Letter pubs.acs.org/JPCL
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 Department of Chemistry, Northwestern University, Evanston, Illinios 60208, United States S Supporting Information *
ABSTRACT: Previous experimental and theoretical studies have demonstrated that nanofabricated synthetic channels are able to pump ions using oscillating electric fields. We have recently proposed that conical pores with oscillating surface charges are particularly effective for pumping ions due to rectification that arises from their asymmetric structure. In this work, the energy and thermodynamic efficiency associated with salt pumping using the conical pore pump is studied, with emphasis on pumps needed to desalinate seawater. The energy efficiency is found to be as high as 0.60 to 0.83 mol/kJ when the radius of the tip side of the conical pore is two Debye lengths and the pump works with a concentration gradient smaller than 1.5. As a result, the energy consumption needed for seawater desalination with 20% salt rejection is 0.32 kJ/L. In addition, the energy consumption can be further reduced to 0.21 kJ/L (20% salt rejection) if the bias voltage is adaptively altered four times during the pump cycle while salt concentration is reduced. If the bias voltage is adaptively increased to higher values, then salt rejection can be improved to values that are needed to produce fresh water that satisfies standard requirements. Numerical analysis indicates that the energy consumption is 4.9 kJ/L for 98.6% salt rejection, which is smaller than the practical minimum energy requirement for RO-based methods. In addition, the pumping efficiency can be further improved by tuning the pump structure, increasing the surface charge, and employing more adaptive bias voltages. The conical pores are also found to more efficiently counteract the concentration gradient compared to cylindrical counterparts. a fluctuating external potential or noise with zero mean to a nonzero directional current,38−40 but the current in this case includes both cations and anions, which are transported in opposite directions. To make a salt pump, net cation and anion fluxes in same direction are needed. In previous work, we explored the possibility and advantage of using conical pores for pumping ions and salt.36 Because of asymmetry of the structure and a charged inner surface, even conical pores with static surface charges are able to selectively pump ions whose charge is opposite the surface charge. Moreover, if the surface charge can be controlled dynamically (allowed to oscillate synchronously with applied external potentials), then more efficient salt pumping can be achieved and the pumping flux is several times larger than that for cylindrical pores. Once net cations and anions can be driven in the same direction, the pump can counteract a concentration gradient. Consequently, this kind of pump can be employed for seawater desalination. Because oceans and seas contain ∼97% of the world’s water, desalination is one of the most important approaches for supplying fresh water to fill the rapidly growing global water gap. However, existing commercial desalination, such as is achieved using reverse osmosis (RO), multistage flash distillation, and electrodialysis (ED),41 is still an expensive solution for generating fresh water even after decades of
I
on transport and pumping are essential processes for the function of living organisms, allowing cells able to communicate with each other or with the outside environment. To achieve this functionality, biological systems have developed ion channels, ion transporters, and ion pumps. The transport properties of biological channels have also inspired research on their artificial counterparts.1−9 Solid-state artificial nanochannels can provide robust and controllable platforms for diverse applications such as current rectification, ion selectivity, and the detection of biomolecules.10−15 Hence, the use of synthetic solid-state nanopores for control of the transport of ions and molecules has attracted extensive attention in recent years, with potential applications in nanotechnology (biosensors, purification), biomedical devices, environment studies, and energy science.16−19 Among the different artificial nanochannels, artificial conical pores with charged inner surfaces have received extensive attention, including both experiments and theoretical simulations.20−33 A primary reason for this is that conical pores can enhance ion transport in one preferred direction and inhibit flow in the opposite direction, that is, current rectification. Both normal rectification and reverse rectification (rectification in the opposite sense) have been found in conical pores depending on geometry of the pores.34−37 Because conical pores demonstrate a preferred direction of ion flow and may be able to counteract diffusive flow, conical pores may be a good starting point for developing electrically driven ion pumps. Indeed, recent experiments have demonstrated that conical pores can convert © XXXX American Chemical Society
Received: May 9, 2017 Accepted: June 7, 2017 Published: June 7, 2017 2842
DOI: 10.1021/acs.jpclett.7b01137 J. Phys. Chem. Lett. 2017, 8, 2842−2848
Letter
The Journal of Physical Chemistry Letters development due to large energy consumption and capital costs. RO is the most energy-efficient desalination technique to date with a commercial performance of 6.48 kJ/L.42 There have been many efforts to improve the performance of RO by developing ultrathin nanoporous membranes.43−46 However, the practical minimum energy of the RO-based method is 5.6 kJ/L,42 so there is not much room for further improvement. New methods other than RO should be proposed to further reduce energy consumption to the theoretical limit of 3.8 kJ/L. In this work, the energy and thermodynamic efficiencies of conical pore-based pumping and desalination are analyzed. Our model for the simulation of conical pores is based on the classical Poisson−Nernst−Planck (PNP) and Navier−Stokes (NS) equations. Within the PNP−NS model, the mobile ions are represented as a continuous charge density described by the Nernst−Planck equation, and the potential profile is solved through the Poisson equation. The influence of electroosmosis is introduced via the Navier−Stokes equation. The PNP−NS model has been widely used to study ion transport in both biological and nanofluidic channels.47−50 In this work, the PNP−NS equations are solved via a finite element method (FEM) using a homemade module DOLPHIN,51 which is based on the Multiphysics Object-Oriented Simulation Environment (MOOSE) package.52 We present a detailed description of the theoretical methodology in the Supporting Information (SI). Also, in the SI, we provide a detailed analysis of the electroosmotic effect. When an oscillating bias voltage is applied, the total ion flux through a pore during the cycle period, τ, can be obtained from the cation flux JC(t) and the anion flux JA(t) using Jnet = ∫ τ0 [JA(t) + JC(t) ] dt, while the net current is given by Inet = |e| ∫ τ0 [JC(t) − JA(t)] dt. Hence the energy consumed in one cycle of pumping can be written as Q = |e| ∫ τ0 [JC(t) − JA(t)] V(t) dt. When the conical pores are connected to two reservoirs with different concentration, energy will be stored by the pump, given by E = |e|Jnet Δμ, where Δμ = kT ln(Chigh/Clow) is the difference in chemical potential between the two reservoirs and Chigh/Clow is the concentration gradient. The thermodynamic efficiency, that is, energy stored in the ions divided by energy consumed per cycle, can then be defined using ζ = JnetΔμ/Q. Note that the efficiency calculation ignores energy lost by dissipation or other processes that are not included in the PNP−NS formalism. The pump performance can also be assessed by calculating the energy efficiency η = Jnet/Q, which measures the amount of ions pumped per unit of energy consumed. Let us now investigate the factors that affect energy efficiency. If we consider the case where the bias voltage has an oscillating rectangular waveform, where the current and voltage are constant for each half cycle, typical ion fluxes for the pump model defined in the SI are shown by Figure S2. Hence, the fluxes of cations and anions can be written as JC = −χJA ≡ J0 (JA = −χJC ≡ J0) for 0 < t < τ/2 (τ/2 < t < τ), where χ is the p u m p i n g / l e a k a g e r a t i o , w h i c h i s i n t r o d u ce d a s
Similarly, the thermodynamic efficiency is ζ=
∫0 [JA>(t ) + JC>(t )] dt τ
∫9 [JA 0).36 Here the energy consumption can be rewritten as Q = (1 + χ)|e|J0V0τ and the net flux is Jnet = (χ − 1)J0τ. Hence, the energy efficiency is η=
χ−1 (χ + 1)eV0
(2)
In principle, χ is a function of surface charge, geometry of the pores, voltage, and concentration gradient. Equation 1 indicates that the energy efficiency can be improved by increasing χ, reducing voltage, or a combination of both. Note that the minimum value of eV0 that leads to a positive flux is Δμ, so in the limit of large χ it is possible for the pump to operate with unit thermodynamic efficiency. The geometry of the system is illustrated in Figure 1a and Figure S1. Here we assume that there are electrodes at the top
τ
χ=
(χ − 1)Δμ (χ + 1)eV0
(1) 2843
DOI: 10.1021/acs.jpclett.7b01137 J. Phys. Chem. Lett. 2017, 8, 2842−2848
Letter
The Journal of Physical Chemistry Letters
Figure 2. Pumping flux (a) and energy efficiency (b) of pumping under different amplitudes of the oscillating bias voltage and concentration gradient. (c,d) Energy efficiency of pumping driven by rectangular bias voltage oscillation when four adaptive steps in altering V0 are employed. The chemical potentials difference (Δμ) is indicated in panel d at each voltage altering point.
L. This energy efficiency can be further improved by tuning the geometry of the pores, as considered later. Figures 1 and 2b indicate that energy efficiency can be improved by employing a smaller V0, but a lower bias voltage can only be effective with smaller concentration gradient. Thus one possible approach for improving the energy efficiency is to adaptively alter V0 according to the concentration gradient. As an example we will consider four adaptive steps in altering V0. Figure 2c,d shows the energy efficiency by applying different V0 for different ranges of concentration gradient. At the beginning of the pumping,V0 = 0.1 kT/e can be employed to pump ions with very high efficiency. With increasing concentration gradient, the energy efficiency drops quickly. When the concentration gradient is >1.1, the energy efficiency for a voltage 0.1 kT/e becomes smaller than that for a voltage 0.2 kT/e. Hence, V0 is changed to 0.2 kT/e until another optimal V0 is required to counteract the larger concentration gradient. In this way, the effective energy efficiency can be improved significantly. The effective efficiency in the concentration range [0:1.5] is ∼1.12 mol/kJ, and thus the energy consumption is reduced to 0.21 kJ/L (20% salt rejection). In this example, V0 is varied only four times with increasing concentration gradient. Further improvement in the energy efficiency can be expected if V0 is switched more often to adapt to the concentration gradient. To increase salt rejection to a higher amount that is needed for desalination to give acceptable fresh water, a higher bias voltage is required to offset the larger concentration gradient. Figure 3 plots the energy efficiency of pumping when V0 is adaptively altered from 0.1 to 4 kT/e. In this case, the pumping is designed to counteract a concentration gradient of 50, that is, 96% salt rejection. The corresponding effective pumping efficiency is found to be 0.18 mol/kJ. Note that V0 is varied only seven times in this example. To estimate what a larger number of stages would give and to determine the energy needed to produce a higher concentration gradient of 140 that is needed for “standard” fresh water, we have developed an empirical curve (the red curve in Figure 3) based on the following argument. Figure 2 indicates that, for a given bias voltage V0, the effective energy efficiency is only 10% smaller than the maximal energy efficiency for that bias voltage when the chemical potential difference [Δμ = kT ln(Chigh/Clow)] is 1 1.66, as expected based on our previous analysis of eq 2. In addition, we see that a larger bias voltage can overcome a larger concentration gradient. For the rectangular voltage wave, the energy efficiency for V0 = 0.5 kT/ e can be as high as 0.60 to 0.83 mol/kJ as long as the concentration gradient is smaller than 1.5 and the effective pumping efficiency is 0.75 mol/kJ. Accordingly, the energy consumption in desalination for 20% salt rejection is ∼0.32 kJ/
0
The red curve in Figure 3 shows that this formula provides an excellent fit to the numerical results and further that the energy efficiency needed to achieve a concentration gradient of 140 is 0.245 mol/kJ. From this, the energy consumption is 4.9 kJ/L 2844
DOI: 10.1021/acs.jpclett.7b01137 J. Phys. Chem. Lett. 2017, 8, 2842−2848
Letter
The Journal of Physical Chemistry Letters
Figure 5. (a) Pumping flux and (b) energy efficiency of pumping by conical pores with different radii. (c,d) Comparison of the performance of conical (r1 = 2λ, r2 = 20λ) and cylindrical pores (radii are 2λ, 4λ, and 7λ). A rectangular shape bias voltage wave is applied, and the amplitude of the bias voltage is 0.5 kT/e.
Figure 3. Energy efficiency of pumping driven by rectangular bias voltage oscillation when V0 is adaptively altered from 0.1 to 4 kT/e. The red curve plots the pumping efficiency for given concentration 1 gradient when the bias voltage (V0) is adaptively altered as 0.85 Δμ and the blue curve is the corresponding energy consumption.
efficiency can be tuned by the radius of the smaller pore and the ratio of the two pore radii. For conical pores r2/r1 > 1, and the pumping efficiency increases with decreasing pore radius due to the stronger surface charge-induced gating effect. In contrast, the radius has little effect on the pumping efficiency of cylindrical pores (r2 = r1). Moreover, the efficiency increases when the ratio between the two radii approaches 1 (i.e., the conical pores reduce to cylindrical ones). However, improvement in efficiency by reducing the radius ratio is not very significant when the radius is
