introduction to materials science, chapter 5, diffusion university of virginia, dept. of materials...
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Introduction To Materials Science, Chapter 5, Diffusion
University of Virginia, Dept. of Materials Science and Engineering 1
Diffusion how atoms move in solids
Diffusion mechanisms Vacancy diffusion Interstitial diffusion Impurities
Mathematics of diffusion Steady-state diffusion (Fick’s first law) Nonsteady-State Diffusion (Fick’s second law)
Factors that influence diffusion Diffusing species Host solid Temperature Microstructure5.4 Nonsteady-State Diffusion – Not Covered / Not Tested
Chapter 5 Outline
Introduction To Materials Science, Chapter 5, Diffusion
University of Virginia, Dept. of Materials Science and Engineering 2
Diffusion transport by atomic motion.
Inhomogeneous material can become homogeneous by diffusion. Temperature should be high enough to overcome energy barrier.
What is diffusion?
Introduction To Materials Science, Chapter 5, Diffusion
University of Virginia, Dept. of Materials Science and Engineering 3
Concentration Gradient Interdiffusion (or Impurity Diffusion).
Self-diffusion: one-component material, atoms are of same type.
Inter-diffusion vs. Self-diffusion
Before After
(Heat)
Introduction To Materials Science, Chapter 5, Diffusion
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Vacancy diffusion
To jump from lattice site to lattice site, atoms need energy to break bonds with neighbors, and to cause the necessary lattice distortions during jump.Therefore, there is an energy barrier.Energy comes from thermal energy of atomic vibrations (Eav ~ kT)
Atom flow is opposite to vacancy flow direction.
Diffusion Mechanisms (I)
Atom migration Vacancy migration
AfterBefore
Introduction To Materials Science, Chapter 5, Diffusion
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Interstitial diffusion
Diffusion Mechanisms (II)
Interstitial atom before diffusion
Interstitial atom after diffusion
Generally faster than vacancy diffusion because bonding of interstitials to surrounding atoms is normally weaker and there are more interstitial sites than vacancy sites to jump to.Smaller energy barrier Only small impurity atoms (e.g. C, H, O) fit into interstitial sites.
Introduction To Materials Science, Chapter 5, Diffusion
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Flux of diffusing atoms, J.
Number of atoms diffusing through unit area per unit time [atoms/(m2s)]
or
Mass of atoms diffusing through unit area per unit time [kg/(m2 s)]
Mass: J = M / (A t) (1/A) (dM/dt)
AJ
Introduction To Materials Science, Chapter 5, Diffusion
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Diffusion flux does not change with time
Concentration profile: Concentration (kg/m3) vs. position
Concentration gradient: dC/dx (kg / m4)
Steady-State Diffusion
BA
BA
xx
CC
x
C
dx
dC
Introduction To Materials Science, Chapter 5, Diffusion
University of Virginia, Dept. of Materials Science and Engineering 8
Fick’s first law: J proportion to dC/dx
Steady-State Diffusion
D=diffusion coefficientdC
J Ddx
Concentration gradient is ‘driving force’
Minus sign means diffusion is ‘downhill’: toward lower concentrations
Introduction To Materials Science, Chapter 5, Diffusion
University of Virginia, Dept. of Materials Science and Engineering 9
Concentration changing with time
Fick’s second law
Nonsteady-State Diffusion(not tested)
Find C(x,t)
2
2
C J
t x
CD
x
Introduction To Materials Science, Chapter 5, Diffusion
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Atom needs enough thermal energy to break bonds and squeeze through its neighbors.
Energy needed energy barrier Called the activation energy Em (like Q)
Diagram for Vacancy Diffusion
Diffusion Thermally Activated Process
AtomVacancyEm
Distance
Ene
rgy
Introduction To Materials Science, Chapter 5, Diffusion
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Room temperature (kBT = 0.026 eV) Typical activation energy Em (~ 1 eV/atom) (like Qv)
Therefore, a large fluctuation in energy is needed for a jump.
Probability of a fluctuation or frequency of jump, Rj
Diffusion – Thermally Activated Process
R0 = attempt frequency proportional to vibration frequency
Swedish chemist Arrhenius
0 exp mj
B
ER R k T
Introduction To Materials Science, Chapter 5, Diffusion
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1. Probability of finding a vacancy in an adjacent lattice site (Chap. 4):
times2. Probability of thermal fluctuation
Calculate Activated Diffusion
The diffusion coefficient = Multiply
0 exp mj
B
ER R k T
.exp v
B
QP Const k T
.exp expm V
B B
E QD Const k T k T
0 exp d
B
QD D k T
Arrhenius dependence.
Introduction To Materials Science, Chapter 5, Diffusion
University of Virginia, Dept. of Materials Science and Engineering 13
Diffusion coefficient is the measure of mobility of diffusing species.
Diffusion – Temperature Dependence
Arrhenius Plots (lnD) vs. (1/T) or (logD) vs. (1/T)
0 0exp expd dQ QD D DRT kT
dx
dCDJ
D0 – temperature-independent (m2/s)Qd – the activation energy (J/mol or eV/atom)R – the gas constant (8.31 J/mol-K) orkB - Boltzman constant ( 8.6210-5 eV/atom-K)T – absolute temperature (K)
0
1ln ln dQ
D DR T
0
1log log
2.3dQ
D DR T
or
Introduction To Materials Science, Chapter 5, Diffusion
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Diffusion – Temperature Dependence (II)
Graph of log D vs. 1/T has slop of –Qd/2.3R,
intercept of ln Do
21
21d T1T1
DlogDlogR3.2Q
T
1
R3.2
QDlogDlog d
0
Introduction To Materials Science, Chapter 5, Diffusion
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Diffusion – Temperature Dependence (III)
Arrhenius plot:Diffusivity for metallic systems
Introduction To Materials Science, Chapter 5, Diffusion
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Diffusion of different species
Smaller atoms diffuse more readily
Diffusion faster in open lattices or in open directions
Introduction To Materials Science, Chapter 5, Diffusion
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Diffusion: Role of the microstructure (I)
Self-diffusion coefficients for Ag
Depends on diffusion path
Grain boundaries and surfaces less restrictive
Introduction To Materials Science, Chapter 5, Diffusion
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Diffusion: Role of the microstructure (II)
Plots are from computer simulations
Initial positions are shown by the circles, paths are shown by lines. See difference between mobility in the bulk and in the grain boundary.
Introduction To Materials Science, Chapter 5, Diffusion
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Factors that Influence Diffusion
Temperature - diffusion rate increases very rapidly with increasing temperature
Diffusion mechanism - interstitial is usually faster than vacancy
Diffusing and host species - Do, Qd is different for every solute, solvent pair
Microstructure - diffusion faster in polycrystalline vs. single crystal materials because of the rapid diffusion along grain boundaries and dislocation cores.
Introduction To Materials Science, Chapter 5, Diffusion
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Summary
Activation energy Concentration gradient Diffusion Diffusion coefficient Diffusion flux Driving force Fick’s first and second
laws Interdiffusion Interstitial diffusion Self-diffusion Steady-state diffusion Vacancy diffusion
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