J. Phys. Chem. B 2006, 110, 26211-26214
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On Enhanced Translational Diffusion or the Fractional Stokes-Einstein Relation Observed in a Supercooled Ionic Liquid K. L. Ngai* NaVal Research Laboratory, Washington, DC 20375-5320 and Dipartimento di Fisica and INFM (UdR Pisa), UniVersita` di Pisa, Largo B. PontecorVo 3, I-56127, Pisa, Italy ReceiVed: August 29, 2006; In Final Form: October 5, 2006
From their experimental studies of the supercooled molecular ionic liquid 1-butyl-3-methylimidazolium hexafluorophosphate (BMIM-HFP), Ito and Richert [J. Phys. Chem. B 2006, in press.] found that the StokesEinstein and the Debye-Stokes-Einstein laws do not hold. Instead, enhanced translational diffusion or fractional Stokes-Einstein and fractional Debye-Stokes-Einstein relations are observed, just like in nonionic glass-forming liquids, including 1,3-bis(1-naphthyl)-5-(2-naphthyl)benzene, o-terphenyl, and sucrose benzoate. The comprehensive measurements made by Ito and Richert have determined the critical parameters that the coupling model needs to explain the observed fractional Stokes-Einstein and fractional Debye-StokesEinstein relations in the supercooled molecular ionic liquid.
1. Background Measurements of dynamics in glass-forming liquids indicate the shear viscosity, η; the self-diffusion coefficient, D;1 and the rotational correlation time,2 τc all slow down dramatically with decreasing temperature. However, the products Dη and Dτc are not constant but increase as the temperature is lowered toward Tg, as seen in 1,3-bis(1-naphthyl)-5-(2-naphthyl)benzene (TNB) and other glass-formers.3-11 Hence, there is enhanced translational diffusion, and the data do not satisfy the Stokes-Einstein (SE)1 and the Debye-Stokes-Einstein (DSE)2 relations. One explanation offered is based upon the spatially heterogeneous dynamics in supercooled liquids.4 It assumes that regions of differing dynamics give rise to the Kohlrausch relaxation function, exp[-(t/τi)1-ni], in ensemble averaging measurements. The decoupling between self-diffusion and rotation occurs because D and τc are averages over different moments of the distribution of relaxation times, with D ∝ 〈1/τ〉 emphasizing fast dynamics, while τc ∝ 〈τ〉 is determined predominantly by the slowest molecules. In order for this explanation to be consistent with the observed monotonic increases of the products Dη and Dτc as the temperature is lowered toward Tg, the breadth of the relaxation time distribution has to increase (or the Kohlrausch exponent, 1-n, has to decrease) correspondingly. However, Richert and co-workers6 recently reported that the dielectric spectra of TNB are characterized by a temperatureindependent width (e.g., (1-nd) is constant equal to 0.50) from 345-417 K. The Tg of TNB is 342 K. Photon correlation spectroscopic and NMR12 measurements all indicate a temperature-independent distribution of relaxation times. Similar results of enhanced translational diffusion and break down of SE and DSE relations but temperature independent (1 - nd) were found in other glass-formers, including o-terphenyl7 and in sucrose benzonate.9 Thus, the data of TNB, o-terphenyl, and sucrose * Corresponding author. Phone: 202-767-6150. Fax: 202-767-0546. E-mail:
[email protected].
10.1021/jp065601c
benzonate contradict the explanation based on spatial heterogeneities, as well as results from molecular dynamics simulations. A different explanation was offered by the coupling model (CM).10 It was pointed out10,11,13 that the breakdown of the SE and DSE relations in glass-forming liquids are special cases of a more general phenomenon that different dynamic observables µ weigh the many-body relaxation differently and have different coupling parameters nµ (i.e., different degrees of intermolecular cooperativity) that enter into the stretch exponents of their Kohlrausch correlation functions,14
〈µ(0) µ(t)〉/〈µ2(0)〉 ) exp[-(t/τµ)1-nµ]
(1)
In the CM, the observed Kohlrausch relaxation time, τµ, is related to the primitive relaxation time, τ0µ, by the relation10,15,16
τµ(T) ) [tc-nµτ0µ(T)]1/(1-nµ)
(2)
where tc ≈ 2 ps for molecular and polymeric glass-formers. Applying eq 2 to each observable µ, we can immediately see that the observable having a larger nµ will bestow a stronger temperature dependence for its relaxation time τµ. This is because the primitive relaxation times of all observables τ0µ, uninfluenced by many-body relaxation dynamics, should have one and the same temperature dependence. To demonstrate enhanced translation diffusion by the CM, we need to know not only either nη for viscosity, nd for dielectric relaxation, or nNMR for NMR, but also nD for diffusion.10 Unfortunately, for translational diffusion, so far only the diffusion constant, D, was available from experiments.3,4,9,12 The complete time dependence of the correlation function for self-diffusion or probe diffusion of TNB, o-terphenyl, and sucrose benzonate have not been measured, and nD cannot be determined. Theoretical arguments have been given before10 to show that nη is larger than nD, but this is by no means certain until experimental data prove it in the future. Nevertheless, assuming nD < nη or nD < nd, from the CM eq 2, when applied to variables η or d for µ
This article not subject to U.S. Copyright. Published 2006 by the American Chemical Society Published on Web 12/02/2006
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and compared with that for variable D, it follows that τRη or τRd has a stronger temperature dependence than τRD.10 This is because in the CM all primitive relaxation times, τ0µ, have same temperature dependence, i.e.,
τ0D(T) ∝ τ0d(T) ∝ τ0η(T)
(3)
Hence, we have an explanation of the breakdown of the SE and DSE relations. This CM explanation holds whether nd and nη are temperature-dependent or -independent, as long as nD < nη or nD < nd. In this manner, we have explained17 quantitatively for TNB the stronger temperature dependences of τRη and τRd than that of τRD by taking a constant nη ) nd ) 0.50 as found by Richert and co-workers by dielectric relaxation measurements6 and assuming nD ) 0.37. In spite of the success, the undesirable and arbitrary assumption of a value of nD less than nη or nd has to be made10,17 in order to explain the breakdown of the SE and DSE relations. A comprehensive experimental study of a supercooled molecular ionic liquid, 1-butyl-3-methylimidazolium hexafluorophosphate (BMIM-HFP), [bmim+] [PF6-], by Ito and Richert18 offers an opportunity to avoid making this assumption. This is because the molecules in ionic liquids are charged, their dynamics of translational diffusion can be fully characterized by ionic conductivityrelaxationmeasurementasafunctionoffrequency.18-21 The coupling parameter nD is obtainable by fitting the frequency dependence of the ionic conductivity relaxation to the Fourier transform of the Kohlrausch function of eq 1. Ito and Richert18 made isothermal conductivity relaxation measurements as a function of frequency on BMIM-HFP. From the data they deduced the electric modulus relaxation time, τM, and the coupling parameter nM. Since τM is proportional to τD and nD is the same as nM,22 and thus, from the conductivity relaxation data of Ito and Richert, we can deduce nD as well as the temperature dependence of τD. In addition, Ito and Richert measured the solvation dynamics and rotational dynamics of a probe molecule as a function of time. From the correlation functions they obtained the solvation and rotational relaxation times, τsol and τrot, and the corresponding coupling parameters, nsol and nrot. They showed that τsol and τrot follow the temperature dependence of η/T obtained by Xu et al.24 for more than ten decades, from subnanoseconds at room temperature to seconds near the glass transition temperature Tg. On the other hand, τM (and hence τD) follows this trend only for temperatures T > 1.2Tg, but its temperature dependence becomes significantly weaker than η/T in the 1.1Tg > T > Tg range. In fact, τD ∝ τM ∝ η0.73/T, i.e., we have a fractional SE or DSE relation in the 1.1Tg > T > Tg range. This deviation is the same as the enhanced translational diffusion or fractional Stokes-Einstein behavior observed in TNB, o-terphenyl, and sucrose benzoate. All parameters to test the CM explanation of the enhanced translational diffusion are known from the experimental data reported by Ito and Richert. The natural course of action is to carry out such a test, and this is the objective of this paper. 2. Enhanced Translation in BMIM-HFP BMIM-HFP is the ionic liquid extensively studied by different groups with different techniques.18,19,24-26 Ito and Richert18 measured the response of BMIM-HFP in the solvation of the solute molecule quinoxaline (QX) at a probe level concentration of around 10-4 mol/mol. The solvation dynamics was monitored as a function of time and the result was fit by the Kohlrausch function18
C(t) ) exp[-(t/τsol)βsol] βsol ) 0.30 ( 0.03
(4)
The characteristic solvation time, τsol, was obtained for 15 different temperatures between 194 and 205 K. The Kohlrauschexponent, βsol, is temperature independent and equal to 0.30 ( 0.03,18 and
nsol ≡ (1 - βsol) ) 0.70 ( 0.03
(5)
The rotational motion of the probe molecule QX in BMIMHFP also was measured by Ito and Richert. The result at each temperature was used to calculate the orientational correlation function of the probe molecule, r(t), which was also fit by the Kohlrausch function,
r(t) ) r(t)0) exp[-(t/τrot)βrot]
(6)
It was found that βrot has no systematic temperature dependence and is equal to 0.37 ( 0.05 or
nrot ≡ (1 - βrot) ) 0.63 ( 0.05
(7)
Thus, within experimental error, βsol and βrot or nsol and nrot are about the same. Ito and Richert measured the dielectric impedance of the BMIM-HFP sample in the frequency range 3 mHz to 3 MHz for temperatures between 182 and 246 K. Ito and Richert followed the common practice of using the frequency-dependent electric modulus, M*(ν) ) 1/*(ν), to characterize the ionic conductivity relaxation of the liquid. Because the dielectric spectra exhibit asymmetrically broadened peaks, they fit them to the empirical Havriliak-Negami (HN) functions27 empirical functions, which have two fractional exponents, R and γ. The product Rγ of the two exponents is equal to 0.4, and Ito and Richert asserted that the HN fits in the frequency domain correspond to a time domain correlation function having the Kohlrausch form
φ(t) ) exp[-(t/τM)βM]
(8)
with the exponent βM ≈ 0.51 ( 0.05, or
nM ≡ (1 - βM) ) 0.49 ( 0.05
(9)
Since the frequency dispersion of M* does not change with temperature, neither does the HN exponents R and γ nor the Kohlrasuch exponent βM. However, fitting the M*(ν) data directly by the one-sided Fourier transform of the Kohlrausch function (eq 5) and emphasizing a good fit to the main peak of the electric loss modulus M′′(ν) and its low-frequency flank yield a larger βM ≈ 0.59 and nM ≈ 0.41. Both the values of nM ≈ 0.5 given by Ito and Richert or the value of nM ≈ 0.41 are smaller than nsol and nrot. Either one of these two possible values of nM can explain the enhanced translational diffusion, as we shall discuss in the next section. In molten, glassy, and crystalline ionic conductors, where there is only one kind of mobile ion, the electric modulus has been proven repeatedly to give a faithful representation of the macroscopic conductivity relaxation due to migration of ions.23 In fact, the measured dc conductivity, σ, is well approximated by the Maxwell relation, o∞/〈τM〉. Here o is the permittivity of free space, ∞ is the high-frequency dielectric constant, and 〈τM〉 ) [Γ(1/βM)/βM]τM and is the mean electric modulus relaxation time of the Kohlrausch function (eq 8). For ionic liquids, both the cation and the anion execute
Studies of a Supercooled Molecular Ionic Liquid
J. Phys. Chem. B, Vol. 110, No. 51, 2006 26213
Figure 1. Temperature dependence of various measured and calculated relaxation times of BMIM-HFP. Solid circles are the electric loss modulus peak time constants τM measured by Ito and Richert (ref 18), the plus signs are the τM values from ref 19, and the dotted line is a VFTH fit to τM(T). Open circles show the solvation times (τsol) based on QX, and multiplication signs represent the probe rotation times τrot(T) for QX in BMIM-HFP. A solid triangle represents the logarithm of viscosity, log(η/P), from ref 24, after 10.3 has been subtracted from it to match the log(τ/s) scale. The other lines are all calculated relaxation times. The dashed line is τ0M(T) calculated with the Kohlrausch exponent βM ≈ 0.51 given by Ito and Richert, the dashed-dotted line is τ0sol, the thick solid line is the calculated τrot(T), and thick dashed line is the calculated τsol.
correlated translation and rotation, and the electric modulus measured will reflect the response of these more complex motions. Comparing the various relaxation times, Ito and Richert found that τrot is nearly identical to τsol at the same temperature within uncertainty. The solvation time constants τsol(T) follow the structural relaxation times in terms of η/T for more than ten decades, from subnanoseconds at room temperature to seconds near the glass transition temperature Tg. Hence,
τsol(T) ≈ τrot(T) ∝ η/T
(10)
The electric modulus relaxation times τM(T) agrees with τsol(T) at higher temperatures. However, at temperatures below about 1.2 Tg, τM(T) becomes less than τsol(T), and the ratio τM(T)/τsol(T) monotonically decreases with decreasing temperature. Due to its weaker temperature dependence, τM(T) follows a fractional Stokes-Einstein law, τM(T) ∝ η0.73/T, similar to the relation between the self-diffusion coefficient D and viscosity, D ∝ η0.74, observed in three nonionic glass-forming liquids TNB,5,12 o-terphenyl,5,12 and sucrose benzoate.9 The two phenomena can be considered as equivalent because in the ionic liquid translational motion of the charged liquid constituents (ions) is required to relax the macroscopic electric field with relaxation time τM(T) proportional to the diffusion time.22 As stated by Ito and Richert, the problem that remains to be explained is the decoupling near Tg of τM(T) from τsol(T) or, equivalently, the fractional Stokes-Einstein law, τM(T) ∝ η0.73/T.
Figure 2. Same as in Figure 1, except τ0M(T) is calculated with the Kohlrausch exponent βM ≈ 0.59, as obtained by fitting directly the electric loss modulus data of Ito and Richertto to the one-sided Fourier transform of the Kohlrausch function as shown in the inset.
3. Explanation from the Coupling Model Just like the nonionic glass-formers TNB, o-terphenyl, and sucrose benzoate, the Kohlrausch stretch exponents of solvation and probe rotation, βsol and βrot, of BMIM-HFP remain constant at temperatures below about 1.2Tg. As discussed in the Introduction, for the nonionic liquids, the decoupling of τM(T) from τsol(T) and the fractional Stokes-Einstein relation τM(T) ∝ η0.73/T cannot be explained by applying an argument similar to that used for viscous liquids, namely that D ∝ 〈1/τ〉, emphasizing fast dynamics, while τc ∝ 〈τ〉 is determined predominantly by the slowest molecules. The CM approach10 embodied by eqs 1-3 is now used in an attempt to explain the data. The inputs are the data of the relaxation times τsol(T), τrot(T), and τM(T) and the constant values of nsol, nrot and nM given before by eqs 5, 7, and 9, respectively, and eq 3 is rewritten as
τ0M(T) ∝ τ0sol(T) ∝ τ0rot(T)
(11)
First, we use eq 2 to calculate the primitive electric modulus relaxation times, τ0M(T), from τM(T). In this calculation, the Vogel-Fulcher-Tammann-Hesse equation that fits the data of τM(T) together with nM )0.49 (eq 7), all given by Ito and Richert, are used. The tc of BMIM-HFP is taken to be equal to 2 ps like in the case of molecular liquids that are not ionic. The calculated τ0M(T) values are shown in Figure 1 as a function of reciprocal temperature together with the experimental data of τM(T), τsol(T), and τrot(T). From eq 11, τ0sol(T) and τ0rot(T) have the same temperature dependence as τ0M(T), but they differ from τ0M(T) in absolute magnitudes by constant factors λsol and λrot, respectively. This difference of the primitive relaxation times by a constant factor is understandable, because they correspond to different observables. While τ0sol(T) and τ0rot(T) are related to solvation and rotation of the probe quinoxaline in BMIM-HFP, τ0M(T) is electric field relaxation originating from diffusion of changed molecules of BMIM-HFP itself.
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Substituting τ0sol(T) ) λsolτ0M(T) and τ0rot(T) ) λrotτ0M(T) into eq 2 and rewriting for the present case, we have
τsol(T) ) [tc-nsolτ0sol(T)]1/(1-nsol) ) [tc-nsolλsolτ0M(T)]1/(1-nsol) (12) and
τrot(T) ) [tc-nrotτ0rot(T)]1/(1-nrot) ) [tc-nrotλrotτ0M(T)]1/(1-nrot) (13) Only λsol and λrot are unknown when calculating τsol(T) and τrot(T) by the equations above. It is worthwhile to point out that the calculated τsol(T) and τrot(T) have a stronger temperature dependence than τM(T) independent of the value of λsol and λrot. This result follows readily from the fact that nsol> nM and nrot> nM. In fact, we can deduce at once from nsol ) 0.70 ( 0.03, nrot ) 0.63 ( 0.05, and nM ) 0.49 ( 0.05 that
τM(T) ∝ [τsol(T)](1-nsol)/(1-nM) ) (τsol)0.59
(14)
τM(T) ∝ [τrot(T)](1-nrot)/(1-nM) ) (τrot)0.73
(15)
and
Thus, for an explanation of the observed stronger temperature dependences of τsol(T) and τrot(T) compared with τ0M(T), all parameters are known from the experiment of Ito and Richert. The fractional power in eq 15 is about the same as observed. The one in eq 14 is a bit too small, but it is not unacceptable considering the uncertainties in both nsol and nM. Quantitative comparison is also made between τsol(T) and τrot(T) calculated by eqs 12 and 13 with the choice of λsol ) 52.5 and λrot ) 10.5, as shown in Figure 1 by the thick solid and thick dashed lines, respectively. Again, the τrot(T) values calculated by eq 13 are in better agreement with the experimental data than the τsol(T) calculated by eq 12. We mentioned before that fitting the M*(ν) data directly by the one-sided Fourier transform of the Kohlrausch function and emphasizing a good fit to the main peak of the electric loss modulus M′′(ν) and its low-frequency flank yield a larger βM ≈ 0.59 and a smaller nM ≈ 0.41. The fit is shown in the inset of Figure 2. This alternative value of nM ≈ 0.41 is also smaller than nsol and nrot and can explain the fractional Stokes-Einstein relaxation and enhanced translational diffusion as well. This is demonstrated in Figure 2, where τrot(T) represented by the thick solid line is calculated by eq 13 with nrot ) 0.63, λ ) 66, and nM ≈ 0.41. 4. Conclusion The comprehensive experimental measurements of conductivity relaxation, solvation dynamics, probe rotation, and viscosity
available for the supercooled ionic molecular liquid, BMIMHFP, have become available. The coupling parameters of all processes are known including that of translational-rotational diffusion, which is obtained from the electric modulus representation of the dielectric measurements. The situation presents an opportunity of critically testing the Coupling Model explanation of the observed fractional Stokes-Einstein and fractional Debye-Stokes-Einstein relations. The test is successful. Acknowledgment. The work was supported by the Office of Naval Research. K.L.N. thanks Professor Pierangelo Rolla for hospitality during his stay at Universita` di Pisa when the paper was written. References and Notes (1) Einstein, A. InVestigations on the Theory of Brownian Motion; Dover: New York, 1956. (2) Debye, P. Polar Molecules; Dover: New York, 1929. (3) Chang, I.; Fujara, F.; Geil, B.; Heuberger, G.; Mangel, T.; Sillescu, H. J. Non-Cryst. Solids 1994, 172-174, 248. (4) Cicerone, M. T.; Wagner, P. A.; Ediger, M. D. J. Phys. Chem. B 1997, 101, 8727. (5) Ngai, K. L.; Magill, J. H.; Plazek, D. J. J. Chem. Phys. 2000, 112, 1887. (6) Richert, R.; Duvvuri, K.; Duong, L. J. J. Chem. Phys. 2003, 118, 1828. (7) Richert, R. J. Chem. Phys. 2005, 123, 154502. (8) Chakrabarti, D.; Bagchi, B. Phys. ReV. Lett. 2006, 96, 187801. (9) Rajian, J. R.; Huang, W.; Richert, R.; Quitevis, E. L. J. Chem. Phys. 2006, 124, 014510. (10) Ngai, K. L. J. Phys. Chem. B 1999, 103, 10684. (11) Ngai, K. L.; Mashimo, S.; Fytas, G. Macromolecules 1988, 21, 3030. (12) Mapes, M. K.; Swallen, S. F.; Ediger, M. D. J. Phys. Chem. B 2006, 110, 507. (13) Ngai, K. L. J. Chem. Phys. 1993, 98, 7588. (14) Kohlrausch, R. Pogg. Ann. Phys. 1847, 12 (3), 393. (15) Ngai, K. L.; Tsang, K. Y. Phys. ReV. E 1999, 60, 4511. (16) Ngai, K. L. J. Phys.: Condens. Matter 2003, 15, S1107. (17) Ngai, K. L. Philos. Mag. In press. (18) Ito, N.; Richert, R. J. Phys. Chem. B 2006 (in press). (19) Rivera, A.; Ro¨ssler, E. A. Phys. ReV. B 2006, 73, 212201. (20) Ito, N.; Huang, W.; Richert, R. J. Phys. Chem. B 2006, 110, 4371. (21) Huang, W.; Richert, R. J. Chem. Phys. 2006, 124, 164510. (22) Ngai, K. L.; Leo´n, C. Phys. ReV. B 1999, 60, 9396. (23) Hodge, I.; Ngai, K. L.; Moynihan, C. T. J. Non-Cryst. Solids 2005, 351, 104. (24) Xu, W.; Cooper, E. I.; Angell, C. A. J. Phys. Chem. B 2003, 107, 6170. (25) Triolo, A.; Russina, O.; Hardacre, C.; Nieuwenhuyzen, M.; Gonzalez, M. A.; Grimm, H. J. Phys. Chem. B 2005, 109, 22061. (26) Yamamuro, O.; Minamimoto, Y.; Imamura, Y.; Hayashi, S.; Hamaguchi, H. Chem. Phys. Lett. 2006, 423, 373. (27) Havriliak, S.; Negami, S. Polymer 1967, 8, 161.
