Scalability of nanopore osmotic energy conversion

Abstract Artificial nanofluidic networks are emerging systems for blue energy conversion that leverages surface charge‐derived permselectivity to induce voltage from diffusive ion transport under salinity difference. Here the pivotal significance of electrostatic inter‐channel couplings in multi‐nanopore membranes, which impose constraints on porosity and subsequently influence the generation of large osmotic power outputs, is illustrated. Constructive interference is observed between two 20 nm nanopores of 30 nm spacing that renders enhanced permselectivity to osmotic power output via the recovered electroneutrality. On contrary, the interference is revealed as destructive in two‐dimensional arrays causing significant deteriorations of the ion selectivity even for the nanopores sparsely distributed at an order of magnitude larger spacing than the Dukhin length. Most importantly, a scaling law is provided for deducing the maximal membrane area and porosity to avoid the selectivity loss via the inter‐pore electrostatic coupling. As the electric crosstalk is inevitable in any fluidic network, the present findings can be a useful guide to design nanoporous membranes for scalable osmotic power generations.

). Red, green, and skyblue colors denote the cis-to-trans ion concentration ratio (ctrans/ccis).dpore is the diameter of the pores.The curves are almost linear and cross zero current at zero voltage when there are no salt gradients irrespective of dpore (red).On the other hand, smaller pores tend to show stronger rectifying behaviors with 1000fold salt gradients (skyblue).Another distinct feature can be seen at the intermediate salt concentration difference, where the polarity of the diode characteristics is inverted in the case of a 20 nm-sized nanopore (green).In addition to the curvatures, larger negative intersects at zero current are found for smaller pores under higher ion concentration ratio ctrans/ccis suggestive of larger diffusion potential difference induced by the salt gradient-mediated ion transport in conduits of stronger permselectivities.The ionic current rectification ratio rrec at ±0.8 V of the Iion -Vb curves recorded for single pores of diameter dpore (after subtracting Vele from the data) under various salt concentration differences noted by ctrans/ccis.Note that rrec is shown in logarithmic scale so that the polarity of the rectifying behavior can be seen by its sign as depicted by the blue and yellow regions as well as the insets describing the actual rectification directions.Scanning electron micrograph of a 200x200 twodimensional array of 20 nm-sized nanopores in a 40 nm-thick SiNx membrane.The entire multipore region of 9 μm square broke after the reactive ion etching process to sculpt the multipores due presumably to the too-narrow structures between the pores to endure the internal stress in the CVD-formed SiNx layer (inter-pore distance was about 45 nm).Scale bar denotes 10 μm.

Figure S2 .
Figure S2.Salt gradient-mediated ionic current (Iion) versus transmembrane voltage (Vb) characteristics of a cylindrical channel in a 40 nm-thick SiNx membrane.The redox potential difference (Vele) is subtracted from Vb (TableS1).Red, green, and skyblue colors denote the cis-to-trans ion concentration ratio (ctrans/ccis).dpore is the diameter of the pores.The curves are almost linear and cross zero current at zero voltage when there are no salt gradients irrespective of dpore (red).On the other hand, smaller pores tend to show stronger rectifying behaviors with 1000fold salt gradients (skyblue).Another distinct feature can be seen at the intermediate salt concentration difference, where the polarity of the diode characteristics is inverted in the case of a 20 nm-sized nanopore (green).In addition to the curvatures, larger negative intersects at zero current are found for smaller pores under higher ion concentration ratio ctrans/ccis suggestive of larger diffusion potential difference induced by the salt gradient-mediated ion transport in conduits of stronger permselectivities.

Figure S3 .
Figure S3.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of 1.5 μm-sized micropore in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S4 .
Figure S4.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of 300 nm-sized pore in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S5 .
Figure S5.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of 100 nm-sized nanopore in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S6 .
Figure S6.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of 30 nm-sized nanopore in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S7 .
Figure S7.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of 20 nm-sized nanopore in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S8 .
Figure S8.Ionic current rectification in a single pore under salinity gradients.The ionic current rectification ratio rrec at ±0.8 V of the Iion -Vb curves recorded for single pores of diameter dpore (after subtracting Vele from the data) under various salt concentration differences noted by ctrans/ccis.Note that rrec is shown in logarithmic scale so that the polarity of the rectifying behavior can be seen by its sign as depicted by the blue and yellow regions as well as the insets describing the actual rectification directions.

Figure S9 .
Figure S9.Estimation of ion selectivity in nanopores.a, The ionic current (Iion) versus transmembrane voltage (Vb) characteristics of two 20 nm-sized pores (inter-pore distance dpp = 1000 nm) in a 40 nm-thick SiNx membrane with a 1000-fold cis-to-trans salt concentration difference.The redox potential Vele was subtracted from Vb.The intersect at zero current defines the diffusion voltage Vdif that represents the degree of permselectivity as Vdif = Sion(kBT/e)ln(ccis/ctrans), where Sion is the selectivity factor denoting perfect cation-selective and non-selective transport in the pore by Sion = 1 and 0, respectively.b, Vdif plotted as a function of the logarithmic salt concentration ratio Ln(ccis/ctrans).The green dashed line is a linear fit, whose slope is used to calculate Sion.

Figure S10 .
Figure S10.Theoretical estimations of ion transport in a 1.5 μm-sized pore in a 40 nm-thick SiNx membrane under 1000-fold salt concentration difference at the cis and trans chambers.a, The ionic current (Iion) versus transmembrane voltage (Vb) characteristics obtained by solving Poisson-Nernst-Planck and Navier-Stokes equations in a framework of a finite element method.The salt concentration at trans (ctrans) was set to 1.37 M NaCl while that at the cis (ccis) was either 1.37 M (grey) or 1.37 mM (red).b-c, The ion concentration distributions around the micropore under Vb = +1 (b) and -1 V (c).Arrows indicate the direction of the water flow induced by electroosmosis due to the negative native charges on the SiNx wall surface.The hydrodynamic flow pushes the high-(low-) concentration electrolyte solution into the pore thus lowering (raising) the pore resistance in a Vb-dependent manner to induce the rectifying behavior shown in (a).

Figure S11 .
Figure S11.Model and conditions used for simulations of the electroosmotically-drived ionic current rectification in 1.5 μm-sized pore in a 40 nm-thick SiNx.a-b, Geometry of the pair-pore system (a, not to scale) and the actual model with meshes (b).c, Boundary conditions for the regions A through M defined in (a).Φ, σ, c, n, Ni, µEO, p, and v are the surface potential, surface charge density of the pore wall and membrane surface, ion concentration, normal vector, ion flux, electroosmotic mobility, pressure, and fluid velocity, respectively.Zeta potential ζ is deduced from σmem = -15 mCm -2 by Graham equation.S1

Figure S12 .
Figure S12.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of two 20 nm-sized nanopores separated by 5000 nm in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S13 .
Figure S13.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of two 20 nm-sized nanopores separated by 1000 nm in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S14 .
Figure S14.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of two 20 nm-sized nanopores separated by 300 nm in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S15 .
Figure S15.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of two 20 nm-sized nanopores separated by 100 nm in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S16 .
Figure S16.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of two 20 nm-sized nanopores separated by 80 nm in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.
Figure S17.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of two 20 nm-sized nanopores separated by 60 nm in a 40 nmthick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S18 .
Figure S18.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of two 20 nm-sized nanopores separated by 30 nm in a 40 nm-thick SiNx membrane under various salt gradients.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S19 .
Figure S19.Rectification ratio of the ionic current characteristics of pair pore systems.a, A sketch of two cylindrical channels of 20 nm diameter in a 40 nm-thick SiNx membrane.The distance between the centers of the nanopores dpp is varied from 5000 nm to 30 nm. b, rrec as a function of the salt concentration ratio ctrans/ccis.Error bars are the standard deviation of the rectification ratio estimated from the ionic current characteristics of the pair nanopores of various dpp.

Figure S20 .
Figure S20.Theoretically-deduced ion concentration distributions around 20 nm-sized pair nanopores in a 40 nm-thick SiNx membrane under 1000-fold salt concentration difference between cis and trans.a-e, Heat maps of the ion concentrations of the pair-pore systems with inter-pore distances dpp of 10000 nm (a), 2000 nm (b), 500 nm (c), 100 nm (d), and 30 nm (e).
Figure S21.Inter-pore distance-dependent ion concentration gradients.a, The ion concentration cion along the axial direction of the leftside of the 20 nm nanopores in FigureS17.Orange and purple dashed lines denote z = -500 nm and +500 nm, respectively.b, The concentration gradients Δcion calculated from cion at z = +500 nm and -500 nm.Note that Δcion first decreases but then decreases with reducing inter-channel distance dpp, which is in fair agreement with the non-trivial dpp dependence of the permselectivity observed in the experiments (see Figure3din the main text).

Figure S22 .
Figure S22.Model and conditions used for simulations of the salt concentration distributions around the salinity gradient-applied pair nanopores.a-b, Geometry of the pair-pore system (a, not to scale) and the actual model with meshes (b).c, Boundary conditions for the regions A through M defined in (a).Φ, σ, c, n, Ni, µEO, p, and v are the surface potential, surface charge density of the pore wall and membrane surface, ion concentration, normal vector, ion flux, electroosmotic mobility, pressure, and fluid velocity, respectively.Zeta potential ζ is deduced from σmem = -15 mCm -2 by Graham equation.S1

Figure S23 .
Figure S23.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of a 2 x 2 array of 20 nm-sized nanopores in a 40 nmthick SiNx membrane under various salt gradients.Inter-pore spacing is 9000 nm.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S24 .
Figure S24.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of a 4 x 4 array of 20 nm-sized nanopores in a 40 nmthick SiNx membrane under various salt gradients.Inter-pore spacing is 3000 nm.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S25 .
Figure S25.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of a 8 x 8 array of 20 nm-sized nanopores in a 40 nmthick SiNx membrane under various salt gradients.Inter-pore spacing is 1286 nm.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S26 .
Figure S26.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of a 10 x 10 array of 20 nm-sized nanopores in a 40 nm-thick SiNx membrane under various salt gradients.Inter-pore spacing is 1000 nm.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S27 .
Figure S27.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of a 20 x 20 array of 20 nm-sized nanopores in a 40 nm-thick SiNx membrane under various salt gradients.Inter-pore spacing is 474 nm.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S28 .
Figure S28.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of a 40 x 40 array of 20 nm-sized nanopores in a 40 nm-thick SiNx membrane under various salt gradients.Inter-pore spacing is 231 nm.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S29 .
Figure S29.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of a 60 x 60 array of 20 nm-sized nanopores in a 40 nm-thick SiNx membrane under various salt gradients.Inter-pore spacing is 153 nm.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S30 .
Figure S30.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of a 80 x 80 array of 20 nm-sized nanopores in a 40 nm-thick SiNx membrane under various salt gradients.Inter-pore spacing is 114 nm.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S31 .
Figure S31.Ionic current (Iion) versus transmembrane voltage (Vb) characteristics of a 100 x 100 array of 20 nm-sized nanopores in a 40 nm-thick SiNx membrane under various salt gradients.Inter-pore spacing is 91 nm.The ion concentration at cis (ccis) and trans (ctrans) denote the salinity difference across the membranes.Red plots are the average Iion estimated from the data obtained by scanning Vb from +1 to -1 V and -1 to +1 V. Error bars show the standard deviations.

Figure S32 .
Figure S32.Optimal salinity difference for gaining maximal osmotic power output from pair-pore membranes.The osmotic power Posm at different salinity gradients normalized by that at ctrans/ccis = 1000 (P1000).Dotted line points to Posm/P1000 = 1.The results show maximum Posm at 200-fold salinity difference irrespective of the inter-pore distance dpp.The trend was also the same for the single-nanopore.b, It suggests the fact that the permselectivity remains to be high upon enlarging ccis from 1.37 mM to 6.85 mM, thereby offering the large osmotic power by the gain in the ionic conductance.

Figure S33 .
Figure S33.Optimal salinity difference for gaining maximal osmotic power output from multipore membranes.The porosity dependence of the maximum osmotic power density Posm/4Amem under different salinity gradients.The results show maximum power density of 14 Wm -2 at 200fold salinity difference with the number of nanopores Npore = 100.

Figure S34 .
Figure S34.Rectification ratio of the ionic current characteristics of two-dimensional arrays of 20 nm-sized nanopores in 40 nm-thick SiNx membranes.Change in the ionic rectifying behaviors upon increasing the membrane porosity.Dashed line points to log10rrec = 0 that denotes no Iion rectification.Blue and yellow colors denote a transition of the behaviors from permselective to non-selective ion transport characteristics upon increasing the number of pores from 100 to 400.
Figure S35.Scanning electron micrograph of a 200x200 twodimensional array of 20 nm-sized nanopores in a 40 nm-thick SiNx membrane.The entire multipore region of 9 μm square broke after the reactive ion etching process to sculpt the multipores due presumably to the too-narrow structures between the pores to endure the internal stress in the CVD-formed SiNx layer (inter-pore distance was about 45 nm).Scale bar denotes 10 μm.