MOLECULAR WEIGHT DISTRIBUTION IN LINEAR STEP-GROWTH POLYMERS

The molecular weight distribution of a linear condensation or addition polymer can be easily calculated if we assume that each functional group has an equal chance of reacting with other groups regardless of the size of the oligomer. This assumption is called Flory's equal reactivity principle. According to this principle, the probability that a given reactive group has reacted is equal to the fraction p of all condensed functional groups of the same type, which is called the extend of the reaction. An oligomer containing x repeat units must have undergone x-1 reactions. The probability that this number of reactions has occured is simply the product of all reaction probabilities, i.e. p^x-1, whereas the probability of finding an unreacted end group is is 1- p.¹ Hence, the total probability, P_x that a given oligomer is composed of exactly x unitis is given by

P_x = (1- p) p^x-1

P_x is equal to the mole fraction, n_x, of x-mers in the reaction mixture of the extend p:

P_x = n_x = N_x / N

Then the total number of x-mers is given by

N_x = N (1- p) p^x-1

where N is the total number of molecules of all sizes. This number is related to the initial number of monomers or total number of units, N₀, by

N = N₀ (1- p)

where DP = N₀ / N equals the average degree of polymerization⁴

DP = N₀ / N = 1 / (1- p)

With this substituition, the total number of x-mers can be written in the form

N_x = N₀ (1- p)² p^x-1

If the weight of the (condensing) end-groups of each molecule is neglected (for example H + OH), the molecule weight of each molecule is directly proportional to the length of the chain x. Hence, the weight fraction w_x can be written as

w_x = x N_x / N₀

The error for condensation polymers will be only significant for low molecular weight polymers.

Then the most probably weight fraction distribution (i.e. the weight-average molecular weight distribution) is given by^6,7

w_x = x (1- p)² p^x-1

This distribution is sometimes called Flory-Schulz distribution. The form of this distribution implies that shorter polymer chains are favored over longer ones. It also implies that the length distribution broadens and shifts to higher molecular weight with increasing extend of reaction (see Figures below).

MOLE FRACTION DISTRIBUTION OF LINEAR STEP-GROWTH POLYMERS

WEIGHT FRACTION DISTRIBUTION OF LINEAR STEP-GROWTH POLYMERS

From the expressions above, the number average molecular weight M_n can be easily calculated:

M_n = m N₀ / N = m / (1- p)

where m is the molecular weight of a mer unit. The weight-average molecular weight M_w can be calculated as follows:³

M_w = ∑_x w_x M_x = m (1- p)² ∑_x x² p^x-1  = m (1+ p) / (1- p) = (1+ p) M_n

where M_x = x·m is the molecular weight of a x-mer. The ratio of weight and number average of the polymer molecular weight, M_w / M_n, is called the polydispersity (PDI) or heterogeneity index (D). It is a measure for the broadness of a molecular weight distribution. The polydispersity is unity if all polymer moelcules are of the same size. The polydispersity for the most probable molecular weight distribution is given by

D = m (1 + p) (1- p) / [m  (1- p)] = (1+ p)

Thus, when the probabilty p is equal to 1 (i.e. all A and B units have reacted at least with one other unit (A-B), the polydispersibility index for the most probable distribution for a linear step-growth reaction approaches 2.