Ratchet-Based Ion Pumps for Selective Ion Separations

The development of a highly selective, membrane-based ion separation technology could significantly improve the sustainability and energy efficiency of water treatment technologies and emerging applications, such as electrochemical ${\mathrm{CO}}_2$ reduction, extraction of valuable metals from seawater, and battery recycling. In this work, we show through computational modeling that an electronic flashing ratchet mechanism can be used for high-precision ion separation. The suggested ratchet-based ion pumps utilize a unique feature of electronic ratchets, frequency-dependent current reversal, to drive ions with the same charge, but different diffusion coefficient, in opposite directions. The model shows that ions whose diffusion coefficients differ by as little as 1% can be separated by driving them in opposite directions with a velocity difference as high as 1.2 mm/s. Since the pumping properties of the ratchet are determined by a time-varying electric input signal, the proposed ion pumps could be instrumental in realizing an efficient, large-scale, and fit-for-purpose system for selective ion separation. Examples of ratchet-driven systems for lithium extraction from seawater, lead removal from drinking water, and water desalination are discussed and analyzed.

Popular Summary

Capabilities to extract a single solute from water can advance the sustainability and efficiency of various applications, such as water desalination and lithium extraction from seawater. Here, the authors explore flashing ratchets, which are devices that use asymmetric and temporally varying electric potentials to drive a net particle current. Computational modeling shows that ratchet-based ion pumps can drive ions with very similar properties in opposite directions. As a result, such devices are promising for removing specific ions, even trace amounts, from a mixed solution. Since the membrane is operated by an electric signal, the target ions can be tuned in real time, paving the way for simple fit-to-purpose selective ion separation systems.

Figure 1

Spatial and temporal components of the potential distribution.

Figure 2

Net velocity of two positively charged monovalent ions $(z=1)$, as a function of the input signal frequency. Shaded area marks the frequency range in which the ions are driven in opposite directions, according to their diffusion coefficients. Ratchet parameters are $L=1\mu \mathrm{m}$, $x_c=0.7$, $V_{\max }=2.5\mathrm{V}$, $\delta =0.25$, $\alpha =-0.5$.

Figure 3

Concentration response, $C(x,t)$, of two positively charged monovalent ions $(z=1)$ with diffusion coefficients $D=1.2\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$ (a) and $D=2\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$ (b). Potential distribution, $U(x,t)$, is shown on the right axis. Filled lobes are guides to the eye to highlight the main contributors to the net ion flux. For the high D ion (b), net flux is in the negative x direction due to ion injection ($\tau =0.26T$ to $\tau =0.28T$). This effect is less significant for the low D ion (a), resulting in a net flux in the positive x direction. Signal frequency is $f=83\mathrm{kHz}$, and all other ratchet parameters are the same as those in Fig. 2.

Figure 4

Net velocity of monovalent cations as a function of their diffusion coefficient and signal frequency. At a given frequency, the direction for ion transport is determined by the diffusion coefficient. All ratchet parameters are the same as those in Fig. 2.

Figure 5

(a) Normalized net ion velocity as a function of normalized frequency, $\mathrm{\Gamma }$ (i.e., characteristic net velocity curves), for $\delta =0.25$ and different symmetry factors, $\alpha$. Velocity reversal is possible only when the time-averaged potential is negative. Dashed line shows the slope of the characteristic velocity curve at the normalized stopping frequency, which defines the separation resolution. (b) Normalized net ion velocity as a function of duty cycle, $\delta$, for $\mathrm{\Gamma }=1$ and different symmetry factors, $\alpha$. Ratchet parameters are $x_c=0.7$, $V_{\max }=2.5\mathrm{V}$ (dimensional values used in the calculation are $L=1\mu \mathrm{m}$, $z=1$, and $D=1\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$).

Figure 6

(a) Separation resolution magnitude as a function of symmetry factor, α, and duty cycle, δ. Light blue region corresponds to input signals that do not yield velocity reversal. (b) Separation resolution magnitude as a function of maximum amplitude, $V_{\max }$. Inset shows the characteristic net velocity curves for several values of $V_{\max }$. Increase in slope with $V_{\max }$ results in a nonlinear increase in the separation resolution. Unless stated otherwise, ratchet parameters are $\delta =0.25$, $\alpha =-0.5$, $x_c=0.7$, $V_{\max }=2.5\mathrm{V}$ (dimensional values used in the calculation are $L=1\mu \mathrm{m}$, $z=1$, and $D=1\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$).

Figure 7

(a) Net velocity of $\mathrm{N}{\mathrm{a}}^{+}$, $\mathrm{L}{\mathrm{i}}^{+}$, $\mathrm{C}{\mathrm{u}}^{2+}$, and $\mathrm{P}{\mathrm{b}}^{2+}$ as a function of input signal frequency. $\mathrm{P}{\mathrm{b}}^{2+}$ and $\mathrm{L}{\mathrm{i}}^{+}$ can be separated from $\mathrm{N}{\mathrm{a}}^{+}$, but $\mathrm{C}{\mathrm{u}}^{2+}$ cannot be separated from $\mathrm{N}{\mathrm{a}}^{+}$ due to their similar zD values. Units for $zD$ in the plot are in ${10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$. Ratchet parameters are $L=1\mu \mathrm{m}$, $x_c=0.7$, $V_{\max }=1\mathrm{V}$, $\alpha =-0.5$, $\delta =0.25$. (b) Net velocity of $\mathrm{C}{\mathrm{l}}^{-}$ and $\mathrm{N}{\mathrm{a}}^{+}$, showing ambipolar transport. Ratchet parameters are $L=1\mu \mathrm{m}$, $x_c=0.6$, $V_{\max }=5\mathrm{V}$, $\alpha =-0.38$, $\delta =0.25$. (c) Conceptual scheme for a flashing ratchet with a sawtooth electric potential in solution.

Figure 8

Illustration of a flashing ratchet driving positively charged (a) and negatively charged (b) particles. Solid blue line illustrates the potential distribution during sequential steps, empty circles mark the position of particles at the beginning of every step, and filled circles mark their position at the end of every step.

Figure 9

Numerical versus analytical solution for a potential distribution described by Eq. (C1). Ratchet parameters $\mathrm{are}L=10\mu \mathrm{m}$, $\alpha =-0.75$, $\delta =0.25$, and $\beta V_{\max }$ between (a) 0.5 to 2 and (b) 5 to 11.

Figure 10

(a) Instantaneous average ion velocity, $\overline{v}(\tau )$, along a time period. Inset shows the integral of $\overline{v}(\tau )$ up to point $\tau$, divided by time period T. (b)–(g) Ion concentration (left axis) and ratchet potential (right axis) at various points along the time period, marked in (a). All graphs show curves for two monovalent ions $(z=1)$ with diffusion coefficients $D=1.2\times {10}^{-5}$ and $2\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$. For high D ions, considerable injection at $0.25T<\tau <0.28T$ results in a negative net velocity. Ratchet parameters are $L=1\mu \mathrm{m}$, $x_c=0.7$, $V_{\max }=2.5\mathrm{V}$, $\delta =0.25$, and $\alpha =-0.5$, and signal frequency is $f=83\mathrm{kHz}$.

Figure 11

(a) Model diagram showing the potential distribution for three spatial periods of the ratchet, and a definition of concentration boundary conditions. (b),(c) Ion flux, N, as a function of concentration ratio between the left and right reservoirs, $C_L/C_R$, with ratchet signal frequencies $f=20\mathrm{kHz}$ (b) and $f=40\mathrm{kHz}$ (c). Insets show the same data set for wider flux and concentration ratio ranges. Ratchets show velocity reversals with frequency, even when driving ions against a concentration gradient. Ratchet parameters are $L=1\mu \mathrm{m}$, $x_c=0.7$, $V_{\max }=1\mathrm{V}$, $\delta =0.25$, and $\alpha =-0.5$. Ion properties are $z=1$ and $D=1\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$. Ion concentration on the side with lower concentration is set to $1\mathrm{mol}/{\mathrm{m}}^3$.

Figure 12

(a) Characteristic net ion velocity curves for different $x_c$. (b) Separation resolution as a function of $x_c$. (c) Characteristic net velocity curves for different ion valence, $|z|$, with $V_{\max }=1\mathrm{V}$. Inset table shows the separation resolution between same-charge ions. $|{\mathcal{R}}^{\ast }|$ is in units of $(\mathrm{cm}/\mathrm{s})/({10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s})$. In (a)–(c), unless stated otherwise, ratchet parameters are $x_c=0.7$ and $V_{\max }=2.5\mathrm{V}$ (dimensional values used in the calculation are $L=1\mu \mathrm{m}$, $z=1$, and $D=1\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$). (d) Net velocity of $\mathrm{N}{\mathrm{a}}^{+}$, ${\mathrm{K}}^{+}$, and ${\mathrm{H}}^{+}$ as a function of input signal frequency. Net velocity curves are calculated from the normalized characteristic curve presented in (a) for $x_c=0.6$, using the appropriate diffusion coefficient ratio for each ion. $\mathrm{N}{\mathrm{a}}^{+}$, $D=1.33\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$; ${\mathrm{K}}^{+}$, $D=1.96\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$; ${\mathrm{H}}^{+}$, $D=9.31\times {10}^{-5}\mathrm{c}{\mathrm{m}}^2/\mathrm{s}$.

Figure 13

Time to extract 99% of lead ions from the feed compartment as a function of the extraction velocity. Inset shows the one-dimensional system model, and the time evolution of the lead ion distribution in the feed compartment, using an extraction velocity of 0.01 cm/s. Time step between two adjacent curves is 20 s.

Figure 14

Net ion velocity as a function of input signal frequency for the extraction of $\mathrm{L}{\mathrm{i}}^{+}$ from $\mathrm{N}{\mathrm{a}}^{+}$ (a) and $\mathrm{P}{\mathrm{b}}^{2+}$ from $\mathrm{N}{\mathrm{a}}^{+}$ (b). Solid, dashed, and dotted lines are for $V_{\max }=1$, $0.85$, and $0.8\mathrm{V}$, respectively. Ratchet parameters are $L=1\mu \mathrm{m}$, $x_c=0.7$, $\alpha =-0.5$, and $\delta =0.25$.