Section: 13 | Pressure-Volume-Temperature Relationships for Polymer Melts |
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 The recommended form of citation is: John R. Rumble, ed., CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), CRC Press/Taylor & Francis, Boca Raton, FL. If a specific table is cited, use the format: "Physical Constants of Organic Compounds," in CRC Handbook of Chemistry and Physics, 103rd Edition (Internet Version 2022), John R. Rumble, ed., CRC Press/Taylor & Francis, Boca Raton, FL.

# PRESSURE-VOLUME-TEMPERATURE RELATIONSHIPS FOR POLYMER MELTS

Christian Wohlfarth

Numerous theoretical equations of state for polymer liquids have been developed. These, at the minimum, have to provide accurate fitting functions to experimental data. However, for the purpose of this table, the empirical Tait equation along with a polynomial expression for the zero pressure isobar is used. This equation is able to represent the experimental data for the melt state within the limits of experimental errors, i.e., the maximum deviations between measured and calculated specific volumes are about 0.001-0.002 cm3/g.

The general form of the Tait equation is:

V(P,T) = V(0,T){1 – C ln[1 + P/B(T)]}     (1)

where the coefficient C is usually taken to be a universal constant equal to 0.0894. T is the absolute temperature in K and P the pressure in MPa. The volume V is the specific volume in cm3/g. The Tait parameter B(T) has the very simple meaning that it is inversely proportional to the compressibility κ at constant temperature and zero pressure:

κ(0,T) = –[1/V(0,T)](dV/dP) = C/B(T)     (2)

The B(T) function is usually given by:

B(T) = B0 exp[–B1(T-273.15)]     (3)

though sometimes a polynomial expression is used:

B(T) = b0 + b1(T-273.15) + b2(T-273.15)2     (4)

The zero-pressure isobar V(0,T) is usually given by:

V(0,T) = A0 + A1(T-273.15) + A2(T-273.15)2     (5)

where A0, A1, A2 are specific constants for a given polymer (the expression T-273.15 is used because fitting to the zero-pressure isobar is usually done in terms of Celsius temperature). Other forms for V(0,T) are also found in the literature, such as

V(0,T) = A3 exp[A4(T-273.15)]     (6)

or

V(0,T) = A5 exp(A6T1.5)     (7)

where A3 and A4 or A5 and A6 are again specific constants for a given polymer.

The Tait equation is particularly useful to calculate derivative quantities, such as the isothermal compressibility and the thermal expansivity coefficients. The isothermal compressibility κ(P,T) is derived from equation (1) as:

κ(P,T) = –(1/V)(dV/dP) = 1/{[P + B(T)][1/C - ln(1 + P/B(T))]}     (8)

and the thermal expansivity α(P,T) as:

α(P,T) = (1/V)(dV/dT) = α(0,T) – PB1κ(P,T)     (9)

where α(0,T) represents the thermal expansivity at zero (atmospheric) pressure and is calculated from any suitable fit for the zero-pressure volume, such as equations (5) through (7) above.

Because polymer melt PVT-behavior depends only slightly on polymer molar mass above the oligomeric region, usually no information is given in the original literature for the average molar mass of the polymers.

Table 1 summarizes the polymers or copolymers considered here and the experimental ranges of pressure and temperature over which data are available. In Table 2 the Tait equation functions, with parameters obtained from the fit, are given for 90 polymer or copolymer melts.

# References

1. Zoller, P., J. Appl. Polym. Sci., 23, 1051–1056, 1979. [https://doi.org/10.1002/app.1979.070230410]
2. Starkweather, H. W., Jones, G. A., and Zoller, P., J. Polym. Sci., Pt. B Polym. Phys., 26, 257–266, 1988. [https://doi.org/10.1002/polb.1988.090260204]
3. Fakhreddine, Y. A., and Zoller, P., J. Polym. Sci., Pt. B Polym. Phys., 29, 1141–1146, 1991. [https://doi.org/10.1002/polb.1991.090290913]
4. Rodgers, P. A., J. Appl. Polym. Sci., 48, 1061–1080, 1993. [https://doi.org/10.1002/app.1993.070480613]
5. Rodgers, P. A., J. Appl. Polym. Sci., 48, 2075–2083, 1993. [https://doi.org/10.1002/app.1993.070501205]
6. Yi, Y. X., and Zoller, P., J. Polym. Sci., Pt. B Polym. Phys., 31, 779–788, 1993. [https://doi.org/10.1002/polb.1993.090310705]
7. Callaghan, T. A., and Paul, D. R., Macromolecules, 26, 2439–2450, 1993. [https://doi.org/10.1021/ma00062a008]
8. Wang, Y. Z., Hsieh, K. H., Chen, L. W.,and Tseng, H. C., J. Appl. Polym. Sci., 53, 1191–1201, 1994. [https://doi.org/10.1002/app.1994.070530906]
9. Privalko, V. P., Arbuzova, A. P., Korskanov, V. V., and Zagdanskaya, N. E., Polym. Intern., 35, 161–169, 1994. [https://doi.org/10.1002/pi.1994.210350206]
10. Sachdev, V. K., Yashi, U., and Jain, R. K., J. Polym. Sci., Pt. B Polym. Phys., 36, 841–850, 1998. [https://doi.org/10.1002/(SICI)1099-0488(19980415)36:5<841::AID-POLB11>3.0.CO;2-9]

## TABLE 1. Names of the Polymers, Abbreviation Used, and Range of Experimental Data Applied in the Determination of the Equation Constants

 Polymer Symbol T/K P/MPa Ref. Continued on next page... Ethylene/propylene copolymer [50 wt%] EP50 413-523 0.1-63 4 Ethylene/vinyl acetate copolymer [18 wt% vinyl acetate] EVA18 385-491 0.1-177 4 Ethylene/vinyl acetate copolymer [25 wt% vinyl acetate] EVA25 367-506 0.1-177 4 Ethylene/vinyl acetate copolymer [28 wt% vinyl acetate] EVA28 367-508 0.1-177 4 Ethylene/vinyl acetate copolymer [40 wt% vinyl acetate] EVA40 348-508 0.1-177 4 Polyamide-6 PA6 509-569 0.1-196 4 Polyamide-11 PA11 478-542 0.1-200 5 Polyamide-66 PA66 519-571 0.1-196 4 cis-1,4-Polybutadiene cPBD 277-328 0.1-284 4 Polybutadiene, 8% 1,2-content PBD-8 298-473 0.1-200 6 Polybutadiene, 24% 1,2-content PBD-24 298-473 0.1-200 6 Polybutadiene, 40% 1,2-content PBD-40 298-473 0.1-200 6 Polybutadiene, 50% 1,2-content PBD-50 298-473 0.1-200 6 Polybutadiene, 87% 1,2-content PBD-87 298-473 0.1-200 6 Poly(1-butene), isotactic iPB 406-519 0.1-196 4 Poly(butyl methacrylate) PnBMA 307-473 0.1-200 4 Poly(butylene terephthalate) PBT 508-576 0.1-200 3 Poly(ε-caprolactone) PCL 373-421 0.1-200 4 Polycarbonate-bisphenol-A PC 424-613 0.1-177 4
 aPhenoxy = Poly(oxy-2-hydroxytrimethyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene). bPolysulfone = Poly(oxy-1,4-phenylenesulfonyl-1,4-phenyleneoxy-1,4-phenyleneisopropylidene-1,4-phenylene). cPolyarylate = Poly(oxyterephthaloyl/isophthaloyl T/I=50/50)oxy-1,4-phenyleneisopropylidene-1,4-phenylene.

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