J . Phys. Chem. 1990, 94, 4105-4712 Stratospheric partial pressures, shown by the dotted line in Figure 1, are a factor of 10-20 below the HCI pressure along the “ice”/hexahydrate coexistence curve, given as log (Pi/),(Torr)) = 10.00 - 2970/T (where subscript i/h denotes the value along the “ice”/hexahydrate coexistence). Assuming that a solubility limit for HCI-in-ice is not reached and that the composition of “ice” coexisting with the hexahydrate does not vary much and its value is that at the eutectic, we have xi/h = 0.0027(0.1285) x(pHCI) = xi/h(pHCI/pi/h)o’s and the HCI content of ”ice” for stratospheric conditions is about 0.009 mol %. The data in Figure 4 for NAT were taken at an HCI pressure a factor of I O below the HCI pressure at the phase boundary, and thus, assuming that the temperature variation of the HCI in NAT solubility is similar to HCI in “ice”, the HCI content of NAT in the stratosphere would be about 0.3 mol %. The bulk solubility of HCI in ice presented here suggests a very slow HCI content of ice particles in the stratosphere. According to Leu,33the heterogeneous reactions of C10N02 and N 2 0 Son ice with very low HCI content is primarily with water, and thus HCI reactions on NAT become very important for activating the chlorine that is contained in the HCI reservoir. The heterogeneous reactions have not as of yet been studied on NAT surfaces; however, Tolbert et reported reactions on sulfuric acid solutions at stratospheric temperatures and noted that dissolved HCI (33) Leu, M. T. Geophys. Res. Leu. 1988, I S , 8 5 5 . (34) Tolbert, M. A.; Rossi, M . J.; Golden, D. M. Geophys. Res. Lett. 1988, I S , 847.
4705
was readily available for reaction at concentrations as low as 0.02 mol % HCI. It is not clear that the reactions involving HCI in ice have been measured under conditions that would ensure that only “ice” were present. Coexisting liquid or HCI hydrate could be present if the HCI pressure is outside the “ice” stability region. In addition, Turco et al.” discussed the requirements imposed upon heterogeneous reactions by the dechlorinating effects of the sedimentation of ice particles in the cold, polar stratosphere. This concern arose because of the recent measurements’0,20suggesting high HCI solubilities in ice. Hanson and Mauersberger2I estimated that 60% of the HCI reservoir could be absorbed on ice clouds with a frozen water content of 2 ppmv. A new estimate for the amount of HCI condensed in ice particles can be made assuming a maximum of 0.01 mol % HCI content. The amount of HCI dissolved in 2 ppmv of ice is then about 0.2 ppbv out of a total HCl content of 2 ppbv. This is a small fraction of the chlorine reservoir, and the requirement that chlorine become activated before sedimentation occurs can be relaxed. The conclusion by Turco, however, that heterogeneous reactions on NAT are the major source of active chlorine is supported by the measurements reported here. More study of heterogeneous reactions on NAT and ice at low HCI pressures are needed to determine how heterogeneous mechanisms might be responsible for the release of active chlorine.’ This is now one of the most important aspects concerning the polar ozone holes. In addition, the HCI adsorption/absorption behavior on sulfuric acid mixtures is needed to determine the HCI affinity for stratospheric aerosols present globally. (35) Turco, R. F.; Toon, 0. 9.; Hamill, P. J . Geophys. Res., in press.
Simulation of Polymer Chain Dynamics with Small Organic Molecules and Their Mixtures S . Havriliak, Jr. Modifiers Research Department, Rohm and Haas Co., Bristol, Pennsylvania 19007 (Received: September 5, 1989)
Experimental simulation of polymer properties or structure with small organic molecules is a well-known technique that appears not to have been applied to the study of polymer chain dynamics. In this work we apply this simulation methodology by examining the dielectric relaxation data on isoamyl bromide and its mixtures with 2-methylpentane reported by Denney et al. These molecules form an interesting pair for study because they are isometric, they form solutions over their entire composition range, and one of them is a simple polar molecule while the other is nonpolar. They also tend to form glasses rather than crystallize when cooled to low temperatures. Denney’s relaxation data are represented in terms of a function proposed by Havriliak and Negami for polymers. The parameters of this function as well as their dependence on temperature were determined by using the multiresponse techniques developed by Havriliak and Watts. The dynamic parameters for the mixtures in the 50-75 mol % range of isoamyl bromide are similar to the parameters for the a-relaxation process of many polymers previously reported. The parameters are discussed in terms of the general Kirkwood-Cole theory of polar liquid relaxation and Mansfield’s specific model for polymer chain dynamics. The experimental results described in this work support the results derived from Mansfield‘s model for polymer chain dynamics, Le., the shape of the dielectric relaxation process when viewed in a complex plane is not due to molecular weight but is due to the nature and relative magnitudes of the intramolecular and intermolecular interactions.
Introduction This paper is another in a series of papers directed at a better understanding of the dynamics of polymer chains above and below their glass transition region. Simulation of polymer properties with small (organic) molecules is a time-honored pursuit in the study of polymer structure property relationships. For example, Liang et a1.I were among the first to examine the idea that polymer ( I ) Liang, C. Y.;Krimm, S.; Sutherland, G. B. B. M.J . Chem. Phys. 1956, 25(3), 543.
chain vibrations in the infrared region may actually be understood by studying a series of linear hydrocarbons of differing molecular weight. More recently Paul2 initiated an understanding of compatibility of polymer blends by measuring the heats of mixing of small molecule solutions. Still more recently, Paul and CruzRamo3 simulated the compatibility of poly(viny1chloride) blends (2) Paul, D. R.; Newman, S.Polymer Blends; Academic Press: New York, 1978; Chapter 1. (3) Cruz-Ramos, C. A.; Paul, D. R. Mocromolecules, in press.
0 I990 American Chemical Society
4706
Havriliak
The Journal of Physical Chemistry, Vol. 94, No. 11, 1990 1.0
,
1
t 0 0
G3
t
00
:
00 00
0
0.0
'
2
00
2.s
3
3.s
4
4.s
s
(1.9
DIELECTRIC CONSTANT Figure 1. Complex plane plot of 33 mol 3'% isoamyl bromide in 2methylpentane at the different tempeatures listed in the legend.
with copolymers of vinyl acetate and ethylene by measuring the heats of mixing of solutions of organic molecules thought to represent local polymer structure. They then compared their results to the dielectric relaxation results of Runt et al.: among others. Even the numerical simulation of the ring-flip mechanism in polycarbonate by Perchak et al.5 is an attempt to understand polymer chain dynamics from a study of ellipses on lattices of arbitrary rigidity. To be sure, polymer chain dynamics have been studied by making measurements on (dilute) solutions of polymers, but this is not quite the same because the polymer chain is still present. What has not been studied is the extent to which small organic molecules and their solutions can simulate the relaxation properties of polymers, in other words, to examine the question, To what extent is the polymer chain responsible for the relaxation processes observed in polymers? Stated conversely, To what extent is the local structure of a polymer system responsible for the dynamics of relaxation processes observed in polymers? A superficial inspection shows that the normalized complex dielectric constant data of 33 or 50 mol 5% isoamyl bromide (IABR) in 2-methylpentane (2MEP) shown in Figures 2 and 3 of the work by Denney et aL6 are remarkably similar to comparable plots for the a process in polymers. Specifically, the loci at high frequencies are linear while the loci at low frequencies are circular arcs, when the relaxation data are viewed in the complex plane. Furthermore, though the data have been normalized for the magnitude of the dispersion, there is a band around the loss maximum that is so characteristic of comparable plots for polymers. Finally, rate plots based on temperature vs loss maximum frequency yield activation energies that are similar to many polymers. The single difference is that the experimental temperatures for the solutions are much lower then they are for polymers. The objective of this study is to analyze the dielectric relaxation data of Denney et al. in terms of the techniques developed expressly for polymers. His data were chosen for such a study for several reason. First, as already mentioned, complex plane plots of the 33 and 50 mol % solutions appear to be similar to the results observed in polymers. Second, the data were made available to me in tabular form so that experimental quantities did not have to be picked off graphs thereby obscuring any analysis of residuals. Third, the measurements were made on liquids in a liquid cell of fixed geometry that is readily calibrated, thereby eliminating the inherent problems of adhesion of the polymer test specimen to the metal test electrodes. Furthermore, cell constants for solid test specimens are difficult to determine. Finally, the electrical measurements were made under conditions of high standards, even (4) Rellick, G. s.;Runt, J. P.; ACS Polym. Mater. Sci. Eng. 1985,.52, 331. (5) Perchak, D.; Skolnick, J.; Yaris, R. Macromolecules 1987, 20, 121. (6) Denney, D. J.; Ring, J. W. J . Chem. Phys. 1966. 44(12). 4621.
TABLE I: Relaxation Function Parameters and TWr Conf-e Limits for Representing t b Dielectric Relnxation Behavior of 33 md % Isoamyl Bromide Solutions in 2-Metbylpentw at Camitant Temperature uaram -167.8 O C -166.6 O C -165.3 'C -164.5 O C -161.6 O C 5.60 5.54 5.48 5.42 €0 5.63 0.0 1 0.01 0.01 SD 0.01 0.01 2.24 2.22 2.20 2.06 €, 2.24 0.01 0.02 SD 0.01 0.01 0.05 8.73 10.39 9.24 11.32 InoVb) 8.15 0.04 0.06 0.06 SD 0.06 0.05 0.70 0.82 a 0.68 0.74 0.77 0.01 SD 0.01 0.01 0.01 0.01 0.70 0.53 0.64 0.62 B 0.73 0.03 0.02 SD 0.03 0.02 0.03 8 X IO-' 5 x 10-7 S D ~ 7 x 10-7 9 x 10-8 8 X lo-' var real 0.0003 0.0001 0.0001 0.0002 0.0001 0.0003 imag 0.0001 0.0004 0.0003 0.0003 std 0.010 0.01 5 0.012 real 0.019 0.0083 0.018 0.018 0.0044 imag 0.003 0.016 mean 3.78 4.14 3.94 4.36 real 3.58 0.52 0.55 0.52 0.52 imag 0.51 cov
real imag DOF
0.5 2.3 20
0.2 3.9 22
0.3 3.5 21
0.4 3.1 20
0.27 3.4
"See text for definitions. by todays standards, Le., guard circuit and wide frequency range. Estimation of the Parameters Dielectric Datu. A complex plane plot for the 33 mol % IABR taken from the work of Denney6 covering the temperature range that define the relaxation process is given in Figure 1. Experimental data at lower temperatures are to the high-frequency side, while data at higher temperatures are to the low-frequency side. The frequency range of the data reported by Denney is 50 Hz to 500 kHz. Plots for the other concentrations, as already noted are similar and are not reproduced here. Constant-Temperature Results. Equation 1' is used to represent the complex dielectric constant, t * ( w ) as a function of radian frequency w = 27rJ wherefis in hertz, at constant temperature:
+
In this expression, e * ( w ) = t'(w) ic"(w) is the complex dielectric constant, t'(u) is the real or storage part of the complex dielectric constant, while ~ " ( w is ) the imaginary or loss part of the complex dielectric constant, and i = (-l)l/*. The other five quantities are parameters of eq 1 and need to be determined from the experimental data. The parameters toand represent the equilibrium and instantaneous dielectric constants, respectively. Both sides of eq 1 are unitless, and the parameters on the left-hand side will be referred to as the magnitude parameters, while those on the right-hand side will be referred to as the dynamic parameters. The parameter T~ represents the relaxation time, while a and p are formally related to the distribution of relaxation times. There are number of interesting features of this expression that should be pointed out. First, when a = p = 1, the locus of points in the complex plane is a semicircle, Le., the familiar Debye expression is obtained. For the case p = 1 and a in the range 0-1 the Cole-Cole8 expression is obtained, Le., the locus in a complex plane is a semicircle. For the case a = 1 and p in the range 0-1 the familiar Cole-Davidson9 expression is obtained, Le., the locus in a complex plane is a "skewed semicircle". In other words eq 1 combines the functional forms of previously known relaxation functions into a single expression and probably rep(7) Havriliak, S., Jr.; Negami, S . Polymer 1967, 8, 161. (8) Cole, R. H.; Cole, K.S.J . Phys. Chem. 1941, 9, 341. (9) Cole, R. H.; Davidson, D. J . Chem. Phys. 1950, 19, 1484.
The Journal of Physical Chemistry, Vol. 94, No. 11. 1990 4101
Simulation of Polymer Chain Dynamics 1.0
,
0.8
-
B 0.6
16.0
I
*
-
0
1 --
0
n
I 3
-
10.0
-
8.0
-
cr,
Q
8
12.0
-
a
s
6
-
E
?
0
0.4
14.0
1 0
4Q
3
0
Y
u
2
2.5
5
5.5
4.5
4
5
5.5
6
s
DIELECTRIC CONSTANT Figure 2. Complex plane plot of 33 mol 5% isoamyl bromide in 2methylpentane at -165.3 O C . Experimental data are represented by the circles, while the calculated values from single- and multiple-temperature analysis are also listed here.
\ 6.0
8.6
8.8
9
9.2
9.4
9.6
9.8
10
TEMPERATURE Figure 4. Plot of the instantaneous dielectric constant with temperature. The short solid lines represent different confidence levels listed in the legend, while the dashed line represents the expectation values based on the parameters in Table I1 and eqs 1-6. 0‘9
: t
0.8
0.8
0.7
t‘
5%
5m4 I I I 5.3 -170 -168 -166 -164
I
a
I
I
I
$
Oe7 0.6
-162 -160 -158 -156
TEMPERATURE Figure 3. Plot of the equilibrium dielectric constant with temperature. The short solid lines represent different confidence levels listed in the legend, while the dashed line represents the expectation values based on the parameters in Table I1 and eqs 1-6.
0.6
0.6
I
’
I
I
I
I
I
I
I
-170 -168 -166 -164 -162 -160 -158 -156
resents the dielectric relaxation process of many polymers within experimental error. In the past graphical methods were used to evaluate the parameters of eq 1 for each temperature. This method was then extended to other temperatures to determine the temperature dependence of the parameters. In this work the method used to determine the parameters of eq 1 at -165.3 OC is based on the multiresponse technique developed by Havriliak and Watts.Io The results at this temperature are given in Table I. Also listed in Table I are a number of important statistical quantities that represent the goodness of fit. Scaled determinant (SD) is the value of the determinant at convergence divided by the number of observations used in the regression. It is useful in determining if two analyses have been brought to the same level of regression. The next quantity in Table I is the variance (VAR), which has two components, a real and an imaginary one. It is defined as the mean sum of squared residuals. In other words the differences between the experimental and calculated quantities (Le., residuals) (IO) Havriliak. S..Jr.: Watts, D. Design, Data, and Analysis, by some friends of Cuthbert Daniel: Malows, C . , Ed.; Wiley: New York, 1986.
TEMPERATURE Figure 5. Plot of the log (relaxation time) in seconds with reciprocal temperature (kelvin). The dashes represent different confidence levels listed in the legend, while the dashed line represents the expectation values based on the parameters in Table I1 and eqs 1-6.
are squared, then summed, and finally divided by the degrees of freedom (DOF). Taking the square root of VAR leads to the model standard error of estimate, listed as MSE in Table I. This parameter is probably the most important statistical quantity in Table I because it is a summary of the fit. MSE has the same statistical significance as does the experimental standard deviation associated with a mean of a replicated measurement. For the compositions in Table I, the real and imaginary parts of MSE have been reduced to about the same level so that the regression process has not favored either of the two quantities. Mean is simply the average of the real or imaginary quantities. The coefficient of variation (COV) is defined as the ratio of MSE to mean times 100. The differences between the real and imaginary COVs reported in Table I are due to the large differences in the mean since it is about 10 times larger. A plot of the calculated
4708
Havriliak
The Journal of Physical Chemistry, Vol. 94, No. 11, 1990
-
t-
I
+2 SIGMA,
-
, b.4.L....DATA , .....
0.4 -170 -168 -166 -164 -162 -160 -158 -156
TEMPERATURE parameter with temperature. The dashes represent different confidence levels listed in the legend, while the dashed line represents the expectation values based on the parameters in Table Figure 6. Plot of the
(Y
I 1 and eqs 1-6.
TABLE II: Relaxation Function Parameters and Tbeir Confidence Limits for Representing the Dielectric Relaxation Bebavior of Isoamyl Bromide Solutions in 2-Metbvlwntane (-165.4 'C) IABR. mol % param 33.3 33.3 50 75 100 I, 5.54 5.53 7.19 10.19 13.51 0.01 0.01 0.02 SD 0.01 0.01 c, -0.024 -0.036 -0.068 -0.097 0.004 0.001 0.002 SD 0.001 2.65 12 2.22 2.19 2.25 2.52 0.02 SD 0.02 0.01 0.01 0.01 -0,021 -0.015 -0.004 C2 -0.014 0.008 0.003 0.002 SD 0.003 11.31 10.121 9.59 I3 9.23 9.31 0.005 0.01 0.01 SD 0.05 0.02 c3 -6.62 -7.42 -8.12 -8.00 0.04 0.05 SD 0.07 0.01 1 .oo I4 0.74 0.747 0.824 0.909 0.005 0.004 SD 0.01 0.005 0.01 0.004 0.0036 c4 0.013 SD 0.001 0.001 0.0005 0.58 0.60 0.55 Is 0.64 0.63 0.01 0.01 0.01 SD 0.02 0.01 CS -0.009 -0.001 2 SD 0.003 0.0019 2.0 X IO4 6 X IO4 4.9 x 10-2 SD" 6.8 X IO-' 4.2 X var real imag std real hag
mean real imag cov'
real" hag DOFa
Ei:
0.00091 0.00023
0.00134 0.00073
0.0023 0.001 1
0.022 0.012
0.010 0.018
0.030 0.015
0.036 0.027
0.048 0.034
0. I5 0.11
3.94 0.52
3.98 0.53
4.9 0.69
5.7 1.1
8.1 2.0
0.25 3.45
0.8 3.0 200 -166.5 -171
0.7 4.0 244 -160 -167
0.8 3.2 245 -154 -160
1.8 5.6 206 -142.9 -1 53
TO T60
u
5r&
0.00010 0.00033
"See text for definitions. 0.150
1
I
methods based on multiresponse techniques." In this method the variation of the complex dielectric constant with frequency and temperature is pooled to form a single data set. The complex dielectric constant is assumed to be given by eq 1, while the temperature dependence of the five parameters is assumed to be given by eqs 2-6. In these expressions Tis in O C , Tois a reference €0 = I , + C i ( T - 7'0) (2)
+ Cz( T - To) = I3 + C3(RK- RK,) = I , + C , ( T - To) fl = Is + C5(T - To)
t,
log, 0.000 L -6
-I
I -
-4
-2
0
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2
= I2
4
LOG RELATIVE TIME, SEC Figure 7. Plot of the 6 parameter with temperature. The dashes represent different confidence levels listed in the legend, while the dashed line represents the expectation values based on the parameters in Table 11 and eqs 1-6. and experimental values at -165.3 "C is given in Figure 2. This method of data reduction was extended to several other temperatures, and the results are also listed in Table I. The regression process would not converge for all the temperatures since at high temperatures the data were to the low-frequency side or at low temperatures the data were to the high-frequency side of the relaxation process. The results for the five temperatures that would converge for the 33.3 mol % are given in Table I. Plots of the five parameters of eq 1 against temperature are given in Figures 3-7. In these plots, the expectation values are not reported; rather their means at f 1, 2,and 3 MSE limits are plotted with temperature. Variable-Temperature Results. The parameters of eq 1 and their dependence on temperature can also be determined by using
(3) (4)
(5)
(6) temperature, chosen to center the experimental data at T - To = 0, RK = 1000/(273 -+ T ) , RKo = 1000/(273 + To),and T~ = I/&, or log, ( T ~ )= -log, cfo). These equations, except for eq 4,assume a linear dependence of the parameter on temperature, while eq 4 is essentially an Arrhenius plot. In this way it is not necessary to have a complete frequency range to define the parameters. Attempts to regress the 33.3 mol % IABR data using a reference temperature of To = -165.3 O C and data in the range -169.8 to -156.9 "C were successful, and the results are listed in Table 11. Also listed in Table I1 are the single-temperature results for the same temperature as the reference temperature. This technique of pooling the data to form a single data set extended the temperature range of the analysis from 6.2 to 12.9 O C , which is a 2-fold increase. The reason for this success is that the data set is balanced because there are (roughly) as many data points to the low-frequency side as there are to the high-frequency side of ( I I ) Havriliak, S., Jr.; Watts, D. Polymer 1986, 27, 1509.
The Journal of Physical Chemistry, Vol. 94, No. 1 1 , 1990 4709
Simulation of Polymer Chain Dynamics
TABLE 111: Results of a Linear Regression of Solution Conductance with Frequency temp, OC 127.1 135.9 144.8 156.3
R2 0.999 0.997 0.980 0.968
slope 2.01 2.06 2.03 2.02
SEOFO
CY
0.02 0.05 0.1 1 0.18
1.01 1.06 1.03 1.02
€0 - e,
= {I
+
(iWTo)")-'
d In
(7)
( T / T ~ )
is the distribution of relaxation times function and
( T / T ~ )
I I I I
I I I I I
I
.---
I I I I
I
+- - - - - +- - - - - +I I I
I
=
~ ; F ( T / T O )+( ~i o T 0 ) - ' d In
I
I
.----
the relaxation process. A comparison of the multiple temperature and single-temperature analysis at the reference temperature shows the results to be identical. The results of multiple temperature regression procedure are also reproduced in Figure 2 . Various other models for the temperature dependence of the parameters were tried. For example, eo or G, was assumed to depend on reciprocal absolute temperature, but there was no effective decrease in the residuals. Linear temperature in O C was assumed because there was no regression results to select any of the other forms and eqs 2-6 represent the simplest forms to represent temperature dependence. The parameter estimates listed in Table I1 are plotted against temperature in Figures 3-7 and are represented by the dashed line. The multiple-temperature analysis appears to trend from the low- to high-temperature limits of the single-temperature estimates as the temperature is raised. The most probable explanation for this systematic shifting of the parameters within the confidence limits from one side to the other is that the singletemperature estimates were unbalanced data sets that prejudice convergence to one side or the other depending on how the data are distributed in the complex plane. Combining all the temperatures to form a pooled data set gives a better balance of the data since about half are in the central region of the dispersion while the other half are equally distributed between the low- and high-frequency regions. This technique is readily extended to the other mixtures, Le., 50 and 75 mol % IABR; these results are given in Table 11. The frequency range for these mixtures are the same as it is for the 33 mol % mixture. Some difficulties were encountered when this method was extended to pure IABR. The frequency range for the 100 mol % IABR is 50 Hz to 3 MHz. Inspection of the data at high frequencies, i.e., greater than 500 kHz, showed small deviations from the linear behavior expected at high frequencies. Attempts to resolve these deviations in terms of a high-frequency dispersion lead to results that is contrary to expected physical behavior, Le., the low-frequency (residual) dielectric constant was lower than the high-frequency (residual) dielectric constant. Another characteristic of these deviations is that they occurred always at high frequencies at the other temperatures, suggesting that their origins are due to experimental errors. For this reason the data set was truncated above 500 kHz. This truncation keeps the data sets set the same for all compositions. The results for this analysis are given in Table 11. Solution conductance data were measured by Denney et al. for the 50 mol % mixture at those temperatures where t ' ( ~ )did not change with frequency. Under these conditions W T
