{"title":"Boltzmann’s Equation","authors":"B. Cantor","doi":"10.1093/oso/9780198851875.003.0004","DOIUrl":"https://doi.org/10.1093/oso/9780198851875.003.0004","url":null,"abstract":"Thermodynamics describes the relationship between heat, work, energy and motion. The key concepts are the conservation of energy and the maximisation of entropy (or disorder) as given by the first and second laws of thermodynamics. Boltzmann’s equation explains how the entropy of a material is related to the disorder of its atoms or molecules, as measured by the probability or the number of equivalent atomic or molecular structures. This chapter examines thermodynamic properties such as internal energy, enthalpy and Gibbs and Helmholtz free energy; physical properties such as specific heat and thermal expansion coefficient; and the application of thermodynamics to chemical reactions, solid and liquid solutions, and phase separation. Ludwig Boltzmann’s early life as the son of a minor tax official in Austria is described, as are: his scientific career in a series of Austrian and German universities; his philosophical arguments with Ernst Mach and the phenomenalists about whether atoms do or do not exist; his increasing moodiness, paranoia and bipolar disorder; and his ultimate suicide while trying to recuperate from depression in Trieste.","PeriodicalId":227024,"journal":{"name":"The Equations of Materials","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121574996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Burgers Vector","authors":"B. Cantor","doi":"10.1093/oso/9780198851875.003.0011","DOIUrl":"https://doi.org/10.1093/oso/9780198851875.003.0011","url":null,"abstract":"When a material is stretched beyond its elastic limit, the atoms and molecules begin to slide over each other. This is called plasticity, and is dominated by the motion of defects in the crystal structure of the material, notably line defects called dislocations. The structure and magnitude of a dislocation is defined by its Burgers vector, which is a topological constant for a given dislocation line in a given material, so there is an effective Burgers equation: b = constant. This chapter describes: the structure of edge; screw and mixed dislocations and their associated line energy; the way in which dislocations are generated and interact under stress, leading to the yield point, work hardening and a permanent set in the material; and the use during manufacturing of deformation processing, annealing, recovery and recrystallisation. Jan Burgers’ early life in Arnhem at the beginning of the 20th century is described, as are: his time as a student with the charismatic but depressive Paul Ehrenfest, who later committed suicide; his appointment as the first Professor of Aerodynamics at Technische Universiteit Delft at a time of massive growth in the aviation industry; his contributions to aerodynamic and hydrodynamic flow as well as major Dutch engineering projects such as the Zuiderzee dams and the Maas river tunnel; his growing disaffection with the commercialisation of science and its use in warfare; and his philosophical dalliance with Soviet communism and then American capitalism.","PeriodicalId":227024,"journal":{"name":"The Equations of Materials","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127072552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Avrami Equation","authors":"B. Cantor","doi":"10.1093/oso/9780198851875.003.0009","DOIUrl":"https://doi.org/10.1093/oso/9780198851875.003.0009","url":null,"abstract":"When materials are heated or cooled, their structure often changes. This is called a phase transformation. Phase transformations are used extensively to modify and control the final microstructure and properties of a material during manufacturing into its final product form. The Avrami equation describes the sigmoidal (S-shaped) way in which the amount of a new phase evolves, initially accelerating as particles of the new phase nucleate and grow, and then decelerating as the old phase becomes progressively exhausted. This chapter explains the development of new phases by nucleation and growth, the mechanisms of precipitation, eutectoid and martensite reactions, and the use of time–temperature–transformation curves to understand and control transformation behaviour. The Avrami equation was derived independently in the mid-20th century by Melvin Avrami at Columbia University, Robert Mehl and his student W. Johnson at Carnegie Tech, and Andrei Kolmogorov at Moscow State University. Avrami was horrified by the development of the atomic bomb at the end of the Second World War and dropped out of society to work as a caretaker on Orcas Island off the West Coast of America, before changing his name and returning as a physicist some years later; Mehl is known as one of the father figures of metallurgical science in the United States; and Kolmogorov made important advances in fields such as trigonometry, probability, topology, turbulence and genetics.","PeriodicalId":227024,"journal":{"name":"The Equations of Materials","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126087351","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Gibbs Phase Rule","authors":"B. Cantor","doi":"10.1093/oso/9780198851875.003.0003","DOIUrl":"https://doi.org/10.1093/oso/9780198851875.003.0003","url":null,"abstract":"Materials are made up of regions of space that are homogeneous in structure and properties, called phases. The number of different phases in a material depends on its temperature, pressure and composition, as given when the material is at equilibrium by the Gibbs phase rule. This was discovered by the American scientist J. Willard Gibbs during his ground-breaking investigations in the late 19th century into the thermodynamics of heterogeneous materials. This chapter explains the differences between solutions, mixtures and compounds; the use of phase diagrams to determine the structure of a material; and the way in which phase transformations can be used to change the structure of a material. Gibbs grew up in an academic family at Yale University in New Haven at the time of the American Civil War. He was the first person to receive an engineering doctorate in the United States, and he later became a fundamental theoretician of thermodynamics, statistical mechanics and vector fields.","PeriodicalId":227024,"journal":{"name":"The Equations of Materials","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114130077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Arrhenius Equation","authors":"B. Cantor","doi":"10.1093/oso/9780198851875.003.0005","DOIUrl":"https://doi.org/10.1093/oso/9780198851875.003.0005","url":null,"abstract":"The Arrhenius equation describes the way in which the speed of a chemical reaction varies exponentially with temperature. This chapter describes the thermodynamics of chemical reactions, the complexity of chemical kinetics, their explanation in terms of atomic and molecular collisions and transitionary activated states, and the concepts of molecularity, reaction order and collision and reaction cross section. Svante Arrhenius was the son of an estate manager at Uppsala University. He was tremendously innovative scientifically, inventing the interdisciplinary fields of physical chemistry, the ionic theory of acids and bases, environmental science, global warming and immunochemistry. He had longstanding feuds with many, more conventional, scientists, particularly his doctoral supervisors, who nearly failed him because they thought his development of ionic theory was neither ‘proper’ physics nor ‘proper’ chemistry. He became Director of the Swedish Academy of Sciences Högskola in Stockholm, where he oversaw the initiation of the Nobel Prizes in Physics, Chemistry, Medicine, Literature and Peace.","PeriodicalId":227024,"journal":{"name":"The Equations of Materials","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115412935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Fermi Level","authors":"B. Cantor","doi":"10.1093/oso/9780198851875.003.0013","DOIUrl":"https://doi.org/10.1093/oso/9780198851875.003.0013","url":null,"abstract":"The Fermi level is the maximum energy of the electrons in a material. Effectively there is a Fermi equation: EF = E\u0000 max. This chapter examines the discrete electron energy levels in individual atoms as a consequence of the Pauli exclusion principle, the corresponding energy bands in a material composed of many atoms or molecules, and the way in which conductor, insulator and semiconductor materials depend on the position of the Fermi level relative to the energy bands. It explains: the concepts of electron mobility, mean free path and conductivity; the dielectric effect and capacitance; p-type, n-type, intrinsic and extrinsic semiconductors; and the behaviour of some simple microelectronic devices. Enrico Fermi was the son of a minor railway official in Rome. He had a meteoric scientific career in Italy, developing Fermi-Dirac statistics for the energies of fundamental fermion particles (such as electrons and protons), discovering the neutrino, and explaining the behaviour of different materials under bombardment from fast and slow neutrons. After initially joining Mussolini’s Fascist Party, he became unhappy at the level of anti-Semitism (his wife was Jewish) and left suddenly for America, immediately after receiving the Nobel Prize in Sweden. At Columbia and Chicago Universities and at Los Alamos National Labs, he played a key scientific role in developing controlled fission in an atomic pile, leading to the development of the atomic bomb towards the end of the Second World War, and the nuclear energy industry after the war.","PeriodicalId":227024,"journal":{"name":"The Equations of Materials","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130842697","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fick’s Laws","authors":"Brian Cantor","doi":"10.1093/oso/9780198851875.003.0007","DOIUrl":"https://doi.org/10.1093/oso/9780198851875.003.0007","url":null,"abstract":"Atoms and molecules are not completely immobile within a solid material. They move by jumping into vacancies or interstitial sites in the crystal lattice. The laws describing their motion were discovered by Adolf Fick in the mid-19th century, modelled on analogous laws for the flow of heat (Fourier’s law) and electricity (Ohm’s law). According to Fick’s first law, the rate at which atoms move is proportional to the concentration gradient, with the diffusion coefficient defined as the constant of proportionality. Fick’s second law generalises the first law to a wide range of situations and is called the diffusion equation. This chapter examines a number of characteristic diffusion profiles; the difference between self, intrinsic, inter- and tracer diffusion coefficients; the Kirkendall effect and porosity formation when different components move at different speeds; and the Arrhenius temperature dependence of diffusion. Fick was a physiologist and derived his laws initially to describe the flow of blood through the heart. He made advances in anatomy, physiology and medicine, developing methods of monitoring blood pressure, muscular power, corneal pressure and glaucoma. He lived at the time of Bismarck’s post-Napoléonic unification of Germany and the associated flowering of German science, engineering, medicine and culture.","PeriodicalId":227024,"journal":{"name":"The Equations of Materials","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127419497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Griffith’s Equation","authors":"B. Cantor","doi":"10.1093/oso/9780198851875.003.0012","DOIUrl":"https://doi.org/10.1093/oso/9780198851875.003.0012","url":null,"abstract":"Most materials fracture suddenly because they contain small internal and surface cracks, which propagate under an applied stress. Griffith’s equation shows how fracture strength depends inversely on the square root of the size of the largest crack. It was developed by Alan Griffith, while he was working as an engineer at Royal Aircraft Establishment Farnborough just after the First World War. This chapter examines brittle and ductile fracture, the concepts of fracture toughness, stress intensity factor and stBiographical Memoirs of Fellows ofrain energy release rate, the different fracture modes, and the use of fractography to understand the causes of fracture in broken components. The importance of fracture mechanics was recognised after the Second World War, following the disastrous failures of the Liberty ships from weld cracks, and the Comet airplanes from sharp window corner cracks. Griffith’s father was a larger-than-life buccaneering explorer, poet, journalist and science fiction writer, and Griffith lived an unconventional, peripatetic and impoverished early life. He became a senior engineer working for the UK Ministry of Defence and then Rolls-Royce Aeroengines, famously turning down Whittle’s first proposed jet engine just before the Second World War as unworkable because the engine material would melt, then playing a major role in jet engine development after the war, including engines for the first vertical take-off planes.","PeriodicalId":227024,"journal":{"name":"The Equations of Materials","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131381393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Gibbs-Thomson Equation","authors":"B. Cantor","doi":"10.1093/oso/9780198851875.003.0006","DOIUrl":"https://doi.org/10.1093/oso/9780198851875.003.0006","url":null,"abstract":"The external surface of a material has an atomic or molecular structure that is different from the bulk material. So does any internal interface within a material. Because of this, the energy of a material or any grain or particle within it increases with the curvature of its bounding surface, as described by the Gibbs-Thomson equation. This chapter explains how surfaces control the nucleation of new phases during reactions such as solidification and precipitation, the coarsening and growth of particles during heat treatment, the equilibrium shape of crystals, and the surface adsorption and segregation of solutes and impurities. The Gibbs-Thomson was predated by a number of related equations; it is not clear whether it is named after J. J. Thomson or William Thomson (Lord Kelvin); and it was not put into its current usual form until after Gibbs’, Thomson’s and Kelvin’s time. J. J. Thomson was the third Cavendish Professor of Physics at Cambridge University. He discovered the electron, which had a profound impact on the world, notably via Thomas Edison’s invention of the light bulb, and subsequent building of the world’s first electricity distribution network. William Thomson was Professor of Natural Philosophy at Glasgow University. He made major scientific developments, notably in thermodynamics, and he helped build the first trans-Atlantic undersea telegraph. Because of his scientific pre-eminence, the absolute unit of temperature, the degree Kelvin, is named after him.","PeriodicalId":227024,"journal":{"name":"The Equations of Materials","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124205197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Scheil Equation","authors":"B. Cantor","doi":"10.1093/oso/9780198851875.003.0008","DOIUrl":"https://doi.org/10.1093/oso/9780198851875.003.0008","url":null,"abstract":"Many materials are manufactured by solidification, either as a final product by casting, or as an intermediate ingot or bar. The Scheil equation describes the partitioning that takes place during solidification and the resulting spatial redistribution of solute, which makes it difficult to maintain a homogeneous material composition, and which leads to unwanted concentrations of harmful impurities. This chapter explains nucleation and growth processes during solidification, the resulting dendritic, faceted, equiaxed and columnar structures depending on thermal conditions and material type, coupled solidification of two-phase eutectic materials, and typical casting methods and associated structures and defects. Very little is known about Erich Scheil, who worked at the Max Planck Institute in Stuttgart in the mid-20th century.","PeriodicalId":227024,"journal":{"name":"The Equations of Materials","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124490157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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