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Dielectric spectroscopy on some aqueous solutions of 3:2 valent electrolytes. A combined frequency and time domain study.
Journal of Molecular Liquids, 36 (1987) 15-35
15
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
DIELECTRIC SPECTROSCOPY ON SOMEAQUEOUS SOLUTIONS OF 3:2 VALENTELECTROLYTES. A COMBINED FREQUENCY AND TIME DOMAINSTUDY. DEDICATED TO PROFESSOR ROBERTH. COLE UDO KAATZE and KARL GIESE* Drittes Physikalisches Institut, Universit~t G~ttingen, BUrgerstraBe 42-44, D-3400 G~ttingen (West Germany)
(Received 26 November 1986) ABSTRACT The complex dielectric spectrum of aqueous solutions of aluminium, indium, and scandium sulfate has been determined between 10 ~ z and 60 GHz using time domain and frequency domain techniques. A difference method especially matched to the study of electrically conducting solutions has been applied in the time domain measurements. Various relaxation spectral functions were f i t t e d to the measured data. The results obtained thereby are discussed with respect to the effects of kinetic depolarization, dielectric saturation and ion complex formation. INTRODUCTION The detailed knowledge of ion-ion and ion-solvent interactions in aqueous solutions is extremely important for our understanding of the molecular structure, kinetics, and dynamics of numerous liquid systems, many of them with considerable relevance in chemistry, biology or medicine. The complex (electric) permittivity spectrum
(1)
c(~) : e'(~) - i~"(v)
of electrolyte solutions enables statements on both types of interactions. I t is capable to directly reflect the presence or absence of dipolar ion complexes and also allows for conclusions on hydration properties (rfs. 1-5). Unfortunately, however, the total dielectric spectrum e(~) of conducting samples combines two different contributions as is usually expressed by the relation
~(~)
: ~d(~) - i ~
(
(~) - i ~ ( ~ )
+
o
One part (~d) of ~ results from polarization processes, the other one (-iO/~om) *Present address: Institut fur Medizinische Physik und Biophysik, Universit~t G~ttingen, GoBlerstraBe 1Of, D-3400 G~ttingen. 0167-7322/87/$03.50
© 1987 Elsevier Science Publishers B.V.
16
from the d r i f t of ionic species (i = ( - i ) 1 / 2 ; m = 2~v; v, frequency; a, specific e l e c t r i c conductivity; ~o = 8"854"10-12AsV-1m-1)" With decreasing frequency (~ ~ 1 GHz) d i e l e c t r i c measurements on electrolyte solutions are thus subject to increasing experimental error since the ion currents tend to more and more exceed the polarization currents to be observed. In the present study we tried to overcome these d i f f i c u l t i e s by combining spot frequency and time domain spectroscopic (TDS) measurements to cover a broad frequency range. With the available equipment frequency domain measurements on the chosen solutions (1.2 < o < 2.6 S/m) were possible between 0.5 and 58 GHz. A TDS method matched to the study of highly conducting liquids has been developed which allowed the complex dielectric spectrum to be determined for the frequency range from 10 MHz to 3 GHz. Recent studies of d i e l e c t r i c effects in aqueous solutions of inorganic salts, including 3:1 valent electrolytes, confirmed a special a b i l i t y of the d-electrons of transition metal ions to interact with ionic ligands (refs. 3-5).For the present investigation we therefore selected aqueous solutions of aluminium, scandium, and indium sulfate since the t r i v a l e n t cations of these salts have do , d8, and d10 electron orbits, respectively. EXPERI~NTAL SECTION Aqueous solutions The sulfates (FLUKA, 98 % or better) were used without further p u r i f i c a t i o n . The solutions were prepared by f i l l i n g preweighed amounts of salts and deionized, d i s t i l l e d and degassed water into suitable flasks. To avoid formation of hydroxy complexes with t r i v a l e n t cations perchloric acid was added to appropriately adjust the pH value of the solutions (AI, 2.1; Sc, 2.4; In, 2.0). The sulfate ion is partly protonated at the ionic strength and pH used in this study. Specific e l e c t r i c conductivity As b r i e f l y indicated in the Introduction section the specific e l e c t r i c cond u c t i v i t y ~ of the solutions has to be accurately known. We used a two-electrode capillary cell (ref. 6) with suitable cell constant to measure o at 0.1, 1, 10, and 100 kHz. A sensitive impedance bridge (GENERALRADIO 1654) has been u t i l i z e d which allowed the resistance of the l i q u i d - f i l l e d cell to be compared with a precise decade resistor (GENERAL RADIO 1433-F). The ring-shaped electrodes of the cell were covered by a deposite of f i n e l y divided black platinum to minimize errors due to electrode polarization effects (ref. 7). The cell constant has been determined with an error of less than 0.2 % using aqueous NaCl and KCI of appropriate concentrations as reference liquids. The temperature was controlled during the measurements to within 0.05 K.
17 Frequency domain measurements At microwave frequencies (0.5 ~ ~ ~ 58 GHz) the complex (relative) permittivi ty ~(~) of the solutions has been measured by transmitting propagating electromagnetic waves of frequency ~ through specimen cells of variable sample thickness (refs. 1,8,9). As schematically shown in Fig. 1, the liquid under test was f i l l e d in a piece of circular cylindrical waveguide or coaxial line for this purpose. A receiving probe, also a circular waveguide or coaxial l i n e , was immersed in the liquid. Combined with high precision ball bush guides (FEINPRDF, G~ttingen) the probe was shiftable free of backlash along the direction z of wave propagation. The wavelength k and attenuation exponent ~ have been interferometrically determined by adjusting a reference signal of calibrated variable amplitude. Five double beam interferometers, each matched to a limited frequency range were used. One was constructed with coaxial line components, the
ziT 8
S+)~
)
2
(Z) = E oe iwte-("*i~)z
---
12
FJ 3
9
10
11
Fig. 1. Schematic representation of the microwave double-beam interferometers used to perform frequency domain measurements between 0.5 and 58 GHz. I , monochromatic o s c i l l a t o r ; 2, frequency counter or meter; 3, uniline or coaxial attenuator; 4, directional coupler or coaxial beam s p l i t t e r ; 5, impedance matching transformer; 6, measuring cell containing the liquid sample; 7, movable probe; 8,9, phase s h i f t e r and variable attenuator, respectively, u t i l i z e d to adjust maximum interference with zero signal at probe position s; 10, calibrated variable attenuator with constant phase s h i f t used to produce zero signal at probe position s + ~; 11, superheterodyne receiver; 12, zero signal indicator.
iB others with standard waveguide devices. The p e r m i t t i v i t y is related to the measured q u a n t i t i e s ~, X, and ~
according
to the f o l l o w i n g equations e'(V) =
~"(~)
-
- \-,~-~--1 + \-~C} mX2 0 ~
(3)
,
(4)
where Xo = (speed of l i g h t ) / v denotes the wavelength in free space and ~c the c u t o f f wavelength of the empty c e l l . The c e l l s were surrounded by a water-jacket to allow the temperature to be c o n t r o l l e d to w i t h i n ± 0 . 1 K . Errors in the complex p e r m i t t i v i t y values measured with t h i s technique are mainly due to the l i m i t e d zero-level s e n s i t i v i t y of the microwave receiver, imperfections of the calibrated attenuator used to adjust the amplitude of the reference signal and temperature f l u c t u a t i o n s . Errors in the determination of the frequency and in the micrometer drive c o n t r o l l i n g the position of the s h i f t able probe are n e g l i g i b l y small. Careful consideration of the sources of e r r o r in the predominant part of the frequency range leads to an uncertainty of ± i % in the values of both, ~'(~) and ~"(~). Due to higher loss the accuracy of the p e r m i t t i v i t y values decreases when going to the high and low frequency end (v > 18 GHz, v < i GHz) of the measuring range (Table i ) . According to equ. 2 the e r r o r in the d i e l e c t r i c part of the loss depends on the s p e c i f i c e l e c t r i c c o n d u c t i v i t y of the s o l u t i o n . To g l o b a l l y characterize the accuracy of the c~ data uncertainties of the 0.1 molar AI2(S04) 3 s o l u t i o n (o = 2.06 S/m, 25°C) are compiled as an example in Table i . TABLE 1
GHz
Z1~'IE ' %
I~E"Ic . %
0.5 0.7 1 2-5 5-18 18-30 30-40 40-58
3 2 1 1 1 2 4 i0
2 2 1 1 1 1 2 3
.
.
.
.
ACdl~ . d % 15 12 5 3 2 2 2 3
The experimental errors in the complex p e r m i t t i v i t y values measured in the frequency domain f o r the 0 . i molar aqueous s o l u t i o n of AI2(S04) 3 at 25°C.
19 Time domain spectroscopy (1) Principle of measurement. In the frequency range 10 I~Hz ~ v s 3 GHz time domain reflectometry (ref. 10) has been used to derive the complex permittivity ~(u) from the Fourier transform of a step-like fast-rising voltage pulse transmitted in a coaxial line and reflected from the specimen c e l l . We utilized the Hewlett-Packard TDS system with a computer connected for automatic data recording and processing. Special emphasis was layed on the matching of the cell construction and measuring method to the study of aqueous electrolyte solutions. (2) Specimen cell.
The construction of the cell is shown in Fig. 2a. The
sample liquid is contained in a circular waveguide of diameter 2a. This waveguide is sealed by an e l e c t r i c a l l y matched polystyrene window against the coaxial feeding line of characteristic impedance Zo = 50 ~. The cell is made of stainless steel and is provided with a thermostatic coat not shown in Fig. 2a.
a)
A
b)
I ~ d
~ IT E(v)Cz
T l -
p
ely) C1
T
tpC 3 1
2
3 A
Fig. 2. Construction of the cutoff cells utilized in the TDS studies (a) and i t s equivalent network representation for plane A-A(b). 1, coaxial feeding line (50 ~; diameter of the outer and inner conductor 7 and 3.04 mm, respectively); 2, polystyrene window (~p = 2.54; 2ap = 8.5 mm; 2b = 2.25 mm); 3, circular waveguide containing the l i q u i d ; d, length defined by equ. (5). At frequencies far below the cutoff frequency of the TM01 mode in the waveguide section, the cell can be approximately represented by a capacitance C, which apparently is equivalent to a short piece of coaxial line with i t s openc i r c u i t termination at distance d from the face of the polystyrene window (Fig. 2a). The relation between C and d is given by C = ~(~)dlCCoZ )
,
(5)
20 where co denotes the speed of l i g h t in free space and Z the characteristic impedance of an empty coaxial line with the radius of the inner and outer conductor being b and a, respectively. Accurate modal analysis (ref. 11) of the stray fields at both sides of the plane A-A separating the coaxial feeding line from the waveguide section leads to the equivalent network representation of the cell shown in part b of Fig. 2. In this lumped c i r c u i t element network the capacitance C1 corresponds to components of the stray f i e l d which are t o t a l l y restricted to the waveguide section, while the series arrangement of C2 and C3 has regard to electrical f l u x lines which intersect plane A-A. Expressed by the capacities Ci , i = I . . . 3, the capacitance C of the cell when f i l l e d with the sample l i q u i d is given by C = e(v)C I +
I
+
,
(6)
where ~ (= 2.54) is the p e r m i t t i v i t y of the polystyrene window. I t is easily P seen by comparison of equs. (5) and (6) that the electrical length d of the equivalent coaxial line section depends on the p e r m i t t i v i t y under test. This behaviour is shown by Fig. 3 where some theoretical curves are presented together with experimental data for a variety of specimen cells f i l l e d with deionized water as test l i q u i d of known p e r m i t t i v i t y (~w(O) = 78.36 at 25°C (ref. 12)). In the case of a sample of such high p e r m i t t i v i t y the cell capacitance is essentially given by ~(~)CI. The effective frequency range of the specimen cell is limited by the cutoff frequency ~ c / ~ r - ~ of the TM01 mode in the c i r c u l a r waveguide. The frequency ~c is defined by the relation v c = Co(2.61a)-1
(7)
As i l l u s t r a t e d by Fig. 4, a rapid change is found from a capacitive load of the feeding line to a predominantly ohmic load when passing the cutoff frequency. In that diagram the apparent complex "void capacitance" C/~(~) is displayed as a function of frequency for two cells with waveguides of 4 and 7 mm diameter, respectively. TDS data measured with deionized water as sample l i q u i d are compared with theoretical curves (ref. 11). At low frequencies C/~(~) approaches the real value d/(CoZ) where d follows Fig. 3. Near the cutoff frequency ~c a high expenditure of numerical calculations is required to evaluate measured reflection coefficients with s u f f i c i e n t accuracy. In addition, the results are exceptionally sensitive to phase errors in this frequency range. For these reasons, the TDS measurements on aqueous electrolyte solutions reported in this communication have been performed with the cell of 4 mm diameter and the evaluation of the recorded signals was restricted to frequencies below 3 GHz. On these conditions the specimen cell essentially represents a simple capacitor. I t
21
1.2
I
I
I
I 0.3
I 0.~ bla
I 0.5
¢(0)
1.1
/ /
m
"" '10 1.0
/ /
0.9
/ 78.36 0.8
0.2
0.6
Fig. 3. The quantity ~d/a displayed as a function of the r a t i o b/a of c e l l r a d i i (Fig. 2) for d i f f e r e n t nonconducting samples of s t a t i c p e r m i t t i v i t y c(O). Full curves have been t h e o r e t i c a l l y derived by modal analysis of the e l e c t r i c a l f i e l d s ( r e f . I i ) . Full and open c i r c l e s represent data measured by time domain and frequency domain techniques, respectively, for a v a r i e t y of c u t o f f c e l l s containing deionized water as sample l i q u i d . shows, however, a weak increase of i t s capacitance with ~ at frequencies close below the c u t o f f frequency ~c" This behaviour can be approximately described by a few f i r s t
terms of a power series in c(~)(~/~c)2.
(3) Difference method.
Within the frequency range of t h i s TDS study the f r e -
quency dependence in the p e r m i t t i v i t y c d of the solutions under consideration is expected to be mainly determined by contributions resulting from reorientational motions of dipolar solute complexes. To increase the s e n s i t i v i t y in the measurement of these contributions we applied a difference method ( r e f s . 13,14), which is suited to reduce the e f f e c t by unwanted reflections of the pulse from points of impedance mismatch in the coaxial l i n e system ( r e f . 15). Aqueous solutions of sodium chloride with i t s s p e c i f i c dc-conductivities exactly adjusted to those of the sample liquids were used as reference with known d i e l e c t r i c properties ( r e f s . 16-18). Let v ( t ) denote the incident s t e p - l i k e voltage pulse and s ( t ) and Sr(t ) the pulses reflected by the c e l l when i t is f i l l e d with the sample and the reference l i q u i d , respectively. Due to the sameness of the dc-conductivities of the
22
100
I
./:it'\ C I
I
I
I I
/~<~
75 Ir
,~
50 ~e~e,e.o.o.e'°4r"
!
25
0
•
d
?V.
0~,
\
/I
I_l
r,~
I
3
!
~iI °- -~°~ v' ~ o ~
.d 2" d.n-Q: °..l.v-v'v'rv I 2
~
4
5
" I
7 GHz 10
Fig. 4. Real part ( f u l l symbols) and imaginary part (open symbols) of the apparent complex void capacitance C/E(v) plotted as a function of frequency v for cells with 4 mm (triangles) and 7 mm (circles) waveguide diameter 2a. The figure symbols indicate data measured by time domain techniques with deionized water as sample. The f u l l curves result from a modal analysis (ref. 11) of the electrical f i e l d configuration. reference and sample l i q u i d , s(t) and Sr(t ) reach the same stationary value for long times i t ÷ ~). At f i n i t e t the difference between the two reflected waveforms allows for the evaluation of the difference between the corresponding reflection coefficients which w i l l be denoted by R(v) and Rr(V) furtheron. According to transmission line theory the relation 2 Rr - R =
(I
÷
ZoYr)2
Zo(Y - Yr) Zo(y _ yr ) 1+ 1+ Z Y
(8)
or
can be used to express this difference by the input admittances Y = imC and Yr = imCr' respectively. With the Fourier transforms of incident and reflected waveforms denoted by V(v), S(v), and Sr(V), the difference Rr - R can be derived from the experimental data with the aid of the equation Rr - R : (Sr - S)/V This expression can be combined with equ. (8) to give
(g)
23 Sr c - cr
S
( i + ZoYr)2
= ~
i~ZoV
(10)
2 - (1 + ZoYr)(Sr - S)/V
At low frequencies, where the specimen cell can be s u f f i c i e n t l y well represented by a lumped capacitor, the dielectric contribution ~d(~) to the permittivity ~(~i of the solution under test is given by the relation CoZ ~d(V) = Edr(~ ) +
d
(C - Cr)
(11)
At frequencies near the cutoff frequency ~c of the TM01 mode in the waveguide section, the p e r m i t t i v i t y of the sample can only be approximately derived from equ. (11). This result, however, can be u t i l i z e d as i n i t i a l value in an iterative determination of ~d(~) from C - Cr(4) Sensitivity of method and experimental accuracy,. The present TDS study aims at an as sensitive as possible detecton of permittivity contributions which are due to dipolar solute complexes in aqueous electrolyte solutions. The sensit i v i t y in these measurements is strongly dependent on electric properties of the cell as can be shown by consideration of two quantities characterizing the difference signal, namely i t s time integral A and the centre of gravity t of A. A and t follow from the values at v = 0 of the spectrum V(u)(Rr(V ) - R(v)) and of i t s derivative with respect to the frequency, respectively. Assuming an incident step voltage pulse of unit height, one obtains 2to(~d(O ) - ~dr(0)) (12)
A ~
(1 + toO/~o)2 and to(~d(O) + Edr(0)) 1 + to~/~ o
+
(~d(0) - ~d(~))~ - (~dr(O) - ~dr(~))~r
,
(13)
~d(0) - ~dr(0)
where t o = Zod/(ZCo) is the charging time of the empty capacitor representing the cell and T and ~r are the mean dielectric relaxation times of the relaxation time distribution functions for the solution under test and the reference l i q uid, respectively. Here and in the following ~(~) denotes a permittivity extrapolated to high frequencies. The second term on the right hand side of equ. (13) reduces to ~ i f differences between the permittivities of the two liquids are solely due to a solute relaxation process of dispersion step Ed(O) - Cdr(0) and mean relaxation time ~. According to equs. (12) and (13) two demands have to be made on the cell capacitance in order to obtain a high s e n s i t i v i t y in the measurements. On the one hand, to~/~ o = 1 is required to reach a maximum value in the area A. On the other hand, the capacitance has to be small enough to allow the f i r s t term on the right hand side of equ. (13) to be smaller than (or comparable to) the
24
second term. For the l a t t e r one contains the information on the differences between the mean relaxation times of the test and reference liquid. With the 4 mm diameter cell acharging time t o = 3ps is realized. Together with the specific conductivities of the present solutions this value leads to 0.4 ~ toO/Co ~ 0.9. To improve the signal-to-noise ratio and to reduce time referencing errors a great number of repetitive scans were in phase added to one another. Each scan was available as a record of 1024 digitized data points with the outset of the interval of measurement defined by that time at which the step-voltage pulse f i r s t passes the sampling head. As a result of these provisions the time referencing error was reduced to a value in the order of one tenth of a sampling interval. To reduce the influence of unwanted reflections delay lines of suitable length have been inserted between the sampling head and specimen cell as well. Most of the signals resulting from undesired reflections on line discontinuities can be shifted out of the time interval of interest thereby. In most part of the frequency range under consideration (0.01 to 3 GHz), the difference method enables measurements with errors smaller than 2.5 % in e' and 5 % in e" Below 0.1 GHz, however, the uncertainty in the ~ values increases d" with decreasing v. I t may be characterized by Ae~ = ± i at 10 MHz. At low frequencies, the fact that the permittivity data are derived from scans of f i n i t e time intervals, may add an additional uncertainty. But a reliable estimation of truncation errors requires knowledge on the asymptotic behaviour at long times of the functions to be determined. This knowledge, of course, is usually not available. RESULTS AND TREATMENT OF DATA In Fig. 5, the real part, ~ ' ( v ) , and the negative imaginary part excluding conductivity contributions, ~"d(V), of the complex permittivity is displayed as a function of frequency v for the 0.1 molar aqueous solution of AI2(SO4)3 at 25°C and also for pure water at the same temperature. A representation of measured data in the complex c',~-plane is given in Fig. 6 for the 0.05 and 0.15 molar AI2(S04)3 solution at 25°C. As indicated by the different figure symbols used for the 0.1 molar AI2(S04)3 solution (Fig. 5) the ~' and c~-values found by frequency and time domain measurements nicely harmonize. In the microwave (ref. 12) and near millimetre wavelength region (refs. 20, 21) the dielectric spectrum of the pure solvent can be represented (ref. 22) by a DEBYErelaxation spectral function (ref. 23) which is defined by the expressi on ~d(V) = ~w(=) +
Ew(O) -
~w(~)
1 + imTw
(14)
25
100 I
t
t
o-o o- o
t
80
....... r - - %
I
0|
I
t
. J¢(-)--
II
I
=
/i\ ^..o "°'' 10 7
10 8
. ~ + f " (2nTI)-~(Z~T.) -I 10 9
101° Hz
10"
V
Fig. 5. Real part E'(~) and dielectric contributions c~(v) to the negative imaginary part ~"(~) of the complex permittivity displayed as a function of frequency v for water (crosses (ref. 12)) and the 0.1 molar aqueous solution of AI2(S04) 3 (circles) at 25°C. Open and closed circles indicate data from TDS and frequency domain measurements, respectively. The multiplication sign identifies the static permittivity ~w(0) of water (ref. 19). The full curves are graphs of the relaxation spectral function given by equ. (15) with the parameter values presented in Table 2. At 25°C the parameters of equ. (14) have the following values (refs. 12,22): extrapolated high frequency permittivity Cw(=) = 5.16 + 0.5; extrapolated static permittivity ~w(0) = 78.36 + 0.05; relaxation time ~w = (8.27 -+ 0.02) ps. The spectra of the solutions clearly show two dispersion (d~'(v)/dv < 0)/dielectric loss (~(v) > 0) regions, indicating that different relaxation mechanisms are present. The characteristic frequency (2~T1)-1 of one process has nearly the same value (~ 20 GHz) as that of pure water (2~w)-1. Obviously, the
26
40
/
30
="a W
I v
I 27 2 ~ '17 3 -_.~-~,,,+
I
I 12.5 _.
I
I
I
20
_.
10
\ ~ . .
0
10
l/
20
30
40
50
60
|_J-
70 •
t
80
"O_~U.11,
90
100
E'(v) Fig. 6. The dielectric spectra of a 0.05 (o) and 0.15 (e) molar aqueous solution of AI2(S04)3 at 25°C shown in a complex plane representation with the frequency v being parameter ("COLE-plot"). The f u l l curves represent the spectral function defined by equ. (15) with the parameter values found by the f i t t i n g procedure. The dashed lines indicate the subdivision of the complex permittivity into solute and solvent contributions. process with relaxation time ~1 reflects the reorientational motions of the solvent molecules. The relaxation .frequency (2~T2)-I of the other dispersion/absorption region is d i s t i n c t l y smaller (~ 0.7 GHz). We assume this l a t t e r contribution to the dielectric spectrum to be due to dipolar ion complexes. This assumption is confirmed by the plots shown in Fig. 7 where E'(v) is displayed as a function of frequency for the 0.1 molar solution of Sc2(S04)3 and also for an aqueous solution of zwitterionic colamine phosphoric acid with nearly the same relaxation frequency (2~T2)-1. The low-frequency dispersion of the l a t t e r liquid clearly results from reorientations of solute zwitterions. The solute and solvent relaxation process may both be subject to relaxation time distributions. We therefore used the relaxation spectral function ~d(V) = a(=) +
~i - ~(~)
i + (i~T1)(l"hl)
E(0) - ~i
+ .... 1 + (i~T2)(l-h2)
(15)
to analytically describe the measured spectra of the solutions. Using this relation a COLE-C0LE relaxation time distribution G(~) (ref. 25) is assumed to underly both processes. This distribution function, i f
TG(T) is plotted versus
I n ( z / z i ) , i = 1, 2, is symmetrically bell shaped around T/T i = 1, where ~I and T2 denote the principal relaxation times. Parameters hi , i = 1, 2, measure the width of the relaxation time distribution. The values for the parameters of equ. (15) have been found by f i t t i n g the relaxation function to the measured spectra using a nonlinear least-squares regression analysis. The results obtained by
2'/
160
c(O)
~
120
so
W
I
i
L-o-o-o-'----~.o~
/
LO _
-
,
,
80 - . . . .
t--
0 100
~
~(~
(21TT2)-I
--
60
(2n'~z1-I
_
200
107
,
(2~i)-I I ~, I
I
10 s
10 9
IC(')--e~'-~ 101° Hz
101~
Fig. 7. Real part e'(~) of the d i e l e c t r i c spectrum of the 0.1 molar aqueous solution of Sco(SOA)3 (e) and of an 1-molar solution of colamine phosphoric acid (o) at 25°C.~ TABLE 2
A12(S04)3
Solute c c(~) cz •z hz c(0) m2
hz
[mol/l] [ps] [ps]
± ± ± ± ± ±
0.5 0.4 1 % 0.02 2 % 2 %
±100
%
In2(S04) 3 Sc2(S04)3
0.05 3.5 75.5 8.2 0.04 90.2 272
0.1 4.5 71.7 8.3 0.03 92.7 232
0.15 4.4 69.1 8.4 0.04 92.9 213
0.03
0.02
0.07
0.1 4.1 75.0 8.3 0.04 93.1 250
0
0.1 4.6 74.5 8.2 0.02 88.8 232
0.04
Values of the parameters of the relaxation spectral function defined by equ. (15) for the solutions of 3:2 valent salts at 25°C.
this f i t t i n g procedure are collected in Table 2. Applications of relation (15) does not imply that only the C0LE-COLE relaxation time distribution allows for a satisfactory description of the measured
28
relaxation processes. Within the limits of experimental error, however, a decision in favour of one of the commonly used relaxation time distribution functions, including the asymmetrical COLE-DAVIDSON function (ref. 26) is not possible. This finding is i l l u s t r a t e d by the results shown in Fig. 8 where for the Sc2(S04) 3 solution the deviations
6c'(v) : ~ '
~(v)-~'
mea:
calc (v)
(16)
and II
~
~ il
6 d( ) : 8e"(v) :
(17)
il
meas(V) - ecalc(V )
are compared for the two model functions given by the equation E1 - ~(~)
~d(V) : c(~) +
N A~n + g I + (imzl)(1-hl) n=2 1 + im%n
(18)
with N = 2 or 3, respectively. The calculated values ~calc(V) and ~calc " t"v") are those found, i f relaxation (18) with N = 2 or 3 is f i t t e d tothemeasuredspectra.
1
I
I
I
-I W
I
--
0
0 - o o o ~ _ _,_~
-I I
0
0 -1 lO 7
10 s
10 9
lO 1°
Hz
1011
Fig. 8. The deviations ~ ' ( v ) and 6c~(v) as defined by equs. (16) and (17), respectively, plotted versus the frequency v for the 0.1 molar solution of Sc2(S04)3 at 25°C. Different figure syn~ols mark different model relaxation functions: e, N = 3 and o, N = 2 in equ. (18).
2g
I t is clearly demonstrated by Fig. 8 that the 6E'(v) and 6~"(u) values are only insignificantly smaller i f the solute relaxation process i s represented b y two DEBYErelaxation terms (N = 3) instead of a single one (N = 2). Thus a clear-cut conclusion on the presence or absence of a small relaxation time distribution is impossible. DISCUSSION Contribution of solvent water to the permittivit~ The principal relaxation times ~i of the solvent water in the solutions of 3 valent sulfates are found very close to the pure water value ~w = 8.27 ps at 25°C (Table 2). The differences ~1 - Zw are too small to be discussed with respect to specific effects of ionic hydration. Also too small to allow for definite conclusions on solute-solvent interactions are the values of parameter hI. Substantial deviations from the pure water value Cw(O) (= 78.36, 25°C), however, emerge in the solvent contribution to the static permittivity of the solutions. These deviations may be due to at least three different effects, namaly kinetic depolarization, dilution and depolarizing internal electric fields as secondary phenomenon, and structure saturation. Kinetic depolarization (refs. 27-31), f i r s t theoretically predicted by ONSAGER (ref. 32), is due to a coupling between hydrodynamic and dielectric effects which in conducting solutions of polar liquids gives rise to a reduction in the solvent contribution to the extrapolated static p e r m i t t i v i t y . The theory of kinetic depolarization has been confirmed by various experimental facts. Among these facts is the strong dependence of the extrapolated static permittivi ty of sulfuric acid on small admixtures of water which has been considered by HALL and COLE (ref. 33). Water molecules in the f i r s t hydration shell around small multivalent cations have a long residence tima (ref. 34). Within the time scale under consideration i t is therefore adequate to treat small cations together with their f i r s t hydration sphere as one rigid body. The effect of kinetic depolarization can then be considered on the basis of the HUBBARD-ONSAGER continuum model for large ion size. Assuming perfect slip boundary conditions on solvent flow at the ion surfaces this theory predicts the kinetic polarization deficiency 6cHO to be given by the relation (ref. 30) 2 two Cw(O) - Cw(~) 6cHO - 3
co
(19)
Cw(O)
With the present solutions of 3:2 valent salts the aCHO values are small (0.69 ~ a~HO % 1.48). Nevertheless, p e r m i t t i v i t i e s corrected for the kinetic polarization deficiency, ~1 + 6~HO' w i l l be used in the following discussion of the other effects nentioned above.
30 We use the BRUGGEMANmixture formula (ref. 35) given by the expression Va = i
~i + 6EHO - ~e [ -
~w(O) ~I/3
l
~w(0) - ~e
I
(20)
\ e l + a~HO/
to calculate the apparent volume fraction va of spheres with p e r m i t t i v i t y Ce(~2) which seem to be solved in the homogeneous dielectric of permittivity ~w(O). The va values are substantially greater than the volume fractions v of solute. We conclude, that a part of the water around the ions is d i e l e c t r i c a l l y saturated and has thus to be reckoned among the particles with permittivity ~ rather than e the d i e l e c t r i c a l l y homogeneous solvent of ~w(O). Since saturation effects are unlikely to exist around the comparatively big sulfate ion, apparent values Z+ of t o t a l l y i r r o t a t i o n a l l y bound water molecules per cation can be calculated for the 3:2 valent salts using the relation Z+ : (va - v)/(2c @w)
,
(21)
where @w (= 18.069 ml/mol at 25°C (ref. 36))denotes the molar volume of water. Most interesting, remarkably different Z+ values are found for the three 3valent cations, namely 13 for Al 3+, 5 for In 3+, and 2.5 for Sc3+. Let us b r i e f l y consider this result in view of the structure of the dipolar ion complexes assumed to be present in the solutions. The formation of these complexes is usuall y described by the EIGEN mechanism (ref. 37) of stepwise association which may be represented by the relation
Mm+ and LI- are shorthand notations for the metal ion and the anionic ligand, respectively. The high Z+ value for the small Al 3+ ion (r + = 0.5 ~ (ref. 38)) following from the analysis of the dielectric spectra may be taken to indicate, that predominantly solvent-separated ion pairs are formed with this cation, namely outer sphere (Mm+(H20)L l - ) or outer-outer sphere (Mm+(H20)2Ll-) complexes. This conclusion is in agreement with the results derived from ultrasonic absorption spectra of aqueous A12(S04)3 solutions (ref. 39). I t is only b r i e f l y mentioned here, that Z+ = 12.7 (ref. 3) had been found for aqueous solutions of AICI 3, a system for which dielectric spectroscopy yields no indications of ion complex formation. I t has thus to be concluded that in solutions of aluminium sulfate the cation is obviously able to form i t s complete shell of dielect r i c a l l y saturated water. The small Z+ values for the somewhat bigger In 3+ and Sc3+ ions (r + = 0.8 for both (ref. 38)) reflect a weaker bonding of their hydration water. The
31 sulfate ion seems to induce a substantial rearrangement of the hydration structure of those cations. Indium, which has d10 outer electron configuration, shows a similar behaviour also in aqueous solutions of i t s chloride, bromide, and iodide. Z+ values between 3.9 and 6.1 had been evaluated for those salts previously (ref. 3). Relaxation process of the solute A f i r s t of a l l surprising result of our d i e l e c t r i c study is the finding of only one dispersion/dielectric loss region in addition to the water relaxation. On the contrary, the ultrasonic absorption spectrum of aqueous solutions of 3:2 valent electrolytes usually reveals more than one relaxation process. An example is shown in Fig. 9 where for comparison with the d i e l e c t r i c losses, E~(v), the ultrasonic excess absorption per wavelength, ~(v)~ - By, is displayed as a function of frequency for the 0.1 molar solution of Sc2(S04)3. The excess absorption is that part of the total absorption per wavelength, ~ , which is due to chemical relaxation processes. The curve for the scandium sulfate solution clearly contains contributions from three relaxation processes. These can be attributed to the three transitions between the different ionic species represented by equ.
(22).
100
I
I
I
I
I
30 10 o
3
r/
1
lO0,
#
I
\~./ ```` / \ . I/y'. ~\.
I
I
I
I
//
--
``\
\\ \
I
-
\~ I
303 I
I0-
~-;../" .~ / 4 / , \
3-
/
I0.3 10 s
1 10 6
/
//
I/@///l//' 10 7 10 e
Y
(2~T2}-
I 10 9
(2.TJ -1
, \\ ,,
-
I\',, 101° HZ 1011
Fig. 9. The ultrasonic excess absorption ~L - By (o (ref. 39)) and the dielect r i c loss ~ = ~" - o/(Co~ ) (e) plotted versus frequency v for the 0.1 molar solution of Sc2(S04)3 at 25°C.
32
As w i l l be outlined below the difference with respect to the solute relaxations in the two types of spectra results from the fact that ultrasonic spectroscopy probes the association/dissociation processes only while dielectric spectroscopy is also sensitive to the reorientational motions of the dipolar species Let again the 0.1 molar solution of scandium sulfate serve as an example since for that system the concentrations of the outer ("o") and of the outer-outer ("oo") sphere complex relative to that of the inner ( " i " ) sphere complex are known (ref. 39). The dipole moments ~ i ' ~o' and ~oo of the three species can be estimated by geometrical considerations. Using the relation c(O) - ~1 = Aci + Aeo + Aeoo (23) 2 c 2 2 ~ci~ i + oUo + Coo~oo the following estimates are obtained for the contributions of the d i f ~ r e n t ion complexes to the static permittivity of the Sc2(S04)3 solution: Aei/(e(O ) - el) Aeo/(~(O ) - ~I)
= 0.44 = 0.19
(24)
aeool(~(O) - e l ) : 0.37 Also by geometrical considerations and based on the DEBYE model ( r e f . 23), which predicts the d i e l e c t r i c ~ l a x a t i o n time m r the reorientational motions of a spherically shaped dipolar p a r t i c l e to be proportional to the t h i r d power of i t s radius, the following reorientation time ratios can be estimated: Zo/Ti = 1.3 Too/Zi = 1.6
(25)
These ratios are by far too small to allow in the dielectric spectrum for a clear separation of the different relaxation processes. To i l l u s t r a t e this statement a COLE representation is given in Fig. 10 for the resulting curve of a superposition of three DEBYEterms. The parameters of these terms follow equs. (24) and (25). Deviations from the DEBYEsemicircle also indicated in that plot are smaller than the usual experimental errors. Dielectric spectroscopy has proven to be a powerful method in detecting dipolar ion complexes including very short-lived species. I t is, however, l i t t l e suited for the identification and study of different s i m i l a r l y sized complexes which might be together present in a solution. Nevertheless, a global statement can be made on the ion complexes of 3:2 valent salts which support our discussion in the preceeding section. The dispersion steps ~(0) - c I of the 0.1 molar solutions of aluminium, indium, and scandium sulfate are 21.0, 18.1, and 14.3, respectively. These data seem to also indicate, that the concentration of
33
contact ion pairs increases in the above series of cations.
I 0.5
I
1.58 _
n
I
0.89 ~ ,
_ -
/
%o.s
8. 9e~/
•
! 0
,
0.5
.16
to.o. 1.0
¢'(v) Fig. 10. Complex plane representation of the spectral function ~(~) = 0.44/ (i + i f ) + 0.19/(1 + i 1.3f) + 0.37/(1 + i 1.6f) ( f u l l curve). The dashed semicircle is the graph of a simple DEBYE relaxation term (f = 2 ~ i ) . ACKNOWLEDGMENT We thank Professor R. Pottel for valuable discussions.
REFERENCES 1 R. Pottel, Die komplexe Dielektrizit~tskonstante w~Briger LGsungen einiger 2:2-wertiger Elektrolyte im Frequenzbereich 0.1 bis 38 GHz, Ber. Bunsenges. Phys. Chem. 69 (1965) 363-378. 2 R. Pottel, The Complex Dielectric Constant of Some Aqueous Electrolyte Solutions in a Wide Frequency Range, in: B.E. Conway and R.G. Barradas (Eds.), Chemical Physics of Ionic Solutions, Wiley, New York, 1966, pp. 581-598. 3 U. Kaatze, Dielectric Effects in Aqueous Solutions of 1:1, 2:1, and 3:1 valent Electrolytes: Kinetic Depolarization, Saturation, and Solvent Relaxation, Z. Phys. Chem. Neue Folge 135 (1983) 51-75. 4 J. Barthel, R. Buchner, and H.-J. Wittmann, Leitf~higkeit und dielektrische Eigenschaften w~Briger CdSO4-LGsungen, Z. Phys. Chem. Neue Folge 139 (1984) 23-37. 5 U. Kaatze, V. LGnnecke, and R. Pottel, Dielectric Spectroscopy on Aqueous Solutions of Zinc(II)Chloride. Evidence of Ion Complexes, submitted for publication. 6 W. Walisch and J. Barthel, Die genaue Messung kleiner Elektrolytwiderst~nde nach der Phasensprungmethode mit einer Tauchzelle, Z. Phys. Chem. Neue Folge 34 (1962) 38-50. 7 R.A. Robinson and R.H. Stokes, Electrolyte Solutions, Butterworths, London, 1959, pp. 87-117. 8 T.J. Buchanan, Balance Methods for the Measurement of Permittivity in the Microwave Region, Proc. IEE Part I l l 99 (1952) 61-66. 9 U. Kaatze, Ein Verfahren der dielektrischen Relaxationsspektroskopie polarer FIUssigkeiten im Mikrowellenbereich, Mikrowellen Magazin 1980, 46-49.
34
10 R.H. Cole, Time Domain Reflectometry, Annu. Rev. Phys. Chem. 28 (1977) 283-300. 11 K. Giese, Theoretical Analysis of Cutoff Type Specimen Cells for Dielectric Measurements of Liquids, to be published. 12 U. Kaatze and V. Uhlendorf, The Dielectric P~operties of Water at Microwave Frequencies, Z. Phys. Chem. Neue Folge. 126 (1981) 151-165. 13 K. Giese, TDS-Studies of Aqueous Ionic Solutions, in: A.M. Bottreau, Y. Dutuit, and C. Marzat (Eds.), Proceedings of the Second International Time Domain Spectroscopy Meeting, University of Bordeaux, 1975, Vol. I I , 10-19. 14 R.H. Cole, S. Mashimo, and P. Winsor IV, Evaluation of Dielectric Behaviour by Time Domain Spectroscopy. 3. Precision Difference Methods, J. Phys. Chem. 84 (1980) 786-793. 15 K. Giese and R. Tiemann, Determination of the Complex Permittivity from Thin Sample Time Domain Reflectometry. Improved Analysis of the Step Response Waveform, Adv. Mol. Relaxation Processes 7 (1975) 45-59. 16 K. Giese, U. Kaatze, and R. Pottel, Permittivity and Dielectric and Proton Magnetic Relaxation of Aqueous Solutions of the Alkali Halides, J. Phys. Chem. 74 (1970) 3718-3725. 17 R. Pottel, Dielectric Properties, in: F. Franks (Ed.), Water, a Comprehensive Treatise, Plenum, New York, 1973, Vol. I l l , 401-432. 18 J.B. Hasted, Aqueous Dielectrics, Chapman and Hall, London 1973. 19 G.A. Vidulich, D.F. Evans, and R.L. Kay, The Dielectric Constant of Water and Heavy Water between 0 and 40°C, J. Phys. Chem. 71 (1967) 656-662. 20 J.B. Hasted, S.K. Husain, F.A.M. Frescura, and JoR. Birch, Far Infrared Absorption in Liquid Water, Chem. Phys. Let., submitted for pulication. 21 J.B. Hasted, S.K. Husain, F.A.M. Frescura, and J.R. Birch, The Temperature Variation of the Near Millimetre Wavelength Optical Constants of Water, submitted for publication. 22 U. Kaatze, The Dielectric Spectrum of Water in the Microwave and Near M i l l i metre Wavelength Region, submitted for publication. 23 P. Debye, Polar Molecules, Chemical Catalog, New York, 1929. 24 H. Bieler, Messungen des dielektrischen Spektrums von konzentrierten w~Brigen L~sungen kleiner zwitterionischer MolekUle im Frequenzbereich I MHz bis 39 GHz, Diplom-Thesis, University of G~ttingen, 1982. 25 K.S. Cole and R.H. Cole, Dispersion and Absorption in Dielectrics. I. Alternating Current Characteristics, J. Chem. Phys. 9 (1941) 341-351. 26 D.W. Davidson and R.H. Cole, Dielectric Relaxation in Glycerine, J. Chem. Phys. 18 (1950) 1417. 27 J.B. Hubbard, L. Onsager, W.M. van Beek, and M. Mandel, Kinetic Polarization Deficiency in Electrolyte Solutions, Proc. Natl. Acad. Sci. USA 74 (1977) 401-404. 28 J.B. Hubbard and L. Onsager, Dielectric Dispersion and Dielectric Friction in Electrolyte Solutions. I , J. Chem. Phys. 67 (1977) 4850-4857. 29 J.B. Hubbard, Dielectric Dispersion and Dielectric Friction in Electrolyte Solutions. I I , J. Chem. Phys. 68 (1978) 1649-1664. 30 J.B. Hubbard, P. Colonomos, and P.G. Wolynes, Molecular Theory of Solvated Ion Dynamics. I l l . The Kinetic Dielectric Decrement, J. Chem. Phys. 71 (1979) 2652-2661. 31 P.G. Wolynes, Dynamics of Electrolyte Solutions, Ann. Rev. Phys. Chem. 31 (1980) 345-376. 32 L. Onsager, personal communication, 1973. 33 D.G. Hall and R.H. Cole, Kinetic Depolarization in Sulfuric Acid, J. Chem. Phys. 68 (1978) 3942-3943. 34 H.G. Hertz, Nuclear Magnetic Relaxation Spectroscopy, in: F. Franks (Ed.), Water, a Comprehensive Treatise, Plenum, New York, 1973, Vol. I I I , 301-399. 35 C.J.F. B~ttcher and P. Bordewijk, Theory of Electric Polarization, Elsevier, Amsterdam, 1978, Vol. I I , 479. 36 K.S. Kell, Thermodynamic and Transport Properties of Fluid Water, in: F. Franks (Ed.), Water, a Comprehensive Treatise, Plenum, New York, 1972, Vol. I , 363-412.
36 37 M. Eigen and K. Tamm, Schallabsorption in Elektrolytl~sungen als Folge chemischer Relaxation. I I . MeBergebnisse und Relaxationsmechanismen fur 2-2wertige Elektrolyte, Ber. Bunsenges. Phys. Chem. 66 (1962) 107-121. 38 B.E. Conway, Ionic Hydration in Chemistry and Biophysics, Elsevier, Amsterdam, 1981, 65. 39 A. Bonsen, W. Knoche, W. Berger, K. Giese, and S. Petrucci, Ultrasonic Relaxation Studies in Aqueous Solutions of Aluminium Sulphate and Scandium Sulfate, Ber. Bunsenges. Phys. Chem. 82 (1978) 678-683.
15
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
DIELECTRIC SPECTROSCOPY ON SOMEAQUEOUS SOLUTIONS OF 3:2 VALENTELECTROLYTES. A COMBINED FREQUENCY AND TIME DOMAINSTUDY. DEDICATED TO PROFESSOR ROBERTH. COLE UDO KAATZE and KARL GIESE* Drittes Physikalisches Institut, Universit~t G~ttingen, BUrgerstraBe 42-44, D-3400 G~ttingen (West Germany)
(Received 26 November 1986) ABSTRACT The complex dielectric spectrum of aqueous solutions of aluminium, indium, and scandium sulfate has been determined between 10 ~ z and 60 GHz using time domain and frequency domain techniques. A difference method especially matched to the study of electrically conducting solutions has been applied in the time domain measurements. Various relaxation spectral functions were f i t t e d to the measured data. The results obtained thereby are discussed with respect to the effects of kinetic depolarization, dielectric saturation and ion complex formation. INTRODUCTION The detailed knowledge of ion-ion and ion-solvent interactions in aqueous solutions is extremely important for our understanding of the molecular structure, kinetics, and dynamics of numerous liquid systems, many of them with considerable relevance in chemistry, biology or medicine. The complex (electric) permittivity spectrum
(1)
c(~) : e'(~) - i~"(v)
of electrolyte solutions enables statements on both types of interactions. I t is capable to directly reflect the presence or absence of dipolar ion complexes and also allows for conclusions on hydration properties (rfs. 1-5). Unfortunately, however, the total dielectric spectrum e(~) of conducting samples combines two different contributions as is usually expressed by the relation
~(~)
: ~d(~) - i ~
(
(~) - i ~ ( ~ )
+
o
One part (~d) of ~ results from polarization processes, the other one (-iO/~om) *Present address: Institut fur Medizinische Physik und Biophysik, Universit~t G~ttingen, GoBlerstraBe 1Of, D-3400 G~ttingen. 0167-7322/87/$03.50
© 1987 Elsevier Science Publishers B.V.
16
from the d r i f t of ionic species (i = ( - i ) 1 / 2 ; m = 2~v; v, frequency; a, specific e l e c t r i c conductivity; ~o = 8"854"10-12AsV-1m-1)" With decreasing frequency (~ ~ 1 GHz) d i e l e c t r i c measurements on electrolyte solutions are thus subject to increasing experimental error since the ion currents tend to more and more exceed the polarization currents to be observed. In the present study we tried to overcome these d i f f i c u l t i e s by combining spot frequency and time domain spectroscopic (TDS) measurements to cover a broad frequency range. With the available equipment frequency domain measurements on the chosen solutions (1.2 < o < 2.6 S/m) were possible between 0.5 and 58 GHz. A TDS method matched to the study of highly conducting liquids has been developed which allowed the complex dielectric spectrum to be determined for the frequency range from 10 MHz to 3 GHz. Recent studies of d i e l e c t r i c effects in aqueous solutions of inorganic salts, including 3:1 valent electrolytes, confirmed a special a b i l i t y of the d-electrons of transition metal ions to interact with ionic ligands (refs. 3-5).For the present investigation we therefore selected aqueous solutions of aluminium, scandium, and indium sulfate since the t r i v a l e n t cations of these salts have do , d8, and d10 electron orbits, respectively. EXPERI~NTAL SECTION Aqueous solutions The sulfates (FLUKA, 98 % or better) were used without further p u r i f i c a t i o n . The solutions were prepared by f i l l i n g preweighed amounts of salts and deionized, d i s t i l l e d and degassed water into suitable flasks. To avoid formation of hydroxy complexes with t r i v a l e n t cations perchloric acid was added to appropriately adjust the pH value of the solutions (AI, 2.1; Sc, 2.4; In, 2.0). The sulfate ion is partly protonated at the ionic strength and pH used in this study. Specific e l e c t r i c conductivity As b r i e f l y indicated in the Introduction section the specific e l e c t r i c cond u c t i v i t y ~ of the solutions has to be accurately known. We used a two-electrode capillary cell (ref. 6) with suitable cell constant to measure o at 0.1, 1, 10, and 100 kHz. A sensitive impedance bridge (GENERALRADIO 1654) has been u t i l i z e d which allowed the resistance of the l i q u i d - f i l l e d cell to be compared with a precise decade resistor (GENERAL RADIO 1433-F). The ring-shaped electrodes of the cell were covered by a deposite of f i n e l y divided black platinum to minimize errors due to electrode polarization effects (ref. 7). The cell constant has been determined with an error of less than 0.2 % using aqueous NaCl and KCI of appropriate concentrations as reference liquids. The temperature was controlled during the measurements to within 0.05 K.
17 Frequency domain measurements At microwave frequencies (0.5 ~ ~ ~ 58 GHz) the complex (relative) permittivi ty ~(~) of the solutions has been measured by transmitting propagating electromagnetic waves of frequency ~ through specimen cells of variable sample thickness (refs. 1,8,9). As schematically shown in Fig. 1, the liquid under test was f i l l e d in a piece of circular cylindrical waveguide or coaxial line for this purpose. A receiving probe, also a circular waveguide or coaxial l i n e , was immersed in the liquid. Combined with high precision ball bush guides (FEINPRDF, G~ttingen) the probe was shiftable free of backlash along the direction z of wave propagation. The wavelength k and attenuation exponent ~ have been interferometrically determined by adjusting a reference signal of calibrated variable amplitude. Five double beam interferometers, each matched to a limited frequency range were used. One was constructed with coaxial line components, the
ziT 8
S+)~
)
2
(Z) = E oe iwte-("*i~)z
---
12
FJ 3
9
10
11
Fig. 1. Schematic representation of the microwave double-beam interferometers used to perform frequency domain measurements between 0.5 and 58 GHz. I , monochromatic o s c i l l a t o r ; 2, frequency counter or meter; 3, uniline or coaxial attenuator; 4, directional coupler or coaxial beam s p l i t t e r ; 5, impedance matching transformer; 6, measuring cell containing the liquid sample; 7, movable probe; 8,9, phase s h i f t e r and variable attenuator, respectively, u t i l i z e d to adjust maximum interference with zero signal at probe position s; 10, calibrated variable attenuator with constant phase s h i f t used to produce zero signal at probe position s + ~; 11, superheterodyne receiver; 12, zero signal indicator.
iB others with standard waveguide devices. The p e r m i t t i v i t y is related to the measured q u a n t i t i e s ~, X, and ~
according
to the f o l l o w i n g equations e'(V) =
~"(~)
-
- \-,~-~--1 + \-~C} mX2 0 ~
(3)
,
(4)
where Xo = (speed of l i g h t ) / v denotes the wavelength in free space and ~c the c u t o f f wavelength of the empty c e l l . The c e l l s were surrounded by a water-jacket to allow the temperature to be c o n t r o l l e d to w i t h i n ± 0 . 1 K . Errors in the complex p e r m i t t i v i t y values measured with t h i s technique are mainly due to the l i m i t e d zero-level s e n s i t i v i t y of the microwave receiver, imperfections of the calibrated attenuator used to adjust the amplitude of the reference signal and temperature f l u c t u a t i o n s . Errors in the determination of the frequency and in the micrometer drive c o n t r o l l i n g the position of the s h i f t able probe are n e g l i g i b l y small. Careful consideration of the sources of e r r o r in the predominant part of the frequency range leads to an uncertainty of ± i % in the values of both, ~'(~) and ~"(~). Due to higher loss the accuracy of the p e r m i t t i v i t y values decreases when going to the high and low frequency end (v > 18 GHz, v < i GHz) of the measuring range (Table i ) . According to equ. 2 the e r r o r in the d i e l e c t r i c part of the loss depends on the s p e c i f i c e l e c t r i c c o n d u c t i v i t y of the s o l u t i o n . To g l o b a l l y characterize the accuracy of the c~ data uncertainties of the 0.1 molar AI2(S04) 3 s o l u t i o n (o = 2.06 S/m, 25°C) are compiled as an example in Table i . TABLE 1
GHz
Z1~'IE ' %
I~E"Ic . %
0.5 0.7 1 2-5 5-18 18-30 30-40 40-58
3 2 1 1 1 2 4 i0
2 2 1 1 1 1 2 3
.
.
.
.
ACdl~ . d % 15 12 5 3 2 2 2 3
The experimental errors in the complex p e r m i t t i v i t y values measured in the frequency domain f o r the 0 . i molar aqueous s o l u t i o n of AI2(S04) 3 at 25°C.
19 Time domain spectroscopy (1) Principle of measurement. In the frequency range 10 I~Hz ~ v s 3 GHz time domain reflectometry (ref. 10) has been used to derive the complex permittivity ~(u) from the Fourier transform of a step-like fast-rising voltage pulse transmitted in a coaxial line and reflected from the specimen c e l l . We utilized the Hewlett-Packard TDS system with a computer connected for automatic data recording and processing. Special emphasis was layed on the matching of the cell construction and measuring method to the study of aqueous electrolyte solutions. (2) Specimen cell.
The construction of the cell is shown in Fig. 2a. The
sample liquid is contained in a circular waveguide of diameter 2a. This waveguide is sealed by an e l e c t r i c a l l y matched polystyrene window against the coaxial feeding line of characteristic impedance Zo = 50 ~. The cell is made of stainless steel and is provided with a thermostatic coat not shown in Fig. 2a.
a)
A
b)
I ~ d
~ IT E(v)Cz
T l -
p
ely) C1
T
tpC 3 1
2
3 A
Fig. 2. Construction of the cutoff cells utilized in the TDS studies (a) and i t s equivalent network representation for plane A-A(b). 1, coaxial feeding line (50 ~; diameter of the outer and inner conductor 7 and 3.04 mm, respectively); 2, polystyrene window (~p = 2.54; 2ap = 8.5 mm; 2b = 2.25 mm); 3, circular waveguide containing the l i q u i d ; d, length defined by equ. (5). At frequencies far below the cutoff frequency of the TM01 mode in the waveguide section, the cell can be approximately represented by a capacitance C, which apparently is equivalent to a short piece of coaxial line with i t s openc i r c u i t termination at distance d from the face of the polystyrene window (Fig. 2a). The relation between C and d is given by C = ~(~)dlCCoZ )
,
(5)
20 where co denotes the speed of l i g h t in free space and Z the characteristic impedance of an empty coaxial line with the radius of the inner and outer conductor being b and a, respectively. Accurate modal analysis (ref. 11) of the stray fields at both sides of the plane A-A separating the coaxial feeding line from the waveguide section leads to the equivalent network representation of the cell shown in part b of Fig. 2. In this lumped c i r c u i t element network the capacitance C1 corresponds to components of the stray f i e l d which are t o t a l l y restricted to the waveguide section, while the series arrangement of C2 and C3 has regard to electrical f l u x lines which intersect plane A-A. Expressed by the capacities Ci , i = I . . . 3, the capacitance C of the cell when f i l l e d with the sample l i q u i d is given by C = e(v)C I +
I
+
,
(6)
where ~ (= 2.54) is the p e r m i t t i v i t y of the polystyrene window. I t is easily P seen by comparison of equs. (5) and (6) that the electrical length d of the equivalent coaxial line section depends on the p e r m i t t i v i t y under test. This behaviour is shown by Fig. 3 where some theoretical curves are presented together with experimental data for a variety of specimen cells f i l l e d with deionized water as test l i q u i d of known p e r m i t t i v i t y (~w(O) = 78.36 at 25°C (ref. 12)). In the case of a sample of such high p e r m i t t i v i t y the cell capacitance is essentially given by ~(~)CI. The effective frequency range of the specimen cell is limited by the cutoff frequency ~ c / ~ r - ~ of the TM01 mode in the c i r c u l a r waveguide. The frequency ~c is defined by the relation v c = Co(2.61a)-1
(7)
As i l l u s t r a t e d by Fig. 4, a rapid change is found from a capacitive load of the feeding line to a predominantly ohmic load when passing the cutoff frequency. In that diagram the apparent complex "void capacitance" C/~(~) is displayed as a function of frequency for two cells with waveguides of 4 and 7 mm diameter, respectively. TDS data measured with deionized water as sample l i q u i d are compared with theoretical curves (ref. 11). At low frequencies C/~(~) approaches the real value d/(CoZ) where d follows Fig. 3. Near the cutoff frequency ~c a high expenditure of numerical calculations is required to evaluate measured reflection coefficients with s u f f i c i e n t accuracy. In addition, the results are exceptionally sensitive to phase errors in this frequency range. For these reasons, the TDS measurements on aqueous electrolyte solutions reported in this communication have been performed with the cell of 4 mm diameter and the evaluation of the recorded signals was restricted to frequencies below 3 GHz. On these conditions the specimen cell essentially represents a simple capacitor. I t
21
1.2
I
I
I
I 0.3
I 0.~ bla
I 0.5
¢(0)
1.1
/ /
m
"" '10 1.0
/ /
0.9
/ 78.36 0.8
0.2
0.6
Fig. 3. The quantity ~d/a displayed as a function of the r a t i o b/a of c e l l r a d i i (Fig. 2) for d i f f e r e n t nonconducting samples of s t a t i c p e r m i t t i v i t y c(O). Full curves have been t h e o r e t i c a l l y derived by modal analysis of the e l e c t r i c a l f i e l d s ( r e f . I i ) . Full and open c i r c l e s represent data measured by time domain and frequency domain techniques, respectively, for a v a r i e t y of c u t o f f c e l l s containing deionized water as sample l i q u i d . shows, however, a weak increase of i t s capacitance with ~ at frequencies close below the c u t o f f frequency ~c" This behaviour can be approximately described by a few f i r s t
terms of a power series in c(~)(~/~c)2.
(3) Difference method.
Within the frequency range of t h i s TDS study the f r e -
quency dependence in the p e r m i t t i v i t y c d of the solutions under consideration is expected to be mainly determined by contributions resulting from reorientational motions of dipolar solute complexes. To increase the s e n s i t i v i t y in the measurement of these contributions we applied a difference method ( r e f s . 13,14), which is suited to reduce the e f f e c t by unwanted reflections of the pulse from points of impedance mismatch in the coaxial l i n e system ( r e f . 15). Aqueous solutions of sodium chloride with i t s s p e c i f i c dc-conductivities exactly adjusted to those of the sample liquids were used as reference with known d i e l e c t r i c properties ( r e f s . 16-18). Let v ( t ) denote the incident s t e p - l i k e voltage pulse and s ( t ) and Sr(t ) the pulses reflected by the c e l l when i t is f i l l e d with the sample and the reference l i q u i d , respectively. Due to the sameness of the dc-conductivities of the
22
100
I
./:it'\ C I
I
I
I I
/~<~
75 Ir
,~
50 ~e~e,e.o.o.e'°4r"
!
25
0
•
d
?V.
0~,
\
/I
I_l
r,~
I
3
!
~iI °- -~°~ v' ~ o ~
.d 2" d.n-Q: °..l.v-v'v'rv I 2
~
4
5
" I
7 GHz 10
Fig. 4. Real part ( f u l l symbols) and imaginary part (open symbols) of the apparent complex void capacitance C/E(v) plotted as a function of frequency v for cells with 4 mm (triangles) and 7 mm (circles) waveguide diameter 2a. The figure symbols indicate data measured by time domain techniques with deionized water as sample. The f u l l curves result from a modal analysis (ref. 11) of the electrical f i e l d configuration. reference and sample l i q u i d , s(t) and Sr(t ) reach the same stationary value for long times i t ÷ ~). At f i n i t e t the difference between the two reflected waveforms allows for the evaluation of the difference between the corresponding reflection coefficients which w i l l be denoted by R(v) and Rr(V) furtheron. According to transmission line theory the relation 2 Rr - R =
(I
÷
ZoYr)2
Zo(Y - Yr) Zo(y _ yr ) 1+ 1+ Z Y
(8)
or
can be used to express this difference by the input admittances Y = imC and Yr = imCr' respectively. With the Fourier transforms of incident and reflected waveforms denoted by V(v), S(v), and Sr(V), the difference Rr - R can be derived from the experimental data with the aid of the equation Rr - R : (Sr - S)/V This expression can be combined with equ. (8) to give
(g)
23 Sr c - cr
S
( i + ZoYr)2
= ~
i~ZoV
(10)
2 - (1 + ZoYr)(Sr - S)/V
At low frequencies, where the specimen cell can be s u f f i c i e n t l y well represented by a lumped capacitor, the dielectric contribution ~d(~) to the permittivity ~(~i of the solution under test is given by the relation CoZ ~d(V) = Edr(~ ) +
d
(C - Cr)
(11)
At frequencies near the cutoff frequency ~c of the TM01 mode in the waveguide section, the p e r m i t t i v i t y of the sample can only be approximately derived from equ. (11). This result, however, can be u t i l i z e d as i n i t i a l value in an iterative determination of ~d(~) from C - Cr(4) Sensitivity of method and experimental accuracy,. The present TDS study aims at an as sensitive as possible detecton of permittivity contributions which are due to dipolar solute complexes in aqueous electrolyte solutions. The sensit i v i t y in these measurements is strongly dependent on electric properties of the cell as can be shown by consideration of two quantities characterizing the difference signal, namely i t s time integral A and the centre of gravity t of A. A and t follow from the values at v = 0 of the spectrum V(u)(Rr(V ) - R(v)) and of i t s derivative with respect to the frequency, respectively. Assuming an incident step voltage pulse of unit height, one obtains 2to(~d(O ) - ~dr(0)) (12)
A ~
(1 + toO/~o)2 and to(~d(O) + Edr(0)) 1 + to~/~ o
+
(~d(0) - ~d(~))~ - (~dr(O) - ~dr(~))~r
,
(13)
~d(0) - ~dr(0)
where t o = Zod/(ZCo) is the charging time of the empty capacitor representing the cell and T and ~r are the mean dielectric relaxation times of the relaxation time distribution functions for the solution under test and the reference l i q uid, respectively. Here and in the following ~(~) denotes a permittivity extrapolated to high frequencies. The second term on the right hand side of equ. (13) reduces to ~ i f differences between the permittivities of the two liquids are solely due to a solute relaxation process of dispersion step Ed(O) - Cdr(0) and mean relaxation time ~. According to equs. (12) and (13) two demands have to be made on the cell capacitance in order to obtain a high s e n s i t i v i t y in the measurements. On the one hand, to~/~ o = 1 is required to reach a maximum value in the area A. On the other hand, the capacitance has to be small enough to allow the f i r s t term on the right hand side of equ. (13) to be smaller than (or comparable to) the
24
second term. For the l a t t e r one contains the information on the differences between the mean relaxation times of the test and reference liquid. With the 4 mm diameter cell acharging time t o = 3ps is realized. Together with the specific conductivities of the present solutions this value leads to 0.4 ~ toO/Co ~ 0.9. To improve the signal-to-noise ratio and to reduce time referencing errors a great number of repetitive scans were in phase added to one another. Each scan was available as a record of 1024 digitized data points with the outset of the interval of measurement defined by that time at which the step-voltage pulse f i r s t passes the sampling head. As a result of these provisions the time referencing error was reduced to a value in the order of one tenth of a sampling interval. To reduce the influence of unwanted reflections delay lines of suitable length have been inserted between the sampling head and specimen cell as well. Most of the signals resulting from undesired reflections on line discontinuities can be shifted out of the time interval of interest thereby. In most part of the frequency range under consideration (0.01 to 3 GHz), the difference method enables measurements with errors smaller than 2.5 % in e' and 5 % in e" Below 0.1 GHz, however, the uncertainty in the ~ values increases d" with decreasing v. I t may be characterized by Ae~ = ± i at 10 MHz. At low frequencies, the fact that the permittivity data are derived from scans of f i n i t e time intervals, may add an additional uncertainty. But a reliable estimation of truncation errors requires knowledge on the asymptotic behaviour at long times of the functions to be determined. This knowledge, of course, is usually not available. RESULTS AND TREATMENT OF DATA In Fig. 5, the real part, ~ ' ( v ) , and the negative imaginary part excluding conductivity contributions, ~"d(V), of the complex permittivity is displayed as a function of frequency v for the 0.1 molar aqueous solution of AI2(SO4)3 at 25°C and also for pure water at the same temperature. A representation of measured data in the complex c',~-plane is given in Fig. 6 for the 0.05 and 0.15 molar AI2(S04)3 solution at 25°C. As indicated by the different figure symbols used for the 0.1 molar AI2(S04)3 solution (Fig. 5) the ~' and c~-values found by frequency and time domain measurements nicely harmonize. In the microwave (ref. 12) and near millimetre wavelength region (refs. 20, 21) the dielectric spectrum of the pure solvent can be represented (ref. 22) by a DEBYErelaxation spectral function (ref. 23) which is defined by the expressi on ~d(V) = ~w(=) +
Ew(O) -
~w(~)
1 + imTw
(14)
25
100 I
t
t
o-o o- o
t
80
....... r - - %
I
0|
I
t
. J¢(-)--
II
I
=
/i\ ^..o "°'' 10 7
10 8
. ~ + f " (2nTI)-~(Z~T.) -I 10 9
101° Hz
10"
V
Fig. 5. Real part E'(~) and dielectric contributions c~(v) to the negative imaginary part ~"(~) of the complex permittivity displayed as a function of frequency v for water (crosses (ref. 12)) and the 0.1 molar aqueous solution of AI2(S04) 3 (circles) at 25°C. Open and closed circles indicate data from TDS and frequency domain measurements, respectively. The multiplication sign identifies the static permittivity ~w(0) of water (ref. 19). The full curves are graphs of the relaxation spectral function given by equ. (15) with the parameter values presented in Table 2. At 25°C the parameters of equ. (14) have the following values (refs. 12,22): extrapolated high frequency permittivity Cw(=) = 5.16 + 0.5; extrapolated static permittivity ~w(0) = 78.36 + 0.05; relaxation time ~w = (8.27 -+ 0.02) ps. The spectra of the solutions clearly show two dispersion (d~'(v)/dv < 0)/dielectric loss (~(v) > 0) regions, indicating that different relaxation mechanisms are present. The characteristic frequency (2~T1)-1 of one process has nearly the same value (~ 20 GHz) as that of pure water (2~w)-1. Obviously, the
26
40
/
30
="a W
I v
I 27 2 ~ '17 3 -_.~-~,,,+
I
I 12.5 _.
I
I
I
20
_.
10
\ ~ . .
0
10
l/
20
30
40
50
60
|_J-
70 •
t
80
"O_~U.11,
90
100
E'(v) Fig. 6. The dielectric spectra of a 0.05 (o) and 0.15 (e) molar aqueous solution of AI2(S04)3 at 25°C shown in a complex plane representation with the frequency v being parameter ("COLE-plot"). The f u l l curves represent the spectral function defined by equ. (15) with the parameter values found by the f i t t i n g procedure. The dashed lines indicate the subdivision of the complex permittivity into solute and solvent contributions. process with relaxation time ~1 reflects the reorientational motions of the solvent molecules. The relaxation .frequency (2~T2)-I of the other dispersion/absorption region is d i s t i n c t l y smaller (~ 0.7 GHz). We assume this l a t t e r contribution to the dielectric spectrum to be due to dipolar ion complexes. This assumption is confirmed by the plots shown in Fig. 7 where E'(v) is displayed as a function of frequency for the 0.1 molar solution of Sc2(S04)3 and also for an aqueous solution of zwitterionic colamine phosphoric acid with nearly the same relaxation frequency (2~T2)-1. The low-frequency dispersion of the l a t t e r liquid clearly results from reorientations of solute zwitterions. The solute and solvent relaxation process may both be subject to relaxation time distributions. We therefore used the relaxation spectral function ~d(V) = a(=) +
~i - ~(~)
i + (i~T1)(l"hl)
E(0) - ~i
+ .... 1 + (i~T2)(l-h2)
(15)
to analytically describe the measured spectra of the solutions. Using this relation a COLE-C0LE relaxation time distribution G(~) (ref. 25) is assumed to underly both processes. This distribution function, i f
TG(T) is plotted versus
I n ( z / z i ) , i = 1, 2, is symmetrically bell shaped around T/T i = 1, where ~I and T2 denote the principal relaxation times. Parameters hi , i = 1, 2, measure the width of the relaxation time distribution. The values for the parameters of equ. (15) have been found by f i t t i n g the relaxation function to the measured spectra using a nonlinear least-squares regression analysis. The results obtained by
2'/
160
c(O)
~
120
so
W
I
i
L-o-o-o-'----~.o~
/
LO _
-
,
,
80 - . . . .
t--
0 100
~
~(~
(21TT2)-I
--
60
(2n'~z1-I
_
200
107
,
(2~i)-I I ~, I
I
10 s
10 9
IC(')--e~'-~ 101° Hz
101~
Fig. 7. Real part e'(~) of the d i e l e c t r i c spectrum of the 0.1 molar aqueous solution of Sco(SOA)3 (e) and of an 1-molar solution of colamine phosphoric acid (o) at 25°C.~ TABLE 2
A12(S04)3
Solute c c(~) cz •z hz c(0) m2
hz
[mol/l] [ps] [ps]
± ± ± ± ± ±
0.5 0.4 1 % 0.02 2 % 2 %
±100
%
In2(S04) 3 Sc2(S04)3
0.05 3.5 75.5 8.2 0.04 90.2 272
0.1 4.5 71.7 8.3 0.03 92.7 232
0.15 4.4 69.1 8.4 0.04 92.9 213
0.03
0.02
0.07
0.1 4.1 75.0 8.3 0.04 93.1 250
0
0.1 4.6 74.5 8.2 0.02 88.8 232
0.04
Values of the parameters of the relaxation spectral function defined by equ. (15) for the solutions of 3:2 valent salts at 25°C.
this f i t t i n g procedure are collected in Table 2. Applications of relation (15) does not imply that only the C0LE-COLE relaxation time distribution allows for a satisfactory description of the measured
28
relaxation processes. Within the limits of experimental error, however, a decision in favour of one of the commonly used relaxation time distribution functions, including the asymmetrical COLE-DAVIDSON function (ref. 26) is not possible. This finding is i l l u s t r a t e d by the results shown in Fig. 8 where for the Sc2(S04) 3 solution the deviations
6c'(v) : ~ '
~(v)-~'
mea:
calc (v)
(16)
and II
~
~ il
6 d( ) : 8e"(v) :
(17)
il
meas(V) - ecalc(V )
are compared for the two model functions given by the equation E1 - ~(~)
~d(V) : c(~) +
N A~n + g I + (imzl)(1-hl) n=2 1 + im%n
(18)
with N = 2 or 3, respectively. The calculated values ~calc(V) and ~calc " t"v") are those found, i f relaxation (18) with N = 2 or 3 is f i t t e d tothemeasuredspectra.
1
I
I
I
-I W
I
--
0
0 - o o o ~ _ _,_~
-I I
0
0 -1 lO 7
10 s
10 9
lO 1°
Hz
1011
Fig. 8. The deviations ~ ' ( v ) and 6c~(v) as defined by equs. (16) and (17), respectively, plotted versus the frequency v for the 0.1 molar solution of Sc2(S04)3 at 25°C. Different figure syn~ols mark different model relaxation functions: e, N = 3 and o, N = 2 in equ. (18).
2g
I t is clearly demonstrated by Fig. 8 that the 6E'(v) and 6~"(u) values are only insignificantly smaller i f the solute relaxation process i s represented b y two DEBYErelaxation terms (N = 3) instead of a single one (N = 2). Thus a clear-cut conclusion on the presence or absence of a small relaxation time distribution is impossible. DISCUSSION Contribution of solvent water to the permittivit~ The principal relaxation times ~i of the solvent water in the solutions of 3 valent sulfates are found very close to the pure water value ~w = 8.27 ps at 25°C (Table 2). The differences ~1 - Zw are too small to be discussed with respect to specific effects of ionic hydration. Also too small to allow for definite conclusions on solute-solvent interactions are the values of parameter hI. Substantial deviations from the pure water value Cw(O) (= 78.36, 25°C), however, emerge in the solvent contribution to the static permittivity of the solutions. These deviations may be due to at least three different effects, namaly kinetic depolarization, dilution and depolarizing internal electric fields as secondary phenomenon, and structure saturation. Kinetic depolarization (refs. 27-31), f i r s t theoretically predicted by ONSAGER (ref. 32), is due to a coupling between hydrodynamic and dielectric effects which in conducting solutions of polar liquids gives rise to a reduction in the solvent contribution to the extrapolated static p e r m i t t i v i t y . The theory of kinetic depolarization has been confirmed by various experimental facts. Among these facts is the strong dependence of the extrapolated static permittivi ty of sulfuric acid on small admixtures of water which has been considered by HALL and COLE (ref. 33). Water molecules in the f i r s t hydration shell around small multivalent cations have a long residence tima (ref. 34). Within the time scale under consideration i t is therefore adequate to treat small cations together with their f i r s t hydration sphere as one rigid body. The effect of kinetic depolarization can then be considered on the basis of the HUBBARD-ONSAGER continuum model for large ion size. Assuming perfect slip boundary conditions on solvent flow at the ion surfaces this theory predicts the kinetic polarization deficiency 6cHO to be given by the relation (ref. 30) 2 two Cw(O) - Cw(~) 6cHO - 3
co
(19)
Cw(O)
With the present solutions of 3:2 valent salts the aCHO values are small (0.69 ~ a~HO % 1.48). Nevertheless, p e r m i t t i v i t i e s corrected for the kinetic polarization deficiency, ~1 + 6~HO' w i l l be used in the following discussion of the other effects nentioned above.
30 We use the BRUGGEMANmixture formula (ref. 35) given by the expression Va = i
~i + 6EHO - ~e [ -
~w(O) ~I/3
l
~w(0) - ~e
I
(20)
\ e l + a~HO/
to calculate the apparent volume fraction va of spheres with p e r m i t t i v i t y Ce(~2) which seem to be solved in the homogeneous dielectric of permittivity ~w(O). The va values are substantially greater than the volume fractions v of solute. We conclude, that a part of the water around the ions is d i e l e c t r i c a l l y saturated and has thus to be reckoned among the particles with permittivity ~ rather than e the d i e l e c t r i c a l l y homogeneous solvent of ~w(O). Since saturation effects are unlikely to exist around the comparatively big sulfate ion, apparent values Z+ of t o t a l l y i r r o t a t i o n a l l y bound water molecules per cation can be calculated for the 3:2 valent salts using the relation Z+ : (va - v)/(2c @w)
,
(21)
where @w (= 18.069 ml/mol at 25°C (ref. 36))denotes the molar volume of water. Most interesting, remarkably different Z+ values are found for the three 3valent cations, namely 13 for Al 3+, 5 for In 3+, and 2.5 for Sc3+. Let us b r i e f l y consider this result in view of the structure of the dipolar ion complexes assumed to be present in the solutions. The formation of these complexes is usuall y described by the EIGEN mechanism (ref. 37) of stepwise association which may be represented by the relation
Mm+ and LI- are shorthand notations for the metal ion and the anionic ligand, respectively. The high Z+ value for the small Al 3+ ion (r + = 0.5 ~ (ref. 38)) following from the analysis of the dielectric spectra may be taken to indicate, that predominantly solvent-separated ion pairs are formed with this cation, namely outer sphere (Mm+(H20)L l - ) or outer-outer sphere (Mm+(H20)2Ll-) complexes. This conclusion is in agreement with the results derived from ultrasonic absorption spectra of aqueous A12(S04)3 solutions (ref. 39). I t is only b r i e f l y mentioned here, that Z+ = 12.7 (ref. 3) had been found for aqueous solutions of AICI 3, a system for which dielectric spectroscopy yields no indications of ion complex formation. I t has thus to be concluded that in solutions of aluminium sulfate the cation is obviously able to form i t s complete shell of dielect r i c a l l y saturated water. The small Z+ values for the somewhat bigger In 3+ and Sc3+ ions (r + = 0.8 for both (ref. 38)) reflect a weaker bonding of their hydration water. The
31 sulfate ion seems to induce a substantial rearrangement of the hydration structure of those cations. Indium, which has d10 outer electron configuration, shows a similar behaviour also in aqueous solutions of i t s chloride, bromide, and iodide. Z+ values between 3.9 and 6.1 had been evaluated for those salts previously (ref. 3). Relaxation process of the solute A f i r s t of a l l surprising result of our d i e l e c t r i c study is the finding of only one dispersion/dielectric loss region in addition to the water relaxation. On the contrary, the ultrasonic absorption spectrum of aqueous solutions of 3:2 valent electrolytes usually reveals more than one relaxation process. An example is shown in Fig. 9 where for comparison with the d i e l e c t r i c losses, E~(v), the ultrasonic excess absorption per wavelength, ~(v)~ - By, is displayed as a function of frequency for the 0.1 molar solution of Sc2(S04)3. The excess absorption is that part of the total absorption per wavelength, ~ , which is due to chemical relaxation processes. The curve for the scandium sulfate solution clearly contains contributions from three relaxation processes. These can be attributed to the three transitions between the different ionic species represented by equ.
(22).
100
I
I
I
I
I
30 10 o
3
r/
1
lO0,
#
I
\~./ ```` / \ . I/y'. ~\.
I
I
I
I
//
--
``\
\\ \
I
-
\~ I
303 I
I0-
~-;../" .~ / 4 / , \
3-
/
I0.3 10 s
1 10 6
/
//
I/@///l//' 10 7 10 e
Y
(2~T2}-
I 10 9
(2.TJ -1
, \\ ,,
-
I\',, 101° HZ 1011
Fig. 9. The ultrasonic excess absorption ~L - By (o (ref. 39)) and the dielect r i c loss ~ = ~" - o/(Co~ ) (e) plotted versus frequency v for the 0.1 molar solution of Sc2(S04)3 at 25°C.
32
As w i l l be outlined below the difference with respect to the solute relaxations in the two types of spectra results from the fact that ultrasonic spectroscopy probes the association/dissociation processes only while dielectric spectroscopy is also sensitive to the reorientational motions of the dipolar species Let again the 0.1 molar solution of scandium sulfate serve as an example since for that system the concentrations of the outer ("o") and of the outer-outer ("oo") sphere complex relative to that of the inner ( " i " ) sphere complex are known (ref. 39). The dipole moments ~ i ' ~o' and ~oo of the three species can be estimated by geometrical considerations. Using the relation c(O) - ~1 = Aci + Aeo + Aeoo (23) 2 c 2 2 ~ci~ i + oUo + Coo~oo the following estimates are obtained for the contributions of the d i f ~ r e n t ion complexes to the static permittivity of the Sc2(S04)3 solution: Aei/(e(O ) - el) Aeo/(~(O ) - ~I)
= 0.44 = 0.19
(24)
aeool(~(O) - e l ) : 0.37 Also by geometrical considerations and based on the DEBYE model ( r e f . 23), which predicts the d i e l e c t r i c ~ l a x a t i o n time m r the reorientational motions of a spherically shaped dipolar p a r t i c l e to be proportional to the t h i r d power of i t s radius, the following reorientation time ratios can be estimated: Zo/Ti = 1.3 Too/Zi = 1.6
(25)
These ratios are by far too small to allow in the dielectric spectrum for a clear separation of the different relaxation processes. To i l l u s t r a t e this statement a COLE representation is given in Fig. 10 for the resulting curve of a superposition of three DEBYEterms. The parameters of these terms follow equs. (24) and (25). Deviations from the DEBYEsemicircle also indicated in that plot are smaller than the usual experimental errors. Dielectric spectroscopy has proven to be a powerful method in detecting dipolar ion complexes including very short-lived species. I t is, however, l i t t l e suited for the identification and study of different s i m i l a r l y sized complexes which might be together present in a solution. Nevertheless, a global statement can be made on the ion complexes of 3:2 valent salts which support our discussion in the preceeding section. The dispersion steps ~(0) - c I of the 0.1 molar solutions of aluminium, indium, and scandium sulfate are 21.0, 18.1, and 14.3, respectively. These data seem to also indicate, that the concentration of
33
contact ion pairs increases in the above series of cations.
I 0.5
I
1.58 _
n
I
0.89 ~ ,
_ -
/
%o.s
8. 9e~/
•
! 0
,
0.5
.16
to.o. 1.0
¢'(v) Fig. 10. Complex plane representation of the spectral function ~(~) = 0.44/ (i + i f ) + 0.19/(1 + i 1.3f) + 0.37/(1 + i 1.6f) ( f u l l curve). The dashed semicircle is the graph of a simple DEBYE relaxation term (f = 2 ~ i ) . ACKNOWLEDGMENT We thank Professor R. Pottel for valuable discussions.
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