Chapter 7 – Complementary Experimental Tools  285

(7.3)

B

k M

C

C

T

j

sat

b

2

1

=

−

(

)

where

k1 is the secondary nucleation rate constant

MT is the density of the crystal suspension

j and b are empirical exponents of ~1 (as high as ~1.5) and ~2 (as high as ~5), respectively

Analytical modeling of the nucleation process can be done by considering the typical free

energy change per molecule ΔGn associated with nucleation. This is given by the sum of the

bulk solution (ΔGb) and crystal surface (ΔGs) terms:

(7.4)

∆

∆

∆

∆

Ω

G

G

G

r

r

n

b

s

=

+

= ⁻

+

4

4

3

2

π

µ

π α

where

α is the interfacial free energy per unit area of a crystal of effective radius r

Ω is the volume per molecule

Δμ is the change in chemical potential of the crystallizing molecules, which measures

the mean free energy change in a molecule transferring from the solution phase to

the crystal phase

Standard thermodynamic theory for the chemical potential indicates

(7.5)

∆µ

σ

=

−



^



^=

k T

C

C

C

k T

B

sat

sat

B

ln

ln

where σ is often called the “saturation.” Inspection of Equation 7.4 indicates that there is a

local maximum in ΔGn equivalent to the free energy barrier ΔG^* for nucleation at a particular

threshold value of r known as the critical radius rc:

(7.6)

r

k T

c

B

= ^2Ωα

σ

In practice, two interfacial energies often need to be considered, one between the crystal and

the solid substrate and the other between the crystal and the solid surface substrate in which

crystals typically form on. Either way, substituting rc into Equation 7.5 indicates

(7.7)

∆

Ω

G

k T

* =



^



^

16

3

3

3

2

πα

σ

B

Thus, the rate of nucleation Jn can be estimated from the equivalent Boltzmann factor:

(7.8)

J

A

G

k T

A

B

n

n

B

=

−



^



^=

−



^



^

exp

exp

∆

α

σ

3

2

where A and B are constants. Thus, there is a very sensitive dependence on nucleation rate

with both the interfacial energy and the supersaturation. This is a key thermodynamic explan­

ation for why the process of crystal formation is so sensitive to environmental conditions and

is considered by some to be tantamount to a black art! Similar thermodynamic arguments

can be applied to model the actual geometrical shape of the crystals formed.