Pattern Formation: from Turing to Nanoscience

Transcription

1 IV GEFENOL School September 2014 Pattern Formation: from Turing to Nanoscience D. - Mallorca - Spain Mallorca - Spain

2 Why is it interesting to study pattern formation in nonequilibrium systems? The Universe is full of spatio-temporal structures with striking similarities. Where do they come from? Can we understand them, including the similarities? Are there unifying concepts? Are quantitative modeling and predictions possible? Nonequilibrium spatio-temporal patterns are technologically important in various domains. In materials, for example, they result from instabilities, self-assembling, self-organization, Their understanding may lead to the tailoring of on-demand devices but also to the control of their lifetime They induced the search for specific mathematical and numerical methods for the understanding of complex systems. Many of them are described by PDEs, for which qualitative insights are needed to corroborate numerical simulations or experimental results, the effect of boundaries, noise, etc. which is by no means an easy task and requires strong interactions between theory, experiment and computation. In these lectures, I will review some basic aspects of instability and pattern formation theory. Their application in materials modeling will be illustrated in a few examples.

3 Summary Introduction, historical background, methods, benefits and perils of self-organization. Instabilities and bifurcations, reaction-diffusion dynamics. Pattern formation, selection and stability in amplitude and model equations. Applications in materials design and nanotechnology: - self-organization of surface atoms in thin film growth - surface roughening and cracks - mechanical instabilities and deformation localization. Conclusion

4 Introduction. Two paradigms of instabilities and spatio-temporal pattern formation. A. Rayleigh-Bénard (and Bénard-Marangoni) convection C.Marangoni 1865, H. Bénard 1900, J.Pearson 1958, E. Koschmieder 1991, G. Ahlers et al

5 1900 H. Bénard: first quantitative measurements 1916 Lord Rayleigh: linear stability analysis (free b.c.) 1926 H. Jeffreys: linear stability analysis (rigid b.c.) Conduction state: =0, = h, = Dimensionless perturbations dynamics (Navier-Stokes for incompressible Boussinesq fluid): P = silicon oil: 10.3 water: 6.99 air: 0.72 α: expansion coefficient g: acceleration of gravity κ: thermal diffusivity ν: kinematic viscosity

6 stress free b.c. =0,1 = =0,1 =0, =0,1 =0 "# =0,1 =0 "#, sin(())exp-../ + eliminate 4 linear evolution: 9 :; <,=,> = ( # ) # +. #. #.=,> + ( # ) # +. # =,> < =,> = ( # ) # +. # =,> +?,=,> instability for A> (CD +E D ) F rigid(no slip) b.c.: =0,1 = =0,1 =0 = QR S Q =0,1 =0? G =1708,. G =3.12 A= stress-free B= rigid (no slip) E D? G = 27 4 )K ~ 657.5,. G = ) 2

7 Busse Koschmieder Ahlers 1965: Schluter. Lorz, Busse : weakly nonlinear analysis 1977: Swift-Hohenberg: amplitude equation, effect of noise : detailed experimental analysis, secondary bifurcations, Pattern competition (rolls, hexagons, squares, ), spatio-temporal chaos, phase dynamics, defects, numerical simulations, SH equation (P>>1, stressfree,elimination of ): Amplitude equation: (multiple scale analysis Newell, Whitehead,Segel) Swift Hohenberg Newell Lyapunov functional relaxational stationary patterns cf. 2d solid P nonlinearitiesin thevelocityfield non relaxational effects spatio temporal chaos (cf. turbulence)

8 hexagons rolls G.Ahlers Bifurcation diagram from SH eq. ε = 0.008, , Dislocations in Rayleigh-Benard convection (Pantaloni et al.)

9 Atacama desert λ~meter Mushroom pore λ~ µ m Coral λ~mm Death Valley Void lattice in Mo λ~ 20 nm Quantum dots lattice λ~ 10 nm Universality? Generic vs non generic aspects? Minimal models?

10 B. Chemical instabilities, patterns and oscillations : Turing, Hopf, 1921 : Bray: chemical clock 1952: Turing instability mechanism (besides his activity as the father of modern computer science) A.Turing ( )

11 Reaction-diffusion equations, linear stability analysis instabilities symmetry breaking linear growth of patterns of different types (6). nonlinear analysis? numerical analysis? pattern selection? Z=ν,\=],unstable uniform Z=ν,\= ],unstable uniform oscillatory(cf.hopf) _ inhibitor,` activator,z>ν,unstable vs spatial modes 2 remaining types: traveling waves require at least 3 equations Kele s science blog

12 1950: Belousov: chemical oscillations in the oxidation of malonic acid, rejected paper for thermodynamic impossibility? 1956: Prigogine & Balescu: nonequilibrium thermodynamics (extending Th. de Donder s work on chemical affinity (1923)) : Zhabotinsky: confirmation of oscillations, chemical waves,.. Prigogine Belousov Balescu Zhabotinsky 1966: Prigogine and the Brussel s school, dissipative structures Concept of dissipative structure : a state reflecting the ability of a system to utilize the dissipation generated out of equilibrium to open novel evolutionary pathways, separated from the "thermodynamic branch" of equilibrium-like states by an instability (I. Prigogine, 1969). Universal role of the distance from equilibrium as a source of order, unifying framework for the understanding of diverse self-organization phenomena.

13 1967: the Brusselator model of autocatalytic chemical reactions (Prigogine, Nicolis and Lefever) Nicolis 1967: concept of synergetics (H.Haken) Concept of synergetics : in a system, consisting of many nonlinearly interacting subsystems, depending on the external control parameters (environment, energy-fluxes) self-organization takes place. Interdisciplinary, far from thermal equilibriun dynamics (H. Haken,1969). 1990: Experimental observation of Turing patterns in a chemical system (Bordeaux group, Texas group) (why so late?) Lefever Haken De Kepper Swinney

14 In the mean time: chemical chaos (1980, Roux-Swinney), spatio-temporal chaos, (exp.: J. Gollub, 1994; G. Ahlers, 1998.) patterns in biological systems (A.T. Winfree, The geometry of biological time, 1980), bifurcation and dynamical systems theory (H.Poincare, 1885; V.I. Arnold, 1978;J. Guckenheimer, P. Holmes, 1983, G. Iooss, D.Joseph, 1980, etc. weakly nonlinear analysis (A. Newell, 1974; J.Swift, P. Hohenberg, 1977,) numerical simulations,(cell. automata, mol. dynamics, PDE, ) Nowadays: Studies and modeling of pattern formation in hydrodynamics (including non Newtonian fluids), chemistry, biology, mechanics, nonlinear optics, insect and bacterial colonies, materials science, ecology, sociology,.

15 The Brusselator model for autocatalytic chemical reactions a _, 2_+` 3_, b+_ `+c, _ f unstable < _=a b+1 _+_ #`+c d _ < `=b_ _ #`+c e ` unstable D X < D Y stable < σ (r,t) = [ε - (q c 2 + ) 2 ] σ (r,t) + v σ 2 (r,t) - u σ 3 (r,t)

16 Codimension 2 bifurcation: Coupling between oscillatory and spatial modes, e. g Brusselator with j k j l = η = ; m 1+a # 1 b G n =b G o b complex spatio-temporal patterns b G n =b G o p G p coupling between Turing and Hopf bifurcations, simultaneous growth of oscillatory and spatial modes, nonlinear couplings make selection, possibility of localized states Ex.1D: < a=za+ 1+-s < t # a 1+-u a # a w+-x? # a <?=y?+4p G # < t #?? #? z a #? (/,{)=?(/,{)exp-p G / + a /,{ exp-ν{+].]. w, x, z monitor pattern selection and stability

17 t x Matkovsky et al. Tozas et al. b b G n b G o Turing mixed Hopf stable uniform DeWit et al. η

18 A. De Wit 3D lamellar domains t Leppanen et al. Turing-Hopf cod 2 x

19 Other Instabilities Faraday instability in granular materials Viscous fingering + chemistry RB convection in liquid crystal Patterns and quasipatterns in nonlinear optical system

20 Materials Instabilities Defect patterning: PSB in Cu during fatigue exp. PbSe/PbEuTe Wrinkling of Cu film on polymer substrate Buckling instability in CNT Ripples on irradiated nanorods Shell structure of hen s egg

21 Deformation instabilities in irradiated Si Self-organized vacancy loops in Cu irradiated by protons SAQD in InAs BCC void lattice in Mo BCC void lattice in Nb

22 Methods: numerical: ab initio, molecular dynamics, Monte-Carlo, celular automata, stochastic methods microscopic: quantum mechanics, statistical mechanics, many body theory, mesoscopic: ODE, PDE, bifurcation theory, dynamical systems, nonlinear dynamics, master equations, numerical analysis, macroscopic: continuum mechanics, feature size models, finite elements calculations, Perils and benefits of self-organization in nanoscience: In the making of new materials and devices: self-assembly of elementary components to generate new phases, structures and patterns for specific applications through instabilities -> desire to use self-organization to control microstructure, In the breaking of materials: materials utilization occurence of deformation instabilities, plastic instabilities, fracture instabilities, defect self-organization, etc. -> desire to use self-organization to control the lifetime of the material

23 Bibliography M.C. Cross and P. Hohenberg, Rev.Mod.Phys. 65, (1993) D. Walgraef, Spatio-Temporal Pattern Formation, Springer Verlag (1996) Lui Lam, Nonlinear Physics for Beginners, World Scientific (1998) I. R. Epstein, J.Pojman, An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns and Chaos, Oxford University Press (1998) R. de Borst, E. van der Giessen, Material Instabilities in Solids, Wiley (1998) J. Moloney, A.C. Newell, Nonlinear Optics, Westview Press (2004) N.M. Ghoniem and D. Walgraef, Instabilities and Self-Organization in Materials, Vol. 1. Fundamentals of nanoscience, Oxford University Press (2008). [draft available in resources ] N.M. Ghoniem and D. Walgraef, Instabilities and Self-Organization in Materials, Vol. 2. Applications in materials design and nanotechnology, Oxford University Press (2008). [draft available in resources ] H. Greenside and M.C. Cross, Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press (2009) and references therein