Lagrangian Formulation of Interfacial Energetics and Higher-Order Electric-Field Response in PLA/PCL/PEG–Nanozeolite Membranes
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Quantum-mechanical descriptions of electrically assisted membrane fabrication require a consistent relationship among the applied electric field, interfacial structure, and electronic energy. This study developed a Lagrangian and variational framework for density functional theory (DFT) modeling of poly(lactic acid)/polycaprolactone/poly(ethylene glycol) (PLA/PCL/PEG)–nanozeolite interfaces relevant to electric-field-assisted nonsolvent-induced phase separation (E-NIPS) under an applied voltage of 10 kV. The formulation integrated a low-energy electromagnetic description with constrained electron-density and orbital stationarity conditions, interfacial interaction energies, and field-induced responses. A contact-dependent analytical potential was constructed by combining a Morse potential with dipolar, quadratic, and fourth-order electric-field contributions. Assuming an electrode separation of 3 cm and a uniform electric field, an applied voltage of 10 kV corresponded to a nominal electric-field strength of 3.333 × 105 V m-1. The combined polymer–zeolite interfacial contact model yielded a minimum interaction energy of -91.30 kJ mol-1 and a nominal-field-induced energy shift of -2.8459 J mol-1. Geometric relaxation increased the calculated polarizability from 310.000 to 313.4775 atomic units (a.u.). Central finite-difference calculations demonstrated the expected second-order dependence of truncation error on the electric-field step size, whereas a reduced quadratic energy model yielded a root mean square error (RMSE) of 0.36189 J mol-1 across six additional electric-field evaluation points. Compared with the relaxed second-order expansion, the fourth-order formulation reduced the maximum truncation error for the C5 configuration from 0.81463 to 0.001035 J mol-1. The theoretical derivations distinguished structural relaxation effects at the polymer–zeolite interface from biases in the estimated response coefficients arising from the omission of higher-order electric-field terms. The resulting equations established a reproducible representation of interfacial energetics and specified the variational potential required for subsequent density-functional extensions. However, quantitative calibration and independent validation of the proposed framework require atomistic DFT reference calculations of electronic energies, electron densities, and electric-field response properties.
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