Brownian transport II: Brownian transport II: Constrained GeometriesConstrained Geometries

• bio-systems, porous media

• artificial submicron devices

• noise rectification mechanisms

P Hänggi and FM Rev Mod Phys 81 (2009) 387P. Hänggi and FM, Rev Mod Phys, 81 (2009) 387

Entropic channelsEntropic channels

)(teFr x ξrrr

& +=Augsburg group, PRL 96, 2006, PRE75, 2008

ξ(t): gaussian, ‹ξι(t)›=0‹ξi(t)ξj(0)›=2D0δijδ(t)

w(x): channel radius

[ ] )(/)()()( xPDxAxDxP ′+∂∂∂Zwanzig, JCP, 1992 [ ] )(/)()()( 0 xPDxAxDxP xxt +∂∂=∂

entropic term)(ln0 xDFx σ−−=)(xAD

σ(x)=2w(x) in 2D, =πw(x)2 in 3D, channel cross section

entropic term)(ln0 xDFx σ)(xAD0

D(x)=D0/[1+w’(x)2]α, α =1/2 (3D), 1/3 (2D)

Reguera, Rubi, PRE64, 2001

A(x)

mobilitymobility

tt

FxF

/)]0()([li

/)( =

&

&μ

txtxxt

/)]0()([lim −=∞→

μ(F) → 1, for F → ∞ theory fails

diff i itdiff i itBurada et al, ChemPhysChem 10, 2009

diffusivitydiffusivity

ttxtxD 2/])()([lim 22 −= ttxtxDt

2/])()([lim=∞→

D(F) goes through a depinning peak,and D(F) → D for F → ∞and D(F) → D0 for F → ∞

Costantini, FM, EPL 48, 1999

Mobility and Diffusion in Mobility and Diffusion in a periodic potentiala periodic potentiala periodic potentiala periodic potential

Model: single Brownian particle in a tilted washboard potential

L=2πwashboard potential

FtxVxxm ++′−−= )()( ξγ &&& μ=v/F v∞=F/γ

γ= 0.01

kT=1

ξ(t): thermal noise ⟨ξ⟩ = 0, ⟨ξ(t)ξ(0)⟩ = 2γ kTδ(t)V(x)=V(x+L): periodic substrate

γ = 0.1

validityvalidityyy

1 fast transverse re-equilibration

Burada et al, PRE75, 2007

1. fast transverse re-equilibration

ty << max{tx,tF}

ty =w2max/2D0 transverse diffusion

tx = L2/2D0 longitudinal diffusionP(δx) ≈ F/D0 exp(-Fδx/D0)

tF=L/F drift

2. smooth channel corrugationsdespite much effort, validity of the reductionist

P(δx) F/D0 exp( Fδx/D0)

2. smooth channel corrugations

w’(x) << max{1,D0/FL}

f l d

validity of the reductionist approach is restricted restricted to special geometries

more stringent for large drives

Laachi et al, EPL 80, 2007

BonaBona--fidefide 2D and 3D channels2D and 3D channelsBonaBona fidefide 2D and 3D channels2D and 3D channels

)(teFr x ξrrr

& +=

Fick Jacobs approach failsFick-Jacobs approach fails

even for F=0

F

r1

r

F

r0

interest not conceptual only

applicationsapplicationsapplicationsapplications

channel networks in natural andchannel networks in natural and

artificial porous media (zeolites,

membranes, etc)

particle ratcheting in 2D and 3D

channels is more effective for sharp

boundaries (eg. magnetic vortices in

type-II superconductors)

Nori, F.M., PRL 83, 1999

2D septate channels2D septate channels

structure-less pores

reflecting walls

stationary regime

P(δ ) ( Fδ /D )

mobility curvemobility curveF-1/2

P(δx) ~ exp(-Fδx/D0)

decreasesdecreases monotonicallymonotonically

F

μμ(∞) = Δ/yL independent of xL

for D0/F << Δ

(F) ( ) F 1/2 l dμμ(F) − μμ(∞) ∝ F-1/2, slow decay

μμ(0) depends on Δ/yL and xL/yL Savelev, FM, PRE 80, 2009

zerozero--drive mobilitydrive mobility

logarithmic dependence on Δ/yL

⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−=

LL

L

yxy ln21

1)0(

π

μL

LL

yyx

<<Δ>>

⎠⎝ LL y

• μ(0)=x2L/2D0τ0 with τ0 a MFPTμ( ) L/ 0 0 0

• map the FPE into a Poisson eqnfor the electrostatic field in a for the electrostatic field in a multistrip chamber (Purcell 1977)

τ0 ⇔ capacitance C0

• CERN report 77/09 for C0

diffusivitydiffusivity

NO excess diffusion peak

NO horizontal asymptote D(F)=D

23

112

)(⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛ Δ−

Δ=

DF

DFD

NO horizontal asymptote D(F)=D0

quadratic Fquadratic F dependence (Taylor’s type)00 12 ⎟

⎠⎜⎝

⎟⎠

⎜⎝ DyD L

• = v(t) is a dichotomic process v∈{0,F}

)(tx&

⎞⎛•

/

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

F

F tFtCττ

ττττ

02

02 exp)(

with τ0+τF=τ, τF /τ =Δ/yL

and τ0=(1-Δ/yL)τ = [(yL-Δ)/2]2/3D0

• ∫∞

=0

)()( dttCFD

3D septate channels3D septate channels

r1important difference: two transverse diffusion times,ty =r1

2/2D0 and MFPT out of the annulus, with r0

y 1 / 0 ,absorbing inner walls (large F regime), tr=r0

2/2D0 ln(r1/r0)

mobility curvemobility curve

μμ(∞) = (r0/r1)2

for D0/F << r0

μμ(F) − μμ(∞) ∝ F-1, faster decayy

μμ(0) easier to compute

zerozero--drive mobilitydrive mobility

linear dependence on pore radius

21

02)0(rrxL

πμ = Berezhkovskii et al, JCP 116 (2002);118 (2003)

1rπ

• μ(0)=x2L/2D0τ0μ( ) L 0 0

• Poisson eqn for a circular disk symmetrically placed between two parallel conducting planes 2xL apart.

τ0 ⇔ capacitance C0

• C0=8r0

diffusivitydiffusivity2

01

0

2

21

20

2

21

20

21

0

ln12

)(⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟⎟

⎠

⎞⎜⎜⎝

⎛=

DF

rr

rr

rrr

DFD

01110 2 ⎠⎝⎠⎝⎥⎦⎢⎣ ⎠⎝⎠⎝ DrrrD

quadratic in F, like in 2D

• = v(t) is a dichotomic process v∈{0,F}

)(tx&

• ⎟⎟⎠

⎞⎜⎜⎝

⎛−=

F

F tFtCττ

ττττ

02

02 exp)(

with τ0+τF=τ and τF=[1-(r0/r1)2]τ = r0

2/2D0 ln(r1/r0)

• ∫∞

=0

)()( dttCFD

Eccentric pore geometriesEccentric pore geometries

yLxxh

xL

• no “free lane” along the x-axis

f l F D /F Δ diff i i t• for large F, D0/F << Δ, diffusion againstcompartment wall is 1D with exit time

τ≈(xh-Δ/2)2/2D0 F

mobilitymobility μ

for large F

v=xL/τ being τ>>xL/F

FxF L

F τμ ⎯⎯ →⎯ ∞→)(

diffusivitydiffusivity

(F)/D

0

yysame as for a driven

free Brownian walker D

(

2

22

2)(

τσ

ττ×= LxFD

F/D0

… for a better estimate Cox formula

Absolute Negative MobilityAbsolute Negative Mobility

•• special channel geometryspecial channel geometry

•• spherical, pointspherical, point--like particleslike particles

•• F F →→ FF00 + F+ F11cos (cos (ΩΩt) t) with F0 << F1

F0

jin the presence

of an ac drive j= x& j

x

of an ac drive j

flows against the

static bias F0

x

F0 y0

Reimann, Hanggi, PRL88, 2002

ANM in a septate channelANM in a septate channelF → F(t) = F0 + F1(t) with F1>F0

F1+F0

(F F )t

v(F)=μ(F) F

-(F1-F0)

v(F) μ(F) F

μ(F) monotonically decreasing function

Q:Q: can v(F1+F0) < v(F1-F0)?ANM?ANM?

QQ ( 1 0) ( 1 0)

A:A: no ANM for symmetric particles as μ(F) decays not faster than 1/F!

Recipe:Recipe: We must “lower” the symmetry of the system!

pore asymmetry →Arrhenius traps

back to the original schemeEichhorn et al, PRL88, 2002 trap δxtδxt

particle asymmetry →ellipsoidal objects ellipsoidal objects

ANM predictedANM predicted

ANM for (ANM for (prolateprolate) ellipsoidal particles: ) ellipsoidal particles: δδxxtt = b= b--aa

exit time longer for exit time longer for F1 || F0

F F ⎟⎞

⎜⎛ −± 0)()( FabFe.g. F0 < F1 ⎟⎟

⎠⎜⎜⎝±=±

0

010

)(exp)(D

Fττ

optimal ANM for

⎟⎞

⎜⎛ −⎟

⎞⎜⎛ − )()( FabFab

⎟⎟⎠

⎞⎜⎜⎝

⎛+<Ω<⎟⎟

⎠

⎞⎜⎜⎝

⎛−

0

00

0

0 )(exp)(expD

FabD

Fab τJ

sensitivity to other parameters

eccentricity D etceccentricity, D0, etc

ConclusionsConclusions

• channels in 2D and 3D show properties that are unaccounted p p

for in Fick-Jacobs approximation

• septate channels as reference solution for the design of

artificial nanodevices. Next: extended particles, chains, etc

• I wish to thank my coworkers atAugsburg University, Germany (P. Hänggi, G. Schmid, P.S. Burada)

Loughborough University, UK (S. Savelev, A. Pototskii)

Perugia University, Italy (M. Borromeo)

RIKEN, Japan (F. Nori, P. Ghosh)