The heat-response family, assembled on one bench
You now hold the whole heat-response family. Deliver heat to a solid and four things happen, one per earlier guide in this rung. The lattice fills with vibration and stores energy — that is specific heat, the thermal sponge. The atoms rattle harder in their bonding-energy wells and, because those wells are lopsided, sit a shade farther apart, so the part swells — that is thermal expansion. The vibration and, in metals, the free electrons carry heat from hot spots to cold — that is thermal conductivity. And if the swelling is fought hard enough while the material is brittle, it cracks — that is thermal shock. Design is nothing more exotic than arranging these four to serve you instead of ambush you.
Two numbers do most of the design work, and it is worth naming them plainly. The first is the expansion coefficient, usually the Greek letter alpha, in units of parts-per-million per degree (x10^-6 /K): steel is about 12, aluminium about 23, silicon a mere 3, a polymer up around 60. The second is the conductivity k, in watts per metre per degree (W/m-K): copper about 400, steel about 50, a glass about 1, a foam about 0.03. Almost every thermal design decision — where to leave a gap, what to bond to what, what to make a heat sink or an insulator from — comes down to reading these two numbers off a table and asking whether you want them high, low, or matched.
Expansion in design: free it, or fight it on purpose
The first rule of expansion is the gentlest: if a part must swell, let it. A steel bridge deck a hundred metres long grows by tens of millimetres between a winter night and a summer afternoon, so we cut it into segments and drop an expansion joint — the finger-comb gaps you feel your tyres thrum over — between them, and mount it on rollers at one end. Railway track once used the same trick with a visible gap every rail length (the old clackety-clack), pipelines loop into deliberate expansion bends, and a saucepan lid is left a touch loose so it does not jam when hot. In every case the design removes the restraint so the material can move for free and no stress builds. The joint is the humble hero of thermal design.
The reason the joint matters is the price of forgetting it, and the arithmetic is brutally direct. Clamp a bar so it cannot expand, warm it by delta-T, and it pushes back with a thermal stress of sigma = E times alpha times delta-T — stiffness times expansion times temperature rise, the formula from the expansion guide. Put real numbers in for a steel rail welded down solid: E about 200 GPa (200000 MPa), alpha about 12 x 10^-6 /K, a summer rise of 40 degrees C. That is 200000 times 12 x 10^-6 times 40 = about 96 MPa of compression, a large fraction of the rail's yield strength — enough, on a badly built line, to buckle the track sideways into a sun kink. Continuous welded rail survives only because it is pre-tensioned and clamped hard enough to hold that force down. Restraint plus temperature equals stress, every time.
Now the delightful inversion: sometimes you want the movement, and you engineer a mismatch on purpose. Bond a strip of high-alpha metal (say brass) back-to-back with a strip of low-alpha metal (say Invar or steel), and on heating the brass side stretches more than the steel side can follow. Unable to slide free, the pair curls — bending toward the low-expansion side like a book warping in the damp. That is the bimetallic strip, and its curl is a clean, repeatable readout of temperature. It snaps old thermostats and toaster timers on and off, trips a circuit breaker when overload heats it, wags the needle of a dial oven thermometer, and flashes a car turn signal. The very effect that buckles a rail, harnessed with two chosen metals, becomes a sensor and an actuator in one.
Matching expansion: seals, chips, and solder joints
The nastiest expansion problem is not one material swelling — it is two joined materials swelling by different amounts. Every time the temperature changes, a thermal-expansion mismatch shears the bond between them, and the bigger the alpha gap, the harder the shove. Classic case: sealing a metal wire through a glass envelope, as in a light bulb or a vacuum feedthrough. Seal it hot, cool it down, and if the metal shrinks more than the glass the joint is wrenched apart and the glass cracks. The fix is not a stronger glue but a matched metal: Kovar, an iron-nickel-cobalt alloy engineered to have almost exactly the alpha of hard borosilicate glass (both near 5 x 10^-6 /K), so the two shrink in lockstep and the seal stays tight from oven to freezer. Match the alpha and the mismatch stress never appears.
THERMAL DESIGN CHEAT-SHEET (representative room-temperature values)
material alpha k design role
(x10^-6 /K) (W/m-K)
---------------- ----------- ------- ------------------------
copper 17 400 heat spreader / heat sink
aluminium 23 240 light heat sink
carbon steel 12 50 structure; rails, pipes
silicon 3 150 chips (very low alpha)
alumina 8 30 substrate / insulator
silicon carbide 4 120 thermal-shock survivor
borosilicate glass 3 1 ovenware (low alpha)
glass-ceramic ~0 ~3 stovetop-to-freezer dish
zirconia (YSZ) 10 2 thermal-BARRIER coating
solid polymer 60 0.2 cheap bulk insulator
polymer foam -- 0.03 best insulator (traps air)
MATCH alpha -> bonded parts and seals survive temperature swings.
HIGH k -> spread / dump heat (sinks).
LOW k -> block heat (barrier coats, foams).
Low alpha + high k + high strength + low E -> best thermal-shock survivor.This same mismatch is the quiet villain inside every phone and laptop. A silicon chip has an alpha of about 3 x 10^-6 /K; the organic circuit board it is soldered to is up around 15 to 18. Every time the device powers up and heats, then cools when idle, the two expand by different amounts and the tiny solder balls joining them are sheared back and forth. That is not a one-time overload but a cyclic one, so the joints die by fatigue — micro-cracks creeping through the solder over thousands of power cycles until a connection opens and the device fails. Engineers fight it by shrinking the chip-to-board distance, choosing substrate materials with alpha closer to silicon, and flowing a stiff underfill epoxy under the chip to spread the shear over the whole footprint instead of concentrating it on the corner joints. Thermal fatigue of solder is one of the top reasons electronics eventually stop working.
Conductivity in design: spread it, block it, or coat with it
When a part makes unwanted heat, you want it gone, and that calls for the high end of the conductivity table. A CPU heat sink and the spreader plate under it are made of copper or aluminium precisely because their free electrons ferry heat away at 400 or 240 W/m-K, pulling the hot spot's warmth out to a wide finned surface where air can carry it off. This is also the deep reason a metal doorknob feels cold and a wooden one feels warm at the very same temperature: the metal's high k yanks heat out of your skin fast, while wood or foam, poor conductors, let your fingertip stay warm. Your hand is not a thermometer of temperature — it is a crude sensor of conductivity, a fact you met head-on in the conduction guide.
There is a beautiful shortcut hiding here for metals. In a metal the same free electrons carry both electric charge and heat, so a good electrical conductor is almost automatically a good thermal conductor — copper leads at both. That proportionality between thermal and electrical conductivity is the Wiedemann-Franz law, and it lets you guess one from the other for any metal. Be honest about its edge, though: it holds only where electrons dominate the heat transport. Diamond wrecks the intuition — it is an electrical insulator yet one of the best heat conductors known, because in diamond the heat rides on phonons, not electrons, marching through a stiff, near-perfect lattice. Wiedemann-Franz is a law about metals, not a law about everything.
Flip to the low end of the table and the goal reverses: keep heat out. The trick is never the solid itself but the gas it traps — a polymer foam, fibreglass batt, or silica aerogel conducts at 0.02 to 0.03 W/m-K not because the solid is special but because it is mostly tiny still-air pockets, and still air is a superb insulator. And the boldest move of all wants low conductivity on a metal part. A jet-engine turbine blade is a nickel superalloy running in gas hotter than its own melting point; it survives only because a paper-thin thermal-barrier coating of zirconia (k about 2 W/m-K) is sprayed over it, insulating the metal so its surface runs 100 to 150 degrees C cooler while the engine burns hotter and more efficiently. Zirconia earns the job partly because, unusually for a ceramic, its alpha is high enough (about 10 x 10^-6 /K) to stay reasonably matched to the superalloy beneath — the mismatch lesson from the last section, applied to a coating that must not peel off.
Surviving temperature change, and picking the material for it
The last property is the one that ends a part's life outright. Splash cold water on a red-hot dish and the surface tries to shrink while the still-hot interior holds it stretched, so tensile stress spikes at the surface — and in a brittle solid that tension finds the worst flaw and runs a crack: thermal shock. From the shock guide we have its figure of merit, a merit index you can read straight off the cheat-sheet: R roughly equal to (strength times k) divided by (E times alpha). To survive temperature swings a material wants to be strong, conductive (so no steep temperature difference builds), floppy (low E, so a given strain costs less stress), and low-expansion (small alpha). That single expression explains the whole kitchen: ordinary soda-lime glass cracks under a hot-cold shock because its alpha is high, while borosilicate ovenware (alpha about 3) shrugs it off, glass-ceramic hobs go from a 500-degree burner to a wet cloth because their alpha is near zero, and silicon carbide kiln furniture endures because its high k and strength dominate the top of the fraction.
- State the thermal function and its hard constraint in one line — 'dump 50 watts without exceeding 80 degrees C', or 'survive a 200-degree quench without cracking', or 'seal to this glass across a wide temperature range'.
- Decide which single thermal property dominates, and turn it into a performance index to maximize: a heat sink maximizes k; an insulator minimizes k; a thermal-shock survivor maximizes (strength times k) / (E times alpha); a seal minimizes the alpha gap to its partner.
- Plot the candidates on the relevant Ashby chart — conductivity versus expansion, or strength versus modulus — and read off the winning corner, the family that lands where your index is best.
- Screen the survivors against everything else — cost, density, maximum service temperature, corrosion, and whether you can actually shape and join it — because the thermal winner is useless if it is unaffordable or unweldable.
That procedure is just materials selection pointed at heat, and it closes both this rung and, honestly, the whole idea of the course: structure sets properties, properties meet a function, and design is the disciplined match between them. Keep the honest edges in view as you go. A thermal-shock number rests on a strength that is really the worst-flaw statistic of a brittle solid, so quote it with Weibull scatter, not as a single guaranteed value. Expansion and conductivity are fixed by bonding, so you choose them by choosing the material, not by treating it. And every thermal property drifts with temperature — specific heat, conductivity, even alpha itself climb or fall as the part heats — so a value read at room temperature is a starting point, not a promise at 800 degrees C. Design with the family, respect its limits, and heat becomes a tool rather than a trap.