In materials science, segregation refers to
the enrichment of atoms, ions, or molecules
at a microscopic region in a materials system.
While the terms segregation and adsorption
are essentially synonymous, in practice, segregation
is often used to describe the partitioning
of molecular constituents to defects from
solid solutions, whereas adsorption is generally
used to describe such partitioning from liquids
and gases to surfaces. The molecular-level
segregation discussed in this article is distinct
from other types of materials phenomena that
are often called segregation, such as particle
segregation in granular materials, and phase
separation or precipitation, wherein molecules
are segregated in to macroscopic regions of
different compositions. Segregation has many
practical consequences, ranging from the formation
of soap bubbles, to microstructural engineering
in materials science, to the stabilization
of colloidal suspensions.
Segregation can occur in various materials
classes. In polycrystalline solids, segregation
occurs at defects, such as dislocations, grain
boundaries, stacking faults, or the interface
between two phases. In liquid solutions, chemical
gradients exist near second phases and surfaces
due to combinations of chemical and electrical
effects.
Segregation which occurs in well-equilibrated
systems due to the instrinsic to the chemical
properties of the system is termed equilibrium
segregation. Segregation that occurs due to
the processing history of the sample (but
that would disappear at long times) is termed
non-equilibrium segregation.
== History ==
Equilibrium segregation is associated with
the lattice disorder at interfaces, where
there are sites of energy different from those
within the lattice at which the solute atoms
can deposit themselves. The equilibrium segregation
is so termed because the solute atoms segregate
themselves to the interface or surface in
accordance with the statistics of thermodynamics
in order to minimize the overall free energy
of the system. This sort of partitioning of
solute atoms between the grain boundary and
the lattice was predicted by McLean in 1957.Non-equilibrium
segregation, first theorized by Westbrook
in 1964, occurs as a result of solutes coupling
to vacancies which are moving to grain boundary
sources or sinks during quenching or application
of stress. It can also occur as a result of
solute pile-up at a moving interface.There
are two main features of non-equilibrium segregation,
by which it is most easily distinguished from
equilibrium segregation. In the non-equilibrium
effect, the magnitude of the segregation increases
with increasing temperature and the alloy
can be homogenized without further quenching
because its lowest energy state corresponds
to a uniform solute distribution. In contrast,
the equilibrium segregated state, by definition,
is the lowest energy state in a system that
exhibits equilibrium segregation, and the
extent of the segregation effect decreases
with increasing temperature. The details of
non-equilibrium segregation are not going
to be discussed here, but can be found in
the review by Harries and Marwick.
== Importance ==
Segregation of a solute to surfaces and grain
boundaries in a solid produces a section of
material with a discrete composition and its
own set of properties that can have important
(and often deleterious) effects on the overall
properties of 
the material. These ‘zones’ with an increased
concentration of solute can be thought of
as the cement between the bricks of a building.
The structural integrity of the building depends
not only on the material properties of the
brick, but also greatly on the properties
of the long lines of mortar in between.
Segregation to grain boundaries, for example,
can lead to grain boundary fracture as a result
of temper brittleness, creep embrittlement,
stress relief cracking of weldments, hydrogen
embrittlement, environmentally assisted fatigue,
grain boundary corrosion, and some kinds of
intergranular stress corrosion cracking. A
very interesting and important field of study
of impurity segregation processes involves
AES of grain boundaries of materials. This
technique includes tensile fracturing of special
specimens directly inside the UHV chamber
of the Auger Electron Spectrometer that was
developed by Ilyin.
Segregation to grain boundaries can also affect
their respective migration rates, and so affects
sinterability, as well as the grain boundary
diffusivity (although sometimes these effects
can be used advantageously).Segregation to
free surfaces also has important consequences
involving the purity of metallurgical samples.
Because of the favorable segregation of some
impurities to the surface of the material,
a very small concentration of impurity in
the bulk of the sample can lead to a very
significant coverage of the impurity on a
cleaved surface of the sample. In applications
where an ultra-pure surface is needed (for
example, in some nanotechnology applications),
the segregation of impurities to surfaces
requires a much higher purity of bulk material
than would be needed if segregation effects
didn’t exist. The following figure illustrates
this concept with two cases in which the total
fraction of impurity atoms is 0.25 (25 impurity
atoms in 100 total). In the representation
on the left, these impurities are equally
distributed throughout the sample, and so
the fractional surface coverage of impurity
atoms is also approximately 0.25. In the representation
to the right, however, the same number of
impurity atoms are shown segregated on the
surface, so that an observation of the surface
composition would yield a much higher impurity
fraction (in this case, about 0.69). In fact,
in this example, were impurities to completely
segregate to the surface, an impurity fraction
of just 0.36 could completely cover the surface
of the material. In an application where surface
interactions are important, this result could
be disastrous.
While the intergranular failure problems noted
above are sometimes severe, they are rarely
the cause of major service failures (in structural
steels, for example), as suitable safety margins
are included in the designs. Perhaps the greater
concern is that with the development of new
technologies and materials with new and more
extensive mechanical property requirements,
and with the increasing impurity contents
as a result of the increased recycling of
materials, we may see intergranular failure
in materials and situations not seen currently.
Thus, a greater understanding of all of the
mechanisms surrounding segregation might lead
to being able to control these effects in
the future. Modeling potentials, experimental
work, and related theories are still being
developed to explain these segregation mechanisms
for increasingly complex systems.
== Theories of Segregation ==
Several theories describe the equilibrium
segregation activity in materials. The adsorption
theories for the solid-solid interface and
the solid-vacuum surface are direct analogues
of theories well known in the field of gas
adsorption on the free surfaces of solids.
=== Langmuir-McLean theory for surface and
grain boundary segregation in binary systems
===
This is the earliest theory specifically for
grain boundaries, in which McLean uses a model
of P solute atoms distributed at random amongst
N lattice sites and p solute atoms distributed
at random amongst n independent grain boundary
sites. The total free energy due to the solute
atoms is then:
G
=
p
e
+
P
E
−
k
T
[
ln
⁡
(
n
!
N
!
)
−
ln
⁡
(
n
−
p
)
!
p
!
(
N
−
P
)
!
P
!
]
{\displaystyle G=pe+PE-kT[\ln(n!N!)-\ln(n-p)!p!(N-P)!P!]}
where E and e are energies of the solute atom
in the lattice and in the grain boundary,
respectively and the kln term represents the
configurational entropy of the arrangement
of the solute atoms in the bulk and grain
boundary. McLean used basic statistical mechanics
to find the fractional monolayer of segregant,
X
b
{\displaystyle X_{b}}
, at which the system energy was minimized
(at the equilibrium state), differentiating
G with respect to p, noting that the sum of
p and P is constant. Here the grain boundary
analogue of Langmuir adsorption at free surfaces
becomes:
X
b
X
b
0
−
X
b
=
X
c
1
−
X
c
exp
{\displaystyle {\frac {X_{b}}{X_{b}^{0}-X_{b}}}={\frac
{X_{c}}{1-X_{c}}}\exp }
(
−
Δ
G
R
T
)
{\displaystyle \left({\frac {-\Delta \,G}{RT}}\right)}
Here,
X
b
0
{\displaystyle X_{b}^{0}}
is the fraction of the grain boundary monolayer
available for segregated atoms at saturation,
X
b
{\displaystyle X_{b}}
is the actual fraction covered with segregant,
X
c
{\displaystyle X_{c}}
is the bulk solute molar fraction, and
Δ
G
{\displaystyle \Delta \,G}
is the free energy of segregation per mole
of solute.
Values of
Δ
G
{\displaystyle \Delta \,G}
were estimated by McLean using the elastic
strain energy,
E
e
l
{\displaystyle E_{el}}
, released by the segregation of solute atoms.
The solute atom is represented by an elastic
sphere fitted into a spherical hole in an
elastic matrix continuum. The elastic energy
associated with the solute atom is given by:
E
e
l
=
24
π
K
μ
0
r
0
(
r
1
−
r
0
)
2
3
K
+
4
μ
0
{\displaystyle E_{el}={\frac {24\pi \,\mathrm
{K} \,\mu \,_{0}r_{0}(r_{1}-r_{0})^{2}}{3\mathrm
{K} \,+4\mu \,_{0}}}}
where
K
{\displaystyle \mathrm {K} \,}
is the solute bulk modulus,
μ
0
,
{\displaystyle \mu \,_{0},}
is the matrix shear modulus, and
r
0
,
{\displaystyle r_{0},}
and
r
1
,
{\displaystyle r_{1},}
are the atomic radii of the matrix and impurity
atoms, respectively. This method gives values
correct to within a factor of two (as compared
with experimental data for grain boundary
segregation), but a greater accuracy is obtained
using the method of Seah and Hondros, described
in the following section.
=== Free energy of grain boundary segregation
in binary systems ===
Using truncated BET theory (the gas adsorption
theory developed by Brunauer, Emmett, and
Teller), Seah and Hondros write the solid-state
analogue as:
X
b
X
b
0
−
X
b
=
X
c
X
c
0
exp
{\displaystyle {\frac {X_{b}}{X_{b}^{0}-X_{b}}}={\frac
{X_{c}}{X_{c}^{0}}}\exp }
(
−
Δ
G
′
R
T
)
{\displaystyle \left({\frac {-\Delta \ G'}{RT}}\right)}
where
Δ
G
=
Δ
G
′
+
Δ
G
s
o
l
{\displaystyle \Delta \,G=\Delta \,G'+\Delta
\,G_{sol}}
X
c
0
{\displaystyle X_{c}^{0}}
is the solid solubility, which is known for
many elements (and can be found in metallurgical
handbooks). In the dilute limit, a slightly
soluble substance has
X
c
0
=
exp
⁡
(
Δ
G
s
o
l
R
T
)
{\displaystyle X_{c}^{0}=\exp \left({\frac
{\Delta \,G_{sol}}{RT}}\right)}
, so the above equation reduces to that found
with the Langmuir-McLean theory. This equation
is only valid for
X
c
≤
X
c
0
{\displaystyle X_{c}\leq X_{c}^{0}}
. If there is an excess of solute such that
a second phase appears, the solute content
is limited to
X
c
0
{\displaystyle X_{c}^{0}}
and the equation becomes
X
b
X
b
0
−
X
b
=
exp
⁡
(
−
Δ
G
′
R
T
)
{\displaystyle {\frac {X_{b}}{X_{b}^{0}-X_{b}}}=\exp
\left({\frac {-\Delta \,G'}{RT}}\right)}
This theory for grain boundary segregation,
derived from truncated BET theory, provides
excellent agreement with experimental data
obtained by Auger electron spectroscopy and
other techniques.
=== More complex systems ===
Other models exist to model more complex binary
systems. The above theories operate on the
assumption that the segregated atoms are non-interacting.
If, in a binary system, adjacent adsorbate
atoms are allowed an interaction energy
ω
{\displaystyle \omega \,}
, such that they can attract (when
ω
{\displaystyle \omega \,}
is negative) or repel (when
ω
{\displaystyle \omega \,}
is positive) each other, the solid-state analogue
of the Fowler adsorption theory is developed
as:
X
b
X
b
0
−
X
b
=
X
c
1
−
X
c
exp
⁡
[
−
Δ
G
−
Z
1
ω
X
b
X
b
0
R
T
]
{\displaystyle {\frac {X_{b}}{X_{b}^{0}-X_{b}}}={\frac
{X_{c}}{1-X_{c}}}\exp \left[{\frac {-\Delta
\,G-Z_{1}\omega \,{\frac {X_{b}}{X_{b}^{0}}}}{RT}}\right]}
When
ω
{\displaystyle \omega \,}
is zero, this theory reduces to that of Langmuir
and McLean. However, as
ω
{\displaystyle \omega \,}
becomes more negative, the segregation shows
progressively sharper rises as the temperature
falls until eventually the rise in segregation
is discontinuous at a certain temperature,
as shown in the following figure.
Guttman, in 1975, extended the Fowler theory
to allow for interactions between two co-segregating
species in multicomponent systems. This modification
is vital to explaining the segregation behavior
that results in the intergranular failures
of engineering materials. More complex theories
are detailed in the work by Guttmann and McLean
and Guttmann.
=== The free energy of surface segregation
in binary systems ===
The Langmuir-McLean equation for segregation,
when using the regular solution model for
a binary system, is valid for surface segregation
(although sometimes the equation will be written
replacing
X
b
{\displaystyle X_{b}}
with
X
s
{\displaystyle X_{s}}
). The free energy of surface segregation
is
Δ
G
s
=
Δ
H
s
−
T
Δ
S
{\displaystyle \Delta \,G_{s}=\Delta \,H_{s}-T\Delta
\,S}
. The enthalpy is given by
−
Δ
H
s
=
γ
0
s
−
γ
1
s
−
2
H
m
Z
X
c
(
1
−
X
c
)
{\displaystyle -\Delta \,H_{s}=\gamma \,_{0}^{s}-\gamma
\,_{1}^{s}-{\frac {2H_{m}}{ZX_{c}(1-X_{c})}}}
[
Z
1
(
X
c
−
X
s
)
+
Z
v
(
X
c
−
1
2
)
]
+
24
π
K
μ
0
r
0
(
r
1
−
r
0
)
2
3
K
+
4
μ
0
{\displaystyle \left[Z_{1}(X_{c}-X_{s})+Z_{v}\left(X_{c}-{\frac
{1}{2}}\right)\right]+{\frac {24\pi \,\mathrm
{K} \,\mu \,_{0}r_{0}(r_{1}-r_{0})^{2}}{3\mathrm
{K} \,+4\mu \,_{0}}}}
where
γ
0
{\displaystyle \gamma \,_{0}}
and
γ
1
{\displaystyle \gamma \,_{1}}
are matrix surface energies without and with
solute,
H
1
{\displaystyle H_{1}}
is their heat of mixing, Z and
Z
1
{\displaystyle Z_{1}}
are the coordination numbers in the matrix
and at the surface, and
Z
v
{\displaystyle Z_{v}}
is the coordination number for surface atoms
to the layer below. The last term in this
equation is the elastic strain energy
E
e
l
{\displaystyle E_{el}}
, given above, and is governed by the mismatch
between the solute and the matrix atoms. For
solid metals, the surface energies scale with
the melting points. The surface segregation
enrichment ratio increases when the solute
atom size is larger than the matrix atom size
and when the melting point of the solute is
lower than that of the matrix.A chemisorbed
gaseous species on the surface can also have
an effect on the surface composition of a
binary alloy. In the presence of a coverage
of a chemisorbed species theta, it is proposed
that the Langmuir-McLean model is valid with
the free energy of surface segregation given
by
Δ
G
c
h
e
m
{\displaystyle \Delta \,G_{chem}}
, where
Δ
G
c
h
e
m
=
Δ
G
s
+
(
E
B
−
E
A
)
Θ
{\displaystyle \Delta \,G_{chem}=\Delta \,G_{s}+(E_{B}-E_{A})\Theta
\,}
E
A
{\displaystyle E_{A}}
and
E
B
{\displaystyle E_{B}}
are the chemisorption energies of the gas
on solute A and matrix B and
Θ
{\displaystyle \Theta }
is the fractional coverage. At high temperatures,
evaporation from the surface can take place,
causing a deviation from the McLean equation.
At lower temperatures, both grain boundary
and surface segregation can be limited by
the diffusion of atoms from the bulk to the
surface or interface.
== Kinetics of Segregation ==
In some situations where segregation is important,
the segregant atoms do not have sufficient
time to reach their equilibrium level as defined
by the above adsorption theories. The kinetics
of segregation become a limiting factor and
must be analyzed as well. Most existing models
of segregation kinetics follow the McLean
approach. In the model for equilibrium monolayer
segregation, the solute atoms are assumed
to segregate to a grain boundary from two
infinite half-crystals or to a surface from
one infinite half-crystal. The diffusion in
the crystals is described by Fick’s laws.
The ratio of the solute concentration in the
grain boundary to that in the adjacent atomic
layer of the bulk is given by an enrichment
ratio,
β
{\displaystyle \beta }
. Most models assume
β
{\displaystyle \beta }
to be a constant, but in practice this is
only true for dilute systems with low segregation
levels. In this dilute limit, if
X
b
0
{\displaystyle X_{b}^{0}}
is one monolayer,
β
{\displaystyle \beta }
is given as
β
=
X
b
X
c
=
exp
⁡
(
−
Δ
G
′
R
T
)
X
c
0
{\displaystyle \beta ={\frac {X_{b}}{X_{c}}}={\frac
{\exp \left({\frac {-\Delta \,G'}{RT}}\right)}{X_{c}^{0}}}}
.
The kinetics of segregation can be described
by the following equation:
X
b
(
t
)
−
X
b
(
0
)
X
b
(
∞
)
−
X
b
(
0
)
=
1
−
exp
⁡
(
F
D
t
β
2
f
2
)
{\displaystyle {\frac {X_{b}(t)-X_{b}(0)}{X_{b}(\infty
\,)-X_{b}(0)}}=1-\exp \left({\frac {FDt}{\beta
\,^{2}f^{2}}}\right)}
erfc
(
F
D
t
β
2
f
2
)
1
2
{\displaystyle {\text{erfc}}\left({\frac {FDt}{\beta
\,^{2}f^{2}}}\right)^{\frac {1}{2}}}
Where
F
=
4
{\displaystyle F=4}
for grain boundaries and 1 for the free surface,
X
b
(
t
)
{\displaystyle X_{b}(t)}
is the boundary content at time
t
{\displaystyle t}
,
D
{\displaystyle D}
is the solute bulk diffusivity,
f
{\displaystyle f}
is related to the atomic sizes of the solute
and the matrix,
b
{\displaystyle b}
and
a
{\displaystyle a}
, respectively, by
f
=
a
3
b
−
2
{\displaystyle f=a^{3}b^{-2}}
. For short times, this equation is approximated
by:
X
b
(
t
)
−
X
b
(
0
)
X
b
(
∞
)
−
X
b
(
0
)
=
2
β
f
F
D
t
π
=
2
β
b
2
a
3
F
D
t
π
{\displaystyle {\frac {X_{b}(t)-X_{b}(0)}{X_{b}(\infty
\,)-X_{b}(0)}}={\frac {2}{\beta \,f}}{\sqrt
{\frac {FDt}{\pi \,}}}={\frac {2}{\beta \,}}{\frac
{b^{2}}{a^{3}}}{\sqrt {\frac {FDt}{\pi \,}}}}
In practice,
β
{\displaystyle \beta }
is not a constant but generally falls as segregation
proceeds due to saturation. If
β
{\displaystyle \beta }
starts high and falls rapidly as the segregation
saturates, the above equation is valid until
the point of saturation.
== In Metal Castings ==
All metal castings experience segregation
to some extent, and a distinction is made
between macrosegregation and microsegregation.
Microsegregation refers to localized differences
in composition between dendrite arms, and
can be significantly reduced by a homogenizing
heat treatment. This is possible because the
distances involved (typically on the order
of 10 to 100 µm) are sufficiently small for
diffusion to be a significant mechanism. This
is not the case in macrosegregation. Therefore,
macrosegregation in metal castings cannot
be remedied or removed using heat treatment.
== Further reading ==
== See also
