JOURNAL OF CHEMICAL PHYSICS
VOLUME 116, NUMBER 19
15 MAY 2002
Dielectric relaxation of electrolyte solutions using terahertz
transmission spectroscopy
M. L. T. Asaki, A. Redondo, T. A. Zawodzinski, and A. J. Taylor
Materials Science and Technology Division and Theoretical Division, Los Alamos National Laboratory,
Los Alamos, New Mexico 87545
共Received 9 July 2001; accepted 20 February 2002兲
We use terahertz 共THz兲 transmission spectroscopy to obtain the frequency dependent complex
dielectric constants of water, methanol, and propylene carbonate, and solutions of lithium salts in
these solvents. The behavior of the pure solvents is modeled with either two 共water兲 or three
共methanol and propylene carbonate兲 Debye relaxations. We discuss the effects of ionic solvation on
the relaxation behavior of the solvents in terms of modifications to the values of the Debye
parameters of the pure solvents. In this way we obtain estimates for numbers of irrotationally bound
solvent molecules, the numbers of bonds broken or formed, and the effects of ions on the
higher-frequency relaxations. © 2002 American Institute of Physics. 关DOI: 10.1063/1.1468888兴
I. INTRODUCTION
is split into two pulses; one pulse excites a dipole antenna
with a 50 ␮m, low-temperature-grown GaAS 共LT-GaAS兲 gap
and the other pulse traverses a variable delay line and gates a
dipole receiver which has a 10 ␮m gap on silicon on sapphire. The THz radiation from the emitter is collimated and
focused, resulting in a 1-cm-diam waist; the sample is placed
at this focal spot. After traversing the sample, the THz beam
is focused onto the dipole receiver. The delay line in the gate
beam is scanned, yielding the temporal waveform of the
electric field of the transmitted THz pulse, as shown for pure
methanol in Fig. 2共a兲. The resultant usable bandwidth of this
THz spectrometer ranges from about 5 to 30–50 cm⫺1.
For each transmission measurement, a liquid sample
共pure solvent or electrolyte solution兲 was loaded into a 100
␮m pathlength crystalline quartz cell that was placed in the
focused beam. Immediately after the sample transmission
data was obtained, the cell was emptied, flushed, and dried in
situ, and a reference transmission scan was taken using the
empty cell. All data were taken at room temperature. In between sets of differing salt-solvent combinations, the cells
were washed with acetone, methanol, and distilled water in
succession, and then baked at 140 °C for a minimum of 30
min before loading new samples.
Such a coherent measurement of the transmission of the
THz pulse through the sample, as described above, directly
yields both parts of the complex dielectric function 关or the
absorption coefficient and index of refraction, as shown in
Fig. 2共b兲兴 over the frequency bandwidth of the THz spectrometer in the following fashion. The complex transmission
function in the frequency domain T( ␻ ) is obtained from
time-domain scans of the full and empty cell by fast Fourier
transforming the scans and dividing the results for the full
cell by those for the empty cell. An expression for T( ␻ ) has
also been derived from a multilayer dielectric model which
describes the transmission and reflection at each interface for
both the full and empty cell in terms of the known complex
dielectric constant of the cell and the cell gap when it is
empty, and the unknown dielectric constant ␧共␻兲 of the
The dynamical response of a solvent to a solute still
remains a poorly understood but important scientific problem. A large number of important processes such as ion association, dissociation, transport, and charge transfer reactions occur in liquids, and all are significantly influenced by
the solvent dynamics. A step towards understanding solvation dynamics is the determination of the dielectric response
of pure solvents to far-infrared radiation, which probes molecular motions that occur on time scales of 1–10 ps. This
can then be followed by the determination of the dielectric
response of those same solvents when they are perturbed by
electrolytes.
Using the relatively new technique of femtosecond terahertz 共THz兲 time-domain transmission spectroscopy,1 the accurate determination of the complex dielectric function over
the frequency range from 5–50 cm⫺1 is relatively straightforward. Such measurements have been performed for polar
and nonpolar solvents including water, alcohols, carbon tetrachloride, liquid benzenes, as well as various solvent
mixtures.2–10 In this work, we obtain corroborating results
for pure water, methanol, and propylene carbonate, and extend our attention to lithium salts dissolved in them. As in
previous studies,4,11,12 we use the Debye-based dielectric relaxation model which describes the dielectric relaxation as
diffusive reorientational motions in the liquid.
To our knowledge, we present the first THz spectroscopy
study of ions in solution. Our use of THz frequencies allows
us to directly probe the higher-frequency relaxations of the
solvents, and in turn, the effect of the presence of ions on
these relaxations. Additional results were obtained from analyzing the behavior of the main 共low-frequency兲 Debye
relaxation.
II. EXPERIMENTAL SETUP
Our experimental set up is a standard femtosecond THz
time-domain transmission spectrometer1 共Fig. 1兲. In this apparatus, the 800 nm, 100 fs output of a Ti:sapphire oscillator
0021-9606/2002/116(19)/8469/14/$19.00
8469
© 2002 American Institute of Physics
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8470
Asaki et al.
J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
TABLE I. Solvent molecules available per ion pair for given approximate
molarities.
⬃Solvent molecules per ion pair
FIG. 1. THz setup: generation and detection.
sample.13 This treatment includes the Fabry–Perot term describing multiple reflections in the cell gap. For each value of
␻, we equated this expression to the experimentally determined value for T( ␻ ) and iteratively solved for ␧共␻兲 in the
sample.
We took data for pure water, methanol, and propylene
carbonate, as well as for solutions of three lithium salts in
these three solvents: lithium triflate 共lithium trifluoromethanesulfonate兲,
lithium
imide
共lithium
bistrifluoromethylsulfonimide兲, and lithium chloride. Lithium
chloride was not soluble in PC at the molarities we chose.
Pure water samples and water used in aqueous solutions was
obtained from a Barnsted NanoPure Water Purification System. All methanol, pure or in solution, was HPLC grade,
purchased from Fisher Scientific and used without further
Approx. molarity
H2 O
MeOH
PC
0.1
0.2
0.5
1.0
2.0
3.0
5.0
555
310
110
55
27
19
10
250
110
50
25
12
8
25
11
6
4
purification. Likewise, anhydrous propylene carbonate 共less
than 0.005% water兲 was purchased from Aldrich and used
without further purification. All salts used in preparing solutions were dried for a minimum of 48 h. Dried salts were
then stored in an argon-atmosphere dry box.
Solutions were mixed by weight; all solutions in PC
were prepared in the argon-atmosphere dry box, while solutions in water and methanol were prepared in open atmosphere using salts just removed from the dry box. Solution
concentrations ranged from approximately 0.1 M to close to
saturation 共0.5 M and higher depending on the salt/solvent
combination兲. The approximate number of solvent molecules
available per ion pair for given molarities is shown in Table
I. These ratios were obtained by simply dividing the grams
of the solvent in a solution by the molecular weight of the
solvent, and likewise dividing the grams of salt in the solution by the formula weight of the salt, to yield the total
number of solvent molecules and ion pairs in a given solution.
III. MODELING THE PURE SOLVENT DATA
A. The Debye model
The extraction of dynamical information from these frequency domain measurements of ␧共␻兲 depends on the model
chosen to describe the dielectric function. The most commonly used model for dielectric relaxation in the GHz frequency regime is the Debye model which assumes that the
polarization induced by applying an external electric field
relaxes exponentially to equilibrium, as exp(⫺t/␶), once the
field is turned off.14 The time constant ␶ for this relaxation
process is the Debye relaxation time. For a given medium,
one or more Debye-type relaxation processes are possible.
For multiple Debye-type relaxation processes, the complex
frequency-dependent dielectric function ␧共␻兲 has the form4
n
␧៝ 共 ␻ 兲 ⫽␧ ⬁ ⫹
FIG. 2. 共a兲 Example waveforms for cell with a pure methanol sample and
empty. 共b兲 Real and imaginary parts of the complex index of refraction, n
and k, as functions of frequency for pure methanol.
␧ ⫺␧
j
j⫹1
兺
1⫺ ␣ ␤ ,
1⫹
i
␻
␶
兲
关
共
兴
j⫽1
j
j
j
共1兲
where ␧ ⬁ is the real part of the dielectric constant at the
high-frequency limit, and ␧ 1 is the low-frequency limit of the
real part of the dielectric constant, also know as the static
dielectric constant (␧ static); the ␧ j are intermediate values of
the real part of the dielectric constant; ␶ j is the Debye relaxation time that corresponds to the jth relaxation process; ␣ j
and ␤ j can both take on a range of values that affect the
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J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
Dielectric relaxation of electrolyte solutions
8471
TABLE II. Values of parameters determined by fitting experimental data to the multiterm Debye model.
Numbers in parentheses refer to the standard deviation in units of the least significant digit used.
␧ static
Solvent
a
H2 O
H2 Ob
H2 O
Present work
PCc 共GHz only兲
PC
Present work
MeOHa
MeOHb
MeOH
Present work
77.97
78.36
共constant兲
78.96
共6.94兲
64.7
65.43
共3.53兲
32.50
32.63
共constant兲
32.59
共2.83兲
␶1
共ps兲
␧2
␶2
共ps兲
8.32
8.24共40兲
6.18
4.93共54兲
1.02
0.18共14兲
4.59
3.48共70兲
7.87共79兲
4.80共10兲
0.18共03兲
3.42共18兲
40.4
38.27
共3.10兲
51.5
48共4兲
7.46
5.41共07兲
6.91
1.24共02兲
3.35共04兲
0.21共01兲
4.41
1.86共04兲
5.91
5.35共41兲
7.09
1.25共55兲
4.90
3.37共44兲
1.12
0.16共11兲
2.79
2.10共45兲
38.76
共1.71兲
4.82共24兲
1.06共17兲
2.98共12兲
0.22共03兲
1.98共04兲
␧3
␶3
共ps兲
␧⬁
a
Reference 11.
Reference 4.
c
Reference 16.
b
characteristics of relaxation.14 For purely exponential 共Debye兲 relaxation, ␣ j ⫽0 and ␤ j ⫽1. If ␣ j is allowed to have a
value between 0 and 1 共0⭐ ␣ j ⬍1, and ␤ j ⫽1兲, then we obtain the Cole–Cole equation. In this case, the results are
similar to the simple Debye model, only now there is a symmetric distribution of relaxation time centered on the discrete
␶ j of the Debye model. Likewise, if ␤ j is allowed to vary
( ␣ j ⫽0 and 0⬍ ␤ j ⭐1兲, we obtain the Cole–Davidson equation which describes an asymmetric distribution of relaxation
times. Finally, if both ␣ j and ␤ j are allowed to vary as mentioned above, then the Havriliak–Negami equation is obtained which incorporates characteristics of both the Cole–
Cole and Cole–Davidson equations.
B. Applicability of the Debye model
Although doubt has previously been cast as to whether
the Debye equations are applicable at frequencies15 above 10
cm⫺1, subsequent analysis of pure solvents in the THz frequency range4 revealed Debye-like behavior out to ⬃35
cm⫺1. Before examining solutions of salts in various solvents, we measured the complex dielectric functions of pure
water, methanol, and propylene carbonate, to demonstrate
that our results are comparable to those obtained previously
by other researchers. For each solvent, we found that the
quality of our fits was not improved by varying ␣ j and ␤ j of
Eq. 共1兲, from 0 and 1, respectively. As a consequence, we
kept ␣ j and ␤ j fixed at those values. Table II tabulates our
Debye parameters and those of others for all three pure solvents.
It should be noted that although our frequency range of 5
to 30–50 cm⫺1 did not cover the main 共low-frequency兲 relaxation for the pure solvents, we were still able to reproduce
the results of other workers. By fitting the pure methanol
data, using Eq. 共1兲 with three terms ( j⫽3), we were able to
match the values reported by others for almost all parameters, within a standard deviation. Figure 3 shows a Cole–
Cole plot of experimental data and the corresponding fit for
pure methanol.
Likewise, by fitting water with two relaxations, we reproduce the results of other researchers in the THz frequency
range.4,10 There is some evidence of three relaxations in propylene carbonate,16 and Bayesian statistical analysis of our
pure PC data corroborated this assumption.17 We therefore
used three relaxations for PC instead of two. This allowed us
to obtain values in good agreement with those of Barthel
et al.16 for the main low-frequency term. However, because
of our lack of data in the GHz range, the main term parameters show large standard deviations 共as high as 30%– 40%兲.
This uncertainty is greatly reduced by fixing the higherfrequency parameters at their best fit values and varying only
␧ static and ␶ 1 during a second iteration of fitting. The resulting standard deviations of ␧ static and ␶ 1 are in the range of
5%–10%.
C. Correlating the Debye model to the physical
environment in the solvent
To what molecular motions do the Debye relaxations in a
pure solvent correspond? A number of motions occur as the
frequency sweeps from the gigahertz up through the terahertz
region: ion pair relaxation and electrolyte conductance occur
roughly at the tens of GHz level and below, while molecular
FIG. 3. Cole–Cole plot for pure methanol, plus the fit to the data.
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8472
Asaki et al.
J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
reorientation encompasses the range from tens of GHz all the
way up into the THz range.18 Our frequency range of 0.2 to
1 THz puts us in an ideal range to probe the faster molecular
solvent motions, as these frequencies correspond to relaxation times from about 0.8 ps down to 0.15 ps.
There are three types of solvent relaxations which we
anticipate finding at the frequencies we use to study pure
polar solvents19: 共1兲 the main relaxation 共low frequency兲
which corresponds to cooperative relaxation of the bulk solvent, 共2兲 large-angle rotations of ‘‘free’’ or single solvent
molecules, and 共3兲 small translations 共characteristic distances
much smaller than a molecular diameter兲 and small rotations.
For example, in methanol, the main relaxation corresponds
to flexing and/or reorientation of chains of molecules. The
second relaxation corresponds to the rotation of a chain-end
or free methanol molecule. The fastest relaxation is characteristic of hydrogen bonding. For the three solvents which
we studied, collective motions of the main relaxation should
occur on a time scale of tens of picoseconds; the inertial
rotations of single solvent molecules should occur on time
scales of several picoseconds; and the small motions should
occur on time scales of hundreds of femtoseconds.
As already pointed out in the previous section, our experimental data indicate that there are three relaxations for
methanol and for PC, and two relaxations for water. Do these
results make sense in light of what we might expect? In
methanol, a solvent which forms hydrogen-bonded
chains,20–22 we can assign the main relaxation to the flexing
of chains of molecules to accommodate the need of a molecule at a chain end to rotate; the pure solvent data yield
␶ 1 ⫽38.76 ps for this process. The second relaxation process,
with ␶ 2 ⫽1.06 ps, can be correlated to the rotation of a chainend molecule 共or ‘‘free’’ molecule兲. And the third relaxation,
with ␶ 3 ⫽0.22 ps, would correspond to the small motions of
a methanol molecule between two hydrogen-bond acceptor
sites.15,19
Water, also a hydrogen-bonded liquid, shows only two
relaxations instead of three. This is because the motions of
water during cooperative relaxations are indistinguishable
from the rotation of free water molecules.23,24 The fits to our
data lead to values for the main and second relaxations of
water of ␶ 1 ⫽7.87 ps and ␶ 2 ⫽0.18 ps.
Even though PC is a nonhydrogen-bonded liquid, at
room temperature it shows three relaxations similar to those
in methanol. The main relaxation can be attributed to cooperative reorientation of the correlated dipoles.25 Our experimental results for the corresponding Debye relaxation time is
␶ 1 ⫽38.27 ps. The second relaxation can be assigned to the
rotation of single solvent molecules as in methanol. We obtained a value of 1.24 ps for ␶ 2 . Finally, the third relaxation,
with a value of ␶ 3 ⫽0.21 ps, is due to the small, fast rotations
and translations, as in the other two solvents.
It is interesting to note that, excepting the merging of ␶ 1
and ␶ 2 in water, the corresponding relaxation values for different solvents are similar. Experimental values for parameters for all solvents and solutions can be found in the Appendix.
FIG. 4. Cole–Cole plots showing individual relaxation terms, as defined by
Eq. 共1兲 for j⫽1, 2, 3, and the corresponding ⌬␧ j for pure methanol and 1.0
M lithium triflate in methanol. Values of ⌬␧ j change substantially with the
addition of ions.
IV. EXPERIMENTAL RESULTS AND DISCUSSION
FOR ELECTROLYTES
When we add a salt to a solvent, we expect to observe
modifications of the Debye parameters because of the interactions of the salt ions with the solvent. We now examine
how the addition of salts affects each of the two or three
relaxations present in the pure solvent in terms of changes to
the Debye parameters of the solvent. We also propose possible mechanisms by which the observed changes take place.
In addition, we will compare how the same parameters vary
from salt to salt and from solvent to solvent.
For all relaxations, ⌬␧ j , defined as ⌬␧ j ⫽␧ j ⫺␧ j⫹1 ,
gives a measure of the population participating in the jth
relaxation. A decrease or increase in ⌬␧ j indicates a decrease
or increase in the ability of molecules to participate in that
relaxation motion. For example, ⌬␧ 1 for an electrolyte solution is smaller than ⌬␧ 1 for the pure solvent because in the
electrolyte solution, fewer solvent molecules are able to participate in the main relaxation. This is shown graphically in
Fig. 4 where the ⌬␧ j for pure methanol and a 1.0 M solution
of lithium triflate in methanol are shown. Clearly, ⌬␧ 1 for
the solution is much smaller than for the pure solvent. It is
also evident from Fig. 4 that ⌬␧ 2 and ⌬␧ 3 exhibit small
changes due to the solvation of ions.
In a similar manner to ⌬␧ j , changes in the ␶ j give information about the structure of the solvent. The speed at
which the jth relaxation takes place can either decrease or
increase.
A. Main term
1. Dielectric decrement of the main relaxation term
Several research groups have shown23,24,26 that the solvent ‘‘static’’ dielectric constant, i.e., the zero-frequency response of the solvent in a solution, ␧ static-solvent should decrease with increasing salt concentration for all salt-solvent
combinations. 共Note: the solvent ‘‘static’’ dielectric constant
is distinctly different from the solution static dielectric constant which may include a low-frequency relaxation term due
to ion pair relaxation.兲 This decrease is attributable to the
combination of three effects: 共1兲 Irrotational bonding of solvent molecules around ions due to the high field strength at
ionic surfaces which prevents solvent molecules in the first
solvation shell 共or beyond兲 from responding freely to an applied external field; 共2兲 excluded volume effects, which refer
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J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
Dielectric relaxation of electrolyte solutions
to the dilution of the polar solvent molecules—in other
words, ions 共with dielectric constant ⬃2兲 take up volume
which would otherwise be occupied by solvent molecules
共with bulk dielectric constant ranging from 32 to 78, in our
case兲; 共3兲 kinetic depolarization due to the relative motion 共in
opposite directions兲 between solvent molecules and ions in
an applied field; this motion causes friction which increases
viscosity thereby decreasing the ability of solvent molecules
to respond to an applied field.
The frequencies used for data acquisition were high
enough to make the contribution of conductivity to the measured permittivity negligible. Thus, when we fit the data with
Debye parameters, we obtained satisfactory values for the
zero-frequency solvent dielectric constant. Indeed, it was
necessary to include the zero-frequency solvent dielectric
constant in order to fit the data.
a. Irrotationally bound solvent molecules. We use the
formalism of Haggis et al.,27 based on the dielectric theory
of mixtures, to calculate the number of ‘‘irrotationally
bound’’ solvent molecules28 associated with an ion pair in
solution. The relationship between the zero-frequency solvent dielectric constant of a solution and the concentration is
given by
␧ solution⫽␧ solvent⫺ ␦ ␧ c,
共2兲
where c is the concentration of ions in moles per liter, and
␧ solution and ␧ solvent refer to the zero-frequency dielectric constant of the solvent in solution and the pure solvent, respectively, and
␦ ␧ ⫽ ␤ 关 ␧ solvent⫺␧ ion兴 V ion
⫹ ␤ 关 ␧ solvent⫺␧ ⬁solvent兴 V solventn irr ,
共3兲
where ␤ is a shape factor set equal to 1.5 as in Ref. 27, ␧ ion
is the dielectric constant in the space occupied by the ions,
V ion and V solvent refer to the partial molar volumes of the ion
pair and solvent molecules, respectively, and n irr is the number of irrotationally bound solvent molecules for which we
solve the equation; and all other ␧ are as defined above. The
subscript ␧ distinguishes ␦ ␧ from the decrement in ␶, ␦ ␶
which appears later. Haggis et al. found that Eq. 共2兲 is applicable to salt solutions in water up to 1.5 M, therefore we
restrict our use of this equation to concentrations lower than
this limit. If we consider the expression for the dielectric
decrement or slope ␦ ␧ given in Eq. 共3兲 term by term, we see
the factors that contribute to the decrease of the zerofrequency solvent dielectric constant. The first term on the
right-hand side of Eq. 共3兲 represents the loss of the contribution of solvent molecules that have been replaced by the
solute, plus the contribution of the solute particles. The second term on the right-hand side of Eq. 共3兲 accounts for the
change in contribution of solvent molecules which go from
the bulk state of the liquid to being irrotationally bound to
the ions. The results for n irr from our data, using Eqs. 共2兲 and
共3兲, are given in column 1 of Table III. Table III also contains, in columns 2 and 3, numbers for solvent molecules in
association with ions found by different methods; the last
two columns contain the decrement value for the ␶ of the
8473
TABLE III. The calculated number of solvent molecules associated with the
ion pair of a given salt. n irr is the number of irrotationally bound solvent
molecules determined using Eqs. 共2兲 and 共3兲. n EM is the number of associated solvent molecules estimated with the use of effective medium theory as
given in Eq. 共4兲. n G98 is the number of associated solvent molecules predicted with the use of semiempirical and ab initio quantum chemical calculations. ␦ ␶ refers to the initial linear slope derived from plots of ␶ 1 vs
increasing concentration, and ‘‘No. of bonds’’ indicates the calculated number of bonds that are broken or formed per ion pair using Eqs. 共8兲 and 共9兲.
Solvent
H2 O
MeOH
PC
Li
Li
Li
Li
Li
Li
Li
Li
Salt
n irr
n EM
n G98
␦␶
No. of
bonds
triflate
imide
chloride
triflate
imide
chloride
triflate
imide
8.5
15.5
4.5
13.5
11.5
19
3.0
1.5
⬃6
⬃18
⬃4.5
⬃12.5
⬃10.5
⬃20
⬃2.5
⬃1.5
7
10
6
7
8
6
4
4
⫺1.00
⫺2.79
⫺1.16
⫺8.37
⫺11.90
⫺40.18
⫹13.82
⫹4.63
⫺4.16
⫺11.62
⫺4.68
⫺0.60
⫺0.88
⫺3.04
⫹4.23
⫹1.39
main term and the number of predicted bonds broken or
formed. These columns will be discussed below.
b. Bound solvent molecules using effective medium
theory. We also compare the predictions of effective medium theory29,30 to our experimental data. We use this approach to model our solutions by a simple two-component
system—a large liquid volume with a dielectric constant ␧ A ,
in which is embedded a number of small regions of dielectric
constant ␧ B , corresponding to the ions with associated solvent molecules. The volume fractions of the two components
are given by ␩ A and ␩ B where ␩ A ⫹ ␩ B ⫽1. We use the low
concentration limit for which ␩ B Ⰶ1, and as mentioned
above, ␧ B is the dielectric constant of the cluster which consists of the ion/ion pair plus the associated irrotationally
bound solvent molecules. Then the dielectric constant of the
solution is given by29,30
冋
冉
冉
1⫹2 ␩ B
␧ solution⫽␧ solvent
1⫺ ␩ B
冊
冊
␧ ion⫹cluster⫺␧ solvent
␧ ion⫹cluster⫹2␧ solvent
␧ ion⫹cluster⫺␧ solvent
␧ ion⫹cluster⫹2␧ solvent
册
. 共4兲
Unlike the model defined by Eqs. 共2兲 and 共3兲, in which
the dielectric constants of the pure solvent, the irrotationally
bound solvent molecules, and the solute particles are all considered in determining the total solution dielectric constant,
an effective dielectric constant ␧ B is used to define the ion共s兲
and their irrotationally bound solvent molecules.
For our calculations, we used ␧ A ⫽␧ static for the pure
liquids, and ␧ B equal to a volume-fraction-weighted average
of the solute dielectric constant (␧ ion⫽2) and the infinite
frequency dielectric constant of the solvent molecules,
␧ B ⫽␧ ion⫹cluster⫽
V solvent•n•␧ ⬁solvent⫹V ion•␧ ion
,
n•V solvent⫹V ion
共5兲
where n is the number of solvent molecules in a cluster and
V solvent and V ion are the molar volumes of the solvent and ion
pair, respectively. Figure 5 shows experimental decrements
in the zero-frequency solvent dielectric constant plotted
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8474
J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
Asaki et al.
FIG. 5. Calculated zero-frequency solvent dielectric constants for fixed numbers of irrotationally bound solvent molecules using effective medium theory.
共a兲–共c兲 show salts in water, 共d兲–共f兲 show salts in methanol, and 共g兲 and 共h兲 shows salts in propylene carbonate. The number of bound solvent molecules n is
shown for each line, and those marked (n irr) and 共G98兲 refer to values obtained from Eqs. 共2兲 and 共3兲 and from GAUSSIAN 98, respectively.
along with the predictions of effective medium theory for
multiple cluster sizes. A predicted decrement is plotted for
each value of n chosen. Those values of n subscripted 共irr兲
and 共G98兲 refer to values obtained from the data using Eqs.
共2兲 and 共3兲 and from GAUSSIAN 98, respectively. In Fig. 5,
plots 共a兲–共c兲 show results for salts in water, 共d兲–共f兲 show
salts in methanol, and 共g兲 and 共h兲 show salts in propylene
carbonate. It can be seen in Fig. 5 that effective medium
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J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
Dielectric relaxation of electrolyte solutions
theory approximates the data only for low concentrations,
i.e., when ␩ B Ⰶ1, as discussed above. The approximate values of irrotationally bound solvent molecules predicted by
effective medium theory are given in column 2 of Table III.
Both of the methods we use to estimate the number of
irrotationally bound solvent molecules assume that irrotational bonding is the major effect, while also taking into
account excluded volume effects. Kinetic depolarization is
neglected, for this effect is expected to be quite small at low
concentrations. At higher concentrations, the conductivity, on
which the kinetic polarization depends, drops off sharply
thereby keeping this term small.23
c. Quantum chemical calculations of associated solvent
molecules. Finally, we also use semiempirical and ab initio
quantum chemical calculations to predict the optimal number
of solvent molecules associated at low concentration with the
cation (Li⫹ ) and with each anion in each solvent. Although
these coordination numbers are not directly comparable with
the values for n irr calculated from the data,24,28 they are still
of interest because they set a lower limit for the values of
irrotationally bound solvent molecules obtained from analyzing the data.
The semiempirical calculations were performed with the
GAUSSIAN 98 package using the parameterized model, revision 3 共PM3兲.31 Additional ab initio calculations were also
performed on select ion-solvent combinations as a spot
check to the PM3 results. These were done with GAUSSIAN 98
using the B3LYP hybrid32 density functional theory approximation and the 6-31G* basis set which includes polarization
functions.
To do the calculations for a particular ion–solvent combination, we first optimized the geometry for the ion and a
single solvent molecule, then for the ion and two solvent
molecules, and so on. We then tabulated the energies for all
configurations, i.e., for the ion alone, for a solitary solvent
molecule, and for the ion surrounded by a range of solvent
molecules, from one to six or more. We used these energies
to obtain optimum association numbers.
The algorithm we used to determine the optimum numbers of solvent molecules associated with each ion is as follows. We considered two interaction energies:
共1兲 The average interaction energy per solvent molecule
for the optimized geometry of one ion surrounded by n solvent molecules,
E⫽
共 E separated⫺E cluster兲
,
n
共6兲
where E separated is the energy when the solvent molecules and
ion are infinitely removed from each other, and E cluster is the
energy when they are in a cluster of optimized geometry.
共2兲 The average interaction energy between the ion and
each solvent molecule when the energy of solvent molecules
interacting with each other has been subtracted,
E⫽
共 E cluster⫺E cluster-ion⫺E ion兲
,
n
共7兲
where E cluster is as above, E cluster-ion is the energy due to the n
solvent molecules plus their associations with each other, and
E ion is the energy due to the ion.
8475
FIG. 6. 共a兲 Four methanol molecules tetrahedrally associated with Li⫹ , 共b兲
five methanol molecules with Li⫹ —four of them closely associated, and one
hydrogen-bonded in a second shell.
We found that the first bound solvent molecule had the
highest interaction energy, followed by the second solvent
molecule, and so on. But, typically, there was a significantly
larger decrease in interaction energy with the addition of the
‘‘nth’’ solvent molecule. For example, the addition of the
fifth methanol molecule to a Li⫹ -methanol cluster resulted in
a significantly lower average interaction energy than for four
methanol molecules for both types of interaction energy. The
optimum association number corresponded to the number of
solvent molecules just before this dropoff in the interaction
energy.
Our choice of optimum association numbers was confirmed by visual inspection of the optimized geometries,
such as those shown in Fig. 6. Figure 6共a兲 shows four methanol molecules tetrahedrally associated with Li⫹ , while Fig.
6共b兲 shows the lithium cation with five methanol
molecules—four of them closely associated, and one hydrogen bonding in a ‘‘second shell.’’ Thus, the optimized geometries were consistent with significant drops in the two types
of interaction energy per solvent molecule. These drops in
energy corresponded to adding solvent molecules which
were in the ‘‘second solvation shell’’ or even farther away
from the ion.
The calculations yielded four solvent molecules in association with the Li⫹ cation as the optimal arrangement in all
three solvents used, in agreement with published calculations
and simulations for the lithium cation in water,33–35
methanol,36,37 and propylene carbonate.38,39 The results for
the cation and anion were combined to give the total expected first shell population per ion pair; these results are
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8476
Asaki et al.
J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
shown in column 3 of Table III. Table III reveals an excellent
agreement between the results for the number of irrotationally bound solvent molecules determined by the three different methods.
2. Relaxation decrement of the main term
As with ␧ static-solvent the relaxation time ␶ 1 of the main
term solvent relaxation, shows a concentration dependence.
Unlike the zero-frequency solvent dielectric constant, which
decreases with increasing concentration, ␶ 1 can either decrease or increase depending on the nature of the solvent.40 A
decrease in ␶ 1 would indicate a loss of structure in the solvent 共such as the breaking of hydrogen bonds in a hydrogenbonding solvent兲 whereas an increase in ␶ 1 would point to an
increase in structure that slows down the response time of the
solvent to an external field.41
We observed a decrease in ␶ 1 in water and methanol,
while an increase was seen in propylene carbonate. This is in
agreement with our expectation that hydrogen-bonded solvents should experience a randomization of structural order,
while polar, nonhydrogen-bonded propylene carbonate
should experience an increase in local order. In water, ␶ 1
appears to be dependent on the average time that an individual water molecule remains in the bulk state, i.e., with an
average of four hydrogen bonds, before reaching a state with
only one 共or zero兲 hydrogen bonds, which allows it to
rotate.24 With the addition of ions, the bonding structure is
disrupted by the solvation of the ions, so the average number
of hydrogen bonds between solvent molecules decreases.
This is reflected in the decrease in time that it takes for a
bulk solvent molecule to reach the state in which it can rotate.
The situation with methanol is slightly different. The
first relaxation in methanol is assigned to flexing of long
chains of molecules. The addition of ions to methanol is
thought42 to reduce chain–chain interactions 共i.e., branching兲
and reduce the effective average chain length, thus reducing
the reorientation time of the relaxing chains.
Propylene carbonate exhibits the opposite response of
water and methanol. The addition of salts to this
nonhydrogen-bonded solvent increases ␶ 1 . A percentage of
the polar molecules lose their capacity for the dipole–dipoledominated motions, which govern reorientation in the pure
bulk solvent, by forming associations with Li⫹ . The triflate
and imide anions are basically unsolvated in PC.43 Figure 7
shows the behavior of ␶ 1 as a function of concentration for
all of our salt-solvent combinations; plot 共a兲 shows results
for salts in water, 共b兲 shows salts in methanol, and 共c兲 shows
salts in propylene carbonate. Column four of Table III gives
the decrement or increment of ␶ 1 .
Using the decrement or increment of ␶ 1 , we can make a
rough estimate as to how many bonds are being formed or
broken in the solvent per ion pair. According to Hasted and
Roderick,42 the rate of change of ␶ 1 as a function of concentration 共at low concentrations兲 can be written as
␶ solution⫽ ␶ solvent⫹ ␦ ␶ c,
共8兲
FIG. 7. Debye relaxation time ␶ 1 for the main term as a function of concentration. 共a兲 ␶ 1 values in water, 共b兲 ␶ 1 in MeOH, and 共c兲 ␶ 1 in PC. Representative error bars are shown.
where once again c is the molar concentration of ions and
␶ solvent and ␶ solution refer to the main solvent relaxation time
of the pure solvent and the solution, respectively. Hasted and
Roderick also use
␶ solution
n sp%
⫽
,
␶ solvent n s 共 p⫹⌬p 兲 %
共9兲
where n sp% and n s(p⫹⌬p)% are the numbers of solvent molecules with s hydrogen bonds at different percentages p% of
broken hydrogen bonds. Here, n sp% is the number of solvent
molecules that on average have less than the optimum number of hydrogen bonds. In the denominator of Eq. 共9兲,
n s(p⫹⌬p)% refers to the same situation when salt has been
added to the pure solvent and a greater percentage of molecules have less than the optimum number of hydrogen
bonds. For example, in water approximately 70% of the water molecules are 4-bonded to other waters at room temperature, the remaining 30% averaging fewer bonds. So, n sp%
would correspond to 30% of available water molecules in the
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J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
pure solvent, and n s(p⫹⌬p)% would correspond to slightly
more than 30% once ions have been introduced because
some of the tetrahedrally bonded waters are now part of solvation shells of the ions. At a given concentration, the total
number of bonds in the solution can then be compared to that
of the pure solvent, and the estimate for the number of broken bonds can be made.
We approach the analysis of the ␶ 1 decrement in methanol in the same fashion as we did with water, however, for
the same molarity, the number of bonds in methanol is only
about 14 of that in water, and this is taken into account in the
n sp% /n s(p⫹⌬p)% ratio. For pure methanol, approximately
75% of the molecules are 2-bonded, the remaining 25% averaging fewer than 2 bonds.44
In pure PC there is no bonded network. After the introduction of the salt in the solvent, ‘‘bonds’’ are formed between the ions and the polar PC molecules, and we can use a
similar formalism as for water and methanol. However, in
this case we begin with no PC molecules forming a network
of bonds; then n sp% would correspond to 100% of the molecules in the pure solvent. With increasing salt concentration,
n s(p⫹⌬p)% is then less than 100%. The calculated numbers of
bonds being broken or formed for all salt-solvent combinations 共in the low concentration limit兲 are given in columns 5
of Table III.
3. Discussion of results
a. Salts in water. In water, lithium imide causes the
greatest decrease in the zero-frequency solvent dielectric
constant as a function of increasing concentration. The effect
of lithium triflate is smaller than that of lithium imide, and
lithium chloride depresses the value of the zero-frequency
solvent dielectric constant even less than lithium triflate. Not
surprisingly, the number of irrotationally bound solvent molecules associated with the salts follows the same trend:
n imide⬎n triflate⬎n chloride . This behavior can be explained by
considering that lithium imide has ⬃16 associated waters,
lithium triflate has ⬃6, and lithium chloride has ⬃4.5. If we
subtract 4 waters for Li⫹ in each case 共the first solvation
shell of Li⫹ from quantum chemical calculations兲, as well as
the first solvation shell of each anion 共from quantum chemical calculations, 3 for triflate, 6 for imide, and 2 for chloride兲, we end up with ⬃6 for imide, ⬃0 for triflate, and ⬃0
for the chloride. Basically this indicates that the triflate and
chloride anions are completely solvated by the first solvation
shell. The larger imide anion 共see Fig. 8兲 has more available
surface area to accommodate water molecules with its
charged oxygens, and the data indicates a much larger solvation number than predicted by quantum chemistry.
When we consider ␶ 1 , we notice that the ordering of
greatest decrease in ␶ 1 is not the same as that for ␧ static-solvent ;
it is now imide ( ␦ ␶ ⫽2.79)⬎chloride ( ␦ ␶ ⫽1.16)⬎triflate
( ␦ ␶ ⫽1.00). Imide and triflate keep their relative positioning
(imide⬎triflate兲, but chloride and triflate switch places 共now
chloride⬎triflate兲. If indeed this phenomenon is real, it suggests that although triflate interacts with more water 共⬃2 per
triflate兲 than chloride 共⬃0.5 per chloride兲, the chloride–water
interaction is stronger than the triflate–water interaction. In
other words, there is less of a hindrance to rotational motion
Dielectric relaxation of electrolyte solutions
8477
FIG. 8. 共a兲 Triflate anion; 共b兲 imide anion.
for water around triflate, even though triflate can accommodate more waters, than for water around chloride. Therefore,
the ␶ 1 of the triflate is less affected. Imide has so many more
waters than chloride 共12 waters for imide to the 0.5 for chloride兲 that it still has a much greater effect than chloride in
decreasing ␶ 1 .
b. Salts in methanol. In methanol, the ordering of the
salts, from greatest to least decrement in ␧ static-solvent is
chloride⬎triflate⬇imide. The same ordering is found for ␶ 1 .
We may postulate that chloride has the greatest decrement in
␧ static-solvent because, with the smallest size, it has a greater
field strength at the ion–solvent interface than either triflate
or imide. Accordingly, lithium chloride has ⬃20 associated
methanol molecules, while lithium triflate and lithium imide
have ⬃13 and ⬃11, respectively. If we subtract the 4 methanol molecules tetrahedrally bound to the Li⫹ as well as the
first solvation shell of each anion 共from quantum chemical
calculations, 2 for chloride, 3 for triflate, 4 for imide兲, we are
left with ⬃14 solvent molecules in the case of lithium chloride, ⬃6 for lithium triflate, and ⬃3 for lithium imide. It is
likely that some of these ‘‘extra’’ solvent molecules—
molecules in excess of the numbers indicated by quantum
chemical calculations—are associated with the cation and
some with the anion. Whatever the pattern of distribution,
these numbers are suggestive of a second solvation shell for
ions in methanol.
Indeed, we would expect ions in water to have fewer
associated solvent molecules than the same ions in methanol
because, in water, the ions are screened much more effectively than in methanol. This is immediately evident when
one considers that the static dielectric constant of pure water
(␧ static⫽78) is more than twice that of pure methanol
(␧ static⫽32). It is also quite possible that the large number of
associated molecules in methanol indicates that methanol
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8478
Asaki et al.
J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
molecules beyond the first solvation shell are affected
strongly due to the chain structure of the solvent.45
The above suppositions are supported by the experimental second shell numbers for lithium triflate 共⬃0 water molecules versus ⬃6 methanol molecules兲, and lithium chloride
共⬃0 water molecules vs ⬃14 methanol molecules兲. However, for lithium imide, the trend is reversed 共⬃6 water molecules vs ⬃3 methanol molecules兲. This may be explained,
at least in part, by noting that the larger methanol molecules
experience more steric hindrance around the oxygens of the
imide anion than do the smaller water molecules.
c. Salts in propylene carbonate. Most or all of what happens in PC can be explained in terms of association or dissociation of the ion pairs. This is because both triflate and
imide are basically unsolvated in PC,43 while Li⫹ is solvated
by four tetrahedrally arranged PC molecules.38,39 So if
lithium triflate were to completely dissociate in PC, then we
could expect the number of irrotationally bound PC molecules to be four. The same argument can be made for
lithium imide upon complete dissociation in PC—four irrotationally bound solvent molecules per ion pair should be
obtained.
Experimentally, we obtain ⬃3.1 and ⬃1.5 irrotationally
bound PC molecules for lithium triflate and lithium imide,
respectively, which indicates that both salts experience association, even at low concentrations. However, lithium imide
associates more strongly than does lithium triflate. In line
with this, ␶ 1 for lithium triflate increases more sharply than
for lithium imide because more PC molecules are held by the
lithium cations in lithium triflate solutions.
roughly constant at all concentrations, the increase in ␶ 2 may
be due to greater viscosity as the number of ions in solution
grows.
2. Salts in methanol
For all three salts, ⌬␧ 2 remains approximately constant
at low concentrations 共0.5 M and below兲. This means that the
population of chain end molecules is not increasing—an ion
either grabs a chain in the middle without breaking it or the
ion binds both ends of a chain that it has broken. At higher
concentrations 共in Li triflate solutions兲 ⌬␧ 2 drops sharply
because at 1 and 2 M concentrations only 25 and 11 methanol molecules, respectively, are available per ion pair 共see
Table I兲.
As with ⌬␧ 2 , the values of ␶ 2 , ⌬␧ 3 , and ␶ 3 remain
roughly constant within our experimental error for solutions
of all three salts, indicating that single molecule rotations
and the smaller, faster hydrogen-bond-related motions of
molecules are basically unaffected by the addition of ions at
low concentrations 共⬍1 M in methanol兲. This may be interpreted as in the case of the fastest relaxation in water—the
nature of hydrogen bonding in methanol is not influenced
greatly by the introduction of ions.
All of this means that the chain structure of methanol is
not greatly disrupted by the addition of ions at low concentrations. In other words, the chains do not break—they
‘‘kink’’ to accommodate solute particles. This is corroborated
by the very low number of bonds broken obtained for the
salts in methanol 共0.6, 0.9, and 3.0 for triflate, imide, and
chloride, respectively兲.
B. Higher-frequency terms
As discussed in Sec. III C, we were able to resolve one
higher-frequency solvent relaxation 共than the main solvent
relaxation term兲 in water and two higher-frequency solvent
relaxations in methanol and PC. The values of ␧ and ␶ for
these terms are shown as functions of concentration in Fig. 9.
Plot 9共a兲 shows ␧ 2 and ␧ ⬁ as a function of concentration for
all three salts in water; plots 9共b兲 and 9共c兲 show ␧ 2 , ␧ 3 , and
␧ ⬁ as functions of concentration for methanol and propylene
carbonate, respectively. Likewise, plot 9共d兲 shows ␶ 2 as a
function of concentration for all three salts in water; plots
9共e兲 and 9共f兲 show ␶ 2 and ␶ 3 as functions of concentration
for salts in methanol and propylene carbonate, respectively.
We now will analyze the results shown in this figure.
1. Salts in water
As we might suspect, and as has been postulated,15 the
small rotations and translations of the water molecules of the
solutions, at all concentrations, do not change much 共within
experimental error兲 from the values of pure water. If these
motions are related to hydrogen bonding, then the constancy
of the parameters indicates that the nature of hydrogen bonding in the solution is essentially the same as for pure water.
However, the data do suggest very gradual increases in ␧ 2 ,
and ␧ ⬁ , and a slightly greater increase in ␶ 2 with increasing
concentration. While the difference ⌬␧ 2 ⫽␧ 2 ⫺␧ ⬁ remains
3. Salts in PC
For lithium imide in PC, ⌬␧ 2 remains constant at a value
about 10% greater than in the pure solvent for concentrations
up to one molar. This means that the dipole–dipole correlated structure of the bulk solvent is not greatly disrupted in
imide solutions. This is not surprising considering that, on
average, only ⬃1.5 PC molecules are required to solvate a
lithium imide ion pair. The relaxation time for the second
term increases above that of the pure solvent by about 10%.
In other words, for low concentration solutions of lithium
imide, the population of PC molecules able to perform inertial rotations is stable at a value roughly 10% greater than in
the pure solvent, and the rotating molecules move at a speed
roughly 10% slower than in the pure solvent.
We find a different situation in lithium triflate solutions
in PC. About twice as many PC molecules 共⬃3 compared to
⬃1.5兲 are needed to solvate lithium triflate ion pairs. Accordingly, ⌬␧ 2 at low concentration is about 25% greater than the
value for pure PC. This indicates that the dipole–dipole correlations in the bulk solvent are more disrupted by lithium
triflate, with the result that more PC molecules are available
for inertial rotations. This is so even though more PC molecules are also involved in solvating ion pairs. We also note
an increase in ␶ 2 for lithium triflate solutions, up to 35%
higher for a one molar solution than in the pure solvent. So
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J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
Dielectric relaxation of electrolyte solutions
8479
FIG. 9. High frequency ␧ and ␶ values as functions of concentration. 共a兲–共c兲 ␧ values and 共d兲–共f兲 the ␶ values. Error bars are shown for all points.
even though more single PC molecules are free to rotate,
they experience much more drag in the solution than in the
pure solvent.
For the third relaxation, ⌬␧ 3 and ␶ 3 remain approximately constant for both salts. This means that small, fast
motions of dipoles with respect to each other remains the
same in the solutions as in the pure solvent, although bulk
PC is not a hydrogen-bonded liquid as are water and methanol.
V. CONCLUSIONS
We have used THz transmission spectroscopy to measure the complex permittivity of water, methanol, and propylene carbonate, as well as solutions of lithium triflate, lithium
imide, and lithium chloride in these solvents. We have assigned physical motions in the solvents to the various Debye
relaxations and interpret the changes in the Debye relaxation
parameters that the introduction of ions induces.
The changes in the main solvent relaxation parameters
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8480
Asaki et al.
J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
共not including ion pair relaxation, which we do not address兲
yield information about irrotationally bound solvent molecules and the number of bonds broken or formed in ionic
solvation. We find good agreement between the three methods used to determine the numbers of irrotationally bound
solvent molecules.
The behavior of ⌬␧ 2 , ⌬␧ 3 , ␶ 2 , and ␶ 3 for methanol and
propylene carbonate and ⌬␧ 2 and ␶ 2 for water help us to
present a coherent picture of solvation in all three solvents.
We demonstrate that hydrogen-bonding behavior in water
and methanol is basically unaffected by the addition of ions
共at our molarities兲 within experimental error. Likewise, the
fastest Debye relaxation in nonhydrogen-bonded propylene
carbonate remains the same in solution as for the pure solvent.
ACKNOWLEDGMENTS
M.L.T.A. gratefully acknowledges C. Londergan for assistance with sample preparation and L. R. Pratt for useful
conversations. This work was supported by the U.S. Department of Energy under Contract W-7405-ENG-36 under the
LDRD program at Los Alamos.
APPENDIX: EXPERIMENTAL DATA FOR PURE SOLVENTS AND SOLUTIONS
Values of parameters determined by fitting experimental data to the multiterm Debye model. Numbers in parentheses refer
to the standard deviation in units of the least significant digit used.
Solvent/
molarity
␧static
H2 O
78.96
共6.94兲
Li triflate in
75.98
共5.90兲
69.44
共6.50兲
65.93
共3.39兲
52.44共49兲
47.44
共2.53兲
0.515 M
0.659 M
1.828 M
2.472 M
0.492 M
0.769 M
1.710 M
2.374 M
75.17
共8.83兲
57.50
共5.74兲
47.20
共3.32兲
32.97共02兲
25.61
共1.07兲
Li chloride in
H2 O
0.179 M
0.559 M
78.63共30兲
71.58
共2.51兲
62.36
共2.96兲
59.78
共3.08兲
56.31共45兲
47.38
共1.50兲
32.59
共2.83兲
2.012 M
2.750 M
4.382 M
MeOH
␶2
共ps兲
7.87共79兲
4.80共10兲
0.18共03兲
3.42共18兲
7.66共67兲
4.83共09兲
0.125共025兲
2.63共35兲
7.34共77兲
4.88共11兲
0.16共03兲
2.96共30兲
7.22共39兲
5.01共00兲
0.15共01兲
2.68共15兲
6.58共07兲
6.53共45兲
5.47共05兲
5.55共08兲
0.29共01兲
0.29共02兲
3.42共05兲
3.48共04兲
7.40共1.02兲
4.82共13兲
0.15共03兲
2.61共32兲
6.47共73兲
4.90共11兲
0.17共03兲
2.79共22兲
5.73共49兲
4.78共10兲
0.16共02兲
2.44共25兲
5.22共00兲
4.69共25兲
5.28共00兲
5.23共00兲
0.30共01兲
0.29共01兲
3.22共03兲
3.08共03兲
7.75共00兲
6.93共24兲
5.07共02兲
4.97共04兲
0.21共01兲
0.19共01兲
3.35共05兲
3.27共08兲
6.37共36兲
5.22共06兲
0.18共01兲
3.04共10兲
5.93共37兲
5.54共08兲
0.24共02兲
3.54共07兲
5.73共05兲
5.30共21兲
5.62共04兲
6.07共00兲
0.23共02兲
0.25共01兲
3.61共15兲
3.64共07兲
38.76
共1.71兲
4.82共24兲
1.06共17兲
␧3
␶3
共ps兲
␧⬁
H2 O
0.098 M
1.341 M
␧2
H2 O
0.099 M
Li imide in
␶1
共ps兲
2.98共12兲
0.22共03兲
1.98共04兲
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J. Chem. Phys., Vol. 116, No. 19, 15 May 2002
Solvent/
molarity
␧static
Dielectric relaxation of electrolyte solutions
␶1
共ps兲
␧2
␶2
共ps兲
␧3
␶3
共ps兲
␧⬁
36.84
共1.66兲
34.34
共4.51兲
24.27
共2.22兲
23.92
共5.33兲
4.68共12兲
1.14共06兲
3.13共02兲
0.14共01兲
1.54共06兲
4.88共24兲
1.42共17兲
3.22共04兲
0.17共01兲
1.68共05兲
4.31共23兲
1.22共33兲
3.22共11兲
0.20共02兲
1.82共06兲
3.94共00兲
1.13共28兲
3.49共11兲
0.29共02兲
2.13共02兲
40.56
共11.3兲
33.98
共10.5兲
31.82
共2.32兲
11.74
共2.60兲
4.69共48兲
1.04共21兲
2.79共03兲
0.04共02兲
0.48共86兲
5.21共72兲
1.73共44兲
3.15共07兲
0.19共02兲
1.86共07兲
5.48共00兲
1.92共05兲
3.01共01兲
0.15共01兲
1.74共06兲
5.57共1.07兲
2.23共53兲
3.08共00兲
0.19共00兲
1.87共03兲
40.03
共1.41兲
21.71
共2.24兲
38.27
共3.10兲
4.57共04兲
0.86共01兲
2.76共02兲
0.11共01兲
1.62共07兲
4.78共19兲
0.94共07兲
2.91共00兲
0.20共01兲
2.03共04兲
5.41共07兲
1.24共02兲
3.35共04兲
0.21共01兲
1.86共04兲
42.42共48兲
45.88
共2.21兲
47.13
共1.61兲
55.55
共3.68兲
56.63
共6.51兲
5.81共02兲
6.04共36兲
1.28共01兲
1.37共14兲
3.34共01兲
3.34共05兲
0.20共015兲
0.20共01兲
1.88共00兲
1.91共03兲
6.10共04兲
1.66共02兲
3.59共02兲
0.24共01兲
2.00共02兲
4.86共04兲
0.71共00兲
3.12共06兲
0.15共02兲
1.87共09兲
4.55共06兲
0.73共09兲
3.21共14兲
0.20共02兲
2.13共03兲
39.21 共30兲
40.33 共99兲
5.83共01兲
5.86共09兲
1.40共01兲
1.42共05兲
3.47共01兲
3.67共03兲
0.20共00兲
0.23共01兲
1.84共02兲
1.97共02兲
42.57
共1.47兲
5.79共04兲
1.41共01兲
3.47共03兲
0.20共01兲
1.85共02兲
8481
Li triflate in MeOH
0.101 M
30.42共61兲
0.486 M
21.75
共3.75兲
19.54
共1.61兲
13.11
共1.69兲
0.943 M
1.787 M
Li imide in MeOH
0.098 M
0.484 M
0.915 M
27.78
共3.56兲
23.38
共11.2兲
17.01 共01兲
1.759 M
7.28
共2.24兲
Li chloride in MeOH
0.075 M
31.06 共63兲
0.409 M
16.86 共96兲
Pure PC
65.43
共3.53兲
Li triflate in PC
0.934 M
57.60 共34兲
53.12
共3.49兲
46.04 共56兲
1.725 M
38.64 共43兲
2.445 M
31.18
共4.48兲
0.100 M
0.485 M
Li imide in PC
0.112 M
0.465 M
0.879 M
1
62.97 共35兲
55.76
共1.22兲
43.63 共53兲
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