9780192806727.pdf

90 STATISTICAL PHYSICS

Of course, relative citation frequencies are no measure of relative importance.

Who has not aspired to write a paper so fundamental that very soon it is known

to everyone and cited by no one? It is nevertheless obvious that there must be valid

reasons for the popularity of Einstein's thesis. These are indeed not hard to find:

the thesis, dealing with bulk rheological properties of particle suspensions, con-

tains results which have an extraordinarily wide range of applications. They are

relevant to the construction industry (the motion of sand particles in cement mixes

[R4]), to the dairy industry (the motion of casein micelles in cow's milk [D4]),

and to ecology (the motion of aerosol particles in clouds [Y2]), to mention but a

few scattered examples. Einstein might have enjoyed hearing this, since he was

quite fond of applying physics to practical situations.

Let us consider Einstein's Doktorarbeit in some detail. His first step is hydro-
dynamic. Consider the stationary flow of an incompressible, homogeneous fluid.
If effects of acceleration are neglected, then the motion of the fluid is described by
the Navier-Stokes equations:

where v is the velocity, p the hydrostatic pressure, and 77 the viscosity. Next, insert

a large number of identical, rigid, spherical particles in the fluid. The radius of

the solute particles is taken to be large compared with the radius of the solvent

molecules so that the solvent can still be treated as a continuum. The solution is

supposed to be dilute; the total volume of the particles is much smaller than the

volume of the liquid. Assume further that (1) the overall motion of the system is

still Navier-Stokes, (2) the inertia of the solute particles in translation and their

rotational motion can be neglected, (3) there are no external forces, (4) the motion

of any one of the little spheres is not affected by the presence of any other little

sphere, (5) the particles move under the influence of hydrodynamic stresses at

their surface only, and (6) the boundary condition of the flow velocity v is taken

to be v = 0 on the surface of the spheres. Then, Einstein showed, the flow can

still be described by Eq. 5.6 provided rj is replaced by a new 'effective viscosity'

77*, given by

where <p is the fraction of the unit volume occupied by the (uniformly distributed)

spheres. Let the hard spheres represent molecules (which do not dissociate). Then

where ./V is Avogadro's number, a the molecular radius, m the molecular weight

of the solute, and p the amount of mass of the solute per unit volume. Einstein

had available to him values for rf/rj for dilute solutions of sugar in water, and <p