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Molecular Weight of Step-Growth Polymers

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Step growth polymerization involves bi or multifunctional monomers. Bifunctional monomers react to form linear step growth polymers, whereas multifunctional monomers react to form non-linear or branched polymers.
As the step-growth polymerization involves step-wise condensation of monomers, the molecular weight also builds up eventually. Consequently, high molecular weight polymers are obtained at the late stages of the polymerization, where 99% of monomers have been consumed.
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For any given polymer, the weight average molecular weight (Mw) is higher than, if not equal to, the number average molecular weight (Mn). The only situation in which the weight average molecular weight and the number average molecular weight are equal is when a polymer consists only of chains with equal molecular weight. However, this never happens in a synthetic polymer, since it is difficult to control the polymerization process up to a molecular level with accuracy to a hundred percent.
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Ziegler–Natta polymerization is another form of addition or chain‐growth polymerization used for synthesizing linear polymers over branched polymers. The catalyst used for polymerization is the Ziegler–Natta catalyst, named after Karl Ziegler and Giulio Natta, who developed it in 1953. This catalyst is an organometallic complex of titanium tetrachloride and triethyl aluminum, with the active form of the catalyst being an alkyl titanium compound. Using the Ziegler–Natta...
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Copolymers are the products obtained from the polymerization of multiple monomer species. So, in a polymer chain itself, there can be multiple repeating units that come from different monomers. The process of synthesizing a polymer from different monomer species is called copolymerization. When two monomers are involved, the polymer is known as a bipolymer. Polymers with three and four monomers are termed terpolymers and quaterpolymers, respectively. Figure 1 depicts the copolymerization of...
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The cationic polymerization mechanism consists of three steps: initiation, propagation, and termination. In the initiation step of the polymerization process, the π bond of a monomer gets protonated by the Lewis acid catalyst, which is formed from boron trifluoride and water. The protonation of the π bond generates a carbocation stabilized by the electron‐donating group. In the propagation step, the π bond of the second monomer acts as a nucleophile and attacks the...
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Polymers02:34

Polymers

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The word polymer is derived from the Greek words “poly” which means “many” and “mer” which means “parts”. Polymers are long chains of molecules composed of repeating units of smaller molecules, known as monomers. They either occur naturally, such as DNA and proteins, or can be constructed synthetically, like plastics. They have varied structural characteristics, such as linear chains, branched chains, or complex networks, that contribute to the...
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Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
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On a Factorization Formula for the Partition Function of Directed Polymers.

Tobias Hurth1, Konstantin Khanin2, Beatriz Navarro Lameda3

  • 1Institute for Mathematics, Freie Universität Berlin, Arnimallee 9, 14195 Berlin, Germany.

Journal of Statistical Physics
|October 23, 2023
PubMed
Summary
This summary is machine-generated.

This study introduces a factorization formula for directed polymer partition functions in a weak disorder regime. The research extends previous findings by showing the error term is small in sub-ballistic regimes for polymers.

Keywords:
Directed polymersPartition functionRandom walk in a random environmentWeak disorder

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Area of Science:

  • Probability Theory
  • Statistical Mechanics
  • Mathematical Physics

Background:

  • Directed polymers in random potentials are a key model in statistical mechanics.
  • Understanding polymer behavior in disordered systems is crucial for various scientific fields.

Purpose of the Study:

  • To establish a factorization formula for the point-to-point partition function of directed polymers.
  • To analyze the behavior of the error term in the weak disorder regime and extend previous results.

Main Methods:

  • Utilizing a factorization formula for the partition function.
  • Analyzing the quotient of partition function and transition probability.
  • Investigating the behavior of the error term in large time intervals.

Main Results:

  • A factorization formula for the point-to-point partition function is proven.
  • The error term is shown to be small uniformly over starting and endpoints in the sub-ballistic regime.
  • Asymptotics for spatial and temporal correlations of limiting partition functions are derived.

Conclusions:

  • The study extends Sinai's results on the smallness of the error term to a broader range of polymer dynamics.
  • The findings provide new insights into the statistical properties of directed polymers in disordered media.