Determining diffusion coefficients of ionic liquids by means of field cycling nuclear magnetic resonance relaxometry D. Kruk, R. Meier, A. Rachocki, A. Korpała, R. K. Singh, and E. A. Rössler Citation: The Journal of Chemical Physics 140, 244509 (2014); doi: 10.1063/1.4882064 View online: http://dx.doi.org/10.1063/1.4882064 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/24?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Primary and secondary relaxation process in plastically crystalline cyanocyclohexane studied by 2H nuclear magnetic resonance. I J. Chem. Phys. 138, 074503 (2013); 10.1063/1.4790397 Intermolecular relaxation in glycerol as revealed by field cycling 1H NMR relaxometry dilution experiments J. Chem. Phys. 136, 034508 (2012); 10.1063/1.3672096 Nuclear magnetic resonance studies on the rotational and translational motions of ionic liquids composed of 1ethyl-3-methylimidazolium cation and bis(trifluoromethanesulfonyl)amide and bis(fluorosulfonyl)amide anions and their binary systems including lithium salts J. Chem. Phys. 135, 084505 (2011); 10.1063/1.3625923 Proton field-cycling nuclear magnetic resonance relaxometry in the smectic A mesophase of thermotropic cyanobiphenyls: Effects of sonication J. Chem. Phys. 121, 554 (2004); 10.1063/1.1740751 Diffusional behavior of polypeptides in the thermotropic liquid crystalline state as studied by the pulse fieldgradient spin-echo 1 H nuclear magnetic resonance method J. Chem. Phys. 113, 7635 (2000); 10.1063/1.1312278

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THE JOURNAL OF CHEMICAL PHYSICS 140, 244509 (2014)

Determining diffusion coefficients of ionic liquids by means of field cycling nuclear magnetic resonance relaxometry D. Kruk,1,2 R. Meier,2 A. Rachocki,3 A. Korpała,4 R. K. Singh,5 and E. A. Rössler2 1

Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Słoneczna 54, PL-10710 Olsztyn, Poland 2 Universität Bayreuth, Experimentalphysik II, 95440 Bayreuth, Germany 3 Institute of Molecular Physics, Polish Academy of Sciences, Smoluchowskiego 17, 60-179 Pozna´n, Poland 4 Department of Biophysics, Jagiellonian University Medical College, Łazarza 16, 31-530 Kraków, Poland and Institute of Physics, Jagiellonian University, Reymonta 4, 30-059 Kraków, Poland 5 Ionic Liquid and Solid State Ionics Laboratory, Department of Physics, Banaras Hindu University, Varanasi 221 005, India

(Received 14 February 2014; accepted 27 May 2014; published online 30 June 2014) Field Cycling Nuclear Magnetic Resonance (FC NMR) relaxation studies are reported for three ionic liquids: 1-ethyl-3- methylimidazolium thiocyanate (EMIM-SCN, 220–258 K), 1-butyl-3methylimidazolium tetrafluoroborate (BMIM-BF4 , 243–318 K), and 1-butyl-3-methylimidazolium hexafluorophosphate (BMIM-PF6 , 258–323 K). The dispersion of 1 H spin-lattice relaxation rate R1 (ω) is measured in the frequency range of 10 kHz–20 MHz, and the studies are complemented by 19 F spin-lattice relaxation measurements on BMIM-PF6 in the corresponding frequency range. From the 1 H relaxation results self-diffusion coefficients for the cation in EMIM-SCN, BMIM-BF4 , and BMIM-PF6 are determined. This is done by performing an analysis considering all relevant intraand intermolecular relaxation contributions to the 1 H spin-lattice relaxation as well as by benefiting from the universal low-frequency dispersion law characteristic of Fickian diffusion which yields, at low frequencies, a linear dependence of R1 on square root of frequency. From the 19 F relaxation both anion and cation diffusion coefficients are determined for BMIM-PF6 . The diffusion coefficients obtained from FC NMR relaxometry are in good agreement with results reported from pulsedfield-gradient NMR. This shows that NMR relaxometry can be considered as an alternative route of determining diffusion coefficients of both cations and anions in ionic liquids. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4882064] I. INTRODUCTION

Nuclear Magnetic Resonance (NMR) relaxometry is a powerful method of investigating dynamics of liquids.1–3 Using commercially available NMR relaxometers based on the Field Cycling (FC) technique one can perform relaxation experiments in a frequency range covering three orders of magnitude: 10 kHz–20 MHz (for 1 H). Special home-built relaxometers allow reaching frequencies down to some 10 Hz.4, 5 The spin-lattice relaxation rates are given as linear combinations of spectral densities which are Fourier transforms of a time correlation function. The latter describes stochastic fluctuations of spin interactions causing the relaxation process, taken at frequencies related to the energy level structure of the relaxing system which are determined by the external magnetic field. 1 H (19 F) relaxation in liquids is mainly caused by intraas well as intermolecular magnetic dipole-dipole interactions. The intramolecular dipolar interactions fluctuate due to molecular rotation, while intermolecular interactions are modulated by relative translational diffusion of the interacting molecules. Recently we have published a series of works6–8 exploiting the potential of FC 1 H NMR relaxometry for neat liquids. A spherical molecule rotating and translating in a viscous medium yields the relationship: τ trans /τ rot = 9, where τ rot is the rotational correlation time of rank two, 0021-9606/2014/140(24)/244509/11/$30.00

while τ trans denotes the corresponding time constant for translational diffusion.9 It has been found that for “real” molecules the ratio is larger, reaching the value of 20–40.6, 10 As τ trans > τ rot the intermolecular relaxation, associated with translational dynamics, dominates at low frequencies while at high frequencies the relaxation contribution associated with rotational dynamics prevails. In consequence, 1 H (19 F) relaxation studies performed versus frequency give access to rotational as well as translational dynamics by means of a single experiment. Yet, 1 H NMR relaxometry as a method of determining diffusion coefficients has gained momentum only recently. For long times the translational correlation function follows a power law, ∝ t−3/2 , that is characteristic of free diffusion. Correspondingly, at low frequencies the relaxation rate follows a universal relaxation law; explicitly, the rate is proportional to square root of the frequency.11–15 From the corresponding slope the diffusion coefficient D can straightforwardly be obtained. We have tested this approach for molecular liquids8 and polymers16 comparing the obtained diffusion coefficients with results of pulsed-field-gradient NMR (PFG NMR).8 The agreement is very good that allows us to state that FC 1 H NMR can be treated as a method complementary to PFG NMR. The range of diffusion coefficients typically accessible by FC 1 H NMR is (10−9 – 10−13 )m2 /s.6 A closed form expression for the translational (intermolecular) spectral density has been derived quite a

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while ago17, 18 assuming that the molecules can be treated as hard spheres, with the nuclei of interest placed in their centers, undergoing Fick diffusion (force-free-hard-sphere model). We have exploited this description to reproduce FC NMR relaxation for molecular liquids in the whole frequency range.6, 7 With this experience we have turned to ionic liquids. Due to the broad range of applications of ionic liquids,19–22 for which knowledge of transport coefficients is of great importance, there are several reports on their diffusion coefficients measured by means of PFG NMR;23–27 At this stage one should stress that PFG NMR measures self-diffusion coefficient of the molecule (or ion), while FC NMR probes the relative diffusion coefficient, D12 = D1 + D2 , where D1 , D2 are self-diffusion coefficients of the participating species (for identical molecules D12 = 2D). The relaxation studies on molecular liquids have been carried out for protons (1 H). The ionic liquids considered in this work are composed of cations containing 1 H, and anions which, in some cases, contain 19 F (BF4 , PF6 ). For such systems, 1 H and 19 F relaxometry is exploited to determine, in a concomitant way, diffusion coefficients of cations and anions. As in the case of 1 H also 19 F relaxation is determined by fluctuations of the dipole-dipole interaction and thus reflects intra- as well as intermolecular contributions. Discussing intermolecular dipolar interaction one should also mention a recent work on intermolecular NOE in liquids,28 in which special emphasis is put on 1 H-19 F NOE for ionic liquids as a source of information on both translational and rotational dynamics. Here we report results from FC NMR on three ionic liquids, the complexity of which (in terms of different NMR nuclei involved) is progressively increased. First, 1 H spinlattice relaxation data for 1-ethyl-3-methylimidazolium thiocyanate (EMIM-SCN) is considered. As the anion does not contain NMR nuclei which are relevant (the only NMR active nucleus is 14 N, but its influence can be neglected due to its small gyromagnetic ratio), the relaxation scenario is the same as for molecular liquids investigated previously,6–8 and the results allow direct determination of the diffusion coefficient of the 1 H containing cation. The next step is 1-butyl-3methylimidazolium tetrafluoroborate (BMIM-BF4 ) for which 1 H relaxation experiments have been performed. Here, the intermolecular relaxation stems from 1 H-1 H dipole-dipole interactions modulated by relative diffusion of cation and 1 H-19 F dipole-dipole interactions modulated by the relative cation-anion diffusion. The third system studied is 1-butyl3-methylimidazolium hexafluorophosphate (BMIM-PF6 ) for which the 1 H relaxation experiments have been accompanied by 19 F measurements. The 1 H relaxation again originates from 1 H-1 H and 1 H-19 F dipolar interactions, while 19 F relaxation probes anion-anion diffusion via 19 F-19 F interactions and complementarily to 1 H the cation-anion diffusion via the 19 F -1 H coupling. The relaxation data, collected in a broad temperature range, are analyzed profiting (if possible) from the universal low-frequency dispersion law resulting from Fickian diffusion. In addition, a full description of the relaxation profile, including all relevant relaxation contributions is applied. The obtained diffusion coefficients are

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compared with results from PFG NMR, which are available in literature.23–27 The paper is organized as follows. In Sec. II the theoretical description of 1 H and 19 F relaxation for the mentioned systems is presented. Section III contains experimental details. In Sec. IV the relaxation data are analyzed and the obtained results discussed, while Sec. V contains concluding remarks. II. SPIN RELAXATION PROCESSES IN LIQUIDS – THEORETICAL BACKGROUND

As anticipated, dipole-dipole interactions are the predominant source of 1 H and 19 F relaxation in the liquids studied. As the dipolar couplings can be of intra- or intermolecular origin, the measured spin-lattice relaxation rate, R1I (ωI ) = T1−1 (ωI ) (ωI denotes the angular frequency of nuclei with spin I) is a sum of both contributions.6, 7, 9 In the simplest case, when the only NMR active nuclei in the molecule are protons, the 1 H relaxation rate, R1H (ωH ) is given as HH HH (ωH ) + R1H,inter (ωH ), R1H (ωH ) = R1H,intra

(1)

explicitly indicating that the interactions causing relaxation are between protons, 1 H-1 H, (ωH = γ H B0 is the proton frequency and γ H denotes gyromagnetic factor). The HH (ωH ) relaxation contribution is described by the wellR1H,intra known BPP expression9, 29–31 HH HH R1H,intra (ωH ) = CDD [Jintra (ωH ) + 4Jintra (2ωH )],

(2)

HH denotes the 1 H-1 H dipolar relaxation constant. where CDD The intramolecular dipolar couplings are modulated by rotational dynamics. This is reflected by the spectral density, Jintra (ω), which for rotational dynamics in liquids typically assumes a Cole-Davidson form,2, 32

Jintra (ω) = Jrot (ω) =

sin[β arctan(ωτCD )] , ω[1 + (ωτCD )2 ]β/2

(3)

with τ rot = βτ CD , where τ rot is the rotational correlation time, while 0 < β ≤ 1 is a phenomenological stretching parameter; for β = 1 Jintra  (ω) becomes Lorentzian; the spectral density is normalized: Jintra (ω)dω = π /2. HH The intermolecular relaxation contribution, R1H,inter 6, 7, 9 (ωH ), is given by an expression analogous to Eq. (2), μ 2 3 0 2 HH NH γH ¯ R1H,inter (ωH ) = 10 4π × [Jinter (ωH ) + 4Jinter (2ωH )], (4) where NH is the number of protons per volume. The intermolecular dipolar interactions are modulated by relative translational diffusion of the participating molecules or ions. Assuming the force-free-hard-sphere model, the spectral density, Jinter (ω), can be expressed as17, 18 Jinter (ω) = Jtrans (ω) 1 = 72 3 d

∞ 0

u2 u2 τtrans du, 81+9u2 −2u4 +u6 u4 + (ωτtrans )2 (5)

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where u denotes an integration variable. The translational correlation time, τ trans , is defined as τ trans = d2 /D12 ; d denotes the distance of closest approach for the interacting species, while D12 is their relative translational diffusion coefficient given as a sum of diffusion coefficients of the participating molecules: D12 = D1 + D2 ; for identical molecules D12 = 2D. When the relaxation is caused by 1 H-1 H intermolecular dipolar interactions and identical species are considered it is convenient to use the notation: D12 = DHH = 2DH ; then the distance of closest approach refers to 1 H-1 H distance, d = dHH . In the low-frequency range, (when ωτ trans  1) the translational spectral density can be approxi3/2 √ mated as Jtrans (ω) = a − 2 3/2π ω, where a denotes a 9D12

constant.8, 11–15, 17, 18 This is a general feature which results from free diffusion controlling translational motion in the liquid at long times. For the case of 1 H relaxation caused by dipolar interactions between protons of identical molecules or ions (DHH = 2DH ) one obtains8, 13, 14   μ 2 √2 + 8 0 2 HH HH γ ¯ R1,H (ωH ) = R1,H (0) − NH 4π H 15 −3/2 √

× π DH H

ωH .

(6)

HH Here the intramolecular relaxation rate, R1,intra (ωH ) is inHH cluded into R1,H (0) as it can be treated at low frequencies as frequency independent due to the fact that the rotational spectral density (cf. Eq. (3)) in contrast to Jtrans (ω) becomes flat in the low-frequency limit, and in addition τ rot  τ trans holds.6, 7, 10 This implies that one can directly determine the diffusion coefficient from the low-frequency slope of the 1 H √ spin-lattice relaxation rate R1H H (ωH ) plotted versus ωH . 1 This will be used to analyze H spin–lattice relaxation data for EMIM-SCN. It is worth to mention that for real, multi-nuclear molecules the intermolecular dipolar interactions are also affected by rotational dynamics as the interacting nuclei are not placed in the molecular centers. This issue has been discussed in Refs. 6 and 8, where it has been demonstrated that due to time scale separation of the rotational and translational dynamics, the rotational contribution to the inter-molecular reHH (0) term, not affecting laxation can be included into the R1,H Eq. (6). The description becomes more complex when the ionic liquid includes 1 H and 19 F containing species, as is the case of BMIM-BF4 and BMIM-PF6 . The 1 H relaxation rate R1H (ωH ) results from the following contributions:8 HH HH HF R1H (ωH ) = R1H,intra (ωH ) + R1H,inter (ωH ) + R1H,inter (ωH ). (7) The first two terms in Eq. (7) have already been defined above HF (ωH ), (Eqs. (2) and (4)). The last term in Eq. (7), R1H,inter 1 19 originates from H- F (cation–anion) dipole-dipole interactions. It is given as the following combination of spectral densities:9, 29–31, 33, 34 μ 2 1 0 HF NF γH γF ¯ [Jinter (ωH − ωF ) R1H,inter (ωH ) = 10 4π (8) +3Jinter (ωH ) + 6Jinter (ωH + ωF )],

where NF denotes the number of 19 F nuclei per volume and ωF is the resonance frequency of 19 F. The intermolecular spectral density, Jinter (ω), is given by Eq. (5), but now the distance of closest approach concerns the 1 H-19 F distance, d = dHF , while the diffusion coefficient D12 = DHF describes the relative diffusion of cation and anion. There are also relaxation contributions originating from 1 H-11 B and 1 H-31 P dipole-dipole interactions for BMIM-BF4 and BMIM-PF6 , respectively. Due to relatively small gyromagnetic factors of 11 B and 31 P and lower concentrations of the nuclei these contributions can be neglected. As γ B /γ F = 0.34 and in BMIMHB BF4 NB = NF /4 one can estimate that for BMIM-BF4 R1H,inter H F ∼ = 0.15R1H,inter (taking into account the spin quantum number of 11 B which is 3/2, while it is 1/2 for the other nuclei involved); analogously for BMIM-PF6 one gets γ P /γ F = 0.43, HP HF ∼ . The relationNP = NF /6 which gives R1H,inter = 0.03R1H,inter HB HF ∼ ship R1H,inter = 0.15R1H,inter can raise some doubts whether HB contribution can be neglected, but, anticipating the R1H,inter HF contribution to the results presented in Sec. IV, the R1H,inter 1 H relaxation in BMIM-BF4 will turn out to be much smaller HH contribution. than the R1H,inter For systems like BMIM-PF6 (as well as BMIM-BF4 ) 1 H relaxation experiments can be complemented by 19 F relaxation studies. 19 F relaxation rate, R1F (ωF ), consists of the following contributions: FF FP R1F (ωF ) = R1F,intra (ωF ) + R1F,intra (ωF ) FF FH (ωF ) + R1F,inter (ωF ). + R1F,inter

(9)

FF The intramolecular relaxation rate, R1F,intra (ωF ), resulting 19 19 from F- F dipolar interactions within the same molecule is given by Eq. (2) in which the index “H” is replaced by inFF denotes then the 19 F relaxation constant. The dex “F”; CDD rotational correlation time, τ rot , in the intramolecular spectral density (Eq. (3)) describes now rotation of the anion. The FP (ωF ), second intramolecular relaxation contribution, R1F,intra is given as FP FP R1F,intra (ωP ) = CDD [Jintra (ωF − ωP ) + 3Jintra (ωF )

+ 6Jintra (ωF + ωP )],

(10)

FP where CDD is the intramolecular relaxation constant for the 31 F- P coupling; again τ rot in Eq. (3) refers in this case to the anion rotation. As far as the intermolecular relaxation is concerned, one has to consider 19 F-19 F and 19 F-1 H conFF FH (ωF ) and R1F,inter (ωF ) respectively. The tributions, R1F,inter FF R1F,inter (ωF ) term is given by Eq. (4) in which index “H” is replaced by index “F,” combined with the intermolecular spectral density of Eq. (5). The distance of closest approach d now refers to the anion-anion distance, d = dFF ; analogously D12 = DFF = 2DF , where DF is the anion diffusion coefficient. FH (ωF ) contribution yields (in analogy Eventually, the R1F,inter to Eq. (8)), 19

FH (ωF ) = R1F,inter

μ 2 1 0 NH γH γF ¯ [Jinter (ωH − ωF ) 10 4π + 3Jinter (ωF ) + 6Jinter (ωH + ωF )]. (11)

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FIG. 1. Chemical structure of ionic liquids investigated (from the left: EMIM-SCN, BMIM-BF4 , BMIM-PF6 ).

Now the distance of closest approach (Eq. (5)) refers to the cation-anion distance, d = dHF , and D12 = DHF . There is also FP 19 31 F- P intermolecular contribution, R1F,inter (ωF ), which can be obtained by replacing NH , γ H , ωH in Eq. (11) by NP , γ P , FF (ωF ) ωP . This term is, however, much smaller than R1F,inter FH (ωF ) due to a lower spin density (NP = NF /6) and and R1F,inter FP FF ∼ . smaller gyromagnetic factor of 31 P: R1F,inter = 0.02R1F,inter III. EXPERIMENTAL DETAILS 1

H spin-lattice relaxation of the ionic liquids: EMIMSCN, BMIM-BF4 , and BMIM-PF6 (cf. Figure 1) has been measured using STELAR FFC 2000 relaxometer which covers the frequency range of 10 kHz–20 MHz (for 1 H). Moreover 19 F spin-lattice FC relaxation experiments have been done for BMIM-PF6 in the corresponding frequency range of 9.4 kHz–18.8 MHz. The relaxation data have been collected in the temperature ranges: 220–258 K (1 H, EMIMSCN), 243–318 K (1 H, BMIM-BF4 ), 258–323 K (1 H and 19 F, BMIM-PF6 ). In the FC experiments the external magnetic field is generated by a solenoid coil connected to power supply via MOSFETs which allow for switching the current. The probehead is equipped with a saddle coil into which glass tubes containing the sample can directly be inserted from the top of the main magnet. The probehead enables temperature control with accuracy better than 1 K. Above room temperature dried air flow is heated, below evaporated liquid nitrogen is used instead. The samples are degassed and sealed; typically it contains about 0.5 ml of liquid. The experiment is conducted as follows. First the sample is polarized in a magnetic field of about 0.47 T. The field is applied sufficiently long to fully polarize the sample (equilibrium magnetization is reached). Then the field is switched to a lower value (referred to as relaxation field) and the magnetization decreases in time towards new equilibrium. After a delay time, τ , a 90◦ pulse is applied at a detection field of about 0.38 T (16.2 MHz for 1 H). The pulse length is about 4.5– 4.7 μs. The magnitude of the acquired free induction decay (FID) is recorded for different τ values leading to a magnetization curve (magnetization versus time). At least four accumulations with different phase are applied to avoid artifacts. At higher frequencies (above 10 MHz), the sample is not initially polarized. Thus, instead of a magnetization decay a magnetization increase up to the value corresponding to equilibrium at the high frequency is recorded versus time. The investigated relaxations are single exponential. Uncertainties of the relax-

ation rates determined from the exponential fits do not exceed 2%. For more detailed information about the principles of FC NMR relaxometry the reader is referred to Ref. 1. The measurement time for one point (one T1 value) depends on the relaxation time itself. The recycle delay (time between the repetition of each sequence) is set to 5T1 . The same rule is applied to the polarization field. For relaxation times of the order of 10–100 ms the measurement of one T1 value takes up to 2 min, for four accumulations, when the magnetization curve consists of about 20 points. The temperature range in which the experiment is performed is limited from the low-temperature side by the switching time of the STELAR relaxometer (about 3 ms), and from the high-temperature side by a noticeable relaxation dispersion (faster dynamics implies a weaker frequency dependence of the relaxation rate). IV. ANALYSIS AND DISCUSSION A. EMIM-SCN

We begin the analysis with the simplest system, EMIMSCN. As explained, in this case there is only one intermolecular relaxation pathway – the cation-cation, 1 H-1 H, dipoledipole interactions. Therefore Eq. (6) linking the diffusion coefficient, DHH , with the low-frequency dependence of R1H √ versus ωH applies. Figures 2(a) and 2(b) shows 1 H spin√ lattice relaxation rate for EMIM-SCN plotted versus νH (ωH = 2π ν H ) in the low-frequency range. The corresponding linear fits from which the cation-cation diffusion coefficients, DHH , have been obtained using Eq. (6) are indicated by solid lines. For EMIM-SCN the 1 H concentration is NH = 4.36 × 10−2 Å−3 (density of EMIM-SCN is 1.11 g/cm3 ). The corresponding diffusion coefficients, DH = DHH /2, are collected in Table I (right part) and shown in Figure 3 (the figure also contains further results which will be discussed later). As explained in Sec. II, Eq. (6) is a limiting expression, valid for ωτ trans < 1 (ω denotes the upper limit of the low frequency range). As the formula stems from Taylor expansion of Jinter (Eq. (5)) truncated to the first order term, it always includes an intrinsic error caused by neglecting the higher order terms which role increases with increasing ωτ trans . On the other hand when one limits the analysis to a narrow frequency range the number of experimental points is small and the relaxation dispersion is less pronounced which leads to another source of uncertainties. Although there is no strict mathematical recipe for determining the optimal frequency range some comments might be useful. The analysis should

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(a)

FIG. 3. Diffusion coefficients, DH , for the cation in EMIM-SCN, BMIMBF4 , and BMIM-PF6 as obtained from FC 1 H NMR and from 19 F in the case BMIM-PF6 compared with literature data from PFG NMR: BMIM-BF4 23 (red open squares), BMIM-BF4 24 (red open circles); BMIM-PF6 25 (green open squares), BMIM-PF6 26 (green open circles), BMIM-PF6 27 (green open triangles (synthesized), and green open stars (commercial). The estimates of the errors of the cation diffusion coefficients are provided in Tables I–III, for EMIM-SCN, BMIM-BF4 , and BMIM-PF6 , respectively.

(b)

√ FIG. 2. 1 H spin-lattice relaxation rate, R1H , versus νH for EMIM-SCN at (a) lower temperatures and (b) higher temperatures; solid lines: linear fits at low frequency along Eq. (6), dashed lines: fits of the full dispersion shown in Figures 4(a) and 4(b).

start from the highest temperature as then the range of linearity is broad and one is unlikely to exceed it. After determining the diffusion coefficient one has to check whether the relationship ωτ trans < 1 holds in the chosen frequency range. For this purpose one has to estimate τ trans assuming a reasonable distance of closest approach, d (being of the order of the molecular diameter). With decreasing temperature the range of linearity shrinks as τ trans becomes longer. When the frequency

range has been determined for the highest temperature, it will not be exceeded at lower temperatures if ω/D (D denotes the diffusion coefficient) did not increase (as τ trans ∝ 1/D when ωτ trans does not increase ω/D does not increase either). In Table I the estimated product of ωτ trans for EMIM-SCN is listed. Moreover, in the Appendix a test which helps to detect possible inconsistencies in the analysis is explained and applied. It requires that the product DR1 (0) remains unchanged with temperature (R1 (0) denotes the relaxation rate expected for ω = 0). Indeed, as shown in Figure 14 (the Appendix), this is the case. Alternatively, according to the description outlined in Sec. II, the relaxation rates can be described for all frequencies covered by Eqs. (1)–(5), i.e., the force-free-hardsphere model is assumed. Attempting the analysis in terms of these formulae we have concluded that the dispersion data obtained in the present temperature range can be well reproduced neglecting the intramolecular relaxation

TABLE I. Diffusion coefficients of cation, DH , in EMIM-SCN obtained from the full analysis of 1 H spinlattice relaxation dispersion, R1H (ωH ), (complemented by dHH and τ trans values) and from the low-frequency √ linear dependence of R1H (ωH ) on νH ; in parentheses uncertainties of the diffusion coefficients are given, ω denotes the upper limit of the low frequency range. Full analysis Temp. (K) 220 223 225 228 230 233 238 243 248 253 258 a

Low frequency slope

dHH (Å)

τ trans (s)

DH (m2 /s)

2.47 2.45 2.42 2.41 2.36 2.35 2.31 2.26 2.22 2.20 2.13

2.52 × 10−7 1.67 × 10−7 1.25 × 10−7 9.41 × 10−8 5.61 × 10−8 4.39 × 10−8 2.35 × 10−8 1.21 × 10−8 8.42 × 10−9 5.30 × 10−9 3.26 × 10−9

1.21 × 10−13 (8%) 1.79 × 10−13 (6%) 2.34 × 10−13 (5%) 3.08 × 10−13 (4%) 4.96 × 10−13 (4%) 6.29 × 10−13 (3%) 1.14 × 10−12 (3%) 2.12 × 10−12 (4%) 2.93 × 10−12 (6%) 4.58 × 10−12 (11%) 6.98 × 10−12 (11%)

DH (m2 /s) 1.25 × 10−13 1.72 × 10−13 2.33 × 10−13 3.10 × 10−13 5.15 × 10−13 6.31 × 10−13 1.17 × 10−12 2.16 × 10−12 2.86 × 10−12 4.45 × 10−12 6.57 × 10−12

(7%) (4%) (3%) (3%) (2%) (2%) (2%) (2%) (2%) (3%) (2%)

ωτ trans a 0.14 0.12 0.17 0.14 0.13 0.13 0.06 0.06 0.06 0.08 0.05

To estimate τ trans , d = 2.5 Å has been used.

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(a)

FIG. 5. 1 H spin-lattice relaxation rates for BMIM-BF4 . Solid lines: fits acHH HF cording to the formula: R1H (ωH ) = R1H,inter (ωH ) + R1H,inter (ωH ), dashed H H lines: R1H,inter (ωH ) contribution only. The parameters are: dHH = 2.41 Å, DHH = DHF = 3.06 × 10−13 m2 /s, dHF = 4.16 Å (243 K); dHH = 2.40 Å, DHH = DHF = 6.02 × 10−13 m2 /s, dHF = 4.16 Å (248 K).

(b)

FIG. 4. 1 H spin-lattice relaxation rate, R1H (ωH ), for EMIM-SCN versus frequency at (a) low temperatures and (b) high temperatures; solid lines: fits of the relaxation dispersion in the whole frequency range.

contribution. Solid lines in Figure 4 show fits obtained by taking into account only the intermolecular relaxation, i.e., HH (ωH ) in Eq. (1). Omitting the insetting R1H (ωH ) = R1H,inter 1 1 tramolecular H- H contribution leads to some discrepancies in the high-frequency and low-temperature range but the effect is small (cf. Figure 4(a)). The fits have been performed for two adjustable parameters, dHH and DHH . The distance of closest approach, dHH , refers to the distance between EMIM cations. It somewhat varies (becomes shorter) with temperature, while one expects that dHH remains temperature independent. This effect has also been reported for molecular liquids and attributed to the simplified model of the fluctuations of the intermolecular dipolar interactions.6 The diffusion coefficients, DH , obtained from the full analysis are included in Table I and Figure 3. They are in good agreement with those determined from the low-frequency behavior analyzed in Figure 2. Small discrepancies (below 6%) between the two sets of results are expected due to the approximate character of Eq. (6) and due to neglecting in the full analysis the intramolecular relaxation contribution (this somewhat influences the overall fits). The discrepancies do not exceed the estimated uncertainties of the diffusion coefficients. The fits of the full relaxation dispersion data are also included in Figure 2 to compare them with the linear fits at low frequencies. As expected, the fits of the full relaxation dispersion also well reproduce the low-frequency range. B. BMIM-BF4

In the next step we consider BMIM-BF4 . Anticipating the results, it has turned out that 1 H relaxation profiles

for BMIM-BF4 can also (like for EMIM-SCN) be reproduced taking into account only the intermolecular 1 H-1 H HH relaxation contribution, R1H,inter (ωH ), (Eq. (7)). Omitting 1 1 the intramolecular H- H contribution again leads to small deviations at high frequencies and low temperatures (cf. Figure 5). Before stating that the analysis of the 1 H relaxation data for BMIM-BF4 can be limited to the 1 H-1 H contribution, one has to inquire into the role of the 1 H-19 F intermolecular relaxation. As the numbers of 1 H and 19 F nuclei per unit volume yield: NH = 4.84 × 10−2 Å−3 and NF = 1.29 × 10−2 Å−3 , respectively, (density of BMIM-BF4 is HF (ωH ) contribu1.38 g/cm3 ), one expects that the R1H,inter HH tion is smaller than R1H,inter (ωH ), but one should figure out how significant it is. Assuming that DHH ∼ = DHF (diffusion of cations and anions is similar) we have fitted the R1H (ωH ) data for BMIM-BF4 at 243 K and 248 K, including HF (ωH ) contribution, i.e., setting: R1H (ωH ) the R1H,inter HH HF (ω ). The results are shown in = R1H,inter (ωH ) + R1H, inter H Figure 5. One sees that the relaxation rates R1H (ωH ) and HH HF (ωH ) (the first one includesR1H,inter (ωH )) do not R1H,inter differ much. As it turns out from the fits that the cation-anion distance of closest approach, dHF , (about 4.2 Å) is larger than the distance between cations, dHH , (about 2.4 Å) one can HF (ωH ) contribution to the overall conclude that the R1H,inter relaxation rate, R1H (ωH ), is, in fact, not significant (less than 10%). The assumption DHH ∼ = DHF has been used only HF (ωH ) contribution. for estimating the role of the R1H,inter Also, as anticipated in Sec. II, the 1 H-11 B contribution is then negligible. Thus, we have fitted the R1H (ωH ) profiles HH (ωH ) relaxation (i.e., attributing them entirely to the R1H,inter HF neglecting the R1H,inter (ωH ) as well as the intramolecular contributions). The experimental data and the fits of the full dispersion are shown in Figure 6. Again, a satisfying interpolation is provided except for the highest frequencies and low temperatures for which contributions from the intramolecular relaxation are expected as in the case of EMIM-SCN (cf. Figure 4). The obtained diffusion coefficients DH = DHH /2 are collected in Table II and compared in Figure 3 with the diffusion coefficient measured by PFG NMR.23, 24 They will be discussed below.

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(a)

(a)

(b) (b)

FIG. 6. 1 H spin-lattice relaxation rate, R1H , for BMIM-BF4 versus frequency at (a) lower temperatures and (b) higher temperatures; solid lines: fits includHH ing only the R1H,inter (ωH ) term.

HF As the R1H,inter (ωH ) contribution has turned out to be insignificant, in analogy to EMIM-SCN the diffusion coefficient of the cation in BMIM-BF4 is also obtained from √ the low-frequency linear dependence of R1H on νH using Eq. (6). The corresponding representation of the R1H (ωH ) data is shown in Figure 7 with the linear interpolation at low frequencies. The obtained diffusion coefficients are included into Table II (which also contains the corresponding ωτ trans values) and Figure 3 – they are in good agreement with those obtained

√ FIG. 7. 1 H spin-lattice relaxation rate, R1H , versus νH for BMIM-BF4 , solid lines: linear fits at low frequency, dashed lines: full fits already shown in Figure 6.

from fitting the full relaxation dispersion; the discrepancies do not exceed 8%. One can see from Figure 3 that the overall agreement between the diffusion coefficients of the BMIM cation in BMIM-BF4 obtained by means of FC 1 H NMR relaxometry and PFG NMR is good. The values from FC NMR relaxometry are somewhat smaller than those from PFG NMR. HF (ωH ) (and This might be explained by neglecting the R1H,inter

TABLE II. Diffusion coefficients of cation, DH , in BMIM-BF4 obtained from the full analysis of 1 H spinlattice relaxation dispersion R1H (ωH ) (complemented by dHH and τ trans values) and from the linear dependence √ of R1H (ωH ) on νH at low frequencies; in parentheses uncertainties of the diffusion coefficients are given, ω denotes the upper limit of the low frequency range. Full analysis Temp. (K) 243 248 253 258 263 268 278 288 298 308 318 a

dHH (Å)

τ trans (s)

2.37 2.38 2.33 2.36 2.32 2.33 2.30 2.27 2.30 2.20 2.22

1.88 × 10−7 1.02 × 10−7 6.06 × 10−8 3.84 × 10−8 2.42 × 10−8 1.61 × 10−8 7.54 × 10−9 3.98 × 10−9 2.45 × 10−9 1.27 × 10−9 8.68 × 10−10

Low-frequency slope DH (m2 /s) 1.49 × 10−13 2.79 × 10−13 4.48 × 10−13 7.25 × 10−13 1.11 × 10−12 1.69 × 10−12 3.51 × 10−12 6.47 × 10−12 1.08 × 10−11 1.91 × 10−11 2.84 × 10−11

(8%) (6%) (3%) (5%) (3%) (4%) (6%) (7%) (9%) (15%) (27%)

DH (m2 /s)

ωτ trans a

1.63 × 10−13 (4%) 2.98 × 10−13 (4%) 4.78 × 10–13 (2%) 7.33 × 10−13 (2%) 1.18 × 10−12 (2%) 1.74 × 10−12 (3%) 3.66 × 10−12 (3%) 6.41 × 10−12 (4%) 1.16 × 10−11 (2%) 1.97 × 10−11 (3%) 2.81 × 10−11 (6%)

0.66 0.41 0.46 0.46 0.29 0.21 0.32 0.26 0.36 0.30 0.21

To estimate τ trans , d = 2.5 Å has been used.

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J. Chem. Phys. 140, 244509 (2014)

(a)

FIG. 8. 1 H spin-lattice relaxation rates for BMIM-PF6 . Solid lines: HH HF fits according to R1H (ωH ) = R1H,inter (ωH ) + R1H,inter (ωH ), dashed lines: H H R1H,inter (ωH ) contribution only. The parameters are: dHH = 2.50 Å, DHH = DHF = 4.11 × 10−13 m2 /s, dHF = 4.13 Å (258 K); dHH = 2.50 Å, DHH = DHF = 1.07 × 10−12 m2 /s, dHF = 4.13 Å (268 K).

HB R1H,inter (ωH )) contribution to the 1 H relaxation as well as different chemical purity of BMIM-BF4 (as in the case of BMIM-PF6 ).

(b)

C. BMIM-PF6

A similar analysis has been performed for BMIM-PF6 . Again one can conclude that the dominating contribution to the overall relaxation rate, R1H (ωH ), stems from the 1 H-1 H inHF (ωH ) can be igtermolecular relaxation and the term R1H,inter nored. This is demonstrated in Figure 8 for 258 K and 268 K; NH = 4.39 × 10−2 Å−3 , NF = 1.75*10−2 Å−3 (density of BMIM-PF6 is 1.21 g/cm3 ) for which we have set for testing: HF (ωH ) contribution is small (about DHH = DHF . The R1H,inter 15%), not only due to smaller NF (compared to NH ), but also, in analogy to BMIM-BF4 , dHF (about 4.1 Å) being larger than dHH (about 2.5 Å). This also implies that the 1 H-31 P contribution is negligible. As it has turned out that the overall relaxation, R1H (ωH ), is predominated by the 1 H-1 H intermolecular contribution, HH (ωH ), the relaxation data, R1H (ωH ), are analyzed atR1H,inter HH (ωH ). The expertributing the relaxation entirely to R1H,inter imental data interpreted in this way are shown in Figure 9.

FIG. 9. 1 H spin-lattice relaxation rate, R1H , for BMIM-PF6 versus frequency at (a) lower temperatures and (b) higher temperatures; solid lines: correHH sponding fits in terms of the R1H,inter (ωH ) relaxation process.

The obtained diffusion coefficients, DH , are listed in Table III and included into Figure 3 in which they are also compared with PFG NMR results.25–27 Table III and Figure 3 also includes diffusion coefficients obtained from the low-frequency √ slopes of R1H versus νH , which are shown in Figure 10. The discrepancies between these two sets of results do not exceed 11%. Again, the overall agreement between the diffusion coefficients, DH , of the BMIM cation obtained by FC 1 H NMR relaxomery and PFG NMR is good. The somewhat smaller values obtained by the first method can be explained, analogously to the case of BMIM-BF4 , by neglecting

TABLE III. Diffusion coefficients of cation, DH , in BMIM-PF6 obtained from the full analysis of 1 H spinlattice relaxation dispersion R1H (ωH ) (complemented by dHH and τ trans values) and from the linear dependence √ of R1H (ωH ) on νH at low frequencies; in parentheses uncertainties of the diffusion coefficients are given, ω denotes the upper limit of the low frequency range. Full analysis Temp. (K) 258 268 278 288 298 308 318 323

Low frequency slope (m2 /s)

dHH (Å)

τ trans (s)

DH

2.45 2.47 2.46 2.40 2.33 2.35 2.30 2.29

1.65 × 10−7 6.49 × 10−8 2.44 × 10−8 1.09 × 10−8 4.49 × 10−9 3.11 × 10−9 1.80 × 10−9 1.23 × 10−9

1.82 × 10−13 4.70 × 10−13 1.24 × 10−12 2.65 × 10−12 6.05 × 10−12 8.87 × 10−12 1.47 × 10−11 2.14 × 10−11

(9%) (4%) (8%) (9%) (5%) (14%) (20%) (13%)

DH (m2 /s) 2.02 × 10−13 4.87 × 10−13 1.27 × 10−12 2.85 × 10−12 5.95 × 10−12 8.29 × 10−12 1.49 × 10−11 1.98 × 10−11

(5%) (3%) (5%) (5%) (1%) (9%) (11%) (3%)

ωτ trans 0.61 0.36 0.17 0.18 0.16 0.13 0.08 0.07

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(a) (a)

(b)

(b)

√ FIG. 10. 1 H spin-lattice relaxation rate, R1H , versus νH for EMIM-BF6 , solid lines: linear fits at low frequency, dashed lines: full fits already shown in Figure 9.

HF the R1H,inter (ωH ) relaxation contribution. Nevertheless, one should also notice that the cation diffusion coefficients obtained by PFG NMR for home-synthesized and commercially purchased BMIM-BF4 also differ27 (cf. Figure 3), which might indicate that a small amount of impurities can affect the diffusion coefficient. As another source of information, in the case of BMIMPF6 we have measured the 19 F relaxation (R1F (ωF )). It is also dominated by the intermolecular relaxation path. Generally, one has to consider the 19 F-19 F and 19 FFF FH 1 H relaxation contributions R1F,inter (ωF ) and R1F,inter (ωF ) 19 (Eq. (11)), respectively. Figure 11 shows F relaxation dispersion data for BMIM-PF6 at 258 K interpreted in terms

FIG. 11. 19 F spin-lattice relaxation rates for BMIM-PF6 . Solid line: FF FH fits according to: R1F (ωF ) = R1F,inter (ωF ) + R1F,inter (ωF ), dashed line: F F F H R1F,inter (ωF ) contribution, dashed-dotted line: R1F,inter (ωF ) contribution.

FIG. 12. 19 F spin-lattice relaxation rate, R1F (ωF ), for BMIM-PF6 versus frequency at (a) lower temperature and (b) higher temperatures; solid lines: fits by a sum of 19 F-1 H and 19 F-19 F intermolecular relaxation rates, R1F (ωF ) FF FH = R1F,inter (ωF ) + R1F,inter (ωF ).

FF FH (ωF ) and R1H,inter (ωF ) contributions; the inof the R1F,inter FF 19 19 19 31 (ωF ) and tramolecular F- F and F- P terms, R1F,intra FP R1F,intra (ωF ), respectively, are ignored. The overall relaxation FF FH (ωF ) and R1F,inter (ωF ) conis decomposed into the R1F,inter tributions. One can see that the larger relaxation contribution stems from 19 F-1 H (anion-cation) interactions, but both FH FF (ωF ) and R1F,inter (ωF ) are comparable. In conterms R1F,inter sequence, the 19 F spin-lattice relaxation data have been analyzed as a sum of the two intermolecular contributions: FF FH (ωF ) + R1F,inter (ωF ), with four adjustable R1F (ωF ) = R1F,inter parameters, DHH , dHH and DHF , dHF . It might be useful to repeat that the relative diffusion coefficients are defined as a sum of self-diffusion coefficients of the participating ions (DH for cations and DF for anions), i.e., DHH = 2DH (DFF = 2DF ) and DHF = DH + DF . The experimental data and theoretical interpolations are shown in Figure 12, and the obtained DH and DF values are collected in Table IV. While at low temperatures the quality of the fit is satisfying one observes at high temperatures and intermediate frequencies a slight increase of the relaxation rates with frequency that likely stems from 1 H-19 F cross-relaxation9 and 31 P chemical shift anisotropy – both effects are not included into our analysis. Thus, the obtained diffusion coefficients are characterized by relatively high uncertainties. Although the analysis of the 19 F relaxation data is more complex, the advantage is that it provides cation-anion as well as anion-anion diffusion coefficients, DHF = DH + DF

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TABLE IV. Diffusion coefficients of cation and anion, DF and DH , respectively, in BMIM-PF6 obtained from the analysis of 19 F spin-lattice relaxation dispersion R1F (ωF ); in parentheses uncertainties of the diffusion coefficients are given. Temp. (K)

dFF (Å)

FF τtrans (s)

258 268 278 288 298 308 318 323

3.21 3.20 3.12 3.14 3.10 3.20 3.15 3.15

4.12 × 10−7 1.65 × 10−7 5.76 × 10−8 2.77 × 10−8 1.10 × 10−8 8.06 × 10−9 5.04 × 10−9 3.85 × 10−9

DF (m2 /s) 1.25 × 10−13 3.11 × 10−13 8.45 × 10−13 1.78 × 10−12 4.35 × 10−12 6.35 × 10−12 9.85 × 10−12 1.29 × 10−11

(23%) (18%) (23%) (30%) (21%) (46%) (82%) (50%)

and DFF = 2DF , respectively. The DF values of the anion are compared with those from PFG NMR25–27 in Figure 13, which also includes the BMIM cation diffusion coefficients, DH , obtained from the 1 H relaxation data (already shown in Figure 3). The anionic diffusion coefficients obtained from FC 19 F NMR relaxometry are somewhat smaller than those from PFG NMR (analogously to the case of cations, as discussed above). The discrepancies might be caused by deficiencies of the interpolation of the relaxation data when neglecting 1 H19 F cross-relaxation and 31 P chemical shift anisotropy effects (cf. Figure 12(b)). One can also see from Figure 13 (PFG NMR) that, the anionic diffusion coefficients for synthesized and commercially purchased BMIM-PF6 differ,27 similarly to cationic diffusion coefficients (cf. Figure 3). It is also interesting to compare the DH values obtained from the analysis of the 19 F relaxation data as DH = DHF − DF with the DH values provided by the 1 H relaxation results (Figure 3). One sees that the DH values obtained from the 19 F relaxation data are somewhat smaller than those obtained from the 1 H relaxation results which might suggest that 19 F relaxation is affected by the cross relaxation and chemical shift anisotropy effects to a larger extend than 1 H relaxation. As anticipated, in Figure 13 the diffusion coefficients of the anion (PF6 ) are also compared with the diffusion coefficients of the cation (BMIM) measured by both PFG NMR

dHF ´ (Å)

HF τtrans (s)

4.20 4.11 4.03 4.01 3.80 3.94 4.01 4.01

8.48 × 10−7 2.12 × 10−7 7.73 × 10−8 3.26 × 10−8 1.54 × 10−8 1.19 × 10−8 7.18 × 10−9 4.12 × 10−9

DH (m2 /s) 1.04 × 10−13 3.98 × 10−13 1.05 × 10−12 2.47 × 10−12 4.70 × 10−12 6.55 × 10−12 1.12 × 10−11 1.95 × 10−11

(17%) (15%) (20%) (27%) (14%) (30%) (45%) (71%)

and FC 1 H NMR. All data sets are close. In other words, both ions are part of a common dynamic process which is actually determined by the phenomenon of the glass transition as demonstrated by depolarized light scattering35, 36 and optical Kerr effect studies.37 V. CONCLUSIONS

In this work FC 1 H and 19 F NMR relaxometry has been used to determine translation diffusion coefficients of the cation for three ionic liquids: EMIM-SCN, BMIM-BF4 , and BMIM-PF6 . Good agreement with values from PFG NMR is obtained for the last two liquids. In this way it has been demonstrated that, as in the case of molecular liquids and polymers also for ionic liquids one can profit from FC NMR as a method of determining diffusion coefficients from the intermolecular relaxation contribution which turns out to be dominant at low frequencies. The cation-cation diffusion coefficient can be straightforwardly determined from the 1 H relaxation dispersion data. Including 19 F NMR relaxometry in the case of BMIM-PF6 also the anion diffusion is accessible and, in addition, one gets anion-cation diffusion from which again the diffusion coefficient of the cation can be calculated. As also confirmed by PFG NMR anion and cation diffusivity are very close, no decoupling of the respective dynamics is observed. ACKNOWLEDGMENTS

The work has been supported by Deutsche Forschungsgemeinschaft (DFG) through Grant No. RO 907/15. APPENDIX: ESTIMATION OF THE UPPER LIMIT OF THE LOW FREQUENCY RANGE

FIG. 13. Comparison of diffusion coefficients, DF and DH (the latter already shown in Figure 3), for anion and cation, respectively, in BMIM-PF6 obtained from 19 F (DF ) and 1 H (DH ) spin-lattice relaxation experiments with literature results – open orange symbols. The estimate of the error of the anion diffusion coefficient for BMIM-PF6 is provided in Table IV.

One sees from Eqs. (4) and (5) that for the intermolecular relaxation rate at zero-frequency, R1, inter (0) one finds 1 , where D and d denote the diffuR1,inter (0) ∝ d13 τtrans = dD sion coefficient and distance of closest approach, respectively. Assuming that d is temperature independent, one can conclude that the product DR1, inter (0) should also remain temperature independent. This criterion can be applied to control the consistency of the analysis based on the universal lowfrequency dispersion law (Eq. (6)). Figure 14 shows the value

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FIG. 14. DH R1H (0) versus reciprocal temperature for EMIM-SCN, BMIMBF4 , and BMIM-PF6 .

of the product calculated for the 1 H relaxation (DR1, inter (0) = DR1H (0)) for all considered liquids for parameters obtained applying Eq. (6). The values are virtually temperature independent as expected. Significant deviations would suggest that the analysis has been extended up to frequencies beyond the linear square root behavior, i.e., an inappropriate assignment of the low frequency range has been made. 1 R.

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