WILLIAM W. GRAESSLEY
2258
The Measurement of Molecular Weight Distribution in Polymers by Cross Linking-Solubility Methods’”
by William W. GraessleyIb Department of Chemical and Metallurgical Engineering, Universitg of Michigan, Ann Arbor, Michigan (Received March 18, 1,904)
The relationship between molecular weight distribution in linear polymers and their subsequent cross linking-solubility behavior has been studied, using samples of polystyrene of different average molecular weights and molecular weight distribution. Films of the polymers were cross linked by y-radiation, and gel fraction as a function of radiation dose was determined by solventJ extraction. The properties of the solubility curves obtained for each sample were sufficient to provide two independent characterizations of -- -the distribution, DP,/DP, and DPJDP,, as well as two radiation parameters, the radiation dose at gelation in the absence of chain scission, and the chain scission parameter.
Introduction This paper describes an investigation to determine whether useful information on molecular weight distribution can be obtained froin the effects of cross linking on the solubility properties of polymers. It is well known that the introduction of random cross links between the chains in a solid polymer alters its solubility properties to the extent that, beyond a minimum cross-linking density, a three-dimensional network is formed, and a uniform solution is no longer formed when the polymer is placed in a suitable solvent. With increasing cross-link density, the amount of the gel phase increases at the expense of sol, and the relative proportion of gel (the “insoluble” fraction) can be determined by solvent extraction. If cross linking alone occurs, the solubility curve depends only on the initial distribution of molecular weights. Equation 1 or its equivalent has been derived by a number of investigatorsz,*for primary chains with large numbers of mers per chain 1- z
=
JomW(n)e-azn dn
High energy radiation is a convenient cross-linking agent for many polymers wince it produces cross links in direct proportion to the total radiation dose ( a = kR). However, main chain scission occurs as well, dso in direct proportion to the total dose. Most polymers therefore require two parameters to describe their radiation behavior : one, the proportionality constant k, relating a and R; the other, a chain scission parameter 0, measuring the number of main chain fractures produced per cross-linked unit formed. These param-
(1)
where a is the cross-link density, the fractional number of mers in the system participating in cross links, (1 - 2 ) = weight fraction of the polymer which is soluble, and W ( n ) is the initial molecular weight distribution, the fractional weight of polymer with degree of polymerization n in the uncross-linked polymer. The Journal of Physical Chemistry
The number-, weight-, and z-average degrees of polymerization are defined in terms of W ( n )in the usual way
(1) (a) Based on a doctoral dissertation submitted to the University of Michigan (1959). and presented in part at the 138th National Meeting of the American Chemical Society, New York, N. Y . , 1960; (b) Department of Chemical Engineering, Technological
Institute, Northwestern University, Evanston, Ill. (2) (a) P. J. Flory, J . A m . Chem. Soc., 69, 30 (1947); (b) A. Charlesby, Proe. Roy. Sac. (London), A222, 542 (1954). (3) A. C. Baskett, Intern. Symp. Macromol. Chem., Milan and Turin, 1954; Ricercu Sci. Suppl., A379 (1955).
nlOLECULAR WEIGHT
O
DISTRIBUTION BY CROSS LINKING-SOLUBILITY METHODS
2259
P 1 I 0
I
0.12
0.08
0.04
0.16
1/R, Mrad-1.
Figure 1. Gel curve for narrow distribution sample N-1.
eters are presumed to be characteristic constants for a polymer and independent of moleaular weight and distribution, The simultaneous occurrence of chain scission changes the solubility behavior by continuously altering the primary molecular weight distribution during cross linking. The modified solubility equation for linear primary chains is4J
made because of the excessive accuracy required of the experimental data to invert eq. 2 or even eq. 1. Calculations of Distribution Parameters from the Solubility Curve. Charlesby and Pinner6 have pointed out that a particularly useful way to analyze the solubilitycross-linking data is to plot [soluble fraction (soluble fractioii)”*]vs. (radiation dose)-I. Figure 1 shows the main features of a typical curve. Below a critical dosage the polymer remains completely soluble, but beyond the gel point dosage Ro, the solubility decreases with a slope So. For large R the solubility (as (1 - z) (1 - x ) ’ ~ ’ )approaches a limiting value 1- with a limiting slope S,. The behavior of the gel curve in this latter region of high cross-linking density may be obtained directly from eq. 2, assuming only that a is linear in R
+
+
Lrn
+
where w(s) = W(.n)e-*“ dn; s = a(x p). This equation has been used to calculate theoretical gel curves for a special clam of initial distributions and for various values of the chain scission ~ a r a m e t e r . ~ The solubility curve as given by either eq. 1 or 2 is dependent primarily on the moments of the initial distribution and only indirectly on the detailed shape of the distribution curve. Thus, solubility measureinents offer some promise of furnishing average molecular weights (actually ratios of_average molecular weights _ _ _ _ _ _ such as DPw/DP, and DP,/DP,) for cross-linkable polymers of unknown (distribution, even though a coinplete determination of the distribution curve cannot be
I,
=
lim [(I - x) R-
m
+ (1 - z)”*J= p
(3)
(4) P. A. Small, J . Polymer Sci.,18, 431 (1955). ( 5 ) ’ M. Inokuti, J . Chem. Phys., 38, 2999 (1963). (6) A. Charlesby and S. 11. Pinner, Proc. Roy. SOC.(London), A249, 367 (1969).
Volume 68, Number 8 August, 1964
WILLIAMW. GRAESSLEY
2260
EF,
z,
and refer to the initial (unirradiated) polymer, and p is a “virtual” gel-point dosage, the radiation dose at which gelation would have occurred in the absence of chain In the presence of chain scission the observed gel-point dose Ro is greater than p ; moreover, p is related to the initial E, __ of the sample by the relationgb: a(ge1 point) = l/DP,. Thus
1 = kDP,
p =
(5)
Although p is not directly measurable, it can be calculated approximately froin the properties of the gel curve iii the vicinity of the gel point. At the gel point, 17: = 0 and eq. 2 beconies
tribution breadths froin essentially monodisperse to somewhat broader than the most probable distribution. Thus, the four equations (eq. 3 , 4 , 8, and 9) provide a method for calculating two radiation parameters, _ _ _ _ p and __ P, and two distribution parameters, DP,/DP, arid DPJDP,, froni the four gel curve properties, Ro, S o , I,, and S,. Strictly eq. 9 requires _ _a _fore_ _ _ _ _sppaking, _ knowledge of DP,+l/DP, for _computing _ _ _ DPJDP,,, but since the tern? containing DP,+I/DP, is small for small P, an approximate _ _ _value, consistent _ _ with _ the apparent trend in DP,jDP, and D P J__ D P @L--should be adequate for obtaining a firm value of DP,/DP,. _ I
Experimental Procedure
Polystyrene wafr chosen for the study because it is easy to prepare and characterize as a linear polymer, and it is cross linked by radiation with few side reactions. One series of samples was prepared by thermal polymerization in bulk to low conversion ( ~ 1 5 % )at various Likewise, the rate of increase of gel with cross-link dentemperatures. Polymers of this type generally consity at the gel point may be evaluated from eq. 2 form fairly closely to the most probable distribution 2a (1 - /3,’2) - w((.p) and are designated hlPD in the tables. A group of (7) = 3p w ( a p ) (p - 1) samples with narrow distribution (designated N in the tables) were prepared by precipitation-fractionation of If ,8 is small and the primary distribution not too one of the bulk polymerized samples. Broad distribubroad, w(a,8) may be expanded in powers of /3 in eq. 6 tion samples (designated B) were prepared by blending and 7, The coefficients of the series will depend on the two samples with widely different average molecular moments of the primary distribution, and these can be weights: B-1 was an equal weight mixture of two MPD expressed as the various DP’s of the system. After samples, and B-2 was the same for two N samples. simplifying and discarding terms of order 82 and higher Molecular weight was measured for all samples by conventional light scattering methods in a Brice1 1 8 Phoenix light scattering photometer. The scattering P Ro so intensity of dilute solutions in benzene was recorded &S a function of concentration and scattering angle, and __ DP, was obtained by the usual Ziinni method. DP, and the distribution curves for the N samples were in which measured by Dr. H. W. McCormick of the Dow Chemical Co. using sedimentation techniques described elsewhere.8 The polymerization and molecular weight data are collected in Table I. The distribution parameters are in Table IV. These last two equations reduce to exact expressions The polymers were cross linked in a cobalt-60 ywhen p = 0, and they are also exact for any 8 in samples radiation source. All irradiations took place in the conforming to~the ‘Lmostprobable” W(n) _ _ _ _distribution: _ central cavity of the source where the dose rate was = azne-an, DP,/DP, = 2, DP,/DP, = 3/2, DP,+~/ approximately 0.8 X lo6rads/hr. ; very precise relative DP, = 4/3. Equations 8 and 9 may thus be regarded as first-order corrections near the gel point for small (7) (a) Charlesby and Pinnera arrlved at the equivalent of eq 3 and amounts of chain scission when the initial distribution 4 by another argument showing that they are generally true even for initially nonlinear chains. They chose to express their result as deviates from the most probable distribution. ComS, = l / k D P , (obtained here by substitution of eq. 5 into)4.qe parison with theoretical gel data calculated for model thereby allowing a calculation for k for samples of known DP,. distributions using eq. 2 indicates they are substantially (b) W. H. Stockmayer, J . Chem. Phys., 12, 125 (1944). correct, for moderate values of p, over a range of dis(8) H. W, McCormick, J . Polymer Sci., 36, 341 (1959).
(Lo
+
I _
+-
~
~
The Journal of Physical Chemistry
226 1
MOLECULAR WEIGHTDISTRIBUTION BY CROSSLINKING-SOLUBILITY METHODS
Table I : Polymerization and Molecular Weight Data for Samples
-Ru Light scatter-
x
10-6Sedimentation
Sample
Preparation
ing
MPD-1
Room temperature
4.93
...
2.05 1.75 1.52 1.44 1.19 3.36 2.69 2.42 1.82 1.77 0.226
...
MPD-2 MPD-3 MPD-4 MPD-5 MPD-6
K-1 N-2 N-3 N-4 5-5 X-6 B-1 B-2
(thermal) 53" (thermal poly.) 60" (thermal poly.) 25" (photo poly.) 65" (thermal poly.) 75" (thermal poly.) Fractionation of MPD-3 Fractionation of MPD-3 Fractionation of MPD-3 Fractionation of MPD-3 Fractionation of MPD-3 Obtained from Don7 Chemical Co. Mixture of MPD-1, MPD-6 Mixture of N-2 andlX-6
3.04 1.46
... ...
... ... 3.68 , . .
2.40 1.77 1.87 0.248
0
0.02
0
0.04
0.06
0.08
1 / R , Mred-1.
Figure 2. Gel curves for several samples having the most probable distribution: A, MPD-2; 0, MPD-4; D, MPD-6.
... ...
doses were obtained by always placing the samples a,t the same location in the cavity. (One series of experiments a t an intensity of 0.08 X lo6rads/hr. showed thak dose rate had no effect on the results.) Films of the polymers, 2-8 mils thick, were cast from benzene solution and evacuated a t elevated temperatures for 1 week or more in an effort to eliminate traces of solvent and dissolved oxygen. Under these conditions sample thickness was found to have no effect on the results. The films were irradiated in vacuo to various totad doses, set aside for about 1 week, and then extracted exhaustively in benzene a t 35'. (Immediate extraction produced erratic solubilities, while delayed extraction gave constant and highly reproducible results.) Other extraction solvents were tried : toluene and cyclohexane gave the same solubilities as benzene, but methyl ethyl ketone appeared to cause a slow, selective rupture of cross links, and equilibrium solubilities were not attainable. Charlesby-Pinner plots of the resulting solubility data are shown in Fig. 1-4.
Results and Discussion The solubility data for all MPD samples gave linear graphs as shown in Fig. 2, which is consistent with eq. 2 requiring that distributions of the form W ( n ) = a2. ne-an have solubility curves of the form
I 0.06
I
0
0.02
0.04
I 0.08
1/R, Mrad-1.
Figure 3. Gel curve for narrow distribution sample N-5.
01
0
I
0.02
I
I
0.04
0.OG
1 / R , Mrad-1.
Figure 4.
Gel curve for broad distribution sample B-2.
Thus p and @ are iminediately obtained froin the slope and intercept of the solubility curve, and its linear form confirms the expected distribution. Gel curves for the N sainples (narrow distribution) and the B samples (broad distribution) were not linear, Volume 68, Number 8 August, 1.968
WILLIAM W. GRAESSLEY
2262
and in fact they were distinguishable by their qualitative behavior: the solubility curves oi the former show a positive (upward) curvature, while the latter show a negative curvature. The solubility parameters required to determine DP,/DP, and DP,/DPw for these samples are shown in Table 11. _
_
I
~~
_
Table I1 : Measured Gel Curve Constants for the Karrow and Broad Distributive Samples
so,
€20,
Sm,
Sample
h'lrrtds
Mrads
Im
lVMrads
S-1
6.19 9.00 12.3 13.4 7.73 19.7
12.2 19.5 29.3 27.6 9.74 23.9
0.37 0.35 0.34 0.34 0.37 0.34
6.00 9.55 11.5 17.0 21.9 60.7
N-3 N-4 N-5 B-1 B-2
independent estimate of k, and a value of (0.55 rt 0.2) X lo-" was found, agreeing with our other results. The source of our disagreement with the other workers is not known. Except for p in SIPD-1 the variations in the radiation parameters from polymer to polymer are small and quite consistent with the experimental errors in the measurements. The slight trend in both p and k with molecular weight appears to be real, but is not large enough to affect the results materially. The distribution results for the N and B samples are shown in Table IV together with a comparison of values Table IV : Comparison of Distribution Parameters from the Gel Curves with Those from Sedimentation Measurements
Sample
~
_- _
- _
N-1 N-3 N-4
_
_
Before with the calculation of DP,/DP, _ _ proceeding __ and DP,/DP,D,it is worthwhile to examine the assumption that k and p are true constants of the polymer: that cross linking and chain scission are both proportional to radiation dose and independent of average molecular weight and molecular weight distribution. Table I11 shows values of p , p, and k , calculated from
Table 111: Calculated Radiation Parameters for Polystyrene
N-5 B-1 B-2 a
1.08 1.12 1.15 1.06 3.26 3.72
_
DP~/DP,--
DP,/DP,,-SedimenGel tation curve
7 -
1.15 1.24 1.48 1.07 3.66 3.94
Sedirnentation
Gel curve
1.06 1.09 1.11 1.05 2.05 2.15
1.19 1.12 1.18 1.04 1.99 1.89
_ _
r,,1/ DPSa
1.1
1.1 1.1 1.0 1.5 1.3
Values in eq. 9 to compute DPJDP,.
_ _ _
____
of DP,/DP, and DP,/DP, from the ultracentrifuge (N samples) or from the known composition of the mixare the approximate tures (B samples). Also shown __ values selected for DP,+I/DP, to compute firm values ~ _ _ of D P J D P , in eq. 9. In practice, it is quite easy to select reasonable numbers since a series of molecular _ _ _ _ _ _ _ _ _ _ weight ratios such as DPw/DPn, DPJDP,, DP,+1/ -_ DP,. . . generally takes on a smooth and systematic series of values. For example, X-4 has a firm (calculnted) value _ -of DP,/DP, = 1.48. An approximate value for DP,/DP, can be obtained by neglecting the term containing DP,+JDP, in eq. 9. ~
Pl
k , rad-1
Sample
B
Mrads
x
MPD-1 MPD-2 MPD-3 MPD-4 MPD-5 MPD-6 N-1 N-3 N-4 x-5 B-1 B-2
(0.49) 0.34 0.38 0.35 0.36 0.35 0.37 0.35 0.34 0.34 0.37 0.34
3.85 10.3 11.0 13.5 13.7 16.3 5.20 7.75 11.5 10.8 6.0 15.4
0.55 0.49 0.54 0.51 0.53 0.54 0.59 0.55 0.50 0.54 0.57 0.46
10"
~
-
_
_
.
I
-
,= 3p -D-P _ _- DP, so
(3)(11.5) (27.6)
=
1.25
_ _ _ _
the data. The average value of the chain scission parameter (excluding MPD-l), p = 0.35 + 0.015, is the same as measured by Schulzgfor polystyrene using high energy electrons. However, the cross-linking constant, k = (0.53 i 0.04) X rad-I, is a t some variance to the value of 1.23 X calculated from the Schulz data, and 1.1 x 10-'1 from Wall's data.IO Swelling measurements were made on the gels to obtain an The Journal of Physical Chemistry
of ~decrease in the series DP,/DP, = 1.48, _ The rate _ D P J D P , N 1.25 suggest that the next term, F,+,/ DP,, should be about 1.1 assuniing the series is smoothly approaching 1.0. When this value is substituted in eq. 9 the new result is DP,/DP, = 1.18, and since ~
~~
(9) A. R.Schula, P. I. Roth, and G. V. Rathmann, J . Polymer Sci., 22, 495 (1956).
(10) L. A. Wall and D. W. Brown, J . Phys. Chem., 61, 129 (1957).
2263
COXTINUOUS ABSORPTION SPECTRA OF HALOGESS AN) ISTERHALOGEK COMPOUNDS
Z z + l / D T_=_ 1.1 _ _is still consistent, 1.18 is the firm value for DP,/DP,. The relatively minor effect of ____ various values for DP,,l/DP, on DP,/DP, is shown by the comparison for sample 5-50 shown in Table V. _ I
Table V Resulting
Assumed _ _
D P , +JDP.
DPJDP, (es. 9)
1.3 1.2 1.1 1.0
1.24 1.21 1.18 1.15
Conclusions The comparisons in Table IV sh0.w that information on the molecular weight distribution of a linear polymer can be obtained from an analysis of its cross linkingsolubility behavior without assumptions as to the particular form of the distribution, if the chain scission pa-
rameter is small. Apparently, _ _ -there is some tendency toward overestiniation of DP,/DP, for narrow distributions and underestimation for broad distributions. This may reflect some systeniatic error in measuring S , because of curvature, or more likely, it may mean that the simple, first-approximation equations are unable to compensate completely for the effect of chain scission. However, the discrepancy is generally small and the numerical agreement quite satisfactory. It - _ _ should be noted also that DP,/DP, and D P J D P , are obtained from quite different portions of the gel curve and thus constitute independent tests of the data. -
_
.
Acknowledgments. The author is grateful to the Yational Science Foundation for predoctoral fellowship support during 1966-1959. Thanks are due also to Dr. R. E. Skochdopole and Dr. H. W. il4cCormick of the Dow Cheniical Company, to Professor Brymer Williams and Professor D. W. McCready for their many helpful suggestions through the work, and to Dr. L. M. Hobbs and Dr. J. A. Manson whose interest originally led to the study.
The Continuous Absorption Spectra of Chlorine, Bromine, Bromine Chloride, Iodine Chloride, and Iodine Bromide
by Daniel J. Seery and Doyle Britton Department of Chemistry, University of Minnesota, Minneapolis 14, Minnesota
(Received March 16, 19Sg)
The continuous absorption spectra for gaseous Clz, Br,, BrCl, ICl, and IBr between 220 and 600 mp litre reported. The experimental data have been fitted to theoretical curves of the type suggested by Sulzes and Wieland. The parameters for these curves are given.
Introduction In the course of shock tube studies of the interhalogen compounds and the dissociation of the halogens, we have found it necessary to know the extinction coefficients for IC1, BrCl, Brz, and C1, in the visible and
near-ultraviolet. Since the literature values for IC1 and BrCl are incomplete O r appear to be incorrect, we have determined the continuous absorption spectra of these compounds, as well as IBr, between 220 and 600 mp. Br2 and Cla have been redetermined in order to Volume 68, Number 8 August, 1964
