Equation of State for Systems Containing Chainlike Molecules

3058

Ind. Eng. Chem. Res. 1998, 37, 3058-3066

Equation of State for Systems Containing Chainlike Molecules Honglai Liu and Ying Hu* Thermodynamics Research Laboratory, East China University of Science and Technology, 130 Meilong Road, Shanghai 200237, China

An equation of state for systems containing chainlike molecules is developed. The equation consists of three contributions. First, as a reference, we adopt the equation for hard-spherechain fluids (HSCF) derived through the r-particle cavity-correlation function of sticky hard spheres. The HSCF equation has a simple form similar to the Carnahan-Starling equation and has been satisfactorily tested by computer-simulation results for chain fluids covering a wide range of chain length. A new mixing rule is derived when the HSCF equation is used for mixtures. Second, as a perturbation, we use directly the square-well potential (SW). Finally, for molecules with specific oriented interactions such as hydrogen bonding, we introduce contribution from chemical association (ASS) that comes naturally from the sticky model used in deriving the reference equation. The equation of state is widely tested by fitting experimental vapor pressures and pVT relations for pure substances, as well as vapor-liquid equilibria, excess enthalpy, and liquid-liquid equilibria for fluid mixtures containing ordinary molecules and chainlike macromolecules. Introduction In recent years, much attention has been given toward the development of equations of state for an assembly of chainlike molecules in the free-space (offlattice) form (e.g., Wertheim, 1987; Dickman and Hall, 1988; Chapman et al., 1990; Walsh and Gubbins, 1990; Chiew, 1990; Li and Chiew, 1994; Yethiraj and Hall, 1991; Amos and Jackson, 1992; Chang and Sandler, 1994; Phan et al., 1993, 1994; Johnson et al., 1994; Ghonasgi and Chapman, 1994; Song et al., 1994; Bokis et al., 1994; Hino et al., 1996). We have made efforts with Professor J. M. Prausnitz in the same line and developed a hard-sphere-chain fluids (HSCF) equation (Hu et al., 1996) through the r-particle cavity-correlation function of sticky hard spheres based on the model of Stell et al. (Cummings and Stell, 1984, 1985; Stell and Zhou, 1989; Zhou and Stell, 1992). The HSCF equation has a simple form similar to the Carnahan-Starling equation and has been satisfactorily tested by computersimulation results for chain fluids covering a wide range of chain lengths. Later, we developed a practical equation of state (Liu and Hu, 1996, 1997) by combining the HSCF equation with a square-well (SW) perturbation term from Alder et al.’s work (1972). For ordinary fluids and fluid mixtures, nonideality is usually attributed to physical forces between molecules. However, for carboxylic acids, alcohols, phenols, amides, water, and their mixtures, as well as some mixtures of nonassociating substances such as acetone and chloroform, self-association or cross-association occurs between molecules due to hydrogen bonding. In these cases, chemical theories of association are always adopted in addition to the physical interactions to account for the nonideal behavior. Earlier work based on equations of state was almost exclusively using a phenomenological approach (e.g., Heidemann and Prausnitz, 1976; Hu et al., 1984; Hong and Hu, 1989). Recently, there has been substantial progress in the * To whom the correspondence should be addressed. Email: [email protected].

statistical mechanical theories of association that provide deeper insight into the structure of fluids containing associated molecules. SAFT equation (Chapman et al., 1989, 1990; Huang and Radosz, 1990, 1991) and shield-sticky theory (Zhou and Stell, 1992) were two examples. When the sticky model is adopted, the above HSCF equation has also been extended to fluids containing associated molecules (Liu et al., 1997). In this work, we present a practical equation of state for chainlike molecules either associated or nonassociated. The equation consists of three contributions: a HSCF equation as a reference, a SW term as a perturbation, and an association term (ASS) that comes naturally from the HSCF equation accounting for the specific interactions. A new mixing rule is derived when the equation is used for fluid mixtures. The equation of state is widely tested by fitting experimental vapor pressures and pVT relations for pure substances, as well as vapor-liquid equilibria, excess enthalpies, and liquidliquid equilibria (LLE) for fluid mixtures containing ordinary molecules or chainlike macromolecules. For LLE calculations of polymer solutions, the usual procedure does not work; therefore, special measure should be taken for the parameters of pure polymers. Satisfactory results have been obtained. Equation of State for Pure Fluids Real fluids can be approximated by square-well-chain fluids with association between molecules. The residual Helmholtz function Ar and corresponding compressibility factor Z can be expressed as sums of hard-spherechain fluid, square-well perturbation, and association terms:

Ar ) Ar(HSCF) + ∆A(SW) + ∆A(ASS)

(1)

Z ) Z(HSCF) + Z(SW) + Z(ASS)

(2)

HSCF Reference Term. The residual Helmholtz function and the compressibility factor for homonuclear HSCFs have been derived through the r-particle cavity-

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Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3059

correlation function (CCF) based on a sticky-point model for chemical association. The r-particle CCF is approximated by a product of effective two-particle CCFs accounting for correlations in nearest-neighbor and next-to-nearest-neighbor segment pairs. The density dependence of the effective two-particle CCF for the nearest-neighbor pair is derived from the rigorous Tildesley-Streett equation (1980). The density dependence of the effective two-particle CCF for the next-tonearest-neighbor pair is obtained by fitting computer simulation data of compressibility factors for linear homonuclear hard-sphere trimers. The CarnahanStarling equation is used for the corresponding monomer system. The final residual Helmholtz function and the compressibility factor for HSCFs are:

βAr(HSCF) (3 + a - b + 3c)η - (1 + a + b - c) ) + N0 2(1 - η) 1+a+b-c + (c - 1) ln(1 - η) (3) 2(1 - η)2 Z(HSCF) )

2

3

1 + aη + bη - cη (1 - η)3

(4)

where η ) πF0rσ3/6 is the reduced density; a, b, and c are functions of chain length:

( r -r 1a + r -r 1 r -r 2a ) r-1 r-1r-2 b ) r(1 + b + b) r r r r-1 r-1r-2 c ) r(1 + c + c) r r r

a)r1+

a2 ) 0.456 96

2

3

(5)

2

3

(6)

2

3

(7)

b2 ) 2.103 86

a3 ) -0.747 45

b3 ) 3.496 95

c2 ) 1.755 03 (8) c3 ) 4.832 07 (9)

As shown from those equations, the HSCF equation has a simple form similar to the Carnahan-Starling equation. It can excellently predict compressibility factors and second virial coefficients for homonuclear HSCFs, including ring and branch molecules covering a wide range of chain lengths, as well as for HSCF mixtures (Hu et al., 1996; Liu and Hu, 1996). SW Perturbation Term. For the perturbation, we directly use the power series obtained by Alder et al. (1972) for a square-well fluid. The Helmholtz function and the compressibility factor arising from square-well interactions between segments are given by

β∆A(SW) N0

9

)r 9

Z(SW) ) r

4

Amn(3x2/π)mηmT ˜ -n ∑ ∑ m n

(10)

4

mAmn(3x2/π)mηmT ˜ -n ∑ ∑ m n

(11)

where T ˜ ) kT/ is the reduced temperature,  is the depth of the square well, and Amn’s are numerical coefficients. Association Term. Assuming associated dimers with bond length L ) σ formed between molecules, the corresponding contributions to the Helmholtz function

and compressibility factor are given by (Liu et al., 1997)

β∆A(ASS)/N0 ) ln(1 - R) + R/2 (ASS)

Z

(

(2e)

)

∂ ln ySiSi+1 1 )- R 1+η 2 ∂η

(12) (13)

where the degree of association R is expressed as

R ) [(2F0∆ + 1) - x1 + 4F0∆]/2F0∆

(14)

∆ ) πωσ3τ-1yS(2e) /3 iSi+1

(15)

where the sticky parameter τ-1 ) eβδ - 1; δ is the sticky or association energy; ω is a surface-fraction parameter responsible for association; and yS(2e) is the iSi+1 effective nearest-neighbor correlation function calculated by

) ln yS(2e) iSi+1 (3 - a2 + b2 - 3c2)η - (1 - a2 - b2 + c2)

+ 2(1 - η) 1 - a2 - b2 + c2 - (1 + c2) ln(1 - η) (16) 2(1 - η)2

Equations 12 and 13 have been satisfactorily tested by computer-simulation data for an associated hard-spheredumbbell fluid (a mixture of hard-sphere dimers and linear hard-sphere quadrimers) at different associating strengths (Liu et al., 1997). Parameters. There are three molecular parameters in this model for nonassociating pure substances, namely, chain length r, cube of segment diameter σ3 and squarewell interaction energy between segments /k. They are regressed from vapor pressures and saturated liquid volumes of pure fluids. For polymers, the ratio of chain length and molar mass r/M is taken to replace the chain length r. Parameters of pure polymers are estimated from densities of melting polymers at different temperatures and pressures. For associating pure substances, there are four molecular parameters that should be regressed, namely, chain length r, square-well interaction energy between segments /k, surface-fraction parameter ω, and association energy δ/k. The cube of segment diameter σ3 is set equal to 19.0 × 10-3 L‚mol-1 referring to alkanes. Theoretically, there are other interactions such as dipole, quadrupole, etc., that have not been considered in the three contributions. Practically, because the parameters are obtained by fitting experimental data, the influences of those interactions are included in those parameters. Results. Figures 1-4 show comparisons of calculated saturated vapor pressures and liquid volumes for cyclopentane, carbon dioxide, methanol, and water, respectively, with their experimental results (Smith and Srivastava, 1986; Angus et al., 1971; Daubert and Danner, 1989). Except for the region near the critical point, the correlation is quite satisfactory. Figures 5and 6 are comparisons of calculated specific volumes for poly(ethyl methacrylate) (PEMA) and poly(ether ether ketone) (PEEK) with experimental data (Rodgers, 1993) at a wide range of temperatures and pressures. Excellent results are also obtained. Equation of State for Fluid Mixtures For fluid mixtures, the Mansoori-Carnahan-Starling-Leland equation (Mansoori et al., 1971) was used

3060 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998

Figure 4. Vapor-liquid equilibria of water. Figure 1. Vapor-liquid equilibria of cycloheptane.

Figure 5. Specific volumes of PEMA. Figure 2. Vapor-liquid equilibria of carbon dioxide.

Figure 3. Vapor-liquid equilibria of methanol.

in our previous work (Liu and Hu, 1997; Liu et al., 1997) for the hard-sphere reference. The merit is that no mixing rule is needed for HSCF mixtures. However, the final equation is then very tedious. In this work, we derive a new mixing rule for HSCF mixtures; as a result, the simple versions of eqs 3 and 4 can still be used. The square-well perturbation term and the association term for mixtures are the same as those in our previous work.

HSCF Reference Term. Equations 3 and 4 are still valid with η ) ∑iπFi0riσi3/6. Parameters a, b, and c are calculated by the following mixing rules:

( ( (

) ) )

a)

∑i xiri 1 +

ri - 1 ri - 1 ri - 2 a2 + a3 ri ri ri

(17)

b)

∑i xiri 1 +

ri - 1 ri - 1 ri - 2 b2 + b3 ri ri ri

(18)

c)

∑i xiri 1 +

ri - 1 ri - 1 ri - 2 c2 + c3 ri ri ri

(19)

Details of derivation of these equations are shown in Appendix A. SW Perturbation Term. Equations 10 and 11 are used with the ordinary mixing rules for the reduced temperature and corresponding parameters:

T ˜ -1 )

K K

K K

∑ ∑φiφj(ij/kT)σij3/∑ ∑φiφjσij3 i)1 j)1 i)1 j)1

ij ) (1 - kij)(ij)1/2, σij ) (1 - lij)(σi + σj)/2

(20) (21)

where kij and lij are adjustable binary interaction

Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3061

Figure 6. Specific volumes of PEEK.

parameters, and φi is the segment fraction of component i.

φi ) xiri/r, r )

∑i xiri

Figure 7. Vapor-liquid equilibria of a hexanes-ethylbenzene system at 338.15 K. k12 ) 0.006 375 0 and l12 ) 0.0.

(22)

Association Term. For mixtures, contributions to the Helmholtz function and the compressibility factor due to association can be derived through chemical association theory as follows:

β∆A(ASS)/N0 )

Z(ASS) )

∑i xi[ln Xi + (1 - Xi)/2]

( )( ) 1

∑i xi X

-

1

2

i

∂Xi

F0

(23)

(24)

∂F0

where

Xi ) (1 +

∑i Fj0Xj∆ij)

-1

(25)

(2e) 3 Fj0∆ij ) πωijτ-1 ij σij Fj0ySiSi+1/3

( ) { [( ) (

F0

∂Xi ∂F0

) -Xi2

∂Xj

∑j Fj0∆ij F0 ∂F

Figure 8. Vapor-liquid equilibria of a 1,1-dichloroethanetetrachloromethane system at 101.325 kPa. k12 ) 0.023 062 5 and l12 ) 0.0.

(26)

+

0

Xj 1 + η

∂ ln yS(2e) iSi+1 ∂η

)]}

(27)

Mixing rules should be used for the cross association between molecules of different kinds,

ωij ) (ωii + ωjj)/2, δij ) (δii + δjj)1/2, σij ) (σi + σj)/2 (28) Binary VLE for Ordinary Mixtures. Figures 7-10 show comparisons of calculated binary vapor-liquid equilibria for hexanes-ethylbenzene at 338.15 K, 1,1dichloroethane-tetrachloromethane at 101.325 kPa, ethanol-1,2-dichloroethane at 333.15 K, and methanolwater at 333.15 K, respectively, with corresponding experimental data (Gmehling, 1977). Satisfactory re-

Figure 9. Vapor-liquid equilibria of a ethanol-1,2-dichloroethane system at 333.15 K. k12 ) 0.052 650 3 and l12 ) 0.0.

sults are obtained with one adjustable interaction parameter kij for both associated and nonassociated systems. Binary Excess Enthalpies for Ordinary Mixtures. Figures 11 and 12 show respectively correlations of excess enthalpies for dichloromethane-tetrachloro-

3062 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998

Figure 10. Vapor-liquid equilibria of a methanol-water system at 333.15 K. k12 ) -0.089 301 2 and l12 ) 0.0.

Figure 13. Liquid-liquid equilibria of a PS-PBD blend. PS: Mn ) 1900, Mw/Mn ) 1.06; PBD: Mn ) 2350, Mw/Mn ) 1.13; k12 ) 0.001 97, l12 ) 0.0.

Figure 11. Excess enthalpy of a dichloromethane-tetrachloromethane system at 298.15 K. k12 ) 0.044 621 and l12 ) 0.0. Figure 14. Liquid-liquid equilibria of a PS-PBD blend. PBD: Mn ) 2350, Mw/Mn ) 1.13; PS: Mn ) 3302, Mw/Mn ) 1.06; k12 ) 0.001 97, l12 ) 0.0.

Figure 12. Excess enthalpy of a cyclopentane-cycloheptane system at 288.15 K. k12 ) -0.016 246 and l12 ) -0.017 866.

methane at 298.15 K and cyclopentane-cycloheptane at 288.15 K (Christensen, 1984). One binary interaction parameter is used for the former, which has positive excess enthalpies. Two adjustable binary interaction parameters must be used for the later, which exhibits both positive and negative values at different composition ranges.

Binary Polymer Blends. Figure 13 shows a comparison of calculated liquid-liquid equilibria for a polystyrene (PS)-polybutadiene (PBD) blend with experimental data (Park and Roe, 1991). Figure 14 shows calculated results for the same system with different molar masses (Roe and Zin, 1980). One temperatureindependent binary interaction parameter is used. Binary Polymer Solutions. For polymer solutions, when the parameters estimated from pVT data of pure melting polymers are used, the liquid-liquid equilibria cannot be satisfactorily correlated. The reason may be that the configurations of polymers in a solvent especially in a nonsolvent are notably different from those in the pure melting state. The molecules in the former state are usually much less extended than those in the latter. For practical purposes, it will be reasonable to incorporate some mixture information into the parameters of pure polymers. We then estimate the squarewell interaction energy /k and a new parameter v* ) rσ3/M from the specific volume of a melting polymer at different temperatures and zero pressure. The ratio of chain length and molar mass of polymer r/M is determined by a reference experimental liquid-liquid critical point of a polymer solution. We adopt lower critical solution temperature (LCST) of a polystyrene (Mw )

Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3063

Figure 15. Liquid-liquid equilibuia of PS-methyl ethyl ketone systems.

Figure 17. Liquid-liquid equilibria of PS-cyclopentane systems.

Figure 18. Liquid-liquid equilibria of PS-cyclohexane systems. Figure 16. Liquid-liquid equilibria of PS-methylcyclohexane systems.

Table 1. Binary Interaction Parameters for PS-MEK and PS-Methylcyclohexane Systems LCST

37 000)-cyclohexane solution as that reference. We then found the following parameters for polystyrene:

r/M ) 0.005 668 9, σ3 ) 134.8434 mL/mol, /k ) 369.467 K Figures 15-18 show comparisons of calculated LCST liquid-liquid equilibria for PS-methyl ethyl ketone and PS-methylcyclohexane solutions and upper critical solution temperature (UCST) liquid-liquid equilibria for PS-cyclopentane and PS-cyclohexane solutions with experimental results (Saeki et al., 1973) where PS has different molar masses. Two temperature-independent interaction parameters k12 and l12 are used. Values are listed in Tables 1 and 2. Although the parameters of PS are obtained by incorporating the information of a mixture with a special solvent (cyclohexane) and a special molar mass (37 000), the correlation for systems with different solvents and different molar masses is quite satisfactory. Discussion and Conclusions The equation developed has a sound theoretical basis because the contributed terms, the HSCF term, the SW term, and the ASS term can all stand the tests of

methyl ethyl ketone

methylcyclohexane

Mw

k12

l12

k12

l12

37 000 97 200 200 000 400 000 670 000 2 700 000

-0.1500 -0.1800 -0.1950 -0.2100 -0.2160 -0.2250

0.4950 0.5016 0.5045 0.5064 0.5073 0.5094

0.0200 -0.0050 -0.0225 -0.0370 -0.0450 -0.0500

0.3416 0.3675 0.3779 0.3858 0.3882 0.3917

Table 2. Binary Interaction Parameters for PS-Cyclopentane and PS-Cyclohexane Systems UCST cyclopentane

cyclohexane

Mw

k12

l12

k12

l12

37 000 97 200 200 000 400 000 670 000 2 700 000

0.0835 0.0853 0.0862 0.0871 0.0873 0.0885

0.2100 0.2100 0.2100 0.2100 0.2100 0.2100

0.0748 0.0754 0.0757 0.0761 0.0762 0.0769

0.2100 0.2100 0.2100 0.2100 0.2100 0.2100

computer simulation results. However, the equation is still semiempirical in nature. It is engineering oriented and is designed for practical purposes. As we have mentioned, we used cavity-correlation functions in

3064 Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998

deriving the HSCF equation. Logically, we should use the same idea for a square-well fluid. However, this would introduce complexities that could not be accepted for practical calculations. On the other hand, suggestion of a dimer for the association is also an oversimplified picture in some occasions. However, considering a more delicate geometry for association would cause unbearable labor. As for LLE calculations of polymer solutions, parameters of polymers should be revised by considering some information of mixtures. Definitely, this measure will deteriorate the calculation of densities of pure polymers. Therefore, the equation developed is a compromise between theoretical rigor and practical efficiency. The above demonstration examples indicate that the equation can be used for a wide range of phaseequilibrium calculations with practical interest. If a temperature-dependent parameter is used, the correlation can be notably improved in some cases. The equation will be helpful for process design and development concerning systems containing both ordinary molecules and polymers. The details of expressions for chemical potentials that are crucial in calculations are shown in Appendix B. Acknowledgment This work was supported by the National Natural Science Foundation of China (29236131, 29506044), the Doctoral Research Foundation of the National Education Commission of China, and the Simulation Sciences Inc. in Brea, USA. Nomenclature A ) Helmholtz function a, b, c ) constants dependent on chain length k, l ) binary interaction parameters M ) molecular weight N0 ) number of molecules p ) pressure, kPa or bar r ) chain length T ) temperature, K V ) molar volume, L/mol v ) specific volume of polymer, mL/g X ) monomer molar fraction x ) liquid-phase molar fraction y ) cavity-correlation function Z ) compressibility factor Greek Symbols R ) degree of association β ) 1/kT ∆ ) strength of interaction  ) square-well interaction energy between segments δ ) association energy φ ) segment fraction η ) reduced density µ ) chemical potential F0 ) molecule number density σ ) segment diameter τ ) sticky parameter ω ) association fraction parameter Superscripts ASS ) contribution due to association HSCF ) contribution of hard-sphere chain fluid

r ) residual properties SW ) square-well perturbation term

Appendix A: Mixing Rules In previous work (Liu and Hu, 1997), the residual Helmholtz function and compressibility factor for chain fluid mixtures are expressed as

βAr(R)0)

βAr

) r˜

N0

K

-

r˜ N0

xiri ln yS(2e) ∑ S i)1

j j+1(i)

-

K

xiri ln yS(2e) ∑ S i)1

j j+2(i)

(A-1)

Z ) r˜ Z(R)0) -

[

K

r˜ - 1 +

xiriη ∑ i)1

d ln yS(2e) jSj+1(i)

K

+

dη

xiriη ∑ i)1

]

d ln yS(2e) jSj+2(i) dη

(A-2)

where Ar(R)0) and Z(R)0) are the residual Helmholtz function and compressibility factor for hard-sphere fluid mixtures, respectively. For a hard-sphere-chain fluid mixture, the effective nearest-neighbor correlation function yS(2e) and its derivative are calculated by eq 16 jSj+1(i) with η ) ∑iπFi0riτi3/6. The next-to-nearest-neighbor correlation function yS(2e) and its derivative are caljSj+2(i) culated by

) ln yS(2e) iSi+2(i)

[

ri - 1 (-a3 + b3 - 3c3)η - (-a3 -b3 + c3) + ri 2(1 - η) -a3 - b3 + c3 2(1 - η)2

]

- c2 ln(1 - η) (A-3)

If the Carnahan-Starling equation of state is used for hard-sphere fluid mixtures,

βAr(R)0) 3η - 1 1 + ) r˜ N0 1-η (1 - η)2 Z(R)0) )

βp 1 + η + η2 - η3 ) rF0 (1 - η)3

(A-4)

(A-5)

Substituting eqs A-3-A-5 and 16 into eqs A-1 and A-2, we recover the simple version of eqs 3 and 4 for the residual Helmholtz function and the compressibility factor with a, b, and c calculated by eqs 17-19. Appendix B: Expressions for Chemical Potentials Thermodynamics gives

βµk ) (∂(βA)/∂Nk0)T,V,Nl0(l*k)

(B-1)

The chemical potential of a component k in a mixture can then be derived. The resulting equation is as follows:

βµk ) ln(βFk0) + βµrk

(B-2)

() (

)

Ind. Eng. Chem. Res., Vol. 37, No. 8, 1998 3065

where

-1

rN0 βµrk ) βµ(HSCF) + β∆µ(SW) + β∆µ(ASS) k k k

(B-3)

βµ(HSCF) ) k (3 + ak - bk + 3ck)η - (1 + ak + bk - ck) 2(1 - η) 1 + ak + bk - ck + (ck - 1) ln(1 - η) + 2(1 - η)2 5+a-b+c + 2(1 - η) (3 + a - b + 3c)η - (1 + a + b - c) + 2(1 - η)2 3 1 + a + b - c ηrkσk

+

[

(1 - η)3

]∑

∂(T ˜ ) ∂Nk0

) 2rkT ˜ -1

N0

∂Xi ∂Nk0

Xj (B-4)

xj

xjrjσj

)

(

bk ) rk 1 +

)

rk - 1 rk - 1 rk - 2 b2 + b3 rk rk rk

(

)

rk - 1 rk - 1 rk - 2 ck ) rk 1 + c2 + c rk rk rk 3

[ [

9

)

rN0 ∂(β∆A(SW)/rN0)

(B-5)

9

(B-7)

∂(β∆A(SW)/rN0)

∑ ∑mAmn(3x2/π)mηm-1T˜ -n m n

∂η

9

(B-9)

4

)

∂(T ˜ )

Xi

2

N0

∂Xi

∂Nk0

(B-13) ∂Xj

+

k0

∂ ln yS(2e) iSi+1

rkσk3

∂η

∑l xlrlσl

)]}

(B-14)

3

{

0 j*k 1 j)k

(B-15)

2(1 - η)2 1 - a2 - b2 + c2 (1 - η)3

+ (B-16)

Literature Cited

4

nAmn(3x2/π)mηmT ˜ -n+1 ∑ ∑ m n

-1

1

-

5 - a2 + b2 - c2 + ) ∂η 2(1 - η) (3 - a2 + b2 - 3c2)η - (1 - a2 - b2 + c2)

(B-6)

∑j

)

[( ) ] 1

∂ ln yS(2e) iSi+1

]( ) ]

∑ ∑Amn(3x2/π)mηmT˜ -n m n

+η

δjk )

∂(β∆A(SW)/rN0) ∂(T ˜ -1) β∆A(SW) β∆µ(SW) ) r + r N + k k 0 rN0 ∂Nk0 ∂(T ˜ -1) ∂(β∆A(SW)/rN0) ηrkσk3 (B-8) r ∂η xjrjσj3

β∆A(SW)

∑i xi

∑j Fj0∆ij N0 ∂N δjk

∑i ∑j φiφjσij3

(B-12)

{ [ (

) -Xi2

j

(

∑i ∑j

∑i φiσik3

-

φiφjijσij3

1 ) ln Xk + (1 - Xk) + β∆µ(ASS) k 2

3

rk - 1 rk - 1 rk - 2 a2 + a3 a k ) rk 1 + rk rk rk

∑i φiikσik3

(B-10)

4

(B-11)

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Received for review November 14, 1997 Revised manuscript received February 27, 1998 Accepted February 28, 1998 IE9708034