How to speed up ion transport in nanopores

The study addresses how to minimize charging and discharging times of nanoporous supercapacitors with subnanometre pores. Conventional assumptions of symmetric charge/discharge times based on RC-circuit analogies fail for EDLCs at voltages above thermal voltage and for nanopores. Rapid voltage steps cause counterion crowding at pore entrances, leading to co-ion trapping and sluggish charging. The research poses an optimization problem: for a given supercapacitor, what time-dependent cell voltage minimizes the time to charge to a target charge and to discharge to zero? The work derives and tests optimized time-dependent voltage protocols—non-linear sweeps for charging and a voltage inversion followed by a sweep for discharging—validated by molecular dynamics simulations and experiments, to accelerate ion transport without sacrificing capacitance.

The paper situates itself within extensive work on supercapacitors, emphasizing the role of nanoporous carbons and ionic liquids in achieving high capacitance and energy density. Prior studies explored charge storage mechanisms, dynamics, and multiscale modeling, including MD and transmission-line models. Earlier MD work (Breitsprecher et al., 2018) showed optimized linear voltage sweeps mitigate clogging versus step potentials. Additional literature covers ionic structure in nanopores, size effects, and equivalent circuit approaches for planar EDLCs. Recent modeling (Lian et al., 2020) highlighted macroscopic diffusion limits in real devices. These works motivate optimizing the applied potential protocol rather than only device geometry to enhance rate performance.

The authors derive a general non-linear charging sweep from co-ion desorption dynamics. They conceptually partition co-ions in a pore into layers and consider incremental voltage steps that each expel a small amount of co-ions. Replacing discrete steps by a continuum yields an implicit optimal sweep formula: t(U) ≈ (l^2/D_e) ∫ [dK/du]/(1+K(U))^2 du, where K(U) is the equilibrium number of co-ion layers at voltage U, l is pore length, and D_e the in-pore co-ion diffusion coefficient. Two closed-form approximations are introduced by fitting K(U): (i) a stretched exponential K(U) = K0 exp[−(γU)^α], leading to an analytical U(t) with a characteristic time τ; and (ii) a low-voltage linear K(U) = K0(1−γU), yielding U(t) = U0 t/(τ0 + t). For discharging, they propose a voltage inversion protocol U(t) = U_ch for t≤0; U_inv + k_inv t for 0<t≤τ_inv; 0 for t>τ_inv, with τ_inv = −U_inv/k_inv, then optimize (U_inv, k_inv). Molecular dynamics simulations: Conducted with ESPResSo (v3.3.1) using constant-potential electrodes via the ICC* algorithm, with Laplace-field superposition for time-dependent potentials. Ions are monovalent WCA spheres (σ=5 Å, ε=1 kJ/mol), carbon atoms WCA (σ=3.37 Å). Temperature T=400 K, Langevin thermostat ζ=10 ps, time step 2 fs. Geometry: two slit pores (one per electrode), accessible pore width w=0.6 nm, pore entrance/closing radii 4 Å/2 Å, electrode separation (bulk) 8 nm. Pore lengths l=12, 16, 20 nm for charging; l=12 nm for discharging. Ion diameter a=0.5 nm. Systems equilibrated 4 ns before charging. Experiments: Fabricated novolac-derived activated carbon (PNC-2h) electrodes with narrow pore-size distribution (average ~0.68 nm) using solvothermal crosslinking, pyrolysis (700 °C), and CO2 activation (1000 °C). Characterization by SEM/BF-STEM and N2 sorption (QSDFT, slit pores). Built symmetric two-electrode full cells with PTFE-bound free-standing electrodes (~100 µm dry), glass fiber separator, EMIM-BF4 electrolyte (room temperature; additional elevated T tests doubled bulk conductivity). Charging protocols: voltage step, linear sweeps (single-cycle voltammetry at specified sweep rates), and piecewise-linear approximations to the derived non-linear sweeps for selected τ parameters. Accumulated charge Q(t) measured over time; for step, fitted Q(t) to bi-exponential form. Discharging experiments implemented voltage inversion protocols analogous to simulations.

• Charging acceleration with non-linear sweeps: The derived non-linear U(t) adapts voltage rate to co-ion desorption, allowing fast early-time voltage changes and slower ramps at intermediate voltages, then near-step increases at high voltages when co-ions are depleted. MD shows non-linear sweeps outperform step voltages and can surpass optimized linear sweeps when τ is chosen appropriately. Optimization of τ via intersection of co-ion desorption and counterion adsorption times yields minimal charging times; too small τ induces co-ion trapping and slow desorption.
• Scaling and analytical forms: Stretched exponential fit K(U)=K0 exp[−(γU)^α] reproduces MD K(U) and yields analytic U(t); low-voltage linear approximation yields U(t)=U0 t/(τ0+t), capturing non-linear time dependence even for linear K(U). Predicted scaling indicates τ(U) ∼ l at U below a threshold U_c ~ γ^−1 (ln K0)^{1/α}, and τ(U) ∼ l^2 at U ≥ U_c. For studied pores (l=12–20 nm), U_c ≈ 1.5 V, weakly increasing with l.
• MD quantitative results: For charging to U=3 V with w=0.6 nm, a=0.5 nm, l=20 nm, non-linear sweeps with sufficiently large τ avoid trapping and outperform optimized linear sweeps; when τ too small (e.g., 4 ns), trapping causes sluggish desorption; optimal τ around where adsorption and desorption times intersect (e.g., τ_opt ≳ 8 ns). The performance gain of non-linear over linear increases with pore length (12–20 nm).
• Experimental charging: With EMIM-BF4 and novolac-derived carbons, slow sweeps (linear at k=8.3 and 2.5 mV/s; non-linear with τ=6 and 20 min) all overtake step-voltage charging. Non-linear (τ=20 min) slightly outperformed linear at k=2.5 mV/s. Step charging fit yielded a slow timescale τ2 ≈ 45 min; a macroscopic diffusion model predicts τ ≈ (L/2+H)^2/D ≈ 17 min for L=150 µm, H=109 µm, D≈10^−11 m^2/s. Optimal sweep rates were likely slower than 0.1 mV/s, implying full optimization would require >8 h.
• Discharging acceleration by voltage inversion: MD demonstrates that applying a negative inversion voltage followed by a linear sweep to zero can reduce discharge times several-fold versus a simple step to 0 V. For U_ch=3 V (l=12 nm, w=0.6 nm), optimal parameters were U_inv ≈ −2.5 V and k_inv ≈ 5.5 V/ns, yielding discharge time ~0.4 ns vs ~1.5 ns for step. Too small k_inv causes overshoot and slow relaxation; too large k_inv approaches step-like behavior. Discharge time vs U_inv exhibits a clear minimum near −2.5 V at optimal k_inv.
• Experimental discharging: Voltage inversion (e.g., U_inv = −2.5 V with practical k_inv) discharges the device faster than a step to 0 V. Transient charge sign reversal observed experimentally (negative intermediate charge), consistent with macropore transport effects not captured in single-pore MD.
• Practical implication: Carefully designed non-linear charging sweeps and inversion-based discharging protocols can markedly accelerate ion transport in subnanometer pores without degrading capacitance, relevant for high-power energy storage, capacitive deionization, and thermocapacitive applications.

The findings directly address the optimization question for time-dependent voltage protocols in nanoporous supercapacitors. By matching the voltage sweep rate to co-ion desorption dynamics, the study avoids counterion-induced entrance clogging and co-ion trapping, cutting charging times below those from previously optimized linear sweeps. The analytic mapping from K(U) to U(t) offers a general strategy applicable across materials, voltages, and pore lengths. For discharging, introducing a controlled negative inversion voltage transient accelerates counterion desorption, likely by transiently depopulating pore entrances and enhancing effusion, providing significantly shorter discharge times than passive diffusion after a step to zero. The results indicate that optimal protocols differ between charging (gentle non-linear ramp) and discharging (inversion plus ramp), underscoring the asymmetry of EDLC dynamics. These advances promise higher power operation without sacrificing energy density, benefiting rapid cycling scenarios and processes prioritizing speed over energy efficiency, such as capacitive deionization and low-grade heat harvesting.

The work introduces and validates optimal time-dependent voltage protocols to speed up ion transport in subnanometre pores. A general non-linear charging sweep derived from co-ion layer depletion yields faster charging than both step and optimized linear sweeps, with gains increasing for longer pores. For discharging, a voltage inversion followed by a sweep to zero minimizes discharge time and outperforms step discharging by several-fold when optimized. Molecular dynamics simulations substantiate the mechanisms and quantitative gains, while experiments with novolac-derived carbon electrodes and EMIM-BF4 qualitatively confirm that slow/non-linear charging sweeps and voltage inversion discharging accelerate device response. Future research should optimize protocols for full devices accounting for multiscale transport, explore asymmetric ion sizes and different electrolytes, integrate real-time feedback to adapt U(t), and investigate temperature and material dependencies to generalize and maximize performance gains.

• Simulations use a simplified single-slit-pore geometry with symmetric WCA ions at T=400 K and a Langevin thermostat; absolute times depend on these parameters. Real electrodes have complex pore networks and additional transport resistances in macropores not captured by the MD setup.
• The co-ion diffusion coefficient D_e is treated as voltage-independent in derivations; real systems may show U-dependent in-pore diffusivities.
• Experimental non-linear and linear sweeps were not fully optimized due to time constraints; optimal sweep rates likely much slower (<0.1 mV/s), so quantitative comparison between non-linear and optimized linear sweeps in experiments remains incomplete.
• Voltage inversion parameters optimized in MD for a model system; optimal (U_inv, k_inv) in full devices may differ due to electrode asymmetries, ion size asymmetry, and macroscopic transport.
• Potential device and electrolyte constraints (e.g., electrochemical stability windows) may limit accessible inversion voltages in practice.