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:

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 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

 

Experimental and Predicted Glass Transition Temperatures of some Polymers

Compound Exper. Tg (K)(a) Predicted Tg (K)(b)
Polycarbonate (PC) 447 435
Poly(tetrahydrofuran) (PTMO) 189 194
Poly(butyl methacrylate) 293 291
Poly(hexyl methacrylate) 268 266
Poly(ethylene terephthalate) (PET) 345 346

a Sigma-Aldrich
b The glass transition temperatures have been calculated with the software 3Ps-Tg from Triton Road

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.
  • Summary

    Glass Transition Temperatures

    can be obtained from quantitative structure-property correlations such as additive atomic or molecular group contribution methods.

  • These methods are based on the assumption that the glass transition temperature can be calculated from weighed additive contributions of the structural groups in the repeat units.

  • It is assumed that the contribution of a structural group is independent of the chemical nature of the adjacent groups.

  • The accuracy of prediction is affected by the size and choice of the individual groups; the larger the groups are, the more accurate is the prediction.

  • Most group contribution methods work well for simple linear polymers but are often unreliable when applied to polymers with bulky side groups or complex molecular architecture (i.e. branched polymers).

  • Group Interaction Modeling

    is one of the most versatile and reliable group contribution methods for the prediction of glass transition temperatures.

  • The glass transition temperature is calculated from the cohesive energy of the polymer normalized to its number of skeletal modes of vibration.

Read More