Numerical annealing models

Two contributions (Bertagnolli et al. 1983, Crowley 1985) in particular established formal mathematical treatments to describe the production and shortening of fission tracks in response to thermal history. The approach by Bertagnolli et al. (1983) did not gain wide acceptance, but remained central to much of the work by the French Besanqon 

Laslett et al. (1987) based their model on a fanning Arrhenius relationship (Fig. 14) between reduced track length r (= l/lo where / is the measured track length and lo is the initial length), log time i), and inverse absolute temperature (7). They derived the following equation for constant temperature annealing that accounted for 98% of the variation in the observed data  

The model attracted considerable criticism and discussion (Crowley 1993a, Carlson 1993a, Green et al. 1993, Carlson 1993b) focusing on two key areas of the model. Firstly, both took issue with the validity of the physical model and its mechanisms. It was argued 

Nevertheless, Carlson s essentially semi-empirical method has gained acceptance, and has been used as the basis for further developments such as the multi-kinetic model of Ketcham et al. (1999). This paper is one of three (Carlson et al. 1999, Donelick et al. 1999, Ketcham et al. 1999) that have addressed many of the earlier criticisms of the Carlson (1990) model. This research group has also produced a substantial annealing data-set of mixed-compositional apatites and established a model to deal with crystallographic effects, both of which have been incorporated into their full annealing model. 

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In numerous applications of polymeric materials multilayers of films are used. This practice is found in microelectronic, aeronautical, and biomedical applications to name a few. Developing good adhesion between these layers requires interdiffusion of the molecules at the interfaces between the layers over size scales comparable to the molecular diameter (tens of nm). In addition, these interfaces are buried within the specimen. Aside from this practical aspect, interdififlision over short distances holds the key for critically evaluating current theories of polymer difllision. Theories of polymer interdiffusion predict specific shapes for the concentration profile of segments across the interface as a function of time. Interdiffiision studies on bilayered specimen comprised of a layer of polystyrene (PS) on a layer of perdeuterated (PS) d-PS, can be used as a model system that will capture the fundamental physics of the problem. Initially, the bilayer will have a sharp interface, which upon annealing will broaden with time. 

Numerical results for the some model polydisperse systems have been reported in Refs. 81-83. It has been shown that the effect of increasing polydispersity on the number-number distribution function is that the structure decreases with increasing polydispersity. This pattern is common for the behavior of two- and three-dimensional polydisperse fluids [81] and also for three-dimensional quenched-annealed systems [83]. 

Abstract In this chapter we review recent advances which have been achieved in the theoretical description and understanding of polyelectrolyte solutions. We will discuss an improved density functional approach to go beyond mean-field theory for the cell model and an integral equation approach to describe stiff and flexible polyelectrolytes in good solvents and compare some of the results to computer simulations. Then we review some recent theoretical and numerical advances in the theory of poor solvent polyelectrolytes. At the end we show how to describe annealed polyelectrolytes in the bulk and discuss their adsorption properties. 

Fig. 19 Main plot SAXS intensity (I) vs momentum transfer for a solution of 51 in acetonitrile (5.1 g L 1). The symbols and the solid line correspond to the experimental data points and the numerical fit using GNOM/DAMMIN simulated annealing, constraining the symmetry to the point group P432 (% = 1.397). Inset reconstructed low resolution particle shape for 51 obtained by the GNOM/DAMMIN fit (semitransparent spheres) superimposed onto the PM3 stationary point (space-filling model, iso-butyl groups substituted by methyl groups)...

Simulation is the modelling of a system with its dynamic processes to gain knowledge, which can be transferred into reality. The most important simulation methods in the field of conformational analysis are molecular dynamics (MD) and Monte Carlo simulations, as well as simulated annealing [56,57], All these approaches are based on extensive numerical calculations and transformations. Therefore, only a brief introduction to these methods and their application to conformational searches is given in the following. 

In order to improve our understanding of the relationships between time, temperature and the observed track parameters in the natural environment, numerous experiments have been conducted over the last twenty-five years. Perhaps the most useful outcome of these annealing experiments has been the development of robust mathematical modeling. 

The adsorption behavior is depicted in Figs. 13 and 14. Our scaling results are in agreement with numerical solutions of discrete lattice models (the multi-Stern layer theory) [61, 62, 111, 117-120]. In Fig. 13, r is plotted as function of / (Fig. 13a) and the pH (Fig. 13b) for different salt concentrations. The behavior as seen in Fig. 13(b) represents annealed PFs where the nominal charge fraction is... 

Equation (44) is an ad-hoc model that is used here for illustrative purposes because it is often useful to think of the glass transition as a combination of a step change in heat capacity with an additional peak that increases in size with increasing enthalpy loss. This is illustrated in Fig. 17. However, it must be stressed that at higher levels of annealing this model cannot be applied. There is no simple analytical expression that can be used and one is forced to use numerical solutions to models such as that given in equation 94. would normally show an Arrhenius dependence on cooling rate ... 

This relationship for the Knudsen regime (low total gas pressure, low sublimation temperature) has been validated by Hottot et al. (2005) who compared with freezedrying experiments conducted with a model BSA formulation with annealing treatment (cf. Fig. 3.19). In the case ofa standard freeze-drying cyde, this successful validation represents a positive result due to numerous assumptions involved in the model and to quite large uncertainties in the corresponding modd parameter values. 

Fig. 20 Determination of the coefficient of bimolecular recombination by performing TDCF experiments with variable delay between the excitation pulse and application of the collection bias, (a) Scheme of the experiment, (b) Experimental TDCF photocurrent transients (open squares) measured on a 200 nm thick layer of slow-dried POFTTiPCBM (1 1) during application of different collection biases Fcoii- The collection bias was applied 150 ns after the laser pulse (t = 0 in this graph). Solid lines show fits to the data using a numerical drift diffusion model with constant electron and hole mobilities. A noteworthy observation is that charges can be fully extracted from these layers within a few hundreds of nanoseconds for a sufficiently high collection bias [171]. (c-f) Q-pre, 2coii> and 2,o, plotted as a function of the delay time for as-prepared and thermally annealed chloroform-cast P3F1T PCBM, and with the pre-bias Fpre set either to 0.55 V (near open circuit) or to 0 V (short-circuit conditions) [172]. Solid lines show fits with an iterative model that considers bimolecular recombination of free charges in competition with their extraction...

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