118
J. Liu et al.
5.2.5 Extension to Multi-component and Multi-phase
Systems
The first PFC model of binary alloys was introduced by Elder et al. [8]. After that, the
developments were the extension of XPFC model to binary alloys by Greenwood et al.
[20] and to multi-component alloys by Ofori-Opoku et al. [21]. Multi-components,
multi-phases (solid, liquid, and different crystal lattices) and corresponding transformation could coexist in their models. Another achievement was to extend full
spectrum of solid-liquid-vapor phase within the PFC framework. Two approaches
were proposed by Schwalbach et al. [22] and Kocher and Provatas [23], respectively.
Based on these developments, multi-component and multi-phase systems were studied, such as eutectic growth [20], solute clustering in Al–Cu–Mg alloys [21], grain
boundary (GB) structures and dynamics in 2D h-BN [24], and also 2D heterostructures of graphene and h-BN [25].
For a binary alloy made up of A and B atoms, the free energy functional can be
written as sum of the free energy functional for two pure systems plus a coupling
term [8]:
F (ρ A , ρ B ) = F A + F B −
dr 1 dr 2 ρ A (r 1 )C AB (r 1 , r 2 ))ρ B (r 2 )
(5.27)
where ρ A = ρ A − ρ
l
A , ρ B = ρ B − ρ
l
B , and C AB is the 2-point correlation function
between atoms A and B. The common method is to introduce a total density field
ρ ≡ ρ A + ρ B and concentration field c ≡ ρ A /ρ, then Eq. (5.27) is rewritten as:
F (ρ A , ρ B ) =
dr
ρ ln
ρ
ρ l
− ρ + ρ
(1 − c) ln(1 − c) + c ln c
−
ρ
2
c
2 C AA + (1 − c)
2 C BB + 2c(1 − c)C AB
ρ
+ ρc
C AA − C AB
ρ lA +
C AB − C BB
ρ lB + ln
ρ lB
ρ lA
(5.28)
To further simplify the expression, two dimensionless order parameters are defined:
n = (ρ − ρ l )/ρ l and ψ = 2c − 1. The derivation procedure is tedious and can found
elsewhere [7, 8]. The following only gives the final result:
F (n, ψ) =
dr
B
l
2
n
2
+
B
x
2
n(2R
2
∇
2
+ R
4
∇
4
)n −
n
3
6
+
n
4
12
+ γ ψ +
ω
2
ψ
2
+
u
4
ψ
4
+
K
2
|∇ψ|
2
(5.29)
The dynamics of ρ A and ρ B is assumed to minimize the free energy F :
J. Liu et al.
5.2.5 Extension to Multi-component and Multi-phase
Systems
The first PFC model of binary alloys was introduced by Elder et al. [8]. After that, the
developments were the extension of XPFC model to binary alloys by Greenwood et al.
[20] and to multi-component alloys by Ofori-Opoku et al. [21]. Multi-components,
multi-phases (solid, liquid, and different crystal lattices) and corresponding transformation could coexist in their models. Another achievement was to extend full
spectrum of solid-liquid-vapor phase within the PFC framework. Two approaches
were proposed by Schwalbach et al. [22] and Kocher and Provatas [23], respectively.
Based on these developments, multi-component and multi-phase systems were studied, such as eutectic growth [20], solute clustering in Al–Cu–Mg alloys [21], grain
boundary (GB) structures and dynamics in 2D h-BN [24], and also 2D heterostructures of graphene and h-BN [25].
For a binary alloy made up of A and B atoms, the free energy functional can be
written as sum of the free energy functional for two pure systems plus a coupling
term [8]:
F (ρ A , ρ B ) = F A + F B −
dr 1 dr 2 ρ A (r 1 )C AB (r 1 , r 2 ))ρ B (r 2 )
(5.27)
where ρ A = ρ A − ρ
l
A , ρ B = ρ B − ρ
l
B , and C AB is the 2-point correlation function
between atoms A and B. The common method is to introduce a total density field
ρ ≡ ρ A + ρ B and concentration field c ≡ ρ A /ρ, then Eq. (5.27) is rewritten as:
F (ρ A , ρ B ) =
dr
ρ ln
ρ
ρ l
− ρ + ρ
(1 − c) ln(1 − c) + c ln c
−
ρ
2
c
2 C AA + (1 − c)
2 C BB + 2c(1 − c)C AB
ρ
+ ρc
C AA − C AB
ρ lA +
C AB − C BB
ρ lB + ln
ρ lB
ρ lA
(5.28)
To further simplify the expression, two dimensionless order parameters are defined:
n = (ρ − ρ l )/ρ l and ψ = 2c − 1. The derivation procedure is tedious and can found
elsewhere [7, 8]. The following only gives the final result:
F (n, ψ) =
dr
B
l
2
n
2
+
B
x
2
n(2R
2
∇
2
+ R
4
∇
4
)n −
n
3
6
+
n
4
12
+ γ ψ +
ω
2
ψ
2
+
u
4
ψ
4
+
K
2
|∇ψ|
2
(5.29)
The dynamics of ρ A and ρ B is assumed to minimize the free energy F :