A New Way of Determining Molecular Weight Distribution, Including

20 A New Way of Determining Molecular Weight Distribution, Including Low Downloaded by UNIV OF MICHIGAN ANN ARBOR on February 18, 2015 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0125.ch020

Molecular Weight, from Equilibrium Sedimentation M A T A T I A H U G E H A T I A and D. R. W I F F Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio 45433 University of Dayton Research Institute, Dayton, Ohio 45424 Formulas leading to determination of molecular weight dis tribution (MWD) from equilibrium sedimentation have been subjected to mathematical analysis. It has been shown that these formulas are a particular case of the Fredholm integral equation of the first kind, which is an improperly posed problem in the Hadamard sense. To cope with this problem the Tikhonov method of regularization has been imple mented, and new computation-oriented equations have been derived. Mathematical analysis shows that very precise re sults can be obtained in case of a narrow molecular weight range. Therefore, the method suggested can be successfully used to determine MWD of a low molecular weight polymer. For a wide MWD or if the distribution is multimodal and very assymmetrical, this method becomes more cumbersome. TiJ"olecular weight determination of a monodisperse macromolecular system from equilibrium sedimentation was devised by Svedberg and Fahraeus i n 1925 ( I ) . They applied the following formula 2RT d\nc (1 - 7ρ)ω dr* 2

(

)

{ l )

Here M is molecular weight, R is the gas constant, Τ is the absolute tem perature i n degrees Kelvin, V is the partial specific volume of the solute, ρ is the density of solution, ω is the angular velocity, r is the distance from the center of rotation, and c is the concentration measured at r. Under 216 In Polymer Molecular Weight Methods; Ezrin, M.; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

20.

GEHATiA

AND

Equilibrium Sedimentation

w i F F

217

ideal conditions Equation 1 can be easily integrated. The integral form w i l l be presented in the following by adopting the Fujita formalism which won a wide acceptance (2) c_

XJf

Co

1 - e-λ"

w

where c is the initial concentration, λ = (1 — Vp)«> (r — r )/2RT, £ = (fb — )/(r — r ) , r and r are the values of r at meniscus and bottom, respectively. Almost simultaneously with the first attempt to determine molecular weight from equilibrium sedimentation, Rinde tried to widen this method to include determination of the molecular weight distribution ( M W D ) of a polydisperse system (3). Unfortunately, this attempt proved to be more complicated and did not result in establishment of a reliable routine. Since the appearance of Rinde's dissertation in 1928, many investigators have tried to determine M W D . Most of these efforts, however, d i d not provide a successful comprehensive technique (4-16). This objective has been accomplished only in a few cases under very limited conditions, such as in case of a Gaussian or near Gaussian M W D , in which only characterizing parameters had to be determined. Scholte (17, 18) deter mined M W D by performing an experimental procedure based on several equilibrium experiments. If it is assumed that the lack of success in developing a comprehensive method leading to M W D determination (a single experiment at a single rotor velocity) stems from mathematical difficulties, the expressions re lating M W D to concentrations or concentration gradients must be more closely analyzed. There are methods other than the classical equilibrium sedimentation which can be used to obtain a molecular weight distribu tion. Two such methods are "equilibrium density gradient" and "sedi mentation velocity." The former method has been developed for aqueous solutions where the density gradient is achieved primarily by use of cesium chloride. In the study of synthetic polymers, which dissolve only in a limited number of organic solvents, the experimental conditions re quired by this method have not been sufficiently developed for all cases. The latter method, that of sedimentation velocity, could be applied to fractionated synthetic polymers for molecular weight determination. H o w ever, the determination of M W D by applying this method may be com plicated by various factors. For example, if the molecular weight is not high enough (M > 100,000), the correction for diffusion becomes cum bersome. The M W D of spherical molecules large enough to neglect diffusion was investigated by Gralen and Lagermalm (19). Methods to correct the resulting curves by introducing the contribution of diffusion 2

0

Downloaded by UNIV OF MICHIGAN ANN ARBOR on February 18, 2015 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0125.ch020

e

2

r2

2

h

a

2

a

2

h

b

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

2

&

218

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

24


16-

11

li Π

l\ 11

ill IT

\

ft

/I

An ' I I

111_

_L

ι

I 11 II I ι I 1 I 1 1 I II

Μ ι 1 I

1

7

1

1

1

1 1

W il

y

I I

I I

1

I

1 1

»!

t

II

\ I ι /

ι

:J

1

-16

30 20 n(s = n/4l. 5; η 1.2,—.41)

40

Figure 1. MWD calculated from Equation 10. The solid line represents the originally assumed distribution f(M), and the dashed line represents the curve / ( M ) resulting from the inverse operation.

are discussed by Hengstenberg (20). According to this analysis such corrections may be complicated. However, when the material under i n vestigation is of sufficiently high molecular weight, the influence of diffu sion can be neglected, and the distribution of the sedimentation constant of the unfractionated material may be directly obtained (21). In any case, to infer M W D from velocity sedimentation experiments, auxiliary measurements must be made to correlate the molecular weight of each fraction with its appropriate sedimentation constant. Since each method has its own limitations, this paper deals only with the technique of ob taining a M W D from an equilibrium sedimentation experiment performed at a single angular velocity.


20.

GEHATiA

AND


w i F F

219

The Mathematical Analysis of Formulas Relating MWD Concentrations or Concentration Gradients

to


M W D can be expressed by several equivalent formulas derived from the theory of equilibrium sedimentation at ideal conditions. In the fol lowing the well-known Fujita formalism (see Equation 2) w i l l be used. This expression related the density of M W D , f ( M ) , to concentrations or concentration gradients, respectively:

- ^

-

H

,

(

J

W

"

The left sides of Equations 3 and 4 depending only on ξ w i l l be de noted as u(£). The expressions i n brackets under the integral sign which are known for every ξ and M w i l l be called kernels and denoted as K(£, Λί). In this way Equations 3 and 4 reduce to a single formula: oo

ιι(ξ) 0.6

= f

Q

Mmax

Κ(ξ, M)î(M)dM

= f

Κ(ξ, M)i(M)dM

η

η

(s-

η/41.

5 ;

n-1,

2,

---,41)

Figure 2. Unimodal MWD using regularization without linear pro gramming. The solid line is the originally assumed distribution, and the crosses are the computed distribution values.


(5)

220

POLYMER

MOLECULAR WEIGHT METHODS


12η

Figure 3. Symmetrical bimodal MWD using regularization with linear pro gramming. The solid line represents the initial distribution, and the circles represent the computed distribution. which is the well-known Fredholm integral equation of the first kind. The upper integration limit ( oo ) has been replaced by M since mathe matically it becomes important later, and beyond M f(M) =0. Since K(£, M ) is always known, only two types of different compu tations may result from Equation 5: computation of u(£) if f ( M ) is known, defined as the "straight operation," and computation of f ( M ) if u(£) is known, defined as the "inverse operation." While the straight operation is a simple integration always leading to reliable results, the inverse operation is a more complicated procedure, and its results are less reliable. It has been shown that the inverse operator of Equation 5 is unstable. A small change in u(£) may cause a very large change and sometimes even uncontrollable oscillations of f ( M ) (22). Hadamard defined this type of mathematical operation an "improperly posed prob lem" ( I P P ) and excluded it from mathematical applications (23). U n fortunately, Equations 3 and 4 are mathematically I P F s , and therefore, their direct application did not result in a comprehensive and reliable method \o determine M W D . To demonstrate the instability of Equations 3 and 4, a symmetrical unimodal distribution f ( M ) was arbitrarily assumed, and the correspond ing u(£) = c/co was determined by applying a straight operation on Equation 3. This newly obtained u(£) function was now used as an input to evaluate back f ( M ) , by applying an inverse operation on the m a x

m a x


20.

GEHATiA

221


AND w i F F

same Equation 3. The parameter λ was taken from a real experiment conducted in the laboratory. The inverse operation was accomplished with the aid of variational calculus which minimizes the following func tional: 1


N[f(M), u « ) ]

M

max

=f [f 0

Κ(ξ, M)W)dM

0

- ιι(ξ)]\ίξ

(6)

By applying the Euler equation, this expression becomes: ^max

b(M) = f

F ( M , x)f(x)dx

Q

(7)

where ι F ( M , x) = f

Q

Κ(ξ, Μ ) Κ ( ξ , *) I,


îl+i — ίΐ—i+l

(15b)

First, one of the terms composing the sum given in Equation 14 was chosen, and then the numerical values of the corresponding a param eters were arbitrarily assumed. Under such conditions a set of linear equations defined by Equation 13 leads to determination of a correspond ing vector f { ,α ,... }== f {«„}. This in turn leads to the determination of the corresponding û { } ( £ ) by implementing Equation 3 or 4. In gen eral the function u{ } determined in this way is different from the function u ( f ) provided by the experiment. Such a difference can be expressed by the following norm: n

αι

2

αη

a n

ttn

ι

(16)

By applying a high-speed digital computer, a search for different sets

closest approximation of the "true" distribution f ( M ) . The method of regularization gave excellent results in the case of a unimodal distribution (41). A symmetrical arbitrarily assumed distribu tion, which previously had led to a "noisy" curve if no regularization was applied, gave very good agreement if determined by implementing E q u a tion 13 (see Figure 2). In the case of a bimodal asymmetrical distribution, the use of regularization proved not to be adequate and this method had to be incorporated into linear programming. Regularization

and Linear

Programming

Even though regularization smooths the "high frequency" noise, a certain amount of "low frequency" noise still exists in the f{* + . . .

(26)

and

ώ + ···

(27)

where 1

-5&t

' = 7* There is no way to measure the "absolute" time appearing i n E q u a tions 26 and 27. The most feasible way to circumvent this problem is to introduce an elapsed time t and an experimental time t*, i.e., 0

t = t + , were mathematically evaluated. Finally, these distributions were superimposed so as to compare the resulting distribu tion with the known f ( M ) function.


W

The equilibrium sedimentation experiment of sample Ρ was analyzed only by applying the computation-oriented expressions. The resulting f ( M ) was compared with the previously obtained f ( M ) . From the curve of f ( M ) the average < M > values were computed and compared with the < M > values computed in the same way from individual distribu tion curves of fractions A , B, and C. These < M > values were also compared with M values obtained in a conventional way from equilibrium sedimentation by applying Equation 2, and with M values determined from velocity sedimentation experiment, Equation 30 (see Table I I ) . W

W

W


20.

GEHATiA

AND


w i F F

231

Results In all calculations the buoyancy factor (1 — Vp) at 35°C was taken as 0.31. The computation-oriented expression Equation 13 was applied in order to obtain the results shown i n Figures 6-9 for sample A , with 0 ^ M < 100,000 and a = 6 χ 10 (Figure 6 ) ; sample B, with 15,000 ^ M ^ 100,000 and « = 2 χ 10 (Figure 7 ) ; sample C, with 100,000 ^ M ^ 180,000 and « = 2 χ 10 (Figure 8 ) ; and sample P, with 0 ^ M ^ 180,000 and

A New Way of Determining Molecular Weight Distribution, Including

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