Entropically-induced asymmetric passage times of charged tracers across corrugated channels

We analyze the diffusion of charged and neutral tracers suspended in an electrolyte embedded in a channel of varying cross-section. Making use of systematic approximations, the diffusion equation governing the motion of tracers is mapped into an effective $1D$ equation describing the dynamics along the longitudinal axis of the channel where its varying-section is encoded as an effective entropic potential. This simplified approach allows us to characterize tracer diffusion under generic confinement by measuring their mean first passage time (MFPT). In particular, we show that the interplay between geometrical confinement and electrostatic interactions strongly affect the MFTP of tracers across corrugated channels hence leading to alternative means to control tracers translocation across charged pores. Finally, our results show that the MFPTs of a charged tracer in opposite directions along an asymmetric channel may differ. We expect our results to be relevant for biological as well synthetic devices whose dynamics is controlled by the detection of diluted tracers.


I INTRODUCTION
The control of the properties of tracer transport along channels or pores is a key issue for a variety of situations. For example, the development of micro-and nanofluidic devices relies on the understanding of the tracer motion in an electrolyte embedded in micro-and nanometric confinement [1]. Moreover, several biological processes such as neuronal signaling, ion pumping, photosynthesis and ATPase, just to mention a few, rely on the transport of ions across membranes or through channels [2]. In general, the structure of the conduit tracers travel through is inhomogeneous and it may present bottlenecks or cavities that can alter the overall transport properties, as it has been recently shown for both charged [3][4][5][6][7][8][9] as well neutral systems [10][11][12][13]. In particular, inhomogeneities in the properties of the channel can lead to rectification [5], diode-like behavior [14,15], or recirculation and negative mobility [8,9]. In many of the aforementioned scenarios the main aim is to control the current of tracers at steady state. However, diverse biological as well as synthetic scenarios are controlled by the recognition of very diluted receptors as it is for diffusion limited reactions and pattern forming systems [16], transport across nuclear [17,18] or plasma [19] membrane or, in general, for detectors of solutes diffusing through porous media [20,21]. Concerning the latter, cone-like pores have been particularly exploited in resistive-pulse sensing techniques to measure properties of diverse particle raging from micro-to nano-metric scales [22][23][24][25].For such systems the time a tracer takes to reach a given target for the first time, namely the Mean First Passage Time (MFPT), constitutes a standard and useful indicator.
In this contribution we study the MFPT of both charged and neutral tracers across channels of varying cross-section characterized by charged channel walls.
Our results show a remarkable dependence of the MFPT on particle charge as well as on channel corrugation. For positively charged channel walls, positive (negative) tracers are depleted (attracted) towards the channel walls and their MFPT is enlarged (reduced). We show that this asymmetric response is especially enhanced when the Debye length, κ −1 is comparable to the channel average section h 0 , consistent with previous results on asymmetric charged tracer motion through inhomogeneous channels [8,9]. Moreover, the MFPT is sensitive to the direction in which the channel is crossed. In particular, such a feature persists also for neutral tracers hence underlying its geometrical origin. We exploit a systematic procedure to approximate the tracer MFTP and develop a framework that allows us to disentangle the geometric (entropic) contribution from the electrostatic (enthalpic) and therefore to identify the interplay between the geometrical constraints and the inhomogeneous distribution provided by the electrostatic interactions. The structure of the text is the following: in section II we derive the 1D effective equation for charged tracers moving in a channel of varying cross-section, in section III we present our results and in section IV we summarize our conclusions.

II THEORETICAL FRAMEWORK
To capture the main features of the interplay between the geometrically induced local rectification provided by the varying-sectionof the channel and the electrostatic field, we study a z − z electrolyte embedded in a channel of varying cross-section, see Fig. 1. We assume that particles are constrained in a channel of varying cross-section whose y-section changes solely along the x-direction and it is constant along z. The channel section accessible to the center of mass of a point-like tracer is 2h(x)L z , being h(x) the half-width of the channel along the y-direction and L z the (constant) width of the channel along the z-direction.
The motion of a suspension of non-interacting charged particles is characterized by a convection-diffusion equation, which in the overdamped regime reads (1) where P q quantifies the probability distribution of tracers of charge eq, where e stands for the elementary charge, D is the tracer diffusion coefficient, β = 1/k B T the inverse thermal energy for a system at temperature T (being k B the Boltzmann constant) and W (x) is the total conservative potential acting on the tracers. We encode the presence of the channel and the electrostatic potential, in the overall potential W (x) defined as: that is periodic along the longitudinal direction x, W (x) = W (x + Le x ), constant along the z direction and confines particles inside the channel. In order to find the electrostatic potential, ψ(x), inside the channel, we should solve the Poisson equation where ǫ corresponds to the medium dielectric constant. The electrostatic potential has to satisfy the boundary condition of constant potential ζ (or prescribed charge density) for conducting (or insulating) channel walls. In Eq. (3) ρ q = ρ 0 exp (−βzeψ(x)) corresponds to the equilibrium ion charge density inside the channel in the absence of tracers. Assuming smoothly-varying channel walls, ∂ x h ≪ 1, we can take advantage of the lubrication approximation, ∂ 2 x ψ(x) ≪ ∂ 2 y ψ(x) and reduce Eq. 3 to a 1D equation for the potential ψ(x). Since the electrostatic field is perpendicular to the channel walls, for channel of varying cross-section we must consider the projection of the electrostatic field along the channel when solving the Poisson equation, as shown in Fig. 2. For smoothly-varying amplitude channels, for which d x h(x) ≪ 1, the projected electrostatic field reduces to where α = arctan (d x h(x)). Therefore, this geometric correction is of second order in d x h(x) and can be neglected in the following. We can exploit the regime d x h(x) ≪ 1, for which the lubrication approximation holds, to simplify Eq. 1. Specifically, we factorize P q (x, t) to arrive at where h 0 is the average amplitude of the channel. Eq. 6 reproduces the well-known Fick-Jacobs approximation [10,[26][27][28][29] that has been exploited in diverse situations, such as entropic resonance [30], cooperative rectification [31] and entropic splitters [32]. Integrating over the channel cross section, we obtain where now the confinement is encoded in the effective potential that, being the integral of the Boltzmann weight over all possible configurations for a given longitudinal position, x, can be interpreted as an equilibrium free energy. In the last step we have taken advantage of the fact that all the quantities of interest are independent of z. Hence, without loss of generality we have assumed eqψ(x, y)e −βeqψ(x,y) dy (9) from Eq. 6 we can define the entropy along the channel In order to keep analytical insight, we assume low salt concentration in the electrolyte and a small potential on the channel walls, ζ, i.e. βeζ ≪ 1. In this regime we can linearize the Poisson-Boltzmann equation, hence the electrostatic potential inside a conducting walls channel (similar results can be obtained for insulating channel walls) reads where κ = 4πℓ B z 2 (ρ + (x) + ρ − (x)) is the inverse Debye length for an electrolyte of valence z and solution ionic strength ρ 0 z 2 , ℓ B = βe 2 /4πǫ stands for the electrolyte Bjerrum length. In this linear regime, when also βeqψ(x, y) ≪ 1, we can linearize Eq. 10 getting where the entropy has a clear geometric interpretation, being the logarithm of the space, 2h(x), accessible to the center of mass of a point-like tracer. Accordingly, we introduce the entropy barrier for neutral tracers, ∆S 0 , defined as which represents the difference, in the entropic potential, evaluated at the maximum, h max , and minimum, h min of the channel aperture. Finally, we can define the total effective free energy difference as which using Eqs. 9,10 leads to

III RESULTS
We will analyze the motion of charged tracers in a channel whose half section along the y-direction is characterized by where h 0 is the average channel section, and h 1 is its modulation amplitude and assume the channel to be flat along the z-direction. φ controls the channel shape with respect to its boundaries fixed at x = 0 and x = L. Accordingly, the maximum and minimum channel apertures In order to characterize the diffusion of tracers in such channels, we study the time tracers take to get at a prescribed channel end for the first time. In particular, we focus on the mean of such a quantity, namely the mean first passage time (MFPT) tracers take to pass across the channel. In the following we assume that one of the ends of the channel, namely the one at x = 0, is in contact with a reservoir of tracers and we measure the MFPT that positive, negative or neutral tracers, t q (x), take to diffuse from a given position inside the channel x to the channel end at x = L. Such a situation corresponds to a reflecting boundary condition on the end of the channel in contact with the reservoir, i.e at x = 0, and to an absorbing condition at the other end, at x = L [38]. Unless otherwise specified, we assume φ = 0 in Eq. 16, for which channel bottlenecks are located at the channel ends. Taking advantage of the 1D projection, Eq. 7, we calculate the x-dependent MFPT, t(x), from [33]: From the numerical solution of this expression, the MFTP of tracers is derived, T q = t q (0). Fig. 3 shows the inverse MFPT for positive and negative tracers across a channel of varying cross-section normalized by the MFPT of neutral tracers, T 0 , whose MFPT does not depend on the Debye length κ −1 . When κ −1 is comparable with the channel average amplitude, h 0 , negative tracers benefit from the attraction to the positively charged and display an enhanced diffusion. On the contrary positive tracers, depleted from the walls, suffer a caging effect due to the entropic barrier resulting in a larger MFPT as compared to that of negative or neutral tracers. Such feature reminds the one observed for tracers in a porous media obtained by coarse-grained numerical simulations [34,35]. Interestingly, such a modulation in the MFPT for charged tracers vanishes for κh 0 ≪ 1 as well as for κh 0 ≫ 1, in agreement with the entropic electrokinetic regime observed for tracers under external forcing [8] and under chemical potential gradients [9]. We can gain insight in the dependence of the MFPT on κ by analyzing the effective barrier experienced by the tracers, quantified by the free energy difference ∆A q as defined in Eq. 15. Fig. 3(b) shows that the dependence of ∆A q on κ is sensitive to tracers' charge. In particular, for κh 0 → 1 the free energy barriers of negative tracers (blue squares) diminishes while the opposite holds for positive tracers (red squares). Such a diverse behavior of ∆A q according to tracer charge explains the different behavior of the MFPT shown in Fig. 3: positive tracers, which experience an enhanced free energy barrier, will take longer to cross the channel as compared to negative tracers.
Using Eqs. 6, 9, 10 we can separately quantify the entropic and enthalpic contributions to the effective free energy difference. As shown in Fig. 3(b), the entropic con-tribution (circles) is mildly affected by variations in κh 0 whereas the enthalpic contribution (diamonds) shows a strong sensitivity. Moreover, while for positive tracers (red points) the enthalpic and a entropic contribution sum up amplifying the magnitude of ∆A q , for negative tracers (blue points) the two contributions have different sign therefore reducing the magnitude of ∆A q .
We can simplify Eq. 17, even in the non-linear regime βeqψ > 1, by assuming ∂ x A q (x) to be piece-wise linear.
Choosing ∂ x A q (x) = ±2 ∆Aq L for x < L/2 or x > L/2 respectively, we can analytically solve Eq. 17 gettinḡ In the limit ∆A q → 0, Eq. 14 leads toT q = L 2 2D in agreement with Eq. 17. When ∆A q = 0, we can substitute the values of ∆A q derived from Eq. 14 into Eq. 18. Fig. 3(a) shows that this approximation provides a very good quantitative agreement with the numerical solutions.
In the linear regime, βeqψ(x, y) 1, we can obtain analytic expressions for the effective free energy barrier experienced by a charged tracer where ∆Θ = tanh (κh min ) κh min − tanh (κh max ) κh max (20) which shows that the enthalpic contribution always vanishes when βζeq → 0 and κh → ∞ or κh → 0, and ∆A q reduces to the entropic barrier, ∆S 0 experienced by neutral tracers. On the contrary, when κh ≃ 1 the enthalpic contribution is relevant and ∆A q retains a dependence the charge: positive (negative) tracers experience an enhanced (reduced) free energy barrier, as shown in Fig. 3(b). We can characterize the dependence of the MFPT on the channel geometry by considering a prescribed electrolyte, κ, and a charged channel with electrostatic potential, ζ, and vary the entropic barrier, ∆S 0 . For φ = 0 (similar results have been obtained for φ = ±π) Fig. 3(c) shows a monotonous increase in the MFPT for all tracers upon increasing ∆S 0 . For neutral tracers such an increase in the MFPT is the signature of the entropic-induced modulation in tracer transport due to the varying-section of the channel. In particular, positive tracers are the most sensitive to the entropic barrier whereas negative tracers are the least sensitive to variations in ∆S 0 , in agreement with the prediction of Eq. 19. For φ = 0, ±π, Fig. 3(c) shows two opposite regimes. While for φ = π/2 the behavior is similar to φ = 0, ±π except that all tracers experience an enhanced dependence on ∆S 0 , for φ = 3π/2, the dependence of the MFPT on ∆S 0 is no longer monotonic and the MFPT is minimized for a non vanishing value of ∆S 0 , in agreement with previous general models that accounted for only enthalpic contributions [36]. Moreover, in the range of values of ∆S 0 for which the effective free energy barrier reduces the MFPT, positive tracers are faster than neutral and negative ones. In contrast, for larger values of ∆S 0 , negative tracers are faster than neutral and positive ones.
The asymmetric response of positive and negative tracers to the channel corrugation can be useful to control their relative motion and positioning along a channel, with relevant applications, such as chemical segregation or particle separation. Hence, it is interesting to quantify the ratio of the MFPT of positive and negative tracers, namely: As shown in Fig. 3(d), for uniform channels, ∆S 0 = 0, positive and negative tracers experience the same MFPT while for increasing ∆S 0 the dependence of τ on ∆S 0 is sensitive to φ. For φ = 0, ±π, negative tracers can be up to ten times faster than positive ones. On the contrary, for φ = 3π/2 the ratio between positive and negative tracers is larger than unity for smaller values of ∆S 0 and eventually is smaller than unity for increasing values of ∆S 0 .
The asymmetric response of positive and negative tracers means that if released homogeneously, they will induce a transient electric current due to the channel inhomogeneous section. This current can be estimated using the MFTPs through the quantity For φ = 0, ±π, Fig. 3(d) shows a non monotonous behavior of ι on ∆S 0 , displaying a maximum for ∆S ∼ 1. In contrast, for φ = 3π/2, the sign of the current changes with ∆S 0 switching from positive currents, for smaller values of ∆S 0 , to negative currents for larger values of ∆S 0 . Since the MFPT displays a non-trivial dependence on the channel geometrical details [37], we have systematically studied the dependence of the MFPT on φ, as shown in Fig. 4(a). In particular, Fig. 4(a) shows that for a large set of values of φ, positive tracers take more time than neutral and negative ones to leave the channel, as depicted for φ = 0. As already anticipated in Fig. 3(c), for φ ∼ π/2 and small values of the entropic barrier, ∆S 0 1, the MFPT of all tracers is smaller than the corresponding value for a flat channel, i.e. for ∆S 0 = 0. Moreover, in this range of parameters, the MFPT of positively charged tracers is smaller than the MFPT of neutral and negative tracers. In contrast, for φ ≁ π/2 and/or for larger values of ∆S 0 tracers MFPT is smaller than the corresponding MFPT for ∆S 0 = 0 and the MFPT of negative tracers is smaller than that of neutral and positive ones. This dependence underlines the sensitivity of the MFPT on the details of the sequence of bottlenecks and apertures. Accordingly, we expect a similar behavior for an asymmetric channel where instead of varying the phase, φ, we modify the relative position of the maximum amplitude of the channel while keeping fixed the channel bottlenecks at the boundaries. Obviously, for a set of connected channels, the dependence on φ will vanish asymptotically.
The involved dependence of the MFPT on the channel geometry rises the question of the difference in the MFPT of a tracer along opposite directions of a prescribed channel. We can exploit the symmetry of the channel under study and consider the ratio between the MFPT of a tracer moving in opposite directions, Γ q , which can be obtained from the results in Fig. 4(a) noticing that, given T q (φ), the MFTP along the opposite direction of the channel is T q (−φ). Accordingly, we quantify this asymmetry with the quantity: that for a constant channel, ∆S 0 = 0, leads to Γ q = 1. Fig. 4.B shows that positive tracers, whose MFPT is more sensitive to φ, experience a remarkable dependence of Γ q on the phase shift, φ, whereas neutral and negative tracers are less affected. Moreover, Fig. 4(b) shows that the modulation of Γ q increases with the entropic barrier, ∆S 0 . In order to quantify such a dependence we can look at the dispersion of the values of Γ q , defined as: The inset of Fig. 4.B shows a monotonic growth of ν q upon increase of ∆S 0 whereas ν q → 1 for ∆S 0 → 0.
If tracers of opposite charge are present at the same time, we can estimate the electric current generated in response to fluctuations as defined in Eq. 22. Fig. 4(c) shows the dependence of ι on the phase φ for different values of ∆S 0 . Interestingly, ι is very sensitivity to the channel geometry, and its sign and magnitude can vary significantly. As shown in Fig. 4(c), for small entropic barriers, ∆S 0 ≪ 1 the amplitude of the deviations in the electric current profile are reduced and the profile is almost symmetric with respect to the phase φ. Hence by tuning the geometry of the channel it is possible to select which tracers has the shortest MFPT. On the contrary, large values of ∆S 0 amplify the difference in the MFPT of tracers with opposite charges and the dependence of ι on φ becomes more complex. Eventually, for larger entropy barriers, ∆S 0 ≃ 10, the enthalpic barrier that positive tracers (depleted from channel walls) have to overcome becomes so large that their MFPT always exceed that of negative tracers, hence leading to a constant sign of ι whose amplitude still retains a dependence on φ.
The asymmetry in the transport properties of the channel shown in Fig. 4(b) resemble that of a diode for which the magnitude of the flux varies upon inverting the boundary conditions. From the MFTP corresponding to each set of boundary conditions we can estimate a rectifying flux which vanishes for symmetric channels, for which T q (−φ) = T q (φ). Fig. 4(d) shows the dependence of Π q on the phase φ. In particular, Π q = 0 for φ = 0, ±π, therefore the asymmetry in the MFPT identifies a direction along which particle can diffuse faster. The non vanishing values of Π 0 for neutral tracers, shows that such an effect has an entropic origin since for neutral tracers there is no enthalpic contribution. The amplitude of the entropic barrier, ∆S 0 strongly affects the values of Π q . In order to quantify such a dependence it is insightful to look at: that, since π −π Π q dφ = 0, captures the overall departure of Π q from the symmetric case characterized by Π q = 0. The inset of Fig. 4(d) shows that µ q has nonmonotonous dependence on ∆S 0 and we can identify a, charge-dependent, optimal value of ∆S 0 that maximizes Π q .

IV CONCLUSIONS
We have studied the mean first passage time (MFPT) of charged and neutral tracers suspended in an electrolyte confined between charged walls. Our data show a remarkable dependence of the MFPT of both charged and neutral tracers on the channel geometry when the double layer is comparable to the channel section. We have found that the MFPT depends on both the amplitude of the channel corrugation, which we quantify through an entropic parameter ∆S 0 (see Fig. 3(c)) as well as on the details of the geometry of the channel captured by φ, (see Fig. 4(a)). In particular, we have found a strong asymmetry in the dependence of the MFTP on φ (see Fig. 4(b)). In particular, by quantifying the difference in the MFPT for tracers diffusing along opposite directions through the corrugated channel, Π q (see Fig. 4(d)) we have found a non monotonic behavior of Π q upon increasing ∆S 0 . For ∆S 0 → 0 or ∆S 0 → ∞ we have Π q → 0 whereas Π q experience a maximum for ∆S 0 ≃ 1 (see inset of Fig. 4(d)). Moreover, for mild values of ∆S 0 and for φ ≃ π/2, we found a reduction of the MFPT of all tracers as compared to the corresponding values obtained for ∆S 0 = 0 (see Fig. 3(c)). Therefore, the corrugation of the channel can reduce the time that the tracers needs to cross it, showing the relevance of entropic constraints as a complementary mechanism to the enthalpic forces identified earlier [36]. Interestingly, these features persist also for neutral tracers, hence underlying the entropic origin of these effects.
For charged tracers, the electrostatic interactions with the charged channel walls provide additional parameters to control the MFPT, namely their positive (negative) charge and the properties of the electrolyte, captured by the Debye length κ −1 . In fact, according to the sign of their charge, tracers are depleted or attracted to the channel walls hence experiencing different enthalpic barriers. When the Debye length, κ −1 and the channel average section h 0 are not commensurate, namely κh 0 ≪ 1 or κh 0 ≫ 1, the MFPT is quite insensitive to the charge of the tracers, similarly to what has been observed in driven systems [8,9].
In the present framework we have dealt with pointlike particles but the extension to finite-size particles is straightforward. As it has been discussed earlier [12,32], it is possible to incorporate the dependence on particle size taking into account that the effective width available to the tracer is reduced by its own size. Accordingly, the formalism developed can be straightforwardly extended changing the integration limits both in the effective free energy barrier, ∆A, and channel opening, ∆S 0 , to h min → h min − R and h max → h max − R. Such a generalization clarifies the impact that tracer size and entropic constraints have on tracer MFTP: larger particles will experience a larger effective entropic barrier as compared to smaller ones, therefore opening a new route for tuning particle MFPT across corrugated channels.
Finally we remark that our results can be used to deduce the shape of a pore by using data coming from resistive pulse sensing experiments [21][22][23][24][25].