Page 270 - Bird R.B. Transport phenomena

254 Chapter 8 Polymeric Liquids

Fig. 8.6-2. Single-molecule bead spring models for
(a) a dilute polymer solution, and (b) an undiluted
polymer (a polymer "melt" with no solvent). In
the dilute solution, the polymer molecule can
move about in all directions through the solvent.
In the undiluted polymer, a typical polymer mole-
cule (black beads) is constrained by the surround-
ing molecules and tends to execute snakelike
motion ("reptation") by sliding back and forth
along its backbone direction.

because of the proximity of the surrounding molecules, it is easier for the "beads" of the
model to move in the direction of the polymer chain backbone than perpendicular to it.
In other words, the polymer finds itself executing a sort of snakelike motion, called "rep-
tation" (see Fig. 8.6-2b).
As an illustration of the kinetic theory approach we discuss the results for a simple
system: a dilute solution of a polymer, modeled as an elastic dumbbell consisting of two
beads connected by a spring. We take the spring to be nonlinear and finitely extensible,
with the force in the connecting spring being given by 4

HQ
c)
F< = (8.6-1)
- (Q/Qo) 2
in which H is a spring constant, Q is the end-to-end vector of the dumbbell representing
the stretching and orientation of the dumbbell, and Q o is the maximum elongation of the
spring. The friction coefficient for the motion of the beads through the solvent is given
by Stokes' law as £ = бтгт^я, where a is the bead radius and r/ is the solvent viscosity. Al-
s
though this model is greatly oversimplified, it does embody the key physical ideas of
molecular orientation, molecular stretching, and finite extensibility.
When the details of the kinetic theory are worked out, one gets the following expres-
sion for the stress tensor, written as the sum of a Newtonian solvent and a polymer con-
tribution (see fn. 3 in §8.4): 5

T = T s (8.6-2)
Here

(8.6-3)
- Х (т р - пкТЬ) = -пкТк у (8.6-4)
н
и
where n is the number density of polymer molecules (i.e., dumbbells), A H = £/4H is a
time constant (typically between 0.01 and 10 seconds), Z = 1 + (3/b)[l - (tr т /ЗпкТ)],
р
and b = HQl/кТ is the finite extensibility parameter, usually between 10 and 100. The

4
H. R. Warner, Jr., bid. Eng. Chem. Fundamentals, 11, 379-387 (1972); R. L. Christiansen and
R. B. Bird, J. Non-Newtonian Fluid Mech., 3,161-177 (1977/1978).
R. I. Tanner, Trans. Soc. RheoL, 19, 37-65 (1975); R. B. Bird, P. J. Dotson, and N. L. Johnson, /. Non-
5
Newtonian Fluid Mech., 7,213-235 (1980)—in the last publication, Eqs. 58-85 are in error.