Some thermoelastic properties of liquid boron at melting point

541

Journal of the Less-Common Metals, 67 (1979) 541 - 550 @ Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands

SOME THERMOELASTIC PROPERTIES OF LIQUID BORON AT MELTING POINT*

D. Sh. TSAGAREISHVILI

and G. V. TSAGAREISHVILI

Institute of Metallurgy, Academy

of Sciences of the Georgian S.S.R.,

Tbilisi (U.S.S.R.)

Summary Approximate values of the atomic volume, the heat capacities at constant pressure and at constant volume, the thermal coefficient of volume expansion, the adiabatic and isothermal bulk modulus, the Griineisen and tne Anderson-Griineisen parameters, the pressure and the temperature derivatives of the adiabatic bulk moduli, the velocities of the pressure and temperature derivatives of the adiabatic bulk modulus and the Debye characteristic temperature of liquid boron at melting point were estimated by means of quasi-thermodynamic correlations. The initial parameters that were used were the experimental values of the heat of fusion of boron and the approximate values of some thermoelastic properties of solid boron at melting point. Graphs showing the correlation between the thermoelastic properties of group IIIa liquid metals were plotted. The good agreement between the values shows the reliability of the estimated values of the thermoelastic properties of liquid boron at melting point.

1. Introduction Interest in determining the thermoelastic properties of liquid metals has increased recently because of their application to modem technology. An experimental study of these properties always encounters some difficulties and the data on the thermoelastic properties of liquid metals in the literature are very few and do not agree. There do not seem to be any data at all on the thermoelastic properties of liquid boron. The quantum-mechanical theories of the liquid state have not yet reached such a level that it would be possible to draw qualitative conclusions and to predict reliably the values of the thermoelastic properties of liquid metals. Therefore, in the present work, we attempted to evaluate the thermoelastic properties with an accuracy which is acceptable for engineering practice. We used the quasi-thermodynamic method developed by *Paper presented at the 6th International Varna, Bulgaria, October 9 - 12,1978.

Symposium on Boron and Borides,

542

Tsagareishvili [l] which enables us to integrate approximately the exact thermodynamic equations. The initial parameters were the values of the thermoelastic properties of solid boron at melting point and the heat of fusion determined experimentally [ 21. The aim of the present paper is to estimate the thermoelastic properties of liquid boron at melting point Z’, .* the molar volume V, the heat capacity C, at constant pressure, the heat capacity Cv at constant volume, the thermal coefficient Q of volume expansion, the adiabatic bulk modulus Bs, the isothermal bulk modulus Br, the Griineisen parameter y, the AndersonGriineisen parameter Ss, the pressure derivative (a&Ja~)~ of the adiabatic bulk modulus, the Debye characteristic temperature 0, the velocity of sound c, the pressure derivative @c/ap)r of the velocity of sound and the thermal coefficient (aBs/aT), of the adiabatic bulk modulus. The indices m and L in the following equations stand for solid and liquid boron respectively.

2. Results The following estimations of the thermoelastic properties of liquid boron at melting point T, are based on the data of the analogous properties of solid boron [ 3 3 . Additional data on the thermoelastic properties of solid boron at T, are given below. The heat capacity Cvm of solid boron at constant volume at T, was estimated by means of the thermodynamic equation [4]

cvrn= cpm-a,

2VmByrn T,

(1)

For solid boron at melting point we have the following values: Cpm = 8.11 cal K-l (g atom)-’ [ 51; V,,., = 4.98 cm3 (g atom)-’ (31; a, = 51.3 X 1O-6 K-l [ 31; BTm = 1.68 mbar [3] (1 mbar = 1012 dyn cm-2); Tm = 2470 K [6] ; from eqn. (1) we obtain Cvm = 6.81 cal K-l (g atom)-1. Then C,“/Cv” = 1.19 and the adiabatic bulk modulus of solid boron at T, will be

141 Bsm = BTm(Cpm/Cvm) = 2.00 mbar

Using the above data it is possible to estimate the Griineisen parameter -ym of solid boron at T, by means of the well-known thermodynamic relation [ 1 J amVmBSm

Trn =

cpm

= 1.51

(2)

To evaluate the volume change A V,/V, of boron on melting an equation which relates A Vf/Vm to the heat AH, of fusion and its sublimation energy E, was used: AV,

-=Vm

AH, Es

(3)

543

(We obtained eqns. (3), (4), (6), (7), (8), (9) and (11) on the basis of the quasi-thermodynamic method developed previously [l] . The deduction of these equations will be published elsewhere [ 71.) The heat of fusion of boron is given by AH, = 12.0 kcal (g atom)-’ [ 21 and its sublimation energy E, = 136.8 kcal (g atom)-’ [8] ; thus from eqn. (3) we find that A V,/V, = 8.8% for boron. The volume change of metal on melting may be evaluated from AV, -= V,

X -47,

(4)

where X is a dimensionless parameter which we shall call the thermochemical parameter of the fusion process which can be determined from

where ASi is the entropy of fusion of metals. As A& = AHfIT, for boron we find that A& = 4.86 cal K-l (g atom)-‘. Hence from eqn. (5) h = 0.60 for boron. After substituting in eqn. (4) the values of the parameters yrn and X, we have AV,/V,,, = 9.9%. Thus the values of A V,/V, obtained from eqns. (3) and (4) do not differ greatly, although they differ significantly from the experimental value of A I’,/ V, = 4.5 f 1% determined by Tawadze et al. [9]. We shah not analyse the discrepancies between the theoretical and experimental values of A VJV, here. An average value of A V,/V, = 0.5 (8.8 + 9.9) = 9.3% is used to calculate the atomic volume V, and the density pL of liquid boron at the melting point: VL = 1.093~V, = 5.44 cm3 (g atom)-l andpL =M/VL = 1.99gcmd3 where M is the mass per gram atom of boron (10.811 g). The adiabatic bulk modulus hL and the isothermal bulk modulus BrL of liquid boron at T, can be calculated using the above values of the moduli Bs m and BTm and the parameter X as follows:

Bs BTL =

BTm exp(--h)

= 0.92 mbar

(7)

Therefore the adiabatic and the isothermal compressibilities of liquid boron at melting point are psL = ( BsL)-’ = 0.68 mbar-’ and pTL = ( BTL)-’ = 1.09 mbar-l respectively. Using the values of BsL and BTL, the ratio of the heat capacity C, L at constant pressure to the heat capacity Cv” at constant volume for liquid boron can be found by means of the well-known formula [4] : CpL/CvL = BsL/BTL = 1.24 The heat capacity CpL at constant pressure of liquid metals can be evaluated from

544

From eqn. (8) we obtain CDL = 9.33 cal K-l (g atom)-’ for liquid boron. Therefore the heat capacity at constant volume will be equal to CvL = CPL/ 1.24 = 7.50 eal K-l fg atom))‘. The relationship between the thermal coefficient (Ye of volume expansion of liquid metals and cr, and X is expressed by h

aL

= LY,

exp i 2

)

from which we obtain (Ye= 62.2 X 10-l K-l. As we now know the values of crL, VL, BsL and CPL we can estimate the Griineisen parameter yL of liquid boron: YL =

(yLVL&

L

CP”

= 1.43

(10)

The Debye temperature OL for liquid metal and the Debye temperature 0, for solid metal are related by QL = 0,

exp -P ( )

(11)

The data on 0, of solid boron are extremely contradictory [ 10 - 121. We considered that the best value for our estimations was the value 0, = 1580 K obtained by Tsagareishvili et al. [ 31 from the standard entropy of boron because it adheres to the rule that 63 increases along the row Li + Be + B -+ C. Then from eqn. (11) we obtain the value OL = 1360 K for liquid boron. As no data indicate to the contrary, we can assume that liquid metals have a qu~i-c~s~line st~c~re near the melting point 1131. In the limits of this approximation it is possible to use the universal quasi-thermodynamic equations not only for solid bodies, as suggested in ref. 1, but also for the estimation of some thermoelastic properties of liquid metals at melting point. Using the above values, the Anderson-G~neisen parameter 6 sL and the pressure derivative ( aBSL/ap), of adiabatic bulk modulus for liquid boron can be estimated from [l] 6s L = 2‘yL

(F),

=

2.86

= (6sL + ,,,iBgL)=

02) 5.34

(13)

The velocity of sound in liquid boron at !I’, can be estimated from the well-known correlation [ 141 L 112 BS = 8484 ms-l CL= (14) ( PL j On the basis of eqn. (14) relationships can be found which allow the estimation of the thermal coefficient (ac,/aT), of the acoustic velocity and its pressure derivative (&@p), for liquid metals at melting point. By differentiation with respect to ~mpera~re at constant pressure eqn. (14) becomes

545

(15) Substituting in eqn. (15) the value 6 sL = 2~~ from eqn. (12), we obtain the following expressions [l]

a lnpL ~

( aT 1

=-a

(17)

L

P

thus we find that

acL

( 1 aTp

1

( 1

= _oLLCL .yL --

2

(18)

In an analogous manner, differentiating eqn. (15) with respect to pressure at constant temperature (near the melting point), we can write

(19) Taking eqns. (12) and (13) and the expression (a ln pL/ap)T account and substituting them in eqn. (19), we obtain

= BTM’ into

(20) Using the values of eL, BTL, cL and yL from eqns. (18) and (20) we find that for liquid boron (ac,/aT), = -0.55 m s-l K-l and (acL/ap)r = 1.52 cm s-l bar-’ . The thermal coefficient of the adiabatic bulk modulus can be determined from eqn. (16); for liquid boron it is equal to (a BsL/iil T)p = -0.29 kbar K-l. The approximate values of some of the thermoelastic properties of liquid boron at melting point that we found are given in Table 1. Analogous data for solid boron are also given in Table 1 because they are very important from the scientific and technical point of view. Because of this, additional calculations on some of the thermoelastic properties of solid boron at T, were performed. The values of 6sm, (aBsm/ap)T and (aBsm/aT), for solid boron were found from eqns. (12), (13) and (16) respectively and the average velocity c, of sound was calculated from the well-known equation [I51 = 10 739 m s-l

(21)

where k and h are Boltzmann’s and Planck’s constants respectively and N is Avogadro’s number. The coefficients (ac,/a !!Jp and (ac,/ap), are easily calculated from the differentiation of eqn. (21) with respect to temperature

546 TABLE 1 Approximate values of some thermoelastic properties of boron at melting point Thermoelastic properties

Units

Solid phase

Liquid phase

V

cm3 (g atom)-l g cmm3 mbar mbar mbar-f mbar-l cal K-l (g atom)-l cal K-l (g atom)-l

4.98 2.17 2.00 1.68 0.50 0.59 8.11 6.81 1.19 1.51 3.02 5.39 1580 51.3 10739 -0.648 -0.604 0.75 -0.31 -1.55

5.44 1.99 1.48 0.92 0.68 1.09 9.33 7.50 1.24 1.43 2.86 5.34 1360 69.2 8484 -0.546 -0.644 1.52 -0.29 -1.98

P

42 BT

Ps PT

CR CV CRJCV

&J (awaph

K K-l ms -1 m s-l K-1 K-l cm s-l bar-’ kbar K-l K-l

0 a x 106 C

$2;)sT,,

10”

x

@CtaP)T @BslaT) (a

In B&TI,

x lo4

at constant pressure and with respect to pressure at constant temperature (near the melting point). As in the quasi-harmonic approximation [l]

rno 1 *m

aT

=

(22)

-Ymam

P

=-

rrn

(23)

BTm the

following relations are obtained:

acrn = -a,c,

t-1

(

-ym --

1

1

(ig-g(~__fi3 From eqns. (24) and (25) the coefficients for solid boron can be determined: (&,/a !Q, = 0.648 m s-l K-l and (k&p)r = 0.75 cm s-l bar-‘. On the basis of these results graphs were plotted to correlate the values A Vt/ for the group IIIa liquid metals (boron, aluVm 9PSLv cL and (a In c&T), minium, indium and thallium) against vohrme VL (Figs. 1 - 4) and of CQ, against psL (Fig. 5). Gallium was omitted from the group IIIa elements because it belongs to the “anomalous” crystals for which the expression A Vf/

547

Fig. 1. A plot of the volume change AVf/V,,, on melting for group IIIa elements against their volume VL in the liquid state at melting point.

4

a

12

u,

cnL&

20

Fig. 2. A plot of the adiabatic compressibility @sL for the group IIIa liquid elements against their volume at melting point.

V, is negative [13]. Experimental data on the thermoelastic properties for liquid aluminium, indium and thallium at their melting points used for plotting the graphs in Figs. 1 - 5 are given in Table 2. These graphs show quite good agreement between the estimated properties of boron and the correlated quantities. Thus we can conclude that the estimated values of the thermoelastic properties of liquid boron at melting point we obtained from quasi-thermodynamic correlations are quite reasonable.

In T1

Al

Element

Experimental

TABLE 2

11.39 16.35 16.01

(161

(cm3 (g atom)-‘))

VL

106

properties

108 (161 120 [17] 140 1171

(K-l)

CYL x

values of some thermoelastic

1.93 2.65 3.24

1141

PsL (mbar-l)

6.0 [18] 2.7 f13J 2.2 [ 181

4730 2315 1660

1141

(m s-l)

CL

0.99 1.29 1.39

(K-l)

(ms-l [I41 0.47 0.30 0.23

_(!!)Px

-(!z) Kp‘l)

indium and thallium at melting point

Av,Wn, (X)

for liquid aluminium,

104

549

I

i0 t

14

12

Fig. 3. A plot of the sound velocity CL of group IIIa liquid elements against their volume at melting point. Fig. 4. A plot of the thermal coefficient (a In CL/~&, of the sound velocity of group IIIa liquid elements against their volume V, at melting point.

Fig. 5. The correlation between the thermal coefficient CYLof volume expansion and the adiabatic compressibility /IsL for group IIIa liquid elements at their melting points.

References 1 D. Sh. Tsagareishvili, The Methods of Calculation of Thermal and Elastic Properties of Inorganic Crystalline Substances, Metsniereba, Tbilisi, 1977. 2 N.-D. Stout, R. W. Mar and W. 0. J. Boo, High Temp. Sci., 5 (4) (19’73) 241- 251. 3 G. V. Tsagareishvili, D. Sh. Tsagareishvili and A. G. Rhvedelidze, Estimation of some thermoelastic properties of /I-rhombohedral boron in wide range of temperatures and pressures, J. Less-Common Metals, submitted for publication. 4 R. Kubo, Thermodynamics, Mir, Moscow, 1970.

550 5 R. A. McDonald and D. R. Stull, J. Chem. Eng. Data, 7 (1962) 84. 6 A. E. Newkirk, Elemental Boron, Boron, Metallo-Boron Compounds and Boranes, Inter-science, New York, 1964, pp. 233 - 299. 7 D. Sh. Tsagareishvili, Bull. Acad. Sci. Georgian SSR, to be published. 8 Ya. J. Gerassimov, V. M. Glazov, V. B. Lazarev, V. V. Jarov and A. S. Pashinkin, Dokl. Akad. Nauk SSSR, 235 (4) (1977) 846 - 849. 9 F. N. Tawadze, I. A. Bairamashvili, G. V. Tsagareishvili and D. V. Hantadze, Stud. Cercet. Metal., 10 (1) (1965) 49 - 52. 10 G. A. Slack, Phys. Rev. A, 139 (1965) 507 - 515. 11 I. M. Silvestrova, L. M. Beliaev, T. Niemyski and Yu, W. Pisarewski, in F. N. Tawadze (ed.), Boron: Preparation, Structure and Properties, Nauka, Moscow, 1974. 12 W. I. Bogdanow, Yu. Kh. Wekilow and G. W. Tsagareishvili, in F. N. Tawadze (ed.), Boron: Preparation, Structure and Properties, Nauka, Moscow, 1974. 13 A. Ubbelode, Melting and Crystal Structure, Mir, Moscow, 1969. 14 G. Webber and R. Stefens, Utilization of physical acoustics in quantum physics and solid state physics, in U. Mezon (ed.), Physical Acoustics, Vol. IV, Part B, Mir, Moscow, 1970. 15 0. Anderson, Dynamics of the frame, in U. Mezon (ed.), Physical Acoustics, Vol. III, Part B, Mir, Moscow, 1968. 16 W. I. Nijenko, Methods of Investigation and the Properties of Boundaries of Containing Phases, Metallurgiya, Moscow, 1976. 17 P. P. Arsentiev and L. A. Koledov, Metallic Meltings and their Properties, Metailurgyia, Moscow, 1976. 18 W. N. Jarkov and W. A. Kalinin, Equations of Solids at High Pressures and Temperatures, Nauka, Moscow, 1968.