Fluctuations in a moving boundary description of diffusive interface growth Rodolfo Cuerno Departamento de Matemáticas & Grupo Interdisciplinar de Sistemas Complejos (GISC) Universidad Carlos III de Madrid [email protected]http://gisc.uc3m.es/~cuerno
43
Embed
Fluctuations in a moving boundary description of diffusive interface growth Rodolfo Cuerno Departamento de Matemáticas & Grupo Interdisciplinar de Sistemas.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Fluctuations in a moving boundary description of diffusive interface growth
Rodolfo Cuerno
Departamento de Matemáticas & Grupo Interdisciplinar de Sistemas Complejos (GISC)
M. Castro: Universidad Pontificia Comillas, Madrid ESM. Nicoli, M. Plapp: Ecole Polytechnique, Paris FRE. Vivo: Universidad Carlos III de Madrid, ES
Experimental:
J. G. Buijnsters: Radboud University Nijmegen, NLF. Ojeda: Tecnatom, Madrid ESR. Salvarezza: INIFTA, La Plata, ARL. Vázquez: Instituto de Ciencia de Materiales de Madrid, ES
Support from MICINN ES
Bacterial colonies
Classic system in the study of fractal growth
Morphology of the colony controlled by: nutrient concentration Cn
Dynamic Renormalization Group study (arbitrary d) of (SMS-like)
Same approach as for randomly stirred fluids D. Forster, D. Nelson & D. E. Stephen, PRA ‘77
Separate Fourier modes into two classes
Solve equation of motion for fast modes perturbatively, e.g.
Average over fast noise components, assuming statistical independence
Perform a large scale approximation
Obtain an equation of motion of the same form with renormalized parameters
Rescale back in order to restore initial wave-vector cut-off
For , obtain a differential parameter flow
Four non-trivial fixed points:
EW:
Morfologically stable:
Galilean:
KPZ: Galilean fixed points is of a “mixed” type
No renormalization
Galilean symmetryNon-linear fixed points
LinearEquilibrium
Non-linearNon-equilibriumNo dimensional
analysis
Shaded regions: G not defined
EWKPZMSG
EWMSG
Unstable; saddle point; stable
DRG fixed point properties
Fixed points and their stability depend on d and
Additional DRG results
Same flow equation for non-linear term (vertex cancellation)for any linear dispersion of the form
Irrelevance higher order linear terms, e.g. n=3, 4
Unstable fixed points in RG flow same scaling behavior as for
1/20 1 2
KPZ irrelevant
Super-ballistic Sub-ballistic
KPZ relevant
SMS, MS-KPZ
Superdiffusive (KPZ)
KS3/2
z z
z zKPZ(d)z
zKPZ(d)
Graphical summary (conjectured)
M. Nicoli, R.C. & M. Castro, PRL ‘09; JSTAT ‘11
Remarks
For any interface-kinetics condition, morphological diffusive instabilities occur at short/intermediate times
These instabilities imply KPZ scaling is (at best) asymptotic and may be unobservable in practice
For fast interface kinetics, KPZ scaling does not occur
It can be also hampered by limited accessible spatial scales
Improvements over small slope approximation needed for improved comparison with experiments
-> phase-field or diffuse-interface formulation of moving boundary problem (M. Nicoli, M. Castro & R. C., JSTAT ’09; M. Nicoli, M. Castro, M. Plapp & R.C., preprint)
Introduce an auxiliary field to track down phases
Couple dynamics to that of the (physical) concentration field
Phase field (diffuse interface) formulation
A. Karma, PRL ‘01, B. Echebarria et al., PRE ‘04
J. S. Langer, ‘86O. Penrose &P. C. Fife, Physica D ‘90
G. Calginap, PRA ‘89
Matching conditions
Equations for bulk (exterior region):
Diffusion equation
Asymptotic expansion (thin interface limit)
Equivalence to moving boundary problem A. Karma & W.-J. Rappel, PRE ‘98
R. J. Almgren, SIAM JAM ‘99
Thus, the thin interface limit retrieves the absorbing barrier limit for
In the limit we obtain e.g. the stationary solutions
and the two model equations are equivalent, provided ( numerical consts.)
This connection allows to perform moving boundary simulations for realistic parameter conditions
A. Karma & W.-J. Rappel, PRE ‘98
R. J. Almgren, SIAM JAM ‘99
Equivalence to moving boundary problem
Phase-field simulations
Kahanda et al. PRL ‘92 Cu ECD
Experiments
Leger et al. PRE ‘98 Cu ECD
Some conclusions/outlook
Morphological transitions in some diffusion-limited-growth systems can be addressed through moving boundary problems; many different contexts
Introduction of noise to account for universality properties of interface fluctuations
Effective interface equations provide interesting evolution problems; need for rigorous results
Phenomenological (vs. universality-based) continuum approach provides: compact description of a variety of (sub)micrometric mechanisms efficient analytical/numerical modelling of global morphological aspects theoretical access to new (interface) phenomena new universal models relevant to general theory of Statistical Mechanics and Non-Linear Science