()() hh = tt = 6 a

Diffusion and velocity relaxation of a Brownian particle immersed in a viscous compressible fluid

confined between two parallel plane walls

Ubbo Felderhof, RWTH Aachen

Many processes in physical chemistry and biology are dominated by the process of diffusion.

In geometry on a small scale, i. e. membranes, thin liquid fims, pores, one has to worry about the influence of geometry

on the diffusion coefficient.

Near a wall diffusion becomes anisotropic and one has to deal with a diffusion tensor dependent on the distance h to the wall.

In bulk Einstein 1905

with friction coefficient Stokes 1850

shear viscosity particle radius

Near a wall with mobility tensor

parallel xy-plane

( )h Dt

D kT

= ζ

6 a ζ = πη

η a

( )

h

=

kT

µ ( )

h

D

t t µ t ( ) h = t ς ( ) h

⁻¹

( )h _xx( )(h _{x x y y}) _zz( )h _{z z}

µ

t =

µ

e e +e e +

µ

e e

0

( ) [1 9 ]

xx 16 h a

µ = µ − h

0

( ) [1 9 ]

zz 8 h a

µ

=

µ

− h Lorentz 1907

0

1 6 a

µ ⁼ πη Higher order correction terms first worked out by Faxén 1925 At present the mobilities and are known very precisely.

Similar results for a particle between two plane walls and

zz

( ) h µ

xx

( ) h µ

( , )

xx h L

µ µ

_zz( , )h L To first order in

a

h

Faxén 1925

( , ) 0[1 1.004 ]

xx 2

L a

L h

µ =µ −

( , ) 0[1 1.452 ]

zz 2

L a

L h

µ =µ − BUF 2005

So far we considered static diffusion tensor

For fast processes it may be necessary to generalize to a frequency-dependent tensor

Again there is an Einstein-type relation

where is the admittance tensor for the geometry h, L at frequency

For applied force the particle velocity is with

( , )h L Dt

( , , )h L

ω

Dt

( , , ) h L ω = kT ( , , ) h L ω

D t y t

( , , ) h L ω yt ω

( )t = Re ωe−^{i t}ω

E E

( )t Re ωe ^{i t}

ω

= −

U U

( , , ) h L

ω

= ω ⋅

ω

U y t E

The diffusion process is related to velocity relaxation by

with velocity correlation function For

( ) (0)t v v

0

( ) ω =

^∞

∫

e^{i tω}

( ) (0)

t dt

D

t

v v

( ) (0) t = (0) (0) e

^{−ζt m}/ ^p

v v v v

_with ^{(0) (0)}

p

kT

= m

v v 1

t

this gives the Einstein relation More generally

Corresponding to admittance In confined geometry

(0) kT D ⁼ ζ

( ) _p ( )

D kT ω i m

ω ζ ω

= − +

( ) ( )

t

p

kT ω i m

ω ζ ω

= − +

y

( ) ω = kT ( ) ω D t y t

0

( ) 1 e

^{i t}

( ) (0) t dt kT

ω =

^∞

∫

ω

y t v v

fluctuation-dissipation theorem

By inverse Fourier transform

( ) (0) ( )

2 kT

i t

t ω e

^ω

d ω

π

∞

−

−∞

= ∫

v v yt

0 t >

This may be used to calculate v( ) (0)t v in confined geometry.

2

3 At high frequency

The behavior at low frequency is of particular interest,

since it is related to the long-time behavior. This is affected by the geometry.

Alder and Wainwright found 1970 in computer simulation

( ) 1

i mp

ω ^≈ −ω

yt 1 as

ω → ∞

( ) (0) t t

⁻3/ 2

v v

_as

t → ∞

This was first understood from kinetic theory, later from hydrodynamics.

The admittance of a sphere in an incompressible fluid behaves at low frequency as

1

2

( ) 1 ( )

t

6 a O

ω a α α

πη ^{⎡ ⎤}

= ⎣ + + ⎦

y

^{α = −i}_η^ωρ

This yields by Tauberian theorem

3/ 2 3/ 2

( ) (0) 1

12 ( )

t kT t

πρ πν

≈ −

v v 1 _as

t → ∞

ν η

= ρ kinematic viscosity Quite generally for

Tauberian theorem small ω behavior large t behavior of of

0

ˆ ( )

^{i t}

( )

f

ω =

^∞

∫

e^ω f t dt

ˆ ( )

f

ω

^{f t( )}

conversely large ω behavior small t behavior of offˆ ( )

ω

_{f t( )} The converse theorem has also played a role in physics.

Following earlier remarks by Lorentz, and work by H. Weyl, there is a famous paper by Mark Kac

„Can one hear the shape of a drum?“

In his paper Kac actually reduces the acoustic problem to a diffusion problem:

Consider the conditional probability of finding a particle at r at time t when it starts out at 0 at time 0

( , | , 0) P r t 0

P

behaves like particle density and therefore satisfies P ²

D P

t

∂ = ∇

∂

The fundamental solution is ²

3/ 2

( , | , 0) 1 exp[ ]

(4 ) 4

P t r

Dt Dt

π

= − r 0

Mean square displacement

i.e. size of probability cloud grows as Kac considered

Write this as integral of decaying exponentials

2 2

( , | , 0) 6 r =

∫

r P ^r t ⁰ d^r = Dt

t

3/ 2

( , | , 0) 1

(4 ) P t

π Dt

0 0 = t > 0

3/ 2 0

( ) exp[ ] 1

(4 )

g t d

λ λ λ Dt

π

∞

− =

∫

then

3/ 2

( ) 1

(4 ) (3 / 2)

g D

λ λ

= π

Γ

(3 / 2) 1

2

π

Γ =

in agreement with Tauberian theorem

(3 / 2) λ

Γ

^{3/ 2}

1 t

large

λ

^small

t

small

λ

^large

t

In this case both types of behavior are realized at the same time.

4

Similarly in a viscous incompressible fluid satisfies the linearized Navier-Stokes equation

2

p

ρ ^∂ t = ∇ − ∇ η

∂

v v ∇ ⋅ = v 0

For d- impulse at t=0 fundamental solution

At the origin

0 t >

( , ) 1 ( , )

t

4

t

= πη ⋅

v r T r P

1 1 3/ 2

( , )

4 3

t t

πη πν

= −

v 0 P all t

3/ 2 3/ 2

1

12 ( ) t ρ πν

= − P

corresponds precisely to the long-time behavior of Brownian particle found from term in

This shows that the velocity correlation function of a Brownian particle is closely related to the Green function of the hydrodynamic

equations of motion.

But the Green function depends on geometry.

One can expect that in particular the long-time behavior is strongly dependent on geometry.

Gotoh and Kaneda (1982) found that in the presence of a single plane wall the long-time behavior is

ω

^{yt( )}

^ω

5 / 2

( ) (0)

x x

v t v t

⁻

v t v

z

^{( ) (0)}

z

t

^{−7 / 2}

I found (2005) that the latter result is incorrect. Both correlation functions behave as

5 / 2

xx

( )

xx

C t ≈ A t

⁻

C

_zz

( ) t ≈ A t

_zz ^{−5 / 2}

with coefficients

A

xx

A

_zz The second coefficient may be

<0, depending on particle mass.

5

( , ) t v r

2 p ( ) ( )t

ρ

^∂t − ∇ + ∇ =

η δ δ

∂

v v P r

Velocity correlation function for a fluid with a single wall

was studied in computer simulation by Pagonabarraga, Hagen, Lowe, Frenkel 1998 It turned out that fluid compressibility has a significant effect.

In bulk compressible fluid one can calculate the velocity correlation function again from the admittance

Result:

t( )

ω

y

3/ 2 5 / 2

( ) (0) 3/ 2

12 ( )

x x

v t v kT t At

πρ πν

− −

≈ +

with a coefficient A that is negative if the fluid is sufficiently compressible, i.e. the decay is not monotonic, but can change sign

BUF, JChemPhys 2005 ( ) 1

t ( )

i mp

ω ⁼ −ω +ζ ω

y with a complicated expression for the

friction coefficient

Zwanzig, Bixon 1970 Bedeaux, Mazur 1974 Metiu et al. 1977

ζ ω ( )

depends on shear viscosity, bulk viscosity, density, compressibility Again a wall causes modification of the behavior, but I found that

the coefficients A_xx and A_zz of the t^-5/2 long-time behavior are independent of compressibility, BUF 2005

(limit to bulk behavior not simple)

For a fluid confined between two walls Pagonabarraga et al.

found a dramatic change of behavior (1997,1998)

( ) ( ) (0)

2

xx x x

C t = v t v ≈ At

⁻ with A<0 no details were shown

They made more elaborate analysis in 2D: fluid between two lines.

In that case

( ) ( ) (0) 3/ 2

xx x x

C t = v t v ≈ At^{− with} A<0 They gave expression for A.

6

Recently I have calculated the coefficient A of the t^-3/2 long-time tail in 3D.

Result:

2 2

5 2 2

0

9 ( )

( ) ( ) (0)

xx x x

2

h L h kT

C t v t v

L c t

π ρ

= ≈ − −

c

0 is the adiabatic (long-wave) sound velocity Note the result is independent of viscosity.

Again the behavior follows from a Tauberian theorem.

The admittance tensor in any geometry can be expressed as

0 0

( , ) ω =

_t

( ) 1 ω ^⎡ ⎣ + A ( ) ( ) ω C ω

_a

( , ) ω ^⎤ ⎦

y r t y F r t

bulk Faxén type coefficents calculated by Bedeaux, Mazur 1974

( , )0

a

ω

F rt

is the reaction field tensor, depends on geometry.

In point approximation

0

0 0 0 0

( , )

ω

lim[ ( , ) ( )]

= → − −

r r

F rt G r r G r r

Bulk Green function is known.

I have calculated for compressible viscous fluid between two planes.

At this gives results mentioned earlier:

0

( −

0

, ) ω G r r

( , , )

0

ω G r r

ω = 0

(0)

0

[1 6 (0)]

xx

aF

xx

µ = µ + πη µ

_zz

⁽⁰⁾ = µ

₀

^{[1 6} + πη ^aF

_zz

^(0)]

( , ) 1 [1 1.004 ]

xx 6 h L a

a h

µ

⁼

πη

^{− at}

h = L / 2

^{Faxén 1925}

( , ) 1 [1 1.452 ]

zz 6 h L a

a h

µ ⁼ πη ^{− at}

h = L / 2

^{BUF 2005}

7

Tauberian theorem is applied to the low frequency behavior

2 2

2 2 2

0 5

1 2 ( )

( , , ) 36 ln ( )

4 3

xx

h L h

F h L X h h O

h L

ω α ξ α α α

πη

⎡ − ⎤

= ⎢ + − + ⎥

⎣ ⎦

iωρ

α η

= −

Here is given by a complicated integral over wavenumber q, coming from Fourier expansion in the xy-plane.

This gives the steady-state results.

The next term leads to cancellation of the bulk t^-3/2 tail.

The mathematical origin of this term is already quite subtle.

Usually the term linear in comes from an integral over wavenumber of the form

X0

2 3

α

h

ω

2 2

2 2

0

( ) ^gq q

f e dq

α q

α

∞

= −

∫

+

cutoff for large q diffusion pole

Expansion in powers of yields

α

1 2

( ) ( )

2 2

f O

g

α

=

π π

−

α

+

α

independent of the cutoff, such a term gives rise to the bulk t^-3/2 tail.

Instead the term comes from a branch cut in the complex q-plane, rather than a simple pole.

In the last term

1

2

( ) exp[ ] [ ]

2 2

f g erfc g

g

α = π π − α α α

2 3

α

h

0 0

c c

η ν

ξ ⁼ ρ ⁼ is the acoustic damping length.

8 The singularity comes from an acoustic diffusion pole

(overdamped sound wave), but with weight q rather than q²

2 ln

α α

The corresponding diffusion coefficient is

The xx element of the reaction field tensor can be expressed as

2 2 0

12 D c L

= ν

0

( , , ) 1 ( , )

xx 4 x

F h L

ω

f q

ω

q dq

πη

∞

=

∫

The function f q_x

( , ) ω

behaves for small

q

and as

_ω

2 2 2 2 2 2 2 2

3 2 2 2 2

2 2 2

2 ( ) 36

( , )

2 12

x

q q q h L h

f q q L q L

α α α ξ

ω α α α ξ

+ − − −

≈ +

+ +

branch cut same as for single wall diffusion pole In the pole term we use the integral

2 2 2

2 2 1

0

exp[ ] 1 ( )

2 q g

gq dq e E g q

α α

α

∞

− =

∫

+

Expansion yields the term, and this gives the t^-2 tail.

Define relaxation functions and from

2ln

α α

9

xx

( ) t

γ γ

zz

^{( )} t

( ) ( )

xx xx

p

C t kT t

m

γ

=

zz^{( )} zz^{( )}

p

C t kT t

m γ

=

(0) 1

γ

xx

=

^γzz(0)=1

0 5 10 15 20 25 30

0 0.01 0.02 0.03

xx( )t

γ

t

10

-3 -2 -1 0 1 2

-12 -10 -8 -6 -4 -2 0

ln | γ

_xx

( ) | t

log t

10

-3 -2 -1 0 1 2

-12 -10 -8 -6 -4 -2 0

log t

10

ln | γ

_zz

( ) | t