CHAPTER 6 – DIFFUSION
6.1 INTRODUCTION
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Many reactions and processes important to the processing of metals rely on the transfer
of mass —> either within a specific solid or from a liquid, a gas, or another solid phase
o This is accomplished by diffusion
Diffusion = material transport by atomic motion
Diffusion couple = joining bars of two different metals together, so there is intimate
contact between the two faces
o Used to demonstrate diffusion
o The couple is heated for an extended period of time (high temp but below the
melting points of the two metals) then cooled to room temp
o Results show the two metals will be separated by an alloyed diffusion zone
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Pure respective metals at the extremities and a region with varied
concentrations of both metals —> metal atoms have diffused
 Can observe a change in concentration over time —> net drift of atoms
from high- to low-concentration regions
o INTERDIFFUSION (aka impurity diffusion) = the process by which one metal
diffuses into another
Diffusion also occurs for pure metals, where all atoms exchanging positions are of the
same type —> this is called SELF DIFFUSION
o No compositional changes with self diffusion, obviously
6.2 DIFFUSION MECHANISMS
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Diffusion from an atomic perspective = stepwise migration of atoms from lattice site to
lattice site
Atoms in solids are in constant motion, rapidly changing positions
For this movement to occur, two conditions must be met:
o (1) there must be an empty adjacent site
o (2) the atom must have sufficient energy to break bonds with its neighbour
atoms and then cause some lattice distortion during the displacement
Energy of motion is vibrational
At a specific temperature, some small fraction of the total number of atoms is capable of
diffusive motion by virtue of the magnitudes of their vibrational energies
o This fraction increases with rising temperature**
Two predominant models for metallic diffusion:
o VACANCY DIFFUSION
 Interchange of an atom from a normal lattice position to an adjacent
vacant lattice site, or vacancy
 Process necessitates the presence of vacancies**
 The extent to which vacancy diffusion can occur is a fn of the number of
these defects present
 There are significantly more vacancies in metals at elevated
temps**
 Diffusing atoms and vacancies exchange positions —> diffusion of atoms
in one directions corresponds to the motion of vacancies in the opposite
direction
 Both self diffusion and interdiffusion occur by this mechanism**
o INTERSTITIAL DIFFUSION
 Atoms migrate from an interstitial position to a neighbouring one that is
empty
 This mechanism is found for interdiffusion of impurities such as H, C, N, O
—> atoms are small enough to fit into the interstitial positions
 Host or substitutional impurity atoms rarely form interstitials and no not
normally diffuse via this mechanism
In most metal alloys, interstitial diffusion occurs much more rapidly than diffusion by
vacancy mode**
o Interstitial atoms are smaller, thus more mobile
o Also, there are more empty interstitial positions than vacancies
The probability of interstitial atomic movement is greater than for vacancy diffusion
6.3 FICK’S FIRST LAW
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Diffusion is a time-dependent process —> the quantity of an element that is transported
within another is a function of time
We often care about rate of mass transfer
o Frequently expressed as DIFFUSION FLUX (J)
 Defined as the mass (or, number of atoms) diffusing through and
perpendicular to a unit cross-sectional area of solid per unit of time
 M = mass
 A = the area across which diffusion is occurring
 t = elapsed diffusion time
o Units of J [=] kg/m2•s or atoms/m2•s
Steady state diffusion in a single (x) direction: flu is proportional to the concentration
gradient (dC/dx)**
o This equation is called Fick’s first law
o D = constant of proportionality, called the diffusion constant
 Units are m2/s
o Negative sign in this eqn denotes the direction of diffusion is down the
concentration gradient (from high to low concen.)
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Fick’s first law may be applied to the diffusion of atoms of a gas through a thin metal
plate for which concentrations (or pressures) of the diffusing species on both surfaces of
the plate are held constant
o Diffusion process will eventually reach a steady-state
 The mass of diffusing species entering the plate on the high-pressure side
is equal to the mass exiting from the low-pressure surface
 At this point, there is no net accumulation of diffusing species in the plate
 This is an example of STEADY-STATE DIFFUSION
o Figure 6.4(b) depicts the concentration profile graph, where C is plotted against
position within the solid, x
 Concentration gradient = the slope at a particular point on this curve
 Concentration profile is assumed to be linear* (so we can use the
expression below to calc.)
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For diffusion problems, we often express concentration in terms of mass of diffusing
species per unit volume of solid —> kg/m3 or g/cm3
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DRIVING FORCE = what compels the rxn to occur
o For diffusion rxns, several forces are possible
o For Fick’s first law, concentration is the driving force**
6.4 FICK’S SECOND LAW: NONSTEADY-STATE DIFFUSION
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Most practical diffusion situations are nonsteady-state —> the diffusion flux and the
concentration gradient at some particular point in a solid vary with time
o There is a net accumulation or depletion of the diffusing species
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Under conditions of nonsteady state, we use partial differential equation known as Fick’s
second law
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o Used for nonsteady state diffusion in one direction
If the diffusion coefficient is independent of composition (verify this for every diffusion
situation), the above eqn simplifies to:
Solutions to this expression  concentration in terms of both position and time
o Sol’ns are possible when physically meaningful boundary conditions are specified
One important solution: semi-infinite solid where surface concentration is held
constant
o A bar of solid is considered semi-infinite if none of the diffusing atoms reach the
bar end during the time of diffusion**
 A bar of length l is considered to be semi-infinite when l > 10√Dt
o Often, the source of the diffusing species is a gas phase
 Partial pressure is maintained at a constant value
o Following assumptions are made:
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Before diffusion, any of the diffusing solute atoms in the solid are
uniformly distributed with concentration C0
 The value of x at the surface is zero and increases with distance into the
solid
 The time is taken to be zero the instant before the diffusion process
begins
o Thus, initial condition:
For t = 0, C = C0 at 0 ≤ x ≤ ∞
o Boundary conditions:
For t > 0, C = Cs (the constant surface concentration) at x = 0
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For t > 0, C = C0 at x = ∞
Application of these boundary conditions to Fick’s second law eqn yields the solution:
o Cx = the concentration at depth x after time t
o The expression erf( /2√Dt) is the Gaussian error fn  you get values from a
mathematical table for various x/2√Dt values
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Cx is a function of the dimensionless parameter x/2√Dt, thus may be determined at any
time and position if the parameters C0, Cs, and D are known
16.5 FACTORS THAT INFLUENCE DIFFUSION
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DIFFUSING SPECIES
o The magnitude of the diffusion coefficient D is indicative of the rate at which
atoms diffuse**
o The diffusing species and the host material influence the diffusion coefficient
TEMPERATURE
o Temperature has a profound influence on the coefficients and diffusion rates
o The temperature dependence of the diffusion coefficient is:
 D0 = a temperature-dependent preexponential [m2/s]
 Qd = the activation energy for diffusion [J/mol or eV/atom]
 R = gas constant = 8.314 J/mol*K (= 8.62 x 10-5 eV/atom*K)
 T = absolute temperatue [K]
o Activation energy – the energy required to produce the diffusive motion of one
mole of atoms
 Large activation energy results in a relatively small diffusion coefficient**
o D0 & Qd can be found in a table for various host materials and diffusing species
 These two parameters are determined experimentally
o It can be noted that the log of the diffusion coefficient has a linear relationship
with the reciprocal of temperature
6.6 DIFFUSION IN SEMICONDUCTING MATERIALS
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Fabrication of semiconductor integrated circuits (ICs) applies stead-state diffusion
Single-crystal silicon is the base material for most ICs
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In order for these IC devices to function properly, very precise concentrations of an
impurity (or, impurities) must be incorporated into minute spatial regions on the silicon
chip —> this is accomplished through atomic diffusion
Typically, two heat treatments are used in this process
o (1) Predeposition step
 Impurity atoms are diffused into the silicon
 Often from a gas phase – partial pressure is maintained constant
 The surface composition of the impurity this remains constant over
time**
 The impurity concentration within the silicon is a function of position and
time according to Fick’s Second Law:
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Predeposition treatments are normally carried out within the temp range
of 900ºC to 1000ºC for typically less than 1 h
o (2) Drive-in diffusion
 Transports impurity atoms farther into the silicon in order to provide a
more suitable concentration distribution without increasing the overall
impurity content
 This treatment is carried out at a higher temp – up to ~1200ºC
 Carried out in an oxidizing atmosphere to form an oxide layer on the
surface
 Diffusion rates through this SiO2 layer are relatively slow, so very few
impurity atoms diffuse out and escape from the silicon**
6.7 OTHER DIFFUSION PATHS
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Atomic migration may also occur along dislocations, grain boundaries, and external
surfaces
o These are sometimes called short-circuit diffusion paths
o Rates are much faster than for bulk diffusion
In most situations, short-circuit contributions to the overall diffusion flux are insignificant
o B/c cross-sectional area of these paths is extremely small**
6.8 DIFFUSION IN IONIC AND POLYMERIC MATERIALS
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IONIC MATERIALS
o For ionic materials, diffusion is more complicated than for metals
o Need to consider diffusive motion of two types of ions that have opposite
charges
o Diffusion usually by vacancy diffusion**
 Ion vacancies occur in pairs
 Vacancies form in nonstoichiometric compounds
 Vacancies are created by substitutional impurity ions having different
charge states from the host ions
o Vacancy behaviour arises from the need to maintain charge neutrality in an ionic
material**
o Associated with the diffusive motion of a single ion is the transference of
electrical charge
 In order to maintain localized charge neutrality in the vicinity of this
moving ion, another species having an equal and opposite charge must
accompany the ion’s diffusive motion**
 Possible charged species include: another vacancy, an impurity atom, or
an electronic carrier (a free electron, or hole)
o Rate of diffusion of these electrically charged couples is limited by the diffusion
rate of the slowest-moving species**
o When an external electric field is applied across an ionic solid, the electrically
charged ions diffuse in response
 This ionic motion gives rise to an electric current
o The mobility of ions is a function of the diffusion coefficient
POLYMERIC MATERIALS
o In polymeric materials, we often care about the diffusion of small foreign
molecules between the molecular chains
 Ex. O2, H2O, CO2, CH4
o A polymer’s permeability and absorption characteristics relate to the degree to
which foreign substances diffuse into the material
o Penetration of these foreign substances can lead to swelling and/or chemical rxns
with the polymer molecules and often a degradation of the material’s
mechanical and physical properties
o Rates of diffusion are greater through amorphous regions than through
crystalline regions**
 Amorphous regions have more “open” structure
o Diffusion mechanism is analogous to interstitial diffusion in metals**
o Foreign molecule size affect the diffusion rate
 Smaller molecules diffuse faster than larger ones
o Diffusion is more rapid for foreign molecules that are chemically inert than for
those that react with the polymer
o The diffusion properties of polymers are often characterized in terms of a
permeability coefficient (PM)
o Steady-state diffusion through a polymer membrane modelled by Fick’s first law:
 J = diffusion flux of gas through the membrane [cm3STP/cm2*s]
 PM = permeability coeff.
 ∆x = membrane thickness
 ∆P = difference in pressure of gas across the membrane
o For some applications, low permeability rates through polymeric materials are
desirable
 Ex. food and beverage packaging, automotive tires
o Polymer membranes are often used as filters to selectively separate one chemical
species from another
 Ex. desalination of water
 In these cases, permeation rate of the substance to be filtered is
significantly greater than for the other substance(s)