DIFFUSION & MECHANICAL PROPERTIES Notes | EduRev

: DIFFUSION & MECHANICAL PROPERTIES Notes | EduRev

 Page 1


 
 
 
 
 
 
 
 
 
 
 
 
 
 
B: DIFFUSION & MECHANICAL PROPERTIES 
 
B1: Diffusion.  
B2: Deformation in Materials. 
B3: Fracture, Fatigue and Creep. 
B4: Strengthening Mechanisms. 
 
Page 2


 
 
 
 
 
 
 
 
 
 
 
 
 
 
B: DIFFUSION & MECHANICAL PROPERTIES 
 
B1: Diffusion.  
B2: Deformation in Materials. 
B3: Fracture, Fatigue and Creep. 
B4: Strengthening Mechanisms. 
 
B1: DIFFUSION 
B1.1 Introduction 
In a sample having non-uniform composition of 
certain kind of atoms, a concentration gradient is set 
in, which leads to migration of atoms from region of 
their higher concentration to the region of lower 
concentration. This phenomenon is referred to as 
Diffusion and continues till atomic distribution 
becomes homogeneous throughout.  
The diffusion process plays an important role in the 
field of metallurgy and fabrication of extrinsic 
semiconductors. The technology of controlled 
diffusion plays a sensitive role in accomplishment of 
desired devices. The metals, in their pure form, are 
soft and do not possess large tensile strength. Their 
properties can be modified by the process of 
alloying in a controlled manner. The desired 
properties of alloys are achieved by proper choice of 
solute (atoms to be diffused) and their amount to be 
diffused. This is accomplished through sophisticated 
techniques and instruments. Further the dynamical 
processes of bio-systems utilize the phenomenon of 
diffusion in wide variety of ways which include the 
role of blood circulation leading to nourishment of 
various body organs and also waste disposal from 
them. The control of heart beat through hormonal 
diffusion and digestion of food through enzymatic 
action are some of the examples. Further the role of 
diffusion in pharmaceutical applications is 
undisputed through the drug therapy curing a 
specific portion of body.   
 
B1.2 Basics of Diffusion 
The process of diffusion involves two components: 
(i) solvent is the medium in which atoms of similar 
or different kinds are to be diffused and (ii) solute 
are the atoms of one or more kinds which are to be 
diffused into the environment of solvent. The solute 
atoms migrate and distribute themselves in solvent 
in process of which they may become interstitial 
impurity (occupying interstitial spaces in solvent 
lattice) or substitution impurity (substituting the 
solvent atom from its regular position in lattice).  
The process of diffusion can be classified into two 
types which are (a) Self-diffusion in which solute and 
solvent atoms are identical. For example diffusion of 
copper in copper (b) Inter-diffusion in which the 
solute atoms are different from the solvent atoms 
as in the case steel where solvent is iron whereas 
solute atoms are those of carbon and many other 
elements. 
The phenomenon of diffusion of solute atoms into 
the solvent environment may occur through the 
following mechanisms:  
(a) Vacancy mechanism where the solute atoms lie 
in the vacancy and moves to other available 
vacancy. Atoms can move from one site to 
Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 150 
 
Page 3


 
 
 
 
 
 
 
 
 
 
 
 
 
 
B: DIFFUSION & MECHANICAL PROPERTIES 
 
B1: Diffusion.  
B2: Deformation in Materials. 
B3: Fracture, Fatigue and Creep. 
B4: Strengthening Mechanisms. 
 
B1: DIFFUSION 
B1.1 Introduction 
In a sample having non-uniform composition of 
certain kind of atoms, a concentration gradient is set 
in, which leads to migration of atoms from region of 
their higher concentration to the region of lower 
concentration. This phenomenon is referred to as 
Diffusion and continues till atomic distribution 
becomes homogeneous throughout.  
The diffusion process plays an important role in the 
field of metallurgy and fabrication of extrinsic 
semiconductors. The technology of controlled 
diffusion plays a sensitive role in accomplishment of 
desired devices. The metals, in their pure form, are 
soft and do not possess large tensile strength. Their 
properties can be modified by the process of 
alloying in a controlled manner. The desired 
properties of alloys are achieved by proper choice of 
solute (atoms to be diffused) and their amount to be 
diffused. This is accomplished through sophisticated 
techniques and instruments. Further the dynamical 
processes of bio-systems utilize the phenomenon of 
diffusion in wide variety of ways which include the 
role of blood circulation leading to nourishment of 
various body organs and also waste disposal from 
them. The control of heart beat through hormonal 
diffusion and digestion of food through enzymatic 
action are some of the examples. Further the role of 
diffusion in pharmaceutical applications is 
undisputed through the drug therapy curing a 
specific portion of body.   
 
B1.2 Basics of Diffusion 
The process of diffusion involves two components: 
(i) solvent is the medium in which atoms of similar 
or different kinds are to be diffused and (ii) solute 
are the atoms of one or more kinds which are to be 
diffused into the environment of solvent. The solute 
atoms migrate and distribute themselves in solvent 
in process of which they may become interstitial 
impurity (occupying interstitial spaces in solvent 
lattice) or substitution impurity (substituting the 
solvent atom from its regular position in lattice).  
The process of diffusion can be classified into two 
types which are (a) Self-diffusion in which solute and 
solvent atoms are identical. For example diffusion of 
copper in copper (b) Inter-diffusion in which the 
solute atoms are different from the solvent atoms 
as in the case steel where solvent is iron whereas 
solute atoms are those of carbon and many other 
elements. 
The phenomenon of diffusion of solute atoms into 
the solvent environment may occur through the 
following mechanisms:  
(a) Vacancy mechanism where the solute atoms lie 
in the vacancy and moves to other available 
vacancy. Atoms can move from one site to 
Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 150 
 
another if there are vacancies and the atom has 
sufficient energy to overcome a local activation 
energy barrier. The activation energy for 
diffusion is the sum of energy required to form a 
vacancy and further cause its motion. 
(b) Interstitial Mechanism is applicable in case of 
small atoms like hydrogen, helium, carbon, 
nitrogen diffuse through the interstitial spaces in 
a crystal. The activation energy for diffusion is 
the energy required for these atoms to squeeze 
through the small voids between the host lattice 
atoms. 
 
B1.3 Macroscopic Model of Diffusion 
B1.3.1 Steady State Diffusion: Fick’s I Law 
The process of diffusion is time dependent which 
implies that one needs to study the quantity of 
solute material that gets transported in solvent is a 
function of time. This time dependent mass transfer 
is often represented by a quantity called diffusion 
flux (J) which is defined as the solute mass 
transferred in unit time through a unit cross-
sectional area placed perpendicular to the direction 
of flow. The diffusion flux can be mathematically 
represented as: 
dt
dM
A
J
1
= 
The units of J are kg/m
2
S or atoms/m
2
s.  
If the diffusion flux does not change with time, a 
steady state condition exists. Common example of 
such a process is diffusion of atoms of gas through a 
metal plate for which the concentration of diffusing 
species (see figure B1.1(a) ) on both sides of plates is 
kept constant. If concentration of diffusing gas 
atoms is plotted as a function of position or depth, 
then the curve is referred to as concentration 
profile, which is shown for the steady state case 
under discussion in figure B1.1(b) as: 
 
 
Figure B1.1: (a) Steady state diffusion across a thin 
plate (b) a linear concentration profile for the given 
diffusion situation. 
 
Mathematically, the concentration gradient is 
expressed as: 
dx
dC
gradient ion Concentrat = 
 
The kinetics of steady state diffusion process is 
described by Fick’s first law which states that the 
diffusion current (J) is directly proportional to the 
concentration gradient ?
?
?
?
?
?
?
?
x
c
. Hence we express this 
mathematically as: 
) 1 . 1 (B
x
c
D J
?
?
- = 
Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 151 
 
Page 4


 
 
 
 
 
 
 
 
 
 
 
 
 
 
B: DIFFUSION & MECHANICAL PROPERTIES 
 
B1: Diffusion.  
B2: Deformation in Materials. 
B3: Fracture, Fatigue and Creep. 
B4: Strengthening Mechanisms. 
 
B1: DIFFUSION 
B1.1 Introduction 
In a sample having non-uniform composition of 
certain kind of atoms, a concentration gradient is set 
in, which leads to migration of atoms from region of 
their higher concentration to the region of lower 
concentration. This phenomenon is referred to as 
Diffusion and continues till atomic distribution 
becomes homogeneous throughout.  
The diffusion process plays an important role in the 
field of metallurgy and fabrication of extrinsic 
semiconductors. The technology of controlled 
diffusion plays a sensitive role in accomplishment of 
desired devices. The metals, in their pure form, are 
soft and do not possess large tensile strength. Their 
properties can be modified by the process of 
alloying in a controlled manner. The desired 
properties of alloys are achieved by proper choice of 
solute (atoms to be diffused) and their amount to be 
diffused. This is accomplished through sophisticated 
techniques and instruments. Further the dynamical 
processes of bio-systems utilize the phenomenon of 
diffusion in wide variety of ways which include the 
role of blood circulation leading to nourishment of 
various body organs and also waste disposal from 
them. The control of heart beat through hormonal 
diffusion and digestion of food through enzymatic 
action are some of the examples. Further the role of 
diffusion in pharmaceutical applications is 
undisputed through the drug therapy curing a 
specific portion of body.   
 
B1.2 Basics of Diffusion 
The process of diffusion involves two components: 
(i) solvent is the medium in which atoms of similar 
or different kinds are to be diffused and (ii) solute 
are the atoms of one or more kinds which are to be 
diffused into the environment of solvent. The solute 
atoms migrate and distribute themselves in solvent 
in process of which they may become interstitial 
impurity (occupying interstitial spaces in solvent 
lattice) or substitution impurity (substituting the 
solvent atom from its regular position in lattice).  
The process of diffusion can be classified into two 
types which are (a) Self-diffusion in which solute and 
solvent atoms are identical. For example diffusion of 
copper in copper (b) Inter-diffusion in which the 
solute atoms are different from the solvent atoms 
as in the case steel where solvent is iron whereas 
solute atoms are those of carbon and many other 
elements. 
The phenomenon of diffusion of solute atoms into 
the solvent environment may occur through the 
following mechanisms:  
(a) Vacancy mechanism where the solute atoms lie 
in the vacancy and moves to other available 
vacancy. Atoms can move from one site to 
Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 150 
 
another if there are vacancies and the atom has 
sufficient energy to overcome a local activation 
energy barrier. The activation energy for 
diffusion is the sum of energy required to form a 
vacancy and further cause its motion. 
(b) Interstitial Mechanism is applicable in case of 
small atoms like hydrogen, helium, carbon, 
nitrogen diffuse through the interstitial spaces in 
a crystal. The activation energy for diffusion is 
the energy required for these atoms to squeeze 
through the small voids between the host lattice 
atoms. 
 
B1.3 Macroscopic Model of Diffusion 
B1.3.1 Steady State Diffusion: Fick’s I Law 
The process of diffusion is time dependent which 
implies that one needs to study the quantity of 
solute material that gets transported in solvent is a 
function of time. This time dependent mass transfer 
is often represented by a quantity called diffusion 
flux (J) which is defined as the solute mass 
transferred in unit time through a unit cross-
sectional area placed perpendicular to the direction 
of flow. The diffusion flux can be mathematically 
represented as: 
dt
dM
A
J
1
= 
The units of J are kg/m
2
S or atoms/m
2
s.  
If the diffusion flux does not change with time, a 
steady state condition exists. Common example of 
such a process is diffusion of atoms of gas through a 
metal plate for which the concentration of diffusing 
species (see figure B1.1(a) ) on both sides of plates is 
kept constant. If concentration of diffusing gas 
atoms is plotted as a function of position or depth, 
then the curve is referred to as concentration 
profile, which is shown for the steady state case 
under discussion in figure B1.1(b) as: 
 
 
Figure B1.1: (a) Steady state diffusion across a thin 
plate (b) a linear concentration profile for the given 
diffusion situation. 
 
Mathematically, the concentration gradient is 
expressed as: 
dx
dC
gradient ion Concentrat = 
 
The kinetics of steady state diffusion process is 
described by Fick’s first law which states that the 
diffusion current (J) is directly proportional to the 
concentration gradient ?
?
?
?
?
?
?
?
x
c
. Hence we express this 
mathematically as: 
) 1 . 1 (B
x
c
D J
?
?
- = 
Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 151 
 
The negative sign indicates that solute atoms 
migrate down the concentration gradient. The 
constant D is called diffusion coefficient which is 
characteristic of solute atoms and solvent 
environment and depends upon the following 
factors: 
(i) temperature of solvent medium 
(ii) mechanism of diffusion 
(iii) type of crystal structure of solvent,  
(iv) imperfections in crystal 
(v)  concentration of diffusing species. 
 
The Fick’s law can be understood to a greater extent 
through a kinetic model. Let’s consider a one 
dimensional flow of solute atoms in normal 
direction through section S as shown in the figure 
B1.2 below: 
 
Figure B1.2: The diagram showing the one 
dimensional flow of atoms through the intermediate 
plane S through their jumping in either of two 
possible directions. 
 
The concentration of solute atoms in adjacent 
planes are indicated by c
1
 and c
2
 (c
1
> c
2
) 
respectively. The atoms on both the planes jump 
randomly to right as well as left. The atoms of plane 
1 which jump to right and those of plane 2, which 
jump to left will cross the section S in opposite 
directions. Since the net flow of solute atoms will be 
towards right so we can write diffusion current as: 
) 2 . 1 (
2
1
2
1
2 1
B v n v n J - = 
In the equation (B1.2), n
1
 and n
2 
represent planar 
densities of solute atoms at planes 1 and 2 
respectively while v represents the jump 
frequency. The factor of half accounts for equal 
probability of jumping in one of the two possible 
directions leading to net rate of flow through 
section S. The concentrations c
1
 and c
2
 are defined 
as number of solute atoms per unit volume at plane 
1 and 2 respectively. Hence we can have relation 
between atomic density and concentration of solute 
atoms at a plane as n = ca. Using this relation in 
(B1.2), we get: 
) 3 . 1 ( ) (
2
1
2 1
B c c av J - =
 
If the concentration does not vary rapidly between 
the planes 1 and 2, then we can write as: 
) 4 . 1 (
2 1
B
x
c
a c c
?
?
- = -
 
Putting (B1.4) in (B1.3), we get: 
) 5 . 1 (
2
1
2
B
x
c
v a J
?
?
- = 
Comparing the equations (B1.1) and (B1.5), the 
diffusion coefficient D in one dimensional case can 
be expressed as: 
S 
c
1
 c
2
 
a 
Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 152 
 
Page 5


 
 
 
 
 
 
 
 
 
 
 
 
 
 
B: DIFFUSION & MECHANICAL PROPERTIES 
 
B1: Diffusion.  
B2: Deformation in Materials. 
B3: Fracture, Fatigue and Creep. 
B4: Strengthening Mechanisms. 
 
B1: DIFFUSION 
B1.1 Introduction 
In a sample having non-uniform composition of 
certain kind of atoms, a concentration gradient is set 
in, which leads to migration of atoms from region of 
their higher concentration to the region of lower 
concentration. This phenomenon is referred to as 
Diffusion and continues till atomic distribution 
becomes homogeneous throughout.  
The diffusion process plays an important role in the 
field of metallurgy and fabrication of extrinsic 
semiconductors. The technology of controlled 
diffusion plays a sensitive role in accomplishment of 
desired devices. The metals, in their pure form, are 
soft and do not possess large tensile strength. Their 
properties can be modified by the process of 
alloying in a controlled manner. The desired 
properties of alloys are achieved by proper choice of 
solute (atoms to be diffused) and their amount to be 
diffused. This is accomplished through sophisticated 
techniques and instruments. Further the dynamical 
processes of bio-systems utilize the phenomenon of 
diffusion in wide variety of ways which include the 
role of blood circulation leading to nourishment of 
various body organs and also waste disposal from 
them. The control of heart beat through hormonal 
diffusion and digestion of food through enzymatic 
action are some of the examples. Further the role of 
diffusion in pharmaceutical applications is 
undisputed through the drug therapy curing a 
specific portion of body.   
 
B1.2 Basics of Diffusion 
The process of diffusion involves two components: 
(i) solvent is the medium in which atoms of similar 
or different kinds are to be diffused and (ii) solute 
are the atoms of one or more kinds which are to be 
diffused into the environment of solvent. The solute 
atoms migrate and distribute themselves in solvent 
in process of which they may become interstitial 
impurity (occupying interstitial spaces in solvent 
lattice) or substitution impurity (substituting the 
solvent atom from its regular position in lattice).  
The process of diffusion can be classified into two 
types which are (a) Self-diffusion in which solute and 
solvent atoms are identical. For example diffusion of 
copper in copper (b) Inter-diffusion in which the 
solute atoms are different from the solvent atoms 
as in the case steel where solvent is iron whereas 
solute atoms are those of carbon and many other 
elements. 
The phenomenon of diffusion of solute atoms into 
the solvent environment may occur through the 
following mechanisms:  
(a) Vacancy mechanism where the solute atoms lie 
in the vacancy and moves to other available 
vacancy. Atoms can move from one site to 
Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 150 
 
another if there are vacancies and the atom has 
sufficient energy to overcome a local activation 
energy barrier. The activation energy for 
diffusion is the sum of energy required to form a 
vacancy and further cause its motion. 
(b) Interstitial Mechanism is applicable in case of 
small atoms like hydrogen, helium, carbon, 
nitrogen diffuse through the interstitial spaces in 
a crystal. The activation energy for diffusion is 
the energy required for these atoms to squeeze 
through the small voids between the host lattice 
atoms. 
 
B1.3 Macroscopic Model of Diffusion 
B1.3.1 Steady State Diffusion: Fick’s I Law 
The process of diffusion is time dependent which 
implies that one needs to study the quantity of 
solute material that gets transported in solvent is a 
function of time. This time dependent mass transfer 
is often represented by a quantity called diffusion 
flux (J) which is defined as the solute mass 
transferred in unit time through a unit cross-
sectional area placed perpendicular to the direction 
of flow. The diffusion flux can be mathematically 
represented as: 
dt
dM
A
J
1
= 
The units of J are kg/m
2
S or atoms/m
2
s.  
If the diffusion flux does not change with time, a 
steady state condition exists. Common example of 
such a process is diffusion of atoms of gas through a 
metal plate for which the concentration of diffusing 
species (see figure B1.1(a) ) on both sides of plates is 
kept constant. If concentration of diffusing gas 
atoms is plotted as a function of position or depth, 
then the curve is referred to as concentration 
profile, which is shown for the steady state case 
under discussion in figure B1.1(b) as: 
 
 
Figure B1.1: (a) Steady state diffusion across a thin 
plate (b) a linear concentration profile for the given 
diffusion situation. 
 
Mathematically, the concentration gradient is 
expressed as: 
dx
dC
gradient ion Concentrat = 
 
The kinetics of steady state diffusion process is 
described by Fick’s first law which states that the 
diffusion current (J) is directly proportional to the 
concentration gradient ?
?
?
?
?
?
?
?
x
c
. Hence we express this 
mathematically as: 
) 1 . 1 (B
x
c
D J
?
?
- = 
Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 151 
 
The negative sign indicates that solute atoms 
migrate down the concentration gradient. The 
constant D is called diffusion coefficient which is 
characteristic of solute atoms and solvent 
environment and depends upon the following 
factors: 
(i) temperature of solvent medium 
(ii) mechanism of diffusion 
(iii) type of crystal structure of solvent,  
(iv) imperfections in crystal 
(v)  concentration of diffusing species. 
 
The Fick’s law can be understood to a greater extent 
through a kinetic model. Let’s consider a one 
dimensional flow of solute atoms in normal 
direction through section S as shown in the figure 
B1.2 below: 
 
Figure B1.2: The diagram showing the one 
dimensional flow of atoms through the intermediate 
plane S through their jumping in either of two 
possible directions. 
 
The concentration of solute atoms in adjacent 
planes are indicated by c
1
 and c
2
 (c
1
> c
2
) 
respectively. The atoms on both the planes jump 
randomly to right as well as left. The atoms of plane 
1 which jump to right and those of plane 2, which 
jump to left will cross the section S in opposite 
directions. Since the net flow of solute atoms will be 
towards right so we can write diffusion current as: 
) 2 . 1 (
2
1
2
1
2 1
B v n v n J - = 
In the equation (B1.2), n
1
 and n
2 
represent planar 
densities of solute atoms at planes 1 and 2 
respectively while v represents the jump 
frequency. The factor of half accounts for equal 
probability of jumping in one of the two possible 
directions leading to net rate of flow through 
section S. The concentrations c
1
 and c
2
 are defined 
as number of solute atoms per unit volume at plane 
1 and 2 respectively. Hence we can have relation 
between atomic density and concentration of solute 
atoms at a plane as n = ca. Using this relation in 
(B1.2), we get: 
) 3 . 1 ( ) (
2
1
2 1
B c c av J - =
 
If the concentration does not vary rapidly between 
the planes 1 and 2, then we can write as: 
) 4 . 1 (
2 1
B
x
c
a c c
?
?
- = -
 
Putting (B1.4) in (B1.3), we get: 
) 5 . 1 (
2
1
2
B
x
c
v a J
?
?
- = 
Comparing the equations (B1.1) and (B1.5), the 
diffusion coefficient D in one dimensional case can 
be expressed as: 
S 
c
1
 c
2
 
a 
Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 152 
 
) 6 . 1 (
2
1
2
B va D = 
If we consider the three dimensional case then the 
atom can jump in any random direction and its 
probability of crossing the section S will be 
6
1
 
instead of half and the diffusion coefficient will 
become: 
) 7 . 1 (
6
1
2
B va D = 
The values of D can range from 10
-20
 –10
-50
 m
2
/s 
which correspond to jumping frequencies ranging 
from 10
-20
Hz to 1Hz. This indicates that the diffusion 
is an extremely slow process. 
 
B1.3.2 Non-Steady State Diffusion: Fick’s II Law 
The diffusion of solute atoms in solvent 
environment obeys the continuity equation given as: 
) 8 . 1 ( 0 B
x
J
t
c
=
?
?
+
?
?
 
The above given equation B1.8 can be interpreted 
as: If we consider a closed solvent volume 
characterized by certain concentration of solute in 
it. Then ant change in concentration of solute as a 
function of time implies inflow (or outflow) of 
solvent through the considered closed volume. This 
equation represents generalized form of 
conservation law. In this particular case, it is 
indicative conservation of solute mass. 
Putting equation (B1.1) in (B1.8), we get: 
) 9 . 1 (
2
2
B
x
c
D
t
c
x
c
D
x t
c
?
?
=
?
?
?
?
?
?
?
?
?
?
-
?
?
- =
?
?
 
The equation (B1.9) is called the Fick’s Second law. 
The solution of equation (B1.9) (i.e. concentration as 
a function of position and time) are possible when 
meaningful boundary conditions are specified.  
The Fick’s second law can be applied to the case of 
one dimensional case flow of solute in semi-infinite 
medium. The basic assumptions for this case are: 
(a) Before diffusion, the diffusing solute atoms in the 
solid are uniformly distributed with the 
concentration C
o
. 
(b) The value of x is zero at surface of semi-infinite 
medium and increases with distance into the 
solid. 
(c) The time is taken to be zero at the instant the 
process of diffusion begins. 
Hence the boundary conditions for this diffusion 
process are: 
) 10 . 1 (
0
0
0 0
0
B
x at C C
x at C C
t For
x at C C t For
s
s
?
?
?
?
?
?
8 = =
= =
>
8 = = = =
 
The application of boundary conditions given by 
equation B1.10 in equation B1.9 gives the solution 
as: 
) 11 . 1 (
2
1
0
0
B
Dt
x
erf
C C
C C
s
x
?
?
?
?
?
?
?
?
- =
-
-
 
Here Cx is the concentration of solute atoms at 
depth x after time t. The 
?
?
?
?
?
?
?
?
Dt
x
erf
2
 denotes the 
Gaussian error function. The equation B1.11 
Prof. J.K.Goswamy’s Notes on Materials Science: Properties of Materials. Page 153 
 
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