Electrokinetic Analysis of Energy Harvest from Natural Salt Gradients in Nanochannels

The Gibbs free energy released during the mixing of river and sea water has been illustrated as a promising source of clean and renewable energy. Reverse electrodialysis (RED) is one major strategy to gain electrical power from this natural salinity, and recently by utilizing nanochannels a novel mode of this approach has shown improved power density and energy converting efficiency. In this work, we carry out an electrokinetic analysis of the work extracted from RED in the nanochannels. First, we outline the exclusion potential effect induced by the inhomogeneous distribution of extra-counterions along the channel axis. This effect is unique in nanochannel RED and how to optimize it for energy harvesting is the central topic of this work. We then discuss two important indexes of performance, which are the output power density and the energy converting efficiency, and their dependence on the nanochannel parameters such as channel material and geometry. In order to yield maximized output electrical power, we propose a device design by stepwise usage of the saline bias, and the lengths of the nanochannels are optimized to achieve the best trade-off between the input thermal power and the energy converting efficiency.


S1 Comparison with Teorell-Meyer-Sievers Model
By using Eq.4 and 5 and in the main context, we obtain the following expression for current-voltage characteristic based on TMS model: where θ = arctan(σ w /eC 0 R). The I(V ) characteristic is then determined by the following relation: where R ch =

S2 Comparison with Charge Regulation Model
Quantitatively in charge regulation model, the relation between surface charge density on the channel wall and the imposed saline concentration, σ w (C 0 ), is calculated in a self-consistent manner. First, a guess value of σ w is proposed for a given C 0 , and then the corresponding surface potential φ s is obtained based on Eq.13 and Eq.14 of the main context 1 . On the other hand, φ s can also be estimated from the Poisson-Boltzmann equation. For example, for cylindrical nanochannel with monovalent ions, it is written as Eq.2 of the main context. By tuning the magnitude of σ w , the values of φ s calculated by the two approaches get to agree with each other. In this way, we arrive at σ w (C 0 ) relation as shown in Fig.S5.
However, quantitatively simulated σ w (C 0 ) by the Charge Regulation Model (solid lines with hollow symbols) does not agree with the fitting values from experiments (dashed lines with solid symbols), as shown in Fig.S5. The former are 1 or 2 orders smaller than the latter, no matter how we tune those parameters of Charge Regulation Model in the reasonable regions. Moreover, the simulated σ w (C 0 ) by the Charge Regulation Model are 1 or 2 orders smaller than the quantities estimated via other experiments (e.g., Effect of Salt Concentration on the Electrophoretic Speed of a Polyelectrolyte through a Nanopore. Phys. Rev. Lett.2007, 98, 238104). We have to conclude that there should be some deficiency when applying the current Charge regulation model directly to SiO 2 nanopore transport. Yet we are not very clear about the mechanism, and then left further exploring to interested readers.

S3 Two-Dimensional Axial-Symmetric Multi-physical Model
As mentioned in the main context, the MoS 2 nanopore system can be modeled as consisting of a cylindrical nanopore and two chamber. As indicated in Fig.S4a, C H and C L are used to designate the bulk concentrations of the concentrated and dilute electrolytes in the two reservoirs, respectively. The distribution of the electric potential φ inside the nanopore is governed by the Poisson equation: Where, φ , ρ e , C + , C − , ε 0 and ε r are the electric potential, net charge density, cation concentration, anion concentration, permittivity of vacuum, dielectric constant of water, respectively. The flux of each ion species consists of the flux distributions from diffusion, electromigration, and convection: Here, J + , J − and u are the cation flux, anion flux and velocity vector, respectively. For the steady-state solution, the ion flux satisfies the following continuity equation: Motion of the incompressible solution is generated by the electric potential and pressure gradient and can be described by the modified Stokes equation and the continuity equation: Here, p and µ are the pressure and viscosity, respectively. In Eq.S7, the inertial terms in the NavierStokes equation are neglected because the Reynolds number for the solution flow in the nanopore is typically very small. In these equations, the transport of the ion species and the solution are strongly coupled. Finally, the current through the nanopore I pore was calculated by integrating the current density over the cross-sectional area of nanopore as follows: Based on the physical model and referring to the Fig.S4(b), the above equations (Eqs.S4-S8) are solved numerically with the boundary conditions as follows: Plane CD : φ = U, C ± = C H Plane EF : φ = 0, C ± = C L Plane AB and HG : Eq.S4-S8 along with the above boundary conditions were solved by using finite element calculations in COMSOL. Fig.S4(b) shows the typical triangular computation meshes used in the simulation. Since the numerical solution accuracy strongly depends on the mesh size, a refined mesh is necessary in the region near the surface where the dependent variable gradients are pronounced. In this study, finer mesh was applied in the region near the charged surface in order to capture the subtle changes in the ion concentration and electrical potential. Solution independence on the mesh size was carefully studied before reporting the final results. The minimum element spacing inside the nanopore is less than 0.1 nm. The calculation results are confirmed to the grid independent solution by solutions obtained from more dense meshes.

S4 Comparison with Two Extreme Cases
In order to evaluate the power generation performance of real SiO 2 nanochannel, we demonstrate the two extreme cases as comparisons: no perm-selectivity and ideal perm-selectivity. In our ∆V σ versus ∆V D model, we have demonstrated the open-circuit voltage coming from RED as follows: For one extreme case: a liquid junction (no EDLs), the surface of the nanochannel is electrically neutral (σ w = 0). And the surface will not attract counterions nor repel co-ions, leading to the concentration of cations and anions is equal in the solution.
In this case, the open-circuit voltage can be simply obtained owing to Λ + = Λ − : When we directly imposed the sea and river at the two ends of nanochannel, we can calculate On the other hand, assuming that SiO 2 nanochannel is the ideal perm-selectivity, there are only cations within nanochannel. The open-circuit voltage can then be obtained by considering Λ − = 0: When we directly imposed the sea and river at the two ends of nanochannel, we arrive at V open = k B T e ln( C max C min ) = 103.4mV (S13) From above calculations, we find the open-circuit voltage (21mV and 103.4mV) in two extreme case are both higher than that results in the real SiO 2 nanochannel as shown in Fig.2a of the main context (15mV). The above analysis also shows good agreement with our qualitative anticipation: the orientation of diffusion voltage ∆V D is from the C max end to the C min end since Cl − diffuses faster than Na + , however the direction of exclusion voltage ∆V σ is from the C min end to the C max end owing to the cation selectivity of SiO 2 nanochannel. The opposite orientations of diffusion voltage ∆V D and exclusion voltage ∆V σ make it reasonable that the amplitude of open-circuit voltage in the real SiO 2 nanochannel is smaller than the ideal perm situation.
If we hope to change the opposite direction of ∆V σ versus ∆V D , we can do some chemical modification on SiO 2 surface to reverse ion selectivity. It certainly enhances the power generation performance, but the costs is greater than the performance improvement. Thus we turn our strategy to the designing of nanochannel geometry parameter and salt concentration at two ends of nanochannel in the main context.