In a good solvent,Rn^3 =^5 (5.16)and we can obtain theKirkwood-Riseman equation(Kirkwood and Riseman 1948 )
½n^4 =^5 (5.17)Staudinger and Notzu considered thefree-draining mode(Staudinger and Nodzu
1930 ), as shown in Fig.5.1b. In this mode, the relative motion of each monomer
incurs the frictional force from its surrounding solvents, which is also quite
applicable to the situation of rigid polymer chains. Accordingly,
½n (5.18)A general equation summarizing all the cases above is known asMark-Houwink
equation
½¼KMa (5.19)wherearanges from 0.5 to 1. From this equation, one can define the viscosity-
average molecular weight, as introduced in Sect.2.4.2(Mark 1938 ; Houwink
1940 ).
5.2 Short Chains
The characteristic feature for the motions of chain-like molecules is that, the
Brownian motion of the whole chain is integrated by the Brownian motion of all
the monomers. Since the motion of each monomer is restrained by the chain
connection of other monomers, the Brownian motion of the polymer as a whole
is slower than the small monomer molecules under comparable conditions. In other
words, the diffusion coefficient of polymers strongly depends on the chain length.
In 1953, Rouse proposed that the ideal chain without volume exclusion (i.e. in
the free-draining mode) could be treated as a coarse-grained bead-spring chain, as
shown in Fig.5.2(Rouse 1953 ). Each spring connecting two beads represents a sub-
molecule along the ideal chain. Its elasticity can be described by the Gaussian
distribution of the distances between two ends of sub-molecules. Therefore, the
total free energy contributed by the entropic elasticity is given by
Eel¼SEi;iþ 1 Sðriþ 1 riÞ^2
<ðriþ 1 riÞ^2 >(5.20)
80 5 Scaling Analysis of Polymer Dynamics