Further remarks on the uniqueness of the

Boltzmann-Matano solution of the diffusion equation

1. J. S. KIRKALDY, Acta Met. 4, 92 (1956). 2. F. E. HARRIS, Acta Met. 9. 389 (1961).

In an early note to this journal the writer(l) pre- sented arguments supporting the idea that parabolic solutions of the one-dimensional diffusion equation with variable diffusion coefficient, subject to initial and boundary conditions corresponding to the infinite diffusion couple, are unique. This argument was put forward to dispel a mathematical misconception quite prevalent among meta~urgists to the effect that para- bolic behavior is an assumption which must be checked by experiment and can at best be only an approxi- mation.

3. C. SEYFERTH,' A&a Met. 16, 259 ‘(19Si). 4. C. SEYFERTH, %. Angew., Math. Mech. 39, 441 (1959).

Eine weitere Bernerkung zur Eindeutigkeit der Boltzmannschen Lijsung der eindimensionalen

DifFusionsgleichung*

Recently, two letters to the Editor have appeared which were critical of my original presentation. I will not comment on the firsU2) since the author’s point regarding finite couples is irrelevant to my discussion of mathematically infinite ones.

The second, due to Seyferth,c3) challenges my state- ment of uniqueness of the parabolic solution by stating a solution which is not parabolic but which ostensibly satisfies the same initial and boundary conditions. I say ostensibly, because in fact his solution corresponds to an initial condition for the concentration consisting of a S-function (i.e. a localized source of solute) super- posed upon the normal step-function. This solution is accordingly irrelevant to the discussion.

Seyferth contends that my “proof” of uniqueness is no proof at all and in the strict mathematical sense he is correct. I was concerned primarily with demon- strating the existence of a parabolic solution which would satisfy the boundary conditions. With no reason to anticipate pathological behavior of the diffu- sion equation, except at d = 0, it was assumed that uniqueness followed directly from existence.

In der Natur erwartet man den eindeutigen Ablauf eines Diffusionsprozesses. Wenn die mathematische Beschreibung hiervon abweicht, sind nicht geniigend physikalische Gegebenheiten in die mathematischen Voraussetzungen aufgenommem worden. In meinen Eindeutigkeitsbcweisen wurde eine Beschr~nkthei~- be~ngun~ fiir die Konzentration vorausgesetzt und auf ihre Not~~endi~keit ( ($1 F&note 3) hingewiesen.

It is regrettable that Seyferth has taken a complete- ly negative approach to our contribution (leaving the impression that he does not agree with our thesis), for his own mathematical researchest4) complement our presentation by rigorously demonstrating that there is at most one solution of the diffusion equation which satisfies the initial condition corresponding to the infi- nite difIusion couple. Thus the parabolic solution, which has been demonstrated to exist, is unique.

Physikdisch- l’echnisches

Institut der Deutschen

Akademie der Wksenschaften

eu Berlin

1. J. S. KIRKALDY, Acta Met. this issue p. 1187. 2. C. SEYFERTE, Acta Met. 10, 259 (1962). 3. J. S. KIRXALDY, Acta Met. 4, 92 (1956). 4. C. SEYFERTH, Math. Naohr. 24, 13 (1962).

+ Received Juno 11, 1962.

~e~~~~~~a~n~ of Metallurgy and J’. S. KIRKALDY

Metallwgical Engineering

M&faster University

Hamilton, Ondario, Canada.

ACTA METALLURGICA, VOL. 10, DECEMRER 1962 1187

If a supersaturated solid solution with a solute oon- centration of a few percent is aged at temperatures

* Received May 7, 1962.

LETTERS TO THE EDITOR

Auf die Einwande von Kirkaldy(l’ zu meiner ersten Bemerkung’s) miichte ich folgendes antwor~n :

Wie ich bereits zum Ausdruck brachte, hat Kirkaldy recht mit der Feststellung, daIi die Substitution il = x/l/t zur L&sung der Differentialgleichung (1) unter den Anfangsbedingungen (2) seiner Veriiffentlichungcs) nicht nur auf eine Naherung fiihrt. Die Boltzmann- sche Losung ist eine exakte Losung dieses Problems. Aus der Existenz einer Losung folgt jedoch noch nicht ihre Eindeutigkeit. Ich habe an einem Beispiel ge- zeigt, daO die Differentialglei~hung unter den von Kirkaldy gemachten Voraussetzungen mehr als tine L&ung haben kann. Die angegebene zweite Losung geht bei x = 0 mit t -+ +O gegen co. Dieses Verhal- ten widersprieht nicht den Anfangsbedingungen ; denn diese enthalten fiir x = 0 keine Vorschrift. Sie wer- den tatsachlich und nicht nur scheinbar erfiillt.

C. SEYFEBTH

References

Guinier-Preston zones in b.c.c. iron*

1188 ACTA METALLURGICA, VOL. 10, 1962

below -0.5Txabs, deviations from random distri- bution of the solute atoms can occur ; these deviations (G.P. zones) are coherent with the matrix. The shape of such zones seems to depend on the ratio of atomic sizes of solute and solvent. They are spheres if this ratio is close to one (Al-Ag) or plates if it differs Tram one &&&I, Al-Zn, C&Be). Plates always form on (100) planes of these f.c.c. lattices. The atoms in those plates can be ordered, as observed in an Al-Zn-

Mg alloy. Almost all observations on zones have been made in f.c.c. lattices.(i) Corresponding effects in b.c.c. lattices have not been reported. However, in supersaturated IS-MO solid solutions, molybdenum- rich clusters grow during aging at low temperatures. These clusters are not completely coherent since a dis- location ring forms, probably after clustering of solute atom-vacancy pairs.(2)

Aging experiments with an iron-3.8 at. % gold

Fxa. I. G. P. zones in an Fe-3.8 at.% Au solid soIution quenched from 1100°C. a, b, c: electron diffraction

w&terns: (a) As quenched (100) in surface (b) Aged 8 hr at 400°C (100) in surface (c) Aged 8 hr at 4OO’C (110) in surface (d) Transmission micrograph of alloy aged 8 hr

at 400%

LETTERS TO THE EDITOR 1189

10011

t (IO01 [OlOl

0 IRON ATOMS IN LAYER A OF (I001

0 I’ ” IV On B ” (1001

0 GOLD ATOMS IN (001)

FIG. 2. Model of a G.P. zone in a b.c.c. Fe-Au solid solution.

solid solution have produced evidence that G.P. zones

can form in a b.c.c. lattice, analogous to those in a

f.c.c. lattice. A homogeneous solid solution of this

alloy can be obtained by quenching from the y-field

(1100”); a y + cc transformation takes place during

quenching.t3) Fig. 1 shows the changes in the trans-

mission electron diffraction patterns of crystals of this

solid solution aged at 400°C. Fig. l(a) is the diffraction

pattern of b.c.c. as-quenched crystals with a (100)

plane parallel to the plane of the foil. Figs. 1 (b) and (c) are patterns of crystals aged 8 hours at 4OO”C, with

{ lOO} and { 1 lo} parallel to the plane of the foil, respec-

tively. Asymmetrical streaks develop on the recipro-

cal lattice points in all (100) directions. These streaks

indicate the formation of gold plates on {lOO},,,,. The

streaks are due to the difference in scattering power

of gold and iron dfA, > fFe) and to the distortion caused

by the difference in atomic radii of the two atoms

rAU/r re= 1.13). The direction of asymmetry of the

streaks indicate that the spacing of the (100) planes

containing zones is larger than in iron in contrast to

the smaller spacing due to smaller copper atoms in

aluminum. A gold zone in iron is shown schematically

in Fig. 2. The size of the zones can be estimated from

a transmission electron micrograph (Fig. Id). The con-

trast is at least partially caused by elastic distortions

around the zones; the images are therefore not sharp.

2oo t@---L- IO IO' IO" 103

[HOURSI

FIG. 3. Hardening in an Fe-3.8 at.% Au alloy aged at 400°C

o-one phase .-two phase

The size of the zones at this stage of aging is about,

25-50 A.

If the alloy is aged for about 100 hr at 4OO”C, nu-

cleation of noncoherent precipitates can be observed.

These particles have a f.c.c. structure and {loo},,,,,

habit, as described for precipitation in an u-Fe 1.14

at. ‘A Au solid solution. t4) The habit plane indicates

that the G.P. zones provide the concentrations of gold

required for nucleation of the f.c.c. particles. The

clusters may grow to a certain size and then transform

to the f.c.c. lattice. According to a direct observation

this transition takes place after the plates have reached

a size of 100-200 A.

Fig. 3 shows the hardness of the alloy aged at 400°C.

The hardness reaches its maximum when the clusters

have reached their maximum size and nucleation of

f.c.c. particles occurs. The decrease in hardness on

further aging is due to a decrease in number of f.c.c.

particles with longer aging times. Hardening due to

formation of zones is possible in iron, as well as in

aluminum alloys.

The author wishes to thank R. D. Schoone of this

Laboratory for his assistance in the experimental work.

Edgar C. Bain Laboratory for

Fundamental Research

E. HORNBO~EN

United States Steel Corporation

Research Center

Monroeville, Pennsylvania

References 1. A. GUINIER, Solid State Physics Vol. 9, p. 345. Academir

Press, New York (1959) 2. E. HORNBOGEN, J. AppZ. Phys. 32, 135 (1961). 3. W. K~STER and E. BRAUN, 2. MetaUk. 41, 238 (1950). 4. E. HORNBOQEN, Acta Met. 10, 525 (1962).

* Received June 11, 1962.

Deformation twins in cobalt*

Hall(l) has reported a {lOi2} composition plane for

deformation twins in large-grained cobalt specimens

that had been compressed in a vise.

During an investigation into plastic deformation in

cobalt crystals of commercial purity, twins have been

produced by bending .i in. dia. single crystal rods.

Two types of twin were observed: normal lenticular

twins and also some very narrow twins. Fig. 1 demon-

strates the two types. There was a tendency for the

narrow twins to form more readily at liquid nitrogen

temperature and for lenticular twins to form at room

temperature, but both have been observed at either

temperature. Analysis of twin traces for the narrow twins was