Dynamics of decomposition of CuCo alloys at the spinodal point

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J O U R N A L OF M A T E R I A L S S C I E N C E L E T T E R S 13 (1994) 1 6 9 9 - 1 7 0 2

Dynamics of decomposition of CuCo alloys at the spinodal point

JUN-MING LIU, Z. G. LIU, Z. C. WU, X. K. MENG

National Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210 008, People's Republic of China

As a typical system for the study of phase separation dynamics, decomposition of CuCo alloys has re- cently attracted much attention due to findings of giant magnetroresistence effect in Cu/Co, Cu/Fe etc. multilayer systems [1-5]. It was reported that a resistance drop in a magnetic field was also observed in decomposed Cug0Col0 alloys annealed at 400- 600 °C [6]. It has been revealed that inter-particle coupling of neighbouring magnetic phases (Co-rich) is responsible for the resistance drop, and the giant magnetoresistance effect depends mainly on two factors. One is alloy composition Co. When Co reaches a critical value, a Co-rich skelton structure will form. In this case no resistance drop will be observed because the magnetic coupling of Co-rich phases cannot be realized. The other factor is the wavelength of the decomposed structure, as a function of which the magnetic coupling between neighbouring Co-rich particles reaches a maximum.

However, the classic theories on phase separation in a quenched system predicted that the wavelength as a function of composition will show an essential singularity at any point on the spinodal curve where the driving force for phase separation becomes zero [7-9]. So far, this problem has not been solved in the phenomenological framework although it was revealed experimentally that this singularity does not exist. In fact, at T = 773 K, for decomposed Cul_xCO x where x = 0.07-0.10, a resistance drop in a magnetic field was observed and the wavelength of the decomposed structure was several nanometers although the spinodal point Csl was 0.075 57 [6, 10].

Therefore, the classic theories seem to be invalid for describing decomposition near the spinodal curve. It then becomes necessary to develop a theoretical model to solve the problem of singularity at the spinodal curve.

In this work we focus on this problem by solving the Cahn-Hilliard equation in a quenched CuCo system. Besides considering the role of non-linear diffusion terms, we also include Fourier-type thermal noise. We show that this problem can be solved successfully.

The free energy AG(Co) of homogenized CuCo alloys was evaluated by calculation of phase diagram ( C A L P H A D ) [11, 12], which can be formulated as:

AG(C0) = (1 - C0)G°u + CoG°co ^- TS mix q- A G ex

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with

A G ex = AiC0(1 - Co) + A2C0(1 - C0)(1 - 2CO) + A3C0(1 - C0)(l - 2C0) 2 +TB1Co(1 - Co) (3) where Co is the average composition of the alloy;

G°u and G°o are the free energies of pure copper and cobalt (fcc); S mix is the ideal entropy of mixing;

AG ~x is the excess free energy; T is temperature (K); R is the general gaseous constant (8.31J m o l - i K - 1 ) ; and A1, A2, A3 and B1 are coefficients of the free energy function (for an fcc solid solution, A~ = 37100, A 2 = 2896, A3 = 3251 and B1 = -5.194. The unit of A G ~x is Jmol-~). Equation 1 does not include the elastic energy because it is very small here.

Equation 1 can be easily expanded into a polynomial:

N

AG(Co) = ~,ai.(Co) i (4) i=0

which is a good approximation of Equation 1 if taking N = 20 except from the case of Co => 0 or 1.

To simplify the simulation, only the one-dimensional case will be considered here. The Cahn-Hilliard equation can be written as [8, 13]:

s c _ s D ( C ) - - + r / T ( X , t)

St S X

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where C is the composition at coordinates (X, t), t is time, X is the position coordinate, D(C) is the diffusion coefficient and K is the gradient energy coefficient. We assume r/T to be a Fourier-type noise with its mean-square amplitude (r/2) satisfying the following fluctuation-dissipation relation [8]:

(r/2) = 2R T. M / r 2 (6) where M is the mobility and r is the distance between two neighbouring atoms (~-0.207 nm for CuCo alloys). A creation of the noise term is introduced below.

For coordinates ( X , t), we define q = C - Co and then formulate D(C) as a polynomial function of C:

N - 2

D(C) = ~, Diq i (7a)

i=0

Di = ~-" M.di+2(aG) (7b)

i! d C i+2

If the solution of Equation I is expressed as:

:/2

q ( X , t) = ~ Qh(t)'exp (ih[3X) (8) -:/2

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where

Qh(t)

is the amplitude of a Fourier wave with wave n u m b e r

h(h 4= 0), J

is the n u m b e r of Fourier waves considered (=256), /3 is a reciprocal of a region L considered here (/3 =

2zr/L).

A t T = 773 K, we evaluated M = 2.998 x 10 -14 mol

# m 2 j - 1 s-~ and K = 3.083 x 10 -17 # m 4 s -1, f r o m the m e a s u r e m e n t s [10], and o b t a i n e d ( t / ~ ) ~ 0.02 s - I . T h e r e f o r e , the noise t e r m can be included by adding J Fourier waves with h t a k e n f r o m - J / 2 to

J/2

per second into the t e r m Qh(t) of E q u a t i o n 8. A n example of the composition fluctuations f r o m t h e r m a l noises is shown in Fig. 1, where the average amplitude is a b o u t 0.02. A t T = 7 7 3 K for t = 10000 s, the noise t e r m does not shows any contribution to the phase separation if the diffusion controlled by E q u a t i o n 5 is not included, which proves that the present technique for creating noise is valid and satisfies the dissipation relation, Equa- tion 6. Substituting Equations 7 and 8 into E q u a t i o n

0.15!

o.loi

0.05 c3 o

-0.05 -0.10 /

-7 !

i

- 0 . 1 5 ~ ~ ^" ^~ -; . . . 0 5 10 15 20 25 30 35 40 45 50 55

X(nm)

Figure 1

An example of stochastic composition fluctuations at T = 773 K, created by the thermal noise term. At ( - - ) t = 5000 s and ( - - - ) t = 10000 s, no fluctuation growth was observed.

5, we obtain [9, 10]:

~Qh(t)/~t

= -(h/3)Z[(Do +

2hZK/3Z)Qh(t)

L N--1 ]

+ ~ 1 DnRh(n, t)

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~ = ~ n + l where

/ -

Rh(1, t) =

J_Qk(t)Qh_k(t) dk

+ o o o -

R b ( i , t) =

J_Rk(i--l,t)Qh_k(t)dk

i = 2 . . . N - 1

T h e time evolution of the amplitudes can be discretized as:

Qh(t + 6 0 = [~Qh(t)/~t]6t +

Qh(t) + Q~ (10) T h e initial fluctuations in the quenched alloy can be created similarly. By combining Equations 8 - 1 0 we can evaluate the composition evolution in the quenched alloy.

Fig. 2 shows evolution of the composition profile with Co = 0.075 and T = 773 K. In the early stage (t = 0 to t = 140 s), the composition profile shows strong wavelength modulation and amplifying with time. This b e h a v i o u r differs f r o m the case without the noise term. F o r the latter, the early stage does not show increasing amplitude, due to the contribution of the gradient energy term. In the inter-medi- ate stage (t = 1200 s and t = 2000 s), s o m e p e a k s of the composition profile a p p r o a c h the equilibrium composition of the Co phase, a c c o m p a n i e d by the disappearance of s o m e other peaks. T h e wavelength, or the distance b e t w e e n two neighbouring

0.12 I 0.08 1 0.04

0 -0.04 -o.o8 I

(a) o

k -- i I

,5 10 15 20 25

X ( n m )

,3 d

0.4 I 0.3 ] 0.2 0.1 0 -0.1 0 (b)

5 10 15 20

X ( n m )

25

0.8- 1 . 0

0 , 6 -

0.4-

~Z (3 _0.2-

-0.2 0 (c)

i . 0 . 8 "

0.6 J 0.4- i

0.2- 0 - -0.2

0 (d)

1'0 1'5 2'0 2~5 5 10 15 2'0 25

X ( n m ) X ( n m )

Figure2

The evolution of composition profile for Cu-7.5 at % Co alloy in decomposition at T = 773 K, including the noise term.

(a) t = ( - - - ) 0, ( - - ) 20 s; (b) t = ( - - - ) 60, ( - - ) 140 s; (c) t = ( - - - ) 260, ( - - ) 1200 s; and (d) t = ( - - - ) 2000, ( - - ) 4000 s.

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