1. Diffuse interface theory
mesoscopic perspective

2. 11/23/2018

3. Equilibrium density functional theory
Free energy of inhomogeneous fluid
e.g. hard-core potential:
Euler – Lagrange equation:
chemical potential
static equation of state
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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
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5. Disjoining potential Interaction energy with solid wall
Energy per unit area
homogeneous fluid–solid
fluid–fluid (distortion)
interaction kernel
Use profile r0 (z–h); compute disjoining potential ms=dF/dh
sharp interface limit:
two-term expansion
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6. Disjoining potential vs
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
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7. Curved interface interfacial energy surface tension
change of chem. pot.
metric factor
variation
variation
displace the interface along the normal
Gibbs-Thomson law
curvature
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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
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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
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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
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11. 1d solutions of the nonlocal equationsm
h/d g
h/d
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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
nonlocal
local
Power tail
Exp. tail
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13. gsl – ggl= g cos q
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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
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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
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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:
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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
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18. Computing dynamic diffuse interface
Contact-line dynamics of a diffuse fluid interface,
D. Jacqmin, J. Fluid Mech (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.
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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
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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 the viscosity/density relation
k(h)
integrate the continuity equation to obtain evolution equation with
diffuse interface
k(h)=
sharp interface
k=h3/3 h
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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
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23. Dewetting pattern
M.Bestehorn & K.Neuffer, Phys. Rev. Lett (2001)
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24. Flow on an inclined plane
M.Bestehorn & K.Neuffer, Phys. Rev. Lett (2001)
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25. Further directions: relaxational theory
Relaxational equations for order parameter
modified continuity equation?
(CH equation in Galilean frame)
suitable for description of non-equilibrium interfaces: interfacial relaxation and interphase transport
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26. Challenges Further directions:
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:
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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, (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 (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, (2001).
L.M.Pismen and B.Y.Rubinstein, Kinetic slip condition, van der Waals forces, and dynamic contact angle, Langmuir, (2001).
L.M.Pismen, Nonlocal diffuse interface theory of thin films and moving contact line, Phys. Rev. E (2001).
A.V.Lyushnin, A.A.Golovin, and L.M.Pismen, Fingering instability of thin evaporating liquid films, Phys. Rev. E (2002).
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