1052
L. UTRACKI AND ROBERT SIMHA
Vol. 67
DILUTE SOLUTION VISCOSITIES OF POLYMERS NEAR THE CRlTlCAL TEMPERATURE; CORRESPONDlNG STATE RELATIONS BY L. UTRACKI' BKD ROBERT SIMHA Department of Chemistry, Cniversity of Southern California, Los Angeles 7, California Received October 29, 1962 The viscosities of ten polystyrene fractions covering a molecular weight range from approximately 7 x l o 3 to 7 X lo6have been measured in cyclohexane over a range of temperatures from above T = e to about T = To,the critical solution temperature. In the first paper we discuss the observed molecular weight dependence of two quantities, namely the intrinsic viscosity a t T , and kl a t and below T = 8. The results for the former show T , to represent a corresponding temperature for the intrinsic viscosity and moreover at T = T,, [?I is inversely proportional to V Z ~ the , critical volume fraction for phase separation. By means of the [q] values we estimate the average segment density within the encompassed coil volume a t T = T , and find it to be equal to u20over the appropriate range of molecular weights. The results for k~ lend further support to the corresponding states principle for solution viscosities previously developed.
I. Introduction Solution viscosities in the dilute and moderately concentrated range have been studied by several authors. These investigations included good as well as poor solvents but never extended t o temperatures lower than e. The purpose of this and the following paper is to explore experimentally this region for the polystyrene-cyclohexane system and to examine the results in the light of current quantitative or semiquantitative concepts, both regarding the limiting values and those a t finite concentrations. Certain predicted effects should become more pronounced for T < 0 and thus can be afforded a more sensitive test. Earlier2 we established the existence of corresponding state relations in good and in &solvents and shall now extend these considerations. 11. Materials and Procedure Seven polystyrene samples were obtained through the courtesy of Dr. H. W. McCormick of theDow Chemical Company. These fractions are identical with those used by McCormick3 and by Debye and his collaborators.4~6 Three others were furnished through the courtesy of Dr. R. Milkovich of the Shell Chemical Company. These latter samples are reported to have M w / M n ratios ranging from 1.16 to 1.8. The polymers were precipitated from toluene solutions by a large excess of methanol and dried in a vacuum oven for two to three days a t 80". Doubly rectified cyclohexane was kept and prior t o each use freshly distilled over sodium wire. The physical properties of our solvent are: v (34') = 1.006 centistokes; 4 3 4 " ) = 0.7649 g./ml.; n Z o1.4263; ~ 1, = 5.6'. The viscosities were determined in a specially devised closed system using viscometers of the Ubbelohde type (Fig. 1). The advantages of this system are a dust-free atmosphere, avoidance of vaporization of the solvent, and the use of solvent vapor instead of air to transport the solution of the upper bulb. The viscometers were calibrated by means of National Bureau of Standards oils. Kinetic energy corrections were applied throughout. The temperature of the water bath was kept constant t o within f0.001' and flow times established by means of stop-watches with an accuracy of =kO.O5 sec. The measurements were carried out in the following manner. About 20 ml. of a filtrated solution (about 1 g./dl.) were introduced through the joint 4 directly to the bottom of the lower bulb which has a volume of ca. 150 ml. This joint is then closed by a Teflon cap,the viscometer is submerged into the bath, and the stopcocks are adjusted. By means of the pressure of the saturated solvent vapor through the entrance B the solution is transported t o the upper bulb (volume ca. 4 ml.). Through the (1) Polish Academy of Sciences, Polytechnic Institute, Lodz, Poland. (2) L. Utracki and R. Simha, J . Polymer. Sci., in press. (3) H. W. McCormiok, J . Collozd Sei., 16, 635 (1961). (4) P. Debye, P. I+, Coll, and D. Woermann, J . Chem. Phys., 33, 1746 (1960). ( 5 ) P. Debye, B. Chu, and D. Woermann, i b i d . , 86, 1803 (1962).
three-way stopcock, the pressure in the apparatus is reduced to atmospheric and the measurements are started. For dilution the appropriate amount of solvent is added through joint C. The requisite concentrations were made up by using a FriedmanLa Mer weight buret. The characteristics of the viscometers are exhibited in Table I. The velocity gradients do not vary much but within this limitation no variations in our results could be observed.
TABLE I CHARACTERISTICS OF VISCOMETERS Viscometer
-CapillaryUpper Length, Diameter, bulb ume, ml. om. mm. 1701-
Viscometer -Constants5B A
A
0.978 5.20 0,00883 10.0 0.6124 B 10.0 5.15 ,00824 1.120 .6124 0.371 5.25 .02763 C 10.1 .4634 a A and B are the constants in the equation v = A t - B/1 where v = kinematic viscosity in centistokes and t = time in sec.
111. Experimental Results The solution viscosities of ten polystyrene samples were measured in cyclohexane a t 36' and lower temperatures extending in most instances below the critical solution temperatures To. The characteristics and critical data of the polymers in cyclohexane are shown in Table 11. TABLE I1 CHARBCTERISTICS O F POLYSTYRENE SAMPLES No.
-vn x
10-3
J!l*/-~fn
to,
OC.
1*E,
%
0.071 7.8 22.3 6.4 1.64 ,118 13.6 12.2 1.80 22.6 6.88 .226 19.8 78.1 1.05 5.6 .256 1.05 22.6 120 .299 22.9 5.0 5 147 1.04 4.6 .347 1.08 25.4 6 221 3.6 ,399 25.9 7 310 1.16 2.8 .562 27.9 1.09 8 523 2.7 .565 1.68 27.9 9 526 2.5 ,651 10 694 1.18 28.7 t, and 212, of a Samples 2, 7 , 9 from Shell; others from Dow. samples 4, 5, 6, and 8 are taken directly from ref. 4. For the remaining samples inter- or extrapolation was used.
1 2 3 4
Typical data are presented in Fig. 2, where the linear portions of the reduced specific viscosity-concentration curves are exhibited for the highest and lowest molecular weight over a range of temperatures. Our measurements extended beyond this range, but will not be used here. Figure 3 shows the intrinsic viscosities as a function of temperature. For the three lowest molecular weights a twofold extrapolation
DILUTE SOLUTIOX VISCOSITIESOF POLYMERS
May, 1963
1053
694 523 526
310 221 147 120
78
0.2
d
14
22.6
I
.-
-
10
g
s
6.4 I
I
I
20
30
I
40
"C.
Fig. 3.-Intrinsic viscosity-temperature curves in cyclohexane for different molecular weights. Short vertical lines indicatc to for each molecular weight, vertical dashed line indicates 8-temperature. Dashed portions, extrapolation of experimental data.
Fig. 1.-Sketch
of viscometer.
0
Description in the text.
o o
350 to 370
O.1 O.lL------
c
and A - 3 4 O C ond 8 - 2 8 O C
J
1
105
104
106
an*
Fig. 4.-Dependence of parameter kl, eq. 1, on molecular weight in cyclohexane a t 34 and 28". Third set of points, data between 35 and 37". The short line on the right-hand side indicates eq. 1 for 36".
earlier authors, we note generally, in very poor solvents, large values of k1 which approach those characteristic of spherical suspensions.6 Moreover there 6.400 -,+_-=m= is a noticeable temperature effect beyond the considerable scatter for the higher molecular weights. The concentration range available below T = 0 is limited and we cannot establish the validity of f corresponding state relations directly as we have done 0 0.2 0.6 I .o I .4 earlier.2 However, we can compare the observed C. molecular weight dependence of k1 with that predicted Fig. 2.-Representative viscosity-concentraficn curves for by the superposition principle and thus indirectly various temperatures. Upper family of lines for Mn = 694,000; examine the validity of t.he earlier general relations lower for M , = 6400. for T < 8. According to the superposition principle is involved, namely in respect to the numerical values we have of To and the intrinsic viscosities a t T,. Finally Fig. 4 shows the kl parameters, defined by the equation where a1 is a constant arising from the expansion of asp/(c[nl) = 1 kl[VIC (1) the master curve in a power series in the reduced a t 34 and 28' as a function of M,. concentration and y the shift factor in the concentration scale. If we amurne that y is a function of molecuIV. Discussion lar weight only, then it is proportional t o v20 the We want to consider here two quantities, namely kl and [q] a t T = T,. In accord with the observations of (6) R. Simha, J . A p p l . P h ~ s .23, , 1020 (1952).
+
L. UTRACKI AND ROBERT SIMHA
1054
critical volume fraction.2 Now from Fig. 3 one derives by least square computations [VI
(T
[n] ( t
=
=
e)
=
9.02
x
10-4
28’) = 10.80 X
~~0.503
(3)
iin0.475
Moreover2 v~C(%)
=;
13.82 X 10’ M n - 0 . 4 7 1
From eq. 2 and 3, therefore
kl (T = 8 ) kl
Vol. 67
the predictions to be made from the superposition principle. I n Fig. 4 are also shown the experimental ICl values for T > 8, covering temperatures between 35 and 37’ and lying generally below those a t 8. If we use the intrinsic viscosities a t 36’ and eq. 2a, a line slightly below that for 34’ is calculated as indicated on the right-hand side of Fig. 4. Very recently, Stern* proposed an empirical relation for kl which can be cast in the form
Mn-0.032
=
kl =
Mn-o.OOB = const.a (4)
(t = 28O) =
D
+
(6)
with D and E independent of molecular weight. This equation is qualitatively similar to our relations inThe present result for T = 0 differs slightly from that asmuch as it predicts an increase of kl with decreasing reported earlier2 due to the difference between the [7]-Mn relation, eq. 3, and that obtained previou~ly.~ molecular weight. However, a fit of the experimental data of R!tcCormick3 and Simha-Zakin7 can be obEquation 4 appears as the solid lines in Fig. 4. I n the tained only over a limited range of molecular weights. light of Fig. 2, the scatter of the experimental points Next, we consider the dependence of the intrinsic in Fig. 4 was to be expected. Kevertheless, we feel viscosity [v] ( T = T C ) , obtained through Fig. 3, that a satisfactory comparison with the predictions as a function of molecular weight. I n Fig. 5 the reof eq. 3 can be made. Here, the constant for the 8ciprocal quantity is plotted and we note that a unique solvent has been obtained by fitting the equation to relationship exists regardless of the variations in the a particular molecular weight, namely 2.21 X 105. absolute magnitude of the temperature between about For 28’, the constant has been defined as the average 8 and 29’. Such an interval produces appreciable of all experimental values, vix., kl = 0.80. changes in the intrinsic viscosity of a given fractisn, Sext, it is of interest to compare the results just particularly in the higher range of molecular weights obtained with those derived from our analytical (see Fig. 3). Vl” conclude that the critical temperaexpression for the master curve,2 which reads ture represents a corresponding state with respect to log (O~JCLVI) = A x Bxo{1 - [ (X/Q - 1)’ C]”z) the intrinsic viscosity. The line connecting the lower series of points in Fig. 5 has been drawn parallel to where A , B, C, and xo are constants. Expansion in the least square line representing vzc as a function of x =c/ y leads to eq. 2 with M n l according to the equation established earlier.2 The correspondence between the two lines permits U I = 2.303[A B(l C)-”’] (5) us to write the equations The bracket represents the initial slopes in Fig. 6 of ref. 2. If we assume on the basis of this figure that (T = T,) = 1.091 x 10-3 ~ ~ 0 . 4 7 1(7) the initial slopes are in good approximation equal for [v] ( T = Tc)vzo = 1.51 toluene and cyclohexane a t 2‘ = 8 and now by assumption include other temperatures as w7ell, we calIt is not surprising to find that the exponent in the culate kl from (4) and (1) Mark-Houwink relation becomes less than 0.5 as the temperature is reduced below 8. However, no explicit h - i [ ~ I = (2.303/~)(A B) (24 results seem to have been reported. It follows from or eq. 7 and previous work2 that the product [y] (T = T,) X c may alternatively be regarded as a reduced k, = const. ;Iln-(a-0.471) concentration variable for the system polystyrenecyclohexane. Moreover, we note from eq. 7 the folwhere C/2 has been neglected compared with unity. lowing relation between co = 1.08/ [7],’t9 the conceizThe logarithmic expression above does not rigorously, tration of incipient overlap of the average coils (as but only in good approximation yield the correct limit existing at infinite dilution) at T = T , and ccrit for C -t 0. The difference, however, is insignificant for our purposes. Substitution of the numerical values2for Ccr1t = 1.51co y , A , and B then gives with eq. 3 assuming a density of 1.08 g./ml. for polystyrene. kl (T 8) 0.94Mn-0.032 (44 If it were permissible to assume that the coil dimensions are independent of concentration a t T = T,, k1 (t = 28’) FS 0.79 this result would imply that the average distance beThe second value may be compared with the average tween molecule centers is reduced by 13% between 0.80, line B, Fig. 4. The first equation 4a shifts the incipient overlap and precipitation. line A as drawn in Fig. 4 downward by four units in We recall that the shift factor y, see eq. 2, for toluene the second place for M n = lo4 and correspondingly a t 30 and 48’ varies as the (-0.64) power of the less for the higher molecular weights. This extent molecular weight.2 If the inverse relationship between of agreement is a11 that can be expected and may be [ a ] ( T = T,) and 2)2c, eq. 7, and the proportionality considered satisfactory. Thus we conclude that the ( 8 ) El. D. Stern, Paper presented at the 142nd Satlonal Meeting of the experimental results for T < 8 are also in accord with American Chemical Society, Atlantic Clty, N. J., September, 1962.
+
+
+
+
+
(7) R. Simha and J. L. Zskin, J . CoZZoid Sc;., 17, 270 (1902).
(9) R. Simha and J. L. Zakin, J . Chem. Phys., 85, 1791 (1960).
DILUTE~ ~ O L U T I OVISCOSITIES N OF POLYMERS
May, 1963
between vz0 and y which was shown for cyclohexane,2 are to be valid for toluene also, this would imply that [a] (T = T,) should vary as the 0.64 power of M,. If this should turn out to be true, and there exists currently no experimental evidence, it would imply a stiff chain a t T = T,. This is not impossible, provided T , for toluene is sufficiently low and the effective internal rotation barrier high. We may use these results to obtain an insight into the state of the solution a t infinite dilution as compared with that prevailing a t the critical point. Specifically, we shall compare the volume fraction v ~ o o of i ~ polymer in the average volume encompassed by an isolated . former will be polymer coil a t T = T , with v ~ ~The estimated by means of the intrinsic viscosity, disregarding the problem of hydrodynamic shielding effects. For a spherical coil we write V2ooil
(%)
=
3 X 100 M , / [ / J N A4n(2)8'2]
1055
e
I/C7lTC v2c
lo4
io5
106
Ll". Fig. 5.-Molecular weight dependence of intrinsic viscosity a t the critical temperature and of critical volume fraction. Upper line, least squares from data of ref. 4; lower line, drawn parallel.
where p is the density and 2 the mean square radius of gyration. Writing the intrinsic viscosity in terms of the parameter 9 as
[a]
=
6"/" 9((,2)*"//Mn
we have for polystyrene ~ z ~ O i l [ 7 ' J=
5.396 X lo-'' 9
(8)
From (7) and (8) one obtains (~20/~2coil)~s4 = 2.798 X 10" 9-l
(9)
(T=T~)
Thus it is immediately evident that the ratio of' volume fractions will be near unity. The actual value depends on the value to be assigned to 9, If we take cf, to be a constant independent of solvent and molecular weight and approximately equal to 2.5 X loz1, the critical ratio is simply 1.1. As an alternative we can use the result 9 = 2.87 X loz1
(10)
which makes 9 a function of solvent and temperature through the expansion factor CY, where a = 1, by definition, for T = e. From eq. 3, 7, and 10, we derive therefore 9 (T
:=
To) = 2.697 X loziLX Mno*O1O5
(~2.J~2ooil)T T~ = 1.038 X Mn-"*"106 (11) Between M , = lo4and lo6,the critical ratio varies from 0.94 to 0.90, or in other words, is constant. For low molecular weights, however, it is not legitimate to continue employing these equations. We have thus found that a t T = T , the average density of segments inside a coil at infinite dilution equals the critical density in & h esolution, within the uncertainties of the calculation$. The actual density in the coil, of course, decreases with increasing distance from the central core. It is not possible, however, to characterize the onset of precipitAtion by the condition of equal segmental volume fractions outside and inside the volume pervaded by the chain, since this volume is a function of concentration.5.Q TJnder the solvent conditions in question, an expansion should take place. (10) L. Utraoki arid R. Simhrt, J , Phys. C h e w , 61, 1058 (1963).
v2c
v2 * Fig. 6.-Relation between volume fraction in solution, v2, and in ~ I , good solvent; C, coil, 1 2 ~ ~ i 1 ;SchematicaJly. A, VZ; B, V Z ~ ~ very V P ~ ~ at ~ I critical , temperature.
We should like to tmggest the tentative and qualitative picture for good and poor solvents described in Fig. 6 . I n the good solvent, vzcoil is assumed to increase continuously with polymer concentration due to osmotic compression. I n the precipitating solvent on the other hand, the initially high volume fraction in the coil is reduced or remains constant, until a t some concentration v2aoil and vz become nearly equal and precipitation sets in. As the solution concentration increases, the distinction between the individual encompassed volumes is lost and what is being compared here are average and mean local concentrations around a given molecular center. Acknowledgment.-The support of this work by grants from the California Research Corporation and the Socony Mobil Oil Company, Inc., is gratefully acknowledged.
