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.
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