Second law of thermodynamics: Intrinsic-nonequilibrium distribution of large ions in charged small nanopores

Technical Summary: Intrinsic-Nonequilibrium Distribution of Large Ions in Charged Small Nanopores

Problem Statement
The paper addresses a fundamental question in statistical mechanics and thermodynamics: whether a macroscopic system immersed in a thermal reservoir can reach a steady state that is intrinsically different from thermodynamic equilibrium, even in the absence of external driving forces. Recent theoretical work suggests that "locally nonchaotic" energy barriers can prevent a system from relaxing to equilibrium, creating a "spontaneous-nonequilibrium domain" (SND). While theoretical "toy models" (e.g., Knudsen gases in gravitational fields) predict such phenomena, they are difficult to test experimentally due to the extreme physical conditions required (e.g., gravitational fields comparable to neutron stars). The authors propose that the strong Coulomb force in charged nanopores, combined with large ions, can serve as a practical experimental analog to these theoretical models. The core problem is to determine if the steady-state distribution of ions in such confined geometries violates the second law of thermodynamics, which dictates that steady states must follow Boltzmann distributions.

Methodology
The study employs a combination of thermodynamic analysis and experimental validation using supercapacitive cells.

1. Theoretical Framework: The authors derive a thermodynamic consistency condition (Equation 5) linking the concentration sensitivity of electric potential ($\delta V = \partial V / \partial c$) and the charge efficiency ($\Lambda = \partial N_{ad} / \partial Q$). They argue that for an equilibrium system, these variables must satisfy a specific relationship derived from the heat-engine statement of the second law of thermodynamics. If the ion distribution is intrinsically non-equilibrium (non-Boltzmann), this relationship should be violated.
2. Experimental Setup: The experiment utilizes microporous carbon electrodes (pore size $\approx$ 1 nm) immersed in aqueous cesium pivalate (CsPiv) solutions. The pivalate ion size ($\approx$ 0.7 nm) is chosen such that the effective pore size is only slightly larger than the ion size ($d_e < 2d_i$), forcing ions into quasi-1D lineups.
3. Procedure:
   • Cells were charged at a constant, slow current to ensure steady-state conditions (verified via rate convergence tests).
   • The electrolyte concentration ($c$) was systematically varied (10 mM to 16 mM) using a liquid replacement technique to simulate the expansion/compression of a "plain" in the theoretical models.
   • Simultaneous measurements of cell potential ($V$) and electrolyte concentration (via liquid conductivity) were recorded during charging.
   • Charge efficiency ($\Lambda$) and the potential sensitivity ($|\delta V|$) were calculated from the data.

Key Contributions and Results

1. Experimental Validation of Intrinsic Nonequilibrium: The primary result is the observation that the measured steady-state ion distribution in the charged nanopores is significantly non-Boltzmannian. The experimental data shows a clear inconsistency with the second law of thermodynamics as formulated for equilibrium systems.
2. Violation of Thermodynamic Consistency: As shown in Figure 5(d), the measured concentration sensitivity of the electric potential ($|\delta V|$) is anomalously large—approximately one order of magnitude greater than the value predicted by the charge efficiency ($\Lambda$) and the second law (Equation 5). In an equilibrium system, these values must balance; their mismatch indicates that the system cannot relax to a standard thermodynamic equilibrium.
3. Quantification of Non-Equilibrium Effects: The analysis reveals that the "intrinsic-nonequilibrium term" in the governing equations (related to the variation of ion diffusion coefficients with potential) dominates the equilibrium term. This suggests that the confinement effect of the nanopore walls acts as a locally nonchaotic energy barrier (SND), preventing the system from achieving the maximum entropy state predicted by conventional statistical mechanics.
4. Confirmation of MD Simulations: The experimental findings align with previous molecular dynamics (MD) simulations which predicted that ion diffusion coefficients in sub-nanometer pores are highly dependent on electric potential, leading to non-equilibrium steady states.

Significance and Claims
The paper claims to provide the first experimental validation of the concept of "intrinsic nonequilibrium" in a macroscopic system driven by Coulomb forces rather than gravity.

• Challenge to the Second Law: The authors assert that their results demonstrate a steady state that differs significantly from thermodynamic equilibrium without external driving forces. This challenges the conventional interpretation of the second law of thermodynamics, specifically the heat-engine statement, which would forbid the production of useful work from a single thermal reservoir in a cycle. The authors suggest that if operated in a specific isothermal cycle, such a system could theoretically produce net work ($W_e > W_{os}$).
• Mechanism: The significance lies in identifying the "locally nonchaotic" nature of ion transport in quasi-1D nanopores as the mechanism preventing equilibration. The confinement restricts particle trajectories such that they cannot explore the phase space chaotically, leading to a steady state that is a local maximum of entropy ($S_{ne}$) rather than the global maximum ($S_{eq}$).
• Modesty of Claims: The authors acknowledge that while the steady state appears "normal" in standard charge curves, the deviation is only revealed when analyzing the coupling between concentration and potential. They do not claim to have built a perpetual motion machine but rather to have experimentally demonstrated a phenomenon that theoretical models predicted but which was previously untestable. The work suggests that the boundaries of the second law may be broader than previously understood, particularly in systems with localized nonchaotic barriers.