J. Phys. Chem. 1980, 84, 649-652 (22) T. Kakefuda and H.-A. Yamamoto, Proc. Natl. Acad. Sci. U . S . A . , 75, 415 (1978). (23) H. Autrup, C. C. Harris, B. F. Trump, and A. M. Jeffrey, Cancer Res., 38, 3689 (1978). (24) J. Deutsch, J. C. Leutz, S. K. Yang, H. V. Gelboin, Y. L. Chiang, K. P. Vatsis, and M. J. Coon, Prm. Nafl. Acad. Sci. U.S.A.,75, 3123 (1978). (25) G. Feldman, J. Rernsen, K. Shindwa, and P. Cerutti, Nature(London), 274, 796 (1978). (26) T. Meehan and K. Straub, Nature (London), 277, 410 (1979). (27) A. R. Boobis, S.A. Atlas, and D. W. Nebert, Pharmacology, 17,241 (1978). (28) D. H. Phillips, P. 1.. Grover, and P. Sims, Chem.-Biol. Interact., 20, 63 (1978). (29) H. W. S. King, M. R. Osborne, and P. Brookes, Chem.-BiOl.Interact., 24, 345 (1979). (30) W. T. Hsu, D. Sagher, E. J. Lin, R. G. Harvey, P. P. Fu, and S. B. Welss, €l/ochem. Biophys. Res. Common., 87, 416 (1979). (31) V. Ivanovic, N. E. Geacintov, H. Yamasaki, and I.B. Weinstein, 6lochemlistry, 17, 1597 (1978). (32) N. E. Geiacintov, A. Gagliano, V. Ivanovic, and I.B. Weinstein, BiochemCtry, 17, 5258 (1978). (33) T. Prusik, N. E. Geacintov, C. Tobiasz, V. Ivanovic, and I.B. Weinstein, Photochem. Photobiol., 29, 223 (1979). (34) T. Prusik and N. E. Geacintov, Biochem. Biophys. Res. Commun., 88, 782 (1979). (35) A. M. Jeffrey, K. W. Jeanette, S. H. Blobstein, I. B. Weinstein, F. A. Belancl, R. G. Harvey, H. Kasai, I. Miura, and K. Nakanishi, J. Am. Chem. fioc.,98, 5714 (1976). (36) I.8. Weinstein, A. M. Jeffrey, K. W. Jennette, S. H. Blobstein, R. G. Harvey, C. Harris, H. Autrup, H. Kasal, and K. Nakanishi, Science, 193, 592 (1976). (37) A. M. Jeifrey, I.B. Weinstein, K. W. Jennette, K. Grzeskowiak, K. Nakanishi, R. G. Harvey, H. Autrup, and C. Harris, Nature (London), 269, 348 (1977). (38) K. Nakaniishi, H. Kasai, H. Cho, R. G. Harvey, A. M. Jeffrey, K. W. Jennette, and I. B. Weinstein, J. Am. Chem. Soc., 99, 258 (1977). (39) M. Koreeda, P. D. Moore, P. G. Wisiocki, W. Levin, A. H. Conney, H. Yagi, and D. M. Jerina, Science, 199, 778 (1978). (40) S. K. Yarig, H. V. Gelboin, B. J. Trump, H. Autrup, and C. C. Harris, Cancerl?es.,37, 1210 (1977).
640
(41) D. M. Jerina and R. E. Lehr in "Microsomes and Drug Oxidations", V. Uilrich, I.Roots, A. Hildebrandt, R. E. Estabrook, and A. H. Conney, Eds., Pergamon Press, New York, 1977, pp 709-720. (42) R. Lavery, A. Pullman, and B. Pullman, Int. J. Quantum Chem., Quantum Bioi. Symp., 5, 21 (1978). (43) J. D. Scribner and S. R. Flsk, Tetrahedron Left., 48, 4759 (1978). (44) R. Lavery, A. Pullman, and B. Pullman, Theor. Chim. Acta, 50, 67 (1978). (45) R. Stewart and M. G. Harris, Can. J . Chem., 55, 3807 (1977). (46) R. E. Christoffersen, Adv. Quantum Chem., 8, 333 (1972). (47) R. E. Christoffersen, D. Spangler, G. G. Hall, and G. M. Maggiora, J . Am. Chem. Soc., 95, 8526 (1973). (48) R. P. Angeli, S. D. Hornung, and R. E. Christoffersen in "The Proceedings of the Third Internation Conference on Computers in Chemical Research, Education, and Technology" (Caracirs, Venezuela, July 1976), E. V. Ludena, N. H. Sabelli, and A. C. Wahl, Eds., Plenum Press, New York, 1977, pp 357-379. (49) R. E. Christoffersen and R. P. Angeli in "New Worlci of Quantum Chemistry" (Proceedings of the Second International Congress of Quantum Chemistry, New Orleans, April 1976), B. Pullman and R. Parr, Eds., Reidel, Hingham, MA, 1976, pp 189-210. (50) L. L. Shlpman in "Carcinogenesis", Vol. 3, P. W. Joines and R. I. Freudenthal, Eds., Raven Press, New York, 1978, pp 139-144. (51) L. L. Shipman in "Polynuclear Aromatic Hydrocarbons", P. W. Jones and P. Leber, Eds., Ann Arbor Science Publishers, Ann Arbor, MI, 1979, pp 569-580. (52)- G. G. Hall, Proc. R. Soc. London, Ser. A , 205, 541 (1951). (53) C. C. J. Roothaan, Rev. Mod. Phys., 23, 69 (1951). (54) R. Destro, T. J. Kistenmacher, and R. E. Marsh, Acta Crysta/logr., Sect. 6 , 30, 79 (1974). (55) L. L. Shipman, Chem. Phys. Left., 31, 361 (1975). (56) L. L. Shipman and R. E. Christoffersen, Chem. Phys. Lett., 15, 469 (1972). (57) P. B. Hulbert, Nature (London), 256, 146 (1975). (58) S. Walles and L. Ehrenberg, Acta Chem. Scand., 23, 1080 (1989). (59) S. K. Yang, D. W. McCourt, and H. V. Gelboin, J . Am. Chem. Soc., 99, 5130 (1977). (60) P. Lianos and S. Georghiou, Photochem. Photobiol., 29, 13 (1979).
Zero-Shear Viscosity of Some Ethyl Branched Paraffinic Model Polymers Raymond L. Arnett" &Men, Colorado 80401
and Charles P. Thomas' Phlllips Petroleum Company, Bartlesville, Oklahoma 74004 (Received March 19, 1979; Revlsed Manuscript Received September 10, 1979) Pubkation costs assisted by Phillips Petroleum Company
The zero-shear viscosities of ethyl branched, polyethylene-likemodel polymers are presented for four degrees of ethyl branching, each with a range of molecular weights, M , and covering a temperature interval. The temperature coefficientof viscosity for these four structures increases exponentially with degree of ethyl branching and allows a precise prediction of that quantity for unbranched polyethylene (29.29 kJ/mol) and for poly(1-butene) (52.9 kJ/mol). For a symmetrical trichain polymer ( M = 114000), the temperature coefficient is twice that, and the viscosity is 100 times that, for a single chain polymer of the same M.
I. Introduction Of the material constants which influence the flow behavior of liquids, particularly polymeric materials, the zero-shear viscosity, vo, is perhaps the most important single constant for characterizing those materials. Polymers of ethylene, with their differing structures depending on the procerrs of polymerization, provide a good example of how flow blehavior is altered by the comparatively slight structural change of including short branches on the main chain. We hLave examined this effect by measuring the 0022-3654/80/2084-0649$0 1.OO/O
zero-shear viscosity of polyethylene-like model polymers covering ranges of ethyl branches, molecular weights, and temperatures. The structures of these polymers lie intermediary between linear, high-density polyethylene on one side and poly(1-butene) on the other. A short extrapolation of our results each way allows estimates of certain properties of vo for each of these end point polymers. Our data on mixtures strongly suggest that the weight-average molecular weight is not the proper average 0 1980 American Chemical Society
650
Arnett and Thomas
The Journal of Physical Chemistty, Vol. 84, No. 6, 7980
averaging the measurements in the constant region or by fitting all the measurements to the inverse hyperbolic sine function as proposed by Eyring4
VISCOSITY (POISE X IO-')
V(K)
= vo
sinh-l PK
(1)
PK
0.01
0.1
1.0 S E A R RATE (SEC-')
Figure 1. Viscosity vs. shear rate for a model polymer: M = 187 600, 20 ethyl branches/1000 C atoms; (0)130 O C , qo = 6.68 X lo5 P; ( 0 ) 150 O C , qo = 4.48 X lo5 P; (A)171 O C , qo = 2.99 X lo5 P; (0) 195 o c , qo = 1.98 x 105 P.
to represent the molecular-weightdependence of viscosities of mixtures. 11. Experimental Section The monodisperse model polymers included in this study are hydrogenated polybutadienes. The preparation of the parent polybutadienes, their subsequent hydrogenation, and the characterization of the resulting polymers have already been describedS2j3Polymers having four levels of ethyl branching were prepared (20,69, 130, and 183 ethyl branches per 1000 total carbon atoms) with several molecular weights for each structure. In addition to these four series of polymers, this study includes a single sample of a long-chain branched polymer. This polymer is a symmetrical trichain or Y-shaped polymer with the atom at the connection of the three equal-length chains being a silicon atom. In addition to the Y shape, this polymer has 20 ethyl branches per thousand carbon atoms on each of the three long chains. We also measured (at three temperatures) the viscosities of binary mixtures of a high- and low-molecular-weight polymer. The mixtures were prepared by dissolving the weighed components, thoroughly mixing the solutions, and allowing the solutions to evaporate to dryness in a nitrogen atmosphere and, at the end, under vacuum. Triplicate preparations were prepared, one for each temperature run. Viscosities were measured with a previously calibrated Weisenberg rheogoniometer using both cone-and-plate and parallel-plate geometries over a range of shear rates for each temperature. The determination of the viscosity in the limit of zero shear rate was not a difficult task for these monodisperse polymers; indeed it was directly measurable for each polymer. The lowest temperature of measurement was set by the melting point of each polymer; we shied away from very high temperatures for fear of chemical decomposition particularly for the highly branched series. The upper limit of shear rate also was set by practical considerations; namely, the loss of sample from between the two platens.
111. Results Typical viscosity data are shown in Figure 1. Two results are at once apparent from this figure: (i) the viscosity in the limit of zero shear rate occurs at easily measured shear rates, K , and holds its level over a decade and a half of increasing rate of shear before becoming shear-rate dependent; (ii) the range of K in the shear-rate dependent portions is far too small to adequately define q ( ~ ) Depending . on the nature of the results obtained (number of observations in the constant portion of the curve), the zero-shear viscosity was obtained either by
The fitting determines both vo and 0. Here, P is a second material constant having the dimension of time. Because of the low values of K , our measurements do not provide a good determination of 0; only the qo values are discussed here. Tables I-IV (see paragraph at end of text regarding suppelmentary material) list the data for each of the four levels of ethyl branching. The first columns (or first rows) give the molecular weights as determined earlier2J from scattered light measurements while the remaining columns (rows) list the zero-shear viscosities a t the several temperatures. The viscosity values listed are means of those determined from cone-and-plate and parallel-plate geometries. The trichain polymer is given as the last entry in Table I. The mixtures we measured were composed of the lowest and highest molecular-weight polymers in Table I. The results of these measurements are listed in Table V (supplementary material) where the weight-average molecular weight, M,,given in the second column is calculated from the M of the separate components and the composition of the sample listed in the first column. The column heading, w2, is the weight fraction of the high molecularweight component.
IV. Discussion In order to correlate all the data (excluding the trichain) in Tables 1-IV, we use the relation In
qo
= a In M + A(T,n)
(2)
where a is a constant and A contains the temperature and structure dependence of qo; n is the degree of ethyl branching (= number of ethyl branches per 1000 total carbon atoms/1000), and T is absolute temperature. In using (2) we first correlate with respect to M, then with respect to T, and lastly with respect to n. We choose this order because it is well established that the dependence of qo on M is that given by (2) with a close to the value 3.4 and because this procedure allows a value of A to be determined for each temperature even though our data do not cover the entire range of T at each M. In order to obtain the best value of a, we fitted (2) to only the four lowest temperatures of the single chain polymers of Table I where a nearly two decade interval of M was available providing the best measure of a. We fitted also the data of Mendelson et ala5from their measurements, at a single temperature, of fractions of a highdensity polyethylene covering nearly as broad a molecular-weight interval. The weighted mean of these five determinations of a is 3.41 f 0.02. This value for a was then used in (2) to determine A(T,n)for all temperatures of all structures including the zero-branched structure.6 Figure 2 shows the results of this work. The figure is equivalent to plots of In qo vs. 1/T at constant M for each structure but, by plotting A(T,n)instead of In vo, the effect of M on qo is eliminated so that all available measurements (regardless of M) contribute to the temperature effect on q,,. These curves in Figure 2 demonstrate a recognized general viscosity behavior of liquids, when a sufficiently broad temperature interval is available, the relations, In qo vs. 1/T, are not linear. If, with Eyring," we consider the
The Journal of Physical Chemistry, Vol. 84, No. 6, 1980 651
Viscosity of Paraffinic Model Polymers
-2 3
1
f
I
8.6
Y --
8.2
I
8.0 0
I
40
80
120
160
1
I
ZOO
240
I 1
1000n = ETHYL BRANCHES/1000 C ATOMS
Figure 3. Dependence of enthalpy of activation for flow on ethyl branch content.
-321 1.8
, 2.0
, 2.2
2.4
'?00/T , 2.6 2.8
3.0
3.2
equate direct determination of a In q O / a(1/2') for poly(1butene). On substituting (4) as the integrand in (3) we have 3523 A(T,n) = -e2.368n + B(n) T (115 5 t ("C) 5200)
3.4
Figure 2. Temperature and ethyl branch dependence of zero-shear viscosity: (A)rr = 0 ; (0) n = 0.02; (V)n = 0.069; (0) n = 0.130; (0)n = 0.183.
flow of liquids at the zero shear limit as a free energy activated process, then the slopes of the several curves in Figure 2 are measures of the enthalpies, AH, of the process as the temperature and degree of ethyl branching change. Thus we express A(T,n) as
A(T,n) = I [ W T , n ) / R l d ( l / T ) + B(n)
(3)
where R is the! gas constant and B the integration constant. Since each structure could not be measured over the same temperature interval, our data do not allow a complete comparison of the enthalpy changes as the number of ethyl branches changes. However, over a small temperature interval the change of AH with T i s slight so that the average vadue, in that interval will serve as a useful characterizatiion of the structure. To this end, the points included in the solid lines in Figure 2 are used to determine each S I R . The results are plotted in Figure 3. The solid line of this figure is calculated from
a,
m ( n ) / R = 3523e2.36sn
(4)
which gives a very satisfactory fit to the four observables. Extrapolation of a / R on one side to unbranched polyethylene yields 3523 f 5 K and on the other to poly(1butene) gives 6368 f 16 K. The numbers following the f signs are the standard deviations of the values preceding them. The high precision of these estimates is a direct consequence of the goodness-of-fitof the derived quantities to our relation 4, of the large number of primary observations involved, and of the precision of those primary observations. We remind the reader that these estimates are obtained from extrapolation of (a In q o / a (1/2'))* (= AHlR) to the structures indicated. Walesa reports a collaborative value for high density polyethylene which is in excellent agreement with our estimate. Wales' value is also shown in Figure 3 as a triangle. We know of no ad-
as a more explicit expression for A but with a restricted temperature interval. Using (5) we reevaluate B for those temperatures for which (5) is valid. Since now every term except B has been smoothed by correlation, B contains all the experimental uncertainties of M and Q, This fact plus the paucity of data to determine B(n)leaves us undecided as to what its form should be. Thus our data, together with those of ref 5, have determined relation 2 as AH, In qo = a In M + -ebn + B(n) RT (115 5 t
(oc)I200; 3 x 103 I M
I 3
x 105)
with a = 3.41 f 0.02, a / R = 3523 f 5 K, b = 2.368 f 0.008, B(0) = -35.78, B(0.02) = -37.04, B(0.069) = -38.11, B(0.13) = -40.88, B(0.183) = -43.54. All the values of B have a standard deviation of about 0.07 except B(0.02) which is 0.02. The straight line portions of the curves in Figure 2, including the extrapolated value for n = 0, are those given by relation 5 with the values of B above. Relation 6 gives a good account of the observed viscosities of ethyl branched macroparaffins for ethyl branch content varying from zero-branched polyethylene to poly(1-butene). Based on considerations of a variety of structures, van Krevelen and Hoftyzer7 arrived at a relation involving the glass temperature, Tg, to describe the whole In qo - 1/T curve. They noted also that at high temperatures the curves approach straight lines; their expression (corresponding to our relation 4) for these portions of the curves for ethyl branched macroparaffins would take the form m ( n ) / R = 3250(1 + ~ n ) ~ Confidence limits for the 3250 value are not given. If that number were 3523 and c were 0.832, then the relation from van Krevelen and Hoftyzer's work would give a good account of the data plotted in our Figure 3. Trichain EJolymer. Having but a single sample of a trichain polymer, we cannot show a molecular-weight dependence; however, Figure 4 displays the rather dramatic effect on AH resulting from making a polymer with three
652
The Journal of Physical Chemistry, Voi, 84, No. 6, 1980 13.0 0
'25
t
t
//
12*0
/'
0
t
/0
/
Arnett and Thomas
1
I
1
I
12 -
/'
IO -
6
-
8-
c
g,5t
0
t
9,0
8.5 1.9
2.0
2.1
2.2
2.3
2l /
0
1000/ T
2.4
2.5
2.6
-2
Flgure 4. Temperature dependence of a symmetrical trichain polymer compared with a single chain of the same molecular weight: (dashed iine) m = 0,single-chain polymer from (6); (points) m = 2, trichain polymer.
equal chains out of a single chain molecule. The points in the figure are ~,,/100 for the trichain polymer while the dashed line represents vo for a single chain polymer of the same molecular weight as calculated from relation 6. Both polymers have 20 ethyl branches per 1000 carbon atoms. Obviously AH for the trichain is not independent of temperature in this interval. An average value of AH/R is 7360 K compared to 3690 K for the single chain polymer over the same temperature interval. Binary Mixtures. All the data in Table V were used to construct Figure 5. In order to have an error smaller than that of a single observation, the 130 and 171" values were converted to 150" and averaged with the directly observed 150" value, The temperature conversion was accomplished through the temperature relation given above for n = 0.02. These average values are plotted (as circles) in Figure 5 against the weight-average molecular weight of each mixture, including those two monodisperse polymers which are the components of the mixture. The solid line in the figure is that calculated from (6) with n = 0.02, t = 150 "C, and M = M,. The deviations of the averaged points from the line are considerable except for the two end points, the pure component polymers. The deviations are, in fact, considerably in excess of the uncertainty of the points. The data themselves in Table V provide the best measure of their uncertainty. After all measurements are converted to one temperature (150 "C) we have four sets of triplicates for mixtures to provide this measure. The design of the experiment together with the subsequent handling of the data ensures that the plotted quantities are averaged over not only the instrument measurements but temperature and sample preparation as well. The error
8
9
10 In M,
11
12
13
Flgure 5. Zero-shear viscosity, at 150 "C,of binary mixtures compared with mnidisperse polymer, n = 0.020 (soli iine) monodisperse polymer from (6); (points) binary mixtures.
measure just mentioned measures all those errors which affect viscosity: temperature, mixture composition, instrument errors. The value for this is 5.86% for the means of three while the root-mean-square deviation of the four means from the line in Figure 5 is 91.7%! We conclude that the data for mixtures describe a shallow S-shaped curve about the straight line drawn. That is to say, we conclude that the weight-average molecular weight, M,, is not the proper average molecular weight to represent the molecular-weight dependence of the viscosities of polydisperse liquids; some other average, a rheological average, say, is required, A definition of this average is yet to be discovered. The single triangle plotted in Figure 5 is the point for the trichain polymer at 150 "C.
Acknowledgment. The experimental portion of this work was carried out entirely in the Research Laboratories of Phillips Petroleum Company. The viscosity measurements were made by J. W. Hutchins. Supplementary Material Available: Tables I-V which contain zero-shear viscosity data (3 pages). Ordering information is given on any current masthead page. References and Notes (1) Phillips Petroleum Co., Denver, Colo. 80202. (2) C. J. Stacy and R. L. Arnett, J . Phys. Chem., 77, 78 (1973). (3) R. L. Arnett and C. J. Stacy, J . Phys. Chem., 77, 1988 (1973). (4) H. Eyring, J . Chem. Phys., 4, 283 (1938). (5) R. A. Mendelson, W. A. Bowles, and F. L. Finger, J . Po/ymer Sci., Part A2. 8. 105 (1970). (6) J. L. S. Wales, Pure Appl. Chem., 20, 331 (1989). (7) D. W. van Kreveken and P. J. Hoftyzer, Angew. Makromol. Chem., 52, 101 (1978).
