Couchman-Karasz Equation

One of the most popular equations for predicting glass transition temperatures (T_g) of amorphous mixtures and random copolymers is the Couchman-Karasz equation.⁽¹⁾ This equation is based on the entropy (S) continuity at the T_g:

S_glass(T,p) = S_liq(T,p)

In the case of blends and copolymers, the specific entropy of the  mixture at T_g is calculated as a weight-fraction-weighted sum of pure-component entropies, and the change of entropy due to mixing,

S_tot(T_g,mix) = ∑_i [ω_i · S_i(T_g,mix) ] + ΔS_mixing(T_g,mix)

where T_g,mix is the glass transition temperature of the mixture in the glass or rubber (liquid) state, ω_i is the mass fraction of component i, and ΔS_mix is the change of entropy due to mixing of the polymer components. It is assumed that the conformational entropy of mixture and that of the components is the same in the liquid and glass state at the glass transition temperature of the mixture, T_g,mix. Hence,

ΔS_mix,glass(T_g,mix ) = ΔS_mix,liq(T_g,mix )

and

∑_i ω_i {S_i,liq(T_g,mix) - S_i,glass(T_g,mix) } = 0

Furthermore, entropy continuity is invoked for each pure component and for the mixture at the glass transtion temperature, T_g,i: S_i,glass(T_g,i ) = S_i,liq(T_g,i ). The specific entropy changes corresponding to heating the individual components and the mixture are

S_i,glass(T_g,mix) =  S_i,glass(T_g,i) + C_pi,glass ln[T_g,mix/T_g,i]

S_i,liq(T_g,mix) = S_i,liq(T_g,i) + C_pi,liq ln[T_g,mix/T_g,i]

Entropy continuity for a multicomponent mixture at T_g,mix yields

∑_i ω_i ΔC_p,i ln[T_g,mix /T_g,i] = 0

which upon rearrangement gives

lnT_g,mix ≈ ∑_i [ω_i · ΔC_pi ln[T_g,i] ]/ ∑_i [ω_i · ΔC_pi ]

where ΔC_pi is the change of the heat capacity when crossing from the glass to the rubber (liquid) state. This is the so called Couchman-Karasz⁽¹⁾ equation for the compositional dependence of the glass transition temperature in miscible multicomponent mixtures via entropy continuity.

The Couchman-Karasz equation has been shown to be very successful for fitting the glass transition temperature of many random copolymers. It can also be used to describe the composition dependence of miscible polymer blends exhibiting negative and positive deviation. However, the Couchman-Karasz equation should not be applied to polymer blends and random copolymers with noticeable specific intermolecular interactions because large deviations from the true T_g of the mixture can be expected.

A suitable example is a blend of polystyrene (PS) and poly(2,6-dimethyl-p-phenylene oxide) (PPO). Both polymers have similar structural groups and no specific interactions are present. The predicted values for such a blend are in excellent agreement with the experimental values (see Figure and Table below).⁵

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References & Notes

1. P. R. Couchman F.E. Karasz, Macromolecules 11, 117–119 (1978)

2. X. Lu and R.A. Weiss, Macromolecules 25, 3242 - 3246 (1992)

3. U. Gaur and B. Wunderlich, J. Phys. Chem. Ref. Data, Vol. 10, No. 4, 1981

4. U. Gaur and B. Wunderlich, J. Phys. Chem. Ref. Data, Vol.11 , No. 2, 1982

5. The values have been calculated with the experimental discontinuos increments in
   specific heat, cited by Wunderlich and Gaur in literature 3 and 4