diffuse interface theory

Diffuse interface theory

mesoscopic perspective

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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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Equilibrium density profile1D static density functional equation

Density profile for a

van der Waals fluid Gibbs surface(defines the nominal thickness)

Q(z) =

surface tension

=Ald –6 /T

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Disjoining potential

Interaction energy with solid wall

Energy per unit area

homogeneous fluid–solid

fluid–fluid (distortion) interaction kernel

Use profile 0 (z–h); compute disjoining potential s=dF/dh

sharp interface limit:two-term expansion

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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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Curved interface

Gibbs-Thomson law

variation

interfacial energymetric factor

surface tension

change of chem. pot.

variation

curvature

displace the interface along the normal

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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

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Equilibrium contact angle

Young-Laplace formula (neglect vapor density,presume the solid

surface in contact with vapor to be dry) sl – gl= cos

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 is modified,since the thickness of the precursor layer is also -dependent

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Interaction of interfaces

Change of surface tension

Shift of chemical potential:

equilibrium

shifted equilibrium

h

liquid

liquidvapor

…valid at h>>d…significant at h~d

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1d solutions of the nonlocal equations

h/d

h/d

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Local (VdW–GL) theory

assume that density is changing slowly; expand

Euler – Lagrange equation:

retain the lowest order distortion term;

€

∇ 2ρ−g(ρ)+μ=0

NB: divergence in the next order with long-range (VdW) interactions

nonlocal local

Power tail Exp. tail

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sl – gl= cos

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Density profiles for a weakly non-wetting fluid

Stationary density profiles.

dashed line: without substrate

Expand in a

First order

Combined solution

phase plane trajectories

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Disjoining potentialUse a solvability condition of the equation for a perturbed profile when the substrate weakly perturbs translational symmetry

Compute chemical potential as a function of nominal thickness

Standard solvability condition:

solvability condition with boundary terms

Inhomogeneity=shift of chemicalpotential

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Evaporation / condensation

• Inner equation for chemical potential:

• Material balance across the layer:

Inner chemical potential:

• Dynamic shift of chemical potential:

c = interface propagation speed

Cahn – Hilliard equation

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Dynamic diffuse interface theory

coupling to hydrodynamics –through the capillary tensor

modified Stokes equation:

explicit form:

€

∇ ⋅(T+S) =0

continuity equation:

compressible flow driven by the gradient of chemical potential

S – stress tensor

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Computing dynamic diffuse interface

Steady flow pattern at a diffuse-interface contact line. The upper fluid has 10-2 density and 10-3 viscosity of the lower fluid.

Contact-line dynamics of a diffuse fluid interface,D. Jacqmin, J. Fluid Mech. 402 57 (2000).

NB: a molecular-scale volume!

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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 curvature2D/3D include weak gravity

• driving potential:

2D/3D use separation between the vertical and horizontal scales to obtain the evolution equation for nominal thickness

€

W = −γ∇2h + μs(h) + V(h)

€

∂h∂t=η−1∇⋅[k(h)∇W]

…to be solved as before

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Computation of the mobility coefficient

solve the horizontal component of Stokes equation:find the horizontal velocity u(z)=(z;h)W

integrate the continuity equation to obtain evolution equation with

k(h)=

k(h)

h

sharp interface k=h3/3

diffuse interface

The function u(z)=(z;h) depends on the viscosity/density relation

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Local vs. nonlocal theoryDifferent original equations; faulted reductionDifferent asymptotic tailsCommon perturbation scheme Common structure of “lubrication” equationsDifferent expressions for mobility coefficient Different expressions for disjoining potential

local nonlocal• Precursor layer also in a weakly non-wetting fluid;• computable static contact angle

€

μs(h)=− A6πh3

−Bh6

€

μs(h)=e−h(a−e−h)

€

θ= 2 μs(h)h0

∞∫ dh

singularityis relaxed

Contact angleis indefinite

matches wellto macroscopics

non-wetting caseis tractable

diffuse interfacetheory

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Dewetting pattern

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

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Flow on an inclined plane

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

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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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ChallengesExperimental: controlled experiments verifying dynamics on nanoscale distancesTheoretical: realistic dynamic description at nanoscale and mesoscopic distancesTheoretical: matching of molecular and continuum descriptionComputational: multiscale computations extending to macroscopic distances

Further directions:

•nonlocal (density functional) theory •relaxational (TDDF) theory•hybrid (continuum - MD) computations

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PublicationsL.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).