Examples Problem
A large piece of glass with a density of 100 g/cm3 and a thickness of 1 cm containing 2.0 at % He is annealed at high temperatures for removing He. The surface concentrations of He for both surfaces are kept at 1.0 at % by a gas source. T

Matsc 503 Kinetics of Materials Processes - Spring 2002 Problem Set #3 - Due Feb. 6 1. The surface of a piece of pure iron is placed in contact with a carburizing gas at 800C in which the activity of carbon is equivalent to a concentration of 0.8 wt % car

Kinetics of Materials Processes - Solutions to Test 2 Problem 1 (35 points)
April 8, 2002 a. Self-diffusion in a pure solid is not correlated, but tracer diffusion is correlated: true or false? True b. Vacancy diffusion in a pure solid is correlated: true

Matsc 503 Kinetics of Materials Processes - Spring 2002 Problem Set #4, Due February 13 1. In the temperature range 70 to 400oC the diffusion coefficient for C in a-Fe is D=0.02exp(-10120/T) cm2/s. If the average vibration frequency of the carbon atoms in

MatSc503 Spring, 2002 Solutions to Homework #1 1) Hydrogen at 1.0 MPa is to be stored at 400oC in outer space, in a thin wall (0.1 mm) spherical iron tank of 0.1 m radius. Please calculate the rate of pressure drop (MPa/s) as a result of diffusion of hydr

Matsc 503 Kinetics of Materials Processes Spring 2002 Problem Set #6 Due April 3
1. To study the diffusion of nickel in the grain boundaries of nickel oxide NiO in the temperature range between 500oC and 800oC, polycrystalline nickel oxide samples were pr

Since the activation for diffusion along extended defects is smaller than that in the bulk, the diffusion along extended defects become increasingly important as temperature decreases Example lnD
Polycrystalline Ag Single crystal Ag 1/T
The ratio of the n

MatSc503 Kinetics of Materials Processes HW #2
Due January 30, 2002
1. Hydrogen in a very large slab of nickel of thickness 1 cm is removed by subjecting it to a vacuum at 600C. The diusion coecient of hydrogen in nickel at 600C is approximately 1.0 106 c

Solutions to Test 1 MatSc 503 - Kinetics of Materials Processes Test date: February 25, 2002
1. A 1-mm-thick sheet of steel at 750C is subjected to conditions that maintain the carbon content at zero on one surface and at 3 at% carbon on the other surface

Homework Problem Set #1
Due January 23, 2002 1. Hydrogen at 1.0 MPa is to be stored at 400oC in outer space, in a thin wall (0.1 mm) spherical iron tank of 0.1 m radius. Please calculate the rate of pressure drop (MPa/s) as a result of diffusion of hydrog

Solution to Homework #4 Problem
1. Delayed 2. Write a defect reaction equation for each of the following processes a Solid solution of CaCl2 in NaCl
CaCl 2 Ca + 2ClCl + V Na Na Nacl
b
Solid solution of SrTiO3 and BaTiO3
SrTiO3 3 SrBa + TiTi + 3OO BaTiO

Examples
(Application of Ficks first law to steady state problems with constant diffusion coefficient (problem 1 and part of problem 3), and Ficks second law to non-steady problems with thin-film and constant surface concentration boundary conditions, and

MatSc503 Kinetics of Materials Processes - Test 1 February 24, 1999 1. (35 points) The phase diagram of a binary system (components A and B) is shown in the figure below. An infinite couple is made with pure B on one side and a finely dispersed twophase a

Solution to Problem #7 of Sample Exam Questions 7. In CaF2 , the predominant ionic defects are the anti-Frenkel defects whose concentrations are controlled from the following defect reaction:
FF V F + Fi'
and the equilibrium constant is given by ' K1 = V

Sample Exam Questions
1. A piece of Steidlium (St) with a dimension 10cm 10cm 1cm containing 1.0 at % Deikium (De) is put into an oven for removing De. The density of St is 5g/cm3. The surface concentration of De is maintained at 0 at%. The diffusion coef

Kinetics of Phase Transformations
(References: J. W. Christian, Theory of Transformations in Metals and Alloys; Porter and Easterling, Phase Transformations in Metals and Alloys)
Change of state, e.g. s l , l v, l s Phase Transformations Change of structu

Coupling of Other Processes with Diffusion
Review of irreversible thermodynamics Generalized linear theory for coupled kinetic processes Thermoelectric phenomena Thermomigration Electromigration Diffusion in multicomponent systems
Irreversible Thermodyna

Self-Diffusion Through Vacancy Mechanism
Self-diffusion: diffusion of atoms in a pure material (e.g., Al atoms in pure Al) or binary and multicomponent compounds in which the diffusion of different atoms is limited to their own sublattices (e.g. Mg and O

(e) Electrons, holes, and ionization states of defects The process of forming intrinsic electron-hole pairs is excitation across the bandgap
null e'+ h
Their concentrations can be approximated as
E n = p = N c N v exp g 2kT
Nc effective conduction band

Kirkendall Effect and Chemical Diffusion
Up to this point, we have considered self-diffusion, tracer atom diffusion, and diffusion of small interstitial atoms in a host lattice of larger atoms. We usually assumed that the diffusion coefficient is a consta

Diffusion in Homogeneous Concentrated Alloys
In a homogeneous concentrated alloy, two tracer diffusion coefficients can be defined, one for each species:
AB DA*
and
AB DB*
They can be measured through a thin-film experiments using tracer atoms. By varying

Thermodynamic Properties of Point Defects
(a) Single vacancy in elementary crystals The enthalpy (energy for constant volume) change by removing an atom inside a crystal and placing it on the surface is called the formation enthalpy (energy) of a vacancy,

Atomic Theory of Diffusion
(See Chapter 2 in Philibert)
The purpose: to relate the phenomenological diffusion coefficient to microscopic parameters such as jump distance and jump frequency Consider a simple problem of diffusion along [001] direction of a

At short times, many terms are required for obtaining the accurate concentration profile. However, at long times, a single term is sufficient as each successive term is smaller than the preceding one. Consider the ratio of the maximum values of the first

Square Root Relationship
For infinite or semi-infinite systems with constant surface concentration, the solution involves only the single dimensionless parameter
x 2 Dt
1. 2. 3. The distance of penetration of any given concentration is proportional to the

(2) Extended initial distributions, e.g., for the initial condition, c
c for x < 0 c( x,0 ) = o 0 for x > 0
co
d
x=0 x
Consider a line source of strength, co d
2 co d 1 2 exp 2(Dt ) 4 Dt
Superposition of the profiles from all elements,
c ( x, t ) =
2 co

Types of Solutions to Ficks Second Law
Error function solutions short-time or infinite systems
l (system size ) > Dt
Trigonometric function solutions long-time or finite systems
l (system size ) ~ Dt
Methods of solution: (1) Fourier transforms; (2) separa

Numerical Example Quasi-steady State
A cubic steel tank of volume 1 liter and wall thickness 0.1 cm is used to store hydrogen with an initial pressure of 8.7 atm. The tank is placed in outer space at 673 K. The hydrogen concentration on the steel surface

Ficks First Law
J = D c x
J flux of diffusing particles (molecules, atoms, ions, point defects, free electrons, electron holes, etc.). Flux has a direction. c x concentration gradient. D diffusion coefficient or diffusivity.
L2 c mole/L3 mole J 2 = D Un