4.1. Brownian motion[[Student version, December 8, 2002]] 107
4.1.4 Friction is quantitatively related to diffusion
Diffusion is essentially a question ofrandom fluctuations:knowing where a particle is now, we seek
the spread in its expected position at a later timet. Section 3.2.5 argued qualitatively that the
samerandom collisions responsible for this spread also give rise to friction. So we should be able to
relate the microscopic quantitiesLand ∆tto friction, another experimentally measurable quantity.
As usual, we’ll make some simplifications to get to the point quickly. For example, we’ll again
consider an imaginary world where everything moves only in one dimension.
Tostudy friction, we want to consider a particle pulled by a constant external forcef.For
example,fcould be the forcemgof gravity, or the artificial gravity inside a centrifuge. We want to
know the average motion of each particle as it falls in the direction of the force. In first-year physics
youprobably learned that a falling body eventually comes to a “terminal velocity” determined by
friction. Let’s investigate the origin of friction, in the case of a small body suspended in fluid.
In the same spirit as Section 4.1.1, we’ll suppose that the collisions occur exactly once per ∆t
(though really there is a distribution of times between collisions). In between kicks, the particle
is free of random influences, so it is subject to Newton’s Law of motion, ddvt = mf;its velocity
accordingly changes with time asv(t)=v 0 +ft/m,wherev 0 is the starting value just after a kick
andmis the mass of the particle. The resulting uniformly accelerated motion of the particle is
then
∆x=v 0 ∆t+^12 mf(∆t)^2. (4.11)
Following Section 4.1.1, assume that each collision obliterates all memory of the previous step.
Thus, after each step,v 0 is randomly pointing left or right, so its average value,〈v 0 〉,equals zero.
Taking the average of Equation 4.11 thus gives〈∆x〉= 2 fm(∆t)^2 .Inother words the particle, while
buffeted about by random collisions, nevertheless acquires a netdrift velocityequal to〈∆x〉/∆t,
or
vdrift=f/ζ, (4.12)
where
ζ=2m/∆t. (4.13)
Equation 4.12 shows that, under the assumptions made, a particle under a constant force indeed
comes to a terminal velocity proportional to the force. Theviscous friction coefficientζ,like the
diffusion constant, is experimentally measurable—we just look through a microscope and see how
fast a particle settles under the influence of gravity, for example.^4
Actually, in practice it’s often not necessary to measureζdirectly: The viscous friction coefficient
for a spherical object is related to itssizebyasimple empirical relation, theStokes formula:
ζ=6πηa. Stokes formula (4.14)In this expression,ais the radius of the particle andηis the “viscosity” of the fluid. Chapter 5
will discuss viscosity in greater detail; for now we only need to know that the viscosity of water at
room temperature is about 10−^3 kg m−^1 s−^1 .It’s generally easier to measure the size of a colloidal
particle (by looking at it), then use Equation 4.14, than it is to measure the particle’s weightfand
use Equation 4.12 to getζ.
(^4) Many authors instead speak in terms of the “mobility,” which equals 1/ζ.