Dewetting patterns Len Pismen Technion Haifa Israel Supported

Dewetting patterns Len Pismen Technion, Haifa, Israel Supported by Israel Science Foundation and Minerva Center for Nonlinear Physics of Complex Systems 3/10/2021 1

1. Perspective 2. Van der Waals droplets: interaction mediated by precursor 3. Vd. W/polar pancakes: fronts and instabilities 4. Non-equilibrium systems: Marangoni motion 3/10/2021 2

Dewetting sequence film instability hole rim instability coarsening 3/10/2021 contact line A micrograph of a film thickness pattern in an evaporating film (Lipson & Leizerson, Technion) 3

Contact line: kinematic view Caterpillar advance (angular shape… that’s a problem) no slip retreat route for a receding fluid no slip multivalued velocity field: stress singularity 3/10/2021 4

Contact line: physico-chemical perspective thermodynamic balance: determines the shape Diffuse interface variable contact angle Stokes equation + intermolecular forces precursor (nm layer) Kinetic slip in 1 st molecular layer breakdown of continuous description? 3/10/2021 interaction with substrate disjoining potential 5

Film evolution – lubrication approximation involves expansion in scale ratio…. Mass conservation: Generalized Cahn–Hilliard equation k(h) – mobility coefficient defined by hydrodynamic model and b. c. Driving potential: surface tension L. M. Pismen and Y. Pomeau, PRE 62 2480 (2000) L. M. Pismen, PRE 64 021603 (2001) 3/10/2021 disjoining potential external forces defined by molecular interaction model 6

Disjoining potential (computed by integrating interaction with substrate across the film) 3/10/2021 7

Mobility coefficient (computed by integrating the Stokes equation across the film) 3/10/2021 8

Examples of disjoining potentials partial wetting precursor thickness complete wetting thin layer Maxwell line thick layer front s-l-g thick layer s-g thin layer 3/10/2021 9

Evolving patterns M. Bestehorn et al (numerics): droplets + precursor; coarsening Lipson et al: evaporating layer pancakes + precursor coexistence 3/10/2021 10

3/10/2021 Courtesy Hans Riegler, MPIKG Golm 11

1. Perspective 2. Van der Waals droplets: interaction mediated by precursor 3. Vd. W/polar pancakes: fronts and instabilities 4. Non-equilibrium systems: Marangoni motion 3/10/2021 12

Coarsening without evaporation – exchange through precursor L. M. Pismen and Y. Pomeau, Po. F 16 2604 (2004) smaller drop in equilibrium with thicker precursor larger drop in equilibrium with thinner precursor flux smaller droplet is sucked in by the big one 3/10/2021 larger droplet is repelled in by the small one 13

Multiscale system region angle scale length scale precursor q hm contact line q hm bulk q a precursor Quasistationary equation length scales differ by many orders of magnitude! bulk a precursor rescaled capillary number 3/10/2021 14

Integral relations Multiply by h–hm; integrate over a large region centered on the droplet. • Dissipation integral • Surface tension • Disjoining potential – =0 ‘ driving force • Speed of motion C = I – 1 3/10/2021 15

Dissipation Integral (2 D) • contact line region: use here static contact line solution • droplet bulk: use spherical cap solution • add up: 3/10/2021 NB: logarithmic factor bulk and contact line contributions cannot be separated in a unique way 16

Dissipation Integral (3 D) • contact line region: multiply local contribution by cos and integrate ( is the angle between local radius and direction of motion) • droplet bulk (spherical cap) • add up: 3/10/2021 NB: logarithmic factor bulk and contact line contributions cannot be separated in a unique way 17

Mass transport in precursor film • negligible curvature • almost constant thickness • quasistationary motion Spherical cap in equilibrium with precursor: film thickness distribution created by well separated droplets: 3/10/2021 18

Migration on precursor layer driving force on a droplet due to local thickness gradient droplet velocity: flux 3/10/2021 19

Ripening through mass exchange flux out of kth droplet radius of a critical droplet evolution equation for droplet radii: 10 droplets 2 droplets 3/10/2021 20

Asymptotics of radii distribution equation for distribution function n diffusional growth (LS) 1 precursor interaction 3 similarity ansatz critical radius general theory: Giron, Meerson & Sasorov, PRE 58 4213 (1998) 3/10/2021 21

Migration & coalescence 3/10/2021 22

Migration & ripening 3/10/2021 23

Experiment? it does not move Lipson et al: interaction suppressed by evaporation (imposes interaction decay length) M. Bestehorn et al (numerics): scale ratio a/d is not sufficiently large 3/10/2021 24

Experiment? it moves Liquid substrate Binary fluid - driven by Marangoni flow 3/10/2021 25

Summary 1 Droplets: the droplet velocities can be computed with the help of integral relations, with no need for computing the shape distortion when motion is slow the coarsening of droplet size distribution proceeds by mass transport through the precursor; this causes an anomalous asymptotic scaling the motion of droplets is driven by the gradient of capillary pressure along their perimeters the droplets motion is suppressed by evaporation 3/10/2021 26

1. Perspective 2. Van der Waals droplets: interaction mediated by precursor 3. Vd. W/polar pancakes: fronts and instabilities 4. Non-equilibrium systems: Marangoni motion 3/10/2021 27

Bistable disjoining potential (Sharma model) Maxwell construction hs– – thin film 3/10/2021 hs+ – thick film 28

Alternative thickness states hs+ – thick film hs – thin film – hs– – thin film 3/10/2021 29

Ridge formation during evaporation ridge evaporation front Maxwell construction evaporation: lower m, thinner film L. M. Pismen, PRE 70 021101 (2004) 3/10/2021 30

Scaling for volatile film Inner (short) outer (long) length scale time scale equation scale ratio 3/10/2021 31

Multiscale expansion scheme front evaporation flux? Outer region 3/10/2021 Inner region evaporation front motion Outer region 32

Inner region (evaporation negligible) (1) propagation speed: e c (2) curvature: ek expand: 1 st order: (1) (2) integrate (1); match to far field outer flux material balance: 3/10/2021 33

Inner region: potential (1) integrate twice 2 constants: j 1– flux through the interface m 1– use solv. cond. of (2) connection between j 1 and c (2) multiply by 3/10/2021 and integrate: 34

Matching conditions potential in the inner region: outer asymptotics: connects inner limits of outer potential and its gradient on both sides use together with formulae for propagation speed and fluxes 3/10/2021 35

Matching conditions: explicit form material balance constants: thickness–potential relation 3/10/2021 36

Outer solution outer equation: linearized thickness–potential relation: stationary outer equation: solve the outer equations on both sides to find relations between inner limits of outer potential and its gradient – close the matching conditions 3/10/2021 Problems to solve: • Straight-line front – quasistationary • Pancake – quasistationary • Straight-line front – comoving frame 37

Front propagation speed Front speed – quasistationary approximation 3/10/2021 38

Larger drops evaporate faster “pancake” r<a r>a 3/10/2021 39

Front speed – comoving frame computation outer equations in comoving frame outer solution curvature-dependent equations in comoving frame front unstable if speed increases with curvature 3/10/2021 40

Front propagation speed Front speed – comoving frame computation (nonlinear dependence on Dm) 3/10/2021 41

Stability of the front stable unstable 3/10/2021 42

Stability of the front 3/10/2021 43

Summary 2 Pancakes: anomalous dependence of the propagation speed on the curvature of the boundary between thin and ``thick'' domains depresses coarsening of the droplet size distribution the zigzag instability bears more similarity to Mullins-Sekerka rather than Rayleigh instability and is not linked to the formation of a ridge 3/10/2021 44

1. Perspective 2. Van der Waals droplets: interaction mediated by precursor 3. Vd. W/polar pancakes: fronts and instabilities 4. Non-equilibrium systems: Marangoni motion 3/10/2021 45

Non-eqilibrium setting convective cooling convective heating cool dewetting warm substrate cold substrate radiative heating film equations 3/10/2021 spreading warm spreading adiabatic substrate 46

Marangoni flow convective heat/cool stationary potential vs. height H –W increased cooling H max height 3/10/2021 47

Marangoni pancakes max height H const contact angle height at V=const |M| 3/10/2021 48

Long-Wavelength Instability in Surface. Tension-Driven Bénard Convection S. J. Van. Hook, M. F. Schatz, W. D. Mc. Cormick, J. B. Swift, and H. L. Swinney Phys. Rev. Lett. 75, 4397– 4400 (1995) 3/10/2021 49

Marangoni instability with surface deformation Golovin, Nepomnyashchy, Pismen GNP-94, Phys. Fluids, 6 34 -48 (1994). (1 +1)d Amplitude equation Surface deformation GNP-97, J. Fluid. Mech. 341 317 -341 (1997). (2 +1) d 3/10/2021 50

Amplitude modulation of short-scale hexagonal pattern Surface depressions evolve into dry spots; elsewhere, hexagonal convection cells remain almost undeformed. GNP-02, Int. J. Bif. Chaos 12 2487 (2002) 3/10/2021 51

Surface elevations localized at penta-hepta defects surface relief Re(A 1) 3/10/2021 pattern |A 1| 52

Courtesy Hans Riegler, MPIKG Golm experiment: P. Lazar, H. Schollmeyer, and H. Riegler, PRL 94, 116101 (2005) theory: A. Yochelis & L. M. P, PRE 72 025301 (2005), still in 1 D, continued with E. Knobloch 3/10/2021 53

Conclusions Patterns: evolution to equilibrium • film/rim instabilities • coarsening non-equilibrium system • film/rim instabilities • suppressed coarsening (evaporation or flow in precursor) • motion, coalescence 3/10/2021 • perpetual motion 54