358 BELL SYSTEM TECHNICAL JOURNAL 



are used for convenience. The two most frequently employed are what are 

 termed the "number average" and the "weight average". If osmotic 

 pressure measurements are made on solutions of the polymer and extra- 

 polated to zero concentration these will lead to a number average molecular 

 weight figure. This average represents what v/e would obtain if we sorted 

 out the molecules according to molecular weight and counted them. Multi- 

 plying each molecular weight (wi)'by the number present (»i) and dividing 

 by the total number of molecules we obtain the number average molecular 

 weight (Mn) or stated mathematically 



— _ 2w<»i _ 1 



^"-"2^-^ (1) 



where /i is the weight fraction of species of molecular weight, nii . 



Usually the osmotic pressure measurements are difficult to carry out and 

 a simpler measurement, that of dilute solution viscosity (DSV) is per- 

 formed. This determination consists in measuring the relative viscosity of 

 a solution of the polymer at one given low concentration and calculating 



{DSV) = ^^^ (2) 



where -qr is relative viscosity and c is the concentration in grams per 100 ml. 

 of solution. A more fundamental quantity usually differing little from the 

 DSV value is the so-called intrinsic viscosity. This is defined as [77] = 



— — . Measurements are made at several concentrations and extrapolated 



Cc-*o 



to zero concentration just as for osmotic pressure. From this value a 

 molecular weight, which may be referred to as a viscosity averagejnolecular 

 weight, can be calculated from the empirical expression [77] = K{MvY where 

 both K and a are constants over a fairly wide range, and which must be 

 independently determined. In some polymer distributions this viscosity 

 average is very close to the weight average defined by 



Mu, = 4^^ = 2/iW.- (3) 



where nii is again the molecular weight of each species and/t is the weight 

 fraction in which it is present in the mixture. In the example of Fig. 1 

 the number averages are indicated by the maxima of the various curves. 

 Here the viscosity and weight averages are identical. 



In polymers an equally important consideration with molecular size 

 distribution is chain molecule structure. It is convenient to distinguish 

 between micro-chain structure and macro-chain structure. By micro-chain 



