Sound velocity and compressibility in liquid metals

J. inorg, nucl. Chem. Vol. 42, pp. 1555-1558 Pergamon Press Ltd., 1980. Printed in Great Britain

SOUND VELOCITY AND COMPRESSIBILITY IN LIQUID METALS S. BLAIRS and U. JOASOO School of Metallurgy, The University of New South Wales, P.O. Box 1, Kensington, Australia, 2033

(First received31 July 1979; accepted[or publication 22 November 1979) Abstract--Sound velocities c(Tm)of liquid metals at the meltingtemperature Tmhave been correlated with equations of the form: c(T,.) = C,,.~(Tm/M)~2,where M is the atomic weight, for the FCC ~liquid (i = 1), BCC ~liquid (i = 2) and CPH--, liquid (i = 3)fusion transitions. C,.,ivaluesappropriateto these phase transitions are recommendedand the correlations find applicationin the calculation of unknown isothermal compressibilitiesfor metallic elements melting by these transitions whence the thermodynamiclimit of their static structure factors, St.(O), may be estimated.

INTRODUCTION Static liquid structure factors S(Q) which are related to measured intensities in X-ray and neutron diffraction experiments allow the pair distribution function g(r) to be obtained via Fourier transforms. In order to obtain a formal relation between S(Q) and the effective pair potential d,(r), the direct correlation function c(r) is utilised and ~k(r) related to c(r) through one of several cluster expansions[l]. Observed structure factors are moderately well reproduced by perturbed and unperturbed hard sphere models of the liquid state[2-3]. A common way of demonstrating this is to compare experimental structure factors with hard sphere forms chosen to match the heights of the principal peaks. It has long been established that a packing fraction y,. = 0.45 is appropriate at the fusion temperature T,~ [4]. Waseda et a1.[5-7] have reported structural information for many liquid metals and metalloids with y,. values ranging from 0.38 to 0.47. Dahlborg et al.[8] have shown that while considerations of geometrical packing are of primary importance in determining S(Q), hard sphere models in general do not yield the correct value for S(O), the long wavelength limit of the static structure factor. S(O) is obtained from the thermodynamic result[9],

S(O) = n kn TI3(T)

(I)

where n is the average particle density, kB the Boltzmann constant and//(T) the isothermal compressibility at TK. While the differences are small on an absolute scale, it implies that the actual direct correlation functions are very different from experiment. As ~b(r) is directly related to c(r) it emphasises the need for an accurate knowledge of S(Q) in the low Q region. Isothermal compressibilities are thus necessary to assist the extrapolation of measured S(Q) to the long wavelength limit whence realistic pair potentials may be obtained. Thus Waseda and Miller[10] smoothly extrapolate observed intensity data at Q less than 0.5 ,~- 1 to zero at Q = 0.0 ]k-,I because isothermal compressibility data for liquid rare earth metals are not known. Isothermal compressibilities are generally obtained indirectly from measurement of sound velocities c(T) in melts[Ill. The data base of measured c(T) is relatively scant and in the absence of experimental values various attempts have been made to develop correlations

whereby c(T) values can be reliably estimated from more accessible property data[ll-13]. In the present communication the inter-relation between c(T,,), T,, and atomic weight M is examined since the latter are readily available for the elements. RESULTS AND DISCUSSION Table 1 lists measured values of the velocity of sound c(T,.) as located in the literature for 32 liquid metals and metalloids at their normal fusion temperatures, c(T,.)* values presented in Table 1 represent recent literature values while other c(T,.) values were the subject of an earlier review[l l]. Isothermal compressibility values /3(T,,) were calculated [11] from c(T,.) using anciiliary liquid density d and isobaric molar heat capacity compilations[14, 15]. Included in Table 1 are the computed values for ST~(O) as calculated from eqn (1) and Yrm =/3(T,.) dc(T,,), the ratio of the isobaric and constant volume molar heat capacities. Srm(O) values in Table 1 are grouped according to their crystallographic structure at 7"= and designated by their Strukturbericht classifications [16], Structure factors ST,.(O)cls calculated from the Carnahan-Starling[17] hard sphere model using Yr, values listed by Waseda[5] and fusion temperatures Tm are also included in Table 1. c(T,~) values were linearly correlated with (T,.IM)~/2 using regression analysis constrained to pass through the origin[18] for the A1, A2 and A3 metallic structures, Fig. 1. Here AI, A2 and A3 represent the Strukturbericht designations for the FCC, BCC and CPH metallic struCtures. The proportionality constants C,,,~ with i = 1, 2 and 3 for the fusion transitions: AI-*L, A2~L, A 3 ~ L (L represents liquid), together with their standard deviations are summarised in Table 2. The looser packed metalloid structures were excluded from the regression analysis owing to the limited c(T,.) data available fbr these aliotropic modifications. Lindemann[19] using the concept of the vibrational instability of crystal lattices related the Debye temperature 00 to M and the molar volume V,. at the fusion temperature:

OD= K,.(T,.I V~ 3 M) '12

(2)

Gschneidner[20] utilised 0o for 64 solid elements obtained from low temperature specific heats to obtain

1555

1556

S. BLAIRS and U. JOASOO Table 1. Sound velocity values c(T,~) and static structure factors St.(O) of liquid metals and metalloids at the melting temperature T., ELEMENT

c(~)

ms

y

~

STm(O)

STRUCTURE

¥/STm(O)

STm(O) c/$

0.0254

-i K

Pb

1821

1,179

600.6~

0.0086

AI

137.1

Ag

2790

1.392

1235.0~

0.0200

AI

75.5

0.0278

AI

4561

1.451

933.5~

0.0170

A1

81.9

0.0278

Cu

3440

1.425

11357.6

0.0214

AI

66.6

0.0254

Au*

2567 ( 2 6 )

1.501

1337.581

0.0128

A1

117.3

0.0278

Co*

4084 ( 2 6 )

1.646

[768

0.0246

AI

66.9

0.0278

Zn

2850

1.246

692.73

0.0138

A3

90.3

0.0254

Cd

2256

1.266

594.26

0.0109

A3

116.1

0.0278

Mg

4065

1.317

922

0.0251

A3

52.4

0.0254

Li

4554

1.117

453.7

0.0292

A2

38.2

0.0254

Na

2526

1.095

371.0

" 0.0230

A2

47.6

0.0254

K

1876

1.122

336.35

0.0228

A2

49.2

0.0254

Rb

1251

1.121

312.64

0.0218

A2

51.4

0.0344

Cs

983

1.087

301.55

0.0212

A2

51.3

0.0344

Ca

2978

1.330

[112

0.0342

A2

38.9

0.0254

Sr

1902

1.192

[041

0.0325

A2

36.7

0.0254

Ba

1331

1.023

[002

0.0350

A2

29.2

0.0254

T1

1650

1.177

0.0101

A2

116.5

0.0278

Fe

4200

1.576

[809

0.0228

A2

69.1

0.0304

Mn*

3364 ( 2 7 )

1.659

[517

0.0338

A2

49.1

0.0278

Pu

1195

1.063

913

0.0133

A2

79.9

576.6

Si

3920

1.543

[685

0.0501

A4

30.8

0.O510

Ge

2693

1.180

[210.5

0.0255

A4

46.3

0.0510

Sn

2464

1.129

505.1

0.0066

A5

171.1

0.0344

In

2337

1.130

429.78

0.0064

A6

176.6

0.0278

Sb

1900

1.144

903.89

0.0196

A7

58.4

0.0431

Bi

1640

1.152

544.59

0.0093

A7

123.9

0.0431

Se*

ii00 ( 2 8 )

1.002

[210.4

0.0432

A8

23.2

Te*

889 ( 2 8 )

1.034

722.65

0.0616

A8

Hg

1511

1.114

234.29

0.0047

All

237.0

0.0278

Ga

2873

1.064

302.9

0.0046

All

231.3

0.0344

S*

1340 (28)

1.141

388.36

0.0625

A17

18.2

16.8

Table 2. Comparison of proportionalityconstants C...i CRYSTAL STRUCTURE AT Tm

Cm,i

m ~/2K-i/2s-1

This work

A2

AI

614.7 -+ 24.3

787.5 ± 33.7

[21]

615.3 -+ 7.2

n**~ 12

n*

734.1 -+ 83.1

6=3

n*- 12

n*=6 607.7 ,+ 43.3

A3

a

572.2 -+ 52.6 5 0 7 . 9 ± ii

n*~ 36

b

457.0 + 40.2

c

859.1 ,+ 16.1

n*~- 5

number of c(Tm) values.

n**

=

number of @D values.

a R

=

813.3 ± 182.9;

b B = 200.6 ± 219.9;

c B = 1272.5 -+ 170.4 m s -1

Sound velocity in liquid metals for the Lindemann melting constant K,. = 138.5-+25,6cmgU2K'12mol-tt3: The range of the standard deviation of K . includes other literature values cited for this quantity[20]. Cho and Puerta[21] have refined K,. and indicate that three different values hold for the fusion transitions A1 ~ L, A2-* L and A3-~ L, Km.~ = 154.7 -+7.2 Kra.2 = 127.7_+ l 1.0 and K,,.3 = 216.0-+ 16.1 cm g~/ZKl12 mo1-~/3, respectively. The assignment of separate K,,.~ for the fusion process of the common metallic structures is a direct consequence of modification of the well known Richard's Rule for the solid-liquid transition and correlation between the fusion entropy and allotropic modification[21-25]. 0o can also be obtained from elastic constant data via

i

I/

!

clTm) ms -1

f

/

1557

the average velocity of sound: c(T,,.) = (kBIh)(4zd3No)'13(V,.) '13 Oo

where h and No are the Planck and Avogadro constants respectively. From eqns (2) and (3) we obtain: c ( Tm ) = Cm.i( T,.I M ) "2

(4)

where C,.,i =K,,,,i(kBlh)(41d3No) '/3. Equation (4) indicates that a linear correlation should exist between c(T,.) and (T,,,/M) ~12. While the estimation of sound velocities in liquid metals from an equation developed from the Debye theory of solid specific heats may be !

,i

A1 FCC

I

A2

I

BCC

/ 4000

//

~/

-4000

. //

/ /

/ 3000

-3000

/

/

/

/

-200/

"2000

/

/

/

/

,~

TlO

/

/

Na•/

/ /

/ /

/ /

•tOO / / ( Tm/MJ1/2 K1/:

6

8

2

4

6

8

/ /

A3 CPH

c l ~ ) . s -1

Metalloids No regression analysis attempted

/ /

.4000

J

-4000 si0

.3OOO

/

3000

z.~

Cd&/ /

/

GP

GP

SnO

/

2000

In0 2000

s#

Bi 0

,P

11

so

.1000

SeO Te0

0

2

(Tm/M)I/2K 112-112 2 !

4

6

8

I

I

J

(3)

|

,

4

6

8

I

I

I

Fig. 1. Correlation of c(T.) and (T,dM) ~/2for liquid metals meltingby the A I ~ L, A2-*L, A3~ L transitions and for metalloids.

1558

S. BLAIRSand U. JOASOO

questioned it is noteworthy that the thermodynamic liquid state relation, eqn (1), is readily expressed in the alternate form: c(Tr.) = {ykHNolST,.(O)}'/2(T,.IM) '/2

(5)

with yISrm(O) constant, application of eqn (4) to the liquid state is appropriate. For comparison, proportionality constants Cm., calculated from Cm.~= K,~.~(kslh)(4cr/3No) ~j3 using the Km.~ obtained by Cho and Puerta[21] are included in Table 2, and represented in Fig. 1 as dotted lines. As is apparent from Table 2 and Fig. 1 agreement between the present correlations of c(Tm) and (Ts/M) m and those predicted using separate Km.~ values[21] characteristic of the A1, A2 and A3 structures is satisfactory, c(T,~) values calculated from the present correlations exceed those suggested by [21] for the A1--:,L and A2-* L melting processes and underestimate those for the A3--+L transition. While it is not possible to resolve the deviations quantitatively, they no doubt result in part from the paucity of ultrasound values for liquid metals as compared to the extensive range of 0o and fusion enthalpies, and in part to errors in c(T,~). Literature c(T,~) have errors typically in the range from 0.2 to 2%. Added support for the correlation of sonic velocity and T~ is suggested by Cantor[29]. However for predicted c(Tm) to agree with observed values a substantial reduction in the molar entropy is necessary. Extraction of the local "structural entropy" [21] associated with phase transformations from the molar entropy of the liquid achieves the necessary reduction. Enderby and March [30] have given a derivation of the Lindemann melting point equation for close packed metals by combining first-order electron theory with the Bragg-Williams theory of an order-disorder transition. They also predicted the approximate constancy of Srm(O) for close-packed metals and suggested the possibility of a systematic difference in the structure factors of metals which are close-packed in the solid state (AI and A3) and those which have the less close-packed A2 structure at Tin. In this connection work by Vosko et al.[31] suggests that the electronic screening is significantly different in the AI and A2 metals. Mean values of Srm(O) from Table 1 together with their standard errors are 0.0185-+0.032 (AI), 0.0166-+0.0032 (A3) and 0.0258_+0.0022 (A2) and supports this suggestion. While the metalloids were excluded from the regression analysis since these elements show excess electronic configurational entropy on fusion and c(Tm) for their various crystal structure are very limited, they have been included in Fig. I. c(T,~) for Hg, Sb, Bi, Sn, Ga and In locate in the proximity of the A1 -~ L and A3 ~ L lines while Si, Ge, Te, Se, and S fall closer to the A2 ~ L line. This behaviour is generally consistent with the liquid state coordination numbers at the nearest-neighbour separation [5]. The theoretical proportionality constants C~.~ = K,..~(kB/h)(41r/3No) u3 shown in Table 2, together with equation 4 are recommended for the calculation of unknown c(Tm) for metals of AI, A2 and A3 structure at the fusion point and for the estimation of their static structure factors Sty(O) in the long wavelength limit. With y/ST~(O) equal to a constant characteristic of the A1, A2 and A3 metallic structures eqn (5) supports the suggestion of a linear correlation between c(T,.) and (Tm/M) m. Mean values of y/ST.(O) from Table 1 together with their standard errors are 90.9-+ 12 (AlL

54.8-+ 7 (A2) and 86.3-+ 18 (A3). The limited number of c(Tr~) values for the A3 and to a lesser extent the A1 structures is reflected in the magnitude of the respective standard errors. By removal of the constraint from the linear regression analysis forcing the correlation of c(Tm) and (Tm/M) ~/2 through the origin, all three structure types lead to a non-zero intercept on the c(T~,) axis. This can significantly influence the value of C,~.~ and would have important consequences when estimating c(T,,) values for other A1, A2 and A3 structures. The resulting non-zero intercepts, B reduce in the order A3, A1 and A2. For completeness, the coefficients of the more general correlation: c( Tm) = Cm.~(Tm/M) "2 + B have been included in Table 2.

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