On the estimation of the Kauzmann temperature from relaxation data

The well known WLF equation describing the relaxation behaviour of glass-forming liquids near the glass transition temperature has been rederived on the basis of Adam and Gibbs' excess entropy model, making use of a novel expression for the entropy of undercooled liquids. It has been shown that C₂ in the WLF equation is a nearly constant fraction of (T_s - T₂), where T_s and T₂ are the reference and the Kauzmann temperatures, respectively. It is demonstrated that the values of T₂ obtained from the relaxation data agree well with those calculated from thermodynamic data. The arguments used provide an explanation for the universality of the WLF constant, C₂.     

Influence of molecular weight on dynamic crossover temperature in linear polymers

Dielectric and light scattering spectra of two linear polymers, polyisoprene (PIP) and polystyrene (PS), were analyzed in broad temperature and frequency range above the glass transition temperature, T_g. The crossover temperature, T_C, was estimated using two approaches: (i) derivative analysis of relaxation times proposed by Stickel and (ii) Mode-Coupling Theory approach. Both estimates provide consistent values. T_C varies with molecular weight (MW) in both polymers, while the ratio T_C/T_g changes significantly with MW in PS only. It appears that the segmental relaxation time at T_C has value τ(T_C) ∼ 10⁻⁷ s for both polymers independent of MW and similar to the value reported for many non-polymeric glass-forming systems. No sign of the dynamic crossover has been observed in the chain relaxation around T_C of the segmental dynamics.     

Crystallization and relaxation dynamics of glass-forming liquids at the Kauzmann temperature

If the entropy extrapolation of supercooled liquids (SCL) suggested by Kauzmann was correct, then they would have the same entropy as their stable crystalline phase at a certain low temperature, below the laboratory glass transition (T_g), known as the Kauzmann temperature (T_K). Extrapolating even further, the liquid entropy would be null at a temperature above absolute zero, violating the Third Law of Thermodynamics and constituting a paradox. Several possibilities have been proposed over the past 70 years to solve this paradox with different degrees of success. Our objective here is to access liquid dynamics at deep supercoolings to test the so-called crystallization solution to the paradox. By comparing the relaxation and crystallization kinetics determined above T_g and extrapolated down to T_K, a possible solution would be that the crystallization time is shorter than the relaxation time, which would mean that a SCL cannot reach the T_K. In this case, the liquid stability limit or kinetic spinodal temperature (T_ks) should be higher than T_K. We tested two fragile glass-forming liquids (diopside and wollastonite) and two strong liquids (silica and germania). For the fragile substances, T_ks ? T_K, hence such a supercooled liquid cannot exist at T_K, and the entropy crisis is averted. On the other hand, the results for the strong liquids were inconclusive. We hope the findings of this work encourage researchers to further investigate the liquid dynamics of different strong glass-forming systems at deep supercoolings.     

Mobility evolution during tri-axial deformation of a glassy polymer

In this study, the evolution of mobility with deformation in a glassy polymer is compared in cases of dilatationally dominated longitudinal and deviatorically dominated uniaxial deformation. The mobility is evaluated via stress relaxation experiments performed at various points along the stress–strain curve, including pre-yield, yield, and post-yield regions. At T_g−5 °C the mobility decreases with deformation in uniaxial extension, but increases with deformation in tri-axial extension. The τ_eff relaxation time calculated for longitudinal deformation exhibits a dramatically different dependence on the excess volume than the relaxation time obtained in structural relaxation experiments. Finally, the criterion for cavitational failure is proposed based on the thermodynamic stability considerations.     

Detection of fast and slow crystallization processes in instantaneously-strained samples of cis-1,4-polyisoprene

Cross-linked samples of natural rubber (NR) and synthetic cis-1,4-polyisoprene (IR) were instantaneously expanded to a predetermined strain ratio, α_s, using a newly-designed high-speed tensile tester. Crystallization behavior after the cessation of deformation was investigated. The high-cycle wide-angle X-ray diffraction (WAXD) measurements could successfully reveal the drastic progress of crystallization within the first a few hundred milliseconds. Quantitative analysis of diffraction intensity clarified coexistence of fast and slow crystallization processes; time constants τ_f and τ_s, and amplitude I_f and I_s, respectively, were estimated for these processes. The values of τ_f were in the range of 50–200 ms, while τ_s ranged between 2.5 and 4.5 s. Almost linear dependence of I_f and I_s on α_s was clarified. The crystallite size in the directions both parallel and perpendicular to the stretching direction decreased with the increase in time-averaged nominal stress. The crystal lattice deformed almost linearly with the average nominal stress. For the fast process, correlation between crystallization and stress relaxation was not recognized, while linear relationship between them was found for the slow process. In every case, strain-induced crystallization was found to be the major origin of stress relaxation. Based on the results, effects of strain on crystallization of polymer melt were discussed.     

The kinetics of the α and β relaxations in isotactic polypropylene

The kinetics of the two mechanically dominant relaxations in isotactic polypropylene are studied by thermal sampling (TS). The Bucci relaxation times are corrected by the method of McCrum. The α and β relaxation processes are found to be governed by the compensation rule. For the α relaxation the isokinetic point occurs at τ_c = 19 μs, T_c = 180°C: the β relaxation isokinetic point occurs at τ_c = 1.3 ms, T_c = 23°C. These large values of τ_c are anomalous: they imply that the approach frequencies (τ^?1_c) are extremely low: 50 kHz and 0.8 kHz for the α and β relaxations respectively. The classical view of the approach frequency gives $\text{kT}\text{h}\text{～6 × 10}{}^{12}\text{Hz}$. If this anomally is substantiated for relaxations in other polymers it will require a complete re-examination of current theories of polymer relaxation mechanisms. It is shown also that the temperature dependence of the difference between the limiting compliances, J_R-J_U, plays a role in thermal sampling and in thermally stimulated creep.     

On the physical significance of bucci relaxation times obtained from thermal sampling

It is shown that the Bucci relaxation time, τ_B, is not in general a true measure of a physical relaxation time in a mechanical or dielectric thermal sampling (TS) experiment. A method is proposed by which τ_B may be corrected to obtain an effective relaxation time, $\text{τ}{}_{\mathrm{<mtext>B</mtext><mtext>1</mtext><mtext>2</mtext>}}$, which is the relaxation time computed from the Bucci equation at 50% relaxation: $\text{τ}{}_{\mathrm{<mtext>B</mtext><mtext>1</mtext><mtext>2</mtext>}}$ when correctly calculated, depends only on the temperature of observation. The calculation is based on two assumptions: (i) that the shape of the distribution of relaxation times (slope of the ramp) has been assessed within the experimental error; (ii) that the shift factor is the same for all relaxation times within the narrow packet activated in the TS experiment. It is shown by experiment that the plot of log τ_B versus T^?1 for a viscoelastic polymer (isotactic polypropylene in the β-region) is curved: the curvature is explained theoretically. The plot of log $\text{τ}{}_{\mathrm{<mtext>B</mtext><mtext>1</mtext><mtext>2</mtext>}}$versusT^?1 yields a straight line, in agreement with the Arrhenius equation. The early TS experiments, which have been interpreted to favour the compensation rule, should be re-analysed since the published analysis is made by Arrhenius plots using uncorrected values of τ_B. The high temperature TS experiments on polystyrene and polypropylene (stated to be anomalous by Lacabanne and co-workers in yielding curved log τ_BversusT^?1 plots) will be found, when analysed by the methods described in this paper, to conform to the Arrhenius equation.     

Prediction of the relationship between the rate of deformation and the rate of stress relaxation in glassy polymers

Kim, et al. (Polymer, 54(15), 3949, 2013) recently reported on the unexpected relaxation behavior of an amorphous polymer in the T_g-region, where the rate of stress relaxation increased with deformation at a strain rate of 1.5 × 10⁻⁴ s⁻¹ but decreased at a strain rate of 1.2 × 10⁻⁵ s⁻¹. This inversion in the ordering with strain rate challenges the underlying structure of the existing nonlinear viscoelastic and viscoplastic constitutive models, where the key nonlinearity is a deformation dependent material clock. The nonlinear stress relaxation predictions of a recently developed stochastic constitutive model, SCM, (Medvedev, et al., J. Rheology, 57(3), 949, 2013) that acknowledge dynamic heterogeneity of the glass have been investigated. The SCM predicts the inversion in the ordering of the mobility with the loading strain rate as reported by the stress relaxation response. The change in perspective on the nonlinear viscoelastic behavior of glassy polymers engendered by the SCM is discussed.     

Physical nature of complex structural relaxation in polysiloxane - PMpTS: α and α′ relaxations

Structural relaxation processes in poly(methyl-para-tolyl-siloxane) (PMpTS) polymers of three molecular weights were studied using dynamic light scattering. Two relaxation processes: the usual α and an additional slow one α′ were observed and studied as function of temperature and molecular weight. Contrary to the structural relaxation, we find that in a plot T-T_g the relaxation times for the α′ process for all molecular weights do not collapse to a single curve. For one of the samples the light scattering correlation functions were compared with the corresponding functions obtained by means of mechanical relaxation, dielectric spectroscopy and computer simulations. The simulations show that the bimodal distribution, i.e. the α relaxation and the slow (α′) process are contained in the correlation functions of most of the probes (optical anisotropy, dipole moment, chain bond, density) in agreement with experimental observations.     

Molecular dynamics study on the viscosity of glass-forming systems near and below the glass transition temperature

Temperature-dependent viscosity is critical to decipher two profound questions in condensed matter physics, namely the glass transition and the relaxation of amorphous solids. However, direct measurement of viscosity over a large temperature range is extremely difficult. Here, using classical molecular dynamics (MD) simulations, we report a novel method to calculate the equilibrium viscosity of supercooled liquid both above and below the glass transition temperature (T_g) and to estimate the nonequilibrium viscosity of glass down to room temperature. Based on the shoving model, we derived an analytical formula showing that the shear viscosity in logarithmic scale changes linearly with the shear-induced variation in shear modulus or potential energy of the glass-forming system. The shear viscosity as a function of steady-state potential energy of liquid under different shear strain rates can be directly calculated in MD simulations; together with its equilibrium potential energy, one can extrapolate the zero-strain-rate equilibrium viscosity. We verified the proposed model by reliably calculating equilibrium viscosity near T_g of four glass-forming systems (Kob–Andersen system, silica, Cu_45.5Zr_45.5Al₉, and silicon) with different fragilities. Furthermore, our model can estimate the nonequilibrium viscosity of glass below T_g; the upper-bound nonequilibrium viscosity of amorphous silica and silicon at room temperature are calculated to be ~10³² and 10²⁵ Pa·s, respectively.     

Dynamics of macromolecular chains. 13C spin relaxation study of short-chain polystyrenes in deutero-chloroform solution

The results of T₁, T₂ and nuclear Overhauser effect (NOE) measurements on molecular weights 510–110 000 at weight fractions 0.05–0.60 in the title system suggest that previously neglected entanglement effects are highly significant for the reorientational processes in polymer chains and that previous T₂ estimates from bandwidths may be too low. T₂ is usually equal to T₁, and deviates only at high concentrations in these systems. The data can be represented by a double exponential reorientational correlation function of the form

$\text{G(τ)= A}\text{exp}\text{(}\text{? τ}\text{τ}{}_{\mathrm{A}}\text{) + B(}\text{? τ}\text{τ}{}_{\mathrm{B}}\text{)}$

where A + B = 1. The correlation times τ_A and τ_B are typically 1.5 × 10^?10 and 1.5 × 10^?9s, respectively. B increases with increasing concentration and molecular weight. The terminal phenyl group rotation is quite free in contrast to the backbone phenyls, and a spinning ratio above 15 has been estimated in a 60% solution for this group. Signal assignments and relaxation times (T₁) are given for 15 different ¹³C signals of these polymers at high concentrations.     

Semidilute solutions of poly(methacrylic acid) in the absence of salt: Dynamic light-scattering study

Water solutions of poly(methacrylic acid) of molecular weight M_w = 3.0 × 10⁴ and M_w = 4.0 × 10⁵, neutralized with NaOH, were investigated by photon correlation spectroscopy. At a low degree of neutralization α, the short-time decay only is observed. When α > 0, a second, much slower, process becomes noticeable. It gains quickly in influence when α increases. The reciprocal values of characteristic relaxation times for both dynamic processes were found to decrease linearly with the square of the scattering vector, showing that both processes are diffusive. The following interpretation of these processes founded on the concentration and angle dependences of the corresponding diffusion coefficients and scattering amplitudes is given: (1) for the low-molecular-weight sample (M_w = 3.0 × 10⁴), D_∫ (fast process) and be attributed to the Nernst-Hartley diffusion coefficient of polyions, and D_s (slow process) to a diffusion of interchain domains (clusters) with a radius of gyration R_G ≈ 50 nm; (2) for the high-molecular-weight sample (M_w = 4.0 × 10⁵), where an overlap of polymer chains occurs, D_f can be attributed to a cooperative diffusion coefficient reflecting the concentration fluctuations due to the polyions and counterions, and D_s to slow concentration fluctuations having a large correlation length (≈ 100 nm). The existence of two diffusive modes in salt-free solutions of polyelectrolytes cannot be explained within the framework of the scaling approach as proposed by de Gennes and Odijk.     

Atomistic molecular simulations of structure and dynamics of crosslinked epoxy resin

Many excellent thermal and mechanical performances of cured epoxy resin products can be related to their specific network structure. In this work, a typical crosslinked epoxy resin was investigated using detailed molecular dynamics (MD) simulations, in a wide temperature range from 250 K to 600 K. A general constant-NPT MD procedure widely used for linear polymers failed to identify the glass transition temperature (T_g) of this crosslinked polymer. This can be attributed to the bigger difference in the time scales and cooling rates between the experiments and simulations, and specially to the highly crosslinked infinite network feature. However, by adopting experimental densities appropriate for the corresponding temperatures, some important structural and dynamic features both below and above T_g were revealed using constant-NVT MD simulations. The polymer system exhibited more local structural features in case of below T_g than above T_g, as suggested by some typical radial distribution functions and torsion angle distributions. Non-bond energy, not any other energy components in the used COMPASS forcefield, played the most important role in glass transition. An abrupt change occurring in the vicinity of T_g was also observed in the plots of the mean squared displacements (MSDs) of the crosslinks against the temperature, indicating the great importance of crosslinks to glass transition. Rotational dynamics of some bonds in epoxy segments were also investigated, which exhibited great diversity along the chains between crosslinks. The reorientation functions of these bond vectors at higher temperatures can be well fitted by Kohlrausch-Williams-Watts (KWW) function.     

Modeling attempt of shape memory performance of epoxy resin with experimental methods

Epoxy resin samples with low cross-linking density were prepared, and their shape recovery rate and glass transition were studied. The results showed that the shape fixity ratio was over 99 % for all of the samples. Without constraint, the final recovery ratio was approximate to 100 % for all of the samples. The temperature with rapid recovery rate for different samples changed in accordance with T _g. Under a constant temperature, the folding angle for the samples of EP80 decreased with the increase in time rapidly, at first, and then tended to level off. Curves can be fitted with the formula of $ y + A_{0} = A/(1 + \exp ((t - t_{0} )/\tau )) $ with R ² higher than 99.9 %. The fitting results demonstrated that the value of τ decreased from 15.1 to 6.39 when the temperature increased from 88 to 98 °C for EP80. The recovery rate decreased a little by extending the holding time from 10 to 60 s. By keeping the testing temperature constant, glass transition temperature (T _g) decreased with the increase in curing agents, and the value of τ reduced with the decrease in T _g. Usually, when temperature was close to T _g, segments of macromolecules were idle to move, and then, the relaxation process, τ, was lengthened and the shape recovery rate decreased accordingly. In a word, τ showed the similar change rules with that of relaxation process of polymers; therefore, the shape recovering process could be predicted with the model of relaxation time and modulus according to relaxation formulas.     

Soft Matter in a Tight Spot: Nanorheology of Confined Liquids and Block Copolymers

The linear frequency-dependent shear rheology and force–distance profiles of molecularly-thin fluids of very different structure were contrasted: a globular molecule octamethylcyclotetrasiloxane (OMCTS), branched alkanes (3-methylundecane and squalane), and a polymer brush in near-theta solution (polystyrene-polyvinylpyridine). In each case the data suggest a prolongation of the longest relaxation time (τ₁) with increasing compression. At frequencies ω > 1/τ₁ the shear response was “solid-like”, but at ω < 1/τ₁ it was “liquid-like”. OMCTS under mild compression exhibited seeming power-law viscoelastic behavior with G′(ω) = G″(ω) over a wide frequency range. Of the branched-molecule fluids, 3-methylundecane exhibited oscillatory force–distance profiles; this confirms prior computer simulations. But squalane (6 pendant methyl groups in an alkane chain 24 carbons long) showed one sole broad attractive minimum. Polymer brushes in a near-theta solvent exhibited changes qualitatively similar to those OMCTS, in particular, a smooth progression of longest relaxation time, generating a transition from “liquid-like” to “solid-like” shear rheology with decreasing film thickness. The common trend of shear response in these systems, in spite of important differences in molecular structure and force–distance profiles, is emphasized.     

Permeable domains of segmented polyurethanes studied with paramagnetic spin probe

Studies on the permeable regions of the dense polyurethane-based membranes were performed by electron spin resonance spectroscopy (ESR) using TEMPO spin probe incorporated into the membrane via diffusion from the vapour phase. The ESR spectra were measured as a function of temperature and microwave power for polyurethanes (PU) varying in the molecular structure and morphology. It was found that the TEMPO spin probe exhibited anisotropic rotation whose anisotropy increased as temperature decreased and was more pronounced for PU with shorter soft segments. The simplified method was used to obtain apparent correlation time τ_c enabling the comparison of the polyurethanes studied. This approach was based on the Arrhenius relation of τ_c vs. 1/T determined from motionally narrowed ESR spectra and on the assumption that this behaviour prevails over a broader temperature range at temperatures generally greater than T_g of a given polymer.     

N.m.r. studies on the effect of water on the glass transition of polystyrene

These investigations are concerned with water-polymer interactions in polymer latices. It is known that water can act as a plasticizer for many solid polymers and cause a reduction in the glass transition temperature, T_g, of the amorphous regions. Experiments were carried out to determine whether pulsed n.m.r. techniques could be used to study the T_g of a polymer suspension and hence the influence of water and electrolyte on it. From T₁ and T₂ proton relaxation measurements as a function of temperature on polystyrene latex systems it was shown that the presence of water lowers the T_g of the polymer particles (by about 10°C), the effect being slightly greater in the presence of concentrated electrolyte. The extent of electrolyte penetration into the particles was deduced by studying relaxation as a function of particle diameter in latices containing paramagnetic Mn²⁺ ions. Using simple theories of relaxation and spin diffusion it was concluded that for all but the smallest particles electrolyte penetration is restricted to a very thin shell of the order of 1 nm. These conclusions were supported by the results of similar measurements on PTFE particles.     

Origin of the divergence of the timescales for volume and enthalpy recovery

Although the relationship between the relaxation timescales of thermodynamic, mechanical, viscoelastic, and dielectric properties in amorphous materials has been studied extensively, no general consensus has been reached. In this work, we examine the relationship between the timescales of volume and enthalpy relaxation for polystyrene using the cooling rate dependence of the glass transition temperature (T_g) obtained from capillary dilatometry and differential scanning calorimetry (DSC). Our analysis suggests that both volume and enthalpy exhibit similar relaxation timescales at temperatures above and below T_g. The divergence of the times required to reach equilibrium noted in the literature at temperatures several degrees below the nominal T_g is attributed to the effects of nonlinearity. The relationship between nonlinearity and dynamic heterogeneity is discussed.     

Broadband dielectric spectroscopic characterization of the hydrolytic degradation of carboxylic acid-terminated poly(d,l-lactide) materials

Broadband dielectric spectroscopy was used to examine carboxylic acid-terminated poly(d,l-lactide) samples that were hydrolytically degraded in 7.4 pH phosphate buffer solutions at 37 °C. The dielectric spectral signatures of degraded samples were considerably more distinct than those of undegraded samples and a T_g-related relaxation associated with long range chain segmental mobility was seen. For both degraded and undegraded samples, a relaxation peak just beneath a DSC-based T_g was observed, which shifts to higher frequency with increasing temperature. Thus, this feature is assigned as the glass transition as viewed from the dielectric relaxation perspective. Linear segments on log-log plots of loss permittivity vs. frequency, in the low frequency regime, are attributed to d.c. conductivity. An upward shift in relaxation peak maximum, f_max, observed especially after 145 d of immersion in buffer, implies a decrease in the time scale of long range segmental motions with increased degradation time.Permittivity data for degraded and undegraded materials were fitted to the Havriliak-Negami equation with subtraction of the d.c. conductivity contribution to uncover pure relaxation peaks. Parameters extracted from these fits were used to construct Vogel-Fulcher-Tammann-Hesse (VFTH) curves and distribution of relaxation time, G(τ), curves for all samples. It was seen that the relaxation times for the α-transition in both degraded and undegraded samples showed VFTH temperature behavior. G(τ) curves showed a general broadening and shift to lower τ with degradation, which can be explained in terms of a broadening of molecular weight within degraded samples and faster chain motions.     

Aggregation behaviour of monosulfonated telechelic ionomers

The behaviour of monosulfonated telechelic polystyrene with various molar masses and different counterions has been investigated using static and dynamic light scattering. For concentrations above the critical micelle concentration, free unimers associate to form well‐defined star‐shaped micelles. The number of polystyrene chains per micelle (n_ag) is independent of the concentration and the type of counterion; n_ag decreases with decreasing molar mass of the polystyrene and increasing polarity of the solvent. The micelles behave like covalently bonded star polystyrene from a structural, dynamic and thermodynamic point of view. © 2000 Society of Chemical Industry