142 Chapter 5. Life in the slow lane: the low Reynolds-number world[[Student version, December 8, 2002]]
5.1 Friction in fluids
First let’s see how the friction formulavdrift=f/ζ(Equation 4.12 on page 107) tells us how to
sort particles by their weight or electric charge, an eminently practical laboratory technique. Then
we’ll look at some odd but suggestive phenomena in viscous liquids like honey. Section 5.2 will
argue that in the nanoworld, water itself acts as a very viscous liquid, so that these phenomena are
actually representative of the physical world of cells.
5.1.1 Sedimentation separates particles by density
If we suspend a mixture of several particle types in water, for example several proteins, then gravity
pulls on each particle with a forcemgproportional to its mass. (If we prefer we can put our mixture
in a centrifuge, where the centrifugal “force”mg′is again proportional to the particle mass, though
g′can be much greater than ordinary gravity.)
Thenetforce propelling the particle downward is less thanmg,since in order for it to go down,
an equal volume of water must moveup. Gravity pulls on the water, too, with a force (Vρm)g,
whereρmis the mass density of water andVthe volume of the particle. Thus, when the particle
moves downward a distancex,displacing an equal volume of water up a distancex,the total change
in gravitational potential energy isU(x)=−mgx+Vρmgx.The net force driving sedimentation is
then the derivativef=−dU/dx=(m−Vρm)g,which we’ll abbreviate asmnetg.All we have done
so far is to derive Archimedes’ principle: The net weight of an object under water gets reduced by
abuoyant forceequal to the weight of the water displaced by the object.
What happens after we let a suspension settle for a very long time? Won’t all the particles
just fall to the bottom? Pebbles would, but colloidal particles smaller than a certain size won’t,
for the same reason that the air in the room around you doesn’t: Thermal agitation creates an
equilibrium distribution with some particles constantly off the bottom. To make this precise, letx
bethe distance from the floor of a test tube filled to a heighthwith a suspension. In equilibrium
the profile of particle densityc(x)has stopped changing, so we can apply the argument that led
to the Nernst relation (Equation 4.25 on page 126), replacing the electrostatic force by the net
gravitational force =mnetg.Thusthe density of particles in equilibrium is
c(x)∝e−mnetgx/kBT. sedimentation equlibrium, Earth’s gravity (5.1)Here are some typical numbers. Myoglobin is a globular protein, with molar massm ≈
17 000gmole−^1. The buoyant correction typically reducesmtomnet≈ 0. 25 mDefining thescale
heightasx∗≡kBTr/(mnetg)≈ 15 m,wethusexpectc(x)∝e−x/x∗.Thusina4cmtest tube,
in equilibrium, the concentration at the top equalsc(0)e−^0.^04 m/^15 m,or99.7% as great as at the
bottom. In other words,the suspension never settles out. In that case we call it an equilibrium
colloidal suspensionor justcolloid.Macromolecules like DNA or soluble proteins form colloidal
suspensions in water; another example is Robert Brown’s pollen grains in water. On the other
hand, ifmnetis big (for example, sand grains), then the density at the top will be essentially zero:
The suspension settles out. How big is “big?” Looking at Equation 5.1 shows that the gravitational
potential energy difference,mnetgh,between the top and bottom must be bigger than the thermal
energy for settling to occur.
Your Turn 5a
Here is another example. Suppose that the container is a carton of milk, withh=25cm.
Homogenized milk is essentially a suspension of fat droplets in water, tiny spheres of diameter