Article pubs.acs.org/JPCB
Dynamic Properties of Glass-Formers Governed by the Frequency Dispersion of the Structural α‑Relaxation: Examples from Prilocaine Z. Wojnarowska,*,†,‡ M. Rams-Baron,†,‡ J. Knapik,†,‡ K. L. Ngai,§,∥ D. Kruk,⊥ and M. Paluch†,‡ †
Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland Silesian Center for Education and Interdisciplinary Research, 75 Pulku Piechoty 1A, 41-500 Chorzow, Poland § CNR-IPCF, Largo B. Pontecorvo 3, I-56127 Pisa, Italy ∥ Dipartimento di Fisica, Università di Pisa, Largo B. Pontecorvo 3, I-56127 Pisa, Italy ⊥ Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Sloneczna 54, Olsztyn PL-10710, Poland ‡
ABSTRACT: General and fundamental properties of glass-formers of various chemical bonding and physical structures have been found in the recent past. These important findings should be key to gain basic understanding of the dynamics at all time scales leading to glass transition. However, the entirety of these general properties has not been found in a single glass-former. For others to appreciate the importance of these properties, they need to collect the supporting experimental data from different glass-formers scattered over many publications. This hurdle may account for the current lack of universal recognition of the importance of these general properties by the research community. In this paper we present experimental studies of the dynamic processes over a broad range of time scales of a single glass-former, prilocaine. Practically the entire collection of fundamental properties has been found in this system. The advance should heighten the awareness of the importance of these properties in anyone’s effort to solve the glass transition problem.
1. INTRODUCTION In the past years a number of general and fundamental properties in the dynamics of glass-formers in both the liquid and glassy states have been found. The notable ones include the following items. (A) The frequency dispersion of the structural α-relaxation is invariant to variations of temperature T and pressure P provided that the relaxation time τα is kept constant.1,2 (B) A secondary relaxation strongly connected to the αrelaxation is universally present,3−13 and the nomenclature, Johari−Goldstein (JG) β-relaxation,14,15 has been used to distinguish it from the other unimportant secondary relaxations without any connection.16,17 (C) The separation in time scale between the α-relaxation and the JG β-relaxation, measured by the ratio of their relaxation times, τα/τβ, is controlled by the full-width at half-maximum (fwhm) of the frequency dispersion of the α-relaxation, or alternatively by size of n18−22 where (1 − n) ≡ βKWW is the fractional exponent of the Kohlrausch correlation function, Φ(t ) = exp[−(t /τα)1 − n ]
tβ(T ,P) ≈ τ0(T ,P)
holds in many glass-formers at temperature T and pressure P. In the CM there is a relation between τ0(T) to τα involving the exponent (1 − n) in the Kohlrausch function given by26 τα(T ,P) = [tc−nτ0(T ,P)]1/(1 − n) ]
(3)
where tc = 1−2 ps for molecular glass-formers. From eq 3 it follows that log[τα(T,P)/τ0(T,P)] is exactly determined by n, and furthermore from relation 2 that log[τα(T,P)/ τβ(T,P)] is approximately determined by n. (E) Over extended temperature range, two different Vogel− Fulcher−Tammann (VFT) functions, VFT1(T) = τ∞1 exp[D1T01/(T − T01)] for T > TB and VFT2(T) = τ∞2 exp[D2T02/(T − T02)] for T < TB are often needed to have a full description of the temperature dependence of τα(T).27−30 (F) The fwhm of the α-dispersion or n shows increase of its magnitude as well as stronger T-dependence on crossing TB from above to below.31,32 (G) The JG β-relaxation is preceded in time by the caged molecules dynamics manifested by the nearly constant loss (NCL) in the susceptibility spectra at lower
(1)
the Fourier transform of which fits the frequency dispersion of the α-relaxation. (D) The primitive relaxation with relaxation time τ0 in the coupling model (CM)23−25 is the precursor of the JG βrelaxation, and the approximate relation © 2015 American Chemical Society
(2)
Received: July 4, 2015 Revised: September 10, 2015 Published: September 10, 2015 12699
DOI: 10.1021/acs.jpcb.5b06426 J. Phys. Chem. B 2015, 119, 12699−12707
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The Journal of Physical Chemistry B temperatures and/or higher frequencies.15,33−35 The frequency dependence of the NCL is a power law, ε″(ν) = Bν−λ with λ ≪ 1. Caged molecules dynamics is not a relaxation process and the NCL has no characteristic time. The caged molecules dynamics regime and the associated NCL is terminated by the onset of the primitive relaxation, which is the precursor of the JG β-relaxation. This is understandable because these intermolecular secondary relaxations effectively move molecules out of the cages. Therefore, in order of magnitude, the primitive frequency, ν0 ≡ 1/(2πτ0), is the lower bound of the NCL, or τ0 is the upper bound of the caged dynamics time regime, a relation verified before in many glass-formers. All the properties from (A) to (G) have not been seen either altogether in a single glass-former or by a single experimental technique. Also some properties were found in a certain class of glass-formers and other properties in another class of glassformer. This paper reports experimental study of a glassforming pharmaceutical, prilocaine, whereby all these properties from (A) to (F) and more are observed mainly by broadband dielectric relaxation with the assists of dynamic mechanical spectroscopy and NMR relaxometry. The situation presented by prilocaine requires any successful glass transition theory to explain all the properties simultaneously and not partly. The advantage of having the experimental data to support all the properties in a single glass-former is the possibility of giving a unified theoretical explanation of their origins. After the presentation of the experimental data, we show that all the properties are either governed by or related to the width of the frequency dispersion of the α-relaxation, and a unified explanation based on the latter is given.
Glentham Life Sciences (purity more than 99%) and it was used without any further purification. The starting material was completely crystalline with the melting point at 310 K. b. Broadband Dielectric Spectroscopy Measurements. Isobaric dielectric measurements at ambient pressure from 10−1 to 107 Hz were carried out using a Novo-Control GMBH Alpha dielectric spectrometer. Using Agilent 4291B impedance analyzer connected with Novo-Control GMBH system, we were able to measure dielectric spectra in the high frequency range from 106 to 109 Hz. For the isobaric measurements, the sample was placed between two stainless steel electrodes of the capacitor with a gap of 0.1 mm. However, in the case of high frequency measurements the sample was placed between two gold-plated electrodes (diameter, 5 mm; gap, 0.05 mm). Using these two methods, the dielectric spectra of prilocaine were collected over a wide temperature range from 153 to 343 K. The temperature was controlled by the Novo-Control Quattro system, with the use of a nitrogen gas cryostat. Temperature stability of the samples was better than 0.1 K. For the pressure dependent dielectric measurements we used a capacitor, filled with the prilocaine sample, which was next placed in the high-pressure chamber and compressed using the silicone oil. Note that during the measurement the sample was in contact with stainless steel and Teflon. The pressure was measured by the Nova Swiss tensometric pressure meter with a resolution of 0.1 MPa. The temperature was controlled within 0.1 K by means of a liquid flow provided by a thermostatic bath. c. Differential Scanning Calorimetry (DSC). Thermodynamic properties of prilocaine were examined using a MettlerToledo DSC 1 STARe System. The measuring device was equipped with liquid nitrogen cooling and a HSS8 ceramic sensor having 120 thermocouples. The instrument was calibrated for temperature and enthalpy using indium and zinc standards. The melting point of prilocaine was determined as the onset of the peak, whereas the glass transition temperature was determined as the midpoint of the heat capacity increment. The sample was measured in an aluminum crucible. All measurements were carried out at the heat rate of 10 K/min. d. NMR Relaxometry. 1H spin−lattice relaxation experiments were performed for prilocaine under ambient pressure using a Field Cycling relaxometer covering a very broad range of 1H resonance frequencies (magnetic fields): from 4 kHz to 40 MHz. Six full relaxation dispersion profiles were collected at 210, 223, 243, 261, 269, and 297 K; the temperature stability was better than 0.1 K. e. Mechanical Spectroscopy. Dynamic mechanical measurements were performed by means of an ARES-G2 rheometer. The sample was placed between aluminum parallel plates with plate diameters of 8 mm and the gap of 1 mm. Shear deformation was applied under conditions of controlled strain and linear viscoelastic response. Experiments were performed upon heating in the temperature range from 219 to 263 K in steps of 2 K. The dynamic shear moduli G′ and G″ at a constant temperature were collected in the frequency range from 0.1 to 100 rad/s.
2. EXPERIMENTAL SECTION a. Sample Studied. The prilocaine (±)-2′-methyl-2(propylamino)propionanilide, N-(2-methylphenyl)-2(propylamino)propanamide, MW = 220.311 g/mol with the chemical structure presented in Figure 1 was supplied from
3. RESULTS Amorphous prilocaine was prepared by rapid cooling of molten sample. To determine the melting point of crystalline sample, differential scanning calorimetry was applied. The DSC curve obtained during heating of the crystalline compound up to 323 K is depicted in Figure 1. An endothermic peak, with an onset
Figure 1. Thermal analysis of crystalline and amorphous forms of prilocaine. All DSC thermograms obtained during heating at a rate of 10 K/min. The inset panel shows the glass transition regions of the quenched sample. 12700
DOI: 10.1021/acs.jpcb.5b06426 J. Phys. Chem. B 2015, 119, 12699−12707
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The α-loss peaks are fitted by the Fourier transform of the Kohlrausch functions, and alternatively by the Havriliak− Negami (HN) functions eq 4 after taking into consideration of the contribution from dc conductivity at lower frequencies.
at 310 K, indicates the melting point of prilocaine. The subsequent heating from the glassy state shows the characteristic signature for the glass transition in the temperature dependence of the heat flow and determine the glass transition temperature at 220 K. a. Dielectric Data at Ambient Pressure. The isothermal spectra of the dielectric loss ε″(ν) taken at ambient pressure and over the broad frequency range from 10−1 ≤ ν ≤ 109 Hz and the temperature range from 153 to 343 K are shown in Figure 2. Consider first the data taken near the glass transition
* (ϖ) = ε∞ + εNH
Δε σ + ε0 iω [1 + (iωτHN)αHN ]βHN
(4)
Exemplary fits at two temperatures in Figure 3a require βKWW = 0.85 at 261 K and βKWW = 0.75 at 223 K. The examples bring
Figure 2. Dielectric loss spectra of prilocaine collected above as well as below the glass transition temperature over the wide range of frequency at ambient pressure conditions.
temperature Tg = 220 K. The slowest process is the dc conductivity, which is followed at higher frequencies by the αloss peak. The fwhm of the α-loss peak is less than 1.5 decades at all temperatures, and smaller than most molecular glassformers. The high frequency flank of the α-loss peak at 221, 231, and 241 K is not a single power law and the entire loss peak cannot be fitted totally by any of the commonly used empirical functions including those of Cole−Davidson, the Havriliak−Negami, and the Fourier transform of the Kohlrausch correlation function. The deviation from any such fit, called the excess wing, is usually found in glass-formers having the α-loss peak with narrow frequency dispersion or small fwhm.36,37 The excess wing originates from the presence of the JG β-relaxation but is not resolved due to the small separation of JG β-relaxation from the α-relaxation, which is caused by the small fwhm (property C), as in this case of prilocaine. The Kramers−Kronig relations obeyed by any relaxation process have not been tested for the excess wing by anyone to show consistency with the interpretation as the unresolved JG β-relaxation. However, evidence of the presence of the JG β-relaxation was obtained by physical aging of other glass-formers with α-relaxation dynamics similar to those of prilocaine.6,17,38 On further decrease of temperature, the dc conductivity and the α-loss peak shifts to lower frequencies. In the glassy state, the new feature that first appears in the experimental frequency window is the power law dependence of the loss, ε″(ν) = Bν−λ with λ < 0.2. This feature, appropriative called the nearly constant loss (NCL) because λ is small, is evident from the few examples of data taken below 211 K. Showing up at even higher frequencies than the NCL is a very fast secondary γ-relaxation.
Figure 3. (a) Superimposed dielectric loss spectra of prilocaine taken at ambient pressure (P = 0.1 MPa), at two different temperatures above Tg. The solid red lines are KWW fits to the experimental data. (b) Changes in FMHM (open diamonds) and αHN, βHN (solid symbols) with temperature for prilocaine.
out the fact that the fwhm of the α-loss peak is not independent of temperature. Fits of all the α-loss peaks by the H−N functions determine the exponents, αHN and βHN, as well as τα. The results are shown by αHN and the product, αHNβHN, in Figure 3b. It is well-known that αHNβHN characterizes the width of the α-loss peak. A smaller value of αHNβHN corresponds to a large width. The fwhm is another measure of the width of the α-loss peak. It has been determined for each α-loss peak, and the values are shown as a function of 1/T in Figure 3b. Both the fwhm and αHNβHN increase monotonically with decreasing temperature and exhibit a broad crossover from a weaker dependence at higher temperatures to a stronger one at lower temperatures at 1/TB = 0.0038 K−1 or TB = 263 K. The values of τα from the fits are plotted against 1/T in the upper panel of Figure 4. The data obtained over more than ten decades cannot be fitted by a single VFT function. Instead, two VFT functions are required to fit as shown in the figure. To substantiate this requirement, we represent the data in terms of [d log τα/d(1/ T)]−0.5 and plot them against 1/T.29,30 The two different linear dependences in the middle panel of Figure 4 indicate there is a change of temperature dependence of τα from one VFT law to another. The intersection of the two linear dependences determines crossover in T-dependence of τα is at TB = 258 K, 12701
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slope of one indicates no breakdown of the Debye−Stokes− Einstein relation. b. Dynamic Mechanical Measurements. The dynamic shear loss modulus G″ in the frequency range from 0.1 to 100 rad/s and at temperatures ranging from 219 to 263 K in steps of 2 K are shown in Figure 5a. The frequency dispersions of these isothermal data are compared in Figure 5b where G″ is scaled by the peak maximum G″max and frequency by the loss peak frequency f max. The solid line is the fit of loss normalized spectrum at 219 K by the Fourier transform of the Kohlrausch function in eq 1 with the exponent (1 − n) ≡ βKWWW = 0.75. The shear modulus relaxation times τα,G determined by the fits are plotted against reciprocal temperature together with the dielectric relaxation times τα,ε in Figure 5c. The solid lines represent the VFT fits of the dielectric τα,ε data in the high and low temperature regions. The inset shows plot of log τα,ε vs log τα,G. The straight line through the data points has slope s equal to 0.92, indicating a slightly stronger temperature dependence of τα,G than τα,ε. c. NMR Relaxometry. The predominant mechanism of 1H relaxation is 1H−1H magnetic dipole−dipole interactions. The relaxation rate R1(ω) (reciprocal spin−lattice relaxation time) at a resonance frequency ω is given by39
Figure 4. (a) τα(T) dependence of prilocaine described by two VFT equations (solid lines). Inset panel: dc-conductivity plotted as a function of α-relaxation time on a log−log scale for prilocaine. (b) Results of the derivative analysis focused on the validity of VFT parametrization. The intersection of the two VFT lines denotes the crossover temperature Tcross = 258 K. (c) Temperature evolution of the fwhm parameter.
R1(ω) = C DD[J(ω) + 4J(2ω)]
(5)
where CDD denotes a dipolar relaxation constant. The quantities J(ω) are the spectral density functions. The ω-dependence of the spectral density functions depends on the dynamics of the motion leading to stochastic fluctuations of the 1H−1H dipolar coupling. The simplest is the Lorentzian spectral density J(ω) = (τ/(1 + ω2τc2)), where τc denotes a correlation time of the fluctuations of the dipole−dipole interaction, It corresponds in dielectric loss to ε″ ∝ ωJ(ω), which is the Debye process (βKWW = 1). The spin−lattice relaxation time in NMR is a result of spin transitions of protons between their Zeeman energy levels and the transitions are caused by magnetic fields fluctuating on a time scale given by the correlation time. The 1 H relaxation dispersion profiles reveal three relaxation
which is practically the same as that of the fwhm. This can be seen by comparing the bottom panel of Figure 4 with the two upper panels. This finding confirms properties E and F of many glass-formers also are found in prilocaine. The dc conductivity, σdc, obtained together with τα from the fits to the spectra at various temperatures are plotted double logarithmically against each other in the inset to Figure 4. The
Figure 5. (a) Imaginary part of the shear response for prilocaine. (b) Shear relaxation loss spectra scaled by the maximum of G″max plotted as a function of frequency divided by the frequency of peak maximum. The solid line indicates the KWW fit of the loss spectrum recorded at 219 K with stretching parameter βKWWW = 0.75. (c) Mechanical and dielectric relaxation times plotted as a function of temperature. The solid line corresponds to VFT fits of experimental data. The inset shows a plot of dielectric relaxation times against shear-mechanical relaxation times. The data were described using single value of the fractional exponent, s = 0.92. 12702
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temperature independent correlation time, τc,1 = 4.3 × 10−6 s, whereas the relaxation constant, CDD,1, decreases with increasing temperature from 2.8 × 105 Hz2 at 210 K down to 3.6 × 104 Hz2 at 297 K (Table 1). This suggests intermolecular origin of this process40 especially because it is not seen in the dielectric data. The main α-relaxation is not observed in the NMR relaxation experiments. This effect is surprising, and it can shed some light on the mechanism of the α-relaxation. The absence of the main process in the NMR relaxation profiles indicates that the process involves protons that are far away from each other, and therefore, the corresponding relaxation constant is small because it depends on r−6, where r is the interproton distance. d. Dielectric Data at Elevated Pressures. Pressure has a different effect on the dynamics of glass-formers than temperature, and deeper insight can be gained by measuring the spectra at elevated pressures. Such dielectric loss data obtained by elevating pressure P from 120 to 480 MPa at constant T = 283 K are presented in Figure 7a. The dc
contributions (motional processes), as shown in the inset of Figure 6. The overall relaxation can be described as a sum of
Figure 6. Τα(T) dependence of prilocaine described by two VFT equations (closed circles), τγ(T) dependence determined from NMR relaxometry measurements (gray diamonds). Inset: 1H spin−lattice relaxation rate dispersion profile for prilocaine at 297 K decomposed into three contributions associated with a fast motional process (dashed-dotted line), an intermediate process (dashed line), and a slow process (dotted line).
three contributions associated with the slow, the intermediate and the fast motional processes: ⎡ ⎤ τc,1 4τc,1 ⎥ + R1(ω) = C DD,1⎢ 2 2 ⎢⎣ 1 + (ωτc,1) 1 + (2ωτc,1) ⎥⎦ ⎡ ⎤ τc,2 4τc,2 ⎥+A + C DD,2⎢ + ⎢⎣ 1 + (ωτc,2)2 1 + (2ωτc,2)2 ⎥⎦ (6)
The two pairs of parameters, CDD,1, τc,1 and CDD,2, τc,2 characterize the slow and intermediate processes, respectively. As the third motional process is too fast to show a distinct dependence of the corresponding relaxation rate on frequency and it is described as a constant (frequency independent) contribution A. The fast process, with estimated time scale shorter than 10−9 s, is not seen in the dielectric measurements. It can be attributed to fast rotation of the methyl groups. The relaxation constant for the intermediate process CDD,2 is temperature independent and has the value 4.5 × 106 Hz2. The correlation time τc,2 is only weakly temperature dependent, as shown in Figure 6. This motional process can be associated with the secondary γ-relaxation detected in the dielectric experiments. The slow process can be described by a
Figure 7. (a) Loss spectra obtained during isothermal measurement carried out at T = 283 K (P = 120−480 MPa). (b) Pressure evolution of structural relaxation times obtained during isothermal compression at 283 K. The solid line denotes the pressure counterpart of the VFT equation. Solid circles show τα(P) in the crystallization range. Inset: pressure dependence of the glass transition temperature for prilocaine.
conductivity, the α-loss peak, and the excess wing all appear in the spectra, as well as the NCL at the highest applied pressure.
Table 1. Fit Parameters of Eq 6 to the NMR Relaxometry Data T [K] 210 223 231 243 261 269 297
CDD,1 [Hz2] 2.8 2.0 1.6 1.1 6.3 5.8 3.6
× × × × × × ×
5
10 105 105 105 104 104 104
(4.5%) (4.5%) (10%) (6%) (7%) (6%) (6%)
τc,1 [s] −6
4.6 × 10 (6.2%) 4.49 × 10−6 (6.0%) 4.45 × 10−6 (13%) 4.37 × 10−6 (8%) 4.14 × 10−6 (9%) 3.93 × 10−6 (7%) 3.73 × 10−6 (6%) 12703
τc,2 [s] −8
3.7 × 10 (11%) 3.5 × 10−8 (9%) 3.2 × 10−8 (8%) 2.9 × 10−8 (7%) 2.2 × 10−8 (7%) 1.9 × 10−8 (7%) 1.60 × 10−8 (6%)
A [Hz] 4.9 (1%) 3.1 (1%) 2.4 (1%) 1.6 (1%) 1.0 (1%) 0.81 (1.5%) 0.48 (1%) DOI: 10.1021/acs.jpcb.5b06426 J. Phys. Chem. B 2015, 119, 12699−12707
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and T is given in Figure 8. From the invariance of the frequency dispersion, it follows that the fwhm of the α-loss peak for any fα is the same whether obtained at ambient pressure or at elevated pressure, and hence the crossovers frequencies fα(PB) and fα(TB) have the same value.
The changes of all the processes on increasing pressure are similar to the effects of lowering temperature at ambient pressure as seen before in Figure 2. In the same manner as described before for the ambient pressure data, the fits of the spectra determine τα as a function of pressure P. The results are given in Figure 7b and the inset shows the change of Tg with pressure. The α-loss peak becomes progressively broader on increasing pressure, like the behavior found at ambient pressure on decreasing temperature. This property is shown in Figure 8a by comparing the frequency dispersion of the α-loss peak at several chosen pressure values.
4. DISCUSSIONS We have listed the apparently general dynamic properties of glass-formers in the Introduction and mentioned that not all of them can be found in the same material. Therefore, it is significant to find all the properties in the same glass-former and relate the characteristics of the properties to the molecular structure. The experimental data of prilocaine presented in the section 3 offer such an opportunity for us to consider all of them from (A) to (F), and to explain them on a theoretical basis. One part of the chemical structure of prilocaine is a phenyl ring with a methyl group resembling methylbenzene, i.e., toluene. The other part is the long and flexible side chain attached to the phenyl ring, which acts like a solvent to screen the strong intermolecular interaction between the rigid phenyl rings. Consequently, the intermolecular interaction in prilocaine is much weaker than toluene.42 Because the intermolecular potential determines the dynamics and thermodynamics of glass-formers, it is clear that the characteristics of the dynamic properties A−F of prilocaine are drastically different from toluene. In fact at temperatures near Tg the fwhm of the α-loss peak in toluene is 2.2 decades42 or n ≡ (1 − βKWW) = 0.48 whereas in prilocaine we have fwhm = 1.5 decade, and n = 0.20. This observed trend is in accord with simulations43 and theoretically with the coupling model (CM)23,24,26 result that a glass-former having a stronger intermolecular interaction has larger fwhm or n. In toluene, the JG β-relaxation is well resolved and widely separated from the α-relaxation in time scale, i.e., large value of log(τα/τ0) ≈ log(τα/τβ). In contrast, the JG β-relaxation in prilocaine is not resolved, but this is consistent with property C because of the small value of n = 0.2 in prilocaine. It is also consistent with the explanation from the CM via eq 3 because the primitive frequency f 0 calculated from it with n = 0.2 in prilocaine (i.e., property D) is too close to fα, as shown in Figures 7a and 8b. Although the JG β-relaxation in prilocaine is not resolved and instead appears in the spectrum as an excess wing, which basically fulfils property B, the relation of its relaxation frequency fβ(T,P) ≈ f 0(T,P) to fα(T,P) of the α-relaxation remains invariant to changing the combinations of P and T at constant fα(T,P). This can be seen from the data in Figure 8b, and it is an example of properties C and D as well as property A1 now also found in prilocaine. As emphasized before in previous studies,1,15 these properties have strong implications on the theoretical understanding of glass transition. Property A is telling us that, at a fixed value of fα(T,P), the frequency dispersion of the α-relaxation is determined by intermolecular interaction because it is invariant to variations of T and P and hence to variations of their conjugate variables, volume, and entropy. Properties B−D indicate the fundamental importance of the primitive relaxation/JG β-relaxation, because it transpires before the α-relaxation and from causality it originates the connection to the α-relaxation in the properties. We have seen already from Figures 3b, 4, and 8a that properties E and F are found in prilocaine. The correlation of the change in temperature or pressure dependence of τα with
Figure 8. (a) Master plot of prilocaine formed by horizontally shifting of spectra recorded at various pressure conditions to overlap that at 440 MPa. The inset presents the pressure evolution of the fwhm parameter. (b) Comparison of the dielectric loss spectra of prilocaine obtained for different temperature and pressure combinations with approximately the same structural relaxation time.
The fwhm of the α-loss peaks measured at all pressure values are presented in the inset of Figure 8a. Although there are some scatter in the values, the two red lines suggests crossover from temperature independent and low values of the fwhm for P below PB ∼ 180 MPa to larger and increasing values of the fwhm above PB ∼ 180 MPa. Going back to Figure 7a, the α-loss peak frequency at PB ∼ 180 MPa fα(PB), is ∼2.5 × 104 Hz. On the contrary, from the ambient pressure data in the upper panel of Figure 4, of value of τα(TB) at TB = 258 K is 1.29 × 10−6 s, and the corresponding value of the α-loss peak frequency at TB = 258 K, fα(TB), is ∼1.2 × 105 Hz. The difference between fα(PB) and fα(TB) by a factor of 4 is within the uncertainties in the determination of TB and PB. Hence it is possible that the frequencies fα(PB) and fα(TB) at the crossover of dynamic properties at PB and TB are actually the same, which was found before in another glass-former.41 This possibility is reaffirmed by the invariance of the frequency dispersion of the α-relaxation in prilocaine to variation of P and T at constant fα, i.e., property A in the Introduction. An example for two combinations of P 12704
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subsections 3a and 3d respectively. For example, βKWW = 0.85 at 261 K and βKWW = 0.75 at 223 K. Although the change of βKWW over these two widely separated temperatures is not so large, it is a property of prilocaine with narrow α-dispersion. In contrast, dielectric studies of other glass-formers having much broader α-dispersion have practically no change of the width of the dispersion. These glass-formers include o-terphenyl (OTP) and trinaphthal benzene (TNB) published by Richert and coworkers45,46 and sucrose benzoate (SB) by Quitevis and coworkers.47 However, over the wide temperature range of OTP, TNB, or SB where there is no change of the α-dispersion, observed is the spectacular breakdown of the Stokes−Einstein (SE) relation and the Debye−Stokes−Einstein (DSE) relation.48 The findings of the α-dispersions of OTP, TNB, and SB are temperature independent and have the impact of invalidating the explanation of the breakdown of the SE and the DSE relations by the spatial heterogeneities of the αrelaxation.49 On the contrary, the alternative explanation based on the coupling model50 remains valid irrespective of the α-dispersion is temperature dependent or not.51 From the broadening of the α-dispersion in prilocaine with decrease of temperature, and the same for the spatial heterogeneities, one may expect the breakdown of the SE and DSE relations. Notwithstanding the width of the α-dispersion (or n ≈ 0.25) of prilocaine is small, and according to the coupling model explanation,52 the prediction is that the degree of breakdown of the SE and DSE relations in prilocaine is minimal. This prediction for glass-formers with small n was verified by the data of silica by Zanotto and co-workers.53 Over the temperature range 1.1Tg < T < Tm, where Tg = 1451 K and Tm = 2007 K, of silica, the diffusion coefficient deduced from viscosity is nearly the same as the self-diffusion coefficient, and thus SE relaxation is obeyed by silica. From molecular dynamics simulations, silica has the small value of n ≈ 0.1.54
the corresponding changes of the width of the dispersion of the α-relaxation or n signals once more the inseparable connection between τα and n or the fwhm of the dispersion of the αrelaxation. Finally, to demonstrate property G in prilocaine, we show the presence of the nearly constant loss (NCL) in the spectra and its relation to the primitive relaxation in Figure 9. Only the
Figure 9. Open symbols: data obtained by shifting the α-loss peak recorded at temperature of 231 K (green closed circles) to lower frequencies to match the data at 221, 211, and 203 K. By this construction we have approximately the complete spectra including the α-loss peaks in the glassy state of prilocaine.
spectra at temperatures near Tg and below it are shown. The data are actually acquired by measurements at frequencies no lower than about log f = −1.5, and shown by solid symbols. The points represented by open symbols are obtained by shifting the α-loss peak at the highest temperature of 231 K (green closed circles) to lower frequencies to match the data at 221, 211, and 203 K. By this construction we have approximately the complete spectra including the α-loss peaks in the glassy state of prilocaine, as well as the value of fα at these three lower temperatures. Each arrow pointing downward at a spectrum indicates the value of log f 0 calculated by eq 3. The power law dependence, ε″(ν) = Bν−λ with λ ≈ 0.17, appearing at 203, 193, 173, and 153 K represents the NCL contribution from caged prilocaine molecules. By inspection of Figure 9, it can be seen that log f 0 is about 1−2 decades lower than the apparent end of the NCL power law regime. This is exactly property G found before in many different glass-formers.11,12,15,23,34,35,44 The caged dynamics regime is terminated by the onset of the primitive relaxation, which is the precursor of the JG βrelaxation. This is understandable because the primitive relaxation effectively moves the entire molecule out of the cages. However, intramolecular secondary relaxation involves motion of a part of the molecule and has the molecules remaining caged. The very fast γ-relaxation of prilocaine found at even higher frequencies than the NCL in the glassy state in Figure 8 is such an example to show the unimportance of the intramolecular secondary relaxation. Herein, we raise some issues that are worth addressing before closing the section. We have mentioned the fwhm of the dielectric α-dispersion broadens (or βKWW decreases) of prilocaine on lowering temperature or elevating pressure in
5. CONCLUSIONS Finding general and fundamental properties in glass-formers of different chemical bondings and physical structures is certainly the key to solve the glass transition problem worth attention from the research community. Notwithstanding, in all glassformer studies in the past no single glass-former exhibits all the properties due to various experimental difficulties or limitations. Multiple glass-formers have to be put together to support the claim that all these general properties exist. Although this should be sufficiently convincing, the drawback is the need for others to examine a tremendous amount of data from different glass-formers published at various places in the literature before they can be convinced. It would be ideal if all or essentially all of the general properties can be found in a single glass-former. If successful, the importance of these general properties would not escape the attention of all researchers. Together with the evidence collected in many other glass-formers, such results from a single glass-former cannot be ignored as the key to solve the glass transition problem. The data of prilocaine presented in this paper provide a realization of this ideal case and are worth notice.
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AUTHOR INFORMATION
Corresponding Author
*Z. Wojnarowska. E-mail:
[email protected]. 12705
DOI: 10.1021/acs.jpcb.5b06426 J. Phys. Chem. B 2015, 119, 12699−12707
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The Journal of Physical Chemistry B Notes
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The authors declare no competing financial interest.
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ACKNOWLEDGMENTS The authors Z.W. and M.P. are grateful for the financial support of the National Science Centre within the Maestro2 project (Grant No. DEC 2012/04/A/ST3/00337). The authors thank Malgorzata Florek - Wojciechowska for the NMR relaxation data.
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