346
NOTES
where a is the condensation coefficient, WA is the Clausing factor for the interior of the cell, and A is the area of the surface of the effusing material. For the cell used in this work WA = 0.5, so eq. 3 reduces to Ps
=
P -
(4)
(&)CUPs
The steady-state pressures and the vapor pressures are given in Table 11. The steady-state pressures were
Table 11: Summary of Pressure Data"
a
T ,OK.
Cell
Pa X 101. mm.
P X 108, mrn.
687 687 685 68 1 717 722 711 702 661 675 661 657 706 698 697 690 700 693
2 2 2 2 5 5 5 5 1 1 1 1
0.602 0.557 0.521 0.354 14.38 16.75 9.70 4.82 0,0860 0.206 0.0940 0.0664 7.81 4.09 3.82 1,708 3.41 2.35
2.09 1.95 1,82 1.23 15.50 17,80 10.46 5.12 0.380 0.910 0.416 0.295 8.30 4.36 4.07 2.14 4.27 2.95
4 4 4 3 3 3
The runs are listed in the order they were made.
calculated using eq. 2, assuming that the copper phthalocyanine was not associated in the vapor phase. The magnitude of 1 / A a was found by plotting PS against caps for the various cells a t one temperature. The points fell on a reasonably good straight line, and thus it seems that a does not change appreciably in the temperature range studied. From the slope of the line the value of 1 / A a was found to be 54 cm.-2. Using this and eq. 4, the values of P were calculated. A plot of log P against 1/T gave a good straight line. A least-squares treatment of the data gives the equation log P,,, = 17.575 - 13900/T with the standard deviation of the slope being 260. Thus for copper phthalocyanine in the range 384-449", the heat of sublimation is 63.6 i 1.2 kcal./niole and the entropy of sublimation (to P = 1 atm.) is 67.2 cal./deg./mole. From 1,lAa it is possible to calculate the condensation coefficient if the area of the solid phase of the effusing material is known, but it is difficult to estimate this when a powder is used. I n this work the nominal T h e Journal of Physical Chemistru
area of the surface was 1.77 If this area is used, a comes out to be about 0.01. Since the true ares is undoubtedly larger than the nominal area, the true value of a is probably less than 0.01. When the crystal structure of copper phthalocyanine is considered, it is not surprising that the condensation coefficient is so low. The molecule is large and flat, and Robertson4 has shown that the planes in adjoining rows of molecules are almost a t right angles to each other. Thus one would expect that in only a small fraction of the collisions the molecule makes with the crystal is the molecular orientation favorable for condensation. (4) J. M. Robertson, J . Chem. Soc., 1195 (1936)
Dynamic Mechanical Properties of Dilute Solutions of Poly-a-methylstyrene
by J . E. Frederick and John D. Ferry Department of Chemistry, University of W i s c o n s i n , Madison, Wisconsin (Received J u l y 25? 1964)
An extensive study of dynamic mechanical properties of dilute polystyrene solutions has been recently reported.' I n this work, viscous solvents were used so the viscoelastic dispersion could be observed in the low audiofrequency range, and measurements a t different molecular weights and concentrations were interpreted by the theories of Zimm,2 R o u s ~ ,and ~ Tschoegl. We now report some additional measurements on poly-a-methylstyrene, undertaken to determine whether the additional steric hindrance of the methyl group would influence the behavior significantly.
Experimental Two samples of poly-a-methylstyrene were generously given us by Dr. P. Rempp of the Centre de Recherches sur les ;\1acromol6culesl Strasbourg. They had been prepared by anionic polymerization and presumably had sharp molecular weight distributions; their weight-average molecular weights, determined a t Strasbourg by light scattering, were 349,000 and 630,000. They were dried in vacuo a t 60" for several (1) J. E. Frederick, N. W. Tschoegl, and J. D. Ferry, J . P h y s . Chem., 68, 1974 (1964).
(2) B. H. Zimm, J . Chem. P h y s . , 24, 269 (1956). (3) P. E. Rouse, Jr., i b i d . , 21, 1272 (1953). (4) N. W. Tschoegl, ibid., 40, 473 (1964).
NOTES
347
days. (Attempts were made to study a third sample with molecular weight 45,000, but this was too low for satisfactory measurements in dilute solution.) Solutions for viscoelastic measurements, in the concentration range from 1 to 3%, were prepared in Aroclor 1248 (:imixture of partially chlorinated diphenyls) as previously described for polystyrenes. Similar solutions at lower concentrations were used for capillary measurements of steady-flow viscosity, from which the intrinsic viscosities in Aroclor 1248 a t 25' were estimated to be 0.8 and 1.0 dl./g., respectively. The solvent viscosity was 2.67 poises a t 25'. 3leasurements were made as before' with the apparatus of Birnboim and Ferry,5 between 0.06 and 400 C.P.S. Data at a single temperature of 25' were sufficient to encompass the important dispersion region, though for some of the solutions measurements were also made at 10' and reduced in the usual manner' to 25' to extend the effective frequency range.
Results and Discussion The results for M = 630,000 a t three concentrations are shown in Figure 1, calculated as the reduced contributions of the polymer to the components of the complex viscosity,6 ?'E = (7' - vlq,)/(q - vlvs) and ?"R = v " / ( q - P ~ ~ J Here . q' and 7" are the real and imaginary components of the complex viscosity of the solution, vs is the solvent viscosity, and v1 is the volume fraction of solvent. Alternative representations as the storage and loss components of the complex shear modulus are plotted in an unpublished report' together with tables of numerical data. The abscissa in Figure 1 is the logarithm of the circular frequency w multiplied by it factor such that, when the data are matched to a pair of theoretical curves, the position of the origin of the latter (shown by a cross on the abscissa axis in each case) provides a determination of the molecular weight.6 Specifically, the location of the cross corresponds to log S / M , where S is a numerical factor given by the theory. The shapes of the frequency dependence of q ' R and )I"R, in the theory of Tschoegl, are specified by two parameters: h , a measure of the strength of the hydrodynamic interaction between different segments of the same polymer molecule, and e, a measure of the departure from Gaussian chain statistics. In the present case, e is taken as 0 because the intrinsic viscosities within their limited precision indicate that d log [ ~ ] / dlog A4 I0.5. The value of h is chosen empirically to give the best fit to the shape of the frequency dependence, between the limits of 0 and which correspond to the special cases of the original Rouse and Zimm theories, respectively. For the theoQ)
0 1%
-0.:
-1.c 0
=sa: -u0
.a
F
o,
-0
-1.c
I
a\
0
-a:
-1.0
Figure 1. Logarithmic plots of ?'E and 7 " for ~ sample with M = 630,000 a t three concentrations as indicated. Pip up (or no pip), 25'; pip left, 10". In the abscissa, c is polymer concentration in g./ml. Curves are drawn for theories of Zimm ( I and 2%) or Tschoegl(3%) with values of h given in Table I.
retical curves which match the data in Figure 1, h = 03 a t 1 and 2% concentration, and h = 15 a t 3y0; there is a shift from Zimm-like toward Rouse-like behavior with increasing concentration, as observed repeatedly for polystyrene. Values of h together with other derived data are given in Table I for the solutions of Figure 1 and also two solutions of the sample of lower molecular weight, the detailed data for which are available elsewhere.' The behavior is Zimm-like for all except the one with the highest M and c. ( 5 ) M. H. Birnboim and J. D. Ferry, J . A p p l . Phys.. 32, 2305 (1961). (6) N. W. Tschoegl and J. D. Ferry, J . Phys. Chem., 68, 867 (1964). (7) J . E. Frederick, Ph.D. Thesis, L'niversity of Wisconsin, 1964.
Volume 66, Number 1
January 1966
KOTES
348
For polystyrene' and polyisobutylene,6 h has been found to decrease with increasing M and/or c, and it is of interest to compare the behavior of poly-amethylstyrene with that of these other polymers under corresponding conditions. For this purpose, the mean square end-to-end distance ( r 2 ) is a more rational measure of molecular size than M , and so this has been calculated for all the solutions concerned from the relation ( r 2 ) = a2((r02)/M)M. The values of ( r o z ) / M were taken from the review of Stockmayer and Kuratas; the expansion factor a was taken as
Table I : Parameters and Derived Calculations"
x
1M 10-8
349 630
Coucn., c wt. x 102,
log
g./cc.
poises
2 3 I
2.88 4.32 1.45 2.88 4.32
20.4 m 42.6 8.9 m 25.0 m 57.6 15
:!
2
log
Mve/
S
IM,,
M
2.368 2.368 2.368 2.368 2.236
5.73 5.82 5.82 5.97 6.00
0.19 0.28 0.02 0.17 0.20
7,
7;
h
log
0
v)
o
D
o
0
'.i
-
15
e
a
eo
0
0
0
OD
2
99
Q 00
63-
0 0 .
2.5 0
I5 0
:',:
,
1
TI
-2.24 -1.98 -2.30 -1.90 -1.62
All a t 25".
unity for poly-a-methylstyrene in A-1248; and for the other solutions a was calculated from intrinsic viscosity data,',6 using eq. 30, 34, and 38 of ref. 4. The comparison is shown in Figure 2 where values of h are located on a logarithmic map of c vs. (r2). The polystyrene and polyisobutylene data are consistent in representing a nionotonic shift of h from m toward 0 (Zinim-like toward Rouse-like behavior) with increasing c and (P). The poly-a-methylstyrene has somewhat higher vaIues of h than does the polystyrene under comparable conditions. Thus, it is necessary to go to slightly higher concentrations or coil sizes to achieve a given change in effective hydrodynamic interaction corresponding to a shift toward Rouse-like behavior. However, there is no striking manifestation of internal stiffness or steric hindrance of the poly-a-methylstyrene, The logarithms of the molecular weights, M,,, obtained from the cross positions in Figure 1 and similar plots for the other solutions are also given in Table I. As usual, the apparent molecular weight from the viscoelastic measurements is somewhat too large, to a degree which increases with increasing concentration. Values of log T ' , the terminal relaxation time, calculated from the relatioq 7 1 = (7 - v17,)M,,/cRTS, are also given in Table I. Recent investigations by Lamb9 and Philippoff l o have shown that at very high frequencies 7' approaches a limiting value qp which is somewhat higher than T h e Journal of Physical Chemistry
w l
-s -1.4
-2.2 -12
-1 1
-10
Log < r 2 > Figure 2. Map of values of parameter h as a function of concentration and mean square end-to-end distance for three polymers. Open circles, polystyrene' i+n A-1248; black circles, poly-a-methylstyrene in A-1248; crossed circles, polyisobutylene in Primol D.6
vS, suggesting that the frequency-dependent contribution of the polymer to the viscosity should appear as 7 - 7- rather than 7 - v1q8. For most of the solutions described here, the difference is probably relatively small. This question will be considered in a later communication. Acknowledgment. This work was supported in part by the U. S.Public Health Service under Grant GM10135. (8) W. H. Stockmayer and M. Kurata, A d e a n . Polymer Sci., 3, 196,
(1963). (9) J. Lamb and A. J. Matheson, Proc. Roy. Soc. (London), A281, 207 (1964), and personal communication. (10) W. Philippoff, T r a n s . SOC.Rheol., in press.
Theoretical Refinement of the Pendant Drop Method for Measuring Surface Tensions
by David Winkel Department of Chemistry, T h e University of W y o m i n g , Laramie, W y o m i n g (Received J u l y 27, 1964)
At present there are relatively few static methods suitable for measuring surface tensions under orthobaric conditions. Two examples are the sessile drop method1a2
