Macromolecules 1988, 21, 147-153
147
The Dielectric Constant Near the Liquid-Liquid Critical Point for Polystyrene in Diethyl Malonate J. L. Tveekrem and S. C. Greer* Department of Chemistry and Biochemistry, The University of Maryland at College Park, College Park, Maryland 20742 D. T. Jacobs Department of Physics, The College of Wooster, Wooster, Ohio 44691. Received June 15, 1987; Revised Manuscript Received August 7, 1987 ABSTRACT: We have measured the dielectric constant as a function of temperature and frequency near the liquid-liquid critical point of a mixture of polystyrene (molecular weight 1.02 X lo5) in diethyl malonate at the critical composition. The range of the frequency was 20 kHz to 1 MHz. For the one-phase region above the critical point, the range of the temperature was 1.1X lod < t < 7.9 X where t is the reduced temperature, IT- TcI/Tc,and T, is the critical temperature. The dielectric constant in the one-phase region near the critical point shows an anomalous increase of about 0.3% above the “background”behavior far from the critical point. This anomaly is consistent with the theoretical prediction of a leading critical exponent (1 - a ) ,where a is 0.11. The amplitude of the dielectric constant anomaly decreases with increasing frequency, an effect not yet considered in the theory. Below the critical temperature, the dielectric constants of the coexisting phases were measured as functions of temperature and frequency. The frequency ranged from 20 kHz to 1 MHz. < t < 3.5 X The difference between the dielectric constants The temperature ranged from 1.4 X of coexisting phases is consistent with a critical exponent @ of 0.325, where @ describes the vanishing of the order parameter at the critical point. The average of the dielectric constants of coexisting phases is nearly linear but shows a critical contribution that can be described by an exponent 2p. Neither above nor below the critical temperature was there any evidence of the theoretically predicted critical anomaly in the dielectric relaxation time. Introduction Modern theories of critical phenomena are based on the idea that the origin of the anomalies in the thermodynamic and transport properties near critical points is the growth of large and long-lived fluctuations in the order parameters.lY2 We present here experiments to test these theories for the description of dielectric properties near a liquidliquid critical point in a polymer solution. One-Phase Region: Temperature Dependence. Both phenomen~logical~,~ and m i c r o s ~ o p i capproaches ~~~ predict that the dielectric constant, e, a t zero frequency in the one-phase region near a liquid-liquid critical point should have the functional form: A 4 t(l-a+A) ... ~ / = p Al A2t (1) The dielectric constant is divided by the mass density, p, to account for the critical anomaly in the density itself, since the density is described by exactly the same function4 (the right-hand side of eq 1). The A;S are parameters that depend on the particular system. The reduced temperature, t, is IT - T,I/T,, where T, is the critical temperature. The exponent a describes the critical divergence of the heat capacity a t constant pressure and composition.6 The correction-to-scaling exponent, A, results from the Wegner expansion for higher order contributions to the critical behavi~r.’,~Calculations give values of 0.11 for CY and 0.50 for A.9 The effect of the (1 - a ) exponent is the addition or subtraction of a term, the magnitude of which decreases as the critical point is neared. The apparent background in the experimental data is a line connecting the data points far from T,. However, this line is strongly influenced by the (1- CY)term, so the true background is never observed. For simplicity, we will denote the size and sign of the critical anomaly as the deviation from the apparent background. In this convention, a positive sign for A3 will cause a decrease in e / p near T , and a negative sign for A , will cause an increase in ~ / p relative , to the apparent background. The thermodynamic analyses of Mistura3 and of Sengers et al.4 predict that A , should have the opposite sign from
+
+
+
+
that of dT,/dE2, where E is the electric field. Only two measurements have been reported for dT,/dE2: for 2,2,4-trimethylpentane + nitrobenzenelo and for aniline + cyclohexane.” For both systems, dTc/dE2is negative, which requires A, to be positive, which leads to a decrease in the dielectric constant near the critical point. Shakhparonov et al.12J3have argued, based on an approximate microscopic theory of fluctuations in the dielectric constant, that the dielectric constant should always decrease near an upper critical solution point or a liquid-vapor critical point. Some have thought that the anomaly in the static dielectric constant should have the same critical exponent as that in the electrical resistivity.14 Measurements of the resistivity near liquid-liquid critical points in nonaqueous mixtures15 give a critical exponent 2p, the value of which is 0.6L9 Measurements of the resistivity in aqueous mixtures have given an exponent (1 - a).16 Measurements of the static dielectric constant near fluid critical points have been reviewed by Sengers et al.4 and by Cohn and Greer.17 The unambiguous determination of the behavior of the dielectric constant requires attention to the effect of gravity near T, and to the influence of the Maxwell-Wagner effect18 and requires correction (as in eq 1)for the behavior of the density. The only unambiguous measurement of 4 t ) at a liquid-gas critical point (for which the density is constant) is that of Pestak and Chanlg for carbon monoxide, for which A , is negative and E increases near T,. There are no reported measurements of a critical anomaly in e(t) near a lower liquid-liquid critical point.20 In every unambiguous measurement near an upper liquid-liquid critical point, A , is positive and c(t) decreases near T,. These unambiguous measurements were made on the systems nitrobenzene + hexane,21benzonitrile + isooctane,18,22 polystyrene + cy~lohexane,2~ and nitroethane + cy~lohexane.~~ We report here new measurements of the static dielectric constant near the upper liquid-liquid critical point in polystyrene + diethyl malonate. The data are precise to < t < 7.9 X 0.004% and are in the range 1.1 X These data are consistent with eq 1,with A , negative and
0024-9297/88/2221-0147$01.50/0 0 1988 American Chemical Society
148 Tveekrem et al.
Macromolecules, Vol. 21, No. 1 , 1988
an increase in the dielectric constant near T,,in contrast to the positive A , observed in every other unambiguous study of a liquid-liquid critical point. In addition, we varied the frequency from 20 kHz to 1MHz and find that A3 decreases with increasing frequency. We have no understanding of the sign of A3 or of its frequency dependence. Two-Phase Region: Temperature Dependence. In the two-phase region below T,, the order parameter, which is the difference in a parameter, p , between the coexisting phases, is expected7s8to vanish a t T, as (pl - p 2 ) = B t q l + BltA + B p + ...) (2)
wD
where p1and p 2 are the values of the order parameter in the coexisting phases, B and B, are parameters depending on the particular system, p is a critical exponent of value 0.325: and t and A are the same as in eq 1. There are a number of possible choices for p at a liquid-liquid critical point,25,26including the mole fraction of either component, the volume fraction of either component, the mass density, and the refractive index. Numerous experimental studies25,26 have found these variables in eq 2 to describe the data satisfactorily, using the predicted value of p and various numbers of terms after the leading critical term. We choose here to regard the dielectric constant as the parameter p . We know of two published experimental studies of the dielectric constants of coexisting phases: a study of nitrobenzene + in which the data did not suffice to test eq 2 and a study of n-heptane + methano127bin which the data were consistent with eq 2. We present new measurements of the dielectric constant of coexisting phases of polystyrene + diethyl malonate as a function of temperature and frequency. The temperature range was 1.4 X < t < 3.5 X and the frequency ranged from 20 kHz to 1MHz. We find the coexistence curve to be consistent with eq 2 with a coefficient which does not depend on frequency. The diameter of the coexistence curve, which is the average of the parameter p (see eq 2) in the coexisting phases, is predictedz8 to have the form (p1 + p z ) / 2 = C1 + C,t C3tl-c‘ + ... (3)
+
where the Ciare parameters depending on the particular system. On the other if the choice of a particular variable for p in eq 3 leads to the (1- a)critical anomaly, then the substitution of any other variable p ‘, which is an analytic function of p , will lead to an additonal term in eq 3: (PI’+ p z ’ ) / 2 = C1’ + C,’t + C3’P + Cq’tl-n + ... (4) Thus the (1- a) term can be masked by the 2/3 term for certain choices of p . We find that the choice of the dielectric constant for the variable p leads to a 2p term in the diameter of the coexistence curve at all frequencies for polystyrene in diethyl malonate. Frequency Dependence. Our study of the frequency dependence of the dielectric constant above and below this liquid-liquid point was an attempt to observe the predicted slowing down of the dielectric relaxation time. The expectation has been that, near a critical point, the rotation of a molecule in an alternating electric field is no longer independent of the motion of other molecules and that this correlation of motion will reduce the frequency at which the molecule can continue to follow the field. The first theoretician to consider this problem was Snider,,l who predicted that the dielectric relaxation will be coupled to the diffusion of fluctuations, from which it follows that the dielectric relaxation frequency, oD,will go to zero near T, as
t3v
t1.89
(5)
where u is the critical exponent which describes the divergence of the correlation length’ near T , and u has the value 0.63.9 Later, G i a n n e s ~proposed i~~ that the dielectric relaxation is coupled to the thermal conductivity, resulting in a different behavior wD
zy
t2v
t1.26
(6)
which is not as strong a slowing down as predicted by Snider. There has been no experimental observation of the dielectric relaxation near a fluid critical point.18*20,27,33 We had hoped to observe this effect in the mixture polystyrene + diethyl malonate in that the high viscosity of the mixture might bring the dielectric relaxation time into an accessible frequency range. The mixture has a viscosity of 45 CPat 0.1 K above Tc.34Snider estimated that wD should be near lo5 Hz for a mixture of organic liquids. We assumed the frequency to be inversely proportional to the viscosity and estimated wD to be near 25 kHz for polystyrene in diethyl malonate. However, we observed no dielectric relaxation in the frequency range 20 kHz to 1 MHz with a resolution of 0.018% in the dielectric constant, for the temperature range 2.9 X < t < 7.9 X Experimental Methods Sample Preparation. The critical temperature for this mixture is extremely sensitive to water as an impurity. For example, a sample prepared without taking care t o exclude water showed a critical temperature near 299 K, whereas samples made with attention to dryness have critical temperatures near 276 K. Thus special care was taken to dry the pure components, the cell, and all implements and to prepare the sample in a dry nitrogen atmosphere. It is also important that the polystyrene be well-characterized and nearly monodisperse. The polystyrene, obtained from Toya ~ ~ a molecular weight of 1.02 x IO5 Soda Manufacturing C O . ,had and a ratio of weight-average molecular weight to number-average molecular weight of 1.02. It was dried in a vacuum over at 318 K for 2 weeks. The diethyl malonate, obtained from Aldrich Chemical Co., was specified to be at least 99% pure. Acids formed by hydrolysis of the ester were removed by extraction with 5% aqueous potassium carbonate. Then the ester was washed with deionized water to remove the potassium carbonate, shaken with sodium sulfate to remove most of the water, filtered to remove the sodium sulfate, and stored over molecular seive to dry it further, The critical composition for this molecular weight of polystyrene in diethyl malonate was determined in our laboratory36 to be near 9.49 weight 7’0 polystyrene. Gruner and G r e e P studied the density of this mixture by using polystyrene of molecular weight 1.07 X lo5 and found a concentration of 9.469 weight % polystyrene to be sufficiently near to critical. By comparison, Hamano et al.3E40reported a critical composition for this mixture of 8.48 weight 7‘0 using polystyrene of molecular weight 2 x lo5. These results are all consistent with the well-known decrease of critical composition as molecular weight increases?l For the present work, we prepared a sample of 9.473 f 0.002 weight % polystyrene in diethyl malonate. We will see from the coincidence of the diameter of the coexistence curve with the data in the one-phase region that this was very near to the critical composition. About 2 cm3 of dry nitrogen were above the sample in the filled dielectric cell. The critical temperature, T,, was determined experimentally from the abrupt change in the behavior of the dielectric constant a t T , and found to be 276.370 K (3.220 “C) This critical temperature is very low compared to other samples of this m i x t ~ r e , ~ ’ indicating that the sample was very dry. Dielectric Constant Measurements. The dielectric cell, shown in Figure 1, is a cylindrical capacitor with twin inner electrodes, allowing the simultaneous measurement of the dielectric constants of coexisting phases. It is a modification of an earlier design which had one inner ele~trode.~’ The electrodes are in a three-terminal mode with the inner electrode at low potential and the outer electrode at high potential. When the
Macromolecules, Vol. 21, No. 1, 1988
Critical P o i n t for Polystyrene in Diethyl Malonate 149
-6,5cmFigure 1. Cylindrical capacitance cell with two electrodes. The parts of the cell are as follows: A, cell lid and first guard ring; B, two inner cylindrical electrodes a t low potential; C, outer cylindrical electrode at high potential; D, guard rings; E, Teflon O-rings. The lid is sealed to the can with a Teflon O-ring. Electrical leads (not shown) to the inner assembly exit the cell through the core of the inner assembly. capacitance bridges are at balance, the guard rings and the thermal shields surrounding the cell are also at the low potential. The capacitance of each empty capacitor is about 3.3 pF. Two different capacitance bridges were used. For high-resolution measurements (0.004 %) a t a fixed frequency (30 kHz), we u sed a General Radio model 1615 bridge,43driven by a 50 mV sine wave,44for which the out-of-balance signal was detected with a PAR Model 5204 lock-in amplifier.45 The fluid sample was so conductive that an external conductance standard had to be added to the bridge to achieve balance.46 T h e standard conductance was made of two 1 MG resistors (with low-temperature dependence) in parallel. The capacitance measurements had to be corrected for the capacitance of the external resistor and the change of that capacitance over the time period of the experiment,& a correction which reduced the accuracy of the General Radio bridge measurements to 0.1 %, even though the precision of those measurements remained a t 0.004%. Indeed, measurements of the sample dielectric constant in the one-phase region using the two capacitors agreed to 0.1%. For lower resolution measurements (0.018%) over a range of frequencies (20 kHz to 1MHz), we used a Hewlett-Packard Model 4277A automatic capacitance bridge.47The accuracy of this bridge was also 0.1%. This bridge was operated on the IEEE-488 interface bus by a Commodore 64 computer4s and a Buscard I1 interface board.49 Temperature Control a n d Measurement. The cell was held in a thermostat with three nested stage^:^^^^ a radiation shield, a heater stage, and cooler stage. Circulating ethanol brought the cooler stage to 2 K below the temperature of interest. An analog controller attached to a resistance heater brought the heater stage to the temperature of interest and held it within 1mK. The space between the stages was evacuated, and all leads were thermally tempered. The radiation shield and the cell passively followed the heater stage in temperature. The cell temperature was measured by a thermistop imbedded in vacuum grease in a hole in the top of the cell. The thermistor resistance was measured by an ac ratio transformer bridges4 The temperature was measured with a precision of 0.001 K, but the accuracy of the calibration on the International Practical Temperature Scale of 1968 was only 0.05 K. Procedures. T h e capacitances of the empty cell, CE,were measured between 273 and 298 K and were reproducible on heating and cooling runs. The General Radio bridge capacitance data were fitted by a quadratic function of temperature, and the Hewlett-Packard bridge data were fitted by linear functions of temperature.4B These functions were then used to calculate the dielectric constant of the sample fluid from its capacitance, Cs: c ( T ) = CS(T)/C,(T). The measurements on the filled cell were all made while cooling but were reproducible from run to run. Once the sample was cooled below T,, it was necessary to heat the cell to about 290 K and shake the cell (i.e., the thermostat) to mix the fluids before beginning another run.
Results and Analysis One-Phase Region: Temperature Dependence. T a b l e I displays the dielectric constant as a f u n c t i o n of
J 5-IO0 -
15
i0
20
25
T-T, IK)
F i g u r e 2. Deviations of the measurements of the dielectric constant as a function of temperature, T, above the critical point of polystyrene in diethyl malonate (Table I), from the apparent background. The "background" was determined from a fit of the 11 points furthest from the critical temperature, T,, to a straight line. The increasing deviation near T , indicates that a critical contribution is present in this mixture near its upper critical solution point.
Table I Dielectric Constant, e, Measured a t 30 kHz with t h e General Radio Bridge, as a Function of Temperature, T, above the Critical Temperature, T,(276.370 K),for a Mixture of 9.473 wt % Polystyrene (Molecular Weight 1.02 X los) in Diethyl Malonate (T- TJ, (T- TJ, K c run K c run 21.856 7.3304 3 2.333 7.8155 2 21.800 7.3315 2 1.669 7.8337 2 21.710 7.3332 1 1.365 7.8421 2 19.908 7.3766 3 0.936 7.8543 2 17.919 7.4239 3 0.790 7.8583 3 16.974 7.4456 1 0.732 7.8604 4 16.940 7.4460 1 0.475 7.8677 3 16.826 7.4499 3 0.449 7.8683 3 14.899 7.4960 3 0.307 7.8727 2 12.223 7.5616 3 0.303 7.8727 6 11.610 7.5767 1 0.297 7.8730 3 9.221 7.6361 7.8764 3 0.201 4 8.757 7.6475 7.8765 1 0.200 4 7.720 7.6733 1 0.175 7.8771 5 6.619 7.7019 7.8784 1 0.132 4 5.204 7.7389 7.8801 2 0.082 4 4.621 7.7541 1 0.074 7.8803 7 3.864 7.7744 2 0.064 7.8805 6 3.348 7.7880 7.8807 2 0.061 4 3.266 7.7901 1 0.042 7.8812 5 3.180 7.7927 7.8813 2 0.034 4 3.089 7.7953 2 0.030 7.8814 5 2.993 7.7979 7.8817 2 0.022 5 2.845 7.8018 2 0.019 7.8818 6 2.700 7.8056 7.8824 2 0.008 5 2.480 7.8114 2 0.003 7.8825 9
temperature a b o v e T,for the critical m i x t u r e of polys t y r e n e + d i e t h y l malonate, as measured b y using the upper electrode and the General R a d i o bridge at 30 kHz. The data t a k e n using the lower electrode show the same behavior and are not given here.46 Figure 2 shows a plot of the deviations of these p o i n t s f r o m a f i t of a s t r a i g h t line to the 11 points f u r t h e s t f r o m T,. The increasing positive deviations near T , d e m o n s t r a t e that t h e r e is a critical anomaly, an i n c r e a s e of about 0.3% above the apparent b a c k g r o u n d dielectric c o n s t a n t .
Table I1 shows the fits of various functions t o t h e data i n T a b l e I. The analysis was done b y using a weighted nonlinear least-squares routine.51 T h e d e n s i t y of t h i s
150 Tveekrem et al.
Macromolecules, Vol. 21, No. 1, 1988
Table I1 Fits of Functions to the Data in Table I for the Dependence of the Dielectric Constant, e, of Polystyrene in Diethyl Malonate above the Critical Temperature on the Reduced Temperature t , Where t = IT- T,I/T,, TIS Temperature, T , Is the Critical Temperature, and p Is the Density in g/cm3 (from Ref 37)" function XU2 A, f u A, f u A, f u A4 f u 17 7.8812 f 0.0003 -7.68 f 0.03 9.3 f 0.4 (1) t = A , + A,t + A3t2 (2) c = A , + Azt + A# + A4t3 3.4 7.8819 f 0.0001 -7.96 f 0.02 20.7 f 0.8 -103 f 7 3.2 7.8832 f 0.0001 -3.14 f 0.06 -2.92 f 0.05 (3) c = A , + A2t + A3t1-' (4) c = A , + APt + A3t1-"+ A4t1-atA 0.84 7.88260 f 0.00008 -5.8 f 0.2 -1.4 f 0.1 2.1 0.2 -5.16 f 0.02 7.5 f 0.3 ( 5 ) t / p = A , + Azt + A3t2 15 7.3630 f 0.0002 (6) r / p = A , + A,t + A3t2 + A,t3 3.0 7.3637 f 0.0001 -5.43 f 0.02 18.0 f 0.8 -96 f 7 -1.46 h 0.04 -2.39 f 0.03 ( 7 ) t i p = A , + Azt + A3t1-' 1.4 7.36463 f 0.00008 (8) c / p = A , + A,t + A,t'-" + A4t1-u*A 0.75 7.36434 f 0.00008 -2.9 f 0.2 -1.6 f 0.1 1.1f 0.2
" T , was taken to be 276.370 K. The exponent 01 was taken to be 0.11 and A to be 0.50 (see ref 9). The standard deviation in T was taken and that in c to be 0.0003. The uncertainty, u, given for each parameter is its standard deviation. The measure of goodness-of-fit is x: (see ref 51). to be 0.001 K
mixture has been measured by Gruner and Greer37using a magnetic float densimeter, with a precision 0.01 % and an accuracy of 0.1%. For the temperature range 1.1X < t < 6.7 X the density shows no critical anomaly. In fits 1through 4,we do not divide the dielectric constant by the density. In fits 5 through 8 we do divide the dielectric constant by the density. Fits 1and 2 use three- and four-term polynomials in the reduced temperature and do not describe the data. Fits 3 and 4 are for eq 1,with three and four terms, respectively (except that t has not been divided by the density). Thus eq 1,using t as the dependent variable and with four terms, provides a satisfactory description of the data. The addition of a fifth term (with exponent 1- a + 2A) does not improve the fit. The variation of T , by 0.002 K does not change the fit. Fits 5 through 8 take t i p as the dependent variable. We do not expect a dramatic effect from dividing by the density, since the density has a critical anomaly of less than 0.01 % , but doing so will remove some of the influence of the background behavior. Fits 5 and 6 use three- and four-term polynomials and do not describe the data satisfactorily. Fits 7 and 8 use eq 1with three and four terms. Fit 7 is already rather good, and fit 8 is slightly better than fit 4. We conclude that our precise (0.004%) measurements of the temperature dependence of the dielectric constant at 30 kHz show a critical anomaly which is an increase of about 0.3% of the background dielectric constant and which is consistent the theoretically predicted function given in eq 1. One-Phase Region: Frequency Dependence. The dielectric constant measurements made a t various frequencies using the Hewlett-Packard bridge had a lower resolution than did the measurements made using the General Radio bridge (0.018% as opposed to 0.004%). The Hewlett-Packard bridge data are not listed here, but are a ~ a i l a b l e .Figure ~~ 3 shows the dielectric constant as a function of frequency at three different reduced temperatures. In general, the dielectric constant has a maximum at low frequencies, remains fairly cosntant between 30 and 500 kHz, and increases again a t high frequencies. We attribute the low-frequency behavior to electrode polarization and the high-frequency behavior to lead capacitances. At no temperature did we ever see the "step" in the dielectric constant as a function of frequency which characterizes a dielectric relaxation. We note also that far above T , ( t = 7.91 x the dielectric constant becomes constant with frequency at 20 kHz, whereas, nearer to T , ( t < the dielectric constant is still decreasing with frequency up to about 60 kHz.
738
i -- . 0
791X10-2 123x10-4
-
791
.
J '
. 7321
A 789
1 1 1 1
I00
LOG FREQUENCY (kPz;
Figure 3. Dependence of the dielectric constant of polystyrene
in diethyl malonate on frequency at three temperatures above the critical temperature, T,. The legend identifies the symbols; t is the reduced temperature (T - T,)/T,. The dielectric constants are high at low and high frequencies because of electrode polarization and lead capacitance, respectively. The dielectric constant is relatively constant between 30 and 400 kHz, except for the increase at 30-60 kHz near T,, which, as discussed in the text, is believed to be due to a frequency dependence of the critical contribution. One possible cause of such behavior is a broad relaxation due to the Maxwell-Wagner effect.ls However, we estimate that the Maxwell-Wagner effect in polystyrene + diethyl malonate should appear at 2 kHz and thus believe that it should not be important in our measurements. Another possible cause of this effect is a frequency dependence of the amplitude of the critical anomaly in the dielectric constant. Indeed, the theoretical predictions are meant to apply to the dielectric constant at zero frequency. To test for such a frequency dependence, we have fitted the dielectric constant divided by the density a t each frequency betwen 30 and 400 kHz to eq 1, using three terms and taking the standard deviation in the dielectric constant to be 0.0013 (0.018%). The amplitudes of the terms in eq 1 as functions of the frequency are given in Table 111. Clearly the absolute value of the amplitude of the critical anomaly, A s , does decrease as frequency increases. The other coefficients, A, and A z , also depend on frequency. We have no explanation for the frequency dependence of the amplitudes. Coexistence Curve: T e m p e r a t u r e Dependence. Having two capacitors, one in each of the coexisting phases, we could measure the dielectric constants of the coexisting phases. These eight data points, as measured with the General Radio bridge, are listed in Table IV and plotted in Figure 4;the temperature range is 1.4 X < t < 3.6 X Also shown in Figure 4 are some of the one-phase
Macromolecules, Vol. 21, No. 1, 1988
Critical Point for Polystyrene in Diethyl Malonate 151
Table I11
The Amplitudes, Ai,of the Terms in Eq 1 for Polystyrene in Diethyl Malonate, as Functions of Frequency, f a f , kHz A1 f 01 A2 f 0 2 A3 f 0 3 30 40 50 60 70 80 90 100 200 300 400
7.3651 f 0.0003 7.3644 f 0.0005 7.3631 f 0.0003 7.3620 f 0.0003 7.3602 f 0.0003 7.3604 f 0.0003 7.3601 f 0.0003 7.3588 f 0.0003 7.3579 f 0.0002 7.3591 f 0.0004 7.3573 f 0.0004
-1.6 -1.7 -1.9 -1.8 -1.5 -1.5 -1.6 -2.7 -3.2 -3.2 -3.3
f 0.2 f 0.2 f 0.1 f 0.1
i 0.1 f
0.1
f 0.1 f 0.2 f
0.1
f 0.2 f 0.2
-2.3 -2.3 -1.9 -1.8 -1.5 -1.5 -1.7 -1.4 -1.0 -1.0 -0.9
f 0.1
0.2
* 0.1
f 0.1 f 0.1 f 0.1 f 0.1 f 0.1 f 0.1 f 0.2 f 0.1
" These fits are based on the data taken with the Hewlett-Packard bridge. The uncertainty in Ai is given as ai,one standard deviation. The details are given in the text.
(b).
I 0 - ------
.E
Table IV The Dielectric Constants, e, at 30 kHz of Coexisting Phases of Polystyrene (Molecular Weight 1.02 X los) in Diethyl Malonate, as a Function of (T,- T),Where T Is the TemDerature and T,Is the Critical TemDerature" 0.004
0.027 0.041 0.052 0.102 0.354 0.688 0.988
7.885 64 7.975 82 7.987 60 8.001 98 8.03563 8.12697 8.19360 8.236 44
7.884 29 7.853 73 7.84568 7.836 77 7.81768 7.776 75 7.76761 7.772 50
x
X
--
-
Y
I
c t-
_-_-_-____-_____
-
0..
.
-
-05-
I
-10
-
1
"These data were taken with the General Radio bridge. The critical temperature of this sample was 276.370 K. The dielectric constant of the upper phase is tu, and that of the lower phase is el. data near T , and the diameter of the coexistence curve (discussed below). The close coincidence of the one-phase data and the diameter near T, indicates that the sample was, indeed, very near to the critical composition. Our coexistence curve can be described by eq 2 if p is taken as the dielectric constant. We did not divide p by the density4 in studying the two-phase region because the density data in the two-phase region37are available only for t < 8 X In fitting eq 2 to the data in Table IV, the exponents were fixed at their theoretical values, T, was fixed at 276.370 K, and the uncertainty in the temperature was taken as 0.001 K. In order to have x,,?be unity when the residuals were randomly distributed, the standard deviation in the dielectric constant had to be taken as 0.003 (0.04%)-10 times its uncertainty in the one-phase region. Then eq 2 described the data if the point at 3.216 K (the point nearest to T,) were omitted and if three terms were used in the fitting equation. The x,,?was 1.2, and the residuals were randomly distributed. The fitting parameters were B = 2.1 f 0.1, B1 = 44 f 6, and B2 = -532 f 78. We believe that the increased scatter in the data in the two-phase region is related to the increase in the equilibration times. We do not know why the point nearest T , is an outlier; we are quite confident that phase separation had occurred when the data point was taken, but it is likely that equilibrium was not reached in the 2 days allowed. Coexistence Curve: Frequency Dependence. The dielectric constant was also measured in the coexisting phases by using the Hewlett-Packard bridge and varying the frequency. The dielectric constant as a function of
frequency in each phase a t a given temperature showed the same general behavior as in the one-phase region (Figure 3). As the frequency was increased, the coexistence curve as a whole was shifted to slightly lower values of the dielectric constant (Figure 4). This shift did not affect the parameters needed to fit the data a t various frequencies to eq 2: for frequencies between 30 and 400 kHz, the parameters were, within the uncertainty, the same as those determined for the data from the General Radio bridge at 30 kHz (Table IV) and also required a standard deviation in the dielectric constant of 0.003 and the exclusion of the point nearest T,. Diameter of the Coexistence Curve: Temperature Dependence. The diameter of coexistence curve for the choice of the dielectric constants of the coexisting phases for p 1 and p 2 is shown in Figure 4. It is clearly not a straight line. We have fitted eq 3 and 4 to the dielectric constant data from the General Radio bridge (Table IV), fixing the exponents and T,,taking the standard deviation in T to be 0.001 K, and omitting the data point closest to T,. The standard deviation for the dielectric constant was taken to be 0.0009 in order to have : x be unity when the residuals were randomly scattered. This standard deviation for e is less than that used for fitting the coexistence curve for the same data set, for unknown reasons. We find
152 Tveekrem et al.
that eq 4 with only the first three terms gives a significantly better fit (x,2 = 0.96) than does eq 3 with three terms (x: = 1.6). The coefficients of the fit to eq 4 are C,’ = 7.9081 f 0.0008, C2’ = 10 f 2, and C3’ = 2.0 f 0.2, where the uncertainty is one standard deviation. The coefficient C1’ should be the same as the coefficient AI in fit 4 of Table 11. The difference of 0.0255 is a reflection of how close the sample composition is to the true critical composition, which cannot be fully assessed since we have no information on the dependence of e on composition. The best choice of fitting function is the same for the data from the Hewlett-Packard bridge (which we do not list here) over the frequency range 30-400 kHz; the coefficients are even the same within their uncertainties, except for C1,which decreases slightly with frequency (e.g., 7.898 f 0.001 a t 400 kHz). We conclude that the curvature in the diameter in Figure 4 is due to a 2P term.
Conclusions The dielectric constant above the critical temperature for polystyrene (molecular weight 1.02 x lo5) in diethyl malonate shows the expected critical anomaly with exponent (1- N). The coefficient of this anomaly is negative, leading to an increase in the dielectric constant of about 0.3% above the apparent background as the critical point is neared. This system behaves differently from other critical mixtures in that the anomaly in the dielectric constant is rather large and in that the effect is an increase in the dielectric constant rather than a decrease. By comparison, the mixture polystyrene (molecular weight 35 800) in cyclohexane has an anomaly in the dielectric constant which is a decrease as the critical point is neared of about 0.007% .23 It is not understood why the amplitude and the sign of the coefficient of the critical anomaly take the values they do. As mentioned the sign of the coefficient of the critical anomaly should be opposite to that of dTc/dE2. The sign of dT,/dE2 will be positive if the field makes the components of the mixture less miscible and negative if the field makes the components more miscible. Let us compare the systems polystyrene + cyclohexane and polystyrene + diethyl malonate. Polystyrene has a small dipole moment of about 0.3 D.52 Diethyl malonate has no “permanent” dipole moment in that the chemical bonds may be assumed to rotate freely, averaging over the bond dipole moments. However, the electric field will align those bond moments and induce a dipole moment of about 0.6 D.53Cyclohexane has no dipole moment. Recall the old adage of chemists that “like dissolves like”. In the case of cyclohexane as the solvent, the field will induce a small moment (due to the polarizability) in the cyclohexane, making the cyclohexane more like the polystyrene and therefore making the two more soluble and producing a negative dTc/dE2. In the case of diethyl malonate as the solvent, the field will induce a rather large dipole moment in the diethyl malonate, making the two components less alike and less soluble, so that dTJdE2 will be positive. However, we know no molecular (as opposed to therm~dynamic~*~) interpretation for the connection between dTc/dE2 and the coefficient of the critical anomaly in the dielectric constant in the one-phase region. In the two-phase region, the difference in the dielectric constants of coexisting phases is consistent with the theoretically predicted exponent 6 , and the average of the dielectric constants shows a curvature which is best described by an exponent 2/3. In the one-phase region, the amplitude of the critical anomaly in t / p decreases with frequency, an effect for
Macromolecules, Vol. 21, No. 1, 1988 which we have no explanation. In the two-phase region, the choice of fitting function and the values of the fitting parameters are insensitive to frequency over the range 30-400 kHz for both the coexistence curve and its diameter. There is no evidence of a dielectric relaxation in our measurements, above or below T,, and thus no evidence of a critical slowing down of a dielectric relaxation time. The dielectric relaxation could have occurred at frequencies above those of our experiment or at an amplitude below our resolution. On the other hand, it seems not unlikely that the dielectric relaxation, like the self-diffusion,2takes place locally in space and is thus not affected by the growing critical fluctuations.
Acknowledgment. We are grateful to R. Kopelman for help in assembling the instrumentation, to R. Bendt for the machining of the cell, and to K. Gruner for helpful discussions and the use of her density data prior to publication. This work was supported by the National Science Foundation under Grant CHE-8413404. Acknowledgment is made to the donors of The Petroleum Research Fund, administered by the American Chemical Society, for the partial support of this research. This work is taken from the Ph.D. dissertation of June L. Tveekrem, presented to the Chemical Physics Program of the University of Maryland at College Park in 1986. Registry No. Polystyrene, 9003-53-6; diethyl malonate, 10553-3.
References and Notes Wilson, K. G. Reu. Mod. Phys. 1983, 55, 583. Hohenberg, P. C.; Halperin, k.I. Reu. Mod. Phys. 1977, 49, 1977. Mistura, L. J. Chem. Phys. 1973,59, 4563. Sengers, J. V.; Bedeaux, D.; Mazur, P.; Greer, S. C. Physcia A (Amsterdam) 1980, 104A, 573. Goulon, J.; Greffe, J. L.; Oxtoby, D. W. J . Chem. Phys. 1979, 21, 4742. Griffiths, R. B.; Wheeler, J. C. Phys. Rev. A 1970, 2, 1047. Wegner, F. Phys. Rev. B: Solid State 1972, 5, 4529. Ley-Koo, M.; Green, M. S. Phys. Reu. A 1977, 16, 2483. Le Guillou, J. C.; Zinn-Justin, J. J.Phys. Lett. 1985,46,L137. Debye, P.; Kleboth, K. J. Chem. Phys. 1965,42, 3155. Beaglehole, D. J. Chem. Phys. 1981, 74, 5251. Shakhparonov, M. I. Russ. J. Phys. Chem. (Engl. Transl.) 1960, 34, 706. Lomova, N. N.; Shakhparanov, M. I. Proc. Acad. Sci. USSR (Transl.) 1957, 134, 899. Kumar, N.; Jayannavar, A. M. J. Phys. C 1981, 14, L785. Shetty, C.; Gunasekaran, M. K.; Vani, V.; Gopal, E. S. R. Pramana 1983,21, 71 and references therein. Anderson, E. M.; Greer, S.C. Phys. Reu. A 1984,30,3129. See also: Gunasekharan, M. K.; Guha, S.;Vani, V.; Gopal, E. S. !I Ber. . Bunsenges. Phys. Chem. 1985,89, 1278. (.‘ohn, R. H.; Greer, S. C. J. Phys. Chem. 1986, 90, 4163. rlioen, J.; Kindt, R.; Van Dael, W. Phys. Lett. A 1980. %A; 445. Pestak, M. W.; Chan, M. H. W. Phys. Reu. Lett. 1981,46, 943. Kaatze, U.; Woermann, D. J . Phys. Chem. 1984,88, 284, saw no critical anomaly in the dielectric constant for the lower critical solution point in 2,6-lutidine + water at 1% resolution. Piekara, A. Phys. Reu. 1932,42,448. Konecki, M. Chem. Phys. Lett. 1978,57,90. There is a disagreement on the sign of the anomaly between these two papers and: Ziejewski, Z.; Piotrowska-Szczepaniak, J.; Ziola, J. Acta Phys. Pol., A 1979, 56, 347. Miller, B. C.; Clerke, E. A.; Greer, S. C. J . Phys. Chem. 1980, 87, 1063. Jacobs, D. T.; Greer, S. C. Phys. Reu. A 1981, 24, 2075. Thoen, J.; Kindt, R.; Van Dael, W. Phys. Lett. A. 1981,87A, 73. Greer, S.C.; Moldover, M. R. Annu. Rec. Phys. Chem. 1981, 32, 233. Kumar, A.; Krishnamurthy, H. R.; Gopal, E. S. R. Phys. Rep. 1983, 98, 57. (a) Ziejewska, Z.; Piotrowska-Szcepaniak, J.; Ziolo, J. Acta Phys. Pol, A 1979, A56, 347. (b) Jayalaxmi, Y.; Guha, S.; Van, V. C.; Gopal, E. S. R. Pramdna 1987, 28, 269.
Macromolecules 1988,21, 153-156 (28) Goldstein, R.; Ashcroft, N. W. Phys. Rev. Lett. 1985,55,2164 and references therein. (29) Widom, B. Proc. Robert A. Welch Found. Conf. Chem. Res. 1972, 16, 161. (30) Buckingham, M. J. Phase Transitions and Critical Phenomena; Domb, C., Green, M. S., Eds.; Academic: London, 1972; Vol. 2, p 18. (31) Snider, N. S. J . Chem. Phys. 1972, 56, 233. (32) Giannessi, C. Phys. Rev. 1980, 22, 706. (33) Kaatze, U.; Woermann, D. Ber. Bunsenges. Phys. Chem. 1982, 86, 81. (34) Gruner, K.; Greer, S. C., to be submitted for publication in Macromolecules. (35) The distributor in the United States is the Varian Instrument Group Service Center in Sunnyvale, CA. (36) Miller, B. C.; Furrow, G. P.; unpublished results. (37) Gruner, K.; Greer, S. C. Macromolcules 1987,20, 2238. (38) Hamano, K.; Kuwahara, N.; Nakata, M.; Kaneko, M. Phys. Lett. 1977, 63A, 121. (39) Hamano, K.; Kuwahara, N.; Kaneko, M. Phys. Rev. A 1979,20, 1135. (40) Hamano, K.; Kuwahara, N.; Kaneko, M. Phys. Rev. A 1980,21, 1321.
153
(41) Dobashi, T.; Nakata, M.; Kaneko, M. J. Chem. Phys. 1980, 72, 6685. (42) Jacobs, D. T.; Greer, S. C. Rev. Sci. Znstrum. 1980, 51, 994. (43) GenRad, Inc., Concord, MA, 01742. (44) Hewlett-Packard model 3311A, Hewlett-Packard, Palo Alto, CA 94303-3308. (45) Princeton Applied Research, Princeton, NJ 08540. (46) Tveekrem, J. L. Ph.D. Dissertation, The University of Maryland at College Park, 1986. (47) Hewlett-Packard model 4277A, Hewlett-Packard, Palo Alto, CA 94303-3308. (48) Commodore Business Machines, Norristown, PA 19403. (49) Batteries Included, Toronto, Ontario, Canada, M5V 1Z1. (50) Thermometrics Series SP, Thermometrics, Inc., Edison, NJ 08817. (51) Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences; McGraw-Hill: New York, 1969. (52) Brandruo. J.: Immereut. E. H. Polvmer Handbook: Interscience: New York, 1976. ' (53) Ketelaar, J. A. A.: van Meers. N. Recl. Trav. Chim. Pavs-Bas 1957, 76; 437. We calculated the dipole moment from the measurements of the dielectric constant as a function of temperature reported in this paper.
Dielectric Relaxation in Dilute Solutions of Poly(organophosphazenes): Evidence of the Localized Electronic Structure of the P=N Double Bonds Shunsuke Uzaki,' Keiichiro Adachi, and Tadao Kotaka* Department of Macromolecular Science, Faculty of Science, Osaka University, Toyonaka, Osaka 560, J a p a n . Received J u n e 15, 1987; Revised Manuscript Received August 10, 1987 ABSTRACT: Dielectric measurements were carried out on dilute benzene solutions of fractionated poly(diphenoxyphosphazene) at a concentration of ca. 0.1%. Loss maxima were observed in the audiofrequency range at 303 K. The relaxation times T determined from the frequencies of these loss maxima increased with increasing (weight-average) molecular weight M, of the fractions in proportion to Mwl,'. This molecular weight dependence and the absolute values of T were in agreement with the prediction of the Zimm theory evaluated by using the experimental intrinsic viscosity values. Thus, poly(ph0sphazene) can be classified as a type A polymer which has components of the dipole moment aligned in the same direction parallel to the chain contour. This strongly suggests that the P=N and N-P bonds are not equivalent in contrast to the current model of the delocalized electronic structure of the P-N backbones. The dipole moment for the parallel component per monomer unit was estimated t o be ca. 6.7 D.
Introduction Dipoles of linear polymers were classified i n t o t y p e s A, B, and C b y Stockmayer.' Type A dipoles correspond to those aligned uniaxially in the direction parallel to the chain contour, and type B dipoles axe perpendicular to the chain. On the other hand, t y p e C dipoles are the ones a t t a c h e d to mobile side groups. The structural formula of the main c h a i n of poly(phosphazene) is usually represented as -(P=N),-.293 If the p o l y ( p h o s p h a z e n e ) molecules a c t u a l l y have t h i s electronic structure, the polymer should be classified as a t y p e A polymer since the c h a i n has n o s y m m e t r y elements s u c h as a mirror plane or a 2-fold rotation axis. On the other hand, if the T electrons on the double b o n d s delocalize, the two bonds become equivalent and hence this polymer s h o u l d not be type A. Allcock2 discussed these two possibilities of the electronic structure of poly(phosphazenes) and suggested that the latter structure is more plausible, because Giglio et al.4 suggested o n the basis of their X-ray crystallographic s t u d y that the bond lengths for the t w o P-N bonds were the same. U n t i l recently, +Present address: Mitsubishi Electric Co., 8-1-1 TsukaguchiHonmachi, Amagasaki 661, Japan. 0024-9297/88/2221-0153$01.50/0
there existed no other experimental evidences to support or disprove this view. In fact, T a n a k a et ale5calculated the molecular orbitals of this p o l y m e r , a s s u m i n g the equivalence of the two P-N bonds. W e have studied this problem b y means of a dielectric m e t h o d involving observation of the dielectric n o r m a l mode process' of dilute solutions of fractionated poly(organophosphazenes). Our preliminary results6 strongly suggested that the localized electronic s t r u c t u r e m o d e l appears t o be more likely. In fact, a recent X-ray analysis carried out b y C h a t a n i and Yatsuyanagi' showed that the two P-N bonds of poly(dich1orophosphazene) are n o t equivalent. If a polymer has t y p e
A dipoles, it should exhibit the dielectric normal mode process due to fluctuation of the
end-to-end vector of t h e molecules.' Then, we c a n easily distinguish type A dipoles from others by the following two characteristic properties. First, the dielectric relaxation due to t y p e A dipoles exhibits a s t r o n g molecular weight d e p e n d e n c e and c a n be described b y t h e bead-spring m o d e l proposed b y Rouse and Zimm.8~9 According to Zimm's theory, the relaxation t i m e s for t h e normal m o d e process in d i l u t e solutions are given by9 T
= O.85Mvs[q]/RT
0 1988 American Chemical Society
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