On the kinetics of partially conserved order parameters: a possible mechanism for pattern formation

Salje, Ekhard K. H.

doi:10.1088/0953-8984/5/27/022

Cited by 26 publications

(24 citation statements)

References 53 publications

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“…A somewhat exotic, but perhaps rather pervasive linear-quadratic coupling mechanism was recently explored by Salje [244] and Salje and Carpenter [245]. The setting of the order parameters is as follows:…”

Section: Order Parameter Couplingmentioning

confidence: 99%

“…The ferroic aspect of Q is expressed by the fourth-order term Q 4 , which stabilizes a strained lattice even without the interaction with P if the harmonic term (1/2)aQ 2 becomes negative. The solutions for the thermodynamic equilibrium (minimum of G) of Equation (31) are known and the entire phase diagram can be derived analytically [244] (Figure 42). Downloaded by [University of Nebraska, Lincoln] at 13:44 05 November 2014 Figure 40.…”

Section: Order Parameter Couplingmentioning

confidence: 99%

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Domain boundary-dominated systems: adaptive structures and functional twin boundaries

Viehland

Salje

2014

Advances in Physics

109

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Domain boundaries typically constitute only a minute fraction of the total volume of a crystal. However, a special (but not unusual) situation can occur in which the domain boundary energy becomes very small. Specifically, the domain size is miniaturized to near-atomic scales and the domain boundary density becomes extremely high. In such cases, the properties of the crystal become dominated by a combination of both the domains and the domain boundaries. This phenomenon differs from most ferromagnetic or ferroelectric materials wherein the motion of the domain boundaries dominates the response. As reported herein, novel emergent phenomena that differ from the properties of either the domains or the domain boundaries may be expected. In this article, we focus on one specific state found in ferroic materials -namely, the adaptive ferroic state. This state can be found, for example, in tweed-like structures in morphotropic phase boundary piezoelectric crystals, ferromagnetic shape memory alloys, and pre-martensitic states. In these materials, the properties of the twin boundaries represent the principal contributors to the functionality of a given system. In fact, further investigations of domain boundary-dominated phenomena could result in novel potential for tailoring functional properties for a desired outcome. It should also be noted that new properties can be designed into twin boundaries that are not the properties of the domains. In this paper, adaptive structures and functional twin boundaries are reviewed, and examples of various observed functionalities (e.g. superconductivity, polarity, and ferroelectricity) and corresponding twin boundary structures are provided. In addition, this review confirms that various theoretically predicted, structurally bridging low-symmetry phases do, in fact, exist. Moreover, the values of the lattice constants of the adaptive state are adjustable parameters that are determined by combinations of cubic, rhombohedral/tetragonal phases, and geometrical invariant conditions. Finally, we show that, in such cases, macroscopic properties are controlled by the unique functionality of the twin walls. Looking forward, domain boundary-dominated phenomena offer an important approach for enhancing the properties of the bulk, and to unique local properties where the "twin is the device". We encourage the community to rethink their approaches to materials by design that have treated the structure as homogeneous and to consider the alternative paradigm where the local structure is different from the apparent average symmetry. PACS: 77 Dielectrics piezoelectrics and ferroelectrics; 64 Equations of state, phase equilibria, and phase transitions; 73 Electronic structure and electrical properties of surfaces, interfaces, thin films, and lowdimensional structures

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Section: Order Parameter Couplingmentioning

confidence: 99%

Section: Order Parameter Couplingmentioning

confidence: 99%

Domain boundary-dominated systems: adaptive structures and functional twin boundaries

Viehland

Salje

2014

Advances in Physics

109

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“…13 We also summarize to what extent the linear stability analysis of these uniformly translating fronts allows us to solve the selection problem, i.e., to determine the basins of attraction of these solutions in the space of initial conditions and for different nonlinearities f , and to what extent it allows us to answer the related question of the convergence rate and mechanism. It will turn out that the linear stability analysis fails to explain how pulled fronts emerging from sufficiently steep initial conditions relax to their asymptotic speed and profile.…”

Section: Stability Selection and Convergence In The Nonlinear Diffusmentioning

confidence: 99%

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Ebert

Saarloos

2000

Physica D: Nonlinear Phenomena

356

596

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Fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes: pulled fronts and pushed fronts. The term "pulled front" expresses that these fronts are "pulled along" by the spreading of linear perturbations about the unstable state. Accordingly, their asymptotic speed v * equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state. The central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towards v * . We show that when such fronts evolve from "sufficiently steep" initial conditions, which initially decay faster than e −λ * x for x → ∞, they have a universal relaxation behavior as time t → ∞: the velocity of a pulled front always relaxes algebraically likeThe parameters v * , λ * , and D are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state. This front velocity is independent of the precise value of the front amplitude, which one tracks to measure the front position. The interior of the front is essentially slaved to the leading edge, and develops universally asis a uniformly translating front solution with velocity v < v * .Our result, which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well-known nonlinear diffusion equation of type ∂ t φ = ∂ 2 x φ + φ − φ 3 , where the invaded unstable state is φ = 0. Even for this simple case, the subdominant t −3/2 term extends an earlier result of Bramson. Our analysis is then generalized to more general (sets of) partial differential equations with higher spatial or temporal derivatives, to PDEs with memory kernels, and also to difference equations such as those that occur in numerical finite difference codes. Our universal result for pulled fronts thus implies independence (i) of the level curve which is used to track the front position, (ii) of the precise nonlinearities, (iii) of the precise form of the linear operators in the dynamical equation, and (iv) of the precise initial conditions, as long as they are sufficiently steep. The only remainders of the explicit form of the dynamical equation are the nonlinear solutions v and the three saddle point parameters v * , λ * , and D. As our simulations confirm all our analytical predictions in every detail, it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts, if they are uniformly translating for t → ∞. An immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions. In addition to our main result for pulled fronts, we also discuss the propagation and convergence of fronts emerging from * Corresponding author. Present address: CWI, Postbus 94079, 1090 GB Amsterdam, Netherlands 0167-2789/00/$ -see front matter © 2000 Else...

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“…Such fronts arise in many convective instabilities in fluid dynamics such as in the wake of bluff bodies [1], in Taylor [2] and Rayleigh-Bénard [3] convection, they play a role in spinodal decomposition near a wall [4], the pearling instability of laser-tweezed membranes [5], the formation of kinetic, transient microstructures in structural phase transitions [6], dielectric breakdown fronts [7], the propagation of a superconducting front into a normal metal [8], or in error propagation in extended chaotic systems [9]. For such front propagation problems, it is known [10][11][12][13][14] that if the initial profile is steep enough, arising, e.g., through a local initial perturbation, the propagating front in practice always relaxes to a unique shape and velocity.…”

mentioning

confidence: 99%

Universal Algebraic Relaxation of Fronts Propagating into an Unstable State and Implications for Moving Boundary Approximations

1998

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We analyze the relaxation of fronts propagating into unstable states. While "pushed" fronts relax exponentially like fronts propagating into a metastable state, "pulled" or "linear marginal stability" fronts relax algebraically. As a result, for thin fronts of this type, the standard moving boundary approximation fails. The leading relaxation terms for velocity and shape are of order 1͞t and 1͞t 3͞2 . These universal terms are calculated exactly with a new systematic analysis that unifies various heuristic approaches to front propagation. [S0031-9007(98) [7], the propagation of a superconducting front into a normal metal [8], or in error propagation in extended chaotic systems [9]. For such front propagation problems, it is known [10][11][12][13][14] that if the initial profile is steep enough, arising, e.g., through a local initial perturbation, the propagating front in practice always relaxes to a unique shape and velocity. Depending on the nonlinearities, one can distinguish two regimes: As a rule, fronts whose propagation is driven (pushed) by the nonlinearities very much resemble fronts propagating into metastable states. This regime is often referred to as "pushed" [10,14] or "nonlinear marginal stability" [13]. If, on the other hand, nonlinearities mainly cause saturation, fronts propagate with a velocity determined by linearization about the unstable state: it is as if they are pulled by the linear instability ("pulled" [10,14] or "linear marginal stability" [13] regime). Some heuristic arguments have been put forward [13] that for large times t the velocity and shape of a pulled front generally relax slowly, as 1͞t. The experimental relevance of such a slow relaxation is illustrated by propagating Taylor vortex fronts. Here the measured front velocities [2] were about 40% lower than predicted theoretically; only later it became clear [15] that this was due to slow transients.In this paper, we identify the general mechanism leading to slow relaxation of uniformly translating fronts, use it to introduce a systematic analysis which allows us to determine all universal asymptotic terms, and point out the implication of the relaxation for the existence of a moving boundary approximation.Our present investigation was, in fact, motivated by an attempt to derive a moving boundary approximation for a thin front propagating into an unstable state [7]. Moving boundary approximations have been applied quite successfully to patterns consisting of domains where the order parameter field varies slowly in space and time due to the coupling to some external field (e.g., temperature in a solidification or combustion front), separated by thin interfacial zones where the order parameter field varies rapidly [16]. Implicit in this method is the assumption that the dynamics on the "inner" interfacial scale and the "outer" pattern scale is adiabatically decoupled in the thin interface limit, so that the boundary conditions for the motion of the interface on the outer scale are local in space and time. However, we find that w...

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On the kinetics of partially conserved order parameters: a possible mechanism for pattern formation

Cited by 26 publications

References 53 publications

Domain boundary-dominated systems: adaptive structures and functional twin boundaries

Domain boundary-dominated systems: adaptive structures and functional twin boundaries

Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts

Universal Algebraic Relaxation of Fronts Propagating into an Unstable State and Implications for Moving Boundary Approximations

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