An optimal error bound for a finite element approximation of a model for phase separation of a multi-component alloy with non-smooth free energy

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Thermodynamic phase-field model for microstructure with multiple components and phases: the possibility of metastable phases

A diffuse-interface model for microstructure with an arbitrary number of components and phases was developed from basic thermodynamic and kinetic principles and formalized within a variational framework. The model includes a composition gradient energy to capture solute trapping, and is therefore suited for studying phenomena where the width of the interface plays an important role. Derivation of the inhomogeneous free energy functional from a Taylor expansion of homogeneous free energy reveals how the interfacial properties of each component and phase may be specified under a mass constraint. A diffusion potential for components was defined away from the dilute solution limit, and a multi-obstacle barrier function was used to constrain phase fractions. The model was used to simulate solidification via nucleation, premelting at phase boundaries and triple junctions, the intrinsic instability of small particles, and solutal melting resulting from differing diffusivities in solid and liquid. The shape of metastable free energy surfaces is found to play an important role in microstructure evolution and may explain why some systems premelt at phase boundaries and phase triple junctions while others do not.Comment: 14 pages, 8 figure

Claw length recommendations for dairy cow foot trimming

The aim was to describe variation in length of the dorsal hoof wall in contact with the dermis for cows on a single farm, and hence, derive minimum appropriate claw lengths for routine foot trimming. The hind feet of 68 Holstein-Friesian dairy cows were collected post mortem, and the internal structures were visualised using x-ray mCT. The internal distance from the proximal limit of the wall horn to the distal tip of the dermis was measured from crosssectional sagittal images. A constant was added to allow for a minimum sole thickness of 5 mm and an average wall thickness of 8 mm. Data were evaluated using descriptive statistics and two-level linear regression models with claw nested within cow. Based on 219 claws, the recommended dorsal wall length from the proximal limit of hoof horn was up to 90 mm for 96 per cent of claws, and the median value was 83 mm. Dorsal wall length increased by 1 mm per year of age, yet 85 per cent of the null model variance remained unexplained. Overtrimming can have severe consequences; the authors propose that the minimum recommended claw length stated in training materials for all Holstein-Friesian cows should be increased to 90 mm

Some Remarks on Functional Analysis

Functional analysis is a key tool in the study of partial differential equations which helps to answer key questions such as existence, well-posedness, and the class in which a solution should belong. We begin these remarks by introducing normed spaces and Banach spaces and then bounded linear operators in normed spaces. Next, we define Hilbert spaces and consider aspects relating to linear operators on Hilbert spaces. With this structure, we are able to consider well-posed of problems and describe the notions of Hölder and Lyapunov stability and Hadamard well-posedness. Finally, we describe how some thermal stress problems can be formulated using abstract operator notation

On the stable discretization of strongly anisotropic phase field models with applications to crystal growth

We introduce unconditionally stable finite element approximations for anisotropic Allen--Cahn and Cahn--Hilliard equations. These equations frequently feature in phase field models that appear in materials science. On introducing the novel fully practical finite element approximations we prove their stability and demonstrate their applicability with some numerical results. We dedicate this article to the memory of our colleague and friend Christof Eck (1968--2011) in recognition of his fundamental contributions to phase field models.Comment: 20 pages, 8 figure

Associations between sole ulcer, white line disease and digital dermatitis and the milk yield of 1824 dairy cows on 30 dairy cow farms in England and Wales from February 2003–November 2004

The milk yields of 1824 cows were used to investigate the effect of lesion-specific causes of lameness, based on farmer treatment and diagnosis of lame cows, on milk yield. A three level hierarchical model of repeated test day yields within cows within herds was used to investigate the impact of lesion-specific causes of lameness (sole ulcer, white line disease, digital dermatitis and other causes) on milk yield before and after treatment compared with unaffected cows. Cattle which developed sole ulcer (SU) and white line disease (WLD) were higher yielding cattle before they were diagnosed. Their milk production fell to below that of the mean of unaffected cows before diagnosis and remained low after diagnosis. In cattle which developed digital dermatitis (DD) there was no significant difference in milk yield before treatment and a slightly raised milk yield immediately after treatment. The estimated milk loss attributable to SU and WLD was approximately 570kg and 370kg respectively. These results highlight that specific types of lameness vary by herds and within herds they are associated with higher yielding cattle. Consequently lesion-specific lameness reduction programmes targeting the cow and farm specific causes of lameness might be more effective than generic recommendations. They also highlight the importance of milk loss when estimating the economic impact of SU and WLD on the farms profitability

Growth and evolution of tetracyanoquinodimethane and potassium coadsorption phases on Ag(111)

Alkali-doping is a very efficient way of tuning the electronic properties of active molecular layers in (opto-) electronic devices based on organic semiconductors. In this context, we report on the phase formation and evolution of charge transfer salts formed by 7, 7, 8, 8-tetracyanoquinodimethane (TCNQ) in coadsorption with potassium on a Ag(111) surface. Based on an in-situ study using low energy electron microscopy and diffraction we identify the structural properties of four phases with different stoichiometries, and follow their growth and inter-phase transitions. We label these four phases α to δ, with increasing K content, the last two of which (γ and δ-phases) have not been previously reported. During TCNQ deposition on a K-precovered Ag(111) surface we find a superior stability of δ-phase islands compared to the γ-phase; continued TCNQ deposition leads to a direct transition from the δ to the β-phase when the K : TCNQ ratio corresponding to this phase regime is reached, with no intermediate γ-phase formation. When, instead, K is deposited on a surface precovered with large islands of the low density commensurate (LDC) TCNQ phase that are surrounded by a TCNQ 2D-gas, we observe two different scenarios: on the one hand, in the 2D-gas phase regions, very small α-phase islands are formed (close to the resolution limit of the microscope, 10–15 nm), which transform to β-phase islands of similar size with increasing K deposition. On the other hand, the large (micrometer-sized) TCNQ islands transform directly to similarly large single-domain β-phase islands, the formation of the intermediate α-phase being suppressed. This frustration of the LDC-to-α transition can be lifted by performing the experiment at elevated temperature. In this sense, the morphology of the pure TCNQ submonolayer is conserved during phase transitions

Precise determination of molecular adsorption geometries by room temperature non-contact atomic force microscopy

High resolution force measurements of molecules on surfaces, in non-contact atomic force microscopy, are often only performed at cryogenic temperatures, due to needing a highly stable system, and a passivated probe tip (typically via CO-functionalisation). Here we show a reliable protocol for acquiring three-dimensional force map data over both single organic molecules and assembled islands of molecules, at room temperature. Isolated cobalt phthalocyanine and islands of C60 are characterised with submolecular resolution, on a passivated silicon substrate (B:Si(111)- ). Geometries of cobalt phthalocyanine are determined to a ~ 10 pm accuracy. For the C60, the protocol is sufficiently robust that areas spanning 10 nm × 10 nm are mapped, despite the difficulties of room temperature operation. These results provide a proof-of-concept for gathering high-resolution three-dimensional force maps of networks of complex, non-planar molecules on surfaces, in conditions more analogous to real-world application