Linear Time Law Research Articles

Heat Conduction at a Variable Heat-Transfer Coefficient

Some practically important problems of unsteady heat conduction with a time-variable relative heat-transfer coefficient are considered. Various approaches to finding a solution to the analytical problem are systematized: decomposition of the generalized integral Fourier transform, the serial expansion of a sought temperature function, and the reduction of the problem to an integral, second-order Volterra equation. It is demonstrated that the solution is reduced to an infinite series of successive approximations of different functional form in all cases, and the main objective of each of these approaches is to find the most advantageous first approximation. Particular cases of the time dependence of the relative heat-transfer coefficients are considered: linear, exponential, power, and root cases. Analytical solutions and numerical experiments are described, and some specific features of the temperature curves of a number of mentioned dependences are revealed. It is established that the picture of the change in the temperature curve for the linear time law of the heat-transfer coefficient becomes appreciably different from the classic case of a constant coefficient, whereas the exponential dependence does not introduce any essential changes.

Investigation of milling processes of semiconductor zinc oxide nanostructured powders by X-ray phase analysis

The processes of mechanical activation of nanostructured zinc oxide powders are investigated by X-ray phase analysis. It was determined, that samples remain in a single phase state during the milling process. The particle size decreases according to the linear time law, and microstrains grow parabolically.

Direct evidence for continuous linear kinetics in the low-temperature degradation of Y-TZP

The kinetics of the tetragonal to monoclinic (t–m) transformation of zirconia in a hydrous environment at 134°C and 3bar pressure was studied. As surface X-ray diffraction, which is conventionally used to explore the progress, has a very limited depth of information, it distorts the quantitative results in a layer-on-layer situation and by itself is ill suited for this reason. Analyzing cross sections is more suitable; therefore, focused ion beam techniques were used to prepare artifact-free cuts. The material was subsequently investigated by scanning electron microscopy, electron backscatter diffraction and Raman spectroscopy. Only the combination of methods makes it possible to resolve the quantifiable details of the process. The transformation starts in the near-surface areas, forms a layer, and the growth of this layer proceeds into the bulk material following a simple linear time law (0.0624μmh−1 for material in the chosen condition), without apparent retardation or limit. The progress yields a gradientless layer with a fixed amount of residual tetragonal zirconia (∼27% for 3Y-TZP in the present conditions) separated from unaffected material by a boundary, which has a roughness only in the grain size range. The kinetics indicates a reaction rate control, where the hydration reaction is the key factor, but is modified by the stepwise access of water to the reaction front opened by the autocatalytic transformation of zirconia with a critical hydration level.

Modeling of shear induced coarsening effects in semi-solid alloys

In long-term rheological shear experiments with semi-solid alloys, coarsening of the particles will falsify the interpretation of the experimental results. The coarsening is intensified by the shear induced convection and the mean size of the particles is changed significantly during the experiments. A simple model has been set up which takes the influence of the convection into account. The resulting growth law has been simplified for diffusion and convection dominated growth. The growth law was verified with shear experiments in a Searl-rheometer with A356 and tin-lead alloys. The experiments demonstrated that under convection the growth follows a linear time law and that the rate constant depends on the root of the shear rate. The correction of experimental results to gain the true viscosity function is demonstrated for a shear jump experiment with A356.

Time‐dependent electrochemical characterization of the corrosion of a magnesium rare‐earth alloy in simulated body fluids

The electrochemistry of the corrosion process of a magnesium rare-earth-alloy is studied in detail in simulated body fluid (m-SBF) over the first 5 days. The aim is to investigate the corrosion mechanism under in vitro conditions. For this purpose we also used electrolytes that contain only some of the components of SBF, they were compared to SBF to investigate the influence of the different ions in SBF. The influence of albumin on the corrosion process was studied with a solution containing m-SBF and albumin in physiological concentration. For this study, impedance spectroscopy series measurements were performed. Additional results were gained from polarization curves. We conclude from the study that the corrosion resistance is significantly lower in m-SBF than in simple isotonic NaCl-solution. Albumin may form a blocking layer on the surface in the first hours of exposure. The formed corrosion layers consisting of amorphous apatite have only a low protective ability. Further results show that the corrosion processes in SBFs follow a linear time-law. The results elucidate critical factors and mechanisms of the electrochemical corrosion process of magnesium rare-earth alloys in SBFs, this understanding is crucial for a successful application of Mg alloys in biomedical applications.

Optical characterization of thin thermal oxide films on copper by ellipsometry

A complete optical characterization in the visible region of thin copper oxide films has been performed by ellipsometry. Copper oxide films of various thicknesses were grown on thick copper films by low temperature thermal oxidation at 125 °C in air for different time intervals. The thickness and optical constants of the copper oxide films were determined in the visible region by ellipsometric measurements. It was found that a linear time law is valid for the oxide growth in air at 125 °C.

The Oxidation Kinetics of Thin Copper Films Studied by Resistivity Measurements

Abstract Resistivity measurements on thin metal films allow to study the kinetics of oxidation. The method is applied to 50 - 60 nm thick copper films deposited on glass substrates under UHV conditions. After annealing at 150°C, the films are exposed to pure oxygen at various temperatures in the range 85 -135°C, and the electrical resistivity is recorded in situ. At these temperatures, the oxygen begins to penetrate into the interior of the films, which results in a relatively steep increase in the film resistivity. A linear time law is valid to good approximation, which can be attributed to the influence of the dissociation of an adsorbed molecular species of oxygen on the reaction velocity. A potential diffusion of oxygen in the grain boundaries is also discussed.

Grain growth kinetics of Si 3N 4 during α/ß-transformation

The growth of ß-Si 3N 4 particles, highly diluted in an oxynitride glass, was analysed by SEM and the growth features of individual crystals were discussed. The observed linear time law for the length growth during α/ß-transformation was correlated with results of a theoreticl analysis taking into account the diffusion in the liquid phase as well as the interface reaction at α- and ß-crystals. The number of initial nuclei was found to be important for a linear time law of growth. Finally, the considerations of the dilute particle system were applied to a Si 3N 4 ceramic. The opposing influence of growth anisotropy of ßSi 3N 4 in the presence of a melt and the packing requirements demanding a reduction of the specific surface area is discussed.

The oxidation kinetics of thin copper films studied by ellipsometry

Thin copper films of about 50 nm thickness were deposited on glass substrates under ultrahigh vacuum conditions and exposed to oxygen at 105°C and various oxygen pressures (0.1–12 Pa). The kinetics of the oxidation process was studied in situ by ellipsometry ( λ = 1152 nm). A layer growth of the Cu 2O oxide phase was observed. The quantitative evaluation led to the conclusion that a linear time law is valid. The important role of adsorbed molecular oxygen species for the reaction velocity became obvious from the pressure dependence of the rate constant.

Amorphization of Ni/Ti multilayers studied by electrical conductance measurements

Ni/Ti multilayers with composition-modulation wavelengths ranging from 35 to 103 Å were annealed at 458 K during approximately 30 h and studied by electrical resistivity measurements and transmission electron microscopy (TEM). The diminution of the electrical conductance during the solid state reaction can be explained by the formation of an amorphous phase at every Ni/Ti interface. The variation of the conductance follows a shifted t 1 2 law for long times. We deduced values of interdiffusion coefficients which are in good agreement with those measured by other methods. A linear time law for short times is expected for an interface limited reaction.

Kinetics of the solid-state reaction for the formation of amorphous ZrCo studied by electrical conductance measurements

Thin-film reactions of Co with Zr have been studied in the temperature range between 473 and 523 K by electrical conductance measurements and cross-sectional transmission electron microscopy (CS-TEM). The reduction of the electrical conductance during the solid state reaction is explained by formation and growth of an amorphous phase at every Zr/Co interface. For long reaction times the growth of the layer thickness follows a shifted $\sqrt t$ law. For short reaction times the measurements show a linear time law, which is expected for an interface limited reaction.

The Effect of Phase Formation on Glass Leaching

The leaching of nuclear waste glasses in aqueous solution is still considered to be the most important process when discussing a potential release of radionuclides. However, in the last few years it has become clear that the release mechanism under static leaching conditions is very complex involving several processes [1]. The kinetics gave no evidence for pure leaching (i.e. the alkali-ion exchange) which should result in a square root time law, or for pure corrosion (i.e. congruent dissolution of the glass) which should result in a linear time law. Moreover, fits using a combination of these two processes gave unsatisfactory results. This is certainly due to additional processes which are not considered by ion exchange and congruent dissolution. Some explanations may be the build up of alteration layers, saturation effects of some elements with phase formation due to solubility limits and disappearance of metastable phases. To describe such a complicated system thermodynamic data are needed and measurements of mass losses as well as analyses of leachates and surfaces layers have to be performed.

The kinetics of the reduction of chromium oxide by hydrogen

The reduction kinetics has been studied as function of hydrogen partial pressures,pH2O/pH2 ratio, gas flow rate, and temperature. The reduction follows a linear time law and is dependent on the gas flow rate below a value of approximately 10 cm · s-1, since the rate is determined by the removal of the water vapor being formed. In this range the reduction rate may be calculated from gas dynamical data. At sufficiently high flow rates the phase boundary reaction is rate determining. The activation energy is 123 kJ · mol-1. The reduction rate is proportional to the square root of hydrogen pressure and decreases with increasing water vapor content.

CURVED PATH METHODS APPLIED TO VERTICALS AND TO WIDE SHOT SPREADS

The ray path‐wave front chart and similar methods used for vertical shots are briefly described, and the possibility and the advantages of applying the linear time law to these methods is pointed out. A short method of fitting the linear depth law to the observed velocity is described. The possibility of extending any method applicable to verticals to wide shot spreads is shown, and a method of correcting straight path dips for curvature in the case of verticals is pointed out.