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Related Concept Videos

Polymers: Molecular Weight Distribution01:10

Polymers: Molecular Weight Distribution

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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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Molecular Weight of Step-Growth Polymers01:08

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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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Radical Chain-Growth Polymerization: Chain Branching01:17

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The skeletal structure of polymers synthesized via radical polymerization is always branched. For example, the polymerization of ethylene by radical polymerization results in a low-density grade of polyethylene with a heavily branched skeletal structure. Here, the radical site abstracts hydrogen from the growing chain, and the radical site shifts from the end (a primary carbon center) to anywhere within the growing chain (a secondary carbon center). Consequently, the part of the chain from the...
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Radical Chain-Growth Polymerization: Mechanism01:09

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The radical chain-growth polymerization mechanism consists of three steps: initiation, propagation, and termination of polymerization. The polymerization initiates when a free radical generated from the radical initiator adds to the unsaturated bond in the monomer. The unpaired electron of the free radical and one π electron in the unsaturated bond creates a σ bond between the free radical and the monomer. As a result, the other π electron in the unsaturated bond converts this species into...
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Radical Chain-Growth Polymerization: Overview01:10

Radical Chain-Growth Polymerization: Overview

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Chain-growth or addition polymerization is successive addition reactions of monomers with a polymer chain. In radical chain-growth polymerization, the reaction proceeds via a free-radical intermediate. The free radical is formed from radical initiators, which spontaneously generate free radicals by homolytic fission. Organic peroxides (such as dibenzoyl peroxide, as shown in Figure 1) or azo compounds are popular radical initiators. A low concentration ratio of radical initiator to monomer is...
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Ziegler–Natta Chain-Growth Polymerization: Overview01:17

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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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Updated: Mar 30, 2026

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
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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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Crossing probability for directed polymers in random media.

Andrea De Luca1, Pierre Le Doussal2

  • 1Laboratoire de Physique Théorique et Modèles Statistiques (UMR CNRS 8626), Université Paris-Sud, Orsay, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 14, 2015
PubMed
Summary
This summary is machine-generated.

We investigated the probability of non-intersecting directed polymers in random potentials. Analytical methods and simulations reveal large time properties of these noncrossing probabilities.

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Area of Science:

  • Statistical mechanics
  • Probability theory
  • Polymer physics

Background:

  • Understanding the behavior of interacting systems in random environments is crucial.
  • Directed polymers in random potentials are a key model in statistical physics.

Purpose of the Study:

  • To calculate the probability that two directed polymers in the same random potential do not intersect.
  • To explore the connection between polymer models and quantum field theory.

Main Methods:

  • Utilized the replica method to transform the problem into the attractive Lieb-Liniger model.
  • Employed the nested Bethe ansatz and MacDonald processes for analytical solutions.
  • Compared analytical results with numerical simulations of a discrete model at high temperatures.

Main Results:

  • Derived analytical expressions for the first few moments of the noncrossing probability.
  • Confirmed analytical findings through numerical simulations.
  • Observed and conjectured several large time properties of noncrossing probabilities.

Conclusions:

  • The study provides a rigorous analytical framework for noncrossing probabilities of directed polymers.
  • The findings offer insights into the large time behavior of these systems.
  • The developed formalism can be extended to analyze more general observables.