J. Phys. Chem. B 2007, 111, 11201-11208
11201
Structure-Dependent DC Conductivity and Relaxation Time in the Debye-Stokes-Einstein Equation G. Power,† J. K. Vij,† and G. P. Johari*,‡ Laboratory of AdVanced Materials, Department of Electronic Engineering, Trinity College, UniVersity of Dublin, Dublin 2, Ireland, and Department of Materials Science and Engineering, McMaster UniVersity, Hamilton, ON L8S 4L7, Canada ReceiVed: March 21, 2007; In Final Form: June 7, 2007
The basis for a modification of the Debye-Stokes-Einstein (DSE) equation between the dc conductivity, σdc, and dielectric relaxation time, τ, has been examined by using broad-band dielectric spectroscopy of LiClO4 solutions in 5-methyl-2-hexanol and 1-propanol and of pure liquids. According to the DSE equation, the log σdc-log τ plots should have a slope of -1. We find that σdc begins to depend upon the structure of an electrolytic solution when a variation of solvent’s equilibrium dielectric permittivity, s, with temperature causes the ion population to vary. As a consequence of this intrinsic dependence, the log σdc-log τ plots do not obey the DSE equation. Inclusion of the effect of change in s on the DSE equation may be useful in analyzing the measured quantities in terms of Brownian diffusion of both ions and molecules in ultraviscous liquids. Proton translocation along a hydrogen bond contributes little to σdc, which appears to be predominantly determined by the ion population in the two alcohols and the solutions. The effect is briefly discussed in the potential energy landscape paradigm of structure fluctuations, and it is suggested that the high-frequency shear modulus measurements of ionic solutions would help reveal the temperature-dependent deviation from the DSE equation.
Introduction There are two types of transport processes in ionic solutions; molecular diffusion and ionic conduction. Rotational and translational diffusion of molecules provides a mechanism for viscous flow and dielectric polarization, and translation diffusion of ions provides a mechanism for dc conductivity, σdc. Both are seen as Brownian motions, but they differ in details, because molecules do not have the same shape, charge-distribution, or size as the ions. Moreover, when dc conduction occurs as a result of proton translocation across the hydrogen bonds in the structure of a material, dc conduction mechanism may become temporally related to the breaking of a hydrogen bond after which a molecule may diffuse, and then its reforming with other neighbors. Interpretations of σdc and molecular diffusion, and their interrelation in liquids, gels, solid-liquid suspensions, and polymers, is not only important for academic and technical purposes but also important for the understanding of osmosis in biological processes, transdermal drug delivery, and the functioning of polymer electrolytes. The Einstein equation had related σdc to the product of the concentration and mobility of electrical charge carriers. Historical development of the subject and an alternative derivation of the relation between the ion and molecular transport in liquids has appeared in a monograph.1 Briefly, Einstein’s relation was extended by Nernst, who replaced the mobility by the translation diffusion coefficient DT and included the temperature-dependence, and then by Stokes, who replaced (DT)-1 by a liquid’s viscosity η. Finally Debye replaced η by the dielectric relaxation * Author to whom correspondence should be addressed. E-mail:
[email protected]. † University of Dublin. ‡ McMaster University.
time, τ, as described earlier.2 The resulting equation is known as the Debye-Stokes-Einstein (DSE) equation (see ref 2 for details). According to this equation, σdc is inversely proportional to τ, i.e., a plot of log σdc against log τ, has a negative slope of unity. Since the 1930s, limited data on a large number of studies have shown that liquids do not follow the DSE equation. Its validity has been discussed recently by using more extensive data and an empirical “fractional DSE equation” been suggested.3 In a related but rarely recognized development, the Debye-Huckeltheoryforionicconductioninaqueoussolutions4-10 was modified by Bjerrum,11 who included an electrostatic term to take into account the extent of electrolytic dissociation. The subject has been described in the undergraduate text books of physical chemistry and in several monographs on electrolytic conduction and ion association,4-10 which may be consulted for details. In recent years, dielectric relaxation spectroscopy, which yields both τ and σdc as transport properties, has been performed over a broad frequency range, and hence an increasing amount of data on τ and σdc has become available over a broad temperature range. This has led to a discussion of the relation between log σdc and log τ. It has been reaffirmed that in most cases the slope (dlog σdc)/(dlog τ) is less than unity.3 On the basis of that finding, the DSE equation has been referred to as fractional Debye-Stokes-Einstein equation, which is seen to be a characteristic feature of liquids.3 A significant deviation from this relation has been observed also when τ and σdc are measured as a function of pressure, P, of a liquid,2,3,12 It has been argued2 that the DSE equation for τ and σdc is likely to be incomplete, because of its two implicit assumptions: (i) τ is directly proportional to the viscosity, η, or inversely proportional to the self-diffusion coefficient, DR (subscript R denoting rotational diffusion), and (ii) the instantaneous shear modulus
10.1021/jp072268j CCC: $37.00 © 2007 American Chemical Society Published on Web 09/01/2007
11202 J. Phys. Chem. B, Vol. 111, No. 38, 2007 G∞ in the Maxwell’s relation, η ) τshearG∞, that was used to equate τshear with τ, is a constant. As pointed out earlier,2 neither is τ proportional to η or τshear nor is G∞ a constant. An organic liquid’s σdc varies with the mobility of the impurity ions in it. When intermolecular hydrogen bonds are present, σdc may also arise from proton translocation along the hydrogen bonds. In the DSE equation, variation of σdc is taken to be entirely due to the variation of mobility of the ions (charged particles), whose Brownian diffusion had been considered in the Einstein equation, and ionic mobility was related to η. But since ions have been known to associate to form ionpairs, which do not contribute to σdc, a change in their population with change in T and P must be taken into account. Thus in terms of Bjerrum’s theory, variation of σdc with T and P also has a structural origin, which need to be included in the DSE equation. From simple consideration of electrolytic dissociation, it was shown that2 log[σdc(T,P)τ(T,P)] ) constant + log R (T,P), where R is the degree of ionic dissociation. For T and P conditions when R ) 1, i.e., the impurity electrolyte is fully dissociated, log[σdc(T,P)τ(T,P)] is equal to a constant if G∞ did not vary with T and P, and τ ) τshear. Therefore, variation of log[σdcτ ] with T and P would indicate a varying degree of ionic dissociation. Moreover, deviation of (dlog σdc)/(dlog τ) from a value of -1 would indicate the variation of log R with log T or with log P. Johari and Andersson2,12 have reported a study of mainly the pressure dependence of σdc and τ of an ultraviscous 50% (w/ w) acetaminophen-aspirin melt. Both substances form hydrogen bonds in their pure states and, additionally, they form hydrogen bonds with each other in the mixture with a temperaturedependent equilibrium. Because of that, the conduction mechanism in the acetaminophen-aspirin binary melt2,12 has been seen as complicated. Also, since no electrolyte was added in that study,2,12 the effect of known types of ions on the dielectric behavior of the mixture was not investigated. Here we report a study of pure liquids and the effect of a dissolved electrolyte on their σdc and equilibrium permittivity. For this purpose, the τ data are taken from an earlier study of 1-propanol and its LiClO4 solution, and new data for LiClO4 solutions in 5-methyl2-hexanol are obtained as a function of T. Experimental Methods The methods for measurements, the equipment, i.e., the capacitor that formed the dielectric cell, temperature monitoring and control device, and broad-band dielectric analyzer, and the sources and purity of the monohydroxy alcohols, 5-methyl2-hexanol13 and 1-propanol,14 and the fast diffusing lithium ion salt LiClO4,14 have been described in several earlier papers which may be referred to for details.13,14 Solutions were prepared by accurately weighing the liquid alcohol and solid salt and studying the solutions immediately after their preparation.
Power et al.
*(ω) ) ∞ +
∆I
+ [1 + (iωτI)RI]βI ∆III σdc ∆II + + (1) RII βII RIII βIII 0iω [1 + (iωτII) ] [1 + (iωτIII) ]
where ω ) 2πf is the angular frequency (f being the linear frequency in Hz), i ) (-1)1/2, ∞ is the limiting high-frequency permittivity, which depends on atomic and electronic polarizability of a material, ∆I, ∆II, and ∆III are, respectively, the dielectric relaxation strengths of the processes designated as I, II, and III, τI, τII, and τIII are the corresponding relaxation times, and RI, RII, and RIII are the symmetric broadness parameters (equivalent to the Cole-Cole15 distribution parameter, but written as 0 < R and Rβ e 1), and βI, βII, and βIII are the corresponding Davidson-Cole16 parameters β with subscripts I, II, and III referring to the three relaxation processes in order of decreasing relaxation time. These parameters R and β quantify the symmetric and asymmetric broadening of the dielectric loss peak relative to the Debye relaxation peak for which R ) β ) 1. We have pointed out that relaxation process I (the slowest, exponential or the Debye-type) observed for 1-propanol carries ∼97% of the total polarization. Relaxation processes II and III which are nonexponential, indicating a distribution of relaxation times, together carry the remaining 3% of the polarization. For these, 0 < R < 1 and 0 < β < 1. The details of these processes have been discussed earlier.13,14,17 Briefly, the relaxation spectra of 0.5 mol % LiClO4 solution in a secondary alcohol, 5-methyl2-hexanol, at 160.6 K17 is shown in Figure 1, where its ′ and ′′ have been resolved into σdc and three relaxation processes I, II, and III and the fit parameters are given. The imaginary part of eq 1 is fitted to ′′, and the height of the real part was adjusted to obtain ∞. The frequency, fm, at the ′′ spectra peak was calculated from,
[ (2 +Rπ2β)] [sin(2Rβπ + 2β)]
fm ) (2πτ)-1 sin
1/R
-1/R
(2)
and τ was taken as equal to 1/2πfm. For this investigation, we use τ of the main relaxation process I, which is Debye-like for both the pure liquid and electrolytic solutions, and not τ of process II, whose significance has been put into question by using electrolytic solutions that tend to break the hydrogen bond structure.13 (For completeness, we include a brief discussion of this aspect at the end of discussion section and thus avoid interruption of the main aspect of this study.) Figure 2 shows the plots of log(τ) and log(σdc) against T for 5-methyl-2-hexanol and 0.5 and 1.0 mol % LiClO4-containing solutions. Figure 3 shows the corresponding plots of log(τ) and log(σdc) for 1-propanol and for its 1.0 mol % LiClO4-containing solutions. Discussion
Results In order to accurately determine the features of the lowest frequency Debye-relaxation spectra whose τ is used in this study, the measured dielectric permittivity, ′, and loss, ′′, were analyzed by including all high-frequency relaxation processes. Approximate values of σdc of the solutions at different temperatures were determined first by using the dielectric loss, ′′, data at low frequencies in the range where it varies linearly with the angular frequency ω. It was then refined by using the summation of relaxation times, as given by the relation for the complex permittivity *(ω),
To test the validity of the DSE equation, we plot log(σdc) against log(τ) for 5-methyl-2-hexanol and its 0.5 and 1.0 mol % LiClO4 solutions in Figure 4 (top), and a straight line of slope -1 is drawn for comparison. The corresponding plots for 1-propanol and its 1.0 mol % LiClO4 solution are shown in Figure 5 (top). In both figures, each data point refers to a certain T of the plots given in Figures 2 and 3. The slope, [dlog(σdc)]/ [dlog(τ)], of the curves at each point of the three data sets is plotted against T in Figures 4 and 5 (bottom panels). For both solutions at low temperatures, this slope is close to -1 for small σdc and τ values, and at high temperatures it systematically
Structure-Dependent DC Conductivity
Figure 1. Resolution of the permittivity and loss spectra of 5-methyl2-hexanol containing 0.5 mol % LiClO4 at 160.6 K. The spectra are resolved into three components I, II, and III, and the dc conductivity. The parameters used are ∆I ) 27.8, τHNI ) 4.59 s, RI ) 1.00, βI ) 0.957, ∆II ) 0.795, τHNII ) 92.7 ms, RII ) 0.894, βII ) 0.499, ∆III ) 0.102, τHNIII ) 1.07 µs, RIII ) 0.524, βIII ) 1.00, ) 2.41, and σ dc ) 1.22 × 10-11 S m-1.
Figure 2. (top) Logarithmic plots of the dielectric relaxation time of pure 5-methyl-2-hexanol and 0.5 mol % and 1.0 mol % LiClO4 solutions corresponding to the Debye-relaxation process against the temperature. (bottom) The corresponding plot of the dc conductivity. τ is in s and σdc in units of S m-1.
deviates from -1 for large σdc and τ values. Note that the scatter in the slope is relatively high for 1-propanol, whose σdc becomes too low to be measured accurately. First we note that there is no significant change in τ when LiClO4 is dissolved in 5-methyl-2-hexanol (Figure 2) and in 1-propanol, but there is a large increase in σdc. The data in Figure 2 show that at 187 K, τ remains at ∼0.6-0.7 ms, but σdc increases from 2.4 × 10-10 S m-1 for pure 5-methyl-2-hexanol to 8.9 × 10-8 S m-1 for 1 mol % LiClO4 solution, i.e., by a factor of ∼370. Correspondingly, at 131-132 K, τ remains at
J. Phys. Chem. B, Vol. 111, No. 38, 2007 11203
Figure 3. (top) Logarithmic plots of the dielectric relaxation time of pure 1-propanol and 1.0 mol % LiClO4 solution corresponding to the Debye-relaxation process against the temperature. (bottom) The corresponding plot of the dc conductivity. τ is in s and σdc in units of S m-1.
Figure 4. (top) Logarithmic plots of the dc conductivity of pure 5-methyl-2-hexanol and 0.5 mol % and 1.0 mol % LiClO4 solutions against the dielectric relaxation time. Each data point refers to a certain temperature in Figure 1. The straight line drawn has a slope of -1. (bottom) The slope of the two plots in the top panel, (dlog10 σdc /dlog10 τ), is plotted against the temperature. τ is in s and σdc in units of S m-1.
0.8-0.5 ms, but σdc increases from 5.0 × 10-10 S m-1 for pure 1-propanol to 5.6 × 10-7 S m-1 for its 1 mol % LiClO4 solution, i.e., by a factor of ∼1100. This difference itself indicates that even when the τ remains relatively unchanged, there is a large effect of addition of ions on σdc, which is inconsistent with the DSE equation. As mentioned earlier here, according to the DSE
11204 J. Phys. Chem. B, Vol. 111, No. 38, 2007
Power et al. nion the number of ions per unit volume, µion the mobility of the ion, and the summation is for all ions of the ith type. In this relation, σdc has an additional dependence on nion, which is structural. For the anions and cations with the same valency of 1 in the simplest ionic equilibrium of the type AB T A+ + B-, it has been deduced that2
log10σdc ) log10(2ecg1) + log10R - log10τ
(4)
where c is the number density of AB molecules prior to ionic dissociation, g1 is part of the pre-exponential factor that converts the Vogel-Fulcher-Tamman equation18-20 for τ to the corresponding equation for µion, and R is the degree of electrolytic dissociation [or (1 - R) the degree of ion association]. The log R term is the structural term and varies with T. Thus for a liquid containing impurities or added electrolytes, the DSE equation becomes
log(σdcτ) ) constant + log R
(5)
and
dlog σdc dlog R dlog T ) -1 + dlog τ dlog T dlog τ
(
Figure 5. (top) Logarithmic plots of the dc conductivity of 1-propanol and 1.0 mol % LiClO4 solution against the dielectric relaxation time. Each data point refers to a certain temperature in Figure 4. The straight line drawn has a slope of -1. (bottom) The slope of the two plots in the top panel, dlog10 σdc/dlog10 τ, is plotted against the temperature. τ is in s and σdc in units of S m-1.
equation, the log(σdc) against log(τ) plot should have a slope of -1. In contrast, the plots in Figures 4 and 5 show that there is a considerable deviation from this value for both the alcohols and the solutions, and this deviation increases with increase in T for pure 5-methyl-2-hexanol and LiClO4 solutions in Figure 4. But the deviation from the DSE equation decreases with increase in T for pure 1-propanol and LiClO4 solution in Figure 5, where the slope is equal to -1 at high temperatures, which is surprising in view of the fact that ultravisocus liquid 1-propanol is seen to have an extensively hydrogen structure, and therefore electrical conduction in it also involves proton translocation along the H-bonds. The DSE equation was originally derived for the mobility of electrically charged entities, and it had been deduced for solids and liquids that were mainly ionic and in which the ion population was taken to remain constant with changing T. The detailed treatment of electrolytic dissociation equilibria came after the development of the DSE equation. Therefore, the effect of ion population variation with T remains to be included in the DSE equation. Bjerrum11 had shown that the dissociation constant Kd of an electrolytic solution varies not only with T but it also varies with the dielectric permittivity of the solvent. It was recently considered2 how the dielectric permittivity can alter the DSE equation Via the ion population change. The formalism has been given earlier,2 but briefly the dc conductivity of a condensed phase is written as,
σdc )
∑ zion,ienion,iµion,i
(3)
where zion is the valency, e the elementary electronic charge,
)
(6)
Equation 6 reduces to the DSE equation when, (i) electrolytic dissociation is complete, i.e., log R ) 0, and, (ii) R remains constant with changing T. With decrease in T, (dlog R)/(dlog T) and (dlog T)/(dlog τ) of an ultraviscous liquid behave in opposite ways. Usually, (dlog T)/(dlog τ) is always negative and becomes very small at T close to Tg as τ increases rapidly with decrease in T. Therefore the product of the two itself varies with T and the sign of (dlog R)/(dlog T), which may be small and close to zero such that the second term in eq 6 itself becomes negligibly small. The degree of ionic dissociation for AB T A+ + Bequilibrium is related to the dissociation constant, Kd, by
Kd )
R2 , or R ) [(1 - R)Kd]1/2 (1 - R)
(7)
A number of expressions have related Kd to the ion-size parameter, ionic charge, and the dielectric permittivity of the solvent, and these have been reviewed and tested variously.4-11,21 As there has been no agreement on the most acceptable relation, we use the Bjerrum equation as a zeroth-order relation between Kd and the electrostatic properties. This gives or equivalently
(
z1z2e2 R2 3000 exp ) Kd ) aionskBT (1 - R) 4πNAaion3 R ) (1 - R)1/2
[
(
z1z2e2 3000 exp aionskBT 4πNAaion3
)]
)
(8a)
1/2
(8b)
where NA is the Avogadro number, aion is the ion-size parameter, z1 and z2 are the valencies of ions 1 and 2, kB is the Boltzmann constant, e is the electronic charge, and s the equilibrium permittivity of the solvent. For the condition that aion remains constant with changing T, eq 8a predicts that log Kd would decrease linearly with increase in 1/sT, as discussed in ref 2. Since the term sT appears in the denominator of an exponential term in eq 8a, Kd and R vary sensitively with sT. Therefore, σdc and τ vary both intrinsically with T and extrinsically with T
Structure-Dependent DC Conductivity
J. Phys. Chem. B, Vol. 111, No. 38, 2007 11205
through the variation of the equilibrium dielectric permittivity, s, in addition to the viscosity. The term sT determines the population of ions or R in an electrolytic solution, and therefore we now discuss how s of liquids varies with T. For a relatively small number of liquids, it varies as 1/T, so that sT remains constant with changing T, but for most liquids s follows the equation based on a statistical theory of dielectric polarization developed by Onsager,22 Kirkwood,23 and Fro¨hlich.24 Accordingly,
s ) ∞ +
[
sT ) ∞T +
]( ) [ ]( )
3s ∞ + 2 2 4πNAF gµ 2 2s + ∞ 3 3kBTM 0 3s ∞ + 2 2 4πNAF g µ02 2s + ∞ 3 3kBM
(9)
(9a)
where ∞ is the limiting high-frequency permittivity of the orientation polarization associated with a certain relaxation process, NA is the Avogadro number, F is the density, M the molecular weight, kB the Boltzmann constant, T the temperature, and µ0 is the vapor-phase dipole moment of a molecule. The quantity g is exactly defined as the sum of the dipole vectors of neighboring molecules.23 It is equal to 1 when there is no intermolecular association and no correlation of dipole vectors. When g > 1, there is a predominantly parallel dipolar correlation and when g < 1, a predominantly antiparallel dipolar correlation. When the ROH molecules associate intermolecularly by Hbonds to form linear chain structures, g > 1. Significance of this analysis, albeit done in the form of a chemical equilibrium between H-bonded and non H-bonded species, lies in both (i) the implication that (non-permanent) linear chains of intermolecularly H-bonded structures may occur on the time average leading to large s values, and (ii) intermolecularly H-bonded ring-dimers form leading to an s close to that of a nonpolar liquid at low temperatures. Both of these affect the relaxation characteristics of the liquid. According to the Clausius Mossotti equation for molecular polarizability (see ref 25 for a discussion of this equation), Rmol ) [(∞ - 1)Vm/(∞ + 2)], or ∞ ) [(Vm + 2Rmol)/(Vm - Rmol)]. Since Rmol is a constant, it follows that for a linear variation of Vm with T, ∞ would increase slightly with decrease in T and the product ∞T would slightly decrease with decrease in T. But the second term on RHS of eq 9a would increase much more rapidly as the terms (3s)/(2s + ∞) and F increase for a fixed value of g, and increase even more as g usually increases with decrease in T. Hence, sT increases with decrease in T. i.e., d(sT)/dT is negative and its magnitude varies with the temperature range of measurements. The quantity d(sT)/dT is low when the transport properties are measured at higher T, and it is high when the properties are measured at lower T. In contrast, s for non-dipolar liquids varies little with change in T and hence sT slightly decreases with decrease in T, as is the case for ∞T. We now consider the effect of change in sT on the σdc of an ionic solution. If sT were to remain constant with changing T, Kd and R in eqs 8a and 8b would remain constant. As the ion population would remain constant in this case, only the ion mobility would change with changing T. This would satisfy the conditions for the DSE equation, and log(σdc) may then vary inversely with log(τ). But since sT for most liquids decreases with increasing T, Kd and R would decrease with increasing T. Variation of the product sT with T is a characteristic property of a liquid. We propose that this variation in Kd and R is structural, varying both with a liquid’s structure and with T,
Figure 6. (top) The plot of Ts for 5-methyl-2-hexanol and 0.5 mol % and 1.0 mol % LiClO4 solutions against the temperature. (bottom) The corresponding plots of log(τσdc) against the temperature. τ is in s and σdc in units of S m-1.
which, as found for the ionic conductivity of solids, is in addition to the variation with T alone. With the above given significance of the sT term, we now investigate how its magnitude varies with T. For 5-methyl-2hexanol and two LiClO4 solutions, s is taken from ref 14, and the product sT is plotted against T in Figure 6 (top). For the pure liquid, sT decreases rapidly with increase in T, but for the 0.5 mol % solution, for which data are available only over a smaller T range, the decrease is less. For the 1.0 mol % solution there is no decrease within at most (2%, the error in determining s. Therefore, according to eqs 8a and 8b, Kd and R for pure 5-methyl-2-hexanol would vary greatly with T, lesser for the 0.5 mol % solution and will remain nearly constant for 1 mol % solution. The plots in Figure 4 (bottom) show that the slope, (dlog10 σdc)/(dlog10 τ) deviates most from -1 and that the deviation becomes less with increase in the amount of LiClO4 in 5-methyl-2-hexanol, as the slope of -1 gradually changes to -0.5 as T is increased and τ decreases. It should be stressed that the variation of the slope with T and/or τ is an additional violation of DSE equation than has been discussed by Roland et al.,3 where this violation was reported for low temperatures with a transition to (dlog10 σdc)/(dlog10 τ) of ∼-0.7. For 1-propanol and 1.0 mol % LiClO4 solution, the s data are taken from ref 13, and sT is plotted against T in Figure 7 (top). Here the behavior is opposite, i.e., with increasing T, the quantity sT for pure 1-propanol remains constant within (2%. But for the 1 mol % solution it decreases almost asymptotically. This means that according to eqs 8a and 8b, Kd and R for 1-propanol would not vary greatly with T, but those for the LiClO4 solution in it would rapidly decrease with increase in T. Therefore for 1-propanol, (dlog R)/(dlog T) is negligible or close to zero, and the DSE equation would appear to be valid. This is evident from the slope of the plot for 1-propanol up to log τ ) -1.7 in Figure 5 (top). For 1 mol % solution, the slope
11206 J. Phys. Chem. B, Vol. 111, No. 38, 2007
Figure 7. (top) The plot of Ts for 1-propanol and 1.0 mol % LiClO4 solution against the temperature. (bottom) The corresponding plots of log(τσdc) against the temperature. τ is in s and σdc in units of S m-1.
of the plot at high temperatures, where log τ values are in the range -6 to -3, is equal to -1. It becomes steeper at lower T or longer τ. Since s becomes more sensitive to T, as the temperature is decreased, one expects that the change in R would be the largest at lower T or longer τ. Although the data at lower T show substantial decrease in the slope in the plot of Figure 5 (bottom), the scatter in the data itself is large and the slope cannot be estimated appropriately without smoothing of the data in the top panel, which was not done. The analysis of the results shows that perhaps the Bjerrum equation for variation of Kd on s may need to be modified, as has been discussed earlier,4-10 or else another equation be used for determining the role of electrolytic dissociation in solutions. However, there are at least four other aspects that would further limit the validity of the DSE equation: (i) Change in the conduction mechanism from ion transport to proton-conduction, (ii) the T-dependent hydrogen-bond equilibrium in the liquid and the consequent proton diffusion contribution to σdc that would itself vary with the ion-population as Coulombic interaction affects hydrogen bonding, (iii) the T-dependence of G∞26,27 in the Maxwell’s relation, τshear ) η/G∞, with the consequence that the relationship between η, D-1, and τ is nonlinear and hence the first term on RHS in eq 5 would not remain constant with changing T, and (iv) the assumption of equality or proportionality of the mechanical and dielectric response times. While variation of G∞ with T could have a minor effect on the constant term in eq 5, the inequality of dielectric and mechanical relaxation times may have a larger effect. Further measurements of σdc, η, τ, and G∞ of liquids would show the relative magnitude of these corrections. All four effects would add to the nonlinearity of log σdc-log τ plots and a slope different from -1. We also consider several related effects. As the ion population varies with T, the liquid’s structure is altered, and G∞ would vary with T. In the theory of molecular interactions, G∞ is determined by the curvature of the intermolecular potential minima which determines the height of the energy barrier for configurational fluctuations and hence τ. In contrast, σdc is a determined by both the height of the potential barrier for migration of ions and the population of ions. Therefore, it is not clear how G∞ and σdc could be related in a simple manner.
Power et al. One possibility may be that the curvature of the potential function varies with the concentration of ions. This then brings us to Goldstein’s potential energy barrier picture28 or the potential energy landscape description.29 In this paradigm, the thermodynamic state point of a pure liquid fluctuates by transitions from one minimum to another minimum in the landscape over the potential energy barriers. The process is thermally activated, and the time scale of vibrations in one minimum is clearly separated from the time scale for transition from one minimum to another. The curvature of the minimum determines its vibrational frequency and hence G∞. In the context of this study, when the ion population remains constant with changing T, fluctuation of the liquid’s state point occurs via transitions between various minima that change the ion positions randomly and thus molecular and ionic configurations change together. But when T is decreased, the ion population decreases, the liquid’s state point shifts to a deeper minima of higher curvature in another part of the potential energy landscape determined by the ion population at that T. Thus a change in ion population with T shifts a liquid’s state point to parts of the landscape corresponding to the new ion population. This means that for a given decrease in the temperature G∞ would increase not only because the state point shifts to a deeper minimum in a given part of the potential energy landscape because of the change in T but also because it shifts to a deeper minimum of another part of the energy landscape determined by the new ion concentration. Effects of electrolytes on G∞ of liquids has not been studied in detail. Electrolytic dissociation in aqueous solutions has been determined by ultrasonic methods, but the velocity of propagation of ultrasonic waves in such solutions does not yield G∞, because the frequency used is too low and the temperatures are such that there is a dispersion of sound wave velocity in liquids. With the current techniques available for performing such studies by Brillouin and other scattering methods, it would be useful to examine how shear modulus of a liquid changes with the electrolyte concentration, and how such changes effect the ionic equilibrium on variation of temperature. We plan to perform such a study. Finally we discuss why we have used both the s and τ values for the Debye relaxation in the analysis given here. This is necessary in the face of suggestions that this relaxation may not represent the molecular diffusion time in the same sense as the non-Debye relaxation for alcohols,30 and that it does not seem to kinetically unfreeze at the calorimetric Tg when the glassy state of some of the alcohols is heated.31,32 Such suggestions have been motivated by an attempt to maintain, (i) that there is a dynamic heterogeneity in the structure of ultraviscous liquids, that assumes that the non-Debye structural relaxation kinetics is due to a distribution of independently relaxing nanoregions which relax at different rates,33,34 and, (ii) that the non-Arrhenius behavior of the relaxation time is (empirically) related with the non-Debye relaxation.35 It was originally suggested that micelle type structures may be present in these pure supercooled alcohols.31 But it has been pointed out that the Debye-relaxation in pure monohydroxy alcohols cannot be attributed a priori to formation of micelles in their liquid’s structure.17 We further point out that presence of micelles in a liquid would cause it to show a non-Newtonian flow, which has not been observed. Also some of the long chain alcohols, such as a variety of isomeric octanols which are expected to show a greater ability to form micelles have indeed shown a non-Debye behavior of their slowest relaxation process.36 Still, it may seem tempting to suggest the possibility
Structure-Dependent DC Conductivity of micelle formation by agglomeration of the hydrogen-bonded molecules, and then to suggest the possibility of difference between the proton conduction within the micelle and in the bulk. But, this would produce an interfacial polarization at the micelle-liquid interface, and there has been no evidence for presence of such micelles from light scattering techniques or simple optical observation. In this context, it is remarkable that a Debye-relaxation has now been observed for the slowest dielectric relaxation process in the ultraviscous state of a dilute solution of di-n-butyl ether in 3-methylpentane, and its total relaxation spectra has been also resolved into three relaxation regions,37 as was done for 1-propanol.30 This demonstrates that hydrogen bonding in a liquid is unnecessary for the occurrence of a Debye-relaxation process. Since its τ also shows a Vogel-Fulcher-Tamman type temperature dependence over a wide temperature range down to a temperature approaching Tg, its features conflict with the above-mentioned view on the dynamic heterogeneity in an ultravsicous liquid’s structure33,34 and the apparent correlation between the nonexponentiality and non-Arrhenius behavior.35 The data for di-n-butyl ether were interpreted in terms of the Anderson-Ullman38 model for environmental fluctuation, as was originally done for monohydroxy alcohols.39 More recently, a different explanation has been given:32 “the Debye process corresponds to a transition among states which differ in energy only in the case of an electrical field,” i.e., “a polarization process that involves states which differ in energy level only if an external field is applied.”32 We point out that this is true in general for the dielectric and mechanical relaxation processes in all materials, in which an applied electrical or mechanical stress produces energetically different states in terms of electrical and elastic dipole orientations, and thereafter removal of the stress returns the system to its random orientation or energy state, a process seen as relaxation, and expressed in terms of energy barrier models. An apparent support for the view that the Debye relaxation differs from structural relaxation has been put forward by comparing the temperature at which τ of a Debye-type dielectric relaxation is ∼100 s against the calorimetric Tg measured by heating at a certain rate. For example, it was argued that31 “more specifically, a detailed DSC study of 1-propanol resulted in Tg ) 96.2 K,53 while the kinetic Tgs are 101.9 K for the Debye process and 97.0 K for the smaller R-process.” This analysis is misleading because the calorimetric Tg of 1-propanol taken from Takahara et al.’s study40 is not from DSC measurements but from an adiabatic calorimetry study where the cooling rate in the Tg range was 0.2-0.4 K/min and “heating” rate, which corresponds to the adiabatic calorimetry, was much slower than the cooling rate. It is well recognized that the Tg measured on heating is lower when the heating rate is lower. In adiabatic calorimetry or in another method using slow heating rates, Tg appears at a relaxation time much longer than 100 s. Moreover, calorimetric and dielectric relaxation times for liquids generally do not agree, as reviewed by McKenna41 and is now generally known. Calorimetric relaxation also has a much broader distribution of times or greater nonexponentiality than dielectric relaxation in general. This raises the issue of different dynamic heterogeneities for the same liquid inferred from the two methods. Similar difficulties of analysis appear in another comparison using data for 2-ethyl-1-hexanol.32 In this study, the temperatures at the extrapolated relaxation time of 100 s in Figure 2, ref 32 yields Tg-kin-R for the Non-Debye relaxation as ∼142 K, Tgkin-D for the Debye relaxation as ∼153 K and Tg from the onset
J. Phys. Chem. B, Vol. 111, No. 38, 2007 11207 temperature of 20 K/min heating DSC scan as ∼147 K and from its midpoint temperature as ∼149 K. The τ data determined from fixed, high-frequency dynamic heat capacity scans were taken to agree with the dielectric τ data for the non-Debye relaxation, but the two sets of data do not follow the same fitting equation and would not lead to the same extrapolated value of T for 100 s at Tg-kin-R. Moreover, dielectric and dynamics heat capacity scans42 of polymers have already shown that dielectric and calorimetric relaxation times differ (see Figure 2 in ref 42). In summary, support for the suggestion that the Debye relaxation represent molecular motions of a type different from the non-Debye process is unreliable, and hydrogen bonding is unnecessary for the observation of a Debye relaxation in an ultraviscous liquid. For these reasons we have used the permittivity and relaxation time values of the Debye relaxation in our analysis. Conclusions As an alternative to empirical “fractional DSE equation” the results of ionic solutions show a need for completing the DSE equation by including the liquid’s structure-dependent electrostatic effects on the electrolytic dissociation constant and then refining it by including the temperature-dependent instantaneous shear modulus and protonic conduction. The new relation between σdc and τ would certainly be more complicated than the current one. Clearly, rotational diffusion of molecules determines τ and translational diffusion of ions determines σdc. But, since ions usually have a different shape and size than molecules, a liquid’s configurational fluctuations due to Brownian diffusion of the two are expected to differ. This difference needs to be considered in the interpretation of σdc and τ in terms of the potential energy landscape. Studies in the future should also consider how the high-frequency shear modulus of ultraviscous liquid changes with temperature to order to investigate the merits of a presumed linear relation between the viscosity and relaxation time and how dissolving electrolytes effect this modulus. Acknowledgment. Postdoctoral Fellowship of Gerard Power for 2005-06 was funded from the Science Foundation of Ireland grant (02/IN.1//I 031). G.P.J. would like to thank Trinity College, Dublin, for their hospitality during his stay there for this study. Note Added After ASAP Publication. This article was published ASAP on September 1, 2007. Reference 35 has been modified. The correct version was published on September 11, 2007. References and Notes (1) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic Press: London, 1986. (2) Johari, G. P.; Andersson, O. J. Chem. Phys. 2006, 125, 124501. (3) Roland, C. M.; Hensel-Bielowka, S.; Paluch, M.; Casalini, R. Rep. Prog. Phys. 2005, 68, 1405. (4) Davies, C. W. Ion Association; Butterworths: London, 1962. (5) Monk, C. B. Electrolytic Dissociation; Academic Press: New York, 1961. (6) Robinson, R. A.; Stokes, R. H. Electrolytic Solutions; Butterworths: London, 1959; pp 396, 551. (7) Fuoss, R. M.; Accascina, F. Electrolytic Conductance; Interscience: New York, 1959. (8) Justice, J.-C. Pure Appl. Chem. 1985, 57, 1091. (9) Barthel, J. M. G.; Krienke, H.; Kunz, W. Physical Chemistry of Electrolytic Solutions: Modern Aspects; Springer: New York, 1998. (10) Bockris, J. O’M.; Reddy, A. K. N. Modern Electrochemistry, 2nd ed.; Plenum: New York, 1998, Vol. I.
11208 J. Phys. Chem. B, Vol. 111, No. 38, 2007 (11) Bjerrum, N. K. Danske Vidensk. Selsk., Mat.-Fys. Medd. 1926, 7, 1. ‘Niels Bjerrum Selected Papers’; Einer Munksgaard: Copenhagen, 1926; p 108. (12) Andersson, O.; Johari, G. P.; Shanker, R. M. J. Pharm. Sci. 2006, 95, 2406. (13) Power, G.; Johari, G. P.; Vij, J. K. J. Chem. Phys. 2002, 116, 4192. (14) Kalinovskaya, O. E.; Vij, J. K. J. Chem. Phys. 2000, 112, 3262. (15) Cole, K. S.; Cole, R. H. J. Chem. Phys. 1941, 9, 341. (16) Davidson, D. W.; Cole, R. H. J. Chem. Phys. 1950, 18, 1417. (17) Power, G.; Johari, G. P.; Vij, J. K. J. Chem. Phys. 2007, 126, 034512. (18) Vogel, H. Phyzik Z. 1921, 22, 645. (19) Fulcher, G. S. J. Am. Ceram. Soc. 1923, 8, 339. (20) Tammann, G.; Hesse, W. Z. Anorg. Allgem. Chem. 1926, 156, 2454. (21) Fuoss, R. M. J. Am. Chem. Soc. 1958, 80, 5059. (22) Onsager, L. J. Am. Chem. Soc. 1936, 58, 1486. (23) Kirkwood, J. G. J. Chem. Phys. 1939, 7, 911. (24) Fro¨hlich, H. Theory of Dielectrics; Oxford University Press: London, 1949. (25) Scaife, B. K. P. Principles of Dielectrics, revised edition; Clarendon: Oxford, 1998. (26) Scarponi, F.; Comez, L.; Fioretto, D.; Palmieri, L. Phys. ReV. B. 2004, 70, 054203. (27) Lind, M. L.; Duan, G.; Johnson, W. L. Phys. ReV. Lett. 2006, 97, 015501.
Power et al. (28) Goldstein, M. J. Chem. Phys. 1969, 51, 3728. (29) Wales, D. J. Energy Landscape with Application to Clusters, Biomolecules and Glasses; Cambridge Univ. Press: Cambridge, U.K., 2003. (30) Hansen, C.; Stickel, F.; Berger, T.; Richert, R.; Fischer, E. W. J. Chem. Phys. 1997, 107, 1086. (31) Wang, L.-M.; Richert, R. J. Chem. Phys. 2004, 121, 11170. (32) Huth, H.; Wang, L.-M.; Schick, C.; Richert, R. J. Chem. Phys. 2007, 126, 104503. (33) Sillescu, H. J. Non-Cryst. Solids 1999, 243, 81. (34) Sillescu, H.; Bo¨hmer, R.; Diezemann, G.; Hinze, G. J. Non-Cryst. Solids 2002, 307-310, 16. (35) Bo¨hmer, R.; Ngai, K. L.; Angell, C. A.; Plazek, D. J. J. Chem. Phys. 1993, 99, 4201. (36) Dannhauser, W. J. Chem. Phys. 1968, 48, 1911. (37) Huang, W.; Richert, R. J. Chem. Phys. 2006, 124, 164510. (38) Andersson, J. E.; Ullman, R. J. Chem. Phys. 1967, 47, 2178. (39) Johari, G. P.; Dannhauser, W. J. Chem. Phys. 1969, 50, 1862. (40) Takahara, S.; Yamamuro, O.; Suga, H. J. Non-Cryst. Solids 1994, 171, 259. (41) McKenna, G. B. Glass formation and glassy behaviour. In ComprehensiVe Polymer Science; Booth, C., Price, C., Eds.; Pergamon: Oxford, 1989; vol. 2, pp 311-362. (42) Beiner, M.; Kahle, S.; Hempel, E.; Schroeter, K.; Donth E. Europhys. Lett. 1998, 44, 321.
