fluctuations in a moving boundary description of diffusive interface growth rodolfo cuerno...
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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
http://gisc.uc3m.es/~cuerno
Coworkers
Modeling:
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
medium (agar) resistence to flagellar motility Ca
Morphological diagram for Bacillus subtilis
M. Matsushita et al.
http://www.phys.chuo-u.ac.jp/labs/matusita/doc/b_pattern.htm
IIIV
IV I
II
IV V
http://www.phys.chuo-u.ac.jp/labs/matusita/doc/b_pattern.htm
http://classes.yale.edu/fractals/
S.G. Alves,S.C. Ferreira Jr. & M.L. Martins, BJP ‘08
Off-lattice clusters
Simplified reaction-diffusion system
Agent based model of structure/metabolic activity of E. coli
Off-lattice cells grow according to Cn and divide, and shove each other
Nutrient concentration held fixed in the bulk; nutrient diffuses within ensuing boundary layer
Low mobility regime J.B. Xavier et al., Environ. Microbiol. ‘07C.D. Nadell, K.R. Foster & J.B. Xavier, PLoS CB ‘10
J. Bonachela et al., JSP ‘11
Cn =0.05 g/l Cn =3 g/l
IV V
Universal interface fluctuations: kinetic roughening
T. Vicsek, M. Cserzö & V.K. Horváth, PA ’90: E. coli
Surface roughness:Surface structure factor:
Roughness (Hurst)exponent
Dynamic exponent
Compact phase
Bacillus subtilis E. coli
T. Vicsek et al., PA ’90J.
Wak
ita e
t al.,
JPSP
’97
“Microscopic” fluctuations influence large-scale morphological properties
Exponent values are (relatively) insensitive to system specifics
C. Ratsch et al., PRL ‘94
F. Tsui et al., PRL ‘96
40 nm
T=530 K
T=680 K
Fractal to compact transition in molecular-beam epitaxy (MBE)
From N. Néel et al., JPCM ‘03
Differences:
Length and time scales
Cell shape/internal structure
Crystalline anisotropies
Similarities:
Diffusive transport (nutrients, adatoms)
Similar morphological transitions
Universality of interface fluctuations
“Material” independent properties
Relevance of “microscopic” fluctuations
A common/analogous description?
Explore a continuum descriptionof interface dynamics that issentitive to fluctuations
Simplified model: interface dynamics vs diffusive growth
Write down a moving boundary problem (with fluctuations) in which transport is by diffusion
A different context: electrochemical deposition
+
Electroneutrality
No anion flux at cathode
Surface diffusion
Butler-Volmer bdry. condition
= same moving boundary problem
Noise amplitudes: local equilibrium approximation
R.C. & M. Castro, PRL ’01; M. Nicoli, M. Castro & R.C., PRE ‘08
Simplified model: interface dynamics vs diffusive growth
slow interface kinetics (reflecting barrier)
fast interface kinetics (absorbing barrier)Noise amplitudes
Other moving boundary problems with fluctuations
Deterministic limit: standard model of thin film production by Chemical Vapor Deposition C.H.J. Van den Brekel & A.K. Jansen, JCG ‘78
Relevant to other systems: electrodeposition; (one-sided) solidification; isolated step in MBE; spreading of precursor fronts; bacterial colonies???
Noise introduced as in two-sided model of solidification A. Karma, PRL ‘93 Dendritic sidebranching
Existence of solutions:
N. K. Yip, J. Non-Lin. Sci. ‘98: Two-sided solidification; fast kinetics condition; spatially correlated noise
V. Barbu & G. Da Prato, Prob. Theor. Rel. Fields ‘02: Two-sided solidification; No Gibbs-Thomson
M. Dudzinski & P. Górka, Appl. Math. Comp. ‘10: Two-sided solidification; polygonal interface
A. Dougherty, P.D. Kaplan & J.P. Gollub, PRL ‘87
(Effective) single interface equation with fluctuations Goal: reduce study to that of a single (effective) stochastic equation
Procedure: projection of dynamics onto interface + small slope approximation
Expectation: physical derivation of Kardar-Parisi-Zhang (KPZ) (noisy Burgers) equation
M. Kardar, G. Parisi & Y.-C- Zhang, PRL ‘86
Paradigmatic of (non-conserved) kinetic roughening systems, e.g. Eden model, PNG, ASEP, …
Recently solved for T. Sasamoto & H. Spohn, PRL ‘10 G. Amir, I. Corwin & J. Quastel, CPAM ‘11 P. Calabrese & P. Le Doussal, PRL ‘11
Project bulk diffusive problem onto the moving boundary
Set up perturbative expansion in surface derivatives (small disturbances)
Neglect multiplicative noise contributions; long wavelength approximation
(cf. C. Misbah, O. Pierre-Louis & Y. Saito, RMP ‘10)
Small slope approximationStochastic
Kuramoto-Sivashinsky equation
Slow interface kinetics (reflecting barrier)
Previously found in other contexts:
Step Dynamics in MBE A. Karma & C. Misbah, PRE ‘93
Erosion by ion-beam irradiationR.C. & A.-L. Barabási, PRL ‘95
Subsequently studied e.g.J. Q. Duan & V. J. Ervin, Nonlin. Anal. ‘01D. Yang, Stoch. Anal. Appl. ‘06B. Ferrario, ibid. ‘08
Stochastic pseudospectral simulation scheme
Can be solved in Fourier space: each mode evolves independently
Diffusive “shadowing” instability “Surface diffusion”
Unstable Stable
Mode dominates -> “cellular” structure
Linear dynamics: Kuramoto-Sivashinsky equation
Small slope approximation
Slow interface kinetics (reflecting barrier) Local morphological instability Chaotic dynamics Disordered asymptotic morphology
(d=1) Kardar-Parisi-Zhang asymptotics
Yakhot’s renormalization mechanism V. Yakhot, PRA ’81cf. also M. Pradas et al., PRL ‘11
M. Nicoli, R. Cuerno & E. Vivo PRE ‘10
(d=2) Kardar-Parisi-Zhang asymptotics
Thin film growth by Chemical Vapor Deposition (CVD) (slow interface kinetics)
SiO2 on Si AFM top view
KPZ scaling
F. Ojeda et al., PRL ‘00
Small slope approximation
Fast interface kinetics (absorbing barrier) Non-local Mullins-Sekerka instability Cusp dynamics Disordered asymptotic morphology
Kinetic roughening properties different from KPZ asymptotics
New equation (¿?): similarities and differences with nKS
CVD growth for fast interface kinetics M. Castro, R.C., M. Nicoli, L. Vázquez, & J. G. Buijnsters, submitted
AFM 1 m2MS+KPZ
6 h
40 min.
Realistic interface kinetics
In experiments a finite interface kinetics is expected:nKS condition
M. Nicoli, M. Castro & R. Cuerno, JSTAT ‘09
Experimentally accessible scales (ECD) effective shape for
Non-KPZbehavior
Effective equations
Generalization
Consider an equation of the form
M. Nicoli, R.C. & M. Castro, PRL ’09; JSTAT ‘11
Take
For asymp. behavior it suffices with most relevant stabilizing term m = 2, but irrelevant terms can be added (m =3, 4, …, and n > m)
Many celebrated limits: ( , m)
Saffman-Taylor = Mullins-Sekerka = (1,3) (fast surface kinetics CVD)
Michelson-Sivashinsky = (1,2)
Kuramoto-Sivashinsky = (2,4) …
1/20 1 2
Super-ballistic Sub-ballistic
SMSMS-KPZ Superdiffusive (KPZ)
KS3/2
An(other) example: stochastic Michelson-Sivashinsky equation
Derived (deterministic case) for premixed flame combustion
D. M. Michelson & G. I. Sivashinsky, Acta Astron. ‘77
Single (large) cusp stationary state
Small cusp creation/annihilation (even by numerical noise)
Stochastic case more meaningful
V. Karlin, Math. Models Meth. Appl. Sci. ‘04 P. Cambray, G. Joulin, I. Procaccia, … ‘90’s
P. Barthelemy, J. Bertolotti & D.S. Wiersma, Nature ‘08Example: Lévy walks
B.J. West, M. Bologna & P. Grigolini ’03 Ch. 8
Previous proposals in the morphologically stable case. E.g.
KPZ equation “Fractional KPZ” equation
These generalizations are “trivial”: exponents are given by dimensional analysis:
Scale invariance ensues if
and the nonlinearity is irrelevant for suitable (equilibrium fluctuations)
P. Kechagia, Y. C. Yortsos & P. Lichtner, PRE ‘01
E. Katzav, PRE ‘03
Kinetic roughening
Proofs in P. Biler, G. Karch & W. Woyczynski, Studia Mathematica ‘99 (deterministic case) J. A. Mann Jr. & W. Woyczynski, Physica A ’01 (noisy case)
Numerics: < zKPZ(d=1)=3/2 (morphologically unstable condition)
= 1 = 1/2
= 1.05z=0.92
= 1.52z=0.44
Numerics: < zKPZ(d=2)=1.61 (morphologically unstable condition)
= 1 = 1/2
= 1.10z=0.90
= 1.55z=0.45
d-independentexponents!!
Numerics: > zKPZ(d) (morphologically unstable condition) = 1.75
d=1, zKPZ(1)=3/2 d=2, zKPZ(2)=1.61
= 1/2, z=3/2 = 0.39, z=1.61KPZ d-dependent exponents!!
Study of scaling behavior
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