Biomedical Engineering Reference
In-Depth Information
eq
Therefore, at equilibrium,
μ
crystal
can be expressed by
μ
mother
.
μ
then can be
expressed as
eq
μ
=
μ
mother
−
μ
(2.7)
mother
For crystallization from a solution, the chemical potential of species
i
is given by
0
i
0
i
μ
i
=
μ
+
kT
ln
ai
≈
μ
+
kT
ln
C
i
(2.8)
where
a
i
,and
C
i
denote the activities and concentrations of solute,
k
is the
Boltzmann constant, and
T
is the absolute temperature.
μ
0
i
denotes the chemical
potential of the solute at standard state (
a
i
=
1). This then gives rise to the
dimensionless thermodynamic driving force:
a
eq
i
)
C
eq
i
)
μ/
kT
=
ln (
a
i
/
≈
ln (
C
i
/
(2.9)
where
a
eq
and
C
eq
are the activities and concentrations of the solute, respectively,
at equilibrium.
Notice that the thermodynamic driving force for crystallization is often expressed
in terms of supersaturation. If we define supersaturation as
σ =
(
a
i
−
a
e
i
)
/ a
eq
(
C
i
−
C
e
i
)
/ C
eq
≈
(2.10)
i
i
Equation 2.9 can then be rewritten as
μ/
kT
=
ln (1
+ σ
)
(2.11)
Inthecaseof
σ
1, Equation 2.11 can be approximated by
)
= σ
μ/
kT
=
ln (1
+ σ
(2.12)
For crystallization from melts at temperatures not far below the melting or
equilibrium temperature, the thermodynamic driving force can also be calculated
from the following equation [20c, 21].
μ/kT
=
H
m
T/
(
kTT
e
)
(2.13)
T
=
(
T
e
−
T
)
(2.14)
where
H
m
is the enthalpy of melting per molar molecule,
T
e
is the equilibrium
temperature, and
T
is supercooling.
2.3.2
Homogeneous and Heterogeneous Nucleation
Nucleation is a process of assembly of atoms or molecules to reach a critical cluster
size (nuclei)
r
c
by overcoming an energy barrier (Figure 2.4a). The nucleation rate
J
describing the number of nuclei successfully generated from the population of
clusters per unit time unit volume is determined by the height of the free-energy
barrier, the so-called nucleation barrier
G
. The occurrence of a nucleation barrier
is attributed to the following two contradictory effects:
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