- No category
Materials Science Cheat Sheet: Formulas & Concepts
advertisement
n= diffraction order Electronegativity on periodic table: low → right Ionic Bonds: + and − ions Cation Bonds: non-metal bond ~ bond where atoms share a pair of electron % Ionic Character of Interatomic bonds: 2 % = (1 − π −0.25(βπΈπ) ) ∗ 100 Bonding Forces: FN = FA + FR ∞ Potential Energy between two atoms: EN = ∫r FN dr = ∞ ∞ ∫r FA dr + ∫r FR dr = EA + ER EN = EA + ER π=wavelength (nm) ddhk=interplanar spacing Bragg's Law: a Interplanar spacing: dhkl = 2 2 2 √h +k +l h,k,l = miller indices of the plane a = unit cell edge length Process for finding Miller Indices: 1. Take reciprocal 2. linearize Ratio (%) of element mass to total alloy mass π1 πΆ1 = ∗ 100 π1 + π2 Burger Vector ( b ): Perpendicular to edge dislocations. Parallel for screw dislocations Fraction of Vacancies: ππ£ = # of vacancies K = Boltzmann constant ππ£ ππ N = total # of atomic sties T = Temperature (Kelvin) = ππ₯π − π ππ ππ£ = energy required to form vacancy Celsius → Kelvin K = °πΆ + 273.15 (DIFFUSION) Hume-Rothery Conditions for Complete Solubility 1. 2. Atomic Packing Factor: APF = volume of ions volume of cell 4 Volume of Sphere: Vsphere = πr 3 3 Volume of cell: Vcell = a3 nA Density for unit cell: p = = V c NA n′ (Ac +AA ) Density of Ceramics: p = # of atoms cenetered on plane Planar Density: PD = π3 3. 4. V c NA Ac = sum of atomic weight of cations in unit AA = sum of atomic weights of aniona in unit Vc = volume of unit cell n' = # of formula units/unit cell NA = avogadro's # Linear Density: LD = 4 3 π∗ π(π 3 ) area of plane # of atoms cenetered on plane area of plane r Ionic Size Ratio: cation ranion Surface Energy: Density of Plane↑ = S.E. ↓ for FCC – most dense = (1 1 1 ) For BCC – most dense = ( 1 1 0 ) Atomic Size Factor: The atomic radii of the two elements must differ by no more than 15%. |π1 − π2 | ∗ 100% ≤ 15% π1 Crystal Structure: Both elements must have the same crystal structure (e.g., both FCC, both BCC). Valency: The elements should have the same or similar valency. A metal tends to dissolve a metal of higher valency more readily than the reverse. Electronegativity: The elements must have similar electronegativities. Large differences may lead to compound formation instead of solid solution. Interstitial Solution: contains small nonmetals (C, H, O, N, B) Interdiffusion: Diffusion of atoms of one material into another material Self-diffusion: Atomic migration in a pure metal BCC has faster diffusion than FCC πππ π ππ πππππ’π ππ π ππππππ π Diffusion Flux: π½ = = ππππ∗π‘πππ π΄π‘ Fick’s First Law: –steady-state diffusion: The rate of mass transfer is proportional to the first derivative of concentration with position (negative sign means high → low concentration) ππΆ πΆ −πΆ π½ = −π· = −π· π΄ π΅ J = diffusion flux, D = diffusion coefficient, ππ₯ ππΆ π₯π΄ −π₯π΅ = concentration gradient, c = concentration, x = diffusion position Fick’s Second Law: –non-steady-state diffusion: The rate of change of concentration over time is proportional to the second derivative of concentration with position ππ₯ ππΆ ππ‘ =π· π2 πΆ ππΆ Solution: πΆπ −πΆπ ENERGY π΄ Attractive Energy πΈπ΄ = − , π΄ = = rate of change of concentration w/ time ππ₯ 2 ππ‘ πΆπ₯ +πΆπ = 1 − erf ( π₯ 2√π·π‘ ) use table for erf( ) π ππΈ π2 π· = π·π ππ₯π(− π ) D = diffusion coefficient ( ), π π π π·π = pre-exponential, R = Gas Constant, Qd = activation energy Diffusion Distance Formula: π₯ = √π·π‘ MECHANICAL PROPERTIES βπΏ πΉ π Strain π = Stress π = (A = area) modulus of Elasticity πΈ = πΏπ Hooke’s Law π = πΈπ and π΄ πΉ π΄ π π π −ππ = π ππ πΈ πΈ πΈ πΈ2 πΈ2 Poisson’s Ratio π = − π₯ = − π₯ Shear Stress π = ππππ ∅πππ π Fracture Toughness πΎπΌπΆ = ππ√ππ a= crack length (when internal do ½ a) , Y = geometry factor, M = momentum πΉ π βπΏ Torsion: π = = Thermal Expansion: = πΌ(π2 − π1 ) π΄π π΄π π Bending Stress: π = πΏ 32π ππ3 = 16πΉ∗πΏ πππ3 Moment: π = πΉ∗πΏ π πΌ∗π π Ohm’s Law: π½ = π ∗ π E-Field: π = π Drift Velocity: π£π = ππ π Electrical Conductivity: π = π|π|π ∗ π POLYMERS (hydrogen + carbon atoms) » Stretching = energy ~~~ break stretch = energy release » Isotactic (same side R), syndiotactic (Opposite side R), Atactic (Random), Stereoisomerism (linked together in the same order), π‘ππ‘ππ ππππ¦πππ π€π‘ Molecular number ππ = = ∑π₯π ππ Weight Avg. ππ€ = ∑π€π ππ π‘ππ‘ππ # ππ ππππππ’πππ ππ = ππππππ ππ π ππ§π πππππ, π€π = π€πππβπ‘ πππππ‘πππ, m = molecular weight Degree of Polymerization π«π· = 4 Yield Strength = when plastic deformation begins | Ductility = amount of plastic deform. @ failure (Measured w/ %EL & %RA) Resilience = area under stress strain to yielding | Toughness = energy absorbed before fracture ~ Area under stress-strain curve Interstitial occurs more rapidly than substitutional (|π§1 | ∗ π)(|π§1 | ∗ π), πΉ = , πΈπ΄ + πΈπ = πΈπππ‘ ππ ELECTRIC RESISTIVITY 1 π π΄ ππ΄ Electrical Resistivity: π = = = , Influence of Temperature on Diffusion π 1 4πππ π΄π π Composites (uper) πΈπ (π’) = πΈπ ππ + πΈπ ππ (lowr) πΈπ (π) = π∗ π (πΈπ πΈπ) πΈπ ππ πΈπ ππ 1 =࡬ πΈπ ππ + 1 πΈπ ππ −1 ΰ΅° Fibers ππ = π , if fiber length, l <15 lc then is it discontinuous 2ππ ππ = shear stress to break away, ππ = volume fraction SEMICONDUCTORS Intrinsic: π = ππ |π|(ππ + ππ ) P-Type extrinsic: π β« π (πππ π π − ) ~~ π΅, π΄π, πΊπ (ππππππ‘πππ ) ~~ π = π|π|ππ Influences: Increase impurity → decrease mobility Increase Temp → decrease mobility π π΄ ππ΄ Ohm’s Law: π = πΌπ Resistivity: π = = Conductivity π = 1 π ππ2 π l −ππ ππ ∗ 100 ππππ’ππ‘πππ% = π΄π −π΄π π΄π ∗ 100 Fracture cuts: strong necking > jagged necking > flat break Mass fraction or weight % Dislocations Dislocation Density = πππππ π ππππππππππ ππππππ ππππ ππππππ Yield Strength of a single crystal = ππ = Electron Mobility ππ = π¨π π ππππππ = ππ + ππ π −π ππππππ ππππππππ π π 2 ππ ππ‘ (πππ) Stress concentrator: ππ‘ = π = 2 α α FAILURE πΎ = specific surface energy, If ππ > ππ ∴ πππππ ππππππππ‘ππ Maximum stress: ππ = 2ππ ΰΆ§ π ππ‘ 2πΈπΎ Critical Stress ππ = ΰΆ§ ππ π π∗π π΄ Transit Time π‘ = Carrier Concentration in semicond. ππ = 1 π¨ −π¨π % Cold work = π πΌ∗π πΏ E. Field πΈ = , π΄ = , π = π 4 ππ΄ A = cross-sect. area, I = current, π = length, Z = # free e- per atom, A = atomic weight ππ π Free Electron Density (ππ3) π = π΄ Drift Velocity π£π = ππΈ Ductility Elongation% = f π Density π½ = ππΈ π πΏ π£π π(ππ +πβ ) e = elementary charge, ππ =electron mobility, πβ =hole mobil Electrical Conductivity: π = π ∗ 1.602 ∗ 10−19 ∗ ππ , n = # free electrons Total Conductivity: π = π(πππ + ππβ ) ~~~~~~~ Determining Donor vs Acceptor, P-Type vs N-Type ~~~~~~~~~ Group 3A = P-Type & Acceptor Group 5A = N-Type & Donor | P-type semiconductors have an extra hole. N-type semiconductors have an extra electron. THERMAL Types of heat transfer: conduction: direct molecular interaction convection: bulk motion of fluid Radiation: Electromagnetic emission ππ ππ Heat Capacity: πΆ = Thermal Conductivity (fourier’s): π = −π ππ ππ₯ OPTICAL βπ π Planck’s Law: πΈ = βπ£ = Index of Refraction: π = π π£ Q = heat energy (J), T = Temp. (K) l = length, q = heat flux (J/m2s), E = energy of photon (J), h = planck’s constant k = thermal conductivity (W/m*k) -34 6.626*10 J, v = frequency, c = speed of light 3E8 dT/dx = temp. gradient (K/m) m/s, π = wavelength Snell’s Law (Refraction of Light): π πππ1 π πππ2 = π2 π1 , π1= incidence angle, π2= refracted angle Critical Angle for Total Internal Reflection (n2 < n1): π ππ(ππ ) = Reflectance and Absorption: πππππ πππ‘π‘ππ πππ‘πππ ππ‘π¦ = πΌπ π π½π π2 π1 Io = incident intensity π½= absorption coefficient l = sample thickness Average Energy = max+πππ 2 Interpolation formula: π = π1 + Polystrene = C8H8 Budadiene = C4H4CH2 = CH2 πΈ−πΈ1 πΈ2 −πΈ1 ∗ (π2 − π1)
0
advertisement
Download
advertisement
Add this document to collection(s)
You can add this document to your study collection(s)
Sign in Available only to authorized usersAdd this document to saved
You can add this document to your saved list
Sign in Available only to authorized users