1. Anandh Subramaniam & Kantesh Balani
DIFFUSION IN SOLIDS
FICK’S LAWS
KIRKENDALL EFFECT
ATOMIC MECHANISMS
MATERIALS SCIENCE
&
ENGINEERING
Anandh Subramaniam & Kantesh Balani
Materials Science and Engineering (MSE)
Indian Institute of Technology, Kanpur
URL: home.iitk.ac.in/~anandh
AN INTRODUCTORY E-BOOK
Part of
A Learner’s Guide
Diffusion in Solids
P.G. Shewmon
McGraw-Hill, New York (1963)

2. Roles of Diffusion Oxidation Creep Sintering Aging Doping Carburizing
To comprehend many materials related phenomenon one must understand Diffusion.
The focus of the current chapter is solid state diffusion in crystalline materials.
Roles of Diffusion
Oxidation
Creep
Metals
Some mechanisms
Sintering
Aging
Precipitates
Doping
Carburizing
Semiconductors
Steels
Many more…
Material Joining
Diffusion bonding

3. Movable piston with an orifice Piston motion
H2 diffusion direction Ar H2
Movable piston with an orifice
Piston motion
Ar diffusion direction
Piston moves in the direction of the slower moving species

4. Kirkendall effect A B Marker motion
Materials A and B welded together with Inert marker and given a diffusion anneal
Usually the lower melting component diffuses faster (say B) A B
Marker motion
Inert Marker – thin rod of a high melting material which is basically insoluble in A & B

5. Concentration / Chemical potential
Diffusion
Mass flow process by which species change their position relative to their neighbours.
Driven by thermal energy and a ‘gradient’
Thermal energy → thermal vibrations → Atomic jumps
Concentration / Chemical potential
Gradient
Electric
Magnetic
Stress

6. Flux (J) (restricted definition) → Flow / area / time [Atoms / m2 / s]
Flow direction

7. A Fick’s I law Assume that only B is moving into A
Assume steady state conditions → J  f(x,t) (No accumulation of matter)
(Truly speaking it is the chemical potential gradient!)
Fick’s first law
Continuity equation
Diffusivity (D) → f(Concentration of the components, T)
Flow direction A

8. A Diffusion coefficient/ Diffusivity
No. of atoms crossing area A per unit time
Cross-sectional area
Concentration gradient
ve implies matter transport is down the concentration gradient
Flow direction A
As a first approximation assume D  f(t)

9. Non-steady state J = f(x,t)
Steady and non-steady state diffusion
D  f(c)
Steady state J  f(x,t)
Under steady state conditions
D = f(c)
Diffusion
D  f(c)
Non-steady state J = f(x,t)
Substituting for flux from Fick’s first law
D = f(c)
If D is constant
 Slope of c-x plot is constant under steady state conditions
If D is NOT constant
If D increases with concentration then slope (of c-x plot) decreases with ‘c’
If D decreases with ‘c’ then slope increases with ‘c’

10. Fick’s II law x Jx
Jx+x
Fick’s first law
D  f(x)

11. RHS is the curvature of the c vs x curve
LHS is the change is concentration with time
+ve curvature  c ↑ as t ↑
ve curvature  c ↓ as t ↑

12. Solution to 2o de with 2 constants determined from Boundary Conditions and Initial Condition
Erf () = 1
Erf ( ) = 1
Erf (0) = 0
Erf ( x) =  Erf (x)
Also
For upto x~0.6  Erf(x) ~ x
x 2, Erf(x)  1
Area
Exp( u2) → 
u →

13. Example where the erf solution can be used
A & B welded together and heated to high temperature (kept constant → T0)
t2 > t1 | c(x,t1)
t1 > 0 | c(x,t1)
t = 0 | c(x,0)
f(x)|t C2
Non-steady state
Flux
f(t)|x
Concentration →
Cavg
If D = f(c)  c(+x,t)  c(x,t) i.e. asymmetry about y-axis
↑ t A B C1
x →
C(+x, 0) = C1
C(x, 0) = C2
AB = C1
A+B = C2
A = (C1 + C2)/2
B = (C2 – C1)/2

14. Temperature dependence of diffusivity
Diffusivity depends exponentially on temperature.
This dependence has important consequences with regard to material behaviour at elevated temperatures. Processes like precipitate coarsening, oxidation, creep etc. occur at very high rates at elevated temperatures.
Arrhenius type

15. ATOMIC MODELS OF DIFFUSION
The diffusion of two important types of species needs to be distinguished: (i) species ‘sitting’ in a lattice site (ii) species in a interstitial void
1. Interstitial Mechanism
Usually the solubility of interstitial atoms (e.g. carbon in steel) is small. This implies that most of the interstitial sites are vacant. Hence, if an interstitial species wants to jump, ‘most likely’ the neighbouring site will be vacant and jump of the atomic species can take place.
Light interstitial atoms like hydrogen can diffuse very fast. For a correct description of diffusion of hydrogen anharmonic and quantum (under barrier) effects may be very important (especially at low temperatures).

16. 2. Vacancy Mechanism
For an atom in a lattice site (and often we are interested in substitutional atoms) jump to a neighbouring lattice site can take place if it is vacant. Hence, vacancy concentration plays an important role in the diffusion of species at lattice sites via the vacancy mechanism.
Vacancy clusters and defect complexes can alter this simple picture of diffusion involving vacancies

17. Hm Interstitial Diffusion1 2
Hm 1 2
At T > 0 K vibration of the atoms provides the energy to overcome the energy barrier Hm (enthalpy of motion)
 → frequency of vibrations, ’ → number of successful jumps / time

18. Substitutional Diffusion
Probability for a jump  (probability that the site is vacant).(probability that the atom has sufficient energy)
Hm → enthalpy of motion of atom
’ → frequency of successful jumps
Where,  is the jump distance

19. Interstitial Diffusion
of the form
D (C in FCC Fe at 1000ºC) = 3  1011 m2/s
Substitutional Diffusion
of the form
D (Ni in FCC Fe at 1000ºC) = 2  1016 m2/s

20. Qsurface < Qgrain boundary < Qpipe < Qlattice
Diffusion Paths with Lesser Resistance
Experimentally determined activation energies for diffusion
Qsurface < Qgrain boundary < Qpipe < Qlattice
Lower activation energy automatically implies higher diffusivity
Core of dislocation lines offer paths of lower resistance → PIPE DIFFUSION
Diffusivity for a given path along with the available cross-section for the path will determine the diffusion rate for that path

21. ← Increasing Temperature
Comparison of Diffusivity for self-diffusion of Ag → single crystal vs polycrystal
Schematic
Qgrain boundary = 110 kJ /mole
QLattice = 192 kJ /mole
Polycrystal
Log (D) →
Single crystal
1/T →
← Increasing Temperature

22. Carburization of steel
Applications based on Fick’s II law
Carburization of steel
Surface is often the most important part of the component, which is prone to degradation.
Surface hardening of steel components like gears is done by carburizing or nitriding.
Pack carburizing → solid carbon powder used as C source.
Gas carburizing → Methane gas CH4 (g) → 2H2 (g) + C (diffuses into steel).
C(+x, 0) = C1
C(0, t) = CS
A = CS
B = CS – C1

23. A 0.2% carbon steel needs to be surface carburized such that the concentration of carbon at 0.2mm depth is 1%. The carburizing medium imposes a surface concentration of carbon of 1.4% and the process is carried out at 900C (where, Fe is in FCC form).
Solved
Example
Data:
The solution to the Fick’ second law:
(1)
(2)

24. From equation (2)

25. Approximate formula for depth of penetration
Let the distance at which
[(C(x,t)C0)/(CSC0)] = ½ be called x1/2
(which is an ‘effective penetration depth’) 
The depth at which C(x) is nearly C0 is (i.e. the distance beyond which is ‘un’-penetrated):
Erf(u) ~ 1 when u ~ 2

26. End

27. Another solution to the Fick’s II law
A thin film of material (fixed quantity of mass M) is deposited on the surface of another material (e.g. dopant on the surface of a semi-conductor). The system is heated to allow diffusion of the film material into the substrate.
For these boundary conditions we get a exponential solution.
Boundary and Initial conditions
C(+x, 0) = 0

28. Diffusion in ionic materials
Ionic materials are not close packed
Ionic crystals may contain connected void pathways for rapid diffusion
These pathways could include ions in a sublattice (which could get disordered) and hence the transport is very selective   alumina compounds show cationic conduction  Fluorite like oxides are anionic conductors
Due to high diffusivity of ions in these materials they are called superionic conductors. They are characterized by:  High value of D along with small temperature dependence of D  Small values of D0
Order disorder transition in conducting sublattice has been cited as one of the mechanisms for this behaviour

29. Calculated and experimental activation energies for vacancy Diffusion
Element
Hf
Hm
Hf + Hm Q Au 97 80
177
174 Ag 95 79
184

30. concentration gradient dc/dx = (1 /  3)/ =  1 /  4
c = atoms / volume
c = 1 /  3
concentration gradient dc/dx = (1 /  3)/ =  1 /  4
Flux = No of atoms / area / time = ’ / area = ’ /  2 1 2
Vacant site 
On comparison with

31. 3. Interstitialcy Mechanism
Exchange of interstitial atom with a regular host atom (ejected from its regular site and occupies an interstitial site)
Requires comparatively low activation energies and can provide a pathway for fast diffusion
Interstitial halogen centres in alkali halides and silver interstitials in silver halides
Steady state diffusion C1
D  f(c)
Concentration → C2
D = f(c)
x →

32. 4. Direct Interchange and Ring