Curvature effects in carbon nanomaterials: Exohedral versus endohedral supercapacitors

Abstract

Capacitive energy storage mechanisms in nanoporous carbon supercapacitors hinge on endohedral interactions in carbon materials with macro-, meso-, and micropores that have negative surface curvature. In this article, we show that because of the positive curvature found in zero-dimensional carbon onions or one-dimensional carbon nanotube arrays, exohedral interactions cause the normalized capacitance to increase with decreasing particle size or tube diameter, in sharp contrast to the behavior of nanoporous carbon materials. This finding is in good agreement with the trend of recent experimental data. Our analysis suggests that electrical energy storage can be improved by exploiting the highly curved surfaces of carbon nanotube arrays with diameters on the order of 1 nm.

Not just graphene: The wonderful world of carbon and related nanomaterials

“…For spherical (onions) or tubular (nanotubes) particles, there is an increase in capacity with decreasing size. 30 Micro-and mesoporous carbons exhibit opposite dependencies of capacitance on pore size.…”

Modern Theories of Carbon‐Based Electrochemical Capacitors

“…Supercapacitors will only compete with existing battery technology if the energy density can be increased while preserving high-power capability. Aside from extensive experimental work, the amount of theoretical effort devoted to the fundamental understanding of EDLCs has also steadily increased [26][27][28][29][30][31][32]. Theoretical approaches range from the early Helmholtz model, to mean-field continuum models, and to modern molecular dynamics (MD) simulations.…”

“…EDLCs are classified into two categories depending on the way that the counterions interact with the carbon surface [28,29]. The first category includes endohedral capacitors where counterions enter inside the pores to establish EDLs ( Figure 5.3a).…”

“…An alternative description is that counterions can only reside on the outer surfaces of carbon particles, exploiting the positive surface curvature. These carbon particles can be approximated as being spherical or cylindrical, as shown in Figure 5.11 [28]. Solvated counterions approach the surface of an electrically charged carbon onion particle to form an exohedral electric double-sphere capacitor (xEDSC).…”

Generalization of the Gouy-Chapman-Stern model of an electric double layer for a morphologically complex electrode: Deterministic and stochastic morphologies

“…These experimental results show three regimes in the capacitance vs pore size data [1,5,14]: (i) shows a nonlinear increase in capacitance, (ii) shows the transition from micropores to mesopores capacitance with a minimum, (iii) shows the anomalous increase and a maximum in capacitance of pores below 1 nm. Huang and coworker [16,17,18] proposed a heuristic model for the capacitance of such a problem with an assumption of cylindrical pores. They included the effect of finite pore sizes and proposed three models for three different pore size regimes: (i) micropore regime (< 2nm)-the electric wirein-cylinder capacitors (EWCCs) model, [17] (ii) mesopores regime ( 2-50 nm)-the electric double cylinder capacitors (EDCCs) model [17] and for pores in macropore regime (> 50 nm) where curvature is not longer significant, the parallel plate capacitor model was proposed.…”