## Schulz-Flory distribution in the event of chain termination through combination

In free-radical polymerization, chain termination can also take place through the union of two growing chains (combination):

For the formation of a polymer chain of a certain length or certain degree of polymerisation e.g. $\mathrm{P.}$ = 1,000 there are the following options:

- An initiator radical only adds one monomer (degree of polymerization
*P.*= 1) combined with a chain radical of the degree of polymerization_{a}*P.*=999_{b}

- A chain radical consisting of 2 monomer units (
*P.*= 2) unites with a chain radical des_{a}*P.*=998_{b}

- A chain radical
*P.*= 3 finds a chain radical with_{a}*P.*= 997 etc. up to_{b}*P.*= 500 and_{a}*P.*=500_{b}

Generally, polymers are made with $\mathrm{P.}$ Structural units obtained by combining the following chain pairs:

1+($\mathrm{P.}$-1), 2+($\mathrm{P.}$-2), 3+($\mathrm{P.}$-3) to $\mathrm{P.}$/2+[$\mathrm{P.}$-($\mathrm{P.}$/2)]

So there exist in total $\mathrm{P.}$/ 2 different combinations that make a polymer with the degree of polymerization $\mathrm{P.}$ can arise from a combination of two chain radicals.

The likelihood of these $\mathrm{P.}$/ 2 different combinations with a certain combination of two chain radicals *P. _{a}*= a and

*P.*= b occurs is twice as large as the product of the individual probabilities

_{b}*w*and

_{a}*w*for the chain radicals a and b. The combination probability

_{b}*w*two different sized radicals with the individual probabilities

_{a + b}*w*and

_{a}*w*is therefore:

_{b}- $${w}_{a+b}=2{w}_{a}{w}_{b}$$

The following applies to the combination of two radicals of equal length:

- $${w}_{a+a}={(w{}_{a})}^{2}$$

If the polymerization of two chain radicals is terminated by combination, the two individual radicals with degrees of polymerization a and b form a macromolecule with degree of polymerization a + b =$\mathrm{P.}$.

From this it follows: b =$\mathrm{P.}$-a

The proportions with which the radicals a and $\mathrm{P.}$-a are represented in the system and thus their individual probabilities *w _{a}* and

*w*are:

_{P-a}- $${w}_{a}={\alpha}^{a}(1-\alpha )$$

- $${w}_{\mathrm{P.}-a}={\alpha}^{\mathrm{P.}-a}(1-\alpha )$$

Substituting into equations (1) and (2), the following applies to the combination of unequal-sized radicals:

- $${w}_{a+(\mathrm{P.}-a)}=2{\alpha}^{a}{(1-\alpha )}^{2}{\alpha}^{\mathrm{P.}-a}=2{\alpha}^{\mathrm{P.}}{(1-\alpha )}^{2}$$

And for the combination of radicals of the same size (a = b =$\mathrm{P.}$/2):

- $${w}_{(\mathrm{P.}/2+\mathrm{P.}/2)}={\alpha}^{\mathrm{P.}/2}{(1-\alpha )}^{2}{\alpha}^{\mathrm{P.}/2}={\alpha}^{\mathrm{P.}}{(1-\alpha )}^{2}$$

Equation (3) describes the probability that any chain radical with the degree of polymerization a with a chain radical with the degree of polymerization $\mathrm{P.}$-a responds. This creates a macromolecule with a + ($\mathrm{P.}$-a) =$\mathrm{P.}$.

Macromolecules are not only created by this combination of chain radicals, but by a number of them as a whole $\mathrm{P.}$/ 2 different combinations. So that you get the probability *w _{P.}* for the formation of a macromolecule with a certain degree of polymerization $\mathrm{P.}$ obtained by any combination, the probabilities of all possible combinations of 1+ ($\mathrm{P.}$-1 to $\mathrm{P.}$/2+$\mathrm{P.}$/ 2 must be added, i.e. equation (3) with $\mathrm{P.}$/ Multiply 2:

- $${w}_{\mathrm{P.}}=\mathrm{P.}{\alpha}^{\mathrm{P.}}{(1-\alpha )}^{2}$$

*w _{P.}* is the probability of the formation of a macromolecule of length $\mathrm{P.}$ by combination or the proportion of macromolecules in length $\mathrm{P.}$ in a polymer in which the chain termination took place exclusively through combination.

Since when two chain radicals combine to form one molecule, the number of macromolecules that arise is only half as large as without a combination, the number is *n _{P.}* of macromolecules with the degree of polymerization $\mathrm{P.}$formed from n macro radicals:

- $${n}_{\mathrm{P.}}=0,5n\mathrm{P.}{\alpha}^{\mathrm{P.}}{(1-\alpha )}^{2}$$

With m = x (1-*α*) [x = number of monomer units in n polymer molecules, (1-$\alpha $) = Fraction of the termination reactions] results in:

- $${n}_{\mathrm{P.}}=0,5x\mathrm{P.}{\alpha}^{\mathrm{P.}}{(1-\alpha )}^{3}$$

The mass fraction is then analogous to the termination through disproportionation / chain transfer *m _{P.}*/ m of the molecules with the degree of polymerization $\mathrm{P.}$ with combination termination:

- $${m}_{\mathrm{P.}}/m=0,5{\mathrm{P.}}^{2}{\alpha}^{\mathrm{P.}}{(1-\alpha )}^{3}\approx 0,5{\mathrm{P.}}^{2}{(\mathrm{ln}\alpha )}^{3}$$

It can be seen from the figure that the distribution becomes more uniform by combining.