Growth mechanism of phases by interdiffusion and diffusion of species in the niobium–silicon system

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Acta Materialia 59 (2011) 1577–1585 www.elsevier.com/locate/actamat

Growth mechanism of phases by interdiffusion and diffusion of species in the niobium–silicon system S. Prasad, A. Paul ⇑ Department of Materials Engineering, Indian Institute of Science, Bangalore 560 012, India Received 14 September 2010; received in revised form 3 November 2010; accepted 7 November 2010 Available online 2 December 2010

Abstract The integrated diffusion coefficient of the phases and the tracer diffusion coefficients of the species are determined in the Nb–Si system by the diffusion couple technique. The diffusion rate of Si is found to be faster than that of Nb in both the NbSi2 and Nb5Si3 phases. The possible atomic mechanism of diffusion is discussed based on the crystal structure and on available details of the defect concentration data. The faster diffusion rate of Si in the Nb5Si3 phase is found to be unusual. The growth mechanism of the phases is also discussed on the basis of the data calculated in this study. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Diffusion; Defect; Intermetallics; Growth mechanism

1. Introduction High-temperature materials play a crucial role on the performance of gas turbines in land-based applications and aircraft engines. At present, Ni-based single-crystal alloys are operating at their limits (around 1150 °C) and further significant increase of the service temperature is almost impossible. Hence there is renewed interest in finding a new material that can withstand high temperatures to increase the fuel efficiency and decrease the emission of unwanted gases. With this goal, many silicide systems have been scrutinized because of their excellent creep and oxidation resistance. Mo- and Nb-based silicides have emerged as suitable materials and extensive studies are being conducted to improve the properties to make them suitable for various applications. Because of their very good strength to density ratio, Nb-based silicides, which can be used above 1350 °C, have attracted the most attention [1– 5]. These materials are basically a mixture of Nb solid solution and Nb5Si3 intermetallic compound with a very small amount of NbCr2 Laves phase. Incorporation of other ⇑ Corresponding author.

E-mail address: [email protected] (A. Paul).

alloying elements, which are mainly partitioned to these phases, helps to achieve a property balance of high-temperature strength, high fracture toughness, high creep and oxidation resistance. These are the same reasons why studies are concentrated on the evolution of the phases, stability and mechanical properties in the Nb/Si and Nb/Nb5Si3 laminate structures [6,7]. It is therefore very important to understand the growth mechanism of the phases and the diffusion mechanism of the species to gain knowledge on the phase evolutions, microstructural stability and the creep properties. To date, most diffusion studies are concentrated on thin films as laminate structures [8–11]. However, diffusion study in the bulk condition is very important to avoid extra complications because of stress or non-equilibrium phase formation. To the best of our knowledge, only one study is available. Milanese et al. [12] conducted bulk diffusion couple experiments to study the growth of phases. They calculated the parabolic growth constants and the integrated diffusion coefficients for the phases NbSi2 and Nb5Si3. However, a basic understanding of the growth mechanism of the phases and the atomic mechanism of diffusion is still lacking. To understand these mechanisms, knowledge on the relative mobilities of the species is very important, but is not available.

1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.11.022

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The aim of the present work, therefore, is first to conduct bulk diffusion couple experiments and determine the diffusion parameters. The integrated diffusion coefficient, ~ int , of the phases, the intrinsic diffusion, Di, and the tracer D diffusion coefficient, Di , of the species i are determined. The growth mechanism of the phases and the atomic mechanism of the species are then discussed based on the data calculated in this study.

Nb

NbSi2

Si

Nb5Si3

2. Experimental procedure Nb (99.98 wt.%) foil with a thickness of 1 mm, (1 0 0) Si wafers (99.999 wt.%) 0.7 mm thick and polished on one side, and silicon flakes (99.98 wt.%) were used for this study. To prepare diffusion couples with pure elements as end members, Nb and Si were cut to dimensions of around 7 mm  7 mm. Nb was polished using standard metallographic techniques to a 1 lm finish. Both Nb and Si pieces were then ultrasonically cleaned for 5 min and clamped in steel fixtures with a minimum pressure to ensure good contact. These samples were then annealed at four different temperatures in the range of 1150–1300 °C for 16 h in vacuum (107 kPa). To gain further insights into the growth process of the Nb5Si3 phase, incremental diffusion couples were prepared. An alloy with an average composition of (Nb–57 at.% Si) was melted in an arc melting furnace under an argon atmosphere. The alloy was remelted five times after turning it upside down for better mixing. The melted alloy was then homogenized at 1300 °C for 50 h in vacuum. The alloy was then cut into pieces 7 mm  7 mm  1.5 mm using electrodischarge machining to couple with Nb after metallographic preparation. Before clamping, TiO2 particles of around 1 lm in size were spread on a bonding surface to study the relative mobilities of the species. These incremental diffusion couples were then annealed in the temperature range of 1200–1325 °C for 24 h in vacuum (107 kPa). The annealed samples were then cross-sectioned using a slow-speed diamond saw. The cross-sectioned samples were again metallographically prepared for further examination. Final polishing was done with 0.04 lm colloidal silica. Nb/ Si diffusion couples were etched with a solution prepared from 10% HNO3, 15% HF and balance lactic acid to reveal the microstructure in the NbSi2 phase. Diffusion couples were then examined by optical microscopy (OM) and scanning electron microscopy (SEM) combined with energy dispersive spectrometry (EDS). The position of the TiO2 particles used as inert markers was located from the Ti X-ray peak. 3. Results and discussion 3.1. Diffusion parameters An optical image of Nb/Si diffusion couple annealed at 1250 °C for 24 h is shown in Fig. 1. Two phases, Nb5Si3 and NbSi2, are formed in the interdiffusion zone. The thick-

30 µm Fig. 1. Optical micrograph of etched Nb/Si diffusion couple annealed at 1250 °C for 24 h. The white dotted line shows the boundary between the duplex morphology in the NbSi2 phase.

ness of the Nb5Si3 phase is much less than the thickness of the NbSi2 phase. It can be seen that the etchant could reveal the grain morphology of the NbSi2 phase only. Duplex morphology in this phase layer is found. Fine and columnar grains are grown on the Nb5Si3 phase side. Much bigger and relatively equiaxed grains are found to grow on the side of Si end member. In fact the phase grows differently from two different interfaces and the presence of duplex morphology indicates the position of the Kirkendall marker plane [13–16]. The average thickness of the NbSi2 and Nb5Si3 phases varies in the range of 47.1–100 and 1.7–4.8 lm, respectively. The error in this measurement is found to be around 3% for the NbSi2 phase and 8% for the Nb5Si3 phase. It is frequently noticed that the diffusion couple fractures at the NbSi2/Si interface because of the stress generated due to very high difference in the coefficients of thermal expansion, a (aNbSi2 ¼ 11:6  106 K1 and aSi ¼ 2:6  106 K1 [17]) and molar volume 6 2 (V NbSi ¼ 8:7  106 and V Si m3 mol1 ). It m m ¼ 12:06  10 is quite possible that the interface is broken during annealing or cooling to room temperature. The parabolic growth constant, kp, of the phases can be calculated from: kp ¼

ðDxi Þ 2t

2

ð1Þ

where Dxi is the phase layer thickness and t is the annealing time. We have limited our study only to the temperature dependence of the growth rate. Milanese et al. [12] studied the growth of the phases with respect to varying time and found that both the phases grow more or less parabolically with time. The kp calculated are shown in Fig. 2 with respect to the Arrhenius equation expressed as:   Q o k p ¼ k p exp  ð2Þ RT where k op is the pre-exponential factor, Q is the activation energy for growth and T is the temperature in K. The activation energies for the growth of the phases are calculated as 285 and 191 kJ mol1 for the Nb5Si3 and NbSi2 phases,

S. Prasad, A. Paul / Acta Materialia 59 (2011) 1577–1585

-13

10

-14

10

-15

10

-16

10

-17

NbSi 2 Nb 5Si 3 Nb 5Si 3 Inc NbSi 2 [12] Nb 5Si 3 [12]

-14

10

-16

10

-18

6.2

6.4

6.6 -4

1/Tx10 ,K

6.8

6.2

7.0

respectively. The data reported by Milanese et al. [12] are shown for comparison. To understand the diffusion mechanism, it is necessary to calculate the diffusion parameters, which are essentially material constants. The phases grow with a very narrow composition range and it is not possible to measure the small concentration gradient to determine the interdiffu~ To circumvent this problem, Wagner sion coefficient, D. [18] introduced the concept of integrated interdiffusion ~ int , which is actually D ~ of a phase, let us say coefficient, D b, integrated over the unknown composition range expressed as: Z N 00i b ~ ~ dN i Dint ¼ D ð3Þ N 0i

~ b (m2 s1) is the interdiffusion coefficient of the where D phase b, Ni is the mol fraction of component i, and N 0i and N 00i are the mol fractions of the phase boundaries. ~ int can be calculated directly from the composition profile D using the expression:

6.4

6.6

6.8

7.0

7.2

1/Tx10 -4,K -1

-1

Fig. 2. Arrhenius plot of the parabolic growth constant for the NbSi2 and Nb5Si3 phases.

~b D int

10

~

k p , m 2/s

10

NbSi 2 Nb 5Si 3 Nb 5Si 3 Inc NbSi 2 [12] Nb 5Si 3 [12]

-12

Dint ,m2/s

10

1579

Fig. 3. Arrhenius plot of the integrated diffusion coefficient for the NbSi2 and Nb5Si3 phases.

and Nb5Si3 phases, respectively. As already discussed, we find fine grains on one side of the marker plane and relatively coarse grains on the other side in the NbSi2 phase. Hence there could be a difference in the contribution from grain boundary and lattice diffusion in different regions of the phase. We actually measure the effective diffusion coefficient, where the growth of the phase could be controlled by lattice as well as by grain boundary diffusion. The activation energy for effective diffusion will be lower when the contribution from the grain boundary diffusion is higher. However, because of grain growth along with the growth of the phase, and the lack of data on defects in the phase, it is almost impossible to quantify the exact contribution from lattice or grain boundary diffusion. Milanese et al. [12] reported a relatively similar value of activation energy for the Nb5Si3 phase (263 kJ mol1), but a relatively high value for the NbSi2 phase (304 kJ mol1). The ratio of tracer diffusivities could be calculated from the known location of the Kirkendall marker plane detected on the basis of grain morphology in the NbSi2 phase. In an A–B binary system, the ratio of diffusivities

2   m   b Pm¼n1 V bm  þ  3  þ  2 b Pm¼b1 V bm þ   m b N  N N  N þ N  N N  N Dx Dxm m m Dx m N N bi  N   N i i Dx i i i i i i m¼2 m¼bþ1 V V b b i i i m m 4 5 þ ¼ þ  þ  Ni  Ni 2t 2t Ni  Ni 

where N and N+ are the mole fractions of element i of the unreacted left- and right-hand side of the end member, respectively, V mm and Dxm are the molar volume and the layer thickness of the mth phase, and t is the annealing time. The molar volume of the NbSi2 and Nb5Si3 phases for the 5 Si3 2 present case is calculated as V NbSi ¼ 8:7 and V Nb ¼ m m 1 3 9:6 cm mol from the lattice parameter data available in the literature [19]. The diffusion data calculated in this study are shown in Fig. 3 along with the data calculated by Milanese et al. [12]. The activation energies calculated in this way are found to be 193 and 250 kJ mol1 for NbSi2

ð4Þ

could be determined using [20,21]: DB V A DB ¼ DA V B DA

" þ R xK # R  þ1 ð1Y Þ Y 1 þ W A N B 1 V m dx  N B xK V m dx ¼ R xK Y R  þ1 ð1Y Þ 1  W B N þ dx A 1 V m dx þ N A xK Vm

where the vacancy wind factor W i ¼ M

2N i ðDA DB Þ

ð5Þ

, Mo is a ðN A DA þN B DB Þ factor which depends on the crystal structure of the system, DA and DB are the tracer diffusion coefficient of elements A o

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and B, DA and DB are the intrinsic diffusion coefficient of elements A and B, Vi is the partial volume  molar  þ of element  i in the phase of interest, Y ¼ N B  N  N B  N B is the B Sauer–Friese variable [22], and xK is the location of the marker plane. It is almost impossible to determine the partial molar volumes for a phase with very narrow composition range and many times partial molar volumes are considered equal to the molar volume, Vm. However, Vm of a phase might change drastically over very narrow composition range, which might lead to a huge error in the calculation of the ratio of intrinsic diffusivities of the elements. On the other hand, we can calculate the ratio tracer diffusivities without knowing the details about the partial molar volumes. It is important to mention that we are neglecting the role of vacancy wind effect here. This is because it is very difficult to determine the crystal structure factor, Mo, in complex crystal structures. It is now important to note that both NbSi2 and Nb5Si3 crystallize as complex structures, hP9 (C40) and tI32 (D8l), respectively. We did not use any marker to locate the Kirkendall marker plane, since it was difficult to produce a successful Nb/Si diffusion couple when inert TiO2 particles were used. However, using etchant, duplex morphology in the interdiffusion zone was revealed in the NbSi2 phase. It has already been demonstrated that the existence of duplex morphology in an interdiffusion zone indicates the location of the Kirkendall marker plane [13–16]. The phase layer grows differently on either side of the marker plane. The ratio of tracer diffusivities in the NbSi2 phase is found to be

DSi DNb

¼ 4:8  1:4

and does not change with temperature. Since no marker plane is found in the Nb5Si3 phase, we could not determine the ratio of diffusivities. Furthermore, since the layer thickness of this phase is very small in the Nb/Si diffusion couple, we were not sure about the overall error involved in the ~ int . It was therefore necessary to conduct calculation of D the incremental diffusion couple experiments, so that only the Nb5Si3 phase grows in the interdiffusion zone with a reasonable thickness. Further it was necessary to use inert markers to locate the Kirkendall marker plane in this phase layer to determine the relative mobilities of the species, since it was not possible to reveal the grain morphology in this phase. As explained in the experimental procedure, Nb was coupled with the Nb–57 at.% Si alloy in the temperature range of 1200–1325 °C for 24 h in vacuum. The interdiffusion zone that developed at 1325 °C is shown in Fig. 4. It can be seen that the Nb–57 at.% Si alloy is actually a mixture of the NbSi2 and Nb5Si3 phases. Further, the interdiffusion zone has grown with two distinct parts. On one side there are Kirkendall pores present and the other side is free from pores. TiO2 particles used as the Kirkendall markers are found very close to the interface of these two parts. This in fact indicates that the layer grew differently at the two different interfaces. In addition, the presence of pores indicates that the diffusion rate of Si is much higher than the diffusion rate of Nb in this phase. Unequal diffusivities lead to vacancy flux in the direction

Nb-57at.Si

Nb5Si3

Nb

(Nb5Si3+NbSi2)

K

Fig. 4. SEM image of the Nb/Nb–57 at.% Si incremental diffusion couple annealed at 1325 °C for 24 h. “K” indicates the location of the Kirkendall marker plane detected from the presence of TiO2 particle.

opposite to that of the faster-moving species. When these vacancies do not get consumed by the sites that act as sinks, they coalesce to form pores. The thickness of the phase varies in the range of 21–36 lm in the temperature ~ int calculated using Eq. (4) range of 1200–1325 °C. The D for the incremental couple is shown in Fig. 3 along with ~ int is a the data calculated for the Nb/Si couple. In fact, D material constant and should not be dependent on the end member compositions. The ratio of tracer diffusion coefficients calculated using Eq. (5) is found to be DSi DNb

¼ 31  15. In this phase too we did not find any differ-

ence in the ratio of diffusivities with temperature. From the knowledge of the integrated diffusion coefficient, the ratio of the tracer diffusion coefficients of the elements and the thermodynamic parameter, one can determine the tracer diffusion coefficient of the species ~ is related to the tracer [21]. The interdiffusion coefficient, D, diffusion coefficient of the species in a A–B binary system as:     d ln aB   ~ D ¼ N A DB þ N B DA W ð6Þ d ln N B where dd lnlnNaAA ¼ dd lnlnNaBB is the thermodynamic parameter, aA and aB are the activity of the elements, and W is the contribution from the vacancy wind effect [23]. It is important once again to mention that here we neglect the role of the vacancy wind effect (i.e. we consider W = 1), which is difficult to determine in these complicated crystal structures. By substituting Eq. (6) into Eq. (3), we can write [24]: Z II   ~b ¼ D N A DB þ N B DA N B d ln aB int I     I ð7Þ ¼ N A DB þ N B DA N B ln aII B  ln aB where I and II represent the two interfaces from which the product phase b had grown from the phases a and B, as shown in Fig. 5. Using the standard thermodynamic relation lB ¼ G0B þ RT ln aB , we can write Eq. (7) as:

S. Prasad, A. Paul / Acta Materialia 59 (2011) 1577–1585

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for DSi and DNb in the NbSi2 phase and is 271 kJ mol1 for both the diffusing species in the Nb5Si3 phase. We ~ int for Nb5Si3 phase would like to mention here that the D is taken as the average of values calculated from Nb/Si and from the incremental diffusion couple. 3.2. Atomic mechanism of diffusion

Fig. 5. Schematic diagram of the driving force calculation for the growth of phase b in the a/B diffusion couple.

  I  N B lII B  lB ¼ þ RT   N D GoB B r ð8Þ ¼ N A DB þ N B DA RT where GoB is the free energy of pure B, lB is the chemical potential of element B, Dr GoB is the driving force for the diffusion of B in the b phase, as shown in Fig. 5. It should be noted that N A Dr GoA ¼ N B Dr GoB . The free energies of the NbSi2 and Nb5Si3 phases are found in Ref. [25]. The free energies of pure Nb and Si are calculated using G = H  TS (where H is the enthalpy and S is the entropy) from the known values of DH o298 (standard enthalpy of mixing at room temperature), DS o298 (standard entropy of mixing at room temperature) and cp ¼ a þ bT þ Tc2 (where a, b and c are constant and T is the temperature in K). Note that enthalpy R T and entropy are related R T c to cp following H ¼ H o298 þ 298 cp dT and S ¼ S o298 þ 298 Tp dT , respectively. The data used in this calculation are listed in Table 1. The driving force for diffusion of elements in different phases at different temperatures are calculated, and are listed in Ta ~ int , Dr Go and DSi , the tracer ble 2. From the knowledge of D Si DNb diffusion coefficients of the species can be calculated and these are shown in Fig. 6a and b. The activation energy for the tracer diffusion coefficients is calculated from the Arrhenius plot and is found to be 187 and 193 kJ mol1 ~b D int



N A DB

N B DA

Table 1 List of thermodynamic parameter related to Nb and Si used for the calculation of driving force for diffusion. Cp (J mol1 K1) Nb Si

DS 0298 (J mol1 K1) 3

5

a

b  10

c  10

23.7 23.9

4.0 2.5

– 4.1

36.4 18.8

In general, it is very difficult to get an idea about the atomic mechanism of diffusion. A certain degree of understanding has been developed for relatively simple crystal structures, such as L1o and L12 [26], based on the tracer diffusion data calculated in several studies. There has been a continuous effort to understand the diffusion mechanism in phases with other crystal structures. The calculation of D helps to elucidate the diffusion mechanism and is also very useful for validating the defect types and their concentration calculated theoretically. In fact, the concentration of different kinds of defects at different compositions and temperatures was calculated in the Nb5Si3 phase by Chen et al. [27]. The phase a-Nb5Si3 is stable below 1700 °C and has D8l structure which basically a tetragonal body centered structure with 32 atoms in an unit cell (tI32) [28]. It consists of four different sublatices: a and b for Nb, and c and d for Si [27], as shown in Fig. 7. All the nearest neighbors are listed in Table 3 [29]. Here, the most important point, relevant to this study, is that only one Si–Si nearest neighbor (NN) bond and 14 Nb–Nb NN bonds are present. Considering this, one will expect that Nb will always have a higher diffusion rate than Si if we consider vacancies to be the only defect present in the structure, and the concentration of vacancies to be in accord with the number of atoms in the different sublattices. However, our experimental result  shows that

Si diffuses D at a much higher rate than Nb DSi ¼ 31  15 . This indiNb cates the presence of complicated defect chemistry in this phase. It was noticed that defect concentrations change dramatically with even very small changes in the composition, as shown in Fig. 8. At stoichiometric composition, the concentration of all the defects is too low to make any significant diffusion. On the Nb-rich side, the concentration of vacancies on the d sublattice and Nb antisites on the c sublattice is much higher than other defects, and could actually play a role in the diffusion of elements. However, the presence of these defects cannot explain why the diffusion rate of Si is so many times higher than that of Nb. On the other hand, at the Si-rich side, the concentration of Si antisites on a and b sublattices for Nb and also vacancies on the same sublattices are much higher than the concentration of other defects. Hence significant diffusion of Si is possible because of exchange of position between these defects. However, again, it does not explain why the diffusion rate of Si is much higher than that of Nb. Diffusion of Nb atoms by exchange with the vacancies present in Nb sublattices should still be higher since the number of Si antisites on the a and b sublattices is much less than the number of Nb atoms present on the same sublattices. Hence, looking

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Table 2 List of driving force for diffusion of elements and diffusion coefficients of phases at different temperatures. Nb5Si3

T (°C)

1150 1200 1225 1250 1275 1300 1325

NbSi2

~ Nb5 Si3  1016 D int

Dr G0Si

1.03 1.42 3.27 4.21 5.78 6.31 9.62

286.8 280.4 276.5 273.4 269.9 266.5 263.1

10

3

DSi

 10

17

 10

1.65 2.41 5.72 7.57 10.7 12.03 18.87

DNb

18

 10

0.54 0.78 1.87 2.48 3.50 3.94 6.18

~ NbSi2  1015 D int

Dr G0Si  103

DSi  1015

DNb  1016

4.3 8.7

73.19 76.28

2.17 4.36

4.78 9.61

15.9

79.49

8.91

12.43

19.7

82.70

9.06

24.70

-13

NbSi 2

-14

10

-15

187 kJ/mol

*

D ,m 2 /s

10

193 kJ/mol

*

D Nb

a

*

10

D Si

-16

6.2

6.4

6.6

6.8

7.0

7.2

1/Tx10 -4,K -1 -15

10

Nb 5Si 3 -16

271 kJ/mol

-17

10

*

D ,m 2 /s

10

271 kJ/mol

Fig. 7. Crystal structure of Nb5Si3, tI32 (D81), showing four different sublattices, a, b, c and d.

-18

10

*

D Nb

b

*

D Si

-19

10

6.2

6.4

6.6

6.8

7.0

7.2

1/Tx10 -4, K -1 Fig. 6. Arrhenius plot of the tracer diffusion coefficients of elements Nb and Si in NbSi2 phase (a) and Nb5Si3 phase (b).

at the numbers, there should always be higher jump probability for Nb than Si. It is quite possible that the difference in the atomic size plays a crucial role. The atomic radii of Nb and Si are 145 and 110 pm, respectively. Since the size of the Si atom is much lower, it is possible that the rate of successful jump of this atom is more than that of Nb atom. However, further probing was very difficult based on the results from the diffusion couples because of the error associated with the data calculated. When the ratio calculated is more than 10, the error in calculations increases drastically and it becomes very difficult to determine systematic small changes with composition. It is also equally difficult to determine the exact composition even using electon probe microanalysis to find whether the phase is Si- or Nb-rich, especially when the phase has a very narrow homogeneity range. Further, dedicated work on simulation is required

Table 3 List of NN atoms in the Nb5Si3 phase. Atom 1

Atom 2

NN atoms

Atom 1

Atom 2

NN atoms

a-Nb a-Nb a-Nb a-Nb b-Nb b-Nb b-Nb b-Nb

a-Nb b-Nb c-Si d-Si b-Nb a-Nb c-Si d-Si

0 8 2 4 4 2 2 3

c-Si c-Si c-Si c-Si d-Si d-Si d-Si d-Si

c-Si d-Si a-Nb b-Nb d-Si c-Si a-Nb b-Nb

0 0 0 8 1 0 2 6

to correlate the concentration of defects with the diffusion rate of the species. It is not possible to deny that Si has a higher diffusion rate than Nb in this phase, which is rather unusual considering the type of NN bonds present in this phase. NbSi2 phase nucleates with the C40 structure, the prototype of CrSi2 [28]. It is a primitive hexagonal structure (hP9) with nine atoms per unit cell. It consists of four layers, as shown in Fig. 9. The atomic arrangement of each of the layer is the same but each layer is rotated by 60° relative

S. Prasad, A. Paul / Acta Materialia 59 (2011) 1577–1585

I

Nb 5 Si 3 p[Nb]

Nb

1583 NbSi 2

II

d

r[Nb]

d

s[Si] d

q[Si] d

Δ x KI 1

K1

Δ x KII1

III

Δ x KII2

K2

Si

Δ x KIII2

Fig. 10. Schematic representation of the growth of the phases in a Nb/Si diffusion couple. K1 and K2 indicate the possible position of the marker II III planes in the phases. DxIK 1 , DxII K 1 , DxK 2 and DxK 2 are the different sublayers grown from different interfaces, p and r denote the flux of Nb, and q and s denote the flux of Si. Fig. 8. Concentration of different defects in the Nb5Si3 as a function of composition at 1500 K [27].

Nb Si

Fig. 9. Crystal structure of NbSi2, hP9 (C40).

to the next layer. Furthermore, each Nb atom is surrounded by five Si atoms, while each Si atom is surrounded by five Nb and five Si atoms. Hence, in the absence of any Nb antisite defects, there should not be any diffusion of Nb. Si can easily diffuse via its own sublattice because of the presence of Si–Si NN bonds. However, our calculation shows that although Si has a higher diffusion rate, Nb also has a reasonable

diffusion rate through the NbSi2 phase DSi DNb

¼ 4:8  1:4 . In fact the sublayer with almost equi-

axed grains next to the Si end member (Fig. 1) was mainly grown because of the diffusion of Nb (this concept will become clearer in the next subsection). This indicates that there must be a reasonable amount of Nb antisite defects present in the structure. However, no theoretical work is available in the literature on the defect chemistry in a disilicide with C40 crystal structure.

growth mechanism of the phases. The main advantage of this technique is that we do not need to have any knowledge of the atomic mechanism of diffusion, which is rather difficult to understand. In this method, the reaction and dissociation of atoms at different interfaces and flux through the phases are related to the thickness of the phase layers. We consider the diffusion couple annealed at 1300 °C for 16 h, as shown in Fig. 1. The reaction and dissociation scheme should be considered as shown in Fig. 10. To begin with, by way of explanation, we consider the presence of the Kirkendall marker plane in both phases. As shown in Fig. 10, Nb from interface I diffused through the Nb5Si3 phase to react with the NbSi2 phase at interface II. In addition, Nb dissociated at interface II and produced NbSi2. The dissociated Nb diffused through the NbSi2 phase to react with Si at interface III and produced NbSi2. Similarly, Si from interface III diffused through the NbSi2 phase to react with the Nb5Si3 phase and produced NbSi2. Further, Si dissociated at interface II from the NbSi2 phase diffused through the Nb5Si3 and reacted with Nb at interface I to produce the Nb5Si3 phase. Dissociation of Si at interface II also leads to the formation of Nb5Si3. Hence, it is clear that a very complicated series of dissociation and reaction processes occurred at interface II. The Nb5Si3 phase had grown by consuming the NbSi2 phase and, at the same time, the same phase was consumed by the growth of the NbSi2 phase. Hence there was mutual competition for growth between these two phases. The ultimate thickness of the phase layers depends on the integrated diffusion coefficient of the phases. The reaction and dissociation schemes can be written as: Interface I 5 8 q Nb þ q½Sid ) q Nb5=8 Si3=8 3 3 Interface II (Nb5Si3 side)

3.3. Growth mechanism of the phases in a Nb/Si diffusion couple using a physicochemical approach The physicochemical approach developed by Paul and co-workers [13–16] is a very efficient tool to explain the

9 16 p Nb1=3 Si2=3 þ p½Nbd ) p Nb5=8 Si3=8 7 7 15 8 q Nb1=3 Si2=3 ) q½Sid þ q Nb5=8 Si3=8 7 7

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S. Prasad, A. Paul / Acta Materialia 59 (2011) 1577–1585

Interface II (NbSi2 side)

40 35

8 15 s Nb5=8 Si3=8 þ s½Sid ) s Nb1=3 Si2=3 7 7 16 9 r Nb5=8 Si3=8 ) r ½Nbd þ r Nb1=3 Si2=3 7 7

K (NbSi2)

NbSi2

30 Nb5Si3

2tv (µm)

25

Kvirtual (Nb5Si3)

20 15 10

Interface III

5 0

-5 -60

2 r Si þ r ½Nbd ) 3 r Nb1=3 Si2=3 p and q are the total moles per unit area of the diffusing species Nb and Si, respectively diffused through the phase Nb5Si3, and r and s are the total moles per unit area of Nb and Si, respectively, diffused through the NbSi2 phase the during  total annealing time. Therefore, at interface II, 16 p þ 8 q moles of Nb5Si3 formed by consuming NbSi2, 7  7  but 87 s þ 167 r moles were consumed by the NbSi2 phase.  Similarly, at the same interface 157 sþ 97 r moles of NbSi2  formed by consuming Nb5Si3 but 157 q þ 97 p moles were consumed by the Nb5Si3 phase. The thickness of the sublayers on either side of the marker planes in the phases can be written: 8 5 Si3 q  V Nb ¼ DxIk1 ð9aÞ m 3     16 8 8 16 5 Si3 pþ q  s þ r  V Nb ¼ DxII ð9bÞ m K1 7 7 7 7     15 9 15 9 2 sþ r  q þ p  V NbSi ¼ DxII ð9cÞ m K2 7 7 7 7 2 ¼ DxIII 3r  V NbSi m K2

Nb5 Si3 m

ð9dÞ NbSi2 m

where V and V are the molar volumes of the Nb5Si3 and NbSi2 phases, respectively. The integrated diffusion coefficient of the phases can be expressed as [13]: i

Nb5 Si3 h 5 Si3 5 Si3 ~ Nb5 Si3 ¼ V m D N Nb q þ N Nb p DxIK 1 þ DxII Nb K1 int Si 2t ð9eÞ

NbSi2   III 2 2 ~ NbSi2 ¼ V m D N NbSi s þ N NbSi r DxII ð9fÞ Nb K 2 þ DxK 2 int Si 2t The ratio of the tracer diffusion coefficients of the diffusing species in each of the phases can be expressed in terms of p, q, r and s as [13]:

DSi

q ð9gÞ ¼ DNb Nb5 Si3 p

DSi

s ð9hÞ ¼ DNb NbSi2 r By solving these equations from the calculated values of the diffusion parameters at 1300 °C, we find: q = 2.45 mol/m2 and p = 0.08 mol/m2. s = 5.071 mol/m2 and r = 1.385 mol/m2.

Nb

-40

-20

0

20

40

60

80

100

x (µm) Fig. 11. Velocity diagram constructed for the Nb/Si diffusion couple annealed at 1300 °C for 16 h. K is the Kirkendall plane in the NbSi2 phase. Kvirtual is the virtual Kirkendall plane in the Nb5Si3 phase.

DxIK 1 ¼ 62:7 lm and DxII K 1 ¼ 58:4 lm III DxII K 2 ¼ 63:45 lm and DxK 2 ¼ 36:16 lm An interesting point should be noted here: DxII K 2 and DxIII K 2 have positive values, which indicates the presence of two sublayers in the NbSi2 phase. On the other hand, DxII K 1 is negative, which indicates that the Nb5Si3 sublayer adjacent to the NbSi2 phase was consumed by the NbSi2 phase. Further insights can be gained from the plot of the velocity diagram. The velocity of the marker plane in the phases can be calculated from:

5 Si3 5 Si3 5 Si3 V Nb ðq  pÞ ¼ 2 t vNb ¼ xNb  xo ð10aÞ K K m   2 2 2 ðs  rÞ ¼ 2 t vNbSi ¼ xNbSi  xo ð10bÞ V NbSi K K m  Nb Si   NbSi  2 x x 5 3 xo x o 5 Si3 2 ¼ K 2t and vNbSi ¼ K 2t are the where vNb K K velocity of the Kirkendall plane in the phases Nb5Si3 and 5 Si3 2 and xNbSi are the distance NbSi2, respectively, and xNb K K of the Kirkendall plane with respect to the initial contact plane, xo ¼ 0. The values are calculated as: 5 Si3 vNb ¼ 1:975  1010 m=s and K 2 vNbSi ¼ 2:78  1010 m=s K 5 Si3 2 ¼ 22:75 lm and xNbSi ¼ 32:1 lm xNb K K

Following, the velocity diagram is plotted as shown in Fig. 11. Here we can see that the 2tvK = xK line intersects the velocity curve only at one point in the NbSi2 phase, which indicates the presence of Kirkendall plane in this phase. K virtual is the virtual Kirkendall plane in the Nb5Si3 phase, which indicates the presence of the marker plane in this phase if this phase is not consumed by the growth of the NbSi2 phase. 4. Conclusion We have conducted a detailed analysis in the Nb/Si system to determine the diffusion parameters of the species,

S. Prasad, A. Paul / Acta Materialia 59 (2011) 1577–1585

and then further discussed the possible atomic mechanism of diffusion. Si is found to have higher diffusion rate in both the NbSi2 and Nb5Si3 phases. It is not surprising to find a higher diffusion rate of Si in the NbSi2 phase, because the number of NN Si bonds is higher than the number of NN Nb bonds. However, to explain the substantial diffusion of Nb one may predict high concentration of Nb antisites to be present in NbSi2 structures. However, simulation work is required first to calculate the concentration of defects and diffusion in this phase and then to validate these findings with the activation energy for tracer diffusion of the species calculated in this work. Theoretical work on defects is presented for the Nb5Si3 phase. However, considering the defect concentrations calculated and number of NN bonds, the much higher diffusion rate of Si compared to Nb is very difficult to explain, unless the size of atoms plays a significant role. Further theoretical work is required to explain the diffusion behavior in this phase. The lack of understanding on the atomic mechanism of diffusion, however, does not affect the understanding on the growth mechanism of the phases, which is very important for practical applications. The physicochemical approach used in the analysis indicates that the NbSi2 phase has duplex morphology because the phase grows differently from two different interfaces. Although we could not find whether the Kirkendall marker plane is present in the Nb5Si3 phase in the Nb/Si diffusion couple, our analysis based on the data calculated from incremental diffusion couple indicates absence of the marker plane because of the consumption of part of the phase layer by the NbSi2 phase at the Nb5Si3/NbSi2 interface. The presence of the Nb3Si phase has been reported in several articles [30–32] in which diffusion was studied in thin films. The absence of this phase in the bulk diffusion couple indicates that this phase should not grow in the temperature range considered in this study or in a lower range in the equilibrium condition. Some other factors must be acting on growth in thin films in order to produce this phase. Acknowledgement A.P. would like to acknowledge the financial support from DRDO, India in carrying out this research.

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References [1] Mendiratta MG, Dimiduk DM. Mater Res Soc Symp Proc 1989;133:441. [2] Mendiratta MG, Dimiduk DM. Scripta Metall 1991;25:237. [3] Subramanian PR, Parthasarathy TA, Mendiratta MG, Dimiduk DM. Scripta Metall 1995;32:1227. [4] Jackson MR, Bewlay BP, Rowe RG, Skelly DW, Lipsitt HA. JOM 1996;48:39. [5] Bewlay BP, Jackson MR, Subramanian PR, Zhao JC. Metall Mater Trans A 2003;34:2043. [6] Gavens AJ, Van Heerden D, Foecke T, Weihs TP. Metall Mater Trans A 1999;30A:2959. [7] Van Heerden D, Gavens AJ, Subramanian PR, Foecke T, Weihs TP. Metall Mater Trans A 2001;32A:2363. [8] Nakanishi T, Takeyama M, Noya A. J Appl Phys 1995;77(2):948. [9] Matteson S, Rothi J, Nicolet MA. Rad Effects Defects Solids 1979;42:217. [10] Cheng JY, Chen LJ. J Appl Phys 1991;69(4):2161. [11] Jang SY, Lee SM, Baik HK. J Mater Sci Mater Elect 1996;7:271. [12] Milanese C, Buscaglia V, Maglia F, Anselmi-Tamburini U. Acta Mater 2003;51:4837. [13] Paul A, van Dal MJH, Kodentsov AA, Van Loo FJJ. Acta Mater 2004;52:623. [14] Paul A, Kodentsov AA, van Loo FJJ. Intermetallics 2006;163:330. [15] Ghosh C, Paul A. Acta Mater 2007;55:1927. [16] Ghosh C, Paul A. Acta Mater 2009;57:493. [17] Son KH, Yoon JK, Han JH, Kim GH, Doh JM, Lee SR. J Alloys Compd 2005;395:185. [18] Wagner C. Acta Metall 1969;17:99. [19] Samsonov GV, Neshpor VS, Ermakova VA. Zh Neorg Khim 1958;3:868. [20] van Loo FJJ. Acta Metall 1970;18:1107. [21] Paul A, Kodentsov A, van Loo FJJ. J Alloys Compd 2005;403:147. [22] Sauer F, Freise V. Z Elechtrochem 1962;66:353. [23] Manning JR. Acta Metall 1967;15:187. [24] Kumar AK, Laurila T, Vuorinen V, Paul A. Scripta Mater 2009;60:377. [25] Fujiwara H, Ueda Y, Awasthi A, Krishnamurthy N, Garg SP. J Phys Chem Solids 2005;66:298. [26] Herzig C, Divinski S. Bulk and grain boundary diffusion in intermetallic compounds. In: Gupta D, editor. Diffusion processes in advanced technological materials. New York/Berlin: William Andrew/Springer-Verlag; 2005. [27] Chen Y, Shang JX, Zhang Y. Intermetallics 2007;15:1558. [28] Schlesinger ME, Okamoto H, Gokhale AB, Abbaschian R. J Ph Equ 1993;14(4):502. [29] Leon-Escamilla EA, Corbett JD. J Solid State Chem 2001;159:149. ¯ gushi T, Nishi K, Nagai H, Numata T, Obara K. J Low Temp Phys [30] O 1980;41:13. [31] Affolter K, von Allmen M, Weber HP, Wittmer M. J Non Crystall Solids 1983;55:387. [32] D’Anna E, Leggieri G, Luches A. Thin Solid Films 1992;218:95.