NMR Relaxometry Accessing the Relaxation Spectrum in Molecular Glass Formers

Abstract

It is a longstanding question whether universality or specificity characterize the molecular dynamics underlying the glass transition of liquids. In particular, there is an ongoing debate to what degree the shape of dynamical susceptibilities is common to various molecular glass formers. Traditionally, results from dielectric spectroscopy and light scattering have dominated the discussion. Here, we show that nuclear magnetic resonance (NMR), primarily field-cycling relaxometry, has evolved into a valuable method, which provides access to both translational and rotational motions, depending on the probe nucleus. A comparison of ¹H NMR results indicates that translation is more retarded with respect to rotation for liquids with fully established hydrogen-bond networks; however, the effect is not related to the slow Debye process of, for example, monohydroxy alcohols. As for the reorientation dynamics, the NMR susceptibilities of the structural (α) relaxation usually resemble those of light scattering, while the dielectric spectra of especially polar liquids have a different broadening, likely due to contributions from cross correlations between different molecules. Moreover, NMR relaxometry confirms that the excess wing on the high-frequency flank of the α-process is a generic relaxation feature of liquids approaching the glass transition. However, the relevance of this feature generally differs between various methods, possibly because of their different sensitivities to small-amplitude motions. As a major advantage, NMR is isotope specific; hence, it enables selective studies on a particular molecular entity or a particular component of a liquid mixture. Exploiting these possibilities, we show that the characteristic Cole-Davidson shape of the α-relaxation is retained in various ionic liquids and salt solutions, but the width parameter may differ for the components. In contrast, the low-frequency flank of the α-relaxation can be notably broadened for liquids in nanoscopic confinements. This effect also occurs in liquid mixtures with a prominent dynamical disparity in their components.

Keywords: dielectric spectroscopy; glass transition; ionic and confined liquids; molecular; nuclear magnetic resonance relaxometry.

Figure 1

(a) ¹H relaxation rate R₁ of partially deuterated glycerol-h₅ at exemplary Larmor frequencies (ω/2π) as a function of reciprocal temperature [35]; a spectral range typical of commercial FC relaxometers is covered. The lines are guides to the eye. (b) Susceptibility spectrum χ″(ω) = ωR₁(ω) as obtained from a home-built instrument together with high-field data. The red solid line represents a fit by a sum of two sub-spectra referring to the translational (dashed line) and rotational (dashed-dotted line) relaxation contributions at 270 K, yielding the indicated parameters (see text for details).

Figure 2

Susceptibility master curves as a function of $\omega {\tau }_{\mathrm{DLS}}\left(T\right)$ ($\tau$ _DLS data from ref. [28]). The plot includes ¹H FC data for glycerol-h₃ (blue closed circles) [36] and glycerol-h₅ (multi-coloured data) [36], and high-field ²H spin-lattice relaxation data for glycerol-d₃ (dark green) and glycerol-d₅ (grey) recorded at 55 MHz (squares) and 40 MHz (triangles) [78]. The master curves include results at temperatures from 180 K to 360 K, the dielectric spectrum at 288 K (orange dashed line) [79], and the PCS spectrum at 220K (magenta dashed line). Black lines represent fits with a model featuring translational and reorientational contributions (see text) for ²H data with reorientational contribution only. Red lines show a fit of the maximum region with a single CD function, yielding a stretching parameter β_CD $\approx 0.47\pm 0.03$.

Figure 3

(a) Master curves of the intramolecular FC susceptibility χ″_intra of glycerol-h₅ as a function of the reduced frequency ωτ_rot [72] (black circles) together with the results from MD simulations (coloured lines) [36]. Data are scaled by the maximum values χ″_max. Inset: simulation results for the intra- and intermolecular relaxation contributions on a reduced frequency scale. The bimodal structure, reflecting translational and rotational dynamics, is well recognized in the intermolecular part. (b) Rotational correlation times τ_rot of glycerol as obtained from NMR [36] and other techniques (DS [1] (blue diamonds), DLS [81] (magenta triangles), and MD simulations [36]) together with inverse self-diffusion coefficients D⁻¹ from static field gradient diffusometry [82] (SFG, purple circles), FC relaxometry on glycerol-h₈ [50], and MD simulations [36].

Figure 4

(a) NMR susceptibility master curves of o-terphenyl (OTP), ethylene glycol-h₄ (EG-h₄) [84], glycerol-h₅ (glyc-h₅), and sorbitol scaled to overlap in the peak region [85]; the r value characterizing the separation of translational and rotational time constants increases from 10 (o-terphenyl) to 26 (ethylene glycol) to 54 (glycerol and sorbitol). In the case of o-terphenyl, the susceptibilities obtained from ²H relaxation [86] and from DLS [87] are included. (b) FC susceptibility master curves of the eight liquids studied so far.

Figure 5

(a) NMR susceptibility master curves of m-tricresyl phosphate (m-TCP) determined from FC ¹H relaxometry (black circles; constructed from data in the range 220 K–383 K), ³¹P NMR relaxometry (blue squares; data in the range 240 K–290 K), and high-field ³¹P NMR (green open squares) [90]. The latter are dominated by the CSA interaction and rescaled in amplitude (right ordinate). Added are the PCS (red line) [4]) and DS (magenta line) [93,94] susceptibilities close to T_g. (b) Rotational correlation times of m-TCP as obtained from different techniques: DS—open green triangles [53]; high-field ³¹P NMR relaxation—blue squares [90]; FC ³¹P NMR—red circles [53]; FC ¹H NMR—black squares [53]; and DLS—violet circles [92]. Reciprocal diffusion coefficient D⁻¹(T) measured by FC relaxometry (open orange stars) [53] and by static field gradient (SFG) diffusometry (solid brown stars) [53]—right ordinate. Inset: dielectric spectra of m-TCP at high temperatures revealing a bimodal spectral shape; included is a FC ¹H NMR spectrum at the corresponding temperature.

Figure 6

Reorientation dynamics of an ionic liquid comprised of 1-propyl-3-methyl-imidazolium cations (C3) and bis(trifluoromethylsulfonyl)imide (TFSI) anions: (a) ²H NMR susceptibility master curve for the selectively deuterated cation, constructed from FC and HF data at the indicated temperatures (colored symbols), together with the PCS susceptibility (black symbols) [111]. The red line is a CD fit of the NMR data. (b) Correlation times of cation reorientation from FC ¹H and ²H relaxometry and from ²H stimulated-echo experiments (STE) [100], and of anion reorientation from FC ¹⁹F relaxometry together with data from DLS [111] and DS [112] studies, the latter for a slightly different ionic liquid consisting of 1-butyl-3-methyl-imidazolium cations (C4) and TFSI anions.

Figure 7

¹H (black circles) and ⁷Li (blue triangles) FC master curves of LiCl-7H2O on a reduced frequency axis, which is defined by the respective peak correlation times τ_peak form KWW fits to the peak regions (red lines). The inset compares the FC peak correlation times with data from shear relaxation [119], DS [117], DLS [117], viscosity measurements [116], and conventional HF NMR [118] for the indicated aqueous salt solutions.

Figure 8

(a) Field-cycling ¹H and ⁷Li susceptibilities compared to DS and shear relaxation susceptibilities at 150 K and 140 K, respectively, plotted over a reduced frequency scale $\omega {\tau }_{peak}$. The data are scaled by their maximum values. (b) $\omega R_1\left(T\right)$ ¹H and ⁷Li FC data, rescaled horizontally by the respective ${\tau }_{DS}\left(T\right)$ from DS data [117]. Most FC data collapse onto a master curve. Deviations occur at low temperatures, where a secondary water process separates from the α-process [120].

Figure 9

The dielectric spectra of (a) 4-methyl-3-heptanol (4M3H) and (b) 5-methyl-3-heptanol (5M3H). The spectra of 4M3H are fitted by a sum of a Debye and a Havriliak–Negami function.

Figure 10

(a) The low-frequency part of the FC ¹H NMR susceptibility master curves of the four octanol isomers. For comparison, the susceptibilities of glycerol-h₃ and o-terphenyl and a CD susceptibility are included. (b) Rotational time constants obtained from DS (open symbols) and FC ¹H relaxometry (solid symbols). Inset: diffusion coefficients determined by FC ¹H relaxometry and from the literature [124,125]. The solid lines are guides for the eye.

Figure 11

(a) ¹H spin-lattice relaxation rate R₁ as a function of the square root of the frequency, ω^1/2. The self-diffusion coefficient D is determined from the linear part of R₁(ω) via Equation (4), as indicated by the straight lines. Inset: magnification of the high-temperature data. (b) Normalized NMR time correlation functions derived from the susceptibility master curves of the octanol isomers (see Figure 9a), glycerol-h₃, and o-terphenyl (OTP) in a double logarithmic representation. The dashed line is a CD function representing the rotational dynamics and the dotted line indicates a long-time t^−3/2 power law reflecting the translational dynamics.

Figure 12

FC ¹H and ²H NMR susceptibilities of ethylene glycol in cylindrical silica pores with a diameter of 3.0 nm. For ¹H, the susceptibility master curve constructed by horizontally shifting the data sets at the indicated temperatures is shown together with a HN fit (red line), yielding the shape parameters α = 0.81 and β = 0.45. For ²H, the susceptibilities at 300 K and 320 K are interpolated with power laws ω^0.8 and shifted horizontally for better visibility. The inset compares the correlation times obtained from the position of the ¹H susceptibility peak at 235 K and the shift factors used for the construction of the master curve with correlation times from HF ²H relaxation and DS studies on ethylene glycol in various silica pores and in the bulk [137,138,142]. DS studies find two dynamic processes, P1 and P2, from which P1 corresponds to the main NMR (α) relaxation process.

Figure 13

(a) Dielectric spectra (DS, solid lines) [143,144] corrected for the Curie factor and scaled in amplitude by a global factor to match the FC ³¹P NMR susceptibility (open symbols) of tributyl phosphate (TBP) at the indicated temperatures [53]. The blue dashed line is the interpolated NMR data from panel (b) re-scaled with ${\tau }_{\alpha }$(T = 170 K). (b) Susceptibility master curve of TBP constructed from FC ³¹P relaxation data at temperatures from 190 K to 240 K on a reduced frequency scale ωτ_peak. A DS spectrum at 170 K [144] and a DLS spectrum at 147 K are included [27], both shifted to the NMR peak. Inset: correlation times of the α- and β-processes from FC ³¹P NMR relaxometry, DS [143], and DLS [27]. The black line is a guide for the eye.

Figure 14

Susceptibility master curve of cyanocyclohexane from FC ¹H NMR relaxometry together with dielectric spectra (black symbols) at the indicated temperatures in K [145]. The data are scaled for the best overlap of the α-relaxation peak. The FC and DS data show a secondary (β-) relaxation at high frequencies. However, the amplitude of the β-relaxation with respect to that of the α-relaxation is significantly higher in FC than in DS.