Quantitative Effects of Factors on Tg

The lower the molecular weight of a polymer sample, i.e., the greater the number of chain ends, the higher becomes the contribution to the free volume when these begin moving consequently Tg decreases as the molecular weight of the sample 

In a polymer sample of molecular weight and density p, the number of chains per unit volume is given by pNAv/Mn, where Nav is Avogadro s number, and so the number of chain ends per unit volume is IpNAvIMn- If 9 is the contribution of one chain end to the free volume then the total fractional free volume due to all chain ends, fc, will be given by 

It can be argued that fc is the extra amount of free volume in a low molecular weight compound, as compared to a polymer of infinite molecular weight, that causes lowering of the glass transition temperature from 7, oo to Tg. So fc will be equivalent to the free volume expansion between Tg and Tg oa, which means that 

Problem 2.29 A polydisperse polystyrene sample was fractionated into four components of various molecular distributions and Tg of each fraction was measured  

Branches present in small numbers on a polymer chain are known to decrease Tg. This effect also can be explained using. the free volume concept. Since branches give rise to chain ends, the above arrSfysis of the effect of molecular weight on Tg can be extended to branching. Thus, if the total number ends per chain is y, then by the analysis as given above one may write 

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