8.6 Thermal conductivity

Definition of thermal conductivity

Thermal conductivity quantifies how well a material conducts heat. In statistical mechanics, it connects the microscopic behavior of particles (their collisions, vibrations, and energy exchanges) to the macroscopic flow of heat we can measure.

Fourier's law

Fourier's law states that heat flux is proportional to the negative temperature gradient:

$q = -k \nabla T$

Here, $q$ is the heat flux (energy per unit area per unit time), $k$ is the thermal conductivity, and $\nabla T$ is the temperature gradient. The negative sign means heat flows from hot to cold.

This law assumes steady-state conditions, a roughly linear temperature profile, and an isotropic material with constant $k$.

Units and dimensions

• SI units: watts per meter-kelvin (W/m·K)
• Dimensional formula: $[M^1 L^1 T^{-3} \Theta^{-1}]$
• Typical values span a huge range: ~0.026 W/m·K for air, ~400 W/m·K for copper, and up to ~2200 W/m·K for diamond at room temperature
• Values depend on material composition, crystal structure, and environmental conditions like temperature and pressure

Microscopic origins

Thermal conductivity emerges from the collective behavior of energy carriers inside a material. Statistical mechanics gives you the tools to connect what individual particles do (scatter, vibrate, collide) to the bulk thermal conductivity you'd measure in a lab.

Phonon transport

Phonons are quantized lattice vibrations, and they're the dominant heat carriers in electrically insulating materials like ceramics and polymers. Phonons propagate through the crystal lattice, transferring thermal energy as they go.

• Scattering processes limit how far phonons travel. Phonon-phonon scattering (especially Umklapp processes) and phonon-defect scattering are the main ones.
• The Debye model provides a useful approximation for the phonon contribution to thermal conductivity, treating the lattice as a collection of oscillators with a cutoff frequency.

Electron transport

In metals and doped semiconductors, free electrons are the primary heat carriers. These electrons carry both charge and thermal energy, which is why good electrical conductors tend to be good thermal conductors too.

• The Wiedemann-Franz law formalizes this: it states that the ratio of thermal conductivity to electrical conductivity is proportional to temperature, with the proportionality constant being the Lorenz number $L_0 = \frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2$.
• Electron-phonon interactions also influence thermal transport, particularly at intermediate temperatures where these scattering events are frequent.

Photon transport

At high temperatures, especially in transparent or semi-transparent materials, photons can carry significant thermal energy through radiative transfer.

• The Rosseland diffusion approximation treats photon transport as a diffusive process, yielding an effective "photon thermal conductivity."
• This contribution matters in porous materials, foams, and high-temperature environments like furnace linings.

Thermal conductivity in solids

Solid materials span an enormous range of thermal conductivities, from ultralow values in aerogels (~0.01 W/m·K) to extremely high values in diamond. The atomic structure of a solid largely determines where it falls on this spectrum.

Crystalline vs amorphous materials

Crystalline solids (metals, single-crystal semiconductors) generally conduct heat much better than amorphous ones. The long-range periodic order in a crystal lets phonons propagate over long distances before scattering.

Amorphous materials (glasses, many polymers) lack this order. Their phonon mean free paths are very short, often on the order of interatomic spacings, which severely limits thermal conductivity.

Temperature dependence

The way thermal conductivity changes with temperature depends on the material type:

• Pure crystals show a peak at low temperatures. Below the peak, conductivity rises as more phonon modes are excited. Above it, Umklapp phonon-phonon scattering increasingly limits transport.
• Metals show a nearly linear decrease at elevated temperatures because electron-phonon scattering intensifies.
• Amorphous materials have weak temperature dependence, often increasing slightly with temperature.
• Superconductors exhibit a dramatic drop in electronic thermal conductivity below the critical temperature, since Cooper pairs don't carry entropy.

Impurities and defects

Any disruption to a perfect lattice scatters phonons and reduces thermal conductivity:

• Point defects (vacancies, substitutional atoms) scatter short-wavelength phonons effectively
• Grain boundaries in polycrystalline materials add thermal resistance at each interface
• Dislocations and extended defects create additional scattering centers
• In thermoelectric materials, intentional doping and alloying are used to reduce lattice thermal conductivity while preserving electrical conductivity

Thermal conductivity in fluids

In fluids, thermal energy transfers through molecular collisions and, in many practical situations, through bulk fluid motion (convection). Statistical mechanics describes the collision dynamics and energy exchange at the molecular level.

Liquids vs gases

• Liquids generally have higher thermal conductivity than gases because molecules are closer together and interact more strongly.
• Gas thermal conductivity increases with temperature: faster molecules collide more often and transfer energy more efficiently. Notably, for an ideal gas, thermal conductivity is independent of pressure (the shorter mean free path at higher pressure is offset by higher molecular density).
• Liquid thermal conductivity typically decreases with temperature as intermolecular forces weaken.

Convection vs conduction

In fluids, you often can't ignore bulk motion. Two dimensionless numbers help you figure out what dominates:

• The Rayleigh number ($Ra$) compares buoyancy-driven convection to thermal diffusion. When $Ra$ is large, natural convection dominates over pure conduction.
• The Prandtl number ($Pr = \nu / \alpha$) compares momentum diffusivity ($\nu$) to thermal diffusivity ($\alpha$). It tells you whether the velocity boundary layer or the thermal boundary layer is thicker.

Forced convection (driven by pumps, fans, etc.) typically dominates heat transfer in engineering applications.

Measurement techniques

Different materials and geometries call for different measurement approaches. Methods fall into two broad categories.

Steady-state methods

These establish a constant temperature gradient and measure heat flow directly:

• Guarded hot plate: the standard for insulating materials. A heated plate is sandwiched between two samples, with guard heaters to prevent lateral heat loss.
• Axial flow method: suited for rod-shaped samples with high conductivity.
• Radial flow method: used for cylindrical specimens or fluids.
• Comparative technique: measures an unknown sample against a reference material of known conductivity.

Transient methods

These track how temperature evolves over time, often yielding faster results:

• Transient plane source (TPS): a sensor placed between two sample pieces measures thermal response to a heat pulse.
• Laser flash: fires a short laser pulse at one face of a thin sample and measures the temperature rise on the opposite face. This gives thermal diffusivity, from which you calculate conductivity using $k = \alpha \rho c_p$.
• 3ω method: effective for thin films and nanoscale materials. An AC current heats a metal line on the sample, and the third harmonic of the voltage signal reveals thermal properties.
• Transient hot wire: widely used for fluids. A thin wire acts as both heater and thermometer.

Applications and engineering

Heat sinks and thermal management

• High-conductivity materials like copper (~400 W/m·K) and aluminum (~237 W/m·K) are standard heat sink materials in electronics
• Heat pipes use liquid-vapor phase change to achieve effective thermal conductivities far exceeding any solid material
• Thermal interface materials (TIMs) fill microscopic air gaps between components, dramatically improving contact conductance
• Advanced approaches include microfluidic cooling channels and nanostructured surfaces

Insulation materials

• Low-conductivity materials like fiberglass (~0.04 W/m·K) and aerogels (~0.015 W/m·K) minimize heat transfer in buildings and industrial systems
• Vacuum insulated panels achieve extremely low conductivity by eliminating gas-phase conduction
• Reflective insulation targets radiative heat transfer specifically
• Phase change materials absorb and release latent heat, providing thermal buffering

Thermoelectric devices

Thermoelectric materials need a specific combination: low thermal conductivity but high electrical conductivity. The figure of merit $ZT = \frac{S^2 \sigma T}{k}$ captures this, where $S$ is the Seebeck coefficient, $\sigma$ is electrical conductivity, $T$ is temperature, and $k$ is thermal conductivity.

Nanostructuring is a key strategy here: introducing grain boundaries and interfaces scatters phonons (reducing $k$) without heavily degrading electron transport (preserving $\sigma$). Applications include solid-state Peltier coolers and waste heat recovery generators.

Theoretical models

Kinetic theory approach

The simplest model treats heat carriers as gas-like particles. This yields the well-known expression:

$k = \frac{1}{3} C v \lambda$

where $C$ is the volumetric specific heat, $v$ is the average particle velocity, and $\lambda$ is the mean free path.

This formula gives good physical intuition: thermal conductivity increases if carriers move faster, travel farther between collisions, or carry more energy per unit temperature change. However, it's a rough approximation for real materials, especially low-dimensional or strongly anharmonic systems.

Boltzmann transport equation

The Boltzmann transport equation (BTE) describes how the distribution function of carriers evolves in phase space under the influence of temperature gradients and scattering:

$\frac{\partial f}{\partial t} + \mathbf{v} \cdot \nabla_r f + \mathbf{F} \cdot \nabla_p f = \left(\frac{\partial f}{\partial t}\right)_{\text{scatt}}$

The relaxation time approximation simplifies the scattering term by assuming the distribution relaxes exponentially toward equilibrium. This makes the BTE tractable for many practical problems, including semiconductor device modeling and thermoelectric materials research.

Quantum mechanical treatments

For complex or nanoscale systems, more sophisticated methods are needed:

• Density functional theory (DFT) calculates phonon dispersion relations and scattering rates from first principles
• Non-equilibrium Green's function (NEGF) method models quantum thermal transport in nanostructures where wave effects matter
• First-principles molecular dynamics simulates thermal transport without empirical potentials
• Quantum Monte Carlo techniques provide high-accuracy predictions for strongly correlated systems

Nanoscale thermal transport

When system dimensions shrink to the scale of carrier mean free paths, thermal transport departs significantly from bulk Fourier behavior. This regime is increasingly relevant for modern nanoelectronics and nanostructured materials.

Ballistic vs diffusive transport

• Diffusive transport is the familiar regime where Fourier's law holds. Carriers undergo many scattering events as they cross the system.
• Ballistic transport occurs when the system size is smaller than the mean free path. Carriers travel from one boundary to the other without scattering.
• The Knudsen number ($Kn = \lambda / L$, where $L$ is the system size) characterizes the transition: $Kn \ll 1$ is diffusive, $Kn \gg 1$ is ballistic.
• In one-dimensional channels at low temperatures, ballistic thermal conductance is quantized in units of $\pi^2 k_B^2 T / 3h$.

Phonon mean free path

The phonon mean free path is the average distance a phonon travels between scattering events. It varies enormously:

• In amorphous materials, mean free paths can be as short as a few angstroms
• In high-quality single crystals at low temperatures, they can reach micrometers or more
• Different phonon frequencies have different mean free paths (spectral analysis reveals this distribution)
• When the sample dimension approaches the mean free path, boundary scattering becomes dominant and thermal conductivity drops below the bulk value

Size effects

• Thin films show thickness-dependent thermal conductivity that can be much lower than the bulk value
• Nanowires and nanotubes exhibit diameter-dependent transport
• Superlattices (alternating thin layers of different materials) exploit interfacial phonon scattering to achieve very low cross-plane thermal conductivity
• These size effects are exploited in thermoelectric materials to reduce $k$ without proportionally reducing electrical conductivity

Thermal conductivity in composites

Composite materials let you engineer thermal properties by combining constituents with different conductivities. The effective thermal conductivity depends on the component materials, their volume fractions, and the microstructure geometry.

Effective medium theory

• The Maxwell-Garnett model works well for dilute suspensions of particles in a matrix
• The Bruggeman model (self-consistent approach) handles higher volume fractions where neither phase is clearly the "host"
• Differential effective medium theory extends these ideas by incrementally adding inclusions
• Near the percolation threshold, a small change in filler fraction can cause a large jump in effective conductivity if the filler is highly conductive

Series and parallel models

These provide rigorous bounds on the effective thermal conductivity of a two-component composite:

• Series (layered perpendicular to heat flow): gives the lower bound. $\frac{1}{k_{\text{eff}}} = \frac{f_1}{k_1} + \frac{f_2}{k_2}$
• Parallel (layered along heat flow): gives the upper bound. $k_{\text{eff}} = f_1 k_1 + f_2 k_2$
• The Hashin-Shtrikman bounds are tighter and account for particle shape and arrangement
• Real composites fall somewhere between these bounds depending on microstructure

Anisotropic thermal conductivity

Many materials conduct heat better in some directions than others. When this happens, thermal conductivity must be described by a tensor rather than a single scalar value.

Layered materials

Graphite is the classic example: its in-plane thermal conductivity (~2000 W/m·K) dwarfs its cross-plane value (~6 W/m·K). This roughly 300:1 anisotropy ratio arises because strong covalent bonds within graphene layers facilitate phonon transport, while weak van der Waals bonds between layers do not.

• van der Waals heterostructures (stacked 2D materials) offer tunable anisotropic thermal properties
• Polymer composites with aligned fillers can achieve engineered anisotropy
• Thermal management designs exploit anisotropy to spread heat laterally while blocking it vertically

Single crystals

Crystal symmetry determines the degree of anisotropy. Cubic crystals (like silicon) are thermally isotropic, but lower-symmetry crystals are not.

• Sapphire ($Al_2O_3$) shows ~35% difference between its a-axis and c-axis thermal conductivities
• Different silicon carbide polytypes exhibit varying anisotropy
• Anisotropic thermal expansion combined with anisotropic conductivity can generate thermal stresses in constrained geometries

Thermal conductivity at interfaces

Every interface between dissimilar materials introduces additional thermal resistance. In nanostructured materials and multilayer devices, these interface resistances can dominate the total thermal resistance.

Kapitza resistance

Kapitza resistance (or thermal boundary resistance) arises from the mismatch in phonon properties across an interface. Even two materials in perfect contact have finite thermal resistance at their boundary.

• At low temperatures, Kapitza resistance follows a $T^{-3}$ dependence
• It's inversely related to interface thermal conductance ($G = 1/R_K$)
• In nanoelectronics and superlattice thermoelectrics, Kapitza resistance at many closely spaced interfaces can significantly reduce overall thermal conductivity

Acoustic vs diffuse mismatch models

Two classical models bracket the expected behavior:

• The acoustic mismatch model (AMM) assumes phonons reflect and transmit specularly at the interface, like light at a glass surface. Transmission probability depends on the acoustic impedance mismatch. This works better at low temperatures where phonon wavelengths are long and interfaces appear smooth.
• The diffuse mismatch model (DMM) assumes all phonons lose memory of their direction upon hitting the interface and scatter randomly. This is more appropriate at higher temperatures where interface roughness scatters phonons diffusely.
• Neither model is fully accurate across all conditions. Hybrid approaches and molecular dynamics simulations provide improved predictions for real interfaces.