102
4 Interfacial Gaseous States
The left panel of Fig. 4.6 shows that, at T m , the solid phase and the liquid phase can
coexist in the bulk because their two free energy components are balanced. On the
surface, in contrast, the two free energy components are not balanced at T m , because
the molecules would lose the same amount of entropy as in the bulk by freezing
but do not lose as much bonding energy as in the bulk because not all the bonds
will be formed at the surface when the bulk of the material freezes. Consequently,
the liquid phase is favored at the surface, at and slightly below T m , and remains so
until T becomes so small that T S, which is smaller than T m S, becomes equal to
U surface . This is the thermodynamic source of pre-melting.
The right panel of Fig. 4.6 schematically shows the relative positions of the four
energy levels of the surface liquid, the bulk liquid, the surface solid, and the bulk
solid. For a hexagonal close-packed structure, for example, an atom in the bulk solid
has 12 bonds and an atom at the surface of the solid has 6 bonds. In a special case
that each atom on average loses 2/3 of its bonds upon melting, an atom in the bulk
liquid will have 4 bonds and an atom at the surface of the liquid will have 2 bonds.
Consequently, U surface, liquid > U bulk, liquid > U surface, solid > U bulk, solid (i.e., −2 > −4 >
−6 > −12, where the zero energy level corresponds to a free atom without any bond),
as shown in the figure. For a material whose molten liquid retains unusually high
number densities of the bonds (retains high degrees of structural order), the order
of the middle two levels may be altered. For example, in a special case that each
atom in a hexagonal close-packed crystal on average only loses 1/3 of its bonds upon
melting, an atom in the bulk liquid will have 8 bonds and an atom at the surface of the
liquid will have 4 bonds. Consequently, in U surface, liquid > U surface, solid > U bulk, liquid >
U bulk, solid (i.e., −4 > −6 > −8 > −12, where the zero energy level corresponds to a
free atom without any bond). From now on, we only consider the former case.
At a temperature below T m , the solid phase is favored to the liquid phase for both
the bulk and the surface if it were not for the entropic component. The inclusion of
the entropic term would lower the free energy of both the surface liquid and the bulk
liquid by T S, which is somewhat smaller than T m S, because T is supposed to
be lower than T m . For small subcoolings, T S can be smaller than U bulk but still
can be larger than U surface (the left panel of Fig. 4.6). Then, in the right panel of
Fig. 4.6, the inclusion of the T S term will bring the free energy of the surface liquid
down to below the free energy of the surface solid but still leave the free energy of
the bulk liquid above that of the bulk solid. In our usual favorite scale of the rather
artificial example of the hexagonal close-packed crystal, this situation corresponds
to T S being larger than 4 (i.e., a shift from −2 to −6) but smaller than 8 (i.e.,
a shift from −4 to −12). Thus, the solid phase can be more stable in the bulk but
the liquid phase can be more stable at the surface at the same temperature for small
subcoolings (i.e., pre-melting is favored). For a more quantitative treatment of the
subject, the thickness dependence of the pre-melting layer will need to be accounted
for [40, 41]. A qualitative conclusion is that the thickness of the pre-melting layer
diminishes with subcooling as the free energy cost per unit mass of retaining a liquid
phase progressively increases with subcooling.
4 Interfacial Gaseous States
The left panel of Fig. 4.6 shows that, at T m , the solid phase and the liquid phase can
coexist in the bulk because their two free energy components are balanced. On the
surface, in contrast, the two free energy components are not balanced at T m , because
the molecules would lose the same amount of entropy as in the bulk by freezing
but do not lose as much bonding energy as in the bulk because not all the bonds
will be formed at the surface when the bulk of the material freezes. Consequently,
the liquid phase is favored at the surface, at and slightly below T m , and remains so
until T becomes so small that T S, which is smaller than T m S, becomes equal to
U surface . This is the thermodynamic source of pre-melting.
The right panel of Fig. 4.6 schematically shows the relative positions of the four
energy levels of the surface liquid, the bulk liquid, the surface solid, and the bulk
solid. For a hexagonal close-packed structure, for example, an atom in the bulk solid
has 12 bonds and an atom at the surface of the solid has 6 bonds. In a special case
that each atom on average loses 2/3 of its bonds upon melting, an atom in the bulk
liquid will have 4 bonds and an atom at the surface of the liquid will have 2 bonds.
Consequently, U surface, liquid > U bulk, liquid > U surface, solid > U bulk, solid (i.e., −2 > −4 >
−6 > −12, where the zero energy level corresponds to a free atom without any bond),
as shown in the figure. For a material whose molten liquid retains unusually high
number densities of the bonds (retains high degrees of structural order), the order
of the middle two levels may be altered. For example, in a special case that each
atom in a hexagonal close-packed crystal on average only loses 1/3 of its bonds upon
melting, an atom in the bulk liquid will have 8 bonds and an atom at the surface of the
liquid will have 4 bonds. Consequently, in U surface, liquid > U surface, solid > U bulk, liquid >
U bulk, solid (i.e., −4 > −6 > −8 > −12, where the zero energy level corresponds to a
free atom without any bond). From now on, we only consider the former case.
At a temperature below T m , the solid phase is favored to the liquid phase for both
the bulk and the surface if it were not for the entropic component. The inclusion of
the entropic term would lower the free energy of both the surface liquid and the bulk
liquid by T S, which is somewhat smaller than T m S, because T is supposed to
be lower than T m . For small subcoolings, T S can be smaller than U bulk but still
can be larger than U surface (the left panel of Fig. 4.6). Then, in the right panel of
Fig. 4.6, the inclusion of the T S term will bring the free energy of the surface liquid
down to below the free energy of the surface solid but still leave the free energy of
the bulk liquid above that of the bulk solid. In our usual favorite scale of the rather
artificial example of the hexagonal close-packed crystal, this situation corresponds
to T S being larger than 4 (i.e., a shift from −2 to −6) but smaller than 8 (i.e.,
a shift from −4 to −12). Thus, the solid phase can be more stable in the bulk but
the liquid phase can be more stable at the surface at the same temperature for small
subcoolings (i.e., pre-melting is favored). For a more quantitative treatment of the
subject, the thickness dependence of the pre-melting layer will need to be accounted
for [40, 41]. A qualitative conclusion is that the thickness of the pre-melting layer
diminishes with subcooling as the free energy cost per unit mass of retaining a liquid
phase progressively increases with subcooling.