Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

DOI: 10.1039/C9SM02144B
(Paper)
Soft Matter, 2020, Advance Article

Ben L. Werkhoven* and
René van Roij

Institute for Theoretical Physics, Center for Extreme Matter and Emergent Phenomena, Utrecht University, Princetonplein 5, Utrecht, 3584 CC, The Netherlands. E-mail: ben_werkhoven@hotmail.com

Received
28th October 2019
, Accepted 27th December 2019

First published on 6th January 2020

We theoretically study the electrokinetic transport properties of nano-fluidic devices under the influence of a pressure, voltage or salinity gradient. On a microscopic level the behaviour of the device is quantified by the Onsager matrix L, a generalised conductivity matrix relating the local driving forces and the induced volume, charge and salt flux. Extending L from a local to a global linear-response relation is trivial for homogeneous electrokinetic systems, but in this manuscript we derive a generalised conductivity matrix G from L that applies also to heterogeneous electrokinetic systems. This extension is especially important in the case of an imposed salinity gradient, which gives necessarily rise to heterogeneous devices. Within this formalism we can also incorporate a heterogeneous surface charge due to, for instance, a charge regulating boundary condition, which we show to have a significant impact on the resulting fluxes. The predictions of the Poisson–Nernst–Planck–Stokes theory show good agreement with exact solutions of the governing equations determined using the finite element method under a wide variety of parameters. Having established the validity of the theory, it provides an accessible method to analyse electrokinetic systems in general without the need of extensive numerical methods. As an example, we analyse a reverse electrodialysis “blue energy” system, and analyse how the many parameters that characterise such a system affect the generated electrical power and efficiency.

The unique properties of nano-fluidic devices derive ultimately from the relatively large surface to volume ratio. These properties make the field of nanofluidics of great importance for transport in porous materials such as porous rocks^{6} and membranes.^{7} Additionally, nano-fluidic devices offer new promising roads to desalination,^{8} DNA translocation^{9–11} and renewable energy harvesting.^{12,13} For instance, they have been used to convert hydrostatic energy into electric power^{14,15} and to harvest energy from mixing salt and fresh water by reverse electrodialysis (RED),^{16,17} pressure retarded osmosis (PRO)^{18–20} or capacitive double layer expansion (CDLE).^{21} All of these nanofluidic devices are based on essentially the same system, composed of a channel with charged walls connecting two reservoirs with different reservoir conditions. Recent advances highlight the great potential for nanofluidics of carbon nanotubes (CNT),^{22} boron nitride nanotubes (BNNT)^{23} and MoS_{2} nanopores,^{24} which exhibit unique properties due to their small size and favourable electric properties.

Within linear response, we quantify the relation between the driving forces, Δp, ΔV and Δμ, and generated fluxes, Q, I and J, by a conductivity matrix G,

(1) |

(2) |

Alternatively, it is also possible to impose the flux instead of the applied potentials. For example, in an electric open-circuit channel the two reservoirs are not electrically connected and therefore no electric current can flow in steady state. In this case the flux I = 0 is imposed instead of the potential drop, but then too eqn (1) can be used. Since Δμ directly generates the current I_{DO} given by eqn (2), the only way to obtain a vanishing I is for the system to develop a potential drop over the channel, commonly referred to as the diffusion potential ΔV_{dif}^{2}, such that the induced electro-osmotic current exactly cancels the diffusio-osmotic current I_{DO}. The total current is simply the sum of the separate contributions, I_{total} = I_{DO}(Δμ) + I_{EO}(ΔV_{dif}) = 0, and we find

(3) |

In this article, we will show how we can obtain the conductivity matrix G from a well-known microscopic linear response theory based by the Onsager matrix L, which we will calculate analytically within the Poisson–Boltzmann formalism. We then show how to extend L, which is in essence a local linear-response equation, to G, which is a global linear-response equation. In order to validate our method, we compare predictions of eqn (1) with solutions of the Poisson–Nernst–Planck–Stokes equations obtained using finite element method (FEM). While FEM results are typically more precise, the great advantage of the proposed method is that these are much easier to implement and do not require complicated numerical techniques, and can thus be more easily used to analyse more complex nanofluidic systems. As an example, we will use the generalised conductivity matrix G to show how to incorporate a charge regulation mechanism with a salinity gradient, and compare predictions of the generated current with experiments on boron nitride nanotubes.^{23} The proposed framework provides a general formalism to investigate all electrokinetic systems as listed in Table 1, but as an example we will use G to analyse an electrokinetic system using reverse electrodyalisis under a wide variety of parameters without the need for extensive numerical calculations with FEM. This analysis highlights the convenience and utility of the conductivity matrix G for nanofluidics and electrokinetic systems in general.

Boundary conditions | System |
---|---|

Δμ = 0, Δp = 0, ΔV ≠ 0 | Electro-osmosis |

Δμ = 0, Δp ≠ 0, I = 0 | Streaming potential |

Δμ ≠ 0, Δp = 0, I = 0 | Membranes/diffusio-osmosis |

Δμ ≠ 0, Δp ≠ 0, I = 0 | Pressure retarded osmosis & desalination |

Δμ = 0, Δp ≠ 0, I ≠ 0 | Mechanical energy conversion |

Δμ ≠ 0, Δp = 0, I ≠ 0 | Reverse electrodyalisis |

Δμ = 0, Q = 0, ΔV ≠ 0 | Capacitive double layer expansion |

(4) |

The disadvantage of eqn (4), however, is that it relates the local driving forces to the fluxes, while eqn (1) relates the global driving forces to the fluxes. Since the global rather than the local driving forces are experimentally imposed or measured, in order for eqn (4) to be useful it must be extended to the same form as eqn (1). This is straightforward if L is constant throughout the channel, since then we can simply integrate eqn (4) along the length of the channel and find that L = G. This is the case when a non-zero Δp and ΔV is imposed, since only under extreme circumstances do these influence the properties of the channel. However, since the properties of the electric double layer are strongly affected by the salinity ρ_{s}, a non-zero Δμ necessarily leads to a laterally varying salinity ρ_{s} and thus a laterally varying L. In that case, therefore, it is no longer clear how to convert eqn (4) to a global equation, except in the case of a small relative change in salinity across the channel. If, however, the salinity changes for example from 20 mM to 500 mM, as is the case for fresh to sea water, a clear method is required to obtain the fluxes from L.

In order to obtain G from a heterogeneous L(z) we start from the condition that all fluxes Q, I and J are, in steady state and for non-leaky channels, constant throughout the channel (independent of z). In order to calculate the fluxes as a function of the global driving forces, we divide the system into infinitesimally small segments of width dz, schematically represented in Fig. 2, and apply the Onsager equation, eqn (4), for each segment

(5) |

(6) |

(7) |

(8) |

(9) |

As stated before, the Onsager matrix L can be determined analytically within Poisson–Boltzmann theory, and we can subsequently use eqn (9) to find the conductivity matrix G.

However, we can significantly simplify eqn (9) by splitting the contributions to L in a volume (L^{vol}) and a surface (L^{surf}) contribution,

L = L^{vol} + L^{surf},
| (10) |

In principle, the induced fluxes Q, I and J can flow via the EDL, represented by G^{surf} or via the region outside the EDL, represented by G^{vol} (each a sequence of many infinitesimally small conductors as in Fig. 2). These two are, in general, connected, represented by the dashed line (to be precise, every infinitesimal conductor is connected to its volume/surface counterpart). We can, however, significantly simplify the system by disconnecting the surface and volume fluxes (i.e. removing the dashed line in Fig. 3), which can intuitively be understood by realising that all radial components of the fluxes are small or negligible (such that the interchange between volume and surface is also small). We expect this simplification to break down for small aspect ratios /R and/or large heterogeneities across the channel.

The advantage of separating the volume and surface contributions is that the total conductance is now determined by the sum of the two separate conductances (note that G^{vol} and G^{surf} themselves can still originate from a laterally heterogeneous L^{vol} and L^{surf} respectively). We can analytically calculate G^{vol}, by evaluating eqn (9) with L^{surf} = 0 (see Section B, ESI,† for a derivation). On the other hand, it is not possible to determine G^{surf} analytically in the same way as G^{vol}. In order to obtain an analytic expression we approximate G^{surf} by L^{surf} evaluated at the average salinity ,

(11) |

G ≈ G^{vol} + G^{surf} ≈ G^{vol} + L^{surf}(ρ_{s} = ),
| (12) |

One significant advantage of the above formalism is that it is straightforward to also incorporate lateral heterogeneities other than a salinity gradient. For example, we will consider BNNTs and CNTs in this article, which obtain their surface charge from the adsorption of an OH^{−} ion. Because OH^{−} carries a net charge, the amount of OH^{−} adsorption depends on the surface charge itself via a mechanism known as charge regulation,^{30–32} and can be expressed as a Langmuir-type relation

σ(z) = z_{σ}Γ(1 + 10^{−pH+pK}e^{−eψ0(z)/kBT})^{−1},
| (13) |

Fig. 4 shows the salinity at the channel axis as determined from FEM solutions of the PNPS equations. Even for a very needle-shaped channel (/R = 25), the in- and outlet salinities clearly differ from the reservoir salinities. The effect becomes more pronounced for shorter and/or wider channels with a small aspect ratio. For example, for /R = 5 the outlet salinity is a factor 4 larger than ρ_{min} (see Section C, ESI†). We denote the inlet salinity as ρ_{in} and the outlet salinity as ρ_{out}, which now explicitly depend on R and due to the entrance effects (see Section C, ESI† for derivation). This correction is similar (although not equal) to the so-called access resistance,^{10} as it also slightly adjusts the salinity gradient. The chemical potential drop over the channel Δμ_{ch} is consequently not equal to the chemical potential difference between the two reservoirs, but actually

(14) |

In this article we (generally) assume a linear profile

(15) |

(16) |

(17) |

In this article we will consider an electrokinetic system as depicted in Fig. 1, with length , radius R, salinity ρ_{s}(z) given by eqn (15) and surface charge σ. The fluid flow is determined by the Stokes equation with an electric body force and the incompressibility condition,^{32}

−∇p + η∇^{2}u + e(ρ_{+} − ρ_{−})E = 0, ∇·u = 0,
| (18) |

−b∂_{r}u_{z}(r = R) = u_{z}(r = R),
| (19) |

(20) |

(21) |

By combining the above equations with the solutions of Poisson–Boltzmann formalism for a 1:1 salt,^{2,38} the full 3 × 3 Onsager matrix can be determined analytically. The majority of the matrix elements of L are already known, although we do find a contribution to L^{surf}_{23}, the non-advective contributions of eqn (17), that appears to have been overlooked in previous studies.^{23,39} It is an important contribution that cannot be ignored, and is in fact required by the symmetry of L. This term is intimately linked to the heterogeneity of the EDL: since the Debye length λ_{D} is a function of z, diffusio-osmosis generates a lateral component to the electric field which contributes to the generated fluxes (see Section G.8, ESI,† for detailed discussion of this subtle contribution). For the sake of completeness, however, we present not just L_{32} but the full 3 × 3 matrix.

Eqn (17) shows the Onsager matrix elements, with the Bjerrum length and λ_{D} = (8πλ_{B}ρ_{s})^{−1} the Debye length, ε the permittivity of water, ψ_{0} the surface potential, z_{σ} the sign of the surface charge, the average ion diffusion constant and the mobility mismatch. The constants P_{i} are positive numbers and function of ρ_{s}, σ and ψ_{0} only. For small surface charge, 2πλ_{B}λ_{D}σ ≪ 1, all these constants scale as P_{i} ∼ σ^{2} ∼ ϕ_{0}^{2}, while for large surface charge, 2πλ_{B}λ_{D}σ ≫ 1, P_{1} ≈ π^{2}/2, P_{2} ∼ σ, P_{3} ∼ |ϕ_{0}|, P_{4} ≈ π^{2}/4 and P_{5} ∼ σ. These constants are solutions to rather involved integrals, and the full expressions and their derivations can be found in Section D, ESI.† Note that L^{vol}_{12} and L^{vol}_{23} change sign if σ changes sign, while L^{vol}_{13} does not. This is directly reflected in Fig. 1, which shows that Q and J are always in the same direction while the direction of I with respect to Q and J depends on the sign of σ.

Most elements are known by specific names, for example in the context of electro-osmosis^{32} and diffusion-phoresis;^{40} L_{11} is inversely proportional to the fluidic impedance , L_{12} is proportional the streaming conductance , L_{13} is proportional to the diffusio-osmotic mobility D_{DO} = k_{B}TL_{13}, L_{22} is the electric conductivity of the channel and L_{23} the diffusio-osmotic conductivity. Since we use the full nonlinear Poisson–Boltzmann equation for non- or weakly-overlapping EDLs to determine L, we expect eqn (17) to break down for salinities exceeding approximately 100 mM, when finite-size effect become significant, or for multivalent ions. Moreover, for strongly overlapping EDLs both the solution to the Poisson–Boltzmann equations, and consequently the Onsager matrix eqn (17), as well as the entrance effects, eqn (16), must be adjusted, for example using the thin-pore limit.^{41}

Even though the formalism presented in this article still applies for strongly overlapping EDLs, the expressions for the Onsager matrix are modified in this regime and as a consequence a full analysis strongly overlapping EDLs is outside the scope of this article. We note, however, that eqn (17) shows excellent agreement for R ≈ 3λ_{D} (see Section E, ESI†), but breaks down for R ≈ 2λ_{D} with λ_{D} the Debye length associated to ρ_{min}.

Fig. 5 shows the dependence of the average fluid velocity ū = Q/(πR^{2}), electric current I and salt flux J on ρ_{max}/ρ_{min} ∈ [1,100], with ρ_{min} = 1 mM, for NaCl from the FEM calculations compared to the predictions of eqn (12) (blue) and eqn (9) (black), both for a short-circuit (a–c) and an open-circuit (d–f) system, for a charge regulation boundary condition with σ(ρ_{s} = 1 mM) = −0.05 e nm^{−2} (eqn (13) with pH–pK = 0.05), b = 0 nm, R = 60 nm, = 1.5 μm, ρ_{min} = 1 mM, D_{Na} = 1.33 × 10^{−9} m^{2} s^{−1} and D_{Cl} = 2.03 × 10^{−9} m^{2} s^{−1} (β = −0.21). Fig. 5 shows that eqn (9) is very accurate in reproducing the FEM results. In all cases, eqn (12) is less accurate than eqn (9) but often surprisingly accurate given its simplifications, especially if ρ_{max}/ρ_{min} ≲ 10 in both short-circuit and open-circuit conditions. The agreement in the open-circuit case thus shows that, even if there are multiple driving forces (i.e. both ΔV ≠ 0 and Δμ ≠ 0) the formalism remains accurate. We have furthermore compared the predictions and the FEM calculations for a non-zero slip length (b = 10 nm), smaller radius (R = 40 nm), higher surface charge (σ = −0.1 e nm^{−2}), smaller channel length ( = 375 nm) and higher minimum salinity (ρ_{min} = 20 mM) and found good agreement for all parameter variations (see Section E, ESI†). In addition, Fig. 5(d) shows that in an open-circuit system ΔV_{dif} changes sign for large Δμ since I_{DO} changes sign in the short-circuit case (I_{DO} changes sign due to the competition between L^{surf}_{23} and L^{vol}_{23}). Moreover, Fig. 5(e) shows that the fluid flow first decreases, than increases and even changes sign with increasing ρ_{max}/ρ_{min}. This is the result of an intricate balance between diffusio-osmosis due to Δμ and electro-osmosis due to ΔV_{dif}. The balance between the diffusio-osmotic and electro-osmotic driving forces depends strongly on β, and is thus very different for KCl (β = 0) than for NaCl (β = −0.21), and additionally depends on z_{σ}. Both of these behaviours are in agreement with experimental observations and interpretations.^{10,42}

Fig. 5 The short-circuit electric current I, open-circuit potential ΔV_{dif}, average fluid velocity and salt flux J as a function of ρ_{max}/ρ_{min}. The red line represents the FEM results, the blue line the prediction of (12) and the black line (9) for σ(ρ_{s} = 1 mM) = −0.05 e nm^{−2} ((13)), R = 60 nm, b = 0, β = −0.21 (NaCl) and ρ_{min} = 1 mM for short-circuited (a–c) and open-circuit (d–f) system. |

Recent experimental advances allow for direct comparison between theory and experiments for these kind of systems. For instance, measurements on osmotic power generation using a single boron nitride nanotubes (BNNT), carbon nanotubes (CNT) and MoS_{2} nanopores, have been shown to surpass older RED technologies based on much thicker membranes.^{43} With the theory presented here, we can directly compare with recent experiments. Fig. 6 shows the (short-circuit) diffusio-osmotic current I_{DO}, for both a constant charge and a charge regulating boundary condition eqn (13), as a function of the salinity ratio ρ_{max}/ρ_{min} for a nanochannel with ρ_{min} = 1 mM, σ(ρ_{s} = 1 mM) = −0.25 e nm^{−2}, R = 40 nm, b = 3 nm and = 1250 nm, which can be directly compared to the diffusio-osmotic current measurements on BNNT by Siria et al.^{23} Here, σ(ρ_{s} = 1 mM) was chosen such that similar I_{DO} values were obtained. First of all, it is evident from Fig. 6 that, especially for large ρ_{max}/ρ_{min}, the charge regulation boundary condition has a significant effect on the predicted electric current. A charge regulation boundary condition (eqn (13)) and the small slip length b = 3 nm of BNNTs^{44} are sufficient to obtain very similar values for I_{DO} (order 0.1 to 1 nA), but with a surface charge more than an order of magnitude smaller than estimated by Siria et al.^{23} Note that the contribution from the slip length, which, for large σ, scales with σ^{2} (see eqn (17) and associated text), becomes increasingly dominant for increasing σ. Even a relatively small slip length of b = 3 nm can therefore significantly affect the predicted fluxes. Note furthermore that σ varies significantly as a function of the channel position z, see inset Fig. 6, which explains why the charge regulation boundary condition gives a larger I_{DO} compared to the constant charge boundary condition, and furthermore emphasises the importance of even a small but finite b.

Fig. 6 The diffusio-osmotic current I_{DO} for KCl as a function of the salinity drop over the channel according to eqn (12) (blue) and eqn (9) (black) for both a constant charge (CC, dashed) and charge regulation (CR, full) boundary condition. For both CC and CR, σ(ρ_{s} = 1 mM) = −0.25 e nm^{−2}, ρ_{min} = 1 mM, R = 40 nm, b = 3 nm and = 1250 nm. The inset shows the surface charge as a function of the lateral position z. |

The surface charge σ(ρ_{s} = 1 mM) = −0.25 e nm^{−2} is much smaller that the value obtained from conductivity measurements on BNNT by Siria et al.^{23} It has recently been shown, however, that the adsorbed OH^{−} contributes significantly to the conductivity and other properties of the channel.^{45} Conduction via the Stern layer is not included in the current model, but an increased conduction will probably only lower the predicted surface charge even more. We have recently developed models for mobile surface charges,^{46,47} and incorporating these in the current theory is subject of future research.

The electrokinetic RED system, schematically represented in Fig. 7, is embedded in an electric circuit and thus allows a non-zero current I = I_{RED} to flow through the system. However, the circuit also contains an (Ohmic) resistance R_{load} that harvests the electric energy, which requires a potential drop ΔV in order for a non-zero current to flow. Assuming that R_{load} can be chosen freely, we will assume that R_{load} is chosen such that the generated electric power is optimised (as opposed to the energy conversion efficiency). It is straightforward to show that the generated power is maximised when R_{load} equals the resistance of the channel ,^{10,17} which fixes the current to half the short-circuit current eqn (2),

(22) |

(23) |

(24) |

Fig. 7 Schematic representation of an RED circuit, where a diffusio-osmotic system is embedded in an electric circuit with a resistance R_{load}. |

On the basis of eqn (23) and (24), we are in the position to use the conductivity matrix G to investigate the effect of system parameters on the RED performance without the need to run intensive FEM or other numerical calculations for each parameter set. As mentioned, two materials have shown great potential for osmotic energy conversion: CNTs and BNNTs. The reason for the success of the former is believed to be related to the small friction of water with the surface, i.e. a large slip length b,^{44,48–50} while for the latter the large surface charge is believed to be main cause,^{23} in addition to the large conductivities shown by both.^{23,34,45,49} For both materials, we assume a charge regulating boundary condition as in eqn (13). There are many parameters to investigate, but here we will focus on 4 main aspects: the surface charge density σ, the slip length b, the minimum salt concentration ρ_{min} and the mobility mismatch β.

As an example we will investigate a nanochannel with R = 40 nm, although it should be kept in mind that for RED a smaller R generally results in a higher P_{RED} and α_{RED}. However, the slip length of CNTs is known to vary with R,^{50} so a constant R allows us to assume a constant b for this analysis. We will use b_{BNNT} = 3 nm as the slip length for BNNTs^{44} and b_{CNT} = 30 nm for the slip length of CNTs.^{44,50}

Fig. 8 and 9 show the RED power and efficiency for ρ_{min} = 1 mM and ρ_{min} = 20 mM, respectively, for both KCl (a and c) (β = 0) and NaCl (b and d) (β = −0.21), for b = 0 (black dotted), b = 3 nm (red dot-dashed) and b = 30 nm (blue full) as a function of σ. The horizontal axis represents the surface charge σ at ρ_{s} = 1 mM. The surface charge of both BNNT and CNT surfaces originate from an OH^{−} adsorption reaction^{23,34,45} and strictly only takes negative values. Positive values are included (H^{+} adsorption), however, for a more complete analysis. There are a few observation we can make from these figures.

Fig. 9 As in caption Fig. 8, but with ρ_{max} = 500 mM and ρ_{min} = 20 mM. |

First of all, a comparison of the black (b = 0), red (BNNT, b = 3 nm) and blue (CNT, b = 30 nm) shows that not only a large but also a moderate slip length b has a significant effect on the electrokinetic properties of the system, as was also noted for mechanical energy conversion.^{51} This confirms that the large slip length of CNTs makes these nanochannels so promising. In addition, Fig. 8 confirms the point emphasised above, that even a small b can have significant effects on the current through the channel, especially for large σ.

Secondly, we see that the predicted power can significantly differ between KCl and NaCl, especially for large ρ_{min}, shown in Fig. 9. Many experiments are performed with KCl, but it is not a priori clear whether these results can be extrapolated to NaCl (the main species of salt for large-scale applications of RED). The difference between these two salts originates from the mobility mismatch, β_{KCl} ≈ 0 and β_{NaCl} ≈ −0.21, which not only affects the resulting fluxes but also breaks the charge inversion symmetry (see eqn (17)). If NaCl is the main constituent of the electrolyte, a positively charged surface is more effective than a negatively charged surface: a negatively charged surface will attract the cations to the surface, but Na^{+} has a lower mobility than Cl^{−}. The EDL thus has a lower overall mobility if σ < 0 than if σ > 0. This provides a general rule that RED systems generate more power at a higher efficiency if z_{σ}β < 0, because then the ion with the highest mobility is the most abundant in the EDL.

A comparison of Fig. 8 and 9 furthermore emphasises the point that the generated power and efficiency do not purely depend on the concentration ratio (i.e. Δμ), but are both a function of the separate salinities ρ_{min} and ρ_{max}.^{17} This is especially true for NaCl, for which the broken inversion symmetry is significantly more apparent for ρ_{min} = 20 mM (Fig. 9) than for ρ_{min} = 1 mM (Fig. 8). Especially if b = 0, the difference between the two cases is very pronounced (compare black dotted line Fig. 8(b, d) and 9(b, d)). The dependence on ρ_{min} can be understood by the fact that L^{vol}_{23}, and consequently G_{23} and I_{DO}, increase with βρ_{s} (see eqn (17)). All slip-length contributions are, however, independent of ρ_{s}, and all scale as bσ^{2} for large σ (see eqn (17)).

We also find that the generated power for ρ_{min} = 20 mM and ρ_{max} = 500 mM is higher than for ρ_{min} = 1 mM and ρ_{max} = 25 mM if b = 0, especially for NaCl with σ > 0. However, the efficiency α_{RED} is nearly an order of magnitude higher for ρ_{min} = 1 mM than for ρ_{min} = 20 mM, even though the chemical potential drop Δμ is the same in both cases. Both can be understood by the increased role played by the volume contributions L^{vol} of eqn (17). These contributions scale with ρ_{s}, so an increased ρ_{min} naturally leads to a larger I_{DO} (if β ≠ 0 via L^{vol}_{23}) and thus a larger P_{RED}. Similarly, the total salt flux J increases with ρ_{min} (L^{vol}_{33} ∝ ρ_{s}) which, in turn, decreases α_{RED} (see eqn (24)).

Finally, note that P_{RED} and α_{RED} develop a minimum, with a minimum value of zero, for NaCl with a small negative surface charge. This minimum shifts to larger values of σ if ρ_{min} increases, since this minimum is given by the value of σ for which the volume and surface contributions to I cancel. If we take the surface charge of CNTs at ρ_{s} = 1 mM to be σ = −(0.03–0.1) e nm^{−2},^{49,50} we even find that CNT are typically not far removed from the minimum observed in Fig. 9. We should note, however, that the location of this minimum depends on systems parameters such as R and b, so this does not mean that CNTs should not be used for RED. It does, on the other hand, stress the important point that β, σ (including its sign) and ρ_{min} are important parameters to keep in mind when optimising a given channel.

Note that our values for P_{RED} are of the same order of magnitude as measurements on BNNTs.^{23} These values are also consistent with measurements on nanopores,^{24} where they found P_{RED} three orders of magnitude higher than for micron-thick membranes, with three orders of magnitude lower. The predictions do certainly depend on the radius R, as RED typically generates more power per unit area and is more efficient for smaller R.^{17} The present analysis, however, emphasis the point that different systems with differing R, ρ_{min}, σ (including its sign), b and β, are optimised differently. There is of course an immense variety when it comes to nanochannels, but the framework presented in this article provides an accessible method with which these channels can be analysed. Moreover, the framework can be further improved, for example for smaller R, because the most restricting assumption of the Onsager matrix presented in this article, eqn (17), is the assumption of non/weakly-overlapping EDLs, meaning that eqn (17) is viable for R ≳ 12 nm for ρ_{min} > 10 mM. There is no general analytic theory for the matrix elements of L for arbitrary λ_{D}/R, but it is possible to take the thin-pore limit (λ_{D} ≪ R) of the Poisson–Boltzmann formalism to obtain analytical solutions.^{41} In addition, the Poisson–Boltzmann formalism typically breaks down for ρ_{s} > 100 mM, but there are theories to improve on Poisson–Boltzmann.^{52,53} Lastly, as already stated, it has been shown that surface conduction plays an important role for BNNTs and CNTs,^{45} which can further affect the (quantitative) predictions of the theory. This will be the subject of future research.

Having established the accuracy of the conductivity matrix G, we used it to analyse Reverse Electrodialysis without the need to use extensive numerical calculations such as FEM. We compared typical values for carbon nanotubes and boron nitride nanotubes, and showed, for example, that such systems behave differently when KCl is used compared to NaCl. Most notably, in the case of NaCl we showed that negatively charged surfaces such CNTs and BNNTs are significantly less effective than positively charged surfaces, especially if salinities like those of fresh and sea water are used. We furthermore emphasised that the produced power does not solely depend on the chemical potential drop across the channel, but on the reservoir salinities separately. We thus found that systems with different surface charge, different type of salt and salinities are optimised differently. Electrokinetic systems present a very large parameter space, too large to fully explore here, but for this reason electrokinetic systems represent a great variability and applicability. The framework presented in this article provides an insightful and convenient method to analyse them.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c9sm02144b |

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