Research ArticleMATERIALS SCIENCE

Optimal transport and colossal ionic mechano-conductance in graphene crown ethers

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Science Advances  12 Jul 2019:
Vol. 5, no. 7, eaaw5478
DOI: 10.1126/sciadv.aaw5478
  • Fig. 1 Potential transport mechanisms in graphene crown ether pores.

    (A) Using partial charges qO = −0.24e, consistent with the electrostatic potential from DFT, the ion transport mechanism is drift-diffusion. In this case, a K+ (purple sphere) finds an empty pore and translocates through it; the pore then remains empty again for several nanoseconds (see the Supplementary Materials). (B) At larger partial charges (qO = −0.54e or qO = −1.0e), the vicinity of the charge separation at the pore rim results in an energetic well deep enough to trap a K+ (light purple). For a current to be present, an incoming K+ (dark purple) knocks out and replaces the trapped K+. This yields a two-step knock-on mechanism reminiscent of some biological potassium ion channels. However, for qO = −0.54e, the mechanism shifts toward a majority drift-diffusion process at moderate (1%) strain due to a shallowing of the free-energy well. Oxygen, carbon, and hydrogen atoms are small red, gray, and white spheres, respectively.

  • Fig. 2 Colossal ionic mechano-conductance.

    (A) Schematic of graphene on (or embedded within) a polymeric matrix support (such as ∼50-nm-thick epoxy resin) with a window where the crown ether pore is located. The membrane can be strained by stretching or bending (e.g., via a piezoelectric actuator offset from the window). Alternative experimental setups are possible, such as metallic regions that bind the graphene and are used to apply strain. (B) Relative change in current versus strain in the crown ether pore (for qO = −0.24e and −0.54e) at different voltages. At lower voltages, the current increases substantially with strain, as shown by the fitted dashed lines. (C) Current without strain (I0) for different values of qO and V. Going from qO = −0.24e to qO = −0.54e, the current increases as highly charged oxygen atoms compensate the loss of waters from the hydration layers. However, going from qO = −0.54e to qO = −1.0e, the current decreases because a further increase in the pore charge makes it harder for an ion to escape the potential well in the pore. The typical drift-diffusion limit—set by an access resistance to an uncharged pore with an approximate radius of 0.1 nm—at 0.25 V is about 0.35 nA. Only the intermediate charge case—the one most analogous to biological systems—approaches this limit: It is about a factor of 2 lower at a 2% strain. Further strain allows it to reach this limit. The smaller charge case can approach within a factor of 6. The range of accessible currents is due to the extensive mechanistic leeway permitted by the intermediate charge. The error bars are the standard error (SE) from five parallel runs.

  • Fig. 3 Free-energy landscape.

    Free-energy profile, ΔFK, for a K+ translocating along the z axis of an 18-crown-6 graphene pore at different strains. The charge on the oxygen atom, qO, is presented at the top of each plot. The peaks and valleys in ΔFK(z) are due to the balance between dehydration energy penalties and electrostatic interactions with the charged pore atoms. For qO = −0.54, there is an additional contribution from the K+ occupying the deep potential well, giving a barrier at z ≈ 0.2 nm when the incoming ion attempts to go into the already occupied pore (this is confirmed by a free-energy profile with only 1 K+ and 1 Cl present; see the Supplementary Materials). The illustrations show the position of the K+ in the peaks and valleys. In general, the free-energy profile flattens with increasing—but still small—strain, tending toward barrierless transport and making it easier for the ion to translocate. The error bands are the SE from five parallel runs.

  • Fig. 4 Optimum ion transport and selectivity.

    Ionic current versus strain in a pore with qO = −0.24e and an applied bias of 0.25 V. For small strain (blue line), the current increases rapidly, commensurate with a decrease in the outer barriers with increasing strain. This gives an overall flattening of the free-energy profile. Transport (and selectivity; see below) is optimal near a 3% strain (green line) when the outer barrier is sufficiently diminished but the central barrier is not too high. Further increases in strain continue to increase the central barrier (dehydration remains unchanged in this strain regime), decreasing the current (red line). Schematics of the free-energy profiles are above each regime. For every value of strain and voltage, the selectivity of K+ over Cl is perfect for all practical purposes, as the negative partial charge of the oxygens on the pore rim create a strong barrier for Cl transport. Separate simulations (each of duration 1 μs with an equal mixture of NaCl and KCl at 1 M Cl) reveal that Na+ does not cross the pore (except for a 0% strain where a single Na+ crossing event occurs during the 1-μs time frame). Because of the near complete exclusion of Na+ in the time frame of the simulation, we can determine a high-confidence lower bound on the selectivity, which goes from 15 at 0% strain to 52 at 3% strain (see the main text for details). In other words, the Na+ and Cl exclusion is maintained, while K+ current changes with strain, and the lower bound on selectivity exactly traces the K+ current in the plot.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/7/eaaw5478/DC1

    Section S1. Simulation methods

    Section S2. Equilibrium free energy and many-body effects

    Section S3. Ion transport mechanism: Knock-on versus drift-diffusion

    Section S4. Ion transport through multiple barriers

    Fig. S1. Models used for charge calculations.

    Fig. S2. Free-energy profile when only one K+ and one Cl are present in the solution.

    Fig. S3. Free-energy dependence on strain for qO = −0.24e near the pore.

    Fig. S4. Dehydration effects and the dielectric constant in confined geometries.

    Fig. S5. Residence time for a K+ translocating through the crown ether pore in 1 M KCl solution with various values of qO, strain, and applied voltage.

    Fig. S6. Delay time between one K+ leaving the pore and another K+ replacing it for various values of qO, strain, and applied voltage.

    Table S1. Geometric parameters and Bader charges from plane-wave DFT.

    Table S2. Oxygen point charges within an 18-crown-6 graphene pore using electrostatic potential fitting (CHELPG and RESP) and Bader analysis.

    Table S3. Force-field parameters to calculate the total bonded energy.

    Table S4. Pore strain, supercell edge lengths (ℓx, ℓy), nominal pore radii (rn), and geometric pore radii (rp) as a function of supercell strain.

    References (4767)

  • Supplementary Materials

    This PDF file includes:

    • Section S1. Simulation methods
    • Section S2. Equilibrium free energy and many-body effects
    • Section S3. Ion transport mechanism: Knock-on versus drift-diffusion
    • Section S4. Ion transport through multiple barriers
    • Fig. S1. Models used for charge calculations.
    • Fig. S2. Free-energy profile when only one K+ and one Cl are present in the solution.
    • Fig. S3. Free-energy dependence on strain for qO = −0.24e near the pore.
    • Fig. S4. Dehydration effects and the dielectric constant in confined geometries.
    • Fig. S5. Residence time for a K+ translocating through the crown ether pore in 1 M KCl solution with various values of qO, strain, and applied voltage.
    • Fig. S6. Delay time between one K+ leaving the pore and another K+ replacing it for various values of qO, strain, and applied voltage.
    • Table S1. Geometric parameters and Bader charges from plane-wave DFT.
    • Table S2. Oxygen point charges within an 18-crown-6 graphene pore using electrostatic potential fitting (CHELPG and RESP) and Bader analysis.
    • Table S3. Force-field parameters to calculate the total bonded energy.
    • Table S4. Pore strain, supercell edge lengths (ℓx, ℓy), nominal pore radii (rn), and geometric pore radii (rp) as a function of supercell strain.
    • References (4767)

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