JOURNAL OF POLYMER SCIENCE: PART A-2 VOL. 8, 89-103 (1970)

Universal Calibration in GPC: A Study of Poly- styrene, Poly-a-Methylstyrene, and Polypropylene

H. COLL, Shell Development Compun y , Emeryville, Culiforniu 94608 and D. K. GILDING*, Shell Chemical Company, Woodbury, New Jersey

Synopsis The theoretical justification for using Mml, or a similar quantity, as a universal

calibration parameter in GPC is reviewed. The equation based on this parameter is applied to transform the primary calibration curve, obtained by means of polystyrene samples, into calibration curves for poly-a-methylstyrene, polypropylene, and linear polyethylene. The Mark-Houwink equations for these polymers, as they are used in the transformation, are discussed. The resulting GPC calibration curves are compared with molecular weights and peak elution volumes of fractionated poly-a-methylstyrene and polypropylene. The same comparison is made with samples of polypropylene and polyethylene having very broad molecular weight distributions. The agreement lies within experimental error,

THEORETICAL

Basis of Universal Calibration Calibration in gel-permeation chromatography ( GPC) means to estab-

lish a functional relationship between the emergence volume V of the peak maxima of a series of nearly monodisperse samples and their molecular weights. Among synthetic polymers such samples are usually difficult to obtain. One exception is a series of polystyrenes (PS) in the molecular weight range of 103-106 with BW/M,, < 1.15 (Waters ASSOC., Framingham, Mass., ArRo Laboratories, Joliet, Illinois).

It is therefore desirable to calibrate a system of chromatographic columns for a given solvent at a specified temperature by means of these PS stan- dards, and to transform this primary calibration curve into a calibration curve for some chemically different polymer the ehromatogram of which is to be subjected to quantitative interpretation. Such a transformation requires the existence of a “universal calibration parameter” which in turn can be related to molecular weights. If we exclude specific inter- actions between the solute molecules and the chromatographic gel as not germane to the GPC process, we may look for some geometric parameter of the dissolved polymer molecules as “universal.”

Benoit and co-workers1*2 have demonstrated that the product of molecu- lar weight and intrinsic viscosity, M[q] , for structurally different polymers, when plotted as a function of V conforms to a single curve. According

* Present address, Nester-Faust Manuf. Corp., Newark, Delaware. 89

@ 1970 by John Wiley & Sons, Inc.

90 COLL AND GILDING

to Einstein’s equation for intrinsic viscosity, M[q] defines an effective hydrodynamic volume of the polymer molecules, and the hydrodynamic volume (or radius) has been referred to as the “universal parameter” of GPC calibration. This conclusion can be given a theoretical basis, at least for linear polymer molecules present in solution as statistical coils, in the limiting case of equilibrium distribution between the moving and the stationary phase in the GPC ~ o l u m n . ~ - ~ We shall briefly review the underlying reasoning.

The emergence volume for a certain molecule is a function of the prob- ability of that molecule entering into pores (the stationary phase) along the chromatographic column. This probability will depend on a number of factors, such as pore-size distribution, solvent, temperature, which, however, will remain unchanged during the experiments. Under practical conditions flow rate and concentration will also play a role, but we shall limit our attention to extremely dilute solutions and flow rates small enough to approach the condition of equilibrium distribution. Finally, the effective size of the polymer molecule remains as the determining factor for the probability of entry into the gel phase, and hence for V. The effective size of a solid spherical molecule is uniquely determined by its radius. In the case of flexible-chain molecules we must introduce an aver- age statistical dimension and ask for the magnitude of change of free energy AG brought about by the deformation of a polymer coil to allow entry into a pore of given dimensions. Meier6 has shown by statistical calculations on the compression of linear polymer coils between parallel plates that AG is primarily determined by the change of configurational free energy, AG,, while the heat of mixing plays only a minor role, and for the moment we shall neglect the latter. Then, from Gaussian statistics, AG = AG, is given by only one statistical dimension of the molecule, and by the distance between the parallel plates which serve here as a model for the confining cavity. The molecular dimension may be expressed as root-mean-square radius of gyration, ( l?2)1/2, or end-to-end distance (h2)*/2 = N1/2b, where N is the number and b the effective length of polymer segments. Casassa3 has calculated the increase of configurational free energy for confinement in slots, cylinders, and spheres. The significant result of all these calculations is that beside the dimensions of the cavity only one other variable, i.e., the statistical dimension of the undeformed coil, determines AG irrespective of tacticity or chemical nature. Hence, we may conclude that in the case of linear molecules, (hz) l l 2 or (B2) ll2 are universal calibration parameters which determine V. With reference to the Flory-Fox equation

and Einstein’s equation for intrinsic viscosity,

GPC CALIBRATION 91

where Q and C are constants, these parameters should also be proportional to the hydrodynamic radius R , defined in terms of [q].

Casassa and Tagami4 have extended the calculations to include regular star-shaped molecules with 4, 8, and 12 branches. The results for these molecules with long-chain branching show that the functional relation between AG, and R2 differs significantly from the one found for linear molecules. Yet, a fair agreement between the results for linear and branched molecules was shown if AGc is expressed in terms of (Nb2)1/2g1'6, where g = i&ranched/@linear. According to the theory of Zimm and Kilb' for the intrinsic viscosity of branched molecules [q]br = g1/2[q]~in [if eq. (1) were also valid for branched molecules, the exponent of g would be "21. Then, from eqs. (1) and (2) it follows that

R , = ( [q]M/C)'I3 = const. X (Nb2)1/2g1/6 (3)

justifying calibration in terms of M [ Q ] . The agreement, however, ap- pears to be fortuitous. It becomes poorer towards the high molecular weight limit of the chromatogram, and also with increasing degree of branching. These conclusions have been presented by Casassa and Tagami.4 Calculations for other types of branched molecules appear to be quite involved and have not come to our attention.

Grubisic et aL2 have reported that even poly(benzy1 L-glutamate) in dimethylformamide, a rodlike (helical) molecule, conforms to calibration in terms of M [ Q ] . A theoretical justification has so far not been given.

Instead of the product M [ Q ] we have preferred to use M [ q ] l f ( ~ ) where

f ( ~ ) = 1 - 2 . 6 3 ~ + 2.86c2,

E = ( 2 ~ - 1)/37

a being the exponent in the Mark-Houwink equation. occurs in the equation of Ptitsyn and Eizner,8

The factor f ( ~ )

[q] = @o f ( E) 63'2 ( R2) 3 /2 /M (4) with Qo = 2.86 x which can be considered a refinement of the Flory- Fox equation. It becomes identical with the latter at the theta point ( C = l ) , except for the somewhat larger constant Qo. The use of eq. (4) has also been suggested by Cantow et al.9 and by Moore and Arrington.lo

We have accepted the equilibrium distribution of polymer coils between the mobile phase and the pores of the gel as adequately described by the statistics of the unperturbed coil. However, by using eq. (4) for estab- lishing a connection with the experimentally determined intrinsic viscosity we make allowance for chain perturbation. For given values of [q] and M the effective dimensions of a polymer coil will be greater in a good solvent where f ( ~ ) < 1 than in a theta solvent. Accordingly, the elution volume will be smaller than follows from a calibration based on M[q] .

In good solvents the change of free energy of polymer coils, upon confinement in a pore, is

One further point deserves to be mentioned.

92 COLL AND GILDING

augmented by a free energy of mixing arising from an inqreased density of polymer segments, an effect which we ignored in the preceding arguments. For a good solvent, characterized by a coil-expansion factor a = 1.5, Meier6 has calculated the heat of mixing to be of the order of 10% of the configurational free energy. This means that the probability of entry into pores of the gel is somewhat decreased in a good solvent : the molecules are larger in a good solvent than in a theta solvent, and also in this regard the factor f (e ) corrects in the right direction.

The arguments as presented imply equilibrium between the moving and the stationary phase in the GPC column. However, it is questionable whether this is a realistic assumption under conventional experimental conditions. The diffusion rates, particularly of large molecules, are too small for equilibration" unless one assumes that the pores accessible to large molecules are quite shallow. The argument that the so1ut.e is in- jected as a rather broad band which allows for extended periods of equilibra- tion with the gel phase is open to criticism, since it implies that extremely small quantities of solute (corresponding to monomolecular bands, for instance) should give markedly different results. As yet a satisfactory conclusion to the problem of diffusion is still outstanding. Influences of diffusion rates, being of a secondary nature in the GPC process, should not require modification of the above concepts of universal calibration. Diffusion coefficients are inversely proportional to the effective frictional radius of the polymer molecules. The latter can be taken as approxi- mately proportional to R,, and hence to M[I,I], or to M [ I , I ] / ~ ( ~ ) . In other words, molecules of the same R , will show approximately the same diffusional behavior.

This argument can presumably be extended to secondary effects in general, such as peak broadening and skewing of the chromatogram as a function of concentration. Phenomena of this kind, although of consider- able importance in quantitative GPC are of no immediate interest in the context of the present investigation since we are primarily concerned with comparisons between chemically different polymers. It is assumed that effects of nonideality should be approximately of the same magnitude for polymers of the same M[9] , irrespective of chemical nature.

Calibration Procedure

A primary calibration curve is first established by means of the available narrow samples of polystyrene for which we may write

log M , = F ( V ) (5)

MI is the molecular weight of PS. The calibration curve of some other polymer, with subscript 2, under a given set of experimental conditions is obtained from

1% M1 C?ll/f(4 = 1% M2 C?lZ/f (€2) (6)

GPC CALIBRATION 93

and the respective Mark-Houwink equations [ q ] ~ = K1MU1, [ q ] 2 = K2Mu2. Substitution in eq. (6) yields

Combining eqs. (7) with (5) established the relation between M2 and V. The calibration curves are parallel if a1 = az, in which case also f ( ~ ) = f ( e 2 ) . The parameters K l and al, for PS, are readily determined from the intrinsic viscosities and molecular weights of the PS standards. In many instances Kz and Q may be found in the literature. If not, they can usually be obtained with reasonable accuracy from measurements on unfrac- tionated samples. It follows from the definition of the viscosity-average molecular weight, Mu = [ZCiM;a/ZCi]l‘a, that K and a remain unchanged if M,, is substituted for M in the Mark-Houwink equations, irrespective of the molecular weight distribution in the polymer samples. In most cases &fu can be estimated with an error not exceeding f5% from aw, the weight average molecular weight, assumed to be accurately known from light scattering measurements. Typically M n / M m = 0.88 f 0.05 for moderately broad samples (2 < &fw/Mn < 6) . Exceptions are samples with highly skewed distributions, such as the log-normal distribution, in a poor solvent, in which case M u will assume a lower value.

If the Mark-Houwink relation for polymer 2 is known for some solvent other than the one used in GPC, measurements of intrinsic viscosity on a series of unfractionated samples in that other solvent can be used for deter- mining Mu. Measurements of [q] in the GPC solvent will then give K z and az. If the Mark-Houwink exponents can be expected to be similar in both solvents (e.g., both solvents being “very good”), only a minimal error from the width of molecular weight distribution should be introduced.

We note that the foregoing procedure is designed to establish calibration curves such as one would obtain with near-monodisperse samples of polymer 2, to the extent to which the PS standards can be considered as near-monodisperse.

It is well known that in good solvents one set of K and a does not suffice to describe the relation between [q] and M over the entire range of molecu- lar weights which may be of interest in GPC, as this is expressed, for instance, by the theory of Stockmayer and Fixman.I2 Then, for better accuracy, the calibration curve of polymer 2 is to be calculated by segments, each segment being characterized by particular values of a, K . In prac- tice, however, in the molecular weight range 104-106 it appears that quite often the errors inherent in the determination of the [q+M relationship are too large to justify this refinement.

Peak Maxima of Polydisperse Samples

In order to document the correctness of the proposed calibration pro- cedure, calculated calibration curves should coincide with results directly

94 COLL AND GILDING

obtained by means of near-monodisperse samples of the polymer under investigation. If these samples are narrow enough for i$fW to be equated with Mn, elution volumes that correspond to the maxima of the chro- matographic peak can immediately be correlated with molecular weight. To prepare such narrow fractions from polymer with an initially broad distribution is impractical and allowance for some polydispersity has to be made.

Berger and Shultz13 have treated the problem of the position of peak maxima for several analytical molecular weight distributions. If the calibration curve, log M versus V , can be approximated by a straight line, the maximum of the chromatographic peak corresponds to aW in the case of the Schulz-Zimm distribution, while the highly asymmetrical log-normal distribution exhibits the maximum of the chromatographic peak at a lower average, via. (A?,,,A?f,,)l/z. Kotliar14 has criticized the use of the log- normal distribution and other highly skewed distribution functions as in- adequate for the description of the molecular weight distribution in polymer samples, particularly in the case of fractionated material. We may then expect the peak maximum of the chromatogram to correspond to a higher molecular weight average than (@w@n)l’z, and in all likelihood not to be far below aW. For fractionated material with aW/A?,, < 2 the viscosity- average molecular weight, being a few per cent below Mm in a good solvent, appears to be an appropriate choice.

In the following then we have illustrated the use of the proposed trans- formation of calibration curves for poly-a-methylstyrene and polypropylene in conjunction with polystyrene standards. A few results on unfrac- tionated samples of polypropylene and polyethylene have likewise been presented.

EXPERIMENTAL

Samples

The polystyrene (PS) samples used were narrow-distribution samples supplied by Waters Associates (Framingham, Mass.), and by ArRo Laboratories (Joliet, Illinois). In establishing the Mark-Houwink rela- tionship we have assumed that A?w = the viscosity-average molecular weight. Some uncertainty exists with respect to the molecular weight of sample 61970. While = 18.9 X lo5 is in line with the other standards with respect to the GPC elution volume, a lower value, a,,, = 16.7 X lo5, agrees better with the plot of intrinsic viscosities versus molecular weight (both numbers have been given by the supplier). The data are sum- marized in Table I.

The poly-a-methylstyrenes (PAMS) were prepared, fractionated, and characterized in our 1ab0ratory.l~

The whole polypropylenes (PP) with one exception, were commercial- grade polymers. Most of the fractions were prepared from the material designated 5220 by a fractional coacervation technique with cyclohexanone

GPC CALIBRATION 95

and dimethyl phthalate as the solvent-nonsolvent mixture. The separa- tions were initially liquid-liquid in nature, but after coalescing, the co- acervate crystallized, and was easily removable from the remaining dilute solution.

The four high-density polyethylenes were commercial-grade samples. Three of them, designated HM, contained about 5% polypropylene.

Solvents

1,2,4-Trichlorobenzene (TCB) was obtained from Dow Chemical The solvent was dried by passing through a column of silica

Prior to use in the chromatograph the solvent was degassed at Company. gel. 165 f 3°C.

Intrinsic Viscosities

Intrinsic viscosities were measured in decalin and in TCB at 135°C. In order to prevent degradation, either 0.1% Ionol antioxidant or 0.05% Ionox 330 with 0.25% dilauryl thiopropionate were added to the solvent. Viscosities were measured by means of Cannon-Ubbelohde semimicro viscometers and by means of Zeitfuchs viscometers.

Light Scattering

Measurements were carried out on PP in a-chloronaphthalene at 15OOC in a modified Brice-Phoenix photometer.o The refractive index increment was determined as -0.191 cc/g (5460 A). The solutions were clarified by filtration through cellulose pads at 150°C.

Gel-Permeation Chromatography

The experiments were carried out with a modified Model 100 chro- mat$graphoof W5ters Associates. The column combination was lo7 A, lo6 A, lo5 A, lo3 A. The flow rate was 1 ml/min. In each case, 2 ml of sample solution, a t a concentration of 0.25%, was injected. Antioxidant was added to the solvent. A 3-in. cartridge of 200 A Bioglass (120-200 mesh; Biorad Laboratories, Berkeley, California) was installed at the end of the column train to remove the nitrogen peak occurring at count 50. This cartridge has no effect on the GPC trace of polymeric species after initial conditioning.

RESULTS AND DISCUSSION

Table I summarizes the molecular weighbs, intrinsic viscosities, and GPC data of the different polymer samples. Intrinsic-viscosity molecular- weight relations were calculated by the method of least squares. The following results found (cf. Fig. 1) rre summarized in eqs. (8)-( 10) ; (dl/g).

Polystyrene (5.1 X lo4 < M < 2 X lo6; TCB, 135°C):

[v] = 1.21 X 10-4M0.707 (8)

TA

BL

E I

Po

lym

er S

ampl

es a

nd D

ata*

App

aren

t n

* z i3

Poly

styr

ene

U z

4190

037

4.11

<

1.10

1.

08

25.0

1.

63

0

Is], d

l/g

E

lutio

n vo

lum

e,

M,.,/

a,,

from

Sa

mpl

e M

. x

10-5

M, x

10-

6 M

,/n;r

, (T

CB

, 135

°C)

5 m

l cou

nts

chro

mat

ogra

m

8 61

970

18.g

b 16

.7(?

)b

<I

.20b

3.

01

21.7

41

9003

8 8.

67

<1.

15

1.98

23

.5

1.77

30

0-13

6.

70

<1.

10

1.57

la

1.60

<

1.06

0.

59

27.1

1.

37

4199

5 0.

98

<1.

06

0.40

28

.3

1.28

41

9004

1 0.

51

<I

.06

0.26

29

.7

1.19

41

9003

9 0.

199

<I .06

0.14

31

.9

1.51

8a

0.

103

<1.

06

0.10

33

.0

1.09

41

9004

0 0.

050

34.6

1.

06

7-11

-11

2.15

" 1 .

15d

0.63

26

.6

6-75

-11

0.87

1

.1

0.33

28

.6

7-23

-111

0.

29

1.1

0.

16

31 .O

7-

25-1

1 0.

14

1.3

0.

10

32 $

4

Z

Poly

-or-

met

h yls

t yre

ne

4.70

0

1.33

Poly

prop

ylen

e fr

actio

ns

I1 1

11

.2e

4.61

21

.9

I1

7.

27

3.61

23

.1

I11

3 4.

58

2.28

24

.0

I4

1.

59

1.05

I1

1 5

1.19

0.

81

26.9

V

6

0.35

0.

36

29.8

I

7

0.18

31

.2

v7

0.

11

0.15

32

.3

PP

436

3.

32

23.1

52

20

4. OO

e 4.

600

4d

2.25

24

.0

Unf

ract

iona

ted

poly

prop

ylen

es

% K

1.06

27

.9

5

5420

3.

06

3.70

>

7 1.

93

25.1

n

5820

2.

16

2.60

>

5 1.

50

26.0

n

HB

242

3 0.

81'

'I C

omm

erci

al p

olye

thyl

enes

1.

56

26.6

Y

1.16

27

.7

>6

0.89

26

.8

12

8 Z i

HM

240

6 0.

53

HM

242

0 0.

50

HM

242

7 0.

37

6 S

uper

scri

pts

refe

r to

the

ent

ire

colu

mn.

R

epor

ted

by s

uppl

ier.

0 F

rm 1i

gh.t s

catte

ring

. d

From

ligh

t sca

tteri

ng a

nd o

smom

etry

. 6 F

rom

mea

sure

men

ts o

f [q

] in

deca

lin a

t 13

5°C

and

[q

] =

1.0

0 X

10-4

M0.*

O. 18

10

From

mea

sure

men

ts o

f [q]

in d

ecal

in a

t 13

5°C

and

[TJ]

=

3.2

6 X

10-4

2.0

1.54

1.

30

1.44

98 COLL AND GILDING

Poly-a-methylstyrene (TCB, 135°C) :

[q] = 1.61 X 10-4M0.672

Polypropylene (TCB, 135°C) :

[ q ] = 1.37 X 10-4M0.75

For the four unfractionated PE samples, under the same conditions, we found

[q] = 4.34 X 10-4M0.724 (11)

Our points for PE seem to agree reasonably well with the data of Boni et a1.l6 These authors characterized their PE fractions in decalin by means of Chiang’s correlation,” [q] = 6.2 X 10-4@w0.70, while we used the corre- lation given by Wesslau,l* [v] = 3.26 X 10-4M0.77. The difference be- tween these two correlations is quite significant. Recalculation of our data by means of Chiang’s equation would shift the points in Figure 1 to the right of Boni’s data points and decrease the slope. In view of the narrow range of Mv covered by our samples, the above Mark-Houwink relationship for PE cannot be claimed to be valid over an extended range of molecular weights.

Samples with Mw < 51,000 were not included because a decreasing slope of log [q] versus log M was indicated a t lower molecular weights, while [q] of these lower samples was not in the range of the PE and PP samples (Fig. 1). In general, our intrinsic viscosities of PS lie slightly above the correspond- ing curve reported by Boni et a1.,16 expressed as [q] = 8.95 X 10-5M,0.727 (dl/g) .

Also here we assumed that M v = Mw. Presumably because of the lower range of molecular weights, a smaller exponent in the Mark-Houwink equation was indicated than in the case of PS.

The expression for PP was derived from all experimental points, in- cluding those of the commercial samples of broad molecular weight dis- tribution. In all cases, was calculated from measurements in decalin (135°C) by means of Chiang’s c~rrelation,’~ [v] = 1.00 X 10-4Mw0.80. The scatter of the experimental points is too large to reveal nonlinearity of the plot in Figure 1. The slope is surprisingly large compared with that for PS, since one would assume TCB to be a better solvent for PS than for PP.

Some question, however, remains with regard to many Mark-Houwink relations reported in the literature. As we noted before, Wesslau’s results1* differ appreciably from those more recently reported by Chiang” for PE, which indicate molecular weights generally higher (by up to 25%). In view of the two most likely sources of error, i.e., chain branching and incomplete clarification of the light-scattering solutions, Wesslau’s expression should

In the case of PS we have assumed that $!ID = M w .

The agreement is good, however. The intrinsic viscosities of PAMS are close to those of PS.

A larger slope is also indicated in the case of the PE samples.

GPC CALIBRATION 99

Fig. 1. PE (from data of Boni et al.l6).

/ \ B STRAIGHT LINES1

I I I # \ / , I I I I I I I I I 20 22 24 26 2a 30 32 34

ELUTION VOLUME, COUNTS

Fig. 2. Chromatograms traced from strip chart.

be given preference, since it correlates given [ q ] with lower molecular weights than Chiang's equation.

Figure 2 shows some typical GPC traces for polymers of narrow and of broad distribution. The position of the peak maximum was defined by drawing tangents through the inflection points of the peak. Some un- certainty as to the exact location of the peak maximum must be conceded in the case of samples of broad distribution. In the last column of Table I

100 COLL AND GILDING

107

+ 106 2

5 M

8

w 3 E

3

2

2 10s

lo" 21 23 25 27 29 31 33

ELUTION VOLUME, COUNTS

Fig. 3. Polts of molecular weight vs. peak elution volume.

apparent ratios of A?lw/Mn are presented. These ratios were calculated directly from the chromatogram. They represent a measure for the broadening of the peaks. Disregarding one sample ( M = 19,900) we found successive broadening with increasing molecular weight. Tailing (skewing) of the peaks was evident in the case of PS standards of high molecular weight.

In Figure 3 the molecular weights of the samples are plotted against peak elution volumes. In the case of PE and PP the molecular weights were viscosity averages determined in decalin: It was assumed that a, is the same in TCB. The molecular weights of the narrow PS and PAMS samples are weight averages. These data are compared with calibration curves calculated by means of eq. (7) after substitution of the respective parameters from the Mark-Houwink equations.

The calibration curve for PAMS (TCB, 135°C) from eqs. (7), (S), and (9) was calculated as

(12)

This curve lies above the PS calibration curve, somewhat outside the range of experimental error indicated by the scatter of points (Fig. 3). The disagreement is small, however. A systematic error of a few per cent in the determination of A?l, by light scattering may be a plausible explanation, since the PAMS and PS samples were characterized in different labora- tories. Some error may also be due to the difference in the molecular

log M ~ A M ~ = -0.058 + 1.021 log M p s

GPC CALIBRATION 101

weight ranges over which the intrinsic viscosities were measured, resulting in a higher exponent (al = 0.707) for PS than for PAMS (0.672). By settingf(e1) = f(e2) in eq. 7, as this would follow from a1 = a2, a slightly better agreement is achieved.

The calibration curve (TCB, 135°C) was calculated as

The results for PP show a greater scatter.

log M p p = 0.0496 + 0.975 log M P S (13)

The agreement with the experimental points seems to be better at law molecular weights. At high molecular weights the peak elution volumes are somewhat larger than calculated. This may perhaps be attributed to differences between PP and PS with respect to concentration effects. Yet, errors in the Mark-Houwink equation appear to be a more likely source for these relatively small differences; this is suggested by the rela- tively large exponent (0.75) for PP. A smaller difference between the intrinsic viscosities of PP and PS at higher molecular weights than shown in Figure 1 would bring the calculated calibration curve for PP into better agreement with the measured values. Possible errors in Chiang's equa- tionlg as well as errors in our own measurements of intrinsic viscosities will be contained in eq. (13). In the case of the unfractionated PP samples, the experimental data may be subject to some error, since we have identi- fied the position of the peak maximum with a,,.

The expression for PE, for the limited range of molecular weights dealt with in this study (TCB, 135"C), was calculated as

log M P E = -0.329 + 0.990 log M P S (14)

Again, any shortcomings of Wesslau's correlation18 and of our own measure- ments of intrinsic viscosities will be reflected in the numerical constants of this expression. The agreement with the experimental points for and broad PE samples is fair. Other authors16-20 have shown that calibration in terms of M[q] is valid for PS and PE in TCB. Hence, our results suggest that the peak maxima, even in the case of broad distributions, should be close to M v . This is to be expected for molecular weight dis- tributions somewhat more asymmetrical than the Schulz-Zimm distribu- tion, provided no distortion of the peak occurs as a result of nonideal behavior in the GPC apparatus. Without further experimental evidence it is hazardous to generalize this conclusion, however, since many polymer samples may not conform to this type of distribution. Furthermore, we have located the peak maximum by means of tangents through the inflec- tion points, yielding a point of intersection which is very much influenced by distortions (skewing) of the peak, as may occur as a result of nonideal behavior.

It should be stated again that the transformation of calibration curves, as expressed by eq. (7) , is not tied to any particular average of the molecu- lar weight. If the primary calibration has been obtained with mono- disperse polymer, then the curve calculated for some other polymer should

102 COLL AND GILDING

likewise refer to monodisperse material. Repeated reference to viscosity- average molecular weight was made because we assumed the peak maxima of polydisperse samples to correspond to i@*, and also because by sub- stitution of the proper @,, in t.he Mark-Houwink equation the parameters K and a remain the same as for monodisperse samples.

CONCLUSION

The results of this investigation, as well as most of the results so far re- ported by other authors, confirm the correctness of the universal calibra- tion based on M[q] or on M[q]/ f (e) . We have given some reasons why, in principle, we prefer the latter expression. Unfortunately, the experi- mental evidence presently available does not allow a conclusive distinction as to which of the two expressions gives better results. A test would re- quire a comparison of two polymers in some solvent such that the expo- nents, a, were significantly different from each other.

It appears that the universal calibration can be used with confidence as long as specific interactions between polymer and gel can be ruled out, and provided the molecular weight of the solute is high enough for random- coil statistics to be valid. Approximations are involved, particularly in the case of branched molecules, but the errors introduced are probably no larger than the uncertainties en countered in conventional molecular weight determinations.

The behavior of rigid molecules, in comparison to flexible chains, deserves further study. The correctness of the Mark-Houwink parameters in the molecular weight range of interest is essential for calculating correct calibration curves. Discrepancies between Mark-Houwink relationships for a given polymer, as they appear in the literature, invite some caution.

The authors wish to thank Miss L. R. Prusinowski for the preparation of the poly- propylene fractions.

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GPC CALIBRATION 103

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Received March 27, 1969