L11_Solid_Diffusion

THE WAY TO
SOMEWHERE…
Sub-topics
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Diffusion
Diffusion processes in industry
RATE PROCESSES IN SOLIDS
At any temperature different from absolute zero all atoms, irrespective
of their state of aggregation (gaseous, liquid or solid), are constantly in
motion
diffusion
Diffusion refers to the net flux of any species, such as ions,
atoms, electrons, holes, and molecules.
Flux = (conductivity) x (driving force)
In the case of atomic or molecular
diffusion, the “conductivity” is
referred to as the diffusivity or
the diffusion constant D
this diffusion constant (D) reflects
the mobility of the diffusing
species in the given environment
The “driving force” for many
types of diffusion is the existence
of a concentration gradient.
The term “gradient” describes the
variation of a given property as a
function of distance
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NON-SPONTANEOUS PROCESSES
1.
For process to be started,
the atoms should have
sufficient energy to
overcome an activation
energy barrier.
2.
Q=activation energy
State 1
Involves reduction in energy
Or
Energy change is negative
State 2
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DIFFUSION
As T of the system is increased, more and more molecules will attain the
activation energy level.
In statistical mechanics, Maxwell–
Boltzmann statistics describes the
distribution of material particles
over various energy states and
probabilities to find a particle in
definite state:
Probability ~ exp(-∆E/RT)
R– Boltzman constant = 1.38
x10 -23 J/atom K
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WHAT IS DIFFUSION?
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SELF-DIFFUSION
C
C
A
D
B
D
A
t
B
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VACANCY OR SUBSTITUTIONAL DIFFUSION
Why, in general, is
the activation energy
for self diffusion higher
for materials of high
melting point?
• Atoms move into the
vacancies places.
• More vacancies are
created at higher
temperature.
• Diffusion rate is higher
at high temperatures.
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ATOMIC DIFFUSION IN SOLIDS
Diffusion is a process by which a matter is
transported through another matter.
Examples:
9 Movement of smoke particles in air :
Very fast.
9 Movement of dye in water :
Relatively slow.
9 Solid state reactions : Very restricted
movement due to bonding.
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Diffusion processes may be divided into two types:
(a) steady state and (b) non-steady state.
HOW FAST DOES DIFFUSION OCCUR?
Cu flux Ni flux
Concentration
of Cu [kg/m3]
Concentration
of Ni [kg/m3]
Position, x
• Concentration Profile, C(x): [kg/m3]
flux in x-dir.
[kg/m2-s]
Diffusion coefficient [m2/s]
dC
Jx = − D
dx
concentration
gradient [kg/m4]
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The flux is defined as the number of
The rate at which atoms, ions, particles atoms passing through a plane of unit
or other species diffuse in a material
area per unit time
STEADY-STATE DIFFUSION
o Diffusion is a time dependent process and the rate of mass transfer is
the diffusion flux (J).
o In a steady-state condition the concentration gradient is constant.
Steady state diffusion takes place at a constant rate - that is, once
the process starts the number of atoms crossing a given interface (the
flux) is constant with time.
This means that throughout the system dc/dx = constant and dc/dt = 0.
Fick’s First law:
Net flow of atoms
Per unit area per
Unit time = J
atoms/cm2s
The diffusive flux
is proportional to
the existing
concentration
gradient.
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STEADY – STATE DIFFUSION PROCESS
A practical example of steady-state diffusion –
the purification of hydrogen gas.
One side of a thin sheet of palladium metal is exposed to the
impure gas composed of hydrogen and other gaseous species such
as nitrogen, oxygen, and water vapor. The hydrogen selectively
diffuses through the sheet to the opposite side, which is
maintained at a constant and lower hydrogen pressure.
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PROBLEM - CONCENTRATION GRADIENT
Problem: A plate of iron is exposed to a carburizing (carbon-rich)
atmosphere on one side and a decarburizing (carbon-deficient) atmosphere on
the other side at 700C.
If a condition of steady state is achieved, calculate the diffusion flux of
carbon through the plate if the concentrations of carbon at positions of 5 and
10 mm beneath the carburizing surface are 1.2 and 0.8 kg/m3, respectively.
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Assume a diffusion coefficient of
3 x 10 -11 m2/s at this temperature.
CONCENTRATION GRADIENT
The concentration gradient shows how
the composition of the material varies
with distance:
c is the difference in concentration
over the distance x
atoms/m2 s
Kg/m2 s
Kg/m3
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PROBLEM - SEMICONDUCTOR DOPING
One way to manufacture transistors, which amplify
electrical signals, is to diffuse impurity atoms into a
semiconductor material such as silicon (Si).
Problem: Suppose a silicon wafer 0.1 cm thick, which
originally contains one phosphorus atom for every 10
million Si atoms, is treated so that there are 400 Patoms for every 10 million Si atoms.
Calculate the concentration gradient (a) in atomic
percent/cm and (b) in atoms/(cm3 x cm)
The lattice
parameter of
silicon is 5.4307 A.
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DIFFUSIVITY
Diffusivity depends on
9 Type of diffusion : Whether the diffusion is
interstitial or substitutional.
9 Temperature: As the temperature increases
diffusivity increases.
9 Type of crystal structure: BCC crystal has lower
APF than FCC and hence has higher diffusivity.
9 Type of crystal imperfection: More open structures
(grain boundaries) increases diffusion.
9 The concentration of diffusing species: Higher
concentrations of diffusing solute atoms will affect
diffusivity.
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DIFFUSION COEFFICIENT
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EFFECT OF TEMPERATURE ON DIFFUSION
A large activation energy results in a relatively small diffusion coefficient.
Temperature has a most profound influence
on the coefficients and diffusion rates.
When the temperature increases, the diffusion coefficient D increases
and, therefore, the flux of atoms increases as well.
At higher temperatures, the thermal energy supplied to the diffusing
atoms permits the atoms to overcome the activation energy barrier and
more easily move to new sites in the atomic arrangements.
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At low temperatures—often below about 0.4 times the absolute
melting temperature—diffusion is very slow and may not be significant.
IMPURITY DIFFUSION INTO SILICON WAFER
the activation
energies in ionic
materials are high
and the rates of
diffusion are low
Doping Silicon with P
SiO2
Impurities are made to diffuse
into silicon wafer to change its
electrical characteristics.
• Used in integrated circuits.
• Silicon wafer is exposed to
vapour of impurity at 1100C in
a quartz tube furnace.
• The concentration of
impurity at any point depends
on depth and time of exposure.
1. Deposit P rich
layers on surface.
silicon
2. Heat it.
3. Result: Doped
semiconductor
regions.
silicon
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DESIGN PROBLEM: INTEGRATED CIRCUIT
INTERCONNECTS
Top layers that serve as the wire
for this device (interconnect).
Diffusion-layer doped silicon that
have been coated with an
interlayer dielectric.
What material can be used
for interconnects?
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DESIGN OF AN IRON MEMBRANE
A cylinder 3 cm in diameter
and 10 cm long contains a gas
that includes 0.5 x1020 N atoms
per cm3 and 0.5 x1020 H atoms
per cm3 on one side of an iron
membrane.
Gas is continuously introduced to the pipe to assure a constant
concentration of nitrogen and hydrogen.
The gas on the other side of the membrane includes a constant 1
1018 N atoms per cm3 and 1 x 1018 H atoms per cm3.
The entire system is to operate at 700 C (at this T iron has the BCC
structure).
Design an iron membrane that will allow no more than 1% of the
nitrogen to be lost through the membrane each hour,
while allowing 90% of the hydrogen to pass through the membrane
20
per hour.
TYPES OF DIFFUSION
In volume diffusion, the atoms move through the crystal
from one regular or interstitial site to another. Because of
the surrounding atoms, the activation energy is large and
the rate of diffusion is relatively slow.
Atoms can also diffuse along boundaries, interfaces, and
surfaces in the material. Atoms diffuse easily by grain
boundary diffusion, because the atom packing is
disordered and less dense in the grain boundaries.
Because atoms can more easily squeeze their way through
the grain boundary, the activation energy is low.
Surface diffusion is easier still because there is even less
constraint on the diffusing atoms at the surface.
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TUNGSTEN -THORIUM DIFFUSION COUPLE
Consider a diffusion couple between pure tungsten
and a tungsten alloy containing 1 at% thorium.
After several minutes of exposure at 2000°C, a
transition zone of 0.01 cm thickness is established.
What is the flux of thorium atoms at this time if
diffusion is due to
(a) volume diffusion,
(b) grain boundary diffusion, and
The lattice parameter of
(c) surface diffusion?
BCC tungsten is 3.165 Å.
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NONSTEADY-STATE DIFFUSION
Concentration of solute atoms at any point in metal changes with
time in this case.
Change of concentration of solute atoms with change in time in different planes
Fick’s Second Law
Ficks second law:
Rate of compositional change is equal to
diffusivity times the rate of change of
concentration gradient.
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NON-STEADY STATE DIFFUSION
• Copper diffuses into a bar of aluminum.
Surface conc.,
Cs of Cu atoms
Cs
bar
pre-existing conc., C o of copper atoms
C(x,t)
t3
t2
to t1
Co
position, x
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SOLUTION
One practically important solution is for a semi-infinite solid in which the
surface concentration is held constant.
Frequently, the source of the diffusing species is a gas phase, the partial
pressure of which is maintained at a constant value.
Furthermore, the following assumptions are made:
1.Before diffusion, any of the diffusing solute
atoms in the solid are uniformly distributed with
concentration of C0 .
2. x at the surface is zero and increases with
distance into the solid.
3. The time is taken to be zero before
the diffusion process begins.
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FICK’S SECOND LAW – SOLUTION
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TABULATION OF ERROR FUNCTION VALUES
Z
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CARBURIZING
Diffusing carbon
atoms
Low
carbon
Steel part
Carbon Gradients
In Carburized metals
• Result:
--hard to deform: C atoms
"lock" planes from shearing.
--hard to crack: C atoms put
the surface in compression.
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INDUSTRIAL APPLICATIONS OF DIFFUSION –
CASE HARDENING
Sliding and rotating parts needs to have
hard surfaces.
• These parts are usually machined with
low carbon steel as they are easy to
machine.
• Their surface is then hardened by
carburizing:
• Steel parts are placed at elevated
temperature (927C) in an atmosphere of
hydrocarbon gas (CH4).
• Carbon diffuses into iron surface and
fills interstitial space to make it harder.
Photograph of a steel gear that
has been ‘‘case hardened.’’ The outer
surface layer was hardened by a high29
temperature heat treatment during
which carbon from the surrounding
atmosphere diffused into the surface.
EFFECT OF TEMPERATURE ON DIFFUSIONEXAMPLE
If diffusivity at two temperatures are determined, two
equations can be solved for Q and D0
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TIME COMPUTATION
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DIFFUSIVITY DATA FOR SOME METALS
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PROBLEM: TIME FOR DIFFUSION
Carburizing: the steel piece is exposed, at an elevated temperature, to an
atmosphere rich in a hydrocarbon gas, such as methane (CH4 ).
Consider an alloy that initially has a uniform carbon concentration
of 0.25 wt% and is to be treated at 950 C. If the concentration of carbon at
the surface is suddenly brought to and maintained at 1.20 wt%,
how long will it take to achieve a carbon content of 0.80 wt% at a position
0.5 mm below the surface?
The diffusion coefficient for carbon in iron at this temperature is 1.6 x 10 -11
m2/s.
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DESIGN PROBLEM
The wear resistance of a steel gear is to be improved by hardening its
surface. This is to be accomplished by increasing the carbon content
within an outer surface layer as a result of carbon diffusion into the
steel; the carbon is to be supplied from an external carbon-rich gaseous
atmosphere at an elevated and constant temperature.
The initial carbon content of the steel is 0.20 wt%, whereas the surface
concentration is to be maintained at 1.00 wt%. In order for this treatment
to be effective, a carbon content of 0.60 wt% must be established at a
position 0.75 mm below the surface.
Specify an appropriate heat treatment in terms of temperature and time
for temperatures between 900C and 1050 C. Use data in Table for the
diffusion of carbon in γ-iron. The gas const = 8.31 J/mol K
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DESIGN OF A MORE ECONOMICAL HEAT
TREATMENT
10 h are required to successfully carburize a batch of 500 steel
gears at 900°C, where the iron has the FCC structure.
We find that it costs $1000 per hour to operate the carburizing
furnace at 900°C and $1500 per hour to operate the furnace at
1000°C.
Is it economical to increase the carburizing temperature to
1000°C?
What other factors must be considered?
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SUMMARY:
STRUCTURE & DIFFUSION
Diffusion FASTER for... Diffusion SLOWER for...
• open crystal structures
• close-packed structures
• lower melting T materials
• higher melting T materials
• materials w/secondary
bonding
• materials w/covalent
bonding
• smaller diffusing atoms
• larger diffusing atoms
• lower density materials
• higher density materials
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