Extended Diffusion Equation Research Articles

Third-order transport coefficient tensor of charged-particle swarms in electric and magnetic fields.

Third-order transport coefficient tensor of charged-particle swarms in neutral gases in the presence of spatially uniform electric and magnetic fields is considered using a multiterm solution of Boltzmann's equation and Monte Carlo simulation technique. The structure of the third-order transport coefficient tensor and symmetries along its individual components in varying configurations of electric and magnetic fields are addressed using a group projector technique and through symmetry considerations of the Boltzmann equation. In addition, we focus upon the physical interpretation of the third-order transport coefficient tensor by considering the extended diffusion equation which incorporates the contribution of the third-order transport coefficients to the density profile of charged particles. Numerical calculations are carried out for electron and ion swarms for a range of model gases with the aim of establishing accurate benchmarks for third-order transport coefficients. The effects of ion to neutral-particle mass ratio are also examined. The errors of the two-term approximation for solving the Boltzmann equation and limitations of previous treatments of the high-order charged-particle transport properties are also highlighted.

Mushy Zone Resolidification in a Temperature Gradient in Multiphase and Multicomponent Alloys

A model for simulating mushy zone resolidification in a temperature gradient is presented. For describing macroscopic mass transport in the liquid phase in the mushy zone, an extended diffusion equation is solved numerically using the Finite Difference Method. Temperature dependent local equilibria at each position in the mushy zone are calculated using the thermodynamic software package ChemApp. The resolidification model treats multicomponent alloying systems and accounts for multiphase equilibria. Simulation results for peritectic Cu-40wt%Al and eutectic Al-5wt%Si-1wt%Mg alloys are compared with microstructures from temperature gradient annealing experiments. It is shown that the model is well suited to predict mushy zone resolidification in multicomponent and multiphase alloys. The predicted evolution of the liquid fraction is qualitatively in full agreement with the observed microstructures, including local remelting at the peritectic temperature prior to resolidification, an effect that was first predicted by the model and confirmed by the experiments.

Microstructural changes and material parameter development

AbstractAt previous GAMM‐meetings the first two authors reported on the spinodal decomposition and coarsening in lead‐containing as well as lead‐free solders. They derived an extended diffusion equation for the corresponding computer modelling on the basis of a multi‐component theory of mixtures. Consequently, they performed numerical studies that are capable of predicting quantitatively the development of SnPb and AgCu microstructures. Knowing this development is quite important from a technological point‐of‐view in order to quantify the current joining capability of the solder material (“Resteigenschaften”). On the basis of these results this paper now attempts to predict the corresponding change of the materials properties, in particular of the elastic constants. This is done by means of a homogenization technique developed by the last two authors. In this paper we will briefly summarize the results related to the extended diffusion equation, present numerical simulations of the coarsening process, outline the theory underlying the homogenization and, finally, show how the material properties behave over time. Wherever possible, alternative analytical solutions for the homogenized constants are also presented (e.g., by laminate theory). (© 2009 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A higher gradient theory for solid mixtures

AbstractDuring use the microstructure of solders will change over time. To enable us to determine the lifetime and reliability of these materials, a theory to predict such changes is needed. In this paper two kinds of changes are considered: The forming of intermetallic compounds due to chemical reactions of two components and spinodal decomposition of a binary alloy followed by coarsening. A theoretical approach is presented and evaluated numerically. For this purpose, an extended diffusion equation is needed and derived within the framework of undisputed concepts of the entropy principle. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Propagators and Time-Dependent Diffusion Coefficients for Anomalous Diffusion

Complex diffusive dynamics are often observed when one is investigating the mobility of macromolecules in living cells and other complex environments, yet the underlying physical or chemical causes of anomalous diffusion are often not fully understood and are thus a topic of ongoing research interest. Theoretical models capturing anomalous dynamics are widely used to analyze mobility data from fluorescence correlation spectroscopy and other experimental measurements, yet there is significant confusion regarding these models because published versions are not entirely consistent and in some cases do not appear to satisfy the diffusion equation. Further confusion is introduced through variations in how fitting parameters are reported. A clear definition of fitting parameters and their physical significance is essential for accurate interpretation of experimental data and comparison of results from different studies acquired under varied experimental conditions. This article aims to clarify the physical meaning of the time-dependent diffusion coefficients associated with commonly used fitting models to facilitate their use for investigating the underlying causes of anomalous diffusion. We discuss a propagator for anomalous diffusion that captures the power law dependence of the mean-square displacement and can be shown to rigorously satisfy the extended diffusion equation provided one correctly defines the time-dependent diffusion coefficient. We also clarify explicitly the relation between the time-dependent diffusion coefficient and fitting parameters in fluorescence correlation spectroscopy.

Path Planning and Obstacle Avoidance Considering Rotary Motion of Load for Overhead Cranes

This paper deals with path planning for autonomous overhead cranes considering load rotation. We present conditions defining interference between rectangular loads and obstacles, and an area in which motion can be performed without interference is determined in configuration space. We then derive a secure path planning algorithm using the potential method based on a 3-dimensional diffusion equation extended to consider rotary motion. A parameter in the extended diffusion equation which suppresses rotary motion is made clear, and its effect is demonstrated. Lastly, we propose a quadratic surface interpolation method for achieving smooth path planning, and its effectiveness is demonstrated through simulation.

Theoretical and experimental investigations of microstructural changes in lead-free solders

Experiments show that the microstructure of solders changes over time. From a materials science point-of-view this phenomenon considerably affects the reliability and lifetime of microelectronic products. It is, therefore, important to quantitatively predict the amount of microstructural change. In the present paper we concentrate on a theoretical and experimental description of spinodal decomposition as well as subsequent phase growth in binary solder alloys. We present experimental results and a theoretical description, which is based on an extended diffusion equation of the phase field type, and which can be interpreted as an generalization of the Cahn–Hilliard equation. Moreover, it takes diffusion of the Fickian type, surface tensions along the phase boundaries as well as local thermo-mechanical stresses into account. In particular we turn our attention to the determination of the required material parameters, which can all be obtained consistently from atomistic models or adopted from literature. As an example the FCC-structured lead-free solder alloy Ag–Cu is considered and numerical results are presented.

Optical cell with periodic resistive heating for the measurement of heat, mass, and thermal diffusions in liquid mixtures

A new technique for the measurement of heat, mass, and thermal diffusions in liquids has been developed. Similar to laser induced dynamic gratings, a temperature grating is created in the sample. Thermal expansion transforms the temperature into a refractive-index grating, which is read by diffraction of a readout laser beam. In a multicomponent mixture an additional concentration grating is formed by thermal diffusion driven by the temperature gradients of the temperature grating. Differently to laser induced dynamic grating experiments we use Joule heating instead of optical heating. For that purpose we have built cuvettes which have a grating of transparent conducting strips on the inner side of one of their windows. If heated by an electric current a temperature grating will build up in the sample. Both the heat equation and the extended diffusion equation have been solved in two dimensions to allow for quantitative data analysis. Our apparatus and method of analysis have been validated by measurements of heat, mass, and thermal diffusions in pure and binary liquids. Heat diffusion can be correctly determined as was shown for pure toluene, pure dodecane, and the symmetric mixture of isobutylbenzene dodecane. Mass and thermal diffusions were studied in the three symmetric mixtures of dodecane, isobutylbenzene, and tetralin. The obtained diffusion and Soret coefficients agree with the literature values within the experimental errors. Uncompensated transient heating effects limit the resolution of the experimental technique.

Generalization of the reaction-diffusion, Swift-Hohenberg, and Kuramoto-Sivashinsky equations and effects of finite propagation speeds

The work proposes and studies a model for one-dimensional spatially extended systems, which involve nonlocal interactions and finite propagation speed. It shows that the general reaction-diffusion equation, the Swift-Hohenberg equation, and the general Kuramoto-Sivashinsky equation represent special cases of the proposed model for limited spatial interaction ranges and for infinite propagation speeds. Moreover, the Swift-Hohenberg equation is derived from a general energy functional. After a detailed validity study on the generalization conditions, the three equations are extended to involve finite propagation speeds. Moreover, linear stability studies of the extended equations reveal critical propagation speeds and unusual types of instabilities in all three equations. In addition, an extended diffusion equation is derived and studied in some detail with respect to finite propagation speeds. The extended model allows for the explanation of recent experimental results on non-Fourier heat conduction in nonhomogeneous material.

Modeling Phase Separation and Coarsening in Ag‐Cu

AbstractThe ongoing miniaturization and environmental trends in microelectronics recently turn the attention to new materials the properties of which still have to be studied. Of fundamental importance for material properties is the temporal development of the microstructure which can experimentally be observed in several solder alloys. Starting with a detailed overview on coarsening phenomena and intermetallic growth as observed in lead‐free solders we concentrate on the description of nucleation and spinodal decomposition in these alloys. An extended diffusion equation of the phase field type is presented in which all required material parameters are determined either from the literature / databases or from calculations based on the Embedded‐Atom Method (EAM). We will restrict to the one dimensional case of diffusion and consider as an example the FCC‐structured lead‐free binary solder alloy Ag‐Cu. Furthermore a numerical study is performed in order to arrive at a quantitative description of temporal micromorphological development. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Path Planning and Obstacle Avoidance Control Considering Rotary Motion of Load for Overhead Crane (1st Report, Path Planning Considering Rotary Motion of Load)

In this paper, the path planning considering rotary motion of load is presented for automous overhead crane. First, we give the condition of interference between a rectangular load and obstacles, and a movable area without interference is decided in the configuration space. Secondly, a secure path planning is derived by the potential method based on extended 3-dimensional diffusion equation considering rotary motion. Furthermore, the parameter of suppressing rotary motion in the extended diffusion equation is clarified, and its effect is demonstrated. Finally, Quadratic curved surface interpolation method is proposed to get the smooth path planning, and the effectiveness of the proposed method is demonstrated through simulation results.

Computer modeling of the coarsening process in tin–lead solders

The temporal progress of phase separation and coarsening observed in eutectic tin/lead (SnPb) solders was investigated using computer modeling. The modeling framework is based on an extended diffusion equation of the phase-field model type that includes classical Fickean diffusion, effects of surface tensions according to the Cahn–Hilliard formalism, as well as diffusive morphology changes due to local thermo-mechanical stresses and strains. The numerical solution of the extended diffusion equation for a eutectic tin/lead microstructure is performed by means of discrete Fourier transforms (DFTs). In particular, a one-dimensional computer modeling procedure will be presented in detail. The phase separation and the coarsening process are discussed, and it is investigated how these depend on different initial conditions and on different temperature levels. The results are in fair agreement with experimental observation. Moreover, the weight and volume fractions of the α- and β-phase, as well as an average “α-particle size” are determined numerically and their temporal development is studied.

A study of the coarsening in tin/lead solders

This paper presents a model, which is capable to simulate the coarsening process observed during thermo-mechanical treatment of binary tin–lead solders. Fourier transforms and spectral theory are used for the numerical treatment of the thermo-elastic as well as of the diffusion problem encountered during phase separation in these alloys. More specifically, the analysis is based exclusively on continuum theory and, first, relies on the numerical computation of the local stresses and strains in a representative volume element. Second, this information is used in an extended diffusion equation to predict the local concentrations of the constituents of the solder. Besides the classical driving forces for phase separation, as introduced by Fick and Cahn–Hilliard, this equation contains an additional term which links the mechanical to the thermodynamical problem. It connects internal and external stresses, strains, temperature, as well as concentrations and allows for a comprehensive study of the coarsening and aging process.

Transport of submicron particles from a leak to a perpendicular surface in a chamber at reduced pressure

At subatmospheric pressure, thermodynamic conditions of semiconductor manufacturing processes change rapidly during pump down or venting operations. Important insights regarding the behavior of particles in real equipment can be obtained using simplified models and idealized experiments under static conditions. Experimental and theoretical studies were performed in order to understand the particle transport into a chamber and the particle deposition behavior on a free-standing wafer at reduced pressure (down to 10 2 Pa). The transport mechanisms taken into account were convection, diffusion and external forces such as sedimentation and thermophoresis. To model these particle transport processes, the analogy between the governing equations of momentum, energy and mass was applied to the extended diffusion equation. In their nondimensional form, the results of the numerical calculation give detailed information about velocity, temperature and particle concentration boundary layer thickness as well as their distributions. In particular, the influence of external forces on the particle concentration field in the vicinity of the surface was investigated. The experimental study consisted of the generation of a monodisperse, fluorescent latex aerosol, the injection of the aerosol through a tiny capillary at the top of a vertical chamber at subatmospheric pressure and the deposition of the particles on a cooled, free-standing wafer in the chambers center. The deposited particles were detected and counted by an optical microscope connected to an image processing unit. This paper presents the results from particle transport and deposition experiments conducted in a chamber while varying the pressure level and the temperature of test surface.

Transport of submicron particles from a leak to a perpendicular surface in a chamber at reduced pressure

At subatmospheric pressure, thermodynamic conditions of semiconductor manufacturing processes change rapidly during pump down or venting operations. Important insights regarding the behavior of particles in real equipment can be obtained using simplified models and idealized experiments under static conditions. Experimental and theoretical studies were performed in order to understand the particle transport into a chamber and the particle deposition behavior on a free-standing wafer at reduced pressure (down to 10 2 Pa). The transport mechanisms taken into account were convection, diffusion and external forces such as sedimentation and thermophoresis. To model these particle transport processes, the analogy between the governing equations of momentum, energy and mass was applied to the extended diffusion equation. In their nondimensional form, the results of the numerical calculation give detailed information about velocity, temperature and particle concentration boundary layer thickness as well as their distributions. In particular, the influence of external forces on the particle concentration field in the vicinity of the surface was investigated. The experimental study consisted of the generation of a monodisperse, fluorescent latex aerosol, the injection of the aerosol through a tiny capillary at the top of a vertical chamber at subatmospheric pressure and the deposition of the particles on a cooled, free-standing wafer in the chambers center. The deposited particles were detected and counted by an optical microscope connected to an image processing unit. This paper presents the results from particle transport and deposition experiments conducted in a chamber while varying the pressure level and the temperature of test surface.

A memory-function approach to the angular correlation functions in anisotropic liquids

A memory-function approach based on the Mori theory is used to test the validity of the extended diffusion equation commonly adopted to calculate angular correlation functions in partially oriented liquids. It is shown that the two methods give the same results, under the assumption of no correlation between angular variables and angular velocities, as far as the long-range terms of the intermolecular potential responsible for the orientational order are significantly smaller than the short-range terms which determine the magnitude of the friction tensor in the isotropic phase. This implies that the extended diffusion equation can give a reasonably accurate description of the rotational motion of the molecules in liquid crystals even at moderately high values of the orientational torque.