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Diffusion, fragmentation, and coagulation processes: analytical and numerical results.

Poul Olesen1, Jesper Ferkinghoff-Borg, Mogens H Jensen

  • 1The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark. polesen@nbi.dk

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 26, 2005
PubMed
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We developed dynamical rate equations for physical processes involving growth, fragmentation, and coagulation. Our findings reveal Bessel-type distributions without coagulation and exponential decay with it, confirmed by simulations.

Area of Science:

  • Physics
  • Physical Chemistry
  • Materials Science

Background:

  • Understanding particle size dynamics is crucial in various physical and chemical processes.
  • Existing models often simplify complex interactions like fragmentation and coagulation.
  • A unified framework for these processes is needed.

Purpose of the Study:

  • To formulate and solve dynamical rate equations for systems with diffusive growth, fragmentation, and coagulation.
  • To analyze the resulting particle size distributions under different process conditions.
  • To validate theoretical predictions with numerical simulations.

Main Methods:

  • Formulation of dynamical rate equations.
  • Exact analytical solution for processes without coagulation.

Related Experiment Videos

  • Mapping the nonlinear model with coagulation to a Riccati equation.
  • Derivation of asymptotic solutions for the distribution function.
  • Numerical simulations for validation.

    • Main Results:

    • Identified Bessel-type size distributions with exp(-x(3/2)) decay for large sizes when coagulation is absent.
    • Derived explicit formulas for expansion coefficients using Airy functions.
    • Obtained a standard exponential decay (exp(-x)) for large sizes in the presence of coagulation.
    • Observed a crossover from Bessel to exponential decay for intermediate sizes.
    • Achieved perfect agreement between theoretical predictions and numerical simulations.

    Conclusions:

    • The developed dynamical rate equations accurately describe particle size evolution under growth, fragmentation, and coagulation.
    • The study provides exact solutions and asymptotic behaviors for these complex processes.
    • Numerical simulations confirm the validity of the theoretical framework and derived solutions.