The Young's modulus (E) of a material is a key parameter for mechanical engineering design. Silicon, the most common single material used in microelectromechanical systems (MEMS), is an anisotropic crystalline material whose material properties depend on orientation relative to the crystal lattice. This fact means that the correct value of E for analyzing two different designs in silicon may differ by up to 45%. However, perhaps, because of the perceived complexity of the subject, many researchers oversimplify silicon elastic behavior and use inaccurate values for design and analysis. This paper presents the best known elasticity data for silicon, both in depth and in a summary form, so that it may be readily accessible to MEMS designers.

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What is the Young's Modulus of Silicon?

Introduction to multi-layer feed-forward neural networks

Wave propagation and group velocity

Carbon dioxide is often promoted as a sustainable solvent, as CO 2 is non-flammable, exhibits a relatively low toxicity and is naturally abundant. However, injudicious use of carbon dioxide in a process or product can reduce rather than enhance overall sustainability. This review specifically examines the use of CO 2 to create greener processes and products, with a focus on research and commercialization efforts performed since 1995. The literature reveals that use of CO 2 has permeated almost all facets of the chemical industry and that careful application of CO 2 technology can result in products (and processes) that are cleaner, less expensive and of higher quality.

Supercritical and near-critical CO2 in green chemical synthesis and processing

In this article, the effects of size and confinement at the nanometre size scale on both the melting temperature, Tm, and the glass transition temperature, Tg, are reviewed. Although there is an accepted thermodynamic model (the Gibbs–Thomson equation) for explaining the shift in the first-order transition, Tm, for confined materials, the depression of the melting point is still not fully understood and clearly requires further investigation. However, the main thrust of the work is a review of the field of confinement and size effects on the glass transition temperature. We present in detail the dynamic, thermodynamic and pseudo-thermodynamic measurements reported for the glass transition in confined geometries for both small molecules confined in nanopores and for ultrathin polymer films. We survey the observations that show that the glass transition temperature decreases, increases, remains the same or even disappears depending upon details of the experimental (or molecular simulation) conditions. Indeed, different behaviours have been observed for the same material depending on the experimental methods used. It seems that the existing theories of Tg are unable to explain the range of behaviours seen at the nanometre size scale, in part because the glass transition phenomenon itself is not fully understood. Importantly, here we conclude that the vast majority of the experiments have been carried out carefully and the results are reproducible. What is currently lacking appears to be an overall view, which accounts for the range of observations. The field seems to be experimentally and empirically driven rather than responding to major theoretical developments.

https://iopscience.iop.org/article/10.1088/0953-8984/17/15/R01/pdf

Effects of confinement on material behaviour at the nanometre size scale

A new technique is developed to determine the dispersion relation and the propagational speeds of waves in dispersive solids. The phase spectrum of a broadband pulse is linearly related to the dispersion relation of the dispersive medium. The method is simpler than the continuous‐wave phase comparison technique. Application is made to measure the phase and group velocities of waves in fiber‐reinforced composite materials and in thin wires. This technique is expected to be applicable to measurements of acoustic or electromagnetic wave speeds in other dispersive media.

On the determination of phase and group velocities of dispersive waves in solids

Effects of high‐pressure CO2 on the glass transition temperature and mechanical properties of polystyrene

A method is proposed for determining the elastic constants of an anisotropic solid from acoustic-wave group velocities measured in off-symmetry directions in a specimen. The method corresponds to a two-stage optimization procedure in which the elastic constants are varied so as to obtain a least-squares fit of measured to calculated group velocities. At each iterative step a numerical minimization process is employed to find the wave normals and velocities of waves having the required ray directions. The method is demonstrated with synthetic data generated for a number of cubic crystals, and to experimental data obtained on single crystals of silicon using the ultrasonic point-source/enpoint-receiver technique. When provided with longitudinal- and transverse-mode velocity data, the method is able to accurately recover all three independent elastic constants. When only longitudinal mode data are provided, ${\mathit{C}}_{11}$ and the combination ${\mathit{C}}_{12}$+2${\mathit{C}}_{44}$ can be accurately recovered, but not ${\mathit{C}}_{12}$ and ${\mathit{C}}_{44}$ individually.

Determination of the elastic constants of anisotropic solids from acoustic-wave group-velocity measurements.

1. Introduction.- 1.1 Goal.- 1.2 Relation to Other Scientific Fields.- 1.3 Plan of the Monograph.- 2. A Quantitative Description of Nature.- 2.1 Synergetics of Natural Phenomena.- 2.2 A Description of Nature.- 2.3 Fundamentals of Quantitative Description.- 2.4 Fundamentals of Physical Laws.- 2.5 The Random Character of Physical Variables.- 2.6 Expression of Natural Laws by Differential Equations.- 2.7 Methods of Empirical Modeling.- 2.7.1 The Role of Models.- 2.7.2 Piecewise Linear Models of Empirical Natural Laws.- 2.8 Introduction to Modeling by Neural Networks.- 2.8.1 Functional Properties of a Neuron.- 2.8.2 Empirical Modeling by a Perceptron.- 3. Transducers.- 3.1 The Role of Sensors and Actuators.- 3.2 Sensors and Actuators of Biological Systems.- 3.2.1 Performance Characteristics of Biological Sensors.- 3.2.2 Structure of Biological Sensors.- 3.2.3 Transduction Characteristics of Biological Sensors.- 3.3 Operational Characteristics of Transducers.- 3.3.1 Transducer Classification.- 3.3.2 Transduction Characteristics.- 3.3.3 Sensor Loading Effects.- 3.3.4 Transducer Field Characteristics.- 3.4 Fabricated Transducers.- 3.4.1 Microsensors and Integrated Sensors.- 3.4.2 Synthetic Bio-sensors and Neurobiology.- 3.5 Transducers in Intelligent Measurement Systems.- 3.6 Future Directions in Transducer Evolution.- 4. Probability Densities.- 4.1 Estimation of Probability Density.- 4.1.1 Parzen Window Approach.- 4.1.2 An Optimal Selection of the Window Function.- 4.1.3 Nearest Neighbor and Maximal Self-Consistency Approach.- 4.1.4 The Self-Consistent Method in the Multivariate Case..- 4.1.5 Numerical Examples.- 4.1.6 Conclusions About Filtering of the Empirical PDF.- 5. Information.- 5.1 Some Basic Ideas.- 5.2 Entropy of Information.- 5.3 Properties of Information Entropy.- 5.4 Relative Information.- 5.4.1 Information of Continuous Distributions.- 5.4.2 Information Gain from Experiments.- 5.5 Information Measure of Distance Between Distributions.- 6. Maximum Entropy Principles.- 6.1 Gibbs Maximum Entropy Principle.- 6.2 The Absolute Maximum Entropy Principle.- 6.3 Quantization of Continuous Probability Distributions.- 6.3.1 Quadratic Measure of Discrepancy Between Distributions.- 6.3.2 Information Divergence as a Measure of Discrepancy.- 6.3.3 Vector Quantization and Reconstruction Measure of Discrepancy.- 7. Adaptive Modeling of Natural Laws.- 7.1 Probabilistic Modeler of Natural Laws.- 7.2 Optimization of Adaptive Modeler Performance.- 7.3 Stochastic Approach to Adaptation Laws.- 7.4 Stochastic Adaptation of a Vector Quantizer.- 7.5 Perturbation Method of Adaptation.- 7.6 Evolution of an Optimal Modeler and Perturbation Method.- 7.7 Parametric Versus Non-Parametric Modeling.- 8. Self-Organization and Formal Neurons.- 8.1 Optimal Storage of Empirical Information in Discrete Systems.- 8.2 Adaptive Vector Quantization and Topological Mappings.- 8.3 Self-Organization Based on the Absolute Maximum-Entropy Principle.- 8.4 Derivation of a Generalized Self-Organization Rule.- 8.5 Numerical Examples of Self-Organized Adaptation.- 8.6 Formal Neurons and the Self-Organization Process.- 9. Modeling by Non-Parametric Regression.- 9.1 The Problem of an Optimal Prediction.- 9.2 Parzen'1. Introduction.- 1.1 Goal.- 1.2 Relation to Other Scientific Fields.- 1.3 Plan of the Monograph.- 2. A Quantitative Description of Nature.- 2.1 Synergetics of Natural Phenomena.- 2.2 A Description of Nature.- 2.3 Fundamentals of Quantitative Description.- 2.4 Fundamentals of Physical Laws.- 2.5 The Random Character of Physical Variables.- 2.6 Expression of Natural Laws by Differential Equations.- 2.7 Methods of Empirical Modeling.- 2.7.1 The Role of Models.- 2.7.2 Piecewise Linear Models of Empirical Natural Laws.- 2.8 Introduction to Modeling by Neural Networks.- 2.8.1 Functional Properties of a Neuron.- 2.8.2 Empirical Modeling by a Perceptron.- 3. Transducers.- 3.1 The Role of Sensors and Actuators.- 3.2 Sensors and Actuators of Biological Systems.- 3.2.1 Performance Characteristics of Biological Sensors.- 3.2.2 Structure of Biological Sensors.- 3.2.3 Transduction Characteristics of Biological Sensors.- 3.3 Operational Characteristics of Transducers.- 3.3.1 Transducer Classification.- 3.3.2 Transduction Characteristics.- 3.3.3 Sensor Loading Effects.- 3.3.4 Transducer Field Characteristics.- 3.4 Fabricated Transducers.- 3.4.1 Microsensors and Integrated Sensors.- 3.4.2 Synthetic Bio-sensors and Neurobiology.- 3.5 Transducers in Intelligent Measurement Systems.- 3.6 Future Directions in Transducer Evolution.- 4. Probability Densities.- 4.1 Estimation of Probability Density.- 4.1.1 Parzen Window Approach.- 4.1.2 An Optimal Selection of the Window Function.- 4.1.3 Nearest Neighbor and Maximal Self-Consistency Approach.- 4.1.4 The Self-Consistent Method in the Multivariate Case..- 4.1.5 Numerical Examples.- 4.1.6 Conclusions About Filtering of the Empirical PDF.- 5. Information.- 5.1 Some Basic Ideas.- 5.2 Entropy of Information.- 5.3 Properties of Information Entropy.- 5.4 Relative Information.- 5.4.1 Information of Continuous Distributions.- 5.4.2 Information Gain from Experiments.- 5.5 Information Measure of Distance Between Distributions.- 6. Maximum Entropy Principles.- 6.1 Gibbs Maximum Entropy Principle.- 6.2 The Absolute Maximum Entropy Principle.- 6.3 Quantization of Continuous Probability Distributions.- 6.3.1 Quadratic Measure of Discrepancy Between Distributions.- 6.3.2 Information Divergence as a Measure of Discrepancy.- 6.3.3 Vector Quantization and Reconstruction Measure of Discrepancy.- 7. Adaptive Modeling of Natural Laws.- 7.1 Probabilistic Modeler of Natural Laws.- 7.2 Optimization of Adaptive Modeler Performance.- 7.3 Stochastic Approach to Adaptation Laws.- 7.4 Stochastic Adaptation of a Vector Quantizer.- 7.5 Perturbation Method of Adaptation.- 7.6 Evolution of an Optimal Modeler and Perturbation Method.- 7.7 Parametric Versus Non-Parametric Modeling.- 8. Self-Organization and Formal Neurons.- 8.1 Optimal Storage of Empirical Information in Discrete Systems.- 8.2 Adaptive Vector Quantization and Topological Mappings.- 8.3 Self-Organization Based on the Absolute Maximum-Entropy Principle.- 8.4 Derivation of a Generalized Self-Organization Rule.- 8.5 Numerical Examples of Self-Organized Adaptation.- 8.6 Formal Neurons and the Self-Organization Process.- 9. Modeling by Non-Parametric Regression.- 9.1 The Problem of an Optimal Prediction.- 9.2 Parzen's Window Approach to General Regression.- 9.3 General Regression Modeler, Feedback and Recognition.- 9.4 Application of the General Regression Modeler.- 9.4.1 Empirical Modeling of Acoustic Phenomena.- 9.4.2 Prediction of the Seismic Capacity of Walls.- 9.4.3 Modeling of a Periodontal Disease Healing Process.- 10. Linear Modeling and Invariances.- 10.1 Relation Between Parametric Modeling and Invariances.- 10.2 Generalized Linear Regression Model.- 10.2.1 An Example of Iterative Determination of a Linear Regression Model.- 10.3 Sequential Adaptation of Linear Regression Model.- 10.4 Transition from the Cross- to Auto-Associator.- 10.4.1 Application of the Auto-Associator to Analysis of Ultrasonic Signals.- 11. Modeling and Forecasting of Chaotic Processes.- 11.1 Modeling of Chaotic Processes.- 11.2 Examples of Chaotic Process Forecasting.- 11.3 Forecasting of Chaotic Acoustic Emission Signals.- 11.4 Empirical Modeling of Non-Autonomous Chaotic Systems.- 11.4.1 Example of Economic Time-Series Forecasting.- 11.5 Cascade Modeling of Chaos Generators.- 11.5.1 Numerical Experiments.- 11.5.2 Concluding Remarks.- 12. Modeling by Neural Networks.- 12.1 From Biological to Artificial Neural Networks.- 12.1.1 Basic Blocks of Neural Networks and Their Dynamics.- 12.2 A Linear Associator.- 12.3 Multi-layer Perceptrons and Back-Propagation Learning.- 12.4 Radial Basis Function Neural Networks.- 12.5 Equivalence of a Radial Basis Function NN and Perceptrons.- 13. Fundamentals of Intelligent Control.- 13.1 Introduction.- 13.2 Basic Tasks of Intelligent Control.- 13.2.1 Empirical Description of a Controlled System.- 13.2.2 General Identification by Non-Parametric Modeling.- 13.3 The Tracking Problem.- 13.4 Cloning.- 13.5 An Empirical Approach to Optimal Control.- 13.5.1 The Theoretical Problem of Optimal Control.- 13.5.2 Experimental Description of Plant Performance and Optimal Control.- 13.5.3 Design of an Intelligent Optimal Controller.- 13.5.4 The Influence of the Environment on Optimal Control.- 13.5.5 The Problem of Phase Space Exploration.- 13.5.6 Numerical Simulations of Optimal Control.- 13.5.7 Summary and Conclusions.- 14. Self-Control and Biological Evolution.- 14.1 Modeling of Natural Phenomena by Biological Systems.- 14.2 Joint Modeling of Organism and Environment.- 14.3 An Operational Description of Consciousness.- 14.4 The Fundamental Problem of Evolution.- A. Fundamentals of Probability and Statistics.- A.1 Sample Points, Sample Space, Events and Relations.- A.2 Probability.- A.3 Random Variables and Probability Distributions.- A.4 Averages and Moments.- A.5 Random Processes.- A.6 Sampling, Estimation and Statistics.- B. Fundamentals of Deterministic Chaos.- B.1 Instability of Chaotic Systems.- B.2 Characterization of Strange Attractors.- B.3 Experimental Characterization of Chaotic Phenomena.- References.

Synergetics of Measurement, Prediction and Control

A method is described to optimally determine the elastic constants of anisotropic solids from wave‐speeds measurements in arbitrary nonprincipal planes. For such a problem, the characteristic equation is a degree‐three polynomial which generally does not factorize. By developing and rearranging this polynomial, a nonlinear system of equations is obtained. The elastic constants are then recovered by minimizing a functional derived from this overdetermined system of equations. Calculations of the functional are given for two specific cases, i.e., the orthorhombic and the hexagonal symmetries. Some numerical results showing the efficiency of the algorithm are presented. A numerical method is also described for the recovery of the orientation of the principal acoustical axes. This problem is solved through a double‐iterative numerical scheme. Numerical as well as experimental results are presented for a unidirectional composite material.

Wolfgang Sachse

Papers

On the determination of phase and group velocities of dispersive waves in solids

Effects of high‐pressure CO2 on the glass transition temperature and mechanical properties of polystyrene

Determination of the elastic constants of anisotropic solids from acoustic-wave group-velocity measurements.

Synergetics of Measurement, Prediction and Control

Optimal determination of the elastic constants of composite materials from ultrasonic wave‐speed measurements