243 research outputs found
The Stefan problem with variable thermophysical properties and phase change temperature
In this paper we formulate a Stefan problem appropriate when the
thermophysical properties are distinct in each phase and the phase-change
temperature is size or velocity dependent. Thermophysical properties invariably
take different values in different material phases but this is often ignored
for mathematical simplicity. Size and velocity dependent phase change
temperatures are often found at very short length scales, such as nanoparticle
melting or dendrite formation; velocity dependence occurs in the solidification
of supercooled melts. To illustrate the method we show how the governing
equations may be applied to a standard one-dimensional problem and also the
melting of a spherically symmetric nanoparticle. Errors which have propagated
through the literature are highlighted. By writing the system in
non-dimensional form we are able to study the large Stefan number formulation
and an energy-conserving one-phase reduction. The results from the various
simplifications and assumptions are compared with those from a finite
difference numerical scheme. Finally, we briefly discuss the failure of
Fourier's law at very small length and time-scales and provide an alternative
formulation which takes into account the finite time of travel of heat carriers
(phonons) and the mean free distance between collisions.Comment: 39 pages, 5 figure
The one-dimensional Stefan problem with non-Fourier heat conduction
We investigate the one-dimensional growth of a solid into a liquid bath,
starting from a small crystal, using the Guyer-Krumhansl and Maxwell-Cattaneo
models of heat conduction. By breaking the solidification process into the
relevant time regimes we are able to reduce the problem to a system of two
coupled ordinary differential equations describing the evolution of the
solid-liquid interface and the heat flux. The reduced formulation is in good
agreement with numerical simulations. In the case of silicon, differences
between classical and non-classical solidification kinetics are relatively
small, but larger deviations can be observed in the evolution in time of the
heat flux through the growing solid. From this study we conclude that the heat
flux provides more information about the presence of non-classical modes of
heat transport during phase-change processes.Comment: 29 pages, 6 figures, 2 tables + Supplementary Materia
Optimal loading of hydrogel-based drug-delivery systems
Drug-loaded hydrogels provide a means to deliver pharmaceutical agents to
specific sites within the body at a controlled rate. The aim of this paper is
to understand how controlled drug release can be achieved by tuning the initial
distribution of drug molecules in a hydrogel. A mathematical model is presented
for a spherical drug-loaded hydrogel. The model captures the nonlinear
elasticity of the polymer network and thermodynamics of swelling. By assuming
that the drug molecules are dilute, the equations for hydrogel swelling and
drug transport partially decouple. A fast optimisation method is developed to
accurately compute the optimal initial drug concentration by minimising the
error between the numerical drug-release profile and a target profile. By
taking the target drug efflux to be piecewise constant, the optimal initial
configuration consists of a central drug-loaded core with isolated drug packets
near the free boundary of the hydrogel. The optimal initial drug concentration
is highly effective at mitigating the burst effect, where a large amount of
drug is rapidly released into the environment. The hydrogel stiffness can be
used to further tune the rate of drug release. Although stiffer gels lead to
less swelling and hence reduce the drug diffusivity, the drug-release kinetics
are faster than for soft gels due to the decreased distance that drug molecules
must travel to reach the free surface
Security Policies as Membranes in Systems for Global Computing
We propose a simple global computing framework, whose main concern is code migration. Systems are structured in sites, and each site is divided into two parts: a computing body, and a membrane which regulates the interactions between the computing body and the external environment. More precisely, membranes are filters which control access to the associated site, and they also rely on the well-established notion of trust between sites. We develop a basic theory to express and enforce security policies via membranes. Initially, these only control the actions incoming agents intend to perform locally. We then adapt the basic theory to encompass more sophisticated policies, where the number of actions an agent wants to perform, and also their order, are considered
Curvature controls beading in soft coated elastic cylinders:Finite wavemode instability and localized modulations
Axisymmetric beading instabilities in soft, elongated cylinders have been observed in a plethora of scenarios, ranging from cellular nanotunnels and nerves in biology to swollen cylinders and electrospun fibers in polymer physics. One of the common geometrical features that can be seen in these systems is the finite wavelength of the emerging pattern. However, modelling studies often predict that the instability has an infinite wavelength, which can be associated with localized necking or bulging. In this paper, we consider a soft elastic cylinder with a thin coating that resists bending, as described by the Helfrich free energy functional. The bending stiffness and natural mean curvature of the coating are two novel features whose competition against bulk elasticity and capillarity is investigated. For intermediate values of the bending stiffness, a linear stability analysis reveals that the mismatch between the current and natural mean curvature of the coating can lead to patterns emerging with a finite wavelength. This analysis creates a continuous bridge between the classical solutions of the shape equation obtained from the Helfrich functional and a curvature-controlled zero-wavemode instability, similar to the one induced by the competition between bulk elasticity and capillarity. A weakly non-linear analysis predicts that the criticality of the bifurcation depends on the controlling parameter, with both supercritical and subcritical bifurcations possible. When capillarity is introduced, the criticality of the bifurcation changes in a non-trivial way
Mathematical Modelling of Tyndall Star Initiation
The superheating that usually occurs when a solid is melted by volumetric
heating can produce irregular solid-liquid interfaces. Such interfaces can be
visualised in ice, where they are sometimes known as Tyndall stars. This paper
describes some of the experimental observations of Tyndall stars and a
mathematical model for the early stages of their evolution. The modelling is
complicated by the strong crystalline anisotropy, which results in an
anisotropic kinetic undercooling at the interface; it leads to an interesting
class of free boundary problems that treat the melt region as infinitesimally
thin
Controlled topological transitions in thin film phase separation
In this paper the evolution of a binary mixture in a thin-film geometry with
a wall at the top and bottom is considered. By bringing the mixture into its
miscibility gap so that no spinodal decomposition occurs in the bulk, a slight
energetic bias of the walls towards each one of the constituents ensures the
nucleation of thin boundary layers that grow until the constituents have moved
into one of the two layers. These layers are separated by an interfacial region
where the composition changes rapidly. Conditions that ensure the separation
into two layers with a thin interfacial region are investigated based on a
phase-field model. Using matched asymptotic expansions a corresponding
sharp-interface problem for the location of the interface is established.
It is then argued that this newly created two-layer system is not at its
energetic minimum but destabilizes into a controlled self-replicating pattern
of trapezoidal vertical stripes by minimizing the interfacial energy between
the phases while conserving their area. A quantitative analysis of this
mechanism is carried out via a thin-film model for the free interfaces, which
is derived asymptotically from the sharp-interface model.Comment: Submitted 23/12/201
Surface induced phase separation of a swelling hydrogel
We present a formulation of the free boundary problem for a hydrogel that accounts for the interfacial free energy and finite strain due to the large deformation of the polymer network during solvent transport across the free boundary. For the geometry of an initially dry layer fixed at a rigid substrate, our model predicts a phase transition when a critical value of the solvent concentration has been reached near the free boundary. A one-dimensional case study shows that depending on the flux rate at the free boundary an initial saturation front is followed by spinodal decomposition of the hydrogel and the formation of an interfacial front that moves through the layer. Moreover, increasing the shear modulus of the elastic network delays or even suppresses phase separation
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Surface induced phase separation of a swelling hydrogel
We present a formulation of the free boundary problem for a hydrogel
that accounts for the interfacial free energy and finite strain due to the
large deformation of the polymer network during solvent transport across the
free boundary. For the geometry of an initially dry layer fixed at a rigid
substrate, our model predicts a phase transition when a critical value of the
solvent concentration has been reached near the free boundary. A
one-dimensional case study shows that depending on the flux rate at the free
boundary an initial saturation front is followed by spinodal decomposition of
the hydrogel and the formation of an interfacial front that moves through the
layer. Moreover, increasing the shear modulus of the elastic network delays
or even suppresses phase separation
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