Polymer Physics

wherevxis the flow rate on the direction ofx. Assuming that the strain happens
homogeneously, the frictional drag force on the flow length dxis

ffBvxdxBsxdx (7.18)

Therefore, the work made by the frictional force is

Efffx

Rð= 2

R= 2

Bsx^2 dx

Rð= 2

R= 2

tKsx^2 dxstR^3 (7.19)

The entropy elasticity of real polymer coils in a good solvent is

EelkTð

R

R 0

Þ^5 =^2 (7.20)

Then the total free energy of the coil becomes

F¼EelþEfR^5 =^2 stR^3 (7.21)

As illustrated in Fig.7.10, there exists a critical value(st). Whenst<(st),
the free energy of the extended coil appears higher than the random coil, then the
random coil is more stable; whenst>(st)*, the extended coil becomes more
stable. Therefore, the free energy barrier between the two coil states determines the
coil-stretch transition to be first-order like. Here the dimensionless value ofstcan
be namedDeborah number(De),

Dest (7.22)

Similar to the Weissenberg number in shear flow, the Deborah number reflects
the competition between the relaxation timetof the polymer coil and the working
times^1 imposed by the external field (Dealy 2010 ; Reiner 1964 ). The criticalDe*

R

F

sτ < sτ*

sτ = sτ*

sτ > sτ*

0 nb

Fig. 7.10 Illustration of the
free energy versus coil
dimension changing with the
product of extensional flow
ratesand the relaxation timet

7.2 Characteristics of Polymer Flow 137