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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. heat capacity (J/mol-K) dQ C dT energy input (J/mol) temperature change (K) • Two ways to measure heat capacity: Cp : Heat capacity at constant pressure. Cv : Heat capacity at constant volume. Cp usually > Cv J Btu • Heat capacity has units of mol K lb mol F Chapter 17 - 1 Dependence of Heat Capacity on Temperature • Heat capacity... -- increases with temperature -- for solids it reaches a limiting value of 3R R = gas constant 3R = 8.31 J/mol-K 0 Cv = constant 0 qD • From atomic perspective: T (K) Adapted from Fig. 17.2, Callister & Rethwisch 3e. Debye temperature (usually less than T room ) -- Energy is stored as atomic vibrations. -- As temperature increases, the average energy of atomic vibrations increases. Chapter 17 - 2 Atomic Vibrations Atomic vibrations are in the form of lattice waves or phonons Adapted from Fig. 17.1, Callister & Rethwisch 3e. Chapter 17 - 3 Specific Heat: Comparison increasing cp Material • Polymers Polypropylene Polyethylene Polystyrene Teflon cp (J/kg-K) at room T 1925 1850 1170 1050 • Ceramics Magnesia (MgO) Alumina (Al2O3) Glass 940 775 840 • Metals Aluminum Steel Tungsten Gold 900 486 138 128 cp (specific heat): (J/kg-K) Cp (heat capacity): (J/mol-K) Selected values from Table 17.1, Callister & Rethwisch 3e. Chapter 17 - 4 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 Chapter 17 - 5 Thermal Expansion Materials change size when temperature is changed initial final Tinitial Tfinal Tfinal > Tinitial f 0 α (T finalT initial ) 0 linear coefficient of thermal expansion (1/K or 1/°C) V V0 v T For isotropic materials αv = 3αl Chapter 17 - Atomic Perspective: Thermal Expansion Thermal expansion arises from an increase in the average distance between the atoms Asymmetric curve: -- increase temperature, -- increase in interatomic separation -- thermal expansion Symmetric curve: -- increase temperature, -- no increase in interatomic separation Chapter 17 -- no thermal expansion Coefficient of Thermal Expansion: Comparison Material increasing • Polymers Polypropylene Polyethylene Polystyrene Teflon • Metals Aluminum Steel Tungsten Gold • Ceramics Magnesia (MgO) Alumina (Al2O3) Soda-lime glass Silica (cryst. SiO2) (10-6/C) at room T 145-180 106-198 90-150 126-216 23.6 12 4.5 14.2 Polymers have larger values because of weak secondary bonds • Q: Why does generally decrease with increasing bond energy? A: deeper and narrower energy “trough” 13.5 7.6 9 0.4 Chapter 17 - 8 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 16.5 x 106 ( C)1 rearranging Equation 17.3b 0 T [ 16.5 x 10 6 (1/ C)](15 m)[ 40C ( 9C)] 0.012 m 12 mm Chapter 17 - 9 Thermal Conductivity The ability of a material to transport heat. Fourier’s Law heat flux (J/m2-s) dT q k dx temperature gradient thermal conductivity (J/m-K-s) T2 T1 x1 heat flux T2 > T1 x2 • Atomic perspective: Atomic vibrations and free electrons in hotter regions transport energy to cooler regions. Chapter 17 - 10 Mechanisms of Heat Conduction Chapter 17 - 11 increasing k Thermal Conductivity: Comparison Material k (W/m-K) • Metals Aluminum 247 Steel 52 Tungsten 178 Gold 315 • Ceramics Magnesia (MgO) 38 Alumina (Al2O3) 39 Soda-lime glass 1.7 Silica (cryst. SiO2) 1.4 • Polymers Polypropylene 0.12 Polyethylene 0.46-0.50 Polystyrene 0.13 Teflon 0.25 Energy Transfer Mechanism atomic vibrations and motion of free electrons (electron motion is more efficient) atomic vibrations (phonons are primarily responsible) vibration/rotation of chain molecules Selected values from Table 19.1, Callister & Rethwisch 3e. Chapter 17 - 12 Thermal Stresses • Occur due to: -- restrained thermal expansion/contraction -- temperature gradients that lead to differential dimensional changes Thermal stress E (T0 Tf ) E T Chapter 17 - 13 Example Problem -- 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 Step 1: Assume unconstrained thermal expansion 0 Tf thermal (Tf T0 ) room Step 2: Compress specimen back to original length 0 compress thermal room Chapter 17 - Example Problem (cont.) 0 The thermal stress can be directly calculated as E(compress) Noting that compress = -thermal and substituting gives E(thermal ) E (Tf T0 ) E (T0 Tf ) Rearranging and solving for Tf gives 20ºC Tf T0 Answer: 106°C 100 GPa E -172 MPa (since in compression) 20 x 10-6/°C Chapter 17 - 15 Thermal Shock Resistance • Occurs due to: nonuniform heating/cooling • Ex: Assume top thin layer is rapidly cooled from T1 to T2 rapid quench tries to contract during cooling T2 resists contraction T1 Tension develops at surface E (T1 T2 ) Critical temperature difference for fracture (set = f) Temperature difference that can be produced by cooling: (T1 T2 ) quench rate k (T1 T2 ) f racture f E set equal • (quench rate) f or f racture Thermal Shock Resistance ( TSR) • Large TSR when f k E f k is large E Chapter 17 - 16 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 f k E Chapter 17 - 17 Magnetic Properties Generation of a Magnetic Field -- Vacuum • Created by current through a coil: B0 H I 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: H NI (A / m) • Computation of the magnetic flux density in a vacuum, B0: B0 = 0H permeability of a vacuum (1.257 x 10-6 Henry/m) Chapter 17 - 18 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 Chapter 17 - 19 Magnetic field vector etc. Chapter 17 - 20 Finally B0 = 0H Chapter 17 - 21 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 Chapter 17 - 22 Generation of a Magnetic Field -within a Solid Material • A magnetic field is induced in the material B applied magnetic field H current I B = Magnetic Induction (tesla) inside the material B = H permeability of a solid • Relative permeability (dimensionless) r 0 Chapter 17 - 23 Generation of a Magnetic Field -within a Solid Material (cont.) M = cmH • Magnetization Magnetic susceptibility (dimensionless) B = 0H + 0M • B in terms of H and M • Combining the above two equations: B B = 0H + 0 cmH = (1 + cm)0H cm > 0 vacuum cm = 0 cm < 0 H permeability of a vacuum: (1.26 x 10-6 Henry/m) cm is a measure of a material’s magnetic response relative to a vacuum Chapter 17 - Origins of Magnetic Moments • Magnetic moments arise from electron motions and the spins on electrons. magnetic moments electron electron spin nucleus electron orbital motion Adapted from Fig. 18.4, Callister & Rethwisch 3e. 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 Chapter 17 - 25 B (tesla) Types of Magnetism (3) ferromagnetic e.g. Fe3O4, NiFe2O4 (4) ferrimagnetic e.g. ferrite(), Co, Ni, Gd ( cm as large as 106 !) (2) paramagnetic ( cm ~ 10-4) e.g., Al, Cr, Mo, Na, Ti, Zr 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. Chapter 17 - 26 Diamagnetic / Paramagnetic random aligned Applied Magnetic Field (H) (2) paramagnetic opposing No Applied Magnetic Field (H = 0) none (1) diamagnetic Chapter 17 - 27 Ferromagnetism Chapter 17 - 28 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 Chapter 17 - 29 Ferrimagnetism • Like ferromagnets, permanent – Consider the mineral magnetite Fe3O4 – Can be written Fe2+O2--(Fe3+)2(O2-)3 Chapter 17 - 30 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 Chapter 17 - 31 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 Chapter 17 - 32 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. Chapter 17 - 33 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 Chapter 17 - 34 Hysteresis and Permanent Magnetization • The magnetic hysteresis phenomenon B Stage 3. Remove H, alignment remains! => permanent magnet! Stage 2. Apply H, align domains H Stage 4. Coercivity, HC Negative H needed to demagnitize! Stage 5. Apply -H, align domains Adapted from Fig. 18.14, Callister & Rethwisch 3e. Stage 1. Initial (unmagnetized state) Stage 6. Close the hysteresis loop Chapter 17 - 35 Hard and Soft Magnetic Materials B -- large coercivities -- used for permanent magnets -- add particles/voids to inhibit domain wall motion -- example: tungsten steel -Hc = 5900 amp-turn/m) Soft Hard magnetic materials: 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.) Chapter 17 - 36 Superconductivity Found in 26 metals and hundreds of alloys & compounds Mercury Copper (normal) 4.2 K Fig. 18.26, Callister & Rethwisch 3e. • TC = critical temperature = temperature below which material is superconductive Chapter 17 - 37 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 T 2 HC (T ) HC (0)1 2 TC Fig. 18.27, Callister & Rethwisch 3e. Chapter 17 - 38 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) Chapter 17 - 39 ANNOUNCEMENTS Reading: Core Problems: Self-help Problems: Chapter 17 - 40 (3) Ferromagnetism • Saturation magnetization Ms: – Maximum possible magnetization – Magnetization when all the dipoles are mutually aligned with the external field Chapter 17 - 41 Chapter 17 - 42 Chapter 17 - 43