Materials Science and Engineering I
Ch. 5 Thermally activated
processes and diffusion in solids
Smith and Hashemi, " Foundations of Materials Science and Engineering " ,
6th Ed., McGraw-Hill 2019
Outline

Rate process and Arrhenius equation

Atomic diffusion in solids

Kirkendall effect

Industrial applications of diffusion processes

Temperature-dependent diffusivity
2
Rate process in solids



Reactions in solids proceeds to achieve a lower energy state.
Reacting atoms must overcome an energy barrier.
Energy of reacting atoms follows a temperature-dependent distribution.
E*
E*
Activation Energy
Er
Reactants
Er = Energy of reactants
E* = Activation energy level
ΔE* = Activation energy
Ep = Energy of products
Energy released
due to reaction
亞穩態
EP
Products
Reaction coordinate
3
Rate process in solids

Boltzmann statistics: Probability of finding an atom/molecule with
energy E in a system of fixed atoms/molecules is given by
∞
1 =
0
𝛼𝑒
𝐸
−( 𝑘𝑇 )
𝐸
𝐸 ∞
−𝑘𝑇
−𝑘𝑇
𝛼𝑒
𝑑𝐸 = 𝛼 (−𝑘𝑇𝑒
) 0 = 𝛼 (𝑘𝑇)
1
𝛼 =
𝑘𝑇
1 −𝐸 / 𝑘𝑇
𝑒
𝑘𝑇
k = 1.38 x 10-23 J/(atomK)
4
Rate process in solids

The fraction of atoms having energies greater than E* in a
system is given by
𝑛(𝐸 > 𝑬∗ )
𝑁𝑡𝑜𝑡𝑎𝑙
−𝑬∗
= 𝐶𝑒 𝑘𝑇
∞
𝐸
−𝑬∗
1
𝑃( > 𝑬∗ ) =
𝑒 −𝑘𝑇 𝑑𝐸 = 𝑒 𝑘𝑇
𝐸∗ 𝑘𝑇
n = number of molecules with energy greater than E*
Ntotal = total number of molecules
k = Boltzmann’s constant
C = Constant
5
Vacancy formation in solids

The fraction of vacancies in a solid at thermal equilibrium
under a temperature T is given by
−𝐸𝑉
𝑛𝑣
= 𝐶𝑒 𝑘𝑇
𝑁
Td
→
nv
d
nv = vacancy concentration
Ev = vacancy formation energy
N = Atomic concentration
→
10
~
23
6
qracaesiloltol
世 a
daiatoa
一
leflive constamt
3
( 些 ) α hv
−𝐸𝑉
𝑛𝑣
= 𝐶𝑒 𝑘𝑇
𝑁
7
Arrhenius equation

The rate of chemical reaction is given by Arrhenius equation.
Rate of reaction = Ce-Q/RT
C’e-E*/kT
AfB → C
R KGJ [B]
=
↓
R
KNAv
Q = Activation energy J/mol
R = Molar gas constant J/molK
T = Temperature in Kelvin
C = Rate constant (Independent of temperature)

=
Ce
ce 舞
Rate of reaction depends upon number of reacting molecules.
8
Arrhenius equation
Arrhenius equation can also be written as
ln (rate)
= ln ( C)
– Q/RT
or
Log10 (rate) = Log10 (C) – Q/2.303 RT

Arrhenius plot
Which is similar to
Y
=
b
+ m X
The equation of a straight line with Y
intercept as ‘b’ and slope ‘m’.
Y
X
b
m
Log10(rate)
(1/T)
Log10(C)
Q/2.303R
9
(After J. Wulff et al., “Structure and Properties of Materials,” vol. II: “Thermodynamics of Structure,” Wiley, 1964, p.64.)
Atomic diffusion in solids


Diffusion is a process by which a matter transports
through another matter.
Examples:

Movement of smoke particles in air: fast

Movement of dye in water: slow

Solid state reactions: very slow
10
Substitutional diffusion

If atom ‘A’ has sufficient energy, it moves into
the vacancy  self diffusion
Activation
Energy of
Self diffusion
=
Activation
Energy to
form a
Vacancy
+
Activation
Energy to
move a
vacancy
𝐸 ∗ = 𝐸𝑉 + 𝐸𝑚

.
Activation energy increases with rising
melting point.
𝐸∗
D = Do exp −
kT
11
Activation energy vs melting point
𝑒𝑉
𝑘𝑐𝑎𝑙
1
= 23.6
= 98.74 𝑘𝐽 / 𝑚𝑜𝑙
𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒
𝑚𝑜𝑙
12
Kirkendall effect
The mass transport of two components with dissimilar diffusion rates in an
alloying or reaction system
W. Wang et al, Chem. Mater. 2013, 25, 1179−1189
13
Kirkendall effect
Unfilled voids are left behind and coalesce into large pores (Kirkendall void)
Zn-Cu alloy (brass) grows in the direction of the faster-moving species (Zn)
JSn
JCu
Jv
JZn
C. N. Liao
JCu
14
Kirkendall effect
15
Interstitial diffusion mechanism



Atoms move from one interstitial site
to another.
The moving atoms are much smaller
than the matrix atoms.
Example: Carbon interstitially
diffuses into BCC  or FCC  iron.
𝐸∗
D = Do exp −
kT
Interstitial atoms
𝐸 ∗ = 𝐸𝑚
Matrix
atoms
16
Steady state diffusion


No change in concentration of solute atoms at different planes in
a system over a long period of time.
No chemical reaction occurs. Only net flow of atoms.
C1
Concentration
of diffusing C2
atoms
Solute atom flow
Distance x
Diffusing
atoms
Unit
Area
Net flow of atoms
per unit area per
unit time = J
17
Fick’s First Law

The flux or flow of atoms is given by
𝑑𝑐
𝐽 = −𝐷
𝑑𝑥


J = Flux or net flow of atoms
D = Diffusion coefficient
dc
= Concentration gradient.
dx
A steady state diffusion condition J = constant
Example: Diffusivity of FCC iron at 500 oC is 5 x 10-15 m2/s and at
1000 oC is 3 x 10-11 m2/s
18
Diffusivity & concentration gradient
Smoothness  diffusivity
Slope  concentration gradient
𝒅𝒄
𝑱 = −𝑫
𝒅𝒙
19
Diffusivity

Diffusivity depends upon

Diffusion type: interstitial or substitutional

Temperature: diffusivity increase with temperature



Crystal structure: BCC has lower APF and higher
diffusivity than FCC
Crystal imperfection: Open structures (grain boundaries)
 high diffusivity
Concentration: Higher concentrations of diffusing solute
atoms will affect diffusivity.
20
Diffusivity data for some metals
21
Temperature-dependent diffusivity
K. N. Tu, JAP, 94, 5451 (2003)
22
Non-steady state diffusion
Concentration of diffusing atoms at any point in metal
changes with time.
 Fick’s second law: Rate of compositional change is equal
to diffusivity times the rate of change of concentration
Plane 2
gradient.
Plane 1

𝑑𝐶𝑥
𝑑
𝑑𝐶𝑥
𝑑𝐶𝑥2
=
𝐷
=𝐷 2
𝑑𝑡
𝑑𝑥
𝑑𝑥
𝑑 𝑥
D is constant
23
Fick’s second law - continuity equation
C(x)
C(x+x)
J(x+x)
J(x)
x
x+x
dC ( x )
J ( x)   D
dx
dC ( x )
C ( x  x )  C ( x ) 
x
dx
dC ( x  x )
J ( x  x )   D
dx
dC ( x )
d 2C ( x )
 D
D
x
2
dx
dx
[ J ( x )  J ( x  x )] At
d 2C ( x )
C ( t  t )  C ( t ) 
D
t
2
Ax
dx
dC
d 2C ( x )
D
dt
dx 2
24
Transport of mass, heat, and electricity
Fick's first law
Mass
Heat
𝑑𝑐
𝐽M = −𝐷𝑀
𝑑𝑥
𝜕𝐶
𝑑2 𝐶
= −𝛻 𝐽⃑𝑀 = 𝐷𝑀 2
𝜕𝑡
𝑑𝑥
Fourier's law
𝜕𝑇
𝜌𝑑 𝐶𝑝
= −𝛻
𝜕𝑡
𝑑𝑇
𝐽𝑄 = −𝑘
𝑑𝑥
Ohm's law
Electricity
𝑑𝑉
𝐽𝐸 = −
𝑑𝑥
𝜕𝑇
=
𝜕𝑡
𝑘
𝑑2 𝑇
𝑑2 𝑇
= 𝐷𝑄 2
𝜌𝑑 𝐶𝑝 𝑑𝑥 2
𝑑𝑥
𝜕𝑉
𝐶𝑞
= −𝛻
𝜕𝑡
𝜕𝑉
=
𝜕𝑡
𝐽⃑𝑄
𝑑2 𝑇
= 𝑘 2
𝑑𝑥
𝐽⃑𝐸
𝑑2 𝑉
=𝜎 2
𝑑𝑥
𝜎 𝑑2𝑉
𝑑2𝑉
=
𝐷
𝐸 𝑑𝑥 2
2
𝐶𝑞 𝑑𝑥
25
Fick’s Second Law – Solution
𝐶𝑠 − 𝐶(𝑥, 𝑡)
𝑥
= 𝑒𝑟𝑓
𝐶𝑠 − 𝐶0
2 𝐷𝑡
𝐶 𝑥, 𝑡 = 𝐶𝑠 − (𝐶𝑠 − 𝐶0 )𝑒𝑟𝑓
2 𝑦 −𝑧 2
𝑒 𝑑𝑧
𝑒𝑟𝑓 𝑦 =
𝜋 0
𝑥
2 𝐷𝑡
Cs
Cs: Surface concentration of element in
gas diffusing into the surface
Cx
C0:= Initial uniform concentration of
element in solid
C(x,t): Concentration of element at
C0
distance x from surface at time t
x: distance from surface
D: diffusivity of solute
t: time
Time = t2
Time= t1
Time = t0
x
Distance x
26
Error function
27
Carburization
Carburization
C%
Low carbon
Steel part
Diffusing carbon
atoms
Carbon Gradients
In Carburized metals
(After “Metals handbook,” vol.2: “Heat Treating,” 8th ed, American Society of Metals, 1964, p.100)
28
Carburization
Consider a gear of 1018 steel (0.18wt%).
Calculate the time required to increase
the carbon content to 0.35 wt % at 0.40
mm below the surface of the gear at
927°C. Assume the carbon content at the
surface to be 1.15 wt% and Diffusivity of
C in  Fe at 927°C =1.28 10-11 m2/s.
The solution of the diffusion equation is
C, %
1.15 %
0.35 %
0.18 %
0.40 mm
29
Dependence of dopant concentration
on Si resistivity
h t t p s: / / w w w . a p l u s p h y sics.c o m / c o u rs e s / h o n o rs / m icr o e / silico n . h t m l
30
Diffusion in Si wafer
31
Effect of temperature on diffusion
If diffusivity at two temperatures are determined, two
equations can be solved for Q and D0
 Example: The diffusivity of silver atoms in silver is 1 x 10-17
at 500 0C and 7 x 10-13 at 1000 0C.
Therefore,



D1000 exp( Q / RT2 )
Q  1 1 
   

 exp
D500 exp( Q / RT1 )
 R  T2 T1  
7  1013
1 
 Q 1
 exp  


17
1  10
 R  1273 773  
Solving for activation energy Q
Q  183KJ / mol
32
Diffusivity data for some metals
33
(After L.H. Van Vlack. “Elements of Materials Science and Engineering.” 5th ed., Addison-Wesley, 1985. P.137.)