Fronts Fronts Zoltán Rácz Zoltán Rácz Institute for Theoretical Physics Eötvös University Homepage: poe.elte.hu/~racz Patterns from moving fronts (1) Importance of moving fronts: Patterns are manufactured in them. Examples: Crystal growth, DLA, reaction fronts. Dynamics of interfaces separating phases of different stability. Classification of fronts: pushed and pulled. (2) Invasion of an unstable state. Velocity selection. Example: Population dynamics. Stationary point analysis of the Fisher-Kolmogorov equation. Wavelength selection. Example: Cahn-Hilliard equation and coarsening waves. (3) Diffusive fronts. Liesegang phenomena (precipitation patterns in the wake of diffusive reaction fronts - a problem of distinguishing the general and particular). Literature W. van Saarloos, Front propagation into unstable state, Physics Reports, (2003) M. C. Cross and P. C. Hohenberg, Pattern Formation Outside of Equilibrium, Rev. Mod. Phys. 65, 851 (1993).

Fronts separating stable and unstable phases crystallization fronts chemical reaction fronts The problems: stableunstable (1) What is the speed of the front? (2) Is there any nontrivial structure in the wake of the front?

precipitate Naturwiss. Wochenschrift 11, 353 (1896) A random experiment Zrínyi, 95 Liesegang phenomena d=1 d=3 d=2 Nontrivial patterns in d=1-3 dimensions agates A B C

Infection front: Fisher-Kolmogorov-Piskunov equation Initial condition x n - concentration of the infected (immune) 1 0 Fixed shape moving with velocity v? Solution exists for arbitrary v stable unstable 02 v stable v*=2 selected velocity

FKP equation: Instability at small velocities Initial condition x n 1 0 Fixed shape Solution exists for arbitrary v v - friction v (friction) smalloscillations around 0 unphysical (n<0) 10

FKP equation: Instability at small velocities Solution 10 Unphysical (n<0) oscillations for small v Small limit: 02 v unstable stable v*=2 selected velocity (marginal stability theories)

FKP equation: Leading edge analysis looking for stationary point in the moving frame Initial condition x n 1 0 Fixed shape Assumption: Leading edge determines the velocity (the front is pulled by the leading edge) linearization Fourier transform

FKP equation: Leading edge analysis II Looking for stationary point in the moving frame ~const Stationary phase Stationarity in the moving frame: Selected velocity: x n 1 0

Conservation laws and moving fronts: Cahn-Hilliard equation x n 1 0 stable unstable x n 1 0 stable unstable stable 1 n 0 F(n) n is conserved: unstable initial condition x n 1 0 stable v

Derivation of the Cahn-Hilliard equation n is conserved: 1 n 0 F(n) The flux of particles should reduce the chemical potential: continuity equation kinetic coefficient From : can be scaled out

Conservation laws and moving fronts: Cahn-Hilliard equation 1 x n 1 0 stable v unstable Standard form: Questions: (1)What is the velocity of the front? (2)What is the wavelength of the pattern left in the wake of the front?

Linear stability analysis of the CH equation Linearization: 1 n 0 F(n) Fourier modes Dispersion Linear instability: It leads to spinodal decomposition Inflection points in F(n) constant in space

Pulled front in the CH equation Linearization + Fourier modes x n 1 0 stable v unstable Looking for stationary point in the moving frame ~const Stationary phase Stationarity in the moving frame: goes to zero at spinodal

Pulled front in the CH equation Results: x n 1 0 stable v unstable Stationary phase Stationarity in the moving frame:

Wavelength observed in the lab Wavelength: distance between the maximal densities x n 1 0 stable v unstable Oscillation in the comoving frame with period In the moving frame: In the laboratory frame: Wavenumber:

Liesegang mintázatok d=1d=3d=2d=3 Achátok Zrínyi, 95 Nemtriviális mintázatok d=1-3 dimenzióban Történeti megjegyzés: Gyógyítja a: kígyómarást, skorpiószúrást, lázat. Megvéd: fert ő zést ő l, villámlástól. Hosszú életet biztosít. Perzsa mágusok, bizánci császárok és reneszánsz uralkodók kedvencei.

Characterization of patterns Experiment Matalon-Packter law a0a0 b 0 ~ a 0 A +BC distance from the interface time of appearence width Width law Spacing law Time law n Other (not general) observations: inverse patterns, fine structure between bands, …

Elméletek W. Ostwald (1897), N.R. Dhar et al. (1925), C. Wagner (1950), S. Prager (1956), Ya.B. Zeldovitch et al. (1960), S. Shinohara (1970), M. Flicker et al. (1974), A B a0a0 b0b0 A + BI 1 I 2 ? 1 2 S. Kai et al. (1982), G.T. Dee (1986), B. Chopard et al. (1994), … A B a0a0 b0b0 c0c0 forrása L. Gálfi and Z.R., PRA 38, 3151 (1988) C ion-szorzat túltelítés nukleáció és növekedés C

Liesegang sávok fázisszeparáció modellje c c A B Szabadenergia: c forrása F(c) A B a0a0 b0b0 c0c0 C PRL 83, 2880 (1999) c0c0 clcl chch Megmaradási törvény: T

c c x xnxn x n ~ Q (1+p ) n p = p(a 0,b 0 ) Termodinamika rögzíti c h és c l -t, és c megmaradásából következik movie A B Cahn-Hilliard egyenlet forrással w n ~ x n chch clcl Liesegang sávok fázisszeparáció modellje II c0c0 C

Távolság-törvény a0a0 b0b0 c0c0 Csapadékképz ő dés feltétele c>c * C c*c* c*c* c*c* c*c* c*c* Megmaradási törvény: Matalon-Packter T. Antal et al., J.Chem.Phys. 109, 9479 (1998) a0a0

Prognosztika: egy sáv megjelenési ideje Physica A274, 50 (1999) c x xnxn chch clcl DaDa DbDb b0b0 c0c0 C a0a0 Paraméterek meghatározása kísérletb ő l c0c0 DfDf hossz-skála: id ő skála: A dinamika részletei:

Solidification (Stefan problem in d=1) T 0 – melting temperature Equations: solidliquid T 0 – Δ supercooledT S ≤ T 0 (solid) (liquid) Boundary conditions: At the solid-liquid (plane) interface: Equilibrium at melting temperature (assumption). Latent heat must be conducted away. D S – heat diff.coeff. in solid D l – heat diff.coeff. in liquid – heat cond. coeff. L – latent heat