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Related Concept Videos

Radical Chain-Growth Polymerization: Chain Branching01:17

Radical Chain-Growth Polymerization: Chain Branching

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...
Radical Chain-Growth Polymerization: Overview01:10

Radical Chain-Growth Polymerization: Overview

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...
Radical Chain-Growth Polymerization: Mechanism01:09

Radical Chain-Growth Polymerization: Mechanism

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 the...
Ziegler–Natta Chain-Growth Polymerization: Overview01:17

Ziegler–Natta Chain-Growth Polymerization: Overview

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 catalyst, high molecular...
Indeterminate Products01:29

Indeterminate Products

Indeterminate forms also arise in the evaluation of limits involving products, particularly when one factor approaches zero while the other tends to positive or negative infinity. This situation, commonly described as a zero-times-infinity form, does not have an immediately interpretable outcome. Depending on how the factors behave relative to one another, the limit of such a product may be zero, infinite, or a finite nonzero value.Product Limits and Algebraic RewritingTo analyze limits of this...
Molecular Weight of Step-Growth Polymers01:08

Molecular Weight of Step-Growth Polymers

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.
The extent of the...

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DNA Nanotubes as a Versatile Tool to Study Semiflexible Polymers
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End-to-end distribution for a wormlike chain in arbitrary dimensions.

Shafigh Mehraeen1, Bariz Sudhanshu, Elena F Koslover

  • 1Department of Mechanical Engineering, Stanford University, Stanford, California 94305, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|July 23, 2008
PubMed
Summary

We developed a new method to accurately calculate wormlike chain statistics in any dimension and flexibility. This approach enhances the precision of end-to-end distribution functions for biophysical applications.

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Area of Science:

  • Polymer physics
  • Statistical mechanics
  • Biophysics

Background:

  • Wormlike chains are fundamental models in polymer physics.
  • Calculating their statistical properties, especially the end-to-end distribution function, is crucial for understanding polymer behavior.
  • Existing methods often face limitations in accuracy or applicability across different chain rigidities and dimensions.

Purpose of the Study:

  • To develop an efficient and accurate methodology for calculating wormlike chain statistics.
  • To provide an exact analytical solution for the end-to-end distribution function in arbitrary dimensions.
  • To enable precise analysis of polymer behavior in various biophysical contexts.

Main Methods:

  • Constructed an exact analytical solution for the wormlike chain end-to-end distribution function in Fourier-Laplace space.
  • Utilized an infinite continued fraction representation for numerical stability and compactness.
  • Employed asymptotic methods (power-law expansion, Rayleigh-Schrödinger perturbation theory) for accurate Fourier-Laplace inversion.
  • Adapted the methodology for calculating the single-chain structure factor.

Main Results:

  • Developed a robust methodology applicable to all chain rigidities in arbitrary dimensions.
  • Achieved enhanced accuracy for the end-to-end distribution function, approaching machine precision.
  • Derived simple, closed-form expressions for the single-chain structure factor, facilitating comparison with experimental data.
  • Demonstrated the utility of the method through chain statistics realizations.

Conclusions:

  • The presented methodology offers a significant advancement in calculating wormlike chain statistics.
  • It provides a versatile tool for biophysical research, enabling more accurate modeling and interpretation of experimental results.
  • The enhanced accuracy and broad applicability make it valuable for diverse polymer science problems.