Coefficient of Thermal Expansion

The coefficient of thermal expansion (CTE) is defined as the fractional increase in length or volume per unit rise in temperature:

α_L = (1/L) · (∂L / ∂T)_p

α_v = (1/V) · (∂V / ∂T)_p = - (1/ρ) · (∂ρ / ∂T)_p

For an isotropic material such as an amorphous polymer, the volumetric expansion coefficient is in average three times the linear expansion coefficient:

α_v ≈ 3 · α_L

The expansion of polymers on heating depends on mostly intermolecular forces, because the bond length between atoms is virtually independent of temperature. Since a polymer has more internal bonds per unit volume than a low molecular weight compound, a polymer has in general a lower expansivity than the related low-molecular liquid (monomer).

Below the glass temperature, the expansivity is much reduced because the polymer chains have less room to "wiggle" due to the closer packing and stronger intermolecular forces. Empirically, it has been found that α_rubber ≈ 2.5 · α_glass.

A simple model for the prediction of CTE values has been developed by Krevelen.⁽¹⁾ He found that the ratio of the molar thermal expansivity, ∂V_m / ∂T, and the van der Waals volume, V_w, is nearly constant:

E_r = ∂V_m / ∂T ≈ 1.0 · 10^-3V_w

E_g = ∂V_m / ∂T ≈ 0.45 · 10^-3V_w

According to Krevelen, the deviations between the calculated and experimental values are in the same range as the deviations between values mentioned by different investigators. Since measured values from different sources show large deviations, the accuracy of Krevelen's method is difficult to assess. In many cases, generic values are the best option. As has been shown by Krevelen and others, the ratio of the molar volume V_m(298) and the van der Waals volume V_w is more or less constant for all polymers and has a value of approximately 1.6¹. Thus

α_r = E_r / V_m(298) ≈ 1.0 · 10^-3V_w / (1.6 · V_w) = 6.5 · 10^-4

α_g = E_g / V_m(298) ≈ 0.45 · 10^-3V_w / (1.6 · V_w) = 2.7 · 10^-4

We compared these two values with the average thermal expansion coefficients of a large number of compounds in rubber and glass state and found that the average CTE values are very similar to those suggested by Krevelen.

The thermal expansion coefficients of a polymer in its rubber state can be estimated with the Boyer-Spencer rule⁽³⁾:

α_r · T_g ≈ 0.164

We found that the Boyer-Spencer rule often underestimates the expansion coefficient of the polymer in its rubber state and that the relation

α_g · T_g ≈ 0.08

gives often better estimates for α_r when combined with the Simha-Boyer rule:⁴

α_r ≈ 0.113 / T_g + α_g = α_g (0.113 + 0.08 ) / T_g = 0.193 / T_g

 The table below lists the volumetric expansion coefficients for some common polymers.

References

1. D.W. van Krevelen and Klaas te Nijenhuis, Properties of Polymers, 4th Ed., Amsterdam (2009)
2. R.F. Boyer and R.S. Spencer, J. Appl. Phys. 15, 398 (1944)
3. R. Simha and R.F. Boyer, J. Chem. Phys., 37, 1003 (1962)
4. James E. Mark, Physical Properties of Polymers Handbook (2007)