凝聚态理论 (4).pdf

4 Physical Behavior of Colloidal Particles
Brownian Motion
Translational Brownian Motion
Small solid particles, about 10-4 cm in size, can be introduced into a drop of liq-
uid and then observed under a microscope. The particle is not at rest in the drop
but is seen to be constantly moving about in a highly irregular way. The motion
is a perpetual motion which seems to violate the second law of thermodynamics.
Although this phenomenon seems to be first reported by a Dutch physician Jan
Ingenhausz (Klafter et al., 1996), it is named as Brownian motion, since it was
first investigated and reported by the Scottish botanist Robert Brown in 1827 as a
physical phenomenon. However, he did not understand the origin of this phenom-
enon. Meanwhile Louis Bachelier developed a theory of Brownian motion in
1900 as a model of stock market fluctuation, but his work did not referred to the
physical nature. After all, it remained for Albert Einstein in 1905 and for Marian
Smoluchowski in 1906 to shed light into the phenomenon as due to random fluc-
tuations to be expected even in thermal equilibrium.
In order to avoid any confusion in the description, consider that small solid
particles are now in a large volume of liquid, which contains many liquid mole-
cules. In the frame fixed to the volume at an absolute temperature T, the mole-
cules move with thermal velocity v, which obeys Gaussian distribution function,
 3  
m 2 mv2
F…v†ˆ exp
…4:1†
2pkBT 2kBT
where m is the molecular mass and kB is Boltzmann's constant. This distribution
is stationary and symmetric about v = 0, where it has a pronounced peak. Thus,
the average vanishes all the time, but the variance is finite, given by
3k T
hv2iˆ B …4:2†
m
44 4 Physical Behavior of Colloidal Particles
Therefore, molecules must exhibit a fluctuating velocity due to mutual colli-
sions. A small foreign solid particle in the liquid is accordingly subject to a fluc-
force due to collisi