4-1 supplement : Thermal Expansion of Glass Thermal expansion determines if a glass will be shock resistant, able to withstand high thermal stresses Small thermal expansion coefficient leads to high thermal shock resistance Large thermal expansion leads to low thermal shock resistance Most materials expand as they are heated Refractory metals and ceramics-Expand less Polymers-Expand more Some materials expand very little SiO2 glass; b-spodumene, Li2O.Al2O3.4SiO2 Complex systems with more than one material must have matched or compensated thermal expansions

Thermal expansion on the atomic scale sufficient to look at the classic harmonic oscillator. For zero energy (at 0K) the oscillator is at the bottom, no movement. This is where we define the energy zero. For finite temperature, the energy is increased by kT. Quantum: almost the same, non-equidistant levels non-symmetric prob. distr. again: note that for the harmonic solid which we usually like to look at, there is no expansion

• Materials change size when heating. CTE: coefficient of thermal expansion (units: 1/K) Sides symmetry for harmonic approximation

Typical Thermal Expansion Coefficients of Materials

Polycrystalline materials undergo phase transformations Thermal expansion changes at each phase transition c-SiO2 has numerous phase changes and numerous volume changes that must be accounted for during heat up of systems using SiO2

For isotropic materials, homogeneous in three directions,… Volume expansion coefficient is 3 times larger than linear expansion Glasses are isotropic Fine grained polycrystals are isotropic

Thermal Expansion of Alkali Borate Glasses Addition of alkali modifier decreases thermal expansion coefficient in alkali borate glasses Modifier in low alkali borate glasses, cross links glass structure Creation of tetrahedral borons Adding bonds to boron, increasing connectivity of network Strengthening the network Rigidity of the glassy network increases Thermal expansion decreases with modifier

4-2 supplement : Morse shows another expression for the potential Any real crystal resists compression to a smaller volume than its equilibrium value more strongly than expansion due to a larger volume. Peter Bruesch Phonons：Theory and Experiments Ⅰ P154

D deionization， >0 Expand it： Morse potential Defined as 0 0, at equilibrium point anharmonic term Harmonic term

Only consider the cubic term： are force constant， Only consider the cubic term： Harmonic term anharmonic term Based on the Boltzman statistics With no cubic term With cubic term, thermal expansion occurs and the coefficient is a constant Deviation from the equilibrium point

Cited from Kittel p89

the solution to he anharmonic approximation the equation for a biatomic motion the reduced mass ① ② The solution is ③ If we consider the higher term in the Fourier expansion, there will be terms with 3ω from③ and ①, and assume sA<<1， ⑤ ④ ⑥ For the fact that

⑦ Form the curve slop equilibrium point , the average kinetics for the biatoms Substitute it in ⑦ Substitute it in ⑤ Implication：the description of molecules’ motion is very complicated with multi atoms. It not only need the basic frequency but also involved with

4-3 supplement : classical quantization