Prediction of Glass Transition Temperature

Several scientists have proposed quantitative structure–property relationships to predict the glass transition temperature of polymers from their chemical structure.^1-6 These methods work well for linear polymers in the absence of bulky side groups, whereas for polymers with large side groups the predictions are often unreliable.
One of the most advanced group contribution method is the so called "Group Interaction Model" of Porter (GIM). Porter suggested following relationship for the glass transition temperature of linear polymers of high molecular weight⁶:

T_g(K) = A · E_coh / N + C · θ₁

where A and C are two constants having the values

A = 0.0513 K·mol/J
C = 0.224

E_coh is the cohesive energy of a repeat unit, and N is the total number of skeletal modes of vibrations defined in a glass just below the T_g. These two parameters can be calculated with group contribution methods:

E_coh = ∑k N_k · F_k

N = ∑k N_k · A_k

F_k and A_k are the group contributions for an increment occurring N_k times in the backbone. Some group contributions for the GIM model parameters A_k and F_k can be found in David Porter's monograph "Group Interaction Modelling of Polymer Properties". Many of the suggested GC values for skeletal modes of vibration are a good starting point, but are not necessary representative for all polymers. In fact, N is the parameter that is most difficult to assign a precise value. One reason is the time and the temperature dependency of this parameter. This makes the calculation of physical properties that depend on N difficult. For example, the coefficient of thermal expansion, α, might change with increasing temperature, since new modes of vibration might become available at higher temperatures.

Another important parameter in Porter's equation is the reference temperature of skeletal mode vibration normal to the chain axis, θ₁. This parameter can be estimated with the equations

θ_1,t = 550 · (14 · N_osc / M)^1/2 (trans conformer)

θ_1,g = 316 · (14 · N_osc / M)^1/2 (gauche conformer)

That is, gauche and trans conformers have different reference temperatures. Luckily, most polymers tend to be either mainly trans or mainly gauche conformers (often trans), so that only one value has to be chosen⁶. However, this assumption might not be true for all polymers.

As Wunderlich has shown, the inverse proportionality between θ₁ and the molecular weight M is also valid for any given series of polymers with side-chains. But no general equation can be found that applies to all classes of polymers with side chains. The reason for this disagreement can be found in the choice of the number of modes assigned to the side chain.⁷

 

References and Notes

1. D.W. van Krevelen and Klaas te Nijenhuis, Properties of Polymers, 4th Ed., Amsterdam (2009)
2. U.T. Kreibich, H. Batzer, Angew. Makromol. Chem., 83, 57-112 (1979)
3. H. Batzer and U.T. Kreibich, Angew. Makromol. Chem., 105, pp. 113-130 (1982)
4. H.K. Reimschuessel, J. of Poly. Sc.: Polymer Chemistry, Vol. 17, 2447-2457 (1979)
5. A. A. Askadskii, Pure & Appi. Chem., Vol 46, pp. 19—27 (1976)
6. D. Porter, Group Interaction Modelling of Polymer Properties, Marcel Dekker, New York (1995)
7. Porter does not use the same convention as Wunderlich for the number of side chain skeletal modes of vibration; only modes that oscillate normal to the main chain axis are included as skeletal modes, whereas Wunderlich assigns the same values to side chain modes as to main chain modes of vibration.