Note on capillary waves in the gradient theory of interfaces

1985 ◽  
Vol 63 (8) ◽  
pp. 1132-1133 ◽  
Author(s):  
Luis De Sobrino
Keyword(s):  
Free Energy ◽  
Dispersion Relation ◽  
Classical Theory ◽  
Capillary Waves ◽  
Gradient Theory ◽  
Gradient Term

The calculation of the dispersion relation for capillary waves in the gradient theory of interfaces has been generalized to the case in which the gradient term in the free energy depends on temperature. The dispersion relation is found to agree with the classical theory.

On capillary waves in the gradient theory of interfaces

1985 ◽  
Vol 63 (2) ◽  
pp. 131-134 ◽  
Author(s):  
Luis de Sobrino ◽  
Jože Peternelj
Keyword(s):  
Dispersion Relation ◽  
Thermal Effects ◽  
Equations Of Motion ◽  
Capillary Waves ◽  
Gradient Theory ◽  
Two Phase ◽  
Long Wavelength ◽  
Phase Solution

We have solved the equations of motion for an inhomogeneous, nondissipative fluid linearized about a two-phase solution in order to determine the dispersion relation for capillary waves of long wavelength. The solution is reasonably rigorous in that no physical assumptions have been introduced. We find that, in accordance with the results of Turski and Langer and contrary to other workers' claims, the dispersion relation agrees with classical capillary theory only if thermal effects are included.

A Mean-Field Free-Energy Lattice Boltzmann Model for Liquid-Vapor Interfaces

2006 ◽  
Author(s):  
Shi-Ming Li ◽  
Danesh K. Tafti
Keyword(s):  
Free Energy ◽  
Lattice Boltzmann ◽  
Mean Field Theory ◽  
Mean Field ◽  
Capillary Waves ◽  
Lattice Boltzmann Model ◽  
Wave Number ◽  
Gradient Theory ◽  
Pressure Equation ◽  
Liquid Vapor

A nonlocal pressure equation is proposed for liquid-vapor interfaces based on mean-field theory. The new nonlocal pressure equation is shown to be a generalized form of the nonlocal pressure equation of the van der Waals theory or the “square-gradient theory”. The proposed nonlocal pressure is implemented in the mean-field free-energy lattice Boltzmann method (LBM) proposed by Zhang et al (2004). The modified LBM is applied to simulate equilibrium interface properties and the interface dynamics of capillary waves. Computed results are validated with Maxwell constructions of liquid-vapor coexistence densities, theoretical relationship of variation of surface tension with temperature, theoretical planar interface density profiles, and the dispersion relation between frequency and wave number describing the dynamics of capillary waves. It is shown that the modified LBM gives very good agreement with the theories. In addition, preliminary calculations of phase transition and binary droplet coalescence are also presented.

scholarly journals Phase-field gradient theory

2021 ◽  
Vol 72 (2) ◽  
Author(s):  
Luis Espath ◽  
Victor Calo
Keyword(s):  
Free Energy ◽  
Field Equation ◽  
Field Gradient ◽  
Phase Field ◽  
Constitutive Relations ◽  
The Body ◽  
Second Grade ◽  
Gradient Theory ◽  
Boundary Edge ◽  
Field Equations

AbstractWe propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a hypermicrotraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a ‘generalized Swift–Hohenberg equation’—a second-grade phase-field equation—and its conserved version, the ‘generalized phase-field crystal equation’—a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions.

Spin density wave – metal (superconductor) competition: A phase segregation scenario

2002 ◽  
Vol 12 (9) ◽  
pp. 61-64
Author(s):  
C. Pasquier ◽  
M. Héritier ◽  
D. Jérome
Keyword(s):  
Free Energy ◽  
Dispersion Relation ◽  
Fermi Level ◽  
Spin Density ◽  
Density Wave ◽  
Spin Density Wave ◽  
Phase Segregation ◽  
Organic Conductor ◽  
One Dimensional ◽  
Unique Parameter

We present a model comparing the free energy of a phase exhibiting a segregation between spin density wave (SDW) and metallic domains (eventually superconducting domains) and the free energy of homogeneous phases which explains the findings observed recently in (TMTSF)2PF6. The dispersion relation of this quasi-one-dimensional organic conductor is linearized around the Fermi level. Deviations from perfect nesting which stabilizes the SDW state are described by a unique parameter t$'_b$, this parameter can be the pressure as well.

The Cahn–Hilliard gradient theory for phase separation with non-smooth free energy Part I: Mathematical analysis

1991 ◽  
Vol 2 (3) ◽  
pp. 233-280 ◽  
Author(s):  
J. F. Blowey ◽  
C. M. Elliott
Keyword(s):  
Free Energy ◽  
Asymptotic Behaviour ◽  
Stationary Solution ◽  
Mathematical Analysis ◽  
Stationary Problem ◽  
Stationary Solutions ◽  
Two Dimensions ◽  
Gradient Theory ◽  
Weak Formulation ◽  
Regularity Results

A mathematical analysis is carried out for the Cahn–Hilliard equation where the free energy takes the form of a double well potential function with infinite walls. Existence and uniqueness are proved for a weak formulation of the problem which possesses a Lyapunov functional. Regularity results are presented for the weak formulation, and consideration is given to the asymptotic behaviour as the time becomes infinite. An investigation of the associated stationary problem is undertaken proving the existence of a nontrivial stationary solution and further regularity results for any stationary solution. Stationary solutions are constructed in one and two dimensions; a formula for the number of stationary solutions in one dimension is derived. It is then natural to study the asymptotic behaviour as the phenomenological parameter λ→0, the main result being that the interface between the two phases has minimal area.

A Mean-Field Pressure Formulation for Liquid-Vapor Flows

2006 ◽  
Vol 129 (7) ◽  
pp. 894-901 ◽  
Author(s):  
Shi-Ming Li ◽  
Danesh K. Tafti
Keyword(s):  
Free Energy ◽  
Mean Field ◽  
Capillary Waves ◽  
Wave Number ◽  
Density Profiles ◽  
Pressure Formulation ◽  
Boltzmann Method ◽  
Relationship Of ◽  
The Relationship ◽  
Liquid Vapor

A nonlocal pressure equation is derived from mean-field free energy theory for calculating liquid-vapor systems. The proposed equation is validated analytically by showing that it reduces to van der Waals’ square-gradient approximation under the assumption of slow density variations. The proposed nonlocal pressure is implemented in the mean-field free energy lattice Boltzmann method (LBM). The LBM is applied to simulate equilibrium liquid-vapor interface properties and interface dynamics of capillary waves and oscillating droplets in vapor. Computed results are validated with Maxwell constructions of liquid-vapor coexistence densities, theoretical relationship of variation of surface tension with temperature, theoretical planar interface density profiles, Laplace’s law of capillarity, dispersion relationship between frequency and wave number of capillary waves, and the relationship between radius and the oscillating frequency of droplets in vapor. It is shown that the nonlocal pressure formulation gives excellent agreement with theory.

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