G. C. BERRYAND T. G F o x
3540
Vol. 86
can use these ratios and eq. 68, 69, 72, and 73 (putting = 0) to deduce the remaining four unknowns in these equations: K , a s h , k,, and m. Froin eq. 35, one can obtain k s h . Equation 12 will then give S s h as a function of the concentration. Equation 67 will yield n, and eq. 71 will give aL (assuming that cys has been estimated by means of eq. 5). From K and k s h , one can (eq. 33) determine the product kextkmc, but not the individual values of these two factors. Likewise, from eq. 34, one can obtain the product uz'Ae, but not uZ1and A € separately. In this way, one can determine, from experimental data, all the constants needed for substitution into the The p in eq. 71-73 is defined by the equation closed-form equations for ARI and ASl (eq. 28 and 54). If 0 is not negligible, the more complicated equations, p = - nrksk, (74) containing this quantity, must be used; another exuz I perimental quantity, such as $,,"'/$ao or bX"l/bX0, is I t is possible that some of the quantities assumed needed, unless one or more of the "unknowns" can be in the foregoing derivations to be independent of temdetermined or estimated in another way. perature do in fact vary with the temperature. If so, Testing of the equations presented in this paper, usappropriate adjustments must be made in these relaing published experimental data, has been begun. The tionships. results will be reported in due course. If the theory The functions deduced for X h and $s show a very and its equations should prove satisfactory, it will be similar dependence on concentration. (Compare eq. possible, from a relatively small amount of experimental 28 with eq. 59, also 31 with 62 and 32 with 63). Theredata, to deduce curves for the variation of the thermofore Xh and xs should vary oppositely, a behavior that dynamic properties of polymer solutions up to quite has often been noted empirically. In certain systems high concentrations. (At very high concentrations the relationship is approximately a rectilinear 0ne.7312,13 certain other factors6 not considered in the present The accuracy of the rectilinearity probably depends development may become important.) It is hoped, on the magnitudes of certain of the parameters. moreover, that the theory will lead to the determination of the various molecular constants which affect the Correlation with Experiment thermodynamic solution properties and so eventually From precise experimental data one can determine x and its dependence on concentration and tempera- to a better understanding of the phenomena and an ability to predict the properties of new systems. ture, hence values of $ao, h,O, and the ratios $ a ' / $ O , An extension of this theory to solutions of graft and J / a " / J / a o , b,'/b,O, and b,"/b:. If p is negligible, one block copolymers has already been published.I4 3!,
(12) H. T a k e n a k a , J. Polymer Sci., 24, 321 (1957). (13) G . Rehage a n d H. M e y s , ibid., SO, 271 (1958).
(14) M. L. Huggins. $bid., C1, 445 (1963).
[CONTRIBUTIOK FROM
THE
MELLONINSTITUTE, PITTSBURGH 13, P E N N S Y L V A N I A ]
Viscometric Tests of Excluded Volume Theories BY G. C. BERRYA N D T. G F o x RECEIVEDAPRIL15, 1964 Some aspects of viscometric tests for excluded volume theories are discussed. Attention is confined t o those systems for which the intrinsic viscosity and the molecular weight are known, possibly as a function of temperature. I t is concluded that definitive evaluation of the various theories requires d a t a a t higher molecular weights than normally studied or over a large temperature span.
Introduction I t is our purpose here to comment briefly on some aspects of viscometric tests for the various theories of the excluded volume effect in polymer coils. We will restrict our attention to studies for which only [77] and M are known, possibly as a function of temperature. Viscometric studies have recently received renewed interest because of new theoretical developments in both the hydrodynamic and thermodynamic aspects of the problem. I t is an over-simplification to consider these aspects separately, but the theoretical developments which attempt to include both effects simultaneously are still a t an early stage. l P 3 M K u r a t a a n d H Y a m a k a w a , J . C h e m . P h y s . , 29, 311 (1958). ( 2 ) 0 B. P t i t s y n and I. E . Eizner, Z h . F i s . K h i m . . 92, 2464 (1958).
(1)
Thermodynamic Effects on the Intrinsic Viscosity The molecular description of the hydrodynamic flow has been understood in terms of equivalent models advanced by Debye and B ~ e c h e Kirkwood ,~ and R i ~ e m a n and ,~ and this description has been utilized to form the basis of approximations to include thermodynamic (excluded volume) effects.1 , 2 , 8 , 9 A principle result of the hydrodynamic calculations (3) See, for example, t h e review: A. Peterlin, M a k r o m o l . C h e m . , 94, 89 (1959). (4) P. Debye a n d A. M . Bueche, J . Chem. P h y r . . 16, 565 (1948) (5) J. G. Kirkwood and J. Riseman, ibid., 18, 565 (1948). (6) H. C. B r i n k m a n , P h y s i c a , 19, 447 (1947). (7) B. H. Zimm, J . C h e m . P h y s . , 1 4 , 269 (1956). ( 8 ) T. G Fox a n d P. J. Flory, J A m , C h e m . Soc., 'IS, 1904 (1951). (9) M . K u r a t a and W. H Stockmayer, A d v a n Polymcr Sci.. 9 , 196 (1963).
VISCOMETRIC TESTS OF EXCLUDED VOLUME THEORIES
Sept. 5, 1964
354 1
is that [ q ] should depend on some equivalent chain volume, a t least for large enough molecular weights6 (more specifically for large enough h).Io I t has long been recognized that the principal effect of increasing thermodynamic interaction is to augment the chain extension, and thus the equivalent volume, by some factor. Accordingly, one writesg
where
a'
=
6'/'hF(h)
A 3 = (S2/M)Oa/' Here, the parameter h and the function F ( h ) are left unspecified except that hF(h) increases with M to within 1% of its asymptotic limit (2.89 X loz3)' for h > 3 and that this limit is apparently reached for most flexible polymer-solvent systems, a t least for M > ca. lo5. The ratio of the mean-square radius Sz to the molecular weight M is to be taken under (denoted by subscript 0). Equation 1 may then be considered to be a defining relation for the hydrodynamic expansion factor a,,. It should be noted that a,,is not necessarily equal to a, the expansion factor for the mean square radius. Three expressions for a, in terms of thermodynamic variables will be considered here. (A fourth,I3 numerically quite close to eq. 3," will not be discussed separately.) These are 1519
a?
-
g/fi
a?
=
alBA-3M1/'
(2)
where al = (134/105) for linear chains. The parameter B is related to the magnitude of the thermodynamic interaction and may be taken as proportional to the function $ 1 ( 1 - O/T) of Flory,I4 for example. Here p may be taken as 3 or "2. Equation 2 with = 3 is the familiar Flory-Fox8 equation, and is the Flory-Fox equation as modified by Kurata The difference rests and Yamakawa' when p = on the approximation that a? = a for the former, and a,,3 = as/'for the latter. Similarly, eq. 3 results from combination of an equation due to Fixman'5 for a ( T , M ) and the above mentioned approximations for CY, as a function of a. Equation 4, due to Stockmayer and Fixman," is similar to eq. 3, but rests on a somewhat different intuitive notion. Evaluation of Experimental Data dependence of a,,3 on BA-3M'/' according to eq. 2 4 may be seen in Fig. 1 . (The curves for p = lie slightly below the corre-
A. Isothermal Data.-The
(IO) T h e notation is chosen to be consistent with recent usage.9.ll In particular, a definition of the hydrodynamic parameter h may be found in ref. 7. (11) W.H.Stockmayer and M. Fixman, J. Polymer S c i . , C1, 137 (1963). (12) P. J. Flory and W. R. Krigbaum, J. C h e m . P h y s . , 18, 1086 (1950). (13) M.Kurata, W.H. Stockmayer, and A. Roig, i b i d . , 8 8 , 151 (1960). (14) P. J. Flory, i b i d . , 17, 303 (1949). (15) M.Fixman, i b i d . , 86, 3123 (1962)!
Fig. 1.-The hydrodynamic expansion factor a,,'V S . the thermodynamic parameter BA -sd%according to Flory-Fox (eq. 2; p = 3), Flory-Fox and Fixman (eq. 3; p = 3), and StockmayerFixman (eq. 4).
sponding curves for p = 3 in each case.) The striking difference of the products B A - 3 predicted by the different relations for any given aV3and V%M is apparent. Also apparent, however, is the functional similarity vs. B A - 3 d % for selected intervals of the curves of aV3. For example, if experimental data are confined to the interval 2 < a,,3 < 5 , as in often the case for polymer with M > lo6 in good solvents, then any of the curves can be represented by a relation of the form ar13= a rnBA-%& to within a few per cent, where a and m are empirical constants. Thus, attempts to analyze the three relations on the basis of experimental aV3vs. M over a restricted range of a,, require precise data if they are to succeed. For example, if in the interval 2 < < 3.6 (which corresponds to some data for polymethyl methacrylate in toluene a t 6Ool6)eq. 2 ( p = 3) is presumed accurate, then the values of B A - 3 computed from eq. 4 will be constant to within *20% over the same interval. A systematic error in computing BA-3 such as could reside in an empirical [7]o v s . M relation, for example, could easily cause this large a discrepancy. T o be sure, estimates of B A according to the various relations 2-4 will yield different values for B A - 3 , but unless one has a priori knowledge of B A P 3 this will not suffice to obtain the correct functional form for a,,. Similarly, attempts to compute A from viscometric data confined to systems where B is large (good sol-
+
(16) T.G Fox, P o l y m e r , 8 , 111 (1962)
G. C. BERRYAND T. G Fox
3542
Vol. 86
vents) by extrapolation t o M = 0 according to apthe relation appropriate for large M before attempting propriate rearrangement of any of the above relato extend the treatment to vanishingly small M . t i o n ~ ~become ~ " ~ suspect ~ ~ without a priori knowledge B. Temperature Dependence.-In some cases [v ] of which is the correct expression. For example, if is known as a function of both molecular weight and eq. 2 ( p = 3) is presumed correct, but the data are temperature. Then, if A 3 (e.g., [?IO), and its temperaplotted in a manner appropriate to eq. 4,11 then nearly ture dependence, are known as well, the value of H computed from any of eq. 2 4 may be examined straight lines will result until fi (or more correctly to yield parameters such as Flory's Gl and 0.I' I n this BA-3V'%) becomes small, but this line will yield too case, B is plotted against 117' and the intercept for large an intercept, and hence too large a value for .A3. B = 0 yields 0 while the slope is proportional to'$1& This may be readily seen by examining the curve for T h e plot should be linear2324and data for different eq. 2 in Fig. 1 over the interval 2 < aV3< 7. A linear molecul2r weight polymers over the same temperarelation will be seen to provide reasonable fit to the ture interval should all superpose to yield a single curve data in this restricted interval (at least to within usual if the correct relation is being used (see, for example,' experimental error) but such a line will not yield the data given in ref. 16). Unfortunately, the curves a, = 1 at B A - ~=~0.ZConversely, if eq. 4 is B us. 1,lT calculated from eq. 2 4 do not usually differ presumed correct and the data are plotted according to much in shape over the temperature spans normally the format appropriate to eq. 2, then data extrapolated available. Thus, evaluation rests on comparison of linearly from large values of d% will yield too low an $1 and/or 0 obtained from this viscometric analysis intercept (too small a value for .A3). In either case, with values obtained elsewhere, such as from light curvature is expected only when passing from high to scattering or osmotic pressure studies. This type of so t h a t the actual shape is low values of Bk3* study does seem to provide promise for eventual quite dependent on 8 . evaluation of eq. 2-4, but it lies outside our stated Attempts to extend these treatments to lower area of interest here, viz., data confined t a viscosity molecular weights, where presumably the curvature and absolute molecular weight, and so will not be mentioned above will be significant and possibly useful pursued further here. 25 in deciding which is the appropriate relation to use in Mark-Houwinl-Sakurada Coefficient extrapolation, meet with another uncertainty. InWe now wish to examine the coefficient v in the emspection of values of BA -3 for a variety of systems over pirical Mark- Houwink-Sakurada equation wide intervals of M , in particular at very low values of M (< cu. lo5),invariably reveals trends in B k 3with [7]= K M ' (5) M regardless of which of the eq. 2-4 is used to compute BA-3.'s This variation of B A - 3 with M may be due in terms of relations 1-4. Combination of eq. 1-5 to an increasing nongaussian character of the chain as yields v as a function of cy? for each of the eq. 2-4, M decreases, and so would not be unexpected. l7 denoted, respectively, by subscripts 2-4 in eq. 6-8.2' It has been found experimentally t h a t [7l0 is proM 4 X lo6 for the portional to hf'' for 2000 This suggests system polystyrene-cyclohexane. that the parameter h F ( h ) in eq. 1 has indeed attained its asymptotic value over the entire span of M . Other(7) wise, according to calculations of Debye and B ~ e c h e , ~ Kirkwood and R i ~ e m a n and , ~ other^,^,^^ [7]0 should 1 1 cy$3 - 1 gradually change from dependence on M1/' to dependv4= -+-ence on dd with decreasing M through the molecular 2 2 ff,3 weight dependence of h F ( h ) . Unfortunately, however, Figure 2 shows v as a function of c y q for the four exthis may not be regarded as a unique conclusion. Other pressions t h a t resultz7 (v4 and v3 ( p = 3) are identical). possibilities include dependence of B and/or A 3 on T h e most striking difference between calculations iM for low M Z 2 or an increasing nongaussian character based on Flory's14 expression for a and those based of the chain which has an unknown effect on the hydroon Fixman's relation l5 (or relations numerically simidynamic equation for [VI,. In any case, the constancy lar13) is the prediction of the latter that v should exof [7]o/ML'2together with variation of B k 3 with M ceed ca. 0.75 for systems for which a, 2 1.3, whereas as calculated from a? indicates t h a t there may be 0.75 only at much the former would predict v effects specific to low molecular weight polymers that larger cy,, and in fact places an upper bound of 0.80 have not yet been identified. I n view of these unon v . Typical "good solvents" have a, in the range certainties, i t would seem preferable t o establish first
0.75 are expected according to eq. 3 and 4 whenever a > 1.5, regardless of what the value of B might be. Clearly, however, the larger b the smaller M is predicted to be for the appearance of v > 0.75. Data in the usual systems apparently require fairly large M (> say lo6) for a q 3to be greater than about 1.8. This means that experimental determination of u becomes difficult for a variety of reasons, including imprecise measurement of M for very large M (>lo7), fairly large shear corrections to [ q ] , which decrease the measured values of [ q ] as M increases, thus yielding too low values for v,
M . 29 The variation of aq3 with temperature discussed above also is reflected in a dependence of Y on temperature; and, provided a large enough temperature interval can be spanned, this dependence can also be utilized to evaluate eq. 6-8. The experimental observation that u tends to 0.5 in good solvents as M decreases is in accord with any of eq. 6-8 i f the molecular weight dependence of hF(h) may be ignored and gaussian statistics assumed to be valid. A considerable body of experimental data does in fact exhibit this type of behavior (see, for example, ref. 9 or 19). The molecular weight M* for which v becomes 0.5 (to within experimental error) depends of course on the product B A P , M* increasing as B A - 3 decreases. The nature of the expected transition may be seen in Fig. 3. Here'a reduced plot of [ q ] v s . A4 is constructed for each of the equations 1,2 and 1,4. It is seen that the transition occurs over a narrow enough span in M to allow a very large span in M to be correlated by two expressions of the Mark-HouwinkSakurada type with two different sets of constants K and v , K and v changing a t some molecular weight M* which depends on B A - 3 (the transition occurs for BA - 3 v ' X X ? ca. unity). Figure 3 may be useful in itself to obtain an estimate for B if A 3 is known (from measurement of [ q l O / d % for example). It could serve as a master curve for 11with one of the theoretical comparison of [?]/@'Avs. 1 relations to obtain B A + and hence B. Thus, the graph of the experimental data is shifted relative to the master curve, Fig. 3 (two units of horizontal shift for every one unit of vertical shift), to obtain B A - 3 as the shift factor required for coincidence of the two graphs. This method has the advantage of allowing one to obtain a simultaneous fit of all of the data by inspection. Equations 1-4 may also be used to obtain an estimate of K and v simultaneously from a single determination of [ v ] and M in a good solvent provided a43is known. This subject has been discussed more extensively by
(28) Several examples of systems for which Y is n e a r u n i t y given by K u r a t a a n d Stockmayers h a v e been o m i t t e d from Fig. 2 since values of Y quoted f o r these systems vary widely. (See M . M. H u q u e , D. A. I. Goring, and S. G M a s o n , Can. J . C h e m . , 36, 952 (1958); A . M . Holtzer, H. Benoit, a n d P. D o t y , J. P h y s . C h e m . , 58, 624 (1954).) I n addition, these are systems for which t h e polymer is normally considered t o behave a s a stiff chain a n d t h e analysis required i s beyond our scope here.
(29) A low shear r a t e viscometer has been constructed according t o t h e design of Zimm a n d Crothers, in connection with a s t u d y on which one of us (G.C R.) is currently engaged. and it will probably he used t o study some high molecular weight polymers ( c a . 107) in a n u m h e r of p w d solvents f o r which extensive investigation a t lower molecular weiKhts h a %heeri perfrmned in an effort t o determine Y a t large a?: 8.H Zimm and 1 ) hl Crotheri.. Proc. N a l i . A c n d . Sci.. 48, 905 (19C,2).
but experimental accuracy seldom permits this. TABLE I 4 Y Ref. Chloroform 3.1-7.9 0.80 a Dichloro4.4-5 3 0 . 7 6 u ethane a Methyl methacrylate Methyl meth2.8-3.5 0.72 acrylate Styrene Toluene 1.4-3.5 0.71 b 1.5-3.5 0.75 19 Styrene Benzene 1.6-2.5 0 66 c Isobutylene Benzene 2 6-3.5 0.71 d Vinyl acetate Bu tanone 2.2-3.2 0.67 e Vinyl acetate Benzene 1.7-2 2 0.62 e Vinyl acetate Trichlorobenzene Vinyl acetate Methanol 13-0.4 060 f a E. Cohn-Ginsberg, T. G Fox, and H. F. Mason, Polymer, 3, 97 (1962). * P. Outer, C. I. Carr, and B. H. Zimm, J . Chem. Phys., 18, 830 (1950). T. G Fox and P. J . Flory, J . Phys. Colloid Chem., 53, 197 (1949). R. 0. Howard, Doctoral Thesis, Massachusetts Institute of Technology, 1952; and G. C. Berry, A. R. Schultz, J. A m . Chem. Soc., 76,3422 (1954). L. M . Hobbs, and V. C. Long, Polymer, 5, 31 (1964). I M. Matsumoto and Y. Ohyanagi, J . Polymer Sci., 46, 441 (1960). Polymer
Solvent
Methyl methacrylate Methyl methacrylate
F. W. BILLMEYER, JR.,
3544
Orofino and (Equation 6 (9 = 3) was also obtained by these authors.)
AND
V. KOKLE
Vol. 86
range (with a different value of BA - 3 for each relation to be sure) with acceptable precision, but still yield quite different correlations a t higher (or lower) molecular weights. Indeed, [ 7 ] - M data on the system polystyrene-benzeneIg correlate with either eq. 2 ( p = 5 / 2 ) or with eq. 4 with about the same precision over the molecular weight span lo4 < M < 2 X lo6, although the values of so obtained differ by a factor of 3. These calculated [ 7 ] os. M curves differ considerably, however, by the time M is ca. lo7. It is suggested that the most reliable evaluation of viscosity-molecular weight relationships may involve examination of [7]-M data as a function of temperature for systems where there is a priori knowledge of the thermodynamic parameter B (as well as A 3 ) as a function of temperature, that is, knowledge of $, and 8. Further, it is suggested that examination of the MarkHouwink-Sakurada coefficient v a t high molecular weight (high CY,,) should provide a basis for a reasonable choice between a t least some of the relations 2-4. Acknowledgment.--We acknowledge the assistance of R. E. Kerwin on some of the numerical calculations required a t certain stages of this evaluation, This study was supported in part by the Research and Technology Division, Air Force Materials Laboratory, under Contract No. AF 33(657)-10661.
Conclusion In summary, we are forced to conclude that no definitive evidence for the unique applicability of any of the eq. 2-4 has yet been exhibited on the basis of combined viscometric and absolute molecular weight data alone. This inadequacy may be traced directly to failure of a n y of the theories to describe adequately the data for low molecular weight ( M < lo5) systems. This uncertainty makes analysis of data restricted to good solvents to determine the thermodynamic parameter B and/or the chain extension parameter A 3 hazardous since the values obtained depend directly on the extrapolation procedure employed which in turn depends on which of the eq. 2-4 are used. Similarly, attempts to extrapolate [ v ] os. M correlations to molecular weights higher than those explicitly studied are liable to yield erroneous results unless the correct set of relations is used. T h a t is, any of the relations 2-4 could very well fit a given set of data in a good solvent over a restricted molecular weight (30) T. A. Orofin0 and T. G F o x , Aeronautical Systems Division Report ASD-TR 61-22. Part I , United States Air Force, Wright-Patterson Air Force Base. Ohio. -
[CONTRIBUTION FROM THE PLASTICS
DEPARTMENT, E. I. DU
PONT DE
NEMOURS AND
CO.,
INC.,W I L M I N G T O N ,
DELAWARE]
The Molecular Structure of Polyethylene. XV. Comparison of Number-Average Molecular Weights by Various Methods' BY F. W. BILLMEYER, JR.,
AND
V. KOKLE
RECEIVED APRIL 2, 1964 Number-average molecular weight was determined for samples of both linear and branched polyethylenes by at least two of the thermodynamic methods of osmometry, cryoscopy, and ebulliometry, the kinetic method
of thermoelectric differential vapor pressure, and the chemical method of end-group analysis. Within the known limitations of each technique, the agreement among methods was excellent, the results of three different methods often falling within a range of less than 570.
Introduction Measurement of the molecular weight of high polymers has been a subject of great scientific and technological importance ever since the macromolecular nature of these compounds was proved in the early 1920's. Very few new techniques for these measurements have been developed since that time, however, the latest of these with a thermodynamic basis being the light scattering method of Debye2 yielding information on particle size as well as the weight-average molecular weight and second virial coefficient. More recently, a kinetic method has been proposed3 for the determination of the number-average molecular weight. and several older techniques have been improved or applied for the first time to random-coil polymers. This paper considers several methods for determining the number-average molecular weight in application ( I ) Previous papers in this series are published in J . A m . Chem SOL.. 76, 6110 ( 1 9 5 3 ) ; 79, 5079 ( 1 9 5 7 ) ; 81, 3219 (1959), J . P o l r m e r Sci., 61, 251 ( 1 9 6 1 ) ; A l , 1921 ( 1 9 6 3 ) . and an article in press; and J . P h y s . C h e m . , 6 7 , 2728 (1963). (2) P. Ilebye. J . A p p i . P h y s , 16, 338 (19441, J P h y s . Colloid C h e m . , 61, 18 (1947). (3) K. Pasternak, P. Rrady, and H . Ehrmantraut. paper presented at the A C H E M A 1961, Frankfurt a m Main. Germany, 1961.
a t elevated temperature to branched and linear polyethylene. These techniques include osmometry, cryoscopy (as described in paper XI11 of this series4) and e b ~ l l i o m e t r y ,the ~ above-mentioned thermoelectric differential vapor pressure m e t h ~ dand , ~ the chemical method of end-group analysis by infrared spectroscopy. The purpose of this research was to define the range of applicability of,each of the above methods and to demonstrate the validity of the results of kinetic and chemical methods in application to the polyethylene system. This involves comparisons among the results of the various methods; other comparisons of this type were presented in paper X I of this seriess for number-average molecular weights determined by osmometry and cryoscopy and in paper XIV' for weight-average molecular weights determined by light scattering and by Archibald ultracentrifugation. ( 4 ) E. J . Newitt and V , Kokle, J , Poiymer Sci., in press. (5) M. Ezrin and G. C . Claver, Preprints, Division of Polymer Chemis try. American Chemical Society, Vol 3. KO.1, March, 1962, p, 308. ( 6 ) V . Kokle, F W. Billrneyer, Jr., L. T. Muus, and E . J. Kewitt. J . Polymer S c i . . 6 1 , 251 (1962) ( 7 ) N E . Weston and F . W . Billmeyer, J r . , J P h y s . C h e m , 61, 2728 (1963).
