本研究根據Helmholtz模型進行模擬水相硫酸和氫氧化鉀以及有機相四乙基四氟硼酸銨/乙腈電解質的碳電極電雙層電容。所提出的模型結合圓柱形孔隙模型模擬大孔及中孔與呈現一常數表面電容(C/S)micro的狹縫狀微孔。碳材料的孔隙結構和孔徑分佈利用非定域密度泛函理論(NLDFT)方法進行分析，接著使用微孔結構碳材料的實驗電容數據，以評估不同的電解質在碳微孔產生的常數(C/S)micro值，並確定微孔的分子篩效應，微孔中得到的常數(C/S)micro值表明了介電常數在電極/電解質間與電解質離子溶劑化層中的溶劑分子數目成正比，而中孔的C/S值隨孔徑減小而降低則是因為孔壁曲率影響效果的增加。對於水相電解質，在微孔的C/S的值比在中孔和大孔較大是由於電解質離子溶劑化層具有較高的介電常數及微孔侷限效應。各個碳材料的預測電容值結果與實驗數據呈現極好的一致性，從而驗證了模型的可靠性，這個模型能夠提供可靠、準確的電容值預測，也闡明了不同的孔隙結構和界面性質對於電雙層形成機制與電容性能的影響。

This study reports on a facile method based on Helmholtz models for simulating the electric double-layer capacitance of various forms of carbon in aqueous H2SO4 and KOH and organic tetraethylammonium tetrafluoroboraote/acetonitrile electrolytes. The proposed method combines cylindrical pore models for macropores and mesopores with the slit-pore model for micropores exhibiting constant surface-based capacitance (C/S)micro. The pore structures and pore size distribution of the carbon are analyzed using a method based on non-local density functional theory (NLDFT). We then used data related to the capacitance of microporous carbon to evaluate the constant C/S values produced by distinct electrolytes in carbon micropores and to determine the molecule-sieving effect of the micropores. The constant C/S values obtained from the micropores suggest that the dielectric constant at the electrode-electrolyte is proportional to the solvent-molecular number of the ion-solvating layer. The C/S values in mesopores decreased with a decrease in pore size due to the effects of wall-curvature confinement. For aqueous electrolyte, the C/S values in micropores are larger than those in mesopores and macropores due to the compactness of the ion-solvating layers, which account for the higher dielectric constant in the micropores. Our simulation results regarding the capacitance values of each carbon are in excellent agreement with experiment data, thereby verifying the reliability of the proposed model. This model is capable of providing reliable, precise predictions of capacitance values and also reveals the mechanism underlying the double-layer formation of distinct pores and the interfacial properties associated with capacitive performance.

參考文獻
[1] Y. Zhou, Y. Kim, C. Jo, J. Lee, C. W. Lee, and S. Yoon, "A novel mesoporous carbon-silica-titania nanocomposite as a high performance anode material in lithium ion batteries," Chem Commun (Camb), vol. 47, 4944-6, 2011.
[2] C. K. Chan, R. N. Patel, M. J. O’Connell, B. A. Korgel, and Y. Cui, "Solution-Grown Silicon Nanowires for Lithium-Ion Battery Anodes," ACS Nano, vol. 4, 1443-1450, 2010.
[3] S. Yoon, J. Lee, T. Hyeon, and S. M. Oh, "Electric Double-Layer Capacitor Performance of a New Mesoporous Carbon," Journal of The Electrochemical Society, vol. 147, 2507-2512, 2000.
[4] B. E. Conway, Electrochemical Supercapacitors : Scientific Fundamentals and Technological Applications. New York: Kluwer Academic Plenum, 1999.
[5] Y. Kumar, G. P. Pandey, and S. A. Hashmi, "Gel Polymer Electrolyte Based Electrical Double Layer Capacitors: Comparative Study with Multiwalled Carbon Nanotubes and Activated Carbon Electrodes," The Journal of Physical Chemistry C, vol. 116, 26118-26127, 2012.
[6] J. J. Yoo, K. Balakrishnan, J. Huang, V. Meunier, B. G. Sumpter, A. Srivastava, et al., "Ultrathin planar graphene supercapacitors," Nano Lett, vol. 11, 1423-7, 2011.
[7] A. Burke, "Ultracapacitors: why, how, and where is the technology," Journal of Power Sources, vol. 91, 37-50, 2000.
[8] P. Simon and Y. Gogotsi, "Materials for electrochemical capacitors," Nat Mater, vol. 7, 845-854, 2008.
[9] L. Bonnefoi, P. Simon, J. F. Fauvarque, C. Sarrazin, J. F. Sarrau, and A. Dugast, "Electrode compositions for carbon power supercapacitors," Journal of Power Sources, vol. 80, 149-155, 1999.
[10] J. P. Zheng and T. R. Jow, "High energy and high power density electrochemical capacitors," Journal of Power Sources, vol. 62, 155-159, 1996.
[11] D. Pech, M. Brunet, H. Durou, P. Huang, V. Mochalin, Y. Gogotsi, et al., "Ultrahigh-power micrometre-sized supercapacitors based on onion-like carbon," Nat Nano, vol. 5, 651-654, 2010.
[12] M. F. El-Kady, V. Strong, S. Dubin, and R. B. Kaner, "Laser Scribing of High-Performance and Flexible Graphene-Based Electrochemical Capacitors," Science, vol. 335, 1326-1330, 2012.
[13] R. Kötz and M. Carlen, "Principles and applications of electrochemical capacitors," Electrochimica Acta, vol. 45, 2483-2498, 2000.
[14] T. L. Makarova, B. Sundqvist, P. Scharff, M. E. Gaevski, E. Olsson, V. A. Davydov, et al., "Electrical properties of two-dimensional fullerene matrices," Carbon, vol. 39, 2203-2209, 2001.
[15] D. Eder, "Carbon Nanotube−Inorganic Hybrids," Chemical Reviews, vol. 110, 1348-1385, 2010.
[16] D. Li and R. B. Kaner, "Graphene-Based Materials," Science, vol. 320, 1170-1171, 2008.
[17] Y. Wang, Z. Shi, Y. Huang, Y. Ma, C. Wang, M. Chen, et al., "Supercapacitor Devices Based on Graphene Materials," The Journal of Physical Chemistry C, vol. 113, 13103-13107, 2009.
[18] M. J. Allen, V. C. Tung, and R. B. Kaner, "Honeycomb Carbon: A Review of Graphene," Chemical Reviews, vol. 110, 132-145, 2010.
[19] A. B. Fuertes and S. Alvarez, "Graphitic mesoporous carbons synthesised through mesostructured silica templates," Carbon, vol. 42, 3049-3055, 2004.
[20] P. Simon and Y. Gogotsi, "Capacitive Energy Storage in Nanostructured Carbon–Electrolyte Systems," Accounts of Chemical Research, vol. 46, 1094-1103, 2013.
[21] H. F. Stoeckli, "Microporous carbons and their characterization: The present state of the art," Carbon, vol. 28, 1-6, 1990.
[22] J. Romanos, M. Beckner, T. Rash, L. Firlej, B. Kuchta, P. Yu, et al., "Nanospace engineering of KOH activated carbon," Nanotechnology, vol. 23, 015401, 2012.
[23] P. Simon and A. Burke, "Nanostructured Carbons: Double-Layer Capacitance and More," Interface-The Electrochemical Society, vol. 17, 38-43, 2008.
[24] M. Karthik, E. Redondo, E. Goikolea, V. Roddatis, S. Doppiu, and R. Mysyk, "Effect of Mesopore Ordering in Otherwise Similar Micro/Mesoporous Carbons on the High-Rate Performance of Electric Double-Layer Capacitors," The Journal of Physical Chemistry C, vol. 118, 27715-27720, 2014.
[25] W. Xing, S. Z. Qiao, R. G. Ding, F. Li, G. Q. Lu, Z. F. Yan, et al., "Superior electric double layer capacitors using ordered mesoporous carbons," Carbon, vol. 44, 216-224, 2006.
[26] K. Xia, Q. Gao, J. Jiang, and J. Hu, "Hierarchical porous carbons with controlled micropores and mesopores for supercapacitor electrode materials," Carbon, vol. 46, 1718-1726, 2008.
[27] D. Saha, E. A. Payzant, A. S. Kumbhar, and A. K. Naskar, "Sustainable Mesoporous Carbons as Storage and Controlled-Delivery Media for Functional Molecules," ACS Applied Materials & Interfaces, vol. 5, 5868-5874, 2013.
[28] S. Yoon, J. Lee, T. Hyeon, and S. M. Oh, "Electric Double-Layer Capacitor Performance of a New Mesoporous Carbon," Journal of The Electrochemical Society, vol. 147, 2507-2512, 2000.
[29] L. Zhang, X. Yang, F. Zhang, G. Long, T. Zhang, K. Leng, et al., "Controlling the Effective Surface Area and Pore Size Distribution of sp2 Carbon Materials and Their Impact on the Capacitance Performance of These Materials," Journal of the American Chemical Society, vol. 135, 5921-5929, 2013.
[30] 劉守悟, 吳旻聰, 江建章, 吳培費, and 劉尚試, "中孔碳材作為電極觸媒基材," 高分子電解質燃料電池之觸媒專刊, vol. 56, 47, 2009.
[31] C.-H. Chang, Y.-C. Hsu, and C.-M. Yang, "Introduction to the Nanocasting Routes toward Three-dimensionally Ordered Mesoporous Materials," CHEMISTRY(The Chinese Chemical Society, Taipei), vol. 66, 127-133, 2008.
[32] 林有銘, 陳立基, 蔡協和, 楊正憲, 林弘萍, and 鄧熙聖, "奈米孔洞碳材特性與超級電容應用," 工業材料雜誌, 2012.
[33] H.-C. Huang, C.-W. Huang, C.-T. Hsieh, and H. Teng, "Electric double layer capacitors of high volumetric energy based on ionic liquids and hierarchical-pore carbon," Journal of Materials Chemistry A, vol. 2, 14963, 2014.
[34] A. Shaheen, Al-Muhtaseb, and A. R. James, "Preparation and Properties of Resorcinol-Formaldehyde Organic and Carbon Gels," Advanced Materials, vol. 15, 101-114, 2003
[35] M.-F. Hsueh, C.-W. Huang, C.-A. Wu, P.-L. Kuo, and H. Teng, "The Synergistic Effect of Nitrile and Ether Functionalities for Gel Electrolytes Used in Supercapacitors," The Journal of Physical Chemistry C, vol. 117, 16751-16758, 2013.
[36] C.-W. Huang, C.-T. Hsieh, P.-L. Kuo, and H. Teng, "Electric double layer capacitors based on a composite electrode of activated mesophase pitch and carbon nanotubes," Journal of Materials Chemistry, vol. 22, 7314, 2012.
[37] H.-Y. Liu, K.-P. Wang, and H. Teng, "A simplified preparation of mesoporous carbon and the examination of the carbon accessibility for electric double layer formation," Carbon, vol. 43, 559-566, 2005.
[38] C.-W. Huang, C.-A. Wu, S.-S. Hou, P.-L. Kuo, C.-T. Hsieh, and H. Teng, "Gel Electrolyte Derived from Poly(ethylene glycol) Blending Poly(acrylonitrile) Applicable to Roll-to-Roll Assembly of Electric Double Layer Capacitors," Advanced Functional Materials, vol. 22, 4677-4685, 2012.
[39] H. Teng and T.-C. Weng, "Transformation of mesophase pitch into different carbons by heat treatment and KOH etching," Microporous and Mesoporous Materials, vol. 50, 53-60, 2001.
[40] T.-C. Weng and H. Teng, "Characterization of High Porosity Carbon Electrodes Derived from Mesophase Pitch for Electric Double-Layer Capacitors," Journal of The Electrochemical Society, vol. 148, A368, 2001.
[41] B. Munkhbayar, M. J. Nine, J. Jeoun, M. Bat-Erdene, H. Chung, and H. Jeong, "Influence of dry and wet ball milling on dispersion characteristics of the multi-walled carbon nanotubes in aqueous solution with and without surfactant," Powder Technology, vol. 234, 132-140, 2013.
[42] A. G. Bannov, N. F. Uvarov, A. V. Ukhina, I. S. Chukanov, K. D. Dyukova, and G. G. Kuvshinov, "Structural changes in carbon nanofibers induced by ball milling," Carbon, vol. 50, 1090-1098, 2012.
[43] W. W. Russel, "The Adsorption of Gases and Vapors. Volume I: Physical Adsorption (Brunauer, Stephen)," Journal of Chemical Education, vol. 21, 52, 1944.
[44] J. Y. Ying, C. P. Mehnert, and M. S. Wong, "Synthesis and Applications of Supramolecular-Templated Mesoporous Materials," Angewandte Chemie International Edition, vol. 38, 56-77, 1999.
[45] S. Brunauer, P. H. Emmett, and E. Teller, "Adsorption of Gases in Multimolecular Layers," Journal of the American Chemical Society, vol. 60, 309-319, 1938.
[46] E. P. Barrett, L. G. Joyner, and P. P. Halenda, "The Determination of Pore Volume and Area Distributions in Porous Substances. I. Computations from Nitrogen Isotherms," Journal of the American Chemical Society, vol. 73, 373-380, 1951.
[47] C. N. R. Rao, A. K. Sood, K. S. Subrahmanyam, and A. Govindaraj, "Graphene: The New Two-Dimensional Nanomaterial," Angewandte Chemie International Edition, vol. 48, 7752-7777, 2009.
[48] R. Ryoo, S. H. Joo, and S. Jun, "Synthesis of Highly Ordered Carbon Molecular Sieves via Template-Mediated Structural Transformation," The Journal of Physical Chemistry B, vol. 103, 7743-7746, 1999.
[49] J. Landers, G. Y. Gor, and A. V. Neimark, "Density functional theory methods for characterization of porous materials," Colloids and Surfaces A: Physicochemical and Engineering Aspects, vol. 437, 3-32, 2013.
[50] P. Tarazona, "Phase equilibria of fluid interfaces and confined fluids Non-local versus local density functionals " Molecular Physics, vol. 60, 573-595, 1987.
[51] N. A. Seaton, J. P. R. B. Walton, and N. quirke, "A new analysis method for the determination of the pore size distribution of porous carbons from nitrogen adsorption measurements," Carbon, vol. 27, 853-861, 1989.
[52] C. Lastoskie, K. E. Gubbins, and N. Quirke, "Pore size heterogeneity and the carbon slit pore: a density functional theory model," Langmuir, vol. 9, 2693-2702, 1993.
[53] T. A. Centeno and F. Stoeckli, "The assessment of surface areas in porous carbons by two model-independent techniques, the DR equation and DFT," Carbon, vol. 48, 2478-2486, 2010.
[54] H. D. Young, Physics. New York: Addison-Wesley Publishing Co., 1992.
[55] P. G. Kostyuk, S. L. Mironov, P. A. Doroshenko, and V. N. Ponomarev, "Surface charges on the outer side of mollusc neuron membrane," The Journal of Membrane Biology, vol. 70, 171-179, 1982.
[56] K. B. Oldham, "A Gouy–Chapman–Stern model of the double layer at a (metal)/(ionic liquid) interface," Journal of Electroanalytical Chemistry, vol. 613, 131-138, 2008.
[57] C. H. Hamann, A. Hamnett, and W. Vielstich, Electrochemistry. New York: Wiley-Vch, 1998.
[58] M. S. Kilic, M. Z. Bazant, and A. Ajdari, "Steric effects in the dynamics of electrolytes at large applied voltages. I. Double-layer charging," Physical Review E, vol. 75, 021502, 2007.
[59] I. Borukhov, D. Andelman, and H. Orland, "Steric Effects in Electrolytes: A Modified Poisson-Boltzmann Equation," Physical Review Letters, vol. 79, 435-438, 1997.
[60] E. Frackowiak, "Carbon materials for supercapacitor application," Phys Chem Chem Phys, vol. 9, 1774-85, 2007.
[61] J. Huang, B. G. Sumpter, and V. Meunier, "Theoretical model for nanoporous carbon supercapacitors," Angew Chem Int Ed Engl, vol. 47, 520-4, 2008.
[62] J. Huang, B. G. Sumpter, and V. Meunier, "A universal model for nanoporous carbon supercapacitors applicable to diverse pore regimes, carbon materials, and electrolytes," CHEMISTRY-A EUROPEAN JOURNAL, vol. 14, 6614-26, 2008.
[63] G. Feng, R. Qiao, J. Huang, B. G. Sumpter, and V. Meunier, "Ion Distribution in Electrified Micropores and Its Role in the Anomalous Enhancement of Capacitance," ACS Nano, vol. 4, 2382-2390, 2010.
[64] H. Marsh and M.-A. D. Díaz-Estébanez, Sciences of Carbon Materials. Spain, 2000.
[65] J. R. Fryer, "The micropore structure of disordered carbons determined by high resolution electron microscopy," Carbon, vol. 19, 431-439, 1981.
[66] J. Guo, J. R. Morris, Y. Ihm, C. I. Contescu, N. C. Gallego, G. Duscher, et al., "Topological Defects: Origin of Nanopores and Enhanced Adsorption Performance in Nanoporous Carbon," Small, vol. 8, 3283-3288, 2012.
[67] J. Chmiola, G. Yushin, Y. Gogotsi, C. Portet, P. Simon, and P. L. Taberna, "Anomalous Increase in Carbon Capacitance at Pore Sizes Less Than 1 Nanometer," Science, vol. 313, 1760-1763, 2006.
[68] J. Chmiola, C. Largeot, P. L. Taberna, P. Simon, and Y. Gogotsi, "Desolvation of ions in subnanometer pores and its effect on capacitance and double-layer theory," Angew Chem Int Ed Engl, vol. 47, 3392-5, 2008.
[69] B. Daffos, P. L. Taberna, Y. Gogotsi, and P. Simon, "Recent Advances in Understanding the Capacitive Storage in Microporous Carbons," Fuel Cells, vol. 10, 819-824, 2010.
[70] C. Largeot, C. Portet, J. Chmiola, P.-L. Taberna, Y. Gogotsi, and P. Simon, "Relation between the Ion Size and Pore Size for an Electric Double-Layer Capacitor," Journal of the American Chemical Society, vol. 130, 2730-2731, 2008.
[71] J. Chmiola, G. Yushin, R. Dash, and Y. Gogotsi, "Effect of pore size and surface area of carbide derived carbons on specific capacitance," Journal of Power Sources, vol. 158, 765-772, 2006.
[72] J. Rouquérol, D. Avnir, C. W. Fairbridge, D. H. Everett, J. M. Haynes, N. Pernicone, et al., "Recommendations for the characterization of porous solids," Pure and Applied Chemistry, vol. 66, 1739-1758, 1994.
[73] K. S. W. Sing and R. T. Williams, "The Use of Molecular Probes for the Characterization of Nanoporous Adsorbents," Particle & Particle Systems Characterization, vol. 21, 71-79, 2004.
[74] ISO-15901-3, Pore size distribution and porosity of solid materials by mercury porosimetry and gas adsorption, Analysis of micropores by gas adsorption, 2007.
[75] F. Stoeckli and T. A. Centeno, "Optimization of the characterization of porous carbons for supercapacitors," Journal of Materials Chemistry A, vol. 1, 6865, 2013.
[76] T. A. Centeno, O. Sereda, and F. Stoeckli, "Capacitance in carbon pores of 0.7 to 15 nm: a regular pattern," Phys Chem Chem Phys, vol. 13, 12403-6, 2011.
[77] T. A. Centeno and F. Stoeckli, "Surface-related capacitance of microporous carbons in aqueous and organic electrolytes," Electrochimica Acta, vol. 56, 7334-7339, 2011.
[78] S. Kondrat, A. Kornyshev, F. Stoeckli, and T. A. Centeno, "The effect of dielectric permittivity on the capacitance of nanoporous electrodes," Electrochemistry Communications, vol. 34, 348-350, 2013.
[79] S. Senapati and A. Chandra, "Dielectric Constant of Water Confined in a Nanocavity," The Journal of Physical Chemistry B, vol. 105, 5106-5109, 2001.
[80] B. E. Conway, J. O. M. Bockris, and I. A. Ammar, "The dielectric constant of the solution in the diffuse and Helmholtz double layers at a charged interface in aqueous solution," Transactions of the Faraday Society, vol. 47, 756-766, 1951.
[81] L. S. Palmer, A. Cunliffe, and J. M. Hough, "Dielectric Constant of Water Films," Nature, vol. 170, 1952.
[82] D.-e. Jiang, Z. Jin, D. Henderson, and J. Wu, "Solvent Effect on the Pore-Size Dependence of an Organic Electrolyte Supercapacitor," The Journal of Physical Chemistry Letters, vol. 3, 1727-1731, 2012.
[83] D. E. Jiang, Z. Jin, and J. Wu, "Oscillation of capacitance inside nanopores," Nano Lett, vol. 11, 5373-7, 2011.
[84] S. Kondrat, C. R. Perez, V. Presser, Y. Gogotsi, and A. A. Kornyshev, "Effect of pore size and its dispersity on the energy storage in nanoporous supercapacitors," Energy & Environmental Science, vol. 5, 6474-6479, 2012.
[85] Y. Marcus, "A simple empirical model describing the thermodynamics of hydration of ions of widely varying charges, sizes, and shapes," Biophysical Chemistry, vol. 51, 111-127, 1994.
[86] A. V. Marenich, R. M. Olson, C. P. Kelly, C. J. Cramer, and D. G. Truhlar, "Self-Consistent Reaction Field Model for Aqueous and Nonaqueous Solutions Based on Accurate Polarized Partial Charges," Journal of Chemical Theory and Computation, vol. 3, 2011-2033, 2007.
[87] G. Feng, J. Huang, B. G. Sumpter, V. Meunier, and R. Qiao, "Structure and dynamics of electrical double layers in organic electrolytes," Phys Chem Chem Phys, vol. 12, 5468-79, 2010.
[88] A. J. Bard and L. R. Faulkner, Electrochemical Methods Fundamental and Application. Canada: John Wiley & Sons, 1980.
[89] F. Rouquerol, J. Rouquerol, and K. Sing, Adsorption by powders and porous solids London: Academic Press, 1999.
[90] O. Stern, "ZUR THEORIE DER ELEKTROLYTISCHEN DOPPELSCHICHT," Zeitschrift für Elektrochemie und angewandte physikalische Chemie, vol. 30, 508-516, 1924.
[91] C. Vix-Guterl, E. Frackowiak, K. Jurewicz, M. Friebe, J. Parmentier, and F. Béguin, "Electrochemical energy storage in ordered porous carbon materials," Carbon, vol. 43, 1293-1302, 2005.
[92] G. Lota, T. A. Centeno, E. Frackowiak, and F. Stoeckli, "Improvement of the structural and chemical properties of a commercial activated carbon for its application in electrochemical capacitors," Electrochimica Acta, vol. 53, 2210-2216, 2008.
[93] G. Gryglewicz, J. Machnikowski, E. Lorenc-Grabowska, G. Lota, and E. Frackowiak, "Effect of pore size distribution of coal-based activated carbons on double layer capacitance," Electrochimica Acta, vol. 50, 1197-1206, 2005.
[94] K. W. Frese, "Calculation of Gibbs hydration energy with the ion-dielectric sphere model," The Journal of Physical Chemistry, vol. 93, 5911-5916, 1989.
[95] Z.-A. Zhang, Y.-Q. Lai, J. Li, and Y.-X. Liu, "Electrochemical behavior of wound supercapacitors with propylene carbonate and acetonitrile based nonaqueous electrolytes," Journal of Central South University of Technology, vol. 16, 247-252, 2009.