Schulz-Flory distribution

A Schulz-Flory distribution (after Paul John Flory and Günter Victor Schulz ) is a distribution function that can be used in the molecular weight distribution of polymers . This distribution is found in many reactions, especially polycondensation and specifically terminated polymerizations . A distribution closely related to the Schulz-Flory distribution is the Schulz-Zimm distribution .

description

The probability for a chain extension or for a chain termination reaction is given . Then the discrete probability of encountering a chain with monomers is equal to the probability of successful reactions and a chain termination reaction: ${\displaystyle p}$ ${\displaystyle 1-p}$ ${\displaystyle N}$ ${\displaystyle (N-1)}$

${\displaystyle p(N)=p^{N-1}(1-p)}$.

This probability corresponds to the mole fraction of the chains with monomers. According to the above formula, the mole fraction follows a geometric distribution . ${\displaystyle x_{a}(N)=p(N)}$ ${\displaystyle N}$

The probability that no chain extension takes place in a reaction is also often introduced - the probability that chain extension takes place is accordingly . Then it arises ${\displaystyle a=(1-p)}$ ${\displaystyle p=1-a}$

${\displaystyle x_{a}(N)=a(1-a)^{N-1}}$,

where is practically an empirically determined constant. ${\displaystyle 0<a<1}$

The weight fraction of chains of length is defined as: ${\displaystyle w_{a}(N)}$ ${\displaystyle N}$

${\displaystyle w_{a}(N)={\frac {\text{Masse aller N-Mere}}{\text{Gesamtmasse}}}={\frac {(Nm_{0})\cdot x_{a}(N)}{\sum _{N=0}^{\infty }Nm_{0}x_{a}(N)}}=a^{2}N(1-a)^{N-1}}$,

where is the mass of an N-mer and is the monomer mass. ${\displaystyle Nm_{0}}$ ${\displaystyle m_{0}}$

Weight percentage according to Schulz-Flory distribution

The mass mean is

${\displaystyle {\overline {M}}_{w}=(1+p){\overline {M}}_{n}}$ and

the number mean

${\displaystyle {\overline {M}}_{n}=m_{0}/(1-p)}$, where (as above) is the monomer mass. ${\displaystyle m_{0}}$

In the Schulz-Flory distribution, the polydispersity is

${\displaystyle Q={\frac {{\overline {M}}_{w}}{{\overline {M}}_{n}}}={1+p}}$.

Hence is . ${\displaystyle \lim _{p\to 1}Q=2}$

Oligomerization of Ethene

Weight fraction for the Schulz-Flory distribution in the oligomerization of ethene with compensation line ( green )

In the oligomerization of ethene to form longer α – olefins, a number of technical processes are implemented, the product mixtures of which are based on a Schulz-Flory distribution. Highly relevant applications for this are the SHOP process or the Ziegler-Alfen synthesis .

The figure shows the proportions of the mass of the resulting products compared to the total mass. The products are sorted according to the number of carbon atoms. It can be seen that with a product mixture that is distributed according to the Schulz-Flory distribution, short-chain α-olefins are preferably formed. The longer the carbon chains, the lower their occurrence in the product mixture. The main products here are those particles that have no more than twelve carbon atoms. It should be noted that the oligomerization of ethene only produces molecules with an even number of carbon atoms. The compensation line ( green ) shows that there is a Schulz-Flory distribution, with all possible states being marked by crosses.

See also

• Controlled Free Radical Polymerization

literature

• Bernd Tieke: Macromolecular Chemistry . 2nd edition, Wiley-VCH, 2005, ISBN 3-527-31379-6 .

Web links

• Schulz-Flory or Schulz-Zimm distribution (SF)
• Schulz-Flory distribution in the event of chain termination through combination

Individual evidence

1. ^ Paul J. Flory: Molecular Size Distribution in Linear Condensation Polymers1 . In: Journal of the American Chemical Society . tape 58 , no. October 10 , 1936, ISSN  0002-7863 , p. 1877–1885 , doi : 10.1021 / ja01301a016 .
2. ↑ Entry on most probable distribution . In: IUPAC Compendium of Chemical Terminology (the “Gold Book”) . doi : 10.1351 / goldbook.M04035 Version: 2.3.1.
3. ^ ^{A b} Wilhelm Keim , Arno Behr, Günter Schmitt: Basics of Industrial Chemistry, 1st edition, Otto Salle Verlag GmbH & Co., Frankfurt am Main, Verlag Sauerländer AG, Aarau 1986, ISBN 3-7935-5490-2 (Salle) , ISBN 3-7941-2553-3 (Sauerländer), pp. 138-141.