Prediction of Glass Transition Temperature
The glass transition temperature can be predicted with group contribution methods. These methods work well for linear polymers in the absence of bulky side groups, whereas for
polymers with large side groups the predictions
One of the most advanced group contribution models is the so called "Group Interaction Model" of Porter (GIM). Porter suggested following relation for the glass transition temperature of linear polymers of high molecular weight(1):
Tg(K) = A · Ecoh / N + C · θ1
where A and C are two constants having the values
A = 0.0513 K·mol/J
C = 0.224
Ecoh 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 Tg. These two parameters can be calculated with group contribution methods:
Ecoh = ∑k Nk · Fk
N = ∑k Nk · Ak
Fk and Ak are the group contributions for an increment occurring Nk times in the backbone. Some group contributions for the GIM model parameters Ak and Fk 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
θ1,t = 550 · (14 · Nosc / M)1/2 (trans conformer)
θ1,g = 316 · (14 · Nosc / 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 chosen(1). 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.(3)
|Compound||Exper. Tg (K)(a)||Predicted Tg (K)(b)|
|Poly(ethylene terephthalate) (PET)||345||346|
References and Notes
- D. Porter, Group Interaction Modelling of Polymer Properties, Marcel Dekker, New York (1995)
- 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.