IC/98/6

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

INVESTIGATION OF THERMO DYNAMIC AND TRANSPORTPROPERTIES OF LIQUID TRANSITION METALS USING

WILLS-HARRISON POTENTIALS

M.A. Khaleque1, G.M. BhuiyanPhysics Department, University of Dhaka, Dhaka 1000, Bangladesh

and

R.LM.A. Rashid2

Physics Department, University of Dhaka, Dhaka 1000, Bangladeshand

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

Thermodynamic properties such as entropy, specific heat capacity at constant pressureand isothermal compressibility have been calculated for liquid 3d. 4d and 5d transitionmetals near melting temperatures. The hard sphere diameter for all such systems is es-timated from the potential profile generated from the Wills and Harrison's prescriptionusing linearized WCA theory of liquid. Evaluated values of entropy and specific heat ca-pacity are found to be in good agreement with the experimental data. Transport propertylike shear viscosity for these liquid metals is obtained using the same potential profile.Lack of experimental data at melting temperatures hampers detailed comparison for allsuch systems. However, for the case of transport property; the results obtained are foundto compare qualitatively well with the available experimental data.

MIRAMARE - TRIESTE

January 1998

1 Corresponding author: Fax: 880-2-865583.2Regular Associate of the ICTP.

I. Introduction:

It is well known that the presence of a strong d-character makes transition metals difficult to handletheoretically. Nevertheless during the last three decadesan impressive progress has been made towards theunderstanding of the interactions of transition metals andphysical properties related to interionic interactions [1-4]. In a novel approach Wills and Harrison (WH)[1] extendedthe pseudopotential theory for simple metals to include thed-electrons. In the theory, s-p and d-band contributions aretreated separately. The S-p band contribution is described bythe empty core model [5], in which the only free parameteris the so-called core radius, rc. The value of rc isgenerally determined as in simple metals by fitting physicalproperties like bulk modulus, equilibrium pressure orresistivity at constant volume. The d-band contributioncontains three main approximations: (i) the Friedelrectangular model for the d-band density of states; (ii) theatomic sphere approximation to evaluate d-d matrix elementsand overlap between the d-states of the neighbouring ions;(iii) the incorporation of the s-d hybridisation throughthe variation of relative occupancies of the s- and d-orbitals. However, the final expression for effectiveinterionic interactions appear to be very simple in form tohandle numerically. This form also allows one toinvestigate the s-p and d-band effects separately.

There are evidences [&,7] that potentials for solidsat lower temperatures if applied without any change fortheir liquids near melting temperature describe the latterreasonably well. This suggests that, temperature dependenceof potentials has very little effect when calculations areperformed near melting. In the same spirit we have appliedWH potentials originally developed for solid to the liquidstate calculations presented in this paper. Application ofthe WH model in the liquid state calculations has provedsuccessful [8-11] in the past.

Previously we have studied the static structure [8] andthermodynamic properties [12] of liquid 3d transitionmetals. There the WH~model and the linearised Weeks-Chandler- Anders on (LWCA) thermodynamic perturbation theory for liquid[13] have been used. Moreover, in those caseswe have suitably adjusted the values of rc keeping all otherparameters unchanged. A good agreement between thetheoretically evaluated values and experimental data werefound. In the present paper we have extended our previous

work to include the liquid 4d and 5d series- A good modelwill describe both the static and dynamic properties ofliquid with the same accuracy and as such parameters of WH'soriginal work are used in ou'r investigations without anychange- For the purpose of completeness we have included the3d series in this study.

Careful analysis of X-ray diffraction data for staticstructure factor of liquid 3d transition metals shows thatthese metals near melting largely resemble thenoninteracting hard : sphere liquid [14]. Moreover, simpleanalytic expressions for static and dynamic properties forhard sphere (HS) liquid are now available. Therefore, a hardsphere reference system may be a natural and convenientchoice as the starting point for the liquid metals. A recentcalculation [15] shows that hard sphere approximation fortransport properties of simple metals reproduces better thecorresponding experimental data obtained from spacelaboratory. In this work the LWCA with HS reference systemis used to determine the effective hard sphere diameter. Theaccuracy of the LWCA theory for liquid metal calculationshas been tested elsewhere [8,13].

A large number of calculations on static propertiesof liquid transition metals are available [9, 16-18] now inthe literature. Except Refs. [9, 16 and 17], all calculationsare confined to the 3d transition metal series only. Butthe calculations for transport properties such as shearviscosity are rare in the literature although somecalculations exist for simple liquid metals [15,19,20]- Theknowledge of viscosity describes not only the viscousproperty of the metal but also is needed in the study ofcritical cooling rates for glass formation [21].

The layout of this paper is as follows. Insection 2, WH theory for effective pair interaction and thelinearised version of the thermodynamic perturbation theoryfor liquid LWCA are briefly discussed. Formulas used in thecalculation of thermodynamic and transport properties havealso been illustrated in this section. The evaluated resultsare presented and discussed in section 3. This paper isconcluded in the same section.

2. Theory:2.1: Interionic Interaction:

The total potential energy per atom of a metallicsystem may be written as [2,3]:

Etat = E(p) + S *(rij) (1}

where E(p) denotes the structure independent part of thspotential and ij)(rij) is the effective interionic potential.Within the pseudopotential formalism the s- and d-bandcontributions to the effective interionic interaction may betreated separately [1]

cf)(r) = $B(r) + 4>d(r) (2)

In the above equation s-band contribution, 4>3, is describedby an empty core potential [5] . Since the Ichimaru-Utsumidielectric screening function [22] satisfies the selfconsistency conditions in the compressibility sum rule andthe short range correlation, it has been used to evaluate(j>3. The d-band contribution, <f>d, consists of two terms, anattractive and a repulsive term (for details see Ref. [8])

(Mr) = 4>«(r) + <j>c(r) (3)

The attractive term, (|>a(r) , arises from the broadening ofthe d-band,

77

and the repulsive term <j>r(r) arises from the shift of thecentre of gravity of the d-band,

(5)71

where Nc denotes the number of nearest neighbours. In orderto account for the effect of the hybridisation a nonintegralvalue of Zs, the s-electron occupancy number, is used in away such that the sum (Zs+Zd) gives the total number of sand d electrons in the two outermost shells per atom. Theinput values involved in our equations are the effectives-electron occupancy number Zs, empty core radius, rc andradius of the d-orbital, rd . From the point of consistencyconsideration one needs to take all these parameters fromthe same work.

Wills and Harrison, in their original work, have notderived the structure independent part of the potentialexplicitly. Recently, Jakse and Brstonnet [23] have proposed a simple

prescription to drive the structure independent part of the potentialenergy for transition metals. This method includes both s- and d-electroncontributions to to ta l E(p).,However, physical propertiesstudied in th is work do not require the structureindependent part of the potential explicitly.

2.2: LWCA Theory:

The Weeks-Chandler-Andersen [24] (WCA) theory describesthe static structure factor for a system of particlesinteracting through a repulsive potential in terms of a hardsphere diameter a. This diameter1 is determined from the so-called blip function:

B(r) = Y«(r) [exp(-pu(r) - exp(-puo(r)] (6)

using the condition j r2 B (r) dr = 0, where u (r) and uo(r)are the soft sphere and hard sphere potentials respectively.p=(kET)'

1, here kB denotes the Boltzmann constant and T isthe temperature. YCT(r) is the cavity function associatedwith the hard sphere distribution function and can beevaluated following Meyer et al [13] (see also [8]).

In the linearised version of the WCA (LWCA), the samecondition is used but the functional form of r2 B (r) isapproximated by triangular functions instead of teethlikefunctions (for details see Ref, [13] } . In this case theequation to determine the effective 0 is a transcendentalequation of the following form [8,13],

2.3: Static Properties:

If liquid transition metals are assumed to benoninteracting hard sphere fluid, thermodynamic quantitiessuch as isothermal compressibility, %T, entropy, S, specificheat capacity, Cp, may be expressed in terms of the packingfraction i] (= npcrVS) in the following way [25],

s_N

where p denotes the number density,

and

(11)

3 1(12)

f 4(1- T}) (2+(13)

C p - CF

PXT

Here av is the thermal expansion coefficient and is given byav = (1/V) (6V/5T) = %TYv. It may be noted that the temperaturedependence of XT is very small for alkali [3] metals. In acalculation Tamaki and Waseda [25] assumed that 91n%T/9T isnegligibly small for liquid 3d transition metals. In thesame spirit, we have also neglected the contribution of

in this work.

2.4: Dynamic Properties:

For a hard sphere liquid the theory for shearviscosity originally developed by Rice and Allnatt [26] maybe expressed in the following form [14]

(2) (3) (a) (15)

where the first and the second terms on the right are thecontribution due to the hard part, and the third term is due tothe soft part of the effective pair potential. The fourthterm is the kinetic energy contribution to the viscosity.Explicitly these terms read,

15

bnpcr

15B.(i) (16)

15(17)

Awnp1 }Jd2

30£(18)

and

(19)

where g(r ) , the pair distribution function, ra ionic mass andv atomic volume. The quantities Bo

a), Bai2) are:

(20)

B® = (21)

where

(22;

3- RESULTS AMD DISCUSSIONS:

The results of calculations for thermodynamic andtransport properties for liquid 3d, 4d and 5d transitionmetals near their melting temperatures are presented in thissection. The thermodynamic properties studied are specificheat, entropy, isothermal compressibility, and the transportproperty studied is the shear viscosity.

The effective pair potentials for liquid transitionmetals are obtained from the Wills-Harrison parameterisation[1] . Figure 1 illustrates potential .<J>(r) for liquid Fe, Ruand W which have been chosen as representative of the 3d, 4dand 5d series. From calculated a it has been observed thatthe effective hard sphere diameter of each system lies at aposition which is located at the left of the minimum of thepotential. For example, for Fe, Ru and W the positions ofthe effective hard sphere diameters are at 2.16 A, 2.36 Aand 2.56 A respectively, whereas the positions of theminimum of the corresponding potentials are at 2.4 A, 2.6 Aand 2.9 A respectively. Packing fractions T\ [=npo3/6) forthe 3d, 4d and 5d liquid metals (27 elements) are lyingbetween 0.38 to 0.50 for all systems except for Mo, Ag, Ta,W, Pt and Au. The values of r[ for these systems are found tobe 0.514, 0.583, 0.514, 0.510, 0.569 and 0.631 respectively.It is well known that the pair distribution function, g(r),is intimately related to the effective pair potential, q>(r),via statistical mechanics. Analysis of experimental results[14] show that for the liquid 3d transition metals, theratio of the positions of the first and second peaks of g(r)is about 0.53 and that for the second and third is about 0.66.The ratio of the positions of the first and second peaks ofthe calculated g(r) for 3d, 4d and 5d metal is about 0.5 andthat for the second and third peaks is 0.69. This suggests•chat our g(r) for transition metals represent the structuralproperty of the actual systems fairly well.

The evaluated results for specific heat at constantpressure, Cp, and entropy, S, for the three series arepresented in Table I. From' the Table it is seen that theagreement between the theoretical and correspondingexperimental values is reasonably good. The maximumdifferences between the calculated and experimental entropyare 2.1 cal/mol.K: for Sc (a 3d metal), 4.7 cal/mol.K for Zr(a 4d metal) and 7.1 cal/mol.K for Pt (a 5d metal). Theminimum difference between the calculated and experimentalentropy is 0.3 for Cr (a 3d metal), Q.I for Ru (a 4d metal)and 0.2 for Os (a 5d metal} . Similarly the maximum and

minimum difference between experimental and calculatedvalues of specific heat capacity for 3d series are 4.2ca/mol.K and 1.2 cal/mol.K, for 4d series 2.8 cal/mol.K and0.3 cal/mol.K and for 5d series 2.6 cal/mol.K and 0.6cal/mol.K. It may be noted that packing fractions for Cu, Zr,Nb, Mo, Ag, Hf, Ta, W, Pt and Au are found to be" high due tolarge hard sphere diameter estimated from the potentialprofile using LWCA method. For this reason it may beinferred that the parameters used to calculate thepotentials warrant further scrutiny and adjustment. Thecalculated values of entropy for Ag and Au are found to beout of acceptable limit. It is well known that for the d~series the electronic contribution to the entropy issignificantly large varying from 1.5-2.2 cal/mol.K for the3d series [27] . If the electronic contribution were added tothe calculated results the agreement would have been better.This part of calculation needs the knowledge of density ofstates which are not available.

From Table I, it is seen that the calculatedvalues of isothermal compressibility agree reasonably wellwith the available experimental data. However, thedifferences between the calculated and experimental valuesfor liquid Co, Ag and La are found to be significant.

Finally, let us turn to the results of shear viscosityfor the liquid 3d, 4d and 5d series. Results are presentedin Table II. It is seen that the contributions arising frompair potentials {\xm and u.(3J) to the viscosity of liquidsare predominant while the contribution due to the kinetic partfV^ is negligibly small. The viscosity measurements of mostof the liquid transition metals are yet to be done and lackof experimental data hampers us to compare our calculatedresults with experimental values. From the few availableexperimental values, it is observed that the theoreticalvalues (except for Ag and Au) are, in general, smaller inmagnitude. From the table, it appears that 5d liquids, •except La, have relatively larger values than the other twoseries. It is interesting to note that elements, exceptnoble metals, having the large values of shear viscosity liein the middle of all three series.

We would like to conclude our work by the following remarks:

(1) Some liquid transition metals in the middle of each ofthe three series are more viscous than the early and latesystems.

(2) Perturbation theory with HS reference system is a goodfirst approximation for the study of static and dynamicproperties of liquid transition metals.

(3) The WH model with present parameterization has not beensuccessful to predict the transport properties of transitionmetal liquids satisfactorily, although it describesthermodynamic properties reasonably well. But most of thecalculated shear viscosity values of each series are foundto be of almost equal magnitude. This reflects that thestrong d-band effect is not so pronounced in the case oftransport properties of liquid transition metals. Resultsalso show that values of viscosities for early systems ofeach series are relatively lower than others. That is, theearly transition metals of each series are less viscous thanothers.

(4) We believe that as far as transport properties of liquidtransition metals are concerned the WH model has to beparameterized in the liquid state for quantitative results.Work along this line is going on and will be reported in due course.

AcknowledgmentsThis work was done within the framework of the Associateship Scheme ofthe Abdus Salam International Centre for Theoretical Physics, Trieste,Italy. Financial support from the Swedish International DevelopmentCooperation Agency is acknowledged.

10

References:

1. J. M. Wills and W. A. Harrison, Phys. Rev. B28(1983) 4363.

2. M. W. Finnis and J. E. Sinclair, Phil. Mag. A50 (1984)45.

3. M. S. Daw and M. I. Baskes, Phys, Rev. B29 (1984) 6463.

4. J. A. Moriarty, Phys. Rev. B49 (1994) 12431.

5. N. W. Ashcroft, Phys. Lett. 23 (1966) 48.

6. S. M. Foiles, Phys. Rev. B32 (1985) 3409.

7. J. M. Holender, J. Phys. Condens. Matter 2 (1990) 1291;Phys. Rev. B41 (1990) 8054.

8. M. A. Khaleque, G.M.Bhuiyan, R.I.M.A. Rashid and S.M.M.Rahman, Phys. Chem. Liq. 30 {1995) 9.

9. J. L. Bretonnet and Decerouche, Phys. Rev. B43 (1991)8924-

10. C. Regnaut, Z. Phys. B76 (1989) 179.

11. C. Hausleitner, G. Khal and J. Hfner, J. Phys: Condens.Matter 3 (1991) 1589.

12. M. A. Rahman, G. M. Bhuiyan, M. A. Khaleque and R.I.M.A.Rashid, to be published.

13. A. Mayer, M. Silbert and W. H. Young, Chem. Phys. 49(1980) 147.

14. Y. Waseda. The Structure of Non-crystalline Materials(McGraw-Hill, New York, 1980).

15. T. Itami and K. Sugimura, Phys. Chem. Liq. 29 (1995)31.

16. F. Aryaseriawar, M. Silbert and M. J. Stott, J. Phys.F:Met. Phys. 16 (1986) 1419.

17. G. M. Bhuiyan, M. Silbert and M. J. Stott, Phy. Rev.B53 (1996)'636.

18. L. Do Phuong, A Pasturel and D. Nguyen Manh, J. Phys:Condens. Matter 5 (1993) 1901.

19. S. M. M. Rahman and L. B. Bhuiyan, Phys. Rev. B33(1986) 7243.

20. Y. Waseda and K. Suzuki, Phys. Stat. Sol. (b) 49(1972) 643; Phys.Stat. Sol.(b) 57 (1973) 351.

21. T.T. Goh, Y. Li and S.C. Ng, Singapore J. Phys. 12(1996) 35.

22. S. Ichimaru and K. Utsumi, Phys. Rev. B24 (1581) 7385.

23. N. Jakse and J.L.Bretonnet J. Phys: Condens. Matter 7(1995) 3803.

24. J. D. Weeks, D. Chandler and H. C. Anderson, J. Chem. Phys., 54(1971) 5237;H. C. Anderson, D. Chandler and J. D. Weeks, Adv. Chem. Phys. 34C1976) 105.

25. S. Tamaki and Y. Waseda, J .Phys . F: Metal Phys. 6(1976) L89.

26. S. A. Rice and A. R. A l l n a t t , J . Chem. Phys. , 34 (1961)2144.

27. A. Mayer, M, J . S t o t t and W.H. Young, P h i l . Mag. 33(1976") 381.

28. R. Hultgren, P.D. Desai, D.T. Hawkins, M. Gleiser, K.K.Kelley and D.D. Wagman, Selected Values of theThermodynamic Properties of" the Elements, (AmericanSociety for Metals, Ohio, 1973).

29. Y. Waseda and S. Ueno, Sci. Rep. Res. Inst. TohokuUniversity, 34A (1987) 1.

30. M. Shimoji, Liquid Metals (Academic Press, 1977) p26.

31. E.A. Brandes, Smithells Metals Reference Book, 6th Ed,(Butterworths, London, 1983),

12

Table I: Ionic number density , p, in AcmVdyne;

and

-3 taken from [17]; Isothermalcompressibility, j T ,in 10"12 cmVdyne; heat capacity at constantpressure, C» , in cal/mol.K and entropy, S, in cal/mol.K for thetransition metals. The eKperimental data for CP and S are taken from Ref.[28]. Experimental values for XT are from [29,30].

Syst

ScTiVCrMnFeCoNiCuYZrNbMoTcRuRhPdAgLaHfTaWReOsIrPtAu

T(K)

183319732173217315331833182317731423182521732741289024452583223818271235124325003269,368334533318268320531423

P

.0391

.0522

.0634

.0726

.0654

.0756

.0786

.0792

.0755

.0287

.0392

.0493

.0586

.0540

.0649

.0626

.0594

.0518

.0258

.0405

.0499

.0580

.0611

.0636

.0607

.0577

.0526

(Cal)

2.822.011.671.763.302.112.161.951.223.041.320.860.581.991.131.571.480.717.671.160.600.480.760.791.010.460.35

X*(Expfc]

3.641.401.311.101.731.050.961.031.49

——————1.322.114.29————————

(Cal)

7.647.607.417.106.847.026.947.138.708.108.868.779.217.197.767.498.0611.457.308.759.209.08S.098.018.03

10.9013.86

Cs(Expt)

10.57

10.09.4

11.011.09.638.657.58.378.08.0

10.010.010.010.08.38.08.28.0

10.08.5

10.08.6

10.08.37.4

S(Cal)

21.8421.7822.3222.8322.7922.6422.9122.3518.3923.4221.6622.1521.0725.4224.0124.3222.6514.3125.6824.2623.6824.0626.0426.1025.5618.4911.28

3(Expt)

23.93

24.0023.1023.8223.9623.8822.9420.2725.8426.4325.9225.6825.6523.9324.7823.9521.7325.4129.2528.9228.3129.0126.3126.7525.6523.37

Table II: Hard sphere diameter o, Shear viscosity-coefficients ]X for three transition metal series nearmelting temperatures. Experimental values are taken from Ref.[31] .

S cT iVCrMnFeCoNiCuYZrNbMoTcRuRhPdAgLaHfTaWReOsI rP tAu

0(A)

2.782.522.342.202.242.162.122.142.323.142.902.682.562.442.362.362.462.783.142.862.702.562.442.402 . 4 42.662.84

\x(cP)

.030

.040

.049

.049

.033

.043

.042

.04 6

.063

.035

.055

.073

.088

.049

.065

.056

.061

.083

.027

.083

.118

.130

.103

.104

.093

.137

.133

(cP)

0.7290.9311.0541.0080.7050.9460.9521.0371.7280.9951.7362.2332.9681.2501.9091.5661.7543.1050.6872.578"3.8814.4373.2663.2232.855.2106.583

M-(cP)

0.5130.8590.9510.97 60.8341.2231.0601.0270.8110.8.091.2221.3982.1431.6451.6371.6701.4050.6930.9522.2062.0403.1843.5303.329"2.5641.6871.850

(CP)

.043

.058

.072

.076

.055

.067 .

.068

.071

.080

.048

.069

.092

.106

.07 5

.091

.081

.082

.089

.040

.105

.144

.160

.139

.141

.126

.150

.132

Meal(cP)

1.3161.8882.12 62.1091.6282.2782.1182.1812.6831.8883.0833.7975.3053.0213.7023.3733.3013.9741.7064.9736.1847.9127.0396.7965.6357.1848.699

Mexp(CP)

_ _ _

5 . 54 . 25 . 03 . 9

3 . 7

4 . 3

14

1.5

1.0

Q>

0.5

0

-0.5

-1.0

\ /

Figure 1: The effective pair potential <j>(r),for Ru and -•-•-•- for W.

for Fe,

15