A. A. Lacey

The origin and interpretation of the signals of MTDSC

Abstract In Modulated-Temperature Differential Scanning Calorimetry (MTDSC) a conventional heating programme is modulated cyclically. The heat-flow signal is split into an underlying and an approximately periodic part. We summarise the present state of the method with regard to chemical reactions and melting. We also give a more extensive treatment of the glass transition.

A mathematical model for Modulated Differential Scanning Calorimetry

Reading and co-workers introduced a new technique a few years ago called Modulated Differential Scanning Calorimetry or MDSC. Here the first part of a theoretical analysis for this technique is given. A simple mathematical model for modulated differential scanning calorimetry in the form of an ordinary differential equation is derived. The model is analysed to find the effect of a kinetic event in the form of a chemical reaction. Some possible sources of error are discussed. A more sophisticated version of the model allowing for spatial variation in a calorimeter is developed and it is seen how it can be reduced to the earlier model. Some preliminary work on a phase change is also presented.

Total Blow-Up versus Single Point Blow-Up

Traditional thermal explosion theory is used to describe reaction initiation in condensed explosives and is limited formally to nondeformable materials. Kassoy and Poland [4] significantly extended this theory to develop an ignition model for a reactive gas in a bounded container in order to describe the induction period. During this induction period there is a spatially homogeneous pressure rise in the system which causes a com- pressive heating effect in the constant volume container. Mathematically this compressibility of the gas is expressed by means of an integral term in the induction model for the temperature perturbation 0(x, t). This model is given by e,-Ak&“+::-1.&j e,(x, t)dx Y x-2 0(x, 0) = d(x) 2 0,

Multidimensional reaction diffusion equations with nonlinear boundary conditions

We consider diffusion models in which distributed nonlinear absorption mechanisms compete with nonlinear boundary sources. Assuming that the nonlinearities are weak, formal asymptotic approximation...

A hyperbolic non-local problem modelling MEMS technology

In this work we study a non-local hyperbolic equation of the form utt = uxx + λ 1 (1− u)2 ( 1 + α ∫ 1 0 1 1−udx )2 , with homogeneous Dirichlet boundary conditions and appropriate initial conditions. The problem models an idealised electrostatically actuated MEMS (Micro-Electro-Mechanical System) device. Initially we present the derivation of the model. Then we prove local existence of solutions for λ > 0 and global existence for 0 λ+ for some constant λ+ ≥ λ−, and with zero initial conditions, it is proved that the solution of the problem quenches in finite time; again similar results are obtained for other initial data. Finally the problem is solved numerically with a finite difference scheme. Various simulations of the solution of the problem are presented, illustrating the relevant theoretical results.

Diffusion models with blow-up

A number of physical situations, including chemical reactions, electrical heating, and fluid flow, give rise to nonlinear diffusion problems. In this paper models are derived and results relating to the blow-up of the solutions are given. Some of the proofs are outlined and the physical significance of the results is noted.

A Mathematical Model for Flash Sintering

A mathematical model is presented for the Joule heating that occurs in a ceramic powder compact during the process of flash sintering. The ceramic is assumed to have an electrical conductivity that increases with temperature, and this leads to the possibility of runaway heating that could facilitate and explain the rapid sintering seen in experiments. We consider reduced models that are sufficiently simple to enable concrete conclusions to be drawn about the mathematical nature of their solutions. In particular we discuss how different local and non-local reaction terms, which arise from specified experimental conditions of fixed voltage and current, lead to thermal runaway or to stable conditions. We identify incipient thermal runaway as a necessary condition for the flash event, and hence identify the conditions under which this is likely to occur.

Macroscopic models for melting derived from averaging microscopic Stefan problems II: Effect of varying geometry and composition

A mushy region is assumed to consist of a fine mixture of two distinct phases separated by free boundaries. A method of multiple scales, with restrictions on the form of the microscopic free boundaries, is used to derive a macroscopic model for the mushy region. The final model depends both on the microscopic structure and on how the free-boundary temperature varies with curvature (Gibbs–Thomson effect), kinetic undercooling, or, for an alloy, composition.

Bounds on solutions of one-phase Stefan problems

The reformulation of one-phase Stefan problems, by use of a Baiocchi-type transformation, as ‘oxygen-diffusion’ problems, makes it possible to compare different solutions. The comparison extends to ‘zero-specific-heat’ cases, which are better known, in two dimensions, as Hele-Shaw problems. Known solutions of Hele-Shaw problems can be used to bound and estimate asymptotic behaviour of solutions to Stefan problems. The use of similar techniques gives rise to some exact solutions of ‘squeeze-film’ problems and some limited results concerning continuity.

Motion by curvature in generalized Cahn-Allen models

AbstractThe Cahn-Allen model for the motion of phase-antiphase boundaries is generalized to account for nonlinearities in the kinetic coefficient (relaxation velocity) and the coefficient of the gradient free energy. The resulting equation is