4.2 Interfacial Gaseous Layers
97
to accommodate the excess amount of gas would have been to slightly increase its
thickness. Given the very large aspect ratio (i.e., flat shape) of a micro gas pancake,
the increase in the total interfacial area when a given amount of excess gas is added to
the system would be minimized by increasing the thickness of the micro gas pancake
by a little bit. Then, the experimentally observed lateral spreading of a micro gas
pancake, which is contrary to the expectation, suggests that there must be a very high
free energy cost to increasing the thickness. This is to say that the disjoining pressure
must be a very steep function of the thickness of the gas film.
The regular occurrence of the Ostwald ripening between two interfacial nanobubbles that had been sitting on a single micro gas pancake first suggests that the gas that
was feeding the growing nanobubble must have come from the shrinking one. Then
it follows that the thickness of the underlying micro gas pancake could not have
been increasing (if the underlying micro gas pancake consumed the gas supplied
by the shrinking nanobubble, then where did the gas that was feeding the growing
nanobubble could have come from?). Second, the same observation suggests that the
thickness of the underlying micro gas pancake could not have been decreasing during
the Ostwald ripening either (if the gas that was feeding the growing nanobubble was
supplied by a shrinking micro gas pancake, then why did the smaller nanobubble
shrink at all?). If the thickness of the micro gas pancake can neither be increasing
nor decreasing, then we may conclude that its thickness must have been constant
during the Ostwald ripening between the two interfacial nanobubbles that had been
sitting on top of it.
The above thermodynamic considerations both suggest that a micro gas pancake
must have a very well-defined thickness. In other words, the disjoining pressure of
a micro gas pancake must be a very steep function of its thickness on both sides
of said well-defined thickness, with a minimum confined to a very narrow range
(a step function that has two steps over a very narrow range centered around the
experimentally observed thickness). Unfortunately, we have no idea as to a potential
mathematical format of the disjoining pressure of a micro gas pancake, let alone as
a function of its thickness. Nevertheless, the qualitative implications of the above
thermodynamic considerations are highly significant; they not only imply that (1)
the thickness of a given micro gas pancake must remain constant with time for as
long as the quasi-static condition is maintained, but also imply that (2) all micro
gas pancakes in the same system (under the same supersaturation of the gas and
were connected to each other through the surrounding aqueous media) must have
the same thickness. We can conclude as such despite our inability to determine a
mathematical expression of the disjoining pressure of the interfacial gaseous film or
to accurately measure its thickness, which illustrates the power and the elegance
of thermodynamics. Thermodynamics cannot, however, elucidate the underlying
molecular mechanisms that limited the thickness of the micro gas pancakes to such
a well-defined thickness.
As we saw in the previous (Sect. 4.1), the van der Waals forces across a micro
gas pancake (solid substrate–gas film–liquid) is expected to be attractive because the
refractive index of a gas must be lower than that of the underlying solid substrate or
that of the liquid on top [10, 11]. Such attractive van der Waals forces and the negative
97
to accommodate the excess amount of gas would have been to slightly increase its
thickness. Given the very large aspect ratio (i.e., flat shape) of a micro gas pancake,
the increase in the total interfacial area when a given amount of excess gas is added to
the system would be minimized by increasing the thickness of the micro gas pancake
by a little bit. Then, the experimentally observed lateral spreading of a micro gas
pancake, which is contrary to the expectation, suggests that there must be a very high
free energy cost to increasing the thickness. This is to say that the disjoining pressure
must be a very steep function of the thickness of the gas film.
The regular occurrence of the Ostwald ripening between two interfacial nanobubbles that had been sitting on a single micro gas pancake first suggests that the gas that
was feeding the growing nanobubble must have come from the shrinking one. Then
it follows that the thickness of the underlying micro gas pancake could not have
been increasing (if the underlying micro gas pancake consumed the gas supplied
by the shrinking nanobubble, then where did the gas that was feeding the growing
nanobubble could have come from?). Second, the same observation suggests that the
thickness of the underlying micro gas pancake could not have been decreasing during
the Ostwald ripening either (if the gas that was feeding the growing nanobubble was
supplied by a shrinking micro gas pancake, then why did the smaller nanobubble
shrink at all?). If the thickness of the micro gas pancake can neither be increasing
nor decreasing, then we may conclude that its thickness must have been constant
during the Ostwald ripening between the two interfacial nanobubbles that had been
sitting on top of it.
The above thermodynamic considerations both suggest that a micro gas pancake
must have a very well-defined thickness. In other words, the disjoining pressure of
a micro gas pancake must be a very steep function of its thickness on both sides
of said well-defined thickness, with a minimum confined to a very narrow range
(a step function that has two steps over a very narrow range centered around the
experimentally observed thickness). Unfortunately, we have no idea as to a potential
mathematical format of the disjoining pressure of a micro gas pancake, let alone as
a function of its thickness. Nevertheless, the qualitative implications of the above
thermodynamic considerations are highly significant; they not only imply that (1)
the thickness of a given micro gas pancake must remain constant with time for as
long as the quasi-static condition is maintained, but also imply that (2) all micro
gas pancakes in the same system (under the same supersaturation of the gas and
were connected to each other through the surrounding aqueous media) must have
the same thickness. We can conclude as such despite our inability to determine a
mathematical expression of the disjoining pressure of the interfacial gaseous film or
to accurately measure its thickness, which illustrates the power and the elegance
of thermodynamics. Thermodynamics cannot, however, elucidate the underlying
molecular mechanisms that limited the thickness of the micro gas pancakes to such
a well-defined thickness.
As we saw in the previous (Sect. 4.1), the van der Waals forces across a micro
gas pancake (solid substrate–gas film–liquid) is expected to be attractive because the
refractive index of a gas must be lower than that of the underlying solid substrate or
that of the liquid on top [10, 11]. Such attractive van der Waals forces and the negative