Landau-Khalatnikov equation

The temporal dependence of the fluctuations can be described by the overdamped equation of motion — the Landau-Khalatnikov equation [36, 37]. It can be understood as follows The equilibrium configuration of the order parameter is determined by the minimum of the free energy, 5T = f dVSf(T/,Vr/) = f dV 6f/6r])5r], which is satisfied for Sf/drj = 0. Here, S/Srj = d/dr] — V d/dVri). If the system is out of equilibrium, Sf/Sr] acts like a generalized elastic force which is balanced by viscous forces. When the macroscopic velocity can be neglected the viscosity is related solely to the rate of change of the order parameter Q,... 

We neglect fluctuations of the order parameter, and assume that there is a simple linear relationship between a torque dg dr and a relaxation rate dr ldt. Physically it means that the steeper potential well g(ri) (larger dg dr ), the faster is relaxatimi (larger dr ldf) of the induced order parameter. Hence the Landau-Khalatnikov equation reads... 

Then, for the high symmetry (high T) phase, f) = 0 and the linearized Landau—Khalatnikov equation for q is given by ... 

Here, the first two terms came from minimisation procedure of the free energy, see Eqs. (8.15) and (11.43) and the viscous term was discussed earlier, see Eqs. (9.31) and (9.32) [21]. In terms of the phase transition theory, Eq. (11.63) may be regarded as the Landau-Khalatnikov equation discussed in Section 6.5.1. It describes the director rotation in magnetic field H with rotational viscosity yi = 2 — 3 and without the director inertia term. In the limit of small (p-angles, it reduces to the linear form ... 

At a low field with ignored term in Eq. (13.10), the dynamics of the director soft mode can be investigated using the Landau-Khalatnikov equation, see Section 6.5.1. The corresponding equation describing the balance of the viscous and elastic torques reads ... 

The switching time of the director tilt angle in the electroclinic effect is independent of the electric field, and is defined only by the rotational viscosity je and the elastic modulus A. The corresponding switching times are derived from the Landau-Khalatnikov equation for the balance of the viscous and elastic torques... 

The dielectric contribution of each of these modes has been worked out using the extended Landau free energy expansion of Sec. 2.5.11. The static electric response of each mode is obtained by minimizing the free energy in the presence of an electric field. The relaxation frequency of the fluctuations in the order parameter is obtained by means of the Landau-Khalatnikov equations, which control the order parameter dynamics. 

Landau-Ginzbuig coefficients 282 Landau-Ginzburg Hamiltonian, critical 303 Landau-Khalatnikov mechanism 564 Landau-Lifshitz theory, phase transitions 366 Landau-Peierls instabilities 285, 647 Langmuir-Blodgett film, atomistic simulations 85 Laplace equation 445... 

The Landau-Khalatnikov (LG) equation associated with the rotational viscosity has the generalized form (Lahiri and Pal Majumder 2011, 2012)... 

We now investigate the nature of the polarization profile that is expected to be influenced by the cross-link harmonic interaction introduced in (6.29). The dynamics of the polarization in response to a time-dependent driving field can be obtained utilizing the Landau-Khalatnikov (Lahiri and Pal Majumder 2011, 2012 Lo 2003) equation of the form... 

Khalatnikov I. M., Introduction to the Theory of Superfluidity (Benjamin, New York, 1965) Kiss G. and Porter R. S., (1978). Rheology of concentrated solutions of poly(Y-benzyl-glutamate, J. Polym. Sci. Polym. Symp. Vol. 65, pp. 193 Landau L. D. and Lifshitz E., Theory of Elasticity (Pergamon, New York, 1964) 3rd. edition Leslie F. M., (1966). Some Constitutive equations for anisotropic fluids, Quat.J. Mech. Appl. Math. Vol. 19, pp. 357. 

---