Landau theory of domain walls for one-dimensional asymmetric potentials

Abstract

Investigation of a great number of physical systems shows that a Landau free energy of the form F(φ)=Hφ+(A/2)φ2+(B/3)φ3+(C/4)φ4 describes a first-order phase transition in an internal or external field H. To study the formation of static domain walls in these systems we include a spatial gradient (Ginzburg) term of the scalar order parameter φ. From the variational derivative of the total free energy we obtain a static equilibrium condition. By solving this equation exactly for different physical parameters and boundary conditions, we obtained different quasi-one-dimensional soliton-like solutions. These solutions correspond to three different types of domain walls between the two different phases which are created in the system. In addition, we obtain soliton lattice (domain wall array) solutions, calculate their formation energy and the asymptotic interaction between the solitons. By introducing certain transformations, we show that the solutions obtained here can be used to study domain walls in other physical systems such as described by asymmetric double Morse potentials. Finally, we apply our results to the specific cases of liquid crystals and the jam phenomena in traffic flows.

Successive phase transitions and kink solutions inϕ8,ϕ10, andϕ12field theories

“…Meanwhile, the structure near the second first-order transition point T I c (II) is similar to that of the asymmetric double well φ 4 model of a first-order phase transition [2].…”

“…In this case, one can also show that the potential has maxima at φ = 0, ± (b 2 + a 2 )/2. From (4), we also find that α 4 /α 2 6 is constrained to satisfy the inequality 1 4 < α 4 α 2 6 < 3 8 .…”

“…At the first-order phase transition, i.e., T = T I c , the potential can always be written in the form (3) with a, b and α 6,4,2 being related by (2). Since there are four degenerate minima, we expect two different kinds of kinks, one connecting a to b (or, equivalently, −b to −a) and another connecting −a to +a, as x goes from −∞ to +∞.…”

“…An asymmetric double well in φ 4 field theory can also describe first-order transitions [2]. However, if one has to capture all symmetry-allowed phases in a lowdimensional phase transition [3] or describe a succession of phase transitions, then one has to consider either multi-component field theories [4] or higher-than-sixthorder single-component field theories [5,6].…”

Coupled Nonlinear Ginzburg-Landau and Mechanics Model for Martensitic Transformations in Polycrystals

“…With the temperature dependent Landau polynomial expanded in strain up to sixth order, the model of temperature induced MT in one-dimensional homogeneous medium qualitatively derived the temperature dependent thermodynamic properties and shape memory behaviors, such as the latent heat in transformation, the stress-strain curves at various temperatures (especially the hysteresis below critical temperature), the lattice softening, the stress effect on the temperatureinduced phase transition, etc. Afterwards, the one-dimensional Landau potential was applied to different structural transitions, and the various solutions of domain walls characterized in soliton, kink or periodon, were also analyzed [Fal83, SS03,SSL01], which are directly associated with the morphology of twinned or single martensite in the subsequent chapters.…”