Moving domain wall

As mentioned above, domain switching in ferroelectrics is accompanied by domain nucleation, moving domain walls and restructuring of dipoles and charges. A characteristic feature of this irreversible process is the appearance of a hysteresis loop in the dependence of dielectric displacement... 

Equation (6.335) is known to have an exact soliton-like solution, or solitary wave (also called a moving domain wall), when a/b < 1, and this solution was reported by Schiller, Pelzl and Demus [246] and Cladis and van Saar loos [43]. It is given by... 

Imposition of no-slip velocity conditions at solid walls is based on the assumption that the shear stress at these surfaces always remains below a critical value to allow a complete welting of the wall by the fluid. This iraplie.s that the fluid is constantly sticking to the wall and is moving with a velocity exactly equal to the wall velocity. It is well known that in polymer flow processes the shear stress at the domain walls frequently surpasses the critical threshold and fluid slippage at the solid surfaces occurs. Wall-slip phenomenon is described by Navier s slip condition, which is a relationship between the tangential component of the momentum flux at the wall and the local slip velocity (Sillrman and Scriven, 1980). In a two-dimensional domain this relationship is expressed as... 

The kinetics of the nonconserved order parameter is determined by local curvature of the phase interface. Lifshitz [137] and Allen and Cahn [138] showed that in the late kinetics, when the order parameter saturates inside the domains, the coarsening is driven by local displacements of the domain walls, which move with the velocity v proportional to the local mean curvature H of the interface. According to the Lifshitz-Cahn-Allen (LCA) theory, typical time t needed to close the domain of size L(t) is t L(t)/v = L(t)/H(t), where H(t) is the characteristic curvature of the system. Thus, under the assumption that H(t) 1 /L(t), the LCA theory predicts the growth law L(t) r1 /2. The late scaling with the growth exponent n = 0.5 has been confirmed for the nonconserved systems in many 2D simulations [139-141]. 

FIGURE 9.5 Movement of a domain wall. The broken lines illustrate the domain wall, which separates two domains with magnetic moments lined up in opposite directions. The moments twist to align with the applied field, H, and the wall moves in the direction of the arrow. 

For a material to be soft magnetically, its domain walls must move easily. The principal obstacles to domain wall movement are inclusions and grain boundaries. Low dislocation contents, residual stresses, and a low interstitial content, are also important. 

A domain wall under an external electric field moves in a statistical potential generated by their interaction with the lattice, point defects, dislocations, and neighboring walls. Reversible movement of the wall is regarded as a small displacement around a local minimum. When the driven field is high enough, irreversible jumps above the potential barrier into a neighboring local minimum occur (see Figure 1.23). 

The origin of the nonlinearity and hysteresis in the films is most likely due to displacement of domain walls [4], If domain walls move in a medium with a random distribution of pinning center, the response of the material can be described, in the first approximation by Rayleigh relations. We next demostrate how optical interferometry can be sued to verify whether this particular model applies to the investigated pzt thin film. In the case of the converse piezoelectric effect, when the driving field E is varied between — Eo and Eo, the piezoelectric strain x is hysteretic and can be expressed by the following Rayleigh relations ... 

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