1.4 Nucleation of Two-Dimensional Solid Layers
27
alkanes above T sf is positive and small in the absolute values, as expected of any
ordinary liquid. The negative surface excess entropy between the melting point and
the surface freezing point shows that the surface is more ordered than the bulk,
consistent with the “frozen” state of the monolayer and the disordered state of the
underlying liquid bed. For n-octadecane, for example, the surface excess entropy
below the T sf , S surface can be calculated using our previous data [25]:
S surface = −∂γ/∂ T ≈ −1
mJ/m
2
· K
−1
≈ −1 × 10
−3 N · K
−1 m
−1
(1.4.7)
Figure (1.7) shows that the surface layer could be subcooled to below T m of noctadecane (T m ≈ 28.1 °C ≈ 301 K) and the small negative slope of ∂γ /∂T continues
(the positive surface excess entropy remains constant) until the nucleation for the
surface freezing transition eventually takes place at the points indicated by the vertical
arrows [25]. The positive surface excess entropy remains constant below T sf before
the eventual transition is consistent with the subcooled and disordered liquid surface
below T sf .
Meanwhile, the bulk heat of fusion of n-heptadecane (C 17 ), n-octadecane (C 18 ),
n-nonadecane (C 19 ), and n + eicosane (C 20 ) are 45 kJ/mol, 60 kJ/mol, 43.3 kJ/mol,
and 45.6 kJ/mol, respectively [22, 31]. Many normal alkanes of intermediate chain
lengths transition from liquid to a bulk rotator phase [16] at T m , as we noted above. A
separate phase transition between the bulk rotator phase and the bulk crystalline phase
takes place a few Kelvins below T m of each normal alkane. However, n-octadecane is
a special case and happens to undergo the liquid–rotator and the rotator–crystalline
transitions at the same temperature of T m [19], and consequently its bulk latent
heat of fusion is larger than that of the other normal alkanes. In other words, the
larger latent heat of fusion observed for n-octadecane can be attributed to the latent
heat of rotator–crystalline transition, which is required to provide thermal energy
to the molecules to rotate around their lattice positions. The bulk heat of fusion of
liquid–rotator phase transition of n-octadecane is expected to be similar to that of
n-heptadecane (C 17 ) and n-nonadecane (C 19 ), or about 44 kJ/mol.
L = T m dS ≈ 44 kJ/mol
(1.4.8)
If we assume that the thickness of the frozen layer to be monomolecular [16,
29], then we can calculate the head group area (α) of n-octadecane molecule from
dividing the molecular volume (v m ) by the molecular length (l ≈ 2.6 nm):
α = v m /l = M w /ρN A l ≈ 2.1 × 10
−19 m
2
(1.4.9)
where M w is the molar weight (254.5 g/mol) [32], ρ is the density (0.78 g/cm
3 ) [32],
and N A is the Avogadro number. From Eq. (1.4.9), we can calculate the number of
n-octadecane molecules in a unit surface area (1 m
2 ) as
α
−1
= 4.76 × 10
18 m
−2
(1.4.10)
27
alkanes above T sf is positive and small in the absolute values, as expected of any
ordinary liquid. The negative surface excess entropy between the melting point and
the surface freezing point shows that the surface is more ordered than the bulk,
consistent with the “frozen” state of the monolayer and the disordered state of the
underlying liquid bed. For n-octadecane, for example, the surface excess entropy
below the T sf , S surface can be calculated using our previous data [25]:
S surface = −∂γ/∂ T ≈ −1
mJ/m
2
· K
−1
≈ −1 × 10
−3 N · K
−1 m
−1
(1.4.7)
Figure (1.7) shows that the surface layer could be subcooled to below T m of noctadecane (T m ≈ 28.1 °C ≈ 301 K) and the small negative slope of ∂γ /∂T continues
(the positive surface excess entropy remains constant) until the nucleation for the
surface freezing transition eventually takes place at the points indicated by the vertical
arrows [25]. The positive surface excess entropy remains constant below T sf before
the eventual transition is consistent with the subcooled and disordered liquid surface
below T sf .
Meanwhile, the bulk heat of fusion of n-heptadecane (C 17 ), n-octadecane (C 18 ),
n-nonadecane (C 19 ), and n + eicosane (C 20 ) are 45 kJ/mol, 60 kJ/mol, 43.3 kJ/mol,
and 45.6 kJ/mol, respectively [22, 31]. Many normal alkanes of intermediate chain
lengths transition from liquid to a bulk rotator phase [16] at T m , as we noted above. A
separate phase transition between the bulk rotator phase and the bulk crystalline phase
takes place a few Kelvins below T m of each normal alkane. However, n-octadecane is
a special case and happens to undergo the liquid–rotator and the rotator–crystalline
transitions at the same temperature of T m [19], and consequently its bulk latent
heat of fusion is larger than that of the other normal alkanes. In other words, the
larger latent heat of fusion observed for n-octadecane can be attributed to the latent
heat of rotator–crystalline transition, which is required to provide thermal energy
to the molecules to rotate around their lattice positions. The bulk heat of fusion of
liquid–rotator phase transition of n-octadecane is expected to be similar to that of
n-heptadecane (C 17 ) and n-nonadecane (C 19 ), or about 44 kJ/mol.
L = T m dS ≈ 44 kJ/mol
(1.4.8)
If we assume that the thickness of the frozen layer to be monomolecular [16,
29], then we can calculate the head group area (α) of n-octadecane molecule from
dividing the molecular volume (v m ) by the molecular length (l ≈ 2.6 nm):
α = v m /l = M w /ρN A l ≈ 2.1 × 10
−19 m
2
(1.4.9)
where M w is the molar weight (254.5 g/mol) [32], ρ is the density (0.78 g/cm
3 ) [32],
and N A is the Avogadro number. From Eq. (1.4.9), we can calculate the number of
n-octadecane molecules in a unit surface area (1 m
2 ) as
α
−1
= 4.76 × 10
18 m
−2
(1.4.10)