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ANALYSIS OF THE MEAN FIELD FREE ENERGY FUNCTIONAL OF ELECTROLYTE SOLUTION WITH NONHOMOGENOUS BOUNDARY CONDITIONS AND THE GENERALIZED PB/PNP EQUATIONS WITH INHOMOGENEOUS DIELECTRIC PERMITTIVITY
Liu, Xuejiao1,2; Qiao, Yu1,2; Lu, Benzhou1,2
2018
Source PublicationSIAM JOURNAL ON APPLIED MATHEMATICS
ISSN0036-1399
Volume78Issue:2Pages:1131-1154
AbstractThe energy functional, the governing partial differential equation(s) (PDE), and the boundary conditions need to be consistent with each other in a modeling system. In electrolyte solution study, people usually use a free energy form of an in finite domain system (with vanishing potential boundary condition) and the derived PDE(s) for analysis and computing. However, in many real systems and/or numerical computing, the objective domain is bounded, and people still use the similar energy form, PDE(s), but with different boundary conditions, which may cause inconsistency. In this work, (1) we present a mean field free energy functional for the electrolyte solution within a bounded domain with either physical or numerically required artificial boundary. Apart from the conventional energy components (electrostatic potential energy, ideal gas entropy term, and chemical potential term), new boundary interaction terms are added for both Neumann and Dirichlet boundary conditions. These new terms count for physical interactions with the boundary (for a real boundary) or the environment influence on the computational domain system (for a nonphysical but numerically designed boundary). In addition, the boundary energy term also applies to any bounded system described by the Poisson equation. (2) The traditional physical-based Poisson-Boltzmann (PB) equation and Poisson-Nernst-Planck (PNP) equations are proved to be consistent with the complete free energy form, and different boundary conditions can be applied. (3) In particular, for the inhomogeneous electrolyte with ionic concentration-dependent dielectric permittivity, we derive the generalized Boltzmann distribution (thereby the generalized PB equation) for the equilibrium case, and the generalized PNP equations with variable dielectric (VDPNP) for the nonequilibrium case, under different boundary conditions. (4) Furthermore, the energy laws are calculated and compared to study the energy properties of different energy functionals and the resulting PNP systems. Numerical tests are also performed to demonstrate the different consequences resulting from different energy forms and their derived PDE(s).
Keywordfree energy functional electrolyte boundary conditions variable dielectric generalized Poisson-Nernst-Planck/Poisson-Boltzmann equations energy law
DOI10.1137/16M1108583
Language英语
Funding ProjectNational Key Research and Development Program of China[2016YFB0201304] ; China NSF[NSFC 21573274] ; China NSF[NSFC 91530102] ; [TZ2016003]
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000431201200023
PublisherSIAM PUBLICATIONS
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Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/30273
Collection计算数学与科学工程计算研究所
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, Natl Ctr Math & Interdisciplinary Sci, State Key Lab Sci & Engn Comp, Beijing 100190, Peoples R China
2.Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China
Recommended Citation
GB/T 7714
Liu, Xuejiao,Qiao, Yu,Lu, Benzhou. ANALYSIS OF THE MEAN FIELD FREE ENERGY FUNCTIONAL OF ELECTROLYTE SOLUTION WITH NONHOMOGENOUS BOUNDARY CONDITIONS AND THE GENERALIZED PB/PNP EQUATIONS WITH INHOMOGENEOUS DIELECTRIC PERMITTIVITY[J]. SIAM JOURNAL ON APPLIED MATHEMATICS,2018,78(2):1131-1154.
APA Liu, Xuejiao,Qiao, Yu,&Lu, Benzhou.(2018).ANALYSIS OF THE MEAN FIELD FREE ENERGY FUNCTIONAL OF ELECTROLYTE SOLUTION WITH NONHOMOGENOUS BOUNDARY CONDITIONS AND THE GENERALIZED PB/PNP EQUATIONS WITH INHOMOGENEOUS DIELECTRIC PERMITTIVITY.SIAM JOURNAL ON APPLIED MATHEMATICS,78(2),1131-1154.
MLA Liu, Xuejiao,et al."ANALYSIS OF THE MEAN FIELD FREE ENERGY FUNCTIONAL OF ELECTROLYTE SOLUTION WITH NONHOMOGENOUS BOUNDARY CONDITIONS AND THE GENERALIZED PB/PNP EQUATIONS WITH INHOMOGENEOUS DIELECTRIC PERMITTIVITY".SIAM JOURNAL ON APPLIED MATHEMATICS 78.2(2018):1131-1154.
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