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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

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Coworkers Modeling: M. Castro: Universidad Pontificia Comillas, Madrid ES M. Nicoli, M. Plapp: Ecole Polytechnique, Paris FR E. Vivo: Universidad Carlos III de Madrid, ES Experimental: J. G. Buijnsters: Radboud University Nijmegen, NL F. Ojeda: Tecnatom, Madrid ES R. Salvarezza: INIFTA, La Plata, AR L. Vázquez: Instituto de Ciencia de Materiales de Madrid, ES Support from MICINN ES

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Bacterial colonies Classic system in the study of fractal growth Morphology of the colony controlled by: nutrient concentration C n medium (agar) resistence to flagellar motility C a Morphological diagram for Bacillus subtilis M. Matsushita et al. III V IV I II

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IV V S.G. Alves,S.C. Ferreira Jr. & M.L. Martins, BJP ‘08 Off-lattice clusters

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Simplified reaction-diffusion system Agent based model of structure/metabolic activity of E. coli Off-lattice cells grow according to C n 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. ‘07 C.D. Nadell, K.R. Foster & J.B. Xavier, PLoS CB ‘10 J. Bonachela et al., JSP ‘11 C n =0.05 g/lC n =3 g/l IV V

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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

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Compact phase Bacillus subtilisE. coli T. Vicsek et al., PA ’90 J. Wakita et al., JPSP ’97 “Microscopic” fluctuations influence large-scale morphological properties Exponent values are (relatively) insensitive to system specifics

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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

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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 description of interface dynamics that is sentitive to fluctuations

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Simplified model: interface dynamics vs diffusive growth Write down a moving boundary problem (with fluctuations) in which transport is by diffusion

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A different context: electrochemical deposition + Electroneutrality No anion flux at cathode Surface diffusion Butler-Volmer bdry. condition = same moving boundary problem

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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

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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

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(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

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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)

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Small slope approximation Stochastic 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 irradiation R.C. & A.-L. Barabási, PRL ‘95 Subsequently studied e.g. J. Q. Duan & V. J. Ervin, Nonlin. Anal. ‘01 D. Yang, Stoch. Anal. Appl. ‘06 B. Ferrario, ibid. ‘08 Stochastic pseudospectral simulation scheme

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Can be solved in Fourier space: each mode evolves independently Diffusive “shadowing” instability“Surface diffusion” UnstableStable Mode dominates -> “cellular” structure Linear dynamics: Kuramoto-Sivashinsky equation

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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 ’81 cf. also M. Pradas et al., PRL ‘11 M. Nicoli, R. Cuerno & E. Vivo PRE ‘10 (d=2) Kardar-Parisi-Zhang asymptotics

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Thin film growth by Chemical Vapor Deposition (CVD) (slow interface kinetics) SiO 2 on Si AFM top view KPZ scaling F. Ojeda et al., PRL ‘00

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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

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CVD growth for fast interface kinetics M. Castro, R.C., M. Nicoli, L. Vázquez, & J. G. Buijnsters, submitted AFM 1 m 2 MS+KPZ 6 h 40 min.

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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

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Non-KPZ behavior Effective equations

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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/ Super-ballistic Sub-ballistic SMS MS-KPZ Superdiffusive (KPZ) KS 3/2

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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

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P. Barthelemy, J. Bertolotti & D.S. Wiersma, Nature ‘08 Example: Lévy walks B.J. West, M. Bologna & P. Grigolini ’03 Ch. 8

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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)

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Numerics: < z KPZ (d=1)=3/2 (morphologically unstable condition) = 1 = 1/2 = 1.05 z=0.92 = 1.52 z=0.44

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Numerics: < z KPZ (d=2)=1.61 (morphologically unstable condition) = 1 = 1/2 = 1.10 z=0.90 = 1.55 z=0.45 d-independent exponents!!

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Numerics: > z KPZ (d) (morphologically unstable condition) = 1.75 d=1, z KPZ (1)=3/2d=2, z KPZ (2)=1.61 = 1/2, z=3/2 = 0.39, z=1.61 KPZ d-dependent exponents!!

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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.

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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

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Four non-trivial fixed points: EW: Morfologically stable: Galilean: KPZ: Galilean fixed points is of a “mixed” type No renormalization Galilean symmetry Non-linear fixed points Linear Equilibrium Non-linear Non-equilibrium No dimensional analysis

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Shaded regions: G not defined EW KPZ MS G EW MS G Unstable; saddle point; stable DRG fixed point properties Fixed points and their stability depend on d and

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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

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1/ KPZ irrelevant Super-ballistic Sub-ballistic KPZ relevant SMS, MS-KPZ Superdiffusive (KPZ) KS 3/2 z z z z KPZ (d) z z KPZ (d) Graphical summary (conjectured) M. Nicoli, R.C. & M. Castro, PRL ‘09; JSTAT ‘11

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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)

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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, ‘86 O. Penrose &P. C. Fife, Physica D ‘90 G. Calginap, PRA ‘89

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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

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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

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Phase-field simulations Kahanda et al. PRL ‘92 Cu ECD Experiments Leger et al. PRE ‘98 Cu ECD

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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

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