Coarsening: transient and self-similar dynamics in 1-D

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1 Coarsening: transient and self-similar dynamics in 1-D ½ h ln(t) Ú ¼º N 1 3 N t 2/ x ¼ 1 ¼ ¼º ½ Ü t Michael Gratton (Northwestern) and Thomas Witelsi (Due) Cahn-Hilliard PDE models: destabilizing 2nd, regularizing 4th order terms Image processing (Perona-Mali), materials science (Ostwald ripening), fluid dynamics (Dewetting), granular materials,... Short-time behavior: linear instability spinodal decomposition or roughening formation of arrays of near-equilibrium localized structures Long-time behavior: Coarsening Energetically favored gradient flow process of re-grouping into fewer structures Coarsening Dynamical System (CDS): ODEs for interacting structures Long-long-time behavior: Stationary distribution (LSW theory) and statistical scaling law for coarsening (Kohn-Otto rate bounds) Short-long-time: Transient dynamics controlled by spatial structure of IC s [submitted to Physica D] [supported by NSF DMS CAREER ]

2 Dewetting: The instability of coatings of viscous fluids on solid surfaces. Very undesirable for most applications (painting, printing, microfluidic devices). Lubrication models of flows of viscous fluid films on hydrophobic solids. Multi-scale dynamics controlled by materials properties. Dewetting: pattern forming instabilities brea-up of layers into droplets Long-time behavior: Coarsening re-grouping into fewer larger droplets Observed non-uniformities in experiments for polymer films on SiO Holes complex patterns (fractals?) polygonal ridges droplets Increasing time [P. F. Green et al, 21]

3 Classical lubrication models for thin viscous films Fluid volume: x, y L z H(x, y, t) Navier-Stoes eqns: velocity field u for viscous incompressible flow Re D u Dt = p + 2 u u = Low Reynolds number creeping flow limit: Stoes flow = p + 2 u u = Asymptotics in the aspect ratio: δ = H/L Boundary conditions at z = and z = h(x, y, t) The Reynolds lubrication equation in 1-D h t = x ( h 3 p ) x h = h(x, t) : J = h 3 x p : p = p[h] : film thicness mass flux dynamic pressure

4 Contributions to the pressure p Π(h) 2 h x 2 1. Surface tension of the free surface (linearized curvature of h) 2. Fluid-solid intermolecular forces: chemical properties of the solid and fluid Wetting/non-wetting interactions described by a potential U(h) disjoining pressure: Π(h) = du dh ex: Scotchgard, Teflon, TurtleWax,... The dewetting model: h t = x ( h 3 x [ ]) Π(h) 2 h x 2 Π(h) = 1 ɛ ( ɛ h ) 3 [ 1 ( ) ɛ n ] h n 1 ɛ > : thicness scale for equilibrium Ultra-thin film (UTF), h(x, t) ɛ Cahn-Hilliard-lie model with degenerate mobility early stage instabilities called spinodal dewetting

5 1-D Equilibrium fluid droplets: localized steady states h = H(x) p[h] = const( p) = Phase plane ½ Ñ Ü d 2 H dx 2 = Π(H; ɛ) p µ µ µ Droplet: homoclinic solution Asymptotic structure for ɛ ¼ Û Ñ Ò ¼ Ü Û (i) Droplet core: parabolic profile for x w H(x) 1 2 p( w2 x 2 ) (ii) Contact line: matching region x w determines wih and contact angle dh dx = A = 2 Π (ɛ) w( p) = Ā x w p Droplet mass mass of core region m( p) = w w H(x) dx 2A3 3 p 2

6 Dynamics of a single near-equilibrium droplet on L x L Initial condition: h(x, ) = H(x; p) Boundary conditions: small mass fluxes imposed Fluxes set a slow timescale, σ 1: J( L) = σ J J(L) = σ J + τ = σt Fluxes drive the slow evolution of the droplet. They change the droplet s position X(τ ) and pressure (mass) P (τ ): h(x, t) = H(x X(τ ); P (τ )) + σh 1 (x, τ ) + O(σ 2 ) Equations for droplet evolution derived from the solvability conditions for the h 1 (x, τ ) eqn: dp = C P (P )(J + J ) dx = C X(P )(J + + J ) The coefficient functions C P (P ), C X (P ) are given by integrals of the equilibrium droplet H(x; P ).

7 Dynamics of arrays of droplets Widely separated droplets: each has a locally constant pressure Differences in the pressures will generate fluxes through the UTF ¼ ¾ Ⱦ ½ Ô È È ¼ ½ ¾ Ü ¼ Can define a potential fcn, J = x V (p) The flux between droplet and its neighbor, + 1: ¼ ½ ¼ Ƚ Ü ¼ V (P +1 ) V (P ) J,+1 = [X +1 w(p +1 )] [X + w(p )] Evolution equations for an array of N droplets, = 1, 2,, N, are dp = C P (P )(J,+1 J 1, ) dx Nonlinear system with nearest-neighbor coupling Interacting particle system with finite-time singularities: some P (Mass ) droplet collapse = C X (P )(J,+1 + J 1, ) [Glasner and Witelsi: PRE 23, Physica D 25]

8 Coarsening Dynamical System (CDS) = Near-equilibrium ODE system + Coarsening rules For t < t 1{dX = dp } = = 1, 2,, N At t = t 1, the soln of ODEs satisfies a detection condition. Stop the ODEs. Remove collapsed drop. Create new IC s at t = t + 1 for remaining N 1 drops via coarsening rules for collision or collapse: n o X (t + 1 ) = X [X, P ; t 1 ] P (t + 1 ) = P[X, P ; t 1 ] = 1,, N 1 For t + 1 t < t 2 { dx = dp } = = 1, 2,, N 1 At t = t 2 another coarsening event detected... (rinse and repeat, N N 1 N 2 )

9 Coarsening via droplet collapse: Collapse: when Mass (P ) for one drop Initial Conditions ½ Evolution of Pressures ÔÑ Ü ½ È ½ ¼ ½ ¾ Ü ¼ ¹½¼ ¼ ½¼ Ü ¼ ¼ ¾¼¼¼¼¼ Ø ¼¼¼¼¼ [PDE results (dots), ODE model (curves)] dp P 4 L P (t) ([T c t]/l) 1/3 Heuristic scaling argument: statistics for N 1 L: mean spacing between drops on fixed domain, L = O(1/N) P : typical drop pressure, P = O(M 1/2 ) M : typical mass M = O(1/N) T c : critical collapse time, T c = O(P 3 L) Assume independent, uncorrelated events: 1 N dn = 1 T c N(t) = O(t 2/5 )

10 Coarsening statistics N N t 2/ t Upper bound on coarsening rate can be rigorously established via two estimates on the energy of the system [Kohn and Otto 22] PDE conserves mass and dissipates energy E = U(h) h2 x dx = O(N 1/2 de ) = h 3 p 2 x dx [Otto, Rump, and Slepcev 26]: Thin film PDE N(t) O(t 2/5 ) [Esedoglu et al]: Coarsening in image processing PDE models Alternative approach to analysis: reduce PDE to CDS and study dynamics of the CDS, ala [Dai and Pego 25]

11 Simplified CDS model V (P +1 ) V (P ) J,+1 = [X +1 w(p +1 )] [X + w(p )] V (P +1) V (P ) X +1 X Dilute limit (narrow drops): X w Define separations: L X X 1 No-drift limit: dx / so X const Focus on droplet masses: P M 1/2 (until deletion) dm = M 1/2 +1 M 1/2 L +1 M 1/2 M 1/2 1 L Conserves mass d Same gradient flow, E = O(N 1/2 ) E = M 1/2 de ( M ) = 1 2 = (M 1/2 +1 M 1/2 ) 2 L +1

12 Even-more-Simplified CDS model dm = M 1/2 +1 M 1/2 L +1 M 1/2 M 1/2 1 L Replace individual separations by average L = L total /N dm = M 1/2 +1 1/2 2M + M 1/2 1 L Replace neighbors by average (mean field pressure) M 1/2 = 1 N M 1/2 dm = 2 ( ) M 1/2 M 1/2 L Discrete mean field model (DMF) Discrete collapse events change N(t) and L (piecewise const) M (t) changes continuously Global coupling through M, no nearest neighbor preference

13 Lifshitz-Slyozov-Wagner (LSW) continuous mean field model Evolution of the distribution of drop sizes: φ(m, t) N(t) = φ(m, t) dm M total = Conservation law for droplets for m > Flux from DMF model φ t + m [ u(m)φ ] = u(m) = dm = 2 L ( 1 m 1 m ) mφ(m, t) dm Mean-field averages 1 m = 1 N A nonlocal model... 1 m φ(m, t) dm Discrete collapse events are averaged out m, L, N all evolve continuously L = L total N

14 LSW model (continued) Stationary distribution self-similar solution for φ φ(m, t) = t 4/5 f(η) η = m t 2/5 N(t) = O(t 2/5 ) 1 (3.18) f.5 (3.17) zmax z Issues about uniqueness/convergence [Niethammer and Pego] Generally good agreement with CDS simulations for t asymptotics But, too long for comparison with real experiments (polymer films) Want to study transient coarsening behaviors: Are there different transients due to differences in IC φ (m) from the two dewetting instability mechanisms? Homogeneous (Spinodal) vs. Heterogeneous (Nucleation).7 t = t = 6.5 t = t = t = 52 t = UTF UTF h.7 t = 132 t = 163 t = 59 h.7 t = 95 t = 154 t = UTF UTF x x

15 Transient coarsening: Initial distributions φ (m) Homogeneous (Spinodal) vs. Heterogeneous (Nucleation) E(t).2-1 E(t) Frequency Frequency m/ m m/ m Simulated nucleated data: CDS simulation 1 5 (A) (B) 6 (A) (B) (C) (C) (D) (E) (F) 3 N 1 4 (G) φ/1 1.5 (D) (E) (F) 1 3 t step t m Staircasing behavior Concentrations in the mass distribution No collapses (plateau) while support of φ bounded away from m = An avalanche (cliff) when a concentration propagates to m =

16 Transient coarsening: Staircasing behavior Observed with CDS, DMF, and other studies of image processing models A sustainable transient: can recur, depends on population structure N t s N Model C CDS CDS, 25% rearrangement CDS, size-ordered t Sub- and Super-coarsening rates for plateaus/cliffs in staircasing is compatible with the Kohn-Otto bound on average coarsening rate 1 N 1 Ct 2/5 1 t N t N/2 t N/4 t N/8 t N/16 t