For the lower temperature, no correction of the values of D was necessary but for 800°C this procedure led to corrected values of D of about 8% lower than the initial long-time value. The greater influence of initial shearing motion at the higher temperature is possibly due to the decreased viscosity of the melt. Table 1 summarizes the measured diffusivity data. Note that the error limit *D/D is not the statistical error of one single experiment but the
523
Table 1 Diffusion coefficients of 0—2.5 at% Al in Ga at different temperatures. *D/D represents the scatter within the number of experiments at the same temperature ¹ (°C)
D (10~4 cm2 s~1)
*D (10~4 cm2 s~1)
*D/D (%)
Number of experiments
130 330 590 690 800
0.267 0.552 0.922 1.17 1.36
$0.013 $0.055 $0.10 $0.060 $0.17
$5 $10 $10 $5 $13
2 4 2 3 5
reproducibility of data within a series of measurements at the same temperature. 3.2. Diffusion of Al in molten In We also investigated the diffusion of Al in In. The initial concentration step was C "0 and 0 C +5 at% Al. Within the standard range of the 1 Flame-AAS, the accuracy of the concentration determinations was about $3%. The resulting statistical error dD/D was between $5 and $10% (short-time experiment). Two experiments were carried out at 620°C and two at 680°C. The results were D"(8.4$1.5)]10~5 cm2 s~1 (at 620°C) and D"(9.3$2.5)]10~5 cm2 s~1 (at 680°C), respectively, where the first value in the parantheses represents the mean value of both experiments and the second one the difference between the single experiments.
4. Discussion 4.1. Temperature dependence In Fig. 2 we have presented the diffusivities of Al in Ga in the form of an Arrhenius plot. One sees that within the error bars it is possible to connect the data by a straight line (dashed) according to D(¹)"D e~*E@RT, (2) 0 with *E"(8.6$0.6) kJ mol~1 and D "(3.3$ 0 0.4)]10~4 cm2 s~1. However, the plot strongly suggests systematic deviations from this relation.
524
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
Fig. 2. Arrhenius plot of the diffusion coefficients given in Table 1 (open squares and error bars).
An alternative relation due to Swalin [9] is D(¹)"A¹2.
(3)
This relation was confirmed by Frohberg et al. in space experiments for tracer diffusion of 124Sn—112Sn [10] and interdiffusion of Sn and In [11]. A fit of our data to Eq. (3) leads to a value of A"(1.1$0.04)]10~10 cm2 K~2 s~1, but has an offset at ¹"0 K of (1.2$0.3)]10~5 cm2 s~1, which does not agree with theory. Thus, our data suggest a less than quadratic temperature dependence of D. A fit to the empirical relation D(¹)"B¹n
determined over rather limited temperature ranges [13—15]. These data are shown together with our results in Fig. 3. The thick line represents Ga selfdiffusivities obtained by Petit and Nachtrieb [13] between 30 and 100°C in capillaries with diameters of 0.5, 0.76 and 1 mm. Other experiments covered broader temperature ranges: about 260 K, using capillaries of 0.7 mm diameter (Larsson et al. [14], dashed line) and about 370 K, using capillaries of 1.6 mm diameter (Broome and Walls [15], medium-thickness solid line). The full squares represent our Al—Ga data (capillary diameter: 1.0 mm) with the thin-line fit to Eq. (4). Our extrapolated curve coincides with the results of Petit and Nachtrieb, but lies significantly lower than the values of the other two groups. No correlation between capillary diameters and diffusion coefficients can be found. Therefore, wall effects do not seem to be important. If we treat Al—Ga as an ideal system, we can apply the Darken equation [16] to our data. This results in a self-diffusion coefficient of Al lower than that of Ga. However, there is some reason to suppose that Al atoms have a higher mobility and therefore a higher self-diffusion coefficient than Ga. The most significant difference between Al and Ga lies in the atomic mass, Ga being about 2.5 times heavier than Al. (Comparisons of atomic volumes at the melting points [17] as well as diffraction data [18] clearly show that
(4)
yields B"(1.52$0.73)]10~9 cm2 s~1 K~n and n"(1.62$0.07), which is shown by the solid curve in Fig. 2. For a more definitive discussion of the temperature dependence of D — with regard to different diffusion models of liquids — the precision of our data is still too low. 4.2. Comparison with data for pure melts of Ga and In A comparison of our data with self-diffusion data using a modified Darken equation [12] is not possible, since neither the self-diffusion coefficients of Al nor the activity of Al in Ga are available. Only the self-diffusivities of Ga have been
Fig. 3. Compilation of self-diffusion coefficients of liquid Ga [13—15] and comparison to the diffusion data of Al in Ga (this work).
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
525
nearest-neighbour distances are nearly the same for liquid Ga and Al.) That means, Al—Ga is not an ideal system and the diffusion of Al is mainly determined by the properties of the Ga matrix, such as its viscosity. Iida and Guthrie [19] showed that the modified Stokes—Einstein equation k¹ D" , 4pkr
(5)
with k¹ the thermal energy and r the radius of the diffusing particle, can be used to correlate D and the viscosity k. Note that the factor of 4 in the denominator is based on a hydrodynamic calculation by Sutherland [20] for the case where the sizes of the diffusing and matrix particles are equal. According to the results of Ashcroft and Lekner [21], the values of r can be deduced from the atomic volume » at the melting point, assuming hard spheres and . a packing fraction of 0.45 using the relation
A B A
B
1@3 » 1@3 4p . " "2.1r, r30.45~1 N 3 A
Fig. 4. Compilation of diffusion coefficients of Al in Ga (this work) together with self-diffusivities deduced from viscosity data for pure Ga [22,23] using Eq. (5).
(6)
where N is the Avogadro’s number. Hard-sphere A radii of Al, Ga and In, calculated from Eq. (6) are between the onset and the maximum of the first peak of the pair-distribution functions deduced from diffraction data of Waseda [18]. The viscosity of liquid Ga was investigated by Spells [22] and more recently by Iida et al. [23]. Using Eq. (5) and r "2.67 A_ , we transformed G! these viscosity data to corresponding values of diffusion coefficients. The result is shown in Fig. 4, by open circles for Ref. [23] and by dots for Ref. [22]. Both are in very good agreement with our data (full squares and thin line) and obviously do not match the data of Refs. [14,15]. A compilation of data for In melts is plotted in Fig. 5. Self-diffusion data for pure In were given by Lodding [24] (solid line) and Careri et al. [25] (dashed line). Using the viscosity data of Culpin [26] and Iida et al. [23], and r "1.43 A_ , we I/ calculated diffusivities from Eq. (5). The results, presented in Fig. 5 by the dots and open circles, respectively, agree well with our experimental data (full squares).
Fig. 5. Compilation of diffusion coefficients of Al in In (this work) and self-diffusivities of pure liquid In [24,25], together with self-diffusivities deduced from viscosity data [23,26] using Eq. (5).
Although viscosities of binary mixtures of Al—Ga and Al—In are not known to the authors (even the published viscosity data of pure liquid Al scattered over half an order of magnitude), we assume that the viscosity of the Ga or In melt will not change significantly on addition of low concentrations of Al. Thus, our results appear to be representative for the behaviour of the pure melts. As a consequence, crystal growers should consider using our data when self-diffusion coefficients of liquid Ga are needed for transport estimates.
526
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527
5. Conclusions and outlook The shear-cell technique has proved to be a successful method to measure diffusion coefficients in multicomponent melts on earth. We have demonstrated with the binary system Al—Ga that the adequate choice of the concentration interval, improvements of the thermal conditions as well as various methods for the detection of disturbing parameters provide values of diffusion coefficients with a scattering between $13 and $5%. The measurements were done at temperatures ranging from 130 to 800°C. Despite this large temperature interval, the values of the diffusion coefficients of Al in Ga change only by half an order of magnitude. Therefore, the limits of error are still too high to definitively decide on the applicability of different temperature dependences predicted by the various liquid diffusion theories. Good agreement was obtained between the diffusion data for Al (maximum addition 2.5 at%) in Ga and the viscosity data of pure liquid Ga on correlation with a Stokes—Sutherland—Einstein relation. Therefore, it is most likely that the diffusion of Al is determined by the properties of the Ga melt. This is in contrast with Ga self-diffusivities obtained by other authors whose values are significantly higher than our Al—Ga data. This may be due to weaker convective transport during our experiments. Because of the different atomic masses of the components, Al—Ga and Al—In melts are suitable systems to investigate thermodiffusive separation. If these measurements are performed with shear cells on the ground, our experience shows that unavoidable radial temperature gradients within the capillary will pose a serious problem. Using different capillary diameters on the ground and comparing the results to kg-data would help to quantify the influence of these radial gradients. Therefore experiments in microgravity will be essential.
Acknowledgements The authors gratefully acknowledge the assistance of T. Stra{er and D. Buck who programmed
the calculation routines, K. Hugle who carried out the AAS measurements and H. Mu¨ller who performed the thermodiffusion experiments. This work was financially supported by DFG. Appendix A The thermodiffusive drift velocity v was esti5) mated from the thermodiffusive coefficient D@ in the system Al—In (determined in our group as D@(Al)"!3.5]10~8 cm2 s~1 K~1) using ¹(x) v "!D@ , 5) x and assuming that (a) the thermodiffusive coefficient of Al—Ga does not exceed that of Al—In, (b) D@ is concentration independent, (c) the temperature gradient in the capillary is constant (which was experimentally determined) and (d) the solute (Al) concentration is low (it always was less than 2.5 at%). The gravitational sedimentation velocity was estimated from the relation v "uF, ' where u is the mobility of the diffusing particles (evaluated from the experimental values of D using the Einstein relation u"D/k¹) and F is the drift force resulting from the different density of the components (evaluated as F"g*m, with *m the difference of the atomic masses of Al and Ga, assuming equal atomic volume and a low concentration of the solute). References [1] G. Mu¨ller-Vogt, P. Bra¨uer, R. Ko¨{ler, Z. Peranic, J. Schlegelmilch, T. Stra{er, W. Trillsam, in: L. Ratke (Ed.), Immiscible Alloys and Organics, DGM-Informationsgesellschaft, Oberursel, 1992, p. 125. [2] R. Ko¨{ler, Thesis, Karlsruhe, 1990. [3] G. Mu¨ller-Vogt, R. Ko¨{ler, J. Crystal Growth 186 (1998) 511. [4] J.E. Hart, J. Fluid Mech. 49 (2) (1971) 279. [5] J.P. Praizey, Int. J. Heat Mass Transfer 32 (1989) 2385. [6] Y. Malmejac, J.P. Praizey, in: T.D. Guyenne, J. Hunt, (Eds.), Proc. 5th European Symp. on Material Sciences Under Microgravity, ESA Sp-222, ESTEC Reproduction Services, Noordwijk, The Netherlands, 1984, p. 147.
P. Bra( uer, G. Mu( ller-Vogt / Journal of Crystal Growth 186 (1998) 520—527 [7] Y. Malmejac, G. Frohberg, in: H.U. Walter (Ed.), Fluid Sciences and Material Science in Space, Springer, Berlin, 1987, p. 159. [8] P. Bra¨uer, Thesis, Karlsruhe, 1995. [9] R.A. Swalin, Acta Metall. 7 (1959) 736. [10] G. Frohberg, K.H. Kraatz, H. Wever, ESA-SP-222, 1984, p. 201. [11] G. Frohberg, K.H. Kraatz, H. Wever, D1-Symp, Norderney. [12] P.G. Shewmon, Diffusion in Solids, McGraw-Hill, New York, 1963, p. 126. [13] J. Petit, N.H. Nachtrieb, J. Chem. Phys. 24 (1956) 1027. [14] S. Larsson, L. Broman, C. Roxbergh, A. Lodding, Z. Naturf. A 25 (1970) 1472. [15] E.F. Broome, H.A. Walls, Trans. Metall. Soc. AIME 245 (1969) 739. [16] L. Darken, Trans. AIME 174 (1948) 184.
527
[17] D.J. Steinberg, Metall. Trans. 5 (1974) 1341. [18] Y. Waseda, The Structure of Non-Crystalline Materials, Appendices 8 and 9, McGraw-Hill, New York, 1980. [19] T. Iida, R.I.L. Guthrie, The Physical Properties of Liquid Metals, Clarendon Press, Oxford, 1993, p. 214. [20] W. Sutherland, Philos. Mag. 9 (1905) 781. [21] N.W. Ashcroft, J. Lekner, Phys. Rev. 145 (1966) 83. [22] K.E. Spells, Proc. Phys. Soc. (London) 48 (1936) 299. [23] T. Iida, Z. Morita, S. Takeuchi, J. Jpn. Inst. Metals 39 (1975) 1169. [24] A. Lodding, Z. Naturf. A 11 (1956) 200. [25] G. Careri, A. Paoletti, M. Vicentini, Nuovo Cimento 10 (1958) 1088. [26] M.F. Culpin, Proc. Phys. Soc. (London) B 70 (1957) 1069.