Thermoelectric investigation of solidification of lead II. Lead alloys

Journal of Crysta] Growth 112 (1991) 563—570

563

North-Holland

Thermoelectric investigation of solidification of lead II. Lead alloys G.H. Rodway and J.D. Hunt Department of Materials, University of Oxford, Parks Road, Oxford OX] 3PH, UK Received 25 September 1990; manuscript received in final form 19 February 1991

Measurement techniques, based on the Seebeck effect method described in Part I, have been adapted to enable distribution and diffusion coefficients to be determined in dilute alloys, via the detection of temperature changes at the solid—liquid interface. Values 2 s’) for tin in a dilute of the distribution coefficient (k = 0.745 ±0.009) and liquid diffusion coefficient (DL = 0.9 (±0.2) x iO~ m lead based alloy were obtained. A dilute lead—silver alloy was also investigated by the same technique, and a value of DL (1.5 (±0.3) x 10 m2 s 1) obtained for the solute. The conditions for breakdown of a planar interface, and the velocity-undercooling relationship for subsequent cellular growth were also evaluated via thermoelectric measurement of the interface temperature. A separate technique was developed for non-invasive measurement of solute redistribution in zone refining, utilizing the effect of impurity concentration on the thermoelectric power of a material. This method was used to obtain an estimate for the effective distribution coefficient of an impurity during zone refining.

(iii) The conditions for breakdown of a planar

1. Infroduction

interface.

In part I, a method was outlined by which the temperature of a moving solid/liquid interface could be measured from the Seebeck EMF gener-

(iv) The velocity—undercooling relationship for cellular growth. Experiments of this type were carried out under

ated across it. The use of this technique to determine the kinetic undercooling—velocity relationship in high purity lead was described. In

conditions in which convection was negligible. In addition, preliminary results have been obtained in a convecting system, enabling solute redistribu-

addition, an undercooling due to the presence of

tion during zone refining to be studied.

small concentrations of solute was observed. Favier and Camel [1] also detected Seebeck emf changes which were attributable to solute build-up at the interface during the freezing of dilute tin alloys, and suggested that Seebeck EMF measurements might be used to study both interface kinetics and phenomena associated with solute additions. In the present work, the latter possibility has been investigated, using dilute lead alloys of known composition. Techniques have been developed and demonstrated by which it is possible to quantify: (i) The distribution coefficient of a dilute alloy, (ii) The diffusion coefficient of a solute in a liquid alloy at its melting point, 0022-0248/91/$03.50 © 1991

—

2. Experimental methods and results 2.1. Distribution coefficient measurement The experimental equipment and technique were as described in part I, except that, in the present case, the specimen wire was of an alloy. Consider a planar, freezing interface in such a wire specimen. If the interface velocity and wire diameter are sufficiently small, both kinetic undercooling and convective stirring will be negligible. For the present experiment let the wire be of a homogeneous, dilute binary alloy, produced by

Elsevier Science Publishers By. (North-Holland)

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Thermoelectric investigation of solidification of lead. II

deliberate solute addition to initially pure material, Under these operating conditions, the interface temperature will be determined solely by the diffuse solute field ahead of the interface. This will lead to a steady-state solute undercooling, ~T5, relative to a stationary interface of: L%T5

=

mC0(k

—

1)/k,

(1)

where k = distribution coefficient m = liquidus slope and C0 = alloy composition. Since ~T5 can be measured by the Seebeck method, and C0 has been imposed by the quantity of solute in the alloying addition, eq. (1) enables m to be determined in terms of k. However, m and k can be evaluated individually, by using eq. (1) in conjunction with the Van ‘t Hoff equation [2], which may be written as: 21

m — RTe ~k

\ —

where R = gas constant, L = specific latent heat of fusion and 7~= equilibrium melting point of material. Substitution into (1) gives: 2C~ 2/Lk. (3) LIT5 = R7~ 1(k — 1) Solving for k: -

— 1.65 °C/at% for higher concentrations, estimated from the phase diagram.

4)0.5]

(4)

~ [B ±(B where

as

2.2. Liquid diffusion coefficient measurement The operating conditions required for this cxperiment were similar to those outlined above: that the interface undercooling is determined by the diffusive solute field ahead of it. The steadystate solute concentration profile, C, in terms of .

~2)

1)/L,

2

up and extruded into wires, along which freezing interfaces were passed uniformly at low velocities. From a set of measurements by this method, an average value of /~T 5was obtained, namely L~iT5= 0.34±0.025°C. Substitution of this value into eqs. (5) and (4) above yields k = 0.745 ±0.009 and m = —1.59 ±0.05°C/at%. This is in good agreement with the value of k = 0.735 and m =

.

distance, .v, ahead of the interface is then given by

[31:

C=C1~1+

1

—

k

k

exp

J~x\

———I

D1

;

.

(6)

where V = freezing velocity and DL = diffusion coefficient of solute in liquid. The solute concentration C~,at the interface at any instant, is related to LIT 5 by:

LIT5

=

m(C~-

~0)’

(7)

=

B

=

2

+

Upon stopping the interface motion, the rate of decay of the concentration profile, and hence of

LLIT/RT~C (5)

C and the on interfacial LIT5, will depend the valuesolute of DL,undercooling and on the initial

Thus k can be calculated from the experimentally measured dependence of LIT5 on C0, via eqs. (5) and (4), and m can then be obtained from (3). The experimental method has the capability to measure small temperature changes in dilute alloys, whilst ensuring that the solid is growing as a single phase, due to the fact that steep positive temperature gradients are maintained at the interface. It can thus provide phase diagram information in regions not accessible by conventional thermal analysis. To demonstrate the above method, the lead—tin system was used, since its phase diagram is well established. A lead alloy containing 0.62 at% tin was made

steady-state concentration profile (itself dependent on DL, via eq. (6)). This connection between L%T5 and DL is the key to determining solute diffusivity values from Seebeck EMF measurements. The experimental procedure was to move the interface slowly in a fine diameter wire until steady state was reached, then to stop the drive motor and record the subsequent decay in solute undercooling as a function of time. DL was then determined from the best fit between experimental and computed decay curves. The computed decay curves were obtained from a time-dependent control volume finite difference

0.

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Thermoelectric investigation of solidification of lead. II

model of solute diffusion away form the interface. Since the interface was planar and perpendicular to the specimen wire axis, the diffusion problem (and hence the model) was one-dimensional. In the equations describing the change in solute content of each control volume within a timestep, the diffusive solute fluxes across the control volume tion gradient dependent terms, following Fick’s first law. In addition, however, the finite relaxation rate of were boundaries the interface simply represented towards itsby rest concentraposition needed to be accounted for if the solute concentration decay were to be modelled accurately. Allowance for this factor, by translation of the modelled domain at the velocity of the decelerating interface, gave rise to terms dependent on the concentration at the control volume boundaries, representing bulk mass transport across them. Incorporation of both the factors described above, into the model, allowed very close matching of computed and experimental decay curves, if appropriate values of DL were chosen, As with the distribution coefficient experiment, a dilute lead—tin alloy was studied. Some experiments were also carried out with the lead—silver system. To check the reliability of the results, each

600 offset (nV)

500

offset (nV)

400

300

Dots represent experimental decay profile The three curves are computed decay profiles for. 9m2c1 (1) D[ =0.5x10 (2) DL—09x109m2s

1

200

100

00

500

time (s)

1000

(3)

1500

Fig. 2. Dependence of fit between experimental and computed decay profiles on value of diffusion coefficient.

experiment was repeated for a range of steady-state velocities. A good fit between computed and experimental results for tin in molten lead was obtained for DL = 1.0 x i0~ m2 s~, as illustrated for four different velocities in fig. 1. To establish the errors in DL, attempts were made to fit lower and higher DL value computed curves to the experimental results. An example of this is shown in fig. 2. As can be seen, for DL = 0.9 x iO~ m2 ~ the fit is very good, but computed and experimental results clearly do not fit well for somewhat higher (1.3 X iO~ m2 s~) and lower (0.5 X io~ m2 s~) values of DL. The total error appears to be about half the difference between these extreme values, giving DL = 0.9 (±0.2) X i0~ m2 ~ for tin in molten lead at 327 °C. Similar sets of results were obtained for a lead—silver alloy. The experimental

400 ~V=1.6~m/

200 ~

1O.5~m/s

V = 63.5ism/s

time (s)

and computed results in this case fitted best for DL = 1.5 (±0.3) x i0~ m2 s~, i.e. significantly higher than for tin. 2.3. Measurement of cellular interface velocity/

G 0

40()

800

1200

Fig. 1. Comparison of experimental (dots) and computed (curves) declines in offset as solute (tin) diffuses away from the interface upon stopping its motion. Plots are 2 sfor — four in each different case, prior velocities as marked and DL = 10 ~m

undercooling relationship For the cellular growth of a very dilute alloy, only the interface slight grooves remains at the almost cell planar, boundaries, containing where

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G. H. Rodway, J.D Hunt

Thermoelectric investigation of solidification of lead. II

solute is rejected laterally. It is thus reasonable to suppose that the average interface temperature obtained from the Seebeck EMF is effectively a measure of the cell tip undercoolings. Theoretical treatments have related this undercooling to the experimental parameters for cellular or dendritic growth. For example Burden [4] gives the relationship: ____

LIT5=

GLDL

mVC’ 0 +A( D1 (k_1)T5F)

undercooling (l000ths°C) 2000

1500

1000

0.5 ,

(8)

where F = y/L, y = interfacial energy/unit area and L = latent heat of fusion/unit volume. A is a factor which depends on the details of the model used, e.g. A = 5.6 if the marginal stability critenon is used, A = 2.8 if the minimum undercooling condition is chosen. Eq. (8) consists of two terms, the first being dominant at high imposed temperature gradients and low velocities, the second under high velocity, high solute concentration conditions. Cellular growth is favoured by k differing greatly from unity, and for this reason silver was chosen as the solute, having a value of k (estimated from the phase diagram) of <0.1 (and m > 5°C/at%). Offsets were measured for interface velocities in the range 15 to 1500 jz rn/s. and the undercool-

500 ‘

. .

velocity (~sm/s)

sôo

iooo

isoo

Fig. 3. Comparison of experimental (dots) and analytical (curves) values for velocity/undercooling relationship during cellular growth of lead containing a small silver addition The analyses follow Burden [4] using the marginal stability (continuous line) and minimum undercooling (dotted line) condi tions.

mental results than does minimum undercooling, though further experiments would be required to be confident of this result.

Slw)

ing plotted against velocity. Using the method of section 2.1, values of k = 0.066 and m = 5.8°C/at% were obtained from the offset for the lowest velocity run (the interface being believed to be planar under these conditions, as will be cxplained later). The value of GL could be estimated from the interface shape model described in part I, values of a few hundred °C/cm being typical under the prevailing experimental conditions. For the present calculations, G L = 300°C/cm was assumed (as this value was in agreement both with the results of a numerical model of heat flow, and with an earlier calibration of the system using a conventional 2thermocouple), DL wasfor taken as si, the valueand measured silver 1.5 >< 10~ m in lead the of method of section 2.2. The by results the comparison are shown in fig. 3. As can be seen, the marginal stability condition appears to be in better agreement with the experi-

400 ôOism/s 300 200

100

_____

STABLE UNSTABLE ioo —20()

3

loglO(w) 5

Fig. 4. Stability of a planar interface in a dilute lead—silver alloy using the analysis of Mullins and Sekerka [5].

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Thermoelectric investigation of solidification of lead II

The value of k obtained for use in the above analysis depended on the lowest velocity run producing a planar, rather than cellular, interface, This assumption is compatible with the observation that the relevant experimental point on fig. 3 deviates significantly from its predicted value if the interface were cellular. In addition, using the above mentioned experimental parameters, the analysis of Mullins and Sekerka [5] predicts interface instability at between 15 and 30 ftm/s, as shown in fig. 4. By this prediction, all experimental measurements except the lowest velocity one should represent cellular growth. Although only a few cells would be expected to be present in the 200 j.tm specimen at the low velocity ranges, these should exhibit much the same undercooling/ velocity relationship as larger arrays.

the specimen tube. The heater was moved along the specimen using a leadscrew and stepper motor arrangement, allowing the repeated passage either of molten zones, or of heated but non-molten zones, along the specimen. The thermoelectric emf across the ends of the specimen was measured in the same way as for the thin wire (non-convecting) specimens. Interpretation of the thermoelectric data in the present experiments was, however, somewhat different from that of the non-convecting case, due to the fact that zone refining redistributes significant amounts of solute. The interaction of the resultant solute concentration profile with the temperature profile due to the heating element, generates an appreciable thermo-EMF. This property was used as an analytical technique, as described below.

3. Thermoelectric investigation of zone refining

3.2. Determination of solute redistribution and kejy via effect on absolute thermoelectric power (A TP)

3.1. Basis of the technique This experiment differed from those described above in that the specimen diameter was much larger (~ 10 mm) so that, if part of the specimen were melted, appreciable convection would occur in the molten zone. The apparatus used was as illustrated in fig. 5. The molten zone was produced by means of an annular heating element enclosing

The ATP (= dE/dT) of an alloy depends on its solute concentration [6,7]. This dependence is fairly small in dilute liquid alloys [8]. It was therefore assumed that it could be neglected in the molten zone of the current experiment, especially as convective mixing should ensure uniform cornposition except in the very thin diffusion layers at the solid/liquid interfaces. It could not, however, HEAT SINK +

CONNECTING WIRE (SAME ALLOY AS SPECIMEN)

IMPURE LEAD SPECIMEN

RECOR~~] I

H~ATER

COOLER

I

~,~QOLER

g~R~ ~///////////////////~///////////~

Fig. 5. Zone refiner designed to allow in-Situ thermoelectric EMF measurement.

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G. H. Rodway, J. D Hunt

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Thermoelectric investigation of solidification of lead. II

be neglected in the solid, in which significant changes in concentration, resulting from zone refining, interact with large changes in temperature due to the heating element, The concentration profile/temperature profile interaction gives rise to an EMF “signature”, characteristic of the solute distribution, when a heated zone is passed slowly at a uniform rate along the specimen. This property of the system can be used as a non-invasive technique to determine the solute concentration profile within a bar. It can be studied entirely independently of the Seebeck EMF at the solid—liquid interfaces by passing a heated but non-molten zone along the bar, The experimental procedure involved first redistributing impurities within a bar of impure lead, by the slow passage of a number of molten zones. A standard temperature hot, but nonmolten zone was then passed slowly along the bar, and the emf signature recorded. Interpretation of these traces required a matching of the experimental results with computed thermo-EMF values at successive points along the bar. As outlined above, the calculated value depends on the interaction of the concentration profile (and hence ATP profile) and the temperature profile: =

f

p+q

[S(x)

dT(x)/dx]

dx,

(9)

p—q

ermo COMPUTED EMF vs. PosmoN, emf nV)

-~-~

0

where is the EMF measured at point p along the bar, for a hot zone extending from p — q to p + q, S(x) is the ATP of the alloy as a function of position, x, along the bar. T(x) was known from the temperature profile measured by a conventional thermocouple embedded in the specimen. The form of S(x) was obtained from calculation of C(x) for the appropriate number of passes and keti value, and from an assumed form of S(C). In the interpretation, S was taken for simplicity as a linear function of C (its true form could be measured experimentally), and C(x) was obtained from a finite difference model of zone refining [9]. The calculation of the thermo-EMF was incorporated into the model via form of eq. (9) using the discretized C(x) and T(x) values, Before any molten zone passes, the EMF profile on passing a heated zone along the bar was almost flat, indicating that no significant concentration gradients initially existed within the bar. After 10 molten zone passes, however, a distinct profile had emerged, as shown in fig. 6. Comparison of this shape with numerical results gave a best fit for ~ = 0.75, as also shown in fig. 6 (the results for kCff = 0.7 and 0.8 being shown for comparison). Using the same parameters and scales, the equivalent EMF signature after 39 passes was computed. Comparison of this result with the corresponding experiment is given in fig. 7; once again a best fit was obtained for k~~=1 0.75. The agreement between the computed and cx-

EXPERIMENTAL EMF vs. POSITION

REPEAT EXPTL EMF vs. POSITION

% Distance along bar Fig. 6. The thermo EMF “signature” of the impurity distribution in a lead bar on passage of a heated zone after 10 molten zone passes. Numerical modelling of the solute redistribution/temperature profile interaction (left) gave a best fit with experiment for a dominant impurity distribution coefficient of k = 0.75. Plots for comparison are k 0.7 (circles) and k = 0.8 (squares).

/

G.H. Rodway. iD. Hunt

Thermoelectric investigation of solidification of lead II

perimental EMF signatures was surprisingly good, considering the assumptions made in the computation, and the fact that experimental conditions were not ideal: Runs had been carried out at various zone velocities and heater powers, with interruptions to some passes to enable other phenomena to be studied, and the material contained a number of different impurities. The results implied that one dominant impurity was present in the material, with k~ff= 0.75, expected to be somewhat higher than the true k value, due to the finite zone velocity and imperfect mixing. Microanalysis of the material indicated that there was

569

one dominant impurity, namely tin, with lesser quantities of others present. The value of k = 0.735, obtained from the phase diagram for tin (see section 2.1) is in good agreement with the value of kCff obtained above. In view of the assumptions and approximations described in the above method, it may be that this degree of agreement is fortuitous. Nevertheless, the result does demonstrate the capabilities of the technique, and the approximations and sources of error could be reduced by a further set of experiments using an alloy containing a single solute, and by a set of calibration experiments for S(C).

Thermo emf(nV)

0

5000

0

COMPUTED

0 0

EMF vs POS~ON 0 4500

0

4000

0

3500

0

3000

of

2500

EXPERIMENTAL EMF vs. POSITION

REPEAT EXP’T’L EMF vs. PO5~ON

0

0°

2000

/: 1500

1000 500 0

—50G.

o°~

0

~

~

~o

~o

ioo

% Distance along bar Fig. 7. Comparison of experimental and computed thermoelectric signatures as for fig. 6, but after 39 molten zone passes. In this case the best fit was also obtained for a value of k = 0.75. This is shown by comparison with the point plots for k = 0.7 (circles) and k 0.8 (squares).

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Thermoelectric investigation of solidification of lead II

3.3. Other experiments in the convecting system In addition to the technique described above, information can be obtained from Seebeck emf measurements during zone movement, The following can be studied: (i) The effect of zone velocity on refining efficiency. At high travel rates the solute undercooling at the freezing interface (if it remains planar) is larger, as the solute concentration gradient ahead of the interface is steeper, and convection in the bulk liquid is correspondingly less effective at redistributing the solute. Comparisons of the interfacial undercoolings at various zone velocities may be obtained by stopping the interface suddenly after uniform motion at each chosen velocity, and recording the change in emf across the specimen. (ii) The actual solute concentration at any given position along the bar. This is done by a method similar to (i), except that a standard zone velocity is used, for which kett is known. From the change in undercooling upon stopping the zone motion, the solute concentration can be calculated, by a method similar to that described in section 2.1. (iii) The effect of operating conditions, such as zone width and heater power, on convective temperature fluctuations within the molten zone, via the amplitude and frequency of fluctuations in the Seebeck EMF signal measured across the specimen. Preliminary experiments using each of the above techniques have been attempted, and early results appear consistent with previously established data on the specimen alloy and the zone refining process.

4. Conclusions

interface kinetics (see part I), since the undercoolings involved could be made much larger. This advantage was reflected, for instance, in the very close correspondence between computed and cxperimental decay curves in the diffusion coefficient measurements, and the good agreement between the experimentally measured distribution coefficient and that obtained from phase diagram information for the lead—tin system. Complicating factors nevertheless existed due to the presence of solute. These included the possibility of planar interface breakdown, and thermoelectric EMFs induced by composition gradients within the solid. However, these factors themselves provided an opportunity to investigate ccllular growth and solute redistribution, and expeniments which demonstrated these phenomena were also carried out. In summary, the thermoelectric techniques described above appear to be a promising analytical tool in the study of pure metal and alloy solidification. With suitable modifications, they should be applicable to the investigation of a wide range of metallic systems.

Acknowledgement The authors are grateful to Alcan International, Banbury, for sponsorship of this project.

References [1] J.J. Favier and D. Camel, J. Crystal Growth 79 (1986) 50. [2] W. Hume-Rothery, The Structures of Alloys of Iron (Pergamon, Oxford, 1966). Woodruff, The Solid—Liquid Interface (Cambridge University Press, Cambridge, 1973). [4] M.H. Burden, D. Phil. Thesis, Oxford University (1973). [5] W.W. Mullins and R.F. Sekerka, J. AppI. Phys. 35 (1964)

[31 D.P.

444.

The purpose of the work described in this paper has been to demonstrate the versatility of thermoelectric measurement techniques in the study of various aspects of alloy solidification. In some ways, solute effects were easier to measure than

[6] D.D. Pollock, Thermoelectncity Theory Thermometry Tool (ASTM, 1985). [7] W.B. Pearson, in: High Purity Metals (ASM, 1962) p. 202. [8] J.J. Favier, PhD Thesis, Grenoble (1977). [9] G.H. Rodway and J.D. Hunt, J. Crystal Growth 97 (1989) 680.