P V T

R e l a t i o n s h i p s o f

S t a t e

o f

a n d

E q u a t i o n s

P o l y m e r s

Junhan C h o Polymer Science and Engineering Department, Dankook University, Seoul 140-714, South Korea Isaac C. Sanchez Chemical Engineering Department, University of Texas at Austin, Austin, TX 78712, USA

A. Introduction B. Isothermal Compressibility Equations C. Empirical or Semiempirical 3-Parameter Equations of State D. Lattice or Quasi Lattice Models E. Continuum Models F. Tables Table 1. Zero Pressure Volume V 0 (T) and Bulk Modulus B0(T) Table 2. Abbreviations of Polymer Names and the Experimental TemperaturePressure Range for Polymer Liquids Table 3. PVT Properties of Other Polymers Table 4. Characteristic Parameters for the Sanchez-Cho Equation of State Table 5. Characteristic Parameters for the Hartmann- Haque Equation of State Table 6. Characteristic Parameters for the Simple Cell Model Equation of State Table 7. Characteristic Parameters for the Flory, Orwoll and Vrij Equation of State Table 8. Characteristic Parameters for the Simha-Somcynsky Equation of State Table 9. Characteristic Parameters for the Sanchez-Lacombe Equation of State Table 10. Characteristic Parameters for the AHS 4- vdW Equation of State G. References

A.

VI-591 VI-592 VI-593 VI-593 VI-594

VI-595 VI-596 VI-597 VI-597 VI-597

VI-598

VI-599

VI-599

VI-600 VI-600 VI-601

INTRODUCTION

Pressure-volume-temperature (PVT) behavior of polymeric materials are important both from the practical and

scientific viewpoint. On the practical side, a general knowledge of PVT properties are often required for processing polymers. On the scientific side, PVT properties are often required as input information on molecular models of polymer solutions. Information on molecular packing is also reflected in PVT properties. Numerous equations of state (EOS) have been developed to describe polymers. One can divide them into two categories: phenomenological and molecular. Equations of state in the first category are for the empirical representation of volumetric data. The majority of equations in this group are the so-called isothermal compressibility equations (1-4). The purely empirical, original Tait equation and its modification by Tammon have been almost exclusively used for organics, including polymers (2,3,5,6). Sanchez et al. re-examined polymer bulk data in a rigorous classical thermodynamic analysis (7,8). A new principle of temperature-pressure (T-P) superposition of compression response was found. Stated briefly, a dimensionless pressure variable is used to superpose compression data as a function of temperature into a universal curve. The governing parameter of compression is the first pressure coefficient Bx(^ dB/dP)Ps=09 of the bulk modulus B. It is related to the asymmetry of the free energy around its minimum, between dilation and compression. For polymers, B\ is around 10, and universal. A new isothermal equation of state was formulated through a Pade analysis of the pressure dependence of the bulk modulus (8). Both the Tait and the Pade equations describe almost perfectly the isothermal pressure dependence of volume. Both can be used to smoothen experimental PVT data. Equations of state in the second category arise from molecular models and the development of a proper partition function (9). Most earlier works on this group are based on the lattice or quasi-lattice description of a fluid. Three types of models have played a dominant role in the past several decades. These are the cell, hole, and lattice fluid models.

In Prigogine's cell model (10-12), any monomer is considered to be enclosed by the cell of surrounding molecules. The configurational potential is separated into the mean potential (between the center of the cell and surroundings), and the cell potential (generated by the movement of the central monomer within the cell). The cell potential was further simplified to be athermal. The Simha-Somcynsky theory (13) incorporated the lattice imperfection (hole) into the simple cell model by Prigogine et al. Flory et al. (14) borrowed some ideas from the cell model (separation of internal and external degrees of freedom) but used a Tonk's gas-type description for the repulsive part of the intermolecular potential. Unlike the cell model, the molecules are not localized in space, and a van der Waals mean field energy is employed. Meanwhile, the lattice fluid model (15,16) introduces lattice vacancies as pseudo-solvent molecules into the incompressible Flory-Huggins model. In this model, an incompressible polymer solution is converted into a compressible bulk polymer liquid. All theoretical equations of state suggest a corresponding states behavior of polymer PVT properties that requires three scaling parameters such as characteristic temperature (7*), pressure (P*), and volume (V*). The derivation of the partition function can in principle be done in continuum space. This approach yields the so called integral equation theory (9). The self consistent integral equations of the interparticle distribution functions are to be determined. For the simplest hard sphere chains, several analytical equations of state were suggested by Chiew (17-22), Wertheim (23-27) and Chapman (28,29). Chiew's equation of state is based on Baxter's solution of the Percus-Yevick integral equation for the adhesive hard sphere (AHS) system. The AHS has a strong, but extremely short ranged attractive interaction at its surface. This peculiar interaction is treated as chemical bonds. Wertheim and Chapmann considered this adhesive attraction as a perturbation to the hard sphere potential in the framework of graph theory. A different and unique work on this subject was performed by Hall and coworkers (30) without recourse to integral equation theory. A particle insertion method was generalized to hard chains in continuum space. To apply these continuum models to real systems, an energetic contribution to pressure should be added on the basis of a perturbation theory. In the simplest approximation a mean field (van der Waals) energy is added (28,31). The incorporation of the energy term turns these off-lattice models into 3-parameter (V*, T*, P*) corresponding states models. There are empirical or semiempirical equations of state that can be included partly in the two categories. The Sanchez-Cho (32), and the Hartmann-Haque (33) equations are considered the most important in applications. These two equations of state also suggest a corresponding states behavior. This article only deals with liquid data (above Tg or Tm). The mathematical expression of the liquid PVT data with either the Pade or the Tait equations are tabulated here for

extracting the 3 characteristic parameters (V*, T*, P*) for given corresponding states equations of state. The characteristic parameters are essential in the application of a given equation of state to real systems. We present, therefore, a compendium of such parameters for the 3-parameter equations of state described above. These parameters have their own physical meaning according to theories at hand. However, these are treated as adjustable parameters in a general nonlinear regression routine to best represent the smoothed experimental values regenerated from the Sanchez-Cho or the Tait equations. We used the volume data over the whole experimental T-P range. All model parameters are determined from the same data sets.

B. ISOTHERMAL COMPRESSIBILITY EQUATIONS The recently found principle of temperature-pressure (T-P) superposition (7,8) implies that the In(V/V0), is a function of the scaled pressure, P/Bo, as is compression, given below:

(Al) where Vo and Z?o are volume and the bulk modulus at zero pressure, respectively. The right hand side of Eq. (Al) is a rigorous power series expression of ln(V/Vo). The additional contributing numbers B\(i > 0) are the coefficients in the series expansion of the bulk modulus B in pressure: (A2) A tractable approximation to the infinite series in Eq. (Al) was suggested by Sanchez et al. They utilized the empirical observation that the bulk modulus B in Eq. (A2) is linear in pressure to high pressures. Expecting that the modulus B diverges as Pu(u < 1), a Pade analysis and the successive integration of B yielded the Pade equation (8) given below: (A3) where the curvature parameter, co, is defined as (A4) A pair of universal values of B i and co (B i = 10.2, co — 0.9) is observed to be excellent in correlating the bulk volumetric data of various polymers.

The historically well known Tait equation falls into the general statement of the T-P superposition in Eq. (Al): (A5) where C is universally 0.0894 (Cutler's constant). Equation (A5) is the modified version, by Tammon, of the original Tait equation, which was empirically formulated for water. One should notice that the product, CB o, comprises the socalled Tait parameter in the literature. Equations (A3) and (A5) yield an accurate representation (within experimental error) of isothermal, volume-pressure relationships. The zero pressure volume, VQ, and modulus, BQ, are additionally required to complete Eqs. (A3) and (A5). The compilation of these parameters are shown in Table 1. In this table, the abbreviation of polymer names are used. The explanation of the abbreviated names and experimental T-P range are listed in Table 2. Several types of fitting functions are used for VQ. Typically used are the polynomial and exponential expressions. For the bulk modulus, the exponential dependence on temperature is almost exclusively used. Some other polymer PV T data that are not used to obtain the characteristic parameters can be found in the literature, including blends and liquid crystalline polymers (38,50,55,59-67).

pressure, as is shown in Eq. (A6). Equation (A6) is to date the most accurate 3-parameter equation of state. The comparison of Eq. (A6) with the empirical Tait or Pade equation over the whole experimental T-P range gives the average deviation in volume less than 0.0004 cm3/g. The Sanchez-Cho equation is useful in extrapolating data to high pressures and investigating the negative pressure region. The three characteristic parameters for the Sanchez-Cho equation are given in Table 4. Hartmann-Haque (H-H) Equation of State The Hartmann-Haque equation of state (33) has the following functional form: (A8) where the dimensionless variables P, V, f are defined as in Eq. (A7). The P* is identified as the isothermal bulk modulus extrapolated to zero temperature and pressure. The zero pressure isobar in Eq. (A8) originates in the SimhaSomcynsky hole model (13) in the next section. The three characteristic parameters for the Hartmann-Haque equation are given in Table 5. The average deviation between Eq. (A8) and regenerated bulk PVT data (data smoothed using the Tait Eq.) is 0.0009 cm3/g or more than a factor of 2 worse than the S-C equation. D.

C EMPIRICAL OR SEMIEMPIRICAL 3-PARAMETER EQUATIONS OF STATE

Equations in this section are in between the compressibility equations and succeeding model equations of state: (a) they are formulated on the basis of phenomenological arguments; (b) they suggest a corresponding behavior or a universal relationship between dimensionless K T and P The Sanchez-Cho and the Hartmann-Haque equations of state are considered the most important equations in this category. Sanchez-Cho (S-C) Equation of State The Sanchez-Cho equation of state (32) was formulated by combining the empirically observed temperature dependence of the zero pressure volume VQ and modulus BQ with the isothermal Pade equation, Eq. (A3):

(A6) where the required universal numbers are B\ = 10.2, co~ 0.9, and b = 9. The dimensionless variables are defined as: (A7) The V * and P* are related to the van der Waals volume and cohesive energy density (32). The Sanchez-Cho equation provides volume as an explicit function of temperature and

LATTICE OR QUASI LATTICE MODELS

The most frequently used model equations of state based on the lattice or quasi lattice phase space are visited in this section. These are various cell models of Prigogine et al., the hole model, and the lattice fluid model. The common feature of the models in this section is the separation of pressure into the thermal (Pth) and internal (Pi) pressures: (A9) where the dimensionless variables are defined as before Eq. (A7). The physical meaning of the characteristic parameters P*, V*, T* vary among the models. But in general, V* is related to the van der Waals volume, T* is proportional to monomer-monomer interaction energy, and P* is related to the cohesive energy density. Simple Cell Model of Prigogine et al. The cell model by Prigogine et al. (10-12) is an extension of the cell model for small molecules by Lennard-Jones and Devonshire (68) to polymers. Each monomer in the system is considered to be trapped in the cell created by the surroundings. The general cell potential, generated by the surroundings, is simplified to be athermal. This turns the simple cell model into a free volume theory. The mean potential between the centers of different cells are described by the LennardJones 6-12 potential. The dimensionless equation of state has the following form: References page VI-601

(AlO) The factor 2 "1Z6 originates in the choice of the hexagonal close packing lattice as a cell geometry. The factors 1.2045 and 1.011 correct the effects of higher coordination shells on the internal energy. Table 6 gives the three characteristic parameters P*, V*, T* for the Prigogine cell model for the polymers in Tables 1 and 2. The average deviation between Eq. (AlO) and regenerated PVT data is 0.0008 cm3/g. Cell Model by Flory, Orwoll, and VHj (FOV) The cell model by Flory et al. (14) has the simpler mathematical form than the Prigogine cell model: (AU) The Lennard-Jones energy is replaced with a uniform background (van der Waals) energy. The characteristic parameters P*, V*, T* for the FOV theory are shown in Table 7. The average deviation between Eq. (All) and regenerated PVTdata is 0.0022 cm3/g. It has often been pointed out that the fitting performance of the FOV theory is less accurate, especially when covering a large pressure range. However, we present the parameters for the whole pressure range. This will provide a guidance when one fits the PVT data for smaller pressure ranges of interest.

this model. Simha and Somcynsky suggested the approximate representation of the S-S equation of state:

(A14) The characteristic parameters P*, V*, T* for the S-S theory are shown in Table 8. The parameters are obtained by fitting the original S-S equation of state in Eq. (A 12) with Eq. (A13), rather than fitting Eq. (A14). The average deviation between Eq. (A12) and regenerated PVT data is 0.0007 cm3/g. Lattice Fluid Model The Lattice Fluid model developed by Sanchez and Lacombe (15,16) introduces vacancies into the classical incompressible Flory-Huggins model. The lattice vacancy is treated as a pseudo particle in the system. The free energy of an incompressible binary polymer solution is then converted to that of the bulk polymer. The equation of state for a polymer is given below: (A15) Table 9 gives the three characteristic parameters P*, V*, 7* for the lattice fluid (LF) theory. As was with the FOV model, the LF model does not correlate PVT data well over large pressure ranges. The average deviation between Eq. (A15) and regenerated bulk PVT data is 0.0033 cm3/g. E. CONTINUUM MODELS

Hole Model by Simha and Somcynsky (S-S) The simple cell model by Prigogine et al. is modified by Simha and Somcynsky (13) to allow for lattice imperfections (holes). The model can be written as the following equation of state:

(A12) where the factor " / ' is the fraction of occupied lattice sites and satisfies

(A13) The factor s/3c in Eq. (A 13) is called the flexibility ratio, which is usually set to 1. Other factors are the same as those in the Prigogine cell model. In order to calculate volume at given temperatures and pressures, one should solve Eqs. (A12) and (A13) simultaneously. The calculational complexity is a drawback of

The widely used models presented in the last section are derived by considering the partition function of a chain system on a lattice or quasi lattice. The lattice or quasi lattice phase space can be further generalized to be a continuum counterpart. This approach yields, in general, the integral equation theory, which focuses on solving the integral equations of the particle-particle distribution functions (9). The system of hard chains has been treated the most, because it is the simplest and provides the possibility of analytical equation of state in some cases. One approach of this category is to solve the integral equations using the Percus-Yevick closure for the system of adhesive hard sphere (AHS) mixtures (17-22). An adhesive hard sphere is a hard sphere that has attractive sites at surface. The attractive interaction on these attractive sites is infinitely strong and infinitesimally short ranged. The Percus-Yevick closure yields an analytical solution for such systems. The adhesive attraction, which resembles the chemical bonding, is used to build up chains by employing the proper connectivity constraints. A different approach of this category is to manipulate the graphs in the cluster expansion of the system of hard spheres with multiple attractive sites (23-29). The attractive interaction is the same as that of the AHS system, and is treated as a perturbation to the reference hard

sphere potential. The steric incompatibility is taken into account by Wertheim to approximate the graph expansion for distribution functions. This approach is called the thermodynamic perturbation theory (TPT) (23-27) or the statistically associating fluid theory (SAFT) (28,29). A unique approach to the hard chain equation of state has been developed by Hall and coworkers (30). Flory's onlattice particle insertion analysis is generalized to the offlattice calculation. This theory is called the generalized Flory (GF) model. The enhanced evaluation of the insertion probability is further manipulated to yield the equation of state. Since the developed equations of state are for athermal chain systems, an energetic contribution to pressure is required in order for these models to be applicable to real systems. The simplest model adds a van der Waals (vdW) term as a perturbation to the reference pressure (28,31): (A16)

where the term with the subscript 0 implies the athermal reference pressure. The dimensionless variables are defined as usual (Eq. (A7)). The specific mathematical forms of the reference equation of state are as follows: (AHS)

(A17) (TPT or SAFT) (A18) (GF)

(A19)

The three characteristic parameters P*, V*, T* for the AHS -I- vdW equation of state are shown in Table 10. The average deviation between the AHS + vdW equation and regenerated bulk data is 0.0012 cm3/g. The parameters for other two equations are relatively close to those for the AHS -f vdW equation, and are not tabulated here.

F. TABLES TABLE 1. ZERO PRESSURE VOLUME (V0(T)) AND BULK MODULUS (B0(T))0 Polymer PDMS PS PoMS PMMA PnBMA PCHMA PEA PEMA PMA PVAc LPE BPE LDPE-A LDPE-B LDPE-C PBD PBD8 PBD24 PBD40 PBD50 PBD87 i-PB PAr PCL PC BCPC HFPC TMPC PET PIB PI8 PI14 PI41

Vo (D (cm3/g) 1.0079 exp(9.121 x 10' 4 O 0.9287 exp(5.131 x 10"4O 0.9396 exp(5.306 x 10' 4 O 0.8254+ 2.8383 x 10~ 4 f+ 7.792 x 10~V 0.9341 4-5.5254 x 10~4/ + 6.5803 x 10 " 7 r 2 + 1.5691 x 1 0 1 V 0.8793+4.0504 x 10" 4 f+ 7.774 x 1 0 " V - 7.7534 x 10" 1 V 0.8756 exp(7-241 x 10"4O 0.8614 exp(7.468 x 10"4O 0.8365 exp(6.795 x 10~40 0.8250 + 5.820X 10~4/ + 2.940x 10 V 0.9172 exp(7.806 x 10"4O 0.9399 exp(7.341 x 10"4O 1.1484 exp(6.950 x 10"4O 1.1524 exp(6.700 x 10"4O 1.1516 exp(6.730 x 10"4O 1.0970 exp(6.600 x 10"4O 0.9352+ 4.9988 x 10" 4 r+1.268 x 10" 7 / 2 0.9352 + 4.9988 x 10"4f + 1.268 x 10~7r2 0.9352 + 4.9988 x 10" 4 f+ 1.268 x 10~V 0.9352+ 4.9988 x 10~4? + 1.268 x 1 0 " V 0.9352+ 4.9988 x 10~ 4 /+1.268 x 10~7r2 1.1417 exp(6.751 x 10"4O 0.73381 exp(1.626 x 10" 5 F 3 / 2 ) 0.9049 exp(6.392 x 10"4O 0.73565 exp(1.859 x 10" 5 J 3 / 2 ) 0.6737+ 3.634 x 10"4f + 2.370x 10 V 0.6111+4.898x 10" 4 f+1.730x 1 0 " V 0.8497 + 5.073 x 10"4f + 3.832x 10~V 0.6883 + 5.90xl0~ 4 f 1.0750 exp(5.651 x 10"4O 0.9352+ 4.9988 x 10" 4 f+1.268 x 1 0 " V 0.9352 + 4.9988 x 10" 4 /+ 1.268 x 10~V 0.9352 + 4.9988 x 10"4/21.268 x 1 0 " V

B • (D (MPa) 1000.00 exp(- 5.701 x 10"3O 2426.17 exp(- 3.319 x 10"3O 2929.53 exp(- 4.114 x 10~30 3215.88 exp(-4.146x 10"3O 2535.79 exp(- 5.344 x 10"3O 3302.01 exp(-5.220x 10"3O 2161.07 exp(-4.839 x 10"3O 2918.34 exp(- 5.356 x 10"3O 2637.58 exp(- 4.493 x 10"3O 2 291.95 exp(-4.346x 10"3O 1976.51 exp(-4.661 x 10"3O 1980.98 exp(-4.699 x 10"3O 2157.72 exp(-4.701 x 10"3O 2199.11 exp(-4.601 x 10"3O 2088.37 exp(-4.391 x 10"3O 1987.70 exp(- 3.593 x 10~30 10915.90 exp(-5.344 x 10"3O 10915.90 exp(-5.344 x 10"3O 10915.90 exp(-5.344 x 10"3O 10915.90 exp(-5.344 x 10~3r) 10915.90 exp(- 5.344 x 10"3O 1873.60 exp(-4.533 x 10"3O 3321.03 exp(-3.375 x 10~30 2114.09 exp(- 3.931 x 10"3O 3467.56 exp(- 4.078 x 10"3O 4064.88 exp(-4.921 x 10"3O 2646.53 exp(-5.156 x 1 0 3 O 2588.37 exp(-4.242x 10"3O 4135.35 exp(- 4.150 x 10"3O 2240.49 exp(-4.329 x 10"3O 10915.90 exp(- 5.344 x 10"3O 10915.90 exp(-5..344 x 10"3O 10915.90 exp(- 5.344 x 10"3O

References page VI - 601

TABLE 1. cont'd Vo (T) (cm 3/g)

Polymer PI56 /-PP 0-PP Phenoxy PSO PEO PVME PEEK PTFE PTHF PMP PA6 PA66 PECH PVC PPO E/P50 EVA18 EVA25 EVA28 EVA40 SAN3 SAN6 SAN15 SAN18 SAN4 SAN70 SMMA20 SMMA60 a

Bo(T) (MPa)

0.9352 + 4.9988 x 10' 4 f+ 1.268 x 10" 7 / 2 1.1606 exp(6.700 x 10"4O 1.1841- 1.091 x 10~4; +5.286 x 10~V 0.76644 exp(1.921 x 10" 5 r 3 / 2 ) 0.7644 + 3.419 x \0~4t + 3.126 x HT V 0.8766 exp(7.087 x 10-4O 0.9585 exp(6.653 x 10"4O 0.7158 exp(6.690 x 10"4O 0.3200- 9.5862 x 10"4f 1.0043 exp(6.691 x 10"4O 1.4075-9.905 x 10~ 4 /+3,497 x 10~6r2 0.7597 exp(4.701 x 10"4O 0.7657 exp(6.600 x 10~4f) 0.7216 exp(5.825 x 10~4r) 0.7196 + 5.581 x 10~5f + 1.468 x 1 0 " V 0.78075 exp(2.151 x 10" 5 T 3 / 2 ) 1.2291+5.799 x l0~5t+ 1.964 x 10~6f2 1.02391 exp(2.173 x 10" 5 T 3 / 2 ) 1.00416 exp(2.244 x 10~5T3/2) 1.00832 exp(2.241 x 10 " 5 J 3 / 2 ) 1.06332 exp(2.288 x 10~ 5 7 3 / 2 ) 0.9233+ 3.936 x 10~4f +5.685 x 10 ~ V 0.9211+4.37Ox 10~4f + 5.846 x 10~V 0.9044 + 4.207 x 10~4* + 4.077 x 10~V 0.9016 + 4.036 x 10~4f+ 4.206 x 10~7?2 0.8871+3.406 x 10" 4 f+ 4.938 x 1 0 " V 0.8528 + 3.616 x 10"4* + 2.634 x 1 0 " V 0.9063 + 3.570X 10" 4 / + 6.532 x 1 0 " V 0.8610 + 3.350X 10~4f +6.98010"V

10915.90 exp(- 5.344 x 10~30 1667.79 exp(- 4.177 x 10' 3 O 1813.20 exp(- 6.604 x 10~30 4025.73 exp(-4.378 x 10"3O 4092.84 exp(~ 3.757 x 10~3r) 2323.27 exp(- 3.947 x 10~30 2413.87 exp(- 4.588 x 10"3O 4340.04 exp(- 4.124 x 10~30 4756.15 exp(~ 9.380 x 10"3O 1997.76 exp(-4.233 x 10"3O 376.7+ 2.134/-7.0445 x 10~ 3 / 2 4213.65 exp(~ 4.660 x 10"3O 3539.15 exp(- 5.040 x 10"3O 2665.55 exp(-~ 4.171 x 10"3O 3290.83 exp(-5.321 x 10~30 2548.10 exp(- 4.290 x 10~30 5447.43 exp(- 8.103 x 10"3O 2105.15 exp(- 4.537 x 10~30 2062.64 exp(-4.734 x 10~30 2052.57 exp(-4.457 x 10 ~30 2294.18 exp(- 4.989 x 10"3O 2682.33 exp(- 4.376 x 10 ~30 2538.03 exp(-4.286x 10"3O 2666.67 exp(- 3.943 x 10~3f) 2689.04 exp(- 3.858 x 10~30 3236.02 exp(~ 4.431 x 10~30 3751.68 exp(- 3.923 x 10~30 2595.08 exp(-4.143 x 10~30 2919.46 exp(-4.611 x 10~3r)

In the mathematical expression of volume, t and Tare in 0 C and K, respectively. See Table 2 for references.

TABLE 2. ABBREVIATIONS OF POLYMER NAMES AND THE EXPERIMENTAL TEMPERATURE-PRESSURE RANGE FOR POLYMER LIQUIDS0 Polymer Monomer or sourceabbreviation based Name PDMS Dimethyl siloxane PS Styrene PoMS* o-Methyl styrene PMMA Methyl methacrylate PBMA* rc-Butyl methacrylate PCHMA* Cyclo hexyl methacrylate PEA Ethyl acrylate PEMA* Ethyl methacrylate PMA Methyl acrylate PVAC Vinyl acetate LPE* Linear polyethylene (PE) BPE* Branched PE LDPE-A Low density PE-A LDPE-B Low density PE-B LDPE-C Low density PE-C PBD* Butadiene PBD8* Butadiene with 8% 1,2 content PBD24* Butadiene with 24% 1,2 content PBD40* Butadiene with 40% 1,2 content PBD50* Butadiene with 50% 1,2 content PBD87* Butadiene with 87% 1,2 content

T (0C) 25-70 115-196 139-198 114-159 34-200 123-198 37-210 113-161 37-220 35-100 142-200 125-198 112-225 112-225 112-225 4-55 25-200

P (MPa)

Refs.

0-100 34,35,36 0-200 37 0-180 37 0-200 38 0-200 38 0-200 38 0-196 34 0-196 34 0-196 34 0-80 39 0-200 38 0-200 38 0-196 40 0-196 40 0-196 40 0-283.5 41 0-200 42

25-200

0-200

42

25-200

0-200

42

25-200

0-200

42

25-200

0-200

42

TABLE 2. cont'd Polymer Monomer or sourceabbrevation based Name PB PAr* PCL PC BCPC* HFPC* TMPC* PET PIB PI8

1-Butene Arylate Caprolactone Carbonate (PC) Bisphenol chloral PC Hexafluoro bisphenol-A PC Tetramethyl bisphenol-A PC Ethylene terephthalate Isobutylene Isoprene with 8% 3,4 content PI14 Isoprene with 14% 3,4 content PI41 Isoprene with 41% 3,4 content PI56 Isoprene with 56% 3,4 content i-PP Isotactic polypropylene a-PP Atactic polypropylene Phenoxy* Phenoxy PSO Sulfone PEO Ethylene oxide PVME Vinyl methyl ether PEEK Ether ether ketone PTFE Tetrafluoro ethylene PTHF* Tetrahydro furane PMP 4-Methyl-i-pentene PA6 Amide 6

P (MPa)

Refs.

133-246 177-310 100-148 151-340 220-280 220-280 210-290 274-342 53-110 25-200

0-196 0-176.5 0-200 0-176.5 0-50 0-50 0-50 0-196 0-00 0-200

43 44 34 44 45 45 45 46 36,47 42

25-200

0-200

42

25-200

0-200

42

25-200

0-200

42

T (0C)

170-297 80-120 68-300 202-371 88-224 30-198 346-398 330-372 62-166 241-319 236-296

0-196 0-100 0-176.5 0-196 0-68.5 0-200 0-200 0-39 0-78.5 0-196 0-196

43 34 44 48 49 34,50 51 52 49 53 54

TABLE 2. cont'd Polymer abbrevation

TABLE 4. cont'd

Monomer or sourcename

PA66 PECH6 PVC PPO E/P50 EVA18 EVA25 EVA28 EVA40 SAN3 SAN6 SAN15 SAN18 SAN40 SAN70 SMMA2O0

Amide 66 Epichlorohydrine Vinyl chloride Phenylene oxide Ethylene/propylene 50% Ethylene/vinyl acetate 18% Ethylene/vinyl acetate 25% Ethylene/vinyl acetate 28% Ethylene/vinyl acetate 40% Styrene/acrylonitrile 2.7% Styrene/acrylonitrile 5.7% Styrene/acrylonitrile 15.3% Styrene/acrylonitrile 18.0% Styrene/acrylonitrile 40% Styrene/acrylonitrile 70% Styrene/methyl methacrylate 20% SMMA6O0 Styrene/methyl methacrylate 60%

0 b

P (MPa)

246-298 60-140 100-150 203-320 140-250 112-219 94-233 94-235 75-235 105-266 96-267 132-262 104-255 100-270 100-271 110-270

0-196 0-200 0-200 0-176.5 0-62.5 0-176.5 0-176.5 0-176.5 0-176.5 0-200 0-200 0-196 0-196 0-196 0-196 0-196

54 34 34 55 56 57 57 57 57 45 45 45 45 45 58 58

110-270

0-200

58

PVT PROPERTIES OF OTHER POLYMERS0

J( 0 C)

Polymer HDPE LDPE HMLPE i-Poly(l-pentene) Poly(vinylidene fluoride) Poly(methylene oxide) Poly(butylene terephthalate) Qiana nylon Natural rubber vulcanizated with 10-28% sulfur Hevea rubber uncrosslinked crosslinked (6% sulfur) crosslinked (11.5% sulfur) PS/PPO blends PS/PVME blends Main chain LCPs Azomethine ether LCP

P (MPa)

Note

Refs.

140-203 121-175 137-200 140-172 175-240 155-185 215-280

0-196.0 0-196.0 0-200.0 0.1 0.1 18.3-150.9 Figure only 20.7-103.4 Figure only

59 59 38 60 61 62 63

270-320 10-82

20.7-103.4 Figure only 0-1013.0

63 64

10-85

0-80.0

30-350 30-198

0-176.5 0-200.0

20-300

0-200.0

SimhaSomcynsky parameters only

Figure only up to nematic

65

55 50 66 67

This table lists polymers not mentioned in the earlier tables.

TABLE 4. CHARACTERISTIC PARAMETERS FOR THE SANCHEZ-CHO EQUATION OF STATE

Polymer

V* (cm3/g)

PDMS PS PoMS PMMA PBMA PCHMA

0.8071 0.8165 0.8368 0.7139 0.7963 0.7710

T* (K) 1375.1 2277.2 2380.6 2184.2 1855.9 2195.1

Polymer

V* (cm Vg)

PEA PEMA PMA PVAc LPE BPE LDPE-A LDPE-B LDPE-C PBD PBD8 PBD24 PBD40 PBD50 PBD87 PB PAr PCL PC BCPC HFPC TMPC PET PIB PI 8 PI 14 PI41 PI56 i-PP a-PP Phenoxy PSO PEO PVME PEEK PTFE PTHF PMP PA6 PA66 PECH PVC PPO E/P50 EVA18 EVA25 EVA28 EVA40 SAN3 SAN6 SAN15 SAN18 SAN40 SAN70 SMMA20 SMMA60

0.7398 0.7332 0.7121 0.6918 0.9491 0.9723 0.9852 0.9937 0.9869 0.9115 0.9308 0.9359 0.9357 0.9408 0.9498 0.9854 0.6839 0.7671 0.6871 0.5971 0.5264 0.7261 0.6199 0.9382 0.9453 0.9366 0.9370 0.9330 1.0116 0.9690 0.7242 0.6655 0.7441 0.8187 0.6395 0.3638 0.8561 1.0089 0.7130 0.6887 0.6269 0.6252 0.7181 1.0582 0.9585 0.9338 0.9241 0.8864 0.8030 0.7896 0.7860 0.7854 0.7853 0.7616 0.7789 0.7408

T* (K)

P* (MPa)

Refs.

See Abbreviations Table in Part VIlI of this Handbook. Abbreviation may not be recognized by international organization.

TABLE 3.

n

T( 0 C)

1747.4 1771.2 1829.0 1696.9 1655.0 1751.9 1865.4 1923.8 1880.5 1633.8 1798.6 1819.0 1842.9 1892.0 1905.6 1924.1 2243.9 1849.0 2070.3 2249.1 1788.2 1908.0 2022.2 2130.2 1921.0 1911.3 1912.7 1854.5 1991.5 1776.2 2103.4 2232.2 1789.1 1861.3 2126.5 1400.7 1843.0 1885.0 3140.3 2195.2 2068.9 2395.4 1810.4 2384.7 1878.9 1848.3 1812.5 1856.1 2185.7 2010.7 2170.4 2208.4 2435.2 2546.6 2105.6 2099.5

9131.8 11257.0 10623.9 10399.9 9943.5 8630.7 8214.5 7983.8 8225.3 9443.2 9136.3 8708.0 8352.0 7757.3 7241.8 6891.1 11557.3 9530.1 12106.1 9931.3 10173.1 10573.0 15278.8 7045.3 7669.6 8219.9 8112.3 8600.1 6118.1 6202.8 12728.1 13286.4 10805.4 9270.5 14335.5 9757.5 8160.2 6452.5 5718.4 8391.9 8899.3 7551.2 11776.9 6421.0 8107.5 7957.5 8622.1 8441.9 7878.5 8896.8 8602.0 8495.6 7772.6 9177.0 8531.3 8858.9

P* (MPa) 6212.9 7867.6 7772.1 9873.0 9025.3 8515.1

TABLE 5. CHARACTERISTIC PARAMETERS FOR THE HARTMANN-HAQUE EQUATION OF STATE Polymer PDMS PS

V* (cm3/g)

T* (K)

P* (MPa)

0.8795 0.8754

1106 1603

1837 2956

References page VI-601

TABLE 6. CHARACTERISTIC PARAMETERS FOR THE SIMPLE CELL MODEL EQUATION OF STATE

TABLE 5. cont'd Polymer PoMS PMMA PBMA PCHMA PEA PEMA PMA PVAc LPE BPE LDPE-A LDPE-B LDPE-C PBD PBD8 PBD24 PBD40 PBD50 PBD87 PB PAr PCL PC BCPC HFPC TMPC PET PIB PI8 PI14 PI41 PI56 /-PP fl-PP Phenoxy PSO PEO PVME PEEK PTFE PTHF PMP PA6 PA66 PECH PVC PPO E/P50 EVA18 EVA25 EVA28 EVA40 SAN3 SAN6 SAN15 SAN18 SAN40 SAN70 SMMA20 SMMA60

V* (cm3/g) 0.8891 0.7582 0.8552 0.8249 0.8038 0.7976 0.7712 0.7368 1.0381 1.0677 1.0769 1.0842 1.0781 0.9855 1.0065 1.0118 1.0113 1.0157 1.0264 1.0818 0.7408 0.8418 0.7474 0.5356 0.5758 0.8010 0.6802 0.9935 1.0199 1.0098 1.0103 1.0076 1.1066 1.0292 0.7784 0.7246 0.8005 0.8836 0.7044 0.3683 0.9165 1.1228 0.7654 0.7559 0.6734 0.6559 0.7906 1.1112 1.0443 1.0171 1.0077 0.9604 0.8615 0.8545 0.8495 0.8448 0.8326 0.8140 0.8453 0.8037

T* (K) 1608 1467 1309 1517 1284 1298 1334 1151 1211 1313 1387 1426 1396 1199 1301 1317 1336 1370 1387 1452 1614 1411 1502 1520 1306 1430 1484 1422 1389 1375 1377 1339 1475 1197 1481 1623 1254 1342 1568 900 1276 1449 2255 1643 1456 1532 1339 1516 1380 1356 1333 1339 1534 1448 1564 1566 1630 1776 1539 1530

P* (MPa) 3096 3819 3268 3038 3076 3654 3680 3817 2804 2510 2516 2509 2538 3333 3109 2989 2891 2742 2557 2066 3709 3013 3644 3641 2920 2907 4071 2976 2733 2925 2889 2990 1852 2383 4357 3972 3504 3245 3668 3251 2894 1670 2264 2453 3775 3595 3079 2247 2521 2474 2646 2716 2840 3010 2957 3024 3138 3578 2812 2912

Polymer

V* (cm3/g)

PDMS PS PoMS PMMA PBMA PCHMA PEA PEMA PMA PVAc LPE BPE LDPE-A LDPE-B LDPE-C PBD PBD8 PBD24 PBD40 PBD50 PBD87 /PB PAr PCL PC BCPC HFPC TMPC PET PIB PI8 PI14 PI41 PI56 /-PP a-PP Phenoxy PSO PEO PVME PEEK PTFE PTHF PMP PA6 PA66 PECH PVC PPO E/P50 EVAl 8 EVA25 EVA28 EVA4 SAN3 SAN6 SAN15 SAN18 SAN40 SAN70 SMMA20 SMMA60

0.9180 0.9148 0.9302 0.7941 0.8931 0.8906 0.8394 0.8321 0.8054 0.7731 1.0978 1.1179 1.1195 1.1267 1.1210 1.0269 1.0488 1.0539 1.0532 1.0573 1.0679 1.1216 0.7745 0.8758 0.7835 0.6676 0.6116 0.8473 0.7193 1.0393 1.0617 1.0516 1.0521 1.0495 1.1470 1.0758 0.8145 0.7593 0.8447 0.9210 0.7465 0.4829 0.9609 1.1652 0.7956 0.7841 0.6997 0.6855 0.8355 1.1653 1.0874 1.0592 1.0509 1.0012 0.8995 0.8950 0.8875 0.8826 0.8696 0.8493 0.8795 0.8362

T* (K) 3254 5167 5214 4783 4237 4873 4167 4190 4318 3781 4124 4309 4430 4537 4461 3811 4171 4212 4263 4357 4399 4600 52.57 4478 4969 5046 4525 4875 5070 4593 4415 4382 4386 4277 4712 3884 4837 5358 4226 4298 5418 4488 4185 4721 7048 5231 4662 4938 4585 4982 4434 4368 4319 4326 4969 4761 5069 5070 5273 5676 4908 4884

F* (MPa) 388.0 599.0 622.7 765.0 683.0 630.1 652.7 772.8 768.3 755.9 588.1 535.2 549.8 544.6 552.3 681.5 655.5 629.9 608.3 575.4 537.9 460.6 776.0 632.0 774.7 762.1 620.7 609.4 868.8 580.0 572.1 611.4 604.1 628.0 425.8 493.4 903.2 844.5 696.8 677.0 797.0 323.9 576.8 400.0 465.4 550.9 757.7 719.2 668.5 453.5 544.5 537.6 572.9 584.9 603.2 640.8 618.8 630.4 653.6 723.1 600.7 622.9

TABLE 7. CHARACTERISTIC PARAMETERS FOR THE FLORY, ORWOLL, AND VRIJ EQUATION OF STATE

TABLE 8. CHARACTERISTIC PARAMETERS FOR THE SIMHA-SOMCYNSKY EQUATION OF STATE

Polymer

V* (cm3/g)

T* (K)

P* (MPa)

Polymer

PDMS PS PoMS PMMA PBMA PCHMA PEA PEMA PMA PVAc LPE BPE LDPE-A LDPE-B LDPE-C PBD PBD8 PBD24 PBD40 PBD50 PBD87 PB PAr PCL PC BCPC HFPC TMPC PET PIB PI8 PI14 PI41 PI56 /-PP a-PP Phenoxy PSO PEO PVME PEEK PTFE PTHF PMP PA6 PA66 PECH PVC PPO E/P50 EVA18 EVA25 EVA28 EVA40 SAN3 SAN6 SAN15 SAN18 SAN40 SAN70 SMMA20 SMMA60

0.8264 0.8277 0.8457 0.7204 0.8087 0.7772 0.7563 0.7503 0.7277 0.7090 0.9818 0.9992 0.9963 1.0025 0.9974 0.9173 0.9404 0.9438 0.9419 0.9443 0.9511 0.9867 0.6991 0.7830 0.7070 0.6065 0.5521 0.7720 0.6452 0.9455 0.9480 0.9415 0.9415 0.9400 1.0072 0.9755 0.7389 0.6847 0.7719 0.8266 0.6642 0.4215 0.8774 1.0203 0.6896 0.6885 0.6321 0.6210 0.7472 1.0650 0.9724 0.9475 0.9416 0.8985 0.8129 0.8087 0.8020 0.7974 0.7869 0.7648 0.7865 0.7494

5184 8118 8463 7717 6794 7700 6599 6703 6894 6449 6548 6710 6774 6896 6809 5521 6418 6432 6453 6514 6474 6838 8470 6754 8039 8287 7360 8156 8215 7396 6573 6622 6613 6514 7011 6351 7869 8664 7147 6607 8667 7088 7006 7079 9182 7865 7192 7752 7360 8377 6870 6770 6759 6766 7897 7622 8053 7999 8369 8701 7521 7558

326.9 405.2 441.5 568.8 509.6 461.4 511.5 641.3 599.2 599.7 537.6 453.1 469.5 456.4 471.0 454.4 505.5 480.9 460.7 425.0 395.9 403.9 651.2 486.2 671.0 611.0 542.7 514.2 851.0 396.0 419.3 450.1 444.0 473.7 397.4 405.9 713.2 738.2 601.6 512.8 832.9 404.9 459.8 395.0 411.0 558.3 524.6 501.8 650.9 360.2 453.6 444.8 480.5 475.5 453.3 503.1 465.0 461.0 474.7 500.9 475.8 491.3

PDMS PS PoMS PMMA PBMA PCHMA PEA PEMA PMA PVAc LPE BPE LDPE-A LDPE-B LDPE-C PBD PBD8 PBD24 PBD40 PBD50 PBD87 PB PAr PCL PC BCPC HFPC TMPC PET PIB PI8 PI14 PI41 PI56 /PP a-PP Phenoxy PSO PEO PVME PEEK PTFE PTHF PMP PA6 PA66 PECH PVC PPO E/P50 EVA18 EVA25 EVA28 EVA40 SAN3 SAN6 SAN15 SAN18 SAN40 SAN70 SMMA20 SMMA60

V* (cm3/g) 0.9592 0.9634 0.9814 0.8369 0.9358 0.9047 0.8773 0.8710 0.8431 0.8126 1.1406 1.1674 1.1664 1.1734 1.1679 1.0766 1.0960 1.1013 1.1003 1.1049 1.1149 1.1666 0.8091 0.9173 0.8156 0.6975 0.6317 0.8794 0.7426 1.0940 1.1094 1.0997 1.1000 1.0970 1.1884 1.1274 0.8529 0.7903 0.8812 0.9632 0.7705 0.4339 1.0087 1.2050 0.8327 0.8195 0.7343 0.7320 0.8602 1.2227 1.1341 1.1040 1.0949 1.0446 0.9416 0.9352 0.9299 0.9255 0.9124 0.8906 0.9186 0.8739

T* (K) 7864 12840 13080 11940 10310 12030 11040 10190 10460 9348 9793 10390 10580 10860 10660 9225 10029 10122 10229 10450 10502 10920 12390 10870 11830 12190 10550 11540 11800 11360 10578 10541 10547 10284 11060 9494 11730 12770 10150 10360 12580 8126 10280 11030 16870 12640 11370 12350 10580 12220 10630 10440 10310 10360 12070 11490 12360 12380 12900 13790 11800 11780

P* (MPa) 501.4 715.9 746.1 926.4 856.0 772.2 830.8 987.0 969.1 947.4 786.4 692.3 716.2 703.6 718.8 815.0 827.0 792.3 763.7 716.8 671.2 603.7 1003.0 784.5 1020.0 987.8 851.0 819.2 1194.0 686.6 711.5 760.1 750.7 786.9 573.0 627.7 1139.0 1116.0 914.5 848.1 1086.0 658.1 725.5 545.3 549.9 706.9 913.1 849.5 929.4 572.0 705.6 697.8 747.2 753.9 764.2 823.8 779.2 785.3 811.8 874.7 764.0 791.1

References page VI-601

TABLE 10. CHARACTERISTIC PARAMETERS FOR THE AHS + vdW EQUATION OF STATE

TABLE 9. CHARACTERISTIC PARAMETERS FOR THE SANCHEZ-LACOMBE EQUATION OF STATE Polymer

V* (cm3/g)

J * (K)

P* (MPa)

Polymer

V* (cm3/g)

T* (K)

P* (MPa)

PDMS PS PoMS PMMA PBMA PCHMA PEA PEMA PMA PVAc LPE BPE LDPE-A LDPE-B LDPE-C PBD PBD8 PBD24 PBD40 PBD50 PBD87 PB PAr PCL PC BCPC HFPC TMPC PET PIB PI8 PI14 PI41 PI56 /-PP a-PP Phenoxy PSO PEO PVME PEEK PTFE PTHF PMP PA6 PA66 PECH PVC PPO E/P50 EVA18 EVA25 EVA28 EVA40 SAN3 SAN6 SAN15 SAN18 SAN40 SAN70 SMMA20 SMMA60

0.9022 0.8929 0.9142 0.7805 0.8789 0.8444 0.8216 0.8189 0.7908 0.7768 1.0734 1.0878 1.0827 1.0878 1.0835 0.9851 1.0167 1.0194 1.0161 1.0169 1.0224 1.0739 0.7632 0.8477 0.7737 0.6651 0.6077 0.8493 0.7102 1.0213 1.0196 1.0141 1.0139 1.0141 1.0958 1.0658 0.8032 0.7496 0.8492 0.8930 0.7310 0.4515 0.9587 1.1153 0.7406 0.7502 0.6803 0.6686 0.8220 1.1652 1.0580 1.0307 1.0259 0.9772 0.8860 0.8841 0.8733 0.8672 0.8556 0.8222 0.8520 0.8125

466 688 725 668 596 675 581 602 606 582 596 601 603 610 606 462 554 553 552 553 547 609 760 589 728 753 680 752 761 623 555 562 561 557 633 570 690 787 656 567 804 630 626 650 785 713 606 656 681 753 612 602 604 600 701 686 712 703 734 726 658 662

288.5 371.5 405.7 516.9 442.1 410.0 450.6 567.5 521.9 501.3 479.8 411.3 429.9 421.4 431.8 440.2 456.0 437.1 422.5 394.8 373.6 377.5 568.7 453.4 574.4 513.6 455.4 432.4 726.1 350.4 392.3 415.6 410.7 433.1 366.4 354.2 607.4 635.3 492.2 463.0 713.7 357.2 385.6 355.7 422.5 513.7 482.4 469.0 554.1 301.4 407.7 397.0 424.1 419.8 391.6 429.7 404.1 400.1 412.1 466.4 434.7 446.0

PDMS PS PoMS PMMA PBMA PCHMA PEA PEMA PMA PVAc LPE BPE LDPE-A LDPE-B LDPE-C PBD PBD8 PBD24 PBD40 PBD50 PBD87 /-PB PAr PCL PC BCPC HFPC TMPC PET PIB PI 8 PI 14 PI41 PI 56 /-P a-PP Phenoxy PSO PEO PVME PEEK PTFE PTHF PMP PA6 PA66 PECH PVC PPO E/P50 EVA18 EVA25 EVA28 EVA40 SAN3 SAN6 SAN15 SAN18 SAN40 SAN70 SMMA20 SMMA60

0.5629 0.6080 0.6222 0.5264 0.5820 0.5719 0.5294 0.5143 0.5111 0.4891 0.6471 0.6801 0.6995 0.7117 0.7017 0.6793 0.6764 0.6832 0.6865 0.6962 0.7069 0.7029 0.4779 0.5249 0.4691 0.4016 0.3627 0.4638 0.4041 0.7079 0.7030 0.6936 0.6943 0.6850 0.6987 0.6935 0.515 0.4541 0.4974 0.5991 0.4164 0.1807 0.6026 0.6857 0.5321 0.4818 0.4707 0.4747 0.4664 0.7592 0.6792 0.6669 0.6589 0.7130 0.5894 0.5662 0.5685 0.5763 0.5911 0.5745 0.5622 0.5344

7331 16542 17194 15173 12106 15573 10289 9659 10989 9713 7955 9336 10596 11372 10761 11913 11417 11794 12248 13112 13574 11078 12060 8788 10088 10381 8598 7446 8344 16891 13661 13326 13369 12371 10082 10648 212153 10866 8139 12258 8666 3156 10303 8842 23165 11705 15781 19538 7365 14530 10625 10405 9791 10665 14537 11971 13512 14760 18577 19851 12948 12855

707.2 731.7 731.6 986.3 868.5 793.6 933.1 1252.0 1105.2 1246.9 1136.2 924.9 845.6 802.4 846.0 867.5 900.7 843.1 792.9 710.5 642.2 680.3 1319.1 943.2 1439.2 1248.0 946.0 1496.6 1984.3 687.8 689.7 756.7 744.3 817.0 691.1 672.6 1377.3 1582.8 1439.8 900.3 1801.4 2185.6 965.0 718.6 533.4 896.1 820.9 673.1 1538.8 718.7 852.7 831.9 934.8 877.9 737.9 902.0 858.1 816.5 678.8 827.2 850.6 883.0

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