Fronts Zoltán Rácz

1. Fronts 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 propagationintounstablestate, PhysicsReports, 386 29-222 (2003) M. C. Cross and P. C. Hohenberg, Pattern Formation Outside of Equilibrium, Rev. Mod. Phys. 65, 851 (1993).

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

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

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

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

6. 0 1 FKP equation: Instability at small velocities Unphysical (n<0) oscillations for small v Small limit: Solution 0 2 v stable unstable (marginal stability theories) v*=2selected velocity

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

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

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

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

11. 1 Conservation laws and moving fronts: Cahn-Hilliard equation Standard form: -1 n stable 1 v unstable 0 x -1 stable • Questions: • What is thevelocity of the front? • What is thewavelength of the • patternleftinthewake of the front?

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

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

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

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

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

17. Characterization of patterns Experiment Time law A + B C a0 b0~ 10-2 a0 Spacing law distance from the interface Width law n Matalon-Packter law width time of appearence Other (not general) observations: inverse patterns, fine structure between bands, …

18. 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), Elméletek S. Kai et al. (1982), G.T. Dee (1986), B. Chopard et al. (1994), … a0 A b0 B forrása A + B I1 I2 ? C a0 1 c0 A ion-szorzattúltelítés b0 C B 2 nukleáció és növekedés L. Gálfi and Z.R., PRA 38, 3151 (1988)

19. Liesegang sávok fázisszeparáció modellje PRL83, 2880 (1999) a0 cforrása A T c0 A b0 C B B c c0 F(c) Megmaradási törvény: ch c cl -1 0 +1 Szabadenergia:

20. c0 C Liesegang sávok fázisszeparáció modellje II A Cahn-Hilliard egyenlet forrással B ch c xn~Q (1+p )n c p =p(a0,b0) cl xn x Termodinamikarögzíti chés cl-t, és cmegmaradásából következik movie wn~ xn

21. Távolság-törvény T. Antal et al., J.Chem.Phys. 109, 9479 (1998) c* a0 c0 b0 c* c* C Csapadékképződésfeltételec>c* c* Megmaradási törvény: a0 c* Matalon-Packter

22. Prognosztika: egy sáv megjelenési ideje Physica A274, 50 (1999) a0 Da Paraméterek meghatározásakísérletből c0 b0 Db C c0 Df hossz-skála: időskála: A dinamika részletei: ch c cl xn x

23. Solidification (Stefan problem in d=1) T0 – melting temperature DS – heat diff.coeff. in solid Dl – heat diff.coeff. in liquid TS ≤ T0 T0 – Δ supercooled – heat cond. coeff. solid liquid L – latent heat Boundary conditions: Equations: (solid) (liquid) At the solid-liquid (plane) interface: Equilibrium at melting temperature (assumption). Latent heat must be conducted away.