Integral Equations with Hypersingular Kernels -- Theory and Applications to Fracture Mechanics

Hypersingular integrals of the type I_{\alpha}(T_n,m,r) = \int_{-1}^{1} \hpsngAbs \frac{T_n(s)(1-s^2)^{m-{1/2}}}{(s-r)^\alpha}ds |r|<1 and I_{\alpha}(U_n,m,r) = \int_{-1}^{1} \hpsngAbs \frac{U_n(s)(1-s^2)^{m-{1/2}}}{(s-r)^\alpha}ds |r|<1 are investigated for general integers $\alpha$ (positive) and $m$ (non-negative), where $T_n(s)$ and $U_n(s)$ are the Tchebyshev polynomials of the 1st and 2nd kinds, respectively. Exact formulas are derived for the cases $\alpha = 1, 2, 3, 4$ and $m = 0, 1, 2, 3$; most of them corresponding to new solutions derived in this paper. Moreover, a systematic approach for evaluating these integrals when $\alpha > 4$ and $m>3$ is provided. The integrals are also evaluated as $|r|>1$ in order to calculate stress intensity factors (SIFs). Examples involving crack problems are given and discussed with emphasis on the linkage between mathematics and mechanics of fracture. The examples include classical linear elastic fracture mechanics (LEFM), functionally graded materials (FGM), and gradient elasticity theory. An appendix, with closed form solutions for a broad class of integrals, supplements the paper

Exploring the Way to Approach the Efficiency Limit of Perovskite Solar Cells by Drift-Diffusion Model

Drift-diffusion model is an indispensable modeling tool to understand the carrier dynamics (transport, recombination, and collection) and simulate practical-efficiency of solar cells (SCs) through taking into account various carrier recombination losses existing in multilayered device structures. Exploring the way to predict and approach the SC efficiency limit by using the drift-diffusion model will enable us to gain more physical insights and design guidelines for emerging photovoltaics, particularly perovskite solar cells. Our work finds out that two procedures are the prerequisites for predicting and approaching the SC efficiency limit. Firstly, the intrinsic radiative recombination needs to be corrected after adopting optical designs which will significantly affect the open-circuit voltage at its Shockley-Queisser limit. Through considering a detailed balance between emission and absorption of semiconductor materials at the thermal equilibrium, and the Boltzmann statistics at the non-equilibrium, we offer a different approach to derive the accurate expression of intrinsic radiative recombination with the optical corrections for semiconductor materials. The new expression captures light trapping of the absorbed photons and angular restriction of the emitted photons simultaneously, which are ignored in the traditional Roosbroeck-Shockley expression. Secondly, the contact characteristics of the electrodes need to be carefully engineered to eliminate the charge accumulation and surface recombination at the electrodes. The selective contact or blocking layer incorporated nonselective contact that inhibits the surface recombination at the electrode is another important prerequisite. With the two procedures, the accurate prediction of efficiency limit and precise evaluation of efficiency degradation for perovskite solar cells are attainable by the drift-diffusion model.Comment: 32 pages, 11 figure

Oxygen Hydration Mechanism for the Oxygen Reduction Reaction at Pt and Pd Fuel Cell Catalysts

We report the reaction pathways and barriers for the oxygen reduction reaction (ORR) on platinum, both for gas phase and in solution, based on quantum mechanics calculations (PBE-DFT) on semi-infinite slabs. We find a new mechanism in solution: O_2 → 2O_(ad) (E_(act) = 0.00 eV), O_(ad) + H_2O_(ad) → 2OH_(ad) (E_(act) = 0.50 eV), OH_(ad) + H_(ad) → H_2O_(ad) (E_(act) = 0.24 eV), in which OH_(ad) is formed by the hydration of surface O_(ad). For the gas phase (hydrophilic phase of Nafion), we find that the favored step for activation of the O_2 is H_(ad) + O_(2ad) → HOO_(ad) (E_(act) = 0.30 eV) → HO_(ad) + O_(ad) (E_(act) = 0.12 eV) followed by O_(ad) + H_2O_(ad) → 2OH_(ad) (E_(act) = 0.23 eV), OH_(ad) + H_(ad) → H_2O_(ad) (E_(act) = 0.14 eV). This suggests that to improve the efficiency of ORR catalysts, we should focus on decreasing the barrier for Oad hydration while providing hydrophobic conditions for the OH and H_2O formation steps

Theoretical Study of Solvent Effects on the Platinum-Catalyzed Oxygen Reduction Reaction

We report here density functional theory (DFT) studies (PBE) of the reaction intermediates and barriers involved in the oxygen reduction reaction (ORR) on a platinum fuel cell catalyst. Solvent effects were taken into account by applying continuum Poisson−Boltzmann theory to the bound adsorbates and to the transition states of the various reactions on the platinum (111) surface. Our calculations show that the solvent effects change significantly the reaction barriers compared with those in the gas-phase environment (without solvation). The O_2 dissociation barrier decreases from 0.58 to 0.27 eV, whereas the H + O → OH formation barrier increases from 0.73 to 1.09 eV. In the water-solvated phase, OH formation becomes the rate-determining step for both ORR mechanisms, O_2 dissociation and OOH association, proposed earlier for the gas-phase environment. Both mechanisms become significantly less favorable for the platinum catalytic surface in water solvent, suggesting that alternative mechanisms must be considered to describe properly the ORR on the platinum surface