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THERMAL PROPERTIES & CONDUCTION - Coggle Diagram
THERMAL PROPERTIES & CONDUCTION
Specific heat at
constant volume
and
constant
pressure
Thermal energy changes with temperature gives an understanding of the heat energy which necessary to raise the temperature of the material
The increase in energy of a system is equal to the amount of heat absorbed by the system minus the amount of work done by the system
dU
=
dQ
-
dW
Heat capacity- defined as the amount of heat absorbed by a system per unit change in temperature
Energy system described by thermodynamics variables pressure, temperature, and volume
Heat Capacity
for infinitesimal process at constant volume
at constant pressure
Relation of heat capacity at constant pressure and volume;
can be written in terms of experimentally measured quantities
where the volume thermal expansion coefficient
and the volume compressibility of a solid
The experimental measurements of specific heat for solids done at constant pressure (atmospheric)
For most solids in general
When T approaching to 0, Cp approaching to Cv
At low temperature, Cp nearly equal to Cv
Dulong and Petit law
: The value of the heat capacity is
3Nk = 3R per mole = 25Jmol^-1deg^-1
(At room temperature)
Harmonic oscillator potential
Other half of Cv comes from the potential energy which holds the atoms at their equilibrium positions in the solid (the energy depends on the 3N positions)
A potential the atoms in a crystal will oscillate with amplitudes small compared to the internuclear distance
Energy of the oscillator
independent of temperature, and classical physics offers no possibility of a temperature dependence.(Agree with the Dulong-Petit law at high temperatures)
But failed to explain Cv at low temperatures
Einstein's Model
Einstein assumed that the atoms are vibrating as harmonic oscillators.
Assumed Planck's(1990) quantization rule for each oscillator (vibrating oscillators (atoms) in a solid have quantized energies)
Occupational of energy level n: (probability of oscillator being in level n)
Average total energy of a solid :
The prediction obtained :
Limiting Behavior of Cv(T)
High T limit :
Low T limit :
Debye's Model (1912)
Developed a more sophisticated (but still approximate) treatment of atomic vibration in solids :
Atoms are considered as harmonic oscillator that produce elastic waves with varying frequencies from
ω = 0 to ωmax
Treat solid as continuous elastic medium (ignore details of atomic structure)
3N normal modes (patterns) of oscillations
The relations between the energy of a phonon E, the angular frequency ω and the wave vector q are:
Debye temperature :
Specific Heat
Successfully describes the temperature dependence of heat capacity not only at high temperatures but also at low temperature
Debye model gives a better representation for the very low energy vibrations
At high temperature limit
At low temperature limit
Thermal Conductivity
Thermal conductivity is thermal heat/energy being transferred in a material
When there is a temperature gradient across a solid, thermal energy is transferred from the hotter to the cooler end.
Heat transfer mechanism
When heated, electrons, holes and phonon obtain energy larger than the average energy.
In metals, electrons, holes and phonons can transfer or conduct thermal energy from the hotter areas to the cooler parts.
In insulations (dielectric materials), only phonon plays a role in delivering energy.
According to Debye
If the vibration of the fixed lattice in the normal mode:
Phonon distribution does not change with time
Thermal waves also remain constant and travels in solids at the speed of speed
Thermal conductivity of lattices becomes infinite
In reality:
Edge scattering: because of specimen size.
scattering due to phonon and defects: Lattice defects such as point defects, isotope inhomogeneity and other types of defects.
Phonon-phonon scattering: crystals becomes anharmonic at temperatures higher than absolute zero.
Phonon scattering mechanisms causes thermal resistance and cause thermal conductivity to be infinite.
There is no thermal expansion.
Adiabatic and isothermal elastic constants are equal.
The elastic constants are independent of pressure and temperature
The heat capacity becomes constant at high temperatures,
T>θ
Two lattice waves do not interact; a single wave does not decay or change form with time
Thermal Expansion & Thermal conductivity(Examples of anharmonic effects)
Harmonic approximation, phonons do not interact with each other, in the absence of boundaries, lattice defects and impurities, the thermal conductivity is infinite.
Anharmonic effects, phonons collide with each other and these collisions limit thermal conductivity which is due to the flow of phonons.
Mechanisms of thermal resistivity
In general, heat(energy) can be transmitted through a crystal by phonons, photons, free e- or holes, and e-holes pair
in metals the free e- are the best conductors of heat.
In insulators phonons are the principal transporters of heat
Anharmonic Effects
Consider the potential energy of 1D monoatomic linear chain of identical atoms
Forces between only nearest neighbors are assumed.
Eqn:
Normal Process or N-process
Consider the three phonon process where phonons with wave vectors
k1
and
k2
collide, annihilate one another, and create a phonon with wave vector
k3
before the interaction, energy was flowing to the right, and after the interaction, the same amount of energy is still flowing to the right.
N-process does not alter the direction of energy flow, so it cannot contribute to the thermal resistance of a crystal.
It is important to appreciate that normal processes do help to maintain thermal equilibrium.
Umklapp Process or U-Process
k
not equal to 0
This process can provide a thermal resistance to phonon flow (provides for a finite thermal conductivity)
The energy flow for
k1
and
k2
was to the right, the energy flow for
k3
is to the left
Energy flow has been reversed
