Yamakawa-Fujii theory

Fig. 22 Theoretical relations between Af2/[ j] and In L/d for rigid rods ellipsoid Simha equation (D-l) for equivalent ellipsoid of revolution Y-F Yamakawa-Fujii theory for straight cylinders. Dashed lines indicate asymptotes to respective solid curves...

The Kirkwood-Riseman theory [93] and the Zimm theory [98] as well predict that n in eq 3.3 under the 0 condition decreases monotonically from 1 to 0.5 with increasing M, and thus fail to derive eq 3.4. Again, we have to invoke complete immobilization of solvent inside the polymer coil to explain this relation by these theories. The Yamakawa-Fujii theory shows that chain stiffness also counteracts the draining effect on Do, but a d/q value different from that needed for [q]ff has to be used in order to obtain a maximum suppression of the draining effect. 

The Yamakawa-Fujii theory [2, 3] was developed by using the Kirkwood-Riseman formalism with the effect of chain thickness approximately taken into account. The following remarks may be in order. The Oseen interaction tensor was preaveraged. Force points were distributed along the centroid of the wormlike cylinder (not over the entire domain occupied by the cylinder). The no-slip hydrodynamic condition was approximated by equating the mean solvent velocity over each cross-section of the cylinder to the velocity of the cylinder at that cross-section (Burgers approximate boundary condition). 

The Yamakawa-Fujii theory neglects the friction on the end surfaces of the wormlike cylinder. Norisuye et al. [23] and Yoshizaki and Yeimakawa [25] estimated them by capping each end of a wormlike cylinder with a hemisphere and a straight spheroid, respectively. The results showed negligible effects on / unless the axial ratio p of the cylinder is smaller than 10, implying that the Yamakawa-Fujii theory of / is applicable down to vety low M. 

Importantly, Yoshizaki and Yamakawa [25] found that, in contrast to /, [77] of a wormlike cylinder undergoes significant end surface effects until the axial ratio p reaches about 50, on the basis of numerical solutions to the Navier-Stokes equation with the no-slip boundary condition for spheroid cylinders, spheres, and prolate and oblate ellipsoids of rotation. They constructed an empirical interpolation formula for [ y] of a spheroid cylinder which reduces to eq 2.37 for p > 1 and to the Einstein value at p = 1. Then, with its aid, Yamakawa and Yoshizaki [4] formulated a modified theory of [77] for wormlike cylinders which agrees with the Yamakawa-Fujii theory [3] for Lj lq > 2.278 and with the Einstein value at Ljd = 1, regardless of dj2q smaller than 0.1. However, no formulation has as yet been made for L/2q < 2.278 and d/2q > 0.1, i.e., for short flexible cylinders. In what follows, the Yamakawa-Yoshizaki modification is referred to as the Yamakawa-Fujii-Yoshizaki theoiy. 

Bushin et al. [38] and independently Bohdanecky [39] showed that the Yamakawa-Fujii-Yoshizaki theory for [77] can be put in an approximate form as... 

Fig. 5-9. Fit of the Yamakawa-Fujii-Yoshizaki theory to [77] data [41] for PHIC in butyl chloride. Solid line, for Ml = 760 nm , = 35 run, and d = 1.5 nm. Dashed line, for rigid cylinders with Ml = 760 nm and d = 1.5 nm.

The best value of d consistent with the [7/] data was found to be 1.6 ( 0.2) nm. The solid line A in Figure 5-10, calculated with q = A2 nm, Ml = 715 nm, and c/ — 1.6 nm, is seen to fit the data points veiy accurately. Its close fit even for Mw above 3x10 appears to indicate that [77] undeigoes less excluded-volume effect than does (5 ) (compare with Figure 5-3). However, this result is an accident due to the defect of the Yamakawa-Fujii-Yoshizaki theory which overestimates [77] near the coil limit. 

We further analyzed the [ ] data in terms of the Yamakawa-Fujii-Yoshizaki (YFY) theory originally developed for worm-like chains [67,68]. According to Bohdanecky [69], the YFY theory could be cast in a much simpler form... 

Yamakawa, H., Modern Theory of Polymer Solutions. New York Harper Row, 1971. Yamakawa, H., and M. Fujii, Macromolecules 7, 128 (1974). 

Further calculations were also performed on values of GPC "slices" using the persistance length theory of YAMAKAWA and FUJII. A best fit was obtained using this model... 

For KP chains, Yamakawa and Fujii numerically obtained [//] values, but the result turned out to be incomplete for short wormlike cylinders for which effects from cylinder ends are significant. Yamakawa and YoshizaM later modified the theory so as to give the Einstein equation for rigid spheres in the limit of L/d = 1, that is,... 

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