The Statistical Mechanics of Solution-Phase Nucleation …
119
tition function, q
IG
(i 1 ,i 2 ) =
1
i 1 !i 2 !
γ
i 1
1 γ
i 2
2
dr
(i1)
1 dr
(i2)
2 . Here, γ k =
m k k B T
2π
3/2 , where m k
is the mass of the k
th component, k B is the Boltzmann constant, is the Plank
constant, and T is the temperature. U is the potential energy of the system.
serves to partition configuration space defining a cluster of size (i 1 , i 2 ). For the
case of initial studies of DNT, the center of the mass of the cluster is defined by R =
1
i 1 +i 2
2
k=1
i k
j=1 r k,j , and the definition of the cluster depends on the radius cutoff
parameter, r cut . The spherical volume of the cluster is v =
4π
3
r
3
cut . We then write, =
2
k=1
i k
j=1
r cut −
r k,j − R
. With, ¯ i = i 1 + i 2 , and ¯
r ( ¯ i) =
r
(i1)
1 , r
(i2)
2
, q
IG
(i 1 ,i 2 ) =
1
i 1 !i 2 !
γ
i 1
1 γ
i 2
2
¯ i
3/2 Vv ( ¯ i−1) a¯ i , where a¯ i constants that appear in the single component case.
We define monomer partition functions as q (1,0) = γ 1 V and q (0,1) = γ 2 V . Here, V is
the volume of the system. The partition function for the full, non-interacting cluster
system is Q, where ln Q =
i 1
i 2
N (i 1 ,i 2 ) ln
q (i 1 ,i 2 )
− N (i 1 ,i 2 ) ln N (i 1 ,i 2 ) + N (i 1 ,i 2 )
.
Following Reiss and Bowles [44], we introduce particle and volume conservation
constraints through Lagrange multipliers and consider the extended partition function, , to apply the stationary phase analysis:
ln = ln Q + βμ 1
i 1
i 1
i 2
N (i 1 ,i 2 ) + βμ 2
i 2
i 2
i 1
N (i 1 ,i 2 )
−βp
V +
i 1
i 2
v (i 1 ,i 2 ) N (i 1 ,i 2 )
.
(15)
Here, v (i 1 ,i 2 ) is a volume of a cluster, μ k is the chemical potential of monomer k, p is
the pressure of the system.
We then have the conditions,
d ln
dN (1,0)
= ln
γ 1 V
− ln N (1,0) + βμ 1 = 0,
d ln
dN (0,1)
=
ln
γ 2 V
− ln N (0,1) + βμ 2 = 0,
d ln
dV
=
N (1,0) +N (0,1)
V
− βp = 0, and
d ln
dN (i 1 ,i 2 )
=
ln
q (i 1 ,i 2 )
− ln N (i 1 ,i 2 ) + βμ 1 i 1 + βμ 2 i 2 − βpv (i 1 ,i 2 ) = 0. Further simplification gives
βμ 1 = ln
N (1,0)
γ 1 V
, βμ 2 = ln
N (0,1)
γ 2 V
, and βpV = N (1,0) + N (0,1) . The equilibrium
population resulting from the stationary phase analysis, N
EQ
(i 1 ,i 2 ) , is ln N
EQ
(i 1 ,i 2 ) =
ln
q (i 1 ,i 2 )
+ βμ 1 i 1 + βμ 2 i 2 − βpv (i 1 ,i 2 ) . It is natural to define the total Helmholtz
free energy, A
TOT
(i 1 ,i 2 ) , by −βA
TOT
(i 1 ,i 2 ) = ln
q (i 1 ,i 2 )
. Collecting terms, we have ln N
EQ
(i 1 ,i 2 ) =
−β
A
TOT
(i 1 ,i 2 ) + pv (i 1 ,i 2 ) − μ 1 i 1 − μ 2 i 2
. It is useful to separate out the translational
component of the free energy, A
trans
(i 1 ,i 2 ) = −k B T ln (γ TOT V ), where γ TOT =
(m1i1+m2i2)kBT
2π
3/2
and A
TOT
(i 1 ,i 2 ) = A (i 1 ,i 2 ) + A
trans
(i 1 ,i 2 ) . The natural definition of the Gibbs
free energy follows, G (i 1 ,i 2 ) = A (i 1 ,i 2 ) + pv (i 1 ,i 2 ) . We then have N
EQ
(i 1 ,i 2 ) =
e
−β
G (i 1 ,i 2 ) +A
trans
(i 1 ,i 2 ) −μ 1 i 1 −μ 2 i 2
. This may be recast to recover our main result that connects
119
tition function, q
IG
(i 1 ,i 2 ) =
1
i 1 !i 2 !
γ
i 1
1 γ
i 2
2
dr
(i1)
1 dr
(i2)
2 . Here, γ k =
m k k B T
2π
3/2 , where m k
is the mass of the k
th component, k B is the Boltzmann constant, is the Plank
constant, and T is the temperature. U is the potential energy of the system.
serves to partition configuration space defining a cluster of size (i 1 , i 2 ). For the
case of initial studies of DNT, the center of the mass of the cluster is defined by R =
1
i 1 +i 2
2
k=1
i k
j=1 r k,j , and the definition of the cluster depends on the radius cutoff
parameter, r cut . The spherical volume of the cluster is v =
4π
3
r
3
cut . We then write, =
2
k=1
i k
j=1
r cut −
r k,j − R
. With, ¯ i = i 1 + i 2 , and ¯
r ( ¯ i) =
r
(i1)
1 , r
(i2)
2
, q
IG
(i 1 ,i 2 ) =
1
i 1 !i 2 !
γ
i 1
1 γ
i 2
2
¯ i
3/2 Vv ( ¯ i−1) a¯ i , where a¯ i constants that appear in the single component case.
We define monomer partition functions as q (1,0) = γ 1 V and q (0,1) = γ 2 V . Here, V is
the volume of the system. The partition function for the full, non-interacting cluster
system is Q, where ln Q =
i 1
i 2
N (i 1 ,i 2 ) ln
q (i 1 ,i 2 )
− N (i 1 ,i 2 ) ln N (i 1 ,i 2 ) + N (i 1 ,i 2 )
.
Following Reiss and Bowles [44], we introduce particle and volume conservation
constraints through Lagrange multipliers and consider the extended partition function, , to apply the stationary phase analysis:
ln = ln Q + βμ 1
i 1
i 1
i 2
N (i 1 ,i 2 ) + βμ 2
i 2
i 2
i 1
N (i 1 ,i 2 )
−βp
V +
i 1
i 2
v (i 1 ,i 2 ) N (i 1 ,i 2 )
.
(15)
Here, v (i 1 ,i 2 ) is a volume of a cluster, μ k is the chemical potential of monomer k, p is
the pressure of the system.
We then have the conditions,
d ln
dN (1,0)
= ln
γ 1 V
− ln N (1,0) + βμ 1 = 0,
d ln
dN (0,1)
=
ln
γ 2 V
− ln N (0,1) + βμ 2 = 0,
d ln
dV
=
N (1,0) +N (0,1)
V
− βp = 0, and
d ln
dN (i 1 ,i 2 )
=
ln
q (i 1 ,i 2 )
− ln N (i 1 ,i 2 ) + βμ 1 i 1 + βμ 2 i 2 − βpv (i 1 ,i 2 ) = 0. Further simplification gives
βμ 1 = ln
N (1,0)
γ 1 V
, βμ 2 = ln
N (0,1)
γ 2 V
, and βpV = N (1,0) + N (0,1) . The equilibrium
population resulting from the stationary phase analysis, N
EQ
(i 1 ,i 2 ) , is ln N
EQ
(i 1 ,i 2 ) =
ln
q (i 1 ,i 2 )
+ βμ 1 i 1 + βμ 2 i 2 − βpv (i 1 ,i 2 ) . It is natural to define the total Helmholtz
free energy, A
TOT
(i 1 ,i 2 ) , by −βA
TOT
(i 1 ,i 2 ) = ln
q (i 1 ,i 2 )
. Collecting terms, we have ln N
EQ
(i 1 ,i 2 ) =
−β
A
TOT
(i 1 ,i 2 ) + pv (i 1 ,i 2 ) − μ 1 i 1 − μ 2 i 2
. It is useful to separate out the translational
component of the free energy, A
trans
(i 1 ,i 2 ) = −k B T ln (γ TOT V ), where γ TOT =
(m1i1+m2i2)kBT
2π
3/2
and A
TOT
(i 1 ,i 2 ) = A (i 1 ,i 2 ) + A
trans
(i 1 ,i 2 ) . The natural definition of the Gibbs
free energy follows, G (i 1 ,i 2 ) = A (i 1 ,i 2 ) + pv (i 1 ,i 2 ) . We then have N
EQ
(i 1 ,i 2 ) =
e
−β
G (i 1 ,i 2 ) +A
trans
(i 1 ,i 2 ) −μ 1 i 1 −μ 2 i 2
. This may be recast to recover our main result that connects