Some Aspects of Drops Impacting on Solid Surfaces J.E Sprittles Y.D. Shikhmurzaev EFMC7 Manchester 2008
Motivation • Drop impact and spreading occurs in many industrial processes. • 100 million inkjet printers sold yearly. • Recent experiments show standard models of drop impact and spreading will be inadequate for microdrops. Photos courtesy of Romain Rioboo
The Moving Contact Line Problem U The classical recipe of: Bulk Incompressible Navier Stokes Solid No-Slip Free Surface Stress balance with capillary pressure. Fails to provide a solution for a drop spreading steadily over a solid substrate.
Flow Of Liquids Over Solids Two Issues: • Allow For A Solution • Describe The Angle Between The Free Surface and the Solid (The Contact Angle). Standard Solution: • Allow Slip Between Solid and Liquid • Let U
Standard Modelling of Axisymmetric Drop Impact and Spreading • Bulk: Incompressible Navier-Stokes • Boundary: Classical conditions apart from no-slip being replaced by Navier Slip: • Contact Angle Related to contact line speed using empirical relation (Jiang et al 79)
Finite Element Modelling:The Spine Method (Scriven and co-workers) Nodes define free surface. The Spine Nodes fixed on solid.
Does The Standard Model Work?Pyramidal Drops (Millimetre Size) Experiment Renardy et al. Simulation Sprittles.
Qualitatively OKQuantitatively Not Standard Recipe’s Problems: • Incorrect Kinematics • Logarithmic Pressure Singularity at the Contact Line. • But perhaps worst of all…. Prediction of Standard Model Experimentally U U, m/s
Experiments answer, NO! “There is no general correlation of the dynamic contact angle as a function of surface characteristics, droplet fluid and diameter and impact velocity.” (Sikalo et al 02) Can one describe the contact angle as a function of the parameters? “There is no universal expression to relate contact angle with contact line speed”. (Bayer and Megaridis 06) U, m/s
As in Curtain Coating (Used To Industrially Coat Materials) U U, cm/s Standard models: Fixed Substrate Speed => Unique Contact Angle Dynamic contact angle as a function of coating speed for different flow rates (Blake & Shikhmurzaev 02).
Liquid Drops Spreading on Solids:Process of Interface Formation Interfaces are shown with finite thickness for representation only. In Frame Moving With Drop Liquid Gas Solid
The Interface Formation Model’s Predictions Unlike conventional models: • The contact angle is determined by the flow field. • No stagnation region at the contact line. • No infinite pressure at the contact line => Numerics easier
Simplest Model of Interface Formation f (r, t )=0 e1 n n θd e2 • Generalisation of standard/classical model In the bulk: On free surfaces: On liquid-solid interfaces: At contact lines:
Qualitative Results:IFM In Frame Moving With Contact Line Liquid Gas Solid Speed U
Increasing impact speed Changes flow field Qualitative Results:IFM In Frame Moving With Contact Line Liquid Gas Solid Solid Speed U
Mock 05 et al - Drop Impact onto Chemically Patterned Surfaces • Pattern a surface to ‘correct’ deposition. Courtesy of Professor Roisman
Flow over a transition between solids of differing wettabilities. Standard model predicts no effect. Chemically Patterned Surfaces What happens in this region? Shear flow in the far field Solid 2 Solid 1
Molecular Dynamics Simulations More wettable Compressed Less wettable Rarefied Courtesy of Professor N.V. Priezjev
Results - Streamlines Solid 2 less wettable Qualitative agreement Sprittles & Shikhmurzaev, Phys. Rev. E 76, 021602 (2007). Sprittles & Shikhmurzaev, EPJ (2008), In Print
Drop Impact on a Hydrophobic (non-wettable) Substrate • Does The Standard Model Work? Rebound on a Hydrophobic Substrate Re=100, We=10, β = 100, .
Does The Standard Model Work? Impact of a Microdrop Radius = 25 mm, Impact Speed = 12.2 m/s Re=345, We=51, β = 100, . Experiment Dong 06. Simulation Sprittles.
Shikhmurzaev Model • Solid-liquid and liquid-gas interfaces have an asymmetry of forces acting on them. • In the continuum approximation the dynamics of the interfacial layer should be applied at a surface. • Surface properties survive even when the interface's thickness is considered negligible. Surface tension Surface density Surface velocity
Shikhmurzaev ModelWhat is it? • Generalisation of the classical boundary conditions. • Considers the interface as a thermodynamic system with mass, momentum and energy exchange with the bulk. • Used to relieve paradoxes in modelling of capillary flow.