145 psi = 1MPa = 10 N / m 2 1 GPa = 10 N / m 2 1) Atomic size factor: difference in atomic radii between two atoms is less than ~15%.
2 2 3
1 J = 1 kg m / s = 1 kPa L = 1 Pa m 2) Crystal structure: crystal structures for metals of both atom types must be the same.
E = hc / λ λ = h / mv = h/p 3) Electronegativity: greater difference in EN will form an intermetallic compound instead of
ρ = nA / V NC A substitutional solid solution.
n = number of atoms associated with each unit cell 4) Valances: a metal will have more of a tendency to dissolve another metal of higher valency
V C volume of unit cell A = atomic weight than one of lower valency.
N A Avagadro’s number (6.023 x 10 23atoms / mol) Edge dislocation is a linear defect that centers around the line that is defined along the end of
(- Qv / kT)
N V N e the extra half-plane of atoms. This is sometimes termed the dislocation line.
N = number of vacancies N = the total of atomic sites Screw dislocations are formed by a shear stress that is applied to produce a distortion where
Q v the energy required for the formation of a vacancy T = temperature (in Kelvins) one region is shifted one atomic distance compared to another portion.
k = Boltzmann’s constant = 1.38*10 -23J/atom K = 8.62*10 eV/atom K . Dislocations are normally mixed and there magnitude is measured by Burgers vector.
N = N pA/ Atomic mass of element p = density N = atoms / m Diffusion / Material transport / Atomic motion = transfer of mass within a solid
weight percent (wt%) C = m / m + m *100% Interdiffusion / impurity diffusion. = Process whereby atoms of one metal diffuse into another
1 1 1 2
atom percent (at%) C 1 = n m1 / nm1 + n m2 *100% n = m / mm Self-diffusion = Diffusion within pure metals, all atoms exchanging positions are same type
ASTM… N = 2 n-1 N = the average # of grains per square inch at 100x n = Grain # Vacancy diffusion= normal atom to vacancy, limited by number of vacancies in substance
Speed diffusion occurs (rate of mass transfer) diffusion flux (J), defined as the mass (or, CES Material Index
equivalently, the number of atoms) M diffusing through and perpendicular to a unit cross- S = CEWt / 12L 3 find material index using minimum weight
sectional area of solid per unit of time. J = M / At A = area J = kg / m s Our free variable would be w.
Frick’s first law J = –D (∆C/∆x) 2 m = ρV = ρw3t 3 w = 2 / ρLt n4w sub w into original equation
Constant of proportionality D (diffusion coefficient, m /s) S = CEmt / ρLt12L = CEmt / 12ρL we want to minimize m
The negative sign in the expression indicates that the direction of diffusion is down the m = 12SρL / CEt but using material properties
concentration gradient, from a high concentration to a low concentration. m = Sρ / E but S was fixed therefore
Driving force, Concentration gradient = ∆C / ∆x m = ρ / E minimize and that is final answer.
D = D eo (-Qd / RT) Crystalline Amorphous Mixed
D o a temperature-independent pre-exponential R = the gas constant Metals Usually (steel, brass) Rarely (metallic glass) Never
Qd = the activation energy for diffusion T = temperature (Kelvins) Ceramics Often (alumina) Often (soda glass) Often (silicon nitride)
Polymers Never (“crystalline” polymers Usually (polyethylene) Sometimes (nylon)
Elastic Modulus F / A o E (D /L L o
F = Force A o Cross sectional area E = Elastic Modulus always partly amorphous)
More dislocations lowers entropy (S), Dislocation is always thermodynamically unstable.
L o undeformed length D = exceLs deformed length
Bragg’s law says that constructive interference will occur if the extra path is a multiple of the Vacancies increase entropy (∆S randomness).
Diffusion is faster for… Diffusion is slower for…
wavelength. Formula - nλ=2dsinθ
spacing between planes = d = a / √(h + k + l )2 2 a = relationship of r = cubic side Open crystal structures close-packed structures
Lowering melting T materials higher melting Tmaterials
Thermodynamics of Vacancies (not covered in text) Materials with secondary bonding materials with covalent bonding
Some vacancies lower “free energy” ∆G = ∆H – T∆S Total energy
2 3 2 Smaller diffusing atoms larger diffusing atoms
∆Gtot =V∆Gsol + 4π r γ = 4/3sπr ∆G sol+ 4π r γ sfor graph surface area increases, surface Engineering Stress σ = F / A 0
energy increases, so volume decreases, volumetric energy decreases)
F = Force applied perpendicular to the specimen cross section (N)
∆G positive causes nothing, ∆G negative driving force, ∆G = equilibrium A 0 the original cross-sectional area before any load is applied (m ) 2
A crystal only has one perfect configuration. Configurational entropy is zero. Vacancies (on 6 2
σ = Engineering Stress (1MPa = 10 N / m )
the other hand) give rise to many configurations. Add n vacancies to a lattice with N atoms. So Engineering Strain ε = (li– l0) /0l = ∆l / 0
as n increases S increases G decreases
l0= original length before any load is applied (m) l = initantaneous length
Create vacancies until G is at a minimum. To find minimum r take derivative. ε = Engineering strain (unitless or m/m) (can also be expressed as a percentage)
X = n* / N = e (-Qv/RT)
v (-Qsol/kT) Shear and Torsional Tests: τ = F / A 0
X sol= e F = Force or load imposed parallel to the upper and lower faces (N)
q = -k (∆T/∆x) ∆x = x a∆T q = -k / ax q = heat flux 2
o o A 0 Area of faces (m ) τ = Shear stress
gradient line is a defined line that can be placed on a graph. Elastic Deformation
∆Q = wρC ∆T
P Stress-Strain Relationship σ = E ε
heat absorbed = wall thickness * density * E = Constant of Proportionality / Modulus of elasticity / Young’s Modulus (GPa or psi) E
heat capacity * temperature difference
typically ranges from 45GPa to 407GPa
w = √(2at) wall width = √ (2 * thermal diffusivity * diffusion time) If there is a constant modulus then it is called elastic deformation. The greater the modulus the
a ≡ k / ρCP a ≡ thermal conductivity / ( density * specific heat )
stiffer the material, or the smaller the elastic strain that results from the application of a given
∆Q = (∆T√[2t])√( kρC ) P stress. Elastic deformation is nonpermanent. E is proportional to dF/dr
V = IR P = VI R = ρL/A ρ = 1 / σ ρ = V*A / I*l J = σε
τ = Gγ G = shear modulus, slope of linear elastic region
Doping: Add rich Al layers on surface, heat it and then you have a doped semiconductor
Poisson’s ratio (v) is the ratio of lateral and axial strains. v = - ε x ε =z- ε /yε z
regions. Moblity is a measure of a charge carrier’s ability to move through a material. E = 2G(1+v)
Since conductivity depends on a materials ability to conduct charge, a higher dopant mobility
Anelasticity is known as the time that a material takes to return to its original length after a
also means a higher conductivity value. stress has been applied to it.
Atomic hard sphere model: spheres representing nearest-neighbor atoms touch one another.
Plastic deformation occurs when strains are over about 0.005, where permanent, non-
Lattice = three-dimensional array of points coinciding with atom positions (or sphere centers). recoverable deformation occurs.
Coordination number = # of nearest-neighbor or touching atoms
The stress corresponding to the intersection of this line and the stress-strain curve as it bends
Atomic packing factor (APF) = volume of atoms in a unit cell / total unit cell volume over in the plastic region is defined as the yield strength σ (MPy or psi).
Simple-cubic Crystal (SCC) # of atoms = 1, Not closed packed
3 The elastic-plastic transition is very well defined and occurs abruptly in what is termed ayield
Face-centered cubic (FCC) γ a = 2R√2 CN = 12 V= 16R √2 ABCABCABC point phenomenon.
# of atoms = 8*1/8 (corners) + 6*1/2 (faces) = 4 Closed packed, APF= 0.74 (max)
After yielding, the stress necessary to continue plastic deformation in metals increases to a
Body-centered cubic (BCC) α a = 4R / √3 CN = 8 maximum, and then decreases to the eventual fracture. The tensile strength TS (MPa) is the
# of atoms = 2 Not close packed, APF = 0.68
stress at the maximum on the engineering stress-strain curve.
Hexagonal close-packed (HCP) a = 2R√2 CN = 12 ABABABAB Resilience is the capacity of a material to absorb energy when it is deformed elastically and
Area of hexagon = (3sin60)a 2 c/a ratio is 1.633 (can deviate) c = height
then, upon unloading, to have this energy recovered.
# of atoms = 6 Closed packed, APF= 0.74 U r ∫σdε = ½ σ ε =y y 2y / 2E (J/m )
Polymorphism: Metal/non-metal with more than one crystal structure.
Toughness is a mechanical term that is used in several contexts; loosely speaking, it is a
Allotropy: polymorphism found in elemental solids measure of the ability of a material to absorb energy up to fracture energy up to fracture.
Single crystal repeated arrangement of atoms without interruption . . -1 3
ρ = Resistivity (Ω m) σ = electrical conductivity (Ω m) p = the number of holes per m
Point (x,y,z) = h l k, ī is negative, Line is [h l k], Plane (h l k) J = current density E =FFermi energy E = Bang Gap V = drift vedocity
Line: start at beginning of line, find vector coords, subtract, multiply by common factor 2 .
µ e constant of proportionality, electron mobility (m / Vs) n = electron concentration or free
Plane: find intercepts in x,y,z, find reciprocals, and multiply to get integers electrons / m (m ) C = Heat capacity Q= Energy A = temperature-independent constant ( Ω .
Crystalline = material where atoms are situated in a repetitive three-dimensional pattern, each .
m) l = length ά =llinear coefficient of thermal expansion (W / mk) E = modulus of elasticity q
atom is bonded to its nearest-neighbor atoms. = heat flux k = thermal conductivity
Polycrystalline = Crystalline solid composed of collection of small crystals (grains) 7 . -1
Metal conductivities on the order of 10 (Ω m)
Grain boundary = atomic mismatch with the region where two grains meet, occur at angles Insulators low conductivities, ranging between 10 -10and 10 -20(Ω m) -1
Cooled Quickly Smaller Grain Boundary Stronger substance -6 4 . -1
Semiconductors intermediate conductivities, generally from 10 to 10 (Ω m)
Noncrystalline/ amorphous = solids which lack systematic/regular arrangement of atoms The energy corresponding to the highest filled state at 0 K is called the Fermi Energy, E .
Imperfections/ Crystalline defect (tend to change the properties of the substance) = a lattice When electrons attain this energy they jump the conduction band.
irregularity. I.e. point defects (one or two atomic positions), linear (one-dimensional), Two final band structures are similar; one band (the valance band) that is completely filled with
interfacial/boundaries (two-dimensional) electrons is separated from an empty conduction band, and an energy band gap lies between
Vacancy (point defect) = normally occupied lattice site whom atom is missing. Impossible to them. Insulator if band gap is greater than 2eV.
create solid without vacancy. Increases entropy (randomness) of crystal, decreasing free energy Onlyelectrons with energies greater than the Fermi energy may be acted on and accelerated in the presence of an electric field. These
Substitution = replaces atom with different atom are the electrons that participate in the conduction process, which are termedfree electrons. Another charged electronic en