NOTES
1685
INTERNUCLEAR SEPARATION, cm X I O ~
Figure 2. Potential energy curves for Fz and 12. Curves are calciilaFed by the method of H. M. Hulburt and J. 0. Hirschfelder, J . Chem. Phys., 9, 61 (1941). See also J. 0. Hirschfelder, ibid., 35, 1901 (1961).
N , is the population of level i, and Pi, is the probability of transition from i to j and varies with e-Eij/RT where Eii is the energy difference between the levels. It then follows that as i increases, N , will decrease, but Pfi will increase because Eii decreases. Thus, under some circumstances, N , , can have a minimum, implying that there is a rate-determining step in the middle of the activation ladder. Consider the case of a diatomic molecule suddenly heated by shock wave. At the onset of dissociation, and before recombination is important Nj-2,j-z
> N7-2,j-l > N j - l , j
Nj,j+I
< Nj+1,j+2
from 0.107 ev for AVI,Zto 0.050 ev for AVI8,l9. The marked difference in anharmonicity indicates that a Pritchard-type “bottleneck” could develop for Fz but is less likely for 1 2 . This difference also helps to explain why the theory of Benson and Fueno, which uses harmonic oscillator transition probabilities, does not appear to be applicable to Fz. Another factor that helps to explain why the dissociation rate constants for FZare lower than Iz is that for the latter, the possibility of nonnearest-neighbor vibrational transitions must be considered. It has been pointed out“ that, when the separations of the vibrational levels are less than the thermal energy, vibrational transitions from and to next-nearest neighbors are important for the activation process. Furthermore, it should be noted that these relatively large separations of the vibrational levels for Fz are close to the thermal energy (1000°K = 0.086 ev) and this can also result in an over-all decrease in Pii.
Acknowledgment. The author wishes to acknowledge several helpful discussions with Dr. H. J. Kolker, of United Aircraft Research Laboratories. (9) For a compilation of the spectroscopic constants of the halogens, see W. H. Evans, T. R. Munson, and D. D. Wagman. J. Res. Natl. Bur. Std., 5 5 , 147 (1955). (IO) The calculations of the vibrational energy levels for FZand I2 were made from a simple two-term expansion using data from ref 9, and thus there is no pretense of high accuracy. An experimental study by R. D. Verma [ J . Chem. Phys., 3 2 , 738 (1960)] gives about 114 vibrational levels for 11. The major difference between Verma’s work and the calculations used here occurs in the region close to the dissociation limit (approximately the top 30 levels) where Verma’s data indicate a sharp increase in the number of levels and a concurrent decrease in the energy separation. Since a similar experimental study was not available for Fz and since the more precise information on the vibrational levels would not affect the arguments presented here, it was decided to present the IZ calculations in their present form. (11) K. E. Schuler and G. H. Weiss, ibid., 38, 505 (1963).
Thus, depending on the rate of increase of Pj,j+l, etc., there WJl be a reduction in the equilibrium population of upper vibrational levels. It should be emphasized (even as Pritchard does) that this situation is a direct result of the fact that diatomic molecules Viscoelastic Behavior of Dilute Polystyrene are anharmonic oscillators. It should further be pointed out thas the probability of this situation deSolutions in an Extended Frequency Range veloping for a given diatomic molecule will depend on just how a,nharmonic the molecule is. A comparison by John D. Ferry, Larry A. Holmes, J. Lamb, of t’he ratios of the zero-order vibrational frequency to A. J. LZlatheson and the first a.nharmonicitg constantg shows that F2 is by far the most antiarmonic of the halogens. The ratio Department of Electrical Engineering, The University, Glasgow W . 2 , we/wexe is about f ve times larger for Iz than for Fz. Scotland, and the Department of Chemistry, University of Wisconsin, Madison, Wisconsin 65Y06 (Received December 12, 1966) This anharmoilicity of F2 is also obvious in the separations and number of the vibrational energy levels as shonm in Figure 2. 1 2 has about 72 v brational 1evels’O below the continuum and t hc ir separations Dynamic viscoelastic measurements of dilute solu. of polymers in highly viscous solvents in the range from 0.0213 ev for AVI,~to 0.016 ev for A V T ~ I , ~ ~tions For a similar dissociation energy Fz has 19 vibrational low audiofrequency give information about levels below the continuum whose separations range relatively long-range segmental motions which are Volume 70, Number 6 M a y 1066
1686
NOTES
described by the theories of Rouse,* Zimm,6 and TschoegLG At much higher frequencies in such solvents, where the segmental motions do not respond to the oscillatory stress, the dynamic viscosity approaches a limiting value which is much less than the steady-flow viscosity but significantly greater than that of the solvent.'^* This feature affects to some degree the interpretation of the low-frequency measurements. I n order to reexamine the entire frequency range, we have combined here some low-frequency and high-frequency data on the same polymer samples.
I
+-
&A
I
Experimental Section Low-frequency data for the storage and loss shear moduli of three polystyrenes with sharp molecular weight distribution, 5-102, 5-111, and 5-108, have already been reported. Another with higher molecular weight, 5-1161, also generously provided by Dr. H. W. RicCormick of the Dow Chemical Co., has now been similarly studied a t a weight concentration of 1% in Aroclor 1248 a t 10.2 and 24.9" in the frequency range from 0.1 to 400 cps, using the apparatus of Birnboim and Ferry. High-frequency measurements were made on solutions of 5-102, 5-108, and 5-1161 by the torsional piezoelectric crystal as previously described ;7 the frequency was 73 kc/sec and the temperature range from 30" down to the temperature a t which the solution viscosity was about 15 poises. The Aroclors 1248 and 1232, used as solvents in these studies, are partially chlorinated diphenyls furnished by the Monsanto Co. The polymer molecular weights are given in Table I.
Table I : Parameters from High-Frequency Measurements, Reduced to 25" Sample s-102
s-111
M 82
239
x
5-108 10-8
-
5-1161
267
1200
2 0.143 0.181 0.68
1 0.143 0.162
Aroclor 1232 Concn, wt I)#, poise qm, poise 7, poise Concn, wt vat poises n,, poises 7, poises
%
...
,..
...
... %
... ... ... ...
Aroclor 1248 2 2 2.60 2.60 3.8 3.79 7.79 12.9
2 2.20 3.15" 15.1
'
I
I
I
1
I
I
2
3
4
5
6
lo9 w a T Figure 1. Dynamic viscosity plotted against logarithm of frequency reduced to 25" for 2% polystyrene S102 in Aroclor 1248: 6 , low-frequency measurements a t 6.4'; 10.0'; P , 25.0'; and a, piezoelectric measurements.
e,
Results At the highest frequencies and lowest temperatures, the dynamic viscosity q'(= G"/q where G" is the dynamic loss modulus and w is the radian frequency) approaches a limiting value qm. This is illustrated in Figure 1 for sample 5-102 a t a concentration of 2%. Here reduction to 25" has been accomplished for measurements a t other temperatures by multiplying the frequency by aT = (q - q J T m / ( q ~ O Tand P multiplying q t by (7 - ~ & / ( q - qa), where T is absolute temperature and p is density, and the subscript 0 refers to the reference temperature. The temperature superposition is less satisfactory than usual, but it is clear that q m is substantially higher than we. I n Table I, similar limiting values of qcnare given for all the solutions of polystyrene in Aroclor studied here, together with the steady-flow solution viscosity q and the solvent viscosity qs. For comparison there are included some results of Philippoff* on a 2cY, solution of polystyrene 5-111 in Aroclor 1248. As previously discussed, the difference 7, - q s is due to a contribution of the polymer chain to the viscosity associated with
-
0.78
1 2.60 2.83 15.4
a The Aroclor 1248 low-frequency data for 5-108 were obtained in a solvent lot with slightly lower viscosity (2.2 poises) than determined a t high that used for the others. The value of frequencies in the solvent lot of 2.60-poise viscosity has been corrected by the ratio 2.2/2.60.
The J o u r d 0.f Physical Chemistry
oi
(1) N. W. Tschoegl and J. D. Ferry, J. Phys. Chem., 68, 867 (1964). (2) J. E. Frederick, N. W. Tschoegl, and J. D. Ferry, ihid., 68, 1974 (1964). (3) J. E. Frederick and J. D. Ferry, ibid., 69, 346 (1965). (4) P. E. Rouse, Jr., J. Chem. Phy3., 21, 1272 (1953). (5) B. H. Zimm, ibid., 24, 269 (1956). (6) N. W. Tschoegl, ibid., 39, 149 (1963); 40, 473 (1964). (7) J. Lamb and A. J. Matheson, Proc. Roy. Soe. (London), A281 207 (1964). (8) W. Philippoff, Trulap. SOC.Rheol., 8 , 117 (1964). (9) M. H. Birnboim and J. D. Ferry, J. A w l . Phys., 32, 2306 (1961).
NOTES
the volume-filling properties of the statistical coil independent of its motions, similar to the effect of rigid spheres in suspension as described by the Einstein equation.‘O I n comparing the frequency dependence of the storage (G’) and loss (G”) shear moduli with the predicit has been tions of molecultu to represent the polymer contribution to the loss as G” - wqs, or G” - olwqB where 211 is the volume fraction of solvent, ordinarily negligibly different from unity ; correspondingly, the normalized dynamic viscosity has been presented :is (q’ - 01q8)/(q - ulqB). This procedure assumes that the entire polymer contribution to the solution viscosity can take part in the relaxation processes. However, since the molecular theories describe only that portion of loss associated with chain motions, it, has been pointed out7 that the appropriate loss contribution should instead be G” - wqm, while the normalized dynamic viscosity should be (q’ - q m ) / ( q - q m ) . I n Figure 2, G’ and G” - wqm are plotted in this manner for the data of Figure 1 at 6.4’; a single temperature is chosen because of the imperfect superposition with reduced variables, due probably to different temperature dependences of ( q - q m ) and q m , that of q m being probably closer to that of qs. The solid curves are drawn from the theory of Tschoeg16 with h = m (dominant hydrodynamic interaction) and e = 0.065; the agreement with the experimental data is much better than originally reported2 where the theory was matched to G” wvlqs (Figure 1 of ref 2). The normalized dynamic viscosity is plotted as (7’ - q m ) / ( q - qm) for two other solutions in Figure 2, and in Figure 3 the storage shear modulus for all three solutions is plotted combining both low- and highfrequency measurements, for comparison with molecular theory. High-frequency measurements made in Aroclor 1232 have been reduced to equivalent values in Aroclor 1248 by multiplying the frequencies by (q 1 ~ ) 1 2 ~ 8 / ( q- qs)123:l. At very low frequencies, substitution of 7, for vlqs makes little difference, but at higher frequencies the frequency dependence is distinctly affected as previously shown.’ Strictly speaking, the nonzero slope for G’ a t high frequencies observed in Figure 3 is inconsistent with a constant limiting value of qm (or direct proportionality of G” to w ) , since according to the phenomenological relations of viscoelasticity these quantities should be connected by the equation“ d log G‘/d log w = F(G”/G’)/(l - d log G”/d log U) (1) where F is a factor not far from unity. If d log G”/d
1687
5-
4-
3 I i
el
I
0-
-0 3 el
-
00
2-
11 -I
I
I
I
I
1
0
I
2
3
4
lo9 w Figure 2. Frequency-dependent polymer contributions to the storage and loss moduli for 2% polystyrene in Aroclor 1248, at 6.4’, plotted logarithmically against frequency: solid curve, theoretical, with parameters from Table 11. I
yI
I
,
,
5
6
1.0
-
c 0.5
0 0
I
2
3
4
loq wa,
-
- vm)
Figure 3. Normalized dynamic viscosity (7’ vm)/(v plotted against logarithm of frequency reduced to 25’ for two polystyrenes. Low-frequency measurements as follows: S-108, b , 9.9’ and 10.2’; e,18.1’; P , 20.0’; 51161, 6, 10.2O; e,24.9’; 0, piezoelectric measurements; solid curves are theoretical.
log w is 1or d log q‘/d log w is zero, then d log G’/d log w should be zero. However, in practice G” (including the contribution of solvent to the loss) is a t least an order of magnitude greater than G’ in this region, so a small deviation from direct proportionality of G” to (10) A. Einstein, Ann. Physik, 19,289 (1906);34, 591 (1911). (11) J. D. Ferry, “Viscoelastic Properties of Polymers,” John Wiley and Sons, Inc., New York, N. Y.,1961, Chapter 4,eq 9 and 20.
Volume 70, Number 6 M a y 1966
1688
NOTES
Table 11: Parameters from Fitting Tschoegl Theory Sample M x 10-3 Concn, wt %
5-102 82 2
S-108 267 2
5-1161 1200
h
m
m
E
0.065 5.02 0.11
0.090 5.73 0.30
2.5 0.135 6.25 0.17
log M v e log ( M v e I M )
1
(11111.11 o
0
0.5
1.0 X
I
0
I
2
I
1
3
4
5
6
i
IO-’
Figure 5. Effective Einstein radius plotted against square root of molecular weight: 0, in Aroclor 1232; 0, in Aroclor 1248.
loq w a T
Figure 4. Storage shear modulus plotted logarithmically against frequency reduced to 25’ for solutions of Figures 1and 3, with same temperature key; curves theoretical. Cross denotes origin of dimensionless theoretical plot.
(or slight dependence of q’ on W) is still consistent with a substantial positive slope of log G’ us. log w . I n addition, the onset of solvent relaxation occurs12 a t a viscosity of about 10 poises a t 100 kc/sec, and the solvent will thus make a small contribution to G’ a t the higher frequencies.
w
Discussion Comparzson with Molecular Theories when q , i s Taken into Account. Satisfactory agreement with the frequency dependence predicted by the Tschoegl theory with the parameters listed in Table I1 is evident in Figures 3 and 4. The shift from Zimm-like (h = m ) to Rouse-like ( h approaching 0) behavior with increasing molecular weight is observed as in the previous analysis2where q m was not taken into account. The molecular weights M,, in Table I1 are obtained by matching the coordinates of the experimental and theoretical curves as previously described ;2 they are still somewhat higher than the true molecular weights.l3 Equivalent Hydrodynamic Sphere from q,. The Einstein equation can be applied quantitatively to polymer solutions if each polymer molecule is regarded as being replaced by an equivalent hydrodynamic The Journal of Physieal Chemktry
sphere, which makes the same contribution to the solution viscosity as does the polymer above the polymer relaxation region.’ Thus the nonrelaxing part of the polymer contribution to the solution viscosity, q, - qs, may be expressed by analogy with the Einstein equation as
- 7s =
(2) where V zis the volume fraction of the equivalent hydrodynamic spheres. This may be rearranged to give qm
qa.
- 7s =
2.5v27,
2.5CN0V,qs
M
(3)
where c is the concentration of polymer in grams per cubic centimeter, N o is Avogadro’s number, ve is the volume of the equivalent hydrodynamic sphere, and M is the molecular weight of the polymer. Equation 3 shows that qm - q s is proportional to c for a given polymer-solvent system, as found experimentally by Philippoff.* Equation 3 may also be used to calculate the radius of the equivalent hydrodynamic sphere, re, (12) A. J. Barlow, J. Lamb, and A. J. Matheson, unpublished work. (13) NOTEADDED IN PROOF.A very recent theory of A . Peterlin (J.Chem. Ph,ys., in press, and private communication) shows that the finite high-frequency intrinsic viscosity can be attributed to an
internal viscosity associated with the RouseZimm submolecules. It follows that the form of the frequency dependence is modified, so that neither G” - qB nor G” - uqm can be expected to fit the Rouse-Zimm-Tschoegl theories precisely.
COMMUNICATIONS TO THE EDITOR
which is plotted aginst ill1/'for solutions of polystyrene in Aroclor in Figure 5; the dependence is linear in the range covered. Earlier high-frequency data in solvents of low v i ~ c o s i t y ~ are ~ ' in ~ general consistent with this type of treatment, but for such solutions it has not yet been possible to determine qm by direct measurement. More extensive measurements of the, Einstein viscosity in several viscous solvents of different solvent power are required to investigate the dependence of q, on the molecular configuration of the polymer in
1689
the various solvents. It would also be interesting to extend the measurements to higher frequencies to see if the onset of the relaxation of qm occurs close to the relaxation region of the solvent.
Acknowledgments. These studies were supported a t Wisconsin by a grant from the U. S. Public Health Service, No. GM-10135, and a t Glasgow by a grant from the National Engineering Laboratory of the Department of Scientific and Industrial Research.
C O M M U N I C A T I O N S T O T H E EDITOR
Kinetics of Iodination of Mercury Dimethyl in Various Solvents
Sir: I n previous calorimetric experiments, it was noted that the iodination of mercury dialkyls took place at markedly different rates in different solvents. We report here measurements on the iodination of mercury dimethyl in five solvents over a temperature range of 23-45 O . The iodination takes place rapidly in all solvents in sunlight, but it was found following rigorous exclusion of light that there exists a genuine dark-reaction. This thermal reaction was followed by mixing known concentrations of iodine and of mercury dimethyl, each in the solvent concerned, in stoppered flasks inside a light-tight temperature-controlled box. The extent of the reaction was determined by removing samples at known times and determining the unused iodine colorimetrically. Tests showed that our sampling technique did not advance the percentage reaction appreciably, but if the solutions were left in the colorimeter for any length of time the superposition of the photochemical reaction, caused by the photometer light source, upon the genuine therinal reaction led to higher rate constants. I n the concentration ranges used to lo-* mole/l. of e:tch reactant) the reaction was found to be first order in each reactant, and a summary of the results obtained is given in Table I. The rate constants vary by a factor of about 100, but to within the experimental error, the activation energy for the reaction is independent of the solvent. The iodination of mercury dimethyl in carbon tetrachloride has been studied
previously by Razuvaev and Savitskii,2 who found a slightly higher activation energy of 9.5 kcal/mole; their rate constants are consistently larger than ours (e.g. 0.116 instead of 0.073 at 28") and it is not clear from their account whether they were aware of the existence of a photochemical reaction. Table I Solvent
Dielectric A H I I O ~ I ( I ~ ) ,k at 2 8 O , 1. constant kcal/mole" mole-' min-1
Cyclohexane 2.05 Carbon tetra- 2 . 2 2 chloride Benzene 2.30 Chloroform 4.64 Ethanol 25.8
-5.8 -5.8 -4.25 -5.1 -1.65
E, kcal/mole
0.069 =t 0.003 7 . 1 & 1.0 0.073 =t 0.004 7 . 7 f 1.0 0.40 f 0.02 1.49 =t 0 . 0 5 6.95 f 0.35
8 . 5 f 1.0 8 . 0 f 1.0 7 . 4 =k 1.0
K. Hartley and H. A. Skinner, Trans. Faraday Soc., 46, 621 (1950).
There is no obvious correlation between the rate constants and any simple physical property of the solvents. However, there is a reasonable correlation between the rate constants and the quantity - [dielectric constant/AH,,1,(Iz) ] which might suggest that a higher rate of reaction is favored by a higher degree of ionic dissociation and by a higher degree of complexing between the I2 and the solvent. The experiments in ethanol were repeated in 0.125 M (1) H.0.Pritchard, Ph.D. Thesis, University of Manchester, 1951. (2) G.A.Rasuvaev and A. V. Savitskii,Dokl. Akad. Nauk SSSR,85, 575 (1953).
Volume 70,Number 6
M a y 1966
