In eccentric septate channels the pores connecting adjacent compartments are shifted off-axis, either periodically or randomly, so that straight trajectories parallel to the axis are not allowed. Driven transport of a Brownian particle in such a channel is characterized by a strong suppression of the current and its dispersion. For large driving forces, both quantities approach an asymptotic value, which can be analytically approximated in terms of the stationary distribution of the particle exit times out of a single channel compartment.

Brownian transport in coaxial compartmentalized channels1–6 is a topic of potential applications to both natural7 and artificial devices.8 These channels are formed by identical compartments connected by narrow openings (pores) centered around the channel axis. At variance with the case of smoothly corrugated channels, also called entropic channels,9 introduced first in Ref. 10 and further investigated in Refs. 11–15, diffusion in compartmentalized, or septate, channels cannot be reduced to an effective one-dimensional (1D) kinetic process directed along the axis. Accordingly, driven transport in such strongly constrained geometries exhibits distinct features, which cannot be reconciled with the known properties of Brownian motion in quasi-1D systems.9,16 Two transport quantifiers best represent the difference between these two channeled flow regimes:

  • Mobility. The response of a Brownian particle in a channel subjected to a dc drive, F, oriented along the axis direction, x, is expressed by the normalized mobility:
    (1)
    where γ is the particle damping constant and
    $\langle \dot{x}(F) \rangle\break =\lim _{t\rightarrow \infty }[\langle x (t) \rangle -x(0)]/t$
    ẋ(F)=limt[x(t)x(0)]/t
    . In entropic channels μ(F) increases from a relatively small value for F = 0,
    $\mu _0$
    μ0
    , up to the free-particle limit,
    $\mu _\infty =1$
    μ=1
    , for F → ∞.14 On the contrary, in coaxial compartmentalized channels μ(F) decreases monotonically with increasing F toward a geometry dependent asymptotic value,
    $\mu _\infty$
    μ
    , equal to the ratio of the pore to the channel cross section.1 
  • Diffusivity. As the Brownian particle is driven across a periodic array of compartment pores, its diffusivity
    (2)
    picks up a distinct F dependence. In entropic channels with smooth pores, the function D(F) approaches the free-diffusion limit for F → ∞,
    $D(\infty )=D_0$
    D()=D0
    , after going through an excess diffusion peak centered at an intermediate (temperature dependent14) drive value. For a particle of mass m, the bulk diffusivity,
    $D_0$
    D0
    , is proportional to the temperature, that is,
    $D_0=kT/m\gamma$
    D0=kT/mγ
    . Such a peak signals the depinning of the particle from the entropic barrier array.17 In coaxial compartmentalized channels, instead, D(F) exhibits a distinct quadratic dependence on F,3,6 reminiscent of Taylor's diffusion in hydrodynamics.18 This observation suggests that the particle never frees itself from the geometric constriction of the compartment pores, no matter how strong F. In the absence of external drives, Einstein's relation16 
    (3)
    establishes the dependence of the transport parameters on the channel compartment geometry and temperature under equilibrium conditions.2,4,6,19

In this paper, we show that the above properties for driven transport in a coaxial compartmentalized channel cease to apply when the pores are shifted off-axis in different directions, according to either a periodic or a random pattern, as it is often the case in real septate channels. Due to the trapping action exerted by the compartment walls, (i) the mobility curve drops to zero inversely proportional to F, and (ii) Taylor's diverging branch of the diffusivity curve levels off to a horizontal geometry dependent asymptote, D = D(∞).

We start considering a massless or overdamped, pointlike Brownian particle free to move in a two-dimensional (2D) periodic eccentric channel. An external force F is applied to the particle parallel to the channel axis, say, in the positive direction. The suspension fluid is unmovable (no microfluidics effects are considered here) and its interactions with the particle are modeled by equilibrium thermal fluctuations with constant temperature T. The channel, straight, oriented in the x direction, and with constant width

$y_L$
yL⁠, is periodically segmented by means of orthogonal cross walls, a fixed distance
$x_L$
xL
apart, each bearing an opening of fixed cross section Δ. As illustrated in Fig. 1(a), the pore centers are shifted off-axis a distance
$y_h$
yh
, alternately, in the ±y direction.

The overdamped dynamics of the suspended Brownian particle in the channel is modeled by the 2D Langevin equation:

(4)

where

${\vec{r}}=(x,y)$
r=(x,y)⁠,
${\vec{e}}_x, {\vec{e}}_y$
ex,ey
are the unit vectors along the x, y axes, and
${\vec{\xi }}(t)=(\xi _x(t),\xi _y(t))$
ξ(t)=(ξx(t),ξy(t))
are zero mean, white Gaussian noises with autocorrelation functions
$\langle \xi _i(t)\xi _j(t^{\prime })\rangle = 2\delta _{ij}\delta (t\break -t^{\prime })$
ξi(t)ξj(t)=2δijδ(tt)
and i, j = x, y. The noise strength
$D_0$
D0
plays the role of bulk diffusivity for a freely diffusing particle. Note that, to simplify our notation and with no loss of generality, we set γ = 1. Equations (4) have been numerically integrated assuming reflecting walls.16 Stochastic averages were obtained as ensemble averages over
$10^6$
106
trajectories with random initial conditions; transient effects were estimated and subtracted. The stationary 2D distributions of the driven particle were used to visualize the transport flows through the channel compartments. For large drives, particle beams crossing the compartments and diffusing around the compartment walls are clearly detectable, see example of Fig. 1(a). Finally, for the sake of an analytical treatment, we remark that Eqs. (4) can be conveniently rewritten in terms of the rescaled units
$t \rightarrow tD_0$
ttD0
and
$F \rightarrow F/D_0$
FF/D0
. A straightforward dimensional argument shows that the particle mobility, Eq. (1), and diffusivity in units of
$D_0$
D0
, Eq. (2), are both functions of
$F/D_0$
F/D0
, only, for any given channel geometry.

For large enough pore eccentricity, namely for

$y_h\break >\Delta /2$
yh>Δ/2⁠, the central lane of the channel is blocked, so that most particle trajectories overcome the compartment walls by diffusing in the transverse direction and eventually crossing the pores at their inner edge. This mechanism is apparent in the large drive regime of Fig. 1(a): a free-particle beam hits the longer side of the compartment wall, delimited by its pore and the channel wall; it forms an “injection” spot on the wall, which lies opposite to the exit pore with respect to the channel axis. The particle leaves the compartment only after transverse diffusing against wall from the injection spot to the exit pore, with random exit time τ. For large drives, such an exit process is largely independent of the drive itself and its average escape time, 〈τ〉, is substantially longer than the compartment crossing time,
$x_L/F$
xL/F
. Under such driving conditions, one can model Brownian transport in an eccentric channel as a renewal process.20 Accordingly, the mobility and the diffusivity at large F are expressed in terms of the first two moments, 〈τ〉 and
$\langle \tau ^2 \rangle$
τ2
, of the τ distribution, namely

(5)
(6)

The τ distribution can be approximated analytically in the regime of large drives, under the condition that the particle is confined against the cavity compartments within a layer of thickness

$\lambda _F=D_0/F$
λF=D0/F⁠, negligible compared to the pore width.1 Transport is, thus, controlled by the free-transverse diffusion of the particle against the compartment walls. For
$\lambda _F \ll \Delta$
λFΔ
the particle escape from a 2D channel compartment can then be modeled as a standard 1D exit process for the Brownian particle diffusing in a segment with reflecting endpoint
$\pm y_L/2$
±yL/2
and absorbing endpoint
$\mp (y_h-\Delta /2)$
(yhΔ/2)
; the particle starting point,
$\pm (y_h-\Delta /2)$
±(yhΔ/2)
, coincides with the y coordinate of the inner pore edge the particle has crossed last. This problem is analytically tractable;21 hence, the explicit expressions for the mobility,

(7)

and the diffusivity,

(8)

with

$\delta _h=y_L/(2y_h-\Delta )$
δh=yL/(2yhΔ)⁠. These analytical predictions reproduce quite closely the large drive regime of the numerical data plotted in Figs. 1(b) and 1(c). Note that, as expected, the agreement improves on increasing the pore eccentricity. In the limits Δ → 0 and
$y_h \rightarrow y_L/2$
yhyL/2
, the saturation value of D(F) attains a minimum, that is
$D(\infty )/D_0 = 2x_L^2/3y_L^2$
D()/D0=2xL2/3yL2
.

The results obtained so far can be extended, at least in principle, to three-dimensional (3D) septate channels. Let us consider, for example, a cylindrical compartmentalized channel with circular pores and longitudinal x-y section coinciding with the 2D channel discussed above, see sketch in Fig. 2(a). Here, the radii of the channel and the pores are renamed

$r_1$
r1 and
$r_0$
r0
, respectively, while the pore distance from the axis is denoted by
$r_h$
rh
. The numerical simulation of this channel requires adding to the Langevin equations (4), a third spatial coordinate z. The ensuing F dependence of μ(F) and D(F), reported in Fig. 2(b), is qualitatively similar to that observed for the 2D channels of Fig. 1.

FIG. 1.

(a) Sketch of an eccentric 2D septate channel directed along the x axis. Each compartment is

$x_L$
xL long and
$y_L$
yL
wide; the cross section of the connecting pores is Δ; pores have been moved of-axis up and down alternately by
$y_h$
yh
. The stationary 2D distribution of a driven particle is shown for F = 50. (b) Mobility curve μ vs
$F/D_0$
F/D0
for different
$y_h$
yh
. The dashed lines represent the asymptotic law derived in Eq. (7). (c) Diffusivity curve
$D/D_0$
D/D0
vs
$F/D_0$
F/D0
for different
$y_h$
yh
. The dashed lines correspond to the saturation values of Eq. (8). Inset. A comparison with diffusion in the coaxial channel geometry,
$y_h=0$
yh=0
(Taylor's diffusion). Other simulation parameters in panels (b) and (c).
$x_L=1$
xL=1
,
$y_L=1$
yL=1
, Δ = 0.05, and
$D_0=0.05$
D0=0.05
.

FIG. 1.

(a) Sketch of an eccentric 2D septate channel directed along the x axis. Each compartment is

$x_L$
xL long and
$y_L$
yL
wide; the cross section of the connecting pores is Δ; pores have been moved of-axis up and down alternately by
$y_h$
yh
. The stationary 2D distribution of a driven particle is shown for F = 50. (b) Mobility curve μ vs
$F/D_0$
F/D0
for different
$y_h$
yh
. The dashed lines represent the asymptotic law derived in Eq. (7). (c) Diffusivity curve
$D/D_0$
D/D0
vs
$F/D_0$
F/D0
for different
$y_h$
yh
. The dashed lines correspond to the saturation values of Eq. (8). Inset. A comparison with diffusion in the coaxial channel geometry,
$y_h=0$
yh=0
(Taylor's diffusion). Other simulation parameters in panels (b) and (c).
$x_L=1$
xL=1
,
$y_L=1$
yL=1
, Δ = 0.05, and
$D_0=0.05$
D0=0.05
.

Close modal
FIG. 2.

(a) Sketch of an eccentric 3D septate channel directed along the x axis. The pores have been moved off-axis up and down alternately by

$r_h$
rh in the x-y plane. (b) Mobility, μ (main panel), and diffusivity curves,
$D/D_0$
D/D0
(inset), vs
$F/D_0$
F/D0
for different
$r_h$
rh
and
$r_0$
r0
. The dashed lines represent the corresponding asymptotic laws, Eqs. (5) and (6), with 〈τ〉 and
$\langle \tau ^2 \rangle$
τ2
obtained by numerical simulation (see text). Other simulation parameters:
$x_L=1$
xL=1
,
$D_0=0.01$
D0=0.01
,
$r_1=1$
r1=1
;
$r_h$
rh
and
$r_0$
r0
are given in the legends. Arrows denote the asymptotic mobility,
$\mu _{\infty }=(r_0/r_1)^2$
μ=(r0/r1)2
, for 3D coaxial septate channels (Ref. 4).

FIG. 2.

(a) Sketch of an eccentric 3D septate channel directed along the x axis. The pores have been moved off-axis up and down alternately by

$r_h$
rh in the x-y plane. (b) Mobility, μ (main panel), and diffusivity curves,
$D/D_0$
D/D0
(inset), vs
$F/D_0$
F/D0
for different
$r_h$
rh
and
$r_0$
r0
. The dashed lines represent the corresponding asymptotic laws, Eqs. (5) and (6), with 〈τ〉 and
$\langle \tau ^2 \rangle$
τ2
obtained by numerical simulation (see text). Other simulation parameters:
$x_L=1$
xL=1
,
$D_0=0.01$
D0=0.01
,
$r_1=1$
r1=1
;
$r_h$
rh
and
$r_0$
r0
are given in the legends. Arrows denote the asymptotic mobility,
$\mu _{\infty }=(r_0/r_1)^2$
μ=(r0/r1)2
, for 3D coaxial septate channels (Ref. 4).

Close modal

However, a direct comparison with the corresponding asymptotic laws in Eqs. (5) and (6) is now more complicated, as simple analytical expressions for 〈τ〉 and

$\langle \tau ^2 \rangle$
τ2 are not available. For large drives,
$\lambda _F \ll r_0$
λFr0
, the escape of the transported particle out of a cylindrical compartment can be modeled as the exit problem of the same particle diffusing in a 2D eccentric annulus, where the outer circle represents the reflecting wall and the inner off-axis circle plays the role of the absorbing pore edge. The particle is initially confined to a circle of the same size as the inner circle, but placed opposite to it with respect to the center of the outer circle (channel axis). In other words, the particle beam creates an image of its source pore. Moreover, in the limit of small pores, the particle distribution along such circle can be taken uniform. We solved such a 2D exit problem by means of numerical simulation; on inserting the values for 〈τ〉 and
$\langle \tau ^2 \rangle$
τ2
, thus obtained, in Eqs. (5) and (6), we produced the asymptotic fits of the mobility and diffusivity data shown in Fig. 2(b).

Finally, we address a last variation of the 2D eccentric septate channel of Fig. 1. Let us assume that the distance

$y_L$
yL of the pores from the channel axis is randomly distributed in the interval
$\pm {\textstyle\frac{1}{2}}(y_L-\Delta )$
±12(yLΔ)
, according to a certain normalized distribution. We simulated the case of truncated Gaussian distributions with different variance σ. Our numerical curves for μ(F) and D(F) are displayed in Fig. 3. As one might have expected, on increasing σ the horizontal asymptote of the mobility drops from
$\mu _\infty =\Delta /y_L$
μ=Δ/yL
(coaxial channel1) to zero, its slope attaining a maximum for a uniform
$y_h$
yh
distribution (i.e., for σ = ∞). Analogously, the diffusivity gets dramatically suppressed with increasing σ: Taylor's divergence, peculiar of the coaxial channels, disappears as soon as σ becomes of the order of Δ/2; it is replaced by a saturation upperbound, D(∞), which is a decreasing function of σ.

FIG. 3.

Mobility in a randomly eccentric 2D septate channel. Here the pores are equally spaced along the axis, but randomly displaced perpendicular to it, according to a Gaussian distribution with variance σ. The mobility, μ, is plotted v

$F/D_0$
F/D0 for different σ. Other simulation parameters:
$x_L=1$
xL=1
,
$y_L=1$
yL=1
, Δ = 0.05, and
$D_0=0.05$
D0=0.05
.

FIG. 3.

Mobility in a randomly eccentric 2D septate channel. Here the pores are equally spaced along the axis, but randomly displaced perpendicular to it, according to a Gaussian distribution with variance σ. The mobility, μ, is plotted v

$F/D_0$
F/D0 for different σ. Other simulation parameters:
$x_L=1$
xL=1
,
$y_L=1$
yL=1
, Δ = 0.05, and
$D_0=0.05$
D0=0.05
.

Close modal

The conclusion of this paper is that driven transport through an eccentric channel is characterized by saturation of both the current and its dispersion. Further increasing the intensity of the driving force would make transport no more efficient. Moreover, the onset of saturation mechanisms and the relevant saturation points are quite sensitive to the geometry of the channel compartments, a property that one has to keep in mind when designing and operating an artificial porous device for particle manipulation.

Moreover, the present study can be easily generalized to deal with the transport of finite size particles. In this case, the current through the pores is largely affected by excluded volume effects, so that transport in a septate channel can become highly sensitive to the size and the shape of the channeled particles.22 

As an example of current technological interest we mention here certain 2D superconducting devices, where magnetic vortices, or fluxon, can be channeled along finely tailored tracks and then forced to either direction by applying one or more tunable currents perpendicular to the channel axis.23 As magnetic vortices behave like classical overdamped particles,8 the dynamics of Eq. (4) can be easily generalized to design new fluxon circuitry.

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