Supporting Information for Conical Nanopores. for Efficient Ion Pumping and Desalination

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Emory Hardy
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1 Supporting Information for Conical Nanopores for Efficient Ion Pumping and Desalination Yu Zhang, and George C. Schatz,, Center for Bio-inspired Energy Science, Northwestern University, Chicago, Illinois 60611, United States, and Department of Chemistry, Northwestern University, Evanston, Illinios, 608, USA. Theoretical model The potential profile ψ is subject to the Poisson equation, 2 ψ(r) = ρ(r) ɛɛ 0 (S1) where ρ(r) = i c i(r)z i e. c i (r) and z i are the concentration and charge of the ith species of ions (C for Cation and A for anion). The current density of each species of ion is given by ( J i (r) = c i (r)u(r) D i c i (r) + z ) i e c i kt ψ(r). (S2) where e, ɛ 0, ɛ, k and T are the electron charge, permittivity of vacuum, the dielectric constant, Boltzmann constant and temperature. The continuity constraint requires J i = 0 at steady- To whom correspondence should be addressed Center for Bio-inspired Energy Science, Northwestern University, Chicago, Illinois 60611, United States Department of Chemistry, Northwestern University, Evanston, Illinios, 608, USA. S1

In steady state, the NS equation reduces to ν 2 u(r) p(r) e i c i (r)z i ψ(r) = 0. If the fluid velocity is neglected (i.e., electroosmotic effects are neglected), Eqs.

2 state and dc i dt = J i in the transient regime. u is the fluid velocity, which is described by the Navier-Stokes (NS) and continuity equations ρ u(r) t u(r) = 0. = ν 2 u(r) p(r) e i c i (r)z i ψ(r) = 0, (S3) ρ, p, ν and u are the fluid density, pressure, viscosity and flow velocity vector, respectively. In steady state, the NS equation reduces to ν 2 u(r) p(r) e i c i (r)z i ψ(r) = 0. If the fluid velocity is neglected (i.e., electroosmotic effects are neglected), Eqs. S1 and S2 reduce to the Poisson-Nernst-Planck (PNP) model. Figure S1: Schematic diagram of the conical pore in cylindrical symmetry (radial and axial coordinates). A conical nanopore of length L and radius r 1 at the tip side and r 2 at the base side of a membrane that connects the two reservoirs. The surface of the pore and part of the mouth (with length l m ) is charged as shown by the dashed line. Unless specified otherwise, r 1 = 2 λ and r 2 = λ are chosen for the conical pores while the radius of the cylindrical pores is r 2 = r 1 = 2 λ. Also, L = W = r = 00 λ, l m = 0 λ. Cylindrical symmetry of the pores is assumed, which reduces the problem into two dimensions, i.e., radial and axial coordinates. The three-dimensional PNP-NS equations are reduced into a two-dimensional domain due to cylindrical symmetry of the nanopore, which significantly reduces the computational S2

3 cost. It s also worth mentioning that reservoirs must be taken into consideration to avoid uncertainty in boundary conditions at the pore mouths. When the sizes of the reservoirs (W and r) are large enough, the numerical results are independent of reservoir size, as should be the case. After solving the coupled Eqs.(S1) and (S2), the current of each ion can be obtained from the flux density using I i = 2πe rj i (r)dr. The solute is assumed to be KCl. Thus the charges of the ions are z C = z A = 1. The dielectric constant of the KCl solution ɛ is 78., and the diffusion coefficients of the cation and anion are set to D A = D C = 2 cm 2 /s. The viscosity is ν = 3 kg/m/s and density is ρ = 00 kg/m 3. In order to accelerate convergence, c i and ψ(r) are divided into two terms, i.e., c i = c 0 i +c 1 i and ψ = ψ 0 + ψ 1, where c 0 i and ψ 0 are the concentration and potential profile at equilibrium (no bias voltage), respectively. 1 In addition, the following dimensionless units ( L, ψ, c i, t, σ, u and P ) are used in the simulation: Length: L = λ L, where λ = Potential: ψ = ψ kt/e ɛ rɛ 0 kt 2C 0 e 2 N A is the Debye length. Concentration: c i = C 0 c i Time: t = t λ 2 /D 0 where D 0 = cm 2 /s Surface charge: σ = σ ɛ 0ɛ rkt eλ. Velocity: u = u u hs, where u hs = ɛ 0ɛ rk 2 T 2 νe 2 λ. Pressure: P = P νu hs /λ Using the dimensionless units in our codes, the PNP equations are rewritten as 2 ψ = 1 z i c i. 2 j (S4) ( D 0 Ci D i t = c i u λu ) hs D c i z i c ψ i. (S) i S3

4 J C J A τ/2 τ 3τ/2 2τ τ τ Figure S2: Left: Schematic illustration of applied oscillating bias voltage; a rectangular wave shape of the bias voltage is used; Right: Illustration of ion fluxes driven by rectangular bias voltage and the NS equation becomes, D 0 ρ u ν t = P + 2 u 1 2 c i z i ψ i (S6) The boundary conditions of the PNP-NS equations are: Upper/bottom boundaries (marked in red in Fig. S1): ψ(t) = V bias (t)/2, ψ(t) = V bias (t)/2: C A = C C = C 0 ; p = 0. Surface of the pore: n ψ = σ ɛ 0 ɛ boundary condition (blue line in Fig. S1); (dashed blue line in Fig. S1)); u = 0, i.e., no-slip Note that a uniform surface charge is employed in this work. Such homogeneously distributed surface charge can be achieved experimentally, see Ref. 2 and reference therein. An experimental study has also demonstrated that electrochemically switchable surface charges (with charge density as high as e/nm 2, which is similar to what we use) can be achieved on a time scale of milliseconds (similar to what we need) by employing functionalized electrodes. 3 Dynamically tuning the surface charge has also been demonstrated theoretically and experimentally by employing metallic pores. 4, In this case, the boundary condition for the pore should be the potential applied to the metallic pore. We employed a uniform surface charge in our work for simplification as it gives better insight into how surface charge affects the performance of pumping. In practical implementations, the pore surface does not have to be uniformly charged. Our results hold as long as the surface is monopolarly charged and S4

5 sufficient gating is achieved. In this work, a rectangular shape bias voltage profile is employed due to computational efficiency, and the ability to model the results analytically. When a rectangular shape is employed, the bias potential only varies near the switching point. This means that the PNP-NS equations have slow convergence but only for a short time. One can imagine using other profiles, even with optimal control, but the computational cost of solving the PNP-NS equations would be significantly increased. r 2 / r 1 30 (a) 30 (b) Energy efficiency (mol/kj) r 2 / r 1 30 (c) 30 (d) Flux (C 0 λ fmol nm -1 s -1 ) r 1 (λ) r 1 (λ) Figure S3: Energy efficiency (a,b) and pumping flux (c,d) of conical pores with different radii: The results in (a,c) neglect electroosmosis (i.e., PNP model) and the calculations in (b,d) include electroosmotic effects (PNP-NS model). Effect of electroosmosis The calculations reported in the main text include electroosmotic effects by coupling the PNP equations to the NS equation. Here, we also compare those results with a simpler model that neglects electroosmotic effects (i.e., the PNP model, where the fluid velocity is assumed to u = 0 instead of solving the NS equation). Fig. S3 shows the effect of electroosmosis on the energy efficiency and pumping flux. In general, electroosmotic effects increase both energy S

6 efficiency and pumping flux. As the surface is charged, the majority ions dominate ion flow through the pore. Consequently, the majority ions drag the solvent. This electroosmotic flow increases the flux of majority ions and suppresses that of the minority ions. Thus, ion selectivity, i.e., the pumping-leakage ratio, is enhanced and energy efficiency is improved. 1.2 (a) J C J A Flux (C 0 λ fmol nm -1 s -1 ) (b) (c) Energy efficiency (mol/kj) (d) t/τ τ/ 8 Figure S4: (a) Fluxes of cations and ions as function of oscillation cycles for (a) τ = 8, (b) τ = 2 7 and (c) τ = 2 6. (d) energy efficiency for different oscillating periods (τ). Effect of oscillating time-scale on pumping efficiencies and fluxes In the main text, the period τ is set as 8, which corresponds to 93 ms for c 0 = 0.1 M. Here, we present results for different τ. Fig. S4 shows the time-dependent ion fluxes and energy efficiencies for three different τ values. When τ = 8, the ion fluxes can reach steady state in a time which is a small fraction of τ. Therefore, the shape of the transient ion fluxes almost follows the rectangular shape of the applied bias-voltage. Consequently, the ion selectivity and pumping-leakage ratio are high. Thus high pumping efficiency can be achieved. When S6

7 τ is reduced to 2 7, the time that ions need to reach steady state takes a larger fraction of the period and lag between the transient ion fluxes and the switching of bias voltage becomes notable. During the transient regime, the ion selectivity is much smaller than in the steady state. As a result, the pumping-leakage ratio is suppressed and energy efficiency drops in return. When τ is further reduced to 2 6, transient phenomena dominate for about 0.1 cycle, and the pumping-leakage ratio and energy efficiency are further suppressed. Consequently, the energy efficiency drops with decreasing τ as shown by Fig. S4(d). Flux (C 0 λ fmol nm -1 s -1 ) Energy efficiency (mol / kj) Surface charge Figure S: The effect of surface charge on the performance of pumping ions. Larger surface charge facilitates ion selectivity, pumping-leakage ratio and ion concentration within the pore. As a consequence, both efficiency and pumping flux are improved. References (1) Liu, Q.; Wang, Y.; Guo, W.; Ji, H.; Xue, J.; Ouyang, Q. Phys. Rev. E 07, 7, 011. (2) Cheng, L.-J.; Guo, L. J. Chem. Soc. Rev., 39, (3) Riskin, M.; Basnar, B.; Chegel, V. I.; Katz, E.; Willner, I.; Shi, F.; Zhang, X. J. Am. Chem. Soc. 06, 128, (4) Chen, Z.; Chen, T.; Sun, X.; Hinds, B. J. Adv. Funct. Mater. 14, 24, () Tagliazucchi, M.; Szleifer, I. J. Phys. Chem. Lett., 6, S7

Numerical Modeling of the Bistability of Electrolyte Transport in Conical Nanopores

Numerical Modeling of the Bistability of Electrolyte Transport in Conical Nanopores Long Luo, Robert P. Johnson, Henry S. White * Department of Chemistry, University of Utah, Salt Lake City, UT 84112,