Hydrodynamic Slip Boundary Condition for the Moving Contact Line in collaboration with Xiao-Ping Wang (Mathematics Dept, HKUST) Ping Sheng (Physics Dept, HKUST)

No-Slip Boundary Condition ?

from Navier Boundary Condition to No-Slip Boundary Condition : slip length, from nano- to micrometer Practically, no slip in macroscopic flows : shear rate at solid surface

No-Slip Boundary Condition ? Apparent Violation seen from the moving/slipping contact line Infinite Energy Dissipation (unphysical singularity) Are you able to drink coffee?

Previous Ad-hoc models: No-slip B.C. breaks down Nature of the true B.C. ? (microscopic slipping mechanism) If slip occurs within a length scale S in the vicinity of the contact line, then what is the magnitude of S ?

Molecular Dynamics Simulations initial state: positions and velocities interaction potentials: accelerations time integration: microscopic trajectories equilibration (if necessary) measurement: to extract various continuum, hydrodynamic properties CONTINUUM DEDUCTIONCONTINUUM DEDUCTION

Molecular dynamics simulations for two-phase Couette flow Fluid-fluid molecular interactions Wall-fluid molecular interactions Densities (liquid) Solid wall structure (fcc) Temperature System size Speed of the moving walls

Modified Lennard-Jones Potentials for like molecules for molecules of different species for wetting property of the fluid

fluid-1 fluid-2fluid-1 dynamic configuration static configurations symmetricasymmetric f-1f-2f-1 f-2f-1

tangential momentum transport boundary layer

The Generalized Navier B. C. when the BL thickness shrinks down to 0 viscous partnon-viscous part Origin?

nonviscous part viscous part uncompensated Young stress

Uncompensated Young Stress missed in Navier B. C. Net force due to hydrodynamic deviation from static force balance (Young’s equation ) NBC NOT capable of describing the motion of contact line Away from the CL, the GNBC implies NBC for single phase flows.

Continuum Hydrodynamic Modeling Components: Cahn-Hilliard free energy functional retains the integrity of the interface (Ginzburg-Landau type) Convection-diffusion equation (conserved order parameter) Navier - Stokes equation (momentum transport) Generalized Navier Boudary Condition

Diffuse Fluid-Fluid Interface Cahn-Hilliard free energy (1958)

is the chemical potential. capillary force density

= tangential viscous stress + uncompensated Young stress Young’s equation recovered in the static case by integration along x

in equilibrium, together with for boundary relaxation dynamics first-order generalization from

Comparison of MD and Continuum Hydrodynamics Results Most parameters determined from MD directly M and optimized in fitting the MD results for one configuration All subsequent comparisons are without adjustable parameters.

molecular positions projected onto the xz plane

Symmetric Coutte V=0.25 H=13.6 near-total slip at moving CL no slip

symmetric Coutte V=0.25 H=13.6 asymmetric Coutte V=0.20 H=13.6 profiles at different z levels

symmetric Coutte V=0.25 H=10.2 symmetric Coutte V=0.275 H=13.6

asymmetric Poiseuille g ext =0.05 H=13.6

The boundary conditions and the parameter values are both local properties, applicable to flows with different macroscopic/external conditions (wall speed, system size, flow type).

Summary: A need of the correct B.C. for moving CL. MD simulations for the deduction of BC. Local, continuum hydrodynamics formulated from Cahn-Hilliard free energy, GNBC, plus general considerations. “Material constants” determined (measured) from MD. Comparisons between MD and continuum results show the validity of GNBC.

Large-Scale Simulations MD simulations are limited by size and velocity. Continuum hydrodynamic calculations can be performed with adaptive mesh (multi-scale computation by Xiao-Ping Wang). Moving contact-line hydrodynamics is multi- scale (interfacial thickness, slip length, and external confinement length scale).