G. C. BERRY
1194
The Viscosity of Polymer-Diluent Mixtures
by G. C. Berry Mellon Institute, Pittsburgh, Pennsylvania
16219 (Received October $6, 1966)
~-
The empirical equation ?I = (Na/G)Xc(X/Xc)“f with X = (S2/M)Op2Z/u2, is applied to data giving the dependence of the viscosity q on chain length 2 and polymer concentration p2. It is shown that this equation, with a = 1.0 until X > X c , and a = 3.4 for larger X , satisfactorily correlates the viscosity of several polymer-diluent systems provided the effect of 9 2 on the friction factor f is taken into account so that ?I is compared at constant f rather than at constant temperature. This can be accomplished through the dependence of the Vogel parameters a and TOon pz; the former are defined through the relation = toexp[l/ a(T - TO)],with f o a constant. Data are examined for 7 as a function of 2, with cpz constant, and 7 as a function of 9 2 , with 2 constant. Finally, the relation of the parameters a and To to free volume parameters is briefly discussed.
1. Introduction The zero shear viscosity 7 of polymer-diluent systems has received extensive study by many investigators. The results of various empirical correlations’-6 may be given by the empirical expression’ 7 =
(Na/6)Xc(X/XJa!:
(1)
where
x = (S2/M)opzZ/uz
small part of the total change in 7 effected by the addition of diluent since the friction factor f usually depends strongly on the diluent concentration cpz. I n the following, the variation of ?I with cpz and 2 will be discussed within the framework of eq 1 and certain formulations for the dependence of on the cpz given below.
2. Constant Many
and
a
= 3.4 for
X
a
= 1.0 for
X
>X, X , or X < X,, respectively. Reference to Figure 2 shows that the first term in the brackets is initially very large, decreasing with decreasing 'pz, whereas the second term jn the brackets is small until (pz becomes very small. Thus, one expects an initially rapid decrease in q as diluent is added, followed by a region in which q decreases less rapidly with cpz, according to the magnitude of a, which in turn depends on the magnitude of the parameter X relative to the constant X,. Figure 3a shows that this is indeed the observed behavior for isothermal q when (p2 is varied a t fixed 2. Turning to quantitative use of the information in Figure 2, we have used the empirically determined values of cy and To to construct Figure 3b in which log 7 is plotted against log (nZW),at constant { rather than constant T for a polystyrene fraction in dibenzyl ether (2, = 1340). Thus, the term
C = [ l / a ( T - To)], - [l/a(T
- To)],
has been added to log q for each cpz. It i s seen that the data transformed in this way describe straight lines with slopes 3.4 and 1 .O, above and below the critical point ((aZ),, respectively, and that satisfactory superposition i s achieved with the curve for cpz = 1 (corrected for variation in { for short chains by a similar method). Data on polystyrene in diethylbenzene also superpose satisfactorily with the curve for cpz = 1, but these data do not extend below (cpzz),. Data for poly(viny1 acetate) in diethyl T h e Journal of Physical Chemistry
I
1
I
I
I
b.
0.
log+,
zw
217') for a polystyrene Figure 3. (a) Log 7 us. log ( ~ 2a t ~ fraction in dibenayl ether (2,= 1340) with 0.1 < (4 < 1.0;" the intersecting straight lines represent the position of the log 7 us. log ((42,) for the same curve for e = 1. ( b ) C data and for data on fractions with p2 = 1, 0. Here, C corrects the data to constant [.
+
phthalate,8 corrected to constant in the same way, are displayed in Figure 4, where again satisfactory superposition is achieved. The relation obtained by Kelley and Bueche, on the other hand, indicates that data of this type (constant (11) T. G
FOX
and A. R. Schultz, private communication.
VISCOSITY OF POLYMER-DILUENT MIXTURES
1197
t
B
O t
-I
'i
exp(B/(fg
+ af(T - TgNj
(6) Here B is proportional to the fractional void volume required for a segmental jump, f, is the fractional free volume a t the glass temperature T,, and af is the expansion factor for the void volume. Simple comparison of eq 2 and 6 yields the relations = to
B = af/a
(74
and I
i
- 3 ~
-4t
I
1
I
I
2
3
4
log
fg
- To)
(7b)
Williams, Landel, and Ferry suggested that approximately B = 1 and af = a1 - a,, where a1 and ag are the expansion factors for the liquid and the glassy states, respectively. A more recent study on a simple all as was assumed liquid" has suggested that af by earlier inve~tigators.~?'We shall employ the latter approximation here in calculating B and f , as functions of cp2. Thus, eq 7 can be used to compute B and f, from the Vogel parameters To and CY provided T , and a1 are known as functions of p2. Such data on polystyrene in dibenzyl ether1*are displayed in Figure 5, with the corresponding Vogel parameters being given in Figure 2. These data indicate that B remains independent of cpz at a value of B = 0.95 over the interval spanned. In addition, f, is essentially independent of cp2 for cp2 > 0.5 since T , - To varies little over the interval. These are entirely reasonable results, and they suggest that the viscous flow unit is essentially unchanged by the addition of diluent, at least until cp2 < ca. 0.2. It has been reported that the glass temperature of polymer diluent mixtures depends only on cp2 and is independent of the specific solvent.18 If this observation is valid, eq 6 indicates that 17 should also be independent of the nature of the solvent a t given cp2, provided the free volume parameters B, f,, and afare independent of solvent. Since B and f, are expected to be essentially independent of cp2, any specific solvent dependence must be attributed to a f = a1. Unfortunately, few studies of the dependence of a~ on p2 or solvent are known to us; the data displayed in
-
4zw
Figure 4. C f log TJ us. log ( ~ 2 2for~ poly(viny1 ) acetate), open circles, and its concentrated solutions in diethyl phthalate, solid circles, where C corrects the data to constant f ; 2, = 2680, 1510, and 920 for pips right, up, and left, respectively.
2, variable cp2) should be plotted as log (17cp2-''')
=
vs.
log (cp2Z) in order to achieve the superposition displayed in Figures 3b and 4. Unfortunately, there is sufficient latitude in the assignment of the parameter a and To for typical data that this difference cannot be reliably evaluated from any data that have come to our attention. Some data on poly(n-alkyl methacrylates) in diethyl phthalate12 do not exhibit the behavior obtained above for polystyrene and poly(viny1 acetate). Here, a is observed to increase rapidly with decreasing cp2, whereas To decreases to a shallow minimum for cp2 ca. 0.8 as cp2 decreases, then becomes independent of cp2. Furthermore, plots of log 17 vs. log cpzZ a t constant l / a ( T - To) and fixed 2 do not exhibit the behavior seen in Figures 3b and 4 but show a shallow maximum as cpz decreases and then decreases rapidly with decreasing pz. Similar effects have been noted by Ferry and co-workers13in the reduction factors for dynamic mechanical properties of poly(alky1 methacrylates). The cause of this behavior must remain a matter for conjecture at present, but it may represent the effects of stereoregularity in the methacrylate polymers.
-
4. Free Volume Parameters The free volume treatment of the temperature dependence of the viscosity leads to the r e l a t i ~ n ~ , ' ~ - ' ~
(12) F. Bueche, J . Appl. Phys., 2 6 , 738 (1955); A4.Teramoto, R. Okada, and H. Fujita, J . Phys. Chem., 67, 1228 (1963). (13) J. W. Berge, P. R. Saunders, and J. D. Ferry, J . Colloid Sci., 14, 135 (1959). (14) A. K. Doolittle, J . A p p l . Phys., 2 2 , 1471 (1951). (15) F. Bueche, J . Chem. Phys., 24, 418 (1956). (16) XI. H. Cohen and D. Turnbull, ibid., 31, 1164 (1959). (17) D. J. Plazek and J. Magill, to be published; see also abstract in Rheol. Bull., 33, No. 4, 6 (1964). (18) T. G Fox, W. B. Schultz, and A. R. Schultz, private communication.
Volume 70, Number 4 A p r i l 1966
G. C. BERRY
1198
3501 300
*
i.50: 200 .
To
-
Tg TO
4
6.0
I
d’
, , ,
t
2 5.2 5’00.0 0.1
.2
0.3
0.4 0.5 0.6 0.7 0.8
0.3
1.0
92
Figure 5. Free volume parameters for polystyrene in diethyl phthalate.”
Figure 5 suggest that a1 depends on p2,l8so it may also depend on solvent. These results also indicate that attempts to use the usual WLF equation to correlate the dependence of 9 on cpz will be successful only over the restricted interval for which T , - To is constant, for 0.5 < yz < 1.0 for polystyrene in dibensyl ether, for example.
5. Conclusions These calculations show that eq 1 and 2 can be used to predict adequately the effect of diluent addition on 9 for typical polymer-diluent systems. I n particular, if the isothermal data are corrected to a constant value of { through the use of eq 2, then 7 is proportional to the parameter X for X < X , or to X3a4for X > X,. Thus, data taken under different conditions of ’pz, T, and 2 all superpose to yield a single curve when
The Journal of Physical Chemistry
plotted as 11 at constant p us. X = (,?/M)ocp2Z/v2. The parameters a and To in eq 2 need only be viewed as empirical constants to be determined from the temperature dependence of 9 at constant cpz to effect the correction of the isothermal data to constant p. We have not here dealt with the question of molecular weight heterogeneity, but the usual considerations may be expected to apply.19*20Thus, Z has been replaced by the weight-average value Z, for polydisperse samples, and the friction factor { is expected to depend on the number-average chain length 2,. chiefly through the dependence of To on 2, in terms of the parameters used here. As a separate consideration, the parameters a and To can be correlated with the parameter in the free volume treatment of the density dependence of the friction factor to yield reasonable results when applied to data on polymer-diluent systems. That is, the fractional void volume required for a segmental jump and the fractional free volume at the glass temperature T , are essentially independent of concentration except when cp2 < 0.5. This suggests that the effective viscous flow unit is essentially unchanged by the addition of diluent over a wide range of concentrations. Similarly, it is anticipated that these parameters will also be independent of Z at a given yz, except in the limit of very low 2, even though both T , and Towill separately depend on 2, for moderately low Z. Unfortunately, we are unaware of any data which give both q(T) and T , as functions of 2 and 9 2 .
Acknowledgments. It is a pleasure to acknowledge the cooperation of Dr. T. G Fox in making available the extensive data cited in ref 8, 11, and 18. This study was supported in part by the Research and Technology Division, Air Force Materials Laboratory, Wright-Patterson Air Force Base. (19) P. J. Flory, J. Am. Chem. Soc., 6 2 , 1057 (1940). (20) V. R. Allen and T. G Fox, J . Chem. Phys., 41, 337 (1964).
