Dynamics in Glass Forming Sulfuric and Nitric Acid Hydrates - The

Article pubs.acs.org/JPCB

Dynamics in Glass Forming Sulfuric and Nitric Acid Hydrates M. Frey, H. Didzoleit, C. Gainaru, and R. Böhmer* Fakultät für Physik, Technische Universität Dortmund, 44221 Dortmund, Germany ABSTRACT: Deuteron nuclear magnetic resonance (NMR) and dielectric spectroscopy are utilized to investigate the dynamics in sulfuric acid hydrates as well as in nitric acid hydrates for various degrees of hydration. Near the glass transition temperature the electrical response is up to four decades faster than the calorimetric one, a feature found also for several other inorganic ionic liquids. The acid hydrates display pronounced super-Arrhenius behavior with fragility indices of the order of 100. The relaxation strength of the acid hydrates increases with increasing temperature, an observation that was rationalized with reference to the degree of molecular dissociation. Spin relaxometry and stimulated-echo spectroscopy revealed an overall isotropic reorientation process featuring a jump angle of about 30°. Finally, the implications of the present results for the understanding of the glass transition of pure ultraviscous water are discussed.

I. INTRODUCTION Hydrated sulfuric acid plays an enormous role in a variety of seemingly disjoint fields ranging from synthetic strategies in chemical technology and energy storage in conventional car batteries to the atmospheric sciences and questions of astrophysical relevance. For instance, sulfuric acid hydrates are thought to be present on Jupiter’s satellites Europe and Ganymede.1,2 Closer to our home planet, dynamic processes in aerosols containing sulfuric and nitric acid are of utmost importance to understand. This is because they are involved in catalyzing the destruction of ozone molecules in polar stratospheric clouds. Pending scientific studies are often conducted with the goal to find out how aerosol particles form and transform under the low-temperature conditions relevant for the upper atmosphere.3−12 To address many of these questions, it is mandatory to know the phase diagrams of the sulfuric acid hydrates13−15 and of the nitric acid hydrates16 which hence have been studied by a number of researchers. In Figure 1, we reproduce the rather complex phase diagrams of these binary systems from which one recognizes that a number of stoichiometric compounds are formed. These can be described by the general formulas H2SO4·nH2O or HNO3·nH2O with n denoting the molar ratio. For brevity, these compositions will be designated as SAnH and NAnH, respectively, in the following. Structural and other properties of many of the crystalline sulfuric acid hydrates (and of the analogous selenium acid hydrates, SeAnH) were thoroughly studied experimentally (for SA2H,17 SA4H,18,19 SeAH4,20 and SA6.5H21,22) as well as theoretically,23,24 and the same holds for the crystalline nitric acid hydrates.16,25−30 Figure 1 shows that usually relatively deep eutectics are found between the stoichiometric compositions. Therefore, some acid hydrates are stable liquids down to temperatures much lower than the melting points of the pure compounds. Consequently, these acid hydrate compositions can easily be undercooled to temperatures which are even lower than those © 2013 American Chemical Society

present in the upper atmosphere. However, some stoichiometric compositions, particularly in the sulfuric acid system, are good glass formers, too. Glass forming aqueous sulfuric acid solutions were already studied by specific heat,14,31 viscosity,32−35 conductivity,36−39 as well as diffusion measurements.40−42 Apart from transport coefficients, also vibrational properties43−47 and structural aspects44,48 of these hydrates were examined. Owing to a generally much stronger crystallization tendency as compared to the sulfuric systems, the supercooled and glassy states of the nitric acid hydrates were studied much less.41,49−53 The calorimetric glass transition temperatures Tg,cal of the presently investigated acid hydrates are in the range from 153 to 164 K.14,43 Glass transitions as well as transport properties were also investigated for various other acid hydrates.51,54−57 Particular attention was directed to phosphoric and phosphorus acids that hold promising potential for use in solid electrolyte devices.58−60 For the present work, we examined the dynamical properties of several glass forming nitric and sulfuric acid hydrates using dielectric spectroscopy61 and various nuclear magnetic resonance (NMR) techniques. One of the motivations for our study is that mixtures of relatively simple (inorganic) water additives such as, e.g., N2H4, H2O2,62 NH3,63 or KOH,64 sometimes present only in subpercent amounts, are effective in suppressing the crystallization in the resulting aqueous solutions. The extent to which such studies can teach us something about the glass formation of H2O itself, a question which has been explored since long,65 is of vital importance in view of the ongoing debate regarding water’s glass transition with its numerous ramifications.62,66−69 Received: July 30, 2013 Revised: September 3, 2013 Published: September 10, 2013 12164

dx.doi.org/10.1021/jp407588j | J. Phys. Chem. B 2013, 117, 12164−12174

The Journal of Physical Chemistry B

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quadrature channel, they were set to zero. For the NMR experiments, the samples were pipetted into L-shaped sample tubes, subjected to freeze−thaw−pump cycles, and then flamesealed hermetically. Proton diffusion measurements were carried out in the stray field of a superconducting magnet. The gradient strength was g = 20 ± 1 T/m.

III. RESULTS AND ANALYSES A. Frequency Dependent Electrical Conductivity. In Figure 2, we show frequency dependent electrical impedance

Figure 1. Concentration−temperature phase diagrams of the binary systems (a) water with nitric acid (adapted from ref 12) and (b) water with sulfuric acid (adapted from ref 15). The acid concentration axis is given on a mole fraction as well as weight fraction scale wacid. The compositions studied in the present work are marked by arrows.

II. SAMPLES AND EXPERIMENTS SAnH with n = 3, 4, and 6.5 and NAnH with n = 1, 2, and 3 were prepared by weighing appropriate amounts of anhydrous H2SO4 (Merck, stated purity 99.7%), D2O (Sigma Aldrich, 99.99%), a 90 wt % aqueous HNO3 solution (Sigma Aldrich, for n = 1, 2), and a 65 wt % aqueous HNO3 solution (Merck, for n = 3). For brevity, the sample with n = 6.5 will be denoted SA6H in the following. For the dielectric experiments the solutions were transferred to a Hastelloy/sapphire cell which is similar to the one described in ref 70. Although the highly alloyed metal is fairly resistant against corrosion by acids, care was taken to minimize the time interval during which the loaded cell was kept near ambient temperature. Thus, immediately after the preparation, the cell was cooled down to T = 250 K within several minutes and then the temperature was progressively changed to 150 K in steps of 5 K. Prior to each frequency sweep, requiring approximately 10 min, the temperature was stabilized within 0.1 K using a Quatro controller. Measurements of the electrical properties were performed in the frequency range 0.1 Hz ≤ ν ≤ 10 MHz using an Alpha analyzer from Novocontrol. The deuteron NMR experiments were carried out at a Larmor frequency of ωL = 2π × 46.46 MHz using home-built spectrometers and employing standard procedures for the acquisition of spin relaxation and stimulated-echo data. Unless otherwise stated, a solid-echo refocusing delay of Δ = 20 μs was utilized. To eliminate the impact of small spurious signals in the

Figure 2. Various representations of the frequency dependent electrical impedance response of SA3H in the temperature range from 155 to 190 K in steps of 5 K. The first, second, and third rows present the real part (on the left-hand side) and the imaginary part (on the right-hand side) of the complex permittivity (frames a and b), of the electrical modulus (frames c and d), and of the conductivity (frames e and f), respectively. The solid lines in frame d are fits using eq 2.

data for SA3H in various representations. Starting from a representation in terms of the complex dielectric constant ε*(v) = ε′(v) − iε″(v), displayed in Figure 2a and b, we plot also the electrical modulus M*(v) = 1/ε*(v), in Figure 2c and d, and the electrical conductivity σ*(v) = 2πiνε0ε*(v), in Figure 2e and f. ε0 denotes the permittivity of free space. Overall, we observe the patterns expected for a highly conducting material. In the dielectric permittivity representation, the conductivity shows up as a prominent power law, and at the lowest frequencies, the signatures of blocking electrodes are observed.71−73 Their impact is largely masked in the modulus representation which reveals clear-cut steps and peaks in its real and imaginary part, respectively. The dc conductivity σ0 = σ′(v → 0) is most simply revealed by the plateau in σ′(v), and the blocking effects are recognized from a decrease of σ′ toward low frequencies (and at sufficiently high temperatures). Except for the ε′(v) and σ″(v) plots, most representations de12165



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αε and γε determine the form of the dielectric relaxation step and that of the corresponding dielectric loss peaks. In our experimental data, a dielectric loss peak cannot be distinguished as an individual spectral feature, see, e.g., Figure 2b, because any loss peak is strongly superimposed by conductivity effects. Naturally, the latter are most easily recognized from the conductivity representation, see Figures 2e and 3f−j. In Figure 3, the dc conductivity σ0 is highlighted by the horizontal lines. For a description of the modulus data (for an example, see Figure 2d), we used another Havriliak−Negami function

emphasize that also a dielectric relaxation process is present in addition to ionic dynamics. Analogous data were obtained for other amorphous acid hydrates as well, and in Figure 3, we summarize the results in

M *(ν) = M′(ν) + iM″(ν) =

ΔM [1 + (2πiντM )αM ]γM

(2)

Here, ΔM is a measure for the amplitude of the modulus step75 and the meaning of the exponents αM and γM is analogous to those defined below eq 1. Figures 2d and 3a−e show that excellent fits are obtained when using eqs 1 and 2, respectively. The modulus spectra of the samples other than SA3H (which are not shown) are fitted equally well. Let us first discuss the width parameters which all turned out to be independent of temperature. From the description of ε′, we find for SAnH that αε = 0.67, 0.61, and 0.55 for n = 3, 4, and 6.5, respectively. Hence, the lowfrequency part of the spectrum broadens with increasing water content. For NAnH with αε = 0.70 (n = 1) and αε = 0.61 (n = 2), similar trends are obtained. For all samples, the highfrequency part is characterized by αεγε = 0.38 ± 0.02. The parameter αM describing all M″ spectra is 0.91 ± 0.03, which means that the shape is very close to the Cole−Davidson form. For the high-frequency power law exponent, one has αMγM = 0.36 ± 0.02 which equals αεγε within experimental error. This is to be expected for M″ [=ε″/(ε′2 + ε″2)] if ε′ and hence the relaxational contribution Δε is sufficiently small. The various time scales determined from the fits to the dielectric constant (τε), those from the dc conductivity (τσ = ε0ε∞/σ0),76 as well as those evaluated from the modulus spectra (τM) will be presented in section IV.A. The temperature dependence of the relaxation strength Δε is presented in Figure 4 for all investigated materials. One observes that Δε of the acid hydrates increases with increasing temperature and concomitantly ΔM decreases (not shown).

Figure 3. The real part of the frequency dependent dielectric constant (frames on the left-hand side) and the real part of the frequency dependent electrical conductivity (frames on the right-hand side) of each of the five acid hydrates are given in one row. Highest and lowest temperatures are given for each set of data, and the spectra were recorded in steps of 5 K. The solid lines in frames a−e are fits using eq 1, describe the data excellently, and yield Δε, τε, αε, and γε as fitting parameters. The solid lines in frames f−j demonstrate how the DC conductivity was determined.

terms of ε′(v), focusing on the dielectric relaxation, and of σ′(v), with the blocking electrode effects hidden for clarity. For all systems, the spectral shapes exhibit a high degree of similarity, but they differ with respect to relaxation strength, Δε, and conductivity. The high-frequency dielectric constants, termed ε∞, are all in the range 4.6 ± 0.9; only for SA6H, ε∞ = 8 is obtained. For a quantitative analysis of the relaxation and conductivity processes, we employed a modified Havriliak−Negami function74 ε′(ν) = ε∞ + Δε Re{[1 + (2πiντε)αε ]−γε } + Aν−B

(1)

Here, for αε = 1, the Cole−Davidson spectral form is obtained, and setting γε = 1 in addition, the Debye function is recovered which corresponds to single exponential relaxation. In eq 1, τε is the dielectric relaxation time and the last term, involving the phenomenological parameters A and B, accounts for the contribution of AC conductivity effects. The shape parameters

Figure 4. Scaled dielectric relaxation strength Δε as a function of temperature. For SAnH, the data are given as Δε/n to demonstrate that Δε is roughly proportional to n. For NA1H and NA2H, the relaxation strengths are similar to each other. They are divided by 3 to allow for a better comparison with the data for SAnH. 12166



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Furthermore, for the sulfuric acid hydrates, the relaxation strength is roughly proportional to the water content, i.e., to n, while both nitric acid hydrates display approximately the same Δε. An overall increase of Δε with increasing temperature is quite unusual for liquids featuring permanent dipole moments. Thus, it appears that the ionic character of the acid hydrates plays an important role. The degree of dissociation αD can be measured directly using Raman scattering which is quantitatively sensitive to the various (ionized) species. In ref 47, which covers temperatures down to ∼210 K, it was reported that for sulfuric acid hydrates αD increases with decreasing temperature, reaching complete dissociation into SO42− and H3O+ ions presumably only at lower T. Due to its tetrahedral symmetry, SO42− does not possess an electrical dipole moment. The other species participating in the dissociation equilibrium, i.e., H3O+, H2O, HSO4−, and H2SO4, exhibit dipole moments between 1.4 and 2.7 D.77,78 Thus, for increasing temperatures, the number of sulfate ions decreases and more and more species are generated that carry a sizable dipole moment, thus rationalizing the observed temperature dependence of Δε for the sulfuric acid hydrates. Information regarding the dissociation of the nitric acid hydrates at low temperature is somewhat scarce;79 still, similar temperature trends are noted.52 Furthermore, near 210 K in the sulfuric acid system, the degree of dissociation αD was reported to increase from about 40% for n ≈ 3 to about 90% for n ≈ 6.5.47 Hence, less HSO4− ions, i.e., less dipolar species (and more SO42− ions) should be present in the hemihexahydrate. Extrapolating from the Raman work47 to lower temperatures, we expect the relaxation strength in SA6H to be smallest, counter to the observations in Figure 4. This could either mean that the extrapolation is not valid80 or that other factors come into play. B. Deuteron NMR Spectra. In Figure 5 we compile solidecho absorption spectra that were recorded under various conditions. Figure 5a reveals the temperature dependence of the spectra for SA3H. At T = 106 K, much below the glass transition of SA3H (see Table 1), a somewhat unusual spectral shape is observed. While the overall width of the spectrum is comparable to the ones for many other substances, its shape does not display the horn-like singularities otherwise often observed. Merely, it shows two pairs of weakly developed peaks, suggesting that two different electrical field gradient (EFG) tensors are present. For a quantitative estimate, the overall extension of the spectra yields the so-called quadrupolar anisotropy parameter δQ = 3e2qQ/(4ℏ). From the separation (in frequency units) Δν = (1 − η)δQ/2π of the inner peaks, the anisotropy parameter ηi can be estimated,81 while the separation of the outer peaks yields ηo.82 Numerical values thus estimated for the 106 K spectrum are provided in Table 1. Furthermore, most of the spectra shown in Figure 5 are relatively smeared and rounded, suggesting that the quadrupolar anisotropy is subjected to a distribution. A distribution of δQ arises in situations in which due to a significant ionicity of the OD bond its length is not welldefined.59 It seems likely that this is indeed the case for SA3H. As temperature is increased (up to T = 168 K), the spectrum partially narrows, with the outer peaks tending to vanish. Heating to above about 175 K, one obtains only relatively sharp motionally narrowed lines (not shown), indicating that the time scale of the underlying molecular motions is now shorter than about 1/δQ ≈ 1 μs.

Figure 5. Deuteron NMR spectra for various acid hydrates. Frame a shows solid echo spectra for three different temperatures extending to well below Tg. The bars and arrows illustrate the determination of the anisotropy parameter δQ from the overall spectral width and of the asymmetry parameters η from the separation of the singularities via Δν = (1 − η)δQ/2π. Frame b presents spectra of NA1H as a function of the pulse separation at a temperature slightly above Tg. The spectra are normalized to their peak intensity. The decrease of the central intensity with increasing Δ demonstrates the presence of small-angle motions. In frame c, we compare the spectra of all five acid hydrates at a common temperature near 160 K. One recognizes that the SAnH spectra are washed out more than those of NAnH.

Table 1. Compilation of Calorimetric and “Dielectric” Glass Transition Temperature Tg for Sulfuric and Nitric Acid Hydratesa SA3H SA4H SA6H NA1H NA2H

Tg,cal (K)

Tg,ε (K)

δQ/2π (kHz)

ηinner

ηouter

γ

164 158 153 155 163

157 151 146 148 155

165 155 165 165 160

0.75 0.75 0.70 0.45 0.50

0.40 0.35 0.20 0.20 0.10

0.35 0.45 0.50 0.35 0.50

The NMR coefficients include the anisotropy parameter δQ, the asymmetry parameter η, and the Cole−Davidson exponent γ. Except for Tg,cal (for SAnH from ref 14 and for NAnH from ref 43), all parameters are from the present study. a

The spectra recorded at 161 K for nitric acid monohydrate reveal essentially a single pair of “singularities”; see Figure 5b. Hence, only a single EFG tensor dominates the behavior of NA1H at this temperature. As one increases the delay Δ between the solid-echo pulses from 20 to 100 μs, the central part of the spectra loses intensity, indicative for small-step reorientations of the OD bond axes.83 12167



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in the sample with the larger water content (NA2H) freezes first upon cooling. We analyzed these spin relaxation times with the goal to evaluate the underlying motional correlation times τ as a function of temperature and composition. To this end, we used the expressions due to Bloembergen, Purcell, and Pound (BPP)85 which relate the spectral density J(ω) to the spin− lattice relaxation rate

The differences between the spectra of SA3H and NA1H, exemplified in Figure 5a and b, are typical also for the line shapes of the other sulfuric and nitric acid hydrates. This is evident from Figure 5c which compiles the spectra for all five acid hydrates at a common temperature of ∼160 K. C. Spin−Lattice and Spin−Spin Relaxation. To obtain detailed insights into the dynamics of the acid hydrates, we measured longitudinal and transversal magnetization decays and fitted them using M1(t) ∝ exp[−(t/T1)μ] and M2(t) ∝ exp[−(t/T2)λ], respectively. Here, T1 is the spin−lattice relaxation time, T2 is the spin−spin relaxation time, and μ and λ denote the corresponding stretching exponents. In Figure 6, we present the results obtained for all samples. In frame a,

2 2δQ̃ 1 = [J(ωL) + 4J(2ωL)] T1 15

(3)

as well as to the spin−spin relaxation rate 2 2δQ̃ 1 = [3J(0) + 5J(ωL) + 2J(2ωL)] T2 30

(4)

Here δQ̃ denotes the fluctuating part of the effective anisotropy parameter. Equations 3 and 4 should be applied only in the liquid or supercooled liquid states.86 In the presented form of these equations, a minor dependence on the asymmetry parameter η was neglected and overall isotropic motions were assumed. While the original BPP approach is based upon exponential correlation functions and corresponding spectral densities, for glass forming materials non-exponentiality is a ubiquitous feature.87 For the analysis we employed an asymmetric Cole and Davidson type of broadening for the distribution of correlation times.88 This leads to a spectral density JCD(ω) =

sin[γ arctan(ωτ )] ω(1 + ω 2τ 2)γ /2

(5)

with γ characterizing the shape of the underlying distribution. The determination of the parameters τ, γ, and δQ̃ then requires three constraints. Two of them can be chosen as T1 and T2 at every temperature and ωLτ ∼ 1 at the temperature of the T1 minimum provides an additional condition. With these provisos, eqs 3 and 4 can be inverted for temperatures above the T2 minimum, since at lower temperatures the transversal dephasing is governed by dipolar interactions and not by motional processes. Similar inversion procedures were applied before.89,90 For the present analysis we have chosen γ and δ̃Q to be independent of temperature. For all acid hydrates we find that the fluctuating part of the coupling constant is δQ̃ = 2π × (143 ± 3) kHz. This is only slightly smaller than the static coupling and implies that an essentially isotropic motional process governs the spin relaxation. For the various samples the width parameters γ are summarized in Table 1 and they indicate that the distribution of correlation times is about 2−3 decades wide. The mean correlation times following from the analysis are in the range of microseconds or longer, and their concentration dependence will be presented in section IV.A below. D. Stimulated-Echo Experiments. Correlation times in the regime longer than microseconds can be accessed directly using the deuteron stimulated-echo technique when carried out in a homogeneous magnetic field. Applying properly phased radio frequency pulses and by adjusting the evolution time tp and the mixing time tm, one can record the two-time correlation function91

Figure 6. Temperature dependent spin−lattice relaxation times T1 and spin−spin relaxation times T2 for SAnH are shown in frame a. Frame b presents the stretching exponents μ which characterize the longitudinal magnetization recovery. Frames c and d contain analogous results for NAnH.

one recognizes that the T1 times of the sulfuric acid hydrates all agree down to about 180 K, indicating that the T1 minimum appears at almost the same temperature, T ≈ 205 K. For lower temperatures, all samples exhibit the same general behavior, but from the pronounced thermal variation of T1 in the 150−160 K range, it is obvious that the dynamics in SA3H freezes first upon cooling and in Sa6H at the lowest temperature, cf. also the corresponding saturation of T2. Similar common trends are exhibited by the stretching exponents μ, as shown for SAnH in Figure 6b: For temperatures T > 180 K, we find μ ≈ 1, then, for 170−180 K, this exponent is hard to determine reliably due to very fast transversal dephasing (T2 ≤ 20 μs), leading to a significant reduction of the magnetization signal. Then, the exponents μ start to decrease, signaling that T1 roughly equals the time constants governing the motional correlation function giving rise to T1.84 This is the hallmark of the breakdown of ergodicity in our samples. Overall, the results for the nitric acid hydrates show the same trends with temperature, see Figure 6c and d. Interestingly, however, for these hydrates the dynamics

F2(t p , tm) = ⟨cos[ωQ (0)t p] cos[ωQ (tm)t p]⟩ 12168

(6)



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Essentially, the ensemble average (indicated by ⟨...⟩) of the mixing time dependent evolution of the quadrupolar frequency ωQ is monitored. ωQ encodes the orientation of an OD bond via 1 ωQ = ± δQ (3 cos 2 θ − 1 − η sin 2 θ cos 2ϕ) (7) 2 with θ denoting its polar angle with respect to the external magnetic field and the azimuthal angle ϕ is defined in the usual fashion. In Figure 7 we show experimental data recorded using this technique. Frame a contains temperature dependent decay

molecular motion can be assessed.83 The data for SA3H shown in Figure 7b indicate a relatively weak tp dependence. Here, the ratio of the decay times, τ2(10 μs)/τ2(90 μs), is ∼1/3, while for many van der Waals and other glass formers, one finds that τ2(10 μs)/τ2(90 μs) is of the order of 1/10.83 A weak evolution time dependence indicates the presence of relatively large molecular jump angles φ. A rough estimate can be obtained if we identify the above decay time ratio with 3/2 sin2 φ. Such an identification is suggested by an approach due to Anderson which assumes an isotropic reorientation process characterized by well-defined jump angles.92 From 3/2 sin2 φ ∼ 0.33, we obtain φ of almost 30°. Then, in Figure 7c, we compare stimulated-echo data recorded for different sulfuric acid hydrates at a common temperature of about 165 K. One recognizes that the rotational correlation times are shorter if the water concentration in the samples is larger. With the just discussed deuteron stimulated echoes, rotational motions can be monitored. Proton Hahn and stimulated echoes in a magnetic field gradient, on the other hand, provide access to the translational self-diffusion coefficient Dt. For the diffusion experiment, the echo decay is given by ⎡ ⎛ 2 ⎞⎤ S2 ∝ exp⎢ −Dt γH 2g 2t p2⎜tm + t p⎟⎥M1(tm)M 2(t p) ⎝ ⎣ 3 ⎠⎦

(8)

if one sets tm = 0 for the analysis of the Hahn-echo measurements. In eq 8, γH denotes the gyromagnetic ratio of the proton and g = ∂B/∂z is the static field gradient along the z direction. The temperature dependence of Dt is shown in Figure 8 for all currently studied acid hydrates. Overall, these data display

Figure 7. Normalized stimulated-echo signals recorded for various sulfuric acid hydrates. (a) Temperature dependent deuteron echo functions, F2, for SA3H. (b) Evolution time dependence of the stimulated-echo data recorded at 166 K. (c) Comparison of two-time correlation functions of SAnH with n = 3, 4, and 6.5. The solid lines are guides to the eye. The dashed lines represent the normalized longitudinal magnetization decay independently measured at 165 K (frame a) or 166 K (frame b).

functions that were acquired for several temperatures. The solid lines in this figure are the results of fits to the data using a stretched exponential function ∝ exp[−(tm/τ2)β] and yielded correlation times τ2 and exponents β. In the experimental situation, the stimulated-echo decay is given by the product term F2(tp, tm)M1(tm)M2(tp) rather than just by eq 6. However, working at constant tp, the M2 term accounting for spin−spin relaxation is irrelevant. The M1 term, due to spin−lattice relaxation, can be determined independently and was corrected for if necessary. In Figure 7 we included a representative T1 curve for a temperature as low as 165 K at which the ratio τ2/T1 is the most unfavorable. However, already at this temperature, the separation of the two time scales is more than one decade and it is even larger for higher T. Data of the type shown in Figure 7a allow one to trace the temperature dependence of the correlation time. On the other hand, from the evolution time dependence such as the one shown in Figure 7b, geometrical aspects of the

Figure 8. Translational diffusion coefficients of viscous acid hydrates studied using field gradient diffusometry (filled symbols, this work) and quasi-elastic neutron scattering (open symbols, refs 41 and 42).

the same general temperature dependence. In Figure 8 we also included diffusion data from quasi-elastic neutron scattering.40−42 Using this technique, c = 3.2 and 5 M aqueous H2SO4 solutions (corresponding to n ≈ 14 and n ≈ 8, respectively) and a 6.1 M aqueous HNO3 solution (corresponding to n ≈ 7) were investigated.93 At lower temperatures, the neutron data recorded for these more highly diluted acid hydrates indicate a somewhat faster diffusion than we observe via NMR. In order to associate a motional time scale τD with the diffusion coefficient measured using NMR, we employ the Debye relationship for the rotational diffusion coefficient94 Dr = kBT/(8πηRH3) = 1/[S (S + 1)τD]. Here RH denotes the hydrodynamic radius of the diffusing species and S the order 12169



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of the associated Legendre polynomial (S = 2 for NMR quantities). Then, the viscosity η was eliminated from the previous expressions by virtue of the Stokes−Einstein relation, Dt = kBT/(6πηRH), and we obtain τD =

2 2 RH 9 Dt

(9)

There are two ways to use this equation if the diffusion coefficient is known. One can approximate the hydrodynamic radius by either the mean molecular or ionic radius and indeed determine τD. Alternatively, one can turn the procedure around and presume compatibility of τD with other known time scales, thereby estimating the effective RH.

IV. DISCUSSION A. Temperature Dependent Time Scales. Let us now discuss the temperature dependence of the various time scales. Covering a wide dynamic range, we determined motional time constants from impedance spectroscopy (τε, τM, and τσ) and from NMR spectroscopy, i.e., from spin relaxation measurements (τT1 and τT2), from the deuteron stimulated-echo technique (τ2), and from diffusometry (τD). For SA3H, all of these time constants are shown in an Arrhenius plot, see Figure 9a, and they agree very well with each other,95,96 except for τM. The parameters entering into these calculations are ε∞ (given in section III.A), γ (given in Table 1), δ̃Q = 2π × (143 ± 3) kHz (given in section III.C), and RH. For the geometrical radii, we used the ones estimated according to the method of Edward97 which gives RH = 2.55 Å for sulfuric acid and RH = 2.15 Å for nitric acid. The resulting time constants τD agree reasonably well with the other relaxation times presented in Figure 9a. For comparison, we provide also the geometrical radii for the ionic species which are 2.48 Å for HSO4−, 2.1 Å for NO3−, and 1.88 Å for H3O+.98 The relaxation times displayed in Figure 9a are strongly deviating from an Arrhenius law which would yield a straight line in the representation of that figure. All currently determined time constants (except for τM) were fitted using the Vogel−Fulcher expression ⎛ C ⎞ τ = τ0 exp⎜ ⎟ ⎝ T − T0 ⎠

Figure 9. (a) The black symbols in this Arrhenius plot represent data for SA3H including relaxation times determined using the various techniques applied in the present work. The solid line is a fit using eq 10 and the parameters given in the text. For comparison, dielectric relaxation times for water (red diamonds, refs 106 and 107) and relaxation times determined for the presently studied hydrates NA1H, NA2H, SA4H, and SA6H are included. For visual clarity, only the fits to the overall relaxation time traces (such as for SA3H) are shown. Furthermore, literature data on aqueous solutions of H2O2 and N2H4 (ref 62) as well as of NH3 (ref 63) and KOH (ref 64) are included. (b) Tg,cal scaled Arrhenius plot of the motional correlation times that were determined using various NMR techniques (open symbols from diffusion data; filled symbols for τ < 10−5 s from T1; longer τ from F2). A common temperature dependence is obvious. At Tg,cal, the correlation times are much smaller than 100 s.

(10)

that they are 7−8 K lower than the calorimetric temperatures, Tg,cal,100 see Table 1. Expressed in terms of relaxation times, the Tg,cal-scaled plot shown in Figure 9b documents that the presently determined relaxation times are about 10−2 s near Tg,cal. The fragility difference for the various hydrates is too small to be clearly resolved in this plot. The relaxation modes that our methods are sensitive to are thus up to about 4 decades faster than the structural ones. This indicates an intermediate decoupling of the dielectric from the calorimetric relaxation modes. This degree of mode decoupling is similar to that reported for a few other inorganic ionic liquids such as 3KNO3·2Ca(NO3)2 (CKN)101 and the fluoride glass former CdF2−LiF−AlF3−PbF2 (CLAP).102 Since the structural relaxation times τs are 102 s near Tg,cal, it is clear that the fragility index ms estimated on the basis of (otherwise unavailable) τs is larger than mε. Thus, overall, we can conclude that the acid hydrates are rather fragile liquids with indices larger than those of CKN and CLAP. B. Acid Hydrates and Other Aqueous Glass Formers. Let us now discuss our findings in relation to other aqueous

with τ0, C, and T0 denoting phenomenological constants. For SA3H the result is included as a solid line in Figure 9a. By repeating this procedure also for the other acid hydrates, we obtain the other lines shown in this plot. The pre-exponential factors are τ0 = 1 × 10−13 s for SAnH and 1 × 10−14 s for NAnH. The parameters C and T0 can be used to quantify the deviations from the Arrhenius law, but it is also customary to use the fragility index99 m=

∂ log10(τ /s) ∂(Tg /T )

T = Tg

(11)

for this purpose. m measures the slope of the relaxation time trace in a Tg-scaled Arrhenius plot, the so-called Angell plot. Using the relaxation times τε as input for eq 11, we find mε = 85 ± 5 for NAnH and for SA6H and mε values near 100 for SA3H and SA4H. When comparing the glass transition temperatures Tg,ε = T(τε = 100s) determined dielectrically and via NMR, we find 12170



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solutions and on water. In the context of its polyamorphism,103 there is currently much debate regarding the glass transition of water. Taking a route into the vitreous state, not by supercooling the liquid (which fails for water due to an obviously inevitable homogeneous nucleation) but by thawing of suitably amorphized ice, a rate dependent glass transition is debated to take place near 136 K.66 Since the preparation of such samples is rather involved, a “chemical” approach was sometimes exploited by preparing a series of aqueous solutions and subsequent extrapolation to 100% of the solvent65,66 or by studying homologous series of supposedly water-like molecules such as the glycols.69 These attempts to approach the properties of pure and ultraviscous water appear most attractive for water-like systems such as the hydrates of hydrogen peroxide or of hydrazine which even show the “correct”104 Tg.62 Hence, it is tempting to interpret also other properties as possible signatures of the hard-to-get-by ultraviscous liquid state of water. For aqueous H2O2 and N2H4 solutions, a fragility index of m ≈ 60 was estimated.62 Other water additives do usually yield composition dependent glass transition temperatures rendering the extrapolation of properties to the limit of pure water more cumbersome. By analyzing a wide range of such data, water’s fragility index m was estimated to be near 40.68,105 This number is compatible with that from studying aqueous ammonia solutions over a wide range of compositions, down to about 1% of NH3 in H2O.63 Apart from the data for the currently studied acid hydrates, in Figure 9a, we compile various relaxation time traces including those obtained at temperatures >250 K for pure water,106,107 for aqueous KOH solution,64 for H2O2 and N2H4 hydrates,62 as well as for NH3 solutions63 all of which were studied over wide composition ranges. For SAnH, an extrapolation of Tg,ε on a 1/ n scale to 100% water content (i.e., n → 0) yields Tg,ε = 135 ± 2 K closely matching the often quoted calorimetric glass transition temperature of low-density amorphous (LDA) ice. Recent dielectric and calorimetric measurements on the glass transition of LDA indicate a Tg value of about 126 K,108 a value relatively close to Tg of ammonia hydrates (Tg = 121 K63). However, taking into account all aqueous systems shown in Figure 9a, it is clear that simple extrapolations to the limit of pure water have to be considered with great caution. This concerns not only the glass transition temperatures but also the fragility indices which are 41 (NH3 hydrates63), ∼60 (H2O2 and N2H4 hydrates62), and of the order of 100 (for the presently studied acid hydrates). For P2O5 even larger indices are reported,60 while direct measurements above the glass transitions of the amorphous ices show that m ranges from 14 (for LDA) to about 25 (for high-density amorphous ice).108 In view of the decoupling of the time scales summarized in Figure 9b from the calorimetric ones, it is clear that the fragility indices determined in this work for the acid hydrates ought to be smaller than the ones referring to the structural relaxation.

respect, the acid hydrates are similar to several other inorganic ionic liquids, such as the molten salt CKN. Comparison with various (NH3, KOH, H2O2, N2H4, and P2O5) hydrates shows that their fragility indices cover a broad range and are all significantly larger than values obtained for pure ultraviscous waters that were produced from pressure amorphized ice. For all currently studied acid hydrates, Δε was found to increase with increasing temperature. This trend was rationalized with reference to the degree of dissociation determined from previous low-temperature Raman studies. From deuteron relaxometry and stimulated-echo spectroscopy, we found that an overall isotropic reorientation prevails. The molecular motion in the acid hydrates features a relatively large jump angle of the order of about 30°. The smeared solid-echo spectra are in accord with a significant ionic character of the deuteron bonds, and from the two-component structure of these spectra, we inferred that two chemically different species exist. These can naturally be ascribed to deuterons in aqueous and acidic environments.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +49-231755-3514. Fax: +49-231-755-3516. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Graham Williams for bringing the work of R. H. Cole and co-workers and J. B. Hubbard to our attention. REFERENCES

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of ionic conductivity relaxation using the electric modulus with a Cole−Davidson distribution. J. Appl. Phys. 1990, 68, 5128−5132. We note that some authors use the static permittivity εs instead of ε∞ to estimate τσ = ε0εs/σ0 (see, e.g.: Howell, F. S.; Bose, R. A.; Macedo, P. B.; Moynihan, C. T. Electrical Relaxation in a Glass-Forming Molten Salt. J. Phys. Chem. 1974, 78, 639−648). In this case, τσ equals τM and its temperature dependence does not solely reflect the variation of σ0(T), but it is also affected by the (in our case rather strong) temperature dependence of εs that may not always be related to charge transport. See, however: Hubbard, J. B. Dielectric Dispersion and Dielectric Friction in Electrolyte Solutions. II. J. Chem. Phys. 1978, 68, 1649−1664; Hubbard, J. B.; Colonomos, P.; Wolynes, P. G. Molecular Theory of Solvated Ion Dynamics. III. The Kinetic Dielectric Decrement. J. Chem. Phys. 1979, 71, 2652−2661 (and ref 61). The approximate character of the relations for the conductivity relaxation time is also discussed by: Isard, J. O. Study of Migration Loss in Glass and Generalized Method of Calculating the Rise of Dielectric Loss with Temperature. Proc. Inst. Electr. Eng., Part B, Suppl. 1962, 22, 4410−4447. (77) Botschwina, P.; Rosmus, P.; Reinsch, E.-A. Spectroscopic Properties of the Hydroxonium Ion Calculated from SCEP CEPA Wavefunctions. Chem. Phys. Lett. 1983, 102, 299−306. (78) This is the gas phase dipole moment of H2SO4. See: Kuczkowski, R. L.; Suenram, R. D.; Lovas, F. J. Microwave Spectrum, Structure, and Dipole Moment of Sulfuric Acid. J. Am. Chem. Soc. 1981, 103, 2561−2566. This value of the dipole moment was more recently confirmed by theoretical computations. See: Kusaka, I.; Wang, Z.-G.; Seinfeld, J. H. Binary Nucleation of Sulfuric Acid-Water: Monte Carlo Simulation. J. Chem. Phys. 1998, 108, 6830−6848. (79) Close to ambient temperature, more information is available. See, e.g.: Redlich, O.; Duerst, R. W.; Merbach, A. Ionization of Strong Electrolytes. XI. The Molecular States of Nitric Acid and Perchloric Acid. J. Chem. Phys. 1968, 49, 2986−2994. Hlushak, S.; Simonin, J. P.; De Sio, S.; Bernard, O.; Ruas, A.; Pochon, P.; Jan, S.; Moisy, P. Speciation in Aqueous Solutions of Nitric Acid. Dalton Trans. 2013, 42, 2853−2860. (80) In fact, at temperatures near 200 K, the degree of dissociation shows a tendency to retrace for compositions corresponding to n ≈ 6.5; see Figure 10 in ref 47. (81) Haeberlen, U. High Resolution NMR in Solids - Selective Averaging. Adv. Magn. Reson. Supplement 1; Academic: New York, 1976; p 29. (82) It is tempting to associate the occurrence of two different couplings to the presence of deuterons located in O−D···O hydrogen bonds which connect (i) a sulfate tetrahedron with a water moiety or (ii) two of the latter, for which the O−O bond lengths differ substantially.19 (83) Böhmer, R.; Diezemann, G.; Hinze, G.; Rössler, E. Dynamics of Supercooled Liquids and Glassy Solids. Prog. Nucl. Magn. Reson. Spectrosc. 2001, 39, 191−267. (84) Schnauss, W.; Fujara, F.; Sillescu, H. The Molecular Dynamics Around the Glass Transition and in the Glassy State of Molecular Organic Systems: A 2H−Nuclear Magnetic Resonance Study. J. Chem. Phys. 1992, 97, 1378−1389. (85) Bloembergen, N.; Purcell, E. M.; Pound, R. V. Relaxation Effects in Nuclear Magnetic Resonance Absorption. Phys. Rev. 1948, 73, 679− 712. (86) See, e.g.: Diezemann, G.; Sillescu, H. Dipolar Interactions in Deuteron Spin Systems. II. Transverse Relaxation. J. Chem. Phys. 1995, 103, 6385−6393. (87) See, e.g.: Ngai, , K. L. Relaxation and Diffusion in Complex Systems; Springer: Berlin, 2011,. (88) Beckmann, P. A. Spectral Densities and Nuclear Spin Relaxation in Solids. Phys. Rep. 1988, 171, 85−128. (89) Rössler, E.; Sillescu, H. 2H NMR Study of Supercooled Toluene. Chem. Phys. Lett. 1984, 112, 94−98. (90) Schildmann, S.; Reiser, A.; Gainaru, R.; Gainaru, C.; Böhmer, R. Nuclear Magnetic Resonance and Dielectric Noise Study of Spectral 12173



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Densities and Correlation Functions in the Glass Forming Monoalcohol 2-ethyl-1-hexanol. J. Chem. Phys. 2011, 135, 174511. (91) Schmidt-Rohr, K.; Spiess, H. W. Multidimensional Solid-State NMR and Polymers; Academic Press: London, 1994. (92) Anderson, J. E. Environmental Fluctuations and Rotational Processes in Liquids. Faraday Symp. Chem. Soc. 1972, 6, 82−88. (93) For the conversion from c into units of M (or mol of solute per liter of solution) into the molar ratio, n = n1/n2 (here the index “1” corresponds to water and “2” to acid), we used c2 = n2/(V1 + V2) = n2NA(n1Mw1/ρ1 + n2Mw2/ρ2)−1 with the density ρ, the molecular weight Mw, and NA denoting Avogadro’s number. In Figure 1, concentrations are given on a molar scale x2 = n2/(n1 + n2) and on a weight scale w2 = n2Mw2(n1Mw1 + n2Mw2)−1. (94) Chang, I.; Sillescu, H. Heterogeneity at the Glass Transition: Translational and Rotational Self-Diffusion. J. Phys. Chem. B 1997, 101, 8794−8801. (95) Indications for a significant isotope effect were not found. See also: Shepley, L. C.; Bestul, A. B. Deuterium Isotope Effect on Glass Transformation Temperatures of Aqueous Inorganic Solutions. J. Chem. Phys. 1963, 39, 680−687. (96) A similar temperature dependence of τε = 1/ωε (with the dielectric loss peak frequency ωε/2π) and τσ = ε0ε∞/σ0 (as given in section III.A) is expected from the Barton, Nakajima, and Namikawa (BNN) relation. This relation states that σ0 = pε0Δεωε. Here p is a numerical constant of order 1. An explicit expression for it in terms of Havriliak−Negami parameters was given by: Tsonos, C.; Kanapitsas, A.; Kechriniotis, A.; Petropoulo, N. AC and DC Conductivity Correlation: The Coefficient of Barton−Nakajima−Namikawa Relation. J. Non-Cryst. Solids 2012, 358, 1638−1643. The BNN relation was found valid for numerous ionic glasses. See, e.g.: Dyre, J. C.; Schrøder, T. B. Universality of AC Conduction in Disordered Solids. Rev. Mod. Phys. 2000, 72, 873−892. (97) Edward, J. T. Molecular Volumes and Stokes-Einstein Equation. J. Chem. Educ. 1970, 47, 261−270. (98) The agreement, seen in Figure 9a, between τD and other orientational time constants indicates that the geometrical radii given in the text provide a good estimate for the dimension of the translationally diffusing molecular species. On the other hand, a direct estimate of RH from the present Dt (for SA3H) and η (for the 67.7 wt % sample of ref 35) using kBT/(6πηDt) like in ref 63 yields RH < 1 Å. This point deserves further study. (99) Böhmer, R.; Ngai, K. L.; Angell, C. A.; Plazek, D. J. NonExponential Relaxations in Strong and Fragile Glass-Formers. J. Chem. Phys. 1993, 99, 4201−4209. (100) In ref 14, end point Tg’s are reported for sulfuric acid hydrates. The usual onset temperatures are typically up to 3 K lower; see Figure 7 in ref 14. (101) Pimenov, A.; Lunkenheimer, P.; Rall, H.; Kohlhaas, R.; Loidl, A.; Böhmer, R. Ion Transport in the Fragile Glass Former 3KNO32Ca(NO3)2. Phys. Rev. E 1996, 54, 676−684 and references cited therein. (102) Hasz, W. C.; Moynihan, C. T.; Tick, P. A. Electrical Relaxation in a CdF2-LiF-AlF3-PbF2 Glass and Melt. J. Non-Cryst. Solids 1994, 172−174, 1363−1372. (103) Poole, P. H.; Grande, T.; Angell, C. A.; McMillan, P. F. Polymorphic Phase Transitions in Liquids and Glasses. Science 1995, 275, 322−323. (104) The exact glass transition temperature as well as the detailed interpretation of the calorimetric signature used for its determination are the subject of considerable debate. (105) Schmidtke, B.; Petzold, N.; Kahlau, R.; Hofmann, M.; Rössler, E. A. From Boiling Point to Glass Transition Temperature: Transport Coefficients in Molecular Liquids Follow Three-Parameter Scaling. Phys. Rev. E 2012, 86, 041507. (106) Rønne, C.; Keiding, R. Low Frequency Spectroscopy of Liquid Water Using THz-Time Domain Spectroscopy. J. Mol. Liq. 2002, 101, 199−218. (107) Buchner, R.; Barthel, J.; Stauber, J. The Dielectric Relaxation of Water between 0 and 35 °C. Chem. Phys. Lett. 1999, 306, 57−63.

(108) Amann-Winkel K.; Gainaru, C.; Handle, P. H.; Seidl, M.; Nelson, H.; Böhmer, R.; Loerting, T. Water’s Second Glass Transition. Proc. Natl. Acad. Sci. U.S.A., accepted for publication, 2013.

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Dynamics in Glass Forming Sulfuric and Nitric Acid Hydrates - The

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