StokesâEinsteinâNernst Relation in Dilute Electrolyte Solutions of

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Stokes−Einstein−Nernst Relation in Dilute Electrolyte Solutions of Lithium Perchlorate in Polyethylene Glycols (200, 300, 400, and 600) ́ Jolanta Swiergiel, Iwona Płowaś, Jan Grembowski, and Jan Jadzẏ n* Institute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 17, 60-179 Poznań, Poland ABSTRACT: The electrical conductivity was measured for dilute electrolyte solutions of lithium perchlorate (LiClO4) in polyethylene glycols (PEG) of different molecular weights (200−600). The results were interpreted in the frame of the Stokes−Einstein−Nernst model. It was found (i) a breakdown of the model in the glycol-based polymeric matrices used and (ii) an increase of the deviation from the model predictions with increasing molecular weight of the matrix. The role of the flexibility of HO− [CH2−CH2−O]n−H chains in the efficiency of charge carriers transport in liquid glycols is discussed.

1. INTRODUCTION The Stokes−Einstein relation 1 is based on the Einstein’s theory of the Brownian diffusion of small particles1 and the Stokes formula2 for the drag on a rigid spherical particle moving in a viscous fluid D=

kBT Cηr

relation is fulfilled, the electrical conductivity has an inversely proportional relationship to viscosity of the medium where the ions are moving. This work concerns the study of electrical conductivity and conductivity relaxation time for the electrolyte solutions composed of lithium perchlorate (LiClO4) dissolved (at the same molar concentration) in polyethylene glycols (PEG) [the synonym names: poly(oxyethylene) POE or poly(ethylene oxide) PEO] of a different degree of polymerization. The obtained data on the transport of Li+ and ClO−4 ions in the polymeric liquids of different molecular weights and viscosities (taken from the literature)5−8 have been analyzed in the frame of the Stokes−Einstein−Nernst model. Polyethylene glycols used in our experiment represent a large family of polymeric compounds of wide applications as solvents in the pharmaceutical and cosmetic industries as well as food additives and plasticizers.9 The glycols are soluble in water and organic solvents over a large range of molecular weights and concentrations. A significance of these mixtures as green reaction media was expertly reviewed by Ji Chen et al.10 In 1979, Armand11 pointed out the potential applications of polyethylene glycols as a universal polymeric matrix for ions in use in batteries of a high energy density, the supercapacitors, and so forth. It is due to suitable physicochemical properties of the glycols, which can be easily controlled by changing the degree of polymerization of the compound or by preparing the mixtures with other liquids, such as water, for example. Besides, the high conductivity of the glycols-based ionic polymers was predicted, taking into account a high internal flexibility of the polymeric chains.11,12 The quantitative estimation of the flexibility of molecular chains is not a simple task but as a measure of that feature can

(1)

In that equation, D is the diffusion coefficient of a particle of radius r, η is the shear viscosity, kB is Boltzmann constant, T is the absolute temperature, and C is a numerical coefficient that depends on boundary conditions at the particle-fluid interface and ranges from 6π for no-slip to 4π for slip boundary conditions. Equation 1 predicts a linear dependence of the diffusion coefficient of a Brownian particle on T/η, provided that the dimension of the particle (r) is temperature independent. Although the Stokes−Einstein relation was derived for rather macroscopic hydrodynamic model, it is widely used in interpretation of the dynamics of molecular sized particles. In particular, the Stokes−Einstein model is used for interpretation of the experimental data related to the ions movement in viscous media. A basis of that interpretation is the Nernst−Einstein law,3 which links the electrical conductivity (σ) with the diffusion coefficient (2)

σT ∝ D

So, eqs 1 and 2 lead to the following Stokes−Einstein−Nernst relation which is often written as4 ⎛ T ⎞m σT ∝ ⎜ ⎟ ⎝η⎠

(3)

The exponent m represents the general case where the temperature is in a different power in the left and in the right side of the equation and numerous literature data show for m the value of about 0.8.4 As shown in eq 3, for the limiting case of m = 1, that is, when the Stokes−Einstein−Nernst © XXXX American Chemical Society

Special Issue: Memorial Issue in Honor of Anthony R. H. Goodwin Received: July 10, 2015 Accepted: September 23, 2015

A

DOI: 10.1021/acs.jced.5b00577 J. Chem. Eng. Data XXXX, XXX, XXX−XXX

Journal of Chemical & Engineering Data

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be taken the slope of the static permittivity of studied liquid vs temperature. Namely, as it was shown by Fröhlich,13 the slope of the permittivity dependence on temperature is proportional to the entropy change in the liquid due to molecular dipoles ordering, forced by the probing electric field. Because the ordering effect, that is, the entropy change, should be dependent on flexibility of the molecular dipoles, in the present paper, we determined the static permittivity of neat PEGs of different molecular weights as a function of temperature. Hence, the entropy increment was determined and compared with the increment determined for some liquids composed of rigid molecules of similar dipole moment as that of polyethylene glycols polymers.

2. EXPERIMENTAL SECTION Materials. Polyethylene glycols (PEGs), H‑(OCH2CH2)nOH, of different degrees of polymerization PEG-200, PEG-300, PEG-400, and PEG-600and lithium perchlorate (LiClO4), purity of 99.7%, were obtained from Sigma-Aldrich and AcrosOrganics, respectively. The compounds were stored in a desiccator over silica gel and the glycols were, additionally, stored for several weeks over the molecular sieves (4 Å). Table 1 contains the molecular weights and the number n of repeated OCH2CH2 units in used polyethylene glycols, as given in the Certificate of Analysis. Table 1. Sample Information, Where n Is the Number of OCH2CH2 Units in Polymer polyethylene glycol

molecular weighta in g/mol

na

PEG-200 PEG-300 PEG-400 PEG-600

203 299 417 580

4.2 6.4 9.1 12.8

Figure 1. Impedance spectra of PEG-300 + LiClO4 electrolyte solution recorded at different temperatures. The mole concentration of the electrolyte equals cLiClO4= 1.94 mmol·dm−3.

solution with the electrolyte mole concentration cLiClO4= 1.94 mmol·dm−3. The spectra of analogous solutions in PEG-200, 400, and 600 are similar, what illustrates Figure 2, where the set

a

Given in the Certificate of Analysis. Checked by infrared spectroscopy.

The molar concentration of lithium perchlorate dissolved in the polyethylene glycols was equal cLiClO4= 1.94 mmol·dm−3. The electrolyte concentration chosen in our experiment is a compromise between the need to limit the ions pairing process, and the level of the electrical conductivity in the solutions which should be distinctly higher from the conductivity background measured for neat glycols. The solutions were prepared by weighing (the balance A&D, model HR - 120) with an accuracy of ±1 × 10−4 g. Impedance Measurements. The complex impedance spectra of neat polyethylene glycols and PEGs + LiClO4 electrolyte solutions, Z*(ω), were recorded with the use of an HP 4194A impedance/gain phase analyzer in the frequency range from 100 Hz to 5 MHz. The measurements were performed for increasing temperature in the range T = (278.15 to 333.15) K with an exception of the solution of PEG-600 where due to a relatively high melting point (mp ≈ 295 K), the range T = (298.15 to 333.15) K was used. The temperature of the measuring cell was controlled with a “Scientific Instruments” device, model 9700, within ±2 × 10−3 K. The details on the used experimental setup can be found in recent paper.14

Figure 2. Example of the best fits of the Debye-type eq 4 (solid lines) to the impedance spectra (at 298 K) of LiClO4 dissolved in (1) PEG200, (2) PEG-300, (3) PEG-400, and (4) PEG-600, cLiClO4 = 1.94 mmol·dm−3.

of the spectra, recorded at 298 K, are presented in the (Z″,Z′) complex plane (the Nyquist plots). The figure shows that the spectra have a form of semicircles with the centers placed on the real impedance axis. So, the impedance spectra of studied solutions are of the Debye-type and can be reproduced with the following equation:15 Z*(ω , T ) =

3. RESULTS AND DISCUSSION Figure 1 presents, as an example, the impedance spectra recorded at different temperatures for PEG-300 + LiClO4

RDC(T ) 1 + jωτσ(T )

(4)

RDC denotes the dc resistivity of the studied sample and it usually is expressed as the dc conductivity (σDC = 1/kRDC, B



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k = S/l is the constant of measuring cell, S and l are, respectively, the electrode surface and the distance between the electrodes). τσ is the charge relaxation time and corresponds to the ionic relaxation process from the ordered movement (the electric current) to the random distribution. 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 εε τσ = 0 s σDC (5) C0 = kε0 denotes the electric capacity of empty measuring cell, k is mentioned above the cell constant, εs is the static permittivity of a sample under investigation and ε0 = 8.85 pF/m is the permittivity of free space. It is interesting to realize that for the electrolyte solutions of not too different static permittivity, according to eq 5, the higher value of the conductivity, the shorter relaxation of the charge. The solid lines in Figure 2 represent the best fit of eq 4 to the experimental impedance spectra of PEGs + LiClO4 electrolyte solutions. As can be seen, eq 4 perfectly reproduces the spectra and this applies to all recorded 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 Figures 3 and 4, respectively.

Figure 4. Charge relaxation time in HO−[CH2−CH2−O]n−H + LiClO4 electrolyte solutions (cLiClO4 = 1.94 mmol·dm−3) as a function of temperature; n = 4.2 (PEG-200), 6.4 (PEG-300), 9.1 (PEG-400), and 12.8 (PEG-600).

Table 2. Direct Current Conductivity σDC of PEGs + LiClO4 Solutions with Mole Concentration of the Electrolyte cLiClO4= 1.94 mmol·dm−3, Under a Pressure of 1013 hPaa T

σDC

K

μS·cm−1

278.15 283.15 288.15 293.15 298.15 303.15 308.15 313.15 318.15 323.15 328.15 333.15

PEG-200

PEG-300

PEG-400

PEG-600

2.891 3.993 5.352 7.000 8.946 11.19 13.73 16.65 19.81 23.31 27.00 30.94

1.389 1.942 2.628 3.462 4.454 5.603 6.907 8.375 9.968 11.71 13.61 15.62

1.501 2.081 2.798 3.652 4.661 5.829 7.123 8.540 10.06 11.68 13.37 15.15

3.391 4.273 5.276 6.427 7.604 8.862 10.23 11.62

a Standard uncertainties u are u(σDC) = 0.3%, u(c) = 0.02, and u(T) = 0.01 K.

Figure 3. Direct current conductivities of HO−[CH2−CH2−O]n−H + LiClO4 electrolyte solutions (cLiClO4 = 1.94 mmol·dm−3) as a fuction of temperature; n = 4.2 (PEG-200), 6.4 (PEG-300), 9.1 (PEG-400), and 12.8 (PEG-600).

Figure 5 shows that in the case of the electrolyte solutions PEGs + LiClO4, this model is not fulfilled and the deviation from the model prediction is clearly dependent on the degree of polimerization of PEG. As presented in Figure 6, the value of the parameter m is not too far from 1 for PEG-200 + LiClO4 solution (m = 0.92), but for PEG-600 + LiClO4 solution the deviation from the model prediction is quite significant (m = 0.76). The conductivity dependence of the length (n) of the polyethylene glycol polymers (Figure 3) as well as its behavior on the viscosity of the polymeric medium (Figure 5), quite clearly show the crucial role of polymer matrix structure in the ionic conductivity process. In our case, the matrix structure is directly related to the flexibility of the polyethylene glycols chains. As was mentioned in the Introduction, one of the easiest method for experimental visualization of the flexibility of the polymeric dipolar chains is to examine the temperature dependence of the static permittivity (εS) of the sample. An exploitation of that dependence for estimation of a susceptibility of molecular dipoles for ordering by probing electric field is based on the theory by Fröhlich.13 Namely, the theory links

The numerical values of σDC for all studied solutions are gathered in Table 2. The conductivity values of the same order of magnitude as obtained for LiClO4 dissolved in PEGs, have been recorded also for solutions of LiClO4 in other solvents, namely, in poly(propylene oxide)16 and polyethylene glycol + p-tertoctyl phenyl ether.17 As results from Figures 3 and 4, both the conductivity σDC and the relaxation time τσ, which correspond to the same density number of ions in the polyethylene glycols solutions (1.94 mmol·dm−3), distinctly depend on the degree of polymerization of the medium where ions are immersed. So, observed in Figure 3 an isothermal decreasing of the conductivity in PEGs + LiClO4 solutions must result from decreasing ions mobility in an increasingly polymerized matrix. Important for our research is the relationship associated with the Stokes−Einstein−Nernst model which, according to eq 3, provides a linear TσDC vs T/η dependence of a slope m = 1. C



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Figure 5. Stokes−Einstein−Nernst dependences (eq 3) for the dc conductivity of PEG + LiClO4 electrolyte solutions (cLiClO4= 1.94 mmol·dm−3). (1) PEG-200, (2) PEG-300, (3) PEG-400, and (4) PEG-600. The values of the exponent m are given on the picture. The viscosity of neat PEGs were taken from refs 5−8 .

Figure 7. Dielectric spectra of PEG-300 + LiClO4 electrolyte solution recorded at different temperatures.

Figure 6. Slopes m of the Stokes−Einstein−Nernst dependences from Figure 5 as a function of number n of oxyethylene groups in HO−[CH2−CH2−O]n−H polymer.

LiClO4 solution. At low frequencies, the effect of the double layers formation by residual ions (in neat PEGs) or by Li+ and ClO−4 ions (in the electrolyte solutions) near the electrodes of the measuring cell, appears. That effect manifests as a giant increase of the capacity of the cell and consequently, as an apparent increase of the permittivity. In the imaginary part of the dielectric spectra (Figure 7b), the presence of the charge carriers leads to the dielectric losses due to the conductivity process. In such a case, the imaginary part of the permittivity depends linearly on the frequency (in log−log scale) with the slope of about −1. The temperature dependences of the static permittivity determined from the real part of the dielectric spectra of studied neat PEGs and PEGs + LiClO4 solutions, are presented in Figure 8. It results from the data that the static permittivities of studied electrolyte solutions are practically equal to the permittivity of neat PEGs, with an exception of PEG-400 where the permittivity of the solution is about 2% higher than the permittivity of neat polymer. The results points out for a weak solvation of Li+ and ClO−4 ions by the polymeric chains of PEGs.24 Figure 9 presents the entropy increment determined for neat PEGs. The negative values of ΔS reflects the ordering of the HO−[CH2−CH2−O]n−H dipoles caused by probing electric field. However, the absolute values of ΔS are unexpectedly low if one takes into account the dipole moment values of the studied compounds. As shown by Sengwa et al.,25 the dipole moments of polyethylene glycols, measured in the neat state of the compounds, equal to about 3.3 D (PEG-200), 3.9 D (PEG300), 4.2 D (PEG-400), and 4.8 D (PEG-600). An increase of the dipole moment of PEGs is roughly proportional to the number n of oxyethylene groups in the polymer chain. The

the temperature derivative of the permittivity and the orientational entropy increment (ΔS) induced by the measuring electric field of intensity E S ( E ) − S0 ε ∂ε ΔS = = 0 s 2 2 (6) 2 ∂T E E S0 denotes the entropy of the liquid in absence of the electric field and T is the absolute temperature. A negative entropy increment, ΔS < 0, corresponds to the dipolar ordering action of the electric field and it is a typical behavior of dipolar liquids. However, for a given molecular dipole moment, the absolute value of ΔS depends mainly on the flexibility of polar molecules. An anomalous entropy behavior (ΔS > 0) was observed for some highly polar mesomorphic liquids in the vicinity of the phase transitions from the isotropic phase to the liquid crystalline phases.18−20 The εS(T) dependences for PEGs used in our experiment are obtained from the analysis of the dielectric spectra of the glycols. Figure 7 presents the dielectric spectra of PEG-300 recorded at different temperatures. The similar spectra were recorded for remaining neat PEGs and for solutions of PEGs + LiClO4 in order to see influence of ions on the permittivity values. As the dielectric relaxation due to the dipole reorientations in studied PEGs occurs in the gigahertz region,21−23 the frequency range used in this work corresponds to the static region of the dielectric properties of the compounds. Figure 7a presents, as an example, the real part of the dielectric spectra of PEG-300 + D



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AUTHOR INFORMATION

Corresponding Author

*Tel.: +48 61 86 95 167. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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DEDICATION

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REFERENCES

The paper is dedicated to the memory of Anthony R. H. Goodwin.

Figure 8. Temperature dependences of the static permittivity of neat polyethylene glycols (points): (1) PEG-200, (2) PEG-300, (3) PEG400, (4) PEG-600, and (×) the glycols with dissolved LiClO4 of the mole concentration 1.94 mmol·dm−3.

(1) Einstein, A. Investigations on the Theory of the Brownian Motion; Dover: New York, 1956. (2) Stokes, G. Trans. Cambridge Philos. Soc. 1856, 9, 5. (3) Hansen, J. P.; McDonald, J. R. Theory of Simple Liquids; Academic Press: New York, 1991; p 399. (4) Voronel, A.; Veliyulin, E.; Machavariani, V. Sh.; Kisliuk, A.; Quitmann, D. Fractional Stokes-Einstein Law for Ionic Transport in Liquids. Phys. Rev. Lett. 1998, 80, 2630−2633. (5) Ž ivković, N. V.; Šerbanović, S. S.; Kijevčanin, M. Lj.; Ž ivković, E. M. Volumetric and Viscometric Behavior of Binary Systems 2-Butanol + PEG 200, + PEG 400, + Tetraethylene Glycol Dimethyl Ether, and + N-Methyl-2-pyrrolidone. J. Chem. Eng. Data 2013, 58, 3332−3341. (6) Zhang, N.; Zhang, J.; Zhang, Y.; Bai, J.; Huo, T.; Wei, X. Excess molar volumes and viscosities of poly(ethylene glycol) 300 + water at different temperatures. Fluid Phase Equilib. 2012, 313, 7−10. (7) Zhao, T.; Zhang, J.; Li, L.; Guo, B.; Gao, L.; Wei, X. Excess properties and spectroscopic studies for the binary system 1,2ethanediamine + polyethylene glycol 300 at T = (293.15, 298.15, 303.15, 308.15, 313.15, and 318.15) K. J. Mol. Liq. 2014, 198, 21−29. (8) Zhang, K.; Yang, J.; Yu, X.; Zhang, J.; Wei, X. Densities and Viscosities for Binary Mixtures of Poly(ethylene glycol) 400 + Dimethyl Sulfoxide and Poly(ethylene glycol) 600 + Water at Different Temperatures. J. Chem. Eng. Data 2011, 56, 3083−3088. (9) Braun, D. B.; Delong, D. J. Kirk-Othmer Encyclopedia of Chemical Technology, 3rd ed.; John Wiley & Sons, Inc.: New York, 1982, 616− 632. (10) Chen, J.; Spear, S. K.; Huddleston, J. G.; Rogers, R. D. Polyethylene glycol and solution of polyethylene glycol as green reaction media. Green Chem. 2005, 7, 64−82. (11) Armand, M. B.; Chabagno, J. M.; Duclot, M. Fast Ion Transport in Solids; Elsevier: New York, 1979. (12) Sandahl, J.; Börjesson, L.; Stevens, J. R.; Torell, L. M. Elastic and Dynamic Properties of a Poly(propylene glycol)-Lithium Perchlorate Electrolyte: A Brillouin Scattering Study. Macromolecules 1990, 23, 163−167. (13) Fröhlich, H. Theory of Dielectrics, 2nd ed.; Clarendon Press: Oxford, 1958. (14) 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, 2271−2274. (15) Debye, P. Polar Molecules; Chemical Catalog Co.: New York, 1929. (16) McLin, M. G.; Angell, C. A. Ion − Paring Effects on Viscosity/ Conductance Relations in Raman − Characterized Polymer Electrolytes: LiClO4 and NaCF3SO3 in PPG(4000). J. Phys. Chem. 1991, 95, 9464−9469. (17) Abarna, S.; Hirankumar, G. Study on new lithium ion conducting electrolyte based on Polethylene glycol-p-tertoctyl phenyl ether and Lithium Perchlorate. Int. J. ChemTech Res. 2014, 6, 5161− 5167. (18) Jadżyn, J.; Czechowski, G. Prenematic Behavior of the ElectricField-Induced Increment of the Basic Thermodynamic Quantities of Isotropic Mesogenic Liquids of Different Polarity. J. Phys. Chem. B 2007, 111, 3727−3729.

Figure 9. Orientational entropy increment ΔS (eq 6) for neat polyethylene glycols as a function of number n of oxyethylene groups in HO−[CH2−CH2−O]n−H polymer. The increments determined26 for dimethyl sulfoxide (○) and acetonitrile (▽) (the molecular dipole moments of about 4 D) are shown in the picture.

compounds of similar dipole moments, as dimethyl sulfoxide or acetonitrile, that is, the compounds composed of small and rigid molecules, exhibit much higher (absolute) values of ΔS (Figure 9). Of course, one must take into account the difference in the dipoles density in the polymeric and monomolecular liquids. However, an unexpected decreasing of the orientational effect (ΔS → 0), observed for increasing dipole moment of HO−[CH2−CH2−O]n−H polymer (increasing n in Figure 9), gives an evidence for an importance of the flexibility of the polymeric chains were the monomers are linked to each other with single chemical bonds. That flexibility of polymers allows them to orient only some parts of the chains, depending on the actual structural possibilities of a given polymeric medium.

4. CONCLUSIONS The paper presents the results of electrical conductivity measurements in dilute solutions of LiClO4 in polyethylene glycols of different molar weights. The main conclusion resulting from the studies concerns the dependence of the conductivity on the degree of polymerization of the polymer matrix. It was found that at the same value of ions density, the electrical conductivity decreases with increasing molar mass of the polyethylene glycol HO−[CH2−CH2−O]n−H, that is, with an increase of n. This effect can be interpreted as the result of decrease in ions mobility in the polymer matrix associated with increased flexibility of longer polyethylene glycols. E



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(19) Jadżyn, J.; Czechowski, G.; Déjardin, J.-L. Dynamics of the SelfAssembling of Mesogenic Molecules in the Prenematic Region of Isotropic Liquid. J. Phys. Chem. B 2008, 112, 4948−4952. (20) Jadżyn, J.; Sokołowska, U.; Déjardin, J.-L. Temperature Behavior of the Electric Field-Induced Entropy Increment within a Homologous Series of Nematogenic Compounds. J. Phys. Chem. B 2008, 112, 9050−9052. (21) Sengwa, R. J.; Kaur, K.; Chaudhary, R. Dielectric properties of low molecular weight poly(ethylene glycol)s. Polym. Int. 2000, 49, 599−608. (22) Sarode, A. V.; Kumbharkhane, A. C. Dielectric relaxation study of poly(ethylene glycols) using TDR technique. J. Mol. Liq. 2011, 164, 226−232. (23) Mali, C. S.; Chavan, S. D.; Kanse, K. S.; Kumbharkhane, A. C.; Mehrota, S. C. Dielectric relaxation of poly ethylene glycol−water mixture using time domain technique. Indian. J. Pure Appl. Phys. 2007, 45, 476−481. (24) Ohki, T.; Harada, M.; Okada, T. Structural and Thermodynamics Aspect of Ionic Solvation in Concentrated Aqueous Poly(ethylene glycol). J. Phys. Chem. B 2007, 111, 7245−7252. (25) Sengwa, R. J.; Sankhla, S. Characterization of ionic conduction and electrode polarization relaxation processes in ethylene glycol oligomers. Polym. Bull. 2008, 60, 689−700. ́ (26) Jadżyn, J.; Swiergiel, J. On Intermolecular Dipolar Coupling in Two Strongly Polar Liquids: Dimethyl Sulfoxide and Acetonitrile. J. Phys. Chem. B 2011, 115, 6623−6628.

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