lecture 7 diffusion - bangladesh university of engineering...
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Lecture 7
DiffusionRef:
1. WD Callister Jr. Materials Science and Engineering: An Introduction, John Wiley & Sons.
2. DA Askeland. The Science and Engineering of Materials, Chapman & Hall.
A.K.M.B. RashidDepartment of MME, BUET
Introduction
Mathematical Description of Diffusion
Diffusion in Solids
Factors that Influence Diffusion
Diffusion Mechanisms
Diffusion and Materials Processing
Diffusion in Ionic Compounds and Polymers
Today’s Topics ...
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The phenomenon of transport of mass through material by atomic motion is called diffusion.
What is Diffusion?
Mass transport in gasses and liquids occurs by a combination of convection (fluid motion) and diffusion.
In solids, convection does not occur, and diffusion is the only available mass transport mechanism.
Solid-state diffusion is relatively slower than liquid-state diffusion
Introduction
Important materials processes occur by diffusion:
Case hardening of steel
Doping of semiconductors
Oxidation of metals
Solid-state formation of compounds
Sintering of powders to form dense and strong objects
Diffusion bonding of two solids
Applications of Diffusion
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Steel gear
case hardened to improve hardness and resistance to fatigue
diffusing excess carbon or nitrogen into outer surface layer
Adapted from chapter-opening photograph, Chapter 5, Callister 7e. (Courtesy of Surface Division, Midland-Ross.)
Result:The presence of C (or, H) atoms makes iron (steel) harder
Heat-treatment temperature, time, and/or rate of heating/cooling can be predicted by the mathematics of diffusion
Heat-treatment almost always involve atomic diffusion
desired results depends on diffusion rate
Phenomenological approach:
Here we ask:
How can we describe the rate and the amount of mass transport that occurs in terms of parameters we can measure ?
This approach is important to our ability to control such processes as carburising, nitriding, tempering, homogenising of casting, and the like.
Atomic approach:
Here we ask:
What is the atomic mechanism by which atoms move?
This approach is important to our understanding of how diffusion mechanisms affect such processes as precipitation hardening.
Approach in Studying Diffusion
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Mathematical Description of Diffusion
Consider the unidirectional diffusion of carbon from 1wt% C steel rod to a pure iron rod, which is butt welded each other to form a “diffusion couple”and heated to 700 oC to allow diffusion at a significant rate.
After some time at 700 oC, the couple is quenched to room temperature, and analyse the carbon content along the rod.
What is the rate at which carbon atoms move to the right?X
C=
Wt.
Fra
ctio
n C
arb
on
Position, Z
0.01 t = 0
t = t
t =
0
Fe + 1wt% C Pure Fe
Iron-Steel Diffusion Couple
The composition profile might look as shown by the curve labeled t = t.
After t = ∞, the composition will become constant at an average value.
D : diffusion coefficient for diffusing species in solid, distance2
time
minus sign denotes the flux component 1 is towards lower concentrations, i.e. “down the concentration gradient”
: concentration gradient,
C is either mass density or atom density
dCdZ
or, number/volumedistance
mass/volumedistance
J : diffusion flux , or rate of diffusion number of atomsarea-time
massarea-time
or,
J =1
A
dM
dtM = mass/numberA = area
dC1
dZJ1 = - D1
According to the Fick’s First Law, the flux of atoms, J, is proportional to the volume concentration gradient, dC/dZ, i.e.,
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Whenever a concentration gradient is present in metal, a diffusion flux will occur.
Our problem now is:How do we actually determine D ?
One cannot measure J or D directly.
We can only measure composition as a function of Z and t.
Concentration gradient (dC/dZ) varies with both position and time, and so does the flux.
Therefore, we must determine a differential equation for the diffusion process,
by performing a mass balance upon a differential volume element perpendicular to the mass flow direction.
dZ
Jin Jout
1 2
Mass balance during carbon transport,
Mass In – Mass Out = Mass Accumulation
Rate mass in = All mass comes into the volume element through plane 1= Flux at 1 x Area at 1 = (JA)1
Rate mass out = All mass comes out the volume element through plane 2= Rate at 1 + change in rate across the volume element= (JA)1 + [ (JA)/Z ] dZ
Rate accumulation = change in volume concentration in the volumeelement with time = (C . A dZ) / t = A dZ (C/t)
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JA - JA + dZ = A dZ (JA)Z
Ct
Then,
Which reduces to:
JZ
- = Ct
This is known as the Continuity Equation.
Note that, our treatment assumed that, mass transport occurred in only one direction.
The equation holds for all material flow processes when no material is generated within the volume element, for example, flow of heat, neutrons, electrons, etc.
For one-dimensional mass diffusion process, using Fick’s first law:
=Ct
Z
CZ
D
This is known as the Fick’s Second Law of diffusion.
This is a partial differential equation with C as the dependant variable
and Z and t as the two independent variables.
A solution to this equation will give C as a function of Z, t and D.
dC1
dZJ1 = - D1
JZ
- =Ct
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Steady-State Diffusion
Diffusion is a time-dependent process, where the quantity of element transported is a function of time
Steady-state diffusion occurs when the rate of diffusion does not change with time
Diffusion in Solids
gas at pressure PA
gas at pressure PB
direction of diffusion of
gaseous speciesPA > PBand constant
thin metal plate
area, A
Steady-state diffusion across a thin plate
CB
CA
xA xB
Position, x
Con
cent
ratio
n of
di
ffusi
ng s
peci
es, C
Linear concentration profile for stead-state diffusion
J = - DdC
dx
DC
Dx
CB - CA
xB - xA
=dC
dx≅
Example: Steady-state diffusion
A steel plate is exposed to a carburising (C-rich) atmosphere on one side and a decarburising (C deficient) atmosphere on the other side at 700 C.
If a condition of steady state is achieved, calculate the diffusion flux of C through the plate if the concentration of C at positions of 5 and 10 mm beneath the carburisingsurface are 1.2 and 0.8 kg/m3, respectively.
Assume a diffusion coefficient of 3x10–11 m2/s at this temperature.
x1 x2
c1c2
carbonrichgas
carbondeficient
gas
steady state = straight line Given data:C1 = 1.2 kg/m3 X
1 = 5 mmC2 = 0.8 kg/m3 X
2 = 10 mmD = 3x10-11 m2/s
C2 – C1
x2 – x1
J = - D
= 2.4 x 10–9 kg/m2s
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Most practical diffusion situations are non-steady
For non-steady state diffusion
diffusion flux and concentration flux vary with time and at different points of solid
net accumulation or depletion of the diffusing species resulted
Because diffusion in solids is slow, diffusion is almost always transient!
Nonsteady-State Diffusion
Distance
Co
nce
ntr
atio
n o
f d
iffu
sio
n s
pec
ies
t 0
t 1 < t2 < t 3
C
t= D
2C
x2
Non-steady state diffusion is described by the Fick’s second law
Concentration profile, C(x), changes with time
C
t= D
2C
x2
Solution to this differential equation for a semi-infinite solid with constant surface concentration can be done
Cs – C0Cx – C0
C0
Cx
Cs
Distance from interface, x
Co
nce
ntr
atio
n, C
assuming that
1. Initial concentration C0
2. X = 0 at the surface, and increases with distance into the solid
3. At the initial time, t = 0
with the boundary conditions that
1. For t = 0, C = C0 at 0 ≤ x ≤ ∞
2. For t > 0, C = CS (constant surface concentration) at x = 0
3. For t > 0, C = C0 at x = ∞
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General solution to this differential equation
Cx – C0
Cs – C0
= 1 - erfx
2(Dt)
Cx is a function of dimensionless parameter x / (Dt)
erf ( ) : Gaussian error function is defined by
erf (z) = e dy-y22
π
z
0
where x /(Dt) has been replaced by the variable z.
The Error Function
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0
Z
erf
(Z
)
z erf (z) z erf (z)
0.00 0.0000 0.70 0.6778
0.01 0.0113 0.75 0.7112
0.02 0.0226 0.80 0.7421
0.03 0.0338 0.85 0.7707
0.04 0.0451 0.90 0.7969
0.05 0.0564 0.95 0.8209
0.10 0.1125 1.00 0.8427
0.15 0.1680 1.10 0.8802
0.20 0.2227 1.20 0.9103
0.25 0.2763 1.30 0.9340
0.30 0.3286 1.40 0.9523
0.35 0.3794 1.50 0.9661
0.40 0.4284 1.60 0.9763
0.45 0.4755 1.70 0.9838
0.50 0.5205 1.80 0.9891
0.55 0.5633 1.90 0.9928
0.60 0.6039 2.00 0.9953
0.65 0.6420
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Example:A steel has a uniform C concentration of 0.25 wt.%. It is carburised at 950 oC using a media having a C content of 1.2 wt.%. How long will it take to achieve a C content of 0.80 wt.% at a position of 0.5 mm below the surface?
0.421 = erf62.5t
Using table, and after interpolation
z = 0.392 = 62.5t
t = 25400 s
C0 = 0.25 % CCs = 1.20 % CCx = 0.80 % Cx = 0.5 mmD = 1.6x10-11 m2/s
Cx – C0
Cs – C0
= 1 - erfx
2(Dt)
Factors that Influence Diffusion
1. Diffusing Species
Magnitude of diffusion coefficient D indicative of the rate at which atoms diffuse
D depends on both the diffusing species as well as the host atomic structure
Depending on this, two categories of diffusion can be recognised:
1. Self-diffusion
2. Inter-diffusion
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Movement of atoms through their own lattice
Occurs in elemental solids, or pure metals
A
B
C
D
B
C
D
Initially After some time
Self-diffusion
Observe the position of labeled atoms after diffusion
A
Atoms of one metal diffuse into another in response to a concentration gradient, resulting a net drift of atoms from higher to lower concentrations (formation of alloy region)
Inter-diffusion
Faster than self-diffusion
Concentration Profile
100 %
0 %
Cu Ni
Initially
Concentration Profile
100 %
0 %
After some time Example: movement of Ni atoms through the
lattice of Cu atoms
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2. Temperature
Has the most profound influence on the coefficients and diffusion rate
Example: Fe in a-Fe
@ 500oC D = 3.0 E(-21) m2/s@ 900oC D = 1.8 E(-15) m2/s approximately six orders higher
Diffusivity increases with T:
D = D0 exp -Qd
RT
D0 = T independent pre-exponentialQd = activation energy [J/mol], [eV/mol]
D = D0 exp -Qd
RT
Dinterstitial >> Dsubstitutional
C in a-Fe Cu in CuC in g-Fe Al in Al
Fe in a-FeFe in g-FeZn in Cu
D has exponential dependence on T
ln D = ln D0 –Qd
R
1
T
log D = log D0 –Qd
2.3R
1
T
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log D = log D0 –Qd
2.3R
1
T
Qd = – 2.3 Rlog D1 – log D2
1/T1 – 1/T2
Thus, knowing two diffusivity data at two different temperatures, the activation energy for the diffusing atom can be calculated.
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3. Role of Microstructure
Diffusivity of atoms depends on the diffusion path.
In general, the diffusivity is greater through the less restricted structural regions
[1] grain boundary[2] dislocation cores[3] external surface
Surface DiffusionActivation energy for diffusion is the lowest, since there are no atoms above the atom of interest; fewer neighbours, fewer bond breaking
Since grain boundaries are relatively more open structure compared to atomic structures inside the grain, the barrier through grain boundary is much less
Grain boundary Diffusion
Pipe DiffusionSince it feels like movement of atoms through a pipe
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Diffusion Mechanisms
Atoms in solids are in constant motion rapidly changing positions.
Diffusion is just the stepwise migration of atoms from one lattice site to other lattice site.
Two conditions for movement:1. There must be an empty adjacent site2. Atom must have sufficient energy to break bonds with neighbour atoms
Atomic vibration Every atom vibrates very rapidly about its lattice position within crystal
At any instant, not all vibrate with same frequency and amplitude
Not all atoms have same energy
Same atom may have different level of energy at different time
Energy increases with temperature
To jump from lattice site to lattice site, atoms need activation energy to break bonds with neighbours, and to cause necessary lattice distortion.
This energy come from the thermal energy of atomic vibration (Qm ~ kT)
The average thermal energy of an atom (kBT = 0.026 eV at RT) is usually much smaller than the activation energy (Qm ~ 1 eV/atom).
So a large fluctuation of energy (when the energy is “pooled together” in a small volume) is needed for a jump.
The probability of such fluctuation, or the frequency of jumps, Rj, depends exponentially on T and defined as
Rj = R0 exp -Qm
kBT
R0 = attempt frequency, proportional to the frequency of atomic vibrationFigure 5.12 A high energy is required to squeeze atoms past one another
during diffusion. This energy is the activation energy Q. Generally more energy is required for a substitutional atom than for an interstitial atom
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Vacancy Diffusion
Involves interchange of an atom from a normal lattice position to an adjacent vacant lattice site or vacancy
Necessitates presence of vacancies
Diffusing atoms and vacancies exchange positions
they move in opposite directions
Diffusion rate depends on:[1] no. of vacancies[2] activation energy to exchange
Both self- and inter-diffusion occurs by this mechanism
Interstitial Diffusion
Atoms migrate from an interstitial position to a neighboring one that is empty
Found for inter-diffusion of small impurity atoms, such as hydrogen, carbon, nitrogen, and oxygen, to fit into interstices in host.
Host or substitutional impurity atoms rarely have interstitial diffusion
Diffusion rate depends on: [1] vacancy concentration[2] jump frequency
Interstitial atoms are smaller and thus more mobile
interstitial diffusion occurs much more rapidly than by vacancy mode
There are more empty interstitial positions than vacancies
interstitial atomic movement have greater probability
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Diffusion FASTER for .... open crystal structures lower density materials lower melting point materials secondary bonded materials smaller diffusing atoms cations
Diffusion SLOWER for .... close-packed structures higher density materials higher melting T materials covalent bonded materials larger diffusing atoms anions
In Summary ....
Diffusion and Materials Processing
Diffusional processes become very important when materials are used or processed at elevated temperatures.
Grain Growth
Materials composed of many grains contains a large number of grain boundaries, which represent a high-energy area because of inefficient packing of atoms.
Grain growth occurs by diffusion of atoms during high-temperature processing and small grains accumulates to form larger fewer grains in order to reduce grain boundary areas.
Since larger grains yield inferior mechanical properties, heat treatment and many other high-temperature processes are carefully controlled to avoid excessive grain growth.
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Figure 5.31 Grain growth occurs as atoms diffuse across the grain boundary from one grain to another
Figure 5.32 Grain growth in alumina ceramics can be seen from the SEM micrographs of alumina ceramics. (a) The left micrograph shows the microstructure of an alumina ceramic sintered at 1350oC for 150 hours. (b) The right micrograph shows a sample sintered at 1350oC for 30 hours. (Courtesy of I. Nettleship and R. McAfee.)
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Diffusion Bonding
It is a method of joining materials in the solid state at elevated temperature.
Two smooth and clean surface are forced together at a high pressure, producing a high atom-to-atom contact. Addition of high temperature causes diffusion of atoms and results atomic bonding. Finally, elimination of voids occurs by a slow volume diffusion process.
Diffusion bonding often used for joining exotic metals (e.g., titanium), dissimilar metals and materials, and ceramics.
Figure 5.33 The steps in diffusion bonding: (a) Initially the contact area is small; (b) application of pressure deforms the surface, increasing the bonded area; (c) grain boundary diffusion permits voids to shrink; and (d) final elimination of the voids requires volume diffusion
Sintering
In the powder process, powders are consolidated using high pressure to form green compact and then heated at high temperature to form the sintered compact. During heating, particles join together and volume of pore between the particles is reduced. Often pressure and temperature are added together (hot isostatic pressing, or HIPing) to form the object.
Most of the high temperature metals (e.g., tungsten carbide cutting tools) and ceramic materials are processed using this technique.
Figure 5.28 Diffusion processes during sintering and powder metallurgy. Atoms diffuse to points of contact, creating bridges and reducing the pore size
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Figure 5.30 The microstructure of BMT ceramics obtained by compaction and sintering of BMT powders. (Courtesy of H. Shirey.)
Figure 5.29 Particles of barium magnesium tantalate (BMT) (Ba(Mg1/3 Ta2/3)O3) powder are shown. This ceramic material is useful in making electronic components known as dielectric resonators that are used for wireless communications. (Courtesy of H. Shirey.)
Diffusion in Ionic Compounds and Polymers
In metals and alloys, atoms can move into any nearby vacancy or interstitial sites. But in other materials, atom movement may be somewhat more restricted.
Ionic Compound
A diffusing ion only enters a site having the same charge.
In order to reach the site, the ion must physically squeeze past adjoining ions, pass by a region of opposite charge, and move a relatively long distance.
Consequently the activation energies are higher and rates of diffusion are lower than for metals
Diffusion of cations are higher than anions (for being smaller in size)
Diffusion of ions also transfer electrical charge. Thus, as temperature increases, diffusion increases and electrical conductivity also increases.
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Polymers
Diffusion of atoms or small molecules occur from one location to another along a long polymer chain. Strong covalent bonds must need to be broken for this to occur.
Diffusion through crystalline polymer is slower than through amorphous polymers, which have no long-range order and consequently have a lower density.
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Lecture 8
Chemical Kinetics