THERMAL PROPERTIES Heat Capacity The ability of a material to absorb heat • Quantitatively: The energy required to produce a unit rise in temperature for one mole of a material. energy input (J/mol) heat capacity (J/mol-K) temperature change (K) • Two ways to measure heat capacity: Cp : Heat capacity at constant pressure. Cv : Heat capacity at constant volume. Cp usually > Cv • Heat capacity has units of

Dependence of Heat Capacity on Temperature -- increases with temperature -- for solids it reaches a limiting value of 3R R = gas constant 3R Cv = constant = 8.31 J/mol-K Cv Adapted from Fig. 17.2, Callister & Rethwisch 3e. T (K) q D Debye temperature (usually less than T ) room • From atomic perspective: -- Energy is stored as atomic vibrations. -- As temperature increases, the average energy of atomic vibrations increases.

Atomic Vibrations Atomic vibrations are in the form of lattice waves or phonons Adapted from Fig. 17.1, Callister & Rethwisch 3e.

Specific Heat: Comparison Selected values from Table 17.1, Callister & Rethwisch 3e. • Polymers Polypropylene Polyethylene Polystyrene Teflon cp (J/kg-K) at room T • Ceramics Magnesia (MgO) Alumina (Al2O3) Glass • Metals Aluminum Steel Tungsten Gold 1925 1850 1170 1050 900 486 138 128 cp (specific heat): (J/kg-K) Material 940 775 840 Cp (heat capacity): (J/mol-K) increasing cp

Other heat capacity contributions Electrons absorbing energy by increasing their kinetic energy Only for free elections Very small for insulating/semiconducting materials Randomization of electron spin at some specific temperature These two factors are (very) small compared to the contribution from the vibrations

Thermal Expansion Materials change size when temperature is changed Tinitial  initial Tfinal > Tinitial Tfinal  final linear coefficient of thermal expansion (1/K or 1/°C) For isotropic materials αv = 3αl

Atomic Perspective: Thermal Expansion Thermal expansion arises from an increase in the average distance between the atoms Symmetric curve: -- increase temperature, -- no increase in interatomic separation -- no thermal expansion Asymmetric curve: -- increase temperature, -- increase in interatomic separation -- thermal expansion

Coefficient of Thermal Expansion: Comparison Polypropylene 145-180 Polyethylene 106-198 Polystyrene 90-150 Teflon 126-216 • Polymers • Ceramics Magnesia (MgO) 13.5 Alumina (Al2O3) 7.6 Soda-lime glass 9 Silica (cryst. SiO2) 0.4 • Metals Aluminum 23.6 Steel 12 Tungsten 4.5 Gold 14.2 a (10-6/C) at room T Material increasing  Polymers have larger  values because of weak secondary bonds • Q: Why does a generally decrease with increasing bond energy? A: deeper and narrower energy “trough”

Thermal Expansion: Example Ex: A copper wire 15 m long is cooled from 40 to -9°C. How much change in length will it experience? Answer: For Cu rearranging Equation 17.3b

Thermal Conductivity The ability of a material to transport heat. Fourier’s Law temperature gradient heat flux (J/m2-s) thermal conductivity (J/m-K-s) T1 T2 T2 > T1 x1 x2 heat flux • Atomic perspective: Atomic vibrations and free electrons in hotter regions transport energy to cooler regions.

Mechanisms of Heat Conduction  

Thermal Conductivity: Comparison Energy Transfer Mechanism Material k (W/m-K) increasing k • Metals Aluminum 247 atomic vibrations and motion of free electrons (electron motion is more efficient) Steel 52 Tungsten 178 Gold 315 • Ceramics Magnesia (MgO) 38 Alumina (Al2O3) 39 Soda-lime glass 1.7 Silica (cryst. SiO2) 1.4 atomic vibrations (phonons are primarily responsible) • Polymers Polypropylene 0.12 Polyethylene 0.46-0.50 Polystyrene 0.13 Teflon 0.25 vibration/rotation of chain molecules Selected values from Table 19.1, Callister & Rethwisch 3e.

Thermal Stresses • Occur due to: Thermal stress  -- restrained thermal expansion/contraction -- temperature gradients that lead to differential dimensional changes Thermal stress 

Example Problem  0  0 D Tf  0 D -- A brass rod is stress-free at room temperature (20°C). -- It is heated up, but prevented from lengthening. -- At what temperature does the stress reach -172 MPa? Solution: T0 Original conditions  0 Tf Step 1: Assume unconstrained thermal expansion  0 D Step 2: Compress specimen back to original length  0  D 14

Example Problem (cont.)  0  The thermal stress can be directly calculated as Noting that compress = -thermal and substituting gives 20 x 10-6/°C Answer: 106°C 100 GPa 20ºC Rearranging and solving for Tf gives -172 MPa (since in compression) 15 15

Thermal Shock Resistance • Occurs due to: nonuniform heating/cooling • Ex: Assume top thin layer is rapidly cooled from T1 to T2 s rapid quench resists contraction tries to contract during cooling T2 T1 Tension develops at surface Temperature difference that can be produced by cooling: Critical temperature difference for fracture (set s = sf) set equal • • Large TSR when is large

Summary The thermal properties of materials include: • Heat capacity: -- energy required to increase a mole of material by a unit T -- energy is stored as atomic vibrations • Coefficient of thermal expansion: -- the size of a material changes with a change in temperature -- polymers have the largest values • Thermal conductivity: -- the ability of a material to transport heat -- metals have the largest values • Thermal shock resistance: -- the ability of a material to be rapidly cooled and not fracture -- is proportional to

Magnetic Properties Generation of a Magnetic Field -- Vacuum • Created by current through a coil: H I B0 N = total number of turns  = length of each turn (m) I = current (ampere) H = applied magnetic field (ampere-turns/m) B0 = magnetic flux density in a vacuum (tesla) • Computation of the applied magnetic field, H: (A / m) B0 = 0H permeability of a vacuum (1.257 x 10-6 Henry/m) • Computation of the magnetic flux density in a vacuum, B0:

Magnetic field vector etc. Electric field can be defined in terms of the force acting on a charge Ie, if a charge q at rest is experiencing electric force FE from the electric field E, FE = qE where E is electric field. Magnetic field could be defined in a similar way, if there is a monopole, but there’s none Instead, we can use the moving charge q with velocity v. The force acted on the moving charge by a magnetic field B is FB = qv x B Then the unit of B is N / (coulomb m / s) = N/(A m) = T = Tesla

Magnetic field vector etc.  

Finally   B0 = 0H

H & B B : magnetic field in the material From external magnetic field + “internal” magnetic field (magnetization) Magnetic induction, magnetic flux density H: “Driving” magnetic influence from external field Magnetic field strength In vacuum, B & H are essentially same (up to 0) H can be defined as H = B/ 0 - M

Generation of a Magnetic Field -- within a Solid Material • A magnetic field is induced in the material B B = Magnetic Induction (tesla) inside the material applied magnetic field H B = H permeability of a solid current I • Relative permeability (dimensionless)

Generation of a Magnetic Field -- within a Solid Material (cont.) • Magnetization M = mH Magnetic susceptibility (dimensionless) • B in terms of H and M B = 0H + 0M • Combining the above two equations: B = 0H + 0 mH H B vacuum cm = 0 cm > 0 < 0 permeability of a vacuum: (1.26 x 10-6 Henry/m) = (1 + m)0H cm is a measure of a material’s magnetic response relative to a vacuum

Origins of Magnetic Moments • Magnetic moments arise from electron motions and the spins on electrons. magnetic moments electron electron nucleus spin Adapted from Fig. 18.4, Callister & Rethwisch 3e. electron orbital motion electron spin • Net atomic magnetic moment: -- sum of moments from all electrons. -- in atom with completely filled shells, there is total cancellation of orbital & spin moments -> cannot be permanently magnetized

Types of Magnetism (3) ferromagnetic e.g. Fe3O4, NiFe2O4 (4) ferrimagnetic e.g. ferrite(), Co, Ni, Gd ( cm as large as 106 !) B (tesla) (2) paramagnetic ( e.g., Al, Cr, Mo, Na, Ti, Zr cm ~ 10-4) vacuum ( cm = 0) (1) diamagnetic ( cm ~ -10-5) e.g., Al2O3, Cu, Au, Si, Ag, Zn H (ampere-turns/m) Plot adapted from Fig. 18.6, Callister & Rethwisch 3e. Values and materials from Table 18.2 and discussion in Section 18.4, Callister & Rethwisch 3e.

Diamagnetic / Paramagnetic none random No Applied Magnetic Field (H = 0) opposing aligned Applied Magnetic Field (H)    

Ferromagnetism  

Antiferromagnetism Antiferromagnetism Another form of magnetic moment coupling between the adjacent atoms/ions Forms antiparallel alignment of magnetic dipoles MnO (Manganese Oxide) O2- ion has no net magnetic moment Mn2+ has net magnetic moment (due to spin) They align so that the adjacent ions are antiparallel, resulting in no net magnetic moment overall

Ferrimagnetism Like ferromagnets, permanent Consider the mineral magnetite Fe3O4 Can be written Fe2+O2--(Fe3+)2(O2-)3

Effect of temperature The atomic thermal motion counteract the coupling forces between the atomic dipole moments -> Decrease in saturation magnetization At Curie temperature Tc, it becomes 0

Domains Domain: small region in ferro/ferrimagnetic solid where there is a alignment along the same direction Very small, smaller than grain Magnetization of entire solid is the vector sum of the magnetization of all domains

Domains & Hysteresis B does not increase linearly with M Recall B = H So  changes with H Initial permeability i. Starting with initially randomly oriented domain, the domains aligning with the magnetic fields grows At maximum saturation, a single domain remains.

Domains & Hysteresis From saturation, reduce H The B-H curve does not retraces the original path – HYSTRTERESIS B decreases “slower” than H and does not become 0 when H = 0 -> remanence (remanent flux density Br) Single domain rotates New domain forms Some domain with original direction remains (remanence) Can be explained in terms of domain Resistance to domain wall motion Hc: coercivity Magnitude need to make B=0

Hysteresis and Permanent Magnetization • The magnetic hysteresis phenomenon B Stage 2. Apply H, align domains Stage 3. Remove H, alignment remains! => permanent magnet! Stage 4. Coercivity, HC Negative H needed to demagnitize! Adapted from Fig. 18.14, Callister & Rethwisch 3e. H Stage 5. Apply -H, align domains Stage 1. Initial (unmagnetized state) Stage 6. Close the hysteresis loop

Hard and Soft Magnetic Materials Hard magnetic materials: -- large coercivities -- used for permanent magnets -- add particles/voids to inhibit domain wall motion -- example: tungsten steel -- Hc = 5900 amp-turn/m) B Hard Soft H Soft magnetic materials: -- small coercivities -- used for electric motors -- example: commercial iron 99.95 Fe Adapted from Fig. 18.19, Callister & Rethwisch 3e. (Fig. 18.19 from K.M. Ralls, T.H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering, John Wiley and Sons, Inc., 1976.) 36 36

Superconductivity Found in 26 metals and hundreds of alloys & compounds Mercury Copper (normal) Fig. 18.26, Callister & Rethwisch 3e. 4.2 K TC = critical temperature = temperature below which material is superconductive

Critical Properties of Superconductive Materials TC = critical temperature - if T > TC not superconducting JC = critical current density - if J > JC not superconducting HC = critical magnetic field - if H > HC not superconducting Fig. 18.27, Callister & Rethwisch 3e.

Summary • A magnetic field is produced when a current flows through a wire coil. • Magnetic induction (B): -- an internal magnetic field is induced in a material that is situated within an external magnetic field (H). -- magnetic moments result from electron interactions with the applied magnetic field • Types of material responses to magnetic fields are: -- ferrimagnetic and ferromagnetic (large magnetic susceptibilities) -- paramagnetic (small and positive magnetic susceptibilities) -- diamagnetic (small and negative magnetic susceptibilities) • Types of ferrimagnetic and ferromagnetic materials: -- Hard: large coercivities -- Soft: small coercivities • Magnetic storage media: -- particulate g-Fe2O3 in polymeric film (tape) -- thin film CoPtCr or CoCrTa (hard drive)

ANNOUNCEMENTS Reading: Core Problems: Self-help Problems:

(3) Ferromagnetism Saturation magnetization Ms: Maximum possible magnetization Magnetization when all the dipoles are mutually aligned with the external field