A fast and robust numerical scheme for solving models of charge carrier transport and ion vacancy motion in perovskite solar cells

Drift-diffusion models that account for the motion of both electronic and ionic charges are important tools for explaining the hysteretic behaviour and guiding the development of metal halide perovskite solar cells. Furnishing numerical solutions to such models for realistic operating conditions is challenging owing to the extreme values of some of the parameters. In particular, those characterising (i) the short Debye lengths (giving rise to rapid changes in the solutions across narrow layers), (ii) the relatively large potential differences across devices and (iii) the disparity in timescales between the motion of the electronic and ionic species give rise to significant stiffness. We present a finite difference scheme with an adaptive time step that is posed on a non-uniform staggered grid that provides second order accuracy in the mesh spacing. The method is able to cope with the stiffness of the system for realistic parameters values whilst providing high accuracy and maintaining modest computational costs. For example, a transient sweep of a current-voltage curve can be computed in only a few minutes on a standard desktop computer.Comment: 22 pages, 8 figure

Systematic derivation of a surface polarization model for planar perovskite solar cells

Increasing evidence suggests that the presence of mobile ions in perovskite solar cells can cause a current-voltage curve hysteresis. Steady state and transient current-voltage characteristics of a planar metal halide CH$_3$NH$_3$PbI$_3$ perovskite solar cell are analysed with a drift-diffusion model that accounts for both charge transport and ion vacancy motion. The high ion vacancy density within the perovskite layer gives rise to narrow Debye layers (typical width $\sim$2nm), adjacent to the interfaces with the transport layers, over which large drops in the electric potential occur and in which significant charge is stored. Large disparities between (I) the width of the Debye layers and that of the perovskite layer ($\sim$600nm) and (II) the ion vacancy density and the charge carrier densities motivate an asymptotic approach to solving the model, while the stiffness of the equations renders standard solution methods unreliable. We derive a simplified surface polarisation model in which the slow ion dynamic are replaced by interfacial (nonlinear) capacitances at the perovskite interfaces. Favourable comparison is made between the results of the asymptotic approach and numerical solutions for a realistic cell over a wide range of operating conditions of practical interest.Comment: 32 pages, 7 figure

Volume exclusion effects in perovskite charge transport modeling

Due to their flexible material properties, perovskite materials are a promising candidate for many semiconductor devices such as lasers, memristors, LEDs and solar cells. For example, perovskite-based solar cells have recently become one of the fastest growing photovoltaic technologies. Unfortunately, perovskite devices are far from commercialization due to challenges such as fast degradation. Mathematical models can be used as tools to explain the behavior of such devices, for example drift-diffusion equations portray the ionic and electric motion in perovskites. In this work, we take volume exclusion effects on ion migration within a perovskite crystal lattice into account. This results in the formulation of two different ionic current densities for such a drift-diffusion model -- treating either the mobility or the diffusivity as density-dependent while the other quantity remains constant. The influence of incorporating each current density description into a model for a typical perovskite solar cell configuration is investigated numerically, through simulations performed using two different open source tools. {

Constrained optimal control of monotone systems with applications to battery fast-charging

Enabling fast charging for lithium ion batteries is critical to accelerating the green energy transition. As such, there has been significant interest in tailored fast-charging protocols computed from the solutions of constrained optimal control problems. Here, we derive necessity conditions for a fast charging protocol based upon monotone control systems theory

IonMonger: a free and fast planar perovskite solar cell simulator with coupled ion vacancy and charge carrier dynamics

Details of an open-source planar perovskite solar cell simulator, that includes ion vacancy migration within the perovskite layer coupled to charge carrier transport throughout the perovskite and adjoining transport layers in one dimension, are presented. The model equations are discretised in space using a finite element scheme and temporal integration of the resulting system of differential-algebraic equations is carried out in MATLAB. The user is free to modify device parameters, as well as the incident illumination and applied voltage. Time-varying voltage and/or illumination protocols can be specified, e.g. to simulate current-voltage sweeps, or to track the open-circuit conditions as the illumination is varied. Typical simulations, e.g. current-voltage sweeps, only require computation times of seconds to minutes on a modern personal computer. An example set of hysteretic current-voltage curves is presented

Water transfer and crack regimes in nano-colloidal gels

International audienceDirect observations of the surface and shape of model nano-colloidal gels associated with measurements of the spatial distribution of water content during drying show that air starts to significantly penetrate the sample when the material stops shrinking. We show that whether the material fractures or not during desiccation, as air penetrates the porous body, the water saturation decreases but remains almost homogeneous throughout the sample. This air-invasion is at the origin of another type of fracture due to capillary effects; these results provide a new insight in the liquid dynamics at the nano-scale. PACS number(s): 47.56.+r, 68.03.Fg, 81.40.N

Intensification of single cell storms prior to lightning onset

Single cell storms in the UK can produce lightning, despite apparently only having developed to towering cumulus rather than cumulonimbus. Such marginal thunderstorms still present severe weather hazards but are difficult to identify and predict and therefore provide a warning. Observations from the Met Office radar mosaic and ATDNet (Arrival Time Difference Network) show that these single cell storms demonstrate a characteristic increase in the area of high reflectivity storm core during the 15 minutes prior to the first lightning. By using the Met Office Unified Model to investigate reflectivity development in modelled storms, a microphysical explanation for the observed reflectivity increase is identified. During a rapid reflectivity increase, the updraft area at the melting layer, the peak updraft velocity and the storm graupel mass increase. The three quantities examined are linked to each other and to the generation of charge within the storm. The production of graupel is promoted by the increase in updraft area and charge separation is enhanced by the faster peak updraft velocity. This explains some of the physical differences between single cell storms that produce lightning and apparently similar storm systems which do not. It also provides a new basis with which to predict lightning hazard for marginal storms

A Review of Lithium-Ion Battery Electrode Drying: Mechanisms and Metrology

Lithium-ion battery manufacturing chain is extremely complex with many controllable parameters especially for the drying process. These processes affect the porous structure and properties of these electrode films and influence the final cell performance properties. However, there is limited available drying information and the dynamics are poorly understood due to the limitation of the existing metrology. There is an emerging need to develop new methodologies to understand the drying dynamics to achieve improved quality control of the electrode coatings. A comprehensive summary of the parameters and variables relevant to the wet electrode film drying process is presented, and its consequences/effects on the finished electrode/final cell properties are mapped. The development of the drying mechanism is critically discussed according to existing modeling studies. Then, the existing and potential metrology techniques, either in situ or ex situ in the drying process are reviewed. This work is intended to develop new perspectives on the application of advanced techniques to enable a more predictive approach to identify optimum lithium-ion battery manufacturing conditions, with a focus upon the critical drying process

A reduced-order strategy for 4D-Var data assimilation

This paper presents a reduced-order approach for four-dimensional variational data assimilation, based on a prior EO F analysis of a model trajectory. This method implies two main advantages: a natural model-based definition of a mul tivariate background error covariance matrix $\textbf{B}_r$, and an important decrease of the computational burden o f the method, due to the drastic reduction of the dimension of the control space. % An illustration of the feasibility and the effectiveness of this method is given in the academic framework of twin experiments for a model of the equatorial Pacific ocean. It is shown that the multivariate aspect of $\textbf{B}_r$ brings additional information which substantially improves the identification procedure. Moreover the computational cost can be decreased by one order of magnitude with regard to the full-space 4D-Var method