THE REDISTRIBUTION OF SOLUTE ATOMS DURING THE SOLIDIFICATION OF METALS*

W. A. TILLER, K. A. JACKSON, J. W. RUTTER, and B. CHALMERSt

A quantitative analysis is made of the redistribution of solute resulting from solidification in the absence of convection of a binary solution for transient and steady state conditions. Diffusion in the liquid is taken into account and shown to be of importance in determining the solute distribution in both the liquid and the solid. It is shown that the distribution for both normal freezing and zone- melting depends on the rate of solidification. When the speed of solidification is increased abruptly, a band of high solute concentration is formed in the solid; the reverse occurs when the speed is decreased abruptly. Values for the length of the constitutionally supercooled zone of liquid adjacent to a growing solid-liquid interface are calculated.

LA DISTRIBUTION DES ATOMES DUN CORPS DISSOUS LORS DE LA SOLIDIFICATION DES METAUX

Une analyse quantitative est faite de la distribution du corps dissous resultant de la solidification’ en absence de convection, d’une solution binaire, dans les cas des regimes transitoires et permanents. 11 est montre que la diffusion dans le liquide, dont il est tenu compte, joue un r6le important dans la determination de la distribution du corps dissous dans.le liquide et dans le solide. 11 est aussi montre que cette distribution depend de la vitesse de solidification, aussi bien dans le cas de la solidification normale que dans celui de la fusion en zones (“zone-melting”). Quand la vitesse de solidification est brusquement augment&e, une bande de haute teneur en corps dissous est formee dans le solide; l’inverse se produit quand la vitesse est brusquement diminuee. Des valeurs de la, longueur de la zone de surfusion de constitution (“constitutional supercooling”) dans le liquide adjacent a la surface de separation entre le solide en croissance et le liquide ont ete calculees.

DIE WAHREND DES KRISTALLISATIONSVORGANGS IN METALLEN AUFTRETENDE VERANDERUNG IN’ DER VERTEILUNG DER GEL&TEN PHASE

Die in Abwesenheit von KonvektionsstrGmen wahrend der.Kristallisation erfolgende Veranderung in der Verteilung der gel&ten Phase in Zwischenstoffsystemen wird fur stationare und nicht-station&r-e Bedingungen quantitativ diskutiert. Die Diffusion in der fltissigen Phase wird in Rechnung gestellt, und es wird gezeigt, dass sie sowohl fur die Einstellung der geldsten Phase in der Flilssigkeit als such im Kristall von Bedeutung ist. Es wird gezeigt, dass die Verteilung sowohl im Fall der normalen Kristallisation als such im Fall der “Zone-Melting” von der Kristallisationsgeschwindigkeit abhangt. Wenn die Kristallisationsgeschwindigkeit pldtzlich ansteigt, bildet sich ein Band erhohter Konzen- tration im Kristall. Das Umgekehrte tritt bei einer plijtzlichen Verringerung der Kristallisations- geschwindigkeit auf. Die Ausdehnung der auf Grund des Konstitutionsunterschiedes unterkiihlten, der Grenzflache fltissig-fest im wachsenden Kristall unmittelbar benachbarten Fltissigkeitsschicht wird berechnet.

Introductioq

It is well known that redistribution of the solute occurs as a crystal solidifies from a melt of impure metal or alloy. For the case in which a single phase solid is formed, the equilibrium concentration of solute in the solid at the interface between solid and liquid is different from the equilibrium concentration of solute in the liquid adjacent to it, the ratio of the former to the latter being denoted by “K”. This ratio is called the distribution coefficient and is expressed diagrammatically by Figure 1.

The resulting segregation has been studied and treated mathematically by a number of investiga- tors. Notable among these studies has been the recent work of Pfann on zone-melting [I]. In his work, he has extended the treatment of normal freezing by Gulliver [2], Scheuer [3], and by Hayes and Chipman [4], to the treatment of the results of

*Received March 16, 1953. tDepartment of Metallurgical Engineering, University of

Toronto, Canada.

ACTA METALLURGICA, VOL. 1, JULY 1953

traversing a relatively long charge of solid alloy by a small molten’ zone (i.e. small with respect to the length of the charge). He showed that it is possible to eliminate or enhance the segregation of the solute by suitable variation of the growth conditions.

These treatments of segregation, whether they apply to ordinary freezing or zone-melting, are incomplete. The above-mentioned authors have expressed, in one form or another, the concentration

FIGURE 1. Portion of binary constitutional diagram for a solute which lowers the freezing point of the solvent.

‘TILLER I-T AI..: REDISTRIBl-‘I-101 OF

in the solid as a function of g, the fraction solidified, and have derived, for freezing, an equation of the form

c = KC*(l - g)l-’

where Co is the initial concentration of solute in the melt, and k is the ratio of the solute concentration in the solid to the concentration in the liquid.’ Equation (-1) has been based on the following assumptions:

1. Diffusion in the solid is negligible. 2. Diffusion in the liquid is complete (i.e.,

concentration in the liquid is uniform at all times). 3. K is constant. The present paper shows that the second assump-

tion & not; valid. According to Frenkel [5] and to Jost [6], the diffusion coefficient is l-10 cm2/day for all liquid metals thus far studied. It will be shown that the rate of diffusion of solute in the liquid, calculated from this value of the diffusion coefficient, is not large enough for the assumption of complete liquid diffusion to be even approximately valid under the conditions used for the production of single crystals or for zone-melting.

Mixing of the solute due to convection will tend to level the solute concentration in the liquid. However, convection can be eliminated by growth in a vertical tube so that the variation of the liquid density with temperature along the specimen will result in the least dense part of the liquid being at the top. Mixing due to natural convection will then be absent; incomplete mixing will also occur for hori- zontal growth under a low temperature gradient, since natural convection will be reduced by a decrease in temperature gradient. For specimens of small cross-section there will be little convection because of the small space available. For specimens of large cross-section, convection must be reIa&eIy rapid to remove compIetely the layer of high solute concentration formed adjacent to the growing interface, before that layer solidifies. Experimentally, it has been observed, for crystals grown horizontally, that a change in the rate of growth produces a marked change in the soIute concentration of the solid. If mixing were complete, the solute concentration of the solid would be independent of the speed.

If complete mixing could be obtained, by either forced or natural convection, equation (1) would be valid. ExperimentalIy, complete mixing will be approached most cIosely by the use of high fre- quency induction heating, which causes stirring of the liquid, and by growth at a slow speed.

The present solute in both

SOLCTE 420

Normal Freezing

paper examines the segregation of normal freezing and zone-melting.

The following assumptions are made: 1. Diffusion in the solid is negligible. 2. The value of k (as defined in the first paragraph)

is constant. 3. Convection in the liquid is negligible.

Since the solidus and liquidus lines in most equili- brium diagrams are slightIy curved, k will vary; however, an average k can be found for the concen- tration range considered. Attention will be directed to the case of k < I; this occurs when the solubility of solute in the solid is less than in the liquid; this corresponds to “rejection” of solute atoms by the solid as it forms, and therefore to a progressiveIy increasing concentration of solute in the remaining liquid.

Let the solute concentration in the liquid be Cc; then the initial layer of the solid to freeze will have a solute concentration of kCc. As the crystal grows the liquid concentration will rise because of the solute rejected from the interface; this will cause the solid concentration to rise also. This increase of concen- tration will continue until a steady state condition is attained. At this point the solid concentration adjacent to the interface will have reached a steady value and the distribution of solute in the liquid near the interface will be constant also.

Let the form of the liquid distribution CL be as shown in Figure 2, Co being the initial concentration in the liquid. Two factors influence the form of CL, and they are: (1) diffusion in the liquid tending to spread the solute uniformly throughout the melt at a rate governed by the diffusion coefficient D cm2/sec; (2) the process of freezing which acts as a source of solute at the interface between solid and liquid-the rate of freezing is measured by the advance of the interface, R cm/set.

G Intcrfacj i / 1 : :

-00 x x+&i

1 -Dw,tancc from Interface. X -

FIGURE 2. Steady state distribution! CL, of solute in the liquid ahead of the freezing solid-liquid interface.

430 ACTA METALLURGICA, VOL. 1, 1953

By consideration of Figure 2, the amount of solute diffusing into unit area of face x is D(dc/dx),, the amount diffusing out at face x + dx is D(dc/ dx)z+dz; therefore, the net flow into a volume element is D(d2c/dx2) per unit volume. If the solid-liquid interface is considered as the origin, and freezing is represented by moving the liquid distribution past it at the rate R cm/set, then the net flow out of the same volume element due to freezing is R dc/dx. Therefore, the differential equation describing a stationary distribution with respect to this CO-

ordinate system is

distribution CL, especially if the distribution ex- tends far into the liquid. An increasing rate of growth, however, will serve to decrease the effect of the convection on two counts: (i) For rapid rates of growth the exponential solute distribution has a large decay constant and does not extend very far ahead of the interface, but falls rapidly to Co. This distribution may lie entirely within the “stagnant” layer adjacent to the interface which would not be affected appreciably by convection. (ii) Since the growth ,is rapid, there will be little time for convection to alter the distribution CL.

The concentration in the solid C,, rises from its initial value of kCo to its equilibrium value and maintains this concentration as solidification con- tinues. When the steady state has been reached, the amount of solute in the region just ahead of the interface must be constant; therefore, the amount of solute leaving the liquid while the interface is advancing by unit distance must be equal to the amount of solute entering the region concerned. Therefore, the concentration of the solid must equal the concentration of the undisturbed liquid, i.e. Co. If C, < Co, there would be an unbalance in the system such that the excess solute would build up the exponential distribution in CL which would in turn build C, up to Cc. If C, > Co, then the solid would be absorbing more than the source is supply- ing, and would thus tend to lower the exponential in CL, which in turn drives C, back to Co.

(2) D&+,2=0

which yields a solution

(3) CL=Caexp -2x’ -l-C0 ( )

where C, is the solute concentration in the liquid at the interface, and the variable x’ denotes distance in front of the interface.

Equation (3) shows that the liquid distribution is exponential and that its “decay constant” is deter- mined by the ratio of the rate of growth to the diffusion coefficient. The value of D as given by Frenkel [5] is approximately l-10 cm2/day for all liquid metals, whereas R can be varied. A measure of the “characteristic distance’,’ of the redistribution, i.e. where CL has fallen to Co + C,/e of its value at the interface, is plotted in Figure 3, for various values of R.

t .09- t .08 8 .07 -

8‘06. 5’ 9 .z Q5-

2 .04- u)

6 .a- “0 g .oz .

r” ” al .

-Oo I 1 3 4 5 5 7 0 9 to I I@

1 -Rate of Growth, R cm/set. _

FIGURE 3. “Characteristic distance,” D/R, of the solute distribution in the liquid, CL, (equation 5) as a function of the rate of growth, R.

The influence of convection on the solute distribu- tion in the liquid will depend both on the tempera- ture differences in the liquid and on the rate of growth of the crystal. Natural convection is caused by the presence of temperature differences in the liquid. This convection will tend to lower the solute

From a consideration of Figure 2, steady state conditions yield C, = Co, and C, = Co/k - Co at the interface. Therefore, equation (3) becomes

(4) CL= CO (

l+$$exp (+)>.

From equation (4), the form of the solid distribution C, up to the equilibrium level may be calculated. The curve of solute concentration in the solid at the beginning of the specimen must satisfy the following conditions:

1. It must rise from kCo at the beginning of the crystal.

2. It must tend asymptotically to Co with dis- tance along the crystal.

3. It must rise continuously from KC0 to CO. 4. From consideration of Figure 4, the area

between Co and C, must be equal to the area be- tween CL and Co for “conservation of solute.” If it is assumed that the rate of approach of C, to CO with distance along the specimen is proportional to (Co - C,) at any distance x measured from the begin- ning of the specimen, it follows that C, is of exponen- _ _

TILLER ET AL.: REDISTRIBl’TION OF SOLL-‘IE 431

tial form. The justification for this assumption is that it appears plausible and is the simplest possible assumption which fits the four boundary conditions. Using this assumption and the four conditions noted above, C, is derived in Appendix I, and is given b>

(5) c, = co ( (1 - k) : [I - exp(- kax)]+k\

where x is the distance measured from the beginning of the crystal. The rise of C, is illustrated in Figure 4, and similarly the “characteristic distance” of the distribution is found to be x = D/kR cm, at which point C, = C,,i(l - k)(l - l/e) + k).

I - Distance -

FIGURE 4. Distribution of solute in the liquid, C,, and in the solid, Cs, in the tirst part of the specimen to solidify. CT shows the solute concentration at the growing interface.

The region before equilibrium conditions are reached may be termed the transient region. In this region the shape of the solute distribution in the liquid can be determined from C, since CT = C,!k for any point x as shown in Figure -1. As the solid- liquid interface moves from w = 0 to x = xE, the area under the liquid distribution C, - Co at an\ value of x is equal to the area Co - C,3 from 0 to x; this must be so in order to conserve solute. The form of CT is given bl

(6) C,=$=Co(LiA[l- exp(- kgx)]+ 4.

It may also be seen that the transient liquid distribution curve C, is exponential with a “decaJ constant” identical to that which prevails under equilibrium conditions (see Appendix II). The distribution C, is given by

w - ; (x - Rt)

where t represents time and x > lit. For a complete solution of the solute distribution

in the solid, the last part of the specimen to solidify must also be considered. The analysis of this region

is complicated by the fact that a finite liquid region, in which the concentration changes at all points, must be considered. No attempt will be made to derive quantitatively the solute concentrations in this region. However, the qualitative form of the solute distribution in the solid will be derived.

Let: x’ = distance measured from the interface into the liquid,

N = distance measured from the beginning of the specimen,

L = length of specimen, x1 = distance from interface to end of

specimen. Assume that L is large compared to D/kR so that

the steady state solute distribution in the liquid CL is established over an appreciable length of the specimen. The transient solute distribution in the solid at the end of the specimen can then be con- sidered independently from that at the beginning.

As the interface approaches the end of the speci- men, the concentration at any x’ < xi must increase. Assume that the concentration at any x’ < x1 remains constant, as shown in Figure 5. Then, since the steady state distribution CL has been established, the amount of solute in excess of Co which would

FIGCRE 5. Assumed solute concentratio?, CL, in last portion of specimen to solidify, used in derivation of equation (8).

have appeared in the solid as the interface moved to x = (L - x1) is given by the integral of C, from x’=xrtox’= m,andis:

(8) R

Co I-$ g exp - 5x1 . ( )

This quantity of solute is represented by the area under the dotted portion of CL for interface position 2 in Figure 5. If this quantity of solute had come out in the solid by the time the interface reached the point x = (L - xl), the solute concentration in the liquid at any x’ < x1 would not have changed. Since a change in solute concentration in the liquid is necessary to cause a change in solute concentration in the solid, the amount of solute above Co which has appeared in the solid by the time the interface has reached x = (L - ~1) must be less than the value

432 ACT.4 METALLURGIC:\, \:OL. I, 1953

given by (8). RJ- comparing (8) to the similar expression giving the integrated solute deficiency

below CO at the beginning of the specimen, the shape of the distribution in the solid at the end of the

specimen can be compared to that at the beginning.

The solute deficiency at the beginning, from a point x = x2 to a point where the stead). state distribution

CL is established, is found from equation (5) by integration, and is given by

(9) 1

CO - k Rw --kg (-h;x,).

Expressions (8) and (9) arc equal when x2 = xl/k.

Therefore, when the interface is at a distance x2 from the beginning of the specimen, a fraction

[l - exp(- kgxl)l

of the solute represented by the area under the steady

state distribution CL is being carried ahead of the

interface and when the interface reaches a distance x1 = kxz from the end of the specimen, more than

this fraction of the solute is still being carried ahead of the interface. If the concentration in the solid is

insignificantly different from Co at a distance x2 from the beginning of the specimen, it will still be

insignificantly diRerent from CO at a distance less

than kxr from the end. For values of k less than

unity, the rise at the end is more rapid than that

at the beginning. If no increase in concentration occurred at any

x’ < x1, as postulated previously, then the concen- tration in the solid would finally reach CO/k at the

FIGURE 6. Solute distribution in the solid, Cs, and solute concentration in the liquid at the growing interface, CT, for the complete crystal. The dotted lines: CL, indicate the solute distribution in the liquid for various Interface posItIons.

end of the specimen. Hence, the concentration in the solid at the end must reach a value greater than CO/k. If CO is large enough, this will result in production of a second phase in a suitable alloy system. For example, let CV be the limit of solid solubility at the eutectic temperature of a binary

eutectic system. \L’hen the second phase is formed, the solute concentration in the liquid must be

greater than the eutcctic composition C,. Since

C, > Co/k at the end of the crystal, then the solute concentration in the last liquid to freeze is greater

than CO/k2. If Co/k2 > Cg, that is, Co > k2CE, then the second phase will form providecl that the neces- sary nucleation occurs. Therefore, if the initial

solute concentration CO is greater than kCv, then the second phase will appear. If the solute is a gas, then the second phase will be gaseous and bubbles of gas will be nucleated.

The form of the solute distribution in the solid

for the entire specimen is illustrated by Figure 6.

The Effect of Changing R

Knowing the characteristics of the solute distri- butions in both the liquid and the solid, it will be

shown that increasing or decreasing the speed of

growth produces transverse bands of higher or lower concentration. CL is given by equation (4) for any growth rate R. If the growth rate is increased,

then the exponential must become steeper, as shown in Figure 7. When the distribution comes again to a

I -x- FIGURE 7. Steady state solute distributions in the liquid

ahead of the interface for rates of growth R, and RI.

steady state for the new growth rate, the concentra- tion at the interface must once again be Co/k. From Figure 7, the areas under the two curves are now different and this difference in area is exactly

what has been added to the solid distribution due to the changed rate of freezing. If R decreases, the amount of solute entering the solid is temporarily reduced (see Figure 8).

In practice, though the rate of freezing, R, can be changed very rapidly, the solute distribution in the liquid cannot change instantaneously from the distribution characteristic of RI to that characteris- tic of Rz. If RP > RI, then the freezing interface rejects more solute than can diffuse down the con- centration gradient characteristic of RI in the time allowed, and thus the concentration at the interface

must increase. The concentration gradient then

becomes steeper, and as the solute distribution changes, the amount of solute diffusing down the gradient will increase until it becomes equal to the amount being supplied as a result of growth, which is governed by R2. The rate of loss of solute, due to diffusion, will ultimately exceed the rate of suppl> due to growth and the solute concentration will then decrease to the value characteristic of RR?.

Between the two steady state conditions, CL must rise (at the interface) by an amount which depends

t i 5, li; 2 I R2,RIC “;-0’ 0 / y------v,,.,r

I -Distance -

FIGURE 8. Change in solute concentration in the solid, C,, as a result of a change in growth rate, R.

on the change in R and the values of RI and R?. If RI is large, then the distribution CL is essentially confined to a narrow region adjacent to the interface and the change to the distribution characteristic of R2 will be accomplished by moving the interface only a very short distance. In this case, the solute concentration in the solid, C,, will rise and fall steeply.

The amount of solute that appears in the solid as a result of the change in speed can be found bJ- integration of equation (4) for the two values of R, and is given by

(10) AC, = Col_kA_D

The rate of change of R is controlled by the ther- mal conductivity of the crystal and the latent heat

FIGURE 9. of distance.

Assumed change in growth rate, R, as a function

of fusion, i.e., how fast the latent heat is conducted away, and on the imposed change in the rate of supply and extraction of heat.

It is thus apparent that the presence of transverse bands [8] differing in concentration from the equili- brium value of Co ma!- be created by rapid changes in the growth conditions of the cr>*stal.

If the heat flow conditions for the system are disturbed abruptly by lowering the heat input, an amount AQ for a time At and back again, then the crystal will react to this sudden discontinuity and a band will result. The crystal will grow rapidly for a very short distance in an accelerating fashion, first positive and then negative, until once again it returns to its initial growth rate when the heat flow conditions are stabilized at their original value. The resulting distribution is represented in Figure 10.

If the distance SAX (Figure 9) is less than the dis- tance to the maximum in the curve, then the solid distribution will travel up AC, (the steepness being determined by RZ - RI), and at the distance $Ax will start down again since the driving force has been reversed. If the rate of rise is the same as the rate

,,-----.._\

t

I’ ,’ l\

l\ ‘..\

9 \

: n__ CO *xi

-Distance -

FIGURE 10. Change in solute concentration in the solid caused by the assumed change in growth rate shown in Fig. 9.

of fall, the decay constant for ACB will be the same on both sides (see Figure 10). The value of AC, will fall below Co since the acceleration is now negative and the total area under (AC, - Co) must be zero to conserve solute. If the rate of growth is once again reversed at Ax, then AC, will start to rise again.

Supercooling

The liquid ahead of an advancing interface may supercool for two different reasons. The interface must be at a temperature below that at which the solid and liquid would be in equilibrium and, therefore, the adjacent liquid is also supercooled.

The second cause of supercooling arises only in impure metals and alloys, and is related to the distribution of solute in the liquid. It was described and discussed qualitatively by Rutter and Chalmers [7] who describe it as “constitutional supercooling,”

434 ACTA METALLURGICA, VOL. 1, 1953

For an alloy, the distribution of solute in the liquid will be given by equation (4). Thus, every point in the liquid ahead of the interface has a definite concentration of solute atoms, and thus a definite liquidus temperature as given by the equilibrium diagram for the alloy under consideration (see Figure 1). There is also a definite temperature gradient in the liquid metal imposed by the growth conditions. If the temperature at a certain point as maintained by the gradient in the liquid is lower than the liquidus temperature at that point as predicted by the equilibrium diagram, then the liquid at that point wiII be constitutionally super- cooled.

If only constitutional supercooling is considered, then the length of the constitutionally supercooled zone may be calculated. The solute distribution is given by equation (4), and from the equilibrium diagram the equilibrium temperature is given by

ill) T, = To - mCs

where To is the melting point of the pure metal, and m is the slope of the liquidus line (see Figure 1) ; m is assumed to be constant for simplicity. Thus, the equilibrium temperature for any point in front of the interface is given by

(12) T, = TO - mG 1 +-y exp

-Distance ahead d the Intwfacc, cm. -

FIGURE 11. Equilibrium liquidus temperature (solid lines),, calculated from equation (12), and specimen tempera- ture (dashed lines) calculated from equation (13), as functions of distance ahead of the interface.

The temperature gradient in front of the interface may be expressed as

03) T = To - + + Gx’

where TO - mCo/k is the temperature at the inter- face, and G is the temperature gradient in the liquid. If the above two equations are pIotted, the

point of intersection other than x’ = 0 will give the length of the supercooled zone. Thus, for T = Tg it is found that

(14) 1 - exp - $x’ = ;G,(IG_ km x’. ( )

Equation (12) and equation (13) are plotted in Figure 11. The graph gives the length of the super- cooled zone for different growth rates. From equa- tions (12) and (13), the critical growth rate and temperature gradient for no supercooling can be calculated by equating the slopes of the two curves at the interface. These critical values are found from a solution of

(15) G e&,1-‘k R D k’

Incubation Distance for Cellular Substructure

A substructure of a cellular nature has been observed and studied in single crystals by Rutter and Chalmers [7]. They are columnar in the direc- tion of growth, and hexagonal in cross-section. Their presence is attributed to the existence of a constitutionally supercooled zone of the melt just ahead of the interface.

As the crystal grows, the liquid concentration at the interface builds up from its initial value Co to its equilibrium value Co/k in a. manner given by equation (7). Thus, the crystal must grow for some distance before the liquid concentration is large enough to produce any constitutional supercooling. This incubation distance 2 can be calculated by equating the slope of the temperature gradient’ in the liquid to the slope of the equilibrium tempera- ture in front of the interface as given by equation (ll), where the concentration is given by equation (7). The incubation distance is then

For tin containing 0.016 per cent lead (CO = .016 per cent, m = 1OOC”/40 per cent, k = 0.1, R = .005 cm/set, G = 75C”/cm, and D = 3 X 10~Qm2/sec), it is found that 2 is of the order of 0.2 cm.

Singb .Puss Zone-Melting

Consider a charge of binary alloy of constant cross-section whose composition CO is constant. Let a molten zone of length 1 traverse the charge, as shown in Figure 12. In the light of the preceding

TILLER ~‘l‘ AL.: REDISTRIBI-TION OF SOLUTE ‘135

theory, the segregation of the solute will depend on the zone-length 1. By varying the size of the zone- length, two cases may arise.

Case 1. 1 > L,, the characteristic length of the distribution in the liquid, i.e. where the concentra- tion has fallen close to the value CO. In this case the

FIGURE 12. Solidification by zone-melting.

liquid distribution will be identical to that for normal freezing, and thus the theory already devel- oped will apply to the segregation of solute for zone-melting.

Case 2. 1 < L,. Under equilibrium conditions the liquid distribution must satisfy equation (a), and therefore must be of exponential form. At the interface the concentration must be Co/k as usual, and the concentration must fall off as R/D; thus the distribution must be as shown in Figure 13. At a

I - Dtstancc -

FIGURE 13. Steady state solute distribution in molten zone.

distance 1 ahead of the interface, the concentration has not fallen to the value Co, but to a value greater than this, consistent with the exponential decay.

Since the area contained under this distribution must be smaller than the area for the zone-length

- o,*tancc -

FIGURE 14. Solute distributions in solid, Cs, and in liquid, C,, in initial transient region for the case of zone- melting. CT gives the concentration at the growing interface as a function of distance from the beginning of the specimen.

required for normal freezing, C,Y must rise more rapidly. Thus, the smaller 1 is, the more rapid must be the rise of the solute concentration in the solid. This deviation from normal freezing is illustrated in Figure 13. By the use of a treattnent similar to that of Appendix I for finding the decay constant in the solid, the distribution C, may be expressed as

(17) C, = CO{ (1 - K)(l - e?) + k)

where

il=k$[l-exp(-_%l)ll.

Puri$ication

From consideration of equation (17), it may be seen that, since the distribution of solute in the solid rises more sharply than for normal segregation, less purification results.

It is now clear that the smaller I is, the smaller will be the quantity of impurity transferred from the first part of the crystal to the last part in one pass. Therefore, the amount of purification per pass will decrease as 1 decreases below the characteristic length L, of the solute distribution in the liquid; hence, one limit on the zone-length should be 1 > L,.

However, in repeated pass zone-melting, if the zone-length 1 becomes larger than the characteristic length L, of the solute distribution in the solid, the average concentration in the first zone-length to be melted will be increased. When this happens, the average concentration may not be much less than the initial CO of the first pass; then the amount of purification per succeeding pass will be decreased. With the view to increasing the efficiency of this method of purification then we should make 1 < L,.

The conclusions from the above work provide the optimum conditions for purification, and they are :

(18) L, < 1 < L,.

Equation (18) illustrates the importance of R, the rate of growth for a given zone-length I; R deter- mines L, and L,, the characteristic lengths of the solute distribution in the liquid and in the solid.

Comparison with Experimental Observations

Banding

The technique of autoradiography as applied by Stewart et al [S] has made observable a form of banding. It was observed that crystals containing

438 ACTA METALLURGICA, VOL. 1, 1953

small concentrations of solute sometimes tend to grow with the solute segregating out into alternate bands of large and small concentrations transverse to the direction of growth. One of these autoradio- graphs showing banding was placed on the densito- meter, and there was found to be a periodic rise and fall about a mean value as shown in Figure 10. Thus, in the light of the preceding analysis, the phenomenon can be explained as a result of fluctua- tions in the heat input to the liquid.

Cellular Substructure

The existence of a zone of supercooling is not in itself the cause of the cellular substructure; it has been pointed out by Winegard and Chalmers [9] that a positive value of dS/dx’, where S is the amount of supercooling, is the criterion for the formation of the substructure. This condition arises as soon as constitutional supercooling occurs.

The theory developed considers the distribution of solute ahead of the interface for diffusion in one direction only. This is complete only if the interface is plane; however, for an irregularly-shaped inter- face, the treatment would require extension to include lateral diffusion. The presence of the cellular substructure would alter the preceding picture, since the centres of the cells extend further into the melt than the bases, thus altering the equilibrium distri- bution of solute in a direction perpendicular to the direction of growth. Rutter and Chalmers [7], in their study of the substructure, observed this transverse segregation of impurities.

Zone-Melting

Though the distribution of solute as given by equation (17) differs from that of Pfann [l], it predicts the same qualitative results; in addition, however, it shows that the rate of freezing is a factor which influences the redistribution of solute. The more rapid the rate of growth, the steeper will be the rise of the solute concentration in the solid. The zone-length also appears as a factor in the redistribution process; the smaller the zone-length, the steeper will be the rise of the solute concentration in the solid. It may be seen from consideration of equation (17) that any result which may be achieved by changing 1 can equally well be brought about by changing R, the rate of growth. The optimum conditions of operation in the purification process by the repeated pass method have been evaluated. Finally, from the point of view of the redistribution of solute, it is also seen that, when the zone-length is greater than L,, there is no difference between zone-melting and normal freezing.

Acknowledgments The authors are indebted to the National Research

Council, Ottawa, and the University of Toronto, for financial assistance during the course of this work.

APPENDIX I

Refer to Figure 4. Let: Co = solute concentration in the melt before

solidification begins. x = distance measured from the beginning

of the specimen. C, = solute concentration as a function of x. k = distribution coefficient.

Assume that the rate of approach of C, to Co with x is proportional to (Co - C,) at any x. Then:

d(Co - CJ _____ = - a(C0 - C,) dx

where a = a constant. The solution of this equation is:

Co - C,= Ae-@+ B

where A and B are constants. Since (Co - C,) -+ 0 as x + w, therefore B = 0, and

C, = CO - Ae-“”

Applying condition 1 noted in the text (p. 430),

C, = KC0 when x = 0,

kCo= Co-A,

A = Co(1 - k).

Therefore,

C, = Co - Co(l - k)eeQz

= Co((1 - k)(l - e-““) + k}.

It will be seen that this equation satisfies conditions 2 and 3.

Condition.4 is satisfied by proper choice of (Y, as follows: The area A, between CL and Co is

1-kD AI=Co-i-~

The area A2 between Co and C, is

Az = ; Co(1 - k).

The condition A1 = AZ yields

kR CY = -.

D Hence

(5) C, = Co{(l - k)[l - exp(- k$x)] +.k}.

APPENDIX II

Derivation of Equation 17)

Refer to Figure 4. The notation used will be the same as that in

Appendix I: in addition, let: C7. = the solute concentration in the liquid at

the interface, expressed as a function of distance .Y from the beginning of the crystal.

C, = solute distribution in the liquid when the interface is at any particular distance .Y = X, from the beginning of the crystal.

xE: = value of x for which the steady state is essentialI\- established, i.e., C, = Co and CT = G./k.

x’ = distance in liquid measured from interface as origin.

A2 = im(CT(xr) - C0]e+*‘dx’

= $ i_A-k/I -exp(-k&)], Therefore,

R o(=-

If the interface moves from x = 0 to x = xi in time t at rate R, then C, may be expressed as

(7) C,= CO(~,-P(l-exp(-k~L)] ’ For any position, xi, of the interface between 0 and

xE conservation of solute must be maintained. Therefore, the area Al between C, and CQ up to the point xi must be equal to the area AT, between C, and Co.

C, is given b,-

Assuming C, to be of exponential form, it may be 3 written as 4.

C, = (C,(xi> - CO)e-uz + CO, - 3.

where Cr(Xi) denotes Cr evaluated at Xi. Then 6.

7.

References

PFAN~‘, W. G. J. Metals, 4 (1952) 747. GULLIVER, G. H. Metallic Alloys (London, Chas. Griffin

and Co., 1922), Appendix. SCWEUER, E. 2. Metallk., 23 (1931) 237. HAYES, A. and CIIIP~~AN, J. Trans. A.I.M.E., 135 (1939)

85. FKENKEL, J. Kinetic Theory of Liquids (Oxford, Oxford University Press, 1946), p. 201. JOST, W. Diffusion (New York, Academy Press, 1952),

p. 479. RUTTER, J. W. and CHALMERS, B. Can. J. Phys., 311 (1953) 15. STEWART, M. T., THOMAS, R., WAUCWOPE, K., WINEGARD,

W. C., and CHALNPERS, B. Phys. Rev., 83 (1951) 657. ~~IN~G.~~, W. C. and CHALNERS, B. To be published.