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Investigation of Dielectric Decrement and Correlation Effects on Electric Double-Layer Capacitance by Self-Consistent Field Model

Part of: Chemistry Partial differential equations, boundary value problems

Published online by Cambridge University Press:  21 July 2016

Manman Ma ,
Shuangliang Zhao  and
Zhenli Xu
Manman Ma*
Affiliation:
Department of Mathematics and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
Shuangliang Zhao*
Affiliation:
State Key laboratory of Chemical Engineering, East China University of Science and Technology, 200237, Shanghai, China
Zhenli Xu*
Affiliation:
Institute of Natural Sciences, Department of Mathematics, and MoE Key Lab of Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240, China
*
*Corresponding author. Email addresses:[email protected] (M. Ma), [email protected] (S. Zhao), [email protected] (Z. Xu)
*Corresponding author. Email addresses:[email protected] (M. Ma), [email protected] (S. Zhao), [email protected] (Z. Xu)
*Corresponding author. Email addresses:[email protected] (M. Ma), [email protected] (S. Zhao), [email protected] (Z. Xu)
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Abstract

The differential capacitance of electric double-layer capacitors is studied by developing a generalized model of the self-consistent Gaussian field theory. This model includes many-body effects of particles near the interface such as ionic sizes, the order of water alignment and electrostatic correlations, and thus can present more accurate predictions of the electric double-layer structure and hence the capacitance than traditional continuum theories. Analytical simplification of the model and efficient numerical method are introduced, in particular, the approximation of the self-Green's function which describes the self energy of a mobile ion. We show that, when the applied voltage on interfaces is small the dielectric effect of the electrode materials plays an important role. For large voltage, this effect is screened, but the dielectric saturation due to the alignment of the nearby water is shown to be essential. For 2:1 electrolytes, abnormal enhancement on the capacitance due to the dielectric electrode is observed, which is due to the interplay of the image charge effect and Born solvation energy in the self energy of ions.

Keywords

Supercapacitormany-body effectsGreen's functioncontinuum theory

MSC classification

Secondary: 65Z05: Applications to physics 65N06: Finite difference methods 92E99: None of the above, but in this section
Type
Research Article
Information
Communications in Computational Physics , Volume 20 , Issue 2 , August 2016 , pp. 441 - 458
Copyright
Copyright © Global-Science Press 2016 

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References

[1] Huang, J., Sumpter, B. G., Meunier, V., Theoretical model for nanoporous carbon supercapacitors, Angew. Chem. Int. Ed. 47 (3) (2008) 520–524.Google Scholar
[2] Fedorov, M. V., Kornyshev, A. A., Ionic liquids at electrified interfaces, Chem. Rev. 114 (2014) 2978–3036.Google Scholar
[3] Rica, R. A., Ziano, R., Salerno, D., Mantegazza, F., van Roij, R., Brogioli, D., Capacitive mixing for harvesting the free energy of solutions at different concentrations, Entropy 15 (4) (2013) 1388–1407.Google Scholar
[4] Fernández, M. M., Ahualli, S., Iglesias, G. R., González-Caballero, F., Delgado, Á. V., Jiménez, M., Multi-ionic effects on energy production based on double layer expansion by salinity exchange, J. Colloid Interf. Sci. 446 (2015) 307–316.Google Scholar
[5] Iglesias, G. R., Fernández, M.M., Ahualli, S., Jiménez, M. L., Kozynchenko, O. P., Delgado, Á. V., Materials selection for optimum energy production by double layer expansion methods, J. Power Sources 261 (2014) 371–377.Google Scholar
[6] Gouy, G., Constitution of the electric charge at the surface of an electrolyte, J. Phys. 9 (1910) 457–468.Google Scholar
[7] Chapman, D. L., A contribution to the theory of electrocapillarity, Phil. Mag. 25 (1913) 475–481.CrossRefGoogle Scholar
[8] Stern, O., The theory of the electrolytic double-layer, Zeit. Elektrochem 30 (1924) 508–516.Google Scholar
[9] Borukhov, I., Andelman, D., Orland, H., Steric effects in electrolytes: A modified Poisson-Boltzmann equation, Phys. Rev. Lett. 79 (1997) 435–438.Google Scholar
[10] Ben-Yaakov, D., Andelman, D., Harries, D., Podgornik, R., Beyond standard Poisson–Boltzmann theory: Ion-specific interactions in aqueous solutions, J. Phys.: Condens. Matter 21 (2009) 424106.Google ScholarPubMed
[11] Li, B., Continuum electrostatics for ionic solutions with nonuniform ionic sizes, Nonlinearity 22 (2009) 811–833.Google Scholar
[12] Kornyshev, A. A., Double-layer in ionic liquids: paradigm change?, J. Phys. Chem. B 111 (2007) 5545–5557.Google Scholar
[13] Rosenfeld, Y., Free-energy model for the inhomogeneous hard-sphere fluid mixture and density-functional theory of freezing, Phys. Rev. Lett. 63 (9) (1989) 980–983.Google Scholar
[14] Wu, J., Density functional theory for liquid structure and thermodynamics, in: Molecular Thermodynamics of Complex Systems, Vol. 131 of Structure and Bonding, Springer Berlin Heidelberg, 2009, pp. 1–73.Google Scholar
[15] Zhao, S., Liu, Y., Chen, X., Lu, Y., Liu, H., Hu, Y., Unified framework of multiscale density functional theories and its recent applications, in: Mesoscale Modeling in Chemical Engineering Part II, Vol. 47 of Advances in Chemical Engineering, Academic Press, 2015, pp. 1–83.Google Scholar
[16] Jiang, D.-e., Jin, Z., Wu, J., Oscillation of capacitance inside nanopores, Nano letters 11 (12) (2011) 5373–5377.Google Scholar
[17] Hasted, J., Ritson, D., Collie, C., Dielectric properties of aqueous ionic solutions. Parts I and II, J. Chem. Phys. 16 (1) (1948) 1–21.Google Scholar
[18] Lyashchenko, A., Zasetsky, A. Y., Complex dielectric permittivity and relaxation parameters of concentrated aqueous electrolyte solutions in millimeter and centimeter wavelength ranges, J. Molecular Liquids 77 (1) (1998) 61–75.CrossRefGoogle Scholar
[19] Booth, F., The dielectric constant of water and the saturation effect, J. Chem. Phys. 19 (4) (1951) 391–394.Google Scholar
[20] Booth, F., Dielectric constant of polar liquids at high field strengths, J. Chem. Phys. 23 (3) (1955) 453–457.CrossRefGoogle Scholar
[21] Paunov, V., Dimova, R., Kralchevsky, P., Broze, G., Mehreteab, A., The hydration repulsion between charged surfaces as an interplay of volume exclusion and dielectric saturation effects, J. Colloid Interf. Sci. 182 (1) (1996) 239–248.Google Scholar
[22] Abrashkin, A., Andelman, D., Orland, H., Dipolar Poisson-Boltzmann equation: ions and dipoles close to charge interfaces, Phys. Rev. Lett. 99 (7) (2007) 077801.Google Scholar
[23] Frydel, D., Oettel, M., Charged particles at fluid interfaces as a probe into structural details of a double layer, Phys. Chem. Chem. Phys. 13 (9) (2011) 4109–4118.Google Scholar
[24] Gur, Y., Ravina, I., Babchin, A. J., On the electrical double layer theory. II. the Poisson-Boltzmann equation including hydration forces, J. Colloid Interf. Sci. 64 (2) (1978) 333–341.Google Scholar
[25] Macdonald, J. R., Theory of the differential capacitance of the double layer in unadsorbed electrolytes, J. Chem. Phys. 22 (11) (1954) 1857–1866.Google Scholar
[26] Wang, H., Varghese, J., Pilon, L., Simulation of electric double layer capacitors with mesoporous electrodes: Effects of morphology and electrolyte permittivity, Electrochimica Acta 56 (17) (2011) 6189–6197.CrossRefGoogle Scholar
[27] Glueckauf, E., Bulk dielectric constant of aqueous electrolyte solutions, Trans. Faraday Soc. 60 (1964) 1637–1645.Google Scholar
[28] Ben-Yaakov, D., Andelman, D., Podgornik, R., Dielectric decrement as a source of ion-specific effects, J. Chem. Phys. 134 (7) (2011) 074705.Google Scholar
[29] Levy, A., Andelman, D., Orland, H., Dielectric constant of ionic solutions: A field-theory approach, Phys. Rev. Lett. 108 (2012) 227801.CrossRefGoogle ScholarPubMed
[30] Levy, A., Andelman, D., Orland, H., Dipolar Poisson-Boltzmann approach to ionic solutions: A mean field and loop expansion analysis, J. Chem. Phys. 139 (16) (2013) 164909.Google Scholar
[31] Li, B., Wen, J., Zhou, S., Mean-field theory and computation of electrostatics with ionic concentration dependent dielectrics, Commun. Math. Sci. 14 (1) (2016) 249–271.Google Scholar
[32] Bikerman, J., Structure and capacity of electrical double layer, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 33 (220) (1942) 384–397.Google Scholar
[33] Hatlo, M., Van Roij, R., Lue, L., The electric double layer at high surface potentials: The influence of excess ion polarizability, EPL (Europhysics Letters) 97 (2) (2012) 28010.Google Scholar
[34] Nakayama, Y., Andelman, D., Differential capacitance of the electric double layer: The interplay between ion finite size and dielectric decrement, J. Chem. Phys. 142 (4) (2015) 044706.Google Scholar
[35] Bonthuis, D. J., Gekle, S., Netz, R. R., Dielectric profile of interfacial water and its effect on double-layer capacitance, Phys. Rev. Lett. 107 (2011) 166102.CrossRefGoogle ScholarPubMed
[36] Bonthuis, D. J., Gekle, S., Netz, R. R., Profile of the static permittivity tensor of water at interfaces: Consequences for capacitance, hydration interaction and ion adsorption, Langmuir 28 (20) (2012) 7679–7694.Google Scholar
[37] Bonthuis, D. J., Netz, R. R., Beyond the continuum: How molecular solvent structure affects electrostatics and hydrodynamics at solid-electrolyte interfaces, J. Phys. Chem. B 117 (39) (2013) 11397–11413.CrossRefGoogle ScholarPubMed
[38] Fahrenberger, F., Xu, Z., Holm, C., Simulation of electric double layers around charged colloids in aqueous solution of variable permittivity, J. Chem. Phys. 141 (6) (2014) 064902.Google Scholar
[39] Netz, R. R., Orland, H., Beyond Poisson-Boltzmann: Fluctuation effects and correlation functions, Eur. Phys. J. E 1 (2000) 203–214.CrossRefGoogle Scholar
[40] Netz, R. R., Orland, H., Variational charge renormalization in charged systems, Eur. Phys. J. E 11 (2003) 301–311.CrossRefGoogle ScholarPubMed
[41] Wang, Z. G., Fluctuation in electrolyte solutions: The self energy, Phys. Rev. E 81 (2010) 021501.Google Scholar
[42] Wang, R., Wang, Z.-G., Effects of image charges on double layer structure and forces, J. Chem. Phys. 139 (2013) 124702.Google Scholar
[43] Xu, Z., Ma, M., Liu, P., Self-energy-modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches, Phys. Rev. E 90 (1) (2014) 013307.CrossRefGoogle ScholarPubMed
[44] Xu, Z., Maggs, A., Solving fluctuation-enhanced Poisson-Boltzmann equations, J. Comput. Phys. 275 (2014) 310–322.Google Scholar
[45] Lu, B.-S., Xing, X., Correlation potential of a test ion near a strongly charged plate, Phys. Rev. E 89 (2014) 032305.Google Scholar
[46] Ma, M., Xu, Z., Self-consistent field model for strong electrostatic correlations and inhomogeneous dielectric media, J. Chem. Phys. 141 (24) (2014) 244903.Google Scholar
[47] Wang, R., Wang, Z.-G., On the theoretical description of weakly charged surfaces, J. Chem. Phys. 142 (10) (2015) 104705.Google Scholar
[48] Gillespie, D., Valiskó, M., Boda, D., Density functional theory of the electrical double layer: the RFD functional, J. Phys. Condens. Matter 17 (42) (2005) 6609–6626.Google Scholar
[49] Jiang, J., Cao, D., Henderson, D., Wu, J., Revisiting density functionals for the primitive model of electric double layers, J. Chem. Phys. 140 (4) (2014) 044714.Google Scholar
[50] Butt, H.-J., Graf, K., Kappl, M., Physics and Chemistry of Interfaces, Wiley/VCH, Berlin, 2003.Google Scholar
[51] Paunovic, M., Schlesinger, M., Fundamentals of Electrochemical Deposition, Vol. 45, John Wiley & Sons, Hoboken, 2006.Google Scholar
[52] Frydel, D., Mean-field electrostatics beyond the point-charge description, Adv. Chem. Phys. 160 (2016) 209–260.Google Scholar
[53] Lin, L., Yang, C., Lu, J., Ying, L., W. E, , A fast parallel algorithm for selected inversion of structured sparse matrices with application to 2D electronic structure calculations, SIAM J. Sci. Comput. 33 (3) (2011) 1329–1351.Google Scholar
[54] Lin, L., Yang, C., Meza, J. C., Lu, J., Ying, L., W. E, , Selinv—An algorithm for selected inversion of a sparse symmetric matrix, ACM Trans. Math. Softw. 37 (2011) 40:1–40:19.Google Scholar
[55] Grahame, D. C., Differential capacity of mercury in aqueous sodium fluoride solutions. I. effect of concentration at 25°, J. Am. Chem. Soc. 76 (19) (1954) 4819–4823.Google Scholar
[56] Onsager, L., Samaras, N. N. T., The surface tension of Debye-Hückel electrolytes, J. Chem. Phys. 2 (1934) 528–536.CrossRefGoogle Scholar
[57] Levin, Y., Dos Santos, A. P., Diehl, A., Ions at the air-water interface: an end to a hundred-year-old mystery?, Phys. Rev. Lett. 103 (25) (2009) 257802.Google Scholar
[58] Wang, R., Wang, Z.-G., Continuous self-energy of ions at the dielectric interface, Phys. Rev. Lett. 112 (13) (2014) 136101.Google Scholar
[59] Jungwirth, P., Tobias, D. J., Specific ion effects at the air/water interface, Chem. Rev. 106 (4) (2006) 1259–1281.Google Scholar
[60] Grahame, D. C., The electrical double layer and the theory of electrocapillarity, Chem. Rev. 32 (1947) 441–501.CrossRefGoogle Scholar