Homogenization by one-phase diffusion

Materials Science and Engineering, 20 (1975) 155--160

© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

Homogenization by One-Phase Diffusion

L.-E. LARSSON and B. KARLSSON Department of Engineering Metals, Chalmers University of Technology, GSteborg (Sweden)

(Received February 10, 1975)

SUMMARY The homogenization of spheres with the solute deposited on the surface has been studied with the finite difference approximation method assuming one-phase diffusion. Obtained results are compared with the Lavender and Jones solution and with results from two-phase diffusion. It is shown that homogeneity (Cram = 0.99 X Cmax) is obtained for D t / a 2 = 0.3. For long soaking times (Dt/a 2 0.05) the one-phase diffusion model will for many practical cases predict the homogenization kinetics also for two-phase systems with sufficient accuracy. The results suggest that large reductions in time required to produce a fully martensitic structure in sintered steels may be obtained by adding an excess a m o u n t of alloying elements. Large deviations between the present and Lavender and Jones results are due to geometry and initial solute concentration.

I. INTRODUCTION During fabrication of alloys, inhomogeneous distribution of alloying elements is quite often encountered. This is experienced in wrought materials as well as in powder produced. In many instances the fluctuations in concentration are small compared with the solubility limits and no separate phases will form. During subsequent homogenization treatments the redistribution of alloying elements proceeds by onephase diffusion. Even in the presence of t w o separate phases the final homogenization will be governed b y a one-phase diffusion process [1]. For simplified b o u n d a r y conditions homogenization by one-phase diffusion may be pre-

dicted by closed analytical solutions [2,3]. To treat more complex cases which more closely resemble the situation encountered in real structures, approximate solutions to the diffusion equation have to be used. The model by Lavender and Jones [4] has proved useful in some cases [ 5 - 7 ]. A wider choice of geometry and boundary conditions is offered by the use of the finite difference approximation to the diffusion equation [1,3]. This has been used by Heckel and coworkers [8 - 10] for the study of mixtures of elemental powders and for dissolution of second phase particles. The objective of this investigation has been to study homogenization by one-phase diffusion. The difference approximation has been applied to a spherical diffusion model for various boundary conditions. The experimental background is the homogenization of powder blends containing a low-melting component by liquid - solid alloying [11,12]. This is characterized by the almost instantaneous spreading of the low-melting alloying component over the surfaces of the particles of unalloyed base material, followed by the penetration of alloying elements along the grain boundaries and in the volume of the base material. This background leads to the selection of the model described below. However, the model is felt to be appropriate to many other technical situations as well.

II. THE MODEL A spherical grain with increased solute concentration on the surface (Fig. 1) has been used as the main geometrical model. A limited amount of information has been obtained with

156

massboundary transport Centerof Solute J across Matrixes/__ ~symmetry No

Boundary: C~+1 = 2 K C ] v - 1 + (1 -- 2K)C~v j = 0,1,2, ...

(7)

K = N2AT

Fig.1. Geometricalmode]usedfor homogenization calculations.

(8)

where j and n are the indices for time and space respectively. N is the number of nodes and A T the time step. Stability of the solution is maintained b y selecting the step lengths in time and space according to: AT = 0.25 1/N 2.

planar geometry. The magnitude of the segregation may be described b y the ratio ~/c, where ~ is the concentration of solute in the boundary and ~ is the average final concentration in the alloy. By substituting: C---

C

x =x

(I)

a

T-

(9)

The accuracy is checked for all j b y integrating the concentration profiles to determine the total amount of solute in the system. By comparison with the initial solute content the mass conservation is monitored. A maximum deviation in mass of 1% was accepted. The number of nodes (N) ranged from 50 to 400 depending on the starting conditions and the magnitude of the gradients.

Dt a2

III. R E S U L T S

Fick's second law transforms into

a C = a2c

2ac + --

AND DISCUSSION

n = 1,~, ..., N -- 1

Concentration profiles for the different starting conditions are shown in Fig. 2 at different stages of homogenization. We note that after D t / a 2 = 0.02 the profiles corresponding to starting conditions Cs > 4 are virtually identical. This is further illustrated in Fig. 3, which shows the concentration at the b o u n d a r y and at the center of the sphere as a function of homogenization time. The curves for Ccenter all coincide, while there are slight deviations between the boundary concentrations for D t / a 2 > 0.02. Homogenization will be considered complete when Ccenter = 0.99 Cboundary. Note that this definition differs from that used b y earlier authors [4] in aiming at a given final amplitude rather than at a given fraction of the original amplitude. Complete homogenization has in all cases been achieved for D t / a 2 = 0.3. This implies that it takes just as long a time to remove a "large" inhomogeneity as a "small" one. To provide a basis for comparison with experimental data obtained during liquid - solid alloying [11] a hypothetical grain b o u n d a r y transport process was simulated, yielding an average surface concentration of the sphere according to:

J = 0,1,2, ...

C s = [ c / c ] Boundary =

aT

ax 2

xax

(2)

"

Boundary and starting conditions are given by: Boundary

X=O

aC _ 0 ax

1

Starting b
1

T~> 0

(3)

T=0

(4)

C = Cs

where I -- b/a is the initial width of the segregation zone in the spherical space considered (Fig.. 1). The equations governing the homogenization process in dimensionless parameters are then in the finite difference approximation

[1]: Center: C~+ z = (1 -- 6K) C/o + 6KC~ i = 0,1,2 ....

(5)

General: C~.1 = K(1 -- 1 / n ) C ~ - z + (1 - - 2 g ) c ~ + g ( 1 + 1 / n ) C ~ + l

(6)

A(1 -- e -~T)

(10)

157

(a)

i

,

I

-0

10

(b)

Dt/a zx10 5 O ~

9

;9

-5

8

.7 -5

128

/5

Dt/Q2 x 10 5

.4

o.~ o:2

o13

3

32766

1

o~

~5

O~

0.7

Q8

D %0

O~

0

0.1

0,2

:

0.3

o

0.4

0.5

1

1

r

~o

(d)

0.6

0,7

0.8

O.g

v

Dt/a2xlO5

t:

9 g

32

8

\

7 6

128

327~8

e 1 9 ~

/ 0

i

0.1

0.2

o~

a4

/

O.5

0,6

0.7

O.8

0.9

1.0

~o §

9

L

4

"2

(c)

i

5

2048 ~

_ ._--~

/al~ 2 ~ . ~ o

o

512

512

/

12~

5

Dt/o2x 106

I"

- . A 128

0

~11~ !

[

4 1 3

12

I

0

1.0

Position

I

0 on

0.1

I

0.2 0.3 rio

Radius

Q.4

o.o

,

Io

Fig. 2. C o n c e n t r a t i o n profiles along radius o f sphere after various h o m o g e n i z a t i o n times. Starting surface c o n c e n trations were: (a) 10, (b) 4, (c) oo (5-function), (d) 2.

with an empirical value of A = 2. The solution to the diffusion equation for this situation is given in reference 2. When the amount of solute that had entered the sphere corresponded to the final average concentration the transport process was interrupted and the sphere allowed

/~

9

8 7

to continue to homogenize, according to eqns. (1) - (9). The time required to reach this stage was found to become independent of B for B > 500 (Fig. 4). As in previous cases the curves for [ c / E ] center all coincide, and homogeneity is achieved for D t / a 2 = 0.3 (Figs. 5 and 6).

x t,O

10 4

(NO 10_I

i

3 2

2

5 x 10-2 i

1 Conter

1

10 Homogemzotion

102

103

104

Tim~Dt/o2K105

Fig. 3. Variation o f the c o n c e n t r a t i o n s at the center and o n the surface o f the sphere as f u n c t i o n o f h o m o g e n i z a t i o n time. N u m b e r s o n curves s h o w the surface c o n c e n t r a t i o n at D t / a 2 = O.

b 10-2L 1

.............................. 10 102 B

103

104

Fig. 4. The t i m e required to absorb solute corresponding to average c o n c e n t r a t i o n for a sphere w i t h surface conc e n t r a t i o n C s = 2"(1 - - e x p [ - - B T ] ).

158 [ J

1

C C

DI x105 a2

I

500

2000 \ \ / / /

3200 \

F- ,/ l~-,,o

32768 \

1638/, \

I

I

I

1/f

1000 ~

I

I

.~

\

*~ ~ ~

Lavender & Jone~/Cmax,t=Q

8192 ~ \~.,.-~ i z • ~ .

0

05

0

Comparisons with other solutions

The model by Lavender and Jones [4] assumes a planar geometry with a sinusoidal variation in concentration of the segregated element. It predicts a decline of the segregation according to the expression: log ( f ) = 0 . 4 3 4 3 7 r 2 D t / a 2 .

(11)

Here fis the ratio between the amplitude of the segregation at time zero and t. Comparing results from the present investigation with eqn. (11) (Fig. 7) shows a very strong effect of geometry and original amplitude. The spherical s y m m e t r y results in large deviations. The agreement is better for planar geometry, especially for small initial segregations. In Fig. 8 comparisons are made between results from one-phase and two-phase homogenization [1]. We note that for large homogenization times the kinetics are satisfactorily described by one-phase diffusion. To study the final stages of homogenization by assuming the surface concentration on the ........

,

........

,

.....

''r

0.01 0,1 0.2 03 Homogenization Time Dt/a 2

sphere to remain constant at ~ and by following the concentration at the center of the sphere represents an attractive simplified system. The solute concentration profiles are in this case described by the expression [2]: c - - c 1 - 1 + 2-a- ~ (--1)" sin _n r_r ~r,=z n a

Co--Cz

- - D n 2 7r2 t

X exp

a2

(12)

where cz is the initial uniform concentration of solute in the sphere, Co is the constant surface concentration, a the radius and t the time. The resulting variation in solute concentration at the center of the sphere is shown in Fig. 8 for cz = 0 and Co = c. The homogenization time is longer than t h a t predicted by the present model.

Iii

. . . . .

C/~max,t= 0

', j , o . \

~2

~1 2

One Phase Diffusion Boundary

~

/A/"

F,G B

, 0.01

~

....... 0.1 Homogenization

,,I

I , /~ . / / I C m L , ' , , ,,.,

I Time --~--,~102

10

C

1 100

Fig. 6. Variation of the concentration at the center and on ~he surface of spheres with surface concentrationC2 = 2 ( 1 - - e x p [ - - B ' T ] ) f o r T ~ T~q. For T ~ Tk~. ethe system was isolated.

C

g, 1

/?~,000s

0 ~

0

0.4

Fig. 7. Comparison between predicted homogenization behaviour based on the equation by Lavender and Jones and based on the present results. Numbers on curves show surface concentrations at Dt/a 2 = O.

I\\

,

PLanar

Spherical

± o

Fig. 5. Concentration profiles along radius of sphere after various homogenization times. Surface concentration: Cs = 2(1 --exp[--700.T]) for T ~ T~q. For t ~ tl~ the system was isolated.

3

2 !

Center

10 Diffusion Time

Ot/o2x10 2

Fig. 8. Comparison between concentrations at the center and surface of sphere obtained during homogenization by one-phase (solid lines) and two-phase diffusion (dashed lines). Letter symbols refer to boundary conditions listed in Table 1.

159 TABLE 1 B o u n d a r y c o n d i t i o n s used d u r i n g c a l c u l a t i o n o f h o m o g e n i z a t i o n b y t w o - p h a s e d i f f u s i o n

F - -

o

~

X

~ o

b

0

Alternative

ca 0 / c

cg 0 / c

c a ~/ff

cfl a/ff

D~ / D a

t ~-_,0 / thorn

A. L i q u i d / s o l i d B. L i q u i d / s o l i d C. L i q u i d / s o l i d Do L i q u i d / s o l i d

0 0 0 0

1.5 3 10 10

1.2 1.2 1.2 3

1.5 3 10 10

~ oo ~o oo

0.24 0.3] 0.34 0.03

E. S o l i d / l i q u i d F. S o l i d / l i q u i d G. S o l i d / l i q u i d

0 0 0

2 4 4

1.2 1.2 1.2

1.5 1.5 1.5

1 1 2

0.26 0.27 0.28

The expressions derived by Arunsing and Dayal [14] are identical with those resulting from eqn. (12) with the concentrations referring to the solvent instead of the solute. Consequences on the hardenability of inhomog e n e o u s steels

Given a certain cooling rate the required a m o u n t of alloying element (c~equired) to produce 100% of martensite in the structure of steel may be estimated from existing CCTdiagrams [11,13]. Composing the alloy such that ~-= Crequ~d would, according to Fig. 3, require a homogenization time of D t / a 2 = 0.3 to produce a fully martensite structure. Production rate and cost of heating will suffer

from this long soaking. The diffusion time may however be reduced b y adding an excess a m o u n t of alloying element. The cost of this may in many cases be much less than the gains that can be made due to shorter heating times. At the end of the homogenization treatment the distribution of alloying elements will not be homogeneous, b u t the structure will consist of 100% martensite. Figure 9 shows the calculated relation between homogenization time required to obtain a fully martensite matrix and the excess amount of alloying element. Adding three times as much solute as needed b y Crequ~ed will decrease the heating time b y almost a factor of 5. Adding further amounts of alloying elements will have less drastic effect on the gain in homogenization time.

03

IV. C O N C L U S I O N S r~ 0.1 i=

3E

0.01

....

~

J

J

,

i

J

i

2

3

4

5

6

7

8

required

Fig. 9. P r e d i c t e d e f f e c t of a d m i x i n g excessive a m o u n t of a l l o y i n g e l e m e n t s o n t h e h o m o g e n i z a t i o n t i m e req u i r e d t o r e a c h 100% m a r t e n s i t e .

Based on the results of this study the following conclusions can be made: 1. Calculations based on one-phase diffusion will with sufficient accuracy in many cases predict homogenization kinetics also for twophase systems at soaking times larger than D t / a 2 = 0.05. 2. Complete homogenization (Cram = 0.99 × Cmax) will in both cases be achieved after D t / a 2 = 0.3. 3. The time required for complete homogenization is independent of the magnitude of the inhomogeneity.

160

4. Large reductions in time required to produce a fully martensitic structure in steel may be obtained by adding excess amounts of alloying elements. With ~ = 3Crequiredthe calculations predict a five-fold reduction in soaking time compared with that necessary when C-----Crequlred.

5. The agreement between the results from this study and simplified models of Crank [2] and Lavender and Jones [4] is poor.

ACKNOWLEDGEMENTS

The financial support by the Swedish Board for Technical Development is gratefully acknowledged.

REFERENCES 1 B. Karlsson and L.-E. Larsson, Homogenization

by two-phase diffusion, 20 (1975) 161. 2 J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford, 1967. 3 H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, Clarendon Press, Oxford, 2nd ed., 1971. 4 J.D. Lavender and F.W. Jones, An investigation on banding, J. Iron Steel Inst. (London), 163 (1949) 14.

5 RoG. Ward, Effect of annealing on the dendritic segregation of manganese in steel, J. Iron Steel Inst. (London), 203 (1965) 930. 6 W. Peter and H. Finkler, Untersuchungen tiber den Diffusionsausgleich yon Kristallseigerungen der Elemente Mangan, Chrom, Nickel, Vanadin, Phosphor, Schwefel und Kohlenstoff an Proben mit k~nstlichen Seigerungen, Arch. Eisenhiittenw., 38 (1967) 775. 7 F. Weinberg and R.K. Buhr, Homogenization of a low-alloy steel, J. Iron Steel Inst. (London), 207 (1969) 1114. 8 R.A. Tanzilli and R.W. Heckel, Numerical solutions to the finite, diffusion-controlled, two-phase, moving-interface problem (with planar, cylindrical and spherical interfaces), Trans. AIME, 243 (1968) 2313. 9 R.W. Heckel, R.D. Lanam and R.A. Tanzilli, Techniques for the study of homogenization in compacts of blended powders, Advan. Exptl. Tech. in Powder Met., Perspectives in P/M, 1970, 5, p. 139. 10 R.W° Heckel and M. Balasubramaniam, The effects of heat treatment and deformation on the homogenization of compacts of blended powders, Met. Trans., 2 (1971) 379. 11 L.-E. Larsson, Homogenization during liquid solid alloying of a FelCuIMn0.3C powder-forged steel, Mater. Sci. Eng., 19 (1975) 231. 12 H. Fischmeister and L.-E. Larsson, Fast diffusion alloying for powder forging using a liquid phase, Powder Met., 17 (1974) 227. 13 Atlas zur W~rmebehandlung der St'~hle, Verlag Stahleisen, D~isseldorf, 1954/56/58. 14 Arunsing and B. Dayal, On estimation of homogenization time in powder metallurgy, Z. Metallk., 61 (1970) 298.