Check of the temperature- and pressure-dependent Cohen–Grest equation

Temperature Dependence of Structural Relaxation in Glass-Forming Liquids and Polymers

Understanding the microscopic mechanism of the transition of glass remains one of the most challenging topics in Condensed Matter Physics. What controls the sharp slowing down of molecular motion upon approaching the glass transition temperature Tg, whether there is an underlying thermodynamic transition at some finite temperature below Tg, what the role of cooperativity and heterogeneity are, and many other questions continue to be topics of active discussions. This review focuses on the mechanisms that control the steepness of the temperature dependence of structural relaxation (fragility) in glass-forming liquids. We present a brief overview of the basic theoretical models and their experimental tests, analyzing their predictions for fragility and emphasizing the successes and failures of the models. Special attention is focused on the connection of fast dynamics on picosecond time scales to the behavior of structural relaxation on much longer time scales. A separate section discusses the specific case of polymeric glass-forming liquids, which usually have extremely high fragility. We emphasize the apparent difference between the glass transitions in polymers and small molecules. We also discuss the possible role of quantum effects in the glass transition of light molecules and highlight the recent discovery of the unusually low fragility of water. At the end, we formulate the major challenges and questions remaining in this field.

Second-order-derivative analysis of structural relaxation time in the elastic model of glass-forming liquids

Recently it was shown [V. N. Novikov and A. P. Sokolov, Phys. Rev. E 92, 062304 (2015)10.1103/PhysRevE.92.062304] that the second derivative with respect to inverse temperature of the structural relaxation time in some supercooled molecular liquids has a sharp maximum. It marks the point at which the apparent activation energy begins to saturate with decreasing temperature. The elastic model of glass-forming liquids expresses the temperature dependence of the structural relaxation time through that of the shear modulus. In this paper, we test whether this model is able to predict the maximum of the second derivative. We confirm its presence in the elastic model by analyzing the temperature dependence of the Brillouin light scattering in salol. This is a very subtle feature of the temperature dependence, which is greatly enhanced when taking derivatives. Its presence in the Brillouin data provides strong support to the elastic model of glass-forming liquids.

Qualitative change in structural dynamics of some glass-forming systems

Analysis of the temperature dependence of the structural relaxation time τ(α)(T) in supercooled liquids revealed a qualitatively distinct feature-a sharp, cusplike maximum in the second derivative of logτ(α)(T)at some T(max). It suggests that the super-Arrhenius temperature dependence of τ(α)(T) in glass-forming liquids eventually crosses over to an Arrhenius behavior at T<T(max), and there is no divergence of τ(α)(T) at nonzero T. T(max) can be above or below T(g), depending on the sensitivity of τ(T) to a change in the liquid's density quantified by the exponent γ in the scaling τ(α)(T)∼exp(A/Tρ(-γ)). These results might turn the discussion of the glass transition in a different direction-toward the origin of the limiting activation energy for structural relaxation at low T.

Dynamics and thermodynamics of polymer glasses

The fate of matter when decreasing the temperature at constant pressure is that of passing from gas to liquid and, subsequently, from liquid to crystal. However, a class of materials can exist in an amorphous phase below the melting temperature. On cooling such materials, a glass is formed; that is, a material with the rigidity of a solid but exhibiting no long-range order. The study of the thermodynamics and dynamics of glass-forming systems is the subject of continuous research. Within the wide variety of glass formers, an important sub-class is represented by glass forming polymers. The presence of chain connectivity and, in some cases, conformational disorder are unfavourable factors from the point of view of crystallization. Furthermore, many of them, such as amorphous thermoplastics, thermosets and rubbers, are widely employed in many applications. In this review, the peculiarities of the thermodynamics and dynamics of glass-forming polymers are discussed, with particular emphasis on those topics currently the subject of debate. In particular, the following aspects will be reviewed in the present work: (i) the connection between the pronounced slowing down of glassy dynamics on cooling towards the glass transition temperature (Tg) and the thermodynamics; and, (ii) the fate of the dynamics and thermodynamics below Tg. Both aspects are reviewed in light of the possible presence of a singularity at a finite temperature with diverging relaxation time and zero configurational entropy. In this context, the specificity of glass-forming polymers is emphasized.

Free volume from positron lifetime and pressure-volume-temperature experiments in relation to structural relaxation of van der Waals molecular glass-forming liquids

Positron annihilation lifetime spectroscopy (PALS) is employed to characterize the temperature dependence of the free volume in two van der Waals liquids: 1, 1'-bis(p-methoxyphenyl)cyclohexane (BMPC) and 1, 1'-di(4-methoxy-5-methylphenyl)cyclohexane (BMMPC). From the PALS spectra analysed with the routine LifeTime9.0, the size (volume) distribution of local free volumes (subnanometer size holes), its mean, [v(h)], and mean dispersion, σ(h), were calculated. A comparison with the macroscopic volume from pressure-volume-temperature (PV T) experiments delivered the hole density and the specific hole free volume and a complete characterization of the free volume microstructure in that sense. These data are used in correlation with structural (α) relaxation data from broad-band dielectric spectroscopy (BDS) in terms of the Cohen-Grest and Cohen-Turnbull free volume models. An extension of the latter model allows us to quantify deviations between experiments and theory and an attempt to systematize these in terms of T(g) or of the fragility. The experimental data for several fragile and less fragile glass formers are involved in the final discussion. It was concluded that, for large differences in the fragility of different glass formers, the positron lifetime mirrors clearly the different character of these materials. For small differences in the fragility, additional properties like the character of bonds and chemical structure of the material may affect size, distribution and thermal behaviour of the free volume.

Free volume of an oligomeric epoxy resin and its relation to structural relaxation: evidence from positron lifetime and pressure-volume-temperature experiments

From positron annihilation lifetime spectroscopy analyzed with the new routine LT9.0 and pressure-volume-temperature experiments analyzed employing the equation of state (EOS) Simha-Somcynsky lattice-hole theory (SS EOS) the microstructure of the free volume and its temperature dependence of an oligomeric epoxy resin (ER6, M(n) approximately 1750 g/mol , T(g)=332 K ) of diglycidyl ether of bisphenol-A (DGEBA) have been examined and characterized by the hole free-volume fraction h, the specific free and occupied volumes V(f)=hV and V(occ)=(1-h)V, and the size distribution (mean, <nu(h)>, and mean dispersion, sigma(h)) and the mean density N(h)'=V(f)/<nu(h)>, of subnanometer-size holes. The results are compared with those from a previous work [G. Dlubek, Phys. Rev. E 73, 031803 (2006)] on a monomeric liquid of the same resin (ER1, M(n) approximately 380 g/mol, T(g)=255 K ). In the glassy state ER6 shows the same hole sizes as ER1 but a higher V(f) and N(h)'. In the liquid V(f), <nu(h)>, dV(f)/dT, and dV(f)/dP are smaller for ER6. The reported dielectric alpha relaxation time tau shows certain deviations from the free-volume model which are larger for ER6 than for ER1. This behavior correlates with the SS EOS, which shows that the unit of the SS lattice is more heavy and bulky and therefore the chain is less flexible for ER6 than for ER1. The free-volume fraction h in the liquid can be described by the Schottky equation h proportional to exp(-H(h)/k(B)T) , where H(h)=7.8 - 6.4 kJ/mol is the vacancy formation enthalpy, which opens a different way for the extrapolation of the equilibrium part of the free volume. The extrapolated h decreases gradually below T(g) and becomes zero only when 0 K is reached. This behavior means that no singularity would appear in the relaxation time at temperatures above 0 K. To quantify the degree to which volume and thermal energy govern the structural dynamics, the ratio of the activation enthalpies E(i)=R[(d ln tau/dT(-1))]i, at constant volume V and constant pressure P(E(V)/E(P)), is frequently determined. We present arguments for necessity to substitute E(V) by E(Vf), the activation enthalpy at constant (hole) free volume, and show that E(Vf)/E(P) changes as expected: it increases with increasing free volume, i.e., with increasing temperature, decreasing pressure, and decreasing molecular weight. E(Vf)/E(P) exhibits smaller values than E(V)/E(P), which leads to the general inference that the free volume plays a larger role in dynamics than concluded from E(V)/E(P). The same conclusion is obtained when scaling tau to T(-1)V(f)(-gamma) instead of to T(-1)V(-gamma), where both gamma's are material constants.

Thermodynamic interpretation of the scaling of the dynamics of supercooled liquids

The recently discovered scaling law for the relaxation times, tau(T,upsilon) = I(Tupsilon(gamma)), where T is temperature and upsilon the specific volume, is derived by a revision of the entropy model of the glass transition dynamics originally proposed by Avramov [J. Non-Cryst. Solids 262, 258 (2000)]. In this modification the entropy is calculated by an alternative route. The resulting expression for the variation of the relaxation time with T and upsilon is shown to accurately fit experimental data for several glass-forming liquids and polymers over an extended range encompassing the dynamic crossover. From this analysis, which is valid for any model in which the relaxation time is a function of the entropy, we find that the scaling exponent gamma can be identified with the Gruneisen constant.

Free volume of an epoxy resin and its relation to structural relaxation: evidence from positron lifetime and pressure-volume-temperature experiments

The microstructure of the free volume and its temperature dependence in the epoxy resin diglycidyl ether of bisphenol-A (DGEBA) have been examined using positron annihilation lifetime spectroscopy (PALS, 80-350K, 10(-5) Pa) and pressure-volume-temperature (PVT, 293-470 K, 0.1-200MPa) experiments. Employing the Simha-Somcynsky lattice-hole theory (S-S eos), the excess (hole) free volume fraction h and the specific free and occupied volumes, Vf=hV and Vocc=(1-h)V, were estimated. From the PALS spectra analyzed with the new routine LT9.0 the hole size distribution, its mean, <Vh>, and mean dispersion, sigma h, were calculated. <Vh> varies from 35 130 A3. From a comparison of <Vh>with V and Vf, the specific hole number N'h was estimated to be independent of the temperature [Nh(300 K)=N'h/V=0.65 nm-3]. From comparison with reported dielectric and viscosity measurements, we found that the structural relaxation slows down faster than the shrinkage of the hole free volume Vf would predict on the basis of the free volume theory. Our results indicate that the structural relaxation in DGEBA operates via the free-volume mechanism only when liquidlike clusters of cells of the S-S lattice appear which contain a local free volume of approximately 1.5 or more empty S-S cells. The same conclusion follows from the pressure dependency of the structural relaxation and Vf. It is shown that PALS mirrors thermal volume fluctuations on a subnanometer scale via the dispersion in the ortho-positronium lifetimes. Using a fluctuation approach, the temperature dependency of the characteristic length of dynamic heterogeneity, xi, is estimated to vary from xi=1.9 nm at Tg to 1.0 nm at T/Tg>1.2. A model was proposed which relates the spatial structure of the free volume as concluded from PALS to the known mobility pattern of the dynamic glass transition at low (cooperative alpha-relaxation) and high (alpha-relaxation) temperatures. We discuss possible reasons for the differences between the results of our method and the conclusion from dynamic heat capacity.

Molecular dynamics study of the thermal and the density effects on the local and the large-scale motion of polymer melts: scaling properties and dielectric relaxation

Results from a molecular dynamics simulation of a melt of unentangled polymers are presented. The translational motion, the large-scale and the local reorientation processes of the chains, as well as their relations with the so-called "normal" and "segmental" dielectric relaxation modes are thoroughly investigated in wide temperature and pressure ranges. The thermodynamic states are well fitted by the phenomenological Tait equation of state. A global time-temperature-pressure superposition principle of both the translational and the rotational dynamics is evidenced. The scaling is more robust than the usual Rouse model. The latter provides insight but accurate comparison with the simulation calls for modifications to account for both the local chain stiffness and the nonexponential relaxation. The study addresses the issue whether the temperature or the density is a dominant control parameter of the dynamics or the two quantities give rise to comparable effects. By examining the ratio /alphatau//alphaP between the isochronic and isobaric expansivities, one finds that the temperature is dominant when the dynamics is fast. If the relaxation slows down, the fluctuations of the free volume increase their role and become comparable to those of the thermal energy. Detectable cross-correlation between the "normal-mode" and the "segmental" dielectric relaxations is found and contrasted with the usual assumption of independent modes.

Slow dynamics of salol: a pressure- and temperature-dependent light scattering study

We study the slow dynamics of salol by varying both temperature and pressure using photon correlation spectroscopy and pressure-volume-temperature measurements, and compare the behavior of the structural relaxation time with equations derived within the Adam-Gibbs entropy theory and the Cohen-Grest free volume theory. We find that pressure-dependent data are crucial to assess the validity of these model equations. Our analysis supports the entropy-based equation, and estimates the configurational entropy of salol at ambient pressure approximately 70% of the excess entropy. Finally, we investigate the evolution of the shape of the structural relaxation process, and find that a time-temperature-pressure superposition principle holds over the range investigated.

Viscosity at the dynamic crossover in o-terphenyl and salol under high pressure

The viscosities of two prototypical glass formers, o-terphenyl and phenyl salicylate (salol), are shown to exhibit a change in their temperature and pressure dependences at a constant value of the viscosity. This is the first evidence of a dynamic crossover in the viscosity induced by pressure. The characteristic value associated with the change in dynamics is material dependent, but independent of temperature and pressure. These results are in accord with the previous finding, for other glass formers, that the dielectric relaxation time assumes a density-independent value at the dynamic crossover.

Cohen-Grest model for the dynamics of supercooled liquids

Recent experiments have established that, at least for van der Waals glass formers, volume fluctuations contribute significantly to the slowing down of the dynamics near T(g). Accordingly, we use the Cohen-Grest (CG) free-volume model to analyze dielectric relaxation data for six van der Waals liquids. The CG equation accurately describes the structural relaxation times over broader ranges of temperature than the more common Vogel-Fulcher relation. Moreover, the CG equation requires two less adjustable parameters when the data span the Stickel temperature T(B) associated with a change in the dynamics. The characteristic temperature T0 of the CG model can be identified with T(B), suggesting that the crossover reflects onset of percolation of the free volume. The CG parameters used to fit the structural relaxation times allow the free volume per liquidlike molecule to be calculated. These results, however, are at odds with free-volume estimates extracted from pressure-volume-temperature data.

Effect of pressure on the dynamics of glass formers

A description of the pressure dependence of the structural relaxation time has been derived from the Adam-Gibbs theory by writing the configurational entropy in terms of the excess heat capacity and the molar thermal expansion. This new equation was tested successfully on dielectric relaxation data for an epoxy compound over a wide range of temperature and pressure.

Pressure dependence of structural relaxation time in terms of the Adam-Gibbs model

A new equation describing the behavior of the structural relaxation time, tau(T,P), as a function of both pressure and temperature, is discussed. This equation has been derived from the Adam-Gibbs theory by writing the configurational entropy, S(c), in terms of the excess thermal heat capacity and of the molar thermal expansion. Consequently, the parameters introduced in the expression are directly related to specific physical properties of the material, such as the thermal expansion coefficient alpha and the isothermal bulk modulus K0. At a fixed pressure, for low pressures, the found equation reduces to a Vogel-Fulcher-Tammann equation of tau versus temperature with the fragility parameter independent from pressure. The equation for tau(T,P) was successfully tested directly by fitting the dielectric relaxation time data for two isothermal and one isobaric measurements on diglycidyl ether of bisphenol-A, carried out in previous experiments. The parameters estimated by the best fit were in reasonable agreement with the values determined from the known physical properties of the material. Finally, the expression for the change versus pressure of the temperatures at which the same value of tau(max) is obtained (e.g., the change versus pressure of the glass transition temperature) agrees with several expressions previously proposed in the literature to provide a phenomenological description of the observed phenomena.