Theory of transport in liquid metals

Chemical Physics 11 (1975) 201-215 Q North-HollandPublishing Company

THEORY OF TRANSPORT IN LlQUlD METALS. III. Calculation of shear viscosity coeffxients of binary alloys’ P. PROTOPAPAS* and N.A.D. PARLEE* Department of Applied Earth Sciences. Stanford Universir), Stanford, California 94305. USA and Division of Earth Sciences & Technolo~. J.J. Davis Associates. McLean. Virginia 22101. USA

Received IO hlarch 1975 Revised manuscript received 21 July 1975

A t.hco~cticaJ procedure for the calculation of the shear viscosity coefftcienrs OTliquid binary aUoys is proposed based on the simulation of an alloy by a hypothetical singlecomponent hard-sphere liquid and treatment of the latter according to a correcbzd Enskog approach. The Enskog reauh is corrected according to a corresponding states principle developed by the authors in a previous work. ‘The pair correlation function at contact necessary for the Enskog calculation is obtained from the solution of the Carnahan and Starling equation of state for single component hard-sphere tluids. The proposed theory is tested numcricaUy by comparing its viscosity results for tifleen atloys with expcrimencal data and with the mrresponding results of the Eyring approach and the Tham and Gubbins theory. The latter theory is implemented numcricaUy with pairconclalion vlues a~ confact satisfying exactly the Carnahan and Starling equation of ~(31~ for binary hard-sphere Ouids. These ~Iucs rcsutt from anatyric and closed-form cxprcssions for the pair-correlation function developed presently through a correction of the corresponding Lebowitz expressions.

1.

Introduction

The basic “philosophy” behind the transport theory pursued in the present series of articles can be summarized as follows: First, the theory must develop a molecular model describing as realistically as possible, the kinetic behavior of the actual liquid. Second, the theory must derive analytic, closed-form, and as exact as possible, expressions linking the transport properties of the model fluid with a few and readily available numerical input data. Unfortunately, the two above goals are not easy to achieve simultaneously. Rigorous mathematical formalisms have been develop ed so far only for the simpler molecular models. For the more sophisticated models, the mathematics be* Worksupported by (he National Science Foundation

and JJ. Davis Associates_ Part 6t the W.D. dissertation of P. Rotopapas submitted to the Department of Applied Sciences of StanfordUniversity.1975.

* Director of Ear&

.

Sciences and Teechnology. J.J. Davis

&&cafes. * Professor of Extmtive Metallurgy.Stanford Uaivenity.

.

come approximate, terribly involved and numerical calculation of the transport coeffkients requires a vast number of input data, the most of which are harder to measure experimentally, than the transport coefficients themselves. The fundamental feature of the molecular model employed by a kinetic theory is tie relationship between the intermolecular potential and intermolecular distance (or separation). (hereafter denoted as”potenGal”). The potential of the model describes the behavior of the molecules during intermolecular collisions (or interactions), which dictate the overall kinetic behavior of the fluid. In ar.other article [l], the authors of the present work have reviewed extensively the different potential models employed in the past by different kinetic theories and they have commented on the relative rigor and numerical precision of these kinetic theories in relation to their potential. For monatomic molecules, as the ones of liquid metalc, the soft- (or elastic-) sphere potential is the most realistic. According to this model the molecules are perfect spheres which are deformed (or flattened) to a greater

202

P. Pmropapas, N.A.D.

Parlee/7lrcory

extent at higher temperatures during a collision, before they bounce back and resume their perfectly spherical shape. Indeed. at higher temperatures the kinetic energy carried by a molecule is larger. its velocity is higher, therefore the impact of a collision is more severe, causing more extensive flattening of the elastic sphere around the area of contact with another sphere or the wail of the container. During Ihe period of deformation from perfect sphere to the mwimum of flattening, *he velocities of the colliding molecules decelerate from their initial levels to zero. Dur‘ing the period of reformation from the maximum flattening t-o perfect sphere, the velocity of each atom involved in the collision accelerates from zero to an op. posile velocity of the initial one, the absolute value of which depends upon the mass ratio of the two molecules involved in the collision. The authors are not aware of any kinetic theory which has developed exact mathematical formalism for perfectly elastic-sphere potential. The simplest and most unrealistic potential for monatomic molecules is the one of hard (or rigid) spheres. According to this model. the I;-‘olecules are perfect spheres which at any temperature, no matter with what velocity they collide, are brought to a full slop without any deformation. The velocity from a given value drops to zero without any deceleration and from zero assumes a constant velocity of opposite direction withaut any acceleration. Again, the absolute value of the velocity of return depends upcn the mass ratio of the two molecules of the collision (only twobody collisions being considered). Enskog [ 2-41, employing the hard-sphere potential, developed expressions for the coefficients of diffusion. viscosity and thermal conductivity in terms of density, atomic weight, and the only parameter of the hard-sphere potential, i.e., the distance of closest approoc/~ between two colliding molecules (or hard spheres). This length is obviously identical with the center-to-center distance between two hard spheres at contact, which length is equal to the sum of the radii of the colliding spheres. In the case of a pure liquid. (which is the only

case examined by Enskog), all hard spheres in the liquid are of equal size and the sum of the radii of two colliding molecules becomes equal to the hard-sphere diameter of the liquid. A wide variety of potential models can be classified between the hard-sphere and soft-sphere models of intermediate mathemtical rigor and a&racy of numerical results. Some of these mo-

of transport in liquid merals. III

T2 r

A

0

a2 ‘

5.2 rB

QE3 T2

’

G.2< r

8.2 <

T, #a., ‘ re . I

Fig. 1. Equivalence of the impact of collision of two softspheres with Lhe impact of collision of two appropriate hardspheres shrinking with increasing lcmperaturc.

dels are the “square well”, the Lennard-Jones 16,121, the weighted Lennard-Jones, the truncated LennardJones, the Sutherland model, etc., which have been discussed extensively in ref. [l]. The presently pursued transport theory attempts to house under a common roof, the respective advantages of the soft-sphere and hard-sphere models, i.e., the realistic nature of the soft-sphere potential and the mathematical simplicity of the hard-sphere potential. This can be achieved by implementing numerically the Enskog formalism for hard-sphere potential with hard-sphere diameters shrinking with increosirrg

temperature. As it is shown in fig. 1, at a temperature T, , the distance of closest approach, AB, between two colliding soft spheres of radii iA and I~, is also the center-to-center distance of two hard-spheres of radii .f..,l and ‘B,l, being sander than the respective softvheres. At a teniperatute T2 highei ihan T1 the impact of collision is more severe, the deformation of the soft-spheresrA a$ cB is more exten&e and the

P. Proropapas. N.A.D.

Parieell%eory

hard spheres at contact rA,2 and rB,2 are smaller than rA I and rA 2, respectively. Thus, the center-tocenter d&ance beiween two colliding soft spheres, as a function of temperature, (which relationship fully describes the soft-sphere potential), is identical with the centerto-center distance of two hard-spheres at contact, shrinking at a proper rate with increasing temperature. Thus, at any specific temperature the simple mathematical formalism of the hard-sphere potential is fully ap plicable, but the description of the liquid over an extensive temperature range corresponds to an “effective” spft-sphere potential and guarantees drastically improved numerical results over the ones of the hard-sphere potential, i.e., hard spheres with diameters independent of temperature. In the tirst paper [5] of this series, (hereafter denoted as I), a closed form expression for the temperature.dependence of the hard-sphere diameters was developed based on a corresponding states principle. This principle was detected [6] after a statistical treatment of all available self-diffusion experimental data on liquid metals. The numerical predictions for the hard-sphere diameters of fifty-two metals were compared [5,6] with corresponding X-ray data and were found to bc in less than 2% discrepancy with the experiment. The predicted hard-sphere diameters of eighteen other metals were compared [7] with corresponding Monte Carlo data and were found to exhibit also less *than 2% departure from the computer experiment. Alder et al. [8] have found, through molecular dynamics computer experiment, that the Enskog theory for the hard-sphere fluid is not exactly correct and needs substantial correction. In the paper I (ref. [S]) their findings were used to correct the Enskog expression for the self-diffusion coefficient of liquid metals. ln the second paper [9] of the present series, (hereafter denoted at II), it was shown that their findings for hard-sphere potential when applied to real liquid metals led to a very good correction of the Enskog expression for the viscosity coefficient of pure liquids but not as good as a correction developed in paper 11 on the basis of a corresponding states principle. This principle was detected [lo] after a statistical treatment of all available shear viscosity experimental data on liquid metals. In the paper of ref. [I], a theory on a single hardsphere poiential, describing the overall molecular interactions in binary liquid meti alloys as a function of

of transport in lipuid metals. III

203

temperature and composition, was proposed and tested. In other words, a theory was developed for simuiating a binary alloy by a hypotheticai single component fluid with an ‘effective” soft-sphere potential. The importance of such a theory for the conduction of numerical calculations on properties of binary alloys is dramatic. Indeed, both equilibrium and transport properties of binary alloys can be calculated from the respective theory for pure liquids, whenever theory pertaining to the property of cencern does no: exist for alloys. In the paper of ref. [I], the structure factor of binary liquid alloys was calculated on the basis of the theory for the structure factor of pure liquids intplemented with the proposed potential. The results were in excellent agreement with the predictions of the Percus-Yevick theory for *he structure factor of binary alloys and with experimental data. In the present paper the shear viscosity coefficient of binary alloys is calculated on the basis of the respective theory for pure metals [9] implemented with . the potential model developed in ref. [I]_ The results are compared with the experiment and the predictions of the Tham and Cubbins (I 21 theory for shear viscosity of binary hard-sphere liquids, which is a straightforward extension of the Enskog theory [4] for singlecomponent hard-sphere liquids. The numerical results of the proposed approach are in far better agreement with the experiment than the Tham and Cubbins results, for reasons explained in the discussion section. The pair correlation functions at constant, which are required for the conduction of numerical calculations through the 7ham and Cubbins theory. represent modifications of the Lebowite [ 131 expressions. The applied correction to the Lebowitz expressions lead to new expressions which satisfy the Carnaban and Starling [ 141 equation of state for binary liquids instead of satisfying the corresponding Percus-Yevick equation. Molecular dynamics calculations by AIdcr [ 151 and Monte Carlo calculations by Rottenberg [ 161 and Smith and Lea (171 have proven that the Camahm and Starling equation of state for binary liquid mixtures is definitely superior to the corresponding Percus-Yevick equation.

204

P. Fforopapas, N.A.D.

Parleej37leory

of

tramport in liquid

melds.

III

2. Theory 27--I

had-sphere rheov for rhe shenr viscosiry of pure liquids

2.1. The

‘lhe Enskog expression

[3,18]

for the viscosity

’

’

’

’

’

2.5 -

Enskog

2.3 -

of

pure liquids is /.rs = 4j.1~[4n g(o)--’ + 0.8+(0.7614)4ng(o)]. j.ra is the viscosity given by

expression

(1)

for pure dilute gases

,

cl, = (5/ 1602)(mkT/n)“2

(2)

I

015

where k is the Boltzmann

-temperature, u

constant, T is the absolute is the hard-sphere diameter and m is

025

mass. In eq. (1). r) is the packing fraction of the liquid, i.e., the volume fraction of the liquid occupied by hard spheres, (I - 17= void fraction), (3)

where n is the atom density (atoms cmd3). Finally,g(a) in eq. (1) is the pair correktion fittction at conract. The best expression for this function is obtained from the solution of the Camahan-Starling equation of state [ 191 with respect to g(u). This solution is g(o)=@-r1)/2(1

.

-5Q3

045

0.55

?

lhe atom

n=;*??.3,

Q35

(4)

The superiority of the Camahan-Starling equation of state over alternative equations has been established on the basis of the molecular dynamics calculations of Alder and Wainwrtght 120) and the Monte Carlo calculations of Verlet and Weiss [21].

Fig. 2. Corresponding for the shear viscosity

states cmrectim of pun? liquids.

of the Enskog

the final expression for the shear viscosity liquid metal becomes: cc= tic/&)

PQ

theory

of a pure

(5)

.

where b//.tQ) is taken from fig. 2 and pa is given by eq. (1). 2.3.

The

proposed

kinetic model for the simuIn?ion of

a binmy liquid metal alloy by component hard-sphere liquid

a hypothetical single-

This model has been introduced in ref. [ 1). Presently we repeat only the mathematical formalism necessary for the numerical calculations. A single-component hard-sphere liquid has three elements of identity completely dete rmining its transport behavior. These elements for the proposed hypothetical single-component hard-sphere liquid are as follows:

2.2. comction

the Enskog result I?.wshear viscos: fry on the b&s of a cort~sponding states principle of

Iti refs.![9] and [IO] the derivation of a numerical correction of the Enskog result for viscosity has been presented in terms of a corresponding states principle., since in the above references Jhia correction was pro ven to be more effective than the molecular dynamics correction of Alder et al. [s]. we employ, in the pres-. ent work, the corresponding states correction excluaively. Fig. 2 iUus$rates the reducedvisc&ity &a as a function~of the dimenaionles3 packing fraction. Thuf .

.

_.:

.I

._

2.3-l. The had-sphere mass The “atomic weight” of the hard spheres of the hypothetical liquid, AW, is assumed to be the average of the a?omic weigfrts of the two metals in the binary AWi (i = 1,2), weighted by their respective atom fractions,x-

_ I’ AW=Xt SAW, +aQ‘AW,

(6)

.

Thus. thk qas4 of a hypathe~ti_sph&e, m~~~W;~=(X~.AWI+X2’AW1UN.

_ . .-

.m, is .._

_-

.

(7) I. :

P. Protopapas, N.A.D. ParIce/l?reory of transport in liquid

2.3.2. The number density It is assumed that all binary liquid metal alloys of truly metallic components are ideal solutions, i.e., exhibit zero excess volume of mixing. Thus, the volume of one mole of ahoy, V, is the average of the molar volumes of the two-component metals in the pure

state, r/p, weighted by their respective atom fractions, V=X,

q+x2

e.

(8)

Thus, the mass density of the alloy is /I = (AW/v)

= (X, *AW, + X2-AW2)/

(9)

where AW and V were taken from eqs. (6) and (8). respectively. The molar volume of the component metal i in the pure state, e, may be expressed in terms of the atomic weight, AWi, and the mass density of this component in the pure state, pp. _

Substitution

of v

p =(X,

(10) from eq. (10) into eq. (9) yields

+X2-AW2/~z).

(11)

The number density of the alloy n, is related mass density through the expression n=plm

=NIIX1-AWII~p

+X2-AW2/~41.

to the

n,,., = (lr/6)n,

o$ =

0.472..

US)

state. The second hypothesis of the theory is that the hardsphere diameter at any absolute temperature T, u, is related to a, and T,,, through the relation

The subscript

m stands for melting temperature

.

Substitution of u, ces the expression

(1’5)

from eq. (15) into eq. (16) produ-

w = 1.288(AW/p,)“’ X [I -O.l12(T/T

m )“‘]X

IO*

(incm).

(17)

The preferred numerical value for pm is the experimentally established melting point mass density of the alloy. In absence of experimental data, pm can be calculated from eq. (1 I) with pi” values corresponding to the

(12)

pectively. We define the number density of the hype thetical singie-component hard-sphere liquid to be equal to n. The number densities of the two-component metals in the alloy are taken from the obvious relations, ni=xiIa.

(13)

The number densities of the two-component metals in the pure state, nr, are calculated from the corre-

mass densities, pip, as

nip = pip/mi = pFN/AWi.

2.4. The Thorn and Gubbins hanSsphere theory for the shear viscosity of multkomponenr I&uLi mixtures as applied to the specific case of binary liquid mixtures

where p and m were taken from eqs. (11) and (‘7) res-

sponding experimental

sphere liquid. This theory advocates two hypotheses: First, all hypothetical fluids at their respective melting temperatures have the same value of packing fraction. nor, namely 0.472, i.e.,

melting temperature of the alloy.

- AW, +X2 -AW2)/

(X,-AW&‘;

205

u = 0, [I - 0.112(T/T,)“*]/o.888

(Xtq+Xx,9).

I+’ = AWi#

metals.III

(14)

Tham and Gubbins [12] expanded the initial Enskog theory for pure liquids to the multicomponent case, through a straightforward generalization of the mathematical procedure described by Chapman artd Cowling [ 181. Their theory, being based on the simplest of the potentials, i.e., the hard-sphere potential, is the most tractable theory for multicomponent mixtures ever developed. Furthermore, this theory treats the physical model is adopts, i.e.. the one of the hard-

sphere potential in an evcct qucmti~ative manner, since this theory achieves the exact wiifrion of a modtied and generalized Boltzmann iniegrodifferential equation. We present here, in the most compact form,

only the mathematical for the conducting

2.3.3. The M-sphere diameter We assume that the hard-sphere theory proposed in ref. [S] for a liquid metal in the pure state is applicable also to the hypothetical single-componcrit hard,’ ,.

formalism absolutely necessary

of numerical

calculations

in the

special case of binary liquid mixtures. The shear viscosity of a binary hard-sphere mixture is given by the expression /i=A+‘B+C,

(18) ~.

206

P. Profo~~ns. N.A. D. Parleellheory of tmnqort in liquid metals. III

where

b12 = 2an&3p.

(1%

(20)

C=~(lrk~t)“2n:gl,(u11)0~1 (21)

+ ~(11~Tm*)“2n22g*2(o*2)042* +~[2nk~rtm2/(mt+mZ)1”2~r1~~g12(01~)0~~.

The quantities !I\: and b$ appearing in eqs. (19) and (20) are given by the expressions b’lb’ = WI&

- K2H12)l(H11ff7-2-

ff21H,2).

(22)

and -

b:b=((HiIX2-HZtKI)/(H:,H12’-HZtHt*)

(23)

The quantities appearing in the right-hand-side of eqs. (22) and (23) have been defined as follows: H,, = ~~~tgll(otlX~k~/m1)1’2 + 8”1”2u:2g,2(u,2X2”kTm2/m1)“” X (5ml t 3m2)/3(m,+ rn#*

HI2 =-16&“,~zgt@t2) X (2rk7h,m2)“‘/3(m,

,

(24)

2.5. The proposed expression for the pair correlation function

at contact

Lebowitz [ 131 has extended the Percus-Yevick equation for the radial distribution function of a single-component hard-sphere liquid to the multicomponent case. Furthermore, he solved exacfl! his generalized equation

and obtained

closed-form

and + m2)=

,

(25)

&2(Q)

= W2at2(1

-o)*I

x KU, + 0~x1 +r)/2) - 3(11, -r13(ol

K1= 5n, +(4n,n/3)[n,~t81,(ol,) + 2n~m&8t@&l(ml

+ m2)J-

(26)

Hz2. Hz1 and K2 are defined through expressions symmetric to eqs. (24). (25) and (26). respectively, i.e., through replacement of 1 by 2 and vice versa. Returning !o es. (20), the symbolnMt2. b,,. etc. stands for the functions: Mi, = m,/2ml

= J/2,

MI2 = m&in1 + fizz), $1 =&Jlu~t/3p: -~ .

__

expressions

for the pair-correlation functions at COntaCt,gij(Uij). His expressions for the special case of a binary mixture of hard spheres becomes: (3 1) 211 (ol I) = I(1 + ~12) + 302(01 - Q/2021(1 -7iP,

and

. and .’‘ . .

(301

M,,.,H,, , b,, and b,,‘are defmed through expressions symmetric to eqs. (27). (28), (29) and (30), respectively. The wmbolgii(uij) in terms of which all three components A, B and C of the shear viscosity, ~1,have been expressed, stand for the pair-correlation function at contact between two hard spheres i and j. These functions are input data for the Tham-Gubbins theory and no particular expressions have been recommended for them. In section 2.5 of the present work, quantitative expressions for the four gii’s have been derived and subsequently have been employed in the numerical calculations of section 3.

:-

- 02)/21.(32)

gz2(u22) andg2,(u2t) are defined through expressions symmetric to eqs. (3 1) and (32). respectively.

The symbols vi, 9 and ui, are defied tJi =

1,2),

(ii

llniU:/6

as

17=‘)t +7l2.

I

(33) -

(34)

and -. (35)

(27)

oij = (ui+ u/)/2

c2b

Unfortunately, the Per&-Ye&k equation for the . radial distribrition fiction is oniy appnxjnmte..lhe studies of Ff. [ 1$16] and 117) have~shown that tie‘ Camahait_Starling dquation of~p_~, for a bin&y hard-

,i. ’ (29)’

‘.

,-

(ii j 7. I

.

,2) :

_‘..

,.

I.

P. Prolopapar.

N.A.D.

Parleelheory

sphere liquid is more accurate than the corresponding Percus-Yevick equation of state but to the best knowledge of the authors of the present work analytic expressions for pair-correlation functions at contact satisfying the Cam&m-Starling equation of state for binary hard-sphere liquids have not yet been derived. Since expressions as accurate as possible for the correlation functions at contact are strongly desired for the conduction of numerical calculations, (but at the same time this paper does not aim toward the development of a rigorous theory for pair-correlations), in the present section a modification of the Lebowitz expressions for pair-correlation functions at contact is introduced which yields expressions satisfying exUC&J the Camahan-Starling equation of state for binary hard-sphere liquids. Let the Percus-Yevick and Camahan-Starling functions be denoted by (A) and plain symbols, respectively. The equation of state for a binary hard-sphere liquid in terms of Percus-Yevick pair-correlation functions at contact can be stated as follows [see Lebowitz, eq. (44)] : .@=PVjNkT=

PjnkT=

1 + (2n/3n)[n:o:tg^tt(o,t)

+2n1”2u:2g~2(011)+“~u~2~~~(~22)l

9

(36)

where the superscript denotes the number of components. Since the virial theorem, through which eq. (36) has been obtained, can bc applied in a~ identical manner for the expression of the Z(” of any theory in terms of its own Q(u~~)‘s, it is also valid that #=

I +(2sr/3n)[n:o:,gll(ull) (37)

+2”,“2o:~t2(ott)+n5o12g22(o2t)l-

lt is assumed that the Camahan-Starling pair-correlation functions at contact are multiples of the corresponding Percus-Yevick expressions by the same coefficient function C, i.e., gij(oij)

(i. i = 1,2).

= CgjjC”ij)

Substitution yields

of gij(uii)‘s

(38)

from eq. (38) into eq. (37)

ZG)= 1+(2n/3n)C[n:u:tb,,(ott)

(3% -+2n1n*~~‘~(u12)+"~0~~21(a22)lDirect comparison value of C, m.’ __

I. .__ -

-,

of eqs. (36) and (39) produces

_. -

-_

the

of rmnsporr in liquid mculr

207

III

(40)

c=(.z(~-l)l(~~;-I).

Ref. [ 141 provides the general expression ofZtn for an m-component liquid. For the special case of a binary hard-sphere liquid this expression becomes: Z(2)=

((I+~+~2)-3rl(Y,+Y2tl)-Y3r13(I-~)-,3(41)

where

There are two 2”) expressions, one obtained through the virial expansion of the pressure, denoted as sv2’ and ane resulting through expansion in terms of compressibility, denoted as2:‘). These two functions are as follows: 5LZ)=[(l

+17+772)-3311(Y* + Yzs)l(l-s)-3.

(46)

and $)

” = ;Q)E - 3Q3 y3( 1 - q)-3

.

(47)

Lebowitz has employed throughout his work the 2:” expression. Thugs. by substitutingi$‘) from eq. (47) in functhe position of Z (2) of eq. (40), the coeff;cicnt tion C is defined through an involved but closed-form expression. The expressions for the pair-correlation functions at contact [with ~i~(Uii) taken from eqs. (3 I) and (32) and C from eq. (4O)j are the most recommended, since they satisfy exucfly the Cam&m-Starling equntion of stale for binary hard-sphere liquids. However. since the philosophy of this series of articles, as it has already been stated, is accumcy through simpficiry, B much simpler intuitive expression for C has also been derived and tested. The basic assum$ion is that the Camahan-Starling pair-correlation functions at contact for binary hard-sphere liquids outperform the COTresponding Percus-Yevick expressions to the same extent that the Camahan-Sarling expression for the paircorrelation function at contact of a single component hard-sphere liquid outperforms. in accuracy, the corresponding Percus-Yevick expression, i.e.,

208

P. Ptotopapas.

N.A.D.

Parlee/77teory

oftran&wrt

in liquid metals. 111

table 1 Comparfson of numerical results for the correction

of the Lebowitz ptir correlation functions obtained through eqs. (55) and (40)

Cleq.

rl

Ch. tw

CIcq. (4011

rl

C[eq. (%)I

C(cq. (4O)l

0.5666 0.5558 0.5309 0.5297 0.5216 0.5168 0.4906 0.4899 0.4966 0.4944 0.475 1 0.4715

1.2886 1.2720 1.2374 1.2358 1.2255

1.2886 1.2720 1.2374 1.2364 1.2255 1.2152 1.1897 1.1890 1.1850 1.1832 1.1737 1.1700 1.1691

0.4517 0.4508 0.4502 0.4498 0.4484 0.4469 0.4462 0.4452 0.4442 0.4428 0.4418 0.4413 a4405

1.1517 1.1509 1.1504 1.1501 1.1488 1.1476 1.1469 1.1461 1.:453 1.1440 1.1432 1.1427 1.1421 1.1411 1.1407 1.1405 1.1403 1.1400 1.1391 1.1388 1.1379 1.1374 1.1367 I.1366 1.1363 1.1353 1.1347

1.1517 1.507 1.1482 1.1487 1.1488 1.1473 1.1451 1.1461 1.1453 1.1440 1.1423 1.1407 1.1421 1.1411 1.1407 1.I405 1.1395 1.1400 1.1374 1.1386 1.1378 1.1354 1.1347 1.1354 1.1355 1.1344 1.I328

0.4281 0.4276 0.4262 0.4259 0.4252 0.4249 0.4247 0.4236 0.4223 0.4207 0.4206 0.4161 0.4152

1.1343 1.1333 1.1327 1.1322

1.I337 1.I333 1.1327 1.1306

0.4704

0.470 1 0.4660 0.4655 0.464 1 0.4635 0.4632 0.4623 0.4613 0.4611 0.46W 0.4590 0.4589 0.4569 0.4554 0.4544 0.4535 0.4526 0.4524

ill=

1.2196 1.1897 1.1890 1.L85S 1.1834 1.1737 1.1702 1.1691

0.43?3

1.1689

1.1688

1.1649 1.1644 1.1631 1.1625 1.1622 1.1614 1.1605 1.1603 1.1597 1.1584 1.1582 1.1564

1.1649 1.164% 1.1629 1.162S 1.1622 1.1612 1.1605 1.1601 I .1594 1.1584 1.1567 1.1564

0.4388 0.4386 0.4384 0.4380 0.4369 0.4365 0.4355 0.4348 0.4340 0.4339 0.4335 0.4322

1.1551

1.1548 1.1541 1.1529 1.1524 1.1524

0.4315 0.4309 0.4297 0.4290 0.4284

1.1542 1.1534

1.1526 1.1524

rg(&~)l&l(ql~.

(48)

g22(u22)

ts5>1

Cleq.

(4011

1.1320 1.1316 1.1395 1.1303

1.1320 1.1297 1.1305 1.1301

1.1297 1.1295

1.1285 1.129s 1.1293 1.1276 1.1269 1.1262 1.1243 1.1227 1.1215 1.1214

0.4136 0.4120 0.4104 0.4099 0.4092 0.4071 0.4062 0.4058 0.4027 0.3982 0.3962 0.3909 0.3896

1.1284 1 .1275 1.1262 1.1261 1.1227 1.1221 1.1’?18 1.1208 1.1 I97 1.1185 1.1182 1.1176 1.1161 1.1155 1.1152 1.1130 1.1099 1.1085 1.1019 1.1041

0.3836 0.3769 0.3613

1.1002 1.0959 1A866

1.1002 1.0959

1.1293

0.4149

1.1208 1.1187 1.1178 1.1180 1.1172 1.1161 1.1155 1.1152 1.1130 1.1099 1.1081 1.1049 1.0949

1.0789

i~*‘=(1+2~+3q*)(l-~j}-~,

(W

and

and s12@12)=

-rl

Ib(a&a&2(~12)~

(49)

and gzl (~2,) are defined through expressions symmetric to eqs. (48) and (49). respectively. z(a) can be obtained directly from the Percus-Yevick equation 6f state for single-component hard-sphere liquids. Similarly with the binary hard-sphere liquid case, there are two Percw-Yevick equations of state for single.component hard-sphere liquids, one obtained through the +ial expansion of the pressure. and one resulting through an cxpahsion in terms of compressibiity. ,;Thes+:.e$ations arc respectively ,. . :-. __. ._ ., . ,,‘ .- -j: -.

is’=c

(1 +q+$)(1

Again

from the virial theorem it is obtained

3’)

-YTj)-3

= 1 + 27rn&(a)l3

Combination

.

(51)

_

(52)

of eqs.‘(SO) and (52) yields for the virial

cass I

&T)=(2+r1)/2(1-7#. Division

of eqs. (4) and (53)jy

~~u)/~~)=(2-~)/(?~-pci+l).

pa& gives -

_.

’

(53) . (54)

Substituti&-&t(#@) &A q&jin eqs. i48). _ -. , -‘ ,,.“_ __I. _ .; -.:,; -. : : _.

:

P. Proropapas, N.A.D. Parke/7heqy

and (49) gives expressions for the gij(oii)‘s intuitively though to be compatible with the expressions obtained through eq. (38). i.e.. intuitively it was anticipated that C=(Z-T))/(1-rM2+r?).

(3%

The two alternative expressions for C. i.e., the ones of eqs. (40) and (55) were tested numerically over a large number of binary liquid metal alloys. It was found

that they are, indeed, identical in all examined cases. Table 1 provides a few typical results.

3. Calculations ‘The viscosity of a binary liquid metal ahoy is calculated through the proposed theory =a fotrow~: (a) The “atomic” mass, number density and hardsphere diameter of the hypothetical single-component hard-sphere liquid are calculated through eqs. (7). (12) and (17) respectively. (b) The packing fraction and the pair-correlation function at contact of the hypothetical single-component hard-sphere liquid are calculated through eqs. (3) and (4). (c) The Enskog viscosity of the hypothetical singlecomponent

hard-sphere

liquid is calculated

through

eq. (1). (d) The proposed viscosity for the alloy is calculated through eq. (5) with @IQ corresponding to the r) value found in the above step b. For the calculation of the number density in step (a), the mass density data of EUiottt and Gleiser [ 111 have been employed. When T is lower than the Tm of one of the component metals i, the pi” used in eq. (14) has been calculated by extrapolation of the density values of i to temperatures below its melting point. The viscosity of a binary liquid metal alloy is calculated through a corrected Tham and Gubbins theory as follows: (A) The atomic mass, number density in the alloy and the hard-sphere diameter of the two-component metals are calculated through eqs. (7), (I 3) and (17). respectively. The same notions about the mass dens-

ities were used as in step (a). (B) The packing fractions and the p.air-correlation functions of the two-component metals in the alloy. are calculated tbrou8h eqs. (33) and (38),~respcctively, _titJ C gi,venIn eq. (55).

of rmnsporr in liquid meIalr III

209

(C) The Than+Gubbins viscosity for a binary alloy is calculated through eq. (18). (D) The corrected Tham-Gubbins viscosity for a binary ahoy is calculated through eq. (5) with ($/I&J) corresponding to the value of q calculated throu& eq. (34).

4. Numerical

results

The relative succes of the proposed approach to the calculation of the shear viscosity coefficient of binary liquid metal alloys is demonstrated over fifteen illustrated cases. In these figures the curve of the proposed approach is labeled as “PP”. In ali fifteen presented examples the results obtained through the correctcd Tham-Cubbins theory have also been irtchrded for comparison. The curves of this theory are marked

as “TG”. Finally, the results of the elttremeIy successful empirical correlation suggested by Eyting and coworker [22.23] have been drawn and identified with the symbol “E”. The experimental results are presented with solid curves, and the names of the experimenters appear on each figure. The Eyring results stem from the very simple equation, logp=

“t logp,

+“z

logpz.

CW

pi is Ihe shear viscosity of the component metd i in the pure state at the temperature at which the alloy viscosity, p, is sought. The numerical values for pi’s, used in the present work to implement eq. (56) are the best experimental data available. These data are given in table 2 with references to their origin. Figs. 3 and 4 illustrate the viscosity of the alloy Pb-Sn at two different compositions over a range 200 degrees centigrade. This experiment has been conducted by Kanda and Colbum [24]. The ProtopapasAndersen-Parlee (PAP) theory [9] for viscosity of pure liquid metals has been proven very successful in predicting the viscosity of both metals Pb and Sn. Thus, it was anticipated successful prediction of the alloy viscosity as well. This is proven to be the case. The Eyring results are equally successful with the PP results and both are better than the TC. This evaluation of the three theoretical results seems to be valid -throughout the presented tifteen comparisons. The alloy Pb-Sn has also been studied expcrimentaliy by

210:

p. Roropapas.

N.A.D.

PwlceiTheory

of transport in liquid metals. III

Table 2 Expctimer+d

viscosity data for pure metals being components

I390 reo

cue)

cdb)

MC0

CC,

pdd)

r(cW

NW

r(cP)

of the alloys studied in the present work

CC)

Snd)

She) r(cP)

Km

u(cP)

N-T)

PWP)

3.5

500

7.70

LOO

3.45

200

3.37

2Do

2.15

3.1 2.75

5.85 6.75 5.15 4.85

300 250 327.3+ 350

2.49

250 300 350 400

3.02 2.10 2.40 2.18

231.9+ 250 300 350

I .a7 1.66

2.43

600 700 800 8.50

3.11 2.18

1.70 1.50

200 250 300 320.9+ 350

1.35 1.24 1.16 1.09 1.0s 1.00

400 450 500 550 600 650

2.2 2.0 I .a5 1.74 I .65 1.57

900 950 1000 1050 10837 1100

455 4.30 4.05 3.80

I .50

3.60 3.40

1.40 1.30

400 450 500 550 600 650 700

1.18 1.11 1.05 1 .oo 0.96

1.44 1.39 I .35 1.31 1.27

1150 1200 1250 1300 1350 1400

450 500 550 600 630.5? 650 700

0.85 0.83

700 750 800 850 900 950

2.23 2.20 1.84 1.69 1.57 1.47 1.38

1.98

0.96 0.93 0.90

400 450 500 550 600 650 700

3.20 3.05 2.85 2.10 2.55

150 800 850 900 950

1.32 1.27 1.22 1.18

750 800 850 900

1.20 1.13 1.06 1.00

750 800 850 900

0.92 0.89 0.86 0.835

1.15

950

0.94

9so

0.810

1000

0.81

1000

1.14

1450

2.40

1000

lOS0

0.89

1050

1.21

1500

2.25

1050

1.12 1.10

1000 1050 1100 1150 1200

0.89 0.85 0.81 0.71 0.73

1000 1050 1100 1150 1200

0.790 0.17 0.75 0.73 0.71

200 250 271+ 300 350

2.25 I .95

400 450 500 550 600 650 700

750 800 850 900 950

0.88

’ hklting Poinl orlhe mclzxl.Viscosities x lower rcmpentures are needed the mcltin ~oinrs of their component mcrak. “)RE~. 1281.% ref. [251.‘)~=f. 1281.d)ref. [24,26],%f_ I24.261.

because

the melting

1.81 1.66 1.52

1.51 1.37

1.265

point of mvly Jloys is lower than

cPr centipoise.

;;;j

350

400

450

500

t PC)

Fig. 3. Tcmpcrature

dependence

of the shear viscosiry of Pb-

43.O%~sn.

‘Fisher and Phillips [ZS]; In figs. 5 and 6,

results

on

alloy compositions are presented over a temperature tige of approximzitely 200 degrees centigrade. iwo

Again the P.P. results are quite satisfactory. Fishei and _Phillips [25).have also measured the viscosity of the ? So8 Cd-Sb. Results on two alloy cbmpositions cp-‘ ._ . _’; : -

Fig. 4. Tempera& 85.0 Sn.

dependence

of the shear viscosity-of Pb-

pear in figs. 7 and 8. !$nce,th~ PAP theory [9j for the viscosity of pure liquid metals fails to predict success _ fully the viscosity of Sb, but itd& well in pre+ing the viscosity_oiCd, it wis anticipated that the predicted alloy.~@osity~ wquld become +ez+singly i&U.. _I : ._ :_ ,-. -. _-_.

P, Roropapas,

N.A.D.

ParIeelTlwory

of tmnsporr in liquid metals. III 3.2

Fig. 5. Temperature depcndenec of the shear I%-29.56% Sn.

Fig. 6.

I

I

lk~‘~peraNre

Pb-74.05%

I

dcpcndence

‘cd-4&59%Sb

:s

-5

t (“Cl

30

I

--__ ---_ I ----_ 1

?;I;

1.6

2LL

viscosii~

of

of tic

shear

X Sn

viscosity

tmx

Fig. 8. Tempemlurc dependence of Lhc shear viscosity of Cd-46.59% Sb. I

I Pb-74.05

’

of

1

I

r.e ’

I

pb-ll.20%Sb

I 350

I

300

sn.

I

’

I

I

-

I

I

400

t (“Cl

Cd - 4.79 4x Sb -Fisher

Fig. 9. Tempcnlurc Pb-11.2ozsb

IJ Phillips

dependence of UICshear viscosity of

,

----RI?

1,

I,,

---

__

z y

a.

.

L(T) Fig. 7. Temperature dependence of the she= viscosity of Cd-4.79% 5%.

1

,

RF!

Z_T_ ;.=.

2.6

--

0

I

/-

4

,

,

Pb-Sb 350%

Ctarlcy

-

_A-’ --

6

12

16

20

Atom 46 Sb

Fig. 10. Composition dependence of (he shear viscosity of Pb-Sb at 350°C.

rate qith~increasing Sb atom fractions. Indeed, this is _’the case.

Crawley [26] has studied egmimencally

the

Pb-Sb alloy. The PAP theory for the viscosity of pure metals (9) is succesftil for Pb and unsuccessful for Sb

212

P. Prologppas. N.A.D.

Patiee/Tlwory

of rransport in liquid metals. III

3.6

3.2

0

I 20

0

I 40

I

I 60

00

100

Atom X Sn

Fig. 14. Composition Cu-Sn at 1188°C.

Atom % Pb Fig. 11. Composilion dependence of the shc;u viscosity of Sb-Pb at 292°C

dependence

of the *ear viscosity of

3.9

Sb- 59.11 X

Cu

3.5 3.1 ---F? --

Bi,“io9 . sauamlo

T.G. --w--E.

!~~~~~~

1.5

I 700

I 550

I

I 900

I

I KJOO

1

I 1100

t PC)

Fig. IS. Temperature dependence sb-59.11%Cu.

atom fractions was expected. However, fig. 9 demonstrates that for atom fractions of Sb up to 11.20% the temperature dependence of the predicted alloy viscosity is excellent. Furthermore, fig. IO proves that for an Sb atom percent up to 18% the disagreement is never greater than 0.2 centipoise (cP). Pluss [27], who has also studied the Pb-Sb alloy, gives even a better testimony for the PP theory. It is shown in fig. 11 that for an Sb atom percent from 13% to 24% the PP theory never departs from his data by’more than Q_1 cP. Figs. 12 and 13 demqnstrate the temperature and Alom X Bi composition dependence of the viscosity of the Pb-Bi Fig. 13. Composition dependence of the shear viscosity of alloy; respectively. The viscosity ofBi has not been Pb-Biat 8OO’C. .s+cess@ly predicted by the PAP theory [9) for pure .. metals.~Hiiw&er,&e inaccuracy of the PP theory for alloy visco& does no~incre~~rponotwically $th (sei table 2 of ref. [9]). Th?s; again increased inaccuracy of the piopqsed allby theory Ah iticxwing S5 irkas@ atom’% of ril.,Fig..l3 shotis that the m.axir ,_ -. :: -. -. .-._ I ._ ; _. .. . .: _ : ... ; ; .._. . .. .: . . _’ ._ .- __I : -._’ ‘ _. I .-, __ -._ .:_.. ._-; . -__ . . .:. _. : _. .:.._ ._: . -, Fig. 12. Temperature dependence Pb48,48% Bi.

._.

of the &ear viscosity of

.-

,

.’

-‘_

.

.

.._-

of the sheax viscosity of

.:

P. Rotopapas.

I

I

I

Pdee/l%eory

KA.D.

I

Cu-Sb 1ooo*c -Bienios a

_

Sauwmold

40

Atom Pig. 16. Composition Cu-Sb at 1000°C.

e

,

I

dcpcndrnce

I

60

I-

’

loo

of the shear viscosily

I

I

7-

6-

80

% Sb of

I I AI-62.3%Sb

I

Bienloe a Soucrrold -----cF! T G.

I 5CQ

I 600

I 700

I 800

I 900

I lGoo

I 1100

-

I 1200

1 (“CI Fig. 17. Temperature Al-62.3% Sb.

dependence

of the shear viscosity

of

mum inaccuracy occurs in the vicinities of SW& and 100% Bi. It is important to realize that the PP and TG theories do not produe accidentally identiml predictions for the end wtnpositions of pure metals, but always do so since both theories reduce to the Enskog theory for pure liqtid metals. Bienias and Sauerwald [28] have studied experimentally the alloy Cu-Sn. Their results appear to be inconsistent. Not illustrated comparisons show that

o/transport

in liquid memk

III

213

the PP theory is in worse agreement with lhe experiment than 0.25 cP for the compositions 17.94%. 24.% and 34.87% Sn, respectively. However, fig. 14 indicates that at 1188°C. the PP tfieory agrees with the experiment within 0.25 cP at any composition. Since both Cu and Sn have been successfully treated by the PAP theory for pure metals, ~JZare included to consider the experimental results of fig. 14 more reliabte than the rest. Bienias and SauerwaId (281 have also measured the viscosity of the Cu-Sb alloy. Two examples of the PP predictions as functions of temperature and composition appear in figs. 15 and 16, respectively. Since the PAP theory for viscositv of pure metals [9] is SuccXsful for Cu but unsuccessful for Sb, the worse disagceement with the experiment appears near the 1W Sb composition range. Finally, fig. 17 presents a single test of the PP fheory in predicting the viscosity of the Al-Sb alloy. Since the corresponding PAP theory [91 for pure metals faifs to predict successfully the viscosity of both Al and Sb metals, the prediction for the alloy could be either very bad or relatively good, depending upon the sign of the viscosity errors of the pure metals. If both errors were positive or negative the allay error would also be positive or negative and larger than either individual error. If the error for the one component is positive and the error for the other pure component is negative, the net result can be either in positive or negative error but to a lesser extent than either one inditiduaI error, since the two latter counter-balance each other. The PAP results for pure Al and Sb appear to be both in positive error (~.+a -P,,,~ > 0). thus, the good agreement with the experiment of fig. 17 indicates that the Bienias and Sauerwald results 1281 are probably high. A large number of additional comparisons of the theory with the experiment have been included in ref. [lOl.

5. Discussion

- conclusions

The numerical results of section 4 safely establish the conclusion that the proposed theory is in excellent agreement with rhe experiment, not only when both metals of the alloy are successfullly treated by tie Protopapas-Andersen-Parlee theory [9] for pure

214

P. Pmlopapar. N.A.D.

ParleelITteory

metals, but also when one of the two component mcjals is “ill-behaved” and its atom percent is less than 50% or 60%. The Ti+n-Gubbins theory [ 121 is modified by the correction of fig. 2, which brings the Enskog viscosity resuiis for pure metals in quantitative agreement with the experiment. Furthermore, this theory

of

rmnsport in

liquid merals. Ill

available for other pure metals of interest, the proposed theory is recommended in that it is calibrated with the best available viscosity data for all pure metals studied experimentally (ref. [9]).

Acknowledgements

has been numerically implemented with pair correlation values at contact in qwntitative satisfaction of the Camahan-Starling equation of state [ 141 for binary hard-sphere lluids. Moreover, the ThamCubbins theory has been equipped with the Prote papas-Andersen-ParL-:ee [5] hard-sphere diameters for liquid metals which are in quantitative agreement with Monte Carlo data 17). Despite all these precautions the Tharn-Gubbins theory is in worse agreement with the experiment than the theory presently proposed, in all fifteen illustrated cases, and in a greater number of additional cases (see ref. [ IO]). This failure may be attributed to the following reasons. First, the Enskog theory for t@e viscosity of binary hardsphere fluids needs a correction which is a function of both the atomic weight ratio, and the atom diameter ratio of the two component metals. This is the case with the Enskog theory for the .nutual diffusion of binary hard-sphere fluids, aa it has been shown

by Alder et al. [35j. Unfortunately,

such a correction

is not known to the authors for the binary viscosity %nd, therefore, they were forced to use the best available correction for the single-component case. A second reason for the relative inaccuracy of the ThamGubbins

theory

is that although

it is reduced

to the.

Thome [ 181 result in the binary case and to the Enskog [ 181 result in the pure metal case, still it may not be an accurate generalization of the Enskog theory for multi-component dense liquids. The Eyring empirical correlation is proven to be extremely accurate in all examined cases. in many cases it is more accurate than the proposed theory; in other cases less. However,

its simplicity

at ttll desired

temperatures.

$inci they constitute essentially the mean value of all .available experimental-data for the metals to which -they-pert&n. When reliable extirimental data are not ‘/-._’

:

,,i. ‘

.‘~

Associates.

References

I l] P. Protopapas and N.A.D. Parlee. On a Theory for the

Derivation of a Single HardSphere Potential describing the overall mokcular interactions in binary liquid metal

alloys as a function of temperature and composition. High Temp. Sciena, submitted for publication. I 21 D. Enskog and K. Svenski. Vet-Akad. Handl. 63. no 4 (1921). , (31 J.V. Hirschfelder. CF. Curtis and M.B. Bird, Molecular theory of gases and Liquids (Wiley. New York, 1964). [4 j D. Enskog. Arkiv. Math. Astronomi o Fys. 16 (1922) 16. [S 1 P.Protopapas. H.C. Anderseir Bnd N.A.D. Parlee. J. Chcm. Phys. 59 (1973) 15. [6] P. Protopapas. Derivation and testing of Lo &stical mechanical models for the calculation’of the hard-sphere

diameter and the self-diffusion coeffkient of liquid metats as functipns of temperature.. ht. Sci. Ihcsis in Mincml

:--

[i]

Engineaing~Stadord Universiiy (1972) University Mi-crofdms Order No. N4460. P. Protbpapas and N;A.D. Pa&e. Hi& Temp. !5ci&e 6

.._

Howevei,

such data arenot always available; The experimental data of table 2 are recommended as quite reliable

:‘

This study was made possible, thanks to generous support, by the National Science Foundation and the Division of Earth Sciences & Technology of J.J. Davis

makes it preferable

to-the proposed theory when accurate experimental data.foi.the viscosity of the pure component metals, -are available,

The authors wish to acknowledge Dr. B.J. Alder for a helpful discussion about the pair correlation function at contact between dissimilar molecules, and for bringing to their attention the paper of reference [ 141. The various contributions by Professor Hans C. Andersen of the Department of Chemistry at Stanford, through innumerable consultations, have been invaluable. The voluntary assistance of Miss Julie Coombes, for recording the computer input data for many of the numerous studied alloys, is deeply appreciated.

/. I.

[fii ~~.‘%k. PbYS.5~

.-

D.hl~:~t&andTB. Wahwrlght, 3. C+m. (1970)3813.~ _ -,

191P. Protopapik. H.C_ An&r+ Phys. 8(1975),17:

:- -‘

and’N.A.@.

I’

:.

Parke. Chcin..

.‘ -. _ [lo] .P. Prolopa& L+titJoin and\testJng of a r&tJitJcaJt&h~~ - anid ih&jiiOf +ris++n JJqriia’~+etals,W.D. DJnerta,. . . . ,._i_..FT_ _-_ -. Z.” 1.

P. Roropapas.

[ I11 1121 [13] 1141 (IS]

[ 16) 117) [IS]

(191 [ZOl 1211

N.A. D. Parleell?wory

lion, Department of Applied Earth Sciences, Stanford University (1974). University hlicrofii. Elliot and Clciscr, Vol. I: and Elliott, Glciser and Ramttkrishna. Vol. II: Thermochcmistrv tmd Steclmaking (AddisokWesle>, Rcsdittg. hlass.-and London). M.K. Tham and K.E. Cubbins. J. C’hem. Phys. 55 (1971) 268. J.L. Lebowitz. Phys. Rev. 133 (1964) A895. G.A. hlensoori. N.F. Cartxthsn. K.E. Starling and T.W. Lelartd, J. Chcm. Phys. 54 (1971) 1523. B.J. Alder. J. Chem. Phys.40 (1964) 2742. A. Rotcnberg. J. Chcm. Phys. 43 11965) 4377. E.B. Smith zmd K.R. Lea, Nzturc 186 (1960) 714:Trans. ramday Sot. 59 (1963) 1535. S. Ch;lpmul and T.G. Cowling. The Mathematical Theory of Non-Uniform Gases (Cambridge University Press, 1970). N.F. Car&an and K.E. Starling, J. Chcm. Phyr 51 (1969) 635. B.J. Alder and T.E. Wainwright. J. Chcm. Phys. 33 (1960) 1439. L. Vcrlet and J. Wcis, Phys. Rev. A5 (1972) 939; petsonal communicstion.5.

of rransport in l&rid

rnerak. IfI

215

[22l

D. Frisch. H. Eyringand J. Kinctid. J. Appl. t’hys. LL (1940) 75. 1231 R.J. Moore. P. Gibbs and H. Eyring. J. Phys. Chem. $7 (19531 .~._ _ 172. _ _ 124) F.A. Kan& and R.P. Colbum. Phys. Cbem. Liquids 1 (1968) 159. [ZS) H.J. Fisher and A. Phillips J. hktals (19.54) 1060. 1261 A.F. Cnwlcy. whys. Chcm. Liquids 2 (1970) 77. 127) M. Pluss, Zeitschrift fur ~norganische und aUgemcine Chcmic93 (1915) 1. 281 A. Bienias and F. Saucrwtld. Zeitsdwift fur anorganische und ttJJgcmcine Cbemic 161 (J927)SI. 291 B.J. AJdcr, W.E. Alley and J.H. Dymond. J. Ckm. Phys. 61 (1974) 1915. 301 D. Oftc snd LJ. Wittenbcrg. Trans. Metal. Sac. AIME 227 (1963) 706. 311 K. Cuing and F. Sauerwald. Zcitschrift fur artorganische und allgemeitte Chcmie 223 (1935) 2M. (321 T.W. Chapman. AICHE J. 12 (1966) 395. 133) R.E. Botz and G.L. Turc, eds. Handbook of Tables for Applied Engineering Scicncc. 1s~ Ed. (Chemical Rubber co., 1970).