^{1}, B. Schmidtke

^{1}, R. Kahlau

^{1}, D. Bock

^{1}, R. Meier

^{1}, B. Micko

^{1}, D. Kruk

^{1,a)}and E. A. Rössler

^{1,b)}

### Abstract

Although broadly studied, molecular glass formers are not well investigated above their melting point. Correlation times down to 10^{−12} s are easily accessible when studying low-T g systems by depolarized light scattering, employing a tandem-Fabry-Perot interferometer and a double monochromator. When combining these techniques with state-of-the-art photon correlation spectroscopy (PCS), broad band susceptibility spectra become accessible which can compete with those of dielectric spectroscopy (DS). Comparing the results with those from DS, optical Kerr effect, and NMR, we describe the evolution of the susceptibilities starting from the boiling point T b down to T g, i.e., from simple liquid to glassy dynamics. Special attention is given to the emergence of the excess wing contribution which is also probed by PCS and which signals a crossover of the spectral evolution. The process is attributed to a small-angle precursor process of the α-relaxation, and the apparent probe dependent stretching of the α-process is explained by a probe dependent contribution of the excess wing. Upon cooling, its emergence is linked to a strong decrease of the strength of the fast dynamics which is taken as reorientational analog of the anomaly of the Debye-Waller factor. Many glass formers show in addition a slow β-process which manifests itself rather universally in NMR, in DS, however, with different amplitudes, but not at all in PCS experiments. Finally, a three-parameter function is discussed interpolating τ α(T) from T b to T g by connecting high- and low-temperature dynamics.

Financial support of the Deutsche Forschungsgemeinschaft (DFG) through the projects RO 907/10,11,15 is acknowledged. The authors thank P. Lunkenheimer (Augsburg) and M.D. Fayer (Stanford) for providing the DS data for glycerol and the OKE data for benzophenone, respectively.

I. INTRODUCTION

II. RESULTS

A. Glassy dynamics

B. Subtle changes in the susceptibility while supercooling: The excess wing

C. Amplitude of the fast dynamics

D. Probe dependence of the susceptibility

E. The β-process

F. Crossover to simple liquid dynamics at higher temperatures

III. DISCUSSION AND CONCLUSIONS

### Key Topics

- Correlation functions
- 27.0
- Nuclear magnetic resonance
- 23.0
- Dielectric relaxation
- 21.0
- Dielectrics
- 18.0
- Light scattering
- 18.0

## Figures

(a) Dynamic susceptibility of o-terphenyl (normalized by relaxation strength of the α-process) as a function of frequency obtained from depolarized light scattering (tandem-Fabry-Perot interferometry and double monochromator); ^{90} relaxation regimes are indicated; dashed lines: “hybrid model” including α- and fast relaxation contribution, dotted lines: corresponding Cole-Davidson fit of α-peak with stretching parameter β CD = 0.64. (b) Spectra of glycerol; ^{73} different colors signal different relaxation regimes (cf. text); flattening of the low-frequency side of the susceptibility minimum is described by a power-law with exponent γ = 0.19 (dotted line).

(a) Dynamic susceptibility of o-terphenyl (normalized by relaxation strength of the α-process) as a function of frequency obtained from depolarized light scattering (tandem-Fabry-Perot interferometry and double monochromator); ^{90} relaxation regimes are indicated; dashed lines: “hybrid model” including α- and fast relaxation contribution, dotted lines: corresponding Cole-Davidson fit of α-peak with stretching parameter β CD = 0.64. (b) Spectra of glycerol; ^{73} different colors signal different relaxation regimes (cf. text); flattening of the low-frequency side of the susceptibility minimum is described by a power-law with exponent γ = 0.19 (dotted line).

The correlation function, C ^{ (2) }(t), obtained from photon correlation spectroscopy (PCS) as well as from tandem-Fabry-Perot interferometry (DM/TFPI) data after Fourier transformation; dash-dotted line: fit by Kohlrausch law including excess wing contribution, solid blue line: correlation function at T = 290 K shifted to coincide with that at T = 207 K; dotted line: amplitude f of slow dynamics. ^{108}

The correlation function, C ^{ (2) }(t), obtained from photon correlation spectroscopy (PCS) as well as from tandem-Fabry-Perot interferometry (DM/TFPI) data after Fourier transformation; dash-dotted line: fit by Kohlrausch law including excess wing contribution, solid blue line: correlation function at T = 290 K shifted to coincide with that at T = 207 K; dotted line: amplitude f of slow dynamics. ^{108}

Time constants τ α(T) for o-terphenyl (OTP), ^{90,113,116} m-tricresyl phosphate (m-TCP), ^{5,95,114} picoline, ^{70} and glycerol ^{3,5,49,73,115} as obtained by different techniques. Solid lines are fits by Eq. (2) .

Time constants τ α(T) for o-terphenyl (OTP), ^{90,113,116} m-tricresyl phosphate (m-TCP), ^{5,95,114} picoline, ^{70} and glycerol ^{3,5,49,73,115} as obtained by different techniques. Solid lines are fits by Eq. (2) .

Minimum scaling (data from Fig. 1 ): reduced susceptibility vs. reduced frequency; below some temperature the scaling fails. (a) o-terphenyl; envelope of high-temperature data interpolated by a sum of two power-laws with exponent b = 0.65 and a = 0.33 in agreement with prediction of MCT. (b) Glycerol, colors correspond to those in Fig. 1(b) ; dotted line: interpolation with sum of two power-laws (adapted from Ref. 73 and 90 ).

Minimum scaling (data from Fig. 1 ): reduced susceptibility vs. reduced frequency; below some temperature the scaling fails. (a) o-terphenyl; envelope of high-temperature data interpolated by a sum of two power-laws with exponent b = 0.65 and a = 0.33 in agreement with prediction of MCT. (b) Glycerol, colors correspond to those in Fig. 1(b) ; dotted line: interpolation with sum of two power-laws (adapted from Ref. 73 and 90 ).

(a) Dielectric loss of glycerol, ^{3} and propylene carbonate ^{84} (scaled by static susceptibility). Emergence of the excess wing contribution (arrow) is recognized; dashed line: phenomenological interpolation. ^{122} (b) Dynamic susceptibility (normalized by relaxation strength of α-process) of m-tricresyl phosphate as obtained by light scattering: ^{108} DM/TFPI spectra and PCS data transformed into the frequency domain; numbers indicate temperature; dashed lines: full interpolation of α-peak, excess wing and fast dynamics with temperature independent parameters β k, γ, and a, respectively.

(a) Dielectric loss of glycerol, ^{3} and propylene carbonate ^{84} (scaled by static susceptibility). Emergence of the excess wing contribution (arrow) is recognized; dashed line: phenomenological interpolation. ^{122} (b) Dynamic susceptibility (normalized by relaxation strength of α-process) of m-tricresyl phosphate as obtained by light scattering: ^{108} DM/TFPI spectra and PCS data transformed into the frequency domain; numbers indicate temperature; dashed lines: full interpolation of α-peak, excess wing and fast dynamics with temperature independent parameters β k, γ, and a, respectively.

Rescaled OKE data vs. reduced time t/τα of benzophenone (BZP), ^{77} pulse-response representation of the dielectric data of glycerol ^{51} and PCS data of m-tricresyl phosphate (m-TCP); ^{108} dashed lines indicate intermediate power-law/excess wing, dotted lines von-Schweidler law and α-process, respectively.

Rescaled OKE data vs. reduced time t/τα of benzophenone (BZP), ^{77} pulse-response representation of the dielectric data of glycerol ^{51} and PCS data of m-tricresyl phosphate (m-TCP); ^{108} dashed lines indicate intermediate power-law/excess wing, dotted lines von-Schweidler law and α-process, respectively.

(a) Susceptibility spectra of the fast dynamics after subtracting α-peak and excess wing contributions from the overall susceptibility; dashed line: power-law ν^{0.33} of the fast dynamics; shaded area: relaxation strength 1 – f rel of the fast dynamics (adapted from Ref. 90 ). (b) 1 – f rel for α-picoline and o-terphenyl (LS), and glycerol (DS) as a function of temperature. ^{90,121}

(a) Susceptibility spectra of the fast dynamics after subtracting α-peak and excess wing contributions from the overall susceptibility; dashed line: power-law ν^{0.33} of the fast dynamics; shaded area: relaxation strength 1 – f rel of the fast dynamics (adapted from Ref. 90 ). (b) 1 – f rel for α-picoline and o-terphenyl (LS), and glycerol (DS) as a function of temperature. ^{90,121}

(a) Comparing susceptibilities of m-tricresyl phosphate as obtained from PCS (circles), from ^{31}P NMR ^{95} (pentagons) and dielectric spectroscopy (diamonds); (b) susceptibility master curves of glycerol compiled from field-cycling ^{1}H NMR (pentagons), ^{134} dielectric spectroscopy (circles; DS) ^{135} and DM/TFPI (crosses) ^{73,135} and PCS (solid circles); ^{108} dashed (blue) lines: interpolations assuming a relaxation described by a Cole-Davidson function (βCD = 0.64) together with a power-law contribution with exponent γ = 0.2; arrow indicates factor 3 between the amplitudes of the excess wing.

(a) Comparing susceptibilities of m-tricresyl phosphate as obtained from PCS (circles), from ^{31}P NMR ^{95} (pentagons) and dielectric spectroscopy (diamonds); (b) susceptibility master curves of glycerol compiled from field-cycling ^{1}H NMR (pentagons), ^{134} dielectric spectroscopy (circles; DS) ^{135} and DM/TFPI (crosses) ^{73,135} and PCS (solid circles); ^{108} dashed (blue) lines: interpolations assuming a relaxation described by a Cole-Davidson function (βCD = 0.64) together with a power-law contribution with exponent γ = 0.2; arrow indicates factor 3 between the amplitudes of the excess wing.

Data of m-tricresyl phosphate from dielectric spectroscopy (DS), photon correlation spectroscopy (PCS) and NMR in the step-response representation normalized by the relaxation strength of α-process and excess wing (data from Fig. 8(a) , see text). ^{108}

(a) Dielectric spectra of dimethyl phthalate (DMP) disclosing two secondary processes in addition to an α-peak. (b) Corresponding relaxation times. ^{141}

(a) Dielectric spectra of dimethyl phthalate (DMP) disclosing two secondary processes in addition to an α-peak. (b) Corresponding relaxation times. ^{141}

Comparison of the susceptibilities of dimethyl phthalate (DMP) from photon correlation spectroscopy (PCS) ^{108} and dielectric spectroscopy (DS); ^{141} the β-process is not probed by PCS, instead an excess wing is observed.

(a) ^{2}H NMR solid-echo spectra at different inter-pulse distances t p (see inset) for type-B glass formers; plastic crystal cyano cyclohexane (CCH), structural glasses toluene (TOL) and ethanol (ETH); spectra shown for comparable correlation times τ β = 10 μs (T < T g). (b) Corresponding spectra of the type-A glass former glycerol not displaying changes. (c) NMR spectra of cyano cyclohexane at T > T g (t p = 20 μs) revealing line-shape changes due to a loss of spatial hindrance of the β-process. ^{92,96,152}

(a) ^{2}H NMR solid-echo spectra at different inter-pulse distances t p (see inset) for type-B glass formers; plastic crystal cyano cyclohexane (CCH), structural glasses toluene (TOL) and ethanol (ETH); spectra shown for comparable correlation times τ β = 10 μs (T < T g). (b) Corresponding spectra of the type-A glass former glycerol not displaying changes. (c) NMR spectra of cyano cyclohexane at T > T g (t p = 20 μs) revealing line-shape changes due to a loss of spatial hindrance of the β-process. ^{92,96,152}

(a) Spin-lattice relaxation time T1 as a function of the time constant τ β of the β-process: plastic crystal cyano cyclohexane, neat structural glass formers toluene and ethanol, and binary structural glass chlorobenzene-d5/decalin (b) corresponding dielectric relaxation strength Δɛβ/Δɛ data from Refs. 49, 92, and 96 , and 152 .

(a) Spin-lattice relaxation time T1 as a function of the time constant τ β of the β-process: plastic crystal cyano cyclohexane, neat structural glass formers toluene and ethanol, and binary structural glass chlorobenzene-d5/decalin (b) corresponding dielectric relaxation strength Δɛβ/Δɛ data from Refs. 49, 92, and 96 , and 152 .

(a) Susceptibility spectra of ethyl benzene obtained by applying DM/TFPI; blue curves: reflect glassy dynamics; red curves: “simple liquid” dynamics approaching the boiling point T b; green dashed line: interpolation with hybrid model assuming MCT relations for the exponents b = β and a; (b) Corresponding time domain representation of the data. ^{154}

(a) Susceptibility spectra of ethyl benzene obtained by applying DM/TFPI; blue curves: reflect glassy dynamics; red curves: “simple liquid” dynamics approaching the boiling point T b; green dashed line: interpolation with hybrid model assuming MCT relations for the exponents b = β and a; (b) Corresponding time domain representation of the data. ^{154}

Reorientational correlation times of molecular liquids obtained by dielectric spectroscopy (open symbols) and light scattering (full symbols); for trinaphthylbenzene (TNB) and o-terphenyl viscosity data are included; ^{155,156} numbers in K: T g; straight dashed lines: high-temperature Arrhenius behavior; solid lines: full fit by Eq. (2) (adapted from Ref. 112 ).

Reorientational correlation times of molecular liquids obtained by dielectric spectroscopy (open symbols) and light scattering (full symbols); for trinaphthylbenzene (TNB) and o-terphenyl viscosity data are included; ^{155,156} numbers in K: T g; straight dashed lines: high-temperature Arrhenius behavior; solid lines: full fit by Eq. (2) (adapted from Ref. 112 ).

Reduced activation energy E coop(T)/E ∞ plotted vs. a reduced temperature scale in order to provide a master curve for all investigated molecular liquids; color code like in Fig. 15 (adapted from Ref. 112 ).

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