Is the Free Energy of Hydrogel the Sum of Elastic Energy and Ionic Energy?

2006 ◽  
pp. 421-430
Author(s):  
R. W. Roos ◽  
J. M. Huyghe ◽  
F. P. T. Baaijens
Keyword(s):  
Free Energy ◽  
Elastic Energy ◽  
Ionic Energy

MODEL FOR OXYGEN ABSORPTION AND STRUCTURAL PHASE TRANSITION IN YBa2Cu3O7−y

1988 ◽  
Vol 02 (08) ◽  
pp. 1017-1023
Author(s):  
C. VAREA ◽  
A. ROBLEDO
Keyword(s):  
Phase Transition ◽  
Free Energy ◽  
Elastic Energy ◽  
Structural Transition ◽  
Lattice Gas ◽  
Structural Phase ◽  
External Source ◽  
Oxygen Absorption ◽  
Low Oxygen ◽  
Oxygen Oxygen

We reproduce, quantitatively, the observed dependence with temperature and O 2 partial pressure of the ordering of oxygen and the concommitant structural transition in YBa 2 Cu 3 O 7−y by means of a layered oxygen lattice gas model in equilibrium with an external source of O 2. The free energy considers, in addition to Cu -mediated oxygen-oxygen interactions, the elastic energy of the crystal. Depending on the coupling between these two terms the transition may turn into 1st order and further ordering along the Cu-O chains may appear for low oxygen content.

Domain Dynamics in a Ferroelastic Epilayer on a Paraelastic Substrate

2002 ◽  
Vol 69 (4) ◽  
pp. 419-424 ◽  
Author(s):  
Y. F. Gao ◽  
Z. Suo
Keyword(s):  
Free Energy ◽  
Energy Density ◽  
Surface Energy ◽  
Elastic Energy ◽  
Evolution Equations ◽  
Nonequilibrium Thermodynamic ◽  
Elastic Field ◽  
Domain Size ◽  
Order Parameters ◽  
Domain Dynamics

This paper models the domain dynamics in a ferroelastic epilayer within the time-dependent Ginzburg-Landau (TDGL) framework. Constrained on a paraelastic substrate of square symmetry, the epilayer has rectangular symmetry, and forms domains of two variants. The domain wall energy drives the domains to coarsen. The spontaneous strains induce an elastic field, which drives the domains to refine. The competition between coarsening and refining selects an equilibrium domain size. We model the epilayer-substrate as a nonequilibrium thermodynamic system, evolving by the changes in the elastic displacements and the order parameters. The free energy consists of two parts: the bulk elastic energy, and the excess surface energy. The elastic energy density is taken to be quadratic in the strains. The surface energy density is expanded into a polynomial of the order parameters, the gradients of the order parameters, and the strains. In this expansion, the surface stress is taken to be quadratic in the order parameters. The evolution equations are derived from the free energy variation with respect to the order parameters. The elastic field is determined by superposing the Cerruti solution. Examples of computer simulation are presented.

Free Energy of Damaged Solids

2014 ◽  
Vol 1049-1050 ◽  
pp. 1741-1746
Author(s):  
Ji Zhang
Keyword(s):  
Free Energy ◽  
Elastic Energy ◽  
Driving Force ◽  
Damage Evolution ◽  
Damage Mechanics ◽  
Solid Mechanics ◽  
Constitutive Models ◽  
Traditional Fields ◽  
State Functions

This paper examines the free energy potentials of damaged solids for the construction of damage mechanics constitutive models. The physical meaning of free energy in solid mechanics is analyzed in contrast with that in traditional fields of thermodynamics; 1D stress-strain curves are used to show the relationships between various thermodynamic state functions in isothermal loading processes; and the role of plastic free energy in damage evolution is discussed both macroscopically and microscopically. It is concluded that plastic free energy, which is a macroscopic representation of some additional microscopic elastic energy, cannot do work during unloading but get released when damage takes place, constituting part of the driving force for damage evolution.

scholarly journals Existence results for diffuse interface models describing phase separation and damage

2012 ◽  
Vol 24 (2) ◽  
pp. 179-211 ◽  
Author(s):  
CHRISTIAN HEINEMANN ◽  
CHRISTIANE KRAUS
Keyword(s):  
Free Energy ◽  
Phase Separation ◽  
Energy Density ◽  
Elastic Energy ◽  
Weak Solutions ◽  
Existence Results ◽  
Diffuse Interface ◽  
Regularization Methods ◽  
Elastic Energy Density ◽  
Damage Processes

In this paper, we analytically investigate multi-component Cahn–Hilliard and Allen–Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form ∫Ω½Γ∇c : ∇c + ½|∇z|2+Wch(c)+Wel(e,c,z)dx with a polynomial or logarithmic chemical energy density Wch, an inhomogeneous elastic energy density Wel and a quadratic structure of the gradient of damage variable z. For the corresponding elastic Cahn–Hilliard and Allen–Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for strain e.

The Elastic Constants of a Nematic Liquid Crystal

1975 ◽  
Vol 30 (2) ◽  
pp. 230-234 ◽  
Author(s):  
Hans Gruler
Keyword(s):  
Free Energy ◽  
Liquid Crystal ◽  
Elastic Constants ◽  
Elastic Energy ◽  
Nematic Phase ◽  
Mean Field ◽  
Field Energy ◽  
Molecular Length ◽  
Solid Crystal ◽  
The Mean

Abstract The elastic constants of a nematic liquid and of a solid crystal were derived by comparing the Gibbs free energy with the elastic energy. The expressions for the elastic constants of the nematic and the solid crystal are isomorphic: k ∝ ∣ D ∣ /1. In the nematic phase ∣ D ∣ and I are the mean field energy and the molecular length. In the solid crystal, ∣ D ∣ and l correspond to the curvature of the potential and to the lattice constant, respectively. The measured nematic elastic constants show the predicted l dependence.

HTSC Isotope Effect with Soft Lattice Modes

2003 ◽  
Vol 17 (13) ◽  
pp. 2527-2538
Author(s):  
Stephen Nettel
Keyword(s):  
Free Energy ◽  
Elastic Energy ◽  
Isotope Effects ◽  
Structural Phase ◽  
Coupling Constant ◽  
Previous Theory ◽  
Electron Phonon Coupling ◽  
The Masses ◽  
Lattice Softening ◽  
Lattice Modes

An attempt is made to understand the isotope effects measured by Zhao et al. in LaSrCuO in terms of a previous theory for high Tc superconductors, a theory based on pair coupling by soft lattice modes. In the theory, the lattice elastic energy originating with the ion-electron correlation limits the gap. Here we note that this elastic energy does not depend directly on the ionic masses. However, the mechanical lattice parameters in the neighborhood of a structural phase transition are so sensitive that the free energy of oscillation and, hence, the masses are able to have an influence. Minimization of the free energy of the solid with respect to the lattice deformation leads to an equation of state, as basis for finding the effects of the ionic mass changes. With the measured isotope effects at several values of doping as input to the calculation, we obtain, as output, very plausible results for the Fermi energy, the lattice softening, and the electron-phonon coupling constant.

Effect of a thermodynamically consistent interface stress on thermal-induced nanovoid evolution in NiAl

2021 ◽  
pp. 108128652098660
Author(s):  
Mohammad Sadegh Ghaedi ◽  
Mahdi Javanbakht
Keyword(s):  
Free Energy ◽  
Stress Distribution ◽  
Elastic Energy ◽  
Helmholtz Free Energy ◽  
Inelastic Strain ◽  
Interface Stress ◽  
Total Stress ◽  
Highly Nonlinear ◽  
Stress Changes ◽  
Nanovoid Growth

In the present work, the effect of a thermodynamically consistent inelastic interface stress on nanovoid evolution in NiAl is studied. Such interface stress is introduced for the solid–gas interface of nanovoids within the concept of the phase field approach. The Cahn–Hilliard (CH) equation using the Helmholtz free energy describes the evolution of nanovoid concentration. The interface stress changes the total stress distribution and affects the elastic stress field. Thus, due to the significant effect of the elastic energy on nanovoid dynamics, it can indirectly affect nanovoid nucleation and growth. The highly nonlinear coupled CH and elasticity equations are solved using the finite element method and the COMSOL code. The coupling appears due to the presence of the nonlinear nanovoid inelastic strain in the total strain, the presence of the nonlinear inelastic interface stress in the stress tensor and the presence of elastic energy in the Helmholtz free energy. Several examples of thermal-induced nanovoid evolutions are presented to investigate the effect of the solid–gas interface stress. The obtained results show the significant effect of the interface stress on the total stress distribution, and consequently a different distribution of thermodynamic driving force which can affect the nanostructure evolution and the deformation. Mainly, the interface stress represents a promotive effect on nanovoid growth which results in a faster nanovoid growth and a larger nanovoid concentration and region.

PHASON ELASTICITY IN EQUILIBRIUM QUASICRYSTAL MODELS

1989 ◽  
Vol 03 (15) ◽  
pp. 1121-1126 ◽  
Author(s):  
LEI-HAN TANG
Keyword(s):  
Free Energy ◽  
High Temperatures ◽  
Elastic Energy ◽  
Thermal Fluctuations ◽  
Zero Temperature ◽  
Long Wavelength ◽  
Random Tiling ◽  
Crystal Model

The phason elasticity of random tiling and quasiperiodic crystal models is discussed. Long-wavelength phasons in the random tiling model are described by a square-gradient free energy of entropic origin. In the quasiperiodic crystal model, thermal fluctuations lead to a transition from the zero temperature singular linear phason elastic energy to a quadratic phason elastic energy at high temperatures.

Effect of Stress on Coherently Grown Au-Ni (001) Solid Solutions

1996 ◽  
Vol 441 ◽  
Author(s):  
G. Abadias ◽  
V. Gehanno ◽  
A. Marty ◽  
B. Gilles ◽  
M. Dynna
Keyword(s):  
Free Energy ◽  
Solid Solutions ◽  
Elastic Energy ◽  
Experimental Studies ◽  
Unit Cell ◽  
Thermodynamic Calculations ◽  
The Stability

AbstractThe effect of epitaxial stress on the stability of Au-Ni solid solutions grown by MBE on Au(001) is investigated. Experimental studies (TEM, GIXD) have shown that the elastic energy of the AuNi alloys is relaxed via two competing ways: twinning and an ordering process in which pseudo-periodic antiphase structures based on the L10 unit cell are found. The stabilisation of a metastable ordered L10 phase in the stressed Au-Ni system is explained using thermodynamic calculations based on the minimisation of free energy including elastic contributions.

Encultured minds, not error reduction minds

2020 ◽  
Vol 43 ◽  
Author(s):  
Robert Mirski ◽  
Mark H. Bickhard ◽  
David Eck ◽  
Arkadiusz Gut
Keyword(s):  
Free Energy ◽  
Error Reduction ◽  
Energy Principle ◽  
Free Energy Principle ◽  
Current Article

Abstract There are serious theoretical problems with the free-energy principle model, which are shown in the current article. We discuss the proposed model's inability to account for culturally emergent normativities, and point out the foundational issues that we claim this inability stems from.

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