1.4 Nucleation of Two-Dimensional Solid Layers
29
critical surface free energy values [23]. On an ordinary foreign substrate, cohesion
within each layer (CH 2 –CH 2 ) is stronger than the adhesion between the layer and the
underlying foreign substrate. In these cases, before the entire surface monolayer of
the liquid solidifies, a two-dimensional monolayer-nucleus (disk) inevitably exposes
around itself a higher energy component of methylene groups, leading to a finite
(non-zero) surface subcooling which is comparable to the bulk subcooling [25]. This
is equivalent to suggest that the line tension around the two-dimensional monolayer
nucleus is responsible for the activation process involved in surface freezing.
We will further postulate that there will be another energy barrier against subsequent bulk freezing even after the entire top monolayer has frozen, and as such, a
second activation process (subcooling) between the frozen surface monolayer and
the bulk freezing may be required. For non-polar compounds such as n-alkanes, the
intermolecular interactions are dominated by the van der Waals forces, as described
by the Lifshitz theory. The solid phase of n-alkanes has a higher density [22] and
hence, according to the Lorentz–Lorenz relationship, a higher refractive index than
the liquid phase. Thus, the Hamaker constant is expected to be positive for the (liquid–
solidified film–vapor) system [36]. This leads to an important conclusion that the van
der Waals interaction across the solid film is attractive. The thickness of such a film
would be limited to a small value until a sufficient subcooling takes place. This is
an example of a negative disjoining pressure (positive chemical potential) which we
will detail in Chap. 4.
The first-order continuum approximation we used above, though correctly capture
the essence, neglects potentially important thickness dependence of the specific interfacial free energy of the film. To quantitatively estimate the propagation of freezing
from the top frozen surface monolayer into the underlying bulk liquid bed, a proper
account for the long-range surface forces will be essential, which is beyond the scope
of this book. Here, we only qualitatively show what is expected once the thickness
dependence of the surface free energy is taken into account. Again, an analogy of a
reversed system may be helpful here.
It is well established that two solid walls that are interacting across a thin liquid
film experience the so-called oscillatory solvation forces, the period of which reflects
the size of the molecules of the thin films confined in between [37]. A schematic
illustration is shown in Fig. (1.9) in which the wall separation is expressed in the
number of diameters of the confined molecules. Here, the height and the depth of
each neighboring repulsive and attractive peaks correspond to the energy barrier
that needs to be overcome in order to increase or decrease the wall separation (e.g.,
when one starts with an initial surface separation of 2 molecular diameters, then
the activation barrier to decreasing the surface separation to 1 molecular diameter is
much greater than the activation barrier required to increasing the surface separation
of 3).
Generally, the amplitude of such an oscillation is large at a small surface separation
and rapidly diminishes as the surface separation increases, and such oscillation is
superimposed on top of the continuum dependence [37]. The underlying physics
is that the disjoining pressure of the thin liquid film exhibits oscillatory behavior.
Simply put, the system free energy is low when the surface separation is a multiple
29
critical surface free energy values [23]. On an ordinary foreign substrate, cohesion
within each layer (CH 2 –CH 2 ) is stronger than the adhesion between the layer and the
underlying foreign substrate. In these cases, before the entire surface monolayer of
the liquid solidifies, a two-dimensional monolayer-nucleus (disk) inevitably exposes
around itself a higher energy component of methylene groups, leading to a finite
(non-zero) surface subcooling which is comparable to the bulk subcooling [25]. This
is equivalent to suggest that the line tension around the two-dimensional monolayer
nucleus is responsible for the activation process involved in surface freezing.
We will further postulate that there will be another energy barrier against subsequent bulk freezing even after the entire top monolayer has frozen, and as such, a
second activation process (subcooling) between the frozen surface monolayer and
the bulk freezing may be required. For non-polar compounds such as n-alkanes, the
intermolecular interactions are dominated by the van der Waals forces, as described
by the Lifshitz theory. The solid phase of n-alkanes has a higher density [22] and
hence, according to the Lorentz–Lorenz relationship, a higher refractive index than
the liquid phase. Thus, the Hamaker constant is expected to be positive for the (liquid–
solidified film–vapor) system [36]. This leads to an important conclusion that the van
der Waals interaction across the solid film is attractive. The thickness of such a film
would be limited to a small value until a sufficient subcooling takes place. This is
an example of a negative disjoining pressure (positive chemical potential) which we
will detail in Chap. 4.
The first-order continuum approximation we used above, though correctly capture
the essence, neglects potentially important thickness dependence of the specific interfacial free energy of the film. To quantitatively estimate the propagation of freezing
from the top frozen surface monolayer into the underlying bulk liquid bed, a proper
account for the long-range surface forces will be essential, which is beyond the scope
of this book. Here, we only qualitatively show what is expected once the thickness
dependence of the surface free energy is taken into account. Again, an analogy of a
reversed system may be helpful here.
It is well established that two solid walls that are interacting across a thin liquid
film experience the so-called oscillatory solvation forces, the period of which reflects
the size of the molecules of the thin films confined in between [37]. A schematic
illustration is shown in Fig. (1.9) in which the wall separation is expressed in the
number of diameters of the confined molecules. Here, the height and the depth of
each neighboring repulsive and attractive peaks correspond to the energy barrier
that needs to be overcome in order to increase or decrease the wall separation (e.g.,
when one starts with an initial surface separation of 2 molecular diameters, then
the activation barrier to decreasing the surface separation to 1 molecular diameter is
much greater than the activation barrier required to increasing the surface separation
of 3).
Generally, the amplitude of such an oscillation is large at a small surface separation
and rapidly diminishes as the surface separation increases, and such oscillation is
superimposed on top of the continuum dependence [37]. The underlying physics
is that the disjoining pressure of the thin liquid film exhibits oscillatory behavior.
Simply put, the system free energy is low when the surface separation is a multiple