Comments on the theory of the solidification of eutectics

Scripta METALLURGICA

Vol. 2, pp. 663-666 1968 Printed in the United States

Pergamon P r e s s , Inc

COMMENTS ON THE THEORY OF THE SOLIDIFICATION OF EUTECTICS

P. H. Spriggs and R. Elliott Department of Metallurgy, Manchester University, Oxford Road, Manchester 13, England.

(Received June 26, 1968; Revised September 11, 1968) In their paper Jackson and hunt (i) have considered the diffusion controlled conditions which can exist in solidifying eutectics where lamellar or rod structures are formed.

Two

formally identical equations are derived for lamellar or rod growth from the solution of the diffusion equation, which involves sine and cosine series for the lamellar structures and Bessel functions of the first kind for the rod structures;

each equation being an appropriate

relationship between the three variables v, AT and I (or R) i.e. for lame!lar

AT = v~Q L + a L m

(i)

and for rods

AT = vRQ R + a R m R

(2)

where the parameters m; QL, a L

QR and a R are defined in (i).

disagree with the derivation of these equations;

The present authors do not

in fact, as differentiation of equation (i)

shows the experimentally observed relationship between the rate of growth (v) and the interlamellar spacing (I) is a consequence of assuming gro%~h to occur with the smallest possible undercooling (AT), thus ~2v = a L QL

(3)

Differentiation of equation (2) yields a similar relationship for rod structures R2v = a R R

(h)

which at present lacks experimental confirmation. The graphical nature of these relationships is sho%~ in Fig. i. The present authors however disagree with Jackson and Hunt's comparison with experimental observations since they attempt deductions which are contrary to their postulates and experimental facts. For lamellar structures equation (3) may be rewritten using equation (i) as (AT) 2 = h m2aLQ L

(5)

V

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THEORY OF THE SOLIDIFICATION OF EUTECTICS

Vol. 2, No. 19

FIG. i

AV

Aor R

The interfacial undercooling as a function of the interlamellar or inter-rod spacing.

Experimental data is available for relationships (3) and (5) and Jackson and Hunt (i) have shown that values of the diffusion coefficient D and the surface free energy ~ m a y be calculated using the definitions of QL and aL with data taken from the phase diagrams. Their further comr~ents are erroneous since the curves in Fig. i labelled Lamellae and Rods involve the same physical constants and the only difference between them is due to the fact that the rods are one-dimensional and the lamellae are two-dimensional which is reflected in the solution of the fundamental diffusion equation.

More specifically, the

quantities D and a are constant along these curves since they are two parameters of the system.

Hence the use of growth conditions other than the extremal should yield the same

values of D and u.

Such growth conditions violate the requirement that the interfaeial

undercooling shall be a minimum for which there is experimental evidence, and hence may not be observable in practice. Furthermore the relationships (3) and (4) cannot now be expected to apply since they are only true for growth under extremal conditions.

Hence the remarks of Jackson and Hunt

(ref. (i) p.l141) regarding the combination of equations (4) and (i) together with experimental data are meaningless since the physical parameters are essentially the same in both cases while the topological configurations of rods and lamellae are distinct.

It is con-

ceivable that the surface energy for a rod structure could be different from a lamellar structure. It is noteworth M that a third shape is possible, that one phase precipitates as globules. This could be either the breakdown of rods of small diameter and large surface energy or a distinct form of precipitate.

The micro duplex structures found in super plastic alloys may

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be such an example.

THEORY OF THE SOLIDIFICATION OF EUTECTICS

The solution of this problem will involve Legendre polynomials and is

receiving attention. REFERENCES 1.

665

K. A. Jackson and J. D. Hunt, A.I.M.E. Trans. 236, 1129 (1966).