The Journal of
Physical Chemistry
0 Copyright, 1985, by the American Chemical Society
VOLUME 89, NUMBER 18 AUGUST 29, 1985
LETTERS Dielectric Behavior of Aqueous NaCl Solutions Paul Winsor, IV, and Robert H. Cole* Department of Chemistry, Brown University, Providence, Rhode Island 0291 2 (Received: March 25, 1985; In Final Form: June 21, 1985) Measurements of dielectric permittivities by TDR methods for frequencies from 50 MHz to 9 GHz are reported for 0.05, 0.10, and 0.5 M solutions in water at 25 OC. These show a linear decrease in low-frequency permittivity and increase in dielectric relaxation time with concentration. The observed decreases are in rough agreement with microwave results at higher concentrations but smaller than results at lower frequencies and concentrations. They indicate contributions from both kinetic depolarization and ionsolvent interactions,with the total larger than predicted by the theory of Hubbard, Colonomos, and Wolynes, but considerably smaller than by Kusalik and Patey. Possible reasons for the differences are discussed. We report here dielectric permittivities of solutions of NaCl in water at 25 OC and concentrations from 0.05 to 0.5 M. These were of interest primarily for comparison with theories of saturation or solvation in ionic fields and of kinetic depolarization of solvent by moving ions and were obtained at frequencies from 50 MHz to 9 G H z by numerical Fourier transformation of time domain reflectometry (TDR) records. Measurements in this range are needed because other studies of 1-1 salts in water have been either at microwave frequencies (above 3 GHz), and for the most part at concentrations greater than 1 M, or at radio frequencies and concentrations below 0.1 M. We discuss the resulting uncertainties of interpretation after comparison with our results. The TDR instrumentation used was basically as described by Perl et al.’ Solutions were placed in a small volume beyond the coaxial line. The cell end of the center conductor in a 3.5” so formed was calibrated with water and methanol as standards using the method of bilinear analysis2to correct for effects of small impedance mismatches in the system and of wave propagation in the cell. Representative results for a 0.5 M NaCl solution are shown in Figure 1 by the complex plane plot of e” vs. e’ and the plot of e’ vs. we”, where e* = 6’ - it” is the complex permittivity and w = 27r frequency. In both cases, the contribution of d.c. conductance to the measured total loss e,’ has been subtracted; this value was obtained from the limiting long time TDR response and confirmed by audio-frequency bridge measurements. The complex plane plot indicates that to 7 GHz the results are consistent with a semicircular locus (dashed curve) and simple Debye relaxation (1) Perl, J. P.; Wasan, D. T.; Winsor IV, P.; Cole, R. H. J. Mol. Liq. 1984, 28, 103. (2) Cole, R. H. IEEE Trans. Instrum. Meas. 1983, IM-32, 42.
0022-3654/85/2089-3775$01.50/0
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function t* = e, (es - t,)/(l iwrD),where e, and t, are limiting low- and high-frequency pennittivities and rDis the Debye relaxation time. In such a case, the plot of e’ vs. wd’ is a straight line with intercept E, for wd’ = 0 and Slope-TD, which provides a simple way of determining these parameters and estimating uncertainties of f l in t”. Values of cs and T D so derived for 0.05,0.1, and 0.5 M solutions are plotted in Figure 2 as a function of specific solution conductance K, in S/m (1 S/m = 10-* mho/cm); similar plots vs. ion concentration are also linear within the precision of the results. For comparison with other results, the dielectric decrement 6 defined by ts= t, 6c, where eW = 78.6 is the static permittivity of water, is 6 = -16.8 L/mol. The indicated increase in T D to fit our data below 7 GH,, or about half the relaxation frequency of water, is for c = 0.5 M about half the value T~ = 8.3 ps for water. The present value of decrement is roughly consistent with, but larger than, values inferred from microwave measurements as quoted by Hasted.) From plots of t, vs. c for KCl and LiCl at concentrations to 1.5 M (Figure 6.2 in his book), one can estimate that 6 N -14 L/mol below 0.5 M, but mean values based on data at much higher concentrations as well give 6 = -11 L/mol. Mandel and Van Beek‘s4 quite different measurements at radio frequencies for concentrations from 0.02 to 0.100 M shown in Figure 2 are only roughly in agreement with our extrapolated values for concentrations above 0.05 M, but presumably include the contribution of any ion atmosphere relaxation as they were made at frequencies well below the Maxwell relaxation frequency f, (GHz) = 1 8 ~ ,(mho/m)/e,. After subtracting estimates of the
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(3) Hasted, J. B. “Aqueous Dielectrics”; Chapman and Hall; London, 1973. (4) Hubbard, J. B.;Onsager, L.; Van Beck, W. M.; Mandel, M. Proc. Natl. Acad. Sci. 1977, 74, 101.
0 1985 American Chemical Society
3776 The Journal of Physical Chemistry, Vol. 89, No. 18, 1985 0.5 M NoCI, 25OC 40 10 G H r
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Figure 1. Upper: Complex plane locus of C” vs. c’ for 0.5 M NaCl in water at 25 OC. Lower: Plot of C‘ vs. wc” for the same data. Numbers by points are frequencies in gigahertz. NoCI/H20
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Figure 2. Upper: Plot of static permittivity cs vs. specific conductance K~ (in S / m ) for NaCl solutions. Filled circles and error bars from data in ref 4. Lower: plot of Debye relaxation time T~ (in ps) vs. K ~ .
increments Eased on a speculative extension of the Debye-Falkenhagen dilute solution theory,5 which gives relatively larger effects at low concentrations, the estimated slope of the line through their four data points for NaCl corresponds to 6 = -42 L/mol, which is nearly the same as the slope of the dashed line in Figure 2. Although there is only qualitative agreement as to magnitude and concentration dependence of the dielectric decrements, all are much larger than earlier estimates of static solvation/saturation effects from electrostatic continuum argument^.^ This kind of discrepancy is partially resolved by Hubbard and Onsager’s (HO) recognition4 of the kinetic depolarization effect from changing fields of moving ions. Their result6 for a continuum model of the ~
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( 5 ) Falkenhagen, H. ”Electrolytes”; Oxford University Press: London, 1934. (6) Hubbard, J B.; Onsager, L. J . Chem. Phys. 1977, 67, 4859.
Letters solvent gives a contribution proportional to specific conductance K, corresponding to 6 = -6.8 L/mol for a slip boundary condition at the ion surfaces and to -10.2 L/mol for a stick boundary condition (both for NaCl in water at 25 “C). Although the former might seem to be the more reasonable choice, Felderhof‘ has pointed out that the theory is correct for the stick, but not for the slip condition unless the ion is highly polarizable. The more recent molecular dynamics treatment by Hubbard, Colonomos, and Wolynes (HWC)* which includes an ion-dipole interaction term in their correlation function potential gives 6 = -9.1 L/mol for NaCl in a treatment corresponding to the H O slip condition in the limit of large ion size and presumably larger by a factor 3 / 2 for the stick condition. The various calculations of the kinetic effect account for 4040% of our measured decrements, leaving differences which could be accounted for by static continuum theories by plausible adjustment of parameters in the models. This not unreasonable situation is, however, completely changed by recent molecular theory as developed by Kusalik and Pateyg using the linear hypernetted chain approximation for calculations of both kinetic and static effects. Their results for the kinetic effect are not greatly different from the HCW calculations for 1:l salts, but the predicted static decrements are much larger. An estimate from Figure 4 of their paper gives a static contribution of ca.-30 L/mol, which is nearly twice our experimental -16.8 L/mol for the total effect. Unless our present results are grossly in error, which we have no reason to suspect, we can only suggest that the origin of the discrepancy may be a concentration effect, as the theoretical calculations were for dilute solutions at concentrations below 0.05 M, while our experimental decrements are based on seemingly linear decrements in the range 0.05-0.5 M. The increase of relaxation time T~ with salt concentrations to 0.5 M is not defined very precisely, as our uncertainty in time resolution is about 1 ps, but we have no reason to doubt its reality. Microwave results quoted by Hasted3 suggest smaller decreases, from T~ = 8.3 ps for water, in the range 0.5-5 M. The differences may be the result of fitting Debye relaxation functions to the data for somewhat different frequency ranges, as data of Pottello for LiCl above 6 M show considerable deviations (as depressed circular arcs in complex plane plots), but the evidence is too limited to test this or other conjectures. From present evidence, the magnitudes of dielectric decrements of moderately dilute NaCl solutions in water are too large to be accounted for by kinetic effects alone, but much smaller than the static decrements calculated by modern statistical mechanical theory, and we expect similar effects for other salts. It is not clear to us whether the discrepancies result from concentration effects above 0.05 M, or from more basic problems of developing a complete theory which “should include all those effects in a self-consistent form”.* Experimentally, one needs both data of high precision from 1 MHz to 1 GHz, to establish “static” permittivities and ion atmosphere relaxation effects more directly at low and moderate concentrations, and more extensive highfrequency coverage, for better definition of the complete relaxation function.
Acknowledgment. Work supported in part by Exxon Research and Engineering Co., and in part by N S F Grant C H E 82035 11 with equipment funding from Brown University’s Materials Research Laboratory funded by NSF. We are also indebted to a referee for helpful comments about current theory which led to a revision of our original conclusions. (7) Felderhof, B. U. Mol. Phys. 1984, 51, 801. (8) Hubbard, J. B.; Colonomas, P.; Wolynes, P. G.J . Chem. Phys. 1979,
67, 4859. (9) Kusalik, P. G.; Patey, G. N. J . Chem. Phys. 1983, 79, 4468.
(10) Pottel, R. In “Chemical Physics of Ionic Solutions”, Conway, B. E.. Barradas, R. G. Eds.; Wiley: New York, 1966; p 581.
