Polymer Physics

convective constraint release in the bulk phase, inducing disentanglement. There-
fore, the motion of single polymer coils in shear flow can be vividly described as
end-over-end tumbling. Chu and coworkers have observed such kind of motion by
means of fluorescence-labeled DNA macromolecules (Smith et al. 1999 ).
Apparently, the extensional flow causes the deformation of polymer coils. The
simplestdumb-bell modelcan be used to describe this deformation. As illustrated in
Fig.7.9, two beads with a distanceR, and the entropic elastic recovery force upon
deformation of polymer coils is

fe¼KR (7.12)

where the modulusK/kTR 2
0

. The driving forces of the flow are balanced with the
frictional drag forcesff, bringing the deformation rate∂R/∂t, with the proportion
factor as the frictional coefficientz,

ff¼B

@R

@t

(7.13)

From the force equilibrium,

fe¼ff (7.14)

one obtains

B

@R

@t

¼KR (7.15)

The solution of (7.15) gives

R¼R 0 expð

t

t

Þ (7.16)

where the characteristic relaxation timet¼KB.
De Gennes provided a scaling analysis on the deformation transition of polymer
coils under elongational flows (De Gennes 1979 ). Under one-dimensional
stretching, the strain rate of polymer chain

s¼

@vx

@x

> 0 (7.17)

Fig. 7.9 Illustration of the dumb-bell model for the deformation of polymer coils

136 7 Polymer Flow