Book contents
- Frontmatter
- Contents
- preface
- 1 Self-organized and self-assembled structures
- 2 Order parameter, free energy, and phase transitions
- 3 Free energy functional
- 4 Phase separation kinetics
- 5 Langevin model for nonconserved order parameter systems
- 6 Langevin model for conserved order parameter systems
- 7 Interface dynamics at late times
- 8 Domain growth and structure factor for model B
- 9 Order parameter correlation function
- 10 Vector order parameter and topological defects
- 11 Liquid crystals
- 12 Lifshitz–Slyozov–Wagner theory
- 13 Systems with long-range repulsive interactions
- 14 Kinetics of systems with competing interactions
- 15 Competing interactions and defect dynamics
- 16 Diffusively rough interfaces
- 17 Morphological instability in solid films
- 18 Propagating chemical fronts
- 19 Transverse front instabilities
- 20 Cubic autocatalytic fronts
- 21 Competing interactions and front repulsion
- 22 Labyrinthine patterns in chemical systems
- 23 Turing patterns
- 24 Excitable media
- 25 Oscillatory media and complex Ginzburg–Landau equation
- 26 Spiral waves and defect turbulence
- 27 Complex oscillatory and chaotic media
- 28 Resonantly forced oscillatory media
- 29 Nonequilibrium patterns in laser-induced melting
- 30 Reaction dynamics and phase segregation
- 31 Active materials
- References
- Index
16 - Diffusively rough interfaces
Published online by Cambridge University Press: 10 February 2010
- Frontmatter
- Contents
- preface
- 1 Self-organized and self-assembled structures
- 2 Order parameter, free energy, and phase transitions
- 3 Free energy functional
- 4 Phase separation kinetics
- 5 Langevin model for nonconserved order parameter systems
- 6 Langevin model for conserved order parameter systems
- 7 Interface dynamics at late times
- 8 Domain growth and structure factor for model B
- 9 Order parameter correlation function
- 10 Vector order parameter and topological defects
- 11 Liquid crystals
- 12 Lifshitz–Slyozov–Wagner theory
- 13 Systems with long-range repulsive interactions
- 14 Kinetics of systems with competing interactions
- 15 Competing interactions and defect dynamics
- 16 Diffusively rough interfaces
- 17 Morphological instability in solid films
- 18 Propagating chemical fronts
- 19 Transverse front instabilities
- 20 Cubic autocatalytic fronts
- 21 Competing interactions and front repulsion
- 22 Labyrinthine patterns in chemical systems
- 23 Turing patterns
- 24 Excitable media
- 25 Oscillatory media and complex Ginzburg–Landau equation
- 26 Spiral waves and defect turbulence
- 27 Complex oscillatory and chaotic media
- 28 Resonantly forced oscillatory media
- 29 Nonequilibrium patterns in laser-induced melting
- 30 Reaction dynamics and phase segregation
- 31 Active materials
- References
- Index
Summary
The existence and dynamics of interfaces played a central role in the description of the domain-coarsening phenomena considered in the previous chapters. In the late stages of domain growth the random forces in the order parameter kinetic equations were suppressed and the interface dynamics was treated deterministically. In this chapter we provide a more detailed treatment of the effects of noise and diffusion on the structure of the interface. One may capture the essential physics of diffusively rough interfaces in a general model often called the Kardar–Parisi–Zhang (KPZ) equation (Kardar et al., 1986).
KPZ equation
Consider a front in a (d + 1)-dimensional system extended along x and moving, on average, in the x1 direction (see Fig. 16.1). The system is assumed to be infinitely extended along x1 and has linear dimension L along x. In contrast to the description in Chapter 7, we neglect the intrinsic thickness of the interface and investigate the effects of diffusion and noise on the dynamics of the interfacial profile. Referring to Fig. 16.1, let h(x, t) be the position of the interface as a function of x at time t, relative to an arbitrarily selected origin. Its mean position at time t is. We assume that the front propagates with velocity v in a direction normal to its interface; noise provides a destabilizing influence on the front while diffusion tends to remove any surface roughness.
A sketch of a portion of the interface profile h(x, t) as a function of x at time t is shown in Fig. 16.2.
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- Information
- Dynamics of Self-Organized and Self-Assembled Structures , pp. 128 - 139Publisher: Cambridge University PressPrint publication year: 2009