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# Polymer Chemistry. The Basic Concepts - Himenz P.C.

Himenz P.C. Polymer Chemistry. The Basic Concepts - Copyright, 1984. - 736 p.
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is, toward the circumference of the rotor-if the particle density is
greater than that of the supporting medium. A distance r from the axis of
rotation, the radial acceleration is given by co2r, where ñî is the
angular velocity in radians per second. The midpoint of an
ultracentrifuge cell is typically about 6.5 cm from the axis of rotation,
so at 10,000, 20,000, and 40,000 rpm, respectively, the accelerations are
7.13 X 104, 2.85 X 10s, and 1.14 X 106 m sec"2 or 7.27 X 103, 2.91 X
104,and 1.16 X 10s times the acceleration of gravity (g's).
The ultracentrifuge has been used extensively, especially for the
study of biopolymers, and can be used in several different experimental
modes to yield information about polymeric solutes. Of the possible
procedures, we shall consider only sedimentation velocity and
sedimentation equilibrium. We shall discuss these in turn, beginning with
an examination of the forces which operate on a particle setting under
stationary-state conditions.
Three kinds of forces must be considered:
1. The force of a molecule subject to radial acceleration is given by
Newton's second law:
636
Frictional Properties of Polymers in Solution
accel
M_
na
(9.91)
2. A buoyant force is given by the product of the volume V of the
particle, the density p of the solution, and the radial acceleration:
M
Fbuoy = Vpco2r = pco2r
(9.92)
3. A force of viscous resistance is proportional to the stationary-
stage velocity vs according to Stokes' law:
Fvls = fvs
(9.93)
4. The stationary-state velocity is rapidly achieved, and vs
corresponds to the
force in item (1) equalling the opposing forces in items (2) and
(3):
co2r = - co2r + fvs
(9.94)
A A @2
Since the velocity is dr/dt, Eq. (9.94) can also be written as
The stationary-state velocity per unit acceleration is a parameter which
characterizes the settling particle and is called the sedimentation
coefficient s:
s ^ (l - -)
(9.96)
co r fNA Ó p2
As a velocity divided by an accleration, s has units of time, and 10-1 3
sec-the
order of magnitude for a typical solute-is called a svedberg (symbol S)
in honor of Svedberg, a pioneer worker in this field.
Equation (9.95) can be integrated to give
ln r = |1 + - ) co2t +const. = sco2t +const.
(9.97)
which shows that the radial location of a species at various times is
described by a straight line in a plot of ln r versus t. The slope of
this line equals ñî2 s. Many biopolymers are simply identified in terms
of their sedimentation coefficient if
Sedimentation Velocity and Sedimentation Equilibrium
637
insufficient information is available to further resolve the cluster of
constants which define s.
In a solution of molecules of uniform molecular weight, all
particles settle with the same value of vs. If diffusion is ignored, a
sharp boundary forms between the "top" portion of the cell, which has
been swept free of solute, and the "bottom," which still contains solute.
Figure 9.13a shows schematically how the concentration profile varies
with time under these conditions. It is apparent that the Schlieren
optical system described in the last section is ideally suited for
measuring the displacement of this boundary with time. Since the velocity
of the boundary and that of the particles are the same, the sedimentation
As with the diffusion coefficient, sedimentation coefficients are
frequently corrected for concentration dependence and reduced to standard
conditions:
1. The concentration dependence of s is eliminated by making
measurements at several different concentrations and then extrapolating
to zero concentration. The limiting value is given by the symbol s°. This
is the sedimentation analog of D°.
Figure 9.13 Location of sedimentation boundary after various times in an
ultracentrifuge: (a) ñ versus r and (b) dc/dr versus r.
638
Frictional Properties of Polymers in Solution
2. The temperature of an experiment affects s° through Eqs. (9.5) and
(9.96), from which it is evident that s° is proportional to (1 -
ð/ð2)/ò?0, all of which are temperature dependent.
3. In view of item (2), experimental (subscript ex) values of s° are
reduced to standard (subscript s) temperature conditions by the
expression
= (bPW ^ .
(1-P/P2)ex %,s
This is the sedimentation analog of Eq. (9.90) for D°.
Diffusion effects are not absent during a sedimentation experiment as
assumed above. As a matter of fact, the longer the experiment proceeds,
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