Soft Matter Theory
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Analysis of Deformation
Chapter 7: Constitutive Equations Definition: In the previous chapters we’ve learned about the definition and meaning of the concepts of stress and strain. One is an objective measure of load and the other is an objective measure of deformation. In fluids, one talks about the rate-of-deformation as opposed to simply strain (i.e. deformation alone or by itself). We all know though that deformation is caused by loads (i.e. there must be a relationship between stress and strain). A relationship between stress and strain (or rate-of-deformation tensor) is simply called a “constitutive equation”. Below we will describe how such equations are formulated. Constitutive equations between stress and strain are normally written based on phenomenological (i.e. experimental) observations and some assumption(s) on the physical behavior or response of a material to loading. Such equations can and should always be tested against experimental observations. Although there is almost an infinite amount of different materials, leading one to conclude that there is an equivalently infinite amount of constitutive equations or relations that describe such materials behavior, it turns out that there are really three major equations that cover the behavior of a wide range of materials of applied interest. One equation describes stress and small strain in solids and called “Hooke’s law”. The other two equations describe the behavior of fluidic materials. Hookean Elastic Solid: We will start explaining these equations by considering Hooke’s law first. Hooke’s law simply states that the stress tensor is assumed to be linearly related to the strain tensor. -
On Soft Matter
FASCINATING Research Soft MATTER Hard Work on Soft Matter What do silk blouses, diskettes, and the membranes of living cells have in common? All three are made barked on an unparalleled triumphal between the ordered structure of building blocks.“ And Prof. Helmuth march. Even information, which to- solids and the highly disordered gas. Möhwald, director of the “Interfaces” from soft matter: if individual molecules are observed, they appear disordered, however, on a larger, day frequently tends to be described These are typically supramolecular department at the MPIKG adds, “To as one of the most important raw structures and colloids in liquid me- supramolecular scale, they form ordered structures. Disorder and order interact, thereby affecting the put it in rather simplified terms, soft materials, can only be generated, dia. Particularly complex examples matter is unusual because its struc- properties of the soft material. At the MAX PLANCK INSTITUTES FOR POLYMER RESEARCH stored, and disseminated on a mas- occur in the area of biomaterials,” ture is determined by several weaker sive scale via artificial soft materials. says Prof. Reinhard Lipowsky, direc- forces. For this reason, its properties in Mainz and of COLLOIDS AND INTERFACES in Golm, scientists are focussing on this “soft matter”. Without light-sensitive coatings, tor of the “Theory” Division of the depend very much upon environ- there would be no microchip - and Max Plank Institute of Colloids and mental and production conditions.” nor would there be diskettes, CD- Interfaces in Golm near Potsdam The answers given by the three ROMs, and videotapes, which are all (MPIKG). Prof. Hans Wolfgang scientists are typical in that their made from coated plastics. -
Machine Learning for Condensed Matter Physics
Review Article Machine Learning for Condensed Matter Physics Edwin A. Bedolla-Montiel1, Luis Carlos Padierna1 and Ram´on Casta~neda-Priego1 1 Divisi´onde Ciencias e Ingenier´ıas,Universidad de Guanajuato, Loma del Bosque 103, 37150 Le´on,Mexico E-mail: [email protected] Abstract. Condensed Matter Physics (CMP) seeks to understand the microscopic interactions of matter at the quantum and atomistic levels, and describes how these interactions result in both mesoscopic and macroscopic properties. CMP overlaps with many other important branches of science, such as Chemistry, Materials Science, Statistical Physics, and High-Performance Computing. With the advancements in modern Machine Learning (ML) technology, a keen interest in applying these algorithms to further CMP research has created a compelling new area of research at the intersection of both fields. In this review, we aim to explore the main areas within CMP, which have successfully applied ML techniques to further research, such as the description and use of ML schemes for potential energy surfaces, the characterization of topological phases of matter in lattice systems, the prediction of phase transitions in off-lattice and atomistic simulations, the interpretation of ML theories with physics- inspired frameworks and the enhancement of simulation methods with ML algorithms. We also discuss in detial the main challenges and drawbacks of using ML methods on CMP problems, as well as some perspectives for future developments. Keywords: machine learning, condensed matter physics Submitted -
Circular Birefringence in Crystal Optics
Circular birefringence in crystal optics a) R J Potton Joule Physics Laboratory, School of Computing, Science and Engineering, Materials and Physics Research Centre, University of Salford, Greater Manchester M5 4WT, UK. Abstract In crystal optics the special status of the rest frame of the crystal means that space- time symmetry is less restrictive of electrodynamic phenomena than it is of static electromagnetic effects. A relativistic justification for this claim is provided and its consequences for the analysis of optical activity are explored. The discrete space-time symmetries P and T that lead to classification of static property tensors of crystals as polar or axial, time-invariant (-i) or time-change (-c) are shown to be connected by orientation considerations. The connection finds expression in the dynamic phenomenon of gyrotropy in certain, symmetry determined, crystal classes. In particular, the degeneracies of forward and backward waves in optically active crystals arise from the covariance of the wave equation under space-time reversal. a) Electronic mail: [email protected] 1 1. Introduction To account for optical activity in terms of the dielectric response in crystal optics is more difficult than might reasonably be expected [1]. Consequently, recourse is typically had to a phenomenological account. In the simplest cases the normal modes are assumed to be circularly polarized so that forward and backward waves of the same handedness are degenerate. If this is so, then the circular birefringence can be expanded in even powers of the direction cosines of the wave normal [2]. The leading terms in the expansion suggest that optical activity is an allowed effect in the crystal classes having second rank property tensors with non-vanishing symmetrical, axial parts. -
Calorimetric Study on Kinetics of Mesophase Transition in Thermotropic Copolyesters
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Directory of Open Access Journals IJRRAS 3 (1) ● April 2010 Talukdar & Achary ● Study on Kinetics of Mesophase Transition CALORIMETRIC STUDY ON KINETICS OF MESOPHASE TRANSITION IN THERMOTROPIC COPOLYESTERS Malabika Talukdar & P.Ganga Raju Achary Department Of Chemistry, Institute Of Technical Education and Research (ITER), S ‘O’A University, Bhubaneswar, Orissa, India ABSTRACT Macromolecular crystallization behaviour, especially those of liquid crystalline polymers, has been a subject of great interest. The crystallization process for the polyester samples under discussion are analyzed assuming a nucleation controlled process and by using the Avrami equation. The study mainly deals with the crystallization process from the mesophase of two thermotropic liquid crystalline polyesters exhibiting smectic order in their mesophasic state. In order to analyze crystallization behavior of the polyesters, crystallization rate constant and the Avrami exponent have been estimated. Total heat of crystallization and entropy change during the transition has been calculated for both the polyesters. Keywords: Crystallization; Mesophase; Nucleation; Orthorhombic; Smectic. 1. INTRODUCTION The transition kinetics of liquid crystal polymers appears to be of great interest in order to control the effect of processing conditions on the morphology as well as on the final properties of the materials. These LC polymers show at least three main transitions: a glass transition, a thermotropic transition from the crystal to the mesophase and the transition from the mesophase to the isotropic melt. The kinetics of phase transition in liquid crystals may be analyzed for three different processes, the three dimensional ordering from both the isotropic melt state and the mesophase and the formation of the mesophase from the isotropic melt. -
Introduction to FINITE STRAIN THEORY for CONTINUUM ELASTO
RED BOX RULES ARE FOR PROOF STAGE ONLY. DELETE BEFORE FINAL PRINTING. WILEY SERIES IN COMPUTATIONAL MECHANICS HASHIGUCHI WILEY SERIES IN COMPUTATIONAL MECHANICS YAMAKAWA Introduction to for to Introduction FINITE STRAIN THEORY for CONTINUUM ELASTO-PLASTICITY CONTINUUM ELASTO-PLASTICITY KOICHI HASHIGUCHI, Kyushu University, Japan Introduction to YUKI YAMAKAWA, Tohoku University, Japan Elasto-plastic deformation is frequently observed in machines and structures, hence its prediction is an important consideration at the design stage. Elasto-plasticity theories will FINITE STRAIN THEORY be increasingly required in the future in response to the development of new and improved industrial technologies. Although various books for elasto-plasticity have been published to date, they focus on infi nitesimal elasto-plastic deformation theory. However, modern computational THEORY STRAIN FINITE for CONTINUUM techniques employ an advanced approach to solve problems in this fi eld and much research has taken place in recent years into fi nite strain elasto-plasticity. This book describes this approach and aims to improve mechanical design techniques in mechanical, civil, structural and aeronautical engineering through the accurate analysis of fi nite elasto-plastic deformation. ELASTO-PLASTICITY Introduction to Finite Strain Theory for Continuum Elasto-Plasticity presents introductory explanations that can be easily understood by readers with only a basic knowledge of elasto-plasticity, showing physical backgrounds of concepts in detail and derivation processes -
D0tc00217h4.Pdf
Electronic Supplementary Material (ESI) for Journal of Materials Chemistry C. This journal is © The Royal Society of Chemistry 2020 Electronic Supporting Information What is the role of planarity and torsional freedom for aggregation in a -conjugated donor-acceptor model oligomer? Stefan Wedler1, Axel Bourdick2, Stavros Athanasopoulos3, Stephan Gekle2, Fabian Panzer1, Caitlin McDowell4, Thuc-Quyen Nguyen4, Guillermo C. Bazan5, Anna Köhler1,6* 1 Soft Matter Optoelectronics, Experimentalphysik II, University of Bayreuth, Bayreuth 95440, Germany. 2Biofluid Simulation and Modeling, Theoretische Physik VI, Universität Bayreuth, Bayreuth 95440, Germany 3Departamento de Física, Universidad Carlos III de Madrid, Avenida Universidad 30, 28911 Leganés, Madrid, Spain. 4Center for Polymers and Organic Solids, Departments of Chemistry & Biochemistry and Materials, University of California, Santa Barbara 5Departments of Chemistry and Chemical Engineering, National University of Singapore, Singapore, 119077 6Bayreuth Institute of Macromolecular Research (BIMF) and Bavarian Polymer Institute (BPI), University of Bayreuth, 95440 Bayreuth, Germany. *e-mail: [email protected] 1 1 Simulated spectra Figure S1.1: Single molecule TD-DFT simulated absorption spectra at the wB97XD/6-31G** level for the TT and CT molecules. TT absorption at the ground state twisted cis and trans configurations is depicted. Vertical transition energies have been broadened by a Gaussian function with a half-width at half-height of 1500cm-1. Figure S1.2: Simulated average CT aggregate absorption spectrum. The six lowest vertical transition transition energies for CT dimer configurations taken from the MD simulations have been computed with TD-DFT at the wB97XD/6-31G** level. Vertical transition energies have been broadened by a Gaussian function with a half-width at half-height of 1500cm-1. -
Low Shear Rheological Behaviour of Two-Phase Mesophase Pitch
Published as: Shatish Ramjee, Brian Rand, Walter W. Focke. Low Shear Rheological Behaviour of Two-Phase Mesophase Pitch. Carbon 82 (2015), pp. 368-380. DOI : 10.1016/j.carbon.2014.10.082 Low Shear Rheological Behaviour of Two-Phase Mesophase Pitch Shatish Ramjee, Brian Rand*, Walter W. Focke SARChI Chair of Carbon Technology and Materials, Department of Chemical Engineering University of Pretoria, Pretoria, South Africa, 0002 Abstract: The low shear rate rheology of two phase mesophase pitches derived from coal tar pitch has been investigated. Particulate quinoline insolubles (QI) stabilised the mesophase spheres against coalescence. Viscosity measurements over the range 10 to 106 Pa.s were made at appropriate temperature ranges. Increasing shear thinning behaviour was evident with increasing mesophase content. At low mesophase contents the dominant effect on the near Newtonian viscosity was temperature but at higher contents it was the shear rate; temperature dependence declined to near zero. The data indicated that agglomeration could be occurring at intermediate mesophase volume fractions, 0.2-0.3. The Krieger-Dougherty function and its emulsion analogue indicated that in this region the mesophase pitch emulsions actually behaved like „hard‟ sphere systems and the effective volume fraction was estimated as a function of shear rate illustrating the change in extent of agglomeration. At the higher volume fractions approaching the maximum packing fraction, which could only be measured at higher temperatures, the shear thinning behaviour -
Cell Biology Befriends Soft Matter Physics Phase Separation Creates Complex Condensates in Eukaryotic Cells
technology feature Cell biology befriends soft matter physics Phase separation creates complex condensates in eukaryotic cells. To study these mysterious droplets, many disciplines come together. Vivien Marx n many cartoons, eukaryotic cells have When the right contacts are made, phase a tidy look. A taut membrane surrounds separation occurs. Ia placid, often pastel-hued lake with well-circumscribed organelles such as Methods multitude ribosomes and a few bits and bobs. It’s a In the young condensate field, Amy helpful simplification of the jam-packed Gladfelter of the University of North cytoplasm. In some areas, there’s even more Carolina at Chapel Hill says her lab intense milling about1–4. To get a sense of intertwines molecular-biology-based these milieus, head to the kitchen with approaches, modeling, and in vitro Tony Hyman of the Max Planck Institute reconstitution experiments using simplified of Molecular Cell Biology and Genetics systems. “You can’t build models that have and Christoph Weber and Frank Jülicher of all the parameters that exist in a cell,” she the Max Planck Institute for the Physics of says. But researchers need to consider that Complex Systems, who note5: “For instance, in vitro work or overexpressing proteins when you make vinaigrette and leave it, you will not do complete justice to condensates come back annoyed to find that the oil and in situ. “Anything will phase separate vinegar have demixed into two different in a test tube,” she says, highlighting phases: an oil phase and a vinegar phase.” Just Eukaryotic cells contain mysterious, malleable, experiments in which crowding agents as oil and vinegar can separate into two stable liquid-like condensates in which there is intense such as polyethylene glycol are used to phases, such liquid–liquid phase separation activity. -
Exploring Thein Meso Crystallization Mechanism by Characterizing the Lipid Mesophase Microenvironment During the Growth of Singl
Exploring the in meso crystallization mechanism rsta.royalsocietypublishing.org by characterizing the lipid mesophase microenvironment Research during the growth of single Cite this article: van`tHagLet al.2016 transmembrane α-helical Exploring the in meso crystallization mechanism by characterizing the lipid peptide crystals mesophase microenvironment during the 1,3,4 2,5 growth of single transmembrane α-helical Leonie van `t Hag , Konstantin Knoblich , Shane Phil.Trans.R.Soc.A peptide crystals. 374: A. Seabrook6, Nigel M. Kirby7, Stephen T. Mudie7, 20150125. http://dx.doi.org/10.1098/rsta.2015.0125 Deborah Lau4,XuLi1,3, Sally L. Gras1,3,8,Xavier 4 2,5 2,5 Accepted: 12 February 2016 Mulet , Matthew E. Call , Melissa J. Call , Calum J. Drummond4,9 and Charlotte E. Conn9 One contribution of 15 to a discussion meeting 1 issue ‘Soft interfacial materials: from Department of Chemical and Biomolecular Engineering, and 2 fundamentals to formulation’. Department of Medical Biology, The University of Melbourne, Parkville, Victoria 3052, Australia Subject Areas: 3Bio21 Molecular Science and Biotechnology Institute, physical chemistry, biochemistry The University of Melbourne, 30 Flemington Road, Parkville, Victoria 3052, Australia Keywords: 4 in meso crystallization, cubic mesophase, CSIRO Manufacturing Flagship, Private Bag 10, Clayton, Victoria local lamellar phase, DAP12. 3169, Australia 5Structural Biology Division, The Walter and Eliza Hall Institute of Authors for correspondence: Medical Research, 1G Royal Parade, Parkville, Victoria 3052, -
2 Review of Stress, Linear Strain and Elastic Stress- Strain Relations
2 Review of Stress, Linear Strain and Elastic Stress- Strain Relations 2.1 Introduction In metal forming and machining processes, the work piece is subjected to external forces in order to achieve a certain desired shape. Under the action of these forces, the work piece undergoes displacements and deformation and develops internal forces. A measure of deformation is defined as strain. The intensity of internal forces is called as stress. The displacements, strains and stresses in a deformable body are interlinked. Additionally, they all depend on the geometry and material of the work piece, external forces and supports. Therefore, to estimate the external forces required for achieving the desired shape, one needs to determine the displacements, strains and stresses in the work piece. This involves solving the following set of governing equations : (i) strain-displacement relations, (ii) stress- strain relations and (iii) equations of motion. In this chapter, we develop the governing equations for the case of small deformation of linearly elastic materials. While developing these equations, we disregard the molecular structure of the material and assume the body to be a continuum. This enables us to define the displacements, strains and stresses at every point of the body. We begin our discussion on governing equations with the concept of stress at a point. Then, we carry out the analysis of stress at a point to develop the ideas of stress invariants, principal stresses, maximum shear stress, octahedral stresses and the hydrostatic and deviatoric parts of stress. These ideas will be used in the next chapter to develop the theory of plasticity. -
Multiband Homogenization of Metamaterials in Real-Space
Submitted preprint. 1 Multiband Homogenization of Metamaterials in Real-Space: 2 Higher-Order Nonlocal Models and Scattering at External Surfaces 1, ∗ 2, y 1, 3, 4, z 3 Kshiteej Deshmukh, Timothy Breitzman, and Kaushik Dayal 1 4 Department of Civil and Environmental Engineering, Carnegie Mellon University 2 5 Air Force Research Laboratory 3 6 Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University 4 7 Department of Materials Science and Engineering, Carnegie Mellon University 8 (Dated: March 3, 2021) Dynamic homogenization of periodic metamaterials typically provides the dispersion relations as the end-point. This work goes further to invert the dispersion relation and develop the approximate macroscopic homogenized equation with constant coefficients posed in space and time. The homoge- nized equation can be used to solve initial-boundary-value problems posed on arbitrary non-periodic macroscale geometries with macroscopic heterogeneity, such as bodies composed of several different metamaterials or with external boundaries. First, considering a single band, the dispersion relation is approximated in terms of rational functions, enabling the inversion to real space. The homogenized equation contains strain gradients as well as spatial derivatives of the inertial term. Considering a boundary between a metamaterial and a homogeneous material, the higher-order space derivatives lead to additional continuity conditions. The higher-order homogenized equation and the continuity conditions provide predictions of wave scattering in 1-d and 2-d that match well with the exact fine-scale solution; compared to alternative approaches, they provide a single equation that is valid over a broad range of frequencies, are easy to apply, and are much faster to compute.