fluctuations in a moving boundary description of diffusive interface growth rodolfo cuerno...

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

cuerno@math.uc3m.es

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