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Diffusion in confined geometries

P. Sekhar Burada,[a] Peter Hanggi,∗[a] Fabio Marchesoni,[b]

Gerhard Schmid,[a] and Peter Talkner[a]

[a] Institut fur Physik, Universitat Augsburg,Universitatsstr. 1, D-86135 Augsburg (Germany)

[b] Dipartimento di Fisica, Universita di Camerino,Via Madonna delle Carceri 9, I-62032 Camerino (Italy)

∗ Fax: (+49)821 598 3222; E-mail: hanggi@physik.uni-augsburg.de

August 18, 2008

Abstract

Diffusive transport of particles or, more generally,small objects is a ubiquitous feature of physical andchemical reaction systems. In configurations contain-ing confining walls or constrictions transport is con-trolled both by the fluctuation statistics of the jitter-ing objects and the phase space available to their dy-namics. Consequently, the study of transport at themacro- and nanoscales must address both Brownianmotion and entropic effects. With this survey we re-port on recent advances in the theoretical and numer-ical investigation of stochastic transport occurring ei-ther in micro-sized geometries of varying cross sectionor in narrow channels wherein the diffusing particlesare hindered from passing each other (single file dif-fusion). For particles undergoing biased diffusion instatic suspension media enclosed by confining geome-tries, transport exhibits intriguing features such as (i)a decrease of nonlinear mobility with increasing tem-perature or, also, (ii) a broad excess peak of the ef-fective diffusion above the free diffusion limit. Theseparadoxical aspects can be understood in terms ofentropic contributions resulting from the restricteddynamics in phase space. If, in addition, the suspen-sion medium is subjected to external, time-dependentforcing, rectification or segregation of the diffusingBrownian particles becomes possible. Likewise, the

diffusion in very narrow, spatially modulated chan-nels gets modified via contact particle-particle inter-actions, which induce anomalous sub-diffusion. Theeffective sub-diffusion constant for a driven single filealso develops a resonance-like structure as a functionof the confining coupling constant.

Keywords: diffusive transport, Brownian motion, en-tropic effects, diffusion enhancement, single file diffu-sion

1 Introduction

Effective control of transport in artificial micro- andnano-structures requires a deep understanding of thediffusive mechanisms involving small objects and, inthis regard, an operative measure to gauge the roleof fluctuations. Typical such situations are encoun-tered when studying the transport of particles in bi-ological cells [1] and in zeolites [2], catalytic reac-tions occurring on templates or in porous media [3],chromatography or, more generally, separation tech-niques of size disperse particles on micro- or evennanoscales [4]. To study these transport phenomenais in many respects equivalent to studying geometri-cally constrained Brownian dynamics [5, 6, 7]. Thefact that the diffusion equation is closely related tothe time evolution of the probability density P (~x, t)

1

to find a jittering particle at a location ~x at timet dates back to Einstein’s pioneering work on themolecular-kinetic description of suspended particles[8].

With this minireview we focus on the problemof the diffusion of small size particles in confinedgeometries. Restricting the volume of the phasespace available to the diffusing particles by meansof confining boundaries or obstacles originates re-markable entropic effects [9]. The driven transportof charged particles across bottlenecks, such as iontransport through artificial nanopores or artificialion pumps [10, 11, 12, 13] or in biological chan-nels [14, 15, 16, 17, 18] is the better known modelwhere diffusion is determined by entropic barriers.Similarly, the operation of artificial Brownian mo-tors [19, 20, 21], molecular motors [22] and molecularmachines [23] also results from the interplay of dif-fusion and binding action by energetic or, more rel-evant in the present context, entropic barriers [24].The efficiency of such nanodevices crucially dependson the fluctuation characteristics of the relevant de-grees of freedom [25]. In addition, the interplay ofdiffusion over entropic barriers and unbiased timeperiodic drives is responsible for certain paradoxi-cal transport effects, like the recent observation ofentropic-diffusion controlled absolute negative mobil-ity [26].

Although we restricted ourselves to small size par-ticle, we remind the reader that entropic forces surelyaffect the dynamics of extended chains diffusing ona periodic two (2D) or three dimensional (3D) sub-strate. A well established example is represented bythe phonon damping of propagating solitons [27]. An-other example of great interest for its potential ap-plications in nanotechnology is the translocation of along polymer molecule through a pore with openingsize comparable with the polymer gyration radius. Inthis case, entropic effects were first predicted to theo-retically describe the conformation changes the chainundergoes to move past a conduit constriction [28]and, then, experimentally observed both in biologi-cal [29] and artificial channels [30].

Another instance of constrained Brownian dynam-ics that rests, indeed, within the scopes of our minire-view, is the so-called single file diffusion. The motion

of an assembly of small size particles in a narrowchannel can be so tightly restricted in the transversedirections that the particles arrange themselves intoa single file. The longitudinal motion of each par-ticle is thus hindered by the presence of its neigh-bors, which act as nonpassing movable obstacles. Asa consequence, interparticle interactions in one di-mension can suppress Brownian diffusion and lead tothe emergence of a new subdiffusive dynamics [31].

The outline of this work is as follows. In Sec. 2we detail the geometry of the channels relevant toour review and set up the mathematical formalismneeded to model the diffusion of a Brownian particleimmersed in a confined suspension fluid. In Sec. 3 weintroduce some exact results for the mobility and thediffusion coefficient of a driven Brownian particle in aone dimensional (1D) periodic potential, results thatwill be handy in the subsequent sections. In Sec. 4we compute explicitly the entropic effects on particletransport in a static fluid filling a periodically mod-ulated channel. In Sec. 5 we address particle trans-port in a suspension fluid flowing along the channelsubject to stationary pumping. In Sec. 6 we reviewnumerical and analytical results for the diffusion ofa single file along a periodically corrugated channelboth in the presence and in the absence of an externaldrive.

2 Channel models

We consider the diffusive dynamics of spherical par-ticles in 2D or 3D pores, or channels, which extendin the x direction and possess a periodically vary-ing cross section. These channels are supposed to besymmetric with respect to a reflection on the channelaxis in 2D (like those sketched in Fig. 1), or to anyrotation about the channel axis in 3D. The channelsare assumed to be delimited by rigid, smooth walls.The half width of the channel is described by the well-behaved boundary function w(x). The channel wallsdo confine the particles inside the channel but, oth-erwise, do not exchange energy with them. In partic-ular, we do not consider the possibility of adsorptionof particles at the walls. Since we will disregard anyrotatory motion of the particles about their centers of

2

~F(a)

(b)

L

~F(c)

y,z

y,z

xx

Figure 1: (Color online) Brownian particles in a nar-row cylindrical channel directed along the x axis andwith periodically modulated boundary w(x) (longitu-dinal section). (a) Pointlike particle suspended in astatic fluid and subjected to a constant driving forceF (Sec. 4); (b) Single spherical particle dragged alongby a laminar flow (Sec. 5); (c) Single file diffusion ofnonpassing driven particles (Sec. 6.2).

mass, we need not specify particle-wall contact forces.Moreover, the radius of the particles is supposed tobe smaller than the minima of w(x) (channel bottle-necks), so that the particles are not restricted to staywithin a confined region of the channel, but rathermay diffuse everywhere along the channel.

2.1 Diffusion equations

The motion of particles that are immersed in a fluidmedium may be influenced by various types of forces.We will here restrict ourselves to the discussion ofsmall radius (almost pointlike) particles, whose pres-ence does not significantly modify the free motion ofthe fluid around them (laminar flow regime). Thesystematic impact of the fluid on the motion of par-ticles at position ~x ≡ (x, y, z) is given by the Stokesforce [32]

~FStokes = −γ[ ˙~x, t − ~v(~x)] (1)

where ~v(~x, t) is the instantaneous velocity of the fluid

in absence of the particle, ~x is the instantaneous par-

ticle velocity, and

γ = 6πηR (2)

is the friction constant, which is determined by theshear viscosity η of the fluid, and the radius R ofthe particle. At the same time the fluid exerts onthe particle also a random thermal force ~Fth. In thefollowing, we only deal the case of fluids with homo-geneous temperature T and velocity fields that arealmost constant on the particle scale R. As a conse-quence, to insure thermalization at temperature T itsuffices to set

~Fth(t) =√

2γkBT ~ξ(t) , (3)

where kB denotes the Boltzmann constant and ~ξ(t)

is a standard 3D Gaussian noise with 〈~ξ(t)〉 = 0 and〈ξi(t) ξj(t

′)〉 = δij δ(t − t′) for i, j = x, y, z. Otherforces acting on the particles, like hydrodynamic in-teractions among different particles and between sin-gle particles and the wall, will be neglected [33, 34].This simplification, in particular, requires a suffi-ciently low particle density.

Finally, an external force ~Fext may act on the par-ticles describing e.g. the gravity force, or in the caseof charged particles, an electrostatic force. We fur-ther specialize our discussion to constant longitudinalexternal forces pointing in the direction of the sym-metry axis of the channel. The dynamics of the centerof mass ~x(t) of a single particle is then governed byNewton’s equation of motion

m~x = ~Fext − γ[~x − ~v(~x, t)] +√

2γkBT ~ξ(t), (4)

where we explicitly allow for a possible time-dependence of the fluid velocity field. For micropar-ticles moving with typical velocities of the order of1 cm/s, the inertial term m~x(t) in Eq. (4) is neg-ligibly small compared to the environmental forces[35]; therefore, provided that the fluid velocity doesnot changes too fast, that is for spectral frequenciesless than a few 100 Hz, one can safely set m = 0(overdamped limit or Smoluchowski approximation).Under these conditions the equation of motion (4)can be simplified as

~x = ~v(~x, t) +1

γ~Fext +

√

2kBT

γ~ξ(t). (5)

3

This Langevin equation is equivalent to the followingFokker-Planck equation [36] for the probability den-sity P (~x, t) of a particle to be found at the position~x at time t:

∂P (~x, t)

∂t= −~∇ · ~J(~x, t). (6)

Here ~J(~x, t) denotes the corresponding probabilitycurrent density

~J(~x, t) = −(

~v(~x, t) +~Fext

γ

)

P (~x, t) +kBT

γ~∇P (~x, t).

(7)These equations have to be supplemented by appro-priate boundary conditions, which will be discussedin the following subsection.

2.2 Boundary conditions

The channel is typically characterized by two bound-ary regions: A transverse boundary naturally resultsfrom the presence of the walls, whereas a longitudi-nal boundary is required to account for the channellength.

The probability flux normal to the boundary in thepresence of a rigid hard wall must vanish. Thus, toprevent pointlike particles from leaving the channelor being adsorbed at the walls, we must impose that

~n(~x) · ~J(~x, t) = 0 ~x ∈ wall, (8)

where ~n(~x) denotes the unit vector normal to the wallat point ~x. Note, however, that the center of mass ofa spherical particle of finite radius R may approachthe wall only up to a distance R, so that for finitesize particles the boundary conditions (8) applies onan appropriate inner surface parallel to the channelwalls [37].

Various kinds of boundary conditions exist thatregulate the inward and outward probability flowsat the ends of a channel [32]. If the channel con-nects large, well mixed particle reservoirs, then con-stant probability densities PL,R may be assigned atthe channel ends. This leads to Dirichlet boundaryconditions of the form

P (xL, y, z) = PL, P (xR, y, z) = PR, (9)

where xL and xR denote respectively the left andright endpoints of the channel. A more detailed de-scription of the particle flow into and out of a channelcan be achieved by relating flux and probability den-sities at x = xR,L, that is [38]

Jx(xL, y, z, t) = −κLP (xL, y, z, t) ,

Jx(xR, y, z, t) = κRP (xR, y, z, t) . (10)

Here, positive (negative) constants κL,R correspondto partially absorbing (emitting) boundaries. As spe-cial cases, reflecting and absorbing boundaries corre-spond to κR,L = 0 and κR,L = ∞, respectively.

Finally, for an infinitely long channel the periodicboundary conditions

P (x, y, z, t) = P (x + L, y, z, t) (11)

are more appropriate [36]. In the case of velocityfields which are constant with respect to time or varyperiodically in time, these boundary conditions allowfor stationary flux carrying solutions [19, 20, 21].

3 Exact results for 1D systems

In order to set the stage, we consider first the idealcase where the diffusion of a particle in a periodi-cally corrugated channel can be assimilated to thediffusion on an energetic landscape represented, forsimplicity, by a 1D periodic substrate V (x) with pe-riod L, namely V (x + L) = V (x). Such a model isoften employed to model, for instance, nanotube [39]and zeolite diffusion [2]. Let us consider a Brownianparticle with mass m, coordinate x, and friction co-efficient γ, subjected to a static external force F anda thermal noise Fth(t). In the notation of Sec. 2.1,we set Fext = −V ′(x) + F .

The corresponding stochastic dynamics is de-scribed by the Langevin equation

mx = −V ′(x) − γ x + F +√

2γkBT ξ(t), (12)

where ξ(t) is the standard Gaussian noise also definedin Sec. 2.1. Moreover, for the purposes of this review,the substrate V (x) can be taken symmetric underreflection, V (x) = V (−x).

4

In extremely small systems, particle fluctuationsare often described to a good approximation by theoverdamped limit of Eq. (12), i.e., by the masslessLangevin equation

γx = −V ′(x) + F +√

2γkBT ξ(t) , (13)

where the inertia term mx has been dropped alto-gether (Smoluchowski approximation).

An overdamped particle is trapped most of thetime at a local minimum of the tilted substrate aslong as F ≤ F3, F3 denoting the depinning thresholdF3 = max{V ′(x)}. Drift occurs by rare noise in-duced hopping events between adjacent minima. ForF > F3 there exist no such minima and the parti-cle runs in the F direction with average speed ap-proaching the no-substrate limit F/γ. This behavioris described quantitatively by the mobility formula:

µ(F ) ≡ 〈x〉F

, (14)

with

〈x〉 ≡ limt→∞

〈x(t)〉t

=L

〈t(L, F )〉 . (15)

Here and in the following, 〈tn(L, F )〉 denotes the n-th moment of the first passage time of the particleacross a substrate unit cell in the F direction.

As the particle drifts subjected to the external forceF , the random hops cause a spatial dispersion of theparticle around its average position 〈x(t)〉. The cor-responding normal diffusion coefficient,

D(F ) ≡ limt→∞

〈x(t)2〉 − 〈x(t)〉22t

, (16)

can be computed analytically by regarding the hop-ping events in the overdamped regime as manifesta-tions of a renewal process, that is [40]:

D(F ) = L2 〈t2(L, F )〉 − 〈t(L, F )〉2

2 〈t(L, F )〉3. (17)

Simple algebraic manipulations lead to explicit ex-pressions for the nonlinear mobility [36, 41, 42, 43],

µ(F ) =D0L

F

1 − e−LF/kBT

∫ L

0I+(x)dx

, (18)

and for the diffusion coefficient [40],

D(F )

D0= L2

∫ L

0I2+(x)I−(x)dx

[∫ L

0I+(x)dx]3

. (19)

Here, I±(x) =∫ L

0e[±V (x)∓V (x∓y)−yF ]/kBT dy and

D0 = kBT/γ denotes Einstein’s coefficient for a freediffusing Brownian particle.

For F → 0 Eqs. (18) and (19) reproduce thezero-bias identity D(0)/D0 = γµ(0) with µ(0) =

L2/(

γ∫ L

0I+(x)dx

)

[36, 44]. Notably, as F ap-

proaches the depinning threshold F3 the mobilitycurve (18) jumps from zero (locked state) up to close1/γ (running state). Correspondingly, the diffusioncoefficient (19) develops a diffusion excess peak, i.e.with D > D0, consistently with numerical observa-tions in Ref. [45]. Both the mobility step and theD peak get sharper and sharper as T is lowered [40].The same conclusions apply in the presence of iner-tia (i.e. for particles of finite mass m) [45, 46], aswell, with the difference that the relevant depinningthreshold shifts towards lower values (proportionallyto γ as γ → 0 [36]).

Finally, we emphasize that the above formulasfor the nonlinear mobility and the effective diffu-sion coefficient retain their analytic structure alsowhen generalized to anomalous (sub)-diffusion on a1D substrate, by merely substituting the normal dif-fusion constant D0 by the fractional diffusion con-stant occurring in the corresponding fractional diffu-sion equation [47].

4 Particle diffusion in a static

fluid

In the limiting case of pointlike particles with zerointeraction radius diffusing in constrained 2D or 3Dgeometries, elastic contact particle-particle interac-tions can be neglected. As long as hydrodynamicallymediated interactions can also be neglected, see e.g.[33, 34], the confining action of the channel walls ismodeled by the perfectly reflecting boundary condi-tions (8).

5

In the presence of a constant external force point-ing in the channel direction, an overdamped Brown-ian particle suspended in a static medium is describedby the Langevin equation (5), or by the correspond-ing Fokker-Planck equation (6), both with zero ve-locity field ~v(~x, t) ≡ 0. For a general choice of theperiodic boundary w(x), there exist no exact analyt-ical solutions to the Fokker-Planck equation (6) withboundary conditions (8). However, approximate so-lutions can be obtained by reducing the problem offree Brownian diffusion in a 2D or 3D channel to thatof Brownian diffusion on an effective 1D periodic sub-strate (Sec. 3). Under such a scheme, narrow channelconstrictions, corresponding to geometric hindrancesin the fully dimensional problem, are modeled as en-tropic 1D barriers.

In the absence of external forces, i.e., for ~Fext =0, particle dynamics in confined structures [seeFig. 1(a)] can be approximately described by theFick-Jacobs kinetic equation with spatially depen-dent diffusion coefficient [48, 49, 50, 51]:

∂P (x, t)

∂t=

∂

∂x

(

D(x)σ(x)∂

∂x

P

σ(x)

)

, (20)

with σ(x) denoting the dimensionless channel cross-section 2w(x)/L in 2D and πw2(x)/L2 in 3D. Equa-tion (20) was obtained [49] from the full Fokker-Planck equation (6) for small amplitude boundarymodulations w(x), on assuming a uniform densityP (~x, t) and integrating out the transverse coordinates

[e.g. for a 2D channel P (x, t) =∫ w(x)

−w(x)dy P (x, y, t)].

At variance with the original Fick-Jacobs equation[48], introducing an x-dependent diffusion coefficientconsiderably improved the accuracy of the kineticequation (20) thus extending its validity to largerw(x) amplitudes [49, 50, 51]. Here, the expression

D(x) =D0

[1 + w′(x)2]α, (21)

with α = 1/3 in 2D and α = 1/2 in 3D, was de-termined to best account for wall curvature effects[50, 52].

In the presence of weak longitudinal drives F , theFick-Jacobs equation (20) can be further extended to

[50, 52, 53]:

∂P

∂t=

∂

∂xD(x)

(

∂P

∂x+

A′(x)

kBTP

)

, (22)

where the free energy A(x) = E(x) − TS(x) is madeup of an energy, E(x) = −Fx, and an entropic term,S(x) = kB lnσ(x). For a periodic channel, A(x) as-sumes the form of a tilted periodic potential. More-over, for a straight channel, w′(x) = 0, the entropiccontribution vanishes altogether and the particle issubject to the sole external drive. Of course, forF = 0 the free energy is purely entropic and Eq. (22)reduces to the Fick-Jacobs equation (20). An alterna-tive reduction scheme based on macrotransport the-ory has been proposed recently in Ref. [54].

Key quantifiers of the reduced 1D kinetics (22) arethe average particle current or, equivalently, the non-linear mobility (14), and the effective diffusion coef-ficient (17) defined in Sec. 3. On generalizing thederivation of Eqs. (18) and (19) to account for thex-dependence of D(x), one obtains [53], respectively,

γµ(f) =1 − e−f

f

∫ L

0

dx

LI(x)

, (23)

and

Deff

D0=

∫ L

0

dx

L

∫ x

x−L

dz

L

D(z)

D(x)

eA(x)/kBT

eA(z)/kBT[I(z)]

2

[

∫ L

0

dx

LI(x)

]3 ,

(24)where

I(x) =eA(x)/kBT

D(x)/D0

∫ x

x−L

dy

Le−A(y)/kBT (25)

and the dimensionless force f is defined as the ratioof the work done by F to drag the particle a distanceL, to the thermal energy kBT , that is

f =FL

kBT. (26)

We stress here an important difference with theenergetic 1D model of Sec. 3. For a particle mov-ing in a 1D periodic substrate V (x), the barriers ∆V

6

separating the potential wells provide an additionalenergy scale besides FL and kBT , so that the particledynamics is governed by at least two dimensionlessenergy parameters, say, ∆V/kBT and FL/kBT [36].In contrast, Brownian transport in a periodically cor-rugated 2D or 3D channel is solely determined bythe dimensionless force f [53]. This can be provenby rescaling the problem units as follows. We mea-sure all lengths in units of the period L and time inunits of τ = L2/D0, that is, twice the time the parti-cle takes to diffuse a distance L at temperature T . Inthese new dimensionless variables the Langevin equa-tion (5) for a Brownian particle in a static mediumreduces to the equation

~x(t) = ~f + ~ξ(t) , (27)

with no tunable constants [55]. As a consequence,the ensuing Brownian dynamics is controlled only bythe parameter f and, of course, by the additionallength(s) that possibly enter the boundary functionw(x).

The f -dependence of the average particle current(23) and the effective diffusion (24) were comparedwith the results obtained by numerical integration ofthe 2D Langevin equation (5) (for details see Ref.[52]). For simplicity, the channel walls were assumedto have the sinusoidal profile

w(x) = a[sin(2πx/L) + κ] , (28)

with a > 0 and κ > 1. Here, a(κ ± 1) are, re-spectively, the maximum and the minimum chan-nel width. Moreover, a/L controls the slope of theboundary function w(x), which in turn determinesthe modulation amplitude of the diffusion coefficientD(x) in Eq. (21). The mobility µ, Eq. (14), andthe corresponding effective diffusion coefficient D,Eq. (16), were computed by ensemble averaging 3·104

simulated trajectories.The most significant results are displayed in Figs. 2

and 3. At variance with the purely energetic 1D mod-els of Sec. 3, the nonlinear mobility decreases uponincreasing the strength of thermal noise. Moreover,an enhancement of the effective diffusion coefficient,with maximum exceeding the free diffusion constantD0, was also observed [53, 55].

0

0.25

0.5

0.75

1

γµ

γµ

0 20 40 60 80

f := FL/kBTf := FL/kBT

0

0.5

1

1 10 100 1000

Figure 2: (Color online) Nonlinear mobility µ vs.driving force in dimensionless units, f = FL/kBT ,for a 2D channel at different temperatures kBT =0.01 (crosses), 0.1 (pluses), 0.2 (squares), and 0.4 (tri-angles). After rescaling, all data sets collapse on onecurve which at low f closely compares with the an-alytic approximation (23) (solid curve). Other sim-ulation parameters are L = 1, γ = 1, and w(x) =[sin(2πx) + 1.02]/2π. The inset shows the devia-tion of the analytic approximation (23) (solid curve)from the numerical results (dotted curve) for large f .Note that the numerical curve approaches the correctasymptotic limit γµ = 1.

At low values of the control parameter f the ana-lytical approximations (23) and (24) match perfectlythe corresponding numerical curves, whereas devia-tions occur at high f . Most remarkably, contraryto the simulation output, the analytical curves forDeff/D0 and γµ fail to attain the correct asymptoticlimit 1 for f → ∞ (see insets in Figs. 2 and 3). Thisoccurs because the assumption of uniform density dis-tribution, introduced in the Fick-Jacobs formalism toeliminate the transverse coordinates, is no more ten-able in the presence of strong drives.

The agreement between theory and numerics im-proves for smooth modulations of the channel walls,i.e. for small boundary slopes |w′(x)|, which can beachieved, e.g., for appropriately small a [52, 55]. Aphenomenological criterion to asses the validity of thestationary state solutions of the Fick-Jacobs equation(22) can be formulated by comparing the different

7

0

0.5

1

1.5

Deff/D

0

0 20 40 60 80

f := FL/kBT

0

1

1 10 100 1000

Figure 3: (Color online) Effective diffusion coefficientDeff vs. f for the same simulation parameters as inFig. 2. Here, too, the rescaled data collapse on onecurve, which asymptotically approaches the correctlimit Deff/D0 = 1 (inset). The analytic approxima-tion (24) (solid curve) fits well the raising branch ofthe numerical data sets.

characteristic time scales of the problem, namely(i) the diffusion times, respectively, in the transverse,

τ(d)⊥ = (2a2/D0)(1 + κ)2, (29)

(ii) and in the longitudinal direction,

τ(d)‖ = L2/2D0, (30)

(iii) and the drift time the applied force F takes todrag the particle one channel unit length L across,

τ(f)‖ = γL/F . (31)

Uniform probability distribution in the transversedirection sets on only if the transverse diffusion mo-tion of the particle is sufficiently fast relative toboth the diffusive and the drift longitudinal motions,

which implies τ(d)⊥ ≪ min{τ (d)

‖ , τ(f)‖ }. For large drives

it suffices to require that τ(d)⊥ ≪ τ

(f)‖ , which leads to

the condition [55]:

f ≤ fc ≡ 1

2(1 + κ)2

(

L

a

)2

. (32)

0

50

100

150

f c

0 0.5 1 1.5

a/L

10

100

0.1 0.2

Figure 4: Critical force fc vs. boundary modulationamplitude a for the 2D channel of Fig. 2. Other sim-ulation parameters are L = 1, γ = 1, and kBT = 0.1.The numerical values of fc (solid dots) were obtainedby requiring deviations of Fick-Jacobs approximationfrom the simulation results smaller than ∼ 1%. Thesolid line is the fitting law fc = c1(L/a)2 + c2 withc1 = 0.792 and c2 = 3.524. The apparent discrep-ancy with Eq. (32) results from the above-mentionednumerical accuracy criterion and is expected to van-ish for higher accuracy. Inset: same data sets on alogarithmic scale.

The critical force parameter fc, above which theFick-Jacobs description is expected to fail, dependson the remaining free parameters of the problem. Forthe boundary function (28), fc is a function of a/L,alone, as illustrated in Fig. 4. For a ≫ L the Fick-Jacobs approximation becomes untenable already forrelatively small forces f , whereas for a ≪ L its valid-ity extends to significantly larger drives.

5 Particle diffusion in moving

fluids

Let us assume now that a small, spherical particle isswept along in the stationary velocity field ~v(~x, t) ofa moving, incompressible fluid [see Fig. 1(b)], rather

than by a constant force, ~Fext, like in Sec. 4. For~Fext = 0, its dynamics is then described by the

8

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Particle diameter [µm]

−1.8

−1.5

−1.2

−0.9

−0.6

−0.3

0.0

0.3

0.6

v e [µ

m�/s

]

40 Hertz, νR=1.0

40 Hertz, νR=0.5

100 Hertz, νR=0.5

x

sin(2πνt)

Figure 5: Average induced particle current ve ver-sus particle diameter. The particle is immersed ina laminar flow confined to a channel with brokenreflection symmetry (inset). The fluid, with viscos-

ity νR ≡ η/ηwater and ηwater = 1.025 · 103Ns/m2, is

pumped periodically back and forth, along the x axisof the channel, with frequency ν. For further details,see Ref. [56].

Langevin equation, in the overdamped limit,

~x(t) = ~v(~x(t), t) +

√

2kBT

γ~ξ(t) , (33)

where the fluid velocity field ~v(~x, t) is to be deter-mined.

To this purpose, we write first the Stokes equa-tions for a stationary incompressible flow in the limitof vanishing Reynolds numbers (the convective accel-eration terms in the Navier-Stokes equations can besafely neglected). One obtains the so-called “creeping

flow equation” [32]: η∆~v(~x) = ~∇p(~x) , where ~p(~x) isthe pressure field responsible for the stationary lam-inar flow of the fluid and η is the absolute dynamicfluid (or shear) viscosity. On introducing the scalarfield Ψ(x, r), with (r, φ) being the polar coordinatesin the transverse plane (y, z), ~v(~x) can be rewrittenas [32]

~v(x, r) = ~∇× (Ψ(x, r)~eφ/r) , (34)

where ~eφ is the unit φ vector. On substituting

Eq. (34) into the creeping flow equation, one obtainsthe following linear homogeneous fourth-order differ-ential equation [32, 56]:

(

r∂

∂r

1

r

∂

∂r+

∂2

∂x2

)2

Ψ(x, r) = 0 , (35)

with boundary conditions

Ψ|r=0 = c , (36)

∂

∂rΨ|r=0 =

∂3

∂r3Ψ|r=0 = 0 , (37)

~∇Ψ(x, r = w(x)) = 0 , (38)

Ψ(x + L, r) = Ψ(x, r) , (39)

where c is an arbitrary constant. Solutions toEq. (35), determine the velocity field ~v(~x) up to amultiplicative factor, which, in turn, can be estab-lished by imposing some additional condition, e.g. forthe pressure drop across a channel unit, i.e. [56]

p(x, r)−p(x+L, r) = −2

∫ L

0

(

∂2

∂r2vx

)

r=0

dx . (40)

Once the velocity field for a particular channel ge-ometry and pressure profile is known, the Langevinequation (33) can be solved numerically.

Particle separation across a micro-channel is a pro-cess of great technological importance [7, 19]. Any in-homogeneity in the spatial distribution of an ensem-ble of non-interacting suspended particles can onlybe caused by the hydrodynamic interaction betweenparticles and walls [37]. The no-flux boundary con-

dition (8) for unbiased particles with ~Fext = ~0 reads:

~n(~x)·(

~v(~x)P (~x, t) − kBT

γ~∇P (~x, t)

)

= 0 , ~x ∈ wall .

(41)In the case of vanishing drift velocity normal to thechannel walls, ~n(~x) · ~v(~x) = 0, only uniform distri-butions are allowed and no particle separation couldever be achieved [37]. However, as anticipated in Sec.2.2, for finite size particles the no-flux boundary con-dition (41) strictly holds on an effective inner sur-face at a distance from the channel walls. Due toits finite size, a particle cannot move steadily along a

9

given flow streamline as this gets too close to thewalls; upon hitting the wall, the particle bouncesinto inner flow streamlines as if they were subjectto a transverse field gradient. These hydrodynamicforces may, indeed, lead to accumulation and deple-tion zones within the channel.

By generalizing such a boundary effect, Kettneret al. [56] predicted by numerical simulation thata micron-sized channel with broken reflection sym-metry can be used to separate particles according totheir size, as illustrated in Fig. 5. A time oscillatingpressure profile, ~p(~x, t) = ~p(~x)f(t), where f(t) is asinusoidal function with frequency ν, was assumed tocontrol the periodic flow of the fluid, back and forthalong an infinitely long channel. Within the creep-ing flow approximation, the ensuing time dependentvelocity field ~v(~x, t) = ~v(~x)f(t) was then obtained interms of the solution ~v(~x) of the unperturbed Stokesequation introduced above, and the correspondingLangevin equation (33) numerically integrated. Lateron, this mechanism was demonstrated experimentallyby Matthias and Muller [57].

6 Single File Diffusion

As the design and the operation of biology inspirednanodevices [19] have become experimentally afford-able, understanding particle diffusion in a 1D sub-strate has been recognized as a key issue in trans-port control [25]. In this context the particle- andparticle-wall interactions play a central role. Pair in-teraction between thermally diffusing particles doesnot affect the normal character of Brownian diffusion,as long as the particles are able to pass one another,no matter how closely they are confined. This holdstrue even when, under appropriate temperature anddensity conditions, attracting particles cluster or con-dense in the substrate wells [58].

Things change dramatically for strictly 1D geome-tries. Let us consider, for instance, an ensemble of Nunit-mass particles moving with preassigned dynam-ics along a segment of length l. If the interparticleinteraction is hard-core (with zero radius), the elasticcollisions between neighboring particles are nonpass-ing – meaning that the particles can be labeled ac-

d

1 4 (×10-2

)t/τd

0

3×102

C(t

)/(D

0ρ)2

0

0

10-1

101

t/τd

10-4

100(b)

-C(t)/(D0ρ)2

Figure 6: (Color online) Velocity autocorrelationfunction C(t) in a sSF at kBT = 1 and for (ρ, γ) =(0.1, 3) (top, orange curve), (0.1, 5) (middle, blackcurve), and (0.05, 5) (bottom, green curve). Inset:log-log plot of the negative C(t) tails; the solid line is

the predicted tail −C(t)/(D0ρ)2 ≃ (t/τd)− 3

2 /(4√

π).

cording to a fixed ordered sequence. The particles arethus arranged into a file where, at variance with thesituation described in Sec. 2, their diffusion is sup-pressed by the presence of two nonpassing neighbors,also movable in the longitudinal direction.

The diffusion of a free single file (SF), i.e. in theabsence of a substrate, has been investigated in de-tail [31]. In the thermodynamic limit (l, N → ∞ withconstant density ρ ≡ N/l) the mean square displace-ment of each file particle can be written as

〈∆x2(t)〉 = |∆x(t)|/ρ (42)

with |∆x(t)| denoting the absolute mean displace-ment of a free particle. For a ballistic single file (bSF),clearly |∆x(t)| = 〈|v|〉t, where 〈. . .〉 is the ensembleaverage taken over the distribution of the initial ve-locities, and therefore

〈∆x2(t)〉 = 〈|v|〉t/ρ. (43)

A bSF particle diffuses apparently like a Brown-ian particle with normal diffusion coefficient D0 =〈|v|〉/(2ρ). For a stochastic single file (sSF) of Brow-

10

nian particles with damping constant γ at tempera-ture T , the equality |∆x(t)| =

√

4D0t/π yields theanomalous diffusion law

〈∆x2(t)〉 = 2DSF

√t/ρ, (44)

where the mobility factor DSF =√

D0/π is relatedto the single particle diffusion constant D0 = kBT/γ.The onset of the subdiffusive regime (44) occurs fort > τd, τd = (D0ρ

2)−1 being the average time a singleparticle takes to diffuse against one of its neighbors[59, 60]. The diffusive regimes (43) and (44) havebeen demonstrated both numerically [61] and exper-imentally [34, 62]. Let x(t) represent the coordinateof one file particle, assumed to be a continuous differ-entiable stochastic process with 〈x(t)〉 = 0. Kubo’srelation [63]

1

2

d

dt〈∆2x(t)〉 =

∫ t

0

C(τ)dτ (t → ∞), (45)

with C(t) ≡ 〈x(t)x(0)〉, best illustrates the role of thedimensional constraint on SF diffusion. In the case ofnormal diffusion, 〈∆2x(t)〉 = 2D0t, the r.h.s. of Eq.(45) converges to the positive value D0 =

∫∞

0C(τ)dτ ,

namely x(t) diffuses subject to Einstein’s law. Thesubdiffusive dynamics (44), instead, is characterizedby∫∞

0 C(τ)dτ = 0, which, in view of Eq. (45), im-plies that C(t) develops a negative power-law tailC(t) ∼ −cβt−β with β = 3

2 and cβ = DSF/4ρ. Nu-merical simulations support this conclusion. In Fig. 6we report a few curves C(t) for different ρ and γ. Thenegative tails are apparent and compare well with theestimate derived from Eq. (45) (see inset). They area typical signature of the collisional dynamics in a SF:As an effect of pair collisions, an initial velocity x(0)is likely to be compensated by a backflow velocity ofopposite sign [60, 64].

6.1 Stochastic single file on a sub-

strate

We focus now on the case of a SF diffusing on a si-nusoidal substrate with potential [65, 66]

V (x) = d[1 − cos(2πx/L)]. (46)

d

101

103

t

102

104

γ1/2 <∆

x2 >

1.02.55.0

101

104

t

1

102

<∆x2 >

0.11.01.52.02.53.0

1 2 3 4d/T

0.1

10

2F

kT=1.0

d=1.0

γ =d =

t

t1/2t1/2

_d2DSF√

d

d/kBT

kBT = 1.0

d = 1.0

(a) (b)

(c)

Figure 7: (Color online) Diffusion of a stochastic sin-gle file in the periodic potential (46) [66]: (a) 〈∆x2(t)〉vs. t for kBT = 1, d = 1 and increasing γ (fromtop to bottom on the l.h.s); (b) 〈∆x2(t)〉 vs. t forγ = 5 and increasing d (from top to bottom on ther.h.s). The t and t1/2 slopes (dashed lines) have beendrawn for reader’s convenience; (c) the diffusion mo-bility DSF vs. d/kBT for γ = 5 and d = 1 (circles)and kBT = 1 (squares). Other simulation parame-ters are: N = 3 × 103, L = 2π, l = 3 × 103L, and allparticles have unit mass.

This variation of the SF model rises naturally in con-nection with quasi-1D situations where the particles(not necessarily suspended in a fluid) can be repre-sented by disks moving along a narrow spatially mod-ulated channel with cross-section smaller than twicethe disk diameter [see Fig. 1(c)]. The confining ac-tion of the channel can be accounted for by modelingthe SF dynamics in terms of a periodic substrate po-tential with a effective strength d. This is the case,for instance, of most nanotubes and zeolite pores [2].

In the simulations of Refs. [60, 66] the i-th particlewas assigned: (i) random initial position, xi(0), andvelocity, xi(0). Upon each elastic collision it switched

11

velocity with either neighbors without altering the filelabeling; (ii) independent Brownian dynamics deter-mined by a viscous force −γxi and a thermal force√

2γkBTξi(t), with ξi(t) uncorrelated standard Gaus-sian noises defined as in Sec. 2.1, in order to guaran-tee thermalization at temperature T .

Numerical evidence led to conclude that the pe-riodic substrate potential V (x) does not invalidatethe sSF diffusion law (44), although the dependenceof the mobility factor DSF on the system parame-ters becomes more complicated. In panels (a) and(b) of Fig. 7 DSF is characterized as a function ofγ and T , respectively. The identity DSF =

√

D0/γ,reported above for V (x) ≡ 0, applies here, too, un-der the condition of replacing D0 with the modifieddiffusion constant D(F ) of Eq. (19). In Fig. 7(a) therescaled curves γ1/2〈∆x2(t)〉 versus t overlap asymp-totically for any damping regime.

The temperature dependence of the mobilityDSF =

√

D(F )/γ, is more interesting. As implicitin the discussion of Sec. 3, this prediction gets moreand more accurate for large γ and increasingly highactivation-to-thermal energy ratios d/kBT . As a con-sequence, the rescaled mobility

√

γ/d DSF turns outto be a function of d/kBT , alone, in good agreementwith the simulation results displayed in Fig. 7(c).

6.2 Driven single files

Finally, we consider the case of a driven sSF, namely,we now assume that all file particles are subjectedto an additional constant force F pointing, say, tothe right (F ≥ 0). We know from Sec. 3 that thediffusion of a single Brownian particle drifting downa tilted washboard potential exhibits enhanced nor-mal diffusion with diffusion constant (19). Extensivesimulation of a driven SF [66] yielded the numericaldata reported in Fig. 8. The kinetic mobility of thefile, defined as µ = 〈x〉/F , turned out to coincidewith the mobility of a single particle under the samedynamical conditions, irrespective of γ (inset of Fig.8). More remarkably, the subdiffusive regime (44)applies also in the presence of bias, though with anF -dependent mobility factor. When plotted versusd, DSF attained a maximum enhancement for d ≥ F ,i.e., in coincidence with the (noise-assisted) depinning

1 2 3 4d

0

1

2

3

2F 0 1 2 3

F/ d

1

γ µ

0.52.0

γ =F=1

F=0

DSF2DSF

Figure 8: (Color online) Diffusion of a driven stochas-tic single file in the periodic potential (46): the mobil-ity factor DSF vs d for kBT = 0.3, γ = 5, and F = 0(squares) and F = 1 (circles). The solid curves rep-resent the law DSF =

√

D(F )/π with D(F ) given inEq. (19). Other simulation details are as in Fig. 7.Inset: the kinetic mobility, µ = 〈x〉/F vs. F/

√d for

γ = 0.5 (circles) and γ = 2 (squares) at kBT = 0.1.The fitting curves are the analytical predictions for asingle Brownian particle based, respectively, on Eqs.(11.194) (low γ) and (11.50) (high γ) of Ref. [36].

of the file from its sinusoidal substrate. Again, theidentity DSF =

√

D(F )/π combined with formula(19) for D0 provided an excellent fit of the simulationdata for large γ. Of course, the mobility enhance-ment at depinning can be revealed also by plottingDSF versus F at constant d.

7 Conclusions and Outlook

With this minireview we presented the state of the artof diffusive transport occurring in systems character-ized by confinement due to either their finite size orto particle interactions in restricted geometries. Con-finement plays a salient role in the Brownian motionof driven particles. Indeed, the entropic effects asso-ciated with confinement may give rise to anomaloustransport features. Some main new phenomena arethe entropy-driven decrease of mobility with increas-ing temperature, the diffusion excess above the free

12

diffusion limit and the resonant behavior of the ef-fective diffusion coefficient as a function of controlparameters such as temperature, external gradients,or geometry design.

The reviewed transport features can thus be im-plemented in the design of new transport control se-tups or protocols. In particular, Brownian transportthrough confined geometries is expected to find appli-cation in nanotechnology, for instance, to operate de-vices for the separation of nanoparticles, to speed upschemes for catalysis on templates, or also to realizeefficient, driven through-flows of microsized agents,or reactants, in miniaturized lab-on-a-chip devices.

Acknowledgements

This work was made possible thanks to the finan-cial support by: the Volkswagen Foundation (projectI/80424, P.H.); the Deutsche Forschungsgemeinschaft(DFG) via project no. 1517/26-1 (P.S.B., P.H.)and via the research center, SFB-486, project A10(G.S., P.H.); the German Excellence Initiative, viathe Nanosystems Initiative Munich (NIM) (P.S.B.,P.H.); and the Alexander von Humboldt Stiftung, viaa Research Award (F.M.).

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