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Charge Relaxation and Stokes−Einstein Relation in Diluted Electrolyte Solution of Propylene Carbonate and Lithium Perchlorate ́ Jolanta Swiergiel,* Iwona Płowaś, and Jan Jadzẏ n Institute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 17, 60-179 Poznań, Poland ABSTRACT: Impedance spectroscopy was used for studies of electrical conductivity and electric charge relaxation in a diluted solution of propylene carbonate and lithium perchlorate (mole fraction xLiClO4 = 6 × 10−5) in the temperature range from 250 to 350 K. It was found that the viscosity behavior of the conductivity as well as the charge relaxation time related to the Li+ and ClO4− ions immersed in propylene carbonate fulfill quite well the Stokes−Einstein hydrodynamic model. An essentially limited ion-pairing process in the studied solution of a low ion concentration and considerably limited dipolar aggregation of propylene carbonate molecules are considered to be essential factors determining the dynamic behavior of the ions immersed in that strongly polar liquid.

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INTRODUCTION

Many papers devoted to the investigation of the direct current conductivity (σDC) in the propylene carbonate-based electrolytes of different natures have been published.1,9−16 The present paper concerns mainly the dynamic aspect of that conductivity, namely, the relaxation of the electric charge curriers due to the changes of the electric stimulus. That relaxation process reveals important information about the nature of translational diffusion of ions in a given medium. In particular, in the simplest case of the Brownian translational diffusion of ions, the relaxation process of the electric current (due to the disappearance of the electric stimulus, for example) is of the exponential type (in the time domain) and can be characterized by the time constant, the charge relaxation time (τσ). In the literature, that quantity is alternatively named the conductivity relaxation time, and it seems to be not erroneous as τσ describes the relaxation from the state of the ordered flow of charges, i.e., from the conductivity state (σDC), to the state of the randomly distributed charges (σ = 0). From the experimental point of view, the relaxational measurements can be performed much easier and more precisely when an alternating electric stimulus is applied to the system. Then, in the frequency domain, the mentioned above relaxation process due to the normal Brownian translation of ions manifests itself as the simplest possible electrical response: the impedance spectrum is of the Debye-type.17 One must realize that there are three different dynamic phenomena which are named in the literature as “ionic conductivity relaxation”. Two of them are connected with dispersion of the conductivity. The first phenomenon occurs when the blocking electrodes of the measuring cell are used. Namely, in the region of low frequency of the electric stimulus (mostly below 1 kilohertz), the ionic double layers are formed and with decreasing of the frequency, the conductivity of the electrolyte gradually goes to zero. The frequency range where that phenomenon occurs depends mainly on the distance

Liquid cyclic alkylene carbonates are polar aprotic solvents of great technological importance.1 In particular, propylene carbonate (4-methyl-1,3-dioxolane-2-one) exhibits physical properties which are very suitable not only for electrochemical purposes (as in lithium batteries)2−4 but also for numerous other purposes.5 The large temperature range of the liquid phase (from melting at about −49 °C to boiling at about 242 °C), a good solubility of a variety of organic and inorganic compounds, and its biodegrability and low toxicity make propylene carbonate an excellent green solvent which is successfully used in many branches of industry, including the production of cosmetics.1 The electrochemical attractiveness of propylene carbonate results from its high static permittivity (εs ≈ 65, at 25 °C) and high molecular dipole moment (μ = 4.94 D). The crucial role of the solvent permittivity in solubility and dissociation of ionic compounds is obvious, but also important is the solvent dipolar polarizability which is proportional to the molecular dipole moment squared. That polarizability is a fundamental physical quantity in the ion solvation process. The experimental data presented in a recent paper6 have shown that the constant of K+ and I− ion association strongly (exponentially) decreases with an increase of the dipolar polarizability of the solvent used. It is well-known that the ion pairs, being electrically neutral entities, can markedly reduce the conductivity of electrolyte solutions; therefore, the conclusions presented in the mentioned paper,6 in spite of their limited experimental basis, may suggest a simple method for reducing the ion-pairing process. Namely, between the different solvents which are suitable for a given ionic compound, the best choice is the solvent composed of the molecules of the highest dipole moment. Propylene carbonate, with its high molecular dipole moment, is one of the most suitable mediums for electrolyte solutions. This is why that solvent, often in mixtures with other solvents, is considered to be a supermedium for lithium-ion batteries,7 and a great effort is underway to reach an optimal interface between the carbonatebased electrolyte and the graphite electrodes; this is currently one of the most important problems in that field.8 © XXXX American Chemical Society

Received: November 17, 2014 Revised: February 3, 2015 Accepted: February 6, 2015

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conductivity are frequency-independent and are equal to their static value

between the electrodes and the type of material used for the electrodes, but also on the nature of the electrolyte. The second phenomenon is observed at high frequencies and is related to the inertia of solvated ions. With increasing frequency of the electric stimulus, the electrical conductivity falls and the crucial factors here are an extension and strength of the ion solvation. The conductivity relaxation corresponding to that process can be expected at the megahertz frequency range or higher, depending on the system under investigation. The third “ionic conductivity relaxation”, which is a subject of the present paper, concerns the relaxation process occurring when the electric field is suddenly removed from the ionically conducting system or when the field is applied to the system in which the ions are randomly distributed. In both cases the ions relax to the new equilibrium state and the above-mentioned charge relaxation time (τσ) is just related to that relaxation process. This paper presents the results of the electrical conductivity and the charge relaxation studies performed for very diluted solution of lithium perchlorate (Li+ClO4−) in propylene carbonate. The aim of the paper is to investigate the temperature behavior of the ions used in the electrolytes of practical significance. In particular, it is interesting to verify the Stokes− Einstein model in the conditions of a very limited ionic pairing and a specific structure of the liquid propylene carbonate. The studies were performed with the use of impedance spectroscopy.

ε′ = εs ,

(2)

and the imaginary parts ε″(ω) = σDC/ωε0 ,

σ ″(ω) = ωε0εs

(3)

show, in the log−log scale, a linear frequency dependence with the slope of −1 or 1, respectively. Both the impedance Z*(ω , T ) =

RDC(T ) 1 + jωτσ(T )

(4)

and the electric modulus M *(ω , T ) = εs−1(T ) −

εs−1(T ) 1 + jωτσ(T )

(5)

show the frequency dependence of the Debye-type with the same charge relaxation time τσ(T ) =

ε0εs(T ) σDC(T )

(6)

In eq 4, RDC denotes the dc resistivity of the studied sample, and it usually is expressed as the dc conductivity (σDC = 1/kRDC, k is the constant of the measuring cell). As a matter of fact, τσ is a well-known Maxwell time constant (τσ ≡ RDCCDC) of an equivalent circuit for studied molecular system, where the capacitor CDC = εsC0 = εskε0 is connected in parallel to the resistor RDC = (kσDC)−1. The physical meaning of the quantity τσ results from more general consideration based on the two principal equations of electrodynamics, namely Gauss’s law ρ div E = ε0εs (7)

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THEORETICAL BACKGROUND ON CHARGE RELAXATION The usefulness of the electric impedance method in studies of the ionic current relaxation is well-known.18 Among four complex quantities which can be used for description of the frequency dependence of different properties of dielectric materials placed in an external electric field, namely, the permittivity ε*(ω) = ε′(ω) − iε″(ω), impedance Z*(ω) = Z′(ω) + iZ″(ω), electric modulus M*(ω) ≡ (1/ε)* = M′(ω) + iM″(ω), and electrical conductivity σ*(ω) = σ′(ω) + iσ″(ω), only the impedance and the electric modulus show relaxational behavior with the time constant corresponding to the electric charge relaxation, τσ. However, only the impedance can be measured in experiment. Here, ω = 2πf is the angular frequency of the electric stimulus, f the frequency, and i = (−1)1/2. It is important to realize that the above-mentioned four complex quantities are alternative representations of the same macroscopic relaxation data and can be easily transformed to each other according to the scheme iωε0 1 = iωC0Z*(ω) = M *(ω) = ε*(ω) σ *(ω)

σ ′ = σDC

and the charge conservation law div j = −

∂ρ ∂t

(8)

where E and j denote the vectors of the probing electric field and the current intensity, respectively, and ρ is the charge density throughout the volume of the material under investigation. In a particular case, when the ionic current fulfills Ohm’s law, j = σDCE, the combination of eqs 7 and 8 gives a simple differential equation describing the charge relaxation in the studied material ∂ρ ρ + =0 ∂t τσ

(1)

(9)

where τσ is a function of the electric properties of the investigated material and is given by eq 6. The solution of eq 9 presents a simple exponential decay of the charge on the time (t)

where C0 = kε0 is the electric capacity of the empty measuring cell; k = S/l, where S and l are the electrode surface and the distance between the electrodes, respectively, and ε0 = 8.85 pF/m is the permittivity of free space. As the dielectric relaxation due to the dipolar rotational diffusion occurs in propylene carbonate at frequencies much higher (gigahertz region19−22) than those used in our investigations, the only dynamic electric phenomena stimulated by the electric field are related to the ion transport in the liquid. In such a case, the experimental behavior of the four electric quantities mentioned above takes the simplest possible form.23−25 In particular, the real parts of the permittivity and

⎛ t⎞ ρ = ρ0 exp⎜ − ⎟ ⎝ τσ ⎠

(10)

Therefore, it results from the above simple consideration that for an exponential decay of the ionic current (in the time domain) or for the Debye-type of the impedance relaxation spectrum (in the frequency domain), the two conditions should be fulfilled. The first (microscopic) condition concerns the ion dynamics, which should be of the normal Brownian translational B

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Industrial & Engineering Chemistry Research diffusion, and the second (macroscopic) condition concerns the fulfillment of Ohm’s law. However, it is quite possible that these two conditions are not independent from each other, i.e., the ohmic behavior of the ionic current in a given medium is a direct consequence of the normal Brownian diffusion of the electric charges in that medium.

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EXPERIMENTAL PROCEDURE Propylene carbonate (PC) was purchased from Sigma-Aldrich with stated purity of min. 99.7% and was stored over molecular sieves (4 Å) for several weeks before measurements. The dc electrical conductivity of neat propylene carbonate was equal to 7.26 × 10−7 S·cm−1, at 298 K. Lithium perchlorate (LiClO4) of purity 99.7% was supplied by Across Organics and was stored in a desiccator over silica gel. The concentration of the ionic compound chosen for our experiment (the mole fraction xLiClO4 = 6 × 10−5) results from the compromise between two experimental requirements. The first requirement concerns the measurement reliability: the ion concentration should be sufficiently high to give the electrical conductivity of the electrolyte solution at least 1 order of magnitude higher than conductivity of the neat solvent (conductivity background). The second requirement concerns an extension of the ion-pairing process which, being temperature-sensitive, leads to changes of the charge carrier density at different temperatures, which can make interpretation of the experimental results difficult. Of course, the significance of that process decreases with decreasing concentration of ions. Therefore, to avoid the ion-pairing process, one should operate with the ion concentration as low as possible. The dc electrical conductivity of studied propylene carbonate and lithium perchlorate solution was equal to 1.87 × 10−5 S·cm−1, at 298 K. The impedance spectra were recorded with the use of an HP 4194A impedance/gain phase analyzer in the frequency range from 100 Hz to 5 MHz and in the temperature range from 253 to 353 K. The details of the apparatus and the measuring capacitor used were described previously.26

Figure 1. Nyquist plots of the impedance spectra recorded for propylene carbonate and lithium perchlorate solution (a) and for the neat solvent (b) at different temperatures. In panel a, the mole fraction of LiClO4 equals 6 × 10−5 and the solid lines represent the best fit of eq 4 to the experimental data (points).

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RESULTS AND DISCUSSION Figure 1 presents the temperature evolution of the impedance spectra recorded for propylene carbonate and lithium perchlorate solution (Figure 1a) and the spectra of the neat solvent (Figure 1b), for comparison. As can be seen from the presented data, the addition of Li+ and ClO4− ions to propylene carbonate in proportion of the two ions per about 2 × 104 solvent molecules causes a decrease of the electrical resistivity of the mixture of more than 1 order of magnitude in comparison to that of the neat solvent. The impedance data are presented in the (Z″, Z′) complex plane (the Nyquist plots) where the spectra have the form of semicircles with the centers placed on the real impedance axis; therefore, the impedance spectra of the studied solution are of the Debye-type and can be reproduced with eq 4. The solid lines in Figure 1a represent the best fit of eq 4 to the experimental spectra. Resulting from the fitting procedure, the dc ionic conductivity, σDC (= 1/kRDC), and the charge relaxation time, τσ, determined at different temperatures, are presented in Figure 2 in the form of Arrhenius plots. In the figure is also presented the plot for the viscosity of the neat propylene carbonate.27,28 For dilute solutions, such as that studied in this paper, that viscosity will be taken as the viscosity of the studied solution.

Figure 2. Arrhenius plots for dc electrical conductivity and the charge relaxation time determined for solution of propylene carbonate and lithium perchlorate (xLiClO4 = 6 × 10−5). The viscosity (η) concerns the neat solvent.27,28 The values of the thermal activation energy are given in the figure.

As results from Figure 2, the presented quantities fulfill the Arrhenius dependence quite well, and the thermal activation energies of σDC and η are very close to each other, while the activation energy related to τσ is distinctly higher. The ratios of the activation energies of σDC and τσ to that of η are 1.03 and 1.23, respectively. The consequences of that fact will appear shortly. C

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Industrial & Engineering Chemistry Research Figure 3 presents the experimental dependences of σDC(η) and τσ(η) for the studied solution of propylene carbonate and lithium

Figure 4. Stokes−Einstein dependences of the dc electrical conductivity and the charge relaxation time (eqs 11 and 12) for solution of propylene carbonate and lithium perchlorate. The slopes of the lines are shown in the figure.

Figure 3. Experimental dependences of the dc electrical conductivity (σDC) and the conductivity relaxation time (τσ) determined for the propylene carbonate and lithium perchlorate solution (xLiClO4 = 6 × 10−5) on the viscosity of the solvent.27,28 The solid lines are the best fits of the function of the y ∝ xm type to the experimental data (points). The values of the exponent m are (practically) equal to the ratios of the corresponding activation energies depicted in Figure 2.

conclusion concerns the charge relaxation time for which we have recently proposed30 the following dependence on translational diffusion coefficient Dtr =

perchlorate. The viscosity has been taken as that of the neat solvent.27,28 The solid lines in Figure 3 represent the best fit of the function y ∝ xm, (x = η and y = σDC or τσ) to the experiment data. As shown in the figure, with increasing viscosity of the medium, the electrical conductivity decreases exponentially while the charge relaxation time shows an inverse behavior: it increases somewhat stronger than linearly. As expected for the quantities exhibiting an Arrhenius behavior, the values of the exponent m in the σDC(η) and τσ(η) relations are very close to the ratio of the corresponding activation energies of the relation partners. It seems to be interesting to compare these experimental results with predictions of the Stokes−Einstein model in which the translating ions are represented by a sphere of radius rion moving in a continuum medium of the shear viscosity η. By including the Nernst−Einstein equation29 to the model, one obtains the following relation between the translational diffusion coefficient (Dtr) and the dc electrical conductivity (σDC): Dtr =

kBT ∝ TσDC 6πηrion

kT 1 ∝ τσ 6πηrion

(12)

as the translational analogue of the Stokes−Einstein−Debye dependence describing the rotational relaxation time of molecular dipoles immersed in a viscous medium. As presented in Figure 4, the slop of τσ versus η/T is very close to 1, as is predicted by eq 12. That fact explains the reason for the difference in the thermal activation energies of τσ and η (Figure 2) and, consequently, the nonlinearity in the τσ(η) dependence (Figure 3). In our opinion, the observed behavior of the charge relaxation time as a function of the viscosity gives strong support to the Stokes−Einstein model. The results presented above can be summarized as follows: both the electrical conductivity and the charge relaxation time, recorded in diluted solution of ionic compound Li+ClO4− in propylene carbonate, fulfill quite well the Stokes−Einstein model, indicating the viscosity as the main factor driving the ion dynamics in that solvent. More precise and more physically based analysis of the presented experimental results can be performed with the use of the Poisson−Nernst−Planck model recently described by Macdonald.31 As the interactions and structure of the molecular medium undoubtedly play an important role in the behavior of the ions immersed in that medium, let us examine one of the most important features of the structure of the dipolar liquid, namely, the molecular organization due to dipolar correlations in propylene carbonate. The problem is interesting as the dipole moment of propylene carbonate molecule is relatively high (μ1 = 4.94 D), so one can expect here rather strong dipole−dipole correlations. An investigation of that problem requires the determination of the static permittivity of propylene carbonate, which can be quite easily done after transformation of the impedance spectra into the electric modulus spectra23 with the use of eq 1. Figure 5 presents the modulus spectra of the studied solution in the complex plane. The spectra form the semicircles with the centers placed in the real axis of the electric modulus and, as seen in the figure, they can be perfectly described with eq 5. It is worth mentioning that the time constant in that equation

(11)

where kB is the Boltzmann constant. The Stokes−Einstein model predicts a linear dependence of TσDC versus T/η with a slope of 1, in log−log scale. However, one must realize that the hydrodynamic Stokes−Einstein model was formulated with the assumption of a much greater size of moving particle if compared to that of the solvent molecules. Namely, the solute-to-solvent size ratio should be sufficiently high for fulfillment of the basic assumption of the particles moving in a continuous medium. Therefore, it is clear that the conclusions resulting from discussion in the frame of the Stokes−Einstein model on the dynamical behavior of the ions in usual solvents (the case of the present paper) can be seen as only a rough approximation. Figure 4 shows that in the studied ionic system the prediction of the Stokes−Einstein model is fulfilled quite well, in spite of the model approximations discussed just above. Practically the same D

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where μ1 is the dipole moment of single molecule (determined in the gas phase or in diluted solutions in nonpolar solvent), VM the molar volume, NA Avogadro’s number, and ε∞ the permittivity measured in frequencies high enough to prevent dipolar reorientation. The factor gk expresses the prevailing type of intermolecular dipolar coupling occurring in a given liquid, namely, gk < 1 corresponds to the antiparallel dipolar correlations leading to the reduction of the apparent dipole moment per molecule, gk > 1 corresponds to parallel association leading to an increase of the moment, and gk = 1 denotes a lack of the dipolar coupling in liquid under investigation. In calculation of the factor gk for neat propylene carbonate, the values of ε∞ and the density were taken from the literature.28,34 The dipole moment of a single carbonate molecule μ1 = 4.94 D was taken from the paper by Takata.35 Figure 7 presents the Kirkwood correlation factor obtained for the propylene carbonate at different temperatures. In the lower-

Figure 5. Electric modulus spectra of propylene carbonate and lithium perchlorate solution resulting from the transformation of the impedance spectra from Figure 1a with the use of equation M*(ω) = iωC0Z*(ω). The mole fraction of the electrolyte equals 6 × 10−5. The solid lines represent the best fit of eq 5 to the experimental data (points).

is equal to the charge relaxation time, τσ, as in the equation describing the impedance spectra.23 The static permittivity, εs, resulting from the modulus fitting procedure, is presented in Figure 6 as a function of the

Figure 7. Temperature dependence of the Kirkwood correlation factor of liquid propylene carbonate. Full points represent the values obtained in this paper, and open points were obtained by Simeral et al.19

temperature range, similar values of gk were obtained by Simeral et al.19 The factor gk is only somewhat higher than unity and practically temperature-independent; this result quite univocally indicates very reduced dipolar coupling occurring in liquid propylene carbonate. Therefore, the liquid, composed of molecules of high polarity, is practically “monomolecular”. That rather exceptional situation certainly results from a specific geometrical structure of propylene carbonate molecules and probably is related to the chirality of the molecules. The steric hindrance, resulting from that specific geometrical properties, quite efficiently reduces the dipolar coupling of the strongly polar molecules. Therefore, it is quite possible that the specific structure of liquid propylene carbonate contributes to the simple dynamic behavior of the ions diluted in that medium that manifests itself, among others, in a good fulfillment of the Stokes−Einstein model.

Figure 6. Temperature dependence of the static permittivity of propylene carbonate and lithium perchlorate diluted solution. Full points are the result of this paper, and the open points are the literature data obtained for the neat propylene carbonate: (○),19 (☆),28 and (△).32

temperature. As can be seen in the results presented in Figure 6, the presence of the ionic compound Li+ClO4− with the concentration of about 6 × 10−3 % (in mole fraction) practically does not disturb the permittivity of the propylene carbonate. Therefore, it most probably concerns also the viscosity (as described in the discussion of the Stokes−Einstein model) and the density, which will be necessary for discussion of the dipolar coupling in liquid propylene carbonate. In polar liquids, the dipolar coupling can be quite easily detected by determination of the Kirkwood correlation factor (gk)33 (εS − ε∞)(2εS + ε∞) 9kBT VM = μ12 gk 2 π 4 N εS(ε∞ + 2) A

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CONCLUSIONS The results presented in this paper have shown that the Li+ and ClO4− ions, dissolved in propylene carbonate in a low concentration (about one ion per 104 molecules of the solvent), exhibit the normal Brownian dynamics, as expected. Both the ionic conductivity and the charge relaxation time fulfill quite well the dependence on the medium viscosity predicted by the Stokes−Einstein model. It is a rare case of such good fulfillment of the Stokes−Einstein law, and in most papers on the ionic

(13) E

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(15) Srivastava, A. K.; Samant, R. A. Ionic conductivity in binary solvent mixtures. 1. Propylene carbonate (20 mass %) + ethylene carbonate at 25°C. J. Chem. Eng. Data 1994, 39, 358. (16) Hanna, E. M.; Al-Sudani, K. Conductance studies of some ammonium and alkali metal salts in propylene carbonate. J. Solution Chem. 1987, 16, 155. (17) Debye, P. Polar Molecules; Chemical Catalog Co.: New York, 1929. (18) Barsoukov, E.; Macdonald, J. R. Impedance Spectroscopy: Theory Experiment & Applications, 2nd ed.; John Wiley & Sons: London, 2005. (19) Simeral, L.; Amey, R. L. Dielectric properties of liquid propylene carbonate. J. Phys. Chem. 1970, 74, 1443. (20) Cavell, E. A. S. Dielectric relaxation in non-aqueous solutions. Part 5. Propylene carbonate (4-methyl-1,3-dioxolan-2-one). J. Chem. Soc., Faraday Trans. 2 1974, 70, 78. (21) Payne, R.; Theodorou, I. E. Dielectric properties and relaxation in ethylene carbonate and propylene carbonate. J. Phys. Chem. 1972, 76, 2892. (22) Pawlus, S.; Casalini, R.; Roland, C. M.; Paluch, M.; Rzoska, S. J.; Zioło, J. Temperature and volume effect on the change of dynamics in propylene carbonate. Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2004, 70, 061501. ́ (23) Swiergiel, J.; Jadżyn, J. Electric relaxational effects induced by ionic conductivity in dielectric materials. Ind. Eng. Chem. Res. 2011, 50, 11935. ́ (24) Jadżyn, J.; Swiergiel, J. Electric relaxational effects induced by displacement current in dielectric materials. Ind. Eng. Chem. Res. 2012, 51, 807. ́ (25) Swiergiel, J.; Bouteiller, L.; Jadżyn, J. Interpretation of the electric impedance spectra recorded for liquids in the presence of ionic and displacement currents. Ind. Eng. Chem. Res. 2013, 52, 11974. (26) Świergiel, J.; Jadżyn, J. Static Dielectric Permittivity of Homologous Series of Liquid Cyclic Ethers, 3n-Crown-n, n = 4 to 6. J. Chem. Eng. Data 2012, 57, 2271. (27) Moumouzias, G.; Ritzoulis, G. Viscosities and Densities for Propylene Carbonate + Toluene at 15, 20, 25, 30, and 35 °C. J. Chem. Eng. Data 1992, 37, 482. (28) Barthel, J.; Neueder, R.; Roch, H. Density, Relative Permittivity, and Viscosity of Propylene Carbonate + Dimethoxyethane Mixtures from 25 to 125 °C. J. Chem. Eng. Data 2000, 45, 1007. (29) Hansen, J. P.; McDonald, J. R. Theory of Simple Liquids; Academic Press: New York, 1991; p 399. ́ (30) Swiergiel, J.; Bouteiller, L.; Jadżyn, J. Compliance of the Stokes− Einstein model and breakdown of the Stokes−Einstein−Debye model for a urea-based supramolecular polymer of high viscosity. Soft Matter 2014, 10, 8457. (31) Macdonald, J. R. Utility and importance of Poisson−Nernst− Planck immittance-spectroscopy fitting models. J. Phys. Chem. C 2013, 117, 23433. (32) Ding, M. S. Liquid Phase Boundaries, Dielectric Constant, and Viscosity of PC-DEC and PC-EC Binary Carbonates. J. Electrochem. Soc. 2003, 150, A455. (33) Böttcher, C. J. F.; Bordewijk, P. Theory of Electric Polarization: Dielectric in Static Fields; Elsevier: Amsterdam, 1992; Vol. 1, p 254. (34) Moumouzias, G.; Ritzoulis, G. Relative Permittivities and Refractive Indices of Propylene Carbonate + Toluene Mixtures from 283.15 to 313.15 K. J. Chem. Eng. Data 1997, 42, 710. (35) Takata, T. Cyclic carbonates, novel expandable monomers on polymerization. Macromol. Rapid Commun. 1997, 18, 461.

transport in liquids, one operates rather with the fractional Stokes−Einstein law. The reason for such behavior of the ions immersed in propylene carbonate may be related to very low ion concentration where the ion-pairing process can be neglected. In addition, it is probable that some role is played here by the exceptional structure of the solvent, being composed of molecules of a very high dipole moment (about 4.94 D) showing practically no dipole−dipole couplings. The effect can have its source in the specific geometry of the chiral molecules of propylene carbonate which makes the dipole−dipole correlations unfavorable.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +48 61 86 95 162. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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