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Debye Process and β‑Relaxation in 1‑Propanol Probed by Dielectric Spectroscopy and Depolarized Dynamic Light Scattering Jan Gabriel,* Florian Pabst, and Thomas Blochowicz Institut für Festkörperphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany ABSTRACT: We revisit the reorientational dynamics of 1-propanol as a prototype of a monohydroxy alcohol and H-bonding system by dielectric spectroscopy (DS) and depolarized dynamic light scattering (DDLS). In particular, we address the question of whether the Debye relaxation, which is seen as a dominant process in DS, is visible in light scattering and discuss how the Johari−Goldstein (JG) β-process, which is also a prominent feature of the dielectric spectrum, appears in photon correlation spectroscopy. For that purpose we performed depolarized photon correlation experiments with an improved setup and performed additional time domain dielectric experiments which gives us the possibility to compare dielectric and light scattering data in a broad temperature range. It turns out that the improved setup allows to unambiguously identify the JG β-process, which shows almost identical properties in DDLS as in the dielectric spectra, but a Debye relaxation is not present in the DDLS data and can be excluded down to a level of 2.5% of the α-process amplitude.

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motions in the glass.13,14 Later on, however, NMR investigations suggested a different interpretation by showing, in particular in the glassy state, that small-angle reorientation of basically all molecules underlies the JG β-relaxation.15 In that case the question arises as to what extent secondary processes are visible in DDLS, which, at least for optically anisotropic molecules, also probes molecular reorientation. Interestingly, secondary relaxations in DDLS are only rarely reported in the literature.16 Previous light scattering work dedicated to comparing secondary relaxations in DS and DDLS comes to the conclusion that while the so-called excess highfrequency wing frequently observed in dielectric spectra is also recovered in light scattering, a Johari−Goldstein (JG) β-peak may not be visible at all in DDLS.17 This is particularly surprising because depolarized light scattering probes an l = 2 correlation function, while dielectric spectroscopy probes l = 1: Φl(t) = ⟨Pl(cos θ(t))⟩, with Pl(x) being the Legendre polynomial of degree l. A straightforward small-angle expansion reveals that in the case of small-angle reorientation an l = 2 correlation function should be even more sensitive than l = 1 to small-angle reorientation, and the relative amplitude of such a process should be even larger in depolarized light scattering compared with its dielectric manifestation.17,18 In the present paper we aim at a detailed comparison of dielectric spectroscopy and depolarized light scattering in 1propanol. For that purpose we improved our light scattering setup to reduce sources of noise and artifacts in the correlation functions and to be able to perform depolarized photon correlation spectroscopy with a weak scatterer. For comparison,

INTRODUCTION The relaxation behavior of monohydroxy alcohols is a longstanding topic.1 In particular, the pronounced Debye-like process observed in the long-time dielectric relaxation in many of these liquids is still debated. While initially the Debye model was designed as a model for structural relaxation,2 it turned out that in monoalcohols the structural relaxation is significantly faster than the Debye peak, while experiments have provided increasing evidence that the latter represents relaxation of transient supramolecular structures.1,3 For a long time, however, it seemed that only dielectric experiments show evidence of the Debye process, and an earlier investigation pointed out that depolarized dynamic light scattering does not probe a Debye relaxation in 1-propanol.4 However, recently, evidence of a Debye-type relaxation in monoalcohols was reported in shear mechanical response,5 where it is seen as a rheological response comparable to that in short-chain polymers, and indications of a Debye-type relaxation are reported in depolarized dynamic light scattering of H-bonding liquids like water6 and supercooled imidazole.7 Therefore, in the present paper we revisit photon correlation spectroscopy of 1-propanol with a particular emphasis on the long time behavior of the correlation functions in order to discern a possibly small Debye-like contribution. At times shorter than the structural relaxation, many supercooled liquids exhibit a secondary relaxation peak in dielectric spectroscopy, which remains visible even below Tg.8 Especially, in the case of molecules that are rigid with respect to the dielectric probe, such a process is named after Johari and Goldstein, who pointed out in their seminal contribution9 that even in rigid molecules so-called islands of mobility10 in an otherwise frozen or slow matrix may cause such secondary relaxations11,12 and that these processes are due to localized © 2017 American Chemical Society

Received: June 22, 2017 Revised: September 2, 2017 Published: September 5, 2017 8847

DOI: 10.1021/acs.jpcb.7b06134 J. Phys. Chem. B 2017, 121, 8847−8853

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avalanche photodiodes operating in quasi-cross-correlation mode to suppress afterpulsing. In a PCS experiment with a substance that shows only weak anisotropic scattering, one faces the problem that the scattering is partially heterodyne because slightest reflections of the incident beam on the light path, e.g., by sample cell windows or the vacuum shroud, will superimpose with the weak scattering signal. The effect of this is twofold: First of all, with a certain degree of heterodyning the experiment becomes much more sensitive to all kinds of vibrations and other artifacts: any shift in the optical path difference between local oscillator and scattered light on the order of the laser wavelength becomes visible in the data. Second, in the case of partial heterodyning it becomes more difficult to obtain the field autocorrelation function g1(t) from the intensity correlation data. In order to treat the first problem, we took a series of measures to improve the setup, like reducing vibrations by replacing the liquid nitrogen cooling system with a gas cooling system, replacing the turbomolecular pumping station by an ion getter pump, and by stabilizing the temperature in the laboratory to better than 0.2 K, which enhances the long time stability of the laser system. In addition, a custom-made sample cell was used that was optimized to reduce reflections of the laser beam as it passes through. Altogether these improvements lead to a data quality that allows us to Fourier transform the time domain light scattering data and obtain a direct comparison with the dielectric permittivity in the frequency domain. In the case of fully homodyne scattering with the scattered field Es being a zero-mean Gaussian variable, the Siegert relation connects the autocorrelation function of the intensity Is of the scattered light g2(t) = ⟨Is(t)Is(0)⟩/⟨Is⟩2 with the normalized field autocorrelation function g1(t) = ⟨E(t) E*(0)⟩/⟨Is⟩ via g2(t) = Λ|g1(t)|2 + 1 with Λ = 0.9 being the coherence factor, which was determined with a suspension of latex spheres. In order to treat partial heterodyning, we follow the derivation of Bremer et al.25 Starting from a very general ansatz for the intensity correlation function of the scattered light

we combine dielectric frequency domain data from the literature8,19 with new time domain data to cover a broad dynamic range, where all three processes are visible (cf. Figure 1). We start by pointing out some experimental details followed

Figure 1. Frequency domain and Fourier transformed time-domain composite dielectric spectra of 1-propanol at 101 K showing the sum of Debye and convolution parts of the α- and β-process.

by the results obtained in PCS and dielectric spectroscopy. The final discussion focuses on the manifestation of the JG βprocess and Debye relaxation in both methods.

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EXPERIMENTAL SECTION 1-Propanol was purchased from Sigma-Aldrich with a purity of 99% and filtered with 20 nm Watson filters, thereby greatly reducing dust particles in the sample for PCS measurements. Time domain dielectric spectra were recorded using a modified Sawyer−Tower setup described elsewhere in detail.20,21 We note that a similar technique was used to obtain time domain data on propanol previously.22 Across a series of samples and reference capacitors a voltage step is applied, and the time-dependent voltage across the reference capacitor is recorded, which directly measures the polarization of the sample under test. The time domain data were Fourier transformed and combined with frequency domain data published in refs 8 and 19. As both data sets were obtained in different laboratories, a systematic temperature difference of less than 0.5 K and a different filling factor of the capacitors were taken into account to match both data sets in the overlapping frequency region. The PCS experiments were performed with a setup already described elsewhere in detail.23,24 The sample is placed in a coldfinger cryostat (Conti by Cryovac) in a rectangular sample cell with antireflection-coated windows. To obtain consistent temperature values in both methods, the temperature was measured right inside the scattering volume with a Pt100 temperature sensor that was previously calibrated in the dielectric setup. As light source in the DDLS experiment a Coherent Verdi G2 laser was used operating at a wavelength of 532 nm. All measurements were taken in vertical−horizontal depolarized mode, where the incident beam is polarized vertically and the scattered light passes through a horizontal polarization filter before detection. The scattered light is detected under an angle of 90° by a single mode optical fiber and is fed into two

g2(t ) = ΛC 2|g1(t )|2 + 2ΛC(1 − C)|g1(t )| + 1

(1)

the degree of heterodyning is characterized by a constant C = ⟨Is⟩/(⟨ILO⟩ + ⟨Is⟩), which quantifies the influence of the local oscillator ILO with respect to the scattered intensity Is. In the limit of fully homodyne scattering C ≈ 1, and the conventional Siegert relation is obtained. In principle, the intercept of the measured correlation function, i.e., its value at the shortest recorded lag time t0, contains information on the degree of heterodyning. If the measured intercept is given by A = g2(t0) − 1 and if g1(t0) ≈ g1(0) = 1 were a good approximation, the relation would be C=1−

1 − A /λ

(2)

However, microscopic dynamics in the picosecond range will inevitably reduce the field autocorrelation function to a smaller value, g1(t0) = λ < 1. Then the degree of heterodyning is given by C=

1−

1 + A /λ − 2A /(λ Λ) (3)

2−λ

and eq 1 can be solved for g1 to yield 8848

DOI: 10.1021/acs.jpcb.7b06134 J. Phys. Chem. B 2017, 121, 8847−8853

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The Journal of Physical Chemistry B g1(t ) =

1 ( (1 − C)2 + (g2(t ) − 1)/Λ − 1) + 1 C

voltage the external field is switched off, and Δϵϕ(t) is recorded at low temperatures between 116 and 97 K. As the dielectric spectra are dominated by the strong Debye process, the long time behavior is very well represented by a single-exponential decay, as shown by the solid lines in Figure 2. At shorter times α- and β-relaxation play a role, so that the overall analysis is better done with a combination data set in the frequency domain. Such a combined data set is shown in Figure 3, where the Fourier transform of our time domain data is joined with

(4)

Unfortunately, a direct comparison of dielectric spectroscopy and depolarized light scattering is not necessarily straightforward, as light scattering in depolarized scattering geometry probes the reorientation of the anisotropy tensor of the molecular polarizability26 while dielectric spectroscopy probes the reorientation of permanent molecular dipole moments, i.e., a vector quantity. Therefore, correlation functions of different Legendre polynomials are probed in both methods: Φl (t ) = ⟨Pl(cos θ(t ))⟩

(5)

namely l = 1 for dielectric spectroscopy and l = 2 for DDLS, with θ(t) being the angle between the positions of the respective molecular axis at times 0 and t. For that reason a comparison of both correlation functions is not straightforward, even if the optical anisotropy and the permanent dipole moment are located in the same molecular entity and the molecular dipole is approximately aligned with the symmetry axis of the polarizability or, alternatively, the molecule is rigid and the reorientation is entirely isotropic. But even under such favorable conditions the relation between Φ1 and Φ2 will depend on the exact geometry of the underlying stochastic process: While, e.g., in the case of isotropic rotational diffusion the correlation times are different by a factor of 3, τ1/τ2 = 3, in the case of reorientation by random jumps the correlation function becomes independent of l,27 so a general comparison is difficult. However, if molecular reorientation is restricted to small angles, what could be expected for secondary processes like the JG β-relaxation, then an expansion of eq 5 to leading order in θ reveals that the relative amplitude of that small-angle process should be larger by a factor of 3 for an l = 2 correlation function compared to l = 1, so that for such a small-angle process for the respective susceptibilities ″ (ω) = 3χLS ″ (ω) χDS

is expected to hold.

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Figure 3. Frequency dielectric spectra from Kudlik et al.8,19 joined with Fourier transformed time-domain data of 1-propanol with fits for Debye, α-, and β-relaxation.

frequency domain measurements previously published by Kudlik et al.8,19 The data can best be described by a superposition of three peaks, one for the Debye relaxation and one for α- and β-process, respectively, as is indicated schematically in Figure 1. The α-relaxation is modeled with a generalized gamma distribution of correlation times:30

(6)

18,28,29

GGG(ln τ ) = NGG(α , β)e

RESULTS AND DATA ANALYSIS Dielectric Spectroscopy. In Figure 2 we show the time dependence of the polarization response acquired with the time domain setup. After equilibration of the sample at constant

−β / α(τ / τ0)α

⎛ τ ⎞β ⎜ ⎟ ⎝ τ0 ⎠

(7)

with α and β defining the peak shape, τ0 being the time constant, and NGG being a normalization factor; for details see ref 30. The JG β-process is described by 1 Gβ (t ) = Nβ(a , b) a b(τ /τm) + (τ /τm)b (8) with a characterizing a symmetric broadening of the peak while b introduces asymmetry resulting in power laws ωa and ω−ab in the corresponding susceptibility spectra. A general advantage of this approach compared to e.g. a Havriliak−Negami function lies in the fact that for a temperature-independent asymmetry b and a ∝ T eq 8 represents a temperature-independent distribution of activation energies,30 which makes it easy to globally fix the temperature dependence of line shape parameters in a physically reasonable way, which is very convenient for the present analysis. Both distribution functions can be integrated either in time domain to yield the relaxation function Φ(t) or in frequency domain to obtain the complex permittivity ε̂(ω). To combine α- and β-process for the present analysis, we use the Williams− Watts (WW) approach.31 Here, one assumes that the same molecular dipoles are involved in both processes which are

Figure 2. Dielectric time-domain measurements of 1-propanol with exponential fits of the strong Debye relaxation. 8849

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(9)

Here, k represents the relative weight of the β-relaxation. Finally the contribution of the Debye process is added: Φ(t ) = ΔεDe−t / τD + Δεαβ ΦWW (t )

(10)

The Fourier transform of this expression was fitted to the data in Figure 3 up to a temperature of 125 K. Above that temperature there is too little information on the α- and βprocess in the data to allow for a reasonable fit of three processes, so that only the Debye peak was used. The solid lines seen in Figure 3 show the results of this procedure. Of course, due to the fact that most of the α-relaxation is hidden beneath the Debye contribution, a fit of the data in general is difficult to do. For that reason most shape parameters like α and β for the α-relaxation were globally fitted and fixed to temperature-independent values. For the JG β-process a symmetric shape was assumed in accordance with ref 32, i.e., b = 1 for all temperatures, and the width parameter a for the βprocess was set to follow a = ζT with a global constant ζ, which is determined at low temperatures and implies a temperatureindependent distribution of activation energies,30 which is a reasonable assumption for the β-relaxation at least at temperatures below the glass transition. Figure 7 shows the time constants τ obtained from the dielectric data (triangles) for the Debye, α-, and β-processes. Consistent with previously published work by Hansen et al.4 and Kudlik et al.,8,19 the temperature dependences of the time constants τDebye and τα are both characterized by Vogel− Fulcher−Tammann laws. As we now include time domain data, this behavior can be followed to low temperatures and it becomes obvious that both time constants approach each other on cooling, similar to previous findings reported in a series of monoalcohols.33 In the present case the low-temperature data even allow to estimate that both processes will merge around 95 K. For the JG β-process Arrhenius behavior with an activation energy of ΔEa/kB ≈ 2500 K is observed below Tg while the temperature dependence becomes somewhat stronger at higher temperatures. The symmetric β-peak is well described by a temperature-independent distribution of activation energies, so that a = ζT with ζ = 2.9 × 10−3 K−1 for the dielectric data. In Figure 8 the shape parameters of the analysis of the dielectric data are seen as full and open diamonds. Full symbols are used for parameters that were individually fitted for each temperature, and open symbols indicate parameters that were obtained in a global fit or by extrapolation. Photon Correlation Spectroscopy. Figure 4 shows for selected temperatures the autocorrelation function of the electric field g1(t) obtained in photon correlation spectroscopy. g1(t) is calculated using eqs 3 and 4. For that purpose 1 − λ, the amplitude of the microscopic dynamics, has to be estimated to obtain a value for the degree of heterodyning C from the measured intercept of the intensity autocorrelation function via eq 3. In principle, 1 − λ could be obtained with Raman scattering. For the present analysis, however, we simply estimate 1 − λ ≈ 0.2, which is a typical value in DDLS found in other molecular and hydrogen bonding liquids.34 The uncertainty in λ finally leads to an overall uncertainty of about a factor of 2 in the correlation times.

Figure 4. Normalized PCS electric field correlation function g1(t) with fits for α- and β-relaxation.

As seen in Figure 4, two correlation decays are immediately obvious in the accessible time window, and both processes get increasingly separated as the temperature is lowered. The fit functions shown with solid lines represent the same model as used for the dielectric data, just without the additional Debye process. In Figure 5 we show the corresponding Fourier

Figure 5. Fourier transform of the PCS electric field correlation functions g1(t) with fits for α- and β-relaxation (solid lines).

transform of g1(t). Here it is most obvious that besides the αrelaxation the β-process is well resolved in the data, very similar to the dielectric spectra (cf. Figure 3). At low frequencies χ″(ω) ∝ ω is observed as expected, and a Debye peak is not resolved. This is seen in more detail in Figure 6, where the correlation function at 127 K is shown. After the correlation decay which is well described by eq 7, a plateau is observed that covers almost 4 orders of magnitude. The variation of the plateau in that time range is smaller than 2.5% of the amplitude of the α-process. Therefore, if a Debye contribution is present but unresolved in the present DDLS data, its magnitude must be below that level.

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DISCUSSION AND CONCLUSIONS In Figure 7 we bring together the time constants obtained by dielectric and photon correlation spectroscopy. As already reported by Hansen et al.4 for the α process, the time constants of dielectric and photon correlation experiments almost coincide, however with some extra uncertainty for the dielectric 8850

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Figure 6. PCS electric field correlation function g1(t) of 1-propanol at 127 K demonstrating that beyond the α process a plateau extends in the PCS data over more than 3 orders of magnitude in time. A possible Debye relaxation in that region that is not resolved in the present data will have to be smaller than 2.5% of the α-relaxation strength. The inset shows a log−log representation of the data.

Figure 8. Model parameters for α- and β-process of 1-propanol from the analysis of dielectric relaxation data (diamonds) and PCS data (circles). Open symbols refer to parameters that were obtained in a global fit. Full symbols: parameters were fitted individually for each temperature. Unless explicitly shown, uncertainties are smaller than the size of the symbols.

and virtually temperature independent, the JG β-relaxation should be a small-angle process. In that case, however, the argument leading to eq 6 would become relevant, and one would expect the secondary peak to be significantly larger in DDLS than in dielectric spectroscopy, which is not observed. Thus, if eq 6 is applicable in the present case, then large angles have to be involved in the reorientational process; as e.g. for the random jump model, the correlation functions of l = 1 and l = 2 become identical.27 We note here that large angle reorientation underlying the β-process above Tg was already reported in some other system, where internal degrees of freedom were involved in the β-relaxation35 and was also found in simulations of a molecular glassformer.36 We note that in this case it would also be likely that not all molecules are involved in that process and that an “islands of mobility” picture would be recovered. However, such a consideration relies on certain assumptions. The most important one is that indeed molecular reorientation is probed in the DDLS experiment. Whether this is the case depends on the dominant scattering mechanism and was considerably debated in the past.28,37,38 At present there seems to be consensus that reorientation of the optical anisotropy is the dominant mechanism if the anisotropy is large28,37 and when long times are considered.38 The latter even holds true in cases where anisotropy is smaller, and even for glycerol with a comparatively small anisotropy reorientation is probed in DDLS in the regime of the α-relaxation.34 However, in the time range of the JG β-process even a small contribution of an interaction induced mechanism may have a non-negligible effect and cannot be entirely excluded, even if on longer time scale reorientation is clearly dominant. A further assumption is that the dipole moment and the symmetry axis of the anisotropic polarizability either are aligned, which is a rather special situation and certainly not the case in propanol, or are in a fixed orientation with respect to each other. In the latter case one additionally has to assume that the reorientational processes considered are isotropic, i.e., without any preferred axis of reorientation, so that, on average, the correlation functions ⟨Pl(cos θ(t))⟩ refer to the same angle θ in both methods. When checking propanol for these assumptions it turns out that both do not really hold: First,

Figure 7. Arrhenius plot of 1-propanol with dielectric relaxation data (triangles) and PCS data (circles). τD and τα data are fitted with a Vogel−Fulcher−Tammann law, and the JG β-process is well described by an Arrhenius law with an activation energy of 2500 K·kB. Unless explicitly indicated, uncertainties are smaller than the size of the symbols.

τα, as the α-process is hidden under the Debye peak. Moreover, it turns out that also the shape parameters are very similar within experimental uncertainty. As the same functional form was used to analyze the data sets in both experiments, the results can directly be compared, as seen in Figure 8. And interestingly, the same holds true for the JG β-process, which now also is resolved in the PCS experiments. Figures 7 and 8 reveal that not only the time constants but also the shape parameters are almost identical. As is typical for a JG β-relaxation above Tg, the relative amplitude k of the β-process increases with temperature (cf. Figure 8, upper graph). Previously this was interpreted in terms of a highly restricted reorientation of all molecules underlying the β-process and a ceasing of this restriction at higher temperatures leading to an increase in amplitude.8 However, such an interpretation is difficult in the present context, as at least at low temperatures around Tg, where k becomes small 8851

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propanol is known to show internal degrees of freedom in dynamic experiments,39 so that it cannot be considered a rigid molecule. Concerning the second assumption, in contrast to previous findings it has recently become clear that the JG βprocess may be partially anisotropic in some molecules,40 so that the same molecular motion may project in a different way onto the optical anisotropy and the dipole moment. Thus, a strict conclusion cannot be drawn at that point. Further experiments with fully rigid molecules will allow for a more straightforward interpretation. Still, the coincidence is striking, as demonstrated in Figure 9. Here, dielectric and light scattering data are directly compared

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (J.G.). ORCID

Jan Gabriel: 0000-0001-7478-6366 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Financial support by the Deutsche Forschungsgemeinschaft under Grant No. BL 923/1 and within FOR 1583 under Grant No. BL 1192/1 is gratefully acknowledged. We are also grateful to Ernst Rössler, Bayreuth, for providing the original dielectric data of propanol from refs 19 and 8 and for making the dielectric time domain setup available to us.

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REFERENCES

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Figure 9. Direct comparison of dielectric (circles) and depolarized light scattering data (diamonds) at similar temperatures, which underlines the finding that α- and β-relaxation appear to be almost identical in both methods with respect to their line shape. Dashed lines show the αβ-part of the fit to the dielectric data.

in susceptibility representation at similar temperatures. For the dielectric spectra the fit is shown in two versions with and without the Debye contribution, and it becomes obvious that regarding the line shape the DDLS spectra are almost identical with the dielectric ones, just without the Debye peak. As the Debye contribution is also observed in other techniques, like the shear modulus, it seems likely that the reorientation of transient supramolecular structures, like e.g. transient chains as suggested in ref 41, may couple in different ways to various experimental probes. For instance, it may couple strongly to the dipole moment due to a significant projection of the dipole moment onto the backbone of the transient chain, while it may be too weak in the anisotropy in order to be detected in the light scattering signal, below a few percent of the α-relaxation amplitude in the present case. The formation of a transient supramolecular structure may render the reorientation related to the α-process slightly anisotropic with respect to the contour of that chain, so that the degree of correlation that remains after the α-process has relaxed is significantly different for the optical anisotropy and the electric dipole moment. This is in agreement with NMR results on butanol, which show that the α-relaxation occurs as a reorientation of the alkyl chain around the backbone of transient chain formed by the hydroxyl group, which itself relaxes on a longer time scale.41 Further DDLS investigations of a series of monoalcohols, which show different expressions of the Debye contribution in the dielectric spectra, will certainly shed more light on this topic. 8852

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DOI: 10.1021/acs.jpcb.7b06134 J. Phys. Chem. B 2017, 121, 8847−8853