Diffuse interface theory

1. Diffuse interface theory mesoscopic perspective

3. Equilibrium density functional theory Free energy of inhomogeneous fluid e.g. hard-core potential: Euler – Lagrange equation: chemical potential static equation of state

4. Equilibrium density profile 1D static density functional equation Q(z) = b =Ald –6/T Density profile for a van der Waals fluid Gibbs surface (defines the nominal thickness) surface tension

5. Disjoining potential Interaction energy with solid wall Energy per unit area homogeneous fluid–solid interaction kernel fluid–fluid (distortion) Use profile r0 (z–h); compute disjoining potential ms=dF/dh sharp interface limit: two-term expansion

6. Disjoining potential vs. layer thickness at different values of the dimensionless Hamaker constant for a weakly non-wetting fluid (nonlocal theory) thickness of the precursor film

7. Curved interface interfacial energy metric factor surface tension change of chem. pot. variation variation displace the interface along the normal Gibbs-Thomson law curvature

8. wave-vector-dependent surface energy Fradin C,Luzet D,Smilgies D,Braslau A,Alba M,Boudet N,Mecke K and Daillant J 2000 Nature 403 871

9. Equilibrium contact angle Young-Laplace formula (neglect vapor density,presume the solid surface in contact with vapor to be dry) gsl – ggl= g cos q But: equilibrium solid surface is covered by a dense fluid layer even when it is weakly nonwetting. Equation for nominal interface position replace integrate: the dependence on c is modified,since the thickness of the precursor layer is also c -dependent

10. Interaction of interfaces liquid Change of surface tension vapor h liquid Shift of chemical potential: equilibrium …valid at h>>d …significant at h~d shifted equilibrium

11. 1d solutions of the nonlocal equations m h/d g h/d

12. Local (VdW–GL) theory assume that density is changing slowly; expand retain the lowest order distortion term; Euler – Lagrange equation: NB: divergence in the next order with long-range (VdW) interactions

13. gsl – ggl= g cos q

14. Density profiles for a weakly non-wetting fluid Expand in a First order Combined solution Stationary density profiles. phase plane trajectories dashed line: without substrate

15. Disjoining potential Use a solvability condition of the equation for a perturbed profile when the substrate weakly perturbs translational symmetry Inhomogeneity= shift of chemical potential Standard solvability condition: solvability condition with boundary terms Compute chemical potential as a function of nominal thickness

16. Evaporation / condensation Cahn – Hilliard equation • Inner equation for chemical potential: Inner chemical potential: c = interface propagation speed • Material balance across the layer: • Dynamic shift of chemical potential:

17. Dynamic diffuse interface theory coupling to hydrodynamics –through the capillary tensor modified Stokes equation: S – stress tensor explicit form: continuity equation: compressible flow driven by the gradient of chemical potential

18. Computing dynamic diffuse interface Contact-line dynamics of a diffuse fluid interface, D. Jacqmin, J. Fluid Mech. 402 57 (2000). NB: a molecular-scale volume! Steady flow pattern at a diffuse-interface contact line. The upper fluid has 10-2 density and 10-3 viscosity of the lower fluid.

19. Multiscale perturbation scheme • 1D find an equilibrium density profile: liquid at z –  at z – • 1D add interaction with solid substrate; compute the disjoining potential • 2D/3D include weak surface inclination and curvature • 2D/3D include weak gravity • driving potential: • 2D/3D use separation between the vertical and horizontal scales to obtain the evolution equation for nominal thickness …to be solved as before

21. Computation of the mobility coefficient solve the horizontal component of Stokes equation: find the horizontal velocity u(z)=Y(z;h)W The function u(z)=Y(z;h) depends on theviscosity/density relation k(h) integrate the continuity equation to obtain evolution equation with diffuse interface k(h)= sharp interface k=h3/3 h

22. Local vs. nonlocal theory • Different original equations; faulted reduction • Different asymptotic tails • Common perturbation scheme • Common structure of “lubrication” equations • Different expressions for mobility coefficient • Different expressions for disjoining potential local nonlocal • Precursor layer also in a weakly non-wetting fluid; • computable static contact angle

23. Dewetting pattern M.Bestehorn & K.Neuffer, Phys. Rev. Lett. 87 046101 (2001)

24. Flow on an inclined plane M.Bestehorn & K.Neuffer, Phys. Rev. Lett. 87 046101 (2001)

25. Relaxational equations for order parameter Further directions: relaxational theory modified continuity equation? (CH equation in Galilean frame) suitable for description of non-equilibrium interfaces: interfacial relaxation and interphase transport

26. Challenges • Experimental:controlled experiments verifying dynamics on nanoscale distances • Theoretical: realistic dynamic description at nanoscale and mesoscopic distances • Theoretical:matching of molecular and continuum description • Computational: multiscale computations extending to macroscopic distances • nonlocal (density functional) theory • relaxational (TDDF) theory • hybrid (continuum - MD) computations Further directions:

27. Publications • L.M.Pismen, B.Y.Rubinstein, and I.Bazhlekov, Spreading of a wetting film under the action of van der Waals forces, Phys. Fluids, 12 480 (2000). • L.M.Pismen and Y.Pomeau, Disjoining potential and spreading of thin liquid layers in the diffuse interface model coupled to hydrodynamics, Phys. Rev. E 62 2480 (2000). • A.A.Golovin, B.Y.Rubinstein, and L.M.Pismen, Effect of van der Waals interactions on fingering instability of thermally driven thin wetting films, Langmuir, 17 3930 (2001). • L.M.Pismen and B.Y.Rubinstein, Kinetic slip condition, van der Waals forces, and dynamic contact angle, Langmuir, 17 5265 (2001). • L.M.Pismen, Nonlocal diffuse interface theory of thin films and moving contact line, Phys. Rev. E 64 021603 (2001). • A.V.Lyushnin, A.A.Golovin, and L.M.Pismen, Fingering instability of thin evaporating liquid films, Phys. Rev. E 65 021602 (2002).