Cluster Approximation for the Farey Fraction Spin Chain

We consider the Farey fraction spin chain in an external field $h$. Utilising ideas from dynamical systems, the free energy of the model is derived by means of an effective cluster energy approximation. This approximation is valid for divergent cluster sizes, and hence appropriate for the discussion of the magnetizing transition. We calculate the phase boundaries and the scaling of the free energy. At $h=0$ we reproduce the rigorously known asymptotic temperature dependence of the free energy. For $h \ne 0$, our results are largely consistent with those found previously using mean field theory and renormalization group arguments.Comment: 17 pages, 3 figure

Chaotic quasi-collision trajectories in the 3-centre problem

We study a particular kind of chaotic dynamics for the planar 3-centre problem on small negative energy level sets. We know that chaotic motions exist, if we make the assumption that one of the centres is far away from the other two (see Bolotin and Negrini, J. Diff. Eq. 190 (2003), 539--558): this result has been obtained by the use of the Poincar\'e-Melnikov theory. Here we change the assumption on the third centre: we do not make any hypothesis on its position, and we obtain a perturbation of the 2-centre problem by assuming its intensity to be very small. Then, for a dense subset of possible positions of the perturbing centre on the real plane, we prove the existence of uniformly hyperbolic invariant sets of periodic and chaotic almost collision orbits by the use of a general result of Bolotin and MacKay (see Cel. Mech. & Dyn. Astr. 77 (2000), 49--75). To apply it, we must preliminarily construct chains of collision arcs in a proper way. We succeed in doing that by the classical regularisation of the 2-centre problem and the use of the periodic orbits of the regularised problem passing through the third centre.Comment: 22 pages, 6 figure

Polymer bonding by induction heating for microfluidic applications

Microfluidic systems are being used in more and more areas and the demand for such systems is growing every day. To meet such high volume market needs, a cheap and rapid method for sealing these microfluidic platforms which is viable for mass manufacture is highly desirable. In this work low frequency induction heating (LFIH) is introduced as the potential basis of a cost-effective, rapid production method for polymer microfluidic device sealing. Thin metal layers or structured metal features are introduced between the device s substrates and heated inductively. The surrounding material melts and forms a bond when cooling. During the bonding process it is important to effectively manage the heat dissipation to prevent distortion of the microfluidic platform. The size of the heat affected zone (HAZ), and the area melted, must be controlled to avoid blockage of the microfluidic channels or altering the channels wall characteristics. The effects of susceptor shape and area, bonding pressure, heating time, etc, on the heating rate have been investigated to provide a basis for process optimisation and design rules. It was found that the maximum temperature is proportional to the square of the susceptor area and that round shaped susceptors heat most efficiently. As a result of the investigations higher bonding pressure was identified as increasing bond strength and allowing the reduction of heating time and thus the reduction of melt zone width. The use of heating pulses instead of continuous heating also reduced the dimensions of melt zones while maintaining good bond strength. The size of the HAZ was found to be negligible. An analytical model, which can be used to predict the heating rate, was derived. In validating the model by numeric models and experiments it was found that it cannot be used to calculate exact temperatures but it does correctly describe the effect of different heating parameters. Over the temperature range needed to bond polymer substrates, cooling effects were found not to have a significant impact on the heating rate. The two susceptor concepts using thin metal layers (metal-plastic bonds) or structured metal features (plastic-plastic bonds) were tested and compared. While the metal-plastic bonds turned out to be too weak to be useful, the bonds formed using structured susceptors showed good strength and high leakage pressure. Based on the knowledge gained during the investigations a microfluidic device was designed. Different samples were manufactured and tested. During the tests minor leaks were observed but it was found that this was mainly due to debris which occurred during laser machining of the channels. It was concluded that induction bonding can be used to seal plastic microfluidic devices. The following guidelines can be drawn up for the design of susceptors and process optimisation: Materials with low resistivity perform better; For very thin susceptors the effect of permeability on the heating rate is negligible; The cross-sectional area of the susceptor should be as large as possible to reduce resistance; The thickness of the susceptor should be of similar dimensions to the penetration depth or smaller to increase homogeneity of heat dissipation; The shape of the susceptor should follow the shape of the inductor coil, or vice-versa, to increase homogeneity of heat dissipation; The susceptor should form a closed circuit; Higher bonding pressure leads to stronger bonds and allows reduced heating times; Pulsed heating performs better than continuous heating in terms of limited melt area and good bond strength. The drawbacks of the technique are explained as well: introducing additional materials leads to additional process steps. Also the structuring and placement of the susceptor was identified to be problematic. In this project the structured susceptor was placed manually but that is not feasible for mass manufacture. To be able to use the technique efficiently a concept of manufacturing the susceptor has to be found to allow precise alignment of complex designs.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Symbolic dynamics for the NN-centre problem at negative energies

We consider the planar $N$-centre problem, with homogeneous potentials of degree -\a<0, \a \in [1,2). We prove the existence of infinitely many collisions-free periodic solutions with negative and small energy, for any distribution of the centres inside a compact set. The proof is based upon topological, variational and geometric arguments. The existence result allows to characterize the associated dynamical system with a symbolic dynamics, where the symbols are the partitions of the $N$ centres in two non-empty sets

The Non-Trapping Degree of Scattering

We consider classical potential scattering. If no orbit is trapped at energy E, the Hamiltonian dynamics defines an integer-valued topological degree. This can be calculated explicitly and be used for symbolic dynamics of multi-obstacle scattering. If the potential is bounded, then in the non-trapping case the boundary of Hill's Region is empty or homeomorphic to a sphere. We consider classical potential scattering. If at energy E no orbit is trapped, the Hamiltonian dynamics defines an integer-valued topological degree deg(E) < 2. This is calculated explicitly for all potentials, and exactly the integers < 2 are shown to occur for suitable potentials. The non-trapping condition is restrictive in the sense that for a bounded potential it is shown to imply that the boundary of Hill's Region in configuration space is either empty or homeomorphic to a sphere. However, in many situations one can decompose a potential into a sum of non-trapping potentials with non-trivial degree and embed symbolic dynamics of multi-obstacle scattering. This comprises a large number of earlier results, obtained by different authors on multi-obstacle scattering.Comment: 25 pages, 1 figure Revised and enlarged version, containing more detailed proofs and remark

Productionisation issues for commercialisation of microfluidic based devices

Purpose Microfluidic or “lab-on-a-chip” technology is seen as a key enabler in the rapidly expanding market for medical point-of-care (POC) and other kinds of portable diagnostic device. In this paper we discuss two proposed packaging processes for large scale manufacture of microfluidic systems. Design/methodology/approach In the first packaging process, polymer overmoulding of a microfluidic chip is used to form a fluidic manifold integrated with the device in a single step. The anticipated advantages of the proposed method of packaging are ease of assembly and low part count. The second process involves the use of low frequency induction heating (LFIH) for the sealing of polymer microfluidics. The method requires no chamber, and provides fast and selective heating to the interface to be joined. Findings Initial work with glass microfluidics has demonstrated feasibility for overmoulding through two separate sealing principles. One is using the overmould as a physical support structure and providing sealing using a compliant ferrule. The other relies on adhesion between the material of the overmould and the microfluidic device to provide a seal. As regards LFIH work on selection and structuring of susceptor materials is reported, together with analysis of the dimensions of the heat affected zone. Acrylic plates have been joined using a thin (<10 μm) nickel susceptor providing a fluid seal that withstood a pressure of 590kPa. Originality/value Microfluidic chips have until now been produced in relatively small numbers. To scale-up from laboratory systems to the production volumes required for mass markets, packaging methods need to be adapted to mass manufacture