Asymptotic solution of a model for bilayer organic diodes and solar cells

The current voltage characteristics of an organic semiconductor diode made by placing together two materials with dissimilar electron affinities and ionisation potentials is analysed using asymptotic methods. An intricate boundary layer structure is examined. We find that there are three regimes for the total current passing through the diode. For reverse bias and moderate forward bias the dependency of the voltage on the current is similar to the behaviour of conventional inorganic semiconductor diodes predicted by the Shockley equation and are governed by recombination at the interface of the materials. There is then a narrow range of currents where the behaviour undergoes a transition. Finally for large forward bias the behaviour is different with the current being linear in voltage and is primarily controlled by drift of charges in the organic layers. The size of the interfacial recombination rate is critical in determining the small range of current where there is rapid transition between the two main regimes. The extension of the theory to organic solar cells is discussed and the analogous current voltage curves derived in the regime of interest

Spin coating of an evaporating polymer solution

We consider a mathematical model of spin coating of a single polymer blended in a solvent. The model describes the one-dimensional development of the thin layer of the mixture as the layer thins due to flow created by a balance of viscous forces and centrifugal forces and due to evaporation of the solvent. In the model both the diffusivity of the solvent in the polymer and the viscosity of the mixture are very rapidly varying functions of the solvent volume fraction. Guided by numerical solutions an asymptotic analysis reveals a number of different possible behaviours of the thinning layer dependent on the nondimensional parameters describing the system.\ud \ud The main practical interest is in controlling the appearance and development of a ``skin'' on the polymer where the solvent concentration reduces rapidly on the outer surface leaving the bulk of the layer still with high concentrations of solvent. The critical parameters controlling this behaviour are found to be $\epsilon$ the ratio of the diffusion to advection time scales, $\delta$ the ratio of the evaporation to advection time scales and $\exp(\gamma)$, the ratio of the diffusivity of the initial mixture and the pure polymer. In particular, our analysis shows that for very small evaporation with $\delta << \exp(-3/(4\gamma)) \epsilon^{3/4}$ skin formation can be prevented

Tear film thickness variations and the role of the tear meniscus

A mathematical model is developed to investigate the two-dimensional variations in the thickness of tear fluid deposited on the eye surface during a blink. Such variations can become greatly enhanced as the tears evaporate during the interblink period.\ud The four mechanisms considered are: i) the deposition of the tear film from the upper eyelid meniscus, ii) the flow of tear fluid from under the eyelid as it is retracted and from the lacrimal gland, iii) the flow of tear fluid around the eye within the meniscus and iv) the drainage of tear fluid into the canaliculi through the inferior and superior puncta.\ud There are two main insights from the modelling. First is that the amount of fluid within the tear meniscus is much greater than previously employed in models and this significantly changes the predicted distribution of tears. Secondly the uniformity of the tear film for a single blink is: i) primarily dictated by the storage in the meniscus, ii) quite sensitive to the speed of the blink and the ratio of the viscosity to the surface tension iii) less sensitive to the precise puncta behaviour, the flow under the eyelids or the specific distribution of fluid along the meniscus at the start of the blink. The modelling briefly examines the flow into the puncta which interact strongly with the meniscus and acts to control the meniscus volume. In addition it considers flow from the lacrimal glands which appears to occurs continue even during the interblink period when the eyelids are stationary

Derivation and solution of effective-medium equations for bulk heterojunction organic solar cells

A drift-diffusion model for charge transport in an organic bulk-heterojunction solar cell, formed by conjoined acceptor and donor materials sandwiched between two electrodes, is formulated. The model accounts for (i) bulk photogeneration of excitons, (ii) exciton drift and recombination, (iii) exciton dissociation (into polarons) on the acceptor-donor interface, (iv) polaron recombination, (v) polaron dissociation into a free electron (in the acceptor) and a hole (in the donor), (vi) electron/hole transport and (vii) electron-hole recombination on the acceptor-donor interface. A finite element method is employed to solve the model in a cell with a highly convoluted acceptor/donor interface. The solutions show that, with physically realistic parameters, and in the power generating regime, the solution varies little on the scale of the microstructure. This motivates us to homogenise over the microstructure; a process that yields a far simpler one-dimensional effective medium model on the cell scale. The comparison between the solution of the full model and the effective medium (homogenised) model is very favourable for the applied voltages that are less than the built-in voltage (the power generating regime) but breaks down as the applied voltages increases above it. Furthermore, it is noted that the homogenisation technique provides a systematic way to relate effective medium modelling of bulk heterojunctions [19, 25, 36, 37, 42, 59] to a more fundamental approach that explicitly models the full microstructure [8, 38, 39, 58] and that it allows the parameters in the effective medium model to be derived in terms of the geometry of the microstructure. Finally, the effective medium model is used to investigate the effects of modifying the microstructure geometry, of a device with an interdigitated acceptor/donor interface, on its current-voltage curve

Nonlinear electrochemical impedance spectroscopy for lithium-ion battery model parameterization

In this work we analyse the local nonlinear electrochemical impedance spectroscopy (NLEIS) response of a lithium-ion battery and estimate model parameters from measured NLEIS data. The analysis assumes a single-particle model including nonlinear diffusion of lithium within the electrode particles and asymmetric charge transfer kinetics at their surface. Based on this model and assuming a moderately-small excitation amplitude, we systematically derive analytical formulae for the impedances up to the second harmonic response, allowing the meaningful interpretation of each contribution in terms of physical processes and nonlinearities in the model. The implications of this for parameterization are explored, including structural identifiability analysis and parameter estimation using maximum likelihood, with both synthetic and experimentally measured impedance data. Accurate fits to impedance data are possible, however inconsistencies in the fitted diffusion timescales suggest that a nonlinear diffusion model may not be appropriate for the cells considered. Model validation is also demonstrated by predicting time-domain voltage response using the parameterized model and this is shown to have excellent agreement with measured voltage time-series data (11.1 mV RMSE).Comment: 40 pages (excluding supplementary material). Submitted to the Journal of the Electrochemical Societ