Atoms in solid materials are constantly vibrating at very high frequencies and with relatively small amplitudes. 
- vibrations are coordinated since adjacent atoms have bonds between them.
- what results are waves, like sound waves, traveling at the speed of sound.
- set number of waves (or phonon modes)
- the higher the number of waves, the higher the heat capacity

Essentially the energy required to heat material:

As heat increases, more vibrational waves cause more scattering of free electrons.

Waves also participate in the transport of energy during thermal conduction.

Rises rapidly, then plateaus.

Electric contribution (very minor): electrons absorb energy by increasing their kinetic energy.  
- only possible for free e- 

alpha l - the linear coefficient of thermal expansion (units of reciprocal temp).  For volume:

(with diff constant alpha v)

Thermal expansion = increase in average distance between atoms.

As temperature rises (E1 -> E5), our average bond distance rises (R1 -> R5). Interestingly, if this trough was symmetric, then no thermal expansion would occur!

The greater the atomic bonding energy, the deeper and more narrow the trough in the graph is, so an increase in temperature will give rise to a smaller increase in interatomic separation.  Thus, the greater the atomic bonding energy, the smaller the value of alpha l (thermal expansion coefficient) 

Ceramics - very strong bonds, so small coefficent
Metals - intermediate
Polymers - weak bonds, so very large coefficient (big thermal expansion)

Where q = heat flux (per unit time per unit area perpendicular to flow direction)

k = thermal conductivity 

dT/dx = temperature gradient throughout the conducting medium 

Thermal conductivity k 
k = k(l) + k(e)
k(l) = k from lattice vibration waves
k(e) = k from free/conducting electrons (they carry energy through kinetic energy to colder regions).  Much more effective in transport of heat energy. 

Have lot of free e, so electron mechanism of heat transport much more efficient, and are really good conductors of heat.  
- Alloying causes impurities, which act as scattering centers, lowering the efficiency of e motion. 

Nonmetallic materials are thermal insulators (lack large # of free electrons, and phonons not really as effective as free e).
- Furthermore, glass & other amorphous ceramics have lower conductivities (bc they have highly disordered/irregular structures, so phonon scattering)

Good thermal insulators (low thermal conductivities). 
- energy transfer accomplished by vibration and rotation of chain molecules.  
- The more ordered and crystalline structure, the higher the thermal cond. (more coordinated vibrations)