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Growth Kinetics of β Ti Solid Solution in Reaction Diffusion


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Growth Kinetics of

-Ti Solid Solution in Reaction Diffusion

Osamu Taguchi


, Gyanendra Prasad Tiwari


and Yoshiaki Iijima


1Materials Science and Engineering, Miyagi National College of Technology, Natori 981-1239, Japan

2Department of Materials Science, Graduate School of Engineering, Tohoku University, Sendai 980-8579, Japan

The growth kinetics of the-Ti solid solution phases in several pure metal diffusion couples have been investigated. The layer growth of -Ti solid solution phases is observed to obey the parabolic law, indicating that the rate controlling process is diffusion. The temperature dependence of the rate constant of the layer growth shows a linear relationship in the Arrhenius plots at higher temperatures. On the other hand, it deviates from the linearity at low temperatures. This deviation from the linearity is shown to be related to the variation of compositional range of the-Ti solid solution with temperatures. Additionally, the activation energy for the layer growth is found to be equal to sum of the activation energy for interdiffusion and the formation enthalpy of-Ti solid solution in the eutectoid temperature range.

(Received September 12, 2002; Accepted October 17, 2002)

Keywords: reaction diffusion, interdiffusion, growth kinetics, growth rate constant, activation energy for diffusion

1. Introduction

Study on growth kinetics of the layer in reaction diffusion provides valuable information for several industrial proces-ses, such as metal cladding, diffusion bonding, combustion synthesis and fabrication of superconducting metallic com-pounds, and so on. Many investigations have been carried out on the layer growth of intermetallic compounds formed in the diffusion couples.1,2)Fundamental questions in the reaction diffusion are whether the growth is diffusion controlled, whether all the phases predicted from the equilibrium phase diagram are present and whether the interface compositions approach to those predicted from the equilibrium phase diagram. The previous investigations have shown that all the equilibrium phases existing in the phase diagram do not always appear in the diffusion zone3–5)and that in some alloy systems, the interface compositions deviate from those in the phase diagrams.6)The growth of the layer thickness,d, of an intermetallic compound after the diffusion time, t, is generally expressed by

d¼ktn ð1Þ

wherekis the growth rate constant andnthe time exponent. In many cases of the reaction diffusion,n¼1=2, indicating that the growth is controlled by the diffusion. However, in some systems non-parabolic growth without an incubation time has been observed at low temperatures, where some interface reaction or short-circuit diffusion such as grain boundary diffusion predominates.7,8)

The present paper reports the reaction diffusion studies between pure metals Ti–X (X = Fe, Ni, Cr, Co, Cu, Ag and Au). It has been shown that the Arrhenius plots ofk2of the -Ti solid solutions, designated as (-Ti(X)), are bent down-ward at lower temperatures. The bending of the Arrhenius plot of k2 are shown to be related to the temperature dependence of the composition range C of the -Ti(X). Furthermore, the activation energies obtained from the Arrhenius plots for k2ðQ

kÞ, the interdiffusion coefficient

~ D

DðQDÞandCðQCÞare related asQk¼QDþQC.

2. Experimental Procedure

Rods of pure metals, Ti, Fe, Cr, Co, Ni and Ag were made by argon-arc melting of titanium sponge of 99.6%, electro-lytic grades of iron (99.9%), chromium (99.4%), nickel (99.9%) and silver granules of 99.99% purity. The grain size of arc-melted metals ranged from 0.5–1 mm. Pure copper rods were made by vacuum melting of oxygen-free Cu chips of 99.99% purity in an alumina crucible and casting into steel mold followed by hot-forging, rolling and machining into 10 mm diameter rods. All the metallic rods were machined to make disc specimen of 8–10 mm in diameter and 5 mm in thickness. The regular square pieces 10 mm in size and 1 mm in thickness were machined from 99.99% Au pure sheets. For grain growth, the specimens were annealed at 1273 K for 3 days in a vacuum furnace evacuated to 3 mPa. The resultant grain size was 1–3 mm.

Flat face of the disc specimens were ground on water-proof abrasive papers and polished on a buff with fine alumina paste. Semi-infinite type pure metal Ti–X diffusion couples were set in a stainless steel holder and pressed by screws. The couples were welded below the eutectoid transition tempera-ture of the each alloy system. The diffusion annealing were performed in an evacuated quartz tube with Ti foils as gettering materials to remove oxygen, nitrogen and hydro-carbons from the annealing atmosphere. The diffusion annealing was carried out between the eutectoid temperature of the system Ti–X and – transition of pure Ti. The temperatures were controlled within1K.

After the diffusion annealing, the couples were cut parallel to the diffusion direction and the section was polished in the same way as described above. The metallographic structure of the diffusion zone was examined optically by using an etchant containing HF (3%), HNO3(5%) and HCl (5%). The

-Ti(X) phases formed in the diffusion zone were etched in dark-brown color. The layer thickness of the-Ti(X) phase greater than 10mmwas measured by an optical microscopy, while the layer thickness less than 10mmwas measured by use of a backscatter electron image of a scanning electron microscope (Jeol 5600LV). An electron probe microanalyser (Shimadzu EPMA-8705) was used to determine the


tration profile of Ti and metal X.

The interdiffusion coefficients in the-Ti(X) phases were determined by the method of Sauer and Freise.9)

3. Results

3.1 Growth kinetics

Figure1shows that the layer thickness (d) of the-Ti(Fe) and -Ti(Au) phases as a function of time as examples. Similar growth plots have been obtained for all other systems studied and discussed here. The thickness of the -solid solution layer phases formed by reaction diffusion is proportional to the square root of diffusion time. Thus the layer growth obeys the so-called parabolic law, viz.

d ¼kt1=2 ð2Þ

It is indicative of the diffusion controlled growth. The Arrhenius plots of k2 of -Ti(Fe), -Ti(Co) and -Ti(Ni) phases are plotted in Figs.2–4. They show good linearity at higher temperatures, while at lower temperatures, the plots deviate downward from the extrapolated high temperature linear line. In the -Ti(Cu), -Ti(Ag) and -Ti(Au), the temperature region between the allotropic transformation point and the eutectoid temperature is narrow. Hence, although the non-linearity in the Arrhenius plots does exist in their cases also, it is difficult to separate high temperature and low temperature regions unambiguously.

The activation energies,Qkfor the layer growth of the

-Ti(X) phases in high temperature ranges are compiled in Table 1.

Fig. 1 (a) Plot of layer thickness of-Ti(Fe) solid solution vs. square root of diffusion time, (b) Plot of layer thickness of-Ti(Fe) solid solution vs. square root of diffusion time.

Fig. 2 Temperature dependence ofk2andDD~for-Ti(Fe) solid solution.

Fig. 3 Temperature dependence ofk2andDD~for-Ti(Co) solid solution.

[image:2.595.325.524.70.376.2] [image:2.595.327.525.440.743.2] [image:2.595.55.286.467.726.2]

3.2 Interdiffusion coefficients


Figure 5 shows the concentration dependence of the interdiffusion coefficient DD~ in the -Ti(Fe) phase. The composition range of the-Ti(Fe) determined by EPMA at each diffusion temperature is also shown in Fig.5. In the -Ti(Fe),DD~ deceases with increasing the concentration of Fe. The Arrhenius relationship ofDD~shows good linearity in the whole of the temperature range. For the others -Ti(X) phases also, the linearity of Arrhenius plots of DD~ is maintained in the entire temperature range of the experiment. Though the diffusion coefficient varies with the composition, the activation energies are independent of the alloy composi-tion. This is clearly shown in Fig.6. Nearly identical slops of all three plot curves indicates that the activation energy is independent of composition despite the variation ofDD~ with composition. The activation energiesQDfor interdiffusion in Table 1 Comparison of values Qk, QD, Q, (QDþQ), QC and



kJmol1 kJmol1 kJmol1 kJmol1 kJmol1 kJmol1

-Ti(Fe) 360 240 120 360 50 290 -Ti(Co) 370 230 130 360 80 310 -Ti(Ni) 400 210 160 370 160 370

-Ti(Cr) 380 150 60 210

-Ti(Cu) 380 150 210 360

-Ti(Ag) 810 160 680 840

-Ti(Au) 600 350 200 550

-Ti(Pd)3) 270 130 90 220

Fig. 5 Concentration dependence ofDD~in-Ti(Fe) solid solution. Fig. 4 Temperature dependence ofk2andDD~for-Ti(Ni) solid solution.

[image:3.595.318.537.73.438.2] [image:3.595.47.292.448.572.2] [image:3.595.314.534.502.768.2]

-Ti(X) phases are listed in Table1. In each alloy system, it is observed thatQk is much higher thanQD.

4. Discussion

The Arrhenius plot ofk2of-Ti(X) phases does not follow

the expected linear trend at all temperatures but shows a downward deviation from the linearity at low temperatures. Further, the activation energy for the layer growth of-Ti(X) phases is much higher than that for interdiffusion. A typical phase diagram of Ti-rich side for Ti–X alloy systems investigated by the present work is shown schematically in Fig.7. The composition range of the -Ti(X) phases, C, varies significantly with the temperature. Therefore, it is likely thatCmay influence the growth of-Ti(X) phases. In the following discussion, an attempt is made to assess the importance of the C in the growth of -Ti(X) phases. Figure8 schematically shows the concentration profile of a diffusion zone in which an intermetallic compound is present. According to Wagner,10,11)XandXare given by

X¼2ðDtÞ1=2 ð3Þ

X ¼2ðDtÞ1=2 ð4Þ

W¼ ðXXÞ ¼2jjðDtÞ1=2 ð5Þ

whereXandX are distance from the Matano plane to= interface and= interfaces, and are proportionality constants, W and D are the layer growth and the inter-diffusion coefficient respectively.

From eqs. (3), (4) and (5), we have

W2=t¼k2¼4ðÞ2D ð6Þ

The value of4ðÞ2for some intermetallic compounds in Al–Cu system was calculated by Funamizu and Wata-nabe12)and by Nohara and Hirano13)for Fe–Mo system. In the present work, the magnitudes of and or the -Ti(Fe), -Ti(Ni) and -Ti(Co) phases have been evaluated from the penetration curve and the interdiffusion coefficient. Assuming Arrhenius type of behavior, the temperature dependence of 4ðÞ2 for these phases are shown in Fig.9. The similarity between the temperature dependence of4ðÞ2andk2is clearly recognized, namely, that the plots are linear at higher temperatures and bend downward at lower temperatures. The values of Q obtained from the

linear part of the Arrehenius plot of4ðÞ2are listed in Table1.

Funamizu and Watanabe14) proposed, on empirical grounds, a relationship, between k2 and DD~ which may be expressed as follows;

ki¼pðDD~iCiÞa ð7Þ

wherepandaare constants.ki,DD~iandCiare respectively the rate constant, interdiffusion coefficient and composition

Fig. 8 Schematic concentration profile in which an intermetallic com-pound formed in a diffusion zone.

Fig. 9 Temperature dependences of4ðÞ2in-Ti(Fe), Ti(Co) and

Ti(Ni) solid solutions. Fig. 7 Schematic phase diagram of Ti–X alloy system.

[image:4.595.77.261.347.576.2] [image:4.595.321.531.495.757.2] [image:4.595.55.284.626.759.2]

range for the ith phase in the reaction diffusion zone. Ge¨sele and Tu15)have theoretically deduced the equation

k2i ¼G ~DDiCi ð8Þ

where G is a constant related to atomic volume. Equations (7) and (8) suggest that k2

i is a function of DD~i and Ci. A comparison of eq. (6) with eqs. (7) and (8) shows that4ð Þ2C are same. Experimentally, it is found that the temperature dependence of 4ðÞ2 andC are same. This fact emphasizes the importance of the compositional range of a phase on it’s growth in polyphase diffusion. Following Funamizu and Watanabe,14)we apply equation (7) to our data. Figure 10 shows the log–log plots of ki and ðDD~iCiÞ1=2. The Fig.10includes data from the present work as well as those obtained by Lamparteret al.3)for-Ti(Pd) phase. Excepting the-Ti(Cr) and the-Ti(Co) systems, the slope is nearly equal to 1. For intermetallic compounds, similar results were reported by Funamizu and Watanabe.14) The value ofn¼1in Fig.10shows that in the eq. (7),a¼

p¼1 for -Ti(X) phases. Hence, the growth rate equation for-Ti(X) phases may be written as

k2¼W2=t¼DD~C ð9Þ

Generally, the temperature dependences ofk2andDD~ obey Arrhenius behavior. If the temperature dependence ofCis also expressed by an Arrhenius equation, eq. (9) takes the form,

k02expðQk=RTÞ ¼D0expðQD=RTÞC0expðQC=RTÞ


where k20, D0, C0 are the pre-exponential terms in the

Arrhenius equations fork2,DD~

andC. Therefore, from eq. (10), we have

Qk¼QDþQC ð11Þ

The temperature dependences ofCfor the-Ti(X) phases from the allotropic transformation temperature to the eutectoid temperature are shown in Fig.11. For the-Ti(Cu) and -Ti(Ag), the value of C obtained from diffusion couple are also shown by open mark. It is obvious that at high temperatures, the temperature dependence of C can be expressed by an Arrhenius type of equation. HenceQCcan be

termed as enthalpy or heat of formation of -Ti(X) solid solution phases.16)The values ofQCare listed in the Table1.

It is noted that for the layer growth of the-Ti(X) phases, the activation energy for the layer growthQkis equal to the sum

of the activation energy for interdiffusion and the formation energy of the-Ti(X) solid solution. Therefore, the effect of the compositional width on the growth of -Ti(X) solid solution is properly described by the eq. (9).

5. Conclusions

Using pure Ti–pure metal X diffusion couple, the layer growth of the-Ti(X) solid solution phase has been studied in several alloy systems in the temperature range between the allotropic transition of Ti and the eutectoid transition in the Ti-rich side. The present results can be summarized as follows;

(1) The layer growth of -Ti(X) phase is predominantly controlled by diffusion.

(2) The second power of the layer growth constant k is directly proportional to the product ofDD~ andC. (3) The layer growth of the-Ti(X) phase is observed to


obey the parabolic law. The Arrehenius plot of the growth rate constant shows a good linearity at higher temperatures. At lower temperatures, a departure from linearity is observed. This behavior is related to the influence of compositional width of the -Ti(X) solid solution on it’s growth rate in the diffusion zone.


Fig. 11 Temperature dependences ofCfor-Ti(X) solid solutions. Fig. 10 Logarithmic plot ofkvsðDD~CÞ1=2 for reaction diffusion in


(4) The activation energy for the layer growth of the -Ti(X) phase is equal to the sum of the activation energy for interdiffusion and the enthalpy of formation of -Ti(X) phase.


1) J. Philibert: Defect Diffusion. Forum95–98(1993) 493–506. 2) K. Hirano and Y. Iijima:Diffusion in solids, Recent Developments, ed.

by M. A. Dayananda and G. E. Murch, (AIME, Warrendale, PA, 1984) pp. 141–166.

3) P. Lamparter, T. Krabichler and S. Steeb: Z. Metallk.64(1973) 720– 725.

4) G. V. Kidson and G. D. Miller: J. Nucl. Mater.12(1964) 61–69. 5) Th. Heumann: Z. Metallk.59(1968) 455–460.

6) T. Nishizawa and A. Chiba: Trans. JIM16(1975) 767–778. 7) C. F. Bastin and G. D. Rieck: Metall. Trans.5(1974) 1817–1831. 8) F. J. J. van Loo and G. D. Rieck: Acta Metall.21(1973) 61–73. 9) F. Sauer and V. Freise: Z. Electrochem.66(1962) 353–363. 10) W. Jost:Diffusion in Solids, Liquids and Gases, (Academic Press Inc.,

New York, 1960) pp. 68–78.

11) G. B. Gibbs: J. Nucl. Mater.20(1966) 303–306.

12) Y. Funamizu and K. Watanabe: Trans. JIM12(1974) 46–50. 13) K. Nohara and K. Hirano: Trans. Iron and Steel Inst. Jpn.37(1977)


14) Y. Funamizu and K. Watanabe: Trans. Jpn. Inst. Light Met.25(1975) 179–185.

15) U. Ge¨sele and K. N. Tu: J. Appl. Phys.54(1982) 3252–3260. 16) R. A. Swalin:Thermodynmics of Solids, (John Wiley and Sons, 1972)


Fig. 2Temperature dependence of k2 and D~ for �-Ti(Fe) solid solution.
Fig. 6Temperature dependence of D~ in �-Ti(Fe) solid solution.
Fig. 9. The similarity between the temperature dependenceof 4ð�� � ��Þ2 and k2 is clearly recognized, namely, that theplots are linear at higher temperatures and bend downward at
Fig. 11Temperature dependences of �C for �-Ti(X) solid solutions.


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