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LIST OF SYMBOLS
a
: lattice parameter.
a
o
: original lattice parameter.
a
(ave)
: average lattice parameter of the array.
B
: bulk modulus.
c
: lattice parameter of bct.
C
: atomic concentration.
C
i
: atomic fraction of i species.
C
j
: atomic fraction of j species.
d
: near-neighbor interatomic distance.
E
i
: energy required to form an interstitial defects.
E
v
: energy required to form an vacancy defects.
k
: Boltzman constant.
n
: number of Frenkel defects.
N
: number of atoms.
N
i
: number of interstitial.
N
v
: number of vacancies.
P
: Probability.
r
: atomic radius.
S
(ave)
: average interatomic spacing of the array.
Si
(ave)
: average spacing about an atom of the i species.
S
ij
: interatomic spacing between atoms of i and j species.
T
: absolute temperature.
V
: atomic volume.
x
c
: number of carbon atoms per 100 iron atoms.
: thermal expansion coefficient.
: shear modules.
: departures from Vegard's law.
a
: change of lattice parameters.
c
: change of c-axis of bct.
L
: change of length.
T
: change in temperature.
1
CHAPTER ONE
Literature outline of basic concepts
1-1 Introduction
An alloy is a substance that has metallic properties and is composed of two or
more chemical elements, of which at least one is a metal. There are two
possible phases in an alloy, intermediate alloy phase or compound, and solid
solution. If an alloy is homogeneous (composed of a single phase) in the solid
state, it can be only a solid solution or a compound. If the alloy is a mixture, it
is then composed of any combination of the phases possible in the alloy [1].
One important characteristic of a metals alloy is its lattice parameters,
which can be measured by observing the diffraction of either X-rays,
neutrons, or electrons [2], or can be predicted on the basis of the crystal
structure of its constituents and their concentrations [3], can be utilized in
many practical applications.
1-2 Solid Solutions
A characteristic property of metals is that if two (or more) are melted together
in suitable proportions a homogeneous solution often results. When cooled
this is called a solid solution because, as in the case of a liquid solution, the
solute and solvent atoms are arranged at random [3]. Random arrangement of
the two kinds of metal atom is always found if the alloy is cooled rapidly
(quenched). In certain solid solution with particular concentrations of solute a
regular atomic arrangement develops on slow cooling [4].
2
1-2-1 Substitutional Solid Solution
The substitutional solid solution is the more general case, that the solute
atoms replace those of the solvent, so that the two kinds of atom are situated
on a common lattice. The substitutional solid solutions may be accompanied
by either an increase or a decrease in cell volume, depending on whether the
solute atom is larger or smaller than the solvent atom [5].
Substitutional solid solutions are of two types:
(i)
Random substitution solid solutions.
(ii) Ordered substitutional solid solutions.
When there is no order in the substitution of the two elements
(as shown in Fig. (1-1a)), the chance of one element occupying any particular
atomic site in the crystal is equal to the atomic percent of that element in the
alloy. In such a case the concentration of solute atoms can vary considerably
throughout the lattice structure. The resulting solid solution is called a random
or disordered substitutional solid solution. If the atoms of the solute material
occupy similar lattice points within the crystal structure of the solvent
material, this is called an ordered solution (Fig. (1-1b)). Such ordering is
common at lower temperatures since greater thermal agitation tends to destroy
the orderly arrangement [6].
Fig. (1-1) (a) Random substitutional solid solution.
(b) Ordered substitutional solid solution.
(after Narula et al [6].)
(a)
(b)
3
In general, the substitutional solid solutions are not formed by all pairs of
metals due to principal physical factors that control the range of solubility in
solid solutions which are:
i- Atomic Size Factor:
Hume-Rothery et al [3] advanced the hypothesis that the size factor is
favorable for solid solution formation when the difference in atomic radii is
less than about 15 percent. If the atomic size factor is greater than 8 percent
but less than 15 percent, the alloy system usually shows a minimum
solubility. If the atomic size factor is greater than 15 percent, solid solution
formation is very limited.
ii- Crystal Structure Factor:
Continuous solid solubility is possible for two metals that have identical
crystal structures, except for the dimensions of the unit cell which are
governed by the atomic size factor. Thus continuous solid solutions are
possible between face centered cubic Cu-Ni or body centered cubic Mo-W,
but not between body centered cubic Mo and face centered cubic Cu [1].
iii- Electronegativity Factor:
If the two kinds of atoms in a solid solution are respectively electronegative
and electropositive, then it is likely that they prefer to form stable structure
rather than continuous solid solutions [1]. Because these structures frequently
have compositional variations over a certain range, it is proper to call such
crystals intermediate phases rather than compounds [3].
iv- Relative Valency Factor:
Continuous solid solutions can occur only between atoms having the same
valency in the alloy. It is generally true that elements of lower
valency dissolve to a larger extent in a higher valency solvents than the
reverse case [7].
4
1-2-2 Interstitial solid solution
The interstitial solid solution is the more restricted case that the solute atoms
fit into the spaces between those of the solvent, so that interstitial type of solid
solution is confined to cases where one atom is very much smaller than the
other (see Fig. (1-2)), and the most important interstitial solutes are carbon,
nitrogen, hydrogen, and boron [4].
The interstitial addition is always accompanied by an increase in the
volume of the unit cell. If the solvent structure is cubic then the single lattice
parameter a must increase, but if it is not cubic, then one parameter may
increase and the other decrease, as long as these changes result in an increase
in cell volume [3,5].
Atoms of radii less than one tenth of a nanometer can squeeze into the
interstitial sites in most metals and according to the argument proposed in
Ref. [8] that if the ratio of the atomic radius of the solute atom, to that of the
solvent, was less than 0.59 a situation favour the formation of interstitial solid
solution (see Fig. (1-3)).
Fig. (1-2) Interstitial solid solution.
(after Narula et al [6].)
Solvent atom.
Interstitial atom.
5
Fig. (1-3) Atom size rules for substitutional and interstitial alloys.
(after Seeger et al. [8]).
1-3 Atomic radius and coordination number
Atomic radius is defined as half the distance between the nearest neighbours
in the crystal structure of a pure element [6]. The size of an atom is affected
by its coordination number. The type of forces existing between the atoms
also affects it, and these forces may differ in unlike compounds or even
different structural arrangements of the same compound. It is important,
therefore, to realize that absolute values of atomic radii are only
approximately known. Furthermore, the same atom can have several different
radii according to its coordination number and the type of compound in which
it occur [1,9].
The atomic radii derived from interatomic distances in the structure of
the elements are structure-dependent parameters, whereas atomic volumes are
much more nearly independent of structure, and this applies even in the face
centered cubic and body centered cubic structures. For example, by equating
ALLOY SIZE RATIO
SIZE RATIO FAVORABLE TO
FORMATION
0.59
0
Range in which
alloying
is non favored.
0.85
1.0
SIZE RATIO FAVORABLE TO
FORMATION
SIZE RATIO
=
RADIUS
ATOMIC
SOLVENT
RADIUS
ATOMIC
SOLUTE
6
the atomic volumes V of an element in the fcc (CN12) and bcc (CN8)
structures:
2
/
a
4
/
a
V
3
bcc
3
fcc
=
=
It is seen that the near-neighbor interatomic distance d, is about 3%
larger in the fcc structure:
03
.
1
2
/
a
3
2
/
a
2
2
/
a
3
2
/
a
d
d
bcc
bcc
3
bcc
fcc
bcc
fcc
=
=
=
And empirically it is indeed found to be some
2
1
2
to 3% larger in
fcc structures, implying that atomic volumes in the two structures are
nearly equal [10].
1-4 Modeling of lattice parameter in solid solutions
1-4-1 Vegard's law
Vegard's law calls for a linear variation of the lattice parameters a, as
a function of atomic concentration C, in substitutional ionic solid solution
between two components A and B of similar structure [10].
B
B
A
A
a
C
a
C
a
+
=
(1.1)
where a
A
and a
B
are the lattice parameters of solvent and solute
respectively.
Vegard's law really applies to solid solutions of salts [11,12], but it did
not predict the lattice parameters correctly in accordance with the
experimental data for metallic solid solutions [10]. This law is proposed by
Vegard in (1921), since then several papers have been written concerning
departures from Vegard's law and means for corrections to match with the
experimentation.
The behavior can be shown in Fig. (1-4), the actual lattice parameter in
curve (a) lies above that for linear relation, there is said to be a positive
7
deviation from Vegard's law, whereas a curve such as that of (b) corresponds
to a negative deviation [3].
1-4-2 Departures from Vegard's law
Numerous attempts have been made to calculate and predict departure from
Vegard's law. Some of the models will be mentioned below.
Jaswon, Henry, and Raynor [13] investigated copper, silver and gold
alloys in an attempt to explain departures from Vegard's law based on strain
energy due to the introduction of a solute into the matrix. Their analysis lead
to the following equation:
[
]
)
a
a
(
)
a
a
(
13
C
A
B
A
B
-
-
-
=
(1.2)
(a) Positive
deviation
(b) Negative
deviation
A
B
Lattice Par
am
eter
B. Atomic %
0
100
Fig. (1-4) Illustration of positive and negative deviation from
Vegard's Law. (after Hume-Rothery et al.[3]).
8
where
is the departures from Vegard's law, the subscript A refers to solvent
and B to the solute, C is the atomic concentration, a
A
and a
B
are the lattice
parameters of solvent and solute respectively, and a (without a subscript)
refers to the lattice parameter of the alloy.
Zen [14] showed that Vegard's law is valid only when the volumes of
the two components are approximately equal. He suggested that if the specific
volumes of the two components are significantly different, then Vegard's law
does not hold, because it is the volumes rather than the lattice parameters of
the two components which are additive.
Zen suggests that Vegard's law should be expressed as:
(
)
{
}
[
]
3
/
1
B
3
A
B
A
C
a
/
a
1
1
a
a
-
-
=
(1.3)
definition of symbols as in equation (1.2).
Dienes [15] has derived a relation between the nearest-neighbor
distance, the composition and the short-range order parameter for binary
alloys Cu-Au and Au-Ag. The model shows that ordering has some influence
on departures from Vegard's law.
d'Heurle, Nowick, and Seraphim [16] have proposed another model to
calculate the lattice parameter, with or without short-range order, involving a
concept of preferred bond distances.
Gschneidner and Vineyard [17] applied second-order elasticity theory
to obtain the equation:
B
A
2
B
A
C
a
)
a
a
(
B
dp
d
2
-
-
=
(1.4)
where
is the shear modulus, P is the pressure, and B is the bulk
modulus. This equation is valid only for dilute solutions, i.e. when
1
C
B
<<
.
9
The magnitude and sign of the departures from Vegard's law, is poorly
predicted for all models [10]. This suggests that the factor(s) mainly
responsible for departures from the law have not yet been considered, and that
the above considerations are only secondary influences which are general to
the problem, giving results that are essentially independent of the various
models [10].
1-4-3 Moreen's model
In 1971 Moreen et al [18] presented a model to predict the lattice parameters
of metallic solid solutions as a function of composition, this method is based
on the hypothesis that the measured lattice parameter of a solid solution alloy
is the average of all the interatomic spacing within a selected region of the
lattice. Except little deviation in some systems, there is a good agreement
between the calculated and experimental values.
In 1985 Ning Yuatao and Xu. Hua. [19] used Miedema's theory of
electric charge shift, to modify the Moreen's model of prediction of the lattice
parameters in solid solutions. The lattice parameters of Cu-Al, Cu-Si and
Ni-Al solid solutions at equilibrium and Ag-Sn, Ag-La and Ag-Gd extended
solid solutions were calculated and found to be in better agreement with the
experimental values.
10
1-4-4 Models on Fe-C system
Numerous studies [20-23] on the variation of lattice parameters of Fe-C solid
solution with carbon concentration have been done experimentally. It has
been shown that in austenite, which is an interstitial solid solution of carbon
in face centered cubic
-iron, the addition of carbon increases the cell edge a;
but in martensite which is a metastable interstitial solid solution of carbon in
-iron, the c parameter of the body centered tetragonal cell increases while
the a parameter decreases, when carbon is added.
In (1990), Liu Cheng et al [24] used least-squares fitting through
literature data; in the case of iron-carbon austenite yields:
c
c
kx
a
)
x
(
a
+
=
o
(1.5)
where
o
a
is the
-iron lattice parameter, k is a constant, and
c
x
is the
number of carbon atoms per 100 iron atoms.
In the case of iron-carbon martensite, they yield the following
relationship:
c
c
x
k
a
)
x
(
a
1
-
=
o
(1.6)
c
c
x
k
a
)
x
(
c
2
+
=
o
(1.7)
where
o
a
in this case is the
-iron lattice parameter, and
2
1
k
,
k
are
constants.
In (1995) Onink [25] during an experimental work reported that, the
observed dependencies of the lattice parameters of austenite on carbon
concentration and temperature can be combined and expressed as:
[
]
T
)
x
k
k
(
)
x
k
a
(
)
T
,
x
(
a
c
c
c
4
3
1
+
+
+
=
o
(1.8)
where
3
k
and
4
k
are constants, and T is absolute temperature. In
addition, the average lattice parameter of ferrite via temperature is given by:
)
T
k
(
a
)
T
(
a
5
1
+
=
o
(1.9)
where
5
k
is a constant.
11
1-5 Point defects
The mathematically perfect crystal is an exceedingly useful concept. In actual
crystals, however, imperfections or defects are always present and their nature
and effects are often very important in understanding the properties of
crystals. Ordinary materials, however contain imperfections of various kinds
that produce both useful properties and also such undesirable effects as
causing the strength to decrease below that of a perfect crystal [26].
The simplest imperfection is a lattice vacancy, which is a missing atom
or ion, also known a Schottky defect, (Fig. (1-5a)).
The probability (P) that a given lattice site is vacant is proportional to
the Boltzmann factor for thermal equilibrium:
)
kT
/
E
exp(
P
v
-
=
, where
v
E
is the energy required to take an atom from a lattice site inside the crystal
to a lattice site on the surface, k Boltzmann constant and T absolute
temperature [27].
Another vacancy defect is the Frenkel defect in which an atom is
transferred from a lattice site to an interstitial position (Fig. (1-5b)), a position
not normally occupied by an atom [9].
(a)
(b)
Fig. (1-5) Formation in a lattice of
(a) a vacancy and (b) an interstitial
12
1-6 Theme of work
Phase diagrams and/or properties of solid solutions are of great importance in
metallurgy and their determination are extensively done experimentally by X-
ray and thermal analysis methods.
As seen from literature outline, theoretical prediction of phase diagrams
via lattice parameters received less attention, only few publications on the
substitutional solid solutions and practically no available literature on the
interstitial solid solution such as the common Fe-C and Fe-N. Thus the theme
of work will highlight prediction of models involved in the calculation of
lattice parameters via atomic radii of the binary elements and their crystal
structures for the substitutional solid solution Ni-Cu and Ge-Si, and the
interstitial solid solution Fe-C, subject to the correlation with concentration of
solvent/solute, and in addition the model is also correlated with the vacancy
defects and thermal expansion that are highly affected by temperature.
13
CHAPTER TWO
Theoretical Aspects
2-1 Introduction
As metals have relatively high densities, they must consist of atoms that are
packed very closely together, so, for a first approximation it is permissible to
consider the atoms of a metal as hard spheres packed together [26]. But it is
found, that the atoms of a given metal may not always appear to have the
same diameter. Nevertheless the concept of atomic diameter has proved to be
a useful one in metallurgy and plays an important role in understanding the
formation of alloys [28].
For the hard sphere model of the atoms, the nearest nighbour distance
in a crystal of pure element is 2r where r is the radius of the atom; and
according to this model the lattice parameter (
o
a
) of a crystal structure
(see Fig. (2-1)) are given by the expressions [26]:
=
=
=
3
r
8
a
Diamond
2
r
4
a
fcc
3
r
4
a
bcc
o
o
o
K
K
K
K
K
(2.1)
14
0
1/2
0
1/2
0
1/2
0
1/2
3/4
1/4
3/4
0
1/4
4r
a
a
a
r
2r
r
a
Fig. (2-1) Some metallic crystal Structures.
(a) - Body centered cubic structure.
(b) - Face centered cubic structure.
(c) - Diamond structure (after Pillai[26]).
(a)
(b)
(c)
a
a
4r
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