N A S A T E C H N I C A L

R E P O R T

A S A T R R-417

c

SOLUTE DIFFUSIONIN LIQUID METALS

by B. N. Bhat

George C. Marshall Space Flight Center

Marshall Space Flight Center, Ala. 35812

N A T I O N A L A E R O N A U T I C S AND S P A C E A D M I N I S T R A T I O N • W A S H I N G T O N , D. C. • O C T O B E R 1973

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Solute Diffusion in Liquid Metals

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George C Marshall Space Flight CenterMarshall Space Flight Center, Alabama 35812

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Dr B N Bhat received his Ph D from the University of Minnesota in Metallurgy and MaterialsScience. This work was authored while he held an NRC resident Research Associateship

16. ABSTRACT

A gas model of diffusion in liquid metals is presented. In this model, ions of liquid metals areassumed to behave like the molecules in a dense gas Diffusion coefficient of solute is discussed withreference to its mass, ionic size, and pair potential The model is applied to the case of solutediffusion in liquid silver. An attempt has been made to predict diffusion coefficients of solutes withreasonable accuracy

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ACKNOWLEDGMENTS

This work was accomplished while the author held an NRC Resident Research Associateship..Stimulating discussions with Mr. Hans F. Wuenscher of George C Marshall Space Flight Center areappreciated.

TABLE OF CONTENTS

Page

INTRODUCTION 1

THE GAS MODEL OF DIFFUSION 1

EXPERIMENTAL EVIDENCE 4Thermotransport 4Clusters m Liquid Metals ... 4Temperature Dependence of Diffusion Coefficients 4Solute Diffusion 4

SOLUTE DIFFUSION IN LIQUID SILVER 4

CONCLUSIONS 9

REFERENCES 11

111

SOLUTE DIFFUSION IN LIQUID METALS

INTRODUCTION

While the mechanism of diffusion in sohdmetals seems to have been well understood, theprocess of diffusion in liquid metals has beenpuzzling. There is no generally acceptedmechanism of diffusion in liquid metals.Several models of diffusion have beenproposed, namely, fluctuation model [1,2],free volume model [3,4], and hole theory [5].These models have been reviewed by Nachtneb[6,7]. All the models have some degree ofsuccess in predicting the diffusion coefficientsof liquid metals. Yet the disparity betweentheory and experiments is large inmany cases A similar situation exists in thecase of solute diffusion in liquid metals, whichhas been reviewed by Gupta [8]. A later workby Ejima, Ingaki, and Kameda [9] also showsthere is no satisfactory agreement between thetheories and experiments in the case ofdiffusion of sulfur, selenium and tellurium inliquid silver.

Inadequacy of the current models isfur ther exemplified by the results ofthermotransport experiment [10,11], wheremass transport occurs because of a temperaturegradient This is a second-order effect andhence provides a sensitive test of the validity ofdifferent models of diffusion. It turns out thatnone of the current models explains the resultsof thermotransport experiments. In fact, thehole theory predicts values for the heat oftransport which are in direct contradiction tothe experimental observations.

Hence, there is a lot to be desired aboutour understanding of diffusion in liquid metals.It does not seem to be a matter of refining the

current theories. But it calls for a basic changein the approach to diffusion in liquid metals.We need a theory which would explain theobserved experimental results. In view of thepartial success of the current theories, the newmodel would combine the features of theprevious models The gas model of diffusion,which is the subject of this paper, seems tosatisfy the above requirements

THE GAS MODEL OF DIFFUSION

The gas model assumes that the ions in aliquid metal act like the molecules in a densegas. Theory of diffusion in gases has been welldeveloped [ 12,13], and hence it can be used todescribe the process of diffusion in liquidmetals. There are some difficulties, however. Inthe kinetic theory of dilute gases it is assumedthat there are only two-body collisions andthat the molecular diameter is small ascompared to the average distance between themolecules. Both these assumptions are valid indilute gases, but in dense gases, theseassumptions must be reconsidered. Enskog wasthe first to make advance in this direction bydeveloping a kinetic theory of dense gasesmade up of rigid spherical molecules. For thisspecial model there are no three-body orhigher-order collisions. By considering onlytwo-body collisions and by taking into accountthe finite size of the molecules, he was able tograft a theory of dense gases into the dilute gastheory developed earlier. He showed that thefrequency of collisions in a gas made up ofrigid spheres differs by a factor x from that ina gas made up of point particles. The diffusioncoefficient (Dj 2) in a dense binary gas mixtureis given by [13].

(D12)12'dense(Dl2 Dilute

(1)

where

ni ,n2 = number density of components1 and 2 respectively,

and

CT i ,ff 2 = m o l e c u l a r diameter of

components 1 and 2 respectively

There is little variation in the value of xfor dilute binary systems, and hence theexpression for [D^l JJK +„ wiN be sufficientlyuiiuieaccurate for the present discussion.

The diffusion coefficient in a dilutebinary gas mixture is given by [13]

D12 =0.002628 VT3 [mt + m 2 ] / 2 m 1 m 2

(2)

where

Dj 2 = diffusion coefficient, in cm2 /sec,

p = pressures in atmospheres,

T = temperature in degrees K,

T,a = kT/e12

0 12 ,e 12 /k = molecular potential energyparameters of 1-2 interaction in angstromsand degrees Kelvin respectively,

mi ,m2 = molecular masses of components1 and 2 respectively,

^12 (Ti j ) = collision integral with thephysical significance that it indicates thedeviation of the model from the idealizedrigid sphere model,

and

a i +a2o12 = —-— = for a rigid sphere model

where ai ,a2 are the ionic diameters ofcomponents 1 and 2 respectively.

When equation (2) is written for a singlecomponent, the expression for self-diffusioncoefficient is obtained

D0.002628-

self p a2 n(1>1)*(T*)(2a)

where m and a refer to the mass and size ofthe diffusing species.

For a liquid metal the pressure p is notthe same as the pressure of a gas One mustconsider the internal pressure which isequivalent to the cohesive energy of the liquidmetal [14]. An accurate value of p cannot bedetermined at the present time. However, thevalue of p would be nearly the same when oneconsiders the diffusivity of different solutes ina single solvent. The value of the collision

(1,1)*integral £1 j 2 is unity for rigid spheres. For a

normal liquid metal, its value is greater thanunity when Tj*2 is unity or less. The value

of ft 12 (Ti*2) decreases with increasingtemperature, but, slowly. (Refer to the tablesin Reference 13).

From the expression (2) it is evident thatthe diffusion coefficient vanes at T3/2 . IfDI 2 is plotted against T3/2 , one would obtain

a straight line if n/j '1 were constant

S i n c e ft j[V1 * (Tj*2) decreases withtemperature, one would obtain a curved linewhich is concave upwards. The degree ofconcavity will depend on the values

of SI 12' , which in turn depends on the typeof pair potential. Conversely, one coulddetermine the pair potential between the atomsfrom a plot of Dj 2 versus T3'2 .

Expression (2) shows the variables insolute diffusion, namely, atomic mass and ionicsize. It predicts that the diffusion coefficientvaries inversely as the square root of thereduced mass, and square of the effectivediameter CTJ 2 . If the pair potentials are nearlythe same for a set of solute elements in a givensolvent, the relative diffusion coefficients ofthese solutes can be determined in a simplemanner by use of expression (2)

Dense gas approach to diffusion in liquidmetals is not totally new. Ascarelh and Paskin[15] have used a dense gas formulation tocalculate the self-diffusion coefficient ofmercury, indium and tin. Their theory employsvan der Waals' concept of a dense fluid inwhich the attractive potential energy is thoughtof as a uniform negative potential. This isjustified for liquid metals on the basis ofpseudopotential theory according to which theattractive part of the interaction energybetween electronically screened ions is

comparable to that found in rare gasliquids[16]. In their free volume model ofdiffusion Cohen and Turnbull do maintain thatthe diffusing atom does indeed move with gaskinetic velocity [3]. The fluctuation in densityas envisaged by Swalm [ 1 ] is quite conceivablein a dense gas. This view is also supported bycomputer simulation of dense gases byRahman[17] and Paskin and Rahman[18].

The above efforts have treated the liquidmetals as essentially homogeneous. There areother efforts that consider a liquid metal maybe heterogeneous. In his theory of significantliquid structures [19] Eynng assumes that aliquid metal consists of an intermixing ofcrystalline and gaseous fractions. More recentlyMcLachlan has presented some evidence aboutthe gaseous fraction in liquid metals [20]

The idea that a liquid may beheterogeneous comes naturally from the theoryof Larson [21,14]. According to his theory aliquid metal is an aggregate of solid-like,liquid-like and vapor-like atoms. Theirproportion is determined by the relativeprobability of occurrence of these atoms.Above the melting point the probability of aliquid-like atom is more than that of asolid-like atom. The probability of solid-likeatoms decreases with increasing temperature.Hence, there will be some solid (less than 50percent) in a molten metal and its percentagedecreases with increasing temperature. A liquidatom is the one which is gas-like in one (ortwo) dimension and solid-like in other two (orone) dimensions Since the diffusioncoefficient of solid-like atoms is much smallerthan that of gas-like atoms, the observeddiffusion coefficient will correspond to that ofgas-like atoms and its value will lie in betweenthe values of diffusion coefficients in solid andgas phase.

EXPERIMENTAL EVIDENCE Temperature Dependence ofDiffusion Coefficients

There are many experimental results thatseem to support the gas model They aredescribed in this section

Thermotransport [10,11]

In a thermotransport experiment where abinary alloy is held in a temperature gradientfor a sufficiently long time, segregation isgenerally observed This segregation can bedescribed by a quantity called the heat oftransport, which is sensitive to the mechanismof diffusion The previous models have failedto explain the heat of transport either inmagnitude or in sign, whereas the gas modelexplains it satisfactorily. Hence it may bedeemed that the gas model is the mostsatisfactory model of diffusion in liquid metals.

Clusters in Liquid Metals [22]

Clusters have been observed in liquidmetals by Egelstaff in neutron diffractionstudies It has also been suggested thatdiffusion could occur through rotation andtranslation of clusters. According to Larson'stheory (which supports the gas model) clustersof solid can exist in liquid metals near themelting temperature The theory furthersuggests that the number and/or size of clusterswill decrease with increasing temperature. Inthe opinion of this writer, the contribution ofthe clusters to the overall diffusion coefficientwill be small, since the cooperative movementof several atoms is a much less likely event ascompared with a binary collision.

Diffusion coefficients of solutes aregenerally reported in the form

D - D0exp(-Q/RT) (3)

where Q is the apparent activation energy. If

12 (T^) with thethe variation of p andtemperature are neglected, one obtains a valuefor the apparent activation energy fromequation (2),

= 3RT (4)

which is a function of temperature Hence aplot of InD versus 1/T will yield a curved linethat is concave upwards. This has beenobserved in several metals[23]

Solute Diffusion

Experimental results of solute diffusionstudies in liquid metals appear to be in goodagreement with the gas model of diffusion.This is discussed in the following section, forthe case of solute diffusion in liquid silver

SOLUTE DIFFUSION IN LIQUID SILVER

Diffusion of solute in liquid silver hasbeen studied in detail. The current models of

diffusion explain the diffusion behavior ofindium, tin, antimony, and gold in silver withsome success[8] by taking into account thevalence of the solute. This approach fails whenit comes to transition element solutes (iron,nickel, cobalt and ruthenium) or solutes ingroup VIA (sulfur, selenium and tellurium).The gas model, however, works satisfactorilyfor all the solutes Table 1 lists the pertinentdata on the solutes along with the references.

Figure 1 shows the diffusion coefficientsof solutes in liquid silver plotted against T3/2 .Many lines appear to be linear and some arecurved and are concave upwards. With theexception of sulfur (which is a nonmetal) theyare all grouped together. There is no particulartrend noticeable (with respect to either valenceor atomic mass or ionic size). The same curvesare replotted m Figure 2 after dividing thevalue of the diffusion coefficient D, 2 by the

factor v(mi + m2)/mi m2 /a?2 , which takesinto account the mass and the ionic size of thesolute element. The modified diffusioncoefficient is denoted by Di 2

D/2 = 0.002628/pn1(2'1)*(Ti*) (5)

In Figure 2, the curves are observed to separateinto three distinct groups

1 Those with the lowest DJ 2 values arethe transition elements; Fe, Ni, Co and Ru.

2. Those with intermediate Di 2 valuesare the elements indium, tin, antimony, whichare to the right of silver in the periodic table;gold, which is immediately below silver; andzinc, which is adjacent to copper, lyingimmediately above silver in the periodic tableAll these solutes have similar inner ionic core,

and their diffusion coefficients (Di 2 ) arecomparable to Di j , the self-diffusioncoefficient of liquid silver

3 Those with the highest valuesfor D'12 are the elements sulfur, selenium, andtellurium, which belong to the group VIA.

The value of Dj 2 depends on the values of

pressure p and collision integral fliV ^ (Ti*2 ),which depend upon the 1-2 pair potential.Figure 2 is thus largely a manifestation of thedifference in the 'pair potential for soluteelements in liquid silver The diffusioncoefficients of sulfur, selenium, and telluriumare widely separated in Figure 1 , but they aremuch closer together in Figure 2 This will bethe case if sulphur, selenium, and telluriumhave similar pair potentials in silver. This is tobe expected since all these elements belong tothe group VIA of the periodic table, have avalence of two, and accept electrons from theconduction band There are small vanations inthe values of Di 2 within each group, but thesedifferences are much smaller than thedifferences in the values of D i 2 between thegroups. The variation in the slope ofthe Di2 versus T3/2 plot is a measure of the

temperature dependence of Jwhich is different for each solute.

(Ti*2),

Similar considerations apply to the solutesin the second group of Figure 2, namely, gold,indium, tin, antimony, and zinc. These soluteshave similar ionic cores. Gold and zinc are theextremes within this group. This may be due tothe fact that they belong to different periods inthe periodic table of elements Indium, tin, andantimony are in the same row as silver andhence their DJ 2 values may be expected to beeven closer. Figure 2 shows that this is indeedthe case. Interestingly, the slope of the

TABLE 1. PERTINENT DATA ON THE SOLUTES

Group

1

2

3

Element

S

Se

Te

Au

Ag

In

Sn

Sb

Zn

Fe

Co

Ni

Ru

AtomicMass

32.1

79.0

1276

197.0

107.9

114.8

118.7

121 8

65.4

55.8

58.9

58.7

101.7

IonicRadius

(A)

1 84

1.91

2.11

1.37

1 26

0.81

0.71

0.62

0.74

074

0.72

069

0.67

Diffusion Coefficientin Liquid Ag at 961° C

(1234°K)(D12)cm2/sec

5.50x 10'5

2.84

2.32

2.27

2.55

3.65

3.75

4.30

3.90

2.35

238

2.69

2.27

Reference

9

9

9

24

25

26

26

26

27

27

27

27

27

curve D'i 2 versus T3 /2 falls fromAg-»In-»Sn->Sb, in that order, suggesting a

valence dependence of fi J2 '

The transition elements, which form thethird group, have the lowest values of D{ 2 . It isa common experience for transition metals tobehave differently from other metals and herethey have the lowest diffusion coefficients(DJ2). This suggests that the values of p

0and £transition metalsother solutes.

(T^Hor either one of them) forare higher than those for

It is clear that one needs to have accurateknowledge of the pair potentials to evaluate

(i i)*the collision integral £1 \f 2 . At the presenttime pair potentials are being evaluated forpure metals, and it may be several years beforethe pair potentials are available for alloys. Inthe meantime, one can obtain a reasonableestimate of diffusion coefficient of a soluteelement which belongs to a particular group asenvisaged in the previous paragraphs Since theaverage of DJ 2 is known for that group, thevalue of DI 2 can be calculated from theexpression

1<T5 x 8.6 -

7.8-

7.0-

CM

o

ZUJ

OLLU.UJOO

6.2-

5.4-

4.6-

3.8-

3.0-

2.2-

6.2 6.6

TEMPERATURE, T3/2 x 10"4

Figure 1 Diffusion coefficients of various solutes in liquid silver.

Se Te

M

UJ

O

o55

QQUJ

QO

8-

7-

6 -

5 -

4 -

3-

2 -

1 -

44 48 52 56

TEMPERATURE T3/2 x KT4

I

60 64 68

Figure 2 Plot of modified diffusion coefficients of various solutes in liquid silver.

D12 a^group

a,2(6)

Table 2 lists the values of Di 2 for the threegroups at the melting point From this tablethe value of Di 2 for the second group is 2 84The value of diffusion coefficient of cadmiumin liquid silver, for example, may now beobtained from equation (6) and is found to be

^. .Cd in Ag= 3.07cm2/secat961°C(1234°K)

'

(7)

Thus, the gas model provides a simple way ofestimating the solute diffusion coefficients

CONCLUSIONS

The diffusion of solutes in liquid metalscan be satisfactorily described by use of a gasmodel in which ions of metals are assumed tobehave like the molecules in a dense gas Thediffusion coefficient of a solute is a function ofits mass, ionic size and pair potential. Themodel has been applied to solute diffusion inliquid silver It is found that the solutes formgroups in their diffusion behavior, dependingon their position in the periodic table. It ispossible to predict the diffusion coefficient ofa solute with reasonable accuracy.

George C. Marshall Space Flight CenterNational Aeronautics and Space Administration

Marshall Space Flight Center, Alabama 35812, May 1973948-95-54-0000

TABLE 2. VALUES OF D', 2 FOR THE THREE GROUPS AT MELTING POINT

Group

1

2

3

Element

S

Se

Te

Au

Ag

In

Sn

Sb

Zn

Fe

Co

Ni

Ru

Diffusion Coefficient at 961°C (1 234°K)D 1 2 a J 2

2D' -

12V(m1 + m2) /m1m2

6.57

4.82

5.05

328

2.97

2.91

274

2.65

2.48

1.42

1.44

1 58

1.53

DJ2 Average

5.48

2.84

1.49

10

REFERENCES

1 Swalin, R.A.: On the Theory of Self-Diffusion in Liquid Metals. Acta Metallurgica, vol. 7,no 11, 1959, pp 736-740

2 Swalin, R.A.. On the Fluctuation Model of Diffusion in Liquid Metals. Zeitschrift furNaturforschung, vol 23a, June 1968, pp 805-813.

3. Cohen, M.H. and Turnbull, D Molecular Transport in Liquids and Glasses, The Journal ofChemical Physics, vol. 31, no 5, Nov. 1959, pp. 1164-1169.

4 Turnbull, D. and Cohen, M.H. On the Free-Volume Model of the Liquid-Glass Transition.The Journal of Chemical Physics, vol. 52, no. 6, Mar. 1970, pp. 3038-3041.

5. Glasstone, S , Laidler, K.J., and Eyrmg, H. The Theory of Rate Processes McGraw-Hill,1941,p 516.

6. Nachtneb, N.H.: Self-Diffusion in Liquid Metals Advances in Physics, vol. 16, no. 62,Apr. 1967, pp. 309-323.

7. Nachtrieb, N.H.: Diffusion in Liquid Metals. Liquid Metals: Chemistry and Physics, Ch. 12,S.Z. Beer, ed., Marcel-Dekkar, Inc., New York, 1972, pp. 507-525.

8. Gupta, Y.P. Solute Diffusion in Liquid Metals. Advances in Physics, vol. 16, no. 62, Apr.1967, pp 333-350

9. Ejima, T.; Ingaki, N., and Kameda, M. Diffusion of Divalent Solutes in Liquid Silver,Transactions Japan Institute of Metals, vol. 9, 1968, pp. 172-180.

10. Bhat, B.N. and Swalin, R.A.: Thermo transport of Solutes in Liquid Silver. Zeitschrift furNaturforschung, vol. 26a, no. 1, 1968, pp. 45-47

11. Bhat, B.N. and Swalin, R.A • Thermotransport of Silver in Liquid Gold. Acta Metallurgica,vol. 20, Dec 1972, pp. 1387-1396

12. Chapman, S and Cowling, TG.. The Mathematical Theory of Non Uniform GasesCambridge, 1952

13. Hirschfelder, J.O.; Curtiss, C.F.; and Bird, R.B.: Molecular Theory of Gases and Liquids.John Wiley, 1964.

14. Larson, D B : Papers on the Theory of Liquid State. North Pacific Publishers, Portland,Oregon, 1970.

11

REFERENCES (Concluded)

15. Ascarelli, P. and Paskin, A.. Dense-Gas Formulation of Self-Diffusion of Liquid Metals.Physical Review, vol. 165, no 1, Jan. 1968, pp. 222-225

16. Harrison, W A Pseudopotentials in the Theory of Metals, Benjamin, 1966.

17. Rahman, A. Liquid Structure and Self-Diffusion. The Journal of Chemical Physics, vol45, no. 7, Oct. 1966, pp. 2585-2592.

18. Paskin, A. and Rahman, A Effects of a Long-range Oscillatory Potential on the RadialDistribution Function and the Constant of Self-Diffusion in Liquid Sodium. PhysicalReview Letters, vol 16, no. 8, Feb. 1966, pp 300-303

19. Breithng, S.M and Eynng, H Significant Structure Theory Applied to Liquid Metals.Liquid Metals: Chemistry and Physics, S.Z. Beer, ed., Marcel-Dekkar, Inc., New York,1972, pp 213-255.

20. McLachlan, D., Jr.: The Gaseous Fraction in Liquid Metals. Advances in Chemical Physics,Vol. 21—Chemical Dynamics. J.O. Hirschfelder and D. Henderson, ed., Wiley-Interscience,1971, pp. 499-516.

21 Larson, D.B.. The Structure of the Physical Universe. Published by the author, PortlandOregon, 1959.

22. Egelstaff, P.A. Neutron Scattering Studies of Liquid Diffusion Advances in Physics, vol.11, July 1962, pp. 203-232

23. Foster, J.P. and Reynik, R J Self Diffusion in Liquid Tin and Indium Over ExtensiveTemperature Ranges Metallurgical Transactions, vol. 4, no. 1, Jan 1973, pp 207-216

24. Gupta, Y.P. Diffusion of Gold in Liquid Silver Acta Metallurgica, vol. 14, March 1966,pp. 297-304

25. Leak, V.G and Swahn, R.A Diffusion of Silver and Tin in Liquid Silver Transactions ofthe Metallurgical Society of the American Institute of Mining, Metallurgical and PetroleumEngineers, vol. 230, April 1964, pp 426-429.

26. Leak, V.G. and Swahn, R A. Diffusion of Heterovalent Solutes in Liquid Silver ActaMetallurgica, vol. 13, May 1965. pp. 471-478

27. Wang, H.S. Solute Diffusion in Liquid Silver. M.S. Thesis, University of Minnesota, 1967.

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