DIFFUSION IN HETEROGENEOUS BIPHASE STRUCTURES J. KU~ERA

Institute of Physical Metallurgy, Czechosl. Acad. Sci., Brno*)

The present paper deals with the determination of temperature dependence of the effective diffusion coefficient in heterogeneous biphase structures using diffusion coefficients measured in both phases and the short-range order parameter. A new interpretation of this short-range order parameter ~r is given as the relative volume of the ordered structure in the sample, V(ora)/V. For the AuCu 3 alloy, the lacking values of the Au diffusion coefficients at temperatures T~< 760 K in the partially ordered structure of this alloy were calculated on the basis of longrange order parameter values and diffusion characteristics which are valid for the perfectly ordered or disordered structures, respectively. For solid solutions of the A B and A B 3 type the relation between the long and short-range parameters is given; numerical calculations of short-range order parameters were carried out for [3-brass and for the Cu3Au alloy with the use of long-range order parameters and heats of formation. An explanation of the diffusion anomaly in ferromagnetic materials close to the Curie temperature is given and the temperature dependence of the relative volume of the paramagnetic phase in c~-iron is calculated. INTRODUCTION

Up to the present several papers have dealt with diffusion in heterogeneous biphase systems. W a g n e r [1] has treated the problem of self-diffusion of one of the components in a heterogeneous two-phase mixture from the depth towards the surface of the sample, occurring e.g. during selective evaporation. In papers [ 3 - 5 ] the processes of diffusion in samples composed of a part containing only the solid solution and of a part containing a heterogeneous composition of c~and 13crystals have been investigated. This work deals with self-diffusion and heterodiffusion, in biphase heterogeneous systems that show a heterogeneity in such a sense that they contain, in the given volume V of the sample, two types of domains with the total volumes V~ and Vz (Va + Vz = V), showing different order or different chemical composition. Such a type of structure appears in partially ordered substitutional solid solutions, in ferromagnetic metals near the Curie point and in alloys in which two types of crystals, e.g. eutectic, exist at the given temperature. The physical model of a structure, upon which the following considerations are based, is the Cohen-Fine model of short-range order [6] in substitutional solid solutions. According to C o h e n and F i n e perhaps the best way of regarding an alloy with short-range order is to think of it as being made up of continuous imperfectly ordered domains of differing degrees of order, statistically distributed. Because of simplicity and the possibility of development of a quantitative diffusion model let us *) Zi~kova 22, Brno, Czechoslovakia.

286

Czech. J. Phys. B 22 (19721,

Diffusion in heterogeneous biphase structures

further assume an extreme case of the Cohen-Fine model, namely a formation of statistically distributed ordered zones, separated by a disordered matrix. Magnetic domains and areas with randomly oriented spins correspond to this model in ferromagnetics; crystals e.g. of phase 1, in the matrix of phase 2, correspond to this model in biphase solid solutions composed of chemically heterogeneous phases.

INTERPRETATION

OF THE SHORT-RANGE

ORDER

PARAMETER

The short-range order parameter a in a binary solid solution A - B introduced by B e t h e [7] can be interpreted in connection with the Cohen-Fine model in a simple way as a relative volume of ordered domains. It holds namely (1)

q

~ --

-

q(dis)

q(ord) - - q(dls) '

where q = QaB/Q, QaB is the number of the pairs AB, which are unlike, Q is the total number of pairs which are AA, BB and AB in the considered volume of the sample. The quantities q(dis) and q(ord), denote q for the completely disordered and ordered states, respectively. Let us denote the total volume of all the ordered domains in the considered sample V(o~d), the total volume of the disordered matrix V(di~). Obviously it holds (2)

V = V(o~d) + V~d,~) 9

In the completely ordered domains we can write for the number of bonds of the type

AB: (3)

Qa.
where Q(o~d)is the number of all the bonds in ordered domains that can be expressed with the use of all bonds Q in the considered sample by the relation (4)

Q(ord) = e g(ord) V

Analogous relations can be written for the disordered matrix, QAB(d,s)

(5)

(6)

=

q(dis)Q(dls) ,

Q~dis) = Q

V

The parameter q can be expressed as follows q = (q(ord)V~ord) + q
(7) Czech. J. phys.

B 22

(1972)

287

J. Kudera On substituting from eqs. ( 3 ) - ( 7 ) to (1) we obtain (neglecting interdomain walls) (8)

V(erd)

a -

V from which it can be seen that the short-range order parameter is given by the relative volume of the ordered domains of the sample. The advantage of this interpretation consists in the fact that the parameter a can be used even in those cases, when the different domains of samples are composed of chemically identical atoms (ferromagnetic metals near the Curie temperature) or differ in their chemical composition (eutectics). RELATION BETWEEN SHORT- AND L O N G - R A N G E ORDER PARAMETERS

For some alloys experimental values of long-range order parameters, e.g. in [8, 9, 10], are at disposal. For this reason, it is useful to find a relation between S and o- for the often occurring cases of structure order that appear especially with stoichiometric compositions AB and AB3. One of the possibilities how to obtain this relation is to use the Fowler-Guggenheim method based on the quasichemical model [11] of solid solution. With the use of this model F o w l e r and G u g g e n h e i m have developed the theory of ordering processes in alloys of the CsC1 type; their results can be immediately used to obtain the dependence a = a(S) for this type of solid solutions. After simple modifications of equations given in [11] (pages 2 2 8 - 2 3 0 ) we get (9)

a(S) = 1 - -

2(1-

1 + {I + (I-

S 2)

S2)(exp[2v/kT]- I)}I/2'

where S 4= 0, i.e. T < Tc, Tc is the critical temperature. After some modifications we obtain for temperatures T > Tc (10)

a :-

1 - exp(-v/kT) 1 +

(-

In Eqs. (9) and (10) (11)

v = VA, -- 89

+ VBB)

is the change of binding energy of one bond in the solid solution, in forming this solution from pure components. The temperature dependence of the short-range order parameter a ( r ) for ~-brass, calculated from Eqs. (9) and (10)is given in Fig. 1. The values of parameter S used in calculations were taken from the paper by C h i p p m a n and W a r r e n [8]. Values of v were determined from the relation (12)

288

v = --

2AHT(ora) , Nz Czech. J. Phys. B 22 (1972~

Diffusion in heterogeneous biphase structures

~ 6

,

't

J

o.5 F

500

300

.~oo

9oo

1Ioo .

T[K]

Fig. 1. Temperature dependence of the short-range order parameter for ]3-brass. Values for curve 1 were calculated from eq. (9), for curve 2 from eq. (10) and for curve (3) from eq. (30). Table 1 Temperature dependence of the short-range order parameter for 13-brass. T [K]

-- AHT(ord) [cal/mol] Ref. [12]

S Ref. [8]

o(s)

296 524 575 618 653 672 690 712 720 727 732 736 741 740 750 800 850 900 950 !000 1050 1100

2900 2900 2900 2900 2900 2900 2900 2900 2900 2900 2900 2900 2900 2866 2864 2850 2830 2824 2810 2797 2784 2770

1.0000 0.9790 0.9154 0-8424 0.7568 0.7147 0.6517 0.5541 0.4865 0.4069 0.3469 0.2132 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1 "000 0"960 0"852 0"746 0"640 0"593 0"529 0"445 0"397 0"349 0"319 0"271 0"241 0"239 0"236 0"221 0"207 0'195 0.184 0.174 0-165 0.157

Czech. J. Phys. B 22 (1972)

289

J. Kudera where N is the Avogadro number, z is the coordination number in the first coordination sphere, AHr(or~ is the heat of formation of ordered 13-brass [12]. For temperatures higher than the critical temperature, the values AHr(or.~ were calculated from the relation (13)

AHTco~d) = AnT(di~) + (AHT(ora) - AHT,di.,)T=T..

Th~ calculated values AHrr

, S and r

are presented in Tab. 1.

To obtain the dependence rr(S) in alloys with the stoichiometric composition AB s it is necessary to derive it in full extent. Let us consider the crystal with N atoms and N lattice sites, Let us denote with cr the lattice sites corresponding to atoms A in a perfectly ordered state. Their number is N = XAN, XA = 1/4 is the atomic fraction of the element A. For the lattice sites 1/ corresponding in a perfectly ordered state to atoms B, it similarly holds N = xBN, x~ = 3/4, Xa + xB = 1. For the elementary cell in a perfectly ordered AuCu 3 alloy it obviously holds a) each site ~ has z = 12 neighboring sites fl, b) each site fl has zxa/x B neighboring sites 7 and z(xn - XA)/X B neighboring sites/L In an ordered state which is characterized by the long range order parameter S the number of atoms A on ~ sites is R~, the number of atoms B on the ~ sites is W~, the number of atoms B on B-sites is R#, and the number of atoms A on fl-sites is Wp. Followig relations hold (14)

N (1 + 3s) a~ = NXA(X A -{- XBS ) = ~i6 VV~ = NxAxB(t -- S)

= ~ ( 1 - - 3 N S).

R~ = NxB(x~ + x aS) = ~3N (3 + s) 3N (1 - s ) . Eqs. (14) satisfy following relations: a) R~ + W~ + Rp + Wa = N independently of the degree of order, b) for S = 1, R, = N/4, W, --- 0, R B = 3N/4, and Wa = 0, c) for S = 0, accroding to the probability theory, R~ = NIl6, W~ = 3N/16, Rp = = 9N/16, Wp = 3N/16. In the state characterized by the long-range order parameter S there exist following bonds in the considered crystal: Q~aa, .~AAI')~'QBB,~BQBB,~#QAB,aI3Q~AJ, and AAB.P~E.g., the index ~ denotes that the atom A is located on the lattice site a etc. T h e numbers of 290

Czech. J. Phys. B 22 (1972)

D i f f u s i o n in h e t e r o g e n e o u s b i p h a s e s t r u c t u r e s

different types of bonds are given by the equations

05)

Q~aA _-- 4Nz__ S(I - S) + z q a a ,

64 QBAflA = Z q A A ,

4 N z (1 -- S)(2 + S) + zqa A ,

Q~

=-g4

Qgg _- ___4Uz2(I + S) + zqAA,

64 4 U z (1 + S) 2 + z q a a ,

Q~g

Q~

4 U z (1 - S) 2 - zqAa, = -ffi

Gg

=

64

2(1 - s ) - 2zq

..

The parameter qA.4 is introduced by the second equation in the system (15) and its most probable value will be determined from the condition for the minimum of the configurational partition function (20). The total number of all the bonds given by equations (15) is nZ/2 and does not depend on the parameter qAA. In a completely ordered state S = 1, qAa = 0 , and for nonzero numbers of bonds it holds (16)

Qnu-

Nz

4

,

QA.-

Nz

4

For the heat formation of the A B 3 type alloy at temperature in the ordered state it holds

(17)

--AHT(~

= QABVAB +

--

Nz 4

V ~

QBBl)BB - - _2 __ 4 ZUAA - -

Nz -4

VAB - -

ZVBB

1 2 (vaa + v . . ) .

Under the assumption that the interaction occurs in the first coordination sphere only, the energy of the system of N atoms in a configuration given by the parameters S and qaa is expressed by the equation (18)

-- W(S,

QAA) "~- Q A A V A A +

QBBVI~e + Q a B V A B 9

For further calculations it is useful to introduce the function g(S, qAA) proportional to the number of distinguishable microstates by which the macroscopic state characCzech. J. Phys. B 22 (1972)

291

J. Kugera terized by parameters S a n d qaa can be realized, n a m e l y (19)

where

q(S, qaa)

K(S)

=

K( S) (Nz/2)!

is the factor o f p r o p o r t i o n a l i t y that need n o t be expressed explicitly in

these considerations. W i t h the use o f quantities

W(S, qaA) and #(S, qAA] the

con-

figurational p a r t i t i o n function [11] m a y be written f o r the given case in the f o r m (20)

Z(T) = g(S, qaa)exp

{-W(S,

qaa)/kr} .

Table 2 Temperature dependence of the short-range order parameter for the AuCu 3 alloy.

292

T [K]

-- ZlHT(ord) [cal/mol]

S

if(S)

3O3 523 523 573 573 623 623 643 643 653 653 657 660 661 663 665 667 670 700 750 800 850 900 950 1000 1050 1100 1150 1200

1710 1707 1707 1680 1680 1587 1587 1514 1514 1462 1462 1437 1420 1419 1390 1386 1385 1380 1350 1295 1240 1187 1130 1075 1017 953 890 810 720

1.00O

1-000

1-040 0-972 0.945 0.944 0.842 0-841 0-744 0.787 0.785 0.775 0.785 0.775 0.686 0.505 0-300 0-216 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

1-058 0.961 0.925 0.924 0-798 0.797 0-690 0.735 0.732 0.721 0.731 0.720 0.629 0.476 0-360 0.328 0.294 0-276 0.249 0.224 0.203 0.183 0.165 0.149 0.133 0.119 0.104 0.089

Czech. J. Phys. B 22 (1972)

Diffusion in heterogeneous biphase structures

The most probable value of the quantity qAa = qAa for this function is determined by the equation (21)

- -

[ln g -

0qaa

= 0,

(W/kT)]

from which, for q • = qaA 64/4N, it follows (22)

4[(1

In q•

+ S) 2 -

qX]

- S) + q•

[(1 -

S) 2 -

q•

[(1 -

S) -

[(2 + S)(1 - S) + q•

q•

[2(1 + S) + q•

4 A H r .... 3RT

= 4v/kT =

and 4[(1 + S) 2 - qX] [(1 - S) 2 - q• q•

- S) + q•

[(2 + S ) ( l -

[(1

S)q•

-

S)

-

q•

4AHr~ord!~

= exp

--

[2(l + S) + q•

-

3RT

J"

Equation (22) holds for 0 < S < 1; for S = 1 the equation does not hold because qX x = q(ord) = 0. 7

tO

T

F

~

T

T

7

1.

2~

o~

~

o

O.5

o i

0

I

]

i

I

____1

300

1

700

500

900

I

11oo 9

r[x]

Fig. 2. Temperature dependence of the short-range order parameter for the

AuCu 3

alloy.

For a completely disordered state (S = 0, T--* oo) Eq. (22) changes into the equation (23) • 4 -- 20q(dis) • 3 + 20qidls) ~ z -- 16q(ais) x + 4 = 0. 3q(dis) On substituting into relation (1) we get for the short range order parameter (24)

~ =

1 + s(a - s)

+ 2q • ,

2q(~is) where q• and q(%is) = 0.3542499 are the roots of Eqs. (22) or (23), resp. Czech. I. Phys. B 22 (1972)

293

J. K u d e r a

The temperature dependence of the parameter a(S) for the AuCu3 alloy calculated on the basis of experimental values S [9] and from the heat of formation AHr(o~d~ is shown in Tab. 2 and in Fig. 2. The values AHr~o~d~for temperatures T ~ Tc were determined in an analogous way as in the foregoing case for [3-brass.

DIFFUSION

IN PARTIALLY ORDERED

A L L O Y S O F CsC1 T Y P E

The ordered structures of the CsC1 type have a cubic body centered structure, in which the atoms of element A are placed in the corners of the elementary cubes, atoms of the element B in the centres of these cubes9 The connection between the diffusion characteristics and the long-range order parameter in the CuZn alloy, with the lattice of the above mentioned type, was found by K u p e r et al. [13]. The results of their measurements can be well explained on the basis of the Cohen-Fine model, if the fact that the diffusion coefficient in an ordered structure at a given temperature is lower than in disordered alloy, D(o~d)(T) < D(dls)(T), is taken into consideration. Let us assume that in an arbitrary section of a sample, perpendicular to the diffusion flux direction, the ratio of areas belonging to the ordered or disordered structure, S(ord)/S(dls) is constant. For diffusion flux I(ora) through the ordered zones having the cross-section S(o~d)the first Fick law holds (25)

I(ord) =

C3C -- D(ord) OX m

in the disordered zones with cross-section S(d~) ~r

(26)

/(dis) = - D(dis) 8X"

Similarly for the total diffusion flux I = /(ord) q- /(dis) in the whole cross-section S' it holds ~c

(27)

I = -D--.

gx

The total surface S' of the section is connected with the areas of ordered or disordered domains by the relation (28)

S' = S(ord) + S(d~).

From equations (25)-(28) we obtain (29) 294

S'D

=

S(ord)D(ord) "-]- S(dis)D(dis ) . Czech. J. Phys. B 22 (1972)

Diffusion in heterogeneous biphase structures

If we express the relative surfaces by the corresponding relative volumes, we get (30)

D

--

V

D(ord) +

V

D(di,) 9

In this relation the total volume of the sample is given by the equation V = V(ord) q+ V~ai,). With the use of the short-range order parameter we may express the effective diffusion coefficient D by the equation

(31)

D

=

D(dls ) +

0"(D(ord ) - -

D(dis) ) .

"I - 8

-10

-12 I

P

~0

r

i

l

12

r

t6

14

18

Fig. 3. Measured (ooo) [13] and calculated (x x X) values of the Cu self-diffusion coefficient in 13-brass with the use of eq. (31). The agreement of measured values of the effective self-diffusion coefficient of copper in 13-brass [13] with the values calculated from the relation (31) is given in Fig. 3 and is a good proof of the model that was used as basis for our assumptions concerning the diffusion in heterogeneous structures, i.e. in solid solutions with partial order. The values of the self-diffusion coefficient of copper in 13-brass were calculated from the relation (32)

D(ord ) =

2'76 exp { - 3 4

O00/RT}

[cmZ/s],

which, on the basis of measurements by K u p e r , was chosen so as to obtain a) all the values D(orm) that would satisfy the inequality D(ora) < D; b) the activation enthalpy AH(o~d)equal to the activation enthalpy obtained using the above mentioned measurements [13] at the lowest temperature, i.e. at the highest degree of order. The values of self-diffusion coefficient in a disordered alloy were calculated from the relation (33) Czech. J. Phys.

O(dis ) =

B

22 (1972)

9"21 x 10 -3 exp { - 2 1

580/RT}

[cm2/s], 295

J. Ku(era

which was chosen so that a) D(dis ) > D f o r t h e w h o l e investigated temperature interval; b) D = D(dis ) for the temperature range near the melting point; c) AH(di~ ) = 21,580 cal/mol be close to the value quoted by K u p e r for the disordered state (22,040 cal/mol).

- - r ~ - -

i

~

i

--1

T

r

1 - -

i

i

- - ~ - - -

-

'L

s

"X o

-20 !__j 8

a 10

t

~ 12

J .... !4

16

!

_ I _._~ ~___NJ 18

20

7[KJ 10'~

-t

Fig. 4. Temperature dependence of the Au self-diffusion coefficient in disordered (D(dls)) , ordered (D(ora)) and partially ordered (D) AuCu 3 alloys. (A • A) [14], (e 9 9 [15] measured values ( o o o ) values calculated from eq. (31).

If, on the contrary, the diffusion coefficients D, D(ord ) and D(a.9 are well known, equation (30) can be used to determine the short-range order parameter a. The comparison of short-range order parameter for [3-brass calculated from relations (9) and (30) is given in Fig. 1. Rather complete data concerning the diffusion in a binary alloy, characterized by order-disorder type of transformation at the stoichiometric composition A B a, are available for the Au-Cu system. These are mainly the coefficients of Au self-diffusion measured in the temperature range of 780-1200 K [14, 15] for the AuCu a alloy. Further diffusion characteristics of Au self-diffusion and of heterodiffusion of gold in copper are available, allowing application of the relations quoted in [16] for getting the temperature dependence of the self-diffusion coefficient of gold in the disordered AuCus structure, (34) 296

D(ais) = 1"70 x 10 -2 exp {-38,735/RT)

[cma/sec]. Czech. L Phys. B 22 (1972)

Diffusion in heterogeneous biphase structures

The frequency factor in this equation was specified in order to obtain Dtais) > D in the temperature range under study (780-1200 K). The equation for the self-diffusion coefficient of gold in the AuCu3 alloy with an Au ordered structure was obtained by taking the activation enthalpy AHAuc,3 = = 46,342 cal/mol from the work by Benci and G a s p a r i n i [17]; the frequency factor 0"185 cmZ/sec was determined so as to get D > D(ord) in the whole temperature range under study. In this way, the results used in Fig. 3 in [24] and in Fig. 4 in [25] are to be changed, which implies a change of interpretation given in the mentioned papers. For the AuCu 3 alloy the characteristics of self-diffusion of gold at temperatures T < 760 K have been lacking up to the present. The theory developed for the diffusion in ordered and partially disordered structures enables us to obtain these lacking data on the basis of the Eq. (30) and with the use of the short-range order parameter values quoted in Tab. 2. In Fig. 4 the results obtained for self-diffusion of gold in the AuCu3 alloy are summarized.

DIFFUSION

ANOMALIES

NEAR

THE

CURIE

TEMPERATURE

While investigating the diffusion in ferromagnetic materials near the Curie temperature, the authors [ 1 8 - 2 1 ] have observed analogous deviations of the temperature dependence of self-diffusion and heterodiffusion coefficients, similarly as in the case of partially ordered solid solution near To. The diffusion anomalies for self-diffusion of iron [19] and heterodiffusion of nickel [20] in a-iron are quoted as an example in

-If

-12

-f3

-f4 __ 8.0

__ 8.5

k 9.0

I 9.5

fO.O

10.5 ,

ff, O

.-c[K1 104

-i

F i g . 5a. D i f f u s i o n a n o m a l y f o r s e l f - d i f f u s i o n i n a - i r o n [19]. Czech. J. Phys. B 22 (1972)

297

J. K u & r a

Fig. 5a,b. Diffusion characteristics of self-diffusion and heterodiffusion in a-iron are given in the following Tab. 3. In Fig. 5 it can be seen that diffusion in the paramagnetic state is more rapid than in the ferromagnetic state, the activation enthalpy of diffusion in paramagnetic material being lower than in the ferromagnetic state. This phenomenon is connected with the fact that the formation and migration of vacancies in the magnetic domains with identically oriented spins is more difficult than the formation and migration of vacancies in areas, in which the orientation of spins is random. -

I0

-ff

1 -12

-f3

- f4

-15 8.0

8.5

9.0

9.5

'tO.O

"10.5

ll.O

.

Lo' [

f1.5

K"]

T

Fig. 5b. Diffusion anomaly for heterodiffusion of Ni in r

[20].

Table 3 Diffusion characteristics in s-iron. Paramagnetic state Diffusion element

Fe Ni

Do

Q

[cm2/s]

[kcal/mol]

1.9

57.2

1-3

56

Ferromagnetic state Do lcm2/s]

Q [kcal/mol]

Refs.

2.0 1.4

60 58.7

[191

[2Ol

To explain the temperature dependence of the self-diffusion coefficient close to the Curie temperature it is possible to apply the suggested diffusion model. At temperatures T < Tc there exist magnetic domains in ferromagnetic state separated by Bloch walls, the thickness of which for iron at room temperature is about 300 lattice con298

Czech. J. Phys. B 22 (1972)

Diffusion in heterogeneous biphase structures

stants, i.e. their total volume compared to the specimen volume is negligible. The diffusion at temperatures T < Tc takes place practically only in magnetic domains and its temperature dependence is given by the Arrhenius equation. At temperatures T -" Tr the volume of disordered areas increases, the diffusion is faster, even in these areas, and the diffusion coefficient increases more rapidly than it would correspond to the Arrhenius dependence for the ferromagnetic state. At temperatures T > Tc a complete decay of magnetic domains takes place, the diffusion goes on in the disordered structure only and again follows the Arrhenius dependence. 4

f.O

o

e ~ o

oO f f9

1-6"

o0

"1

f

l o., 9

I

900

fO fO00

f'lO0

1200

fix1 Fig. 6. Temperature dependence of the relative specimen volume with r a n d o m oriented spins in a-iron, e e e Ni-heterodiffusion, 9169169 Fe-self-diffusion.

It is obvious that the relative volume of disordered areas the short-range order parameter by the relation (35)

V(dis-) =

1 --

V(ais)/V is connected with

O"

V and it can be easily determined, with the use of measuerd values of D and the values calculated from diffusion characteristics quoted in Tab. 3, from Eq. (31). The temperature dependence of the relative volume of disordered areas determined on the basis of measured values for self-diffusion and heterodiffusion of Ni in a-iron is given in Fig. 6. It can be seen from this figure that the behaviour of V(dis~/Vis, within the errors, very similar for values obtained from self-diffusion and heterodiffusion measurements. This result proves the correctness of the applied model for partially ordered structure which does not depend on the method by which we are detecting it, i.e. with the use of self-diffusion or heterodiffusion characteristic measurements. It is obvious that for the given temperature the mean value of V(ord)/V or V(ai~)/V,resp., represents a constant in the sample with time fluctuations in the given point x, y, z. A very interesting investigation in this field was made by the group of workers [22] at the Kyoto University (Japan), who made a study of order-disorder phase transition dynamics with the use of a computer. They used the numerical display Czech. J. Phys. B 2Z (1972)

299

J. Ku~era

IBM 2250 N H K in which each bond is represented by a luminous homothetic abscissa; its two states "light" or " d a r k " correspond e.g. in ferromagnetics to two signs of spins. It turned out that at temperatures T > Tc the orientation of spins is quite random. With decreasing temperature, greater and greater agglomerations with the same spin orientation appear, that form stable magnetic domains at temperatures

T
CONCLUSION

The present paper proves that the Cohen-Fine model of ordered solid solutions can be, in general, successfully applied to the problem of diffusion in heterogeneous biphase structures that appear in partially ordered solid solutions and in ferromagnetics near the Curie temperature. From the point of view of these applications, it is important that a relation between the effective diffusion coefficient and the short-range order coefficient was found D = D(dis) + (D(o~d)

- -

D(dis)) 0",

which enables us to calculate this coefficient, if we know the diffusion coefficients in both phases (e.g. D(als) and D(ora)) for the ease of a partially ordered solid solution, and the short-range order parameter. The application of the obtained relation to other structures than the partially ordered solid solutions is possible using the expression for the short-range order parameter O" ~

V(~

V which, at the same time, makes possible a simple interpretation of this parameter as a relative volume of the corresponding phase in the specimen. With regard to the fact that the experimentally available quantity in some particular cases is not the short-range order parameter, but the long-range order parameter, a relation for a(S) of the type CsC1 or AuCu a was given (or derived, resp.) on the basis of the quasi-chemical model for solid solutions. Numerical calculations of short-range order parameters with the use of long-range order parameters and the heats of formation were carried out for the Cu-Zn and AuCu3 alloys. 300

Czech. J. Phys. B 22 (1972)

Diffusion in heterogeneous biphase structures

The proposed model of diffusion in heterogeneous biphase systems enabled us to explain the diffusion anomaly in ferromagnetic materials near the Curie temperature which, up to the present, has not yet been explained. Further, this model allows to determine the temperature dependence of the relative volume of the paramagnetic phase, if the diffusion coefficients in the ferromagnetic and paramagnetic structures in the vicinity of Curie temperature are known. In the present work this was done for or-iron. The suggested diffusion model in heterogeneous biphase systems can be further developed especially from the point of view of biphase solid solutions, e.g. from the point of view of diffusion in eutectics. The author wishes to express his thanks to Professor L. M. S l i f k i n for providing a xerox-copy o f a part of Mr. Alexander's thesis [15] which facilitated the evaluation of diffusion characteristics in Cu3Au alloy, and to Mr. M. S z o b for recalculating some equations. Received 15. 6. 1971. References

[1] [2] [3] [4l [5] 16] [7] [8] [9] [10] [11] :[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

W a g n e r C,: Unpublished work, cited according to [2], p. 174. A d d a Y., P h i l i b e r t J.: INSTN, Sacley, PUF, Paris 1966. B t i c k l e H., D e a s c a m p s J.: Rev. Met. 48 (1951), 569. B t i c k l e H.: MetaUforschung 1 (1946), 145. M o s s e M., L d v y V., A d d a Y.: C.R. Acad. Sci. 250 (1960), 3171. C o h e n J. B., F i n e M. E.: J. Phys. Radium 23 (1962), 749. B e t h e H. A.: Proc. Roy. Soc. London. A 150 (1935), 552. C h i p m a n D., W a r r e n B. E.: J. Appl. Phys. 21 (1950), 696. K e a t i n g D. T., W a r r e n B. E.: J. Appl. Phys. 22 (1951), 286. W a l k e r B. C.: J. Appl. Phys. 23 (1952), 118. M u t o T., T a k a g i Y.: The Theory of Order-Disorder Transition in Alloys. Sol. St. Phys. 1 (1955), Ed. F. Seik and D. Turnbull H u l t g r e n R., O r r R. L., A n d e r s o n P. D., K e l l e y K. K.: Selected Values of Thermodynamic Properties of Metals and Alloys. J. Willey, New Yor/London/Sydney 1963. K u p e r A. B., L a z a r u s D., M a n n i n g J. R., T o m i z u k a C. T.: Phys. Rev. 104 (1956), 1536. B e n c i S., G a s p a r i n i G., G e r m a g n o l i F., S c h i a n c h i G.: J. Phys. Chem. Sol. 26 (1965), 687. A l e x a n d e r W. B.: Thesis, Univ. of N.C., USA, 1968, private communication. K u ~ e r a J., M i l l i o n B.: Met. Trans. 1 (1970), 2603. B e n c i S., G a s p a r i n i G.: J. Phys. Chem. Sol. 27 (1966), 1035. B o r g R. J., B i c h e n a l l C. E.: Trans. A I M E 218 (1960), 980. B u f f i n g t o n F. S., H i r a n o K., C o h e n M.: Acta Met. 9 (1961), 434. H i r a n o K., A v e r b a c h B. L., C o h e n M.: Acta Met. 9 (1961), 440. H i r a n o K., A g a r w a l a R. P., A v e r b a c h B. L., C o h e n M.: J. Appl. Phys. 33 (1962), 3049. O g i t a N., N e d a N., M a t s u b a r a T., M a t s u d a H., Y o n e z a w a F.: J. Phys. Soc. Jap. 26 (1969), Suppl. 145. Cited by J. J a n t a , Czech. J. Phys. B 20 (1970), 449. M i l l i o n B., K u ~ e r a J.: Kovov6 mat. 1, VII, (1969), 17. M i l l i o n B., K u ~ e r a J.: Acta Met. 17 (1969), 339, K u ~ e r a J., M i l l i o n B.: Diffusion Processes in Partially Ordered Alloys. A paper presented at T. Graham Memorial Symposium, Glasgow 1969.

Czech. J. Phys. B 22 (1972)

301