Observation of the slow, Debye-like relaxation in hydrogen-bonded liquids by dynamic light scattering Yangyang Wang, Philip J. Griffin, Adam Holt, Fei Fan, and Alexei P. Sokolov Citation: The Journal of Chemical Physics 140, 104510 (2014); doi: 10.1063/1.4867913 View online: http://dx.doi.org/10.1063/1.4867913 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/10?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dynamics of supercritical methanol of varying density from first principles simulations: Hydrogen bond fluctuations, vibrational spectral diffusion, and orientational relaxation J. Chem. Phys. 138, 224501 (2013); 10.1063/1.4808034 A statistical model of hydrogen bond networks in liquid alcohols J. Chem. Phys. 136, 094514 (2012); 10.1063/1.3690137 Hydrogen bonded network properties in liquid formamide J. Chem. Phys. 132, 014506 (2010); 10.1063/1.3268626 Collective contributions to the dielectric relaxation of hydrogen-bonded liquids J. Chem. Phys. 120, 11692 (2004); 10.1063/1.1751392 A nest of structures in dynamics of cellulose diacetate in N,N-dimethylacetamide in quiescent solution state studied by dynamic light scattering J. Chem. Phys. 109, 11027 (1998); 10.1063/1.477741
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THE JOURNAL OF CHEMICAL PHYSICS 140, 104510 (2014)
Observation of the slow, Debye-like relaxation in hydrogen-bonded liquids by dynamic light scattering Yangyang Wang,1,a) Philip J. Griffin,2 Adam Holt,2 Fei Fan,3 and Alexei P. Sokolov1,2,3 1
Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA 3 Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, USA 2
(Received 15 January 2014; accepted 25 February 2014; published online 13 March 2014) The slow, Debye-like relaxation in hydrogen-bonded liquids has largely remained a dielectric phenomenon and has thus far eluded observation by other experimental techniques. Here we report the first observation of a slow, Debye-like relaxation by both depolarized dynamic light scattering (DLS) and dielectric spectroscopy in a model hydrogen-bonded liquid, 2-ethyl-4-methylimidazole (2E4MIm). The relaxation times obtained by these two techniques are in good agreement and can be well explained by the Debye model of rotational diffusion. On the one hand, 2E4MIm is analogous to the widely studied monohydroxy alcohols in which transient chain-like supramolecular structure can be formed by hydrogen bonding. On the other hand, the hydrogen-bonded backbone of 2E4MIm is much more optically polarizable, making it possible to apply light scattering to study the dynamics of the supramolecular structure. These findings provide the missing evidence of the slow, Debye-like relaxation in DLS and open the venue for the application of dynamic light scattering to the study of supramolecular structures in hydrogen-bonded liquids. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4867913] I. INTRODUCTION
Water, monohydroxy alcohols, and several other hydrogen-bonded liquids display a pronounced Debye-like low-frequency dielectric relaxation.1–11 Despite extensive studies of more than half a century, the molecular origin of this process still remains an open question. It is widely believed that the slow Debye-like relaxation is a distinct signature of hydrogen-bonded supramolecular structures, and understanding the nature of this relaxation is a key step for a complete elucidation of the dynamics of hydrogen-bonded liquids. Dynamic light scattering (DLS) is a powerful experimental method for investigating the dynamics of glass-forming liquids, providing information complimentary to dielectric and dynamic mechanical spectroscopy techniques.12, 13 So far, however, this technique (DLS) has not been able to detect a Debye-like process in hydrogen-bonded liquids in the neat state. In particular, no slow Debye-like relaxation has been found in water14 and monohydroxy alcohols6, 14 by DLS. Similarly, studies employing other non-dielectric techniques such as calorimetry15 and dynamic mechanical spectroscopy16 were, until recently, unable to identify the slow, Debye-like relaxation process in monohydroxy alcohols. It was therefore believed that this Debye-like relaxation is a purely dielectric phenomenon. However, this traditional view has been challenged by recent nuclear magnetic resonance5 and mechanical measurements.17 The emerging experimental results from non-dielectric techniques have provided some essential ingredients for the explanation of the slow Debye-like relaxation. a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2014/140(10)/104510/8/$30.00
In this context, the missing evidence from DLS appears more puzzling than ever. In this article we report the first observation of the slow Debye-like relaxation in a model hydrogen-bonded liquid, 2ethyl-4-methylimidazole (2E4MIm), by depolarized photon correlation spectroscopy (PCS) and Brillouin-Raman spectroscopy. Unlike monohydroxy alcohols, a slow Debye-like relaxation can be detected in 2E4MIm by both broadband dielectric spectroscopy (BDS) and depolarized DLS, due to the much higher optical polarizability of the hydrogen-bonded imidazole supramolecular backbone. The relaxation times for the slow Debye-like process from BDS and DLS are consistent with each other and can be well described by the Debye model of rotational diffusion. This finding provides the missing evidence of the slow Debye-like relaxation in DLS and opens the door for the application of dynamic light scattering to the study of supramolecular hydrogen-bonded structures in the neat liquid state.
II. EXPERIMENT
The 2-ethyl-4-methylimidazole (2E4MIm) compound was obtained from Sigma-Aldrich and used as received. Similar to monohydroxy alcohols, 2E4MIm can form linear chainlike supramolecular structure through hydrogen bonding (Fig. 1). Broadband dielectric spectroscopy (BDS) measurements of 2E4MIm were carried out on a Novocontrol Concept 80 system in the frequency range 10−2 –107 Hz. Depolarized photon correlation spectroscopy (PCS) measurements of 2E4MIm were performed in right angle geometry in an Oxford Optistat cryostat with temperature stability of ±0.1 K.
140, 104510-1
© 2014 AIP Publishing LLC
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FIG. 2. Heat flow (red dotted line) and its temperature derivative (blue solid line) measured during (a) cooling and (b) heating scans, at 10 K/min.
Parallel plates of small diameter (4 mm) were used to minimize the contribution of instrument compliance. FIG. 1. Illustration of the hydrogen-bonded structures of 2-ethyl-1-hexanol (2E1H) (a model monohydroxy alcohol) and 2-ethyl-4-methylimidazole. The hydrogen, carbon, oxygen, and nitrogen atoms are represented by white, gray, red, and blue sticks, respectively. The dashed lines stand for the intermolecular hydrogen bonds. Both molecules are capable of forming chain-like supramolecular structures.
The laser wavelength was 647 nm and the power was 75 mW. The scattered light was collected with an optical fiber, split between two avalanche photodiode detectors, and analyzed using an ALV-7004/FAST multi-tau digital correlator. In addition to the PCS measurements, Brillouin-Raman measurements were also carried out to access the dynamics at shorter timescales and higher temperatures. The depolarized light scattering spectra of 2E4MIm were measured in the backscattering geometry, using a tandem Fabry-Perot interferometer (Sandercock Model) and a Raman spectrometer (Jobin Yvon T64000). A solid-state laser with wavelength = 532 nm and power = 75 mW was used for the measurements. Differential scanning calorimetry measurement of 2E4MIm was performed on a TA Instruments Q2000 device with aluminum hermetic pans. The experiment was carried out in the temperature range 333–193 K at cooling/heating rate of 10 K/min. The glass transition temperature (Tg ) was determined from the mid-point of the step in heat flow, which corresponded to the maximum in the derivative of heat flow with respect to temperature. The Tg s from cooling and heating scans are 232 and 235 K, respectively (Fig. 2). Rheological measurements of 2E4MIm were performed on an AR2000ex rheometer (TA Instruments) with 4 and 25 mm parallel plates. Creep measurements were used to determine the zero-shear viscosity (η) of 2E4MIm. η can be obtained from the creep compliance J(t) in the long-time limit: η = lim t/J (t). In addition to creep experiments, dynamic t→∞ mechanical measurements of complex modulus were carried out to obtain the mechanical spectra in the frequency domain.
III. RESULTS A. Broadband dielectric spectroscopy
Representative dielectric spectra of 2E4MIm are presented in Fig. 3, where the real and imaginary parts of permittivity (ε and ε ), derivative of ε (ε der ), and loss modulus (M ) are shown as a function of frequency ( f ). Unlike monohydroxy alcohols, 2E4MIm has high intrinsic proton conductivity, due to its autodissociation.18 As a result, the dc conductivity dominates the dielectric loss spectra. To assist our analysis, the derivative spectra [ε der = (− π /2)∂ε /∂ln f ] are used.19 A typical spectrum of 2E4MIm consists of three processes (Fig. 3): a fast relaxation in the high frequency region, a pronounced slow Debye-like relaxation in the intermediate frequency region, and electrode polarization (EP) effect20 at low frequencies. The complex dielectric permittivity ε∗ of 2E4MIm can be modeled by the following equation: ε∗ (f ) = ε − iε = ε∞ +
ε1 [1 + (i2πf τHN,1 )α1 ]β1
ε2 [1 + (i2πf τHN,2 )α2 ]β2 σ + + A(2πf )n , i2πf ε0 +
(1)
where ε∞ is the dielectric permittivity at infinite frequency, εj (j = 1, 2) is the dielectric relaxation strength, τ HN,j is the relaxation time, α j , β j are the shape parameters, σ is the dc conductivity, n is the slope of EP’s high frequency tail, and A is related to the amplitude of EP. The use of an EP term improves the accuracy of our analysis, but is not essential, because the main relaxation processes and EP are well separated. The relaxation frequency of the fast process coincides with the peak frequency of the electrical loss modulus
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J. Chem. Phys. 140, 104510 (2014) TABLE I. Fit parameters for the slow Debye-like relaxation in dielectric measurements. T (K)
τ HN,1 (s)
ε 1
α1
β1
263.15 260.65 258.15 255.65 253.15 250.65
0.00252 0.00491 0.0120 0.0243 0.0590 0.140
3.79 5.32 6.20 8.14 10.8 12.7
0.89 0.83 0.93 0.93 0.92 0.95
0.89 1.0 0.91 0.99 0.94 0.95
in polyalcohols and similar materials. It has been frequently reported that an ultra-slow Debye-like relaxation can show up in the vicinity of conductivity relaxation frequency in polyalcohols24, 25 and polymers.26 Although the nature of this process has not been completely settled, it is highly likely that it originates from surface polarization effects due to presence of impurities or voids,27–29 and is not an intrinsic feature of the material itself. In contrast to the close connection of Debye-like relaxation and conductivity relaxation in polyalcohols, the frequency of the slow, Debye-like relaxation in 2E4MIm is approximately 1–2 orders lower than the conductivity relaxation frequency. More importantly, as we demonstrate below, the observation of the same process in three different techniques (BDS, PCS, and Brillouin-Raman) rules out any connection between the slow Debye-like relaxation and conductivity relaxation in 2E4MIm. FIG. 3. Representative spectra of 2E4MIm at 258.15 K: (a) real part of dielectric permittivity ε ( f ); (b) imaginary part of dielectric permittivity ε ( f ); (c) derivative of ε ( f ), ε der = (−π /2)∂ε /∂ln f ; and (d) imaginary part of electrical modulus M ( f ). Inset: Frequency dependence of real part of conductivity σ ( f ). (Black) solid curves: Fittings of experimental data using Eq. (1).
M (Fig. 3(d)) as well as the onset of dc conductivity (inset of Fig. 3). This process is therefore assigned to the conductivity relaxation of 2E4MIm. It is widely believed that the proton hopping within the hydrogen-bonded chain structure plays an important role in imidazole-based molecules. Because of its high intrinsic conductivity, the high-frequency dielectric behavior of 2E4MIm resembles that of typical ionic conductors such as ionic liquids21 and phosphoric acids.22 It should be emphasized that the possibility of observing dipolar structural relaxation in highly ionic molecules has remained an open question.23 It is possible that the fast relaxation has a dipolar contribution. The assignment of this process to conductivity relaxation is mainly due to the ionic nature of 2E4MIm and its close connection to dc conduction (inset of Fig. 3). Further experiments, however, are required to understand the exact nature of the fast relaxation (i.e., ionic vs. dipolar). A pronounced slow relaxation can be observed in ε ( f ) ( f ) (Fig. 3(c)) at intermediate frequencies, (Fig. 3(a)) and εder and the shape of this slow relaxation was resolved by simultaneously fitting ε and ε using Eq. (1). It was observed that this slow relaxation is Debye-like, with α in the range 0.83– 1.0 and β in the range 0.89–1.0. The fitting parameters are provided in Table I. It is important to recognize the difference between the slow Debye-like relaxation in 2E4MIm and those observed
B. Photon correlation spectroscopy
The normalized intensity correlation functions (ICF) of 2E4MIm are shown in Fig. 4. The decay of the ICF can be well described by the superposition of two KohlrauschWilliam-Watts (KWW) stretched exponential functions for the normalized electric field correlation function (g1 ): I (0)I (t) − 1 = g2 (t) − 1 = γ [g1 (t)]2 I 2 = γ {ϕ1 exp[−(t/τKWW,1 )βKWW,1 ] + ϕ2 exp[−(t/τKWW,2 )βKWW,2 ]}2 ,
(2)
where g1 (t) and g2 (t) are normalized electric field correlation function and normalized intensity correlation function (ICF), respectively, γ is the spatial coherence factor, ϕ j (j = 1, 2) is the relative relaxation strength, τ KWW,j is the relaxation time, and β KWW,j is the stretching parameter. The slow relaxation is Debye-like, with β KWW in the range 0.77–0.86, whereas the fast relaxation is highly stretched, with β KWW around 0.5. A detailed summary of the corresponding fit parameters is provided in Table II. Fig. 4(b) shows that the use of a single Kohlrausch-William-Watts (KWW) stretched exponential function is insufficient for fitting the experimental correlation functions. C. Brillouin-Raman spectroscopy
The PCS results are corroborated by further BrillouinRaman measurements. Similar to the time-domain PCS experiments, the susceptibility spectrum χ (ω) from BR
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FIG. 5. Imaginary part of the depolarized light scattering susceptibility spectra of 2E4MIm at 400 and 450 K. The red and black solid lines represent the fits of experimental data by Eq. (3).
FIG. 4. (a) Depolarized photon correlation spectroscopy measurements at temperatures T = 240, 245, 250, 255, 260, 265, 275, 285, and 295 K (from right to left). (b) Fitting of the electric field correlation function (EFCF) at 260 K by one (red solid line) and two (blue short-dashed line) KWW functions. The slow relaxation (green dashed line) is Debye-like and approximately 1.5 decades slower than the structural relaxation process (orange dashed-dotted line). The residuals of the fits are displayed in the bottom panel, with open (red) circles for the single-KWW fit and (blue) crossed circles for the double-KWW fit.
TABLE II. Fit parameters for the first-order autocorrelation function in PCS measurements.a Slow (Debye-like) relaxation T (K) 295 285 275 265 260 255 250 245 240 a
Fast (structural) relaxation
ϕ1
τ KWW,1 (s)
β KWW,1
ϕ2
τ KWW,2 (s)
β KWW,2
0.81 0.78 0.79 0.83 0.79 0.82 0.86 0.83 0.82
1.1 × 10−6 6.2 × 10−6 4.3 × 10−5 5.4 × 10−4 2.2 × 10−3 1.1 × 10−2 6.4 × 10−2 0.47 4.4
0.86 ± 0.03 0.86 ± 0.03 0.84 ± 0.01 0.79 ± 0.01 0.85 ± 0.01 0.84 ± 0.01 0.77 ± 0.02 0.82 ± 0.02 0.84 ± 0.02
0.19 0.23 0.21 0.17 0.21 0.18 0.14 0.17 0.18
2.3 × 10−8 2.2 × 10−7 1.5 × 10−6 1.5 × 10−5 7.3 × 10−5 3.7 × 10−4 1.7 × 10−3 0.019 0.19
0.50 ± 0.04 0.50 ± 0.04 0.50 ± 0.04 0.51 ± 0.04 0.52 ± 0.04 0.53 ± 0.04 0.50 ± 0.04 0.49 ± 0.04 0.50 ± 0.04
Fitting error: contrast: 1%; relaxation time: 1%.
experiments consists of a slow, Debye-like relaxation and a fast, highly stretched relaxation (Fig. 5). Moreover, the relaxation times and shape parameters of these two processes are also consistent with those from the PCS measurements (Tables II and III and Fig. 6). Here, it is important to point out that previous BR measurements failed to identify a slow, Debye-like relaxation in hydrogen-bonded liquids, including water.6, 14 The depolarized scattering intensities I(ω) were converted to the susceptibility spectra χ (ω) by using the fluctuation-dissipation theorem: χ (ω) = I(ω)/[n(ω) + 1], where n(ω) = [exp (¯ω/kB T) − 1]−1 is the Bose factor. Fig. 5 shows the χ spectra of 2E4MIm at 400 and 450 K, where two relaxation processes are clearly visible. The slow relaxation has a Debye-like shape, while the fast (alpha) relaxation is stretched. In order to extract their relaxation times and shape parameters, we fit the light scattering susceptibility spectra by the following equation: χ1 χ2 ν , + − iχ ω χ = − 3 (1 + iωτ1 )βCD,1 (1 + iωτ2 )βCD,2 (3) where stands for imaginary part, the first and second ColeDavidson (CD) terms describe the slow Debye-like relaxation and alpha relaxation, respectively, and the last term accounts for the low-frequency tail of the fast dynamics. The fit parameters are summarized in Table III. IV. DISCUSSION A. Glass transition temperature
To further explore the nature of the relaxation processes observed by BDS and depolarized PCS, their relaxation times are shown as a function of inverse temperature in Fig. 6. The mechanical structural relaxation time (τ η ), evaluated from the Maxwell relation: τ η = η/G, is also presented as a reference. The viscosity η was determined from the creep measurement and the glassy modulus G = 0.425 GPa was determined from the dynamic mechanical measurement close to Tg and assumed to be a constant. As a first observation, the
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TABLE III. Fit parameters for the χ spectra in Fig. 5.a Slow (Debye-like) relaxation T (K) 400 450 a
Structural (α) relaxation
Fast relaxation
χ 1
β CD,1
τ 1 (ns)
χ 2
β CD,2
τ 2 (ns)
χ 3
ν
4.1 2.8
0.83 0.84
0.76 0.12
1.1 1.1
0.38 0.38
0.059 0.011
0.12 0.14
0.3 0.3
Fitting error: amplitude: 5%; relaxation time: 1%.
relaxation time of the fast (alpha) process from PCS matches τ η , suggesting that the origin of this process is the structural relaxation of 2E4MIm. Fitting τ α with the Vogel-FulcherTammann (VFT) equation and extrapolating the relaxation time to 100 s yield a dynamic glass transition temperature (Tg ) of 231 K. This agrees well with the differential scanning calorimetry measurement, which gives a Tg of 232 K on the cooling scan. In addition, the conductivity relaxation time, τ σ , which is ∼2 orders of magnitude faster than the dielectric τ slow , matches τ α from PCS and τ η from rheology. According to the classical theory, the ionic transport should be closely connected to the structural relaxation.30 The assignment of the fast process in PCS to structural relaxation is consistent with the coincidence of τ σ and τ α . Therefore, it is clear that the Debye-like relaxation in 2E4MIm is slower than the structural relaxation and not responsible for the glass transition. B. Derivative analysis
We have shown that the light scattering intensity correlation function (ICF) measured for 2E4MIm is not well de-
FIG. 6. Comparison of the temperature dependence of the different relaxation processes probed by broadband dielectric spectroscopy (BDS), rheology (RH), depolarized photon correlation spectroscopy (PCS), and BrillouinRaman spectroscopy (BR). τ σ is the conductivity relaxation time from BDS. τ slow is the relaxation time for the slow Debye-like process. τ η is the segmental relaxation time from the mechanical measurement, calculated using the Maxwell relation, τ η = η/G. The viscosity η was determined from the creep measurement and the glassy modulus G = 0.425 GPa was determined from the dynamic mechanical measurement. τ α is the structural relaxation time from PCS and BR. Dashed lines are the fits of the τ slow and τ α from optical measurements by the Vogel-Fulcher-Tammann (VFT) equation: τ = τ 0 exp[B/(T − T0 )], where τ 0 , B, and T0 are fit parameters. The inset shows that the relaxation times of the slow process from BDS and PCS can be collapsed onto a master curve by vertically shifting the BDS data by ∼0.48 order (a factor of 3).
scribed by a single stretched exponential decay corresponding to the characteristic structural relaxation process. Instead, we have demonstrated that the ICF can be precisely fit with a superposition of two decays – a faster, stretched exponential decay, and a slower, “Debye-like” component that accounts for a majority of the total decay of the ICF. The faster relaxation time is nearly identical to the structural relaxation time calculated from rheological measurements, and due to this agreement we attribute it to the molecular reorientational α process and not to secondary relaxations or the excess wing. These results for 2E4MIm are unique, because for nearly all supercooled liquids studied via depolarized DLS, the ICF has been shown to be composed of one stretched exponential function corresponding to the structural relaxation process. Additionally, for measurements close to the calorimetric glass transition temperature, the excess wing can also contribute to the decay of the ICF, and it typically accounts for a small percentage of the measured decay – similar to dielectric relaxation measurements.31, 32 In order to more thoroughly demonstrate that two relaxations are present in the ICF of 2E4MIm and to precisely determine the stretching exponents corresponding to these two relaxation processes, we have analyzed a quantity associated with the derivative of the ICF for both 2E4MIm and 2-ethyl-1hexanol (2E1H), a model monohydroxy alcohol. If the measured ICF is composed of one stretched exponential decay, such that ICF = {a · exp [ − (t/τ α )β ]}2 , it is straightforward to show that d ln(ICF) 2β + (β − 1) log10 t. = log10 Z = log10 − dt τα β (4) Thus the KWW stretching exponent can be isolated and determined by analyzing the slope of the line on a log-log plot fitted to the quantity −dln(ICF)/dt calculated from the experimental data. Just as a single KWW function is shown to fit the ICF data of 2E1H in Fig. 7(a), it follows accordingly that a single straight line is sufficient to describe the data for 2E1H in Fig. 7(b). The stretching exponent for 2E1H was determined to be β = 0.6 from the slope of this line. At short times, the ICF reaches a plateau and the statistical noise dominates the signal. In contrast, it is clearly seen in Fig. 7(b) that the quantity Z calculated from the ICF measured for 2E4MIm has two distinct regions where lines of quite different slope are required to interpolate the data. This immediately demonstrates that the decay of the ICF of 2E4MIm proceeds in two distinct steps corresponding to two different relaxation processes. The slope of each line corresponds to the stretching exponent of each individual relaxation in a manner similar to Eq. (4),
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FIG. 7. (a) Normalized intensity correlation function (ICF) for 2-ethyl-1-hexanol (2E1H) at two representative temperatures. (b) Derivative analysis of ICF for 2E4MIm and 2E1H.
although some approximations are necessary to quantify these stretching exponents. For the case of 2E4MIm, we consider the correlation function as being a sum of two separate decays, where ICF = [a1 1 (t) + a2 2 (t)]2 = {a1 · exp[−(t/τ1 )β1 ] + a2 · exp[−(t/τ2 )β2 ]}2 , with the first term corresponding to the faster relaxation and the second term corresponding to the slower relaxation. In order to determine the stretching parameter β 2 of the slow relaxation, we make the approximation that for t ≥ τ 2 , 1 = 0. After this approximation is made, we retrieve the case of a single-step correlation function, and Eq. (4) can be used to determine the stretching parameter β 2 from the slope of the data in Fig. 7(b) at longer times. This slope corresponds to β ≈ 0.8 as is shown in Fig. 7(b). In order to determine β 1 from the slope of the data at shorter times, we make the approximation that for t ≤ τ 1 , 2 = 1, such that ICF ≈ {a1 · exp[−(t/τ1 )β1 ] + a2 }2 . Taking the derivative, we find that in this time window d ln(ICF) Z = log10 − dt t β1 a2 ≈ − log10 1 + exp a1 τ1 2β1 + (β1 − 1) log10 t, (5) + log10 τ1 β1 and since the first term of Eq. (5) is approximately constant for t ≤ τ 1 , we can further approximate that in the vicinity of the decay of the fast relaxation component, 2β1 d ln(ICF) Z = log10 − +(β1 −1) log10 t. ≈ C +log10 dt τ1 β1 (6) Once again we see that the slope of Z plotted against log10 (t) in this time interval is directly related to the stretching expo-
nent of the fast decay, which we determined to be β ≈ 0.5 as seen in Fig. 7(b). This derivative analysis clearly demonstrates that there are two decays contributing to the ICF measured in 2E4MIm. Moreover, it shows that they correspond to a slow relaxation component with β ≈ 0.8 and a faster relaxation component with β ≈ 0.5. This result is consistent with the fitting of ICF using Eq. (2) (Table II). C. Comparison of BDS and DLS
We now turn to the slow Debye-like relaxations in BDS and DLS. The relaxation times from the two measurements are comparable on absolute scales and exhibit similar temperature dependence. According to the Debye-Stokes-Einstein relation,12, 33 the rotational diffusion rate (τ −1 ) of a spherical particle in a viscous medium is inversely proportional to the cube of its hydrodynamic radius R, τ −1 =
l(l + 1)kB T , 8π ηR 3
(7)
where l is the tensor rank of the probed relaxation, kB is the Boltzmann constant, T is the absolute temperature, and η is the viscosity of the medium. It follows that the ratio of the first and second-rank rotational correlation time should be equal to 3. It is well known that BDS and depolarized DLS probe the first- and second-rank rotational correlation, respectively.34 Therefore, for the same rotational diffusion process, the dielectric and light scattering relaxation times should differ by a factor of 3, if the angle for each reorientational jump is small.35–37 The inset of Fig. 6 shows that the BDS and DLS relaxation times for the slow, Debye-like relaxation can be collapsed onto a master curve by shifting the dielectric logτ data by ∼0.48 order (a factor of 3). This coincidence not only confirms that the slow processes observed by BDS and DLS are of the same origin, but also suggests that the slow, Debye-like relaxation in 2E4MIm proceeds through small-angle jumps
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TABLE IV. Hydrodynamic radii for structural and slow Debye-like relaxing units. T (K) 295 285 275 265 260 255 250 245 240
τ α (s) 10−8
3.1 × 3.0 × 10−7 2.0 × 10−6 2.0 × 10−5 6.9 × 10−5 4.9 × 10−4 2.2 × 10−3 2.5 × 10−2 0.25
τ slow (s) 10−6
1.2 × 6.7 × 10−6 4.7 × 10−5 6.1 × 10−4 2.4 × 10−3 1.2 × 10−2 7.3 × 10−2 0.52 4.8
η (Pas)
Rα (Å)
Rslow (Å)
Rslow /Rα
10.3 56.4 443 5.54 × 103 2.43 × 104 1.27 × 105 8.10 × 105 6.50 × 106 6.83 × 107
1.4 ± 0.1 1.7 ± 0.2 1.6 ± 0.2 1.5 ± 0.2 1.3 ± 0.1 1.5 ± 0.2 1.3 ± 0.1 1.5 ± 0.2 1.4 ± 0.1
4.8 ± 0.5 4.8 ± 0.5 4.6 ± 0.5 4.6 ± 0.5 4.4 ± 0.4 4.3 ± 0.4 4.2 ± 0.4 4.0 ± 0.4 3.8 ± 0.4
3.4 2.8 2.9 3.1 3.3 2.9 3.2 2.7 2.7
(i.e., Brownian rotational diffusion). However, it is important to keep in mind that in addition to the reorientation of entire supramolecular structure, scission and recombination should also contribute to the observed rotational diffusion.5 Earlier investigations failed to identify the Debyelike relaxation in monohydroxy alcohols by dynamic light scattering.6 Our own depolarized PCS measurement on 2ethyl-1-hexanol (2E1H) seems to confirm this conclusion (Fig. 7). It is important to understand why the slow Debye-like relaxation can be resolved by depolarized PCS in 2E4MIm, but not in monohydroxy alcohols. Recent studies have shown that the structural relaxation in monohydroxy alcohols mainly arises from the motion of the alkyl chains.8 It is well known that the optical polarizabilities of alkyl groups are much higher than that of the hydroxyl group. As a result, in the case of monohydroxy alcohols, the fluctuations of optical polarizability caused by the motion of supramolecular chains will simply be overwhelmed by the motion of the alkyl groups, making it difficult to detect the slow Debye-like relaxation by DLS. This stands in sharp contrast to the dielectric measurement, where the dielectric relaxation is dominated by the polar, supramolecular backbone. Compared to the monohydroxy alcohols, the imidazole-bonded supramolecular backbone is much more optically polarizable and its “side chains” consist of only short methyl and ethyl groups. This special structure makes it possible to observe the slow Debye-like relaxation by both BDS and DLS techniques. From the Debye-Stokes-Einstein relation (Eq. (7)), one can further estimate the average size of the hydrogen-bonded supramolecular chain for 2E4MIm (Table IV). Using the alpha relaxation time from PCS measurements and the viscosity from rheology, the hydrodynamic radius (R) of the structural relaxing unit is estimated to be approximately 1.5 ± 0.2 Å, which roughly corresponds to the radius for the rotation of a single 2E4MIm molecule about an axis through its center of mass and perpendicular to its dipole moment. The apparent radius for the slow Debye-like relaxation is estimated to be approximately 4.4 ± 0.4 Å. Because of the contribution of scission and recombination to the observed rotational diffusion, Rslow should not be literally interpreted as the size of the supramolecular chain. Rather, it should be considered as the lower bound. Therefore, from the ratio of Rslow and Rα , we can conclude that the average length of the hydrogenbonded chain-like structure is at least about three repeating units of 2E4MIm. Additionally, because Rslow is much larger
than the size of 2E4MIm, it is unlikely that the slow, Debyelike is the structural relaxation. This conclusion is consistent with our earlier argument presented in Sec. IV A. Lastly, it is worth mentioning that the Debye-like relaxation of 2E4MIm appears to be slightly more stretched than those of water and monohydroxy alcohols. This is presumably due to the fact that 2E4MIm has two tautomers: 2ethyl-4-methylimidazole and 2-ethyl-5-methylimidazole. Our measurement of neat imidazole shows that its Debye-like relaxation is indeed much narrower.38 V. SUMMARY AND CONCLUDING REMARKS
In summary, we have shown that a slow, Debye-like relaxation can be observed in 2-ethyl-4-methylimidazole by broadband dielectric spectroscopy, depolarized photon correlation spectroscopy, and Brillouin-Raman spectroscopy. This result provides the missing evidence of the slow, Debye-like relaxation in DLS. While monohydroxy alcohols are widely regarded as model systems for the study of supramolecular structure in hydrogen-bonded liquids and have received enormous attention during the past several decades, the present study suggests that our understanding of hydrogen-bonded liquids can benefit tremendously from studying not only monohydroxy alcohols but also other model systems. In the case of dynamic light scattering measurements, highly optically polarizable hydrogen-bonded structures such as imidazole chains clearly have advantages over the hydroxylbonded monohydroxy alcohols. It is also worth mentioning that the slow, Debye-like relaxation has been recently observed in a supramolecular polymer based on the thymine and diamidopyridine motif39 and a close connection between the dielectric Debye-like relaxation and mechanical chain relaxation was demonstrated. By exploring the rich supramolecular chemistry developed during the past several decades, it is possible to find model hydrogen-bonded liquids where the motion of supramolecular structures can be probed by several non-dielectric experimental techniques, including dynamic light scattering and rheology. Lastly, it is worth noting that imidazole-based molecules are also model systems for proton transport.18, 40 It is widely believed that the proton hopping within the hydrogen-bonded chain-like structure of imidazole41 gives rise to high proton conductivity, similar to the well-known Grotthuss mechanism42, 43 in water. The direct observation of the supramolecular structure of 2E4MIm
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by DLS also offers a new powerful experimental approach to the study of proton transport in hydrogen-bonded liquids.
ACKNOWLEDGMENTS
The authors thank A. L. Agapov and J. R. Sangoro for helpful discussions. This work was supported by the NSF Chemistry Program (CHE-1213444). 1 M.
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Kreuer, S. J. Paddison, E. Spohr, and M. Schuster, Chem. Rev. 104, 4637 (2004). 19 M. Wübbenhorst and J. van Turnhout, J. Non-Cryst. Solids 305, 40 (2002). 20 A. Serghei, J. R. Sangoro, and F. Kremer, in Electrical Phenomena at Interfaces and Biointerfaces: Fundamentals and Applications in Nano-, Bio-, and Environmental Sciences, edited by H. Ohshima (John Wiley and Sons, Inc., Hoboken, New Jersey, 2012). 21 J. R. Sangoro, A. Serghei, S. Naumov, P. Galvosas, J. Kärger, C. Wespe, F. Bordusa, and F. Kremer, Phys. Rev. E 77, 051202 (2008). 22 Y. Wang, N. A. Lane, C.-N. Sun, F. Fan, T. A. Zawodzinski, and A. P. Sokolov, J. Phys. Chem. B 117, 8003 (2013). 23 P. Griffin, A. L. Agapov, A. Kisliuk, X.-G. Sun, S. Dai, V. N. Novikov, and A. P. Sokolov, J. Chem. Phys. 135, 114509 (2011). 24 Y. Yomogida, A. Minoguchi, and R. Nozaki, Phys. Rev. E 73, 041510 (2006). 25 R. Bergman, H. Jansson, and J. Swenson, J. Chem. Phys. 132, 044504 (2010). 26 S. Zhang and J. Runt, J. Phys. Chem. B 108, 6295 (2004). 27 R. Richert, A. Agapov, and A. P. Sokolov, J. Chem. Phys. 134, 104508 (2011). 28 R. Casalini and C. M. Roland, J. Chem. Phys. 135, 094502 (2011). 29 M. Paluch, S. Pawlus, and K. Kaminski, J. Chem. Phys. 134, 037101 (2011). 30 P. G. Wolynes, Ann. Rev. Phys. Chem. 31, 345 (1980). 31 P. J. Griffin, A. L. Agapov, and A. P. Sokolov, Phys. Rev. E 86, 021508 (2012). 32 A. Brodin, R. Bergman, J. Mattsson, and E. A. Rössler, Eur. Phys. J. B 36, 349 (2003). 33 P. Debye, Polar Molecules (Dover, New York, 1929). 34 G. Williams, Chem. Soc. Rev. 7, 89 (1978). 35 J. G. Powles, Molecular Relaxation Processes (Academic Press, New York, 1966). 36 C. De Michele and D. Leporini, Phys. Rev. E 63, 036702 (2001). 37 D. Kivelson and S. A. Kivelson, J. Chem. Phys. 90, 4464 (1989). 38 Y. Wang, P. J. Griffin, and A. P. Sokolov, “Collective dynamics and proton transport in liquid imidazole” (unpublished). 39 N. Lou, Y. Wang, X. Li, H. Li, P. Wang, C. Wesdemiotis, A. P. Sokolov, and H. Xiong, Macromolecules 46, 3160 (2013). 40 K.-D. Kreuer, Chem. Mater. 8, 610 (1996). 41 A. Li, Z. Cao, Y. Li, T. Yan, and P. Shen, J. Phys. Chem. B 116, 12793 (2012). 42 N. Agmon, Chem. Phys. Lett. 244, 456 (1995). 43 D. Marx, ChemPhysChem. 7, 1848 (2006).
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THE JOURNAL OF CHEMICAL PHYSICS 140, 104510 (2014)
Observation of the slow, Debye-like relaxation in hydrogen-bonded liquids by dynamic light scattering Yangyang Wang,1,a) Philip J. Griffin,2 Adam Holt,2 Fei Fan,3 and Alexei P. Sokolov1,2,3 1
Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA 3 Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, USA 2
(Received 15 January 2014; accepted 25 February 2014; published online 13 March 2014) The slow, Debye-like relaxation in hydrogen-bonded liquids has largely remained a dielectric phenomenon and has thus far eluded observation by other experimental techniques. Here we report the first observation of a slow, Debye-like relaxation by both depolarized dynamic light scattering (DLS) and dielectric spectroscopy in a model hydrogen-bonded liquid, 2-ethyl-4-methylimidazole (2E4MIm). The relaxation times obtained by these two techniques are in good agreement and can be well explained by the Debye model of rotational diffusion. On the one hand, 2E4MIm is analogous to the widely studied monohydroxy alcohols in which transient chain-like supramolecular structure can be formed by hydrogen bonding. On the other hand, the hydrogen-bonded backbone of 2E4MIm is much more optically polarizable, making it possible to apply light scattering to study the dynamics of the supramolecular structure. These findings provide the missing evidence of the slow, Debye-like relaxation in DLS and open the venue for the application of dynamic light scattering to the study of supramolecular structures in hydrogen-bonded liquids. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4867913] I. INTRODUCTION
Water, monohydroxy alcohols, and several other hydrogen-bonded liquids display a pronounced Debye-like low-frequency dielectric relaxation.1–11 Despite extensive studies of more than half a century, the molecular origin of this process still remains an open question. It is widely believed that the slow Debye-like relaxation is a distinct signature of hydrogen-bonded supramolecular structures, and understanding the nature of this relaxation is a key step for a complete elucidation of the dynamics of hydrogen-bonded liquids. Dynamic light scattering (DLS) is a powerful experimental method for investigating the dynamics of glass-forming liquids, providing information complimentary to dielectric and dynamic mechanical spectroscopy techniques.12, 13 So far, however, this technique (DLS) has not been able to detect a Debye-like process in hydrogen-bonded liquids in the neat state. In particular, no slow Debye-like relaxation has been found in water14 and monohydroxy alcohols6, 14 by DLS. Similarly, studies employing other non-dielectric techniques such as calorimetry15 and dynamic mechanical spectroscopy16 were, until recently, unable to identify the slow, Debye-like relaxation process in monohydroxy alcohols. It was therefore believed that this Debye-like relaxation is a purely dielectric phenomenon. However, this traditional view has been challenged by recent nuclear magnetic resonance5 and mechanical measurements.17 The emerging experimental results from non-dielectric techniques have provided some essential ingredients for the explanation of the slow Debye-like relaxation. a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2014/140(10)/104510/8/$30.00
In this context, the missing evidence from DLS appears more puzzling than ever. In this article we report the first observation of the slow Debye-like relaxation in a model hydrogen-bonded liquid, 2ethyl-4-methylimidazole (2E4MIm), by depolarized photon correlation spectroscopy (PCS) and Brillouin-Raman spectroscopy. Unlike monohydroxy alcohols, a slow Debye-like relaxation can be detected in 2E4MIm by both broadband dielectric spectroscopy (BDS) and depolarized DLS, due to the much higher optical polarizability of the hydrogen-bonded imidazole supramolecular backbone. The relaxation times for the slow Debye-like process from BDS and DLS are consistent with each other and can be well described by the Debye model of rotational diffusion. This finding provides the missing evidence of the slow Debye-like relaxation in DLS and opens the door for the application of dynamic light scattering to the study of supramolecular hydrogen-bonded structures in the neat liquid state.
II. EXPERIMENT
The 2-ethyl-4-methylimidazole (2E4MIm) compound was obtained from Sigma-Aldrich and used as received. Similar to monohydroxy alcohols, 2E4MIm can form linear chainlike supramolecular structure through hydrogen bonding (Fig. 1). Broadband dielectric spectroscopy (BDS) measurements of 2E4MIm were carried out on a Novocontrol Concept 80 system in the frequency range 10−2 –107 Hz. Depolarized photon correlation spectroscopy (PCS) measurements of 2E4MIm were performed in right angle geometry in an Oxford Optistat cryostat with temperature stability of ±0.1 K.
140, 104510-1
© 2014 AIP Publishing LLC
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FIG. 2. Heat flow (red dotted line) and its temperature derivative (blue solid line) measured during (a) cooling and (b) heating scans, at 10 K/min.
Parallel plates of small diameter (4 mm) were used to minimize the contribution of instrument compliance. FIG. 1. Illustration of the hydrogen-bonded structures of 2-ethyl-1-hexanol (2E1H) (a model monohydroxy alcohol) and 2-ethyl-4-methylimidazole. The hydrogen, carbon, oxygen, and nitrogen atoms are represented by white, gray, red, and blue sticks, respectively. The dashed lines stand for the intermolecular hydrogen bonds. Both molecules are capable of forming chain-like supramolecular structures.
The laser wavelength was 647 nm and the power was 75 mW. The scattered light was collected with an optical fiber, split between two avalanche photodiode detectors, and analyzed using an ALV-7004/FAST multi-tau digital correlator. In addition to the PCS measurements, Brillouin-Raman measurements were also carried out to access the dynamics at shorter timescales and higher temperatures. The depolarized light scattering spectra of 2E4MIm were measured in the backscattering geometry, using a tandem Fabry-Perot interferometer (Sandercock Model) and a Raman spectrometer (Jobin Yvon T64000). A solid-state laser with wavelength = 532 nm and power = 75 mW was used for the measurements. Differential scanning calorimetry measurement of 2E4MIm was performed on a TA Instruments Q2000 device with aluminum hermetic pans. The experiment was carried out in the temperature range 333–193 K at cooling/heating rate of 10 K/min. The glass transition temperature (Tg ) was determined from the mid-point of the step in heat flow, which corresponded to the maximum in the derivative of heat flow with respect to temperature. The Tg s from cooling and heating scans are 232 and 235 K, respectively (Fig. 2). Rheological measurements of 2E4MIm were performed on an AR2000ex rheometer (TA Instruments) with 4 and 25 mm parallel plates. Creep measurements were used to determine the zero-shear viscosity (η) of 2E4MIm. η can be obtained from the creep compliance J(t) in the long-time limit: η = lim t/J (t). In addition to creep experiments, dynamic t→∞ mechanical measurements of complex modulus were carried out to obtain the mechanical spectra in the frequency domain.
III. RESULTS A. Broadband dielectric spectroscopy
Representative dielectric spectra of 2E4MIm are presented in Fig. 3, where the real and imaginary parts of permittivity (ε and ε ), derivative of ε (ε der ), and loss modulus (M ) are shown as a function of frequency ( f ). Unlike monohydroxy alcohols, 2E4MIm has high intrinsic proton conductivity, due to its autodissociation.18 As a result, the dc conductivity dominates the dielectric loss spectra. To assist our analysis, the derivative spectra [ε der = (− π /2)∂ε /∂ln f ] are used.19 A typical spectrum of 2E4MIm consists of three processes (Fig. 3): a fast relaxation in the high frequency region, a pronounced slow Debye-like relaxation in the intermediate frequency region, and electrode polarization (EP) effect20 at low frequencies. The complex dielectric permittivity ε∗ of 2E4MIm can be modeled by the following equation: ε∗ (f ) = ε − iε = ε∞ +
ε1 [1 + (i2πf τHN,1 )α1 ]β1
ε2 [1 + (i2πf τHN,2 )α2 ]β2 σ + + A(2πf )n , i2πf ε0 +
(1)
where ε∞ is the dielectric permittivity at infinite frequency, εj (j = 1, 2) is the dielectric relaxation strength, τ HN,j is the relaxation time, α j , β j are the shape parameters, σ is the dc conductivity, n is the slope of EP’s high frequency tail, and A is related to the amplitude of EP. The use of an EP term improves the accuracy of our analysis, but is not essential, because the main relaxation processes and EP are well separated. The relaxation frequency of the fast process coincides with the peak frequency of the electrical loss modulus
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J. Chem. Phys. 140, 104510 (2014) TABLE I. Fit parameters for the slow Debye-like relaxation in dielectric measurements. T (K)
τ HN,1 (s)
ε 1
α1
β1
263.15 260.65 258.15 255.65 253.15 250.65
0.00252 0.00491 0.0120 0.0243 0.0590 0.140
3.79 5.32 6.20 8.14 10.8 12.7
0.89 0.83 0.93 0.93 0.92 0.95
0.89 1.0 0.91 0.99 0.94 0.95
in polyalcohols and similar materials. It has been frequently reported that an ultra-slow Debye-like relaxation can show up in the vicinity of conductivity relaxation frequency in polyalcohols24, 25 and polymers.26 Although the nature of this process has not been completely settled, it is highly likely that it originates from surface polarization effects due to presence of impurities or voids,27–29 and is not an intrinsic feature of the material itself. In contrast to the close connection of Debye-like relaxation and conductivity relaxation in polyalcohols, the frequency of the slow, Debye-like relaxation in 2E4MIm is approximately 1–2 orders lower than the conductivity relaxation frequency. More importantly, as we demonstrate below, the observation of the same process in three different techniques (BDS, PCS, and Brillouin-Raman) rules out any connection between the slow Debye-like relaxation and conductivity relaxation in 2E4MIm. FIG. 3. Representative spectra of 2E4MIm at 258.15 K: (a) real part of dielectric permittivity ε ( f ); (b) imaginary part of dielectric permittivity ε ( f ); (c) derivative of ε ( f ), ε der = (−π /2)∂ε /∂ln f ; and (d) imaginary part of electrical modulus M ( f ). Inset: Frequency dependence of real part of conductivity σ ( f ). (Black) solid curves: Fittings of experimental data using Eq. (1).
M (Fig. 3(d)) as well as the onset of dc conductivity (inset of Fig. 3). This process is therefore assigned to the conductivity relaxation of 2E4MIm. It is widely believed that the proton hopping within the hydrogen-bonded chain structure plays an important role in imidazole-based molecules. Because of its high intrinsic conductivity, the high-frequency dielectric behavior of 2E4MIm resembles that of typical ionic conductors such as ionic liquids21 and phosphoric acids.22 It should be emphasized that the possibility of observing dipolar structural relaxation in highly ionic molecules has remained an open question.23 It is possible that the fast relaxation has a dipolar contribution. The assignment of this process to conductivity relaxation is mainly due to the ionic nature of 2E4MIm and its close connection to dc conduction (inset of Fig. 3). Further experiments, however, are required to understand the exact nature of the fast relaxation (i.e., ionic vs. dipolar). A pronounced slow relaxation can be observed in ε ( f ) ( f ) (Fig. 3(c)) at intermediate frequencies, (Fig. 3(a)) and εder and the shape of this slow relaxation was resolved by simultaneously fitting ε and ε using Eq. (1). It was observed that this slow relaxation is Debye-like, with α in the range 0.83– 1.0 and β in the range 0.89–1.0. The fitting parameters are provided in Table I. It is important to recognize the difference between the slow Debye-like relaxation in 2E4MIm and those observed
B. Photon correlation spectroscopy
The normalized intensity correlation functions (ICF) of 2E4MIm are shown in Fig. 4. The decay of the ICF can be well described by the superposition of two KohlrauschWilliam-Watts (KWW) stretched exponential functions for the normalized electric field correlation function (g1 ): I (0)I (t) − 1 = g2 (t) − 1 = γ [g1 (t)]2 I 2 = γ {ϕ1 exp[−(t/τKWW,1 )βKWW,1 ] + ϕ2 exp[−(t/τKWW,2 )βKWW,2 ]}2 ,
(2)
where g1 (t) and g2 (t) are normalized electric field correlation function and normalized intensity correlation function (ICF), respectively, γ is the spatial coherence factor, ϕ j (j = 1, 2) is the relative relaxation strength, τ KWW,j is the relaxation time, and β KWW,j is the stretching parameter. The slow relaxation is Debye-like, with β KWW in the range 0.77–0.86, whereas the fast relaxation is highly stretched, with β KWW around 0.5. A detailed summary of the corresponding fit parameters is provided in Table II. Fig. 4(b) shows that the use of a single Kohlrausch-William-Watts (KWW) stretched exponential function is insufficient for fitting the experimental correlation functions. C. Brillouin-Raman spectroscopy
The PCS results are corroborated by further BrillouinRaman measurements. Similar to the time-domain PCS experiments, the susceptibility spectrum χ (ω) from BR
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FIG. 5. Imaginary part of the depolarized light scattering susceptibility spectra of 2E4MIm at 400 and 450 K. The red and black solid lines represent the fits of experimental data by Eq. (3).
FIG. 4. (a) Depolarized photon correlation spectroscopy measurements at temperatures T = 240, 245, 250, 255, 260, 265, 275, 285, and 295 K (from right to left). (b) Fitting of the electric field correlation function (EFCF) at 260 K by one (red solid line) and two (blue short-dashed line) KWW functions. The slow relaxation (green dashed line) is Debye-like and approximately 1.5 decades slower than the structural relaxation process (orange dashed-dotted line). The residuals of the fits are displayed in the bottom panel, with open (red) circles for the single-KWW fit and (blue) crossed circles for the double-KWW fit.
TABLE II. Fit parameters for the first-order autocorrelation function in PCS measurements.a Slow (Debye-like) relaxation T (K) 295 285 275 265 260 255 250 245 240 a
Fast (structural) relaxation
ϕ1
τ KWW,1 (s)
β KWW,1
ϕ2
τ KWW,2 (s)
β KWW,2
0.81 0.78 0.79 0.83 0.79 0.82 0.86 0.83 0.82
1.1 × 10−6 6.2 × 10−6 4.3 × 10−5 5.4 × 10−4 2.2 × 10−3 1.1 × 10−2 6.4 × 10−2 0.47 4.4
0.86 ± 0.03 0.86 ± 0.03 0.84 ± 0.01 0.79 ± 0.01 0.85 ± 0.01 0.84 ± 0.01 0.77 ± 0.02 0.82 ± 0.02 0.84 ± 0.02
0.19 0.23 0.21 0.17 0.21 0.18 0.14 0.17 0.18
2.3 × 10−8 2.2 × 10−7 1.5 × 10−6 1.5 × 10−5 7.3 × 10−5 3.7 × 10−4 1.7 × 10−3 0.019 0.19
0.50 ± 0.04 0.50 ± 0.04 0.50 ± 0.04 0.51 ± 0.04 0.52 ± 0.04 0.53 ± 0.04 0.50 ± 0.04 0.49 ± 0.04 0.50 ± 0.04
Fitting error: contrast: 1%; relaxation time: 1%.
experiments consists of a slow, Debye-like relaxation and a fast, highly stretched relaxation (Fig. 5). Moreover, the relaxation times and shape parameters of these two processes are also consistent with those from the PCS measurements (Tables II and III and Fig. 6). Here, it is important to point out that previous BR measurements failed to identify a slow, Debye-like relaxation in hydrogen-bonded liquids, including water.6, 14 The depolarized scattering intensities I(ω) were converted to the susceptibility spectra χ (ω) by using the fluctuation-dissipation theorem: χ (ω) = I(ω)/[n(ω) + 1], where n(ω) = [exp (¯ω/kB T) − 1]−1 is the Bose factor. Fig. 5 shows the χ spectra of 2E4MIm at 400 and 450 K, where two relaxation processes are clearly visible. The slow relaxation has a Debye-like shape, while the fast (alpha) relaxation is stretched. In order to extract their relaxation times and shape parameters, we fit the light scattering susceptibility spectra by the following equation: χ1 χ2 ν , + − iχ ω χ = − 3 (1 + iωτ1 )βCD,1 (1 + iωτ2 )βCD,2 (3) where stands for imaginary part, the first and second ColeDavidson (CD) terms describe the slow Debye-like relaxation and alpha relaxation, respectively, and the last term accounts for the low-frequency tail of the fast dynamics. The fit parameters are summarized in Table III. IV. DISCUSSION A. Glass transition temperature
To further explore the nature of the relaxation processes observed by BDS and depolarized PCS, their relaxation times are shown as a function of inverse temperature in Fig. 6. The mechanical structural relaxation time (τ η ), evaluated from the Maxwell relation: τ η = η/G, is also presented as a reference. The viscosity η was determined from the creep measurement and the glassy modulus G = 0.425 GPa was determined from the dynamic mechanical measurement close to Tg and assumed to be a constant. As a first observation, the
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TABLE III. Fit parameters for the χ spectra in Fig. 5.a Slow (Debye-like) relaxation T (K) 400 450 a
Structural (α) relaxation
Fast relaxation
χ 1
β CD,1
τ 1 (ns)
χ 2
β CD,2
τ 2 (ns)
χ 3
ν
4.1 2.8
0.83 0.84
0.76 0.12
1.1 1.1
0.38 0.38
0.059 0.011
0.12 0.14
0.3 0.3
Fitting error: amplitude: 5%; relaxation time: 1%.
relaxation time of the fast (alpha) process from PCS matches τ η , suggesting that the origin of this process is the structural relaxation of 2E4MIm. Fitting τ α with the Vogel-FulcherTammann (VFT) equation and extrapolating the relaxation time to 100 s yield a dynamic glass transition temperature (Tg ) of 231 K. This agrees well with the differential scanning calorimetry measurement, which gives a Tg of 232 K on the cooling scan. In addition, the conductivity relaxation time, τ σ , which is ∼2 orders of magnitude faster than the dielectric τ slow , matches τ α from PCS and τ η from rheology. According to the classical theory, the ionic transport should be closely connected to the structural relaxation.30 The assignment of the fast process in PCS to structural relaxation is consistent with the coincidence of τ σ and τ α . Therefore, it is clear that the Debye-like relaxation in 2E4MIm is slower than the structural relaxation and not responsible for the glass transition. B. Derivative analysis
We have shown that the light scattering intensity correlation function (ICF) measured for 2E4MIm is not well de-
FIG. 6. Comparison of the temperature dependence of the different relaxation processes probed by broadband dielectric spectroscopy (BDS), rheology (RH), depolarized photon correlation spectroscopy (PCS), and BrillouinRaman spectroscopy (BR). τ σ is the conductivity relaxation time from BDS. τ slow is the relaxation time for the slow Debye-like process. τ η is the segmental relaxation time from the mechanical measurement, calculated using the Maxwell relation, τ η = η/G. The viscosity η was determined from the creep measurement and the glassy modulus G = 0.425 GPa was determined from the dynamic mechanical measurement. τ α is the structural relaxation time from PCS and BR. Dashed lines are the fits of the τ slow and τ α from optical measurements by the Vogel-Fulcher-Tammann (VFT) equation: τ = τ 0 exp[B/(T − T0 )], where τ 0 , B, and T0 are fit parameters. The inset shows that the relaxation times of the slow process from BDS and PCS can be collapsed onto a master curve by vertically shifting the BDS data by ∼0.48 order (a factor of 3).
scribed by a single stretched exponential decay corresponding to the characteristic structural relaxation process. Instead, we have demonstrated that the ICF can be precisely fit with a superposition of two decays – a faster, stretched exponential decay, and a slower, “Debye-like” component that accounts for a majority of the total decay of the ICF. The faster relaxation time is nearly identical to the structural relaxation time calculated from rheological measurements, and due to this agreement we attribute it to the molecular reorientational α process and not to secondary relaxations or the excess wing. These results for 2E4MIm are unique, because for nearly all supercooled liquids studied via depolarized DLS, the ICF has been shown to be composed of one stretched exponential function corresponding to the structural relaxation process. Additionally, for measurements close to the calorimetric glass transition temperature, the excess wing can also contribute to the decay of the ICF, and it typically accounts for a small percentage of the measured decay – similar to dielectric relaxation measurements.31, 32 In order to more thoroughly demonstrate that two relaxations are present in the ICF of 2E4MIm and to precisely determine the stretching exponents corresponding to these two relaxation processes, we have analyzed a quantity associated with the derivative of the ICF for both 2E4MIm and 2-ethyl-1hexanol (2E1H), a model monohydroxy alcohol. If the measured ICF is composed of one stretched exponential decay, such that ICF = {a · exp [ − (t/τ α )β ]}2 , it is straightforward to show that d ln(ICF) 2β + (β − 1) log10 t. = log10 Z = log10 − dt τα β (4) Thus the KWW stretching exponent can be isolated and determined by analyzing the slope of the line on a log-log plot fitted to the quantity −dln(ICF)/dt calculated from the experimental data. Just as a single KWW function is shown to fit the ICF data of 2E1H in Fig. 7(a), it follows accordingly that a single straight line is sufficient to describe the data for 2E1H in Fig. 7(b). The stretching exponent for 2E1H was determined to be β = 0.6 from the slope of this line. At short times, the ICF reaches a plateau and the statistical noise dominates the signal. In contrast, it is clearly seen in Fig. 7(b) that the quantity Z calculated from the ICF measured for 2E4MIm has two distinct regions where lines of quite different slope are required to interpolate the data. This immediately demonstrates that the decay of the ICF of 2E4MIm proceeds in two distinct steps corresponding to two different relaxation processes. The slope of each line corresponds to the stretching exponent of each individual relaxation in a manner similar to Eq. (4),
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FIG. 7. (a) Normalized intensity correlation function (ICF) for 2-ethyl-1-hexanol (2E1H) at two representative temperatures. (b) Derivative analysis of ICF for 2E4MIm and 2E1H.
although some approximations are necessary to quantify these stretching exponents. For the case of 2E4MIm, we consider the correlation function as being a sum of two separate decays, where ICF = [a1 1 (t) + a2 2 (t)]2 = {a1 · exp[−(t/τ1 )β1 ] + a2 · exp[−(t/τ2 )β2 ]}2 , with the first term corresponding to the faster relaxation and the second term corresponding to the slower relaxation. In order to determine the stretching parameter β 2 of the slow relaxation, we make the approximation that for t ≥ τ 2 , 1 = 0. After this approximation is made, we retrieve the case of a single-step correlation function, and Eq. (4) can be used to determine the stretching parameter β 2 from the slope of the data in Fig. 7(b) at longer times. This slope corresponds to β ≈ 0.8 as is shown in Fig. 7(b). In order to determine β 1 from the slope of the data at shorter times, we make the approximation that for t ≤ τ 1 , 2 = 1, such that ICF ≈ {a1 · exp[−(t/τ1 )β1 ] + a2 }2 . Taking the derivative, we find that in this time window d ln(ICF) Z = log10 − dt t β1 a2 ≈ − log10 1 + exp a1 τ1 2β1 + (β1 − 1) log10 t, (5) + log10 τ1 β1 and since the first term of Eq. (5) is approximately constant for t ≤ τ 1 , we can further approximate that in the vicinity of the decay of the fast relaxation component, 2β1 d ln(ICF) Z = log10 − +(β1 −1) log10 t. ≈ C +log10 dt τ1 β1 (6) Once again we see that the slope of Z plotted against log10 (t) in this time interval is directly related to the stretching expo-
nent of the fast decay, which we determined to be β ≈ 0.5 as seen in Fig. 7(b). This derivative analysis clearly demonstrates that there are two decays contributing to the ICF measured in 2E4MIm. Moreover, it shows that they correspond to a slow relaxation component with β ≈ 0.8 and a faster relaxation component with β ≈ 0.5. This result is consistent with the fitting of ICF using Eq. (2) (Table II). C. Comparison of BDS and DLS
We now turn to the slow Debye-like relaxations in BDS and DLS. The relaxation times from the two measurements are comparable on absolute scales and exhibit similar temperature dependence. According to the Debye-Stokes-Einstein relation,12, 33 the rotational diffusion rate (τ −1 ) of a spherical particle in a viscous medium is inversely proportional to the cube of its hydrodynamic radius R, τ −1 =
l(l + 1)kB T , 8π ηR 3
(7)
where l is the tensor rank of the probed relaxation, kB is the Boltzmann constant, T is the absolute temperature, and η is the viscosity of the medium. It follows that the ratio of the first and second-rank rotational correlation time should be equal to 3. It is well known that BDS and depolarized DLS probe the first- and second-rank rotational correlation, respectively.34 Therefore, for the same rotational diffusion process, the dielectric and light scattering relaxation times should differ by a factor of 3, if the angle for each reorientational jump is small.35–37 The inset of Fig. 6 shows that the BDS and DLS relaxation times for the slow, Debye-like relaxation can be collapsed onto a master curve by shifting the dielectric logτ data by ∼0.48 order (a factor of 3). This coincidence not only confirms that the slow processes observed by BDS and DLS are of the same origin, but also suggests that the slow, Debye-like relaxation in 2E4MIm proceeds through small-angle jumps
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TABLE IV. Hydrodynamic radii for structural and slow Debye-like relaxing units. T (K) 295 285 275 265 260 255 250 245 240
τ α (s) 10−8
3.1 × 3.0 × 10−7 2.0 × 10−6 2.0 × 10−5 6.9 × 10−5 4.9 × 10−4 2.2 × 10−3 2.5 × 10−2 0.25
τ slow (s) 10−6
1.2 × 6.7 × 10−6 4.7 × 10−5 6.1 × 10−4 2.4 × 10−3 1.2 × 10−2 7.3 × 10−2 0.52 4.8
η (Pas)
Rα (Å)
Rslow (Å)
Rslow /Rα
10.3 56.4 443 5.54 × 103 2.43 × 104 1.27 × 105 8.10 × 105 6.50 × 106 6.83 × 107
1.4 ± 0.1 1.7 ± 0.2 1.6 ± 0.2 1.5 ± 0.2 1.3 ± 0.1 1.5 ± 0.2 1.3 ± 0.1 1.5 ± 0.2 1.4 ± 0.1
4.8 ± 0.5 4.8 ± 0.5 4.6 ± 0.5 4.6 ± 0.5 4.4 ± 0.4 4.3 ± 0.4 4.2 ± 0.4 4.0 ± 0.4 3.8 ± 0.4
3.4 2.8 2.9 3.1 3.3 2.9 3.2 2.7 2.7
(i.e., Brownian rotational diffusion). However, it is important to keep in mind that in addition to the reorientation of entire supramolecular structure, scission and recombination should also contribute to the observed rotational diffusion.5 Earlier investigations failed to identify the Debyelike relaxation in monohydroxy alcohols by dynamic light scattering.6 Our own depolarized PCS measurement on 2ethyl-1-hexanol (2E1H) seems to confirm this conclusion (Fig. 7). It is important to understand why the slow Debye-like relaxation can be resolved by depolarized PCS in 2E4MIm, but not in monohydroxy alcohols. Recent studies have shown that the structural relaxation in monohydroxy alcohols mainly arises from the motion of the alkyl chains.8 It is well known that the optical polarizabilities of alkyl groups are much higher than that of the hydroxyl group. As a result, in the case of monohydroxy alcohols, the fluctuations of optical polarizability caused by the motion of supramolecular chains will simply be overwhelmed by the motion of the alkyl groups, making it difficult to detect the slow Debye-like relaxation by DLS. This stands in sharp contrast to the dielectric measurement, where the dielectric relaxation is dominated by the polar, supramolecular backbone. Compared to the monohydroxy alcohols, the imidazole-bonded supramolecular backbone is much more optically polarizable and its “side chains” consist of only short methyl and ethyl groups. This special structure makes it possible to observe the slow Debye-like relaxation by both BDS and DLS techniques. From the Debye-Stokes-Einstein relation (Eq. (7)), one can further estimate the average size of the hydrogen-bonded supramolecular chain for 2E4MIm (Table IV). Using the alpha relaxation time from PCS measurements and the viscosity from rheology, the hydrodynamic radius (R) of the structural relaxing unit is estimated to be approximately 1.5 ± 0.2 Å, which roughly corresponds to the radius for the rotation of a single 2E4MIm molecule about an axis through its center of mass and perpendicular to its dipole moment. The apparent radius for the slow Debye-like relaxation is estimated to be approximately 4.4 ± 0.4 Å. Because of the contribution of scission and recombination to the observed rotational diffusion, Rslow should not be literally interpreted as the size of the supramolecular chain. Rather, it should be considered as the lower bound. Therefore, from the ratio of Rslow and Rα , we can conclude that the average length of the hydrogenbonded chain-like structure is at least about three repeating units of 2E4MIm. Additionally, because Rslow is much larger
than the size of 2E4MIm, it is unlikely that the slow, Debyelike is the structural relaxation. This conclusion is consistent with our earlier argument presented in Sec. IV A. Lastly, it is worth mentioning that the Debye-like relaxation of 2E4MIm appears to be slightly more stretched than those of water and monohydroxy alcohols. This is presumably due to the fact that 2E4MIm has two tautomers: 2ethyl-4-methylimidazole and 2-ethyl-5-methylimidazole. Our measurement of neat imidazole shows that its Debye-like relaxation is indeed much narrower.38 V. SUMMARY AND CONCLUDING REMARKS
In summary, we have shown that a slow, Debye-like relaxation can be observed in 2-ethyl-4-methylimidazole by broadband dielectric spectroscopy, depolarized photon correlation spectroscopy, and Brillouin-Raman spectroscopy. This result provides the missing evidence of the slow, Debye-like relaxation in DLS. While monohydroxy alcohols are widely regarded as model systems for the study of supramolecular structure in hydrogen-bonded liquids and have received enormous attention during the past several decades, the present study suggests that our understanding of hydrogen-bonded liquids can benefit tremendously from studying not only monohydroxy alcohols but also other model systems. In the case of dynamic light scattering measurements, highly optically polarizable hydrogen-bonded structures such as imidazole chains clearly have advantages over the hydroxylbonded monohydroxy alcohols. It is also worth mentioning that the slow, Debye-like relaxation has been recently observed in a supramolecular polymer based on the thymine and diamidopyridine motif39 and a close connection between the dielectric Debye-like relaxation and mechanical chain relaxation was demonstrated. By exploring the rich supramolecular chemistry developed during the past several decades, it is possible to find model hydrogen-bonded liquids where the motion of supramolecular structures can be probed by several non-dielectric experimental techniques, including dynamic light scattering and rheology. Lastly, it is worth noting that imidazole-based molecules are also model systems for proton transport.18, 40 It is widely believed that the proton hopping within the hydrogen-bonded chain-like structure of imidazole41 gives rise to high proton conductivity, similar to the well-known Grotthuss mechanism42, 43 in water. The direct observation of the supramolecular structure of 2E4MIm
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by DLS also offers a new powerful experimental approach to the study of proton transport in hydrogen-bonded liquids.
ACKNOWLEDGMENTS
The authors thank A. L. Agapov and J. R. Sangoro for helpful discussions. This work was supported by the NSF Chemistry Program (CHE-1213444). 1 M.
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