4.3. Other random walks[[Student version, December 8, 2002]] 111
10 -710 -6105 106diffusion constant,cm2 s-1molar mass, g/molea
10 -1310 -12105 106 107sedimentation time scale,smolar mass, g/moleb
Figure 4.7:(Experimental data with fits.) (a)Log-log plot of the diffusion constantDof polymethyl methacrylate
in acetone, as a function of the polymer’s molar massM.The solid line corresponds to the functionD∝M−^0.^57.
Forcomparison, the dashed line shows the best fit with scaling exponent fixed to− 1 /2, which is the prediction of
the simplified analysis in this chapter. (b)The sedimentation time scale of the same polymer, to be discussed in
Chapter 5. The solid line corresponds to the functions∝m^0.^44 .For comparison, the dashed line shows the best fit
with scaling exponent fixed to− 1 /2. [Data from (Meyerhoff & Schultz, 1952).]
the molar mass of each unit, so we predict that
If we synthesize polymers made from various numbers of the same units, then
the coil size increases proportionally as the square root of the molar mass.(4.16)
Figure 4.7a shows the results of an experiment in which eight batches of polymer, each with a
different chain length, were synthesized. The diffusion constants of dilute solutions of these polymers
were measured. According to the Stokes formula (Equation 4.14 on page 107),Dwill be a constant
divided by the radius of the polymer blob; Idea 4.16 then leads us to expect thatDshould be
proportional toM−^1 /^2 ,roughly as seen in the experimental data.^7
Figure 4.7 also illustrates an important graphical tool. If we wish to show thatDis a constant
timesM−^1 /^2 ,wecould try graphing the data and superimposing the curvesD=AM−^1 /^2 for various
values of the constantA,and seeing whether any of them fit. A far more transparent approach is to
plot instead (logD)versus (logM). Now the different predicted curves (logD)=(logA)−^12 (logM)
are allstraight lines of slope−^12 .Wecan thus test our hypothesis by laying a ruler along the observed
data points, seeing whether they lie on any straight line, and, if so, finding the slope of that line.
One consequence of Idea 4.16 is that random-coil polymers are loose structures. To see this note
that, if each unit of a polymer takes up a fixed volumev,then packingNunits tightly together
would yield a ball of radius (3Nv/ 4 π)^1 /^3 .For large enough polymers (Nlarge enough), this size
will be smaller than the random-coil size, sinceN^1 /^2 increases more rapidly thanN^1 /^3.
Wemade a number of expedient assumptions to arrive at Idea 4.16. Most importantly, we
assumed that every polymer unit is equally likely to occupy all the spaces adjacent to its neighbor
(the eight corners of the cube in the idealization of Figure 4.6). One way this assumption could fail
is if the monomers are strongly attracted to each other; in that case the polymer will not assume a
random-walk conformation, but rather will pack itself tightly into a ball. Examples of this behavior
(^7) See Section 5.1.2 and Problem 5.8 for more about random-coil sizes.