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Scripta Materialia 60 (2009) 377–380

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Determination of diffusion parameters and activation energyof diffusion in V3Si phase with A15 crystal structure

A.K. Kumar,a T. Laurila,b V. Vuorinenb and A. Paula,*

aDepartment of Materials Engineering and Centre for Electronics Design and Technology, Indian Institute of Science,

Bangalore 560012, IndiabLaboratory of Electronics and Production Engineering, Helsinki University of Technology, FIN 02015 TKK, Finland

Received 16 October 2008; revised 4 November 2008; accepted 4 November 2008Available online 17 November 2008

Diffusion parameters, such as the integrated diffusion coefficient of the phase, the tracer diffusion coefficient of species at differenttemperatures and the activation energy for diffusion, are determined in V3Si phase with A15 crystal structure. The tracer diffusioncoefficient of Si was found to be negligible compared to the tracer diffusion coefficient of V. The calculated diffusion parameters willhelp to validate the theoretical analysis of defect structure of the phase, which plays an important role in the superconductivity.� 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Interdiffusion; Intermetallic; Superconductors; Metal silicide; Tracer diffusion

Intermetallic compounds with the A15 crystalstructure have attracted much attention, since most ofthe compounds with this structure are superconductors.Studies in this group of materials have to date concen-trated to a great extent on the Nb3Sn and V3Si phases[1–11]. Both theoretical and experimental studies havebeen conducted to determine the defect concentrations,which strongly influence on the atomic mechanism ofdiffusion [12,13]. It is already known that mainly antisitedefects are present in the structure to accommodate thedeviation of the composition from stoichiometry [13].Further development on the theoretical understandingof the atomic mechanism of diffusion was not possiblebecause of the lack of sufficient reliable data on the dif-fusion parameters in different systems. To the authors’knowledge, V3Ga is so far the only phase for which dataon tracer diffusion coefficients are available [14].Although there are plenty of articles available on thegrowth kinetics of the Nb3Sn phase, it was not possibleto calculate the diffusion parameters in this phase, sincethe experiments were conducted with the main aim ofimitating the bronze technique (i.e. interaction betweenNb and Cu(Sn) bronze alloy to grow Nb3Sn at the inter-face), which is one of the popular manufacturing routes

1359-6462/$ - see front matter � 2008 Acta Materialia Inc. Published by Eldoi:10.1016/j.scriptamat.2008.11.003

* Corresponding author. Tel.: +91 80 2293 3242; fax: +91 80 23600472; e-mail: aloke@materials.iisc.ernet.in

for producing this superconductor. This is not an idealmethodology to determine the diffusion parameters thatare required to validate the theoretical analysis in thebinary Nb–Sn system.

In this study, we have investigated the V3Si phase inorder to determine the diffusion parameters via the dif-fusion couple technique. This technique not only helpsto determine interdiffusion or integrated diffusion coeffi-cients, but also to indirectly extract the data on the tra-cer diffusion coefficients following systematic Kirkendallmarker experiments and from the knowledge on thermo-dynamic parameters required for the analysis. The tracerdiffusion data calculated following this technique arefound to be comparable with the data calculated follow-ing the conventional tracer method [15]. The aim of thisstudy is therefore to determine the integrated diffusioncoefficient of the phase, the tracer diffusion coefficientsof the species at different temperatures and the activa-tion energy for diffusion.

Vanadium (99.98 wt.%) and silicon (99.999 wt.%)supplied by Alfa Aesar are used in this study. An alloywith average composition of V 29 at.%–Si was melted(containing a mixture of V5Si3 and V3Si phases) to cou-ple with pure V, so that only one phase V3Si grows in theinterdiffusion zone. The alloy was prepared in an arcmelting furnace under an argon atmosphere and re-melted three times to ensure homogeneity. Further itwas equilibrated at 1400 �C for 24 h under vacuum

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378 A. K. Kumar et al. / Scripta Materialia 60 (2009) 377–380

(�10�6 mbar). Slices 7 � 7 � 1 mm3 were cut by slow-speed diamond saw and bonding faces were groundand polished following standard metallographic prepa-ration up to 1 lm finish. Polished specimens were thenclamped together and annealed at eight different temper-atures in the range of 1100–1350 �C for 24 h in vacuum(�10�6 mbar). Prior to annealing, titanium dioxide orzirconium dioxide particles were applied on to one ofthe bonding surfaces as inert Kirkendall markers tostudy the relative mobilities of the species. After theexperiment, the bonded specimen was cross-sectionedby slow-speed diamond saw and the interdiffusion zonewas examined by scanning electron microscopy (SEM)combined with energy dispersive spectrometry (EDS).

The interdiffusion zone of the diffusion couple an-nealed at 1350 �C for 24 h is shown in Figure 1. Theequilibrated alloy with nominal composition V29 at.%–Si has two phases, V5Si3 and V3Si, as can beseen in the micrograph. Titanium dioxide/zirconiumdioxide particles were found at the V/V3Si interface,which indicates that V diffuses at a much faster ratecompared to the negligible rate of Si diffusion throughthe product phase. The thickness was measured at differ-ent places in the interdiffusion zone and then the averagewas taken to calculate the diffusion parameters. Thephase grows within a very narrow homogeneity rangefollowing the phase diagram [16] and it is not possibleto determine the vanishingly small concentration gradi-ent to calculate the interdiffusion coefficient. To circum-vent this problem, Wagner [17] introduced the conceptof integrated diffusion coefficient ðeDintÞ, which can beexpressed in terms of composition of V as:

eDint ¼Z N 00V

N 0V

eD dN V ¼ eDDNV ; ð1Þ

where eD (m2 s–1) is the interdiffusion coefficient, NV isthe composition of V and DNV ¼ N 00V � N 0V is the narrowhomogeneity range of the product phase (V3Si). Since

JSi

J V

V Si3

I

3V Si5

V Si3+ V

XΔ V Si3

II

V Si3

3V Si5

V Si3 V

TiO /ZrO2 2

Figure 1. Interdiffusion zone of the diffusion couple annealed at1350 �C is shown. The schematic diagram explains the flux of thespecies V and Si.

just one phase grows in the interdiffusion zone betweenthe end-members (N�V ¼ 0:71 and NþV ¼ 1), the inte-grated diffusion coefficient for the growth of the productphase V3Si can be calculated following:

eDV 3Siint ¼

NV 3SiV � N�V

� �NþV � NV 3Si

V

� �NþV � N�V

DxV 3Si

� �2

2t; ð2Þ

where NV 3SiV ¼ 0:75 is the average composition, DxV 3Si is

the thickness of the V3Si phase, and t (s) is the annealingtime. The calculated integrated diffusion coefficient ofthe phase at different temperatures is plotted withrespect to inverse of temperature (1/T), as shown inFigure 2, following the Arrhenius equation:

eDV 3Siint ¼ D0 exp � Q

RT

� �; ð3Þ

where D0 (m2 s–1) is the pre-exponential exponent, Q(J mol–1) is the activation energy and R (=8.314 J mol–1 K–1) is the gas constant. The activation energy for dif-fusion is calculated to be 272 kJ mol–1.

Further, the interdiffusion coefficient, eD is related tothe tracer diffusion coefficient of the species,D�Si and D�V , by [15]:

eD ¼ NSiD�V þ NV D�Si

� �W

d ln aV

d ln N V

� �; ð4Þ

where NSi (=0.25) and NV (=0.75) are the compositionof Si and V in the V3Si phase, d ln aV

d ln NVis the thermody-

namic parameter, aV is the activity of element V, andW is the vacancy wind effect and can be considered as1 [15]. By replacing Eq. (4) in Eq. (1) and following Fig-ure 1, we can write:

eD int ¼Z II

INSiD�V þ NV D�Si

� �NV d ln aV

¼ NSiD�V þ NV D�Si

� �NV ln aI

V � ln aIIV

� �; ð5Þ

where I and II represent the interfaces from which theproduct phase grows, as shown in Figure 1.

Further following Figure 3, Eq. (5) and from thestandard thermodynamic relation, lV = GV + RTlnaV,we can write:

Figure 2. The integrated and tracer diffusion coefficient at differenttemperatures is plotted with respect to the Arrhenius equation.

3V Si5G V Si3

G

VG

μVII

μVI

XV

GΔ Gr V

o

Figure 3. Schematic representation of the determination of thethermodynamic parameter.

A. K. Kumar et al. / Scripta Materialia 60 (2009) 377–380 379

eDint ¼ N SiD�V þ N V D�Si

� �NV lIIV � lI

V

� �RT

¼ � N SiD�V þ N V D�Si

� �NV DrGoV

RT; ð6Þ

where GV is the free energy of pure V, lV is the chemicalpotential of V, DrG

oV is the driving force for the diffusion

of V, as shown in Figure 3. Since the mobility of the spe-cies of V is much higher than the mobility of Si as indi-cated from the Kirkendall marker experiment, Eq. (6)can further be rewritten as:

eDint ¼ �N SiD�VNV DrG

oV

RT: ð7Þ

The values of DrGoV at different temperatures are calcu-

lated from the data available in Ref. [16] and with the

Table 1. Integrated diffusion coefficient, driving force and tracerdiffusion coefficient of V at different temperatures.

Temperature(K)

eDV 3Siint

ð�10�17 m2 s�1ÞDrGo

VðJ mol�1Þ

D�Vð�10�17 m2 s�1Þ

1373 0.554 �13,274 2.5421423 1.032 �12,932 5.0351473 3.692 �12,591 19.1511523 4.889 �12,249 26.9531548 8.136 �12,079 46.2361573 10.789 �11,909 63.1931598 15.649 �11,740 94.4481623 21.304 �11,570 132.511

B

1 23

4 5

6

78 9

1011

12

A

B

Figure 4. The nearest neighbor of atoms (a) B

help of Thermocalc software, and are listed in Table 1along with the values of integrated diffusion coefficientand the values of tracer diffusion coefficients of elementV. The change in tracer diffusion coefficient with thechange in temperature is shown in Figure 2. The activa-tion energy for the tracer diffusion coefficient is calcu-lated as 295 kJ mol–1.

What has been stated above shows that we are able tocalculate only the tracer diffusion coefficient of V follow-ing the diffusion couple method However, since the inertmarkers were found at the V/V3Si interface, it was notpossible to determine the tracer diffusion coefficient ofSi. In fact, the position of the markers indicates thatthe diffusion rate of Si is negligible compared to the dif-fusion rate of V. Since we have studied the diffusionexperiments at relatively high temperature, it is expectedthat diffusion occurs mainly following the lattice diffu-sion mechanism. Similar behavior was found in theV3Ga phase [14], where the tracer diffusion coefficientof V following the tracer method could be determinedin the temperature range in which the lattice diffusionmechanism is operative. However, the tracer diffusioncoefficient of Ga could not be determined, since the dif-fusion rate of Ga was also found to be negligible in thiscase. Although the defect concentrations in this V3Siphase have not yet been determined with a reasonabledegree of confidence, the defect concentrations calcu-lated in the Nb3Sn phase provide enough indication toexplain this particular behavior [13]. It can be clearlyunderstood from the A15 crystal structure with A3Bstoichiometric ratio, as shown in Figure 4, that in a per-fect crystal each A atom is surrounded by 10 A atomsand 4 B atoms, whereas each B atom is surrounded by12 A atoms. So it can be clearly visualized that A atomscan diffuse by exchanging position with the vacanciespresent in its own sublattice. However, the diffusion ofatom B will not be possible even if there are vacanciesavailable at the next neighboring position, because thesevacancies are situated in the sublattice of element A andtherefore B atoms would have to move in the wrongsublattice. Diffusion of B is only possible dependingon the presence of antisite defects, BA (i.e. atom B occu-pies the sublattice position that is actually designated foratom A). In this case, vacancy at the sublattice that isdesignated for atom A and BA can exchange positionwithout disturbing the order of the structure. In differentphases the relative concentrations of different defectsmight vary, enabling the different ratios of diffusion

1

23 4

5 6

7

8

9

1011

12

13

14A

and (b) A are shown in a A15 structure.

380 A. K. Kumar et al. / Scripta Materialia 60 (2009) 377–380

coefficients of the elements to be found. The negligiblediffusion rate of element Si in V3Si phase indicates thatthe concentration of Si antisite defects (SiV), like Sn anti-site defects (SnNb) in Nb3Sn [13], must be too less toachieve a measurable diffusion rate of Si.

A.P. would like to acknowledge the financial sup-port received from CSIR, India (No. 22(409)/06/EMRII).

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