Chapter 5 Thermal Conductivity Chapter 5 Thermal Conductivity Thermal Conductivity � What is thermal conductivity? � What is heat? � Heat is the spontaneous flow of energy from an object at a higher t[.]
Chapter 5: Thermal Conductivity Thermal Conductivity • • • • • What is thermal conductivity? What is heat? Heat is the spontaneous flow of energy from an object at a higher temperature to an object at a lower temperature Let’s say that you have a solid with a temperature gradient across it dT/dx (ie One end is at a higher temperature than the other) The amount of energy transmitted across per unit area per unit time, jV, is then: dT jU = − K dx Flux of thermal energy (per unit area per unit time) T2 T1 Heat flow (T1 > T2) in x-direction Thermal Conductivity Coefficient, K (units: power per unit length per unit temp) Thermal Conductivity • • • • What conducts heat in a solid? Heat flow is a random process – in the example above, energy flows in, and doesn’t proceed directly to the other end (it diffuses randomly throughout the solid) We will see that for solids heat flow is largely determined by phonons (lattice vibrations carry the energy throughout the lattice), but free electrons can carry heat as well (this is important for metals) From the kinetic theory of gases, we know that Thermal conductivity • K = Cvl Average particle velocity Mean free path of a particle between collisions Heat capacity/unit volume We are going to show that this is true for phonons as well Theory • If we assume that we have n phonons moving in the x direction (there is an average drift from the hot end to the cold), then the flux (no of phonons/unit area/unit time) is: n concentration of phonons (phonons/unit volume) • • These phonons will carry a certain amount of energy as heat By how much does the temperature change by each one of these phonons moving a distance lx (on the average)? T2 T1 Heat flow (T1 > T2) in x-direction Phonons act like particles, with an overall flux towards the cold end (energy is radiating from hot to cold) Each of these moves an average distance of lx before colliding with another phonon Theory • • As the phonon moves a distance lx, it will reduce the temperature by ∆T (it carries energy in the form of heat) This ∆T is: dT dT ∆T = lx = v xτ dx dx • • Where vx is the average velocity in the x-direction, and τ is the average time between collisions Therefore, the amount of energy carried by each phonon is: dT Energy = c∆T = cvxτ dx Heat Capacity, in J/K Theory • Putting it all together, we have: Negative sign: energy flows in opposite direction as ∆T Net flux of energy = - (flux of phonons) x (energy/phonon) jU = −(n < v x >)(c∆T ) dT = −(n < v x >)(v xτc ) dx dT ≈ −n < v x > cτ dx dT = − n < v > cτ dx (we can this last step if we assume that the avg velocity components are equal: + + = and therefore that: = 1/3 ) Thermal Conductivity Heat capacity/unit volume = concentration x heat capacity • Therefore, using our results that l = vτ, and C = nc, we have: Mean free path = avg velocity x avg time between collisions 1 dT dT dT = − Cvl = −K jU = − n < v > cτ 3 dx dx dx • So, our result for K = 1/3 Cvl works, assuming that v is a constant for phonons (which it is for the acoustic limit: ω = vK) Thermal Conductivity Values • • • • • • • • See page 126, 133 for thermal conductivity values Highest thermal conductivities at room temperature: Cu, Ag, Au (in these cases, there is a large thermal conductivity from the electronic component – free electrons transport heat in addition to phonons) Average conductivities: ~ W cm-1 K-1 What does the thermal conductivity depend upon? At room temp, where C ~constant, the phonon mean free path, l is important What determines the phonon mean free path? For a material with purely harmonic interactions, and a perfect lattice (ie no defects such as dislocations), there would be nothing to stop the phonons – mean free path ~ size of crystal However, in the real world, there is a much smaller, finite mean free path for phonons Phonon mean free paths • • • • • • • • Real materials have phonons which can scatter from lattice imperfections, and each other At each one of these collisions, the phonon momentum must be conserved How does the mean free path depend upon temperature? Experiment finds that l ~ 1/T How can we explain this? The no of phonons at a given temperature is ~ kT Therefore, the collision frequency goes up in proportion to ~ kT This means that l has to decrease by a factor of 1/T (there are more phonons around to scatter off of one another) K = K’ + K’’ K’ K K’’ Quartz NaCl T (C) K l (Å) 0.13 40 -190 0.50 540 0.07 23 -190 0.27 100 (K in W cm-1 K-1) Thermal Conductivity at low temperatures • • • • • • What does the thermal conductivity look like at low temperatures? Thermal conductivity: K = 1/3 Cvl At very low temperatures where there are few phonons around, K ~ CvD (D is the diameter of the specimen) The heat capacity ~ T3, so the thermal conductivity ~ T3 Physicists have actually measured how the thermal conductivity varies as the size of the crystal – in these cases, the phonons propagate throughout the entire crystal and scatter from the edges Often, other effects come into play before this happens Chemical impurties (act like lattice imperfections – cause phonon scattering and lowers l) Distribution of isotopes (see figure 19 in Chapter 5) Amorphous structures Crystal grain boundaries