Comment on the free energy functional of block copolymer melts in the strong segregation limit

1990 ◽  
Vol 23 (8) ◽  
pp. 2413-2414 ◽  
Author(s):  
Takao Ohta ◽  
Kyozi Kawasaki
Keyword(s):  
Free Energy ◽  
Block Copolymer ◽  
Energy Functional ◽  
Free Energy Functional ◽  
Copolymer Melts ◽  
Strong Segregation

scholarly journals Concentrically layered energy equilibria of the di-block copolymer problem

2002 ◽  
Vol 13 (5) ◽  
pp. 479-496 ◽  
Author(s):  
XIAOFENG REN ◽  
JUNCHENG WEI
Keyword(s):  
Free Energy ◽  
Block Copolymer ◽  
Unit Disk ◽  
Energy Functional ◽  
Singular Limit ◽  
Free Energy Functional ◽  
Number Of Layers

We prove the existence of energy equilibria of the di-block copolymer problem in the unit disk. They consist of concentrically layered micro-domains rich in one of the two monomer building units. We construct them by solving the proper singular limit of the free energy functional. The same limit also explains how under a dynamic law of the free energy, circular interfaces of non-equilibria may move to the origin and vanish, or collapse to each other, thereby reducing the number of layers.

scholarly journals On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

scholarly journals Droplet minimizers for the Gates–Lebowitz–Penrose free energy functional

Nonlinearity ◽  
2009 ◽  
Vol 22 (12) ◽  
pp. 2919-2952 ◽  
Author(s):  
E A Carlen ◽  
M C Carvalho ◽  
R Esposito ◽  
J L Lebowitz ◽  
R Marra
Keyword(s):  
Free Energy ◽  
Energy Functional ◽  
Free Energy Functional

scholarly journals Droplet minimizers for the Cahn-Hilliard free energy functional

2006 ◽  
Vol 16 (2) ◽  
pp. 233-264 ◽  
Author(s):  
E. A. Carlen ◽  
M. C. Carvalho ◽  
R. Esposito ◽  
J. L. Lebowitz ◽  
R. Marra
Keyword(s):  
Free Energy ◽  
Energy Functional ◽  
Free Energy Functional

scholarly journals Homogenization of composite ferromagnetic materials

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150365 ◽  
Author(s):  
François Alouges ◽  
Giovanni Di Fratta
Keyword(s):  
Free Energy ◽  
Ferromagnetic Material ◽  
Energy Functional ◽  
Ferromagnetic Materials ◽  
Rigorous Derivation ◽  
Free Energy Functional ◽  
Landau Free Energy ◽  
Ferromagnetic Body ◽  
The Media

The objective of this paper is to perform, by means of Γ - convergence and two-scale convergence , a rigorous derivation of the homogenized Gibbs–Landau free energy functional associated with a composite periodic ferromagnetic material, i.e. a ferromagnetic material in which the heterogeneities are periodically distributed inside the media. We thus describe the Γ -limit of the Gibbs–Landau free energy functional, as the period over which the heterogeneities are distributed inside the ferromagnetic body shrinks to zero.

Classical and semiclassical dynamics in statistical environments with a mixed dynamical and statistical representation

2016 ◽  
Vol 18 (3) ◽  
pp. 1771-1785 ◽  
Author(s):  
Kazuo Takatsuka ◽  
Kentaro Matsumoto
Keyword(s):  
Free Energy ◽  
Real Time ◽  
Energy Functional ◽  
Basic Theory ◽  
Chemical Dynamics ◽  
Functional Surface ◽  
Spatial Gradients ◽  
Free Energy Functional ◽  
Statistical Representation ◽  
Semiclassical Dynamics

We present a basic theory to study real-time chemical dynamics embedded in a statistically treated large environment. It is shown that dynamically treated molecules should run on the free-energy functional surface, if and only if the spatial gradients of temperature functional are all zero.

scholarly journals Cubic microlattices embedded in nematic liquid crystals: a Landau-de Gennes study

2021 ◽  
Author(s):  
Razvan-Dumitru Ceuca
Keyword(s):  
Free Energy ◽  
Additional Term ◽  
Energy Functional ◽  
Free Energies ◽  
Cubic Symmetry ◽  
Surface Anchoring ◽  
Free Energy Functional ◽  
Different Types ◽  
Embedded Particles ◽  
Anchoring Energy

We consider a Landau-de Gennes model for a connected cubic lattice scaffold in a nematic host, in a dilute regime. We analyse the homogenised limit for both cases in which the lattice of embedded particles presents or not cubic symmetry and then we compute the free effective energy of the composite material. In the cubic symmetry case, we impose different types of surface anchoring energy densities, such as quartic, Rapini-Papoular or more general versions, and, in this case, we show that we can tune any coefficient from the corresponding bulk potential, especially the phase transition temperature. In the case with loss of cubic symmetry, we prove similar results in which the effective free energy functional has now an additional term, which describes a change in the preferred alignment of the liquid crystal particles inside the domain. Moreover, we compute the rate of convergence for how fast the surface energies converge to the homogenised one and also for how fast the minimisers of the free energies tend to the minimiser of the homogenised free energy.

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