Fronts Fronts Zoltán Rácz Zoltán Rácz Institute for Theoretical Physics Eötvös University Homepage: poe.elte.hu/~racz Patterns.

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1 Fronts Fronts Zoltán Rácz Zoltán Rácz Institute for Theoretical Physics Eötvös University E-mail: racz@poe.elte.hu 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, 386 29-222 (2003) M. C. Cross and P. C. Hohenberg, Pattern Formation Outside of Equilibrium, Rev. Mod. Phys. 65, 851 (1993).

2 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?

3 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

4 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

5 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

6 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)

7 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

8 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

9 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

10 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

11 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?

12 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

13 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

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

15 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:

16 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.

17 Characterization of patterns Experiment Matalon-Packter law a0a0 b 0 ~ 10 -2 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, …

18 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

19 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 -1 0 +1 Megmaradási törvény: T

20 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

21 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

22 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:

23 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