CONDUCTIVITY RELAXATION IN AQUEOUS ELECTROLYTE SOLUTION
3287
elaxation in a Concentrated Aqueous Electrolyte Solution b
Ambrus,
Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland +
20910
Moynihan,* and P. B. Macedo
VGreous State Laboratory, Catholic University of America, Washington, D . C. 20017
(Received November 16,1970)
Publication costs assisted by the Ofice of Naval Research
Capacitance and conductance measurements were made in a highly concentrated aqueous Ca(NO& solution (6.15 water:salt ratio, 9.03 m) as a function of frequency and temperature. The fl*equencyrange covered was 1-1 X 1Q8 Ha; the temperature range was from -91.5 to - 61 .go. The data were analyzed in terms of the eleckic modulus, n/r (= l/t, where 6 is the complex permittivity). The frequency dependence of the electric rnodulus shows the presence of more than one relaxation. The relaxation process observed at the lowest freqttencies has been identified with the conductance process involved in long-range charge transport and has heen analyzed in terms of a log Gaussian distribution of relaxation times. The sources of the lowstrength relaxation process observed at higher frequencies cannot be unambiguously identified, but may be water molecule reorientation or localized ion-jumping mechanisms.
Introduction Previous studies of electric relaxation in aqueous electrolyte solutions have been confined to the highfluidity region in the vicinity of room temperature. These include investigations of the relaxation of tl,e ionic atmosphere in dilute solution (Debye-Falkenhagen effect),’ the dielectric relaxation of the polar water mo1ecules,2s3 and dispersion effects connected with ionic association reaction^.^ I n addition, it has been pointed out recently4z5 that electric relaxation eff ects associated with the long-range ionic diffusion process may lead to an observable frequency dispersion in the dielectric constant and loss. As an initial effort in an investigation of electric relaxation in aqueous electrolytes at low temperatures and low fluidities, we report below the results of a study of a glass-forming6 concentrated solution of calcium nitrate in water. We begin, however, with a summary of the phenomenology of the frequency dependence of admittance in ionic conductors. The complex permittivity,
-
ie”
7,
--
E)
E’
= eOEs/uo
(31
I-, has been termed the conductivity relaxation time4 and characterizes the decay of the electric field E in the conductor t o zero under the constraint of constant displacement vector via the dc conductance process
E
=
Eoexp( - t/r,)
(4) From eq 1-3, the electric modulus may then be shown to be4 A4 = A$.(
Conductivity Belaxalion e = E’
e, bear the same relationship to one another as do the modulus and the compliance in mechanical relttxation experiments. The simplest representation of the equivalent circuit of an ionic conductor is a simple parallel conductance and capacitance combination that exhibits a frequency independent conductivity, co,and relative permittivity, eS. A characteristic time may be written for this system
1
+iw7aiwr, ) =
of a system is defined
- i(u/weo)
(1)
where w is the angular frequency, e’ the relative permittivity or dielectric constant, e” the dielectric loss, c the specific conductance, and eo the permittivity of F cm-l). An alternative free space (8.854 >< parameter of considerable utility in the analysis of electrical relaxation phenomena in conducting materials has been termed4 the electric modulus, M .
The electric modulus, N , and the complex permittivity,
(I) H. Falkenhagen, “Electrolytes,” Clarendon Press, Oxford, 1934,
p 182 ff.
(2) (a) J. B . Hasted, D. M. Ritson, and C. H. Collie, J. Chem. Phys., 16, 1 (1948); (b) G. H. Haggis, J. B. Hasted, and T. J. Buchanan, ibid., 20, 1452 (1952); (e) F. A. Harris and C . T. O’Konski, J . Phys. Chenz., 61, 310 (1957). (3) (a) R. Pottel, Ber. Bunsenges. Phys. Chem., 71, 136 (1967); (b) R. Pottel, “Chemical Physics of Ionic Solutions,” B. E. Conway and R. G . Barradas, Ed., Wiley, New Yorlr, N. Y., 1966, pp 581589; (c) R. Pottel, Ber. Bunsenges. Phys. Chem., 69, 363 (1965); (d) K. Geise, U. Kaatze, and R. Pottel, J. Phys. Chem., 74, 3718 (1970). (4) V. Provenxano, L. P. Boesch, V. Volterra, C. T, Moynihan, and P. B. Macedo, J . Amer. Ceram. SOC.,in press.
The Journal of Physical Chemistry, Yol. 7 6 , No. 26, 1972
J. H. AMBRWS, c. T. ~ ~ O Y N I H A NAND , P. B. MACEDO
3288 37here
Ms
(6)
1/65
Hence, when treated in the modulus notation, the conducting material exhibits a Debye-like or single relaxation time dispersion, identical in form with that used t o describe shear relaxation processes in liq~ids.~,’The limiting low- and high-frequency values of the real and imaginary parts of the modulus are
Xim Ill’
=
urul
and (ru2)and ( l / ~ are ~ ) defined similarly. (It should be noted that e, in eq 9, in the case of a distribution of conductivity relaxation times, no longer has its e’).) conventional meaning (es = limw--ro The characteristics of a conductivity relaxation with a distribution of relaxation times can be reproduced by an electrical circuit consisting of a series connection of parallel RC elements, each element containing a conductance Gjshunted by a capacitance C.8
(7) lim M” = lim M” = 0 wru>>l
OTUl
(9)
If there are nj conductances with value GI, and if the total number of conductances in the circuit is N ( = B j n j ) , the electric modulus M of the circuit is
where r j = C/G,, and Go is a circuit constant (C, = net capacitance of circuit when the dielectric material in each of the elements is replaced with a vacuum). The correspondence between eq 8 and 14 becomes clear if one makes the identifications
and replaces the integration in eq 8 with a summation. An equivalent circuit of the above type has been used previously t o interpret the frequency dependence of the dielectric constant and loss of polycrystalline semic o n d u c t o r ~ . ~The ~ ~ treatment suggests that the observation of a conductivity relaxation characterized by a distribution of relaxation times in a conducting liquid may be ascribed to a distribution of microregions of varying conductivity, the distribution presumably arising from local density or composition fluctuations, or, in alternative terms, to a distribution of free energy (5) C . T. Moynihan, R. D. Bressel, and C. A. Angell, J. Chem. Phys., 5 5 , 4414 (1971). (6) C. A. Angell and E. J. Sare, ibid., 52, 1058 (1970). (7) T. A. Litovitz and C. M . Davis in “Physical Acoustics,” Vol, IIA, W. P. Mason, Ed., Academic Press, New Y o r k , N. Y . , 1965, pp 281-349. (8) J. Volger, Progr. Semicond., 4, 205 (1960). (9) C. G. KOOPS, Phy3. Rev., 8 3 , 121 (1951).
The Journal of Physical Chemistry, Vol. 76, No. 82, 19Y.8
ELAXATION IN AQUEOUS
ELECTROLYTE SOLUTION
3289
barriers for the ionic diffusion or electron hopping process.4 Dielectric Relaxation I n the case in which a conducting dielectric may undergo a relaxation involving molecular mechanisms whidh do not contribute t o the long-range conductance process (e.q.? reorientation of permanent dipoles), an additional term or terms must be added to the electric modulus expression for the conductivity relaxation. I n the single relaxation time case in which there is only oine dielectric relaxation, eq 5 would then become 0 . d -J I 10-2 io-’
I
I
I
1
10
102
f
The first term refers to the conductivity relaxation and the second term to the dielectric relaxation
M,
=
l/€,
103
104
IH4
Figure 1. Frequency dependence of capacitance and conductance of model circuit for ionic conductor with blocking electrode capacitance. Component values for circuit in inset: C, = 1 pF, C, = lo-* pF, G, = IO+ mho.
and
&I,
1
= 1
lim
E’
ora>>l
ro is the dielectric relaxation time for constant displacement vector and is related to the conventional dielectric relaxation time for constant electrical field, T E , by
I n the more general case either or both terms in eq 15 may have to be modified to allow for a distribution of relaxation times; likewise, additional terms would need t o be added in the event that there is more than one dielectric relaxation, Since T~ in eq 15 is the time constant for the decay of the electric field t o zero in the dielectric medium, only those dielectric relaxations which have r D 1 pF) in comparison to the capacitance, C,, of the solution pF or less). The admittance of the circuit itself o i Figure 1is
Y =
+ w 2 ( C , + C,lZ + W~C~~G,
Gp2
For the case in which C, >> C,, the high- and lowfrequency limits of the net capacitance and net conductance of the circuit are (10) (a) G. D. Robbins, J. Electrochem. Soe., 116, 813 (1969); (b) 6. D . Robbins and J. Braunstein, “Molten Salts: Characterization and Analysis,” G. Mamantov, Ed., Marcel Dekkar, New Yorlr, N. Y., 1969, pp 443-478. (11) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” 2nd ed, Butterworths, London, 1965, pp 93-95. The Journal of Physical Chemiatrg, Vol. 76,No. $9, 1972
J. H. AMBRUS,C. T. MOYNIHAN, AND P. I4, MACEDO
3290 lim C = C,, lim C = C, w-+ m
a-+m
lim G = G,, lim G = 0 03 m
(18)
W-0
Hence at low frequencies the capacitance will commence a rather rapid increase from C, t o the much higher value, C,, as the frequency is decreased, while the conductance will drop off from Gp t o zero. Fortunately, for the case in which 6, >> C,, the time constant, for the dispersion associated with the doublelayer capacitance Tel
= (C,
+ C,)/G,
will be much larger than the conductivity relaxation time 7 ,
=
Cp/Gp
Thus the dispersion due t o the double layer capacitance will occur a t much lower frequencies than the conductivity relaxation (and any subsequent dielectric relaxations) and will in general not obscure any effects of interest. This is shown in Figure 1, where the frequency dependence of the net conductance and capacitance of tnhecircuit in the inset have been calculated using a set of component values typical for an experiment of the type presented here. It should be noted that an rtd.ditional effect arising from the large difference in the magnitudes of C, and C, is that the measured value of Q attains its high-frequency limit at much lower frequencies than does the measured value of c. I n the modulus notation the circuit of Figure 1 yields a particularly simple expression
Figure 2. Dielectric cell: A, Teflon spacer; B, electrode lead, mates with female BNC connector; C, shield for electrode leads; D, O-ring joint; E, brace between electrode shields; F, Delrin insulator; G, cylindrical electrode, 3.8 cm long, 0.79 cm 0.d.; electrode separation 0.3 cm.
liquid nitrogen was used for temperature control (to within =t0.lo). where Co is again n circuit constant corresponding t o The schematic of the dielectric cell is shown in Figure the vacuum capacitance of a dielectric cell. The 2. Except for the plastic insulators and spacers and double-layer capacitance thus makes a constant conthe rubber 0 ring, the cell is constructed entirely of tribution to the real part of the modulus (which contrinickel-plated stainless steel. The exterior of the cell bution may be reglected when C, >> C,) and malies no and the electrode lead shields are connected to the contribution t o the imaginary part. neutral of the admittance bridge and serve t o eliminate any stray capacitances due t o fringing effects. The Experimental Seetion vacuum capacitance, CO,of the cell was approximately Reagent grade calcium nitrate tetrahydrate (Mal0.7 pF. The absence of stray capacitance in the cell linckrodt AR) and deionized mater were used t o prepare and circuit was confirmed by the agreement between a solution with a N20:Ca(Pu'O& mole ratio of approxithe Co values obtained using air and benzene as calmately 6. The exact HzO:Ca(n'03)2mole ratio (6.15) ibrating dielectric media in the cell. The CO value was determined from the density (1.6033 g ~ m at - ~ used in data reduction was determined by measuring 30") using the data of Ewing and Mikovsky.12 the air capacitance of the assembled cell just before the The dielec~triecell was thermostated in a mechaniexperiment. Co was remeasured after the experiment, cally stirred methanol bath in a 4-qt dewar flask. Temwithout disturbing the electrode assembly and found perature was measured with a calibrated copper-cont o agree with the initial value of CO within j=0.4Oj,. stantan thermocouple. Above -778" the bath temperature was controlled to within 3t0.05O by periodically (12) W. W. Ewing and R. J. Mikovsky, I . Amer. Chem. Sac., 7 2 , adding small amounts of Dry Ice. Below -78" 1390 (1950). The Journal
Of
I'hhyaical Chemistry, Vol. 76, No. 99,1979
CONDUCTIVITY RELAXATION IN AQUEOUS ELECTROLYTE SOLUTION
3291
I
Table I : Specifications €or Admittance Bridges Bridge
f range, Hz
Cole-Gross General Radio 1615A Berberian-&k
500-1 X 10' 500-1 X lo6 1-500
C range, pF
Resuits and Discussion The glass transition temperature for Ca(NO&. 6.l5I-I2O, determined at a heating rate of 15-17 deg/ I n our experiments 0.5-1 hr was remin, PS - 7 9 O . G quired for heating and equilibration between successive temperatures, so that all measurements reported for temperatures above -79" may be considered t o have been taken on a sample in internal equilibrium. The relative permittivities, E', and the conductivities, ( T , were calcuhtecl from the measured conductances and capacitances uact the equations -I-
@/Ca; u = eoG/Co
2X to 1 X IO-* (f0.5%) 1 x 10" to 1 x 10-10 ( i l o j , ) 1 x 10-4 to 5 x 10-12 (i~%)
0-100f 0.05 0-10,000*0.01 0-100 f 0.1
Depending on 1 he frequency and conductance range, one of three different bridges was used to measure capacitance, C, and conductance, 6: (1) a three-termind transformer arm ratio bridge based on the design of Cole and Grosfi;13 ( 2 ) a General Radio Model 1615A bridge used in the three-terminal mode; or (3) a threeterminal bridge designed by Berberian and Colei4 for low-frequency measurements. The ranges and accuracies of these bridges are given in Table I. Frequencies of the oscillators used were accurate t o f1%. During measurements, the dielectric cell containing the sample was first quenched in the precooled bath to a temperature in the vicinity of its glass transition temperature. The sample was annealed at this temperature for approximately 1 hr, then cooled t o the lowest temperature at which measurements were t o be made. A series of frequency-admittance measurements was then performed at increasingly higher temperatures. To ensure that the sample had come to thermal equilibrium at each temperature, we waited until the conductance and capacitance readings were constant at a given frequency for at least 10 min before commencing measurements. The measurements were terminated when the sample crystallized. Onset of crystallization was easily detected by the erratic behavior of the conductance.
E'
Q range, 0-1
(20)
I n Figures 3 and 4 the experimental values of E ' and u are plotted as a function of frequency at various temperatures. The very rapid rise in e' with decreasing frequency at the higher temperatures (-72.3' and up) and the lowest frequencies may be associated with electrode polarization, since the conductivity appears to be frequency independent in these regions (compare Figure 1). The frequency dependence of E' and u at frequencies higher than the polarization frequency at
1- 6 W 4
\ -61.9"
-72.3"
t 5 I
10
io2
lo3
lo4
io5
ioE
i1 io7
f IHzl
as Figure 3. Relative permit>tivity, e', of Ca(Pu'03)2.6.1SHz0 a function of frequency and temperature.
temperatures above -72.3' and at all frequencies below this temperature can then be associated with the conductivity or dielectric relaxations of the solution. I n Figures 5 and 6 the data of Figures 3 and 4 are shown plotted in the modulus form, where M' and M" have been calculated from eq 2. At the higher temperatures M' approaches zero at low frequencies, indicating that the initial dispersions in M' and 36" are due to a conductivity relaxation. The fact, that M' approaches zero at low frequencies also indicates that electrode polarization has a negligible effect on the data analysis in the modulus notation (cf. eq 19). M' does not level off, however, at higher frequencies, even at the lowest temperatures at which experiments were performed, so that the relaxation processes present are spread over a considerable range in frequency. The M" plots of Figure 6 are somewhat more informative in this regard. The low-temperature 116" plots show the presence of at least two electrical relaxation processes, one associated with the large asymmetric low-frequency peak and a second associated with a smaller high-frequency peak, only part of which is visible in the frequency range available in this experiment. As a first attempt at deciding what portion of the initial, low-frequency dispersion in illr and 114'' should be associated with the conductivity relaxation, we com(13) R. H. Cole and P. M.Gross, Rev. Sci. Instrum., 20, 252 (1949). (14) J. G . Berberian and R. H. Cole, ibid., 40, 511 (1969). The Journal of Physical Chemistry,Vol. 76,No.22, 1978
J. H. AMBRUS,C. T . MOYNIHAN, AND P. B. MACEDO
0 -61.90
v -6420 0 -66.9"
0.02
0 -69.2"
v -12.3"
e -74.8"
-n.v
M"
a-19.80 0 -83.8" + -9l.5"
o.a
0.00
1
10
f lHzl
Figure 6. Imaginary part of the elect.ric modulus, M", of Ca(N03)2.6.15Hz0as a function of frequency and temperature.
a single relaxation time treatment were adequate, one would expect the calculated M' curve to match the experimental data at low frequencies, but eventually t o fall under the experimental curve as additional dielectric relaxations began to contribute to M ' . As shown in Figure 7, however, the calculated curve for M' exceeds the experimental curve at low frequencies, showing that the single relaxation time treatment predicts too rapid an increase in M' and that the conductivity relaxation needs t o be described in terms of a distribution of relaxation times. As a second attempt at characterizing the initial conductivity relaxation, we have assumed that it may be described in terms of a log Gaussian distribution of relaxation times7
tiHz1 Figure 4. Conductivity, u, of Ca(NOa)~.6.16Hz0 as a function of frequency and temperature.
1
f lHrl
Figure 6. Real part of the electric modulus, M', of Ca(NIO3)Z.6. lAHsO as a function of frequency and temperature.
pared the experimental M' and M" curves with those predicted on the assumption that the conductivity relaxation exhibited single relaxation time behavior. I n this case the conductivity relaxation would make no contribution t o the frequency dependence of E', so that e, may be taken as the value of 'E in the vicinity of low-frequency shoulder in the plots of Figure 3. The relaxation time, T ~ rnay , be calculated from eq 4, using the experimental value of the limiting low-frequency conductivity? ao. The siqgle relaxation time curves for 29.1' and lll''j calculated from eq 5, are compared in Figure 7 with the experimental data for -M.2".15 If The Journal o j Physical Chemistry, Vol. 76, N o . 22, 1972
7,' is the most probable relaxation time, and b is a constant inversely proportional to the width of the distribution function at half-height. Using eq 21 for g(T,) in the dispersion functions of eq 8, an iterative leastand rCr'was persquares fit on the parameters b, &Is, formed simultaneously on the values of M' and M" up to the peak frequencies in Figure 6 for each of the temThe width peratures between -61.9 and -77.8". parameter, b , was found to be virtually the same for each of the seven temperatures, the average value being 0.585 0.010. The data therefore were fitted a second time t o obtain M , and 7', values for the constant value b = 0.585. The least-squares values of M , ( = l / e s ) and 7,' obtained in this fashion are given in Table 11. The statistical uncertainties in n/I, and T,' were in
*
(15) As is evident from Figure 3, the value of e. for - 64.6' might be reasonably taken anywhere in the range 40-60. The single relaxation time plot for Figure 7 is for eg = 50, hut plots using fs anywhere in the 40-60 range lead to the same conclusion, that the single relaxation time treatment does not account for the data.
CONDUCTIVITY RELAXATION IN AQUEOUS ELECTROLYTE SOLUTION
3293
I____.-_^_
Table I1 : Parameters from Fit of Log Gaussian Distribution Functions to Initial Gonductivity Relaxation in Ca(NOa)o.6.15H20 (TU')
T.o
c
-61.9 -64.2 -66.9 -69.2 -72.3 -74.8 -77 *8
b
102MB
fa
0.585 0.585 0.585 0.585 80.586 0.585 El. 585
7.3 7.2 7.4 7.6 7.8 8.3 8.4
13.8 13.9 13.5 13.2 12.8 12.0 11.8
TU',
UO.aalad,
W,0XPt.
ea% (7u)
see
0-1 om-1
0-1 cm -1
(oalod)
2.05 x 5.0 x 1.62 x 4.7 x 2.9 x 1.16 X 1.13 X
10-6
2.9 1.18 3.5 1.19 1.85 4.4 4.5
10-5 10-4 10-4 IO+'
10 .2
x 10-7 x 10-7 x 10-8 x 10-8 x lo-$ x 10-10 x 10-11
3.2 x 1.27 X 3.8 x 1.30 x 1.95 x 4.3 x
10-7
60
IO-'
60
10-8
58 57 55 52 51
10-8
10-9 10-10
-
1 0 ' ~-
c
I
IC
D
.t; 1 0 . ~-
10"-
0
1
A
-
s
106L.. 4.8 _
4.I7
4.I8
_ 4.9103/T50 I I
5.l1 _
5.2
5.3
flH4 Figure 7. Comparison of experimental and calculated electric modulus dispersion curves for Ca(N03)2.6.15H20at -64.2': (-) single relaxation time curve for eg = l/Ms = 50, 7, = eoe8/oo = 3.48 X sec; (- - - - - - -) fit using log Gaussian distribution function over frequency range 0.5-50 kHz; inset; blowup of M' plot in low-frequency dispersion region.
Figure 8. Arrhenius plots of dc conductivity, 00, and average conductivity relaxation time, , of Ca(NQ3)2.6.15Hz0.
a log Gaussian distribution of relaxation times, the average relaxation times needed in eq 10 and 12 are16 (7,)
the respective ranges 1-2 and 2-4%, except for the lowest temperature fitted (- 77.8"), where they were 4 and S%, respectively. The log Gaussian fit to the M' and M" data, for -64.2" is also shown in Figure 7. A moderately sensitive test of this procedure in accounting for the initial conductivity relaxation is the ability t o back-calculate (using eq 10 and 12) the limiting low-frequency values of the relative permittivity ) ~ no, from the paramand conductivity, cs ( T , ~ ) / ( T ~and eters in Table 11. (These parameters are obtained almost entirely from a fit to the data in regions where E ' and u exhibit strong-frequency dependence and differ markedly from the limiting low-frequency values.) For
=
7,'
(7v2)/(7,)2
exp(l/4b2)
=
2.087,'
= exp(1/2b2) = 4.33
b
=
0.585
b = 0,585
The calculated values of E , ( ~ , ~ ) / ( Tand ~ ) ~ uo are given in Table 11, along with the experimental values of no. The experiment'al and calculated values of no are in agreement within 10% or better, which is within the agreement expected from the statistical or experimental uncertainties in b, e,, T,', and uO,exp. As shown in Figure 3, E ~ ( T , , ~ ) / ( T , ) ~cannot be measured precisely because of electrode polarization effects, but the values calculated in Table I1 can be seen to correspond closely (16) J. H. Simmons and P. B. Macedo, J . Chem. Phys., 53, 2914 (1970).
The Journal of Physical Chemistry, V o l . 76, I f o . 222,1972
J. H. AMMRRUS, c. T. n/IOYNIHAN,
3294
AND
P.
Table 111: Fulcber Eqiiation Parameters for Transport Properties of Ca(NO8)2.6.15 HzO Solution A
B
TO
8Ld dev in In w
-4.260
965.75 533.18
161 161
0.04 0.05
Praperty
T range,
W
l/VO,
O C
-61.9 t o -74.8 -61.9 to -77.8
cl C l x l
( ~ ~ see 1 ) ~
- 30.925
to the limiting low-frequency values toward which the plots appear to be headed just before the rapid rise duo to electrode I n Figure 8 are shown Arrhenius plots of (7,) and l/uo. If one ignores the temperature dependence of E ~ which is small compared to the temperature dependence of the rehxation time, l/uo and ( T ~ are ) predicted from eq 12 t o have identical activation energies. The Arrhenius plots in Figure 8 exhibit noticeable curvature, so that the apparent activation energies must be considered t o be temperature dependent. The data for l/ao and (7,) were fitted t o the Fulcher equation 'E
Inw = A
B +T - To
,
(22)
where w is a transport property, and A , B, and To are constants. It was found that within the statistical uncertainty of t)he fit, the same value of T Ocould be used for both l / m and ( T J . The Fulcher equation parameters are given in Table ITI. (It should be noted that the Fulcher equation tends to give an incorrect description of the temperature dependence of transport properties when applied to data extending over many orders of magnitude.l7*l8 Hence the parameters in Table I11 should not be used to extrapolate the data beyond the temperature limits indicated.) The activation energy, E , calculated from the Fulcher equation is
E
B
-
nlnw - BT2 n(l/T) (T - To)2
(23)
so that near equality of the B terms for l/uo and ( T ~ indicates that the apparent activation energies are likewise nearly the same. The apparent activation energy for (7,) increases from 32.8 to 60.0 kcal/mol over the tcmperaturerange -61.9 to -77.8'. Although our log Gaussian fit to the low-frequency portions of the M ' and M" curves of Figures 5 and 6 gives a satisfactory account of the low-frequency dispersion in E' and u due to the conductivity relaxation and yields an average conductivity relaxation time which exhibits the expected correlation with the dc conductivity, this is no guarantee that the distribution of conductivity relaxation Limes is in fact correctly described by a log Gaussian function. Rather, as is evident from eq 9-13, any distribution function which accounts for the low-frequency portion of M' and MI' plots mill yield correct values of the average relaxation time, since only the long relaxation times (which give rise t o The Journal of Physical Chemistry, Vol. '76,No. 22, 1972
)
the lowfrequency dispersion in M' and M") malie any appreciable contribution t o the average timc5 Inspection of Figure 6 shows that the smaller secondary relaxation peak which is evident in the megahertz region for the lower temperature curves is of considerable breadth and that the relaxation times associated with this peak have a much smaller temperature dependence than the conductivity relaxation time. I n view of these complications, we feel that an attempt to accurately dissect the M' and M" plots into contributions from the conductivity relaxation and contributions from other relaxation processes and hence t o determine a genuine distribution function for the conductivity relaxation times would be premature. This high-frequency relaxation peak is similar in nature t o the secondary peaks ( p relaxations) which Johari and Goldstein'$ have found in a wide variety of simple molecular or ionic glass-forming liquids and which they suggest are endemic to the glassy state. One possible source of the secondary relaxation peak in the Ca(NO&- 6.15H20 solution is the reorientation of polar water molecules. I n this solution, however, one expects most of the water molecules to be bound up in the hydration spheres of the Ca2i ions.2o Previous dielectric relaxation studies for dilute aqueous electrolyte solutions (cf. for instance, ref 3d) indicate that the reorientation of water molecules bound in the hydration spheres of high-field cations cannot give rise t o observable dielectric relaxations, since, if the metal ion-oxygen bond bisects the E-0-E angle, rotation of a water molecule about this bond involves no change in dipole orientation. Hence for the Ca(NO&. 6.15H20 solution water molecule reorientation may not be contributing t o the observed high-frequency relaxation. A second possible source of the secondary relaxation are localized ion-jumping mechanisms similar to those which give rise t o high-frequency electrical relaxations in doped alkali halide crystals.21 I n the future we hope t o report the results of similar (17) (a) P. B. Macedo and A. Napolitano, J . Chem. Phys., 49, 1887 (1968); (b) R. Bose, R. Weiler, and P. B. Macedo, Phys. Chem. GZasses, 11, 117 (1970).
(18) H. Tweer, N. LLtberge, and P. B. Macedo, J . Amer. Cerum. Soc.,
71, 121 (1971).
(19) (a) G. P. Johari and M. Goldstein, J . Phys. Chem., 74, 2034 (1970); (b) G. P. Johari and M.Goldstein, J . Chem. Phys., 53, 2372 (1970). (20) C. T. Moynihan and A. Fratiello, J . Amer. Chem. Soe., 89, 5546 (1967). (21) J. S. Dryden and R. J. Meakins, Discuss. Faraday Soc., 23, 39 (1957).
MOLECULAR STRUCTURE AND SHEAR VISCOSITY studies currently in progress for a number of concentrations in the glass-forming region of the Ca(NW2II2O system. These studies will hopefully clarify the r
