THE JOURNAL OF CHEMICAL PHYSICS 129, 184509 共2008兲
Universal critical-like scaling of dynamic properties in symmetry-selected glass formers Aleksandra Drozd-Rzoska, Sylwester J. Rzoska,a兲 and Marian Paluch Institute of Physics, Silesian University, ul. Uniwersytecka 4, 40-007 Katowice, Poland
共Received 31 March 2008; accepted 24 September 2008; published online 14 November 2008兲 Evidence for a possible general validity of the critical-like behavior of dielectric relaxation time or viscosity , ⬀ 共T − TC兲− with → 9 and TC ⬍ Tg on approaching glass temperature 共Tg兲 is shown. This universal behavior is found in various systems where the vitrification is dominated by a selected element of symmetry. The supporting evidence was obtained on the basis of the distortion-sensitive, derivative-based analysis of 共T兲 data for a rodlike liquid crystalline compound 共E7兲, orientationally disordered crystals 共plastic crystals兲, a colloidal nanofluid system, polymer melt 共polystyrene兲, oligomeric liquid 共EPON 828兲, and low molecular weight glass formers 共glycerol, threitol, sorbitol, and 1-propanol兲. Results presented explain the puzzling experimental artifacts supporting the dynamical scaling model 关R. H. Colby, Phys. Rev. E 61, 1783 共2000兲; B. M. Erwin, R. H. Colby, J. Non-Cryst. Solids 307–310, 225 共2002兲兴. It is suggested that spin-glass-like systems may be linked to the discussed pattern. © 2008 American Institute of Physics. 关DOI: 10.1063/1.3000626兴 I. INTRODUCTION
Amorphous materials have a wide range of applications ranging from food processing and pharmacy industries1 to material engineering2–4 and geophysics.2,5 However, despite decades of studies, there is still no coherent picture of phenomena underlying the vitrification process.6–8 The nonDebye distribution of the primary 共structural, ␣-兲 relaxation times 共共T兲兲 and their non-Arrhenius temperature evolution are considered as “universal” hallmarks of vitrification on approaching the glass temperature 共Tg兲.6–8 For describing the temperature evolution of dynamic properties the Vogel–Fulcher–Tammann9 共VFT兲 relation is most often used, namely,6–8
冉 冊
共T兲 = VFT exp 0
冉 冊
C D TT 0 = VFT exp , 0 T − T0 T − T0
共1兲
where T0 is the VFT singular temperature linked to the ideal glass temperature and DT denotes the fragility strength coefficient. The VFT equation portrays also temperature dependences of viscosity 共共T兲兲, the diffusion coefficient 共d共T兲兲, and electric conductivity 共共T兲兲. It is noteworthy that two dynamical domains associated with the crossover at the hypothetically universal time scale at 共TB兲 = 10−7⫾ s 共Refs. 10 and 11兲 and linked to different values of T0 and DT 共Ref. 12兲 have been found. The originally empirical VFT equation9 can be derived from some basic models for glass transition physics, such as the free volume approach6 or the Adam and Gibbs 共AG兲 model.6,13 One of the basic artifacts of the AG model is the concept of cooperatively rearranged regions 共CRRs兲. In the last decade such multibody structures, also known as dynamical heterogeneities, have become fundamental in a兲
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searching for origins of common patterns observed on vitrification.6,14,15 In the year 2000 Colby16 proposed the dynamical scaling model 共DSM兲, in which the concept of CRR was directly recalled. Its basic output was the critical-like relation for the evolution of dynamic properties, namely,16,17
共T兲 = 0
冉 冊 冉 冊 冉 冊 共T兲 0
z
= 0
T − TC TC
= 0
T − TC TC
−
−z
,
where TC ⬍ Tg . 共2兲
The DSM links evolutions of 共T兲 or 共T兲 on T → Tg to the CRR “pretransitional fluctuations.” They are associated with the “critical point” hidden below Tg and described by the correlation length 共T兲 = 0关共T − TC兲 / TC兴−, with = 3 / 2 for the universal “critical exponent” and z = 6 for the decay-time related dynamical exponent. This yielded the universal value of the power exponent = z = 9 for all supercooled liquids and polymers.16,17 The validity of Eq. 共2兲 was tested for polyvinyl-methylether and polymethyl-methacrylate, with the use of the linearized log10 versus log10共T − TC兲 plot, giving the values = 10.0, TC = Tg − 16 K, and = 8.5, TC = Tg − 9 K, respectively.16,17 Additionally, in Ref. 16 results of the analysis via Eq. 共2兲 for 33 glass formers 共Table I兲 were given. Finally, the general validity of such simple criticallike description for polymeric glass formers up to 100 K above Tg was suggested. For nonpolymeric glass formers, particularly low molecular weight ones, a variety of values of extending well above = 20 were reported.16,17 In this case, the general validity of Eq. 共2兲 supplemented by the activation term was postulated, namely,16–18
129, 184509-1
© 2008 American Institute of Physics
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J. Chem. Phys. 129, 184509 共2008兲
Drozd-Rzoska, Rzoska, and Paluch
共T兲 = 0
冉 冊 冉 冊 T − TC TC
−9
exp
Elow kT
for Tg ⬍ T ⬍ TA , 共3兲
where TA is the caging temperature for which an increasing number of molecules is transiently localized by their neighbors. For T ⬍ Tg the description of dynamic properties solely by the Arrhenius term 共T兲 ⬃ exp共Elow / kT兲 was stated. The temperature TA was estimated from the intersection of the high temperature extrapolation of the behavior given by Eq. 共2兲 or 共3兲 and the low temperature extrapolation of the Arrhenius behavior 共T兲 = exp共Ehigh / kT兲 assumed for the domain T Ⰷ Tg. For instance, in poly共vinyl acetate兲 the latter was claimed to be valid for T / Tg ⬍ 0.75 and the parametrization via Eq. 共2兲 for T / Tg ⬎ 0.95, yielding TA / Tg ⬇ 0.84.17 The temperature TA was estimated at approximately 20 K above the dynamic crossover temperature TB at which all molecules are hypothetically caged by their neighbors.17,18 The model by Colby and Erwin16–18 links glass transition to a virtual critical point and suggests that different low molecular weights and polymeric glass forming materials belong to the same universality class of dynamic critical phenomena.19 It is noteworthy that the critical-like behavior is well evidenced for the high temperature dynamical domain as predicted by the mode-coupling theory 共MCT兲, namely,6,7,20
共T兲 = MCT 0
冉
T − TCMCT TCMCT
冊
−⬘
for T ⬎ TCMCT + 20 K, 共4兲
where ⬍ 共TCMCT兲 = 10−7⫾1 s,10 TCMCT is associated with the ergodic-nonergodic crossover, and the power exponent 1.4⬍ ⬘ ⬍ 3.5 is the function of coefficients describing the form of ⬙共f兲 curves.6,7,20 This paper focuses on the possibility of the critical-like description for the low temperature dynamical domain, i.e., in the immediate vicinity of Tg共 ⬃ 100 s兲.6–8 Here the unique advantages of the DSM 共Refs. 16–18兲 are worth recalling: 共i兲 the clear link to dynamic critical phenomena, 共ii兲 the slowing down of dynamics on T → Tg related to a virtual quasicontinuous phase transition hidden slightly below Tg, 共iii兲 the simple and system-independent 共universal兲 parametrization of 共兲 or 共T兲, and 共iv兲 the direct link to the cooperative dynamics. Regarding the latter issue, it is theoretically advised that only experimental methods directly coupled to the four-point correlation function can yield a clear response from the dynamical heterogeneities in glass forming systems.15,21 This is the case of such methods as the photon correlation spectroscopy,22 4D-NMR,23 nonresonant dielectric hole burning,24 or nonlinear dielectric spectroscopy.3,5,25,26 However, the DSM suggests that there is a direct influence of dynamic heterogeneities on the structural relaxation time or viscosity, related to the two-point correlation function.21,27 In the opinion of the authors this behavior may be explained by the mean-field character of the DSM. In Refs. 16 and 17 the exponent = 8.5– 12 was reported for a set of glass forming polymers. Consequently, the general validity of the DSM and Eq. 共2兲 for such systems was suggested. However, the evidence collected in Ref. 17 also
contains ⬇ 21 for cis-polyisoprene or ⬇ 14.6 for atactic polypropylene, in clear disagreement with the DSM Eq. 共2兲. Additionally, in Refs. 16 and 17 the possibility of using a simple critical like Eq. 共2兲 for some low-molecular glass formers can be found, namely, for 1-propanol 共 ⬇ 11.9兲, selenium 共 ⬇ 11.5兲, or propylene carbonate 共 ⬇ 11.5兲. This occurs despite the suggestion that such systems have to obey the “activated” Eq. 共3兲.16 Worth recalling is also the case of a low-molecular liquid Aroclor for which the exponent ⬇ 48 was reported in Table I in Ref. 16. However, Casalini et al.28 presented indisputable evidence for the exponent = 11.84 and TC = 201 K in the range Tg共=246 K兲 ⬍ T ⬍ Tg + 60 K for Aroclor 1254 共pentachlorobiphenyl-PCB54兲. In Ref. 29 Richert carried out state-of-the-art residual and derivative-based analysis using high-resolution dielectric relaxation time data for supercooled poly共vinylacetate兲. He concluded:29 “…the VFT equation matches dielectric data in the ⬎ 10−7 s regime more closely than the scaling law,…, physical relevance of a divergence temperature 共TC共⬍Tg兲兲,…,is considered questionable.” It is noteworthy that basing on the same data Colby and Erwin16,17 suggested a clear validity of the critical-like description via Eq. 共2兲 with the exponent ⬇ 12. Probably puzzling artifacts collected above are the reason why the DSM was hardly recalled over the last years, despite its hypothetically impressive advantages. The present paper shows the evidence that such a description may be valid for a wide group of glass forming materials, independent of their microscopic features. The only prerequisite is that the vitrification should occur in a system dominated by a single element of symmetry. This behavior is shown for a liquid crystal, orientationally disordered crystals 共ODICs兲 共plastic crystals兲, colloidal nanofluids, and spin-glass-like systems. The discussion is supported by the distortion-sensitive derivative-based analysis.12 II. EXPERIMENTAL
Results presented in this paper are based on the authors’ measurements of the dielectric relaxation time in a liquid crystalline 共LC兲 compound, an ODIC glass former, as well as in oligomeric, polymeric, and low-molecular weight glass formers. These studies were conducted in 共1兲 LC eutectic mixture E7, 共2兲 epoxy resin EPON 828 共DGEBA兲, and 共3兲 low-molecular compounds such as glycerol, threitol, and sorbitol and 1-propanol. These results were supplemented by relaxation time data taken from external references, namely, 共i兲 ethanol in the supercooled liquid and plastic phases,30 共ii兲 polystyrene,31 共iii兲 colloidal fluid–isostearate nanoparticle salt,32 共iv兲 ferrofluid containing 5% of Fe1−xCx 共x = 0.2– 0.3兲 spherical nanoparticles 共d = 5.3 nm, Tg = 36 K兲,33 and 共v兲 Fe30Ag40W30 mechanically alloyed nanogranular system 共Tg = 19.7 K兲.34 Temperature dependences of 共T兲 were obtained using the Novocontrol BDS 80 impedance analyzer, equipped with the Quattro temperature unit. In the case of glycerol 关HOCH2共CH共OH兲兲CH2OH; propane-1,2,3-triol兴, threitol 关HOCH2共CH共OH兲兲2CH2OH, 共2R , 3R兲-1,2,3,4-butanetetriol兴, and sorbitol
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Universal critical-like scaling of dynamic properties
关HOCH2共CH共OH兲兲4CH2OH; 共2R,3R,4R,5S兲-hexanel,2,3,4,5,6-hexol兴 experimental 共T兲 data were recalled from the authors’ earlier studies.35 Samples of E7 were purchased from Merck and EPON 828 from Fluka. All liquids were carefully degassed prior to measurements. E7 is an eutectic LC mixture containing 51% if 4-pentyl-4-cyanobiphenyl 共5CB兲, 25% of 4-n-heptyl-4⬘-cyanobiphenyl 共7CB兲, 16% of 4-n-octyloxy-4⬘ cyanobiphenyl 共80CB兲, and 8% of 4-n-pentyl-4⬘-cyanoterphenyl 共5CT兲.36 All components of E7 are rodlike LC compounds. E7 exhibits the isotropic-nematic 共I-N兲 phase transition at TI-N ⬇ 61 ° C. It can easily form a glassy nematic below Tg ⬇ −62 ° C.36 EPON 828 共DGEBA; C18H21ClO3; diglycyl ether of bisphenol A兲 is an epoxy resin, easily vitrifying at any cooling rate. Its dynamics was tested in details earlier.37 1-propanol 关propan-1-ol; C3H8O兴 most often crystallizes on cooling. However, this may be avoided for a carefully degassed sample and fast cooling well below the melting temperature. III. THE DERIVATIVE-BASED ANALYSIS
Results presented below employ the linearized derivative-based transformation of 共T兲 data, recently proposed in Ref. 12. It was shown that for the plot 关d ln / d共1 / T兲兴−1/2 versus 1 / T the domain of the validity of the VFT Eq. 共1兲 is manifested by a linear dependence, namely,12
冋 册 冋 册 d ln d共1/T兲
Ha共T兲 R
−1/2
=
−
−1/2
= 共Ha⬘兲−1/2 = 共DTT0兲−1/2
T0共DTT0兲−1/2 = A − B ⫻ 共1/T兲, T
共5兲
where R is the gas constant and Ha共T兲 is the apparent activation enthalpy. This analysis recalled the “Stickel plot:” d log10 / d共1 / T兲 versus T or 1 / T,38,39 which was introduced to estimate the dynamic crossover temperature TB in supercooled liquids and polymers.7,10,11 However, Eq. 共5兲 links results of the derivative-based transformation to the apparent activation enthalpy Ha共T兲. It also enables the calculation of the optimal set of parameters: DT = 1 / AB and T0 = B / A. Consequently, the final fit can be solely reduced to 0 prefactor.12,40 A similar reasoning can be proposed to identify the domain of the validity of a critical-like description 关Eqs. 共2兲 and 共4兲兴, namely,12,40 T2 T2 TC 1 = = − T = A − BT. d ln /d共1/T兲 Ha⬘
共6兲
Originally, the analysis via Eq. 共6兲 was introduced for testing the validity of the MCT critical-like description 关Eq. 共4兲兴.12 However, the plot T2 / 关d ln / d共1 / T兲兴 versus 1 / T should yield a linear dependence in the domain of the validity of both Eqs. 共2兲 and 共4兲. The subsequent linear regression analysis can yield optimal values of power exponents ⬘ , = 1 / B and singular temperatures TCMCT共⬃ ⬇ 10−7 s兲, TC共⬃ Ⰷ 100 s兲 = A / B. Singular 共“critical”兲 temperatures can also be determined from the plot via the condition T2 / Ha⬘ = 0.12,40
In Ref. 12 the application of a similar derivative-based analysis for the isothermal, pressure paths was also discussed. IV. RESULTS AND DISCUSSION
The existence of a critical point underlying the vitrification process is considered within the AG model,6,13 the “entropic droplets” model by Kirkpatrick, the Thirumalai and Wolynes model,14 the spin-glass analogy models,6 the general model for liquid-liquid transitions by Tanaka,41 as well as the DSM.16–18 The unique feature of the latter is a simple critical-like relation for portraying 共T兲 or 共T兲 behavior on directly recalling the dynamic critical T → T g, phenomena.16,19 However, the superiority of the DSM critical-like description over the VFT parameterization in polymers and low-molecular liquids, recommended in Refs. 16–18, seems to be questionable 共see Introduction兲. In this section the evolution of relaxation times in materials belonging to different classes of glass formers not tested so far in respect to the application of the critical-like Eq. 共2兲 is presented. Studies of the validity of such a description are supported by the distortion-sensitive analysis recalled above. A. Liquid crystals
For LC compounds a set of liquid mesophases between the isotropic liquid and the solid crystal phase exists. This is associated with a gradual freezing of selected elements of symmetry.19,36 The isotropic phase does not exhibit a long range ordering but a local orientational ordering associated with pretransitional phenomena. This can be detected even 100 K above the weakly discontinuous I-N phase transition.40 The nematic mesophase is endowed with a longrange orientational ordering. Moreover, mesophases with additional limited translational ordering may also appear.19,36 It is noteworthy that rodlike liquid crystals have been a subject of long standing immense attention in condensed matter physics.36 Despite these efforts, only recently a complex dynamics, in some aspects similar to that of supercooled molecular liquids, was found.2,42–46 They probably started from Letz et al.43,44 who noted that for a model fluid of hard ellipsoids of revolution, a decoupling of orientational and translational degrees of freedom can occur. This has led to the unique appearance of two ergodic-nonergodic critical crossover temperatures 共TCMCT兲. The upper one coincided with the orientational freezing at the I-N transition. The lower one was linked to the translational caging and was located well below TI-N.43,44 These predictions were confirmed in BDS 共Refs. 2, 40, and 42兲 and optical heterodyne detected optical Kerr effect45,46 studies for the isotropic phase of rodlike nematic liquid crystals 共NLCs兲, including 5CB and 80CB. However, these studies were related to the isotropic phase, terminating at the I-N transition. Hence, the question for dynamics on approaching the glass transition arises. Regretfully, such studies are scarce. This is probably due to the limited number of LC compounds which can vitrify on “normal” cooling.2,36
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184509-4
Drozd-Rzoska, Rzoska, and Paluch
FIG. 1. The evolution of dielectric relaxation times, for the ␦-mode 共circles兲 and the tumbling mode 共stars兲, in E7 mixture of rodlike NLC. The inset shows the derivative-based analysis 关Eq. 共5兲兴 focused on the validity of the VFT Eq. 共1兲. The solid curves are linked to the critical-like parametrization 关Eq. 共2兲兴, which results are shown in Fig. 2. The dashed curve describing data in the isotropic phase is for the MCT Eq. 共4兲.
Below novel results on the temperature evolution of the primary relaxation time in LC mixture E7 which vitrifies in the nematic phase36,47 are presented. Generally, relaxation modes detected in BDS studies on rodlike NLC are associated with the rotational fluctuations around the short axis of the molecule 共␦-relaxation兲 and the tumbling related to the long axis of the molecule. The ␦-relaxation occurs at much lower frequencies than the tumbling mode process. However, in the nematic phase both modes can be detected in the experimentally convenient range of frequencies f ⬍ 1 GHz.36,47 The separation of these modes becomes clear on cooling toward Tg.47 These facts attracted the attention of researchers to the LC mixture E7, strengthen by its significant practical applications 共Refs. 36 and 47 and refs therein兲. Recently, Brás et al.47 carried out the analysis of the temperature evolution of the peak frequency of dielectric loss curves in E7. The possibility of the VFT parametrization was shown with DT = 5.3 and T0 = 184.9 K in the isotropic phase as well as for the nematic phase: DT = 8.2 and T0 = 161.0 K 共the ␦-mode兲 and DT = 2.7 and T0 = 190.1 K 共the tumbling mode兲.47 Figure 1 presents the temperature evolution of the main relaxation processes in E7 obtained from the authors’ measurements. Results of the derivative-based analysis via Eq. 共5兲 shown in the inset confirm the possibility of the VFT parametrization already indicated in Ref. 47. However, the obtained plot can also suggest that a sequence of VFT equations may offer a better description than a single VFT relation suggested in Ref. 47. It is worth recalling that on the Stickel plot38,39 or the equivalent 共Ha⬘兲−1/2 versus 1 / T plot12,40 the VFT behavior is manifested by a sloping straight line. A nonsloping line indicates the Arrhenius behavior.7,11,12,28,39 The decrease in the slope is linked to the increase in DT as well as the decrease in T0 values.12 One can conclude that in “typical” supercooled liquids the value of DT always increases on passing from the high temperature to the low temperature dynamical domain, with the crossover at TB.12 Re-
J. Chem. Phys. 129, 184509 共2008兲
FIG. 2. Results of the derivative-based analysis focused on the validity of the critical-like parametrization of 共T兲 in E7. Values of the exponents and as well as singular temperatures TCMCT and TC were obtained by the linear regression analysis via Eq. 共6兲. H⬘a = d ln / d共1 / T兲 is related to the apparent activation enthalpy. The main part of the figure is for the ␦-mode and the inset for the tumbling mode. Note that Tg = 211 K in E7 共Refs. 36 and 47兲.
sults presented in the inset in Fig. 1 show that in E7 the coefficient DT exhibits an anomalous decrease on T → Tg. A similar behavior was observed earlier in the isotropic phase of 5CB,40 on approaching the weakly discontinuous I-N transition. This was associated with the influence of pretransitional fluctuations.40 Consequently, for E7 the question arises whether below Tg a virtual nematic-smectic A transition, damped due to the earlier vitrification, can be considered. Results presented in Figs. 1 and 2 show that the evolution of 共T兲 in E7 can be well portrayed by the critical-like Eq. 共2兲 on T → Tg and by the MCT critical-like Eq. 共4兲 in the high temperature isotropic phase. It is noteworthy that for the isotropic phase of E7 TC共MCT兲 ⬇ TI-N − 60 K, whereas for 5CB and 80CB TC共MCT兲 ⬇ TI-N − 30 K.2,40 However, the value of TCMCT ⬇ 271 K is approximately the same for 5CB,40 80CB,2 and E7. For the low-temperature domain the distortion-sensitive analysis 共Fig. 2兲 indicated the preference for the critical-like description with the exponent ⬇ 8.7 and the singular temperature TC ⬇ Tg − 10 K for the domain Tg ⬍ T ⬍ Tg + 50 K, in fair agreement with the basic DSM prediction 关Eq. 共2兲兴.16,17 In studies on E7 carried out so far the VFT description with a different set of parameters for the ␦-mode and the tumbling mode was postulated 共Ref. 47 and references therein兲. Hence, the question concerning the relationship between the “single” ␣-relaxation observed in “classical” 共nonmesogenic兲 glass formers and the relaxation modes observed in the supercooled nematic LC appeared. On the basis of the BDS and the specific heat spectroscopy studies it was concluded in Ref. 47 that: “the tumbling mode has to be linked to the relaxation which is responsible for glassy dynamics in the nematic liquid crystal E7.” Results presented in Figs. 1 and 2 show the possibility of the critical-like description 关Eq. 共2兲兴 on T → Tg with approximately the same values of and TC for both relaxation modes in E7. In the opinion of the authors this may suggest that both the tumbling mode and the ␦-relaxation may be associated with the ␣-relaxation.
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184509-5
Universal critical-like scaling of dynamic properties
FIG. 3. Results of the derivative-based analysis 关Eq. 共6兲兴 in ethanol focused on the critical-like parametrization 关Eqs. 共2兲 and 共4兲兴 for ␣-relaxation processes. The inset presents such behavior for the liquid phase in the extended range of temperatures. The analysis employs 共T兲 data from Ref. 30.
B. ODICs
This section presents the temperature behavior of dielectric relaxation times in solid, ODICs 共plastic crystals兲. They are molecular materials for which reorientational motions of the molecules in the solid crystalline state can occur. This is due to the fact that molecules forming such materials provide a weak steric hindrance for reorientation. For plastic crystals the molecular centers of mass display long-range translational order, however, the molecules are dynamically disordered in orientation 共Refs. 48 and 49 and references therein兲. First of all, we recall the results of our 共T兲 evolution in earlier studies on ODIC 关共CH3兲3C共CH2OH兲兴0.70关共CH3兲2C共CH2OH兲2兴0.30, which do not exhibit phase transitions on cooling towards Tg under atmospheric pressure.48,49 The clear preference for the critical-like parametrization described by ⬇ 9 and TC ⬇ Tg − 10 K in the domain Tg ⬍ T ⬍ Tg + 100 K + TB ⬇ T for 共共TB兲 = 1.02⫻ 10−7 s兲 was noted.48 Ethanol is a unique material which can vitrify in the supercooled liquid or plastic crystal states. In Ref. 30 the evolution of relaxation times in both phases of ethanol, including the non-Arrhenius dependence of the secondary -relaxation, were discussed. In each case the possibility of VFT parametrization was shown. Figure 3 employs data taken from Ref. 30 for the derivative-based analysis focused on the validity of the critical-like parametrization 关Eq. 共2兲兴. The obtained results indicate the possibility of such description with a similar set of parameters = 11– 13 and TC ⬇ 86 K ⬇ Tg − 12 K on T → Tg both in the supercooled and in the plastic phases. The inset in Fig. 3 shows results of the derivative-based analysis in the extended range of temperatures, giving evidence for an additional critical-like domain associated with ⬘ ⬇ 4.1, TC⬘ ⬇ Tg + 40 K for 125⬍ T ⬍ 250 K. Figure 4 shows that a critical-like description may be used also for the -relaxation process within the experimental error. In the opinion of the authors the similarity in the super-Arrhenius behavior near Tg for ␣- and -relaxations
J. Chem. Phys. 129, 184509 共2008兲
FIG. 4. Results of the derivative-based analysis 关Eq. 共6兲兴 focused on the validity of Eq. 共2兲 for -relaxation in ethanol. The analysis employs data from Ref. 30. H⬘a = d ln  / d共1 / T兲 is related to the apparent activation enthalpy.
may be considered as the consequence of the coupling model by Ngai,50 where the secondary relaxation is derived from the scaling of the primary relaxation. C. Low-molecular weight liquids
Low-molecular weight supercooled liquids and polymers are probably the most tested group of glass forming materials. In Figs. 5 and 6 the evolution of ␣-relaxation time is discussed for the homologous series of secondary alcohols: glycerol 共Tg ⬇ 188 K兲, threitol 共Tg ⬇ 225 K兲, and sorbitol 共Tg ⬇ 268 K兲.6,7,35 Figure 5 presents their structures and results of the derivative-based analysis focused on critical-like description. Experimental data were taken from the authors’ earlier studies where the parametrization via the VFT 关Eq. 共1兲兴 was only discussed.35 It is clearly visible that in glycerol for T ⬍ 300 K the simple critical-like description is improper. One can only consider an apparent power exponent which
FIG. 5. The linearized and derivative-based analysis 关Eq. 共6兲兴 focused on the validity of the critical-like description 关Eq. 共2兲兴 for the homologous series of secondary alcohols: glycerol, threitol, and sorbitol, which structures are given in the figure. Note the appearance of the critical-like description with the power exponent → 9 in threitol and sorbitol.
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184509-6
Drozd-Rzoska, Rzoska, and Paluch
FIG. 6. The evolution of primary relaxation time in the homologous series of alcohols: glycerol, threitol, and sorbitol. The solid curves are related to the critical-like Eq. 共2兲, with parameters obtained via the derivative-based transformation 关Eq. 共6兲兴, which results are shown in Fig. 5. For glycerol the MCT parametrization 关Eq. 共4兲兴 in the high temperature domain is also shown via the dashed curve. The inset presents results of the linearized derived-based analysis for glycerol via Eq. 共7兲, focused on the validity of the “activated” critical-like DSM Eq. 共3兲.
values → 25 on T → Tg. However, the critical-like behavior clearly manifests for the high temperatures domain, T ⬎ 295 K. The obtained values TCMCT ⬇ 255 K ⫾ 5 and ⬘ = 3.5⫾ 2 may be linked to the MCT predictions 关Eq. 共4兲兴. They are in fair agreement with results of earlier analysis based on the linearized plot 1/⬘ versus T.6,7 However, for threitol and sorbitol the derivative-based analysis in Fig. 6 revealed the emergence of a clear critical-like description on T → Tg, absent for glycerol. The value of the power exponent → 9 as well as the range of validity of Eq. 共2兲 rises, if the length of the molecule increases. We link this behavior to the emergence of an uniaxial symmetry of molecules in the homologous series. In Fig. 3 the critical-like behavior with the exponent ⬇ 12.5 for the supercooled ethanol was shown. Basing on results presented in Figs. 5 and 6, one may expect that for monohydric alcohols with a better uniaxial symmetry than for ethanol the power exponent may shift toward ⬇ 9. This expectation is clearly confirmed by the results for supercooled 1-propanol which are presented in Fig. 7.
J. Chem. Phys. 129, 184509 共2008兲
FIG. 7. The critical-like parametrization of ␣-relaxation time in supercooled 1-propanol on T → Tg, based on values of the critical temperature and the power exponent from the distortion-sensitive analysis which results are shown in the inset. The BDS based estimation glass temperature 共Tg = 105 K兲 = 100 s is in fair agreement with earlier results 共Ref. 6兲.
tray 共T兲 data on T → Tg. For T ⬍ Tg + 80 K a continuous increase in the power exponent up to ⬇ 20 occurs, as shown in the inset in Fig. 9. It is worth recalling that Erwin16–18 postulated an empirical “activated-critical-like” Eq. 共3兲 for portraying 共T兲 behavior in low-molecular glass formers where the application of a simple critical-like Eq. 共2兲 was not possible. However, we noted significant problems in the practical usage of Eq. 共3兲, particularly regarding the estimation of the optimal set of parameters and the range of its validity. Hence, we propose the following preliminary derivative-based analysis of data: T2 TC 1 T2 = = − T = A − BT. d共ln − C兲/d共1/T兲 Ha⬘
共7兲
The plot based on Eq. 共7兲 should yield a linear dependence for the optimal selection of the empirical value of coefficient
D. Oligomeric and polymeric glass formers
Figure 8 presents the evolution of the primary relaxation time for epoxy resin EPON 828 共Tg = 255 K兲, an oligomeric glass former.25,37 Results of the distortion-sensitive analysis focusing on the validity of the critical-like description are given in Fig. 9. The linear dependence is clearly visible only for T ⬎ 280 K and can be associated with the MCT criticallike behavior. Values of the parameters obtained in this hightemperature domain agree with the results reported in Ref. 37 which employed on 1/⬘ versus T plot. For EPON 828 the critical-like description cannot por-
FIG. 8. Temperature evolution of the structural relaxation time in epoxy resin EPON 828. The inset shows results of the derivative-based analysis focused on the activated critical-like DSM description 关Eq. 共7兲兴. The solid curve portraying 共T兲 data for T → Tg is linked to Eq. 共3兲, employing parameters given in the inset. The dashed curve is associated with the MCT Eq. 共4兲, using parameters estimated in Fig. 9.
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FIG. 9. Results of the derivative-based analysis 关Eq. 共6兲兴 in EPON 828 focused on the validity of the critical-like description 关Eqs. 共2兲 and 共4兲兴. The inset shows the evolution of the “apparent” power exponent in Eq. 共2兲, on cooling toward the glass transition.
C = Ea / k, where the activation energy Ea can be related to Elow in Eq. 共3兲. Insets in Figs. 6 and 8 present the results of the derivative analysis of 共T兲 via Eq. 共7兲 for EPON 828 and glycerol, respectively. The main parts of Figs. 6 and 8 show results of 共T兲 parametrization via Eq. 共3兲, using parameters estimated via the mentioned procedure for glycerol and EPON 828, respectively. They confirm the supporting ability of Eq. 共7兲 for the optimal analysis of experimental 共T兲 via Eq. 共3兲. Figure 10 presents the possibility of the critical-like parametrization of 共T兲 on T → Tg for polystyrene, a polymeric glass former. The analysis is based on experimental data taken from Ref. 31, where solely the VFT parametrization of the peak frequency of dielectric loss curves 共f P兲 was discussed. A superior simple critical-like parametrization 关Eq. 共2兲兴 on T → Tg is visible both for the low-molecular weight and for the high-molecular weight polystyrene.
FIG. 11. Results of the derivative-based analysis 关Eq. 共6兲兴 focused on the validity of Eq. 共2兲 for the nanostearate based colloidal nanofluid on approaching the glass transition. The analysis employs 共T兲 data from Ref. 32.
E. Colloidal nanofluid
Colloidal systems can serve as a valuable model for studying nano- and microscale parallels of atomic and molecular systems.51 They also attract the attention due to a redundant degree of freedom and dispersity. The link to a hard spheres fluids causes that colloidal systems are also considered as a valuable experimental model system for studying glass transition phenomena. Worth recalling here are studies showing the validity of the idealized MCT scenario.6 In Ref. 32 the dynamics of relaxation times for a solvent-free nanoparticle colloidal fluid consisting of hard silica nanoparticle cores surrounded by a corona of flexible and charged chains was studied. The inset in Fig. 11 presents the evolution of the slow, structural relaxation process in the isostearate nanoparticle fluid, for which the VFT parametrization with DT ⬇ 9, T0 ⬇ 149 K, and Tg ⬇ 206 K was reported in Ref. 32. The derivative-based plot in the main part of Fig. 9 shows that the same 共T兲 data can be also described by the ⬇ 8.8 and critical-like Eq. 共2兲, with TC共⬇186 K兲 = Tg − 10 K. The solid curve portraying 共T兲 data in the inset is based on these parameters. The obtained critical-like description extends up to T共 ⬇ 4 ⫻ 10−6 s兲 ⬇ Tg + 70 K. F. Spin-glass-like systems
Spin glasses are considered as one of the basic glass forming model systems.6 One of basic empirical artifacts for experimental spin-glass-like systems is the critical-like dependence of dynamic properties, including the relaxation time,6,33,34,52,53
共T兲 = 0共T − Tg兲− and FIG. 10. Results of the derivative-based analysis 关Eq. 共6兲兴 of dielectric relaxation times in vitrifying polystyrene 共PS兲 with different molecular weights. It is based on experimental data from Ref. 31 which are recalled in the insert. The solid curve in the inset is for the parametrization via the critical-like Eq. 共2兲, which parameters given in the main part of the figure.
= 10 – 12.
共8兲
The appearance of such critical slowing down is often considered as a hallmark indicating the glassy phase in spin-
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FIG. 12. Results of the derivative-based analysis focused on the critical-like description 关Eq. 共6兲兴 for the magnetic relaxation time evolution in a nanoparticle, weakly interacting, ferrrofluid liquid, with 5% solution of single-domain particles of the amorphous alloy Fe1−xCx 共x = 0.2− 0.3兲. In Ref. 33 Tg共=TC兲 = 36 K and = 10.3 were reported. The lower inset shows the alternative possibility of parametrization via the VFT relation 关Eqs. 共1兲 and 共5兲兴, omitted in Ref. 33. The upper inset presents experimental 共T兲 data taken from Ref. 40 portrayed by the critical-like Eq. 共2兲, with parameters obtained in the main part of the plot. In Ref. 33 TC = Tg = 36 K and = z = 10.2⫾ 0.9 were reported from the direct fit via the critical-like equation.
glass-like systems. It is also used as a tool for estimating the glass temperature.33,34,52,53 In this section we discuss 共T兲 evolution in two spin-glass-like systems, namely, the nanoparticle ferrofluid33 and the solid nanogranular system.34 For these systems the relaxation time is related to magnetic properties. Experimental data are shown in the upper insets in Figs. 12 and 13.33,34 Solid curves portraying 共T兲 data were plotted on the basis of the distortion-sensitive analysis of data given in the main parts of Figs. 12 and 13. The exponents → 9, slightly lower in comparison with the direct fit of 共T兲 data in Refs. 33 and 34, were obtained. Results of the derivative-based analysis shown in lower insets in Figs. 12 and 13 indicate the possibility of an alternative parametrization via the VFT equation, not considered in Refs. 33 and 34.
FIG. 13. The analysis of the magnetic relaxation time evolution in the mechanically alloyed Fe30Ag40W30 nanogranular system, exhibiting a low temperature spin-glass-like phase. The upper inset shows data from Ref. 34, with the critical-like parametrization 关Eq. 共2兲兴 based on parameters determined from the derivative-based analysis in the main part of the plot which employs Eq. 共6兲. The lower inset shows results of the derivative-based analysis focusing on the VFT description 关Eqs. 共1兲 and 共5兲兴. In Ref. 34 TC = Tg = 19.7 K and = z ⬇ 10.3 were reported from the direct fit via criticallike equation.
V. CONCLUSIONS
This paper discusses the critical-like parametrization in glass forming systems for the low-temperature dynamical domain on T → Tg. In the Introduction such behavior was recalled in respect to the DSM.16–18 However, a similar description , ⬀ 共T − Tg⬘兲−r was also postulated earlier by Hill et al.54,55 and Murthy.56,57 The temperature Tg⬘ was called as the “glass temperature” at which = ⬁.57 The empirical coefficient exponent r ranged from 6 to 27 in different materials.54–57 In Refs. 54–57 the validity of such a description with r ⬇ 9.7 and Tg⬘ = Tg − 15 K was also suggested for glycerol. The analysis was based on the linearized log-log plot for eight experimental points in the temperature range Tg ⬍ T ⬍ Tg + 60 K.54 In Ref. 16 a similar analysis yielded r = ⬇ 13.5 and Tg⬘ ⬇ Tg − 10 K for glycerol. However, the distortion-sensitive analysis presented in this paper shows a clear inadequacy of such a description for glycerol. Pathmanathan and Johari58 compared the critical-like description postulated in Refs. 54 and 55 and the VFT parametrizations. They concluded that:58 “…a preference for the power law description and the theoretical implication of its use remain unjustified,” because of the inadequacy of the critical-like description for portraying experimental data for T ⬎ 2Tg⬘. However, the range of temperatures T ⬎ 2TC 共TC ⬍ Tg兲 is well above the domain of the validity of the DSM criticallike description. Moreover, the range T ⬎ 2TCMCT 共TCMCT Ⰷ Tg兲 is practically beyond the reach of high temperature investigations. Results presented in this paper show that a simple critical-like and approximately universal description of dynamic properties seems to be possible for a variety of glass formers, ranging from low molecular weight liquids and polymers to liquid crystals, plastic crystal, and colloidal fluids. Such behavior can be expected in vitrifying systems dominated by a single selected element of symmetry. The value of the power exponent → 9, i.e., it approaches the universal value advised by the DSM, when the presence of the dominating symmetry becomes more visible. This simple
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critical-like behavior seems to be insensitive to some basic microscopic features of the given material, such as intermolecular interactions. For instance, it occurs in “uniaxial” alcohols 共threitol, sorbitol, 1-propanol兲 with the hydrogen bonding, in a rodlike LC material 共E7兲 or polymer 共polystyrene兲 where the van der Waals interactions are important and in the colloidal nanofluid where the Coulombic interactions are significant. We would like to stress than in each case the detection of the universal critical-like behavior was supported by the distortion-sensitive analysis,12,40 also enabling a reliable estimation of the range of such description. The universal critical-like description was not possible for low-molecular weight glycerol and oligomeric EPON 828 due to the lack of the uniaxial symmetry. For these materials the derivative-based analysis yielding an optimal set of parameters for the “activated, critical-like” Eq. 共3兲 was proposed. The dynamics of spin-glass-like systems where the critical-like Eq. 共2兲, with TC ⬇ Tg, is a well known empirical artifact,52,53 was also discussed. The application of the derivative-based analysis 关Eq. 共6兲兴 for a ferromagnetic nanofluid and a nanogranular magnetic system, confirmed the validity of such a description with exponents = 9.3⫾ 1 and = 9.5⫾ 1, close to the DSM prediction. In this case, the relaxation of the magnetic moments located within the threedimensional network is limited to the one-dimensional arrangement 共ordering兲 “up and down.” The question arises whether such a restriction can be the reason for the shifting of TC toward Tg? It is noteworthy that a similar critical-like behavior of dynamic properties is also reported for vortex glasses in superconducting materials.59,60 Hence, the next question appears if the emerging link between the uniaxial symmetry and the “critical-like” vitrification may also be of importance in this case? The link between the glass transition and the hypothetical critical point hidden in the glass domain is considered by a set of “glass transition” models. However, the experimental evidence indicating for the presence of such a critical point was weak. In this paper the emergence of the universal critical-like scaling with the same power exponent → 9 and the singular temperature TC close to Tg for representatives of various groups of glass forming systems was shown. The results presented can explain the puzzling artifacts associated with the evidence related to the DSM16–18 mentioned above. They also may be of importance for practical applications since the link between the universal behavior and symmetry can facilitate predictions of the evolution of dynamic properties near glass temperature. ACKNOWLEDGMENTS
This research was carried out with the support of the CLG NATO Grant No. CBP.NUKR.CLG 982312 and of the Ministry of Science and Higher Education 共Poland兲 Grant No. N202 147 32/4240. 1
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