60 research outputs found
Coarsening dynamics in one dimension: The phase diffusion equation and its numerical implementation
Many nonlinear partial differential equations (PDEs) display a coarsening
dynamics, i.e., an emerging pattern whose typical length scale increases
with time. The so-called coarsening exponent characterizes the time
dependence of the scale of the pattern, , and coarsening
dynamics can be described by a diffusion equation for the phase of the pattern.
By means of a multiscale analysis we are able to find the analytical expression
of such diffusion equations. Here, we propose a recipe to implement numerically
the determination of , the phase diffusion coefficient, as a
function of the wavelength of the base steady state .
carries all information about coarsening dynamics and, through the relation
, it allows us to determine the coarsening exponent. The
main conceptual message is that the coarsening exponent is determined without
solving a time-dependent equation, but only by inspecting the periodic
steady-state solutions. This provides a much faster strategy than a forward
time-dependent calculation. We discuss our method for several different PDEs,
both conserved and not conserved
Dynamic effects induced by renormalization in anisotropic pattern forming systems
The dynamics of patterns in large two-dimensional domains remains a challenge
in non-equilibrium phenomena. Often it is addressed through mild extensions of
one-dimensional equations. We show that full 2D generalizations of the latter
can lead to unexpected dynamical behavior. As an example we consider the
anisotropic Kuramoto-Sivashinsky equation, that is a generic model of
anisotropic pattern forming systems and has been derived in different instances
of thin film dynamics. A rotation of a ripple pattern by occurs in
the system evolution when nonlinearities are strongly suppressed along one
direction. This effect originates in non-linear parameter renormalization at
different rates in the two system dimensions, showing a dynamical interplay
between scale invariance and wavelength selection. Potential experimental
realizations of this phenomenon are identified.Comment: 5 pages, 3 figures; supplemental material available at journal web
page and/or on reques
Quantitative analysis of the debonding structure of soft adhesives
We experimentally investigate the growth dynamics of cavities nucleating
during the first stages of debonding of three different model adhesives. The
material properties of these adhesives range from a more liquid-like material
to a soft viscoelastic solid and are carefully characterized by small strain
oscillatory shear rheology as well as large strain uniaxial extension. The
debonding experiments are performed on a probe tack set-up. Using high contrast
images of the debonding process and precise image analysis tools we quantify
the total projected area of the cavities, the average cavity shape and growth
rate and link these observations to the material properties. These measurements
are then used to access corrected effective stress and strain curves that can
be directly compared to the results from the uniaxial extension
Kinetic roughening in a realistic model of non-conserved interface growth
We provide a quantitative picture of non-conserved interface growth from a
diffusive field making special emphasis on two main issues, the range of
validity of the effective small-slopes (interfacial) theories and the interplay
between the emergence of morphologically instabilities in the aggregate
dynamics, and its kinetic roughening properties. Taking for definiteness
electrochemical deposition as our experimental field of reference, our
theoretical approach makes use of two complementary approaches: interfacial
effective equations and a phase-field formulation of the electrodeposition
process. Both descriptions allow us to establish a close quantitative
connection between theory and experiments. Moreover, we are able to correlate
the anomalous scaling properties seen in some experiments with the failure of
the small slope approximation, and to assess the effective re-emergence of
standard kinetic roughening properties at very long times under appropriate
experimental conditions.Comment: Journal of Statistical Mechanics: Theory & Experiment, in pres
Strong anisotropy in two-dimensional surfaces with generic scale invariance: Nonlinear effects
We expand a previous study [Phys. Rev. E 86, 051611 (2012)] on the conditions for occurrence of strong anisotropy in the scaling properties of two-dimensional surfaces displaying generic scale invariance. In that study, a natural scaling ansatz was proposed for strongly anisotropic systems, which arises naturally when analyzing data from, e.g., thin-film production experiments. The ansatz was tested in Gaussian (linear) models of surface dynamics and in nonlinear models, like the Hwa-Kardar (HK) equation [Phys. Rev. Lett. 62, 1813 (1989)], which are susceptible of accurate approximations through the former. In contrast, here we analyze nonlinear equations for which such approximations fail. Working within generically scale-invariant situations, and as representative case studies, we formulate and study a generalization of the HK equation for conserved dynamics and reconsider well-known systems, such as the conserved and the nonconserved anisotropic Kardar-Parisi-Zhang equations. Through the combined use of dynamic renormalization group analysis and direct numerical simulations, we conclude that the occurrence of strong anisotropy in two-dimensional surfaces requires dynamics to be conserved. We find that, moreover, strong anisotropy is not generic in parameter space but requires, rather, specific forms of the terms appearing in the equation of motion, whose justification needs detailed information on the dynamical process that is being modeled in each particular case.Partial support for this work has been provided by MINECO
(Spain) Grant No. FIS2012-38866-C05-01. E.V. acknowledges
support by Universidad Carlos III de Madrid
Kardar-Parisi-Zhang asymptotics for the two-dimensional noisy Kuramoto-Sivashinsky equation
We study numerically the Kuramoto-Sivashinsky (KS) equation forced by
external white noise in two space dimensions, that is a generic model for e.g.
surface kinetic roughening in the presence of morphological instabilities.
Large scale simulations using a pseudospectral numerical scheme allow us to
retrieve Kardar-Parisi-Zhang (KPZ) scaling as the asymptotic state of the
system, as in the 1D case. However, this is only the case for sufficiently
large values of the coupling and/or system size, so that previous conclusions
on non-KPZ asymptotics are demonstrated as finite size effects. Crossover
effects are comparatively stronger for the 2D case than for the 1D system.Comment: 5 pages, 3 figures; supplemental material available at journal web
page and/or on reques
Crack Front Segmentation and Facet Coarsening in Mixed-Mode Fracture
A planar crack generically segments into an array of "daughter cracks" shaped
as tilted facets when loaded with both a tensile stress normal to the crack
plane (mode I) and a shear stress parallel to the crack front (mode III). We
investigate facet propagation and coarsening using in-situ microscopy
observations of fracture surfaces at different stages of quasi-static
mixed-mode crack propagation and phase-field simulations. The results
demonstrate that the bifurcation from propagating planar to segmented crack
front is strongly subcritical, reconciling previous theoretical predictions of
linear stability analysis with experimental observations. They further show
that facet coarsening is a self- similar process driven by a spatial
period-doubling instability of facet arrays with a growth rate dependent on
mode mixity. Those results have important implications for understanding the
failure of a wide range of materials
Unstable nonlocal interface dynamics
4 pages, 2 figures.-- PACS nrs.: 68.35.Ct, 05.45.−a, 47.54.−r.Nonlocal effects occur in many nonequilibrium interfaces, due to diverse physical mechanisms like diffusive, ballistic, or anomalous transport, with examples from flame fronts to thin films. While dimensional analysis describes stable nonlocal interfaces, we show the morphologically unstable condition to be nontrivial. This is the case for a family of stochastic equations of experimental relevance, paradigmatically including the Michelson-Sivashinsky system. For a whole parameter range, the asymptotic dynamics is scale invariant with dimension-independent exponents reflecting a hidden Galilean symmetry. The usual Kardar-Parisi-Zhang nonlinearity, albeit irrelevant in that parameter range, plays a key role in this behavior.This work has been partially supported through Grants No. FIS2006-12253-C06-01 and No. FIS2006-12253-C06-06 (MEC, Spain) and No. S-0505/ESP-0158 (CAM, Spain). M. N. acknowledges support by Fundación Carlos III (Spain) and by Fondazione Angelo Della Riccia (Italy).Publicad
Unified moving-boundary model with fluctuations for unstable diffusive growth
17 pages, 9 figures.-- PACS nrs.: 81.10.-h, 68.35.Ct, 64.60.Ht, 81.15.Gh.-- MSC2000 code: 82C24.-- ArXiv pre-print available at: http://arxiv.org/abs/0812.4160MR#: MR2496824 (2010c:82073)Final publisher version available Open Access at: http://gisc.uc3m.es/~cuerno/publ_list.htmlWe study a moving-boundary model of nonconserved interface growth that implements the interplay between diffusive matter transport and aggregation kinetics at the interface. Conspicuous examples are found in thin-film production by chemical vapor deposition and electrochemical deposition. The model also incorporates noise terms that account for fluctuations in the diffusive and attachment processes. A small-slope approximation allows us to derive effective interface evolution equations (IEEs) in which parameters are related to those of the full moving-boundary problem. In particular, the form of the linear dispersion relation of the IEE changes drastically for slow or for instantaneous attachment kinetics. In the former case the IEE takes the form of the well-known (noisy) Kuramoto-Sivashinsky equation, showing a morphological instability at short times that evolves into kinetic roughening of the Kardar-Parisi-Zhang (KPZ) class. In the instantaneous kinetics limit, the IEE combines the Mullins-Sekerka linear dispersion relation with a KPZ nonlinearity, and we provide a numerical study of the ensuing dynamics. In all cases, the long preasymptotic transients can account for the experimental difficulties in observing KPZ scaling. We also compare our results with relevant data from experiments and discrete models.This work has been partially supported by UC3M/CAM (Spain) Grant No. UC3M-FI-05-007, CAM (Spain) Grant No. S-0505/ESP-0158, and by MEC (Spain), through Grants No. FIS2006-12253-C06-01, No. FIS2006-12253-C06-06, and the FPU program (M. N.).Publicad
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