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Diffuse interface theory. mesoscopic perspective. Equilibrium density functional theory. Free energy of inhomogeneous fluid. e.g. hard-core potential:. Euler – Lagrange equation:. chemical potential. static equation of state. Equilibrium density profile. - PowerPoint PPT Presentation
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Diffuse interface theory
mesoscopic perspective
04/19/23 2
04/19/23 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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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).