## 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:

αL = (1/L) · (∂L / ∂T)p

αv = (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:

αv ≈ 3 · αL

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 αrubber ≈ 2.5 · αglass.

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, ∂Vm / ∂T, and the van der Waals volume, Vw, is nearly constant:

Er = ∂Vm / ∂T ≈ 1.0 · 10-3Vw

Eg = ∂Vm / ∂T ≈ 0.45 · 10-3Vw

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 Vm(298) and the van der Waals volume Vw is more or less constant for all polymers and has a value of approximately 1.6(1). Thus

αr = Er / Vm(298) ≈ 1.0 · 10-3Vw / (1.6 · Vw) = 6.5 · 10-4

αg = Eg / Vm(298) ≈ 0.45 · 10-3Vw / (1.6 · Vw) = 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):

αr · Tg ≈ 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

αg · Tg ≈ 0.08

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

αr ≈ 0.113 / Tg + αg = αg (0.113 + 0.08 ) / Tg = 0.193 / Tg

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

### Coefficient of Thermal Expansion at 293 K

Polymer Exper. CTE (1/K) Calc. CTE (1/K)
Poly(methyl methacrylate) 1.8 - 2.5 2.41
Polystyrene 2.0 -2.9 2.14
Poly(vinyl acetate) 2.8 2.81
Natural Rubber, unvulcanized 6.6 6.1
Polyisobutylene 5.5 5.9
The CTE values have been calculated with the software 3Ps-Tg from Triton Road using the GIM method, and the experimental values have been taken from reference (4).

##### 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)
• ## Summary

### Thermal Expansion

The volumetric expansion coefficient is in average three times the linear expansion coefficient.

• A polymer has a lower expansivity than the related low-molecular liquid (monomer).

• Empirically, it has been found that αrubber ≈ 2.5 · αglass.

• The ratio of the molar volume at 298K and the van der Waals volume is approximately 1.6.

• ### Definitions

Specific Thermal Expansivity:
e = ∂v / ∂T   [cm³/g-K]

Molar Thermal Expansivity:
E = ∂Vm / ∂T   [cm³/mol-K]

Thermal Expansion Coefficient:
αV = (1/V) (∂V / ∂T)   [1/K]

Linear Thermal Expansion Coefficient:
αL = (1/L) (∂L / ∂T)   [1/K]

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