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An exact solution of the nonlinear Poisson-Boltzmann equation in parallel-plate geometry

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Abstract

The Poisson-Boltzmann (PB) equation is a fundamental theoretical tool in understanding electric double layers (EDLs) at solid-liquid interfaces. Because of the intrinsic nonlinearity, finding exact analytical solutions of this equation is very difficult, and hitherto only very few exact analytical solutions are known. In this work, a new explicit exact solution for the nonlinear PB equation in parallel-plate geometry is derived in terms of Jacobi elliptic functions. A comparison of the sought solution with the finite element numerical simulation ensures correctness of the solution. We further found that the new solution is numerically consistent with the two existing solutions derived by Behrens & Borkovec (Phys. Rev. E 60:7040, [25]) and Johannessen (J. Math. Chem. 52:504, [36]) in spite of different expressions of the three solutions. This suggests equivalence of the three solutions. In addition, based upon the new solution, we suggest a method of determining the electrostatic potential profile inside the EDL with the experimental data of disjoining pressure.

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Acknowledgements

We would like to acknowledge the support from the National Key Research and Development Program of China (No. 2017YFB0603500) and the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No.51721004).

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Authors and Affiliations

  1. Key Laboratory of Thermo-Fluid Science and Engineering of MOE, School of Energy and Power Engineering, **’an Jiaotong University, **’an, 710049, China

    Wenyao Zhang, Qiuwang Wang, Min Zeng & Cunlu Zhao

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  1. Wenyao Zhang
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  2. Qiuwang Wang
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Correspondence to Cunlu Zhao.

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Zhang, W., Wang, Q., Zeng, M. et al. An exact solution of the nonlinear Poisson-Boltzmann equation in parallel-plate geometry. Colloid Polym Sci 296, 1917–1923 (2018). https://doi.org/10.1007/s00396-018-4394-8

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  • DOI: https://doi.org/10.1007/s00396-018-4394-8

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