Entropic transport of finite size particles
Wolfgang Riefler, Gerhard Schmid, P Sekhar Burada, Peter Hanggi
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Entropic transport of finite size particles
W Riefler, G Schmid, and P H¨anggi Institut f¨ur Physik, Universit¨at Augsburg,Universit¨atsstr. 1, D-86135 Augsburg, Germany
P S Burada
Max-Planck Institute f¨ur Physik komplexer Systeme,N¨othnitzer Str. 38, 01187 Dresden, Germany
Transport of spherical Brownian particles of finite size possessing radii R ≤ R max through narrow channels with varying cross-section area is considered. Applying theso-called Fick-Jacobs approximation, i.e. assuming fast equilibration in orthogonaldirection of the channel axis, the 2D problem can be described by a 1D effectivedynamics in which bottlenecks cause entropic barriers. Geometrical confinementsresult in entropic barriers which the particles have to overcome in order to proceed intransport direction. The analytic findings for the nonlinear mobility for the transportare compared with precise numerical simulation results. The dependence of thenonlinear mobility on the particle size exhibits a striking resonance-like behavioras a function of the relative particle size ρ = R/R max ; this latter feature renderspossible new effective particle separation scenarios.
I. INTRODUCTION
The diffusive behavior of Brownian particles depends mainly on their size, the interactionbetween them, and the environment where they are situated in. If, in addition to thesecharacteristics, particles are confined within narrow, tortuous structures such as nanopores,zeolites, biological cells and microfluidic devices, the restriction of the space available for theparticles will cause entropic barriers that will have strong impact on the diffusive behavior(cf. Ref. [1] and references therein). Effective control schemes for transport in these systemsrequire a detailed understanding of the diffusive mechanisms involving small objects and,in this regard, an operative measure to gauge the role of fluctuations. The study of thesetransport phenomena is, in many respects, equivalent to an investigation of geometricalconstrained Brownian dynamics. As the role of inertia for the motion of the particles throughthese structures can typically be neglected, the Brownian dynamics can safely be analyzedby solving the Smoluchowski equation in the domain defined by the available free space uponimposing reflecting boundary conditions at the domain walls.However, solving the boundary problem in the case of nontrivial, corrugated domainspresents a difficult task. A way to circumvent this difficulty consists in coarsening the de-scription by reducing the dimensionality of the system considering only the main transportdirection, but taking into account the physically available space by means of an entropicpotential [1–3]. The resulting kinetic equation for the probability distribution, the so-calledFick-Jacobs equation, is similar in form to the Smoluchowski equation, but contains now en-tropic contributions leading to genuine dynamics which distinctly differs from those observedfor purely energetic potentials.The driven transport of particles across bottlenecks [1–4], such as ion transport throughartificial nanopores or artificial ion pumps [5–8] or in biological channels [9–12], are strikingexamples where the diffusive transport is regulated by entropic barriers. In addition, geo-metrical confinements and entropic barriers play also a prominent role in the context of theStochastic Resonance phenomenon [13–17].Our objective with this work is to investigate the mobility of noninteracting sphericalBrownian particles in channels with varying cross-section width. In particular, we are inter-ested in the influence of the particle size on the transport within a periodic entropic potentialexhibiting barriers which arise from the geometrical restrictions.The paper is organized as follows: in section 2 we introduce the model and define thetheoretical and numerical problem. Further on, in section 3 we present the basic principlesof the Fick-Jacobs approximation allowing for reducing the two-dimensional problem to anone-dimensional one. The results are presented in section 4. Finally, we give the mainconclusions in section 5.
II. MODELLING
Transport through pores or channels (like the one depicted in Fig. 1) may be caused bydifferent particle concentrations maintained at the ends of the channel, or by the applicationof external forces acting on the particles. Here, we exclusively consider the case of forcedriven transport of spherical particles of radius R . The external force ~F = F ~e x is pointingparallel to the direction of the channel axis. As small deviation from this assumption does notaffect our results, certainly not within the limits of validity of the Ficks-Jacob approximation.Moreover, we shall assume low concentrations of spherical particles such that particle-particleinteractions and all hydrodynamic interaction effects can consistently be neglected. A. Dynamics inside the channel
In general the dynamics of a suspended Brownian particle is overdamped [18] and welldescribed by the Langevin equation: η R d ~r d t = ~F + p η R k B T ~ξ ( t ) , (1)where ~r denotes the center position of the spherical particle in the two-dimensional channel, k B the Boltzmann constant, T the temperature and ~ξ ( t ) is the standard 2D Gaussian noisewith h ~ξ ( t ) i = 0 and h ξ i ( t ) ξ j ( t ′ ) i = 2 δ ij δ ( t − t ′ ) for i, j = x, y . The friction coefficient η R isgiven by Stokes’ law: η R = 6 πνR (2)and depends on the shear viscosity ν of the fluid and the particle radius R . In addition toEq. 1 the full problem is set up by imposing reflecting boundary conditions at the channelwalls. The boundary of the 2D periodic channel which is mirror symmetric about its x -axisis given by the periodic function y = ± ω ( x ) with ω ( x + L ) = ω ( x ) where L is the periodicityof the channel. ω min and ω max refer to the half of the maximum and minimum channel width,respectively.To further simplify the treatment of this problem, we introduce dimensionless variables.We measure all lengths in units of the periodicity of the channel, i.e. x = x ′ L . As a unit oftime τ we choose twice the time it takes the largest transporting particle to diffusively coverthe distance L which is given by τ = L η max / ( k B T ), hence t = τ t ′ . The largest transportableparticle is the particle with radius R max = ω min . Accordingly, the friction coefficient of aparticle of radius R is then given by η = ρη max with the ratio of spherical particle radiibeing ρ = R/R max and η max = 6 πνR max .Summarizing, the Langevin equation (1) reads in dimensionless variables:d ~r ′ d t ′ = fρ ~e x + r ρ ~ξ ( t ′ ) , (3) Lw min w max ~F R yy xx FIG. 1: (Color online) Sketch of the 2D periodic channel with periodicity L , the minimum halfchannel width ω min and the maximum half channel width ω max . The spherical Brownian particleof radius R is subjected to the force ~F . where the dimensionless force parameter [2, 3] f = LFk B T . (4)For the sake of better readability, we shall skip all the primes in the following and proceed,if not mentioned explicitly otherwise, with dimensionless variables.The corresponding Fokker-Planck equation for the time evolution of the probability dis-tribution P ( ~r, t ) takes the form [19] ∂P ( ~r, t ) ∂t = − ~ ∇ ~J ( ~r, t ) , (5)where ~J ( ~r, t ) is the probability current: ~J ( ~r, t ) = 1 ρ (cid:16) f~e x − ~ ∇ (cid:17) P ( ~r, t ) . (6) B. Boundary conditions
As the particles are confined by the channel structure, the probability current has tovanish at the boundaries. Due to the finite size of the particles their center position canaproach the boundary only up to its radius. Consequently, the position vector ~r of a particlewith radius R never approaches the channel walls and is restricted to only a portion ofthe inner channel area, cf. Fig. 1. The effective boundary function ω eff ( x ), which serves as yy xx FIG. 2: (Color online) Sketch of the original tube geometry given by the boundary function ω ( x )and the effective boundary function ω eff ( x ) for the center of the spherical particle. Hereby, theeffective boundary function depends on the radius of the particle which is given in dimensionlessunits by ρ : ρ = 0 .
27 (red dotted line), ρ = 0 .
81 (blue dashed line). boundary for the center of mass, exhibits the distance R from the original, true boundaryfunction ω ( x ). Consequently, the “no-flow” boundary conditions for the center of massdynamics read: ~J ( ~r, t ) · ~n = 0 , for ~r ∈ effective boundaries , (7)where ~n denotes the normal vector field at the effective channel walls. For the considered2D channel structure, the boundary condition becomesd ω eff ( x )d x (cid:26) f P ( x, y, t ) − ∂P ( x, y, t ) ∂x (cid:27) + ∂P ( x, y, t ) ∂y = 0 , (8)at y = ± ω eff ( x ).Note, that the effective boundary function exhibits a complex dependence on the particle’sradius R and could not be given explicitly. If the curvature of the channel wall function ω ( x ) is larger than that of the particle, the effective boundary function exhibits a kink, cf.Fig. 2.For an arbitrary form of ω ( x ), the boundary value problem defined by Eqs. (5), (6) and(8) is very difficult to solve. Despite the inherent complexity of this problem an approxi-mate solution can be found by introducing an effective one-dimensional description wheregeometric constraints and bottlenecks are considered as entropic barriers [2, 4, 20–25]. III. FICK-JACOBS APPROXIMATION
The 1D equation is obtained from the full 2D Smoluchowski equation upon the elimina-tion of the transversal y coordinate assuming fast equilibration in the transversal channeldirection. A. The Fick-Jacobs equation
The marginal probability density along the axis of the channel is defined by P ( x, t ) = Z ω eff ( x ) − ω eff ( x ) P ( x, y, t )d y . (9)Assuming fast equilibration in y -direction the 2D probability distribution becomes P ( x, y, t ) = P ( x, t ) Q ( y | x ) , (10)with the local equilibrium distribution Q ( y | x ) of y , conditional on a given x . If there isno force component in transversal channel direction (as it is in our case), the conditionaldistribution Q ( y | x ) is uniform and reads due to the normalization condition: Q ( y | x ) = 1 / (2 ω eff ( x )) . (11)Then on integrating the full 2D Smoluchowski equation (5) and making use of Eqs. (9),(10) and (11), the Fick-Jacobs equation for the spherical particle is obtained: ∂ P ( x, t ) ∂x = 1 ρ ∂∂x D ( x ) (cid:26) d A ( x )d x + ∂∂x (cid:27) P ( x, t ) , (12)with the dimensionless free energy A ( x ) = − f x − ln ω eff ( x ). For a periodic channel thisfree energy assumes the form of a tilted periodic potential with the bottlenecks formingentropic potential barriers. Note, that for a straight channel, i.e. constant effective boundaryfunction, the entropic contribution vanishes and the particle is solely driven by the externalforce.Introducing the x -dependent diffusion coefficient D ( x ) in Eq. 12 considerably improvesthe accuracy of the kinetic equation, extending its validity to more winding structures [21–25]. The expression for D ( x ) (in dimensionless units) D ( x ) . = 1 (cid:2) ω ( x ) / d x ) (cid:3) / , (13)has been shown to appropriately account for curvature effects of the confining walls [22]. B. Nonlinear Mobility
Besides the effective diffusion coefficient, the average particle current, or equivalently thenonlinear mobility serves as key quantity of particle transport through periodic channels.For any non-negative force the average particle current in periodic structures can be obtainedfrom Ref. [26–28] h ˙ x i = h t ( x → x + 1) i − , (14)where h t ( a → b ) i denotes the mean first passage time of particles starting at x = a toarrive at x = b . Within the Fick-Jacobs equation (12), the mean first passage time can bedetermined: h t ( a → b ) i = ρ Z ba d x exp( − f x ) /ω eff ( x ) Z x −∞ d y exp( f y ) ω eff ( y ) . (15)The nonlinear mobility µ ( f ) is defined by µ ( f ) = h ˙ x i /f and can be obtained as µ ( f ) = 1 ρ · − exp( − f ) f Z d zI ( z, f ) , (16)where I ( z, f ) = exp( − f x ) /ω eff ( x ) Z xx − d y exp( f y ) ω eff ( y ) . (17)In case of a straight channel with ω min = ω max , an exact analytical solution of the full2D Smoluchowski equation (5) is known and the nonlinear mobility equals the free mobility(i.e. without geometrical constrictions) µ = µ free = 1 /ρ = R max /R (for straight channels) . (18)Consequently, the influence of the confinement can be expressed by the ratio of nonlinearmobility for the transport through the channel and the one for the unrestricted case: µ ( f ) µ free = 1 − exp( − f ) f Z d zI ( z, f ) . (19) IV. PRECISE NUMERICS FOR A TWO-DIMENSIONAL CHANNELGEOMETRY
The nonlinear mobility, predicted analytically within the Fick-Jacobs approximation, hasbeen compared with Brownian dynamic simulations performed by a numerical integrationof the full 2D Langevin equation (3), using the stochastic Euler algorithm. As randomnumber generator we used the Box-Muller- and MT19937-algorithm from the
GSL library.The sinusoidal shape of the considered two-dimensional channel is described by ω ( x ) := a sin(2 πx ) + b , (20)with the two dimensionless channel parameters a and b . In physical units, these two pa-rameters are given by aL and bL , respectively. Note that ω ( x ) may also be regarded as thefirst terms of the Fourier series of a more complex boundary function. Due to the symmetrywith respect to the x -axis, the boundary function could be given in terms of the maximumhalf-width ω max = b + a and the aspect ratio of minimum and maximum channel width ǫ = ω min /ω max (with ω min = b − a ), i.e. ω ( x ) = ω max − ω min (cid:20) sin (2 πx ) + ω max + ω min (cid:21) , (21) ω ( x ) = ω max − ǫ ) (cid:20) sin (2 πx ) + 1 + ǫ − ǫ (cid:21) . (22)To ensure, that the spherical particles of radius ρ stay within this channel geometry, theintegration was carried out performing “no-flow” boundary conditions at the channel walls.By averaging over 10 simulations we obtain the steady-state average particle current h ˙ x i = lim t →∞ h x ( t ) i t , (23)and the nonlinear mobility µ = h ˙ x i /f . A. Nonlinear mobility: force and temperature - dependence
Figure 3 depicts the nonlinear mobility as a function of the scaling parameter f for twodifferent particle radii and a fixed channel geometry: ω ( x ) = 0 . / (2 π ) sin(2 πx ) + 1 . / (2 π ).Strikingly, the transport through such channel structures is distinctly different from the oneoccurring in one-dimensional periodic energetic potentials [2–4, 29]. This phenomenon is dueto the different temperature dependence of the barrier shapes. Decreasing the temperature inan energetic periodic potential decreases the transition rates from one cell to the neighboringone by decreasing the Arrhenius factor [30] and, therefore, reduces the nonlinear mobility.For the periodic channel system, a decrease of temperature results in an increase of the . . . µ (cid:14) µ f r ee µ (cid:14) µ f r ee ff . . .
95 20 40 60
FIG. 3: (Color online) Graph for the scaled nonlinear mobility as function of the force parameter f .In the Langevin simulation the different symbols correspond to different particle radius of ρ = 0 . ρ = 0 . .
01. The Fick-Jacobs results, Eq. (19), correspond to the solid lines. The boundary functionreads: ω ( x ) = (0 . / π ) sin(2 πx ) + 1 . / π . dimensionless force parameter f , cf. Eq. (4) and consequently, in a monotonic increase ofthe nonlinear mobility.Due the geometrical restrictions, the nonlinear mobility is always smaller than the mo-bility for the free case, cf. Fig. 3. With increasing scaling parameter, the nonlinear mobilitytends to that of the free case, i.e. µ → µ free for f → ∞ .A comparison of the analytics obtained by means of the Fick-Jacobs-approximation withthe precise numerics enables one to determine validity criteria for the Fick-Jacobs approx-imation, for further details see Ref. [3, 4]. According to them the applicability of theFick-Jacobs approximation for the transport of point particles depends on the smoothnessof the geometry and the scaling parameter f . For finite size particles, the diameter of theparticles should become an additional parameter in the validity criteria. In particular, theparticle’s radius determines the maximum width of the effective channel structure. A largerparticle leads to a smaller maximum effective width. Since for a fast equilibration in theorthogonal tube direction the timescale for the orthogonal diffusion process must be smallerthan the timescale for the drift [3], a smaller maximum channel width favors the valid-ity of the Fick-Jacobs approximation. Thus, with increasing particle size the range of theapplicability of FJ increases, cf. Fig. 3.0However, it turned out, that the transport phenomena presented below, occur for channelstructures for which the validity criteria is not fulfilled. Therefore, we stick in the following,to the numerical results only. B. Particle size
Surely, the transport of particles through small channel systems depends on the size ofthe particles. In particular, the effect of the size on the nonlinear mobility is two-fold.Firstly, the friction coefficient depends on the particle size, resulting in a ρ -dependence ofthe nonlinear mobility even for the case of unconstrained motion (free case), cf. Eq. (18).With increasing radii ratio ρ , the mobility declines. Secondly, there is the influence of thegeometrical confinement. The extent of the bottleneck (2 ω wmin ) gives a limit to the sizeof particles able to travel through the channel. In our scaling the largest sphere possibleto overcome the geometry’s bottleneck has a radii ratio of ρ = 1. Considering particles ofdifferent sizes, the effective bottleneck (2 ω effmin ) will be smaller for spheres of higher diameter.It is intuitive that a small bottleneck hinders the transport.Figure 4(a) depicts the nonlinear mobility as function of the radii ratio ρ for differentgeometries. For a straight channel one observes the nonlinear mobility of the free case µ free which depends reciprocally on the radii ratio ρ . In presence of geometrical restrictions, i.e.for varying cross-section width, the nonlinear mobility is smaller than µ free , cf. Fig. 4 (a).Moreover, upon decreasing the bottleneck half-width (i.e. decreasing the aspect ratio ǫ ) ofthe structure, the nonlinear mobility decreases [31].Deviations from the 1 /ρ -dependence show another effect of the geometrical confinement,cf. Fig. 4 (a). In order to focus on the geometrical effect, we consider the scaled nonlinearmobility, i.e. the nonlinear mobility relative to the nonlinear mobility in free case: µ/µ free ,cf. Fig. 4 (b). This is equivalent to the consideration of the nonlinear mobility of a pointparticle moving in the effective channel geometry defined by the effective boundary function ω eff ( x ), which still depends on the parameter ρ . With increasing ρ , the maximum half-widthof the effective geometry shrinks. As a consequence, the sojourn time, the particle spendson average in a bulge of the channel structure decreases with increasing ρ and the mobilityof the point particle in the effective geometry grows. This behavior causes the maximum inthe scaled nonlinear mobility and the shoulder in the dependence of the nonlinear mobility1 . . . µ (cid:14) µ f r ee µ (cid:14) µ f r ee . . . ρρ (b)(a)0 . . µµ ω max = . π ǫ = 0 . ǫ = 0 . ǫ = 0 . FIG. 4: (Color online) The numerically obtained non-linear mobility µ (a) and the scaled nonlinearmobility µ/µ free (b) as function of the radii ratio ρ = R/R max for different channel geometries( constant-width-scaling : ω max = const), cf. Eq. (22). on the radius, cf. Fig. 4. C. Role of the channel structure
The confinement by the considered channel geometry can be altered by systematicallychanging the parameters ω max and ω min or ω max and ǫ in the boundary function, Eq. (21) orEq. (22) respectively. While for Fig. 4 we examined a constant maximum half-width ω max = const and varied the aspect ratio ǫ which is equivalent to vary the half-width ω min atthe bottleneck, cf. the constant-width-scaling in Ref. [31], it is instructive to also considerboth, the constant-bottleneck-scaling ω min = const as well as the constant-ratio-scaling ǫ = ω min /ω max = const.As we keep the bottleneck-width constant and decrease the maximum half-width, thesojourn times the particles spends in the bulges decreases causing the mobility to increaseand approach the maximum value for a straight channel ( ω min = ω max and ǫ = 1), cf. Fig. 52 . . . µ / µ f r ee µ / µ f r ee . . . ρρ (c)0 . . µµ ω min = π ω max = π ω max = π ω max = π (a) 0 . . ρρ (d) ǫ = 0 . ω max = . π ω max = π ω max = π (b) FIG. 5: (Color online) The nonlinear mobility (a), (b) and scaled nonlinear mobility (c),(d) aredepicted for different scalings of the geometry: constant-bottleneck-scaling, i.e. ω min = const, in(a) and (c) ; constant-ratio-scaling, i.e. ǫ = ω min /ω max = const, in (b) and (d). (a). In contrast, within the constant-ratio-scaling , where the bottleneck half-width scaleswith the maximum half-width, the nonlinear mobility µ depends for small radii only slightlyon ω max , cf. Fig. 5 (b). However, with increasing particle size, i.e. radii ratio ρ , the nonlinearmobility µ shows a striking dependence on the maximum half-width.As pointed out already, with increasing particle radius the effective maximum half-widthdecreases. The effective maximum half-width will exhibit a linear dependence on the radiusif the particles curvature is larger than that of the channel’s boundary function (in physicalunits), ω effmax = ω max − R , for 1 /R > − d ω ( x max ) / d x (cid:2) ω ( x max ) / d x ) (cid:3) / , (24)where x max denotes the x -values for which the boundary function assumes a maximum. Forlarger particle radii the effective boundary function shows a kink and the effective maximumhalf-width ω effmax decreases faster than linearly with the radii ratio ρ . As the sojourn timesthe particle spend in the channel’s bulges depends mainly on the effective maximum width,3a nonlinear dependence of the nonlinear mobility is observed for larger particle radii causinga peak in the scaled nonlinear mobility, cf. Fig. 5 (c) and (d). V. CONCLUSIONS AND OUTLOOK
We studied the transport of finite Brownian particles through channels with periodicallyvarying width. For point size particles it was shown previously [3, 4], that the transportthrough such channels could be approximately described by means of the so-called Fick-Jacobs equation which is based on the assumption of a fast equilibration in orthogonaltransport direction. Validity criteria for the capability of this approximation include adependence on the channel shape and predict an upper limit for the force value. In caseof spherical, finite size particles the maximum force value up to which the Fick-Jacobsequation could be applied depends also on the size of the particle. By comparison of theapproximative result for the nonlinear mobility and the numerical ones we have shown, thatthe equilibration assumptions holds for a wider force-range in case of larger particles thanit is the case for smaller ones.In addition, we pointed out, that the transport of finite, spherical Brownian particlesin channel geometries with highly corrugated channel walls exhibits some striking featureswhich may allow for the development of newly separation devices which extends the func-tionality of the sieves. In particular we found, that the nonlinear mobility of Brownianparticles in such channel structures deviates from the one-over-size dependence predictedby the Stokes law for Brownian particles moving in an environment without geometricalconstrictions. Instead, there is an optimal particle size for which the nonlinear mobility ascompared to the free mobility exhibits a maximum value.Our present study also implicitly used a small concentration of spherical particles suchthat both, effects of particle-particle interactions and forces between particle-particle andparticle-walls due to hydrodynamic interactions can safely be ignored. These complicationswould require totally new and extensive studies that are beyond this present study. More-over, as emphasized with the abstract already, we assumed throughout perfect sphericalsymmetry. Deviations from such spherical symmetry would also impact the viscous frictionlaw behavior [32] and, as well, may give rise to additional, new entropic effects. All suchcomplications are beyond the work presented here; all these latter complications, however,4open up avenues for interesting future investigations.
Acknowledgments
This work has been supported by the Volkswagen foundation (project I/83 902), theMax-Planck society and by the German Excellence Initiative via the Nanosystems InitiativeMunich (NIM).
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