Flory-Schulz Distribution

< List of probability distributions < Flory-Schulz Distribution

What is the Flory-Schulz distribution?

The Flory-Schulz distribution, based on Paul Flory and G. V. Shulz’s [1] 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 [2].

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 [3].

Flory-Schulz distribution properties

The probability mass function (pmf) for the Flory-Schulz distribution is [1]

a²k(1 − a)^{k − 1}, 0 < a < 1,

where k is the number of monomers in the chain.

The cumulative distribution function (CDF)

1 – (1 – a)^k (1 + ak)

The exponential distribution is a good approximate for the Flory-Schultz distribution when k is large [4] or when a tends to 1 [5]. The empirically determined constant a is related to the fraction of unreacted monomer remaining [6].

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 [9]

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)

with a probability generating function (PGF) of

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 [5].

References

1. Flory, P.J. Molecular Size Distribution in Linear Condensation Polymers. J. Am. Chem. Soc. 1936, 58, 1877–1885.
2. Weissner, T. (2022). Toward Enhancing the Synthesis of Renewable Polymers: Feedstock Conversions and Functionalizable Copolymers. Dissertation.
3. Pan, J. (Ed.) (2014) Modelling Degradation of Bioresorbable Polymeric Medical Devices. Elsevier Science.
4. Polymers: Molecular Weight and its Distribution
5. 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
6. IUPAC, Compendium of Chemical Terminology, 2nd ed. (the “Gold Book”) (1997). Online corrected version: (2006–) “most probable distribution”. doi:10.1351/goldbook.M04035
7. 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. 2017, 44, 1153–1164. [Google Scholar]
8. Gallardo, D.I.; Gómez, Y.M.; Castro, M.D. A flexible cure rate model based on the polylogarithm distribution. J. Stat. Comput. Simul. 2018, 88, 2137–2149. [Google Scholar]
9. 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