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Electrical Circuit Modelling of Nanofluidic Systems

Nanofluidic systems exhibit transport characteristics that have made technological marvels such as desalination, energy harvesting, and highly sensitive biomolecule sensing possible by virtue of their ability to influence small currents due to the selective transport of ions. Traditionally many of these applications have relied on the use of nanoporous membranes. The immense complexities of membrane geometry often impede a comprehensive understanding of the underlying physics. To bypass the associated difficulties, here we consider the much simpler nanochannel array comprised of numerous nanochannels and elucidate the effects of interchannel interactions on the Ohmic response of the array. We demonstrate that a nanochannel array is equivalent to an array of mutually independent but identical unit-cells whereby the array can be represented by an equivalent electrical circuit of unit-cell resistances connected in a parallel configuration. We show that the total resistance of the system scales inversely to the number of channels. We further deconstruct the unit-cell to be a combination of multiple contributing resistances connected in series. We validate the theoretical model underlying these electrical abstractions using numerical simulations and experiments. Our approach to modeling realistic nanofluidic systems by their equivalent electrical circuit provides an invaluable tool for analyzing and interpreting experimental measurements, characterization of surface charge properties of newly developed materials, and a method for the design and development of function-specific nanofluidic devices.


INTRODUCTION
With the ever-increasing demand for fresh-water and clean renewable energy [1] combined with the everincreasing appearances of global warming effects, it has never been more apparent and more urgent to improve the performance (efficiency, power input, etc.) of waterdesalination and energy harvesting systems that utilize nanoporous ion-selective membranes. In recent decades, research has been divided into two main thrusts: one, improving the material properties of membranes, and two, improving our fundamental understanding of the phenomena occurring at these very small scales. To achieve the first objective, scientists tune the material properties of macroscopically large membranes and compare how they impact their electric response and process efficiencies. Towards the second objective, the much simpler, if not entirely realistic, the single nanochannel setup is adopted. This scenario allows the probing of the fundamental physics of various nanofluidic and electrokinetic effects to a higher resolution. However, several phenomena that emerge from the scaling up from a microscopically small single nanochannel system to a macroscopically large membrane system are yet to be fully expounded. Here we will demonstrate how the nanochannel array serves as the intermediate of these two different scenarios and how the response varies as the number of channels, , increases from one (single channel system) to an arbitrary number. Since our approach focuses on the system's electrical response, our results hold primarily for the processes of electrodialysis [2,3] (ED) and reverse electrodialysis [3][4][5][6][7][8] (RED) used for water-desalination and energy harvesting, respectively.
To be commercially viable, ED and RED systems must operate with large ionic fluxes and efficiently filter the desired charges to be transported. Thus, such systems traditionally use macroscopically large membranes [see reviews [4,9] (and references therein) for a list of various polymer based membranes that are currently being used for ED and RED]. Nonetheless, in recent decades new materials [SiN [10][11][12][13], silica [14], natural clays [15,16], processed wood [17], metal-organic frameworks (MoS 2 [18], 2 5 [19], BN [5,20], layered double hydroxides [21] and others [22]), layered and aggregated polymers [23][24][25][26][27][28][29][30], carbon nanotubes [31], and graphene and other carbonbased 2D materials [2,32,33]] have been developed and shown to have ED and RED capabilities. While all these materials are not yet commercially viable, they are currently being improved so that they might, one day, replace conventional membranes. The reason that all these membranes can be used for ED and RED is that they all share the two key features needed for ED and REDthe smallest characteristic length in the system is nanometric and the surface is inherently charged.
At sufficiently low concentrations and large surface charge densities, the electric double layers within the pores overlap, such that the membranes are ion-selective, capable of perfect coion exclusion (discussed further below). Thus, once perfect coion exclusion has been achieved, ED and RED are robust processes that are virtually independent of the material itself. The feasibility of ED and RED emanates from the large pore densities of the nanoporous membranes that allow for relatively large fluxes. This is essentially a parallelization process whereby all the pores participate in ion transport. However, thus far, improving the efficiency of such systems has primarily relied on trial-and-error and empirical investigations of material properties and geometric configurations. The conventional reliance on this approach stems from the difficulties arising from the irregular porosity and random tortuosity of the membrane [see Figure 1(a)-(c) for membrane schematic]these geometric features do not allow for a straightforward analysis of the fundamental and microscopic physics occurring at the smallest scale. Consequently, most analyses have been limited to simplified one-dimensional (1D) models that do not account for crucial microscopic details [34].
In the foregoing decades, the single nanochannel system [35][36][37][38] was introduced as the simplest tractable model for its larger cantankerous counterpartthe membrane. Similar to the membrane, nanochannels are ionselective at low concentrations, exhibiting perfect coion exclusion. Additionally, nanochannels benefit from two important advantages. First, their simple geometry is easy to fabricate, easy to comprehend, and easy to analyze (experimentally and theoretically), whereby the fundamental physic becomes more apparent [39][40][41]. Second, additional applications that cannot be realized with conventional membranes are immediately contrived with small nanochannels. The well-defined, deterministic geometries of these engineered nanochannels make them highly amenable to chemo and bio-sensing [42,43] and DNA sequencing [30]  fluid-based electrical circuits (diodes [44][45][46][47][48][49][50], transistors [18,[51][52][53][54], and more [55,56]). Nonetheless, while single-channel systems are indeed favorable for some applications, they are inadequate for other applications. Specifically, the disadvantage of the single nanochannel is that the relatively low fluxes it generates make it impractical to use without scaling uptheir low throughput makes them irrelevant for realistic ED and RED applications.
The natural bridge between macroscopically large membrane systems and microscopically small single nanochannel systems is the ordered nanochannel array system [28,29,44,[57][58][59][60][61][62] [ Figure  1(d)-(f)]. The nanochannel array promises the advantages of both systems: regularized and simple geometry combined with full potential for parallelization to upscale the fluxes. However, the physics of nanochannel arrays is yet to be discerned; namely, the interactions between multiple nanochannels are not fully understood. The purpose of this paper is to elucidate the governing physics and delineate the electrical response of multichannel systems by demonstrating that these complicated systems can be represented by a simple, equivalent electrical circuit. Specifically, we will show that an array of microchannel-nanochannels [ Figure 2 Our goal is to show that an ordered nanochannel array leads to the partitioning of the fluidic domain composed of the reservoirs and individual nanochannels into an ordered array of unit-cells. The system's response can, in turn, be represented by a simple electrical circuit composed of components connected in series and parallel configurations. We have divided this paper to reflect this goal. In Sec. 2, we discuss the concept of the unit-cell, the equivalent circuit of which is a series combination of three different resistance contributions. We discuss each of these terms separately. Then, we elucidate the conditions when the unit-cell resistance can be extended to describe the equivalent parallel circuit of an array system. In Sec. 3, we present numerical simulations and experimental results that confirm our theoretical prediction. The outcomes of our results are discussed thoroughly in Sec.4. We conclude with short remarks in Sec. 5.

ELECTRICAL RESISTANCE OF NANOCHANNELS AND MICROCHANNELS
In our approach, the first step in analyzing array systems is identifying and defining the unit-cell. The unitcell geometry, as well as its ion-selective capability, needs to be defined. This is discussed in Sec. 2.1, where we provide the general expression for the unit-cell resistance, unit-cell , as a function of the geometry and the surface charge density. Section 2.2 expands upon a resistance contribution that we have termed the field focusing resistance, FF , and its relation to the more commonly studied access resistance [63][64][65][66][67], access . In Sec. 2.3, we discuss the two limits of vanishing and ideal selectivity, where unit-cell is shown to be a series combination of contributing resistances. The equivalent circuit of the 2 × 2 array in (a) is a ladder circuit of 4 unit-cell resistances ( unit-cell ) connected in parallel. This circuit can further be abstracted as the total resistance total of the array. (d) The equivalent/total resistance of the unit-cell, unit-cell , is a series of resistances: the nanochannel resistance (denoted in red), nano , the microchannel resistances (denoted in blue), micro , and the resistance arising from the field-focusing of flux lines from a larger geometry into a smaller geometry at the two interfaces (denoted in green), FF . Section 2.4 demonstrates the relation between the total resistance of the array, total , to unit-cell and to the total number of independent unit-cells, , that constitute the array. Specifically, we will show that this is equivalent to a simple circuit of unit-cell resistances connected in a parallel configuration. We note that in the equivalent resistance modelling presented herein do not presuppose the existence of an electrical circuit (a common feature in impedance modeling). Rather, the results of this work are based on solving the full Poisson-Nernst-Planck equations and through an analysis of the solution, do we find that the response is an electrical circuit [68,69]. A brief derivation for the field-focusing resistance is given in Secs. 1.8-1.10 of the Supplementary Information.

The unit-cell resistance
The unit-cell [ Figure 2(b)] is a single nanochannel system where two cuboidal microchannels are bridged by one nanochannel of arbitrary cross-sectional geometry. The microchannels have a height, , width, , and length, .
Here, for the sake of simplicity, we have depicted the nanochannel as a cuboid such that the nanochannel has a height, ℎ, width, , and length, However, our results hold for any nanochannel geometry whose cross-sectional area is nano and length so long as the aspect ratio / nano 1/2 is large ( / nano 1/2 ≫ 1). Also, in the main text, we assume that the microchannels are identical. In the Supplementary Information, where we present the detailed derivation, we present the more general scenario where the microchannels are not identical, and the nanochannel is of arbitrary crosssectional geometry.
For the case of a charged nanochannel with a surface charge density, , it has been shown from the exact analytical solution [68] of the Poisson-Nernst-Planck (PNP) equations that the total Ohmic resistance of the unit-cell is given by The resistances of the nanochannel and microchannels are given by The resistance of an ionic circuit [Eq. (1)] depends on the geometry of the system and the intrinsic properties of the conducting medium-the bulk electrolyte concentration, 0 , the resistivity, res (defined below), and the excess counterion concentration (defined below). Both nanochannel and microchannel resistances scale with their lengths divided by their respective cross-sectional areas. For the microchannel, this area is micro = . In contrast, the nanochannel area nano depends on the details of the nanochannel cross-section itself. For example, for the long cuboidal nanochannel given in Figure 2(b), we have nano = ℎ while for a long cylindrical nanochannel of radius we have nano = 2 . It should be noted that the total resistance due to the microchannels, micro , is not limited to the contribution of the microchannel, micro . Rather, one needs to account for the resistances associated with the microchannel-nanochannel interfaces. We denote this resistance FF and discuss the term thoroughly in the following subsection (Sec. 2.2). The expression for the field focusing resistance, FF , is long and complicated and depends on the geometric parameters of the nanochannel, as well as those of the microchannels (see Supplementary Information for an exact expression). Additional features of FF are discussed below. The factor 2 in Eq. (2) represents the contribution of both (inlet and outlet) microchannels which are identical here (see Supplementary Information for non-identical microchannels). The transport number is an important characteristic of ion transport, and it represents the degree of the system's selectivity [48,70]. The resistivity is given by res = ℜ /( 2 0 ) where ℜ is the universal gas constant, is the (absolute) temperature, is the Faraday constant, is the diffusion coefficient of the ions, and 0 is the bulk concentration of the ions in the reservoirs (at the two ends of the microchannels). Note that here we have assumed that the electrolyte is KCl which can be appropriately assumed to be a symmetric binary electrolyte-i.e., both ionic species have the same valences ( + = − − = 1) with equal diffusion coefficients ( ± = ). The average excess counterion concentration is the average concentration at every cross-section due to the effects of the surface charge. It is given by where nano is the perimeter of the nanochannel's crosssection. Note that if the surface charge density is negative, σ s <0, the excess concentration is positive, Σ s >0, and viceversa.
In this work, our analysis focuses on the Ohmic resistance of nanochannel-microchannel systems, Ohmic . Nonetheless, in the literature, one often finds discussions on the electric response in terms of the Ohmic conductance of a system, ℎ . The two quantities are, however, reciprocal to each other such that ℎ = ℎ −1 whereby either description is appropriate. However, depending on the physical nature of the problem, there are occasions when one is more appropriate than the other. We present two examples one for each scenario. Under certain circumstances, when = 0 one finds This equation is commonly used to describe the common nanochannel ohmic response of any ion-selective system [35,[71][72][73]. At high concentrations (Σ s /c 0 ≪1), the conductance is linear with the concentration, G unit-cell (ΣR micro =0)~ c 0 (bulk property), while at low concentrations ( / 0 ≫ 1), the conductance is concentration-independent, unit-cell ( micro =0)~0 0 (commonly termed surface-conductance).
Special care should be taken with assuming micro = 0, as this is tantamount to assuming that the effects of the microchannels are completely negligible and that the system is quasi-1D [35,69] ( FF = 0). In contrast, when micro ≠ 0, it is beneficial to use the resistance description. We will demonstrate in Sec. 2.3 that in this situation, Eq. (1) can be reduced to a simple sum of electrical resistancesi.e., serially connectedand its physical interpretation in terms of resistance is preferential over conductance. Finally, we wish to emphasize the differences between Eq. (5) and another commonly used, but incorrect model, for the a nanochannel-only system. This model is given by and is referred to as the super-position model as it assumed that the total conductance is a super-position of the bulk conductance and the "surface conductance" that is linear and independent of the bulk concentration. The mathematically and conceptually simplistic form of Eq. (6) also suggests that the resistances associated with the bulk and the surface charge are in a parallel configuration. In the two extreme limits of / 0 ≫ 1 and / 0 ≪ 1, Eqs. (5) and (6) match identically, while in the intermediate range of / 0~1 these two models differ. The difference in these two models is that Eq. (6) is an empirical model that lacks a solid theoretical foundation while Eq. (5) represents the exact solution to the PNP equations. Refs. [68,69] provide a thorough discussion on the shortcomings of Eq. (6) and why it various derivative models are erroneous.

Field-focusing resistance
Before discussing how the electrical response can be written in terms of series and parallel circuits, it is beneficial to discuss the physical meaning of the field focusing resistance, FF , that was introduced above. In the case of a single (isolated) circular pore of radius, cylinder , surrounded by an infinitely large microchannel/reservoir, one recognizes FF to be access = res /(4 cylinder ) which is the classical access resistance solution [63][64][65][66][67]. However, the access resistance is somewhat restrictive as it does not account for the effects of the finite size of realistic reservoirs and interchannel interactions stemming from the presence of multiple pores in the system. When one calculates access , it is assumed that micro is negligible. Indeed, if all the characteristic lengths ( , , ) tend to infinity, then micro tends to zero. Neglecting the effects of micro is sometimes valid and sometimes not. In general, special care should be taken before neglecting micro in multichannel systems. Consider an array whose total height and width are array and array , respectively. If these lengths are large, it is tempting to neglect micro . However, such an assumption is oversimplifying. Consider the case of an array of × cells. Then both array and array need to be divided by and , respectively. If also tends to infinity, then the ratio array / can lead to a finite valuethis is the height of the unit-cell, (a similar argument holds for ) such that the ratio / is not negligible. Hence, the assumption of neglecting micro needs to be evaluated for every physical scenario. Here we consider the most general scenario and retain micro . In this light, it is important to consider the conventional method [74] of assessing the diameter of a nanopore using the classical expression for access . While this gives a reasonable estimate of the pore diameter of a single isolated channel, when characterizing a realistic multichannel system, a better estimate is attained by retaining the contribution of micro and employing the accurate expression for FF for a cylindrical nanochannel presented herein (See Supplementary Information).
To the best of our knowledge, there are only two known solutions for FF . The first is for the classical access resistance, access . The second, derived by us in a set of past works [68,69,69,75] has been investigated for cuboidal nanochannels interfacing with cuboidal microchannels. In fact, Eq. (1) was explicitly derived for cuboidal nanochannels. One of the novelties of this work is to demonstrate that Eq. (1) holds for nanochannels of arbitrary cross-sectional geometries and to provide an expression for FF that can be consistently calculated for any geometry. The mathematical justification and general expression for FF , for any geometry is given in the Supplementary Information. In Table S1 of the Supplementary Information we provide FF for the highly investigated cases of arrays of cuboidal channels or cylindrical pores.
In the next subsection we will analyze Eq. (1) in the two distinct limits that depend on the normalized average excess counterion concentration, / 0 : / 0 ≪ 1 and / 0 ≫ 1. At these limits, Eq. (1) can be described as a set of serially connected resistances. In both instances, it can be observed that the expression for FF is independent of / 0 . This can be attributed to how FF is calculated [68] (see Supplementary Information).

Series circuit
We now demonstrate that Eq. (1), combined with Eq.
(2), represents a simple electrical circuit of resistances connected in a series configuration [ Figure 2(b)]. The physical interpretation of a series of resistances is that ionic fluxes are conserved at every cross-section in the system (Kirchhoff's law).
We will consider the two distinct limits that depend on the normalized average excess counterion concentration, / 0 : / 0 ≪ 1 and / 0 ≫ 1. The former limit of / 0 ≪ 1 corresponds to the case that the excess counterion concentration within the nanochannel is negligible compared to the bulk concentration. This case is commonly referred to as the limit of vanishing selectivity, where the nanochannel does not filter any of the coions and the electric response of the system depends only on bulk properties (i.e., this is the bulk response of the system and is independent of the surface charge density). In this scenario, it can be shown [68] that ≅ 1 2 such that the unit-cell resistance is given by can be shown that = 1. This is the all-important limit of ideal selectivity where the nanochannel exhibits perfect coion exclusion. Ideal selectivity lies at the heart of ED and RED systems. In this scenario, the unit-cell resistance is given by The three substantial differences between Eqs. (7) and (8) are the following. First, in Eq. (7) the factor 1 2 is due to the equal but oppositely directed transport of coions and counterions. Twice the number of charge carriers in the conducting medium halves the resistance. Second, in Eq. (8) the nanochannel resistance is modified by the 0 / term. This term captures the nanochannel's ability to ideally exclude coions. This term also leads to the third significant difference. For a nanochannel system, at least one of its characteristic lengths (ℎ, , or both) is substantially smaller than the length, , such that nano dominates unit-cell .
(7)] and unit-cell (ideal) [Eq. (8)] was verified experimentally in Ref. [69]. In that work, the Ohmic resistance of four different single microchannel-nanochannel systems (i.e., four different unit-cell configurations) was measured over a wide range of concentrations. The microchannel geometry was kept constant across all systems ( micro = ), as well as the nanochannel height and length. The only geometric parameter that was varied was the nanochannel width, , such that both nano ( ) and FF ( ) varied (see Ref. [69] for values for the geometry). Figure 3 shows the excellent correspondence of the experiments and theoretical predictions of the log 10 -log 10 plot of the resistance, unit-cell , versus the bulk concentration, 0 . The following observations are essential: -Equation (1)  be observed that when the width is varied, the ratio of nano to micro is varied such that nano 0 / is not necessarily the dominating resistance but instead micro is the dominating contribution.

Parallel circuit
Thus far, in this work and our past works, we have considered a single channel system, i.e., the unit-cell [ Figure 2(b)]. However, the total electrical resistance total of a multichannel array system consisting of channels remains to be addressed [ Figure 2(a)]. Here we will discuss the total − dependence. To that end, we leverage our understanding from single channel systems. Namely, in single channel systems it is clear that the walls provide boundary conditions of no-flux for electric field lines, ionic flux, electrical current density, etc. Mathematically, such a boundary condition can be written as j⋅n=0 where j is the generalized flux and n is the unit vector normal to the boundary. However, the boundary condition j⋅n=0 is not unique to solid walls. It also describes the boundary condition for planes of symmetry. Thus, for the array in Figure 2(a), the inner planes ensure flux conservation within every independent unit cell. Hence, unit-cells are mutually independent "building blocks" of the array.
Notably, the independence of each unit-cell suggests that the equivalent electrical circuit of an array of identical unit-cells can be described as an array of = × (equal to the total number of nanochannels) unit-cells connected in parallel. In the nanofluidic system, this means that the unit-cells act as independent and identical paths for ionic fluxes across a potential drop, . We emphasize that in a multichannel system, it is unit-cells, rather than individual nanochannels that are mutually independent. Thus, the total resistance is given by Thus far, we have established that unit-cell [Eq. (1)] holds for a single unit-cell system. In the following section, we will establish that Eq. (9) holds for multichannel systems. Note that this total~− 1 scaling differs substantially from other works. One line of works [64,[77][78][79][80] claims a sublinear decrease of the resistance total~− 1/2 , while another line of works [81,82] provides another model (see Sec. 4 for a thorough discussion on both models).
Our model differs from these models in several manners. First, other models do not use the concept of unitcells. Rather, they model a single nanochannel and the classical access resistance and thereafter, modify the resistance of their "single" channel system to account for interactions. However, such an approach implicitly allows for the exchange of flux between adjacent and non-adjacent unit-cells, the modelling of which is subsequently subjected to numerous simplifications. These other models also suffer from additional unnecessary complexities when accounting for both the bulk conductance as well as the conductance associated with the surface charge conductance (see the last paragraph of Sec. 2.1). It should be stated that the condition of no-flux between neighboring cells (nanochannels and microchannels) imposed in our model will be shown to be substantiated by direct simulations in the next section. Finally, we reiterate the important statement that the concept of the unit-cell holds so long as the effects of convection are negligible. However, once convective effects are apparent, additional considerations are needed [83].

RESULTS
Our results are presented in the three following subsections. In the first two sub-sections (Secs. 3.1 and 3.2), we utilize numerical simulations to demonstrate that total~− 1 [Eq. (9)], while in the third sub-section (Sec. 3.3) we utilize the past experimental work of Ref. [83] to confirm Eq. (9).
The numerical simulations are divided into 2D (Sec. 3.1) and 3D simulations (Sec. 3.2). The reason for this, rather, artificial division is primarily for demonstration purposes -2D arrays are simpler to visualize and easier to comprehend. These 2D arrays are degenerate cases of the more general 3D solution where either ℎ = or = (naturally, a 1D system occurs when both ℎ = and = ). Hence, one should expect that the findings in 2D should also hold in 3Dindeed, they do.
For both 2D and 3D numerical simulations, we retained the unit-cell geometry and varied the number of cells, . The details of the numerical simulations can be found in Sec. 2 of the Supplementary Information.
We also note here that there is a "substantial" difference between the "conditions" of the simulations and the experiment. Numerical simulations focus on the case of vanishing selectivity [Eq. (7)], while experiments were conducted in the regime of ideal selectivity [Eq. (8)]. There are several reasons for this.
It is now well established that when a membrane or nanochannel is highly selective, the − is not linear at all voltages. Rather, two additional regimes exist above the low-voltage-low-current Ohmic response given by Eq. (8). These are the diffusion-limited limiting current regime [84,85] and the over-limiting current regime dominated by electroconvection [86][87][88][89][90]. The experiments of Ref. [83] were conducted several years ago with the distinct purpose of delineating the effects of electroconvection on the − of highly-selective nanochannel arrays. In such systems, when electroconvective effects appear, it is known that the unitcell approach breaks down. Hence, we reiterate that our approach is limited only to the Ohmic response [Eqs. (7) and (8)], and our analysis will focus only on the Ohmic data from Ref. [83]. If one chooses to neglect the effects of electroconvection, one can still simulate nanochannel arrays under the appropriate conditions of ideal selectivity. However, simulations of such systems are inherently computationally costly as they account for the sharp Donnan potential drop at the microchannel-nanochannel interface that occurs over the Debye length = [( 0 ℜ )/(2 2 0 )] 1/2 ( 0 and are the vacuum and relative permittivity, respectively). Instead, one can simulate systems under the conditions of vanishing selectivity. This can be rationalized based on the following arguments. Note that Eq. (1) was derived for the general scenario that / 0 can take any value. The two scenarios of vanishing selectivity ( / 0 ≪ 1, [Eq. (7)]) and ideal selectivity ( / 0 ≫ 1 [Eq. (8)]) were derived from this general model. Thus, from the mathematical point of view, these two scenarios are identical in that they are derived from the same equations and boundary conditions. Once this is recognized, simulating one scenario teaches something regarding the other. Consider a cylindrical nanochannel where nano and nano are known (our argument holds for any geometry). Note that the dimensionless number / 0 = −4̃( / ) 2 depends on two dimensionless numbers: ̃= /( 0 ℜ / ) and / . If / 0 ≪ 1, then either |̃| ≪ 1, / ≪ 1, or both. In the limit |̃| → 0 and / ≪ 1 the coion and counterion concentrations are equal everywhere outside of the Debye layer. Thus, the PNP equations are reduced to the simpler Laplace equation [91], whereby the dependence on the Debye length vanishes. Thus, it is easier and computationally efficient to simulate the Laplace equation, which is independent of surface charge effects as well as the effects of boundary layers that form on the order of the Debye length.  Table 1 and Table 2. Figure 4(a) shows the 2D distribution of the electrical potential, , of three cells out of an array of = 50. The potential has been normalized by the thermal potential th = ℜ / ≈ 0.0257 (at room temperature = 298 ). Observe that all three cells have the same distribution. The computed streamlines of the electric field, = − , are depicted in blackone should remember that there is no flux across a streamline. This visualization immediately identifies the streamlines that act as apparent internal walls within the multichannel system. For the sake of clarity, these calculated streamlines have been highlighted in green-these streamlines have not been determined a priori. In fact, in the array simulations, we only impose boundary conditions at the outer edges/surfaces of the array, and all in-bulk results are determined directly by the numerical calculations.

Parallel circuits in 2D
To ensure that our numerical simulations correspond to the theoretical analysis, we undertook several comparisons (in 2D, = ). In Figure 4(b), we have plotted the electric potential along the centerline of a unit cell [marked by the dashed purple line in Figure 4(a)] versus that calculated by simulations of the array. The excellent correspondence confirms the parallel circuit abstraction presented in Figure  2(d) and the solution for (given in the Supplementary  Information). We then calculated the electric current, , for a potential drop,this is the current-voltage response ( − ). Figure 4(c) shows that numerically computed − versus the analytical − of a single unit-cell system multiplied by . The perfect correspondence confirms the prediction of Eq. (9). Figure 4(d) shows the total resistance total has the predicted −1 dependence.

Parallel circuits in 3D
Here, we repeat our analysis for a 3D array, as shown in Figure 5(a). We can create arrays in two perpendicular and independent directions, denoted by and such that = × . For example, for an array of = 50 one can imagine arrays of 2 × 25 or 5 × 10; however, the exact details are unimportant once the unit-cell geometry has been identified for a given arraythe only important parameter is the total number of independent cells, .
Visualization of the potential field in 3D is substantially more challenging. Therefore, we have presented the potential along planes passing through the center of the nanochannels that are parallel to the reservoir walls. We see that the streamlines in these planes are similar to those presented in Figure 4(a). Near the bulk/electrodes, the streamlines are parallel to each other, while near the microchannel-nanochannel interface, they are focused. The difference is in the focusing itself. In a 2D scenario where ℎ ≪ the potential profile is logarithmic [73] [Figure 4(b)] while in a 3D scenario where ℎ ≪ and ≪ the poten- -tial profile scales inversely with the distance from the nanochannel [75,92]. Thus, one should expect that in the 3D scenario, the changes will be more localized to the nanochannel interface. Indeed, Figure 5(b) demonstrates that the profile change along the centerline, ( , = 0, = 0)/ th , near the interface is much sharper. The excellent correspondence between theory and simulations is yet another confirmation of our predictions. We further demonstrate that numerical simulations recapitulate our prediction of the formation of certain virtual internal planes that act as planes of no-fluxthese are the boundaries of the unit-cells. In Figure 5(a), we have plotted a number of streamlines whose initial points are in these planes of no-fluxthese are the green streamlines. Once again, these streamlines are not calculated a priorithey are a result and indication that the system has inner planes of symmetry. Figure 5(c) shows the − response calculated from simulations versus the analytical − of a single unit-cell system multiplied by the number of unit-cells, . Figure  5(d) shows the total resistance total versus from which the predicted −1 dependence [Eq. (9)] can be observed. In the main text, we have considered cuboidal nanochannels [as shown in Figure 2(b)]. In the supplement, we repeat the analysis for a cylindrical channel of radius cylinder ( Figure  S4). Our analysis remains entirely unchanged.

Experiments on 3D arrays
We now demonstrate using the experimental work of Ref. [83] that nanochannel arrays follow the −1 scaling. To that end, we use their data and further extend their analysis. It is necessary to discuss the difference in how the geometry is here relative to how it is defined in Ref. [83]. This is related to the aforementioned comment that identifying the unit-cell is of utmost importance. Figure 6(a) is a schematic of the multichannel array system used in Ref. [83], segmented to highlight the corresponding unit-cells. The geometry of the unit-cells in Figure 6(a) is, in fact, half that of the general unit-cell considered in our above analysis [ Figure 2(b)]. As a result, the analytical expression for varies. Since we provide the final solution here in terms of our approach means unaltered once is calculated correctly for the suitable unit-cell geometry. The differences between the expressions for the "full-cell" and "half-cell" are further dealt with in the Supplementary Information. Also, it should be noted that Ref. [83] focuses on the effects of interchannel spacing and the effects of electroconvection on the overall electrical response of multichannel systems. Therefore, they were not interested in the Ohmic response of the devices and the dependence of on N. Here, we extend the analysis of their data to delineate the Ohmic response and verify Eq. (9). The experiments of Ref. [83] were conducted on several multichannel array systems with the same height, , and length, , but with a varying number of identical nanochannels, , such that the interchannel spacing, , varied from device to device. Note that the interchannel spacing, is equal to the width of the corresponding unitcell. The array spacing is given in the legend of Figure 6(c), and more details can be found in Table 1 of Ref. [83].
For each configuration, numerous − curves were measured. The mean − are plotted in Figure 6(b), where the inset focuses on the low-voltage Ohmic response. Figure  6(b) demonstrates the expected result that increasing leads to an increase in the total current. This makes physical senseas increases, total decreases, as the total area through which the flux is transported increases correspondingly. Figure 6(c) shows the current per channel, channel = / from which it can be observed that channels with larger had larger currentsthis too is consistent with the corresponding reduction of micro and FF .
Using the data from Ref. [83], we calculate total (measured) ( ) = / . Then, using Eq. (8), we isolate from the single-channel system ( = 1) and find that = 5.05[mol/m 3 ] (such that / 0 = 168.3) and the calculated surface charge density is = -0.043[C/m 2 ]. This value extracted from the single-channel system is then used to calculate the unit-cell resistances of all the other array systems, as predicted by Eq. (8) as a function of the interchannel spacing, unit-cell (theory) ( ). In Figure 6(d), we calculate the ratio total (measured) ( )/ unit-cell (theory) ( ). We demonstrate that this ratio varies as −1 , as predicted by Eq (9).

DISCUSSION AND FUTURE DIRECTIONS
In the two preceding sections, we presented the electrical circuit model for a nanochannel array, and numerical simulations and experiments that verified this model. In the following section, we will discuss the outcomes of the results presented in this work and their relation to several recent works that have covered a wide range of topics related to the physics and applications of nanochannel arrays.
Esfandiar et al. [93] considered an array of = 200 channels in a parallel, line array configuration [similar to that of Figure 6(a)]. Esfandiar et al. [93] upscaled from a single channel system to a 200-channel system to increase the sensitivity of their measurements. In their Figure S3, they show that at high concentrations (i.e., vanishing selectivity), the conductance per channel of the 200-channel configuration was equal to that of the single channel. This is unsurprising since, at this limit, the nanochannel resistance dominates the unit-cell response, and thus the multiplication of the unit-cell conductance by = 200 is rather intuitive. At low concentrations (i.e., ideal selectivity), the conductance per channel of the 200-channel configuration was two orders of magnitude smaller than that of a single channel. This, too, is unsurprising since, at this limit, the nanochannel is no longer the dominant resistance. In fact, for such a highly packed system, the effects of micro and FF are prevalent, especially if micro,200~2 00 micro,1 . Also, with regard to Esfandiar et al. [93], it should be mentioned that this experimental work considered sub-nanometer channels, and were relatively successful in utilizing a continuum mechanics model to rationalize their results. Such approaches have been used by others as well [31,71,72,[94][95][96]. Why continuum models work in a regime where they expected to fail warrants further investigation.
Recently, Lucas and Siwy [57] explored the possibility of designing ionic circuits based on nanochannel arrays. They simulated four different array configurations (3 × 3,2 × 3,1 × 3 and 1 × 1) of long cylindrical nanochannels. Thus, depending on the configuration, one can identify several distinct unit-cells. For example, in the 3 × 3 configuration, there are three distinct unit-cells: the central unit-cell, the corner unit-cells, and the center-face unit-cells. Accordingly, in their Figure 2(a), one can observe that the current per channel depends on the esoteric unit-cells within each system. In contrast, in this work, we have considered the much simpler (ordered) array comprised of identical unit-cells. While future works should consider how to extend our approach to systems with multiple unit-cells, it is important to note that the universality of our approach of electrical circuit modeling still holds.
While our work focuses on arrays of long nanochannels, the qualitative nature of the dependence of FF should hold for arrays of short nanochannels (i.e., when the diameter and pore length are of the same order of magnitude). Yazda et al. [20] recently fabricated an array within a hexagonal boron nitride/silicon nitride membrane. They measured the dependence of the generated osmotic current on the pore spacing and reported that the current increased with interchannel spacing. This, too, can be discerned with the understanding that highly isolated channels have large currents. Such a result is consistent with this work as well.
Gadaleta et al. [77] suggested, in the notation of this work, that as → ∞ the total conductance, which is reciprocal to the resistance ( total = total −1 ), goes to zero such that →∞ ( total / ) → 0. Such a suggestion is in contradiction with Eq. (9), which yields total = total −1 = unit-cell . The origin of this inaccuracy in Gadaleta et al. [77] is in their identification and calculation of unit-cell conductance. Specifically, total and unit-cell in their description were dependent on whereas in this work unit-cell is independent of . Using of approximate scaling laws to analyze experimental data can lead to the over-or underestimation of surface charge properties of newly developed materials. This is one shortcoming that this work strives to resolve.
In Figure 1 of Morikawa et al., [82] an electrical circuit is prescribed for a system of N nanochannels under the presumption that all the nanochannels operate as current sources connected in parallel and thereafter connected to the microchannel resistances (the effects of field focusing are not considered but the effects of a load resistance is considered). In the notation of this work, the electrical circuit can be described as total = ( nano / + 2 micro + load ). Their model/analysis has two interesting peculiarities. First, they assume that the entire model is dominated by an infinitely large load and the effects of the nanochannels and microchannels are negligible. This is rather surprising as this results in an electrical current that is independent of the nanochannel and microchannel geometries. Another peculiarity is that at the limit of an infinite number of channels, → ∞, the resistance is once more independent of the nanochannel geometry. In contrast, our Eqs. (1) and (9) do not exhibit such a peculiar behavior.
It has been recently suggested that nanochannel arrays undergo a breakdown of electroneutrality [97] whereby the counterion concentration is unable to balance the effects of the surface charge. The origin of the breakdown can be associated with replacing the boundary condition of no-flux (j⋅n=0) between two neighboring cells to be a boundary condition of zero-electric potential. However, as suggested in Ref. [91] and shown here [in Figure 4(a) and Figure  5(a)], two neighboring cells do not exchange electric-field flux, and the potential is not zero on the planes of symmetry. As a result of which, the suggested breakdown is inconceivable.
Nanochannel arrays have the potential to revolutionize the scope and ability of nanofluidics in improving RED energy harvesting systems. Yet, a conceptual problem needs to be settled concerning the parallelization of single-channel systems. Siria et al. [6] suggested that the single pore RED systems can reach current densities up to 0.8 [kWhm -3 ] and that one could reach substantial energy yields via parallelization. In contrast, Wang et al. [7] argued that parallelization would not yield the promised large fluxes due to the appearance of concentration polarization associated with multichannel systems. It should be stated that the analysis of both works is presented from different perspectivescurrent densities versus currentsand that both are correct. Our model and analysis, given below, can resolve this dichotomy by providing a more complete picture. Wang et al. [7] attribute the changes to concentration polarization that appear upon upscaling from a single channel to a multichannel system. We emphasize that the effects of concentration polarization are always apparentregardless of the number of channels in the system. Note that Eq. (1) [as well as Eqs. (7) and (8)  In all figures, the green arrow points in the direction of increasing lengths. The default geometry of the unit-cell, given by subscripts '1', is as described in Table 2.
To better understand the issue of currents versus current densities, we can consider either Eq. (8), which is relevant to RED processes, or Eq. (7), which is not relevant to RED but has the added benefit that it is surface charge independent. Regardless of the scenario, to increase the ionic currents, one must decrease the resistance as much as possible. Hence, one should decrease micro as much as possible. This suggests reducing FF as much as possible. In the limiting case, one has FF = 0, when the ratios / and ℎ/ approach unity. This scenario corresponds to a 1D membrane system -suggesting that membranes are the ultimate tool for large-scale RED. Here, the ratio nano / micro is maximal, allowing for enhanced fluxes. However, one still needs to account for the effects of micro which is often neglected. From the practical standpoint, in multichannel systems, one can never approach nano / micro → 1 as this would imply that there are no nanochannel walls present in the membrane to provide the surface charges necessary to achieve high selectivity [this corresponds to the scenario where there is a sparse presence of grey space in Figure 1 (e)]. Yet, Siria et al. [6] are correct in stating that isolated systems have larger currents and current densities. We now demonstrate this.
In Figure 7, we consider a cuboidal microchannel where = and cuboidal nanochannels where = ℎ. Figure 7(a) considers a scenario where the nanochannel geometry is kept constant (i.e., = ℎ = .) and the microchannel geometry is varied. Observe that as grows, the nanochannel becomes more isolated from its neighbors and the current increases. This is because micro is decreasing while FF is reaching its lowest value (the square equivalent of access ). Naturally, dividing the current by the constant nanochannel area does not change the trend of the current density, = / nano [ Figure 7(b)]. Physically, increasing the microchannel geometry flanking a nanochannel of a given geometry leads to higher ionic fluxes. Figure 7(c) considers a scenario where the microchannel geometry is kept constant (i.e., = = .) and the nanochannel geometry is varied. Once more, it can be observed that as ℎ/ → 1 such that nano / micro → 1, the current is increased. Once again, this is associated with the decrease of FF . In contrast to the previous scenario, in Figure 7(d) it can be observed that as nano is increased, the current density decreases such that the highly isolated channels have larger current densities and closely packed channels have smaller current densities. The smaller nano is, the larger will be. Thus, it is not surprising that when the size of the system is decreased, current densities improve. The findings related to optimizing the current and current densities can be summarized rather succinctly: -Enhanced current densities occur for highly isolated systems (This corresponds to a case where - Figure 1(e) has a large area of grey membrane material). However, the harvested current is limited by the total size of the system.
-Large total currents appear in highly packed (and non-isolated) systems. In such systems, while the current is maximized, the current density is correspondingly minimized. Hence from a practical point of view, one should define a priori what one is trying to maximize, given the additional constraints of space, thermal and mechanical stability. In recent years there has been an increased interest in the effects of surface charge regulation (SCR) whereby the surface charge (density) can be modulated by adsorption dynamics of ions onto the surface [71,72,[98][99][100][101] (see our recent works [39,102,103] for a thorough discussion). In these works, it shown that the surface charge density is a function of the bulk concentration, pH and other system parameters. The effective concentration dependance of surface density can be roughly written as ( 0 )~0 . Since the average excess concentration, , is linear with the surface charge density [Eq. (4)], the nanochannel resistance [ nano 0 / in Eq. (8)] is modified accordingly. In Figure 8 we demonstrate how the total resistance changes when microchannels (full unit-cell) and SCR are accounted for. First. we plot the nanochannel only resistance unit-cell ( micro =0) , given by Eq. (5) without SCRthis is the red line with a slope of = 0 at low concentrations. We get the expected behavior that at high concentrations, the resistances scale inversely with the bulk concentration, and at low concentration the resistance saturates to a value that is independent of the concentration and determined by the surface charge density. After adding SCR, we find that the low concentration slope is now − (blue dashed line). The magenta dotted line shows the schematic behavior of Eq. (1) , that accounts for the microchannels and field focusing resistances (as shown in Figure 3), but without the effect of SCR. We once more note that at low concentrations, the resistance no longer saturates to a constant value but rather the slope is -1 due to the effects of the microchannels and field focusing. Finally, we show that when SCR is added to Eq. (4) we have a more complicated resultthis is dark blue dashed-dotted line . At high concentrations, the response is determined by the bulk nanochannel resistance with a slope of -1. At intermediate values, the response is still determined by the nanochannel, but now this resistance is determined by SCR such that the slope is − . At even lower concentrations, the microchannels and field focusing effect dominate the response and curve reaches a slope of -1. Finally, it should be noted that our model is focused on characterizing the steady-state response of the system. However, in many scenarios the response is transient in nature. However, transient characterizations [104] are nontrivial and are thus left to future works.

CONCLUSIONS
This works addresses the open question of relating the electrical response of a single nanochannel system to the response of an array. Understanding and elucidating this upscaling is of particular importance to desalination and energy harvesting systems that utilize nanoporous membranes that are comprised of any number of pores. Here, we propose to treat this random geometry as a simplified array whose analysis can be conducted straightforwardly. Notably, the results provide remarkable physical insightsnamely, that nanochannel arrays can be represented as a simple electrical circuit comprised of resistances connected in series and parallel.
Our starting point is the microchannel-nanochannel array presented in Figure 2(a), whereby we observe that it is comprised of "unit-cells" [Figure 2(b)]. The resistance of each unit-cell can then be sub-divided into three separate contributions [ Figure 2(d)]: 1) nanochannel resistance, nano , 2) microchannel resistance, micro , and 3) field focusing resistance, FF (a generalization of access resistance, access , that accounts for interchannel interactions that are not apparent in highly isolated single-channel systems). The electrical resistance of the unit-cell, unit-cell , is then represented as an electrical circuit of serially connected resistances. We then show that the total resistance, total , is that of an electrical circuit comprised of unit-cell connected in parallel such that total = unit-cell / [ Figure 2(d), Eq. (9)]. While this appears to be relatively intuitive and utilizes the simplest of electrical circuits imaginable, to our dismay and surprise, to the best of our knowledge, the model presented in this work [and confirmed by numerical simulations (Secs 3.1 -3.2) and experiments (Sec. 3.3)] has not been previously presented or discussed.
A few important comments are warranted. First, while it appears that we have used a simple abstraction to represent the electrical circuitthis is not the case. In fact, the presented abstraction represents the exact solution to the Poisson-Nernst-Planck equations in the appropriate limits of Ohmic conductance. Hence, the abstraction presented in this work is not merely another hypothesized electrical circuitit is the accurate equivalent circuit, and it supersedes all other suggested models. Second, another consequence of this model is that the analysis of a multichannel system is now reduced to the problem of studying the much simpler unit-cell problem. Third, we provide a new expression for FF that can be computed for any desirable geometry. Fourth, this work has focused on the Ohmic resistance (or Ohmic conductance); however, so long as the effects of electroconvection remain suppressed, the current-voltage response can be generalized to larger voltages when the currents are diffusion-limited. Fifth, without going into additional details given in the above discussion (Sec. 4), this model provides theoretical predictions that can rationalize previous experimental works.
Thus, our approach of electrical circuit modeling and the underlying theoretical models serve as a tractable analytical method to aid the accurate interpretation of experiments, analysis of ED/RED systems of scale, and the design of specific nanofluidic circuitry.