Die Makromolekulare Chemie 116 (1968) 241-249 (Nr. 2792)

From the Institute of Physical Chemistry, University of Uppsala, Uppsala, Sweden

Determination of Molecular Weight Distributions by Gel-Permeation Chromatography

By HANS VINK

(Eingegangen am 22. April 1968)

SUMMARY: The theoretical basis for the determination of molecular weight distributions of polymers

by gel-permeation chromatography has been investigated, using the previously developed theory of partition chromatography. Expressions determining the position and broadening of the concentration distribution in the elution curve have been deduced and the molecular weight dependence of the parameters characterizing the process has been considered. A method is proposed for determining the molecular weight distribution by a convolution procedure.

ZUSAMMENFASSUNG: Die theoretischen Grundlagen fiir die Bestimmung der Molekulargewichtsverteilung von

Polymeren mittels Gel-Permeations-Chromatographie werden untersucht und zwar aus- gehend von der friiher entwickelten Theorie der Verteilungschromatographie. Ausdriicke fiir die Lage und Breite der Elutionskurven werden abgeleitet, und die Molekulargewichts- abhangigkeit der in diese Ausdriicke eingehenden Parameter wird untersucht. Eine Me- thode zur Bestimmung von Molekulargewichtsverteilungen mittels eines Faltungsverfah- rens wird vorgeschlagen.

The use of gel-permeation chromatography for the determination of molecular weight distributions of polymers has come into widespread use in recent years. The method appears to be one of the most promising for a rapid determination of molecular weight distributions and considerable progress in the experimental technique and in the understanding of the nature of the process has recently been madel-17). In the present work an attempt is made to treat the problem on the

basis of general chromatographic theory and to give the theory of the process a more rigorous foundation. In this treatment the gel-permeation chromatography is considered as a special case of partition chromato- graphy. Thus, the basic process is the partition of the polymer between the stationary. gel phase and the mobile solution phase. In order t o get a separation of the polymer on the molecular weight basis the partition coefficient of the polymer has to be molecular weight dependent. This is in general the case and usually the smaller molecules have higher partition

241

H. Vma

coefficients than the larger ones. This depends on the fact that larger molecules have difficulties in entering the gel matrix and therefore have a lower solubility in the gel phase.

In the present treatment some idealizing assumptions about the pro- cess are made. Thus, we assume that specific sorption effects are absent and that the partition coefficient is independent of concentration. The latter condition is in general satisfied when the solution in the column is sufficiently dilute.

1. Basic Theory Symbols$ dl

V

Y = partition coefficient Di = longitudinal diffusion coefficient D2 = diffusion coefficient in the gel V1, Va = volumes per unit of interphase area of mobile and stationary phase,

d = interphase area per unit length of the column a = length of the column 0 = voV1 = rate of (solvent) flow through the column P9 P2 = the mean and the variance of the concentration distribution in a peak,

6, $2 = time derivatives of p and 112

w = vv = velocity of the peak D = spreading coefficient pv, pzv = the mean and the variance with respect to the efflux volume.

= translational velocity of mobile phase

respectively

respectively

V = peak mobility

According to the treatment in18919) the basic laws of linear partition chromatography may be formulated as follows :

pz(t) = 2Dt + const. (2) V2Y

D = D 1 +

2D2V1 (-& + $7 (3)

Eq. (1) gives the relative peak velocity (peak mobility) in the column under steady state conditions. It is valid for peaks of arbitrary form. Eq. (2) gives the spreading of the peak in the column in terms of its

second moment around the mean (the variance). Also this formula is valid for peaks of arbitrary form. The spreading is characterized by a spreading coefficient D, given by Eq. (3).

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Molecular Weight Distributions by Gel-Permeation Chromatography

As these laws are concerned with the situation in the column they are not directly applicable to normal experimental conditions, in which the concentration distribution is determined as a function of the efflux volume. However, it may be shown20) that the moments of the concen- tration distribution with respect to the efflux volume may be expressed in terms of the moments in the column. As a good approximation the following equations hold :

The meaning of these formulae becomes apparent when one observes that a/@ is the time it takes for the peak to pass through the column.

2 . Determination of Molecular Weight Distributions

The original chromatographic theory was based on the moments of the concentration distributions and was not concerned with the form of the concentration distribution curves. For the determination of molecular weight distributions it is essential to have some information about the form of the concentration distribution curve for a single solute. It is obvious that the form of a concentration peak is dependent ou the condi- tions under which it enters the column. However, certain information may be obtained from the general form of the expressions in Eqs. (2) and (3). Thus, from the fact that the formulae are invariant under the reversal of the velocity of the mobile phase, i t may be concluded that an originally symmetrical peak will remain symmetrical. From numerical calcula- tions21) it further follows that the skewness of an asymmetrical peak in general diminishes with time. Observing further that the spreading of the peak is governed by the diffusion law (2), we may as a good approxi- mation represent it by a GAussian curve. Thus, for a polymer of uniform molecular weight Mi, applied to a narrow zone at the entrance of the column, the elution curve takes the form :

where the following symbols have been used : V = effluxvolume mi = mass of the applied sample (m.w. = Mi) Ei = elution volume of sample i (Ei = pv according to (4)).

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H. VINK

For the parameter pi, which determines the breadth of the peak, we get in accordance with (2), (4), and ( 5 ) :

where Ei/O is the time it takes for the peak to travel through the column and 7 is a constant, which takes care of the initial breadth of the peak.

If the polymer sample applied to the column contains molecules of different molecular weight, the composite elution curve may be obtained by integrating (6)

C(V)

The relation for W W

P I

over the entire molecular weight range. m

mass balance has the form : CQ

1 (v - w2 C(V)dV = / exp [ - 4p 1 dV / m(M) dM =j m(M) dM

0 - 0 3 0 0 (9)

where the last equality holds because the integral in the second member over the normalized GAussian function is unity.

Hence, Eq. (8) may be normalized by dividing each member in (8) by the corresponding member in (9). We get

where c(V) is the normalized elution curve and w(M) the molecular weight distribution of the polymer sample. It is more expedient to express the molecular weight distribution in terms of the elution volume. We then get

W

1 c(v)= j. -

0

where dE dM w(E) - = w(M)

If the elution curve c(V) has been experimentally determined, the re- maining problem is to obtain the molecular weight distribution wl(E) by

Molecular Weight Distributions by Gel-Permeation Chromatography

solving the integral eq. (11). The problem is amenable to solution by a convolution procedure. The principles of the method were already pointed out by C L A E S S O N ~ ~ ) in connection with adsorption analysis, and it has recently been successively applied to the determination of molecular weight distributions by diffusion and sedimentation methods 23-26), and in connection with measurements of dipole moments 27-28).

To bring Eq. (11) to the form of a convolution integral we introduce new variables which eliminate the parameter p in (11). Thus, we make the transformations

I n these formulae we will consider D as a constant. It will be shown

Instead of Eq. (11) we now get: later that this is a reasonably good approximation.

n

(15)

where we have the relations

It should be noted that the transformations (13) and (14) bring about a dilation of the scale in the reference frame and thus implicitly take into account the nonuniform broadening of the elution curve. To see this we differentiate Eq. (14), which gives

Taking 16yl = vg which is the standard deviation of the GAussian function in (15), we get as the deviation of the efflux volume around the point V = E

-

16V( = - 6 J D / ~ ' E J E + & (19)

245

H. VINK

This is in agreement with Eq. (7), where v@ is the standard deviation of the GAussian function in (11).

When the correction term TO, representing the breadth of the initial zone, is very small or vanishes Eqs. (13)-(14) should be replaced by the following, more simple transformations :

Instead of (19) we then get :

This shows that the transformations (20) -(21) are less satisfactory when the breadth of the initial zone becomes appreciable.

The range of variation of the variables x and y is determined by Eqs. (13) and (14). However, in (15) the integration may formally be carried out over the entire real axis, as the functions involved are identically zero outside the interval indicated in (15). Thus, we get the convolution:

1 1 ci(y) = - 2 yii- (exp [ - - 4 (y - x).] q(x) dx

h - w

We define the FOURIER transform f(E) of a function f(x) by

?(E) = [f(x) e*Cxdx

- W

and the inverse transform by

f(x) = - 2 x Jw?(t) e-iExd< (25)

Observing that the FOURIER transform of e-x2’4 is 2xe-E2 we get the

--a,

following transformation of the convolution (23)

E d E ) = e-E’@(<) (26)

Hence +(v = e EPc^1(4) (27)

and p(x) is obtained by the inverse transformation (25).

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Molecular Weight Distributions by Gel-Permeation Chromatography

To perform the computations the experimentally determined function c(V) is first converted to cl(y) according t o (16). To do this it is necessary to know values of the parameters D and 7 . An approximate value of D may be obtained from measurements with a uniform substance of a not too low molecular weight, which does not specifically interact with the gel. Then 7 is obtained from the breadth of the initial zone. Denoting the latter by 6 we get:

s = 2 m = 1 / 8 I ) 7 ' (28)

When cl(y) has been determined the integral eq. (23) has to be solved by evaluating the FOURIER transforms numerically. This may be done by using BEEVERS and LIPSON sticks or by a computer. It should be observed that the numerical treatment of the problem is considerably facilitated due to the fact that the FOURIER transform of the kernel of the integral eq. (23) has a simple analytical form.

After the distribution function wl(E) has been obtained from (17), the remaining problem is to express it as a function of molecular weight. For this the elution volume has to be determined as a function of molecular weight. This may be done by calibrating the column with polymer frac- tions of known molecular weight. However, some information may also be obtained from theoretical considerations, and we will in the following examine the molecular weight dependence of the parameters occurring in the chromatographic theory.

From Eqs. (1)-(3) it follows that the only molecular weight dependent parameters are the partition coefficient y and the diffusion coefficients D1 and D2. Further, the partition volume V2 might also depend on the molec- ular weight, as the gel will likely be of a non-uniform structure. This effect may be treated along the lines indicated in ref. 19), but, in a more direct, though less accurate way, this effect may be included in the molecular weight dependence of y.

Considering the molecular weight dependence of the longitudinal dif- fusion coefficient D1 we observe that it represents two effects, the BROWN- ian diffusion and the diffusion due to the irregularity of flow (eddy dif- fusion). The latter effect is in general predominant, and as it is molecular weight independent, we may conclude that D1 is very nearly a constant.

For the molecular weight dependence of y and D2 various relations may be considered. We will here use the following tentative relations :

y = kiM-OL (29)

Da = kzM-6 (30)

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H. VINK

From (l), (4), and (29) we get for the elution volume:

v2 E = a O ( 1 + k l - - M - ) V VI

For the spreading coefficient we get from (3), (29), and (30) :

We observe that in the expression for D the molecular weight depend- ence of y and Dz partially cancel. Considering further that a substantial part of the spreading coefficient is made up of the essentially constant term D1, we find that the molecular weight dependence of D is slight, which justifies the neglect of this dependence in the treatment above.

Finally, we will indicate another approach to the problem, which from the theoretical point of view has great advantages. If, instead of deter- mining the solute concentration as a function of the efflux volume, it is determined inside the column at a fixed time and as function of the posi- tion in the column, the original formulae (1)-(3) may be used directly Then, instead of the integral eq. ( l l ) , we get:

where X is the position coordinate in the column and, according t o (1)

p = m t + const. (34)

For p we get: p = D ( ~ + T ) (35)

where t is the time of measurement and T determines the breadth of the initial zone. Assuming as before that D is constant, we see that p is a constant in (33), which means that the broadening of the peak due t o diffusive effects is uniform. Thus, apart from the smearing effects due to the diffusive broadening and the finite breadth the initial zone, c(X) gives directly the molecular weight distribution w1(p) as a function of the posi- tion in the column. For the elimination of the smearingeffects Eq. (33) may be solved with the help of FOURIER transforms as indicated above. However, the experimental difficulties in determining tbe solute concen- tration inside the column seem to cancel the theoretical advantages of this approach.

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Molecular Weight Distributions by Gel-Permeation Chromatography

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