The Poisson–Boltzmann (P–B) equation is of fundamental importance in understanding solid–liquid electrolyte interfaces that are present in many fields. Due to the nonlinearity, it is usually challenging to find the explicit exact solutions of the P–B equation. The present work reports several perturbation solutions for the nonlinear P–B equation in the Cartesian and spherical coordinates. The new solutions contain a perturbation parameter from the high-order-accuracy approximation of the hyperbolic sine function and thus can apply to high zeta potential conditions. The comparison of the perturbation solutions with the traditional Debye–Hückel solutions and the full numerical solutions validates the robustness and accuracy of the perturbation solutions. The perturbation solutions are explicit and analytical and then can be used for a fast calculation of the EDL potential and interaction energy in versatile applications.

泊松-玻尔兹曼（PB）方程对于理解许多领域中存在的固液电解质界面至关重要。由于非线性，通常很难找到PB方程的精确解。本工作报告了笛卡尔坐标系和球坐标系中非线性PB方程的几种摄动解。新的解包含双曲正弦函数的高阶精度近似的摄动参数，因此可以应用于高Zeta势条件。将扰动解与传统的Debye-Hückel解和完整的数值解进行比较，验证了扰动解的鲁棒性和准确性。