## Kelley-Bueche Equation

### And Effect of Plasticizers on Glass Transition Temperature

It is well known that the glass transtion temperature of an amorphous polymer is reduced by the addition of low-molecular weight miscible compounds such as plasticizers. A low molecular weight compound also causes the glass-rubber transition to occur over a wider range of temperatures than the unplasticized  polymer.

In practice, one would like to be able to predict the Tg of a miscible blend like a polymer and plasticizer from the properties of the components. A number of approaches have been proposed to estimate glass transition temperatures of mixtures from the thermo-physical properties of the pure components. All these relationships have in common that they are based on the assumption of additivity and continuity of certain properties at the Tg. One of the most popular equations for predicting glass transition temperatures of amorphous mixtures is the Kelley-Bueche equation (1961):(2)

Tg,mixi [φi · Tg,i · Δαi] / i [ φi · Δαi]

where Tg,mix and Tg,i are the glass transition temperature of the mixture and of the components, φi is the volume fraction of component i and Δαi(Tg,i) is the change of the coefficient of thermal expansion (CTE) when crossing from the glass to the liquid (rubber) state.

The Kelley-Bueche equation is often used to describe the effect of plasticizers on the Tg of polymers. However, the equation above can only be applied to mixtures where the plasticizer is completely miscible with the host polymer. Furthermore, the equation is only applicable to blends with weak polymer-plasticizer interaction, i.e. van der Waals forces. If specific interactions between the polymer and plasticizer are present, the equation above has to be modified to account for these interactions.(4)

Typically, plasticizers are low molecular weight compounds having a very low Tg. In some cases, a miscible high(er) molecular weight compound of low Tg can be used as a platicizer.

The classic example of a polymer-plasticizer blend is PVC-Phthalate. Unfortunately, these blends show an unusual behavior above 40 % plasticizer content. For this reason, the calculation should be restricted to blends with less than 40 percent plasticizer.

### Glass Transition Temperature of PVC-DMP blends

Weight Fraction DMP Exper. Tg (K)(5) Predicted Tga,c (K) Predicted Tgb,c (K)
0 357 357 357
0.03 343 344 346
0.07 328 329 333
0.15 307 303 309.5
0.20 294 290 297
0.275 280 272 279.5
0.375 255 254 261
The glass transition temperatures have been calculated with the software 3Ps-Tg Triton Road.
aThe experimental discontinuous increment is ΔαPVC = 0.342 · 10-3 1/K.
bThe calculated discontinuous increment is ΔαPVC = 0.347 · 10-3 1/K (3Ps-Tg, Krevelen method)
cA reasonable value for most plasticizers is Δα ≈ 8 · 10-4 1/K

### Glass Transition Temperature of PVC-DMP blends

##### References
1. If the polymer is semi-crystalline, the plasticizer also reduces the melting temperature and/or the degree of crystallinity.
2. F.N. Kelley and F. Bueche, J. Polym. Sci., 50, 549 (1961)
3. J. R. Fried, Polymer Science and Technology, 1st Edition, New Jersey 2003
4. X. Lu and R.A. Weiss, Macromolecules 25, 3242 - 3246 (1992)
5. Scandole et al., Polymer Bulletin 6, 653-660 (1982)
• ## Summary

### Kelley-Bueche Equation

describes the composition dependence of the transition temperature of amorphous mixtures.

• The equation is often used to describe the effect of plasticizers on the glass transition temperature of polymers.

• The Kelley-Bueche equation should only be applied to miscible blends with relative weak specific intermolecular interactions.

• ### Plasticization

The addition of a low-molecular weight miscible plasticizer decreases the glass transition temperature and modulus of the polymer or polymer system.
Typical plasticizers have a Tg in the range of -100°C to -60°C.

• ### One- & Two-Phase System

A blend of polymers consists of two or more phases if two or more glass transition temperatures (Tg) are observed, whereas a single composition dependent Tg is often taken as evidence for a miscible blend (one-phase system).

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