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Related Concept Videos

The Chain Rule: Problem Solving01:23

The Chain Rule: Problem Solving

The thermal expansion of a metal rod shows the application of the Chain Rule when one physical quantity depends on another that varies with time. As the rod is heated, its length changes according to linear thermal expansion, while the temperature of the system varies quadratically with time.For linear thermal expansion, the length L of the rod depends on temperature T such that the rate of change of length with respect to temperature is constant:where L0 = 2 m is the initial length of the rod,...
Debye–Huckel–Onsager Conductance Equation01:28

Debye–Huckel–Onsager Conductance Equation

The Debye-Hückel-Onsager equation is a cornerstone of physical chemistry, providing a method to determine the molar conductance (Λm) and molar conductance at infinite dilution (Λ°m) for uni-univalent electrolytes.Uni-univalent electrolytes are electrolytes that dissociate in solution to produce one cation with a +1 charge and one anion with a –1 charge per formula unit.This equation addresses two crucial phenomena: the asymmetry effect and the electrophoretic effect. According to this equation,...
Maxwell-Boltzmann Distribution: Problem Solving01:20

Maxwell-Boltzmann Distribution: Problem Solving

Individual molecules in a gas move in random directions, but a gas containing numerous molecules has a predictable distribution of molecular speeds, which is known as the Maxwell-Boltzmann distribution, f(v).
This distribution function f(v) is defined by saying that the expected number N (v1,v2) of particles with speeds between v1 and v2 is given by
Carrier Transport01:21

Carrier Transport

The generation of electrical current in semiconductors is fundamentally driven by two mechanisms: drift and diffusion. These processes are essential for the functionality and performance of semiconductor-based devices.
Drift Current:
The drift of charge carriers is started by an external electric field (E). Charged particles, such as electrons and holes, experience an acceleration between collisions with lattice atoms. For electrons, this results in a drift velocity (vd) given 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...
Mean free path and Mean free time01:22

Mean free path and Mean free time

Consider the gas molecules in a cylinder. They move in a random motion as they collide with each other and change speed and direction. The average of all the path lengths between collisions is known as the "mean free path."

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Related Experiment Video

Updated: May 10, 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

Modified diffusion equation for the wormlike-chain statistics in curvilinear coordinates.

Qin Liang1, Jianfeng Li, Pingwen Zhang

  • 1School of Mathematics and Computational Science, Xiangtan University, Xiangtan, Hunan 411105, People's Republic of China.

The Journal of Chemical Physics
|July 5, 2013
PubMed
Summary

This study introduces a new term in polymer theory, enhancing the understanding of polymer conformation and structure using the wormlike-chain model. The findings detail a novel coupling effect and surface energy penalty for polymers on curved surfaces.

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Area of Science:

  • Polymer Physics
  • Statistical Mechanics
  • Soft Matter Physics

Background:

  • The propagator is crucial for analyzing polymer conformation and structure.
  • Existing models often simplify polymer behavior, particularly on complex geometries.

Purpose of the Study:

  • To derive the partial diffusion equation for the polymer propagator in a curvilinear coordinate system.
  • To identify and document previously undocumented terms in polymer dynamics.
  • To investigate the behavior of wormlike chains on curved surfaces.

Main Methods:

  • Utilized the wormlike-chain statistical-physics model.
  • Derived partial differential equations for the propagator in curvilinear coordinates.
  • Incorporated surface curvature effects for polymers on curved surfaces.

Main Results:

  • A novel term coupling the rotating local coordinate frame with an orientation differential operator was identified.
  • The external-field term for polymers on curved surfaces requires a surface curvature energy penalty.
  • The derived equations provide a more accurate description of polymer dynamics.

Conclusions:

  • The study reveals new physics in polymer dynamics, particularly concerning local frame rotations and surface interactions.
  • The findings offer a more comprehensive theoretical framework for studying polymer conformation and structure.
  • This work advances the understanding of polymer behavior in complex environments.