5.4. Problems[[Student version, December 8, 2002]] 171
0.010.1104 105 1060.2molar mass,gmole−^1intrinsic viscosity,m
3 kg-^1
Figure 5.13:(Experimental data.) Log-log plot of the intrinsic viscosity [η]Θfor polymers with different values
for the molar massM.The different curves shown represent different combinations of polymer type, solvent type,
and temperature, all corresponding to “theta solvent” conditions.Open circles:polyisobutylene in benzene at 24◦C.
Solid circles:polystyrene in cyclohexane at 34◦C.The two solid lines each have logarithmic slope 1/2, corresponding
to a 1/2 power law. [Data from (Flory, 1953).]
Write an expression^6 for the relative change (η′−η)/η.
c. We want to explore the proposition that a polymerNsegments long behaves like a sphere with
radiusαLNγfor some powerγ. αis a constant of proportionality, whose exact value we won’t
need. What do we expectγto be? What then is the volume fractionφof a suspension ofcsuch
spheres per volume? Express your answer in terms of the total massMof a polymer, the massm
permonomer, the concentration of polymerc,andα.
d. Discuss the experimental data in Figure 5.13 in the light of your analysis. Each set of points
joined by a line represents measurements taken on a family of polymers with varying numbersN
of identical monomers, with each monomer having the same massm. The total massM=Nm
of each polymer is on thex-axis. The quantity [η]Θon the vertical axis is called the polymer’s
“intrinsic viscosity”; it is defined as (η′−η)/(ηρm,p), whereρm,pis the mass of dissolved polymer
pervolume of solvent. [Hint: Recallρm,pis small. Write everything in terms of the fixed segment
lengthLand massm,and the variable total massM.]
e. What combination ofLandmcould we measure from the data? (Don’t actually calculate it—it
involvesαtoo.)
5.9 T 2 Friction as diffusion
Section 5.2.1′on page 166 claimed that viscous friction can be interpreted as the diffusive transport
of momentum. The argument was that in the planar geometry, when the flux of momentum given
byEquation 5.20 leaves the top plate it exerts a resisting drag force; when it arrives at the bottom
plate it exerts an entraining force. So far the argument is quite correct.
Actually, however, viscous friction is more complicated than ordinary diffusion, because mo-
(^6) The expression you’ll get is not quite complete, due to some effects we left out, but its scaling is right whenφis
small. Einstein obtained the full formula in his doctoral dissertation, published it in 1906, then fixed a computational
error in 1911.