L. UTRACRI AND ROBERT SIMHA
1056
Vol. 67
MOLECULAR %’EIGHT AND TEMPERATURE DEPENDENCE OF INTRINSIC VISCOSITIES I N VERY POOR SOLVENTS B Y L.
U T R A C K I ’ AND
ROBERT SIMHA
Department of Chemistry, University of Southern California, Los Angeles 7, California Received October 89, 1968 Measurements of intrinsic viscosities of polystyrene between the 6 and the critical temperatures in cyclohexane for molecular weights varying between 6 X lo3and 6 X lo6are compared with theories of dilute solutions. The predicted relation between intrinsic viscosity and expansion factor a is obeyed for molecular weights iW, above lo6. For smaller M,, however, the ratio bola,,, with bo a characteristic dimension of a bead and ao the length of a link, decreases with decreasing M,. This cannot be quantitatively interpreted in terms of reduced hydrodynamic interaction. Moreover, bola0 is temperature and solvent dependent. The temperature coefficients of intrinsic viscosity a t and below 6 as a function of molecular weight can be interpreted in terms of the solvent effect alone, without allowing for the additional contribution of an internal rotation barrier. On the other hand, the dimensions, as calculated from the intrinsic viscosity at 6, indicate the existence of an appreciable barrier. Both results can b&brought into accord by the picture of freely jointed segments, each containing five monomer units, or of a restricted angle of 1 4 2 ’ for free rotation around C-C bonds. The exponent a in the Mark-Houwink relation decreases with increasing molecular weight for temperatures sufficiently below e, in accord with theoretical predictions. Although the expressions used throughout are valid for a near unity, they also appear to describe the intrinsic viscosity of toluene solutions, possibly by a compensation of hydrodynamic and thermodynamic factors.
I. Introduction The average molecular size and the intrinsic viscosity of polymers has been investigated by many authors and existing statistical and hydrodynamic theories have been compared with experimental results. However, these investigations were restricted to good solvents or to 8-temperatures. I n the present paper the experimental results reported in the preceding paper2 as well as some data pertaining to polystyrene solutions in good solvents are compared with recent theoretical work. 11. Discussion We require two quantities for our purposes, namely the extension factor a! and the intrinsic viscosity [ q ] . The statistical mechanical theory for the former is most adequately developed for poor solvents where a: is close to unity. For this case the expression3 ( a 3- a:)(l
[7lS =
Here is the mean square separation of chain ends in the 8-solvent and it is assumed that the hydrodynamic interaction between the beads is large. For small molecular weights, therefore, this equation has to be revised (see below). From. eq. 1 and 2, the ratio [~7]/[7]0should be a linear function of MI’/” a t a given temperature. This is confirmed in Fig. 1 for three temperatures below 6. The three lines are drawn by using the limiting values of bolaoderived below. By means of eq. l b and 2 one can write
We shall make use of eq. 3 to calculate a-factors by means of measured intrinsic viscosities. From eq. 2 and 3 me obtain an expression for the factor defined by the Fox-Flory equation
viz. x = ( G / ~ ) ~ ’ ~ ( b ~ /a ~6/T)M1” )~(l
(1)
where bo3is proportional to the volume of a bead and a. the length of a link, may be expanded t o
+ + 1/3x2+ O(xa) 22
=
1
+ 2.0532
(1b)
with an error smaller than 0.2%. The validity of eq. 1 has been shown5 for 0 < x 5 0.2. The intrinsic viscosity is represented by the equation3 [VI = [7ls[l
+ 1.55 + Oe2)l 2
=
2.87
x
(2’)
1021~--0.74
Finally we derive for the temperature coefficient
d In [7]/dT
=
+
2.14(b0/~~)~(8/T~)M~’~a:-~ d In (G)3’Z/dT (4)
(la)
Actually, for lx’ 5 0.2, the range of most interest here, one can replace eq. l a by a linear term cy3
(3)
[ V I = [rlsa2*28
+ 1/30?) = ( 4 / 3 ) ” / ” ~
with
a3 = 1
2.87 X loz1X (2)3’2/f1.1
(2)
with (1) Polish Academy of Sciences, Polytechnic Institute, Loda, Poland. ( 2 ) L. Utracki and R. Simha, J . Phus. CEem.. 67, 1052 (1963). (3) M. Kurata, H. Yamakawa, and H. Utiyama, Makromol. Chem., 34, 139 (1959). (4) M. Kurata, W. H. Stockmayer, and A. ROW,J . Chem. Phus., 33, 151 (1960). (6) Y. Ohyanaai and M. Matsumoto, J . Polumdr Sci., 64. S3 (1951).
assuming b,/ao to be a constant. I n order to obtain numerical values and apply a sensitive test to the behavior of the quantity bo/ao, we plot in Fig. 2 the ratio ( a 3- l)/’[(l- 6 / T ) M n L / ’ ]
=
2.053(G/~)~’/” ( b ~ / a o(5) )~
as a function of Mn for the polystyrene-cyclohexane system a t 2.528, and 30’. Analogous plots for benzene solutions a t 2506.’ and toluene solutions a t 2508 and 30°9 are also shown. It is assumed that B(to1uene) = 160°K. and 6(benzene) = 100°K. The horizontal portions of the lines are simply drawn as averages for a given temperature through all points corresponding t o molecular weights larger than 106. No meaningful (6) T. G Fox, Jr., and P.J. Flory, J . A m . Chem. Sac., 73, 1915 (1951). (7) W. R. Krlgbaum and P. J. Flory. J . Polymer Sci., 11, 37 (1933). (8) H. W. McCormick, J . Collozd Set., 16, 635 (1961). (9) R. Simha and J , 1,. Lakin. ibzd., 17, 270 (1962).
IKTRINSIC VISCOSITIES OF POLYSTYRENE IX POOR SOLVENTS
May, 1963
1057
I
0.9
0.8
0.7
< i; '-I
I
I
I
I
I
I
I
I
100
200
300
400
500
600
700
800
900
of eq. 2 a t three temperatures.
Fig. 1.-Application
Dashed circles indicate extrapolation below T,.
0
0
---
0
0
P
/ /
D
0
Q
8
O L O _ I '
J '
/
0
0
I -
12
ac
0
/--
8
0
@
/ *
0
0
---
>-f@
8
n
c _
/--%--p
*'
Q
@
a
IC IO
8
.-
e
z fr
4Id
8
BENZENE TOLUENE
25' C 25' C
B Q
TOLUENE
30' C
c(
o CYCLOHEXANE c) CYCLOHEXANE
28OC 30°C
D
0
CYCLOHEXANE
25OC
B e-
z
8
6
pl
P
' A
8
$
Z
d0
I
I
100
Fig. 2.-Expansion
300
I
500
I
700
I
900
A
'
I
1800
I
2300
factors as functian of molecular weight and temperature. Full lines, according to eq. 5. Dotted portions, hydrodynamic correction.
L. UTRACKI A N D ROBERT SIMHA
1058
6t
Fig. 3.-Temperature
o o
coefficients of intrinsic viscosities.
distinction can be made between 28 and 30' and the single line C encompasses both sets of data. The following observations may be noted: First, there is a temperature effect, the ratio bolaoincreaxing with decreasing solvent power. The basis for this conclusion is restricted by the narrow temperature range in cyclohexane. Moreover, eq. 5 should not apply to toluene and benzene, except a t very low molecular weights, because x is not sufficiently small. It is noteworthy that it is obeyed, nevertheless, even for large molecular weights, where x is close to 2. I n toluene, the ratio bo/aoactually increases with increasing temperature. There is nothing surprising, of course, in finding both the effective volume of a bead and the segment length to vary with temperature and solvent. Secondly, eq. 5 breaks down for small M . Here the assumption of complete hydrodynamic shielding is no longer valid and eq. 2 is modified to readY
+
+
trll = trllOS(X1 T , 6) t1 p ( X ) x 0 ( X Z ) l 0") X is the hydrodynamic shielding parameter proportional to M112 and X represents the ratio of the shielding functions for a specified value of X a t T and 0 , respectively. Thus, for T < e, we expect S > 1, due to the increased compactness of the coil. The function p ( X ) has been t a b ~ l a t e d . I~n order to compare eq. 2" with our experimental data, Fig. 2 , we proceed as follows: For X = 10, p ( X ) = 1.5, i e . , smaller by 3% than p( a ) = 1.55. On the basis of Fig. 2 we assign this value to a molecular weight of lo5 and thus fix the
and A and B
Vol. 67
t = 28" C. t = 34' C.
Full lines calculated from the first term of eq. 4.
numerical value of the proportionality factor between X and Al,l/z. From eq. l b and 2'' we obtain the exponent of a which replaces the value of 2.26 in eq. 3. Assuming 8 = 1, the dotted lines in Fig. 2 are calculated. For benzene and toluene this procedure has no theoretical foundation but is purely empirical. In cyclohexane, the trend is qualitatively correct, but the calculated decrease is not sufficiently rapid. A choice of 8-values larger than unity increases the discrepancy. For X = 1, the exponent in eq. 3 varies between 2.20 and 2.09 for X = 10 and 1.6, neglecting quadratic terms in x . The choice of a smaller "boundary" value for X would have improved the agreement somewhat. The observed and calculated temperature coefficients a t 28 and 34' are compmed in Fig. 3 as a function of molecular weight. Equation 4 consists of two terms, the first describing the solvent effect and the second the influence of the internal rotation barriers. The former contains the quantity (bo/ao)3which we have just found to vary with temperature. This requires the addition of a term (1.544x/a8)d la (bo/ao)3/dT to the right-hand side of eq. 4. It has the same M'/2dependence as the first one and vanishes a t T = 0. Assuming from Fig. 2 a temperature coefficient of (- 3Yc) per degree, this expression becomes positive for T < e and is less than 20% of the first term. However, on approaching 8, the temperature coefficient
INTRIXSIC YISCOSITIESOF POLYSTYRENE IN POOR SOLVENTS
May, 1963
1059
36' C. Q Q
-
36" C. 30" C. CLcn
0
L3'
L.
e
22O
c.
2.104
PP
106
105
Fig. 4.--R/lolecular weight dependence of intrinsic viscosity above and below &temperature. Dashed lines indicate extrapolation helow T,.
could well be less than assumed above. The numerical factor of 2.14 in eq. 4 is based on the exponent 2.26 in eq. 3, but should decrease with decreasing molecular weight. This effect amounts to 10% for the lowest molecular weight and will be disregarded. Similarly, a-a may be computed from eq. 3 with a constant exponent. For the evaluation of the second term, several models may be investigated. A periodic potential of the polymethylene type with an energy difference E >. 0 between the gauche and trans position yieldslo rO2= (2/3)NZO2(1 2eE'RT) (6)
4 contains no disposable parameter and is represented by the solid lines in Fig. 3. At t = 34' the experimental results are quite well described by this term and the barrier height E must be assumed to be small, ie., less than 100 cal./mole. At 30°, the inclusion of the positive correction term would improve the agreement. Again, E must be small. On the other hand, eq. 6 may be used directly to evaluate E. From eq. 2 and 6 and our data we find atT=e
where 10 is the length and N the number of C-C bonds. This results in a negative and molecular weight independent contribution to the total temperature coefficient. The first term on the right-hand side of eq.
which cannot be reconciled with the observed temperature coefficients and eq. 4. One can formally account for all observationsi by representing the molecule as a chain of freely jlointed large segments. About 5 monomer units per such segment must then be postulated. Alternatively, a box potential with unrestricted
+
(10) A. V. Tobolsky, "Properties and Structure of Polymers," John Wiley and Sons, Inc., New York, N. Y., 1960, Append. D and H.
-
r02/(2NZJ2 = 5.07; E/(RB)= 1.96
L. UTRACKI AND ROBERT SIMHA
1060
Vol. 67
25' C. 30' C. 0 CYCLOHEXANE 2 8 O C. o TOLUENE o TOLUENE
b
IQ6
105
104
M" of theoretical, eq. 7, and experimental molecular weight dependence of intrinsic viscosity in good and poor solvents.
Fig. 5.-Comparison
rotation around a C-C link within an angle 6 may be used. This angle turns out to be about 84'. The dependence of intrinsic viscosity on molecuIar weight is given from eq. l b and 3 as In
[VI = A + (1/2) 1nM + 0.753 In [l + B(l O/T) &I"']
with
A
B
=
In [2.87 X loz1(?)"'//M]
(7)
= 2.053(6/~)~"(bo/~o)~
Here the logarithm should strictly be expanded to the first order term. However, we shall find it useful later on to use eq. 7 as shown. Moreover, B will be treated as independent of molecular weight. The error thus committed for low molecular weights amounts to about 15% in toluene where the effect is largest. We shall consider this as a second-order correction. By comparison with the Mark-Houwink relation [77] =
KM"
(8)
we obtain for the exponent a a = 0.5
For z
>0
+ 0.773 x / ( l + 2.0532)
(9)
0.5 6 a 5 0.877
The infinity occurring for z < 0 has, of course, no physical reality, since from eq. l b it occurs for cr = 0. The exponent a becomes zero and the intrinsic viscosity approaches that of a compact spherical particle when z = -0.28. This value cannot be taken literally,
because the approximations are not valid for such xvalues and a compact configuration is sterically impossible. Equation 9 indicates a decrease of the exponent a with decreasing 2 or T , which is more rapid below than above e. Figure 4 exhibits the experimental molecular weight dependence of the intrinsic viscosity. Whereas for 36 and 30' the plots are linear, the curvature becomes increasingly pronounced as the temperature is lowered. Figure 5 illustrates the successful application of eq. 7 to data for polystyrenetoluene mixtures at 2503 and 30°9 and cyclohexane solutions a t 28'. The parameter A is assumed to have a universal value for all solvents and B is takeii from Fig. 2. Equation 1 suggests the introduction of a reduced temperature difference T
2/zorit
= (1 - @ / T ) / ( l-
O/TJ
neglecting the temperature dependence of From eq. 2 and 10 we have col,/[17ls = 1
+
KT
(10) bo/ao.
(11)
where K is a constant for a given polymer-solvent system and the left-hand side is therefore constant for a specified value of T. This is illustrated in Fig. 6. Here as in Fig. 2, systematic deviations appear for low molecular weights. This deviation is strongest a t the lowest temperature T = To,T = 1,where 121 is largest. The lower portion of Fig, 6 shows the relation of the parameter K to malecular weight. The average value is -0.141,
THERMODYNAWIC PROPERTIES OF CALCIUM HYDRIDE
May, 1963
-* Q
Y
1061
I-'
7=0
Y
0.98
0
v
I
u
7=I 1
0
0
Q 100
300
I
I
500
700
-
900
1/2
Mn Fig. 6.-Reduced
temperature representation of intrinsic viecosities, eq. 10 and 11. The lowest line (points with cross line) indicates the use of the above data for the evaluation of the parameter K .
Since T,/e = 1, this would imply that the critical temperature T , is the arithmetic mean of 8 and the hypothetical temperature To a t which the intrinsic viscosity becomes molecular weight independent. Acknowledgment.-The support of this work hy grants from the California Research Corporation and the Socony Mohil Oil Company, Inc., is gratefully acknowledged.
We have thus examined the validity of eq. 2 and 1 in various ways. The representation in Fig. 6 cowers the whole range of molecular xeights and temperatures and provides a check on the observed molecular weight dependence of Toas well as the temperature dependence of h3. We note finally from eq. 10 that a = 0 for a reduced temperature T~ = 2 for both toluene and cyclohexane.
THERMODYNAXIC PROPERTIES OF CALCIUM HYDRIDE] BY R. W. CURTISASD PREMO CHIOTTI Institute for Atomic Research and Department of Metallurgy, Iowa State University, A m e s , Iowa Received October Si, 1962 The equilibrium hydrogen pressure over a system consisting of calcium and calcium hydride was measured in the temperature range 600-900". The data below 780' can be represented by log P(atm.) = -9610/T 7.346 and that above 780" by log P(atm.) = -8890/T 6.660. The change in slope a t 780" is to be expected on the basis of a solid state transformation existing in calcium hydride. The hydrogen dissociation pressures for the temperature range 600-780" were combined with known thermodynamic data for calcium along with known data on the calcium-hydrogen system to give the following thermodynamic relations: AF0cam = -42,278 31.52T (440-780') and AF0ca=2 = -41,410 24.787' 1.93T log T (25-440').
+
+
+
+
Introduction previous work on the calcium-hydrogen system jncludes the measurement of equilibrium hydrogen presOver the system consisting Of (1) Contribution No. 1236. Work was performed in the Amen Labor&tory of the United S t a t e @Atamia Energy Cernmisaion,
+
hydride, and hydrogen by Treadwell and Stecher2"and . ~results ~ of their investigations Johnson, et ~ 1 The were expressed as (2) (a) W. D. Treadnell and J. Steoher, Helv. Chem. Acta, 36, 1820 (1957); (b) W. C. Johnson, M. F. Stubbu, Ad E, Sidwell, and A. Peahukas, J . Am. Chem. ~ o c , ,61, 318 (1939),
