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DIFFUSION IN SOLIDS
WHY STUDY DIFFUSION?
• Materials often heat treated to improve properties
• Atomic diffusion occurs during heat treatment
• Depending on situation higher or lower diffusion rates
desired
• Heat treating temperatures and times, and heating or cooling
rates can be determined using the mathematics/physics of
diffusion
Example: steel gears are “case-hardened” by
diffusing C or N to outer surface
Courtesy of P. Alpay
DIFFUSION IN SOLIDS
ISSUES TO ADDRESS...
• Atomic mechanisms of diffusion
• Mathematics of diffusion
• Influence of temperature and diffusing species on
Diffusion rate
Courtesy of P. Alpay
DIFFUSION
Phenomenon of material transport by atomic or particle
transport from region of high to low concentration
•
•
•
•
What forces the particles to go from left to right?
Does each particle “know” its local concentration?
Every particle is equally likely to go left or right!
At the interfaces in the above picture, there are
more particles going right than left Î this causes
an average “flux” of particles to the right!
• Largely determined by probability & statistics
Courtesy of P. Alpay
DIFFUSION DEMO
• Glass tube filled with water.
• At time t = 0, add some drops of ink to one end
of the tube.
• Measure the diffusion distance, x, over some time.
to
x (mm)
t1
t2
t3
xo
x1
x2 x3
time (s)
Courtesy of P. Alpay
PROCESSING USING DIFFUSION (1)
• Case Hardening:
-- Example of interstitial
diffusion is a case
hardened gear.
-- Diffuse carbon atoms
into the host iron atoms
at the surface.
Fig. 5.0,
Callister 6e.
(Fig. 5.0 is
courtesy of
Surface
Division,
MidlandRoss.)
• Result: The "Case" is
--hard to deform: C atoms
"lock" planes from shearing.
--hard to crack: C atoms put
the surface in compression.
Courtesy of P. Alpay
PROCESSING USING DIFFUSION (2)
• Doping Silicon with P for n-type semiconductors:
• Process
0.5mm
1. Deposit P rich
layers on surface.
magnified image of a computer chip
silicon
Fig. 18.0,
Callister 6e.
2. Heat it.
3. Result: Doped
semiconductor
regions.
silicon
light regions: Si atoms
light regions: Al atoms
Courtesy of P. Alpay
PROCESSING USING DIFFUSION (3)
Driving force of sintering process = reduction of energy associated with particles surfaces
raw powder synthesis
pressing or casting
densification
grain growth
Two processes
in competition:
densification and coarsening
Porosity = 0
Transparent material
coarsening
Porosity ≠ 0
Translucent material
(if no refractive index mismatch between grains)
Reducing scattering losses implies control of ceramic microstructure
Temperature
“Green”
Ceramic Sintering
Factors affecting
solid-state sintering: . Temperature, pressure
. Green density
. Uniformity of green microstructure
. Atmosphere composition
. Impurities
. Particles size and size distribution
Sintering mechanisms
DIFFUSION: THE PHENOMENA (1)
• Self-diffusion: In an elemental solid, atoms
also migrate.
Label some atoms
After some time
C
C
A
D
D
A
B
B
B*: self diffusion mobility
Self – diffusion coefficient in a simple cubic lattice
δ: interatomic distance
v: jump frequency
Courtesy of P. Alpay
Tracer atom studies may not give the true self-diffusion coefficient, Di*
(12.6)
f: correlation coefficient
Di*: true self-diffusion coefficient
DIFFUSION: THE PHENOMENA (2)
• Interdiffusion: In an alloy or “diffusion couple”, atoms tend
to migrate from regions of large to lower concentration.
Initially (diffusion couple)
After some time
Adapted from
Figs. 5.1 and
5.2, Callister
6e.
100%
0
Cu
Ni
100%
Concentration Profiles
0
Concentration Profiles
Courtesy of P. Alpay
FICK’S LAW AND DIFFUSIVITY OF
MATERIALS
DEFINITION OF FLUXES – FICK’S FIRST LAW
If the concentration of component A is given in mass units, the rate equation
for diffusion is:
(12.1)
WAx: mass flux of A in the x-direction, kg (of A) m-2s-1
ρA: mass concentration of A, kg (of A) m-3 (of total material)
DA: diffusion coefficient, or diffusivity of A, m2s-1
(intrinsic diffusion coefficient)
In the materials field, the rate equation is usually written in terms of molar
concentrations:
(12.2)
jAx: molar flux of A in the x-direction, mol (of A) m-2s-1
CA: molar concentration of A, mol (of A) m-3 (of total material)
Fick’ s first law of diffusion states that species A diffuses in the
direction of decreasing concentration of A, similarly as heat flows by
conduction in the direction of decreasing temperature, and momentum is
transferred in viscous flow in the direction of decreasing velocity.
Eq. (12.1) and (12.2) are convenient forms of the rate equation when
the density of the total solution is uniform (in solid or liquid solutions, or in
dilute gaseous mixture).
When the density is not uniform:
ρ: density of the entire solution, kg/m-3
(12.3)
ρA*: mass fraction of A
C: local molar concentration in the
solution at the point where the gradient is
measured
(12.4)
DIFFUSION IN SOLIDS
1. Self-diffusion
Self – diffusion data apply to homogenous alloys in which there is no gradient
in chemical composition.
DIFFUSION IN SOLIDS
2. Diffusion under the influence of a composition gradient
ƒ Vacancy mechanism
ƒ Ring mechanism
ƒ Interstitialcy mechanism
Flux of gold atoms (according to an observer sitting on a plane moving
with the solid’s velocity):
Total flux of gold atoms (according to an observer sitting on an unattached
plane in space):
(12.7)
The accumulation of gold as a function of time within a unit volume:
(12.8)
(12.9)
(12.10)
Accumulation of nickel in the same unit volume:
(12.11)
If the vacancy concentration within the unit volume is constant, then the
volume is approximately constant and:
(12.12)
(12.13)
Only in terms of the gold concentration gradient:
(12.14)
Accumulation of gold solely in terms of diffusion coefficients and
concentration gradients:
(12.15)
Fick’s first law in its simplest form, for gold:
(12.16)
∼
D indicates that the flux is proportional to the concentration gradient but
side-steps the issue of bulk flow.
The accumulation of gold within the unit volume:
(12.17)
The unsteady-state equation for unidirectional diffusion in solids (Fick’s
second law):
(12.18)
Equation (12.18) is used to obtain analytical solutions for diffusion problems
(12.19)
DAu and DNi: Intrinsic diffusion coefficients
∼
D : Interdiffusion, mutual diffusion, or chemical diffusion
DIFFUSION IN SOLIDS
3. Darken’s equation
(12.20)
nAx: flux of A atoms passing through an unit area in the x-direction
nA: the number of A atoms per unit volume
BA: mobility of A atoms in the presence of the energy gradient
N
_ 0: Avogadro’s number
GA: partial molar free energy of A (chemical potential of A)
(12.21)
ax: activity of A
(12.22)
KB: Boltzmann’s constant = R/N0
XA: Mole fraction of A
aA= XA (for a thermodynamically ideal solution)
(12.23)
NERNST-EINSTEIN EQUATION
Gibbs – Duhem equation:
(12.24)
And only if BA = BA* and BB = BB*, then Eq. (12.19) can be written as:
(12.25)
If the system is ideal:
(12.26)
DIFFUSION IN SOLIDS
4. Temperature dependence of diffusion in solids
(12.27)
(Arrhenius equation)
According to Zener, if the jump process is an activated one, jump frequency
ν can be described by:
(12.28)
ν0= vibrational frequency of the atom in the lattice
Z = coordination number
∆G: = free energy of activation required for the atom to jump from one site into the next
(12.29)
Sherby and Simnad developed a correlation equation for predicting selfdiffusion data in pure metals:
(12.30)
K0 depends only on the crystal structure
V = normal valence of the metal
TM = absolute melting point
D0 is approximated as 1 × 10-4 m2 s-1 for estimation purposes
(12.31)
DIFFUSION IN CERAMIC MATERIALS
The general equation for the transport of the species in the x-direction
under the influence of an electric field and a concentration gradient in the
x-direction:
(12.32)
In the absence of a concentration gradient, the flux is:
(12.33)
The current density:
(12.34)
(12.35)
The electrical conductivity σ:
(12.36)
(12.37)
This equation is known as the “extended” Nernst-Einstein equation
(12.38)
ti: transference number of species i
(12.39)
For NaCl:
(12.40)
For CoO:
Thermodynamic factor can be calculated by:
P°O2: standard oxygen pressure, 1 atm
DIFFUSION IN ELEMENTAL SEMICONDUCTORS
DIFFUSION IN LIQUIDS
1. Liquid state diffusion theories
Hydrodynamical theory:
The force on a sphere moving at steady-state in laminar flow:
and since B = V∞/F,
(12.41)
Stokes – Einstein equation
(12.42)
Sutherland – Einstein equation
(12.43)
A is a constant
M = atomic weight or molecular weight
TM = melting point
D* = self-diffusion coefficient
VM = molar (or atomic) volume
If it is assumed that the atoms in the liquid are in a cubic array, then for materials
generally the simple theory predicts that:
(12.44)
Hole theory: This theory presumes the existence of holes or vacancies
randomly distributed throughout the liquid and providing ready diffusion
paths for atoms and ions. The concentration of these holes would have to
be very great in order to account for the volume increase upon melting,
thus resulting in much higher diffusion rates in liquids than in solids just
below the melting point.
Eyring theory:
If the liquid is considered quasi-crystalline, and the atoms are in a cubic
configuration, then an expression relating D* and η results:
(12.45)
Fluctuation theory:
In this theory, the ‘extra’ volume (over that of the solid) in the liquid is
distributed evenly throughout the liquid, making the average nearest
neighbour distance increase.
Reynik’s fluctuation model:
(12.46)
(12.47)
DIFFUSION IN LIQUIDS
2. Liquid diffusion data
Diffusion in liquid metals:
Diffusion in molten salts and silicates:
Diffusion in common liquids:
DIFFUSION IN GASES
Based on the kinetic theory of gases, self-diffusivity of spherical A atoms
diffusing in pure A can be predicted by:
(12.48)
For the diffusivity of two unequal size spherical atoms A and B, kinetic
theory predicts:
(12.49)
KB: Boltzmann’s constant, 1,38×10-16 ergs molecule-1 K-1
Chapman – Enskog theory:
For monoatomic gases
(12.50)
The collision integral can be calculated by:
(12.51)
where A = 1.06036, B = 0.15610, C = 0.19300, D = 0.47635, E = 1.03587,
F = 1.52996, G = 1.76474, and H = 3.89411.
(12.52)
(12.53)
(12.54)
DIFFUSION THROUGH POROUS MEDIA
(12.55)
void
fraction
effective
diffusivity
tortuosity
Tortuosity is a number greater than one and it ranges from values of 1.5 -2.0
for unconsolidated particles, to as high as 7 or 8 for compacted particles.
An analysis of the flow of molecules through a cylindirical pore radius r
under a concentration gradient yields
(12.56)
Knudsen diffusion coefficient
mean free path of the gas
concentration of the molecules, cm-3
collision diameter of the molecule, cm
If r and λ are of the same order of magnitude, or if the pore diameter is
only one order larger, then Knudsen diffusion is presumably important.
If the ratio DAB/DK is small, then the ordinary diffusion is the rate
determinig step. Conversely, if DAB/DK is large, it is presumed that Knudsen
diffusion controls.
(12.57)
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