Diffusion induced grain boundary migration in the Cu–Zn system

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Diffusion induced grain boundary migration in the Cu–Zn system Bathula Sivaiah, S.P Gupta⁎ Department of Materials and Metallurgical Engineering, IIT Kanpur-208016, India

AR TIC LE D ATA

ABSTR ACT

Article history:

Diffusion induced grain boundary migration (DIGM) has been studied in the Cu–Zn system

Received 14 June 2007

by exposing polycrystalline Cu to Zn vapor with a Cu-38wt.% Zn alloy as the source of Zn.

Received in revised form

The time and temperature dependence of the migration distance has been studied in the

24 August 2007

temperature range 350 to 600 °C. The composition-distance profile was obtained along the

Accepted 28 August 2007

thickness of the sheet specimen to determine the diffusivity, kDbδ at each temperature. Similarly, the quantity of Zn was determined behind the migrated grain boundary to

Keywords:

calculate the coherency strain energy and the total chemical free energy change. It was

Grain boundary migration

observed that a part of the total free energy change was used for volume diffusion ahead of

Grain boundary diffusion

the migrating grain boundary. The effective free energy change was calculated and it was

Diffusion of Zn in copper

observed that the fraction of the total free energy change used for volume diffusion

Kinetics of DIGM

increased as the transformation temperature increased. A plot of instantaneous rate of migration vs. composition behind the grain boundary has indicated that the coherency strain energy acts as the driving force for DIGM. The fine-grained layer followed a parabolic growth behavior. The diffusion coefficients calculated from the thickness of the fine-grained layer are of the same order of magnitude as those calculated from the rate of migration of the grain boundary. The diffusivity as well as the activation energy calculated from the kDbδ vs. 1 / T plot corresponds to that of grain boundary diffusion of Zn in Cu. © 2007 Elsevier Inc. All rights reserved.

1.

Introduction

When two metals are placed in contact with each other, they tend to diffuse into each other provided that they have mutual solubility. The diffusion of one metal along the grain boundaries of the other metal causes the grain boundaries to migrate. The phenomenon is termed diffusion induced grain boundary migration (DIGM). DIGM was first reported by den Broeder [1] while carrying out experiments on Cr–W diffusion couples. DIGM has now being reported in a large number of binary systems. Metallographic examination has been carried out on a number of systems, for example, Fe–Zn [2,3], Cu–Zn [4–8], Cu–Cd [9] etc. The region swept by the migrating grain boundaries gets enriched with solute with the rest of the

polycrystalline grains maintaining the original alloy composition, during DIGM. The kinetics of DIGM was first studied in the Fe–Zn system by Hillert and Purdy [2]. Since then it has been studied in the Fe–Zn [3], Cu–Cd [9] and Cu–Zn systems [4,10–13]. The coherency strain energy has been proposed to be the driving force for DIGM by Hillert [14], who formulated a relationship in terms of the modulus of elasticity, E, misfit parameter, η and the difference in compositions between the leading and trailing grains across the migrating grain boundary. This was based on the idea initially proposed by Sulonen [15]. By the addition of a third element to binary Mo–Ni alloys, it has been demonstrated that the coherency strain energy acts as the driving force for DIGM [16]. By the addition of 19.4 at.% Pd to a

⁎ Corresponding author. E-mail address: [email protected] (S.P. Gupta). 1044-5803/$ – see front matter © 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.matchar.2007.08.031

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Fig. 1 – Microstructure showing grain boundary migration in the early stages in a specimen annealed for 1 day at 480 °C.

Cu–Ni alloy, the lattice parameter has been made equal to that of pure Cu [17] and DIGM carried out in polycrystalline Cu. It, therefore, has been reasoned that DIGM should not occur in the system. However, since DIGM did occur along copper boundaries, it was concluded that coherency strain energy might not act as the driving force for DIGM. However, by ignoring the fact that the composition of Ni and Pd on the migrating grain boundaries may not have the same value as in the Cu–Ni–Pd ternary alloy in contact with copper initially, the conclusion does not appear to be very sound. The current investigation has been carried out to study the microstructure that develops, the rate of grain boundary migration, the thickness of the fine-grained layer, atomic transport mechanism and the mobility during DIGM in polycrystalline Cu exposed to Zn vapor in a larger temperature range.

2.

Experimental Procedure

Diffusion induced grain boundary migration in the Cu–Zn system was studied by the diffusion of Zn over sheets of polycrystalline Cu. A Cu-37.33 at.% Zn alloy was used as the source of Zn. The alloy was prepared by encapsulating 0.5 mm thick sheets of Cu (99.97% purity in silica tubes with the required amount of Zn (99.99% purity) under a vacuum of 2 × 10− 6 Pa. The silica tubes were placed in a vertical furnace. The temperature of the furnace was raised gradually from 400 °C to 700 °C over a period of 1 week, which was followed by keeping the silica tubes for two months at 700 °C. The temperature of the furnace was gradually lowered to about 400 and the alloy quenched in water. The annealed sheet specimens of Cu measuring 6 × 10 × 0.75 mm3 with a relatively large grain size were cut using a diamond wafering machine. The sheet specimens of Cu were polished on both sides with conventional polishing techniques. Individual specimens of Cu were encapsulated along with 0.2 mm thick Cu–Zn alloy sheets in silica tubes under a vacuum of 2 × 10− 6 Pa. The weight of the Cu–Zn alloy was five times the weight of Cu specimens to provide sufficient vapor

pressure of Zn for diffusion into Cu. The diffusion induced grain boundary migration was carried out in the temperature range 400–600 °C. A number of specimens were diffusion annealed at each temperature for periods ranging from 1 to 30 days. A few specimens were also transformed at 350 °C. Each specimen was quenched in a mixture of ice and water maintained at 5 °C. These were prepared for metallographic examination and quantitative evaluation of the migration distance and thickness of fine grain layer was carried out. Each specimen was polished till the large grain size corresponding to the original Cu sheets appeared. This happened to be the most difficult part of the experiment, as it required a number of polishing and etching steps before the appearance of the surface on which the migration distance during DIGM could be recorded. The thickness of the fine-grained layer was measured directly using an image analyzer fitted with the required software. A bar of the appropriate length was projected on to the microstructure (at 500×) and the thickness of the finegrained layer at a regular spacing of 1 cm measured. The average thickness of the fine-grained layer was determined from 50–60 such measurements. The composition analysis was carried out from one edge of the specimen to the other along the sheet thickness. This was done primarily to determine v/kDbδ values at each temperature. A scanning electron microscope (FEI-make) with an EDAX attachment was used to analyze the composition of Zn along the sheet thickness as well as across a grain boundary.

3.

Results and Discussions

The results of the metallographic examination, migration distance and thickness of the fine-grained layer will be presented in this section.

3.1.

Microstructure

Microstructural examination has revealed that the grain boundaries started migrating due to the diffusion of Zn

Fig. 2 – An optical photomicrograph showing extensive migration of a grain boundary in a specimen annealed for 15 days at 440 °C.

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along them as illustrated in Fig. 1 at the position marked “A” from a specimen annealed for one day at 480 °C. The migration distance was observed to increase with an increase in the time of the diffusion anneal as shown in Fig. 2 from a specimen annealed for 15 days at 480 °C. The coherent twin boundaries of the original Cu did not show migration as is apparent from Fig. 3, which has been derived from a specimen annealed for two days at 570 °C. However, migration of grain boundaries was observed. It has been a common observation that certain grain boundaries migrate preferentially during Zn diffusion in Cu as illustrated in Fig. 4 derived from a specimen annealed for two days at 480 °C. This is primarily due to the grain boundary misorientation. A random grain boundary with a higher angle of misorientation will generally have a higher energy in comparison to low angle tilt boundaries and those in a CSL orientation. A random grain boundary with a higher angle of misorientation will migrate preferentially. This can be illustrated using Dupre′ equation represented by the following equality: g1 g g ¼ 2 ¼ 3 sinX1 sinX2 sinX3

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Fig. 4 – A random grain boundary with higher energy has preferentially migrated during DIGM as analyzed using the Dupré equation, 2 days at 480 °C.

ð1Þ

where γsi are the specific grain boundary energies at the triple point and Ωsi are the dihedral angles opposite the corresponding grain boundaries meeting at a grain edge. Analysis carried out at the point “A” in Fig. 4 illustrates that the dihedral angles Ω1, Ω2 and Ω3 (see the sketch) are 113°, 119° and 128° opposite boundaries having energies γ1, γ2 and γ3, respectively. This would lead to: g1 g2 g3 ¼ ¼ sin113- sin119- sin128with the result that γ1 N γ2 N γ3. Examination of the microstructure reveals that the grain boundary with specific energy γ1 has migrated substantially in comparison to the other two boundaries. In the above illustration it has been assumed that since the grain size of the Cu sheets was large and the dihedral angle between the boundaries will be maintained at the polished surface.

Fig. 3 – An optical photograph illustrating that the coherent twin boundaries do not migrate during a diffusion anneal for 2 days at 570 °C.

Metallographic examination of the specimen has revealed preferential migration of the grain boundary segment where a coherent twin boundary meets a grain boundary, Fig. 5. In general, the specific grain boundary energy, γgb, is less than the grain boundary energy of the segment where the coherent twin boundary meets the grain boundary with specific energy γgt i.e., γgt N γgb N γtb (twin boundary). Analysis carried out at the point A in Fig. 5 reveals that the dihedral angles Ωgb, Ωgt, and Ωtb before migration are 125°, 68° and 167°, respectively. Applying the equality proposed by Dupré at the point A would lead to the following results: ggb ¼ 3:64gtb ggt ¼ 4:12gtb The stacking fault energy, γSF, of a Cu-14 at.% Zn alloy is 25 mJ m− 2 [18] at 25 °C. The coherent twin boundary energy will be 2γSF = 50 mJ m− 2. The grain boundary energies will be γgb =182 mJ m− 2 and γgt = 206 mJ m− 2. The dihedral angles measured after ′ = 111.5 , Ωgt ′ = 94° and Ωtb ′ = grain boundary migration are Ωgb ′ = 108 mJ m− 2 and γgt ′ = 116 mJ m− 2. 154.5°. This would give γgb The above example illustrates that γgt N γgb and there is substantial reduction in grain boundary energies during migration. The difference in the two energies will act as a part of the driving force for DIGM. Annealing twins are observed to nucleate and grow on migrating grain boundaries as well as at the triple points. The formation of twins is shown in the photomicrograph of Fig. 6 where a number of them have formed. The formation of twins during DIGM is consistent with the observation made by Lopez in the Ag–Cd system [19] and by Gupta et al. in the Cu–Cd system [9]. The nucleation of twins during DIGM will occur on the same crystallographic planes in the fcc lattice as the annealing twins. These occur by the motion of a/6b112N Shockley partial dislocations on the {111} planes of the fcc lattice. Annealing twins were also observed to nucleate and grow along with the DIGM marked “t” in Fig. 7. The following

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Fig. 5 – Microstructure showing preferential migration of a grain boundary/twin boundary segment from a specimen annealed for 7 days at 480 °C.

criterion has to be satisfied for the formation of annealing twins near the triple point [20]: A2 g2 þ A3 g3 N A2 g2V þ A3 gV3 þ gtb  Atb

ð2Þ

where, A2 and A3 are the grain boundary areas enclosing the twin near the triple point with the specific interfacial energies γ2 and γ3, respectively before migration. The specific grain boundary energies after DIGM are γ2′ and γ3′ for the same boundaries and Atb is the area of the twin boundary. Using the Dupre′ equation it can be easily shown that when twins nucleate near the triple point, γ2′ b γ2, γ3′ b γ3 and there will be a decrease in the dihedral angle between grain boundaries with the energies γ2 and γ3. If we assume that the average dihedral angle between well-equilibrated boundaries is 120°, then the dihedral angle between the same grain boundaries should be

Fig. 6 – Nucleation and growth of twins near a triple point during DIGM, 2 days at 480 °C.

less than 120° when twins form near the triple point during DIGM. The dihedral angle was measured from specimens treated at a number of temperatures. The dihedral angle ranged from 91 to 116° with an average value of 110° and, therefore, it is consistent with the inequality presented in Eq. (2).

3.2.

Rate of Migration

The rate of migration of the grain boundary was determined from the time-dependent migration distance measurement at each temperature. The migration distance was measured from the initial position of the grain boundary to the leading edge of the curved boundary interface. About 40 measurements were made from each specimen from different grain boundaries.

Fig. 7 – Microstructure showing nucleation of twins marked “t” on the migrating grain boundaries, 7 days at 600 °C.

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Fig. 9 – Log x vs. log t plot at all transformation temperatures.

Fig. 8 – The average migration distance, x, vs. time of anneal plotted on a linear scale with the error bar for specimens transformed at 400 °C.

This was necessary as the migration distance is dependent upon the grain boundary structure. The low angle grain boundaries and those with CSL (coincident site lattice) orientations generally have lower diffusivity and hence lower mobility. On the other hand large angle random boundaries have relatively higher diffusivities and mobilities. The migration distance data, therefore, are representation of the average value averaged over a spectrum of grain boundaries. The average migration distance has been plotted against the time of anneal in Fig. 8 with the error bar at 400 °C in a linear plot. The migration distance, x, increases monotonically with the time of transformation. However, the slope of the curve at any position of the plot decreases with increasing time of anneal in agreement with the work of Li and Hillert [3,4] on the Cu–Zn and Fe–Zn alloys as well as that of Kuo and Fournelle [21] for an Al–Cu alloy. In some recent publications on the DIGM in symmetric boundaries of copper it has been suggested that the x vs. time graph is linear when exposed to Zn vapor [10]. The rate of transport of solute through symmetric or CSL boundaries is generally very low initially leading to a small migration distance in specimens treated for shorter times. However, with the diffusion of solute through the grain boundary during DIGM, the character of the boundary will change after some migration as it takes up curvature. The rate of diffusion will enhance as the original symmetrical grain boundary deviates from its exact CSL orientation. The larger the deviation from a CSL orientation the higher will be the rate of transport of solute as it will gradually become a random boundary. King and co workers [5–8] have shown a much smaller rate of migration of CSL boundaries in comparison with random boundaries during DIGM in Cu exposed to Zn vapor. The time independent rates of migration reported by Yamamoto et al. [12], Schmelzle et al. [10] and Goukon et al. [13] are either due to a change in character of the initial symmetric grain boundaries or if the time of anneal used was small at lower transformation temperatures in non-symmetric boundaries to this.

The migration distance data are plotted on a log x vs. log t scale in Fig. 9. The data points pass through a straight line at each transformation temperature in the range 400 to 600 °C and can be represented by the following equation: x ¼ Ktn

ð3Þ

where n is the time exponent and k is the material's constant. The value of n was calculated at each temperature from the slope of the straight line and has been observed to be 0.45, which is close to the value of 0.5 for parabolic growth. The average rate of migration of the grain boundaries was determined at each temperature as a function of the time of transformation from Eq. (4): m¼

xn t

ð4Þ

The rate of migration is plotted against the time of diffusion anneal at each temperature, Fig. 10. At the highest transformation temperature of 600 °C, it decreases from 10.9 × 10− 11 m s− 1 in a specimen treated for 1 day to 2.3 × 10− 11 m s− 1 after 15 days. Similarly, at 400 °C, it decreased from 2.1 × 10− 11 m s− 1 after two days to 0.4 × 10− 11 m s− 1 after 30 days of diffusion anneal. These rates of migration are the same order

Fig. 10 – The rate of grain boundary migration vs. the time of anneal on a log v vs. log t scale.

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thickness of the fine-grained layer at 600 °C has been observed to be significantly higher. The rate of growth of the finegrained layer can be obtained by differentiating “L” with respect to time to yield the following expression: vL ¼ nL=t

ð6Þ

The value of n in this case is also close to 0.45 at each temperature indicating a parabolic growth behavior. The rate of growth of the fine-grained layer has been observed to decrease from 5 × 10− 11 m s− 1 after 1 day to 1.35 × 10− 11 m s− 1 after 15 days of diffusion anneal at 440 °C as can be observed from vL vs. time plot in Fig. 12.

3.4.

Fig. 11 – The thickness of the fine-grained layer, L, vs. the time of anneal plotted on a log–log scale.

of magnitude as the values reported by Li and Hillert [4] for the diffusion of Zn in polycrystalline Cu.

3.3.

Growth of Fine-grained Layer

The thickness of the fine-grained layer, L, was measured for each specimen by using an image analyzer equipped with a software to measure the distance from the edge of the specimen to the extreme position where the fine-grained layer structure terminated. The thickness of the fine grained layer was observed to increase with increasing time of anneal at each temperature, as shown in Fig. 11, in a log L vs. log t plot. The data follow a straight line at each temperature and can be represented by the following equation: L ¼ kl tn ;

ð5Þ

The thickness of the fine-grained layer increased from 16.4 μm after 10 days to 26.4 μm after 30 days at 400 °C. The

Fig. 12 – The rate of growth of the fine-grained layer, VL, vs. the time of anneal on a log VL vs. log t scale.

Concentration-distance Profile

The concentration profile of Zn was determined using a scanning electron microscope through the thickness of the specimen. As an illustrative example, the concentration profile at 400 °C from a specimen diffusion annealed for 30 days is shown in Fig. 13. This profile fits very well with the solution to the diffusion equation after Cahn [22], initially applied by Hillert and Purdy [2] and subsequently by Li and Hillert [3,4] to calculate the diffusion co-efficient kDbδ in polycrystalline Fe and Cu exposed to Zn vapor:  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Cosh Z v=kDb d XaB  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; ¼ X0B Cosh Z0 v=kDb d

ð7Þ

where Z is the distance from the center to any position along the surface and Z0 is the value of Z at the edge of the surface. XαB and XB0 are the concentrations of solute at positions Z and Z0, respectively. Db is the grain boundary diffusion coefficient, k is the ratio of solute in the boundary and the matrix layer next to it and δ is thickness of the grain boundary. Eq. (7) was used to fit the profile and values of v/kDbδ obtained at each temperature. The value of kDbδ was calculated from the procedure followed by Li and Hillert [3,4] by using the instantaneous rate of grain boundary migration at each temperature as shown in Table 1. In a relatively recent publication, Schmelzle et al. [10] have carried out DIGM on

Fig. 13 – The concentration profile of Zn through the sheet thickness, 30 days at 400 °C.

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Table 1 – Instantaneous rate of migration and the diffusivity values during DIGM T, °C

Time, s (× 105)

Migration distance, x, μm

υ, m s− 1 (× 10− 11)

vi / kDbδ, m− 2 (×1010) Eq. (7)

kDbδ, m3 s− 1 (× 10− 21) Eq. (7)

v / kDbδ, m− 2 (×1010) Eq. (8)

350 400 440 480 510 540 570 600

16.41 25.92 12.96 12.96 8.64 8.64 0.864 –

19.8 25.4 29.4 35.4 34.5 40 17.7 –

1.21 0.98 2.28 2.73 4.0 4.63 20.5 –

4.9 2.0 1.82 0.91 0.67 0.42 0.11 –

0.24 0.5 1.24 3 5.93 10.8 18.9 –

7.1 2.5 1.9 1.23 1.11 0.54 1.5 –

kDbδ, m3 s−1 Layer thickness, (×10− 21) Eq. (8) kDbδ, (×10)− 21 Eq. (9) 0.17 0.39 1.2 2.21 3.6 8.5 13.6 –

– (L30 ⁎ /L3⁎ )0.133 (L15/L1)0.505 (L15/L1)1.29 (L15/L1)2.23 (L15/L1)6.33 (L15/L1)12.4 (L15/L1)20

⁎Lt2 — thickness of the fine -grained layer after a long time t2 (days). Lt1 — thickness of the fine-grained layer after a small time t1 (days).

symmetric and asymmetric boundaries during diffusion of Zn in Cu using a Cu-30 wt.% Zn alloy as the source of Zn. Assuming a steady state rate of migration of the grain boundaries and a linear relationship between the solute concentration and the exponential of the distance along the specimen thickness, the following solution has been obtained: ln XaB ðZÞ ¼ ln X0B 

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v=kDb dZ;

ð8Þ

where Z is the position on the specimen surface along which the concentration measurements have been carried out, XB0 is the composition of solute at position Z = 0. According to Eq. (8), a plot of ln XαB vs. Z will give a straight line whose slope pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi will be  v=kDb d. The same concentration vs. distance data used to calculate the diffusivity, kDbδ, through Eq. (7) has been applied to calculate the diffusivity through Eq. (8). The log XαB vs. Z data are plotted in Fig. 14 at 350, 400 and 440 °C. The

Fig. 14 – The composition profile shown in a plot of log XZn vs. distance, Z.

calculated values of kDbδ using Eq. (8) are also shown in Table 1. The two sets of values are of the same order of magnitude as shown in the semi-log plot of kDbδ vs. the reciprocal of absolute temperature, Fig. 15. The activation energy for the diffusion of Zn in Cu has been calculated from the slope of the best straight line that fits the data in the two cases to be 100 ± 10 kJ mol− 1, which is approximately half the value of 190.9 kJ mol− 1 for the volume diffusion of Zn in Cu [23]. Following the work of Li and Hillert [4], the diffusivity, kDbδ, during growth of the fine-grained layer can be calculated from the following equation: kDb d ¼

    1 L4 L 16 t2 Large t Small

=

ð9Þ

A ratio of 15 was used for the large and small times of transformation at all temperatures except at 400 °C where a

Fig. 15 – The diffusivity, kDbδ vs. 1/T plotted on a semi-log scale.

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Table 2 – Coherency strain energy, total chemical free energy change, effective free energy change and mobility during DIGM T, °C

E, GPa

XCB (× 10− 2)

XBnf (×10− 2)

Instantaneous rate of migration, m s−1,(×10− 11)

ΔGɛ, J mol− 1

−ΔGCm, J mol− 1

−ΔGmeff, J mol− 1

M, m4 ( J s)− 1 (×10− 17)

350 400 440 480 510 540 570 600

123 115.5 108.6 104.3 101.5 98.8 96.3 93.8

0.79 1.2 1.4 1.61 1.78 1.95 2.18 2.42

1.43 1.95 2.51 2.73 3.29 3.47 3.63 4.0

1.21 2.8 4.2 5.1 5.7 6.6 7.9 9.63

0.29 0.63 0.8 1.01 1.21 1.41 1.72 2.38

41.7 69.5 86.3 105.7 122.5 140 163.6 205.1

34.3 43.7 68.4 73.4 102.8 107.9 107.5 94.4

0.25 0.46 0.43 0.5 0.4 0.44 0.53 0.73

ratio of 10 was used. The diffusivity values were calculated and the data are shown in Table 1 and plotted in Fig. 15. The diffusivity values are within an order of magnitude of the value calculated from the concentration profile using Eqs. (7) and (8). The activation energy calculated from the slope of the straight line is 120 kJ mol− 1, which is approximately 0.6 times the activation energy for volume diffusion of Zn in Cu [23]. The diffusivity values compare very well with those reported by Hassner et al. [24] and Klotsman et al. [25] for the grain boundary diffusion of Zn in Cu. The activation energy for grain boundary diffusion of Zn in Cu reported to be 96.25 kJ mol− 1 [24] agrees very well with the results of this investigation, and therefore, suggests that the diffusion induced grain boundary migration in the Cu–Zn system occurs by the diffusion of Zn along the grain boundaries of Cu.

3.5. Coherency Strain Energy and the Chemical Free Energy Change In order to calculate the coherency strain energy during DIGM, the composition of the coherent layer behind the migrating grain boundary was determined at each temperature using an EDAX system attached to the scanning electron microscope. The coherency strain energy was calculated from the relationship originally formulated by Hillert [14]:

boundaries are comparable to values reported by Li and Hillert [4]. As a comparison, Li and Hillert [4] have reported a composition of 1.1 to 1.9 wt.% Zn in the region behind the boundary from a specimen diffusion annealed for 12 hours at 400 °C which compares well with 1.2 at.% Zn obtained in this investigation at 400 °C when DIGM was carried out with the Cu–38 wt.% Zn alloy. They used a Cu–30.5 wt.% Zn alloy in their experiment. Similarly, a composition value of 1.5 to 2 wt.% Zn has been reported at 500 °C behind the boundary region which compares well with the 1.78 at.% Zn obtained in this investigation at 510 °C. The surface energy of the curved interface can be estimated from the specific surface energy, γgb, of the Cu–Zn alloy and the radius, r, of the migrated grain boundary using Eq. (11): a =r DGg ¼ gVm

ð11Þ

The specific grain boundary energy will be 620 to 645 mJ m− 2 from the published value of 595 mJ m− 2 at 850 °C for a Cu 30 wt.% Zn alloy and dγgb / dT = −0.1. Taking the radius of the curved boundary to be 50 μm, the interfacial energy, ΔGγ associated with the migrated grain boundary will be 0.09 J mol−1. This value is lower than the coherency strain energy at all transformation temperatures. The chemical free energy change during DIGM can be calculated from the composition of the region behind the grain

 a 2 a XB  X0B =ð1  mÞ DGe ¼ Eg2 Vm

ð10Þ   where E is the modulus of elasticity, g ¼ a10 da=dxB is the misfit parameter, Vαm is the molar volume, ν is Poisson's ratio and XαB is the composition of the region swept by the migrating grain boundary. Since DIGM has been carried out in pure Cu, the value of XB0 (the initial composition) is zero. The value of XαB was recorded from five different regions of the same specimen and the data averaged. The value of η has been calculated to be 0.0589 from the lattice parameter vs. composition data compiled by Pearson [26]. The molar volume has a value of 7.115 × 10−6 m3 mol− 1 and the Poisson's ratio for Cu is 0.343. The temperature dependent values of the modulus of elasticity have been reported by Lord and Orkney [27] and were extracted from their data. The coherency strain energy was calculated at each transformation temperature and is given in Table 2. The coherency strain energy increases with increasing temperature of the diffusion anneal due to an increased concentration of solute in the region swept by the migrating grain boundaries and has values in the range 0.29 to 2.38 J mol− 1. The concentrations of Zn behind the grain

Fig. 16 – A plot of log v vs. log XZn. The slope of the line is 1.9.

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migrating grain boundary. The effective free energy change has been given by the following expression:     2 a

1  Xnf Eg2 Vm C nf B X ¼ RT 1  X ln  ; DGeff m B ð1  mÞRT B 1  XCB

Fig. 17 – The total free energy change, ΔGmC, and the effective free energy change, ΔGmeff, plotted against the instantaneous rate of migration.

0 boundary and the difference in the free energy of pure Cu, GA , and the partial molar free energy of Cu in the solid solution, − GαA. According to the free energy of a sub-regular solid solution given by Hillert and Waldenström [28], the free energy expression can be written as:



Gam ¼ XaA G0A þ XaB G0B þ RT XaA ln XaA þ XaB ln XaB h i     2 þXaA XaB L0 þ XaA  XaB L1 þ XaA  XaB L2

ð12Þ Pa

free energy change,DGCm ¼ G A  G0A ¼  The chemical  dGa α Gam  XaB dXmB  G0A , can be obtained after differentiating Gm with respect to XB. The driving force for DIGM can be written as the following expression: h  2  2 i DGCm ¼ RT ln XaA þ XaB L0 þ XaB 3XaB  4 XaB L1  2 h  4 i þ XaB 5  16XaB þ 12 XaB L2

ð13Þ

The temperature-dependent interaction parameters, L0, L1 and L2 have been reported by Spencer [29] in the course of an assessment of the phase boundaries of the Cu–Zn system:

where XBnf is composition of the matrix ahead of the leading interface at a position λ, which represents the width of the penetration zone. For low rates of migration of the grain boundary, XBnf can be taken as the peak value of the composition profile. Using the value of XBnf, the effective free energy change was calculated at each temperature, Table 2. eff value includes the intrinsic and extrinsic drags The ΔGm experienced by the migrating grain boundary due to both the solvent and solute atoms and varies from 34.3 to 107.9 J mol− 1 in the temperature range of DIGM carried out in the Cu–Zn system. At lower temperatures of DIGM, the difference in the C eff and ΔGm is small. However, as the transformavalues of ΔGm tion temperature increases the difference in the two values increases with a maximum difference obtained at 600 °C. This trend is expected, as the role of volume diffusion of Zn in Cu will enhance as the transformation temperature increases. Assuming that the rate of migration is proportional to the driving force during DIGM, the rate of migration data were plotted against the composition behind the migrating grain boundary on a log–log scale, Fig. 16. There appears to be a linear relationship between the two and it can be represented by the following expression:  n ð15Þ m ¼ k XCB The exponent “n” has been obtained from the slope of the line to be 1.9 and agrees with the value reported by Li and Hillert [4] during DIGM in the Cu–Zn system. It is consistent with the concept of coherency strain energy acting as the driving force. The total chemical free energy change and the effective free energy data were plotted against the instantaneous rate of migration of the grain boundary from the same specimen used to determine the composition behind the grain boundary, Fig. 17. The total free energy change follows a linear relationship with the rate of migration. The slope of both the straight lines is 1 indicating that the rate of migration is directly proportional to the total driving force. The effective

L0 ¼ 41661:1 þ 12:83485T; J mol1 L1 ¼ 6160:7  2:9274T; J mol1 L2 ¼ 14034:4  7:30663T; J mol1 The chemical free energy changes were calculated at each temperature by substituting the composition values in Eq. (13) and are given in Table 2. The magnitude of the chemical free energy change ranges from 41.7 to 205.1 J mol− 1 in the temperature range 350 to 600 °C during alloying of Zn in Cu. These values are about two orders magnitude higher than the coherency strain energy at each temperature. It is well understood that a part of the total free energy change is used for the volume diffusion of Zn in Cu ahead of the migrating grain boundary. In a relatively recent publication Kajihara and Gust [30] have attempted to derive an expression for the effective free energy change considering that volume diffusion of solute does take place ahead of the

ð14Þ

Fig. 18 – Mobility vs. 1/T, plotted on a semi-log scale.

1150

M A TE RI A L S C H A RAC TE RI ZA T ION 5 9 ( 2 00 8 ) 1 1 4 1–1 1 5 1

driving force data shows considerable scatter primarily from a large scatter in the composition data of both XCB and XBnf.

3.6.

Mobility

The mobility of the grain boundaries during DIGM has been determined from the relationship expressed by the following equation: M¼

a mVm eff DGm

ð16Þ

The instantaneous rate of migration has been obtained from specimens treated for five days at all transformation temperatures except at 350 °C where the specimen diffusion annealed for 19 days has been used. The ΔGeff m values have also been calculated from the composition of the region behind the grain boundaries from the same specimens and are given in Table 2. The calculated values of the mobility are of the order of 10− 18 m4 (J s)− 1 as given in Table 2 and shown in Fig. 18 on a semi-log scale of log M vs. the reciprocal of absolute temperature. Yamamoto and Kajihara [11] have reported mobility values in the range 10− 15 to 10− 20 m4 (J s)− 1 when DIGM was carried out in the temperature range 300 to 500 °C in pure copper when exposed to Zn vapor. The activation energy calculated from the semi-log plot of M vs. 1/T has been reported to be 177 kJ mol− 1, which is close to the activation energy for volume diffusion of Zn in Cu. The mobility data points of this investigation do not fall on a straight line in the whole range of temperature. Instead, the data can be best described in terms of the grain boundary migration in dilute alloys during grain growth. According to the theory of grain growth in dilute alloys after Lücke and co-workers [31,32] and Cahn [33], there are two velocity extremes when grain growth experiments are carried out in a wide range of temperature. In the low velocity (low temperature) regime, the solute drag is directly proportional to the rate of migration. The velocity and hence the mobility of grain boundaries will increase initially as the temperature increases. However, since the drag also increases with increasing rate of migration, there will be a transition region. At higher temperatures, the solute drag is inversely proportional to the velocity and occurs under the influence of high driving forces. The mobility of the grain boundary is expected to increase again in this regime with increasing temperature. The migration of the grain boundary is controlled by the grain boundary diffusion of solute in the high velocity regime and by volume diffusion of solute in the low velocity regime. In the log M vs. 1/T plot of Fig. 18, the slope of the straight line drawn through the data points at higher temperatures gives an activation energy of 80 kJ mol− 1 which is less than half of the activation energy of 190 kJ mol− 1 for volume diffusion of Zn in Cu. The mobility data of Yamamoto and Kajihara [11] in the temperature range 300 to 400 °C belong to the low velocity regime as apparent from the reported activation energy of 177 kJ mol− 1.

4.

Conclusions

DIGM has been observed to occur in the Cu–Zn system when polycrystalline copper was exposed to zinc vapor using a Cu-

38 wt.% Zn alloy as the source in the temperature range 350– 600 °C. The migration of the grain boundaries followed a parabolic growth behavior. A fine-grained layer was observed to develop on the surface of the specimen. The growth of the fine-grained layer was also parabolic with time. The diffusivity values, kDbδ, calculated using the concentration profile and the rate of migration of the grain boundaries yielded values in the range 10− 20 to 10− 22 m3 s− 1. By taking the value of the grain boundary thickness, δ, as 0.5 nm the diffusion coefficient values are observed to fall in the range 10− 11 to 10− 13 m2 s− 1 and lie between those of the grain boundary self-diffusion coefficients of Zn and Cu. The diffusivity values also agree with those reported for the grain boundary diffusion of Zn in Cu. The diffusion coefficients are also five to eight orders of magnitude higher than the volume diffusion coefficients of Zn in Cu. The coherency strain energy and the total chemical free energy change have been calculated from the composition of the region behind the migrating grain boundaries at all temperatures. The instantaneous rate of migration of the grain boundaries follows a linear relationship with the composition behind the grain boundary when plotted on a log–log scale. From the observed slope of 1.9, it can be concluded that coherency strain energy is the driving force for DIGM in the Cu–Zn system. The mobility of the grain boundary does not follow a linear relationship with the reciprocal of the transformation temperature in the whole temperature range when plotted in a semi-log scale. It follows a linear relationship in the high temperature regime. However, a transition region exits between the two extremes of low and high velocities, which is due to solute interaction with the migrating grain boundaries.

REFERENCES [1] den Broeder FJ. Interface reaction and a special form of grain boundary diffusion in the Cr–W system. Acta Metall 1972;20:319–32. [2] Hillert M, Purdy GR. Chemically induced grain boundary migration. Acta Metall 1978;26:330–40. [3] Li C, Hillert M. A metallographic study of diffusion induced grain boundary migration in the Fe–Zn system. Acta Metall 1981;29:1949–60. [4] Li C, Hillert M. Diffusion induced grain boundary migration in Cu–Zn. Metall 1982;30:1133–45. [5] Chen FS, King AH. Misorientation effects upon diffusion induced grain boundary migration in the Cu–Zn system. Acta Metall 1988;36:2827–39. [6] Chen FS, King AH. The misorientation dependence of diffusion induced grain boundary migration. Scripta Metall 1986;20:1401–4. [7] Chen FS, King AH. On the nucleation of diffusion induced recrystallization. Scripta Metall 1987;21:649–54. [8] Chen FS, Dixit G, Aldykiewicz AJ, King AH. Bicrystal studies of diffusion induced grain boundary migration in Cu/Zn. Metall Trans 1990;A21:2363–7. [9] Gupta BK, Madhuri MK, Gupta SP. Diffusion induced grain boundary migration in the Cu–Cd system. Acta Mater 2003;51:4991–5000. [10] Schmelzle R, Giakupian B, Muschik T, Gust W, Fournelle RA. Diffusion induced grain boundary migration of symmetric and asymmetric b011N {110} tilt boundaries during diffusion

M A TE RI A L S C H A RAC TE RI ZA T ION 5 9 ( 2 00 8 ) 1 1 4 1–1 1 5 1

[11]

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[31] Lücke K, Detert K. A quantitative theory of grain boundary motion and recrystallization in metals in the presence of impurities. Acta Metall 1957;5:628–37. [32] Lücke K, Stüve HP. Recovery and recrystallization of metals. NewYork: Interscience Publishers; 1963. p. 171. [33] Cahn JW. The impurity drag effect in grain boundary motion. Acta Metall 1962;10:789–98. Glossary

Terms k Db δ γgb v Ω γtb γgt γSF Ai x n k L kL

Dimensions m2s− 1 m J m− 2 m s− 1 J m− 2 J m− 2 J m− 2 m2 m m s− 1 m

Materials constant for fine-grained layer growth

XαB XB0 E η v Vαm a0 dggb dT

ΔGγ

Concentration of solute at a position z from the center Concentration of solute at the surface (z0)

Pa m3 mol− 1 m J m− 2 K− 1 J mol− 1

r m J mol− 1 ΔGɛ Gαm J mol−1 L0,L1 and L2 J mol− 1 ΔGCm J mol− 1 Gi0 = A, B J mol− 1 ¯ aA G

J mol− 1

ΔGmeff ~a XCB,XB

J mol− 1

XBnf M

Segregation ratio Grain boundary diffusion coefficient Grain boundary thickness Grain boundary energy Rate of grain boundary migration Dihedral angle Coherent twin boundary energy Twin boundary/grain boundary energy Stacking fault energy Grain boundary area Migration distance Time exponent Material's constant Thickness of the fine-grained layer

m4( J s)− 1

Modulus of elasticity Misfit parameter Poisson's ratio Molar volume Lattice parameter of Cu Temperature coefficient of the grain boundary energy Free energy associated with the curved boundary

Radius of curvature of the boundary Coherency strain energy Free energy of solid solution Interaction parameters Total chemical free energy change Free energy of the ith component in the pure state Partial molar free energy of A in the α phase Effective free energy change Composition of Zn in the region left behind the migrating grain boundary Composition of Zn at a position λ from the leading interface Mobility of the grain boundary during DIGM