scholarly journals Distributed Optimal Control of the Cahn–Hilliard System Including the Case of a Double-Obstacle Homogeneous Free Energy Density

2012 ◽  
Vol 50 (1) ◽  
pp. 388-418 ◽  
Author(s):  
M. Hintermüller ◽  
D. Wegner
Keyword(s):  
Optimal Control ◽  
Free Energy ◽  
Energy Density ◽  
Free Energy Density ◽  
Distributed Optimal Control

On nematic surface energies

1997 ◽  
Vol 8 (3) ◽  
pp. 293-299 ◽  
Author(s):  
SANDRO FAETTI ◽  
EPIFANIO G. VIRGA
Keyword(s):  
Free Energy ◽  
Liquid Crystal ◽  
Energy Density ◽  
Continuum Theory ◽  
Free Energy Density ◽  
Optic Axis ◽  
Principal Curvatures ◽  
Equilibrium Equations ◽  
Line Integral ◽  
Surface Energies

We review the main outcomes of a continuum theory for the equilibrium of the interface between a nematic liquid crystal and an isotropic environment, in which the surface free energy density bears terms linear in the principal curvatures of the interface. Such geometric contributions to the energy occur together with more conventional elastic contribution, leading to an effective azimuthal anchoring of the optic axis, which breaks the isotropic symmetry of the interface. The theory assumes the interface to be fixed, as for a rigid cavity filled with liquid crystal, and so it does not apply to drops. It should be appropriate when the curvatures of the interface are small compared to that of the molecular interaction sphere. Also, interfaces bearing a sharp edge are encompassed within the theory; a line integral expresses the energy condensed along the edge: we see how it affects the equilibrium equations.

scholarly journals Fluctuations and phase transitions of uniaxial and biaxial liquid crystals using a theoretically informed Monte Carlo and a Landau free energy density

2019 ◽  
Vol 31 (17) ◽  
pp. 175101
Author(s):  
Stiven Villada-Gil ◽  
Viviana Palacio-Betancur ◽  
Julio C Armas-Pérez ◽  
Juan J de Pablo ◽  
Juan P Hernández-Ortiz
Keyword(s):  
Monte Carlo ◽  
Free Energy ◽  
Phase Transitions ◽  
Liquid Crystals ◽  
Energy Density ◽  
Free Energy Density ◽  
Landau Free Energy ◽  
Biaxial Liquid Crystals

A New Elastic Theory of Nanocomposites With Incoherent Interface Effect Based on Interface Energy Density

2019 ◽  
Vol 87 (2) ◽  
Author(s):  
Yin Yao ◽  
Zhilong Peng ◽  
Jianjun Li ◽  
Shaohua Chen
Keyword(s):  
Free Energy ◽  
Energy Density ◽  
Free Energy Density ◽  
Present Theory ◽  
Interface Energy ◽  
Surface Relaxation ◽  
Interface Fracture ◽  
Reinforced Composites ◽  
Stress Jump ◽  
Interface Free Energy

Abstract A continuum theory of elasticity based on the concept of interface free energy density is proposed to account for the effect of incoherent interfaces in nano-phase reinforced composites. With the help of the lattice model, the corresponding interface energy density is formulated in terms of the surface free energy densities of two bulk materials forming interfaces, the lattice relaxation parameters due to the spontaneous surface relaxation and lattice misfit parameters yielded by interface incoherency, while the stress jump at interfaces is formulated with an interface-induced traction as a function of interface free energy density. Compared with existing theories, the interface elastic constants difficult to determine are no longer introduced, and all the parameters involved in the present theory have definite physical meanings and can be easily determined. The coupling effects of characteristic size and interface structure in nanoparticle-reinforced composites are further analyzed with the present theory. It is found that both the decrease of nanoparticle size and the increase of interface incoherence will lead to the decrease of interface fracture toughness and increase of effective bulk and shear moduli of nanocomposites. All these results predicted by the present theory are consistent well with those obtained by previous experiments and computations, which further indicate that the present theory can effectively predict the mechanical properties of nanomaterials with complex interfaces, such as nano-phase reinforced composites and nano-scale metal multilayer composites.

Correlation Functions in the Isotropic Phase of Nematics

1984 ◽  
Vol 39 (6) ◽  
pp. 537-541
Author(s):  
E. Govers ◽  
G. Vertogen
Keyword(s):  
Free Energy ◽  
Energy Density ◽  
Correlation Functions ◽  
Free Energy Density ◽  
Isotropic Phase ◽  
Orientational Correlation ◽  
Orientational Correlation Functions

The orientational correlation functions in the isotropic phase of nematics are calculated starting from the expression of De Gennes for the free energy density of this phase.

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