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Chapter 5
Chapter 5: Diffusion
Diffusion : the movement of particles in a solid from an area of high concentration to an area of low concentration, resulting in the uniform distribution of the substance
Diffusion is process which is NOT due to the action of a force , but a result of the random movements of atoms ( statistical problem )
- Diffusivity and 2 Fick’s laws
- Atomistic mechanisms of diffusion
- Temperature dependence and Arrenius plot
- Industrial applications
- carburized steel,
- dopants in silicon
Chapter 5
5.1 Steady-State Diffusion
Recall: Solvent – the majority atom type (or host atoms): Solute – the element with lower concentration Substitutional – a solid solution in which the solute atoms are replaced by solute Interstitial – solute atoms are located in gaps between host atoms
Consider diffusion of solute atoms (b) in solid state solution (AB) in direction x between two parallel atomic planes (separated by ∆x)
- if there is no changes with time in CB at these planes – such diffusion condition is called steady-state diffusion
For a steady-state diffusion,
flux (flow) , J, of atoms is:
dx
dC
J =− D
J – flux of atoms, atoms/(m^2 s): the number of particles which pass through a unit area in a unit of time;
D – diffusivity or diffusion coefficient, m 2 /s
dC/dx – concentration gradient, atoms/m 4
Chapter 5
Fick’s first law of diffusion
For steady-state diffusion condition ( no change in the system with time ), the net flow of atoms is equal to the
dx diffusivity^ D^ times the diffusion gradient^ dC/dx
dC
J =− D
⎟⎟ ×
m m
atoms
dx
dC
s
m
D
ms
atoms
J
3
2 2
Diffusivity D depends on :
- Diffusion mechanism
- Temperature of diffusion
- Type of crystal structure (bcc > fcc)
- Crystal imperfections
- Concentration of diffusing species
‘-’ sign: flux direction is from the higher to the lower concentration; i.e. it is the opposite to the concentration gradient
Chapter 5
Non-Steady-State Diffusion
In practice the concentration of solute atoms at any point in the material changes with time – non-steady-state diffusion
For non-steady-state condition, diffusion
coefficient, D - NOT dependent on time:
dx
dC
D
dx
d
dt
Second Fick’s law of diffusion: dC^ x x
The rate of compositional change is equal to the diffusivity times the rate of the change of (^2) the concentration gradient
2
x
C
D
dt
dC x
If D ≠ D(x) , in 1D case: =
2 2
2 2
2
z
C
y
C
x
C
D
dt
In 3D case: dC^ x
Change in concentration in 2 semi-infinite rods of Cu and Ni caused by diffusion, From G. Gottstein “Physical Foundations of Material Science”
Chapter 5
Error function
Curve of the error function erf (z) for
Dt
x
z
Chapter 5
Q: Consider the gas carburizing of a gear of 1018 steel (C 0.18 wt %) at 927°C. Calculate the time necessary to increase the C content to 0.35 wt % at 0.40 mm below the surface of the gear. Assume the C content at the surface to be 1.15 wt % and that the nominal C content of the steel gear before carburizing is 0.18 wt %. D ( C in γ iron) at 927°C = 1.28 × 10-11^ m^2 /s
Chapter 5
5.2 Atomistics of Solid State Diffusion
- Diffusion mechanisms: 1. Vacancy (substitutional) diffusion – migration of atom in a lattice assisted by the presence of vacancies
Ex.: self diffusion of Cu atoms in Cu crystal
2. Interstitial diffusion – movement of atoms from one interstitial site to another neighboring interstitial site without permanent displacement any of the atoms in the matrix crystal lattice
Ex.: C diffusion in BCC iron
Chapter 5
Vacancies
The simplest point defect is the vacancy (V) – an atom site from which an atom is missing
Vacancies are always present; their number NV depends on temperature ( T )
vacancy
kT
E
V
V
N N e
−
= ×
NV - # of vacancies
N - number of lattice sites
EV – energy required to form a vacancy
k – Boltzmann constant k = 1.38 ×10-23^ J K-1^ ; or 8.62 ×10-5^ eV K-
T – absolute temperature
Chapter 5
Possible mechanisms of self-diffusion and their
activation energy
- Neighboring atoms exchange sites
- Ring mechanism
- Vacancy mechanism
- Direct interstitial mechanism
- Indirect interstitial mechanism
3 1 eV 1 eV 2 eV
6 0.2 eV 3.4 eV 3.6 eV
4 0.6 eV 3.4 eV 4 eV
1 8 eV - 8 eV
Migration Formation Total
Chapter 5
Anisotropy Effects
From G.Gottstein “Physical Foundations of Material Science”
Chapter 5
Carbon diffusion in Fe
Jump frequency Γ [s -1] of an atom is given by:
kT
G (^) m
e
−
Γ=ν×
Usually ν ≅ 10 13 [s -1] vibrational frequency of the atom
There is a fundamental relationship between the jump
frequency Γ and the diffusion coefficient D which is
independent of mechanism and crystal structure:
2 2
D = ×Γ =
λ – the jump distance of the diffusing atom τ =1/Γ – the time interval between two jumps
J = J MN - J NM
J = CMA ×Γ× × − CNA ×Γ× ×
a^2
D
Chapter 5
Carbon diffusion in Fe
Can be derived from an atomistic considerations of the diffusion processes
J = J MN - J NM
a^2
D
2 2
D = ×Γ =
C atoms are located on the octahedral interstitial sites (black circles)
Only ¼ of possible jumps of C atoms lead to flux in +x Only 2/3 of all C atoms can jump in x direction
Chapter 5
Bolzmann’s equation
kT
E E
e
−^ −
Probability
- On the basis of statistical analysis , Bolzmann’s results showed that the probability of finding a molecule or atom at an energy level E*^ > the average energy E of all the molecules or atoms in a system at a particular temperature T, K, was:
where k (^) B = 1.38×10-23^ J/(atom K) - Boltzmann’s constant
- The fraction of atoms or molecules in a system having energies > E* at a given T (where E* is much greater than the average energy of any atom or molecule:
kT
E
total
C e N
n −* = ×
Chapter 5
Arrhenius Rate Equation
The rate of many chemical reaction as a function of temperature as follows :
RT
E (^) A
Rate of reaction C e
−
_ _ = ×
C – rate constant, independent of T
EA – activation energy
R – molar gas constant
R = 8.314 J mol -1^ K -1; or 1.987cal mol -1K -
T – absolute temperature
Chapter 5
Typical Arrhenius Plot
If we rewrite in natural log plot:
RT
E
ln rate = ln C − A
ln rate = a
ln C = b
T
x
y = b + m × x
or in logarithmic log plot:
RT
E
rate C A
log 10 =log 10 −
Chapter 5
Q.: The diffusivity of Ag atoms in solid silver metal is 1.0X10^17 m^2 /s at 500o^ C and 7.0x10-13^ m^2 /s at 1000o^ C. Calculate the activation energy (J/mole) for the diffusion of Ag in Ag in the T range 500 to 1000 oC.
Chapter 5
Q: If boron, B, is diffused into a thick slice of Si with no previous B in it at a temperature of 1100°C for 5 h, what is the depth below the surface at which the concentration is 10^17 atoms/cm^3 if the surface concentration is 10^18 atoms/cm3? D = 4 × 10-13^ cm^2 /s for B diffusing in Si at 1100°C.
Chapter 5
Summary
- Diffusion : the movement of particles in a solid from an area of high concentration to an area of low concentration, resulting in the uniform distribution of the substance
- Diffusion rate in a system will increase with temperature :
RT
E
o
A
D D e
−
= ×
- Fick’s first diffusion law : for steady-state diffusion condition (no change in the system with time), the net flow of atoms is equal to the diffusivity D times the diffusion gradient dC/dx
- Fick’s second diffusion law : The rate of compositional change is equal to the diffusivity times the rate of the change of the concentration gradient
dx
dC
J =− D
2
2
x
C
D
dt
dC x
