Transport properties of nanopores in electrolyte solutions: the diffusional model and surface currents

Ion transport through single-cation selective nanopores in thin polymer foils is examined experimentally and described by a diffusional model based on the reduction of the three-dimensional Smoluchowski equation into a one-dimensional equation of Fick–Jacobs type. The model enables semi-quantitative predictions of the transport properties of nanopores of various shapes and surface charge properties even when bulk electrolyte values of various parameters are used. The experimental conductivity data clearly indicate the presence of a surface current component not described by the bulk-type diffusion. The values of the measured surface conductivities depend, among others, on the properties of the channel's internal surface. These surface currents play a substantial role in the rectification processes and are partially responsible for the high-cation selectivity of nanopores. Both theory and electrolytic conductivity measurements show that asymmetric nanopores partially rectify the current with a preferential direction of cation flow from a pore of high surface charge density towards a pore of low surface charge density and/or from the narrow towards the wide opening of the pore.


Introduction
The issue of transport of matter confined in nanometre-scale channels, both biological and artificial, has been a concern of biologists and biophysicists for a long time [1]- [10]. One of the reasons for the increased interest is the need to understand how biological channels function, because the majority of physiological processes is based on mass transport through biochannels. Additionally, nanochannels begin to play a very important role in biotechnology, where, for example, they function as biosensors suitable for single-molecule detection [11], as well as acting as nanopumps capable of transporting water [12] or ions [13] against their concentration gradients.
Related to these issues is the recent vigorous dispute on whether the continuous theories are sufficient for the description of transport through nanosized channels. This issue is of importance for the understanding of the role of electrostatics and geometry in ionic conduction both in biological channels [4] and in synthetic nanotubes [14], more so since there have been several attempts reported to apply continuous description to nanoscale systems [15,16] (cf also [4]) with seemingly good agreement for some of the experimental data. We would like to point out, however, that at least some of these results were obtained at the cost of assumptions and approximations dangerously oversimplifying the physics of the considered nanosystems, which leaves the problem of application of diffusional models for the description of nanotransport still open [17]. The question therefore arises whether the Smoluchowski equation is sufficient for at least a semi-quantitative description of these rather complicated phenomena occurring on a nanoscale level. A positive answer would give substantial support to the adherents of the continuous description, much simpler than the more detailed, microscopic or semimicroscopic theories [4,18]. Moreover, such models would be able to give semi-quantitative (at least) suggestions for planning more efficient synthetic biosensors, wet nanopumps, wet nanodiodes, etc.
From the experimental point of view, understanding the role of electrostatics and geometry in determining the ion movement through narrow channels is hindered in the case of biochannels by a very complex channel structure [4,19] and also by their vulnerability, which excludes more demanding experiments. There is a need to design and construct synthetic systems with known geometry and chemical structure, which would be much easier to understand and describe by means of basic physical phenomena. Such a system would also give us the possibility to perform a wide-range of experimental studies not applicable to biochannels. The nanopores suitable for the studies have to function as analogues of biochannels, at least in what concerns the important features of their transport properties such as ion-current fluctuations, rectification and pumping. Indeed, such systems do exist nowadays: earlier studies [20] together with our recent experimental results [13], [21]- [24] show that (i) the time series recorded for the nanofabricated pores in polymeric membranes produce ion-current fluctuations of almost identical statistical characteristics to that of biochannels, (ii) the synthetic nanopores are ion selective, (iii) the synthetic nanopores, which are asymmetric and conical in shape, rectify the cation flow from the narrow entrance towards the wide opening of the pore as well as (iv) pumping cations against their concentration gradients.
The aim of this work is to examine in detail the performance of the diffusional model (called further as 'bulk model'), which (i) is based on the Smoluchowski equation and the Fick-Jacobs approximation [25] and (ii) reproduces correctly the physics of the experimental setup. All values of the (phenomenological) parameters used throughout this paper are taken from various independent measurements under the bulk-solution conditions. We are aware that this is but a crude approximation; nevertheless, for the lack of any more realistic data, we presume that such a model enables a reliable checking of the applicability of the diffusional approach to ion transport through nanopores. The lack of free parameters reduces the danger of 'forcing' the model to fit the data. We want to compare the model predictions with experimental data for ionic conduction of single synthetic nanopores of various sizes and shapes, specifically designed to enable verification of the theoretical model. We believe that these results may also help in designing nanopores with require transport properties (cf section 4). The first short report on this approach (for cylindrical pores only) was published in [26]. In addition, we have recently used this model for the description of the effect of asymmetric nanodiffusion in the presence of concentration gradients [27,28].
We shall concentrate here on average transport properties of the nanopores in the absence of concentration gradients, studied by single-pore current-voltage characteristics. Special emphasis will be given to pores with asymmetric-rectifying I-U curves. The issue of transient transport properties (ion-current fluctuations) [21] will not be considered here.
We have mainly analysed channels with the opening diameter 8 nm, therefore slightly wider than those used in our former studies [13,21] (diameter ≈2 nm in the narrow part where one may expect that ions move in a single file). Wider channels are more probable to be well described by continuous-type theories, whereas in very narrow channels, the random-walk type models (in a fluctuating landscape) seem to be more proper.
The paper is organized as follows. Section 2 describes the experimental setup and experimental data. Section 3 presents the diffusional model and the calculations of the potential inside the pores. Comparison of model predictions and experimental data is discussed in section 4. Section 5 contains comments, discussion and conclusions.

Setup
We are interested in stationary ionic currents through very narrow pores created in a thin dielectric film. The diameter of the pores is between several nm and some tens of nm [13,21]. The whole setup is immersed in an electrolytic solution (cf figure 1). The channel walls possess surface charge whose density can be controlled. To prepare the model pores, the track-etching technique was used. It is based on irradiation of a dielectric film, e.g. a polymer with energetic heavy ions and subsequent chemical etching of the latent ion tracks [29]. The number of pores created in this way is determined by the number of ions which penetrates the foil [30]. In this paper, we present studies of cylindrical pores created in films of polyimide (Kapton 50 HN, DuPont) and polyethylene terephthalate (PET) (Hostaphan RN12, Hoechst) of 12 µm thickness. Singleion irradiation was performed at the Gesellschaft für Schwerionenforschung in Darmstadt (the linear accelerator UNILAC) using Xe and U ions of 11.4 MeV u −1 energy. Ion tracks in PET were developed in 0.5 M sodium hydroxide at 40 • C. Kapton was etched at 50 • C in sodium hypochlorite with 13% active chlorine content, buffered with boric acid to pH 9.4 [31]. The conical pores were prepared as described in [22]. These etching procedures assure the required shape of the pores as well as formation of carboxylate groups on the surface of the membrane and inside the pores [31,32]. The maximum charge density measured at pH 8 is equal to about −1.5 e nm −2 for PET and −1.7 e nm −2 for Kapton [22,32]. The charge can be neutralized by lowering the electrolyte pH. At pH close to the isoelectric point (IEP) of track-etched polymers, estimated on the basis of pKa of the carboxylate groups (for Kapton IEP is at pH 2 and for PET it is at pH 3), the net surface charge is zero. The polymer membrane containing one nanopore was placed between two chambers of a conductivity cell, connected to the home-built current-to-voltage converter of pA sensitivity. The current-voltage characteristics were measured by applying a voltage-ramp (National Instrument card) with 0.005 Hz ramping frequency and amplitude of 1 V [22]. Series of ion-current recordings at KCl concentrations from the range 0.001 to 3 M were performed. The ion current was measured at pH 8 and 2, when the etched-polymer net surface charge was negative and zero, respectively. The experimental error was estimated by measuring the current, when repeating the procedure of mounting the membrane into the cell and filling the chambers with the electrolyte, and it was equal to 3-8%.
The external electrodes were Ag/AgCl wires of 2 mm thick, about 2 cm long and located at 0.5-1 cm from the foil [22]. No changes in the measured current were observed when the distance was changed within this range. This suggests that practically-within experimental error-the whole voltage-drop U occurs on the channel, and corrections from the resistance of the bulk solutions outside the pore are negligible. Estimations through Kirchhoff's law confirm this observation, at least for pores for which the average diameter is smaller than about 1000 nm.

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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT The applied external field was changed adiabatically and it can be considered as static. We have studied cylindrical and conical pores, assuming their perfect geometry. We are aware of the polymer surface roughness [22], which may lead to changes in the real pore shape. We checked however that introducing small-range randomness, even of relatively high amplitude (up to 10% of the local value), either in the geometrical shape of the pore or in the surface charge distribution results in the variations of the calculated current not exceeding 3%.

Data
An ion can approach the channel's surface at most at the distance of its radius. The so-called crystallographic radii are 0.133 nm for K + and 0.181 nm for Cl − [1,18]. Hydrated ions are bigger, r ion is about 0.16 nm but there are no clear-cut data in the literature whether ions in the double layer are hydrated or not. We checked that the calculated currents are little influenced by the change in the values of r ion from 0.13 to 0.3 nm.
The data on the concentration dependence of the diffusion constants for ions D i (which are the most basic parameters in the diffusional description) are scarce and inaccurate. Commonly used are the infinite dilution values D i ∼ 2 × 10 9 nm −2 s −1 (i = +, −), both for K + and Cl − . Nernst-Einstein formula relating the ion's mobility µ i with D i and temperature (β = 1/k B T ) is approximate and valid for very dilute solutions only. However, in the light of the rather complicated dependence of both the mobility and diffusion coefficient on concentration, we found that best results (better than for D i = const) are obtained by the application of relations D i = µ i /βe and χ i = Fc i µ i |Z i |e (F is Faraday's constant and Z i e the ion's charge). Conductivities of ions present in the electrolyte solution are related to the measured electrolyte conductivity χ = χ + + χ − by the relation χ i = t i χ, where t i are the so-called transference number of cations (t + ) and anions (t − ). For bulk KCl, t + ≈ t − ≈ 0.5. The conductivity χ = χ(c), taken both from the tables in [33] and from our measurements, fits the empirical formula (χ in S/m, c in M = moles litre −1 ): χ(c) = − 0.83836 + 6.71282c 1/2 − 4.84314c + 14.1213c 3/2 − 4.90256c 2 for c 0.15M, = 0.04445 + 11.55555c for 0.01 c 0.15M, (2.1) = 0.004445 + 113.55555c for 0.001 c 0.01M.
The non-zero surface charge of the pore brings about the formation of the so-called double layer [9,34,35] consisting of counterions (K + in our case) from the solution, which neutralize the carboxylate groups on the surface. The interactions of cations with the surface charges are therefore strongly dependent on the distance from the surface. We shall approximate this dependence by the use of the screened Debye potential V(r) ∼ r −1 exp(−r/ l D ). For water solutions of 1:1 electrolytes at room temperature, the screening (Debye) length l D can be approximated by the formula [6] with c 0i being the ion concentrations in M and Z i their valencies. It is important to point that the formula for l D (i) is exact in extremely diluted solutions only and (ii) was derived under the assumption that the ions in the solution, K + and Cl − , are also screened. Inside narrow pores, the screening geometry is, however, different from the spherical symmetry assumed in the calculation of ion-ion screened interactions which lead to (2.2). Moreover, concentration of the negative ions is depleted due to the strong repulsion by negative surface charges. This effect is significant at low concentrations and in very narrow parts of a channel. In such situations, the relation (2.2) may not be valid.
Throughout this paper, we assume the following sign convention: the ion current is counted positive when positive ions flow in the direction from the narrow opening of the cone (or high negative surface charge in cylindrical pores) towards the wide entrance of the pore (low surface charge in symmetric pores) (from z = 0 to z = L, cf figure 2).

Surface current
As a result of strong interactions between the charges on the pore walls and the ions in the solution, there is a surface (called also double or diffuse) layer [34,35] formed close to the pore walls. The surface layer contains mainly counterions to the charges on the pore walls. The counterions move along the applied field and contribute to I. Both the literature [34]- [36] and our experimental data show that this effect is especially strong in very narrow pores and at lower concentrations when the double-layer thickness becomes larger. Therefore, the total (net) measured electric current I contains two components: the surface current I s (composed of counterions) flowing through the surface layer and the 'normal' non-selective volume ('bulk') current I b , composed in turn from the cation and anion contributions: As I b is proportional to the volume of the channel and I s to the channel's surface, the latter is negligible in wide channels, but must be taken into account in our case.   At low pH, with zero net surface charge, we do not have the surface current contribution. This fact enables separation of the surface currents, as shown in figure 3. The dashed lines interpolate through experimental points. The distances between the lines are interpreted as κ s . In this way, the values of surface conductance were determined as κ s = 235 ± 15 pS for Kapton pore of 34.5 nm radius and κ s = 13.8 ± 4.8 pS for PET pore of 6.6 nm radius at concentrations of c > 0.1 M where surface conductance is rather stable. For low concentrations (below 0.1 M), the surface conductance diminishes. This effect seems to suggest that for concentrations higher than 0.1 M, the surface layer is saturated by K + ions and κ s is a constant. For c < 0.1 M, surface conductance gradually becomes proportional to the concentration.
Saturated surface current and surface conductance should be proportional to the pore surface, i.e. to the radius of the pore: κ s = 2πr 0 Lχ s , χ s being the surface conductivity. Therefore, for Kapton cylindrical pores κ s /r 0 = 6.8 ± 0.4 pS nm −1 , χ s = (9 ± 0.5) × 10 −5 pS nm −2 , and for PET ones κ s /r 0 = 2.0 ± 0.7 pS nm −1 , χ s = (2.65 ± 1) × 10 −5 pS nm −2 , i.e. more than three times lower. This difference cannot be explained by the higher surface charge of Kapton pores (about 1.7 e nm −2 in comparison with about 1.5 e nm −2 in PET pores). In our opinion the rather high difference of surface currents in Kapton and PET pores can be explained by the differences in the surface mobilities µ s . The latter, in turn, might result from the differences in the surfaces of both types of pores [21,22]. The roughness of the PET surface could be responsible for the lower surface conductivity of the pores. Assuming this hypothesis to be true, we would have µ Kapton s : µ PET s ≈ 3.2, i.e. the Kapton surface mobility of K + ions to be more than three times higher than that of PET.

9
Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT The presence of 'dangling ends'-parts of polymer chains sticking from the pore surface (cf [21])-may also contribute to the instabilities of the surface currents (note that the accuracy of the measurements in the PET pores is much worse than in the Kapton ones). On the other hand, it is the presence of the random motions at the surface which makes the PET pores more similar to the biological channels [21] and substantiates the use of these pores instead of more stable Kapton channels.

Model
The simple single-ion-motion model presented in some of our previous papers [13] explains the behaviour of asymmetric nanopores in a qualitative way, by invoking the so-called ratchet principle [37]. The ratchet model was also applied for biological channels and it describes the action of biological pumps (ATPases) [38]. However, the single-ion approach is not well suited for calculating the ion current. One of the problems of the single-ion approach is the difficulty in introducing interactions of the ion with other ions in the solution and with water. Another drawback of this model is that it does not take into account either thermal fluctuations or fluctuations resulting from the small size of the pore, which significantly influence the ionic movement. Another way of introducing these would be to add the noise component directly into the ion's equation of motion. The resulting stochastic differential equation is, however, very difficult to solve numerically. The simplest way to overcome these difficulties is to apply the Smoluchowski equation, which offers a mean-field approach via generalized diffusion [4,16,25].
We shall discuss the stationary electrodiffusion, in which all parameters are the same as in the bulk solution (i.e. without the influence of the pore's presence). We therefore call our model the bulk model. We want to stress that we did not include in our approach, the surface currents or the electro-osmotic flows [34,35], which would require the use of additional parameters (zeta-potential, surface mobilities, etc) the values of which are unknown.

Potential
To avoid misinterpretations, we shall speak of the potential energy φ i = eZ i V of an ion of valence Z i in the electric potential V , where e is the elementary charge and V is measured in volts (V).
The whole electric potential is the sum of external and internal contributions: In our measurements the external electric field is generated by macroscopic electrodes of potentials V 1 and V 2 located at macroscopic distances from the membrane, and immersed in the electrolytic solution. The field is therefore uniform and, as we mentioned above, practically the whole voltage drop occurs on the foil. Elementary calculations based on Kirchoff's law and the conservation of the total current I flowing through the channel give: inside the channel, and V 1 or V 2 outside the channel, Q(z) being the resistance of the channel of length z. The internal field is generated by the charges located on the channel walls, of charge density ρ(z), dependent in general on the location along the channel axis. We assume that the internal field depends only on z and r (distance perpendicular to the axis) and does not depend on the angle θ (cf figure 2 for explanation of the notation). Therefore, the internal part of the electric potential inside the electrolyte-filled channel, related to this field, is: where is the distance between points x and x , and denotes the dielectric constant (for water solutions we use = 80.1 0 , 0 being the vacuum permittivity), λ = 1/ l D is the inverse Debye (screening) length, factor h(z ) π −π dθ gives the number of charges per unit length on the pore's circumference. Note that λ = λ(z, r), due to the dependence of the screening length on concentration, equation (2.2).
To ensure the accuracy of computation of V int (z, r), and therefore of the ionic currents below 1%, the integration steps z and x = h(z) tan( θ) should be less than 0.5 nm inside the pore and 0.005 nm at the pore's entrances, r less than 0.2% of h(z) near the pore's surface to 3% of h(z) inside the pore, and z less than 0.05 nm nearby the pore's entrances to 10 nm near the pore's centre.

Current
As we have shown above, the contribution of the surface current I s to the total (net) current through a narrow channel is significant and must be taken into account. The calculation of the surface and electro-osmotic components would require the use of full three-dimensional Helmholtz-Boltzmann and Boltzmann-Poisson equations in a rather complicated geometry (cf [4]- [7]). Therefore, further calculations involve only the bulk current I b (equation (2.3)).
There have been several approaches to the issue of diffusion in confined geometries and/or in irregular channels reported recently (for a review see [7]). We shall use the so-called Fick-Jacobs equation [25], in which the boundary problem in two or three dimensions is projected onto an averaged one-dimensional (Smoluchowski) equation, with concentration and (in our case) electric potential averaged over the channel's cross-section A(z) = πH 2 (z): dr r a(z, r, θ)/A(z) otherwise, (3.5) with H 0 l D and In practice, one may put H 0 ≈ 0-the contribution from H 0 to the calculated current does not exceed 0.1% in our experimental conditions. According to Zwanzig [25], in the local equilibrium approximation, the local average concentration c satisfies the one-dimensional Smoluchowski equation (in this case equivalent to the so-called Fick-Jacobs equation [25]): with the current density In one dimension, the stationary overall current is just the current density multiplied by the pore's cross-section: The stationary effective Smoluchowski equation therefore reads as with the solution (we neglect here the dependence of D i on the concentration) as (3.10) i.e.
This is the desired general expression for a stationary flow. Zwanzig in his paper [25] does not give explicitly the formulas for the stationary current, but equation (3.11) is a straightforward consequence of his version of the Fick-Jacobs equation. A special case of equation (3.11) can be found in the paper by Reguera and Rubi [25]. Similar results were obtained in [39].
Neglecting the contributions from the outside of the channel (they give very small corrections to the calculated currents), using V int (z, r, θ) = 0 and V eff (z) = 0, for z = −0 and z = L + 0, as well as assuming the Nernst-Einstein formula to hold (i.e. D i = µ i /βe, χ i (c) = Fc i µ i |Z i |e = t i χ(c)), we get for the electric current I i = Z i FJ i : . (3.12)

Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT
For V int = 0 and χ 0 = χ L , being the channel's average cross-section (for a cone A = πr 0 r L ). This formula was used to calculate the cone diameter. The standard procedure (cf e.g. [35]) of finding the solution of the effective Smoluchowski equation is the iteration of equations (3.10) and (3.11). In our case, the recalculation of both formulae is required at every step ((3.2) and (3.3)), including the computation of concentration profiles c(z, r) across the channel, the latter due to the presence of the concentration-dependent screening length.

Results
Cylindrical pores are simpler for the modelling, therefore also for the first verification of the model. For symmetrical electrolyte conditions on both sides of the membrane, the internal potentials V int (z, r) and V i,int (z) depend on z only near the pore's entrances and are practically constant inside the pore. For sufficiently long pores (L l D ), the contributions from the changes of the potential near entrances are negligible. They introduce an error smaller than 0.1% in the calculated values of the current for pores of length L l D . The whole potential can then be approximated by its intrinsic value V * i . Subsequently, the formula (3.12) for overall bulk current simplifies to: For an asymmetric electrolyte configuration (different concentrations and/or pH) on both sides of the foil, the dependence of the potential inside the pore on z has to be taken into account. In the case of the concentration gradient, the gradient of the potential becomes strongly nonlinear. This results from the nonlinear character of the screened electrostatic potential in the formula (3.3), caused by concentration dependence of the screening length (equation (2.2)). For the pH gradient, however, the potential gradient can be approximated with a high accuracy by a linear form (again, we neglect the end effects): We assume the convention that the solution with high pH is on the side with a working electrode, which determines the sign of the applied potential difference. In this case the integral (3.21) can be calculated analytically, and the overall net current becomes On the other hand, conical pores with nonzero surface charge, due to their inherent geometric asymmetry, are characterized by strongly asymmetric internal potential (both V int (z, r) and V i (z)), as well as by the nonlinear distribution of the external voltage U(z) (cf equation (3.2)) also in symmetric electrolyte conditions. Figure 4 shows comparison of the channel conductance measured and calculated from equation (4.1) (after taking into account the surface currents) for cylindrical channels of different diameter. It can be seen that the model presented in section 3 overestimates the conductance of the pores the more, the lower the concentration. The formula (4.1) shows that this effect becomes stronger for higher values of |V * i |, which results in turn from longer screening lengths l D for lower concentrations. We presume that this effect originates from depletion of negative charges inside the channel (section 2.2). The quantitative description of this observation would require solving of the three-dimensional Poisson-Boltzmann equation near the curved surface, which is an extremely difficult task (cf [7]) 3 .
The concentration dependence of the currents through conical pores resembles that by cylindrical pores (figure 4), therefore we will not analyse that dependence here in detail. The calculated currents for conical pores with tip diameters r 0 > 6 nm are again in a semi-quantitative agreement with our experimental data. The bulk model breaks down for very narrow conical pores (r 0 about 1-3 nm), therefore the direct comparison with the data published earlier [13] is impossible. Still, for pores with r 0 > 6 nm, the predictions of the bulk model, treated as the semi-quantitative hints, seem to be of practical value.
Note that there are no fitted parameters in the calculation of I b . The experimental values of pore conductivity were measured independently.
In symmetric situations (cylindrical pores and symmetric electrolyte conditions on both sides of the membrane), the internal potential changes only the value of the pore's conductivity, the I-U curves remain symmetric. This change occurs either in the conical pores or when the solutions on both sides of the foil are different. Figure 5 shows the measured (circles) and calculated (lines) currents I(U) (i) through a cylindrical channel recorded with a pH difference, inducing asymmetric surface charge on both sides of the membrane, and (ii) through a conical channel in symmetric electrolyte conditions on both sides of the membrane. In both cases the dependence of ion current on the external voltage becomes nonlinear and asymmetric. The rectifying character of the I-U curve reflects the asymmetry of the internal potential-for higher currents, the cations flow (i) from the side with high pH towards the compartment with low pH, i.e., towards the cylinder's end with zero charge, or (ii) from the narrow towards the wide opening of the pore. The full lines represent the calculated bulk current I b and the dashed lines represent the current corrected by I s = −κ s U with κ s = 25 pS for the cylindrical pore, and for the conical pore by I s = χ s,cyl (S cone /L)U, where χ s,cyl is the surface conductivity determined for cylindrical pores and S cone the pore's surface. The values of the surface conductance are taken from our measurements for cylindrical pores reported in the section 2.3.
Note that corrections from surface currents improved the agreement of the model with the data only for the lower branch of the I-U. For U > 0 the measured current is both higher and more nonlinear than I b + I s . This implies that the surface currents also take part in the rectification. The insets show the net rectification current, I = |I(−U)| − |I(U)|. Again the measured net currents (squares) are both higher and more nonlinear than the calculated ones (full line). These The examples shown in figures 5(a) and (b) were chosen to show both the similarities and the differences between the rectification by cylindrical channels with surface charge gradients (electrical asymmetry) and conical channels (geometrical asymmetry). At first glance both pictures look similar. However, the currents shown in figure 5(b) are much higher than those in figure 5(a). This is due to the fact that the effective length of a conical pore is much shorter than the thickness of the membrane. Almost the whole voltage drop occurs at the tip of the cone (cf [27], figure 4).
Similar results were found for cylindrical and conical pores with other radii. The rectification becomes stronger for lower concentrations and for narrower pores. dependences of γ r on cone sizes (in symmetric pH and concentration conditions) are shown in figure 6. Again, the more narrow is the pore, the higher is the rectification effect.
Our results suggest that the rectification phenomenon results from the ratchet-like [37] potential asymmetry. This further supports our first [13,17] ratchet interpretation of ion-current rectification.

Comments and conclusions
The results presented above show that the diffusion approach together with 'bulk-solution values of the parameters'can be a very useful semi-quantitave guide for the construction of nanochannels of desired transport properties. The model predicted and described the possibility of constructing an ionic rectifier based on a cylindrical pore with asymmetric charge distribution, as well as vanishing of the rectification for wider conical pores. It does not however render quantitatively all the aspects of transport phenomena observed with nanopores. It is possible to obtain a better fit of the calculated currents to the experimental data by appropriate changes in the local values of e.g. ion mobilities and other physical parameters (cf [27]), as well as by adjusting the pore shapes. However, there are no independent data on such quantities, and a model with adjustable parameters would make it very difficult to answer the main question posed in this paper, i.e. whether a continuous modelling can be used for the description of the transport phenomena occurring on a nanoscale.
The main (and the easiest to identify) source of a 'non-diffusive' behaviour seems to be the non-negligible presence of the surface current I s , which depends on the pore material. Subtracting this contribution, we get the purely electrodiffusional data, so that the model becomes universal for small pores with different properties-our synthetic pores become thus the model system for nanoscale transport.
It is to be noted that, in principle, the 'non-diffusive' contribution present in our data may contain the electro-osmotic currents, which we estimate to be small, within the overall experimental error-as discussed in [40], the electro-osmotic contribution, important in microchannel transport [34,35], is negligibly small in nanopores.
The evidence (both experimental and theoretical) of the substantial role of surface currents in nanoscale ionic transport is another main result of this work. Characteristic features of these currents seem to be the saturation for concentrations, c > 0.1 M, and substantial role in the rectification of ionic currents in pores with asymmetric internal potentials. It seems safe to assume that, for cylindrical pores without charge gradients, I s is proportional to the area of the internal surface of the pore. However, it is difficult to say anything firm about the dependence of I s on the pore's size in asymmetrical situations (cones, concentration or pH gradients). Our results also suggest that in PET channels, characterized by a much higher surface roughness, the surface mobility is about three times lower than in smooth Kapton channels.
Surface currents, which consist solely of the counter-ion transport (cations K + in our case), seem also to be the plausible explanation of the strong ion selectivity of charged nanopores, reported earlier [13,21,22], and well-known for biological membrane channels [1].
Unfortunately, there is no model for surface currents which would be both reliable and effective. The description proposed in [35] cannot be applied directly to the situation considered in this paper. It is possible that I s could be determined-with a good accuracy-by the diffusional model of the type described in section 3 with appropriate values of the physical parameters, such as ion mobilities, screening length, etc. There are however no experimental values of these parameters. They could be determined by solving the three-dimensional Poisson-Boltzmann equation near the pore's surface (a rather difficult task by itself), but this would require, in turn, a more detailed knowledge of the properties of the polymer surface, its polarization, etc [41] (cf footnote 3). The same can be said about the electro-osmotic flow.
The rectification effect is related solely to the asymmetry of the internal potential inside the channel. Both the measurements and the model calculations for neutral conical channels (i.e. at pH 2), and for symmetric cylindrical channels, gave a linear dependence of I(U) = −κ b U, κ b being the bulk conductance. This supports our earlier interpretations of the rectification and pumping as resulting from the ratchet-like mechanisms [13].
As we have said above, the bulk model investigated in this paper does not render quantitatively all aspects of transport phenomena occurring in nanoscale wet systems. On the other hand, the continuous model considered here does give satisfactory semi-quantitative predictions of the transport properties of nanopores of various shapes and surface charge properties. This seems to support the point of view of the adherents of this type of description [4]. In our opinion the partial success of the diffusional models can be related to the fact that their statistical basis, the original Smoluchowski equation [42], describes the evolution of the (conditional) probability density for finding the Brownian (diffusing) particle performing the random walk in an external field, and therefore is able to describe even the motion of single ions (particles) in nanoscale.
It will certainly be intriguing to investigate the main reasons for limitations of the continuous approach. A correct three-dimensional theory, taking into account as much geometrical and electrostatic details as practically possible, together with further measurements in narrow pores of various geometries and various distributions of internal fields is needed to settle this intriguing problem. Further studies should also answer the question if and how the pore differentiates between various cations. Systematic studies with monovalent cations from lithium to caesium as well as polyvalent cations will give information on the influence of the internal pore potential on the cations movement. One should also check whether the potential of the pore does not change when various cations pass through the pore.