Related Experiment Video
Updated: Jul 19, 2026
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
Renormalization group analysis of polymer cyclization with non-equilibrium initial conditions.
1Department of Physics, Sam Houston State University, Huntsville, TX 77341, USA. PHY_BAF@shsu.edu
The European Physical Journal. E, Soft Matter
|October 10, 2006
Summary
We developed a renormalization group approach for polymer cyclization. The reaction rate for ring initial conditions is related to equilibrium conditions, with a sub-leading logarithmic term significant at early times.
Area of Science:
- Polymer Physics
- Chemical Kinetics
- Statistical Mechanics
Background:
- Polymer cyclization is a fundamental process in polymer science.
- Understanding the kinetics of ring formation is crucial for various applications.
- Previous studies have explored the time dependence of reaction rates, but a comprehensive renormalization group approach is needed.
Purpose of the Study:
- To develop a renormalization group (RG) approach for polymer cyclization under ring initial conditions.
- To analyze the time dependence of the reaction rate at short times, much shorter than the polymer relaxation time.
- To investigate the relationship between reaction rates for ring and equilibrium initial conditions.
Main Methods:
- Renormalization group (RG) analysis.
- Theoretical modeling of polymer dynamics.
- Analysis of Rouse and Zimm chain dynamics.
Main Results:
- The leading time dependence of the reaction rate for ring initial conditions is k(t) ~ t^(-delta), related to equilibrium conditions k(t) ~ t^(1-delta).
- An effective exponent delta approaches 5/4 for long Rouse chains.
- A sub-leading logarithmic term, proportional to (ln t)/t, is identified, significant at early times and for smaller renormalized reaction rates.
- For Zimm dynamics, the leading rate is k(t) ~ 1/t, with a logarithmic correction term.
Conclusions:
- The RG approach provides a unified framework for understanding polymer cyclization kinetics.
- The relationship between ring and equilibrium initial conditions is confirmed and quantified.
- Logarithmic corrections are important, especially for smaller chains and in the Zimm dynamics case, potentially leading to a Landau pole.
More Related Videos
Related Concept Videos
01:17Ziegler–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...
01:09Radical 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...
02:35Free-Radical Chain Reaction and Polymerization of Alkenes
The conversion of alkenes to macromolecules called polymers is a reaction of high commercial importance. The structure of the polymer is defined by a repeating unit, while the terminal groups are considered insignificant. The average degree of polymerization represents the number of repeating units in the polymer molecule and is denoted by the subscript n.
01:03Step-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...
Many natural and synthetic polymers are produced by...
00:57Cationic Chain-Growth Polymerization: Mechanism
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 generated carbocation,...
01:08Molecular 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...
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...