What is high temperature ? 0. Introduction : the brittle-plastic - - PowerPoint PPT Presentation

What is high temperature ?

• 0. Introduction: the brittle-plastic transition from above and below...
• 1. Why do lattices form and maintain their stability? The interatomic

(Lennard-Jones) potential, elasticity, and theoretical strength. (Anharmonicity.)

• 2. What is the relation between temperature and lattice vibrations?

Thermally-activated motion and the Arrhenius equation. Homologous Temperature.

• 3. What are point defects, vacancies ? Why is there an equilibrium concentration
• f vacancies? (on to diffusion--> how do those defects enable diffusion?

deformation?)

As the temperature starts to rise in the crust, towards the brittle- plastic transition, what happens to the material? What deformation processes become increasingly possible? Or the inverse, coming from below, as the temperature starts to cool, what happens to the deformation mechanisms?

Creep processes are irreversible, constituting “yield”, or a deviation from the linear elastic part of the stress strain curve. The smallest scale irreversible process is diffusion, which becomes possible when thermal motion/energy is vigorous enough. The higher the temperature, the faster it is...

1) a material with “flaws”

• r Griffith cracks

at “high temperature” these stress concentrations cannot develop, because they are relaxed by diffusion or other “creep” processes... at high pressure, the cracks cannot

• pen,

rocks deform by slower smoother processes than in the schizosphere...

• 1. Why do lattices form and maintain their stability? The interatomic

(Lennard-Jones) potential, elasticity, and theoretical strength. (Anharmonicity.) ~1920 Bragg & son (W.L. and

• W. H. ) -> x-ray scattering in

metals produced diffraction patterns : a crystal lattice ! ~ 1926, the theoretical strength crisis ! (Griffith-- brittle glass) (Frenkel-- ductile metals)

Interatomic Potential two atoms attractive potential = coulombic/ionic/metallic/van der wals

“Anharmonicity is the deviation of a system from being a harmonic oscillator.” - wikipedia http://en.wikipedia.org/wiki/John_Lennard-Jones

repulsive potential = pauli exclusion (overlapping orbitals) repulsion attraction repulsive potential attractive potential

harmonic approximation

Uia = Eaa x r 12 − x r 6 the depth of the well is proportional

solid gas b = 2r

r x

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Periodic model, theoretical strength (Frenckel derivation... )

= shear modulus

E = Fx F = dE dx τ = dF dA

τ = τmax sin 2πx b τ ≈ τmax2πx b τ = µγ γ ≈ x a

τmax = µ 2π b a τmax ≈ µ 2π

a b

b

• 2. What is the relation between temperature and lattice vibrations?

Thermally-activated motion and the Arrhenius equation.

k = A exp −Ea RT ln(k) = −Ea T 1 T + ln(A) Ea = −R

• ∂ ln k

∂ 1

T

• P

low temperature high temperature

1/T ln(k)

• Ea/R

( )=( )( )

high temperature high temperature high temperature # collisions => reaction total # collisions probability of rxn success energy barrier thermal energy http://en.wikipedia.org/wiki/Arrhenius_equation

low temperature high temperature Karato, p 179 Salts Oxides

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k = A exp −Ea RT

• so what do we define as “high temperature”?

increasing activation energy

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=> a fraction of the melting temperature (of the pure phase), called the homologous temperature, T/Tm

• 3. What are point defects, vacancies ? Why is there an equilibrium

concentration of vacancies? (on to diffusion--> how do those defects enable diffusion? deformation?)

~1 nm

silicon metal, atomic force microscopy http://www.omicron.de/

2-D 3-D... random walks...

Sconf = −kB ln W W = (N + nv)! N!nv!

• G(T, P, N, nv) = G0(T, P) + nvgf

v − kBT ln W

G(T, P, N, nv) = G0(T, P) + nvgf

v − kBT ln

(N + nv)! N!nv!

• ∂G

∂nv

• (T,P)

= gf

v − kBT ∂

∂nv ln W ∂G ∂nv

• (T,P)

= gf

v − kBT ∂

∂nv ln (N + nv)! N!nv!

• ∂G

∂nv

• (T,P)

= gf

v − kBT ln

• nv

N + nv

• ∂G

∂nv

• (T,P)

= gf

v − kBT ln (Xv)

• where
• ∂G

∂nv

• = 0
• Xeq

v = exp

• gf

v

kBT

• The equilibrium concentration of vacancies

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ln increasing T

Maxwell-Boltzmann Statistical Thermodynamics:

What is the melting temperature ?

12 atoms 12 atoms 12 atoms Xv = (1/12)3 ~ 5.7e-4 at 1340 K in gold metal. at some thermal state, there is enough kinetic (thermal) energy and a large enough vacancy concentration that the lattice begins to break down... so what is “high temperature” from a material perspective? when creep (i.e. diffusion) can be fast enough to relax any stress concentrations... T/Tm >0.5-0.6... (but of course, this depends on the time scale of the stress change...)

SUMMARY !!!

• 1. Elasticity is small strain, not sensitive to activation energy, but very

sensitive to lattice spacing.

• 2. The theoretical strength is much larger than observed strength...
• 3. Temperature increases the probability that atoms can jump over the

activation energy “barrier”... Arrhenius relation

• 4. Vacancies exist and have equilibrium concentrations (i.e. they are not

“imperfection”) that are exponential functions of temperature.

• 5. “Melting” can be thought of as a critical concentration of vacancies...
• 6. “High T” is some fraction of the melting temp, rule of thumb: T/Tm > 0.5

creep processes are irreversible, constituting “yield”, or a deviation from the linear elastic part of the stress-strain curve. the smallest scale irreversible process is diffusion, which becomes possible when thermal motion/energy is vigorous enough. The higher the temperature, the faster it is...

1) a material with “flaws”

• r Griffith cracks

at “high temperature” these stress concentrations cannot develop, because they are relaxed by diffusion or other “creep” processes... at high pressure, the cracks cannot

• pen,

rocks deform by slower smoother processes than in the schizosphere...

http://www1.eere.energy.gov/geothermal/geomap.html

estimated temperatures at a depth of 6 kilometers

Temperature, °C

The Structure and Physical Properties of the Earth’s Crust, Geophysical Monograph 14, editor John G. Heacock, pp. 169 – 184, 1971.

The Thermal Structure of the Continental Crust

DAVID D. BLACKWELL

measured heat flow at surface

http://smu.edu/geothermal/heatflow/

Theoretical strength

• Anharmonic model of the atomic bond

Stress concentrations

Brittle-plastic transition

Experimental results of brittle-plastic transition