Molecular Weight Distributions from Sedimentation ... - ACS Publications

21 Molecular Weight Distributions from Sedimentation Equilibrium Experiments Ε. T. A D A M S , Jr. and P E T E R J . W A N

Downloaded by UNIV OF MONTANA on January 25, 2016 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0125.ch021

Texas A & M University, College Station, Texas 77843 D O N A L D A. S O U C E K Illinois Institute of Technology, Chicago, Ill. 60616 GRANT H. BARLOW Molecular Biology Department, Abbott Laboratories, Inc., North Chicago Ill. 60064 Since 1928 it has been realized that molecular weight dis tributions (MWD's) of polymeric solutes could be obtained from sedimentation equilibrium experiments. Two currently used methods developed by Donnelly and by Scholte to obtain MWD's require ideal, dilute solutions and the use of ultracentrifuge cells with sector-shaped centerpieces. These restrictions are removed in this paper. Equations analogous to those of Donnelly and Scholte are presented for nonsector-shaped centerpieces. A method of correcting for nonideal behavior is also presented along with a method of application of this nonideal correction to the measurement of MWD's by Donnelly's or by Scholte's methods, using centerpieces of any shape. T n 1924 Svedberg and Rinde ( J ) published the first paper on the ultracentrifuge. Four years later Rinde (2) advocated the idea of obtaining a molecular weight distribution ( M W D ) from sedimentation equilibrium experiments. H e tried to obtain the distribution of radii from colloidal gold sols. Subsequent methods proposed for obtaining M W D ' s from sedimentation equilibrium experiments were largely based on Rindes (2) pioneering work. There have been a number of attempts to obtain M W D ' s from sedimentation equilibrium experiments, using the concen tration (c) or concentration gradient ( d c / d r ) distribution at sedimenta tion equilibrium. In some cases (3-^5) specific models of the M W D , such as the most probable distribution, were used. Wales and his associates 235 In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.


236

P O L Y M E R M O L E C U L A R WEIGHT METHODS

(6-9) attempted to avoid specific models for the M W D ; they also tried to include nonideal behavior. These methods, as well as a recent one by Sundelof (JO) based on the Fourier convolution theorem, depended on data (c or dc/dr vs. r) obtained from one sedimentation equilibrium experiment; sometimes the experimental error caused negative weight fractions to be obtained for some of the polymeric components. This then was the state of the art until two separate and elegant breakthroughs, the first by Donnelly ( J J , 12) and the second by Scholte (13, 14) showed new ways to solve the problem. Donnelly pointed out that the concentration distribution of the polymeric solutes at sedimen tation equilibrium was i n the form of a Laplace transform. H e showed methods for obtaining analytical expressions for the Laplace transform. Once these expressions were available, one could obtain the M W D from the inverse of the Laplace transform. While this method is theoretically rigorous, it appears to be applicable only to unimodal distributions. D o n nelly's (11, 12) method requires only one sedimentation equilibrium experiment. Refinements to the Laplace transform method and also to the Fourier convolution theorem method have been reported by Provencher and Gobush (15, 16). Scholte's (13, 14) method uses sedimen tation equilibrium experiments at different speeds on the same solution to.obtain M W D ' s . H e showed that his method could be applied to b i modal and trimodal distributions. Both Donnelly's ( J J , 12) and Scholte's (13, 14) methods were restricted to ideal Θ solutions and to ultracentrifuge cells with sector-shaped centerpieces. W e show here how both of these restrictions can be removed. The ideal case is developed first, followed by a method for obtaining M W D ' s under nonideal conditions from sedimentation equilibrium experiments using cells with sector- or non-sector-shaped centerpieces. Ideal

Solutions Sedimentation equilib

Basic Sedimentation Equilibrium Equation.

rium experiments are performed at constant temperature. The condition for sedimentation equilibrium is that the total molar potential, IH, for all components i be constant everywhere i n the solution column of the ultracentrifuge cell. Mathematically this can be expressed as

Mi

=

Mi

+

Φι

=

Mi

2 —

=

c

o

n

s

t

a

n

(!)

t

2xRPM . . ω = — — — = angular velocity Here m is the molar chemical potential of component i, Φι = — M

In Polymer Molecular Weight Methods; Ezrin, Myer; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

i W

2

r /2 2

21.

Sedimentation

ADAMS, JR. ET A L .

237

Equilibrium

its molar centrifugal potential, the molecular weight of component i, and r the distance (from the center of rotation) i n the solution column of the ultracentrifuge cell. The quantity r is restricted to values between and including r , the air-solution meniscus radial position, and r , the radial position of the cell bottom. For simplicity we shall assume that the solution is incompressible. It is not necessary to make this assump tion, but the treatment is easier to follow if we do so. Excellent treatment of compressible solutions w i l l be found in the monograph by Fujita (17) or the review by Casassa and Eisenberg (18). A t constant temperature and for incompressible systems, one notes that the following relations apply for a solution containing q polymeric solutes.


m

b

Hi

=

(2)

c , P, T)

/(Ci, c , · .

q

2

(θμί/θΡ) , c = M-Vi

(3)

τ

The chemical potential of component i can be expressed as μ; = μ\ + RT In

(4)

y id

Here Ci is the concentration of component i in grams per liter, and t/* is the activity coefficient of component i on this concentration scale. The quantity m° is the standard state chemical potential of component i and is a function of temperature only. The standard state of solute compo nent i is chosen so that In t/t —> 1 as c —> 0. In Equation 4, R is the uni versal gas constant, 8.314 Χ 10 ergs/(deg mole), and Τ is the absolute temperature. The quantity In y is a function of the concentration of all q solutes; thus {

7

4

In y

{

= f(c

h

c , · . ., c ) 2

(5)

q

W e w i l l denote the solvent by the subscript 0. Usually In y* is expressed as a Maclaurin series in which only the first term is retained—i.e.,

In

V i

= Σ ( ^ ^ " ) V c ^ C y + . . . = M£B y

ijCj

+ ...

(6)

Here (d In yi/dc )° ^ is the value of the derivative at infinite dilution of the solutes. For ideal solutions In yi is taken to be zero at all concen trations. In this case Equation 1 can be expressed as j

T}Ck

j


238

POLYMER

MOLECULAR WEIGHT METHODS

but \dP

άμ and dP =

w

2

pd(r )/2 2

so that MjQ. - t>,- )

Cr

Co -

m

=

(C ~

(23)

Crjdx

The conservation of mass equations are used with refractometric optics. The absorption optical system w i l l give absorbance (optical density) which is directly proportional to concentration. Unless conservation of mass equations are used, one can only obtain concentration differences from refractometric optics. The Rayleigh optical system w i l l give infor mation proportional to c — c ; thus Equation 20 or 23 would have to be used to obtain c . Note also that the initial concentration c is needed. This must be measured by differential refractometry, by boundary-form ing experiments, or from ultraviolet light absorption. For sector-shaped centerpieces the substitution of Equation 17 into Equation 18a leads to r

fm

Tm

0

^

-

°> = l - ^ ' - A M , )

( 2 4 )

W h e n Equation 24 is substituted into Equation 17, the result is

C i {

® -

1 - exp(-AM ) t


( 2 5 )

21.

243

Sedimentation Equilibrium

ADAMS, JR. ET A L .

The summation of Equation 25 over all i solute components, followed by division by c , leads to 0

m

~c~o~

=

1 -

i

=

exp(-AM ) t

( 2 6 )


Here /« = Co»/co is the weight fraction of component i. N o w differentiate Equation 26 with respect to £ to obtain 1 dc(Ç) _ ~ Γ.

-άθ(ξ)

_

.

"

W

_ " ^

AW/,-exp(-AM^) 1 - βχρ(-ΛΜ,)

v

'

(

ξ )

( 2 7 j

Equations 26 and 27 are the equations needed for obtaining M W D ' s when sector-shaped centerpieces are used. The corresponding equations for the Yphantis centerpiece are = 0) =

ctf

(28) idx

Ci

(S) =

β(ξ) = M.

ε°

Molecular Weight Distributions from Sedimentation ... - ACS Publications

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