Intro+to+Materials+Outline

=An Introduction to Materials=

Our history is divided by the types of materials available at the time: the Stone Age, Bronze Age, Iron Age, and Modern Age.

toc The beauty of scientific perspective can be shrouded with elegant complexity. Upon understand some fundamental atomic concepts, the rest of materials science involves understanding and comparing properties of four general classes of materials: ceramics, metals, polymers and electronic materials.

Phases - gas, amorphous, crystalline Metals, Ceramics, Polymers, E-Mats, Composites Materials Properties || P-Sets Quizzes || Phase Diagrams Materials Selection ||
 * ~ Theory ||~ Practice ||~ Application ||
 * Atomic

=Informal Topics=
 * Learning Styles
 * Academic Integrity
 * Engineering Ethics
 * Confirmation Bias

Research Labs
There are three requirements for making measurements:
 * 1) Sample
 * 2) Source of Energy
 * 3) Detector

Example: To measure the size of a grain in metals (sample), one can use a SEM (energy source) and capture images with the on-board SiLi (detector).

Laboratory equipment could be color coded as follows:
 * 1) Green - Sample preparations (sputter coater, desicators, sample holders, etc.)
 * 2) Red - Energy sources (lasers, x-ray tubes, sonicators,etc.)
 * 3) Blue - detectors (light mircoscope, microphones, balances, etc.)
 * 4) White - maintenance equipment (computers, servers, phones)

=Materials Science - An Introduction= A common question posed to materials scientists is "What is materials science?" Since MatSci is interdisciplinary, typically integrating fields of metals, ceramics, polymers and electronic materials into new fields such as biomaterials, are many ways of describing it. An engineer can master and apply the fundamentals of materials science after answering four core questions:
 * 1) What is stuff made of? - We answer this by understanding what __matter__ is and the various way by which matter takes form.
 * 2) How is stuff classified? - There are ways in which structured matter is organized, __classified__ and communicated.
 * 3) How does stuff behave? - The bulk of materials science is about understanding the vast number of materials __properties__ available.
 * 4) How can I use stuff? - __Applying__ the various properties and understanding their interactions equips an engineer with ability to predict materials properties.

Materials science simply becomes an evolving process of understanding how matter creates various materials, classification of those materials, mastering properties and applying those properties to any material.

Levels of Materials Science
The //science// of materials science originates from the atom. All definitions of an atom and its properties are described at the **primary level** which captures four key fundamental sciences displayed below, each supported by major scientific organizations. Materials science has evolved from physics and chemistry into the area of biology through biomaterials, a relatively new field. Modern concerns in global climates and sustainability demand research in environmentally safe materi als, which signifies a more recent trend towards the earth sciences.

American Assoc. of Petroleum Geologists American Geophysical Union - EOS ||
 * ~ Fundamentals of Science ||~ Definition ||~ Organization - Publication ||
 * = Physics || how particles interact; classical and modern physics || American Physical Society - Physics Today ||
 * = Chemistry || atomic interactions; organic and inorganic chemistry || American Chemical Society - Chemical and Engineering News ||
 * = Earth Sciences || original and present dynamic processes; geology and geophysics || Geological Society of America -
 * = Biology || study of living things; cellular, molecular, modern, developmental, genetics... || several ||

At a **secondary leve**l, materials science expands these fundamentals into several key areas used to describe matter at all length scales: mechanical behavior (including mechanics of materials, fracture mechanics), thermodynamics, electrical properties (e-props), nuclear, transport phenomena ( including diffusion/kinetics). At this level, the complex behavior of the atoms and bulk properties are considered. For instance, the presence of defects, voids and cracks has significant effect on practical material properties. Such special cases are not detailed in the fundamental sciences of physics and chemistry. Rather these phenomena are covered more generally in the field of mechanical behavior or materials. These principles are typically explored in the 3rd or 4th undergraduate years.

The final **tertiary level** is one of specializations in the key areas of modern material science: metals, ceramics, polymers, electronic materials (e-mats) and biomaterials (b-mats). The las two entries of e-mats and b-mats are essentially "composite" field of the first three primary fields.

The three levels of materials science are illustrated in the table below:

Applied Specialties ||= metals - ceramics - polymers - e-mats - b-mats || Complex Behavior ||= mechanical behavior - thermodynamics - transport phenomona - e-props || Fundamental Sciences ||= Physics - Chemistry - Biology ||
 * ~  ||~ Corresponding Fields ||
 * ~ Tertiary Level -
 * ~ Secondary Level -
 * ~ Primary Level -
 * ||= The atom ||

The **tools** that are used to link aspects of the fundamental science are typically man-made //constructs// (i.e. vector analysis, symmetry, tensors, phase diagrams), //conventions// (i.e. right-hand rule) and //notations// (i.e. Miller indices), based on the principles of **mathematics** and predictive power of **probability**. These tools essential establish scientific jargon that enable us to communicate efficiently.

**Science Pioneers**
Aristotle - truth attainable by pure thought Pliny - first encyclopedia, collection of "facts," many were false Ptolemy - earth at the center of the universe, orbits eccentric epicycles Nicholas Capernicus (Polish) - sun at the center of the universe Tycho Brae (Danish) - improved astronomical tools (quadrants), observed an exploding super nova - the heavens change Johannes Kepler (German) - Tycho Brae's successor, 3 Laws: elliptical orbits, equal sweeps of area, (period)^2/(distance)^3 = constant Galileo Galilei (Italian) - observed imperfections in heavens (crater's in moon, sun spots), rolling ball experiment Issac Newton - Integral and Differential Calculus, Laws of Optics, Laws of Motion, Universal Laws of Gravitation (all accomplished within 18 months after the Plague) Henry Cavendish - made torsion balance to measure the gravitational constant (G) American Association for the Advancement of Scientists - Science National Academy of Science - Issues Sigma Xi - American Scientist
 * American Science Organizations**

**Scientific Method**
Pattern Recognition -> Hypothesis (guess) -> Predictions -> Observation and Experiments (measurements)
 * 4 Types of Scientific Questions**
 * 1) Existence questions - what phenomena exist
 * 2) Origin questions - how phenomena began
 * 3) Process questions - how nature works
 * 4) Applied questions - how to manipulate phenomena
 * Limitations to Answering Scientific Questions**
 * 1) Experimental error: is present in all measurements
 * 2) Uncertainty principle: by Heisenberg, any observation changes phenomena
 * 3) Chaos and turbulent behavior: unpredictable behavior of natural phenomena, i.e. weather, water flow
 * 4) Speed of light: we have to wait for light to reach the outer limits of our universe

The fundamental principles of science can all be expressed in the properties of the atom.
 * Fundamental Principles of Science**

Here are the fundamental laws that govern almost all of science. These conservation laws enable us to quantitatively describe the universe and to predict physical phenomena with great accuracy within out measurement limitations. A law is wrong if it is experimentally disproved. That said, we can never prove that a law is right. We are only able to disprove it.
 * Fundamental Laws**


 * Useful Derived Units || Conservation Law ||
 * x ~ p || momentum ||
 * t ~ E || energy ||
 * theta (rotation) ~ omega || angular momentum ||
 * delta (phase change) ~ C || electrical charge ||

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Classical descriptions of forces as defined by Newton have been presented. However, the gravitational force is relatively weak, generally experienced between massive objects, and is easily overcome by other types of forces at smaller length scales. These stronger forces also greatly influence all of the natural processes observed in the universal. There are 4 types of forces: Gravitational forces, Electrostatic forces, Strong forces, Weak forces
 * Fundamental Forces**

Energy is the ability to do work. Work is a force over a distance W = F*d. There are **7 types of energy**: mechanical, chemical, electrical, nuclear, radiant, thermal and acoustic. Of these seven types, there are different forms by which energy is released:
 * Forms of Energy**


 * **Kinetic energy** relates to motion and is promotional to the the square of the velocity.
 * There different storage forms of **potential energy**: gravitational, chemical, magnetic, electrical, elastic
 * **Waves** can transfer energy over long distances. There are several types: transverse (ocean), compressional (sound). Ex. Only 34 minutes after the 9.0 earthquake hit Japan in 2011, seismic activity was detected in southern Florida and water levels perturbed.
 * **Heat** was believed to come from the release of a fluid like medium from substances called "caloric." This theory was disproved when water was boiled by the drilling of a cannon barrel due to friction. It was later understood that heart was related to the vibrated of atoms..The quantity heat energy is related to the amount of material used. Heat transfers between objects three ways: conduction, convection, radiation
 * Light is a form of **radiant** energy. Light was completely described through electromagnetism.
 * **Mass** was discovered by Einstein to relate to energy. Most of the enery is locked into stable matter, but scientists have discovered ways to unleash the nuclear energy.


 * ~ Energy ||~ Related Course ||~ Equation ||
 * Kinetic || Physics I || K.E.= (1/2)mv^2 ||
 * Potential || Physics II || P.E.= mgh ||
 * Wave || Physics III ||  ||
 * Heat || Thermodynamics ||  ||
 * Light || Optics, Physics II ||  ||
 * Mass || Relativity || E = mc^2 ||


 * Thermodynamic Laws Governing Energy**


 * 1) Matter is neither created or destroyed (conservation of energy). The amount of emery available remains constant.
 * 2) Entropy increases. There are limits to energy conversion in reverse. Energy conversion of heat to work s not highly efficient. Energy must be expended to reset engine cycles, which accounts for some heat (energy) loss. Also, energy is lost as heat travels throughout the system. % Efficiency = (T_hot - T_cold)/T_hot) * 100. The wider the temperature range between the heat reservoir and the heat sink, the better the efficiency. Clausius defined entropy to establish a quantitative property of the 2nd law: the ratio between heat energy and temperature (delta_S = delta H / T), Entropy is either constant or increasing. That is the ratio of heat energy to temperature for an object a t T_cold is always greater than at T_hot. This randomness and disorder can be proved statistically, as demonstrated by Boltzmann, where the probability of a disordered system is always higher that the percentage of all possible ordered configurations. .The 2nd Law gives insight that time has a forward, irreversible direction.
 * 3) Absolute zero exists.
 * 4) Zeroth Law - You can't reach absolute zero.

The Thermodynamic Laws have yet to explain the "emergent sciences" that is higher ordered systems based on the complex interaction of its sub-components (sand dunes made by sand grains, thought generated by electrical pulses between neurons, etc.)

=Matter= [\r.L2-L6]

Introduction to Waves
[\r.]

The Wavefunction
[\Volger Chemistry]] [\Susskind - New Revolutions]

The Atom
You will see that all materials properties are related to their atoms and their states of matter.

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What does an atom look like? The Bohr model was the first attempt to describe the atom in terms of energy and position. However, the Bohr model was unable to explain certain phenomena with the electrons. The wave-mechanical model was adopted to describe the position of electrons as a //probability density//, the position and momentum of which are uncertain according to Heisenberg's Uncertainty principle.

How do you track electrons in an atom? Four quantum numbers are used to describe the state of an electron.

What is the structure of energy states in an atom? The Bohr model accounts for the principal quantum number. The wave-mehanical model accounts for all other quantum numbers and thus refined the Bohr model. There are general rules associated with the structure of energy states and the filling of electrons in the those those states by Pauli's exclusion principle. Orbitals can hybridize in order to lower energy. Hybridization is a common phenomenon in covalent bonding.
 * ~ Name ||~ Symbol ||~ Description ||
 * principal || n || gives the shell (K, L, M, N, O); how far away from the nucleus ||
 * azimuthal || l || gives the shape of the sub-shell (s, p, d, f); how the electron travels around the nucleus; range is 0 to n-1 ||
 * magnetic || m_l || in the presence of a magnetic field, the sub-shells split ||
 * spin || m_s || related to the spin moment of the electron, either up or down ||

Bonding
Many properties are related to the type of bonding between atoms, which involve the interaction of electrons between shells of two or more atoms. It is common to view force or potential responses of two atoms approaching each other from an infinitite distance. [\i.force and protential diagrams as function of r]. At first the force is low, nearly zero an infinite distance. As atoms appoach each other, an attractive force builds between the nucleus and electrons of each. At a certain distance, the force is dramatically reduced. The state of equilibrium at which the atoms are separated is considered //bonding//. The nuclei resist being pushed further together by a repusive force associated with the electric repulsion between nuclei. This process can be described in terms of potential energies. The minimum of the potential-distance curve is the bonding energy, which represent the minimum energy required to separate two atoms to infinite distance. While materials have more than two atoms, the bonding energy is proportional to several properties. For example, materials with high bonding energies typically have high melting temperatures. The mechanical stiffness is associated with the shape of the force/potential-interatomic separation curve. Stiff materials have highersteeper slopes on the force curve. The thermal expansion is inversely related to the depth of the through in the potential diagram, having higher bonding energies.

For more on bonding, see the @Bonds section.

States of Matter
Phases

[\radial probability] We are able to change matter from one state to another (that is forcing atoms into periodic and non-periodic arrangements) by varying //temperature// or //pressure//. We will illustrate transformations using a //phase diagram//.
 * Gases - pressure, not density
 * Liquids - density
 * Solids - Amorphous, Crystalline (single and polycrystalline)
 * Plasma - electrified gas; atoms are suppressed to extreme pressures, atoms are so excited the electrons are stripped away and generate current

Single crystals are perfectly arranged atomically in space. Polycrystalline materials account for most solids. These compromise of nucleated solids which each grow atomically as liquid converts to solid. The solids form grains, which grow and terminate when the grains impinge on each other. The intersection of these grains called grain boundaries which are often a mismatch in atomic periodicity.

Phase Diagrams and Kinetics I (Simple)
Basic properties: melting, evaporation, sublimation

Critical point [\r.Dickerson p. 669]

[\Unary systems - water] [\i.Water Diagram]

Composition
The composition, or concentration, of elemental constituents is often expressed in terms of (i) weight percent or (ii) atomic percent. These values take advantage of dimension of mass, [M] and amount, [N] to group matter into measurable quantities.
 * weight percent (wt%): the weight of an element relative for the bulk weight
 * atomic percent (at%): the moles of an element relative to the total moles of the bulk.

Crystal Structures
The arrangement of atoms is considered in a solid material. A crystaline material refers to the periodic, repeatable arrangement of atoms distributed in space over a long range order. Materials lacking such long range order are called noncrystalline or amorphous materials. The structure of atomic arrangements is viewed as hard sphere representing atoms, superimposed on a lattice construct. The lattice is simply a 3D construct of points arranged in space at which the atomic spheres are centered. The most fundamental periodic lattice is called the unit cell. If repeated in space, the structure of the material can be represented, so most material analysis occurs at the unit cell level. Various unit cells are mentioned below [\unit cells].

Many ionic materials contain more than one atom. Most multicomponent ceramics are characterized by crystal structures with two atoms of equal amounts (AX), two atoms of different amounts (A_m X_p), or three different atoms (A_m B_n X_p). Examples of AX crystal structures are rock salt, cesium oxide (like BCC but two different atoms).

Silica and carbon have unique structures. Silica is a series of tetreahedra connected by bridging oxygens. Carbon exists in various polymorhpic forms and does not easily fall within any of the tradiational materials classifications (metals, ceramics, polymers). The forms of carbon are diamond, graphite and fullerenes.

Nature and Symmetry
Let us discuss a topic your are familiar with. In so doing, we will compare the the symmetry of regular solids to nature. Unlike the infinite regular polygons (2D), there are only 5 regular solids (3D).


 * ~ Shape ||~ No. of Faces ||~ No. of Vertices ||~ No. of Edges ||~ Shape of Face ||~ No. of Faces per Vertices ||~ In Nature ||
 * Tetrahedron || 4 || 4 || 6 ||< equilateral triangle ||= 3 || silicon lattice ||
 * Cube || 6 || 8 || 12 ||< square ||= 3 || simple cubic, pyrite ||
 * Octahedron || 8 || 6 || 12 ||< equilateral triangle ||= 4 || embedded in perovskite ||
 * Dodecahedron || 12 || 20 || 30 ||< pentagon ||= 3 || grain ||
 * Icosahedron || 20 || 12 || 30 ||< equilateral triangle ||= 5 || grain ||
 * 1) of edges = (# of faces x # of edges/face)/2
 * 2) of verticies = (# of faces x # of sides/face)/# of times each vertex is shared

s = number of sides in each region c = number of edges per vertex V = verticies E = edges F = faces

E = cV/2 V = sF/c

Euler Characteristic We can predict the relationship between vertices, edges and faces.
 * 1) Draw any continuous sweeping curve (eyes closed)
 * 2) Place dot @ the ends
 * 3) Place a dot @ each intersection
 * 4) Count the number of dots (V)
 * 5) Count the number of edges E), line segments
 * 6) Count the number of regions (F) (don't forget the outside).

V - E + F = 2

The Euler characteristics predicts that the only regular solids are restricted by 2(c+s) > cs

Crystal Systems
Unit cells are defined by 6 lattice parameters: 3 length axes (a, b, c) and 3 interstitial angles (alpha, beta, gamma). The conditions of these lattice parameters defines a crystal system, i.e. cubic, hexagonal tetrahedral, rhombohedral, orthorhomic, monoclic, triclinic.

[\i.crystal systems]

[\Seven systems, Fourteen Bravais Lattices]

[\Nomenclature - positions, directions, planes, Miller indices] [\Structures - BCC, FCC, HCP] [\Brief discussion on tensors, anisotropy and properties] The directionality of material properties is termed anisotropy and corresponds to crystallographic symmetry. Properties that lack directionality are termed isotropic. [\Example of elastic and piezoelectric tensors] We will illustrate the effect of crystal structures on a phase diagram using the more Fe-C system.

Crystallography
It becomes important to identify specific points, directions and planes in crystal systems. The following conventions are commonly used in materials science:
 * Point coordinates - A point P in a unit cell has coordinates q r s, that represent fractional lengths less than or equal to 1 along each axis. Ex. 1/2 3/4 1. Values are not separated by any punctuation.
 * Crystallographic Directions - Miller indices are used to define a vector originating from the origin. Fractions are eliminated by multiplying with the least common denominator and reduced by common factor. Three integers represent directions projected along the x, y, z axes within brackets and not separated by punctuation. Ex. [2 1 2]. Negative directions are indicated with a bar above the value.
 * Crystal planes - Miller indicies (hkl) are used to describe planes for all but hexagonal crystal systems. The indices of a plane (not passing through the origin) are determined by the intercept of the lattice parameters, reciprocals of the intercepts, reductions if needed, and enclosed in parentheses without punctuation or separations. A family of planes are those with equivalent atomic packing and are indicated with braces {hkl}.

code Problem Set 1 Determine following: code In x-ray diffraction, x-ray hit a solid with atomic spacings close to the X-ray wavelength. The atoms in the material scatter the waves in all directions, sometimes at angles the constructively interfere or destructively interfere. If to waves hit two planes of periodic atoms, Bragg's Law is satisfied if the extra distance traveled by the wave penetrating the deeper plane an integer multiple of the wavelength. Using a diffractometer, a //diffraction pattern// can result from a powder sample. Powder consists of single grains in various orientations, which increases the probability of yielding a diffraction pattern.
 * 1) the linear density of [100] in BCC.  Note: there are two atoms at each corner of the [100] direction; "half" the atoms are projected on the direction which equals one atom centered on the [100]
 * 2) the planar density of (110) in FCC.  Note: there are four atoms at the corners of the (110) plane; a "quarter" of the atoms are projected on the plane, which equals one atom centered on the (110)
 * 3) the volumetric density, i.e. "atomic packing factor" for BCC and FCC.

Defects
Perfect single crystals are rare in nature. Most crystals are imperfect containing defects in various spatial dimensions:
 * 0D - Point defects
 * vacancy: an absence of an atom in the lattice. The concentration of vacancies is determined thermodynamically through an Arrhenius--type equation since the presence of vacancies increases entropy. Therefore, increased temperature increases number of vacancies.
 * self-interstitial: an atom in the interstitial position distorts the lattice due to its size.
 * impurity: are often added when alloying to improve mechanical properties or corrosion resistance. For metals and ceramics they can exist as (i) interstitial impurity atoms or (ii) substitutional impurity ions.
 * 1D - Linear defects: include edge and screw dislocations
 * 2D - Interfacial defects: includes grain boundaries, twin boundaries, stacking faults and phase boundaries. Grain boundary energy is a function of the angle of misorientation. High angle grain boundaries are more chemically reactive than the grains.

Point defects are common in metals. In ceramics, point defects may exist for atoms for more than one type of atom, i.e. cations and anions. The defect structure in ceramics is bounded by the electroneutrality, which must be maintained. Ways in which defect structures are accomplished in ceramics are the following: Stoichiometry is maintained with either a Frenkel or Schottky defect whereby the ratio of cations and anions is predicted by the chemical formula. The concentration of Frenkel and Schottky defects also follows and an Arrhenius behavior as a function of temperature.
 * Frenkel defect: a cation-vacancy/cation-interstitial pair. This may be viewed as a diffusion of a cation from a lattice position to an interstitial. This this case, no change in charge is observed.
 * Schottky defect: the removal of a cation-vacancy/anion vacancy pair. This can occur as a migration of atoms from the interior to the bulk surface.

The formation of appreciable solid solutions of two atoms has several requirements. Solubility requirements are listed below: [\r.Callister]: = = =Classes of Materials= The common classes of materials and structures are described below:
 * Atomic size factor: Difference in atomic radii must be within 15%
 * Crystal structure: same crystal structures
 * Electronegativity: probability of intermetallic formation increases with the highly electropositive and highly electronegative elements.
 * Valences: metal will tend to dissolve another metal of higher valency

Metals - BCC, FCC, HCP Ceramics - NaCl, corrundum, (crystalline), glass (amorphous) Polymers - lamellae, spherulites E-mats - perovskite, diamond cubic, zinc-blende [\Composites -] [\Biomaterials -]

Metals
Annealing metals 0.5-0.75 of the melting temperature involves three stages: (i) recovery, (ii) recrystallization and (iii) grain growth.
 * 1) Recovery: increased atomic motions reduces dislocation density
 * 2) Recrystallization: new nucleation sites generate new, strain-free grains, which reduces the material hardness and strength
 * 3) Grain Growth: atomic diffusion at the grain boundaries leads to an increase in the average grain size and simultaneously grain boundary motion; grain boundary motion opposes the direction of atomic diffusion.

Ceramics
Crystalline ceramics have a limited number of effective slip systems, which renders them brittle. For noncrystalline ceramics, plastic deformation occurs by by viscous flow.

[\glass]

[\glass-ceramic [\cystalline cerarmic] According to Kingery, there are essentially two types of engineering ceramics: 1.) close-packing of oxygen atoms with metal ions in interstices (generally oxides) 2.) those characterized by directional covalent bonds which form open networks with low coordination numbers, thus lower density.

Polymers

 * 1) PET
 * 2) HDPE

Electronic Materials
[\r.structure from Callister, Chapter 12]

Ohm's Law V = IR J = sigma*E

Concepts -electrons filled energy states of the atomic shells and sub-shells s, p, d, f -as atoms approach each other, electrons are perturbed by other electrons and the nuclei of other atoms. -When atoms collide, electrons in the sub-shells may split into discrete electron states of lower and higher energies to form //electron energy bands//. -the typical band gap diagrams can represent the energy states available for electrons between two atoms at equilibrium atomic spacings. -[\looking at a E vs interatomic spacing diagram suggests squeezing the atoms together (via lattice strain ?) can widen band gaps]. -There are 4 types band gap diagrams at 0K: (i) can show partially filled band; the highest filled energy states is termed the //Fermi Energy//, E_f. This structure is found in metals. i.e. copper. (ii) overlapping bands of filled and empty states (represented by skewed bands); seen in Magnesium (iii) semiconductor - the filled band of electrons (valence band) is separated from an empty band (conduction band) by a **small** band gap (iv) insulator - the filled band of electrons (valence band) is separated from an empty band (conduction band) by a **large** band gap
 * ~ Type ||~ Conductivity (Ohm-m)^-1 ||~ Material Example ||
 * Conductor || 10^7 || Metals; Cu, Mg ||
 * Semicondutor || 10^-6 - 10^4 ||  ||
 * Insulator || 10^-10 - 10^-20 ||  ||

Only //free electrons// above the Fermi energy can be acted on by an electrical field which are generally excited into those higher states by available energy. Conduction occurs when an electron is promoted above the Fermi energy. For metals, this promotion is easier since it can occur withing the valance band (having both filled and empty states) or promoted due to overlap between the valence and conduction bands. The antithesis of a free electron in semiconductors is a //hole//, existing below the Fermi energy. [\Conduction is a misnomer because conduction can still occur in the valence band for metals] Electrons is insulators and semi-conductors must be supplied with enough energy to cross a bandgap.

Electron mobility Dealing with electrons above the Fermi level, for a perfect crystal, electrons move in the presence of an electric field. QM says these electrons do not interact with atom, which implies the electric current should increase with time. However, current reaches a constant value at the instant of applied voltage due to scattering at crystal imperfections, i.e. impurties atoms, vacancies, interstitial atoms, dislocations, etc. //Drift velocity// v_d = mu_e*E gives an average electron velocity along the direction of the applied field. The //electron mobility// mu_e is a proportionality constant indicating the frequency of scattering events. The total resistivity in a metal is due to contributions from temperature, impurities and plastic deformation represented by an equation known as //Matthiessen's rule//; p_tot = p_t + p_i + p_d. The thermal resistivity is a linear relationship. Thus strengthening a metal by cold-working or solid solution alloying is a tradeoff in electrical conductivity.

Semiconductors //Intrinsic semiconductors// have electrical properties inherent to the host material electronic structure. //Extrinsic semiconductors// have electronic structures modified by impurities by at least 1 impurity in 10^12 atoms.

Intrinsic semiconductors such as Ge, Si, GaAs, InSb, ZnTe, CdS have more insulative properties as elements separate in groups on the periodic table. Such elements widen their electronegativities and become more ionic. Intrinsic conductivity involves the mobility of all charge carriers, including electrons and holes. Somehow, the mobility of holes is less than electrons (mu_h < mu_e), therefore, sigma = n_i * |e| * (mu_h + mu_e), where n_i is the //intrinsic carrier concentration//.

Extrinsic semiconductors have impurities like Si, which has a valence of 5, where four electrons are covalently bonded and one electron is loosely bound and can become a free electron easily. Such //donor// atoms provide electrons have binding energies within the forbidden bandgap, and they are easily excited into the conduction band. This shifts the Fermi energy up into the bandgap. Consequently, once promoted to the conduction band, donor atom electrons do not leave a hole in the valence band. Donor electrons are easily excited at room temperature. Since the number of electrons in the donor states are much greater than the holes in the valance band (n>>p), sigma ~ n_i * |e| * (mu_e). A material with donor electrons is considered n-type, extrinsic semiconductor (in other words, semiconductor with impurities having electron donors). The opposite p-type material is created with a trivalent impurity in which an //acceptor state// near the valence band accepts and electron and creates a hole that can conduct. Since p>>n in p-type materials,sigma ~ p_i * |e| * (mu_p)

Generally, carrier concentration increases with temperature as electrons become more excited. This dependence differs for //extrinsic// semiconductors. For a given dopant concentration, the carrier concentration is constant for a range of temperatures since the number dopants have mostly been ionized near room temperature. At higher temperatures, the intrinsic behavior activates as valence electrons finally excite across the bandgap. Below a certain temperature is a "freeze-out region" where charged carriers are not excited. Charge carrier mobility is reduced logarithmically with increasing dopant concentration since the number of scattering sites increases. Charge carrier mobility also decreases with increase temperature due to the increase in scattering activity.

The Hall effect can be used to measure the charge carrier concentration, type and mobility by measuring the //Hall coefficient//, R_H.

R_H = 1/(n * |e|)

mu_e = |R_H| * sigma

Composites
A combination of materials to exceed the strength of its separate components.

Examples - plywood, concrete, teeth

Phase Diagrams and Kinetics II (Complex)
The creation of new materials and monitoring of microstructure under various conditions are plotted on phase diagrams.

Principles
[\Plots of T or P vs composition; phenomena are phase tranformations vs. solutibilty]

Raoult's Law [\r.Dickerson p. 675]

First Intro to Properties: Colligative Properties [\r.Dickerson p. 678] //Colligative properties// characterize dilute solutions of non-volatile solutes and volatile solvents.
 * 1) "Lowering of vapor pressure of the solvent by the solute"
 * 2) "Elevation of the boiling point of the solvent by the solute"
 * 3) "Lowering of the freezing point of the solvent by the solute"
 * 4) "The phenomenon of osmotic pressure"

[\Binary isomorphous - Cu Ni] [\Methods for interpretations of phase diagrams - identify phases (liquius, solidus, solvus), determine phase compositions (tie line), determine phase amounts (lever rule) [\Equilibrium cooling vs. Non-equilibrium cooling - ex. rawt Al non-equilibrium cooling (Patterson discussion)] [\Binary eutectic - Cu Ag, microstructure insets upon cooling (pure alpha, alpha with some beta, eutectic, primary alpha in eutectic)] [\Determine phase amounts in primary-eutectic composition]

Invariant Points
You were introduced to the eutectic, which is the reaction that occurs when liquid transforms into two solid phases upon cooling:

L --> alpha + beta

Here are other invariant points that are reversible by heating, but are displayed for cases upon cooling [\r.Hummel's intro book to complete the chart]:


 * Invariant Type || reactant(s) || --> || product(s) ||
 * Eutectic || L ||  || alpha + beta ||
 * Eutectoid || delta ||  || gamma + epsilon ||
 * Peritectic || dela + L ||  || epsilon ||

Sample Phase Systems [\r.see Callister for diagrams]
Ceramics
 * Al2O3-Cr2O3 - both have same crystal structure, isomorphous, substitional solid solution
 * MgO-Al2O3 - strange spinel intermediate displayed as a phase instead of a vertical line; there is a fixed comp. where the spinel is stoichiometric. Others it is non-stoichiometric.
 * ZrO2-CaO - partially stabilized zirconia
 * SiO2-Al2O3 - crystobalite, mullite

Metals The iron-iron carbide phase diagram is arguably the most important of all binary alloy systems due to the industry to steel and cast iron. [\r.Callister] [\i.Fe-C Diagram]
 * Ferrite - (alpha iron) pure iron at room temperature having a BCC crystal structure. This BCC structure limits the solubility of interstitial carbon. Ferrite is ferromagnetic below 768 C [\assumed Curie temperature].
 * Austenite - at 912 C, ferrite makes a polymorphic transformation to FCC austenite (gamma iron). At 1394 C FCC austenite reverts to a BCC form known as delta iron). Melting occurs at 1538 C for pure iron. Austentite is not stable below 727 C. However, the FCC structure can accommodate more interstitial carbon than BCC ferrite (up to 2.14 wt%). Austenite is non-magnetic.
 * Cementite - The melting temperature is suppressed with the introduction of carbon up to 6.7 wt%. At 6.7 wt% C, the iron carbide intermediate compound (Fe3C) is known as cementite. Cementite is hard and brittle. Actually, cementite is metastable and the Fe3C will eventually convert to graphite over many, many years.

There are three types of ferrous alloys: pure iron, steel and cast iron.
 * pure iron: 0-0.008 wt% C
 * steel: 0.008-2.14 wt% C; carbon concentrations rarely exceed 1.0 wt%
 * cast iron: 2.14-6.70 wt% C; commercial cast iron typically contain < 4.5 wt%

Key microstructure for ferrous alloys are presented as follows upon cooling at various compostions, forming eutectoid (or pearlite), hypoeutectoid and hypereutectoid microstructures:
 * Pearlite: forms upon cooling through the //eutectoid// phase transformation from gamma iron (primary phase) --> alpha + Fe3C (lamella). The layer thicknesses are approximately 8 to 1. The mechanical properties are between the soft ferrite phase and the hard cementite phase. [\for some reason, C diffuses out of alpha ferrite into cementite; maybe since ferrite is BCC with less interstitials]
 * //Hypoeuctectoiod// alloy: occurs between 0.022-0.76 wt%. Cooloing as austenite, at alpha ferrite nucleates at the grain boundaries. Further cooling grows the ferrite until the euctectoid temperature (T_e) is crossed. This ferrite found prior T_e is known as //proeutectoid// ferrite. Below T_e, eutectoid pearlite (containing again ferrite/cementite) forms around the proeuctectoid alpha ferrite.
 * //Hypereutectoid// alloy: occurs between 0.76-2.14 wt%. Similarly, while cooling, austenite, cementite forms between grains. Below T_e, gamma_iron transforms to eutectoid pearlite. The resulting cementite below T_e is proeutectoid cementite and coats the grains, making it the weak link mechanically.

=Materials Properties= Properties are in essence the response of a particular material subjected to specific stimulus. Based on what we have discussed regarding the atom and the physical states that arise from the physical arrangement of atoms, one can perhaps appreciate that materials properties stem from a series of macroscopic, microscopic and nanoscopic phenomena. For example, //bonding// is a nanoscopic phenomenon, and it is the bonding between atoms and molecules that attributes in part to wide variation of materials properties.

Acronym used to recall common properties observed in all materials is METCOM. [\chart]

Mechanical - strength (tensile, compressive, shear, {yield, flexural, ultimate tensile}), ductility, modulus, toughness, elasticity, fatigue, hardness, density (actually physical property) Electrical - piezoelectricity, conductivity, resistivity, dielectric constant Thermal - expansion, conductivity, diffusivity, "temperatures" (melting, glass transition, Debye), "heats" (of reaction, of vaporization, of fusion, specific _, ___ capacity), "points" (boiling, triple, flash, softening, eutectic, curie,) Chemical - solubility, surface tension, surface energy, activity, corrosion, Optical - color, reflectivity, absorptivity, transmittance, refractive index, luminescence, phosphorescence, scattering Magnetic - remanence (hysteresis), susceptibility, permeability, ferro-, ferri-. Superconductors: mercury at 0 K, niobium under liquid helium, copper oxide under liquid nitrogen

[\see Gibson, Cellular Solids for review/comparison/chats on density, thermal cond., elastic prop, plastic fract.]

Notice, all of these properties combine all of the fundamental physical sciences including physics, chemistry and biology, which illustrates the interdependent nature of the materials science subject. Mastering materials properties requires reference to fundamental science. A strong background in the following can ease the mastery of materials properties:

Note: the astute reader may wonder how acoustic properties fit among those listed above. While acoustics are not commonly classified in materials science curriculum, one may consider this field as a unique subset of vibrational physics.

Some key materials properties will be discussed.

Elastic Modulus
Various loads yield different states of stress

[\r.Ashby text Fig. 1] [\Define stress, strain] 1 N/m^2 is small. 100000 (10^5) N/m^2 is equivalent to 1 atm. Therefore, a common convention is the megapascal (10^6), where 0.1 MPa = 1 atm.

Strain is a material response to stress and a dimensionless term. It emerges as various terms depending on the state of stress.


 * ~ stress state ||~ strain ||~ symbol ||
 * tensile || tensile || [\epsilon_t] = (L-L_0)/L_0 ||
 * compressive || compressive || [\epsilon_c] = (L-L_0)/L_0 ||
 * shear || shear || [\gamma] = w/L_0 ||
 * pressure || dilational || [\capital delta] = (V-V_0)/V_0 ||

[\r.Ashby text, scan p.50 Fig. 4.3 on different stresses, strains and slopes] As stress is applied, the strain response remains elastic up to the elastic limit, beyond which the material deforms permanently. This stress at which the material yields is known as the //yield strength//. Within the linearly elastic regime Hooke's law is followed, and the stress and strain proportional to an elastic constant called the //elastic modulus//. The elastic modulus can be viewed as the stiffness of a material, or its resistance to deformation. Similarly, each stress state has a specific term for modulus. There are 4 possible stress states: //tensile/compressive, shear, and hydrostatic pressure//.


 * ~ stress state ||~ strain ||~ symbol ||~ modulus ||
 * tensile || tensile || [\epsilon_t] || Young's, E ||
 * compressive || compressive || [\epsilon_c] || Young's, E ||
 * shear || shear || [\gamma] || Shear, G ||
 * pressure || dilational || [\capital delta] || Bulk, B or K ||

The elastic moduli are are related by the following equations:

G = E/(2(1+v); K=E/(3(1-2v)

where v = -[\epsilon_t]/[\epsilon] is the Possion's ratio which account for the shrinkage of an element in the y and z directions as the x direction is stretched. when v~1/3, K~E when v~1/2, K>>E Moduli are measured dynamically by measuring the velocity of sound waves propagating through a material. v ~ (Ep)^1/2

[\i.illustrate stress-strain curve. show linear elastic and plastic regions; show resilience, area up to yield stress; show toughness, area up to fracture stress; discuss secant and tangent modulus for non-linear elastic curves]

Ductility, Toughness, Elasticity
[\r.Ashby text, scan p.53 Fig. 4.4 on stress-strain curves for ceramics, metals, polymers]

The area beneath the stress-stain curve is the elastic strain energy. This stored energy per unit volume is released with the stress is relaxed. It is similar to the energy observed when a catapult is released.

Other Strain Stimuli - Magnetic, Electric and Thermal Fields

 * Stimulus Field || Material Property || symbols ||
 * Stress || Elastic modulus || E, G, B ||
 * Magnetic || Magnetostrictive constant ||  ||
 * Electric || Piezo-electric constant || d, e, g, h ||
 * Thermal || Expansion coefficient || [\alpha_t] ||

Magnetism

We will discuss diamagnetism, papmagnetim, ferromagnetism and ferrimagnetism [\r.Callister] Oscillating charges are the source of magnetic forces. An applied magnetic field or //magnetic field strength// (denoted by H related to turns and current in a solenoid) can generate another magnetic field by //magnetic induction//, or //magnetic flux density// (denoted by B). H and B are field vectors characterized by magnitude and direction in space and linearly related by the //permeability//, mu. The //permeability// or //relative permeability// is a measure of the level a material can be magnetized. The //magnetic susceptibility//, chi_m, is related by chi_m = mu_r - 1.

The dielectric displacement, D and and the magnetic field B are similar: B = mu_0*H + mu_0*M D = epsilon_0*E + P


 * The source of the magnetic moment originates from the orbit of electrons around atoms (a moving charge generating a current loop) and the spin of an electron about its axis**. The //Bohr magneton// is the most fundametnal magnetic moment with mu_B = 9.27 x 10^-24 A*m^2, each electron having a spin up or down of +/- mu_B. Since opposing magnetic moments can cancel, atoms with filled shells cannot be magnetized. The various forms of magnetization are described below:
 * Diagmagnetism: a weak, non-permanent magnetism dependent on an external field. It occurs by the change of orbital motion due to an induced field. The induced magnetization is small, thus mu_r < 1. The magnetic moment is __opposite the direction of the induced field__. Diamagnetism is found in all materials, but the effect is small and often ignored. Some materials have atoms with incomplete cancellations of magnetic moments giving rise to //paramagnetism//. With a field, these moments __aligned in the direction with the induced field__ and enhance the magnetization. mu_r >> 1.
 * Ferromagnetism: Some metallic metals, i.e. transition metals and some rare earth metals, have a permanent magnetic moment without a field mainly due to electron spin. There is a small orbital contribution. //Domains// are volumes of material where these spins couple in alignment. This coupling is disrupted if temperature rise above the //Curie temperature//, T_c. Since H << M, B~ mu_0*M. The //saturation magnetization// M_s occurs when all magnetic dipoles in the material are aligned by an external field.
 * Antiferrimagnetism: A type of magnetism that occurs in nonferromagnetic materials. When the spin of neighboring ions align in opposite directions, the solid having no net magnetic moment. Example, MgO where O^-2 ions have no magnetic moment, but the spin moment of the Mg^+2 ions are aligned antiparallel. Above the //Neel temperature//, the antiferromagnetic effect diminishes and the material becomes paramagnetic.
 * Ferrimagnetism: magnetic ceramics. The source of magnetism is due to an incomplete cancellation of magnetic moments among the ions, i.e. cubic ferrites.

Unlike diamagnetic and paramagnetic materials, the flux density, B and the magnetic field strength, H are non-linear. Under an applied field, the size of domains coalesce non-linearly to a point of saturation. When the field is reversed, a hysteresis occurs when the B field lags behind the H field resulting is a remanent flux density or //remanence//, B_r. The lag is due to the resistance of the domain walls to redirect oppositely in the applied field. Due to some domains still oriented is other directions at H=0, a remaining magnetization exists. To force the field to zero magnetic flux, a coercive field must be applied to drive the field in the opposite direction. H_c, the //coercivity//, indicates the amount of field required to drain the remnant field. The full cycle of driving the field in both directions can generate a //hysteresis loop//.

=Materials Science Applications= This section applies the knowledge acquired about materials classes and properties to real-world material science engineering problems.

code Problem Set 2 A student in Dr. Perkins lab is performing stress-strain analysis on a special type of epoxy cured below its T_g. The curve is plotted below. A strange peak was observed so named Perkins peak.

Q1 - State your opinion of this problem. Does it appear difficult? Why? Q2 - What is the most potential cause of the anomalous Perkins peak? code



Phase Diagrams and Kinetics III (applied)
A suggested guide for independent study of phase diagrams is a text by Ashby (see reference).

Classic Binary Systems
[\(Fe-C, Al-Cu, Cu-Ni, Cu-Zn, Pb-Sn, NiO-MgO, MgO-Al2O3, Al2O3-SiO2)]

Ternary Phase Diagrams Constructing ternary phase diagrams is dictated by three laws of thermodynamics. The //Gibbs phase rule// indicates the number of phases able to coexist for a given system at equilibrium. [\r. see callister for review]

P + F = C + N

P = number of phases F = degree of freedom; number of external variables needed to define a state (i.e. temperature, pressure, composition) C = components; elements and stable compounds N = number of noncompositional variables (i.e. temperature and pressure)

You can use Gibbs phase rule to determine how many variables are required to determine the compositions of a number of phases.

[\Important Complex Systems - Bioglass, super alloy] [\i.Bioglass phase diagram]

Ferrous Metals
[\r.Callister]

Steels
They are used in high temperature, corrosive environments, i.e. steam boilers, aircrafts, missiles. ||  ||
 * ~  ||~   ||~   ||~ Description ||~   ||
 * Low Alloy || Low-C || Plain || Low carbon steels are the most highly produced. They have compositions < 0.25 wt% C. They are strengthened by cold-working. Microstructures contain ferrite and pearlite making the alloys soft and weak, but highly ductile and tough. They are machicable, weldable and the least expensive. They are used for automobile components, I-beams and sheets for buildings and bridges. ||  ||
 * " || " || High strength || High strength, low alloy (HSLA) steels have other elements up to 10 wt% to improve corrosion resistance, i.e. Cu, V, Ni, Mo. ||  ||
 * " || Med-C || Plain || These steels have compositions between 0.25 - 0.60 wt% C. They are heat treated to improve mechanical properties by austenizing, quenching and tempering. Tempering forms martensite. ||  ||
 * " || " || Heat treatable || Heat treating offers a combination of highs strength, wear resistance and toughness. They have low hardenabilities, high strengths compared to low-carbon steels, but lower ductility and toughness. They are used in railway wheels and tracks, gears, crankshafts. ||  ||
 * " || High-C || Plain || These steels have compositions between 0.60-1.4 wt%. They are the hardest, strongest, least ductile steels. They are used in wear resist applications and can keep a sharp cutting edge. ||  ||
 * " || " || Tool || Combinations of Cr, V, W, Mo, form highly wear-resistant compounds. These are used as cutting tools, dies and high-strength wire. ||  ||
 * High Alloy ||  || Tool || " ||   ||
 * " ||  || Stainless || Stainless steels are resistant to corrosion. Concentrations of >=11 wt% Cr is required for corrosion resistance; Ni and Mo enhances corrosion resistance. There are three classes, martensitic, ferritic, austenitic:
 * martensitic - can be heat treated such that martensite is the primary component; magnetic.
 * ferritic - contains BCC alpha ferrite. It is not heat treatable; magnetic.
 * austenitic - the gamma iron phase field is extended to room temperature. [\?] Not heat treatable; not magnetic.

Cast Irons
Cast irons contain above 2.14 wt% C, but in practice contain between 3.0-4.5 wt% C.


 * ~  ||~ Description ||
 * Gray Iron || Contain 2.5-4.0 wt% C and 1.0-3.0 wt% Si. Graphite content exists in the forms of flakes surrounded by alpha ferrite or pearlite matrix. The sharp points on the the flakes can serves as stress concentrators; thus have poor mechanical properties. They are used in applications that dampen vibrational energy, high wear resistance and have low viscosity for casting intricate shapes. ||
 * Ductile (Nodular) || Adding Mg or Ce creates new microstructures having spherical or nodular particles instead of flakes. Moderate cooling can make surrounding matricies with pearlite. Slow cooling can yeild ferritic ducile cast iron having strengths comparable to steel. ||
 * White Iron || For < 1.0 wt% Si and fast cooling rates, most carbon exists as cementite instead of graphite. The fracture surface has a white appearance and thus its name. The high fraction of cementite makes it hard but brittle and non-machinable [\particularly due to the proeutectoid cementite between the grains, a path of least resistance for crack propagation]. There are limited applications for need of high wear resist surfaces. ||
 * Malleable Iron || Heating white iron between 800 and 900 C in a neutral atmosphere for a long time decomposes the cementite into graphite which forms clusters surrounded by ferrite or pearlite. It is used for connecting rods, transmission gears, differential cases, flanges and pipe fittings. ||
 * Compacted Graphite Iron || This recently developed material has a microstructure and properties intermediate between gray iron and ductile nodular iron. These properties are accomplished by transforming carbon into graphite with the presence of Si. The chemistry is complicated. This material is used is the folloing applications: diesil engine blocks, exhaust manifolds, gearbox housings, flywheels, high speed train disk brakes. ||

Non-Ferrous Alloys
Steels can be substituted with other materials due to its high density, relatively low electrical conductivity and susceptibility to corrosion. The various alloys will be mentioned below. //Cast alloys// are brittle and cannot sustain deformation by shaping. The mechanical properties of //wrought alloys// can altered by cold working or heat treatments (including precipitation hardening and martensitic transformation).

Copper
Copper has high ductility and electrical conductivity. It is alloyed to improve other mechanical properties as brass, bronze or beryllium copper. This native Cu is used in jewelry
 * Brass - brasses are copper alloys that contain zinc as a substitutional element. The FCC alpha phase is stable for concentrations up to 35 wt% Zn, which is soft. The BCC beta' phase is harder and stronger than the alpha phase.
 * Bronze - bronzes are copper alloys contain various elements such as Sn, Al, Si and Ni. They are stronger than brass.
 * Beryllium copper - these are the most common heat-treatable copper alloys containing 1.0-2.5 wt% Be. They have high tensilte strengths, electrical conductivity, wear resistance and corrosion resistance. They are used in landing gear bearings, springs and surgical instruments.

Aluminum
Aluminum has relatively low density (2.7 g/cm3) than steel (7.9 g/cm3), has high electrical/thermal conductivity and corrosion resistance. Al has high ductility and can be rolled into thin sheets. Al is thermally restricted due to a low melting temperature of 660 C. Mechanical strength is improved by cold working and alloying (with Cu, Mg, Si Mn, Zn), which can reduce corrosion resistance. A new class of Al-Li alloys have been developed for aircraft and aerospace applications, having low densities, high specific moduli, great fatigue and toughness properties.

Magnesium
Magnesium has the lowest density of all structural metals (1.7 g/cm3) and is used in lightweight applications, i.e. aircraft and aerospace. Mg is mostly cast because it is difficult to deform at low temperatures. It is ignitable as a powder. Mg has replaced many plastics of comparable density for its strength, stiffness and recyclability i.e. hand-held devices, computer devices, automobiles.

Titanium
Titanium has low density, high specific strengths, high melting point and high elastic modulus. However, Ti is has lower corrosion resistance at elevated temperatures. They are used in aircraft, aerospace, surgical instruments.

Refractory Metals Refractory metals have high melting temperatures between 2468 - 3410 C i.e. Nb, Mo, W, Ta. W is used as filaments in incandescent bulbs. Ta is chemically resistant in nearly all environments below 150 C.

Superalloys
These alloys have the greatest combination of properties and are used turbine components at high temperatures. The main element is typically Co, Ni, Fe and alloyed with Cr, Ti or the refractory elements (Nb, Mo, W, Ta).

Noble Metals The noble metals are Ag, Au, Pt, Pa, Rh, Ru, Ir, Os. They are typically soft, ductile and oxidation resistant. Ag and Au are solid-solution strengthened with Cu; sterling silver contain 7.5 wt% Cu.

Other Alloys Ni is corrosion resistant and often plated on other metals as a protective layer. Lead and tin alloys are weak and soft having low melting temperatures and corrosion resistance. Zn is soft and suspectibile to corrosion. Zn is used to coat and protect steels by preferential corrosion. Zr is abundant, ductile and corrosion resistant. They are also neutron transparent and used in cladding for U fuel in nuclear reactors.

In 1896, Charles-Edouard Guillaume discovered low expansion iron-nickel alloys and won the 1920 Nobel Prize. The low expansion is not due to the symmetry of the potential energy vs. interatomic spacing curve as presumed. Rather, this property is related to the ferromagnetic states of each metal, which compensate volume expansions as the metals are heats above the curie temperatures. Three unique compositions are described below:
 * Invar: 64 wt% Fe-36 wt% Ni; a = 1.6 x 10^-6 C^-1
 * Super-Invar: 63 wt% Fe, 32 wt% Ni, 5 wt% Co; a = 0.72 x 10^-6 C^-1
 * Kovar: 54 wt% Fe, 29 wt% Ni, 17 wt% Co; a ~ borosilicate glass.

Materials Design and Selection
[\Ashby Plot]

[\r.Manuel course]