Graphene oxide has become an attractive electrode-material candidate for supercapacitors thanks to its higher specific capacitance compared to graphene. The quantum capacitance makes relative contributions to the specific capacitance, which is considered as the major limitation of graphene electrodes, while the quantum capacitance of graphene oxide is rarely concerned. This study explores the quantum capacitance of graphene oxide, which bears epoxy and hydroxyl groups on its basal plane, by employing density functional theory (DFT) calculations. The results demonstrate that the total density of states near the Fermi level is significantly enhanced by introducing oxygen-containing groups, which is beneficial for the improvement of the quantum capacitance. Moreover, the quantum capacitances of the graphene oxide with different concentrations of these two oxygen-containing groups are compared, revealing that more epoxy and hydroxyl groups result in a higher quantum capacitance. Notably, the hydroxyl concentration has a considerable effect on the capacitive behavior.
This paper will devote to construct a family of fifth degree cubature formulae for n-cube with symmetric measure and n-dimensional spherically symmetrical region. The formula for n-cube contains at most n 2 + 5n + 3 points and for n-dimensional spherically symmetrical region contains only n 2 + 3n + 3 points. Moreover, the numbers can be reduced to n 2 + 3n + 1 and n 2 + n + 1 if n = 7 respectively, the latter of which is minimal.
We introduce a piecewise P2-nonconforming quadrilateral finite element. First, we decompose a convex quadrilateral into the union of four triangles divided by its diagonals. Then the finite element space is defined by the set of all piecewise P2-polynomials that are quadratic in each triangle and continuously differentiable on the quadrilateral. The degrees of freedom (DOFs) are defined by the eight values at the two Gauss points on each of the four edges plus the value at the intersection of the diagonals. Due to the existence of one linear relation among the above DOFs, it turns out the DOFs are eight. Global basis functions are defined in three types: vertex-wise, edge-wise, and element-wise types. The corresponding dimensions are counted for both Dirichlet and Neumann types of elliptic problems. For second-order elliptic problems and the Stokes problem, the local and global interpolation operators are defined. Also error estimates of optimal order are given in both broken energy and L 2 (Ω) norms. The proposed element is also suitable to solve Stokes equations. The element is applied to approximate each component of velocity fields while the discontinuous P1-nonconforming quadrilateral element is adopted to approximate the pressure. An optimal error estimate in energy norm is derived. Numerical results are shown to confirm the optimality of the presented piecewise P2-nonconforming element on quadrilaterals.Mathematics Subject Classification. 65N30, 76M10.
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