Extrinsic Point Defects: Impurities
¾ Substitutional and interstitial impurities
¾ Solid solutions, solubility limit
¾ Entropy of mixing, ideal solution model
¾ Enthalpy of mixing, quasi-chemical model
¾ Ideal and regular solutions
¾ Solubility from Gibbs free energy
¾ Interaction between impurities and vacancies
References:
Porter and Easterling, Ch. 1.3
Ragone, Thermodynamics of Materials, Ch. 3.3
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Impurities
Impurities (extrinsic point defects) - atoms which are different from the host.
Even the most pure materials contain some impurities and the concentration of the
impurities is, in most real materials, comparable to or exceeds the concentration of the
equilibrium intrinsic point defects.
Example: Very pure metals 99.9999% - one impurity per 106 atoms; equilibrium vacancy
concentration at Tm - one vacancy per ~104 atoms
May be intentional or unintentional
Examples: carbon added in small amounts to iron makes steel, which is stronger than pure iron
boron added to silicon change its electrical properties
Alloys - mixtures of components
Example: sterling silver is 92.5% silver – 7.5% copper alloy. Stronger than pure silver.
Material is described in terms of "impurity concentration” rather than “alloy composition” if the
amount of component B in matrix A is small
2 types:
substitutional impurities
substitute the host atoms in
the lattice
interstitial impurities
located in interstitial
lattice sites
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Solid solutions
Solid solutions are made of a host (the solvent or matrix) which dissolves the minor
component (solute). The ability to dissolve is called solubility.
¾ Solvent: in an alloy, the element or compound present in greater amount
¾ Solute: in an alloy, the element or compound present in lesser amount
¾ Solid Solution:
9 homogeneous
9 maintains crystal structure
9 contains randomly dispersed impurities (substitutional or interstitial)
¾ Second Phase: as solute atoms are added, new compounds / structures are formed, or
solute forms local precipitates
¾ Solubility Limit of a component in a phase is the maximum amount of the component
that can be dissolved in it
Example: Cu and Ni are mutually soluble in any amount (unlimited solid solubility), while C has
a limited solubility in Fe.
Whether the addition of impurities results in formation of solid solution or second phase
depends the nature of the impurities, their concentration, temperature, pressure…
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Substitutional and interstitial solid solutions
Substitutional solid solutions:
Max solute concentration = 50 at%
e.g. Cu-Ni (unlimited solid solubility)
Factors for high solubility:
¾ Atomic size factor - atoms need to “fit” ⇒ solute and solvent atomic radii should be
within ~ 15%
¾ Crystal structures of solute and solvent should be the same
¾ Electronegativities of solute and solvent should be comparable (otherwise new
intermediate phases are encouraged)
¾ Generally, more solute goes into solution when it has higher valency than solvent
Interstitial solid solutions:
Normally, max. solute concentration ≤ 10%
e.g. ≤ 0.1 at% of C in α-Fe (BCC).
Factors for high solubility:
¾ For fcc, bcc, hcp structures the voids (or interstices) between the host atoms are
relatively small ⇒ atomic radius of solute should be significantly less than solvent
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Equilibrium concentration of impurities ?
In contrast with intrinsic point defects, the concentration of impurities is often fixed and
cannot attain the value that corresponds to the equilibrium at given T and P
The equilibrium in a phase with impurities is usually described in terms of their solubility
Instead of the formation energies and entropies used for the intrinsic defects, the solubility
limit is defined by enthalpies and entropies of mixing
Gibbs free energy of a binary solution:
Let’s consider a binary solution of A and B atoms that have the same crystal structures in their
pure states and can be mixed in any proportions - has unlimited solid solubility, e.g., Cu + Ni
1 mol of homogeneous solid solution contains XA mol of A and XB mol of B. XA and XB are
the mole fractions of A and B in the alloy.
XA + XB =1
Two steps of mixing:
1. Bring together XA mol of A & XB mol of B
G
GB
Gstep1
GA
X AG A
X BG B
0
G 2 = G1 + ΔG mix
ΔG mix = ΔH mix − TΔ Smix
G1 = XAGA + XBGB
XB
2. Mix A & B to make a homogeneous solution
1
ΔGmix is the change of the Gibbs free energy
caused by the mixing
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Gibbs free energy of mixing
ΔG mix = ΔH mix − TΔ Smix
ΔH mix = H 2 − H1
- heat of mixing of the components (heat of formation of a solution)
Δ Smix = S2 − S1
- difference in entropy between mixed and unmixed states (entropy of
formation of a solution)
Model systems:
ideal solution – interactions between atoms A-A, B-B and A-B are identical, and ΔHmix = 0.
The free energy change upon mixing is only due to the change in configurational entropy:
ΔG idmix = −TΔ Smix
statistical or quasi-chemical model – heat of formation (ΔHmix ≠ 0) is evaluated by counting
bonds between atoms of different type. The assumption is that the interatomic distances and
bond energies are independent of composition.
regular solution – random arrangement of atoms in a solution is assumed (no clustering or
compositional ordering)
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Configurational entropy of mixing
Δ Smix = S2 − S1
S1 = k Bln1 = 0 - there is only one way the atoms
Therefore Δ Smix = S2
can be arranged before mixing
if we have Ntot objects and Nspec of them are “special,” the number of
ways the objects can be arranged (number of microstates) is
N tot !
Ω=
N spec !(N tot − N spec )!
Remember for vacancies we had Ntot = N – number of lattice sites, Nspec = n – number of vacancies
For mixing of NA particles of type A with NB atoms of type B: Ω =
ΔS mix = k B lnΩ = k B ln
(N A + N B )!
N B! N A !
(N A + N B )!
N B! N A !
Using Stirling formula: ln N ! ≈ N ln N − N
ΔSmix = k B [ln(N A + N B )!-lnNA !-lnNB!] = k B [(N A + N B )ln(N A + N B ) - (N A + N B ) - N A lnNA + N A - N BlnNB + N B ] =
⎡
⎛ NA ⎞
⎛ N B ⎞⎤
⎜
⎟
⎟⎟ ⎥ = − R [X A lnX A + X B lnX B ]
= − k B ⎢ N A ln ⎜
+ N B ln ⎜⎜
⎟
⎝ NA + NB ⎠
⎝ N A + N B ⎠⎦
⎣
for 1 mol, NA + NB = Na
X A = N A /( N A + N B )
X B = N B /( N A + N B )
NA = XANa, NB = XBNa, NakB = R
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Gibbs free energy of ideal solution
id
for ideal solution ΔHmix = 0 and ΔG mix = −TΔ Smix = RT[X A lnXA + X BlnXB ]
G id = G 2 = G1 + ΔG idmix
G1 = XAGA + XBGB
G id = X A G A + X BG B + RT[X A lnXA + X BlnXB ]
GB T
Gid
1
0
ΔG
id
mix
GA T
T1
GB T
1
2
T2
0
XB
1
How is the position of the minimum of the Gibbs
free energy curve related to the solubility limit?
How does it change when T increases?
GA T
ΔGidmix
2
0
XB
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
1
Enthalpy of mixing: quasi-chemical model
ΔG mix = ΔH mix − TΔ Smix
ΔH mix > 0 - mixing is endothermic (heat absorbed)
ΔH mix < 0 - mixing is exothermic (heat released)
quasi-chemical model: ΔHmix is only related
to the bond energies between adjacent atoms
if there are PAA, PBB, PAB bonds of each
type, the internal energy of the solution is
E = PAA E AA + PBBE BB + PABE AB
if z is the coordination number,
B
A
B
A
B
B
A
A
A
A
B
A
3 types of bonds:
A – A bond (energy EAA)
A – B bond (energy EAB)
B – B bond (energy EBB)
B
N A z = 2PAA + PAB
N B z = 2PBB + PAB
A
A
PAA =
B
N A z PAB
−
2
2
PBB =
N B z PAB
−
2
2
N z
N z
E + E BB ⎞
⎛N z P ⎞
⎛N z P ⎞
⎛
E = ⎜ A − AB ⎟E AA + ⎜ B − AB ⎟E BB + PABE AB = A E AA + B E BB + PAB ⎜ E AB − AA
⎟
2
2
2
2
2
2
2
⎝
⎠
⎠
⎝
⎠
⎝
energy of
energy of
unmixed components
mixing, ΔEmix
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Enthalpy of mixing of a regular solution
E + E BB ⎞
⎛
ΔH mix = H 2 − H1 ≈ ΔE mix = PAB ⎜ E AB − AA
⎟
2
⎠
⎝
- the solution is ideal: ΔHmix = 0
Enthalpy of mixing (heat of formation):
If
If
If
E AA + E BB
2
E + E BB
E AB > AA
2
E + E BB
E AB < AA
2
E AB =
- ΔHmix > 0 – atoms tend to be surrounded by atoms of the same type
- ΔHmix < 0 – atoms tend to be surrounded by atoms of different type
for small differences between EAB and (EAA+EBB)/2 (and for high T) we can still consider a
random arrangement of atoms in a solution (regular solution model). Then
PAB = zNtotXAXB
ΔHmix
ΔHmix = ΩXAXB
and
Ω>0
0
XB
⎛
⎝
where Ω = zN tot ⎜ E AB −
E AA + E BB ⎞
⎟
2
⎠
Ω
1
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Gibbs free energy of a regular solution
ΔGmix = ΔHmix − TΔ Smix = ΩXAXB + RT(XAlnXA + XBlnXB )
0
Ω<0
Ω < 0, ΔHmix < 0 – exothermic mixing, favorable at all T
Δ H mix
For high |Ω| and low T, PAB → max - an ordered alloy could be
formed – the assumption of random mixing is not valid, solution
is not regular, ΔHmix ≠ ΩXAXB
- TΔSmix
Δ G mix
0
XB
Ω > 0, low T
Ω > 0, high T
1
For Ω > 0, ΔHmix > 0 – mixing
(formation of A-B pairs) is avoided
at low T.
Δ H mix
0
0
Δ G mix
At high T entropy helps to mix.
At low T clustering may occur –
solution is not regular.
Δ H mix
Δ G mix
- TΔSmix
- TΔSmix
0
XB
1
0
XB
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
1
Solubility from Gibbs free energy
G
T1
β
The closer is the minimum
of the Gibbs free energy
curve Gα(XB) to the axes
XB = 0, the smaller is the
maximum concentration of
B in phase α.
T
liquid
α
α+l
liquid
0
T1
XB
1
β+l
α
β
α+β
0
XB
What is the temperature dependence of the solubility of B in A?
Have to find the temperature dependence of the minimum of Gα(XB)
G reg = X A G A + X BG B + ΩX A X B + RT[X A lnX A + X BlnX B ]
dG
=0
dX B
- minimum of G(XB)
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
1
Solubility from Gibbs free energy
G reg = X A G A + X BG B + ΩX A X B + RT[X A lnX A + X BlnX B ] =
(1 - X B )G A + X BG B + Ω(1 - X B )X B + RT[(1 - X B )ln(1 - X B ) + X BlnXB ]
⎡
(1 - X B ) + lnX + X B ⎤ =
dG
= -G A + G B + Ω − 2Ω X B + RT ⎢- ln(1 - X B ) −
B
⎥
(
)
dX B
1
X
XB ⎦
B
⎣
- GA + GB + Ω − 2Ω XB + RT[- ln(1- XB ) + lnXB ] =
⎛ X ⎞
G B - G A + Ω(1 − 2XB ) + RTln⎜⎜ B ⎟⎟ ≈ G B - G A + Ω + RTln(XB ) = 0
⎝ 1 - XB ⎠
if XB is small (XB → 0)
⎛ G − GA + Ω ⎞
XB ≈ exp⎜ − B
⎟
RT
⎝
⎠
G(XB) → min
- solid solubility of B in α increases exponentially with T
(similar to vacancy concentration)
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Solubility from Gibbs free energy
XB
XB vs. T
T vs. XB
T
supersaturation with B
→ phase separation α and β
→ precipitation of B-rich intermediate
phase or compound
T
⎛ G − GA + Ω ⎞
XB ≈ exp⎜ − B
⎟
RT
⎝
⎠
Te
XB
α+l
liquid
β+l
α
T1
β
α+β
0
XB
1
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei
Interaction between impurities and vacancies
vacancy-impurity complexes
Impurity atoms may interact with extrinsic point defects (vacancies, divacancies, self-interstitials,
etc.) and form complexes if there is an attractive interactions between the defects.
We can calculate the equilibrium concentration of impurity-vacancy complexes in a way similar
to our analysis of concentrations of vacancies and divacancies.
neqc ≈
neqv nimp z
N
⎛ hbc
exp ⎜⎜
⎝ k BT
neqc − equilibriu m number of complexes
⎞
⎟⎟
⎠
neqv − equilibriu m number of indivitual vacancies
v
ntot
= neqv + neqc
v
nimp z
⎛ hbc
ntot
exp ⎜⎜
≈ 1+
v
neq
N
⎝ k BT
nimp − number of impurities
z − coordinati on number
⎞
⎟⎟
⎠
hbc − binding enthalpy of an impurity - vacancy complex
Thus, for positive binding energies, impurities can substantially increase the total vacancy
concentration, especially at low T
Example: at T = 700 K, in an fcc metal with impurity fraction of 0.5% (nimp/N = 5×10-3) and a
binding enthalpy of 0.1 eV, n v
0.1 eV
⎛
⎞
−3
tot
n
v
eq
≈ 1 + 5 × 10 × 12 exp ⎜
⎟ = 1.315
−4
⎝ 0.86173 × 10 eV/K × 700 K ⎠
University of Virginia, MSE 6020: Defects and Microstructure in Materials, Leonid Zhigilei