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Related Concept Videos

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: 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: 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...
Step-Growth Polymerization: Overview01:03

Step-Growth Polymerization: Overview

Step-growth or condensation polymerization is a stepwise reaction of bi or multifunctional monomers to form long-chain polymers. As all the monomers are reactive, most of the monomers are consumed at the early stages of the reaction to form small chains of reactive oligomers, which then combine to form long polymer chains in the late stages. Hence, the reaction has to proceed for a long time to achieve high molecular weight polymers.
Many natural and synthetic polymers are produced by...
Curvilinear Motion: Polar Coordinates01:27

Curvilinear Motion: Polar Coordinates

In polar coordinates, the motion of a particle follows a curvilinear path. The radial coordinate symbolized as 'r,' extends outward from a fixed origin to the particle, while the angular coordinate, 'θ,' measured in radians, represents the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the particle.
The particle's location is described using a unit vector along the radial direction. Deriving the particle's position with respect to time...

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Related Experiment Video

Updated: May 20, 2026

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
06:55

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level

Published on: September 26, 2016

Ring polymer dynamics in curved spaces.

S Wolf1, E Curotto

  • 1Department of Chemistry and Physics, Arcadia University, Glenside, Pennsylvania 19038-3295, USA.

The Journal of Chemical Physics
|July 12, 2012
PubMed
Summary
This summary is machine-generated.

This study extends ring polymer dynamics to curved spaces using stereographic projection. New algorithms improve energy conservation, and simulations show excellent agreement with exact results for particle-in-a-ring models.

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Last Updated: May 20, 2026

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Area of Science:

  • Computational Chemistry
  • Theoretical Physics
  • Statistical Mechanics

Background:

  • Ring polymer dynamics is a powerful method for simulating quantum systems.
  • Simulating systems in curved spaces presents unique computational challenges.
  • Existing methods may struggle with energy conservation in curved manifolds.

Purpose of the Study:

  • To extend the ring polymer dynamics approach to curved spaces.
  • To develop novel algorithms for integrating Hamilton's equations on curved manifolds.
  • To improve energy conservation in symplectic integrators.

Main Methods:

  • Formulation of ring polymer dynamics using stereographic projection coordinates.
  • Simulation of a particle in a ring (T(1)) model with various potentials.
  • Development and application of new integration algorithms for curved spaces.
  • Analysis of position-position autocorrelation functions.

Main Results:

  • The proposed stereographic projection method successfully extends ring polymer dynamics to curved spaces.
  • New algorithms demonstrate improved energy conservation for symplectic integrators.
  • The position-position autocorrelation function computed in the embedding Euclidean space R(2) yielded superior statistical properties.
  • Excellent agreement was achieved between simulated and exact results for all tested potentials and temperatures.

Conclusions:

  • The developed methods provide an accurate and efficient approach for simulating quantum systems on curved manifolds.
  • The new integration algorithms enhance the reliability of simulations in curved spaces.
  • This work offers a robust framework for applying ring polymer dynamics to complex curved geometries.