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

The BLM’s Duty to Incorporate Climate Science into Permitting Practices and a Proposal for Implementing a Net Zero Requirement into Oil and Gas Permitting

Almost one quarter of all U.S. carbon dioxide (CO2) emissions come from fossil fuels extracted from public lands, and these resources are managed by the Bureau of Land Management (BLM). This article argues that the BLM has a statutory duty to respond to climate change, which includes the duty to avoid exacerbating climate change. The article then moves the legal discussion from aspiration to action by proposing a legal strategy, using the existing legal framework, by which the BLM can achieve net zero emissions from all new mineral development activity. While the article focuses on oil and gas development, the same methodology could be applied to coal mining, tar sands, and other sources of GHG emissions

Evolution of an elliptical bubble in an accelerating extensional flow

Mathematical models that describe the dynamical behavior of a thin gas bubble embedded in a glass fiber during a fiber drawing process have been discussed and analyzed. The starting point for the mathematical modeling was the equations presented in [1] for a glass fiber with a hole undergoing extensional flow. These equations were reconsidered here with the additional reduction that the hole, i.e. the gas bubble, was thin as compared to the radius of the fiber and of finite extent. The primary model considered was one in which the mass of the gas inside the bubble was fixed. This fixed-mass model involved equations for the axial velocity and fiber radius, and equations for the radius of the bubble and the gas pressure inside the bubble. The model equations assumed that the temperature of the furnace of the drawing tower was known. The governing equations of the bubble are hyperbolic and predict that the bubble cannot extend beyond the limiting characteristics specified by the ends of the initial bubble shape. An analysis of pinch-off was performed, and it was found that pinch-off can occur, depending on the parameters of the model, due to surface tension when the bubble radius is small. In order to determine the evolution of a bubble, a numerical method of solution was presented. The method was used to study the evolution of two different initial bubble shapes, one convex and the other non-convex. Both initial bubble shapes had fore-aft symmetry, and it was found that the bubbles stretched and elongated severely during the drawing process. For the convex shape, fore-aft symmetry was lost in the middle of the drawing process, but the symmetry was re-gained by the end of the drawing tower. A small amount of pinch-off was observed at each end for this case, so that the final bubble length was slightly shorter than its theoretical maximum length. For the non-convex initial shape, pinch-off occurred in the middle of the bubble resulting in two bubbles by the end of the fiber draw. The two bubbles had different final pressures and did not have fore-aft symmetry. An extension of the fixed-mass model was considered in which the gas in the bubble was allowed to diffuse into the surrounding glass. The governing equations for this leaky-mass model were developed and manipulated into a form suitable for a numerical treatment

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

Homogenization of the Equations Governing the Flow Between a Slider and a Rough Spinning Disk

We have analyzed the behavior of the flow between a slider bearing and a hard-drive magnetic disk under two types of surface roughness. For both cases the length scale of the roughness along the surface is small as compared to the scale of the slider, so that a homogenization of the governing equations was performed. For the case of longitudinal roughness, we derived a one-dimensional lubrication-type equation for the leading behavior of the pressure in the direction parallel to the velocity of the disk. The coefficients of the equation are determined by solving linear elliptic equations on a domain bounded by the gap height in the vertical direction and the period of the roughness in the span-wise direction. For the case of transverse roughness the unsteady lubrication equations were reduced, following a multiple scale homogenization analysis, to a steady equation for the leading behavior of the pressure in the gap. The reduced equation involves certain averages of the gap height, but retains the same form of the usual steady, compressible lubrication equations. Numerical calculations were performed for both cases, and the solution for the case of transverse roughness was shown be in excellent agreement with a corresponding numerical calculation of the original unsteady equations