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

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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

coefficient is readily measured.

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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