Stability of finite and infinite fluid films and rivulets

1. Javier Diez Instituto de Fisica Arroyo Seco Universidad Nacional del Centro de la Provincia de Buenos Aires Tandil, Argentina Lou Kondic Department of Mathematical Sciences Center for Applied Mathematics and Statistics New Jersey Institute of Technology Stability of finite and infinite fluid films and rivulets

2. Motivation • Numerous applications involving (often unstable) nano-, micro- and macro- films and rivulets • Coatings, printers’ instability, lab-on-a-chip, electro-chemically driven flows,… • Typically, infinite fluid domains are considered; however, the nature usually deals with finite objects… • Obvious questions: • Are finite effects important? • If they are, how to understand them? • In `real’ problems, which instability mechanism is relevant?

3. Overview • Some experimental results • instabilities • Modeling partial wetting • Linear stability analysis • Numerics and why (quasi) 2D is not always enough • What is to be done: work in progress

4. Experiment: Breakup of a rivulet • Fluid rivulet beads up into drops Fluid: PDMS; viscosity: 20 St; jet diameter: 0.046 cm; Solid: glass coated by FC-725 or EGC-1700

5. Main features of instability • Instability develops on a long time scale (minutes, or longer) • Breakup propagates from the ends of the fluid rivulet • Capillary instability is not observed • Different instability mechanism from infinite rivulet case (Davis, JFM ’80, Langbein, JFM ’90, Roy and Schwartz JFM ’99, Yang and Homsy, PoF ‘07) • Number of drops and the time scale depend on fluid and solid properties (viscosity, contact angle, fluid volume) more details: Gonzalez etal, Europhys. Lett. ‘07

6. Modeling assumptions • Inertial effects can be ignored • Capillary number is small • Wetting effects can be modeled using disjoining pressure model • Lubrication approximation is appropriate (even for large contact angles) • Main part of the talk: ignore transverse curvature of a rivulet and consider a semi-infinite film instead

7. Disjoining pressure model Modify Laplace Young condition at the liquid/air interface To include solid-liquid interaction (Frumkin-Deryaguin model) * * use: Fourth order nonlinear PDE for the fluid height:

8. Thin film equation with disjoining pressure model capillarity disjoining pressure * Scales: h: fluid thickness : fluid density : fluid viscosity : surface tension g : gravity

9. Linear stability analysis of 1D equation • Consider a (semi)infinite film: ignore end effects and transverse curvature • Do: Linear stability predicts existence of stable and unstable regions in considered parameter space There is more… Thiele etal PRE ’01, Colloids and Surfaces ’02; Diez & Kondic, PoF ‘07

10. Absolute Stability and Metastability Energy-based analysis shows different response to finite size perturbations Even films that are linearly stable may be unstable with respect to finite size perturbations Important: emerging distance between the drops is different for the two instability mechanisms Thiele etal PRE ’01, Colloids and Surfaces ’02; Diez & Kondic, PoF ‘07 Nucleation dominated Surface dominated Relevance: stability of nanometric films; spinodal and nucleation type of instability

11. Connection of finite and infinite cases • Recall that experiment considers fluid configuration of finite length • What is the connection of the stability properties of a finite and infinite film? • Compute both and compare • Methods: fully nonlinear time-dependent simulations: Diez & Kondic, JCP ‘02

12. 2D (x,z) simulations of 'infinite' film • Consider fluid in domain of length L with periodic boundary conditions perturbed by small harmonic perturbations • Emerging distances between drops center around LSA result

13. 2D (x,z) simulations of a finite film

14. 2D (x,z) simulations of a finite film • Consider fluid in domain of length L (no perturbations) • Very different distance between drops result! • Finite length of the film modifies significantly the instability mechanism • Finite length of a film leads to the distances between drops that are similar to those resulting from an infinite film exposed to finite size perturbations

15. 2D (x,z) simulations of a finite film with surface perturbations yes perturbations no perturbations

16. 2D (x,z) simulations of a finite film in metastable regime 1 drop or 2 drops?

17. Connection of finite and infinite film instabilities • Finite size effects act typically as a finite-amplitude perturbation of an infinite film • The outcome is that resulting distance between the drops may be significantly different for the two cases • Addition of surface perturbations of a finite film may lead to change of regime from nucleation-dominated to surface-dominated, if the perturbations have time to grow: emerging lengthscales depend on the film size (Diez & Kondic, PoF ’07)

18. Current work • Good news • Similar geometries may help us understand rivulet instability using films results • Bad news • Stability analysis of the films predicts stability for the film thicknesses comparable to the rivulet ones • Bring back the transverse curvature and consider a `real’ fluid rivulet • Preliminary computational results show promising agreement with the experiments • To be done: • understand the connection between infinite and finite case, similarly as done for films • Predict instability mechanisms on micro- and nano- scale

19. 3D Simulations and Experiment: Fluid volume dependence Large fluid volume: Small fluid volume:

20. Connection between stability of films and rivulets • Good news • Similar geometries may help us understand rivulet instability using films results • Bad news • Stability analysis of the films predicts stability for the film thicknesses comparable to the rivulet ones • Bring back the transverse curvature and consider a `real’ fluid rivulet • Computational methods: fully implicit finite-difference based time-dependent simulations: Diez & Kondic, JCP ‘02 • Consider parameters similar to the experimental ones • Compare quantitative features of simulations and experiments, and the importance of transverse curvature

21. 3D Simulations: basic features • Simulate only ¼ of the domain • Formation of drops • Instability propagates from the end(s)

22. 3D Simulations: parametric dependence • Simulate thick strip • Drops still form • Distance between drops increased • Smaller contact angle may lead to stability

23. Comparison of 2D and 3D simulations of finite films and rivulets with experiment • Both 2D and 3D simulations show instability propagating from the fluid ends (`end-pinching') • 3D simulations are in good (quantitative!) agreement with experiment regarding number of drops produced • 3D simulations show different range of instability compared to 2D • Dimensionality of the problem important! - transverse curvature present in the 3D simulations modifies the stability properties

24. To do list • Consider in more detail configurations that may lead to an interplay between end-pinching and capillary instability • Extend the analysis carried out for semi-infinite films to rivulets • Relate observed instability to other systems where propagating instabilities are observed (fluid drops: Stone & Leal JFM ’86, ’89; printer’s instability, etc etc)