Microscopic flow around a diffusing tagged
particle: effects of fluctuations on the
hydrodynamics description
R. Vuilleumier
Département de chimie
Ecole normale supérieure – Paris
Workshop on Molecular Dynamics – Warwick, June 5, 2009
Diffusion in a Lennard-Jones fluid
I
Friction from flow around the
moving particle
I
Relationship between diffusion
and viscosity
D=
10000 identical LJ particles
(T ∗ = 1.5, ρ∗ = 0.5)
kB T
cπηR
I
Flow boundary conditions?
I
Direct estimation of the flux?
Stokes’flow
A sphere with a (small) velocity v1 in an incompressible fluid of
viscosity η
I
velocity field
(slip boundary conditions – lab. frame):
f
1
1
~v (~r ) =
cos θ ~er −
sin θ ~eθ
4πη r
2r
-4
-2
0
I
drag force f :
2
f = 4πηRv1
I
-4
friction ξ = 4πηR
diffusion D =
kB T
ξ
4
=
kB T
4πηR
(Stokes-Einstein relation)
-2
0
2
4
Velocity field: linear response
Infinitesimal force
force f applied on particle 1 since t = −∞ along x
Averages of observables
Z
hAif
0
≈ hAi0 + βf
dthv1,x (t)Ai0
−∞
≈ hAi0 + f × β h∆r1,x Ai0
∆r1,x : displacement along x of particle 1 from t = −∞ to t = 0
A ≡ v1 : β h∆r1,x v1 i0 is the mobility
Lenard-Jones fluid
Above the critical point: T ∗ = 1.5, ρ∗ = 0.5 (σ = 1)
2
g(r)
1,5
1
0,5
0
0
2
4
6
r
8
10
12
10000 identical LJ spheres → L = 27.14 σ
Velocity field: diffusing particle frame
ρ~v (~r ) = βh∆r1,x (~vi − ~v1 ) δ(~r −~r1i )i
Use of symmetry ?
Velocity field: diffusing particle frame
ρ~v (~r ) = βh∆r1,x (~vi − ~v1 ) δ(~r −~r1i )i = f (r ) cos θ ·~er + g (r ) sin θ ·~eθ
Same symmetry as the Stokes flow around a sphere
Velocity field: diffusing particle frame
ρ~v (~r ) = βh∆r1,x (~vi − ~v1 ) δ(~r −~r1i )i = f (r ) cos θ ·~er + g (r ) sin θ ·~eθ
0
-0,2
vr(r)
vθ(r)
v
-0,4
-0,6
-0,8
-1
0
1
2
3
4
5
r
6
7
8
9
10
Velocity field: diffusing particle frame
ρ~v (~r ) = βh∆r1,x (~vi − ~v1 ) δ(~r −~r1i )i = f (r ) cos θ ·~er + g (r ) sin θ ·~eθ
-2
0
-1.5
-0,2
v
-1
vr(r)
vθ(r)
-0,4
-0.5
-0,6
0
0.5
-0,8
1
-1
1.5
0
1
2
3
4
5
r
6
7
8
9
10
2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Laboratory frame: backflow
-4
-4
-4
-2
-2
-2
0
0
0
2
2
2
4
4
-4
-2
0
t=1
2
4
4
-4
-2
0
t = 10
2
4
-4
-2
0
t = 100
2
4
Density map
δρ(~r ) = βh∆r1,x δ(~r − ~r1i )i = h(r ) cos θ
-3
0,25
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
-2
0,2
δρ
-1
0,15
0
0,1
1
0,05
2
0
0
2
4
r
6
8
10
3
-3
-2
-1
0
Force felt by the particle = applied force
1
2
3
Hydrodynamic flow – Boundary conditions
Fit of an hydrodynamic flow with slip boundary conditions
0
1
0,8
-0,2
vr(r)
vθ(r)
vr(r)
vθ(r)
-0,4
v
v
0,6
0,4
-0,6
0,2
-0,8
0
-1
0
2
4
6
r
8
Labo. frame
10
12
0
2
4
6
r
Part. frame
The normal velocity is non zero at contact!
8
10
12
Boundary conditions – normal velocity
Correlation between the particle velocity and the presence of a
particle at contact
I
velocity lower than the average tagged particle velocity
I
apparent normal velocity at contact
Hynes, J. T., Kapral, R. et Weinberg, M., JCP 70, 1456 (1979).
Dissipation through viscosity
Velocity field in the long range part
-0,9
-0,92
v
-0,94
vr(r)
vθ(r)
-0,96
-0,98
-1
-1,02
4
6
8
r
10
12
14
I
Momentum flux due to viscosity corresponds to only a force
f = 0.76 injected in the fluid
I
Is there a dissipation term other than viscosity?
Equation of motion for the velocity field
At long times (laboratory frame):
X
∇β hvi,α ·vi,β i1 (~r ) −
hfj→i,α i1 (~r ) − ∇β hvi,α ·v1,β i1 (~r ) = hf1→i,α i1 (~r )
j6=1
Equation of motion for the velocity field
At long times (laboratory frame):
X
∇β hvi,α ·vi,β i1 (~r ) −
hfj→i,α i1 (~r ) − ∇β hvi,α ·v1,β i1 (~r ) = hf1→i,α i1 (~r )
j6=1
Equation of motion for the velocity field
At long times (laboratory frame):
X
∇β hvi,α ·vi,β i1 (~r ) −
hfj→i,α i1 (~r ) − ∇β hvi,α ·v1,β i1 (~r ) = hf1→i,α i1 (~r )
j6=1
Equation of motion for the velocity field
At long times (laboratory frame):
X
∇β hvi,α ·vi,β i1 (~r ) −
hfj→i,α i1 (~r ) − ∇β hvi,α ·v1,β i1 (~r ) = hf1→i,α i1 (~r )
j6=1
0,05
0
2
4πr . <viv1>(r)
-0,05
-0,1
I
Momentum flux through a sphere
of radius r
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Contribution from the term
hvi,α · v1,β i1 (~r )
-0,15
-0,2
-0,25
0
5
r
10
15
Role of the central particle fluctuations
Asymptotic behavior of the velocity autocorrelation
function
1
c(t) ≈ a0 t
−3/2
with
η+ρD
η
0,1
a0 =
2kB T ρ1/2
: viscosity only
4πη 3/2
0,01
c(t)
I
0,001
I
a0 =
T ρ1/2
2kB
:
4π(η + ρD)3/2
including fluctuations
0,0001
1e-05
0,1
1
10
t
100
Evolution equations
The evolution equations are modified
ρ
d~v
dt
dδρ
dt
~ + η∇2~v + ρD∇2~v
= −∇P
~ v (~r )
= D∇2 δρ − ∇ρ~
~v =0
In regions where ρ ≈ cte., the solution still satisfies ∇~
Renormalisation of viscosity
Hydrodynamique avec f = 1 et η → η + ρD
-0,8
0,02
0,015
-0,85
vr(r)
vθ(r)
0,01
v
δρ
-0,9
0,005
-0,95
0
-1
0
2
4
6
r
8
10
12
D = D0 +
14
-0,005
0
kB T
1
4π(η + ρD) R
Bedeaux, D. et Mazur, P, Physica 73, 431 (1974).
2
4
6
r
8
10
12
14
Diffusion mechanism
Most mobile configurations
I
Diffusion constant is the average of ∆r1,x · v1,x
I
How variations of this quantity correlate with the local
structure ?
Computation of
h∆r1,x · v1,x δ(~r − ~r1i )i
Most mobile configurations in a high density LJ fluid
T ∗ = 0.785, ρ∗ = 0.8
-2
0.01
-1.5
0.005
-1
0
-0.005
-0.5
-0.01
0
-0.015
-0.02
0.5
1
1.5
2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Elongated (de-structured) and squeezed solvation shell
Water self diffusion at ambient conditions
0
1
vr(r)
vθ(r)
0,8
-0,2
-0,4
v
v
0,6
vr(r)
vθ(r)
0,4
-0,6
0,2
-0,8
0
-1
0
5
10
15
r (A)
Labo. frame
20
25
0
5
10
15
20
r (A)
Part. frame
Slip boundary conditions even for hydrogen bonded system!
25
Displacement inside the cage
0.1
6
0,08
0.08
0.06
4
δρ(r)
a.dg(r)/dr
0,06
0.04
2
0,04
0
0
δρ
0.02
0,02
-0.02
-2
-0.04
-4
-0.06
0
-0,02
-0.08
-6
-0.1
-6
-4
I
I
I
I
-2
0
2
4
-0,04
0
5
10
6
The water molecule seems displaced in its cage.
Except in its close vicinity
The first shell is also destructured
Characteristic retardation time τret = 350 fs
15
r (A)
20
25
Rotation of the water molecule coupled to diffusion
Computation of h∆~r1 · ~b1 i
0,08
force
0,06
<∆r.b>
O
0,04
H
0,02
0
0
H
2
4
6
8
10
t (ps)
In liquid water, meanfreepath = 0.04 Å
Conclusions
Summary
I
I
Hydrodynamic is valid already for r = 4 with slip boundary
conditions
Effects of the central particle fluctuations:
I
I
Boundary conditions with non-zero normal velocity
An additional dissipation term
Some perspectives
I
Ions in water, hydrophilic and hydrophobic interactions
I
Solute-solvent coupling in transport phenomena
I
Glasses, silver halides, gaussian core particles, ionic liquids. . .
I
Diffusion mechanism of an excess proton in water, in
membranes
Table of contents
Introduction
Diffusion of a particle
Velocity field and density map
Boundary conditions
An extra dissipation
Velocity autocorrelation function
Diffusion mechanism
Water self diffusion
Conclusions