Unified symmetry breaking in confined electrolytes: charge, chemical potential, and the nonlinear capacitance of hollow nanoparticles

Abstract We study the nonlinear electrostatic response of electrolyte-filled, hollow charged nanoparticles, modeled as nanocapacitors with finite wall thickness and curved geometry. Using the linearized Poisson–Boltzmann (LPB) approximation, we derive analytical expressions for the electric double layers (EDLs) and compute the differential capacitance $$C_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> as a function of topology, wall thickness, and electrolyte concentration. Remarkably, we identify two forms of symmetry-breaking that yield the same macroscopic capacitance: one due to variations in surface charge density (charge symmetry breaking), and another due to differing bulk chemical potentials (chemical potential symmetry breaking). In the first case, increasing the wall charge alters the internal and external EDL structures without affecting the overall capacitance—provided the Debye length and geometry remain fixed. In the second case, two distinct electrolytes (e.g., 1:1 at concentration $$\rho _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> and 2:2 at $$\rho _0/4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> ) yield indistinguishable EDLs and capacitance, despite differing significantly in chemical potential. These phenomena reflect a deeper electrostatic invariance governed by confinement, topology, and the violation of the local electroneutrality condition (VLEC), rather than by absolute charge or thermodynamic potential. Our findings establish a unified framework for understanding nonlinear capacitance in nanoconfined systems and suggest design principles for energy storage and biological applications, where control of electrostatic response is critical.