14 Equations of state for polymer systems

Equations of State for Fluids and Fluid Mixtures J.V. Sengers, R.F. Kayser, C.J. Peters, H.J. White Jr. (Editors) © 2000 International Union of Pure and Applied Chemistry. All fights reserved

14 EQUATIONS OF STATE FOR POLYMER SYSTEMS Stephen M. Lambert a, Yuhua Song and John M. Prausnitz b

Department of Chemical Engineering and Chemical Sciences Division Lawrence Berkeley Laboratory University of California, Berkeley, CA 94720, U.S.A. 14.1 Introduction 14.1.1 Overview 14.1.2 Organization 14.2 Equations of State for Pure Polymer Liquids 14.2.1 Cell Models 14.2.1.1 Prigogine's Cell Model (PCM) 14.2.1.2 Flory-Orwoll-Vrij (FOV) 14.2.1.3 Perturbed-Hard-Chain (PHC) 14.2.1.4 Further Modifications and Discussion 14.2.2 Lattice-Fluid (LF) Models 14.2.2.1 Sanchez-Lacombe (SL) 14.2.2.2 Costas and Sanctuary (CS) 14.2.2.3 Panayiotou-Vera (PV) 14.2.2.4 Mean-Field Lattice-Gas (MFLG) 14.2.2.5 Further Modifications and Discussion 14.2.3 Hole Models 14.2.3.1 Simha-Somcynsky (SS) 14.2.3.2 Further Modifications and Discussion 14.2.4 Tangent-Sphere Models 14.2.4.1 Generalized Flory Theories 14.2.4.2 Statistical Associated-Fluid Theory (SAFT) 14.2.4.3 Perturbed Hard-Sphere-Chain (PHSC) 14.3 Extension to Mixtures 14.3.1 Flory-Orwoll-Vrij (FOV) 14.3.2 Perturbed-Hard-Chain (PHC) 14.3.3 Lattice-Fluid (LF) 14.3.4 Mean-Field Lattice-Gas (MFLG) 14.3.5 Simha-Somcynsky Hole Model 14.3.6 Statistical Associated-Fluid Theory (SAFT) 14.3.7 Perturbed Hard-Sphere-Chain (PHSC) 14.3.8 Further Developments of Sphere-Chain Models 14.3.9 Cubic Equations of State for Polymer Mixtures 14.3.10 Associating Systems (Specific Interactions) 14.4 Conclusion References a Present address: PE Biosystems, 850 Lincoln Centre Drive, Foster City, CA 94404 b To whom correspondence should be addressed

523

524 14.1 INTRODUCTION 14.1.1 Overview This chapter provides a summary of equations of state (EOS) for molten polymers and for mixtures of polymers with solvents or other molten polymers. This review compares their theoretical foundations and indicates their usefulness for calculating thermodynamic properties, especially for phase equilibria. Attention is restricted to polymer liquids. No significant attention is given here to glassy polymers or to crystallinity. To describe polymers and their mixtures with solvents and other polymers, equations of state can be divided into four categories: cell models, lattice-fluid models, hole models and tangent-sphere models. The cell and lattice-fluid models provide different adaptations of the incompressible-lattice model of polymer mixtures; however, each incorporates compressibility in a different manner. Hole models combine both methods of incorporating compressibility introduced by cell and lattice-fluid models. Finally, recent advances in statistical thermodynamics have brought to the forefront tangent-sphere models of chain-like fluids. These models abandon lattice origins; they model polymers as freely-jointed tangent-spheres where unbonded spheres interact through a specified intermolecular potential. 14.1.2 Organization This chapter is organized as follows: Section 14.2 discusses the theoretical basis of various EOS for pure liquid polymers. Attention is restricted to amorphous polymers above their glass transition and crystallizable polymers above their melting temperatures. Emphasis is given to the description of pressure-volume-temperature (p VT) properties because these are a primary source of pure-component parameters. Extension of these EOS to mixtures is discussed in Section 14.3. Where possible, quantitative comparisons are cited from literature on various applications of EOS in interpreting, correlating and predicting thermodynamic properties and phase equilibria of mixtures containing polymers. The final section summarizes achievements and deficiencies of equations of state for polymers and polymer solutions. The technical literature pertaining to EOS is very large and increasing at a rapid rate. Therefore, this review is in no sense exhaustive nor is it comprehensive or complete. The authors have necessarily been forced to select those published articles, which, in their view, are most important while neglecting others that different authors might have selected. The authors, therefore, request the reader's sympathetic understanding if some worthy published articles have been omitted as a result of imperfect judgment and because of the inevitable need to keep this review from exceeding a reasonable length. The manuscript for this review was completed in early 1996. 14.2 EQUATIONS OF STATE FOR PURE POLYMER LIQUIDS This section discusses the theoretical foundations of several equations of state to describe the pVT behavior of liquid polymers. Excellent reviews have been presented by Rodgers (1) and Curro (2). The more recent publication by Rodgers (1) focuses primarily on the ability of various EOS to fit p FT data and presents a compilation of pure-component parameters. It briefly describes the theoretical development of a limited number of EOS chosen for parameter regression. Conversely, Curro's review emphasizes the theoretical development and fundamental statistical-mechanical underpinnings of the cell and hole models. Since tangent-

525 sphere models have been presented for polymers only recently, there is no previous review concerning their performance versus that of older models. The review here is somewhere in between, its focus is on the assumptions and approximations that allow the description of chain-like molecules. Compilations of polymer pVT data and EOS parameters are given elsewhere (1,3). Dee and Walsh (4) have compared cell, hole and lattice-fluid models. 14.2.1 Cell Models

14.2.1.1 Prigogine's Cell Model The physical picture of Prigogine's cell model (2,5,6) for pure chain-like molecules is essentially the same as that of the incompressible lattice model: the system contains N molecules each composed of r segments (or "mers") arranged on a lattice (with coordination number z) having a total of rN sites. One molecule occupies r neighboring lattice sites. While the physical view of the cell theory is similar to that of the lattice theory, the cell model differs through its description of inter- and intra-molecular forces. At finite temperatures, each mer is displaced from its central position in the lattice site due to thermal fluctuations. However, its displacement is restricted by spring-like forces exerted on it by mers in neighboring lattice sites to which it is connected and dispersion forces exerted on it by all other mers in the system. As a result, each mer is effectively confined to a "cell" whose volume is characterized by d, the bond-length between successive mers in a chain; R, the (average) distance between nearest-neighbor m e r s of different chains; and R*, the separation distance at the minimum of the (as yet unspecified) pair potential between unbonded mers. At zero temperature, R = d because the mers then all lie at equidistant central positions on the lattice. If we ignore interactions between mers which are not nearest neighbors, R* = R = d because the system must be in the state of lowest energy. While d and R* are molecular constants, R (and hence the cell volume) increases with increasing temperature. Further, since the definition of a segment is somewhat arbitrary, it is convenient to define d = R*. Hence the cell volume is v - R 3 and the characteristic volume of a mer is v* - (R*)3. The total volume of the system is V = Nrv. Since v > v*, free volume is introduced into the lattice model. The equation of state for this system is found from differentiation of the configurational partition function Q

P = kBT(81nQ 8V )r,u

(14.1)

\

where Q can be written as (see, for example, reference 7)

Q=Qcoo[]'tqro,'expl k k~r)

142>

where Vf is the "free volume" available to the center of mass of a molecule as it moves in volume V, A is the de Broglie wavelength, qrot, vib is the rotational and vibrational partition function of the r-mer molecule in the "mean-field" of all other chains in the system, Qcombis the combinatorial factor (the number of ways of arranging N r-mers on a lattice with Nr lattice sites), E o is the potential energy of the system with every mer located at the central position of its lattice site and k B is the Boltzmann constant. Although factoring the partition function in

526 this manner assumes the independence of each contribution, only the combinatorial factor is assumed independent of volume. Because we are interested in the equation of state, we are not concerned here with Qcomb" The temperature and density dependence of qrot,vib is difficult to determine, especially for large polyatomic molecules. The extent to which a large asymmetric molecule can rotate or vibrate depends on the mean-field exerted on it by the surrounding fluid as well as on internal modes unaffected by the surrounding fluid. Consequently, qrot,vib Can be factored into an internal part which depends only on temperature and an external part which depends on volume: qrot, vib =qint (T)qext(V) (14.3) Since the internal part depends only on temperature, it does not appear in the equation of state. To obtain an expression for qextconsideration must be given to the degrees of freedom of large asymmetric molecules. First, consider a rigid r-mer molecule; all bond lengths, bond angles and torsional angles are fixed. This rigid molecule has three degrees of freedom, the same as that for a spherical molecule (one for each translational coordinate). Second, consider an r-mer molecule which is completely flexible (i.e., the molecule does not have any restriction on bond length, bond angle or torsional angle), this molecule has a maximum of 3r degrees of freedom. The number of degrees of freedom for a real large asymmetric molecule lies somewhere in-between. To approximate the number of external degrees of freedom while leaving them unspecified, a parameter c is introduced such that the total number of effective external degrees of freedom per r-mer molecule is 3c, such that 1< ¢ < r. When coupled with Prigogine's approximation that extemal rotational and vibrational degrees of freedom can be considered equivalent translational degrees of freedom, we write

A~

qrot,vib='~-qint qext =qi.t ( T )

144,

where 3(c-1) reflects the number of external rotational and vibrational motions, i.e., those rotational and vibrational motions that are affected by the presence of neighboring molecules. Equations (14.2) and (14.4) summarize the fundamental aspects of the cell model. To proceed further, we need to specify the dependence of free volume Vf and potential energy E 0 on cell volume v. Subsequent variations of Prigogine's cell model differ primarily in their choice of expressions for these quantities. Prigogine's approximation (6) for Vf as a function of reduced cell volume, defined as V - v/v*, is based on a face-centered cubic arrangement of segments around a central segment: Vf=Nrv*7 ( V '/3 - 2-'/6) 3

(14.5)

where 7/is a geometrical constant. The dependence of E 0 on V is derived from assuming various potentials (including harmonic, Lennard-Jones, and square-well potentials) (5,6) between segments of the chain with segments in nearest-neighbor cells. The commonly-used form of Prigogine's cell model is that from the Lennard-Jones potential:

1. ~ A B~ Eo=~vrqz~-~--Uj

(14.6)

527 where e is the well depth at the minimum of the potential. The product qz is given by qz = r ( z - 2 ) + 2 where z is the lattice coordination number; q represents the external surface area of the r-mer. Thus qz is the number of unbonded nearest-neighbor segments surrounding a single chain. Constants A and B are from the Lennard-Jones potential, given a specific geometry (A = 1.2045 and B = 1.011 for a face-centered cubic lattice). Equation (14.6) gives the summation of the potential energy felt by each segment of a chain from surrounding nonbonded nearest-neighbor segments multiplied by the total number of chains. Substituting Equations (14.3)-(14.6) into Equation (14.2) and differentiating according to Equation (14.1) gives the equation of state for Prigogine's cell model (PCM): v

2

--~'1/3 __ 2-1/6 t - - ~

A

(14.7)

--

where ~ , V and 7' are, respectively, reduced pressure, volume and temperature defined as p _ _p_ = prv_____~* v,~= - - v p, qze v*

T = T = ckaT T* qze

(14.8)

and characteristic quantities are denoted by *. 14.2.1.2 F l o r y - O r w o l l - Vrij (FO V)

The Flory-Orwoll-Vrij (FOV) equation of state for pure chain-like fluids (8) is a simplified version of the Prigogine cell model. Instead of using a face-centered cubic geometry, Flory assumes a simple cubic geometry which gives Vf = Nrv* y (~1/3 _ 1)3

(14.9)

where y is a geometrical constant pertaining to a simple-cubic lattice. For the potential energy E o, Flory proposed the simpler expression Eo

=

rNsrl 2v

(14.1 O)

where s is the mean number of external contact sites per segment of molecule and ~7 reflects the mean interaction between a pair of nonbonded segments in the r-mer-chain liquid. Substitution of Equations (14.9) and (14.10) into the partition function yields the equation of state:

ff• v

1 -V1/3_l ~,~

_

"~ 1/3

(14.11)

where the reduced pressure, volume and temperature are defined as:t

t Flory et al. (8) define the total number of external degrees of freedom per molecule as 3rc; hence in the FOV equation, c is defined per segment, whereas in the PCM (Section 14.2.1.1) and PHC (Section 14.2.1.3) equations, c is defined per molecule. This difference in notation affects primarily the definitions of characteristic quantities for each equation of state.

528 p_p_=2(v*)Zp

P=p*

fp = v__v_V/rN

V

v* v * - 7

-

T

2v*ckBT

r=7=

(14.12)

Characteristic parameters are denoted by an asterisk and V = rv . 14.2.1.3 Perturbed-Hard-Chain (PHC) Theory

The Perturbed-Hard-Chain (PHC) theory (9), based on PCM, reexamines the expression used for qrot~vib" Although 3c remains the total number of external degrees of freedom per molecule, the expression used by Prigogine [Equation (14.4)] does not satisfy some important boundary conditions. First, the ideal-gas limit should be obeyed: Vf V lim--vq~xt= (14.13) v-~=A ~ £ Second, when free-volume disappears as the system approaches closest-packing volume V0 the molecule has no external degrees of freedom; therefore, lim V--V~qext f, =0 v~vo A ° Third, for simple fluids (r = c = 1) at all densities, qoxt= 1. Camahan and Starling give an expression for the free volume (10,11)

(14.14)

.

(14.15)

v

L

A

where x = n G / 6 is a numerical constant and V is the reduced volume (14.16) /q~* /qj* while v* is the characteristic hard-core volume per r-mer segment.tt For a large molecule at liquid-like densities, any expression for qext should also obey Equation (14.4). With these boundary conditions in mind, Beret and Prausnitz (9) proposed the function "~v= v = V / N

q,x,_~-~-) _

f

(14.17)

where Vf is given by Equation (14.15). Equation (14.17) meets all necessary boundary conditions. Unlike the expressions used by PCM and FOV, where the potential energy depends only on volume, the expression used by PHC theory depends on volume and temperature. PHC theory uses the potential field calculated by Alder et al. (12) for molecules whose intermolecular forces are represented by the square-well potential: Eo=leq w ( T,V)

(14.18)

tt Note that for the PHC equations, v = V/N is the volumeper molecule, whereas in the PCM and FOV equations v is def'medas the volumeper segment.

529 where the product sq equals the characteristic potential energy per molecule (energy parameter 6 is per segment and q is proportional to the surface area of the molecule); W is an analytical function of reduced volume, V, and reduced temperature, T , defined as (14.19) T* sq The resulting equation of state is

PV 1 ÷4(r/V)-2(r/~)2 (l-r/V) 3 T c where

Anm a r e

1 4 MfmAnml( 1

b~Vn~__lm~__l ~,m-I ~ "~

known coefficients (9) and where the reduced pressure is

_ p p(rv*) p= - -

p*

(14.20)

(14.21)

eq

The PHC EOS is slightly different from other models because r and v* always appear as a product. The three characteristic quantities are c, eq and rv*, rather than r, a / a n d v* which are used in the FOV EOS and PCM EOS. The relationship between c and the characteristic quantities, p*, T* and rv* is

c=p* (rv* )

(14.22)

k.v*

14.2.1.3 Further Modifications and Discussion Despite their differences, the PCM, FOV and PHC equations of state all demonstrate a common feature of most EOS for chain-like fluids: three-parameter corresponding states. In all cases, a pure chain-like molecule is characterized by three quantities: a segment pairinteraction energy, a segment volume and a number of segments per molecule (or in the case of the PHC EOS, the number of external degrees of freedom per molecule). Combinations of these parameters lead to definitions of characteristic volume, temperature and pressure. Pressure-volume-temperature (pVT) data are commonly used to determine these parameters for specific polymers. However, other thermodynamic quantities are o~en used that are related to p VT properties and that are typically measured at low pressures: the coefficient of thermal expansion a, the isothermal compressibility Zr, and the thermal pressure coefficient y = a / Z r .

Expressions for these quantities can be computed using

appropriate thermodynamic identities. For polymers these expressions lead to. shortcut methods for evaluating characteristic parameters. Using the FOV EOS for example, the characteristic volume v* can be computed using experimentally determined specific volume and a at a single temperature T:

v*=v[lq

aT 1-3 3(1+ ~T)J

(14.23)

530 Using the above calculated value of v*, and specific volume at temperature T at a low, typically atmospheric, pressure the characteristic temperature can be evaluated from the zeropressure isotherm of the EOS , 7V4/3 T : ,/3------~_ 1

(14.24)

Lastly, using the isothermal compressibility (or the thermal pressure coefficient) the characteristic pressure can be estimated from ,

p =

(zTv 2

='/TV2

(14.25)

14.2.2 Lattice-Fluid Models 14.2.2.1

Sanchez-Lacombe

Like the cell models, the Sanchez-Lacombe (SL) (13,14) equation of state for chain-like fluids is also based on the incompressible-lattice model but compressibility is introduced in a different manner. The lattice is occupied by both r-mers and vacant lattice sites. The total number of lattice sites in the system, N r , is N r =N O+ r N (14.26) where N Ois the number of vacancies. The close-packed volume of a molecule, assumed to be independent of temperature and pressure, is rv* where v* is the volume of a single mer and is equal to the volume of one lattice site. The total volume of the system is V

=

(No+rN)v*

(14.27) A reduced density is defined as the fraction of occupied lattice sites =-

1

=

pv*

v

=-

v* v

=

rN

(14.28)

N o +rN

where p = N r / V is the density of segments, v = V / N r is the volume per segment and ~ is reduced volume. The energy of the lattice depends only on nearest-neighbor interactions. For a pure component, the only non-zero interaction energy is the mer-mer pair interaction energy 6. Vacancy-mer and vacancy-vacancy interaction energies are zero. The SL model assumes random mixing of vacancies and mers; therefore, the number of mer-mer nearest-neighbors is proportional to the probability of finding two mers in the system. The lattice energy is

(14.29) where G*= z 6 / 2 . The configurational partition function for this system is written

531 Q=QcombeXp(-E/ kB T)

(14.30)

The combinatorial factor Qcombis identical to that of the Flory-Huggins incompressible-lattice partition function (15) where the solvent is replaced by a vacancy

Qcornb=( ~_~N ( N o + r N ) ! No!N!

1

(14.31)

(No+rN) u(r-1)

where # is the flexibility parameter of the r-mer and cr is a symmetry number; both of these are assumed to be constants and therefore do not appear in the equation of state. Substitution of Equations (14.29) and (14.31) into Equation (14.30) and differentiation according to Equation (14.1) yields the Sanchez-Lacombe (SL) lattice-fluid equation of state

.1 .

E1+01n/1111 . .

1

(14.32)

where the reduced (-) and characteristic (*) pressure and temperature are defined by = p_fl_= pv*

f =--_--T _ kBT

(14.33)

T* p* e* e* Equation (14.32) shows that p V T data for polymer liquids are relatively insensitive to polymer molar mass. As polymer molecular weight increases, the 1/r term becomes insignificant. In the limit of infinite polymer molecular weight, Equation (14.32) suggests a corresponding-state behavior for polymer liquids, illustrated in Figure 14.1. The lines for each reduced pressure were calculated using equation (14.32) in the limit r---~ oo. The points are experimental p VT data for several polymers which are reduced by appropriate characteristic parameters. A reduced pressure ~ = 0, is essentially atmospheric pressure; /3 = 0.25 is a pressure the order of 1000 bar.

14.2.2.2 Costas and Sanctuary (C&S) The SL equation of state uses the Flory approximation of lattice thermodynamics. The equation of state developed by Costas and Sanctuary (16,17) utilizes Guggenheim's approximation (18) for Qcombwhich includes chain connectivity through the surface area parameter, q. That parameter is related to the number of mers per molecule and the coordination number z such that zq - r(z-2)+2 is the number of contact sites around a chain, excluding contacts between neighboring mers of the same chain. Guggenheim's approximation for the combinatorial factor is

(14.34)

532

0.96

-

•

~

~.,

0.94 ~

.I~

o.ot 0.84

-

,', P c H M A

~.

• PMMA

0.80

-

0.50

" o

k~

,E = 0

al.

Q PDM5 PPO I

0.55

I

0.60 Reduced

I

0.65

,

,

,

!

,

0.70

,

v

0.7"5

__

Temperature,

Figure 14.1 Corresponding states behavior of polymer p VT data according to the Sanchez Lacombe (Lattice-Fluid) equation of state (14). Abbreviations in legend: PS, polystyrene; PoMS, poly(o-methyl styrene); PcHMA, poly(cyclohexyl methacrylate); PMMA, poly(methyl methacrylate); PnBMA, poly(n-butyl methacrylate); LDPE, low-density polyethylene; PIB, polyisobutylene; PVAc, poly(vinyl acetate); PDMS, poly(dimethyl siloxane); PPO, poly(propylene oxide). Like the SL EOS, the equation of Costas and Sanctuary assumes random mixing, but not random mixing of mers and vacancies; rather, it assumes random-mixing of r-mer and vacancy contact sites. Consequently, the lattice energy is proportional to the probability of finding a pair of nearest-neighbor non-bonded mers:

E=_2Nqe,(_ qN _12=_Nq6,I,o(1-/1,)] 2 LNo +qNJ L(1-A~) where Nq = No + qN is the total number of contact sites and

(14.35)

(14.36) The resulting Costas and Sanctuary (C&S) EOS is

533

)I1E 1 12

l rln(1-1/Zln(1-~ --~ 1-)t/~J

(14.37) ~L k, ~') 2 k. r where pressure, temperature and volume are all reduced by characteristic parameters identical to those defined for the SL equation of state [Equation (14.33)]. The C&S equation of state [Equation (14.37)] reduces to the SL EOS [Equation (14.32)] in the limit z ~ oo or 3.---> 0.

14.2.2.3 Panayiotou and Vera (PV) Panayiotou and Vera (19) use the quasi-chemical approximation (18) to account for nonrandom mixing of holes and mers. In a random mixture, the number of nearest-neighbor vacancy (0)-mer (1) contacts is proportional to the product of their respective number of surface contacts. Hence, in a random mixture:

o:z

N°'

2

q

l N O+qN No l/qN/ N o +qN

z

= -~Nq(1-0)0

(14.38)

where the superscript (o) denotes a random mixture and O=qN/(No+qN ) is the surface fraction of r-mers; 0 is related to the reduced density by

0 =

(q/r)~

1+ (q/r-

(14.39)

1)/5

Due to attractive forces, the number of vacancy-mer contacts decreases as temperature decreases and the number of vacancy-vacancy and mer-mer contacts increases. The quasichemical approximation estimates the effect of temperature and attractive forces on N for a real mixture according to Nol -- N~011-"Ol

;

/-'Ol =

l + 4i - 4(1- O)O[exp@[kBT)- I ]

(14.40)

where F01 < 1 is called the vacancy-mer non-random factor and e is the mer-mer free energy of interaction ~= ch + T6s

(14.41)

which is composed of an enthalpic contribution, oh, and an entropic contribution, cs. The vacancy-vacancy and mer-mer non-random factors,/-'00 and/'11, are related to/-'01 by

OFol + (1-0)Foo

=

1

(14.42)

OF,,

=

1

(14.43)

(1-0)F0,

+

The inclusion of non-random mixing causes two modifications of the lattice-fluid partition function. First, the lattice energy is now a function of temperature as well as the number of vacancies:

534

e

-- - N ~

= -N. rl~

(14.44)

Second, because the number of nearest-neighbor contacts is different from that in a random mixture, the combinatorial factor must also indicate that the number of allowed configurations in the non-random mixture is different from that in the random mixture. The lattice-fluid partition function becomes: Q=QcombQ~exp(-E/kBT)

(14.45)

where Qeombis Guggenheim's approximation for the combinatorial factor for a random mixture [Equation (14.34)] and Q ~ is the contribution to the combinatorial factor due to non-random mixing of vacancies and mers. According to the quasi-chemical approximation, Q ~ takes the form N;°l ! Noo ! Qr~

, 2!

=

(14.46)

N~I! Noo! NOl

!

The resulting Panayiotou and Vera (PV) EOS is p__~v = fin + In ~ + q-/r- 1 _ ZlnFo ° T v 2 The reduced and characteristic parameters are defined by

p _ pvo P=p,-(z/2)~

v ~=

Nrvo

r_ r-r*

k.r

(z/2)e

(14.47)

(14.48)

where vo is the characteristic volume of a lattice vacancy which is assumed constant for all fluids; hence, vo is not an adjustable parameter but there are two adjustable parameters associated with the interaction energy, ~'h and c,, in Equation (14.41). Since e is assumed to be temperature dependent, p* and T* are also temperature dependent. Equation (14.47) reduces to the C&S EOS [Equation (14.37)] if non-randomness in mixing is neglected, i.e., 1-'oo =/~11 =FOl = 1. 14.2.2. 4 Mean-Field Lattice-Gas (MFLG)

The Mean-Field Lattice-Gas (MFLG) (20-23) equation of state parallels the development of the SL equation of state in that a pure fluid is portrayed as a binary mixture of vacant and occupied sites. However, the starting point is not the partition function but an expression for A ~ A, the Helmholtz energy of mixing, proposed by Kleintjens and Koningsveld (24): A =0oln~bo--+T' ln~bld" + g~o~l N~Rr rl

(14.49)

535 where R is the gas constant, ~ the fraction of sites occupied by vacancies of volume v0 , ~bl is the fraction of sites occupied by r~-mers and g is the binary interaction parameter that is a function of composition and temperature:

g=a-~ flO +fill T

(14.50)

1-y~bl where a, flo, and /31 are constants and 2 = 1 - o~/o-0. Here O'l/cr0 is the ratio of surface areas for a segment and a vacancy. In the MFLG model, the vacancy volume v0 is a constant for all fluids. The chain length of a molecule is adjustable or estimated from its molecular volume where estimates for v1 can be obtained from Bondi (25). Likewise, y can also be estimated from Bondi's method assuming that the surface area of a lattice vacancy is given by that of a sphere of volume v0. Hence, the MFLG model contains at least three and a maximum of five adjustable parameters for each component. From Equation (14.49), the equation of state is: PV°=ln#° +/1-~3¢' +#2~-aI RT

( fl° +(1fll y/ Tqkl)(1) 2 Y) 1

(14.51)

Because the MFLG EOS treats a vacancy as a second component, the fraction of occupied sites ~l =l-~bo is identical to the reduced density ,~ in the SL, PV and C&S EOS. While the MFLG EOS has been compared to other lattice-fluid EOS (20) and to cell and hole models (26) for correlating the properties of normal fluids, its ability to correlate the properties of a variety of polymers has not been explored. At present, parameters are available for only a few systems.

14.2.2.5 Further Modifications and Discussion Additional refinements in lattice-fluid theories are concemed with higher-order approximations for lattice thermodynamics. Examples are the Born-Yvon-Green (BYG) approximation (27,28) and the Lattice Cluster theory (29-34). While the BYG approximation has been used to correlate polymer pVT properties (28), the Lattice Cluster theory has not. For a compressible lattice (i.e., lattice with vacancies) a version of the Lattice Cluster theory has been applied to blends. This version has the added feature of representing configurations where one monomer may occupy more than a single lattice site (29,35,36). These higher-order approximations provide little improvement for the correlation of p VT properties of pure polymer liquids. High and Danner (37-39) have constructed a group-contribution lattice fluid (GCLF) equation of state based on the random-mixing version of PV EOS, Equation (14.47). Unlike the PV equation of state, the interaction-energy parameter for a pure fluid is assumed to be independent of temperature. Group contributions were obtained from parameter fits of homologous series of low-molar-mass compounds and used for both solvents and polymers. Since the primary goal was the prediction of solvent activities in polymer solutions, no consideration was given to the ability of the GCLF EOS to reproduce the properties of pure polymers.

536 14.2.3 Hole Models Hole models of polymer liquids combine two methods for incorporating compressibility into the lattice model. The first method is to include lattice vacancies and the second is to vary the cell volume. The addition of vacancies to the cell-model partition function, Equation (14.2), yields the partition function for the hole model:

Q(N'V'T)=Qc°mb(N'y)[Vf(V"Y'co)]UCexp

kB T

]

(14.52)

where y and co are variables related to the total volume; y is the fraction of occupied lattice sites

y_-

Nr

N o +Nr and co is the "cell" volume or volume per lattice site

(14.53)

V (14.54) N O+Nr Through y, the combinatorial factor is given a dependence on volume. Mixing r-mers with vacancies provides an additional source of entropy. The density is altered through changes in the cell volume and through addition or removal of vacancies. The original cell model is recovered when y = 1. Flory's approximation for Qcomb[Equation (14.31)] is most commonly used. Combining Equations (14.53) and (14.54) gives co-

Nrco

V=~ (14.55) Y which indicates the dependence of volume on the independent variables y and co. In order to determine an additional relation between V, co and y, one of the independent variables is treated as an order parameter. The additional relation is obtained by minimizing the Helmholtz energy with respect to the order parameter. The choice of y or co as the order parameter is a key assumption in the development of hole models. In Equation (14.52), the free volume and lattice energy are functions of the lattice-site volume co (which is analogous to v in the original cell models), and the fraction of occupied sites y. The free volume is affected by the presence of vacant lattice sites because segments neighboring vacant lattice sites gain some additional space to translate, rotate or vibrate. The lattice energy is affected because mers can not only interact with other mers, but also interact with vacancies. As a result, the dependence of the free-volume and lattice energy on y and co requires specification.

14.2.3.1 Simha-Somcynsky (SS) Simha and Somcynsky (40,41) choose the volume per lattice site, co, as the order parameter. Minimization of the Helmholtz energy with respect to co is used to derive an additional equation relating V,co and y:

537

kBT ~

)

N,v,r =

3o9 )N,V,T

=0

,14s6,

At constant V, co and y are linearly related by Equation (14.55); therefore, Equation (14.56) can also be written as cqnQ/

=0

(14.57)

Although Equations (14.56) and (14.57) are equivalent, Equation (14.57) is more convenient because the derivative with respect to y is easier to obtain algebraically. The equation of state is computed from:

p=kBT(°qnQ ~ =kBT( dy ~( ~nQ ] =_kBT( y__'~(oqnQ'] t. oV ; ~,~ ~ t.-dF ;t ---@--) ,<,~,~ t v ;t. @ ) ,<,~,.

(14.58)

To determine bulk thermodynamic properties, the order parameter co is held constant in the differentiation of Equation (14.58). To use Equation (14.58), we need an expression for Vf; two expressions have been proposed. The first one averages linearly over single modes of motion or free-lengths with respect to the fraction of occupied sites:

Vf=v*[y(~ 1/3-21/6)+(1- y)~l/3 ~

(14.59)

#

where c~ =co~v* is the reduced cell volume. The second averages linearly over the free volumes:

Vf=v,[y(gl/3 _21/6)3 +(l_y)~]

(14.60)

The first term in both expressions represents a solid-like contribution that is analogous to the free-volume expression in Prigogine's original cell model, Equation (14.5). The second term in both expressions represents a gas-like contribution that is reminiscent of the free volume of an ideal gas. Both expressions assume a face-centered cubic lattice. Equation (14.59) is the more commonly used, but both yield similar results. For the lattice energy, the expression derived using the Lennard-Jones potential with a face-centered cubic lattice is used but weighted by the fraction of occupied sites ~ [A Bq eo=1Y+vzq [-v- J

(14.61)

where, as before, zq = r(z - 2) + 2. Reduced quantities are defined by

V v cksT pry* Tfi= (14.62) Nrv* v* qze qze The volume per segment, v = V/Nr, is different from the volume per cell, co. The reduced . . . .

lattice site volume c~ is related to the reduced volume, F by c~ = yF. Adopting Equation (14.59) for the free volume, the resulting equation of state is

fiF_ (y~)I/3 2 ( A f -(y,~)1/3 _ 2,/6y ~f t Yv' (YF) 3

(14.63)

538 Applying the condition given by Equation (14.57) gives

y

--"

(yV)1/3-2 v6y

6T ' ( y ~ 2

(yV)4

(14.64)

Hence, given reduced temperature and pressure, Equations (14.63) and (14.64) are solved simultaneously for y and ~. The ratio c/r which appears explicitly in Equation (14.64) can be calculated from the characteristic quantities by c p'v* = (14.65) r kBT* The SS EOS, given by Equations (14.63) and (14.64), also produces a three-parameter corresponding-states behavior. 14.2.3.2 Further Modifications and Discussion

Nies and Stroeks (42,43) modify the hole model by adopting approximations for lattice thermodynamics that are essentially identical to those used with the LF equations of state. The first modification (42) uses Guggenheim's expression for the combinatorial factor, Equation (14.34), in the hole-model partition function, Equation (14.52) for random mixing of holes and mers. The lattice energy is also modified because Guggenheim's expression for r-mers introduces the surface area parameter q. Rather than assuming proportionality to the fraction of occupied sites [Equation (14.61)], the lattice energy is set proportional to the fraction of nearest-neighbor mer-mer contacts. A second modification (43) considers non-random mixing of holes by using the quasi-chemical approximation to calculate the combinatorial factor and number of mer-mer nearest neighbors. As in the lattice-fluid EOS, introduction of these approximations provides relatively minor improvement in the ability of the hole model to represent p VT data of polymer liquids, but, when extended to mixtures, improved correlation of miscibility behavior is observed (see Section 14.3.3). The SS EOS, Equations (14.63) and (14.64), has a computational disadvantage. Given temperature and pressure, two quantities (y and V) are unknown, requiting the simultaneous solution of two equations. To circumvent this disadvantage, Zhong et aL (44) proposed a simple expression which relates the occupied-site fraction to an exponential function of temperature y=l-exp(- c.~/ (14.66) ~, 2rTJ The origin of this expression comes from Schottky's theory of lattice vacancies in imperfect crystals. The argument of the exponential of Equation (14.66) is proportional to the energy required to create a vacancy in the polymer liquid. Equation (14.66) replaces Equation (14.64) and the equation of state, Equation (14.63) remains unchanged. Since this expression assumes that y is independent of pressure, it is applicable only to dense fluids such as polymer liquids. Parameter c appears in Equation (14.66) because of the definition of the reduced temperature [see Equation (14.62)]. Zhong et aL (44) make an additional modification by using a bodycentered cubic lattice (z = 8) rather than a face-centered cubic lattice. Zhong et al. (45) also proposed an open-cell model which modifies the SS EOS by proposing another expression for the free volume. The concept of the open cell takes into

539 account that segments may possess enough energy (through collisions with other segments) to move into a neighboring unoccupied cell. The proposed expression for the free volume is

Vf :Vf'[l+nh/3e-`'/kBr]

(14.67)

where Vf' is the original expression for the free volume used in the SS EOS [Equation (14.59)], nh is the number of neighboring positions available to a segment to move and e" is the energy needed to "push" the mer into that site. The adjustable parameter fl accounts for the influence of spatial structures and the shape of molecules on the free volume. Both nh and ~" can be related to properties of simple fluids (such as argon) and to other hole-model variables such as the fraction of occupied sites and the nearest-neighbor interaction energy. (Note that ~" is different from the mer-mer interaction energy ~.) The open-cell model gives slightly better representation of polymer pVT properties but at the expense of an additional adjustable parameter ,8.

14.2.4 Tangent-Sphere Models Within the last ten years, there has been increased interest in developing equations of state for polymers and polymer mixtures that no longer rely on a lattice description of molecular configurations. Although a rigorous statistical-mechanical treatment for polymers in continuous space is difficult because of their asymmetric structure, large number of internal degrees of freedom, and strong coupling between intra- and inter-molecular interactions, a relatively simple model has emerged which portrays chain-like fluids as freely-jointed tangent hard spheres (46-53). A hard-sphere chain (HSC) equation of state can be used as the reference system in place of the hard-sphere reference used in most existing equations of state for simple fluids. Despite their simplicity, hard-sphere-chain models take into account some significant features of real chain-like fluids such as excluded-volume effects and chain connectivity. A common feature of HSC equations of state is that they express the properties of the chain fluid in terms of the properties of simpler fluids, usually unbonded hard spheres or dimers. A fortunate feature of some hard-sphere-chain-based theories is that the reference equations of state can be extended to hard-sphere chain mixtures without using mixing rules. Only attractive terms require mixing rules. Extension of HSC-based equations of state to mixtures is discussed in Section 14.3. To describe the properties of a real polymer, it is necessary to introduce attractive forces by adding a perturbation to an HSC equation of state. Because the influence of attractive forces on fluid structure is weak, a van der Waals type or other mean-field term (e._g. square-well fluids) is usually used to add attractive forces to the reference hard-sphere-chain equation of state. Although numerous details are different, most hard-sphere-chain-based equations of state follow from statistical-mechanical perturbation theory; as a result, the equation of state can be written as

1468 pkBT

pkBT ref+ pkBT pert

where p is the pressure, ,o = N/V is the number density, N is the number of molecules, V is the volume of the system, and T is the absolute temperature. In Equation (14.68), the firstterm represents the reference equation of state, here taken as a fluid of hard-sphere chains

540

(p/pkBT)r~ f =(p/PkaT)HsC, and the second term is the perturbation to account for attractive forces. The next section presents the forms of the reference and perturbation terms for three different HSC based equations of state for pure chain-like fluids. 14.2.4.1 Generalized Flory (GF) Theories In the Generalized Flory theories developed by Hall and co-workers (48,49), the transition from lattice to continuous space is accomplished by replacing the site occupation fraction (bof chains on a lattice with the volume fraction ~7 of chains, i.e., the packing fraction for hardsphere chains, in the so-called osmotic equation of state (48), P

1

pk~r Hsc

= 1 -

lnpr(r/) +

if. [lnpr(x)]dx 7/o

(14.69)

where Pr is the insertion probability, defined as the probability of inserting an r-mer into a random configuration of r-mers at volume fraction r/without creating an overlap. Here, an rmer refers to a series of r identical tangent, freely jointed, hard spheres with diameter d and ~7=(~/6)rpd 3 . The problem of obtaining an accurate equation of state is reduced to developing reasonable estimates for the chain insertion probability, Pr" Dickman and Hall (48) first derived their estimates for P r from the probability of randomly inserting r monomers into the fluid in a manner similar to that in the original Flory and Flory-Huggins lattice theories (15,54). For the generalized Flory (GF) theory, pr is written as

pGF=[pl]Ve(r)/v'(')

(14.70)

where p~ is the insertion probability for the hard-sphere monomer fluid and v~ (r) is the r-mer exclusion volume, defined as the volume which must be free of the center of another r-mer to avoid an overlap. Combining Equations (14.69) and (14.70) gives P

loF =1+ve(r)(Z~-1)

pksr HSC

ve(1)

(14.71)

where Z~ is the compressibility factor of the hard-sphere monomer fluid at the same volume fraction ~7 as that of the chain fluid. According to the definition of exclusion volume (48), the monomer exclusion volume is

ve(1)=~-d 3

(14.72)

However, due to chain connectivity, the r-mer exclusion volume vo(r) ~ rVe(1) . For instance, the dimer exclusion volume is given by

ve(2)=2ve(1)-vlz(d )

(14.73)

541 where Vl2(d) is the intersection of the excluded volumes of two tangent spheres. The r-mer exclusion volume can be evaluated exactly only up to trimers (48):

ve(2)=9rC d3 (14.74) 4 ve( 3)=0.982605d 3 (14.75) Because v~(r) is difficult to evaluate for r > 3, Dickman and Hall suggested an approximate form appropriate for linear chains: ve(r)=v~(3)+(r-3)[v~(3)-v~(2)]

(14.76)

Using the Carnahan-Starling equation of state for the monomer fluid, Z1, Dickman and Hall (55) compared the GF equation (14.71) with Monte Carlo results for hard-sphere chains; agreement is only fair. The deficiency, as they indicated, was the assumption of randomly inserting r monomers into the fluid, which was not able to account explicitly for the chain connectivity, although it was partially taken into account through the use of exclusion volumes (55). Honnell and Hall (49) incorporated chain connectivity resulting from nearest-neighbor correlations into the GF model through the use of a dimer equation of state by Tildesley and Streett (56). The Generalized Flory-Dimer (GF-D) theory, provides an estimate for P r :

('-'~1)[ve(r)-ve(1)]/[ve(2)-ve(1)]

p~rF-O=p1 P2

(14.77)

where P2 is the probability of inserting a dimer into a dimer fluid at the same volume fraction, r/without creating an overlap. With Equation (14.77) the resulting equation of state is

IokBT/HSC Ve(Z)-Ve(1)J Lve(2)-ve(1)

(14.78)

where Z 1 and Z2 are the compressibility factors of monomer and dimer fluids, respectively, at the same volume fraction 77 as that of the chain fluid. The dependence of Z1 on r/is given by the Carnahan-Starling equation and the dependence of Z2 on r/is given by the Tildesley-Streett equation. Hall and co-workers (49),(57) compared the GF-D equation of state to both Monte Carlo and molecular-dynamics simulation results for hard-sphere chains up to 32-mers; agreement is good and the overall accuracy is superior to that of the GF equation of state, Equation (14.71). Figure 14.2 shows results for 32-mers (57). Recently, Hall and co-workers (58) extended their molecular-dynamics results for hard-sphere chains up to 192-mers. Both the GF and GF-D theories are only for hard-sphere chain fluids. However, they can be generalized to include attractive forces. Yethiraj and Hall (59) extended both the GF and GF-D theories to square-well fluids. In general, the attractive contribution to the

542

300.0

J

250.0

// 200,0

150.0

E c,.) 100.0

oF

/

50.0

L

%.o~

'

o.~o

'

o.lo

'

o.;o

'

o.io

0.50

Volume Fraction

Figure 14.2 Compressibility factor vs. volume fraction (packing fraction i/) for freely jointed 32-mers (hard-sphere chains). Points are simulation results of Denlinger and Hall (57). The curves are predictions of the generalized Flory (GF) and generalized Flory-Dimer (GFD) equations of state (49). compressibility factor has the same form as the hard-sphere-chain contribution; e.g., the GF-D contribution for chains with attractive forces is (59)

(pr°[

=

,OkBT)at t

] [

Ve(r)-ve(1) att Ve(2)_Ve(1 ) Z2

-

Ve(r)~Ve(2)lzaa V2(2)_Ve(1)J 1

(14.79)

where Z~ and Z~~ are the compressibility factors for square-well monomers and dimers, respectively. The total compressibility factor for square-well chains is a sum of the HSC and attractive contributions. Yethiraj and Hall (59) used the integral equation theory of fluids with the mean-sphericalapproximation (MSA) closure to obtain equations of state for square-well monomers and

543 dimers; comparisons showed that the GF-D theory is in excellent agreement with simulation data for square-well 4-mers, 8-mers, 16-mers, and 32-mers (60). Later, Vimalchand et al. (61) derived an equation of state for square-well monomers based on a local composition model, and Thomas and Donohue (62) adapted the same approach to develop an equation of state for square-well dimers of different bond lengths. 14.2.4.2 Statistical Associated-Fluid Theory

The Statistical Associated-Fluid Theory (SAFT) (63) is based on the first-order theory of Wertheim's thermodynamic perturbation theory of polymerization (TPT) (53) for hard-sphere chains as the reference system. Wertheim's theory was developed by expanding the Helmholtz energy in a series of integrals of molecular-distribution functions and the association potential. Using some physical arguments, Wertheim showed that many integrals in this series are zero which results in a simplified expression for the Helmholtz energy. Wertheim applied his theory to hard-chain fluids and developed first-order and second-order thermodynamic perturbation theories (TPT1 and TPT2). The first-order theory is applicable only to flexible chain molecules. In the second-order theory, effects of chain conformation and branch structure are taken into account, but a detailed knowledge of the triplet distribution function is required (64). According to the first-order theory the hard-sphere-chain equation of state is P Hsc = 1 + rZbpg(d +) - ( r - l ) p pkBT

Op

J

(14.80)

where g ( d ÷) is the radial distribution function of hard spheres at contact, as calculated from the Carnahan-Starling equation, r is the number of tangent hard-spheres per molecule, and b is the second virial coefficient of hard spheres prior to bonding to form chains: b=2---~-~d 3 (14.81) 3 In Equation (14.80), the first two terms are the non-bonding contributions and the last term reflects chain connectivity. Equation (14.80) obeys the ideal-gas law in the limit p ~ 0. To model a real fluid, a perturbation term is needed in addition to Equation (14.81) to include attractive forces. Chapman et al. (63) adopted the dispersion term derived by Cotterman et al. (65) from molecular-simulation data for Lennard-Jones fluids. For the dispersion term, Huang and Radosz (66) used a series initially fitted by Alder et aL (12) to molecular-dynamics data for a square-well fluid

p =rZZflDap pk BT pe~ ~ p

u

7? p

(14.82)

where r/cp = 0.7405, is the packing fraction of hard spheres at the closest packing, and u=~(l+e/kBT )

e/kB=lO K

(14.83)

where eis the well depth of the potential. In Equation (14.82), Dap are universal constants which have been refitted to accurate p VT, internal energy, and second virial coefficient data for

544 argon (67). The effective hard-sphere diameter d depends on temperature as suggested by the Barker-Henderson perturbation theory (68)

d=cr[1-Cexp(-3c/kBT)],

(14.84)

C=0.12

where o-is the effective hard-sphere diameter at zero temperature; it is also called the distance of separation at minimum potential energy. The SAFT equation of state has been applied to describe thermodynamic properties of polymers as well as solvents (66,69). In general, it was found to correlate experimental data with good accuracy, except at low pressures and high temperatures, or at high pressures and low temperatures. For polymers, experimental SAFT EOS parameters are independent of molar mass, as required by theory.

14.2.4.3 Perturbed Hard-Sphere-Chain Theory The reference part of the perturbed hard-sphere-chain (PHSC) equation of state is based on a generalization (51) of the Percus-Yevick integral-equation theory for hard-sphere chains as obtained by Chiew (47). A simple van der Waals-type term is used for the perturbation. The PHSC equation of state for pure fluids is (70)

P pk~

= 1 + rZbpg(d +) - ( r - 1 ) I g ( d + ) - I 1 -

r2ap k~

(14.85)

where the first three terms represent the reference equation of state for hard-sphere chains, and the last term is the van der Waals perturbation for attractive forces. Parameters r and b are the same as those in SAFT's reference equation of state, Equation (14.80). The third parameter a represents the strength of the attractive forces between two non-bonded segments. In the PHSC theory both b and a are temperature dependent; they are given by (70)

a(T)=~-~-cr36Fa (kBT/c )

(14.86)

b( T)=~_d3 ( T)=~_~_cr3Fo( kBT/6 )

(14.87)

Equations (14.86) and (14.87) result from using the Song-Mason method (71) which scales b and a in terms of two potential parameters 6 and or, 6 is the well depth of the non-bonded segrnent-segrnent pair potential and cr is the distance of separation at the minimum of the potential. In Equations (14.86) and (14.87), Fa and Fb are two universal functions of a scaled temperature kaT/s. Since accurate experimental values of c and o-exist for methane and argon, the thermodynamic properties of these two fluids are used to determine the singlesphere universal functions Fa and Fb (70); they can be accurately represented by the following empirical formulas (72):ttt I"1"I"A simplified version of the PHSC equation is presented here (72). In previous publications (70), the temperature dependences of the universal functions were scaled by a parameter s which is a function of chain length r only. In the present model, this scale factor is removed and the universal functions [Equations (14.88) and (14.89)] were determined from the thermodynamic properties of argon and methane over a wider temperature range. Removal of s(r) from the universal functions

545

F, (kBT/e )=l .8681 exp[- O.O619(kBT/e) ]+0.6715 exp[- 1.7317(kBT/e) 3/2]

(14,88)

F~(kBT/e)=0.7303 exp[- 0.1649(kBT/e) 1/2]+0.2697 exp[- 2.3973(kBT/e) 3/2]

(14.89)

Song et aL (70) have applied the PHSC equation of state to describe thermodynamic properties of normal fluids and polymers. The EOS parameters are available for a wide variety of polymer and pure fluids. They also compared the PHSC EOS with the SAFT and LF EOS; both of which are also three parameter equations of state. Figures 14.3 and 14.4 illustrate the quality of correlation for two common polymers, polystyrene and high-density polyethylene. The PHSC EOS performs much better than the LF EOS (especially at high pressures), equally as well or slightly better than the SAFT EOS. Finally, to close the discussion on pure polymer liquids, we point out a significant caveat in the determination of equation-of-state parameters of polymers: Parameters determined from fitting polymer p VT data often differ significantly from those determined from extrapolation of parameters for lower molar-mass analogs [except for the SAFT EOS, in which the size parameter for polymers was forced to be the same as that for lower molecular-weight analogs (66)], as illustrated in Figure 14.5 for the characteristic temperature) (T* = e./kB) of

1

. --

0

4

~

_~

P.sc

.-"

. . - f ~

.~

100

..-"

2130

,

i000

_=

=

~~--~~~~~~~®

-~>

0.92

-'"

*

'

'

120

_

. . . . . . . . . .

140

160

180

200

'

-

220

Temperature (°C) Figure ]4.3 The pVT properties of polystyrene. "['he symbols are experimental data (169), and the curves are fits using the perturbed hard-sphere-chain (PHSC), the lattice-fluid (LF), and the statistical associating-fluid theory (SAFT) equations of state (70).

allows use of simpler combining rules. The simpler rules do not sacrifice accuracy in fitting thermodynamic properties of pure (non-associating) fluids to obtain equation-of-state parameters. All figures cited on the PHSC EOS in this review are directly taken from previous publications (70,73,74) and conference presentations (75-78). Since two different versions of universal functions have been used, calculations shown in Figures 19-22 and 26 contain the parameter s in the universal functions, whereas calculations in Figures 23-25 and 27 are based on Equations (14.88) and (14.89).

546

1.35

:-- pHscl "'-"

~.. eg

LF SAFT

Jo

I

ibar

1.30

-

400

1.25

,._.., ©

;> O . .,..~ O

o..,

1.20

r./3 ~

1

8

0

0 ") Cg'~

1.15

f

t

!

,40

120

Temperature (°C) Figure 14.4 The p VT properties of high-density polyethylene (HDPE). The symbols are experimental data (170), and the curves are fits using the perturbed hard-sphere-chain (PHSC), the lattice-fluid (LF), and the statistical associating-fluid theory (SAFT) equations of state (70).

800 F

....

I ....

~ i

Ill

....

PHSC

LF

~

o ,,

7 0 0

•

600

--

O

O O

_

-

30

-

n-Alkanes Poly(ethylene)

Ooo

_

400

~

R

C

C

~

[]

200

100

[]

,

o.oo

....

%

5 0 0 -v

I

,

r

,

I

o.os

,

,

,

,

I

olo

,

,

,

,

!

,

,

o15

,

,

I

0.2o

,

,

,

,

o2s

(l/M) tn (mot/g)tn

Figure 14.5 Comparison of the characteristic temperatures of normal alkanes determined from the thermodynamic properties of the saturated liquid (open points) and polyethylene determined from p VT data (filled points) using the PHSC (171) and LF (13,14) equations of state. Characteristic parameters for polymers determined from p VT data are significantly different from those estimated by extrapolation from lower-molecular-weight analogs.

547 polyethylene for the PHSC and LF EOS. In both cases, extrapolation of the characteristic temperatures of the normal alkanes to infinite molar mass" leads to a significantly different characteristic temperature than that determined from fitting polymer p VT data. This difference can have a profound effect on the ability of the equation of state to predict and/or correlate the experimental data for mixtures of polymers with volatile solvents. 14.3 EXTENSION TO MIXTURES In this section we discuss the extension to mixtures of equations of state developed in Section 14.2. The extension of lattice-based equations of state (e.g., the cell model, lattice fluid and hole model) are more succinctly presented by considering binary mixtures; extension to ternary (and higher) mixtures is possible but the mathematical procedure is complex. On the other hand, the tangent-sphere fluid equations of state (e.g., SAFT and PHSC) are easily formulated for any number of components.

14.3.1 Flory-Orwoll-Vrij (FOV) If we assume that the configurational properties of the mixture are given by the customary one-fluid assumption, the partition function for the FOV equation of state for mixtures of chain-like fluids is similar to that for pure fluids: Q = (constant)Qcomb(~-1) 3u~-eexp -k---~ where E - o is the lattice potential energy, F = v / *v

(14.90) is the reduced volume of the mixture,

v=V/(r~N~ +raN2) is the volume per lattice site, and v* is the characteristic hard-core segment volume. Chain-length parameter F and external-degrees-of-freedom parameter g are averages of the pure-component parameters: N=NI+N 2 _

(14.91)

r l N 1+ r z N 2

r=-

-6- e l r l N l

(14.92)

N

(14.93)

+CzrzN2 FN

Two additional assumptions are required. First, the hard-core segment volumes of each component are arbitrarily assumed to be equal, i.e., v 1 =v2*=v* . Consequently, differences in molecular size are reflected only in the chain length ri for each component. For example, in a binary mixture, the characteristic molecular volume of species i is

Vi* =r iv*,

and

ra/rl -- V]/VI*. This leads to the definition of segment fractions ~bi for each component

¢~ = 1 - ~

N1V1

x1

N1V1 + N2V2

Xl +(r2/rl)X2

(14.94)

548 w h e r e x i -- N i / N is the mole fraction of component i.

Second, the intermolecular energy depends on random mixing of surface contacts between molecules. Counting of pair interactions is most easily accomplished by the definition of contact-site fractions: 0~=1_02_

~b~

~ +(~/n)O~

(14.95)

where s i is the number of contact sites per segment of component i.tttt The lattice energy of the mixture is E--0-

Nllr]ll +N22/722 +N12/']12

(14.96)

v where No is the number of i-j contacts and each contact is characterized by the energy - rlo. I v . In a random mixture

2Nil + N12

"-

SlrlN1

(14.97)

2N22

+ N12

-'-

s2r2N2

(14.98)

N12

=

SlrlNl02-- s2r2N201

(14.99)

Substitution of Equations (14.93)-(14.94) into Equation (14.92) gives" (14.100)

- E = Nrs

where ~ is an average surface area parameter (14.101)

S'-~S1 +~2S 2

and Art = rh~ +/722-- 2/712" The Equation of state for the mixture is obtained by differentiation of Equation (14.90) according to Equation (14.1). The result is identical to the equation of state for a pure fluid, Equation (14.11), but now the characteristic pressure and temperature are functions of composition, pure-fluid characteristic parameters and a binary parameter: X12 = S l A r l / 2 ( v * ) 2" P -~lPl +~2P2-~lO2X12

(14.102)

T *-

(14.103)

P

tttt s ~ q / r where q is the number of contact sites per molecule. For a simple long chain, q is related to r by zq=r(z-2)+2; however, for model flexibility, Flory preferred to avoid using an explicit lattice geometry. Therefore, the relation between s and r remained unspecified.

549 Binary parameter Y12 an interchange-energy density that is analogous to the interchange

=szArl/2(v*)2

energy in the incompressible lattice model. Note that X12 ¢ X21 .Other important thermodynamic properties (especially the Helmholtz energy of mixing and chemical potential) can be calculated from the partition function. The Helmholtz energy of mixing is

•

I

Ami~'A=-TAmi~'S+(N1V~+N2V;)qklp

-

1)+~bzp*

1

1

I

qklO2Xlz V

(14104,

The last three bracketed terms in Equation (14.104) represent the energetic contributions to the Helmholtz energy (or, ignoring the PAn~V term, the enthalpic contributions to the Gibbs energy of mixing). Of these three terms, the first two arise from differences between reciprocal reduced volumes and disparities in characteristic parameters of the pure fluids. The last term represents the exchange-energy contribution from nearest-neighbor interactions. The entropy of mixing is the sum of the combinatorial term and a residual term. Amix.S-Ami x.S c°mb-t-Amix.S R

(14.105)

An~.SC°mb=kB[N,lnqkl+N21n~b2]

(14.106) -1 (PzP2ln|•\vv2 In ~'~-1 + T:• ,--1/3

An~S"=3(N~Vl• +N:V:• )

(14.107)

The combinatorial term is identical to the combinatorial entropy of mixing for an incompressible-lattice mixture according to the Flory-Huggins approximation. The residual term is a contribution from the equation of state. The chemical potential of component 1 (typically a solvent), obtained from the differentiation of Equation (14.104) with respect to N 1, can be divided into combinatorial and residual terms in a similar manner: A,t/1 =A]./1 °mb +Aft R

z~

= rlPl*V*[3~n[~g3

(14.108)

+

-

+

(14.110)

The residual chemical potential is associated with the Flory 2" parameter

Z:kB T¢)------~2

(14.111)

Hence, an equation-of-state description of the mixture leads to a composition dependence of the Flory Z parameter. Reasonably good agreement with experiment is shown in Figure 14.6 for polyisobutylene and benzene (79).

550 A modification of the residual chemical potential of the solvent [Equation (14.110)] is made by appending the term -TQ12V~* O~2 ; Q12, analogous to x12, represents the entropy of interaction between unlike segments. This term is independent of density and affects only the chemical potential and not the equation of state. If excess-volume data are used to determine X12, the residual chemical potential of the solvent is under-predicted as shown by the dashed curve in Figure 14.7 for natural rubber and benzene (80). By adjusting Q~2, a fair representation of Z can be obtained without affecting the representation of volumetric properties. This method of data reduction was also used for polyisobutylene with benzene (79), cyclohexane (81) and n-pentane (82), and polystyrene with methyl ethyl ketone (83), ethyl benzene (84) and cyclohexane (85). The role of compressibility effects for the miscibility of polymer blends has also been investigated using the FOV equation of state, McMaster (86) used a generalized version of the FOV EOS to show qualitatively how differences in pure-component thermal-expansion coefficients can lead to lower-critical-solution-temperature (LCST) behavior in polymer blends. In addition, the effects of molecular weight, pressure and polydispersity were also considered qualitatively. Kammer et al. (87) added a parameter reflecting differences in segment size and illustrated its effect on blend miscibility. In addition to differences in interaction energy and compressibility effects, upper critical-solution temperatures (UCSTs) and LCSTs in polymer blends were shown to be sensitive to the segment size parameter.

I

I

I

i

1

t

i

q

I

Experimental

/

t/~'C)

I-3

ID

1.2 I'1

•

10 24.5

0

25

/_ / / /

o °_

I-0 0"9

0"8

-

0"7

--

jo ©

o Q

0"6 0"5 0"4i

~ol 0"0

, 0-1

g.~g.~g.,o,.~

Segment Fraction,

0"6 0-7 0'8 19 , , ,

I

I-0

Polymer

Figure 14.6 The reduced residual chemical potential Z of benzene in poly(isobutylene) as a function of polymer segment fraction. The curve is calculated using the FOV EOS with X~2 = 42 J/cm3 at 25°C (79).

551

°'I o

o-s

Experimental t/(°c) 10 © 0 25

/

40 0" 4 ~ . ~ . ~

i

~

7 . - -~'~

~

0"3

0"2

'

~ ~ ~ I

I

0;2'

0t

3

0 -!4

_

0 - !5

0 - 6!

0 - 7:

0 " !8

0 't9

Segment Fraction, Polymer

Figure 14.7 The reduced residual chemical potential Z of benzene in natural rubber as a function of segment fraction polymer. The dashed curve is calculated using the FOV EOS at 25°C with X~2 = 1.40 cal/cm3 and Q12 = 0. The solid curve is calculated with the same X12 and Q12 = - 0.018 J/(cm3 K) (80).

Rostami and Walsh (88) considered the effect of pressure and molecular weight on the UCST in polymer blends using the FOV EOS. Figure 14.8 shows reasonable prediction of the increase in UCST upon increasing pressure from atmospheric to 100 MPa for polybutadiene (M = 2350) mixed with polystyrene (M = 1200) using two binary parameters (Y12 and Ql2). Y12 was determined from fitting experimental enthalpies of mixing and was the same for all molecular weights of either polymer. Q12 was determined by fitting the maximum cloud-point temperature at atmospheric pressure and varied with the molecular weight of both polymers. Prediction of the change in critical solution temperatures with pressure can only be achieved using an equation-of-state theory. 14.3.2 Perturbed-Hard-Chain (PHC)

The perturbed-hard-chain (PHC) equation of state was extended to mixtures by Donohue and Prausnitz (89), by Kaul et al. (90), and by Cotterman et al. (65). Here we only briefly discuss Henry's constant from the PHC equation of state. Let 1 denote solvent and 2 polymer. Henry's constant for the solvent in the polymer is defined by H12 = '

lim fl wl~O W 1

=

lim RT exp wl~O M l V 2

(14.112) kBT

wheref~ is the fugacity and wl is the weight fraction of the solvent; M1 is the molar mass of the solvent, R is the gas constant, and v2 is the specific volume of the polymer. In Equation

552

373

363

P/

i\

I I

iI

353

~9

e'_

E

363

o

333

1

0,7

i

0,t+

t

0.6

i

0,8

Segment Fraction, Polystyrene Figure 14.8 Experimentally measured cloud points for poly(butadiene)/poly(styrene) blends at 0.1 MPa (O) and 100 MPa (0). The binodals (--) and spinodals (- - -) at each pressure were calculated using the FOV EOS (88). (14.112), !~ c , gAa, and #sv are the chemical potentials from the hard-chain part, attractive part, and second-virial coefficient part, as calculated from the PHC equation of state for mixtures (90). Ohzono et al. (91) and Iwai and Arai (92) applied the PHC equation of state to correlate weight-fraction Henry constants of hydrocarbon vapors in molten polymers. Figure 14.9 compares calculated and experimental Henry constants of normal alkanes in polypropylene (92). With one adjustable binary parameter k~2, Henry constants can be well correlated over a considerable range of temperature.

553

16

15

14

13

I 12

448

I

1

473

498

523

T (K)

Figure 14.9 Henry constants of hydrocarbons in polypropylene. Points are experimental data and curves are calculated using the PHC equation of state (92). 14.3.3 Lattice Fluid (LF) The LF EOS for mixtures is similar in many ways to other adaptations of lattice-based equations of state to mixtures. These similarities and a discussion of applications to real mixtures is provided here. An additional and more comprehensive review of the applications and shortcomings of LF EOS to mixtures has recently been provided by Sanchez (93). The LF theory for mixtures also requires assumptions concerning the characteristic volume v* of a lattice site for the mixture and counting of pair interactions. The original formulation of the SL equation of state for mixtures (14,94) uses assumptions similar to those used in the FOV equation of state. First, the molecular volume of each component is conserved: the molecular volume of a component, determined from its pure-fluid properties r°v * is equal to its molecular volume in the mixture riv*. Consequently, the chain length ri of

a component in the mixture differs from that rio for the pure fluid. In addition, the total number of pair interactions in the close-packed mixture is equal to the number of pair interactions of the components in their close-packed pure states. Since the SL EOS assumes random mixing of segments, these assumptions lead to the following expressions for the characteristic volume and characteristic interaction energy:

554 v = Y,,~v*

(14.113)

i

c

= ZZf~q~6ij ij

= Zqk~e~i-kBTEZd~zo i

ij

(14.114)

where ~b° are closed-packed segment fractions determined from the canonical segment fractions ~bi"

~b° -

(~bi/v*)

(14.115)

J

r/N/

~bi=y, rjN j

(14.116)

J

and Zo.=(ei* + e * - 2 6 * ) / k B T .

With v* and 6* defined by Equations (14.113) and

(14.114), Equations (14.28) and (14.33) define characteristic quantities. The equation of state for the mixture is identical to Equation (14.32); however r, which appears explicitly in Equation (14.32), becomes an average value for the mixture given by 1=~. ~bi • ri

(14.117.)

One parameter g* characterizes a binary mixture; all others are related to pure components. Alternatively, the characteristic pressure of a binary mixture can be expressed in terms of purecomponent characteristic quantities as: •

--

*

--

*

P =~lP, +q~2P2-¢,¢2AP where Ap* = p* + p*

*

- 2pl 2 , Ap*

(14.118) (or equivalently P,2) replaces e 0. as the binary parameter.

Because the mixing rule for v Equation (14.113) is" V * =~-"~Z ~ i ~ j V O . i j

,

•

is somewhat arbitrary, a common alternative (95) to

*

(14.119)

* provides a second binary parameter. Other approximations can be used where vii* = v i* and v U to count the number of pair interactions. For example, random mixing of contact sites rather than random mixing of segments can be assumed (9,96) as well as non-random mixing (19,95), but e o* typically remains as the essential binary parameter. Like the FOV EOS, the LF model (and its variations) has been used to correlate the composition dependence of the Flory Z parameter (residual chemical potential of solvent) (19),(95). In these studies, it was found that a binary entropic-interaction parameter analogous

555 to the Q12parameter in the FOV equation of state, so that calculated solvent residual chemical potentials are of the same order as those measured experimentally, was also needed. The general character of liquid-phase miscibility can be obtained by studying the spinodal condition. According to the LF EOS, that condition can be written as (14): 1

1 +~rz~al_~ X+ ~ 2 T p * z r >0

(14.120)

where

x

= Ap*v*/k~r d(1/T) d~

and

Zr

(14.121)

d(1/r)

d(~/T)

d~

d~

(14.122)

is the isothermal compressibility of the mixture:

TP*XT--V[1/(V-1)+ I/r- 2/VT~'

(14.123)

The first bracketed term in Equation (14.120) is the combinatorial entropy contribution, ~ ( is an energetic contribution and

~/2Tp*;(v/2

is an entropic contribution from the equation of

state. Figure 14.10 illustrates the general behavior of these three terms as functions of temperature. The last term makes an unfavorable contribution to the spinodal and favors demixing. Its magnitude increases with increasing temperature and diverges as the liquidvapor critical temperature is approached. Hence, according to the LF theory, every polymer solution in equilibrium with its vapor should exhibit a LCST prior to reaching its liquid-vapor critical temperature. In addition to mixtures of polymers with normal solvents, the LF EOS has also been applied to polymer-gas systems. Sanchez and Rodgers (97,98) used the LF EOS to predict gas solubility at infinite dilution. The physical properties of the gas and polymer dominate gas solubility and the gas-polymer interaction plays a secondary role. Using no adjustable parameters, gas solubility was quantitatively predicted for several hydrocarbons and chlorinated hydrocarbons in non-polar polymers, such as polystyrene, poly(1-butene) and atactic polypropylene. For polar polymers, such as poly(vinyl acetate) and poly(methyl methacrylate), the solubilities of hydrocarbons were greatly overestimated, but polar and aromatic gases were correlated reasonably well. Similarly, Pope et al. (99) attempted to predict sorption isotherms of nitrogen, methane, carbon dioxide and ethylene in silicone rubber. Model parameters were determined only from pure-fluid properties without using any mixing data. Figure 14.11 shows a typical isotherm for ethylene. In general, the experimental isotherm is under-predicted. From the equation of state, expressions for the partial molar volumes and Henry's constant at infinite dilution for the gases can be derived. In general, the Henry's constant is under-predicted, but a good estimate of the partial molar volume at infinite dilution was obtained.

556

• ~

,~

o

UCST

LCST Tc

Temperature

Figure 14.10 Schematic behavior of the three terms in the spinodal equation derived from the Sanchez-Lacombe (LF) EOS as a function of temperature (14). The horizontal line represents the combination contribution to the spinodal. The dashed curve represents the sum of energetic contribution (pX) and the equation-of-state contribution (p~t2Tp* ZT/2). When this sum lies below the combinational contribution, the mixture will phase separate. Intersections of the dashed curve ahd horizontal line represent upper and lower critical solution temperatures.

200 E o n_

E

150

-

oO~

o

13_

100

~

o ff o o ~

50

.

00

e e~"" 10

20

30

•experimental

40

50

60

/

70

Pressure (bar) °

Figure 14.11 Comparison of theoretical predictions (no adjustable parameters) using the SanchezLacombe EOS with experimental results for ethylene sorption in silicone rubber (99).

557 A novel application of the LF equation of state is provided by describing the effect of a compressed-gas diluent on the behavior of a glassy polymer (100-103). Compressed gases can act as plasticizers when dissolved in a glassy polymer matrix by lowering the polymer's glasstransition temperature (Tg). The glass transition of a polymer or polymer/diluent mixture can be determined by using the Gibbs-Di Marzio criterion, which states that at the glass transition the polymer is essentially frozen and has zero configurational entropy. The sorption of the compressed fluid by the polymer and Tg can be calculated by simultaneously solving the EOS, the condition of equilibrium partitioning of the diluent between the polymer and gas phase, and the Gibbs-Di Marzio criterion. Figure 14.12 shows results for poly(methyl methacrylate) (PMMA)/CO2 (101). Figure 14.12(a) illustrates good agreement between experimental and calculated Tg depression as a function of concentration of dissolved CO2. However, Figure 14.12(b), which plots Tg against CO2 pressure, shows an unexpected phase behavior, first discovered with the model, called retrograde vitrification (by analogy with retrograde condensation). For example, at 100°C and

125 (a) .........

(.)

75

_

25

_

O3 I--

-25

50

0

1O0

150

cm 3 (STP) C02/g Polymer

125

I

I

I

1

(b) 75 0o v

EYJ !--

25-

-25

0

10

20

30

40

50

Pressure (bar)

Figure 14.12 (a) Calculated vs. experimental glass-transition temperature depression of PMMA as a function of CO2 solubility. (b) Glass-transition-temperature depression of PMMA as a function of CO2 pressure showing retrograde vitrification behavior (101).

558

el

©

r,. ~a e-

1.00

0.04.

-

.....,

._=

~I

0.03 bh

-

L)

/ o.io

-o.o2

-~ p...,

-.~ -0.01 r-

-

;;~

0.01

"

0

"

-

-

I

I0

Pressure

o.oo

-

2~0

3o

"-3

(MPa)

Figure 14.13 Concentrations Of coexisting phases for PEG/supercritical-CO2 mixtures at 323 K as a function of pressure. The upper and lower curves correspond to the left and right vertical axes, respectively. Curves were calculated using the PV (LF) EOS using one adjustable binary parameter.

a CO2 pressure of 3.0 MPa, the polymer is a liquid. Decreasing the temperature causes the polymer to undergo a liquid-to-glass transition, as expected; however, a further decrease in temperature causes the glass to become a liquid again. This effect results from the competition between polymer-segment mobility, which declines with decreasing temperature, and diluent solubility, which increases with decreasing temperature. The LF EOS has also been used to describe the phase behavior of supercritical fluidpolymer systems. These systems differ from the above systems in that the polymer may completely dissolve in the gaseous supercritical-fluid (SCF) phase, or partition between a polymer-rich phase and a SCF-rich phase. Figure 14.13 shows an example: poly(ethylene glycol) (M = 400) mixed with CO2 up to pressures of approximately 26 MPa at 50°C (104). The upper curve (left-hand axis) corresponds to the mass fraction of CO2 dissolved in the polymer-rich phase and the lower curve (right-hand axis) corresponds to the mass fraction of polymer dissolved in the SCF-rich phase. The curves are calculated using the random-mixing version of the PV EOS with one adjustable binary parameter. The random-mixing version of the PV EOS has also been used to correlate partitioning of polystyrene in supercritical CO2 and supercritical ethane (105). The EOS verified an observed linear relationship between the polymer partition coefficient and the logarithm of the polymer molecular weight. The Sanchez-Lacombe version of the LF EOS was used to study the cloud-points of poly(methyl methacrylate) in supercritical chlorodifluoromethane (106-108), poly(ethylene-comethacrylate) copolymers in supercritical propane and chlorodifluoromethane (106,107) and polycaprolactone in supercritical chlorodifluoromethane (108). Phase behavior in the presence of cosolvents such as acetone and ethanol was also investigated (107). Cloud-point data can be modeled reasonably well at zero and high cosolvent compositions, but not at intermediate concentrations. This is not surprising because specific interactions between cosolvent

559 molecules play a large role in the phase behavior at these concentrations, but are not accounted for in the model. The LF EOS has also been applied to understand the role of compressibility in the miscibility of polymer blends. The spinodal condition for polymer blends becomes highly sensitive to the interaction-energy term and equation-of-state contributions, because the combinatorial contribution is very small. Using the SL EOS for mixtures, the UCST behavior in polystyrene/poly(ot-methyl styrene) and polystyrene/poly(methyl methacrylate) blends was linked to positive values for Ap*, whereas the LCST behavior in poly(methyl methacrylate)/poly(c~-methyl styrene) blends was a manifestation of differences in purepolymer characteristic temperatures which arises through the function ~ (109). Similar analyses were used to investigate the miscibility of polycarbonate with styrene-methyl methacrylate and styrene-acrylonitrile copolymers (110) and polymethacrylates with styrenemaleic anhydride copolymers (111). Finally, An et al. (112-117) used the multicomponent form of the SL EOS to investigate the effect of polydispersity on pure-polymer p VT properties and blend miscibility. Three types of molar-mass distributions were considered for the description of polydispersity for each component. An et al. examine assumptions often made to justify the application of equations of state for monodisperse polymers to experimental results where the polymers may have a significant degree of polydispersity. One assumption is that the thermodynamic properties of a pure polydisperse polymer are identical to those of the corresponding monodisperse polymer when the average degree of polymerization is sufficiently high (115). Similarly, the effect of polydispersity on the spinodal curves for a blend of polydisperse polymers with an LCST decreases as the average degree of polymerization increases. However, for specific average degrees of polymerization, polydispersity generally decreases the stability of polymer blends when compared to the corresponding monodisperse blends (117). 14.3.4 Mean-Field Lattice-Gas (MFLG)

The Mean-Field Lattice Gas (MLFG) EOS for mixtures is analogous to the Lattice-Fluid EOS in that a binary mixture is modeled as a ternary mixture where one of the components is a lattice vacancy. In addition, the volume vo per lattice site is assumed constant for all pure fluids and the mixture. The Helmholtz energy for the mixture with holes is a straightforward extension of Equation (14.49): AmixA

NrRr

=

~boln~bo+ ~blln~bl+ ~b2ln~b2+gol~O~l +go2~o~2 rl rz

+glZq}l~2-O(~l)-O(~2)

(14.124)

where goi=Oti 4 flOi + flli/T

(14.125)

g12 =amix 4 fl0mix + fllmix/ T

(14.126)

7"i=l-cri/O- 0

(14.127)

1-n~ -r2~2

560 where ai, /~0i 'S, and ill; 's are pure-component MFLG parameters, determined from the thermodynamic properties of pure fluid i. Parameter Yi is related to the contact-surface areas cri of segments of component i and holes. Parameters amix, fl0mix, and fllmix are binary parameters which can be adjusted to match the desired thermodynamic properties of the mixture. The term O(~bi) refers to the Helmholtz energy of pure fluid i. Application of the MFLG EOS has been limited primarily to sorption and swelling of polymer in the presence of supercritical fluids (22). Figure 14.14(a) shows sorption of CO2 by PMMA at 42°C to pressures of approximately 30 MPa. The binary parameters in the MFLG EOS were adjusted to give a best fit of the sorption data. Figure 14.14(b) shows the predicted values of the volume change on mixing as a function of pressure. At high pressures, the MFLG EOS over-predicts the volume change due to overestimation of the pure CO2 density at high pressures and an overestimation of the volume change of the polymer due to swelling by the gas. In a similar application, the MFLG EOS was extended to describe sorption and swelling of crosslinked poly(dimethyl siloxane) elastomers (118) by adding an elastic term of Flory (15) to the Helmholtz energy of mixing. Figure 14.15 shows the weight fraction of CO2 sorbed by the elastomer and its fractional volume change at 42°C for CO2 pressures to 35 MPa. Unlike uncrosslinked polymers (where the sorption increases steadily with pressure), the restricted swelling capacity due to the presence of crosslinks limits the amount of CO2 which can be dissolved in the polymer. This restricted capacity is adequately represented by the modified equation of state.

14.3.5 Simha-Someynsky Hole Model The Simha and Somcynsky equation of state was extended to mixtures by Jain and Simha (119,120). The configurational partition function for hole models, Equation (14.52), is only slightly modified. For a binary mixture, the fraction of occupied sites is y-

rlN 1 + r2N2 No + rlNI + rzN2

(14.128)

where N O is the number of vacant lattice sites. The lattice energy is written

Eo

I

-zNy(qz) 1.011

(yv) 4

2.409

(yv) 2

(14.129)

561

(b)

-s0!

-"~ E

'~-

•

•

-too-!

~°°i

\

too T

0.70

:

0.~0

1.00

60

Weight Fraction, Polymer

120

tSO

2

0

Pressure (bar)

Figure 14.14 (a) Sorption of CO2 by PMMA at 315 K. (b) Volume change on mixing of PMMACO2 mixture as a function of pressure. Curves calculated using the MFLG EOS (22); points are experimental data of Liau and McHugh (21). Here, mole refers to mole of liquid-phase mixture. 0.60

(a)

04 o c o

0.45

o

g "

0.50

0.15

0.00

Fo

~~o Pressure

t.-.. >

~o

2oo

~o

200

(bar)

1.o

>

<.2.

o.8

(b)

g c

~(.2

o.6

--~ 0 >

0.4

C

o ._

0.2

0.0

5o

~oo Pressure

(bcr)

Figure 14.15 Experimental data and model calculations at 42°C for: (a) fraction of CO 2 absorbed by crosslinked poly(dimethyl siloxane) and (b) fraction volume change of PDMS due to swelling by CO2 as a function of CO2 pressure. Calculations were performed using the MFLG EOS with an elastic contribution to account for crosslinking (118).

562

where /qz ,

and (v'/ are the com ositional,y-averagea-contact-site, interaction-energy

and segment-volume parameters calculated as follows:

(qz)

= ZqlX1 + gq2x 2

()(/4

e* v*

= oZe*v

I)(I E*

V*

=.

,-~2 * *2 Ui ,~llVl -

(14.130)

,,4

+ 20102s12vlz + ~2c22v2 +

- * *2 20102ElEV12

+

(14.131)

,-~2 * *2 U:2~'zzV2

(14.132)

F~ii * and v* are pure-component characteristic parameters, 6~2 and v~2 are binary parameters, and 0 i are surface-contact site fractions defined as: where

xlql ; zqi =ri(z-2)+2 (14.133) Xlq 1 + x2q 2 From the above expressions, dimensionless and characteristic pressure, temperature and (specific) volume parameters for the mixture can be defined as 01 = 1-82

=

'

p*

(qz-------~

v

T*

(qz)(e.*)

~ = (v*)

(14.134)

where (r)=xlr ~+x2r2 and (c)=xlq +XzC2 . With the dimensionless variables defined in Equation (14.134), the equation of state for the mixture is identical to Equation (14.63). Like the pure-component equation of state, the occupied-site fraction y is computed from the minimum of the Helrnholtz energy with respect to y, yielding an equation similar to Equation (14.64) except that r and c in Equation (14.64) are replaced by their compositional averages:

ln(1-y)] (r))I(r)-I 3--~) L (r) + y

= -

(1/3)(yv)l/3-21/6y (y~)1/3 _ 21/6y

-

y(2.409 6T~. (y~)2

3.033/ (y~)4

(14.135)

Equation (14.135) and the equation of state for the mixture are solved simultaneously to determine v and y, given temperature, pressure, composition, pure-component parameters and binary parameters. The Helmholtz energy of the mixture is

A/NkBT = XllnX, + x21nx2 - ((r)- l)ln[(z- lif e] + ln(y/(r)) + -~r~C1- y)ln(1- y) v/ -3(c)ln~{v*~l/3I(y'v)l/3-2-1/rY]~-[\ / j

(14.136)

-~[~f~-)-~(c)yI2'409 (yf~)4]l'Oll

The first five terms in Equation (14.136) result from using Flory's combinatorial factor for a (random) ternary mixture of two r-mers and vacancies. The first two terms of Equation (14.136) look like the ideal entropy of mixing because the Helmholtz energy is expressed per molecule rather than per segment. The calculation of compositional derivatives to determine

563 quantities such as the chemical potential and spinodal conditions is difficult because the equation of state is given by two simultaneous equations. As a result, numerical techniques were developed (120). Nies and Stroeks (43,121) have modified the SS EOS for mixtures just as they did for pure fluids, i.e., by incorporating higher-order approximations of lattice thermodynamics. In their first modification (121), Guggenheim's combinatorial factor replaces Flory's combinatorial factor for randomly mixing r-mers and lattice vacancies. In a later version (43), they go further by incorporating the quasichemical approximation for nonrandom mixing.

14.3.6 Statistical Associated Fluid Theory (SAFT)

For mixtures whose pure components are described by hard-sphere-chain reference systems, the general form of the equation of state is the same as that for pure fluids

pk B T= pk B T ref

14137 pk B T

pert

Using SAFT, extension of the reference equation of state, Equation (14.80), to mixtures is straightforward:

IPkBT )P

m

- l + p ~ xixjrirjbogij(dij )ref

i,j

(14.138)

xi(r i - 1) p "

0[3

where x i = N i / N is the number fraction of molecules, ri is the number of hard spheres per molecule of component i= 1,2,...,m. In Equation (14.138), b,y is the second virial coefficient of hard-sphere mixtures and go.(d~) is the i-j pair radial distribution function of hard-sphere pairs ij at contact, as calculated from the Boublik-Mansoori-Carnahan-Starling (BMCS) equation for hard-sphere mixtures: go(dij+)=i 1 +-3 ~'0 -~1 ~:~ - 7"/ 2 (1-/7)2 2 (1-17) 3

(14.139)

where 7?is the packing fraction for hard-sphere mixtures m

/7=-~i xiribi

(14.140)

where bi is the second virial coefficient of a hard sphere of component i, given by Equation (14.81). The variable r/is a particular form of the reduced density. A more general reduced density is

~'Y 4~, bg )

,

(14.141)

564 No mixing rules are necessary. Further, a combining rule for bO. is not necessary because hardsphere diameters are additive 2zc d3 bij =--~--..ij;dij = (d i + dj)/2

(14.142)

However, mixing rules are needed to extend Equation (14.82) to non-athermal mixtures. Huang and Radosz (122) used van der Waals one-fluid (vdW1) mixing rules to calculate the molecular-size parameter r for mixtures m

r=~ xixjr~j

(14.143)

U where r0=(ri + rj)/2

(14.144)

For the molecular-energy parameter u for mixtures, Huang and Radosz (122) have used two methods to derive the mixing rules; the van der Waals one-fluid approximation (vdW1) rooted in conformal-solution theory and a volume-fraction (vjO approximation similar to groupcontribution equations of state. Both sets of mixing rules have been tested to correlate vaporliquid phase-equilibrium data of binary polymer solutions as well as fluid mixtures (122-127). The vdW1 mixing rule for the molecular-energy parameter u for the mixture is

u kl3V

ij

xixjrirjcr3ij(Uo• / kBT) m

(14.145)

Z xixjrirjcr3ij ij

where the size and energy parameters crO. and ug are given by

~u=(~i + ~,j)/2 uu=(uiiu~) 1/2( 1 - k o.)

(14.146) (14.147)

where k/j is an adjustable binary parameter that is non zero only when i ~ j . The volume-fraction mixing rule to calculate the molecular-energy parameter u for the mixture is given by

kBT=~ qOi~oj~~BT)

(14.148)

where the volume fraction ~oi is defined as xiri°'3 (fit--

m 3 Ejxjrjcrj

(14.149)

Wu and Chen (127) have applied the SAFT EOS with the vf mixing rule to correlate vapor-liquid phase-equilibrium data for some polymer solutions. Figure 14.16 shows comparison of calculated activity coefficients of cyclohexane in polystyrene (M=440000) at

565 44°C using the SAFT EOS with experimental data. Also shown are results from other models, including the Bonner-Prausnitz EOS (128), the UNIFAC-FV activity coefficient model (129), and the group-contribution lattice fluid (GCLF) model (130). The SAFT EOS appears to give the best correlation. Its superiority to the UNIFAC-FV model is not surprising because that predictive model contains no adjustable binary parameters. Figure 14.17 shows a comparison of calculated results for gas solubility in polyethylene using the SAFT EOS with experimental data; here Wu and Chen (127) used their generalized binary parameter correlation with temperature. The SAFT EOS was also applied to supercritical and near-critical solutions of polymers (123-126); the vdW1 mixing rule was used. Figure 14.18 shows experimental and calculated pressure-temperature cloud-point curves for alternating poly(ethylene-propylene) (PEP) + propylene for different PEP molecular weights, taken from the paper by Chen et aL (124). The SAFT EOS can correlate the experimental data shown in this figure with use of the binary parameter kU that depends on, and can be correlated against, the polymer molecular weight. Also shown in this figure are the predictions for PEP10k (M=10000) + propylene and PEP15k + propylene using the previously established empirical correlation for kU (124).

8.0

-t

6 7.0

6.0

•

\ d~\ a ~. ~ \

b[~~\ \\

5.0

a . b -- c ...... d ~

Data of K r i g b a u m and G e y m e r (1959) ] BP EOS (fitted) [ U N I F A C - F V (predicted) / G C L F EOS (fitted) / SAFT EOS (fitted) J

\\

C.

2 <

3.0

"--.-- ..

\

2.0

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Weight Fraction, Solvent

Figure 14.16 Comparison of experimental data (172) with calculated activity coefficients of cyclohexane in polystyrene (M=440000) at 317.15 K from various molecular-thermodynamic models (127).

566 0.05

~[-

.

0.04:

E

.=_ rG

003!

~ •

.0_ .,,,,, L #,,,

002

,,,,.

._= 0 0t //// /

0 O0

•

/

Data

of

et al.

Lundberg

(1969)

SAFT EOS

', . . . . . . . . 0

, ......... , ......... 200 4-00

, ......... 600

800

Pressure (bar)

Figure 14.17 Comparison of experimental (173) and calculated solubilities of methane and nitrogen in polyethylene using the SAFT EOS at 461.4 K. The binary parameter is calculated from a temperaturedependent correlation (127). 5O0

,A

26k

400

lSk lOk a~

_

5.91<

"

300

~

t0k

200

7~o

-

0 -t50

-I00

-50

0

50

lO0

150

200

250

Temperature (°C)

Figure 14.18 Experimental (124) and calculated p-T phase boundaries for alternating poly(ethylenepropylene) (PEP) + propylene for different PEP molecular weights. Calculations use the binary interaction-parameter correlation developed for this system. For the PEP10K + propylene system, SAFT predicts a steep upper critical solution temperature (UCST) curve with an upper critical end point (UCEP) near-80 °C (123).

567

14.3.7 Perturbed Hard-Sphere-Chain (PHSC) The PHSC EOS, Equation (14.85) is extended to mixtures as follows P .=l+9~xixj~rjbugo(dij)-Zxi(r i-l) 9kBT i,j i

ii(dii)-I

P xixjrirja~i kBT "

(14.150)

As with Equation (14.85), the first three terms in Equation (14.150) represent the reference equation of state for hard-sphere-chain mixtures and the last term is a van der Waals-type perturbation for attractive forces. For each unlike pair of components (i 4: j), additional parameters, b o. and a u , are needed for the mixture. Their physical meanings are similar to those for pure fluids; b u is the cross-second virial coefficient of hard-sphere mixtures and a u reflects the strength of attractive forces between two unbonded segments. In principle, a combining rule is not necessary for calculating b o. because hard-sphere diameters are additive dij(T)--[di(T ) +

dj(T)]/2

(14.151)

The effective hard-sphere diameters d i and d j , for pure fluids i and j at temperature T, respectively, are calculated from Equation (14.87). An expression for b o. follows from this additivity 2~ 3 bij(r):-~--dU(r)= -~l(b)/3+by3~

(14.152)

An expression for a~j can be obtained by extending Equation (14.86) to mixtures: 2~r 3 F.a (kBT / % ) au(T)=--~crueu

(14.153)

where cr,j is given by Equation (14.146) and c• which is associated with the pair potential between unlike segments; cru is given by Equation (14.146) and %. is given by eij=(,ziiejj )1/2(1 - k~)

(14.154)

Therefore, only one binary adjustable parameter k,j is required. The PHSC EOS has been shown to represent all common types of fluid phase diagrams of binary polymer mixtures, including vapor-liquid equilibria (76) and liquid-liquid equilibria (73). However, these predictions are for model mixtures. Vapor-liquid equilibria for real systems are represented quantitatively but such representation is not difficult, even a cubic equation of state can do that as shown by Tassios and co-workers (131,132). A much more difficult test for an EOS is quantitative representation of liquid-liquid equilibria. Song et al. (73) first used the PHSC EOS for a molecular study of liquid-liquid phase equilibria (LLE) of model binary polymer solutions and blends with special attention given to the effects of polymer molecular weight and pressure. Figure 14.19 shows the liquid-liquid phase diagram for several polymers of varying molecular weight dissolved in their own monomers. Because the monomer and polymer

568 segments are identical, their segment size and energetic parameters are identical and the pairinteraction parameter k]2 is zero. The only difference is the chain length of the two components, leading to the LCST (73). However, as soon as a significant asymmetry in the pair interactions is introduced, both UCST and LCST behavior appear in solvent-polymer mixtures, as illustrated in Figure 14.20. A nonzero value of k]2 introduces asymmetry in the pair interactions. It also shows that increasing polymer molecular mass decreases the LCST and increases the UCST; eventually they merge to form hourglass-shaped phase diagrams upon increasing polymer molecular weight (73). Figure 14.21 illustrates that pressure can have a large effect upon the type of phase diagram, especially for LCSTs, for a 3-mer/300-mer mixture. As pressure rises, LCSTs increase significantly and UCSTs decrease slightly. At pressures well above the critical pressure of the solvent, LCSTs are no longer present below the critical temperature of the solvent, but UCSTs remain. Figure 14.21 also shows that hourglass-shaped phase diagrams can be formed upon decreasing pressure (73).

1.0

t]

~ i ~ , ,

I ~ , i

J , , , I , , , , I

J'

~ 1 , ~ '

~1

' ~

-

0.9

0.8

T T 1

-_,_.. . . . . . . . . . . . . . . . . . . . . . . . . . . . ~

2.22._2..:..:..;

....

_ _ _ _ _ _

~

---

r., = 1 0 0

..... ,-~ = 50o

0.7

=

--

......... r~ = 1 0 0 0

-

--,-~-

-

2

5000

--

-

0.6 _

0.5

Ii =

1

-

"

e22/iEll = 1 . 0

O"22/O"11 = 1.0

-

2

-

k12=0

-

-,

0.4 0.0

P/Pcl =0

-

, , , I , , , ~ I , , , , I , , j , I , , , , I , , , , I , , ,

0.1 Segment

0.2 Fraction

0.3

, I , , t~s

0.4

Polymer

Figure 14.19 Liquid-liquid phase diagram for several different-chain-length polymers mixed with their monomers, calculated from the perturbed hard-sphere-chain (PHSC) equation of state. Since the polymer segments are identical to that of the monomer, only lower critical solution temperature (LCST) behavior is observed at high temperatures approaching T¢1,the gas-liquid critical temperature of the pure monomer (73).

569 1.0

0.8

i T

LI

/

---

['r:

,00[

0.6

0".4 -

e....,_/~1 = 1 . 0

-_-

o'~/~,

=

.

~:oo~5 pm,:o v

0.2 lr

iT,vvlv

'''[~v''l'~'~'l

0.1

0.0

~ I I'I''

0.2

v ' ] v ~ ~ T ! ' ~ : ~-]

0.3

0.4

0.5

S e g m e n t Fraction P o l y m e r

Figure 14.20 Monomer-polymer mixtures exhibiting both UCST and LCST behavior, calculated from the PHSC EOS. As the molecular weight of polymer rises, the LCST decreases and the UCST increases and eventually merge to form hourglass-shaped phase diagrams (73).

1.0

,

,

,

i

~

,

,

,

i

,

,

,

,

rI = 3

1

,

,

,

,

i

,

,

,

,

i

,

,

,

,

r,.=300

e,-~/~l = 1.25

cr~/ott = 1.0

~ =0 0.8 .,.,.-.

T

T I

0.6

:,'- ........ .\,,

..................... 112; ........ ' ......

...........

.....

-

I o.4

0.00

///.-" . . - ' ° " ........ ~:o . ............ . . . .:. . . . . . ~

0.05

0.10

0.15

0.20

0.25

0.30

Segment Fraction Polymer

Figure 14.21 The effect of pressure on the phase diagram for a 3-mer/300-mer mixture, calculated from the PHSC EOS. As pressure falls, the UCST and LCST can merge to form hourglass-type phase diagrams (73). Tel and Pc1 are the gas-liquid critical temperature and pressure of the 3-mer.

570 Song et aL (75) show that the PHSC EOS can quantitatively correlate experimental liquidliquid equilibrium data for some polymer blends. They relaxed the hard-sphere additivity rule for calculating b o. to introduce a binary size parameter )tU • bo (T)=

3

(14.155)

CroFb( k B T / z i j )

where o'0.-~_1 (cri + crj)(1- ~:/)

(14.156)

Thus each binary has two adjustable parameters 2,0. and k 0 . Figure 14.22 shows comparison of calculated UCST coexistence curves with experimental data for polystyrene (PS)-polybutadiene (PBD) blends. Parameters )tlz and klz are determined by matching the experimental UCST (133) with that calculated at the molecular weight of polystyrene (Mps) =2300. Agreement is excellent but very sensitive to the choice of binary parameters. The dashed curve for Mps=3450 is not a correlation but a prediction (75).

200--ililIIii~]llllli~j~llJi4141~i[illllii~ljli~i,ili_ 180 -

16o:-

' "'.,,,

..-""

-

=,....,

12o[- ,,]. / . I-/ 100

[i

/

,

! !

I P

s

L

0.0

0.2

0.4

0.6

0.8

1.0

Weight Fraction, Polystyrene

Figure 14.22 Phase diagrams for polystyrene (PS)-polybutadiene (PBD) blends. Points are experimental data (133). Curves are from the PHSC EOS. The experimental critical temperature and composition at the molecular weight of polystyrene Mes = 2300 (0) are used to determine binary parameters k12 and Z12. The pure parameters of PBD are slightly adjusted to achieve better agreement with mixture data. For Mps =3450 (&), the dashed curve at is a prediction (75).

571 The PHSC EOS was applied to correlate vapor-liquid equilibrium data for polymer solutions by Gupta et al. (76). Figure 14.23 compares theory and experiment for the chloroform/polystyrene system. A binary size-interaction parameter, 212, has been adjusted at 50 °C and then prediction was made at a lower temperature, 25°C. Here binary energyinteraction parameter, k12 has been set equal to zero. Similar results were obtained for many other binary systems. The PHSC EOS can successfully correlate and predict VLE data as shown. However, quantitative representation of polymer-solvent liquid-liquid equilibria is not possible when the pure-component parameters for the polymer are obtained from p VT data. To represent such equilibria it is necessary to use some mixture property to obtain purecomponent polymer parameters (78). Shown in Figure 14.24 is a Shultz-Flory plot (reciprocal of the UCSTs and LCSTs vs. [rl/2+(2r) -1] where r = r2/rl) for t-butyl acetate and polystyrene. The intercepts on the vertical axis (i.e., r ~ m) are the reported upper and lower theta temperatures (i.e., the upper and lower critical solution temperatures for a polymer of infinite molecular weight). For liquid-liquid equilibria, it was found that the pure polymer parameters determined from p VT data could not be used to correlate the molecular weight dependence or the upper or lower theta temperatures, even by adjusting both binary parameters 212 and k12. For the t-butyl acetate/polystyrene system, polymer parameters and one binary interaction parameter (k]2) were determined from the reported upper and lower theta temperatures, one UCST and by setting the segment diameter of the polymer equal to that of the solvent. The UCSTs for other molecular weights and the LCSTs for all finite polymer molecular weights are predicted. In Figure 14.24, the pressure of the system is the vapor pressure of the solution; the upper and lower theta temperatures for this system have also been measured at elevated pressures up to 50 atm. With parameters determined from low pressure data, Figure 14.25 shows that the PHSC provides an excellent prediction for behavior at high pressure. 80

I

I

i

1

f

60

50°C

40

_

qL...o-, o-. -..e- •

20

I,

,0 , t

o"

0

25 °C -

/

predicted

1

0

~

!

I

,

f

0.8

0.2

0.4

0.6

Weight

Fraction,

Solvent

1

Figure 14.23 Vapor-liquid equilibria for polystyrene-chloroform solutions; molecular weight of polystyrene Mps = 290000 . The points are experimental data (174). The curves are from the PHSC EOS. The experimental data at 50 °C are used to determine the binary parameter (~,]2= -0.0234). k]2 is zero (76).

572 0.0045

0.0040

--

~

1

0

-

i

o

0.0035 " "" -

[.-,

~'~

0.0030

UCST •

LCST []

o

F!

......

S a e k i , et al. ( 1 9 7 6 ) Bae, et al. ( 1 9 9 1 ) Calculated I

"i

....................~__... 0.0025

i

.....

"-------....

,d

c1

~,

0.0020

0.00

0.01

0.02

0.03

r -m +

0.04

0.05

0.06

(2r)4

Figure 14.24 Shultz-Flory plot of upper and lower critical solution temperatures of t-butyl acetate/polystyrene solutions. The points are experimental (175,176) and the curves are calculated using the PHCS EOS (78). Here where 2 is the polymer. 501

.

i

I

i

140

i

40[-

...-"" [] 1110200v

oG v

J

30

.-"

Expt. Calc.

/

" []

eu 0L

....

.6800

201-

1010

t

1.0

I

2.0

I

3.0

t

4.0

t40

5.0

Pressure (bar) Figure 14.25 Pressure dependence of the upper (left axis) and lower (fight axis) theta temperatures for t-butyl acetate/polystyrene solutions. The lines are predictions using the PHSC EOS (78), using parameters determined from low-pressure data.

573

1.o~,.~,,~.,,:~,,=,~,=~,l,x-,,~,=,,i~<,,,,..r Data are fr°m Braun et al" ( 1 ~

"j

'~'~

....

:~

...... .-'"'"

¢)

0._.

0.8

o-'""

.-

=

¢b ~m Ca~ =-

....: ..... oo

o

o

o

......- /

o., F......

/ o ° °

E

. . . . M o..~- l---~--~'c

°

o

°

°

o

00~ . . . . 0.0

i ......

//

"

.

./

....... . = , , 0 . C o

...........

/.

....

f'"" o ..-.

o

;

/ "

~'

•

.

;

,'"" 4 ....

0.2

I .....

-;I

0.4

.....

t ~ ....

!,,./,

0.6

~.,,

,~

I , ,-,,

0.8

i ....

1.0

Volume Fraction, Styrene Component 1

Figure 14.26 Comparison of theoretical miscibility maps with experimental data (177) for mixtures of type containing poly(styrene-co-butyl methacrylate) random copolymers. The PHSC theory predicts that immiscibility is caused by LCST behavior. Data: (O) miscible at 25 and 180°C; (A) miscible at 25°C but immiscible at 180°C; (O) immiscible at 25°C and 18°C (74). Relative to other theories, the PHSC theory has a possible important advantage: the spherical-segment diameter of one component need not be the same as that of another component. Further, within a given component, the segment diameters need not all be the same. It is this feature which makes the PHSC theory attractive for mixtures containing copolymers. Hino et al. (74) have applied the PHSC EOS to calculate coexistence curves and miscibility maps for some copolymer mixtures. Figure 14.26 compares theoretical miscibility maps with experiment for mixtures of poly(styrene-co-butyl methacrylate) random copolymers differing in copolymer compositions. In this system, the miscible area decreases as temperature increases because immiscibility is caused by LCST-type phase behavior. The PHSC theory is able to predict miscibility maps of systems where immiscibility is caused by LCST behavior (74). The PHSC theory is applicable to multicomponent mixtures containing polymers or copolymers without further approximations. Song et al. (77) have used the PHSC EOS to calculate liquid-liquid phase diagrams of ternary mixtures containing polymers. Figure 14.27 shows the LCST phase diagram of a temary mixture, which can be calculated only by using an equation of state. This system consists of one polymer, one solvent, and one non-solvent. In this model mixture, the only difference among the three components is their molecular mass. Since immiscibility is caused by LCST behavior, an increase in temperature raises the size of the two-phase region.

574 1 Solvent

. . . . . . .

•

.

Nonsolvent 2

Polymer 3

p/pc1---0.1

rl=2

r2=i " r3=500

cr11=a22=a33

El I=E22=E33

K12=KI3=K23"-0

Figure 14.27 Phase diagram for a model system at constant temperatures and pressure for a temary system with partial miscibility of components 2 and 3 (e.g., a nonsolvent and a polymer) and complete miscibility of both with component 3 (a good solvent for the polymer). The PHSC theory predicts that immiscibility is caused by LCST behavior (77). Tel is the gas-liquid critical temperature of component 1. 14.3.8 Further Developments on Sphere-Chain Models

Since the pioneering work of Dickman and Hall (48), sphere-chain models have become a very active field in polymer thermodynamics. Numerous authors have contributed toward finding new equations of state, modifying the existing equations of state and extending the results for pure components to mixtures. Hall and co-workers have extended their generalized Flory-dimer theory for hard-sphere chains to mixtures (134,135). Honnell and Hall (134) derived a GF-D equation of state for binary hard-sphere-chain mixtures using a conformal-mixing rule, such that the mixturecompressibility factor is set equal to the mole-fraction average of the compressibility factors of the pure components at the same total packing fraction of mixtures; however, calculations are reported only for the case where all segment diameters are the same. Wichert and Hall (135) derived a generalized Flory equation of state for hard chain-hard monomer mixtures, where the monomer diameter is different from the chain-segment diameters. These equations of state compare favorably with computer-simulation results. Schweizer and Curro (136-138) developed a reference-interaction-site model (RISM) integral-equation theory for polymer liquids and mixtures, based on the RISM integralequation theory proposed by Chandler and Andersen (139) for small, rigid-molecular fluids.

575 Application to polymer systems focused on local structure and radial-distribution functions for nodel chain molecules. The RISM theory requires considerable numerical computation and no 2nalytical equation of state for polymers is likely to be forthcoming in the near future. Ghonasgi and Chapman (140) developed an improved SAFT (ISAFT) equation of state for aard-sphere chains by utilizing available information about the dimer; the same result was also 9btained by Chang and Sandler (141). With the site-site correlation function at contact for hard _-_tispheres obtained by Chiew (142) from the Percus-Yevick integral-equation theory, the ISAFT equation of state is in better agreement with simulation data than the SAFT equation of state for hard-sphere chains, especially at low densities. Further, by adding a correction term to Chiew's expression for the site-site correlation function at contact for hard dispheres, the ISAFT equation of state shows good agreement with simulation data at high densities as well. Ghonasgi and Chapman (143) also developed an extension of the SAFT theory to polymer solutions and blends; a polymer is modeled as a flexible chain made of bonded spherical segments; non-bonded segments interact through the Lennard-Jones potential. These studies stressed comparison between theory and simulation; they were not yet directed at reduction to practice for engineering applications. Banaszak et al. (144,145) extended Wertheim's first-order thermodynamic perturbation theory (53) to sticky chains, square-well chains, and Lennard-Jones chains. Their purpose was to examine the effect of chain connectivity on those parts of the compressibility factor and critical temperature which are due to attraction. They demonstrated that the effect of chain connectivity is to reduce the attraction contribution to the compressibility factor and critical temperature. In addition to the development by Song et al. (51) using Chiew's result for hard-sphere chains and mixtures, Malakhov and Brun (146) extended Chiew's theory to the general case of a solution of heterochains differing among themselves in both length and primary structure. Mitlin and Sanchez (50) considered an improved formulation for hard-sphere chains under the Percus-Yevick approximation; their result is the same as Chiew's for linear hard-sphere chains. However, it was shown that their theory failed in the low-density limit for cyclic molecules such as triangle molecules.

14.3.9 Cubic Equations of State for Polymer Mixtures Cubic equations of state have been widely used for engineering applications. Among the most popular are variations on the Redlich-Kwong equation (147) [especially the SoaveRedlich-Kwong equation (148)], and the Peng-Robinson equation (149); they can often successfully correlate vapor-liquid equilibria for normal fluids and mixtures. However, application of cubic equations of state to polymer mixtures is not immediately obvious because the conventional method of evaluating pure-component cubic-equation-of-state parameters requires vapor-liquid critical properties and vapor pressures which are not available for polymers. Orbey and Sandier (150) applied the Peng-Robinson (PR) equation of state, coupled with the mixing rule proposed by Wong and Sandier (151), to correlate vapor-liquid equilibrium data for some polymer solutions. For pure solvents, they used the conventional method to determine EOS parameters from critical properties and the Pitzer acentric factor. For polymers, however, to determine EOS parameters, they chose an arbitrary fixed value of 10-7 MPa for the vapor pressure and then used experimental melt densities or the densities of the glassy polymers (if the temperature of interest was below the glass transition). As expected, the EOS

576 parameters a and b determined in this way for polymers are at least slightly molecular-weight dependent. To extend their approach to mixtures, Wong and Sandier (151) mixing rules give equation of state parameters a and b in the mixture, such that, while the low-density quadratic mole fraction dependence of the second virial coefficient is satisfied, the excess Helmholtz energy at infinite pressure from the equation of state is set equal to that given by an appropriately chosen liquid-phase activity coefficient model. Orbey and Sandier used the Flory-Huggins expression. Figure 14.28 shows a correlation for partial pressure of chloroform in polystyrene (M = 290,000), taken from Figure 1 of their publication (152). Kontogeorgis et al. (131,132) have used the van der Waals (vdW) equation of state to correlate vapor-liquid equilibrium data for polymer solutions. They proposed a method for evaluating the two vdW EOS parameters, a and b, for polymers from two low-pressure volumetric data. Both a and b (assumed temperature independent) can be expressed analytically using two experimental molar volumes each at a different temperature. The pressure in the vdW EOS is set equal to zero. Both a and b are linear functions of polymer molecular weight. For application to polymer solutions, they used the van der Waals one-fluid mixing rules for both parameters a and b and the classical combining rules: a~i=(aiiajj ) 1/2(1-ku) (14.157)

b~j(T)=[bg (T) + b s(T)]/2

(14.158)

where k o. is the binary adjustable parameter. With only one adjustable binary parameter, the van der Waals equation of state is capable of correlating the equilibrium pressures for various polyethylene (PE) and polyisobutylene (PIB) solutions with good accuracy. However, unrealistic, large negative values for k O. parameters are always required, indicating that their procedure, while empirically successful, has no physical basis. Sako et aL (153) developed a three-parameter cubic equation of state based on the generalized van der Waals theory coupled with Prigogine's and Beret's assumptions for external degrees of freedom as equivalent transitional degrees of freedom, using parameter c. Upon using the c parameter, their cubic equation of state is perhaps more applicable to polymers than the conventional cubic equations of state which model all molecules as spheres. The Sako et aL equation of state was successfully used to describe ethylene-polyethylene phase equilibria at high pressures. These studies show that it is simple to correlate low or intermediate-pressure vapor-liquid equilibria for many-common polymer solutions using at least one binary parameter, which is often temperature-dependent. Because no polymer exists in the vapor phase, only the chemical potential of the solvent is required; the chemical potential of the polymer plays no role. However, calculation of liquid-liquid equilibria (or vapor-liquid equilibria at very high pressures) requires the chemical potentials of both solvent and polymer. Comparison of calculated and observed vapor-liquid equilibria at normal pressures provides no meaningful test for a theory because essentially any theory can pass that test. However, a quantitative comparison of calculated and observed liquid-liquid equilibria provides a more meaningful test of the theory. At present, we have no EOS that can quantitatively represent liquid-liquid equilibria for polymer-solvent systems using only one adjustable binary parameter.

577 0.80

0.70

s

0.60 e

13. s v

"~

0.50 -

a a e o

(9

13.. '~ 1=:

o

a a

13_

E

50 °C

o

0.40

a a

0.30-

0 L_. 0 m c-

O

0.20 -

0.10

0

0.0

I

I

I

I

0.2

0.4

0.6

0.8

1.0

Weight Fraction, Chloroform

Figure 14.28 Correlation of partial pressure of chloroform in polystyrene (M=290000) by the PengRobinson equation of state with the Wong-Sandler mixing rule using the Flory-Huggins theory as a boundary condition. Experimental data are shown by (O,Q) (174).

14.3.10 Specific Interactions Specific interactions (energetically-favorable, orientationally-dependent interactions), such as hydrogen bonding, sometimes play an important role in polymer thermodynamics. In polymer blends, specific interactions between dissimilar components are often the only means for promoting compatibility. Polymer blends are inherently unstable because the stabilizing effect of the entropy of mixing is small compared to the destabilizing effect of dispersion forces and differences in compressibility. Further, specific interactions between molecules of the same component (i.e., self-association) can destabilize a polymer/solvent or polymer/polymer mixture. This section briefly describes a few methods for the incorporation of specific interactions into equations of state for chain-like fluid mixtures. The incorporation of specific interactions is accomplished in one of three ways: assigning a temperature dependence to appropriate pairinteraction energies, chemical association models, and appending an association term to the equation of state.

578 Ten Brinke and Karasz (154) presented a simple method for incorporating specific interactions between components into the Flory-Huggins incompressible-lattice model. This model was later incorporated into the LF EOS by Sanchez and Balasz (155) and that LF EOS was later extended to polydisperse polymer mixtures by An et al. (117). The ten Brinke and Karasz model was incorporated into the LF EOS by replacing the pair interaction energy c 0. in Equation (14.114) by an interaction free energy f,7 :

f ij=o°o.+6o%-kB T

1+c%

]

1 + co~/exp(-&~//kBT)

(14.159)

where &o is the additional (attractive) interaction energy due to the formation of a specific interaction and co0. is the ratio of statistical degeneracy of orientations of segments i and j that are non-specific to those that are specific. Equation (14.159) illustrates the competition between the energetic gain when a specific interaction is formed and the entropic penalty needed for the molecules (or segments) to achieve a proper orientation. The LF EOS with this modification can qualitatively demonstrate closed partial-miscibility loops. Also, it can produce a wider spinodal curve for polystyrene/poly(vinyl methyl ether) blends, in better agreement with experiment, but at the expense of additional adjustable parameters. The ten Brinke and Karasz model was also incorporated into the PHSC EOS (73), but in a slightly different manner. In this case the interaction energy %., is replaced by an average interaction, ~-0

e-,j~ig.+(ij&ij

(14.160)

where (0 is the fraction of/j pair interactions given by 1

(u-1 + co/jexp(- &~//kBT)

(14.161)

Parameters coo and &O have the same physical meaning as above. The modified PHSC EOS can qualitatively represent closed partial-miscibility loops (73) and improve representation of the spinodal curve for polystyrene/poly(vinyl methyl ether) blends (75). A second method for including specific interactions into the equation of state is provided by a chemical theory. Association complexes are considered to be new species (or a distribution of species) in the mixture (156,157). All association reactions are characterized by an equilibrium constant. For example, using the LF EOS, Panayiotou and Sanchez (156) considered self-association interactions between solvent molecules (A), characterized by an equilibrium constant K a , while cross-association between solvent (A) and polymer (B) is characterized by an equilibrium constant KAB :

Ai+Aj< KA >Ai+j B+Ai< KAB .~BAi

579 Hence, solvent self-association complexes are represented by homopolymers of A and polymer-solvent complexes are represented by graft copolymer of A and B. The association complexes, when formed, interact with each other only through dispersion forces. It is also assumed that the parameters which characterize pure molecules A or B remain unchanged regardless of whether the polymer exists as a free molecule or in a complex. To illustrate, let us consider hydrogen bonding. The equilibrium constants can be expressed in terms of Gibbs energies of hydrogen-bond formation: KA=exp(- AG°/RT)

(14.162)

K ~ =exp(- AG°B/RT)

(14.163)

The Gibbs energies of hydrogen-bond formation, AG° and AG° can further be decomposed into the energy, volume and entropy changes for hydrogen-bond formation and hence provide three additional parameters for each type of association. Figure 14.29 shows excellent representation of the effects of cross-association on the enthalpies of mixing, volume of mixing and vapor pressures of mixtures of poly(ethylene oxide), or poly(propylene oxide) with chloroform. Using the LF EOS and a more general description of association interactions, Graf et al (157), considered blends of methacrylate-based and styrene-based polymers and copolymers with poly(vinyl phenol). The third method for incorporating specific interactions is summarized by:

PkBT = pkB T

/ phys

pkB T

14164 assoc

where the first term is the original EOS without effects from association and the second term is added to include the effects of association interactions, such as hydrogen bonding. This method differs from the others where the form of the original equation of state is modified or altered in some way. Here, the form of the original EOS is unaltered. Two association terms are currently available. One association term is an extension of the combinatorial expression of Veytsman (158-160) and the other comes from the associating form of the Statistical Associated-Fluid Theory (SAFT) (46,63,66,122,161,162). The association contribution of Veytsman (158), originally introduced into the LF EOS by Panayiotou and Sanchez (159) and recently added to the PHSC EOS by Sch/ffer et al. (163), does not invoke the existence of separate associated species, but instead, emphasizes the enumeration of all possible distributions of hydrogen bonds between segments. Additionally, this model makes a distinction between hydrogen-bonding donor and acceptor groups. Veytsman's result was originally derived for an incompressible mixture. For application to an EOS, it is also necessary to make an additional assumption: the probability that the acceptor and donor group are within the spatial proximity to form a hydrogen bond, is proportional to density. The modified Veytsman model requires three parameters associated with the energy, entropy and volume of hydrogen-bond formation. In some cases, the association volume can be neglected (160,163). Veytsman's method was also used by Panayiotou and Sanchez (159) to describe the enthalpy of mixing, volume of mixing and vapor pressures of mixtures of poly(ethylene oxide) with chloroform. In addition, the association model [Equation (14.164)] was used to show the effect of including hydrogen-bonding interactions between a modified polystyrene (PS) and poly(vinyl methyl ether) (PVME), as shown in Figure 14.30. This figure shows the LCSTs for

580

--

03

0

....

i i i l l l l l

1

E

0 "D

-30

LU

-r

0

J

1

1

I

I

I

I

t

0.5

0.0

1.0

Weight Fraction, Polymer

-1.0

i

i

I

i

[

I

I

1

I

1

!

i

[

I

i

I

i

0

03

E E -0.5 0

LU

>

i

!

0.0 0.5 1.0 Weight Fraction, Polymer 10<.~,,

I

I

I

i

I

I

i

I

] ~

~

. 0 0.0

~

\,,,, . . . .

,

0.5

,

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Weight Fraction, Polymer

Figure 14.29 Experimental and calculated (a) enthalpy of mixing, (b) volume of mixing, and (c) vapor pressures for mixtures of chloroform and poly(ethylene oxide) at 5.53°C. Solid lines calculated using the Sanchez-Lacombe (LF) EOS with a cross-association model. The dashed curve is the LF EOS without association (156). In part Co), mole refers to mole of polymer-solvent mixture.

581

200

G 170

150 (3.0

0.2

0.5

Molar Percentage, Carbinol Content

Figure 14.30 LCST's for the system poly(vinyl methyl ether) + modified polystyrene versus the carbinol content of the modified polystyrene (159). The carbinol in the modified polystyrene allows hydrogen bonding between the two polymers and greater compatibility. Points are experimental and the curve is calculated using the LF EOS coupled with the association model of Veytsman (158). several PS/PVME blends; the PS was modified by copolymerizing with a small number of vinyl phenyl hexafluoro-dimethyl carbinol monomers. The effect is that the -OH group in the carbinol acts as a proton donor while the -O- group in PVME acts as an acceptor. Hydrogenbonding parameters for the -OH group in the carbinol were set equal to corresponding parameters of the -OH group in 1-alkanols. An altemative to the Veytsman association term is provided by the derivation of the SAFT equation of state for chain molecules (46,53,63,161,162,164-167). To arrive at the EOS for chain molecules, the SAFT model assumes that the interaction energy between specific sites on two spherical segments is of covalent strength. However, this assumption can be relaxed to represent specific interactions by assuming that, when two sites on spherical segments of different molecules (including chain molecules) are in the proper orientation, their interaction is given a square-well potential. The depth of this square-well potential is typically greater than a potential-well depth for dispersion interactions. Additional assumptions include: 1) the repulsive cores prevent more than two segments from bonding at a single site; 2) one site cannot simultaneously bond to two sites on another segment; and 3) double bonding between two sites on different segments is not allowed. Intramolecular association was originally prohibited, but has been considered recently by Ghongasi et al. (168). Like the Veytsman model, the SAFT model can also distinguish between donor and acceptor sites. Two parameters, the association energy and the association volume, characterize the interaction between donor and acceptor sites; these parameters are related to the depth and width, respectively, of the square-well potential.

582 Application of the SAFT association term to real polymer systems has been limited; most published work focuses on application to small molecules and their mixtures (63,66,122) and comparison with simulation (161,162). Sch~ifer et al. (163) appended the SAFT association term to the PHSC EOS to describe the effect of solvent self-association on LLE for 1alkanol/polyethylene mixtures and cross-association on VLE for chloroform with poly(ethylene oxide) and poly(propylene oxide).

14.4 CONCLUSION This review of the polymer-solution literature indicates that there has been much progress in the thermodynamics of polymer solutions since the initial ground-breaking work of Flory and Huggins, more than 50 years ago. That initial work was closely tied to the filled-lattice concept which immediately limited consideration to those mixtures where there is no volume change upon mixing at constant temperature and pressure. To relieve that severe limitation, much attention has been given toward developing an equation of state suitable for polymers, polymer/solvent mixtures and polymer blends. While the limitation of the constant-volume assumption of filled-lattice therrnodynamics was recognized early, a major stimulus to develop an equation of state for polymer-solvent systems was provided over 30 years ago when experimental results indicated the common existence of a lower critical-solution temperature which lies above the upper critical-solution temperature. Classical lattice thermodynamics cannot explain such phase behavior. A first necessary test for any proposed equation of state for mixtures is that it must be able to reproduce phase behavior that includes both lower and upper critical-solution temperatures where the lower critical-solution temperature is larger than the upper critical-solution temperature. All proposed equations of state discussed here can meet that test, at least in principle. Upon introducing specific attraction between unlike components (e.g., hydrogen bonding), filled-lattice theory can be modified to give a lower critical-solution temperature which lies below the upper critical-solution temperature. Such modification can also be applied to an equation of state with the same result. As discussed in the early part of this review, for fluids of chain molecules, equations of state can be derived along several lines: cell theories, where the volume of a cell is not constant but depends on temperature and pressure; lattice-fluid or hole theories, where a lattice contains occupied and unoccupied lattice sites (holes); and free-standing theories (independent of any geometric construct such as cells or lattices) where the volumetric properties of an assembly of chains are described using the integral theory of fluids. Analytical results can be obtained from the integral theory of fluids for hard non-attracting chains. For attractive chains, it is necessary to use numerical methods or else to introduce attractive forces as a perturbation about results for the non-attractive chains. All of these theories can successfully describe the p VT properties of pure liquid polymers using (typically) three or four adjustable molecular parameters which, respectively, reflect chain length, segment size, potential energy between two nonbonded segments and (sometimes) chain floppiness as a measure of external degrees of freedom. However, extension to mixtures shows serious deficiencies in all of these theories. First, these theories fail to account for long-range correlation; when one segment of a chain moves, that movement influences the movement of other segments in the same chain. Second, these theories are for simple chains and therefore do not account for the effect of chain branching.

583 Third, and perhaps most serious, all of these theories are of the mean-field type and fail to account for the contribution of fluctuations in density and composition. It has been known for many years that the mean-field approximation is valid only at high densities and high temperatures. Therefore, when the theories described here are used in the critical region (where fluctuations are large), poor results are often obtained; if model parameters are fixed using critical data (coordinates of upper or lower critical-solution temperatures), the calculated two-liquid region is too narrow; but if parameters are fixed from LLE data remote from critical, erroneous predictions are obtained for critical temperatures and compositions. For polymer/solvent mixtures, reliable VLE can be calculated easily (except when the solvent is near its critical temperature) because such calculation requires only the chemical potential of the solvent; the chemical potential of the polymer does not enter the calculation because polymers are nonvolatile. However, even for VLE calculations, results are often best when the concentration of the polymer is low; binary parameters obtained from dilute polymer-solution VLE are often different from those obtained from concentrated polymersolution VLE. Accurate calculation of LLE is much more difficult for polymer/solvent mixtures because typically, the polymer is dilute in one liquid phase and concentrated in the other liquid phase. Mean-field approximations are not good for a dilute phase. If binary parameters are found for a solution dilute in polymer, they are likely to be different from those found for a solution where the polymer concentration is large. Mean-field approximations introduced in most equations of state are not valid for dilute solutions. Distinguishing between intramolecular and intermolecular segment-segment interactions is not as important in concentrated solutions as it is in dilute solutions. Further, unlike VLE, LLE are often sensitive to the polymer's polydispersity. These problems are not readily observed from model calculations but they become clear when calculated results are compared with experiment. Errors introduced by mean-field approximations are evident when the top of the liquid-liquid coexistence curve is insufficiently flat and whenever calculations of phase equilibria are compared with experiment over a large range of polymer concentration, from very dilute to appreciable polymer concentration. Given a set of molecular parameters, it is possible to fit experimental data for one equilibrium liquid phase but, to fit the other equilibrium liquid phase, a different set of molecular parameters is often required. These problems are minimized when equations of state are used for polymer blends. For mixtures of polymers, good results can often be obtained because errors introduced by mean-field approximations tend to cancel. In general, two conceptually different mean-field approximations are invoked during the development of equations of state for chain molecules, regardless of their framework. To calculate the entropy of a collection of polymer molecules, even in the absence of attractive forces, the number of available configurations must be properly calculated without neglect of correlations between segments of a chain that are not nearest neighbors along the chain. It is this neglect which is the essence of the first mean-field approximation. A second mean-field approximation is associated with calculating the potential energy; in that approximation, chain connectivity and the correlation between segments are improperly ignored. The inadequacy of mean-field approximations was recognized many years ago; it is indicated in the classic text by Flory published in 1953. At present we do not have a useful analytical equation of state for mixtures that overcomes this inadequacy. An alternative to an analytical equation is provided by molecular simulation calculations but these are not as yet practical for routine calculations. The current literature has reported several simulation calculations for non-attractive mixtures of chains; only very recently have such calculations

584 been made for pure-polymer liquids with attractive forces. It is likely that we will soon see simulation calculations also for mixtures of attractive chains and solvents. Such simulations are likely to provide better agreement with experiment over the entire composition range. As computers become more powerful and as more efficient computing strategies become available, simulation calculations may eventually replace analytical equations of state. However, for the near future, analytical equations of state will remain useful for application provided that their use is restricted to narrow ranges of composition or else, provided that they are modified semi-empirically to overcome limitations imposed by the mean-field approximation. It is likely that several reasons may be responsible for the failure of EOS theories to represent liquid-liquid equilibria with good accuracy. One reason may follow from the conventional procedure for finding EOS constants for the pure polymer. That procedure is based on reduction ofpVT data for pure liquid polymers; the characteristic energy parameter obtained from perturbed hard-sphere-chain equations is too large [e.g., Figure 14.5], perhaps because pure liquid polymers have branched chains with extensive entanglement. The characteristic energy parameter, therefore, includes the energy of disentanglement. However, the calculated free energy of mixing of a polymer with a solvent makes no provision for disentangling the pure branched polymer prior to dissolution. Better results for liquid-liquid equilibria may perhaps, be obtained if the mixture calculations include a contribution which reflects the disentanglement of branched polymer chains that accompanies the mixing process. That contribution may be important for polymer-solvent mixtures but probably tends to cancel for polymer-polymer blends. Finally, there is a geometric consideration that is usually overlooked. In a polymer chain, the distance between bonded segments is probably appreciably less than the collision diameter which characterizes the interaction between non-bonded segments. Therefore, the parameter o-used in the reference part of the EOS (i.e. the part for non-attracting chains) is not the same as that used in the perturbation part o f the EOS (i.e. the part which corrects for attractive forces). To refrain from using an excessive number of parameters, it is customary to ignore this difference in the two characteristic length parameters. That procedure does not affect the ability of an EOS to fit pure-polymer data but it is likely to have a significant effect on liquidliquid equilibria for polymer/solvent systems.

Acknowledgment This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy under Contract No. DE-AC03-76SF0098 and, in part, by the Donors of the Petroleum Research Fund, administrated by the American Chemical Society. The authors thank Koninklijke Shell Labs (Amsterdam, The Netherlands) and E.I. du Pont Nemours and Co. (Philadelphia, PA) for additional financial support; Ren6 Reijnhart (Shell) and D.T. Wu (du Pont) for prevailing encouragement, and Doros Theodorou for valuable advice.

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