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Analytical solutions describing the phase separation driven by a free energy functional containing a long-range
Isamu Ohnishi1, Yasumasa Nishiura, Masaki Imai
1Department of Computer Sciences and Information Mathematics, Faculty of Electro-Communications, The University of Electro-Communications, Chofu, Tokyo, 182, Japan.
Chaos (Woodbury, N.Y.)
|June 5, 2003
Summary
This study analyzes the Gibbs free energy of diblock copolymer melts, determining the period of microphase separation structures. It rigorously estimates error terms and analyzes system dynamics, revealing sensitivity to initial data.
Area of Science:
- Polymer Physics
- Materials Science
- Statistical Mechanics
Background:
- Diblock copolymer melts exhibit microphase separation into periodic structures.
- The Gibbs free energy functional models this phenomenon, with critical points indicating thermodynamic equilibrium.
- Periodic structures like lamellar and spherical phases are experimentally observed equilibrium states.
Purpose of the Study:
- Characterize the periodic structure of the global minimizer for diblock copolymer melts in the strong segregation limit.
- Mathematically determine the asymptotic expansion of the period with respect to interfacial thickness (epsilon).
- Investigate the dependency of proportionality constants on polymer length ratio and quench depth.
Main Methods:
- Variational problem analysis with long-range interaction.
- Rigorous mathematical estimation of higher-order error terms for the period's asymptotic expansion in one dimension.
- Linear stability analysis of homogeneous steady states to derive the most unstable wavelength.
- Numerical investigation of the gradient flow equation for time evolution.
Main Results:
- Complete determination of the principal part of the asymptotic expansion of the period with respect to interfacial thickness.
- Clear determination of proportionality constant dependencies on polymer length ratio and quench depth.
- Identification of many local minimizers for the free energy and sensitivity to initial data in the system's time evolution.
Conclusions:
- The study provides a rigorous mathematical framework for understanding microphase separation in diblock copolymers.
- Results offer insights into the formation and stability of periodic structures.
- The system's dynamics exhibit complex behavior, including sensitivity to initial conditions, relevant to spinodal decomposition.