# Flory-Schulz distribution

The Flory-Schulz distribution, based on Paul Flory and G. V. Shulz’s  work on chain polymerization models, was developed to describe relative ratios of molecule lengths after a polymerization reaction — the process where a chemical reaction links monomer units to form long chains. The distribution describes the relative ratios of different length polymers that are found in an ideal step-growth polymerization — such as ethylene oligomerization .

While different types of polymer weight distributions can generate polymer chains, including the log-normal distribution, the Flory-Schulz distribution is thought to be the most probable one .

## Flory-Schulz distribution properties

The probability mass function (PMF) for the Flory-Schulz distribution is 

a2k(1 − a)k − 1, 0 < a < 1,

where k is the number of monomers in the chain.

1 – (1 – a)k (1 + ak)

The exponential distribution is a good approximate for the Flory-Schultz distribution when k is large  or when a tends to 1 . The empirically determined constant a is related to the fraction of unreacted monomer remaining .

Other properties include:

• Mean = (2/a) = 1
• Mode = 1 / (log(1 – a))
• Variance = (2 – 2a) / a2.

## Other applications

The Flory-Schulz distribution isn’t limited to applications in chemistry. For example, when applied to disease survival / cure rate contexts, M can be considered as a random variable with support on set {0, 1, 2, …} [7, 8]. In that context, the PMF can be shifted to 

P(M = m)= η2(m + 1)(1 – η)m , m = 0, 1, 2, …, 0 < η < 1

(note the use of η = a and m = k)

The distribution has also been used to investigate how proteins, which can co-aggregate into amyloid fibrils, are related to pathologies such as Alzheimer’s disease .

## References

 Flory, P.J. Molecular Size Distribution in Linear Condensation Polymers. J. Am. Chem. Soc. 193658, 1877–1885. [Google Scholar]

 Weissner, T. (2022). Toward Enhancing the Synthesis of Renewable Polymers: Feedstock Conversions and Functionalizable Copolymers. Dissertation.

 Pan, J. (Ed.) (2014) Modelling Degradation of Bioresorbable Polymeric Medical Devices. Elsevier Science.

 Prigent, S. eta al. Size distribution of amyloid fibrils. Mathematical
models and experimental data. Retrieved April 28, 2023 from: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=1fd6b3146d0d4051f8afcb0d85a32e66083cf64e

 IUPACCompendium of Chemical Terminology, 2nd ed. (the “Gold Book”) (1997). Online corrected version: (2006–) “most probable distribution“. doi:10.1351/goldbook.M04035

 Gallardo, D.I.; Gómez, H.W.; Bolfarine, H. A new cure rate model based on the Yule-Simon distribution with application to a melanoma data set. J. Appl. Stat. 201744, 1153–1164. [Google Scholar]

 Gallardo, D.I.; Gómez, Y.M.; Castro, M.D. A flexible cure rate model based on the polylogarithm distribution. J. Stat. Comput. Simul. 201888, 2137–2149. [Google Scholar]

 Azimi, R. et al. A New Cure Rate Model Based on Flory–Schulz Distribution: Application to the Cancer Data. Mathematics 2022, 10(24), 4643; https://doi.org/10.3390/math10244643

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