A B C D
C. Imitation of opal
Since Einstein was addressing a more general question than
the color of the atmosphere, his results have a wider interest.
Smoluchowski had originally considered a more dramatic
effect on light, brought on by fluids in a very different state. We've
been discussing gases and have mentioned liquids. In our usual experience,
a fluid is generally either one or the other, depending on whether its
temperature is above or below its boiling point-for example, water is a dense
liquid below 100oC and a much less dense gas above. But if we
increase the pressure on a fluid, we find that its boiling point rises and the
density difference between its liquid and gaseous forms decreases. Once
the pressure is high enough, the "boiling point" becomes almost
meaningless because the density difference completely disappears. Any
light waves passing through the fluid under these conditions, whether the waves'
frequencies are high or low, turn out to be strongly scattered. This
effect is called critical opalescence. The "critical"
refers to the state of the fluid, at or near the exact pressure and temperature
that gas and liquid become the same; "opalescence" refers to the
scattered light's resemblance to the light scattered by opals.
Smoluchowski showed that near these conditions, a slight
expenditure of energy could drastically change the fluid's density. Huge
density fluctuations throughout the fluid are therefore practically
inevitable. These huge density variations would cause huge variations in
the speed of any light passing through them, thus producing the strong
light-scattering of critical opalescence. Einstein's effort to calculate
the exact effects on the light waves revealed a connection between critical
opalescence and the appearance of the daylit sky. These phenomena are
different, but both involve the scattering of light due to variations in fluid
density.
While Einstein's analysis was adequate for understanding
the light from the sky, it was actually limited in its applicability to critical
opalescence itself, because it didn't account for the following. If the
molecules in one part of a fluid move together into a smaller volume to make the
fluid more dense there, they have to come from the space around that volume,
thus making the fluid in that space less dense, unless other molecules
from elsewhere take their place. In other words, spontaneous compression
in one place means spontaneous expansion in another place; density fluctuations
in different parts of the fluid are interdependent. Since the molecules
ordinarily don't travel very far in a short time, the interdependence will
mostly be between small regions of fluid close to one another, and the effect of
this interdependence on light scattering is small. But near the fluid's
critical state, the density variations are much larger, and the molecules travel
much greater distances going from one region to another. A density change
in one part of the fluid can thus affect other parts of the fluid much further
away, and omitting the effect of this on light makes Einstein's analysis quite
inaccurate for critical opalescence itself.
In analyzing the color of the sky, though, Einstein was
able to go further than Rayleigh, by using information about fluid density to
indirectly account for how fluids' molecules are distributed. A few
decades after Einstein, Zimm again considered molecules directly as Rayleigh had
done, but this time with new mathematical techniques to account for the
likelihoods of different molecular distributions. One difference between
these likelihoods and Einstein's likelihoods for density variations is that the
newer techniques implicitly account for the way density changes in different
locations balance out. Zimm's more general analysis implies Einstein's
formula whenever a density change in one region of fluid has little effect
beyond that region's immediate neighborhood, but will give quite different
results otherwise.
(.....continued)
A B
C
D
