Force induced dispersion in heterogeneous materials David Dean - - PowerPoint PPT Presentation
Force induced dispersion in heterogeneous materials David Dean Laboratoire dOndes et Matire dAquitaine, Universit de Bordeaux and CNRS In collaboration with: Thomas Guerin , LOMA, Universit de Bordeaux Phy Physic ical al Rev
David Dean Laboratoire d’Ondes et Matière d’Aquitaine, Université de Bordeaux and CNRS In collaboration with: Thomas Guerin, LOMA, Université de Bordeaux Phy Physic ical al Rev eview iew Let Letters ers, 115, 020601 ( , 115, 020601 (2015) 2015)
enhanced dispersion
t→∞ D(t)
Fokker Planck equation on medium with variable isotropic diffusivity Corresponding Ito SDE Mean squared displacement Effective diffusion constant- important for reaction rates, mean first passage times ..
κ0
κ1
✏0 ✏1
Effective dielectric constant: Maxwell 1873, Rayleigh 1892, Maxwell-Garnett 1904, Bruggeman 1935 –old problem
Laplace’s equation in dielectric medium
Correspondance between effective diffusivity and effective dielectric constant Flat spatial average
Wiener variational bounds 1910 (improved bounds by Hashin and Shtrikman 1962) What you might naively expect as equilibrium density is uniform In one dimension
harmonic mean Duality result in two dimensions if
then De = exp(ln κ) geometric mean (Dykhne 1971, Keller 1960s) A part from these exact results there is a huge literature on approximative methods – effective medium, perturbation theory, renormalization, homogenization
External applied, force e.g. gravity, electric field Local Einstein relation mobility/conductivity
t→∞
t→∞
Effective drift Effective dispersion/ diffusion
5 10 15 20 0.8 1 1.2 F L Dii Dxx Dyy Vx /( F)
x / L y / L (x,y) 0.5 0.5 0.5 0.5 0.5 1 1.5
(a) (b)
x / L Ps(x,y) 0.5 0.5 0.5 0.5 0.6 1 1.4 1.8
(c) (d)
10
1
10
2
10 10
1
F L Dii Dxx Dyy
F
Non monotonic behavior
with F Huge increase in dispersion in direction
Steady state pdf in periodic cell
Find explicit expressions for dispersion coefficients for Fokker-Planck equations with arbritary periodic diffusivities and drifts Generalize and extend know results for diffusion with applied force plus periodic potentials in one dimensions (based on first passage time arguments Riemann et al 2000 and Lindner and Schimansky-Geier 2002) to higher dimensions. Recover results from homogenization theory for stationnary incompressible flows (Brenner 1980, Schraiman 1987 Majda and Kramer 1999)
General method from FP in 1d by Derrida 1983 – extension to higher d Dean et al 1996 Here start with SDE – also allows computation of finite time corrections
periodic
Ito SDE
Square and average –important to do it this way!
h(Xi(t) Xi(0))2i h2(Xi(t) Xi(0)) Z t dt0Ai(X(t0))i + h ✓Z t dt0Ai(X(t0)) ◆2 i = 2h Z t dt0κii(X(t0))i
Unit cell Diffusion in infinite periodic cell
Unit cell with Periodic boundaries
X(t) mod(Ω)
In steady state this is in equilibrium
Has PDF obeying FP equation ∂P
Ps(x) steady state distribution HPs(x) = 0 Jsi(x) = − ∂ ∂xj (κij(x)Ps(x)) + Ai(x)Ps(x)
Steady state current Effective drift Stratonovich 1953 Diffusion coefficient
H ˜ P 0(x, y, 0) = δ(x − y) − Ps(x)
Pseudo Green’s function of H
Vi = Z
Ω
dxJsi(x) = Z
Ω
dx Ps(x)Ai(x)
De
ii =
Z
Ω
dx κii(x)Ps(x) + Z
Ω
Z
Ω
dxdy ˜ P 0(x, y, 0)Ai(x)[Jsi(y) − ∂ ∂yj (κij(y)Ps(y))]
λ>0
Expansion in terms of left and right eigenfunctions
Ω
Ω
Ω
Hfi(x) = ✓ Ai(x) − Z
Ω
dyAi(y)Ps(y) ◆ Ps(x) − 2 ∂ ∂xj (κij(x)Ps(x))
Ω
dx fi(x) = 0
Recovers results from homogenization theory for incompressible flows
Define Gives Action of H on f Orthogonality from right/left eigenfunction expansion
Dii = Z
Ω
dx κii(x)Ps(x) + Z
Ω
dxgi(x)[Ai(x)Ps(x) − 2 ∂ ∂xj (κij(x)Ps(x))]
Ω
Ω
Ω
Useful to check numerical methods, compare results with f and g Define Action with adjoint of H Orthogonality condtion
ii + Cii
Ω
Generalizes DD and G. Oshanin (2014) (periodic potentials) and DD and T . Guerin 2014 (diffusivity) – cases with no current
10 20 30 40
t
1.05 1.1 1.15 1.2 1.25 1.3
D(t)
Leading finite time correction – next order decays as exp(−λ1t)/t
Great interest in generalization of Stoke’s Einstein Relation for driven
2000 and Lindner and Schimansky-Geier 2002.
Vi = Z
Ω
dx Ps(x)Ai(x)
i (x) + κij(x)βFj
∂Vi ∂Fi = β Z
Ω
dx Ps(x)κii(x) + Z
Ω
dx ∂Ps(x) ∂Fi Ai(x).
H ∂Ps ∂Fi + ∂ ∂xj (βκjiPs)
∂Ps(x) ∂Fi
Z
Ω
dx ∂Ps(x) ∂Fi
∂Vi β∂Fi = Z
Ω
dx Ps(x)κii(x) − Z
Ω
dx Z
Ω
dyAi(x) ˜ P 0(x, y; 0) ∂ ∂yj (κji(y)Ps(y)) . Perturbation of drift due to force and local Einstein relation differentiate wrt Fi
= 0
= 0
Differentiate steady state FP eq. wrt Fi Conservation of probability has periodic bcs Can compute ∂P ∂Fi with pseudo Green’s function P’
Ω
Ω
Stoke’s Einstein recovered when
∆i = 0
Violation in general when steady state has a non-zero current
Mean first passage time Variance of first passage time Riemann et al 2002
I+(x) = exp (Γ(x)) κ(x) Z 1
x
dx0 exp (−Γ(x0))
I(x) = exp (−Γ(x)) Z x
1
dx0 exp (Γ(x0)) κ(x0)
0 dx κ(x)I±(x)2I⌥(x)
0 dx I±(x)3
0 dxI±(x)
Effective drift Effective diffusion constant Effective potential – if periodic no current
A(x) = dk dx + κ(x)βF κ(x) = 1 κ−1 P
k ak exp( 2πkix L
)
D(F) = 1 κ−1 " 1 + 2β2F 2 X
k>0
|ak|2 β2F 2 + 4π2k2
L2
#
D(0) = κ−1−1
D(∞) = 1 κ−1 " 1 + 2 X
k>0
|ak|2 # = κ−2 κ−13
for this case Express inverse diffusivity as Fourier series Force dependent diffusion constant zero force Diffusion constant saturates at large force Stokes Einstein
Becomes dependent on spatial structure of diffusivity
x / L y / L
(x,y)
0.5 0.5 0.5 0.5 0.5 1 1.5
(a) (b) (c) F
10L 10L
F = 0
κ(x, y) = κ0[1 + 0.95 cos(2πx/L)]
Diffusion clouds from simulations
Steady state Ps is not flat for non-zero force
Dxx = κ−1−1 Dyy = κ
Dij = (κ−1)−1 ( δij + FiFj |F · ex|2 " κ−2 (κ−1)2 − 1 #) ,
At large F
∂tp(x, y, t) = ∂x[κ(x, y)∂xp − h κ(x, y)p] + ∂yκ(x, y)∂yp
with h = βFx
∂x [h κ(x, y) Ps(x, y) ] ≈ 0
At large h
) Ps(x, y) ' C(y)κ−1(x, y)
At large forces equilibriation in the direction x is rapid
Ps(x|y) = 1 κ(x, y) Lx κ−1(y) .
g(y) = L−1
x
Z Lx dx g(x, y)
Period in x direction
∂tπ(y, t) ' Z Lx dx ∂y{κ(x, y)∂y[π(y, t)Ps(x|y)]}
In two dimensions but for arbitrary diffusivity Quasi-static applroximation for x given y
∂tπ(y, t) = ∂2
y[κe(y)π(y, t)] − ∂y{[∂yln κ(y)]κe(y)π(y, t)}
κe(y) = 1/κ−1(y)
Dxx = [βFR(L)]2 W(L) Z L dy W(y) W(L) − R(y) R(L) 2 e−ln κ(y)
Order F2 contribution to diffusion coeff in direction
R(y) = Z y du eln κ(u); W(y) = Z y du κ−1
e (u)eln κ(u)
κ(x, y) = κ(x)
when this F2 term vanishes – saturation in 1d Generic quadratic enhancement
dimension.
dimensions.
General points - perspectives Media with varying mobility/diffusivity