Concept Of Fictive Temperature Research Articles

A density scaling conjecture for aging glasses.

The aging rate of glasses has traditionally been modeled as a function of temperature, T, and fictive temperature, while density, ρ, is not explicitly included as a parameter. However, this description does not naturally connect to the modern understanding of what governs the relaxation rate in equilibrium. In equilibrium, it is well known that the relaxation rate, γeq, depends on temperature and density. In addition, a large class of systems obeys density scaling, which means the rate specifically depends on the scaling parameter, Γ = e(ρ)/T, where e(ρ) is a system specific function. This paper presents a generalization of the fictive temperature concept in terms of a fictive scaling parameter, Γfic, and a density scaling conjecture for aging glasses in which the aging rate depends on Γ and Γfic.

Memory effects in glasses: Insights into the thermodynamics of out-of-equilibrium systems revealed by a simple model of the Kovacs effect.

Glasses are interesting materials because they allow us to explore the puzzling properties of out-of-equilibrium systems. One of them is the Kovacs effect in which a glass, brought to an out-of-equilibrium state in which all its thermodynamic variables are identical to those of an equilibrium state, nevertheless evolves, showing a hump in some global variable before the thermodynamic variables come back to their starting point. We show that a simple three-state system is sufficient to study this phenomenon using numerical integrations and exact analytical calculations. It also brings some light on the concept of fictive temperature, often used to extend standard thermodynamics to the out-of-equilibrium properties of glasses. We confirm that the concept of a unique fictive temperature is not valid, an show it can be extended to make a connection with the various relaxation processes in the system. The model also brings further insights on the thermodynamics of out-of-equilibrium systems. Moreover, we show that the three-state model is able to describe various effects observed in glasses such as the asymmetric relaxation to equilibrium discussed by Kovacs, or the reverse crossover measured on B_{2}O_{3}.

Heat capacity spectroscopy by using stochastic temperature modulated calorimetry: Time-temperature superposition and fictive temperature at the glass transition of poly (vinyl acetate)

Complex heat capacity spectra were measured by stochastic temperature modulated calorimetry using a conventional differential scanning calorimeter on poly (vinyl acetate) in the glass transition region during cooling. It is shown, that the real and imaginary part of the complex heat capacity spectra measured above and below vitrification can be scaled to a master curve by use of the frequency-temperature superposition principle. After the transition from a structurally equilibrated state to the non-equilibrated glass (vitrification), the shift factors deviate from the Vogel-Fulcher-Tammann-Hesse equation. The vitrification is described in the concept of fictive temperature using the Adam-Gibbs-Scherer equation for the temperature dependence of the relaxation time.

Modeling enthalpy relaxation using the Mittag-Leffler function

Understanding the structural relaxation plays an essential role in the application of amorphous solids. Most of the current models use the fictive temperature concept to describe structural relaxation. The fictive temperature is assumed to evolve with temperature through the Kohlrausch-Williams-Watt (KWW) function. In this work, we use the Mittag-Leffler (ML) function to replace the KWW function. We will demonstrate the similarity and difference of the two models. Both the continuous model in the integral form and the discrete model with multiple relaxation processes in differential form are presented. We provide a numerical algorithm to obtain the relaxation spectra of the discrete models. It is shown that the spectrum of the model with the KWW kernel is asymmetric while the spectrum of the model with the ML kernel is symmetric. We further demonstrate that the continuous model and the discrete model are numerically equivalent. The performances of the models with the KWW kernel and the ML kernel are compared in details. It is showed that both models can capture the same trend on the effects of the cooling rate and annealing time on the enthalpy response. However, the magnitude and location of the enthalpy overshoot predicted by the two models are different. Thus, this work provides another choice for the relaxation function.

Impact of micro-alloying on the plasticity of Pd-based bulk metallic glasses

Micro-alloying was performed using additions of Co and Fe to monolithic Pd40Ni40P20 bulk metallic glass. Compression tests showed a plastic strain of 13% for the Co addition (1at.%), whereas the Fe addition (0.6at.%) led to immediate failure after reaching the elastic limit. Plasticity is not reflected by the high Poisson’s ratio of 0.4 since it remained unaffected by minor alloying. Applying the fictive temperature concept to analyze the impact of minor alloying suggests that the total amount of free volume frozen-in during vitrification is less important for the mechanical properties of bulk metallic glasses than its local distribution.

Glass transition and cooperative rearranging regions in amorphous thermoplastic nanocomposites

AbstractThe aim of this work was to study the effect of nanofillers on the structural relaxation phenomena occurring in amorphous poly(ethylene‐terephthalate)/poly(cyclohexane‐dimethanol terephthalate) copolymer (PET/PCHDMT) nanocomposites in correspondence with the glass transition temperature. PET/PCHDMT nanocomposites were prepared by melt mixing with an organic modified montmorillonite at different processing temperatures. Differential scanning calorimetry analysis revealed that addition of the organic modifier alone causes a decrease of the glass transition temperature and an increase of the specific heat discontinuity. Nanocomposites showed a higher glass transition temperature and a lower specific heat discontinuity compared with samples obtained by adding organic modifier to PET/PCHDMT. Both effects were more relevant for samples processed at lower temperatures. Therefore, the glass transition temperature was studied by introducing the concept of fictive temperature and relaxation time. It was found that nanocomposites have a higher apparent activation energy and an increased size of cooperatively rearranging regions compared with neat PET/PCHDMT. Both effects are more relevant for nanocomposites processed at lower temperatures. All the discussed effects are explained by considering the enhanced confinement of PET/PCHDMT macromolecules, due to the presence of intercalated lamellae of organofiller. The efficiency of intercalation is increased by decreased processing temperature, which involves an increase of the nano‐confinement area of the polymer. Copyright © 2012 Society of Chemical Industry

Simulation of temperature history-dependent phenomena of glass-forming materials based on thermodynamics with internal state variables

In ref. [1], a profound ansatz for modelling the various phenomena associated with the glass-transition was presented. It extends classical theories, such as the concept of fictive temperature or order parameters, for example, in two ways. Firstly, temperature excitations as well as mechanical loadings are accounted for in a single consistent approach. Secondly, it is not formulated in the θ– p– V space, but rather in the θ – T ̲ ̲ – E ̲ ̲ space, taking the tensor character of stress T ̲ ̲ and strain E ̲ ̲ into account. Hence, it is a three-dimensional theory where the basic thermodynamic potential, the Gibbs free energy, depends on the stress tensor, the temperature and a set of internal variables. In the current article, expressions for the enthalpy, the entropy, the thermal expansion and the specific heat are derived. The specific heat and the thermal expansion behaviour are looked at for various temperature excitations. It is shown, that the results are in accord with experimental observations in literature.

Volume and enthalpy relaxation of a-Se in the glass transition region

Abstract Volume and enthalpy relaxation studies of amorphous Se have been performed in the glass transition region by mercury dilatometry and differential scanning calorimetry. For simple temperature jump experiments, as well as for more complex thermal history the volume and enthalpy relaxation data can be described by a single set of kinetic parameters for Tool-Naraynaswamy-Moynihan (TNM) model [Δh∗/R = 42.8 kK, ln(ATNM/s) = −133]. Slightly different non-linearity and non-exponentiality parameter were found for volume [x = 0.42, β = 0.58] and enthalpy [x = 0.52, β = 0.65] relaxation data. Similar results were obtained also for Adam-Gibbs-Scherer (AGS) model. The activation energy of viscous flow in the glass transition range is identical with the effective activation energy for relaxation process. A self-consistent data evaluation shows that at moderate departure from equilibrium, volume and enthalpy in amorphous selenium relax in the same way as expressed by TNM and AGS models. Both volume and enthalpy change can be interpreted within the same fictive temperature concept.

Temperature of systems out of thermodynamic equilibrium

Two phenomenological approaches are currently used in the study of the vitreous state. One is based on the concept of fictive temperature introduced by Tool [J. Res. Natl. Bur. Stand. 34, 199 (1945)] and recently revisited by Nieuwenhuizen [Phys. Rev. Lett. 80, 5580 (1998)]. The other is based on the thermodynamics of irreversible processes initiated by De Donder at the beginning of the last century [L'Affinite (Gauthier-Villars, Paris, 1927)] and recently used by Moller et al. for a thorough study of the glass transition [J. Chem. Phys. 125, 094505 (2006)]. This latter approach leads to the possibility of describing the glass transition by means of the freezing-in of one or more order parameters connected to the internal structural degrees of freedom involved in the vitrification process. In this paper, the equivalence of the two preceding approaches is demonstrated, not only for glasses but in a very general way for any system undergoing an irreversible transformation. This equivalence allows the definition of an effective temperature for all systems departed from equilibrium generating a positive amount of entropy. In fact, the initial fictive temperature concept of Tool leads to the generalization of the notion of temperature for systems out of thermodynamic equilibrium, for which glasses are just particular cases.

Fictive temperature measurement of amorphous SiO2 films by IR method

The structure and properties of amorphous materials, in general, change with their thermal history. This is usually explained using the concept of fictive temperature, i.e., the temperature at which the super-cooled liquid state turned into a glassy state. In earlier studies, a simple IR method was used to determine the fictive temperature of silica glasses, both bulk and fiber. In the present study the applicability of the same technique for thin amorphous silica films on silicon was examined. It was found that the IR absorption as well as reflection peak wavenumber of the silica structural band can be used to determine the fictive temperature of amorphous silica films on silicon with an unknown thermal history. Specifically, IR absorbance spectra of an amorphous silica film of thickness greater than 0.5μm grown on silicon can be taken before and after etching a thin surface layer of 20–30nm and the peak wavenumber of the difference signal can be compared with the pre-determined calibration curve to convert the peak wavenumber to the fictive temperature. For a film thicker than ∼2μm, IR reflection peak wavenumber can be converted directly to the fictive temperature of the film by using the calibration curve.

Physical aging and time–temperature behavior concerning fracture performance of polycarbonate

The effects of physical aging on fracture and yielding behavior are polycarbonate are considered. Two groups of Bisphenol A-based polycarbonate, consisted of extruded PC sheets (thickness of 0.25 mm) and injection molded PC bars (thickness of 3.18 mm) are used. These samples were annealed at various temperatures ranging from 60 to 120 °C, for different times varying up to 240 h. For PC sheets the essential work of fracture (EWF) method was used to analyze fracture behavior. The results are compared to the strain energy density with aging time and aging temperature in the ranges investigated. This effect is confirmed by the change in fracture toughness, as measured by three-point bending tests. The concept of fictive temperature ( T f) was used to characterize the degree of aging in the sample. T f of a glass in an aged state at a time t is defined as the temperature at which the volume would be equal to the equilibrium volume at T f if the sample were instantaneously removed to that temperature. Differential scanning calorimetry (DSC) was used to determine T f. The variations of T f with aging time and aging temperature are in agreement with both the strain energy density measurement and the three-point bending tests. These results contradict the effects of aging on fracture toughness observed by the essential work of fracture approach. The latter showed anomalous regions of increasing fracture toughness with aging, leading to spurious conclusions. The brittle–ductile transition in fracture behavior is analyzed by an activation energy approach. Aging increases the brittle–ductile transition temperature and the effect is more pronounced for the lower molecular-weight sample. Fracture tests also showed a decrease in the entropy with aging, confirming the results observed previously from tension and compression tests.

Glasslike dynamical behavior in hierarchical models submitted to continuous cooling and heating processes.

The dynamical behavior of a kind of models with hierarchically constrained dynamics is investigated. The models exhibit many properties resembling real structural glasses. In particular, we focus on the study of time-dependent temperature processes. In cooling processes, a phenomenon analogous to the laboratory glass transition appears. The residual properties are analytically evaluated, and the concept of fictive temperature is discussed on a physical basis. The evolution of the system in heating processes is governed by the existence of a normal solution of the evolution equations, which is approached by all the other solutions. This trend of the system is directly related to the glassy hysteresis effects shown by these systems. The existence of the normal solution is not restricted to the linear regime around equilibrium, but it is defined for any arbitrary, far-from-equilibrium, situation.

Structural Relaxations of Glassy Polystyrene and o-Terphenyl Studied by Simultaneous Measurement of Enthalpy and Volume under High Pressure

The heat capacities, thermal expansivities, and isothermal compressibilities of polystyrene and a mixture of o-terphenyl (67%) and o-phenylphenol (33%) (abbreviated as OTP−OPP) were measured around their glass transition temperatures at atmospheric pressure and high pressure (20.8 MPa for polystyrene and 28.8 MPa for OTP−OPP). The measurements were performed simultaneously with a novel calorimeter recently developed by our group. This calorimeter works under constant pressure up to 100 MPa and in the temperature range between 80 and 380 K. The heat capacity gap ΔCp, thermal expansivity gap Δα, and isothermal compressibility gap Δκ at Tg were determined precisely for both samples. The Prigogine−Defay ratio Π ≡ ΔCpΔκ/(TV(Δα)2) is significantly larger than unity for both samples, indicating that more than two internal parameters are necessary to specify the glassy state thermodynamically. The enthalpy and volume relaxations of the polystyrene and OTP−OPP were also measured simultaneously around Tg after the temperature- and pressure-jump processes. The relaxation paths determined in the ΔHc−ΔVc plane was inconsistent with the fictive temperature concept in the structural relaxation even in the vicinity of the equilibrium state with the volume relaxing faster than the enthalpy.

Development of density distributions in injection molded amorphous engineering plastics. Part II

AbstractThe experimental results obtained on the gapwise density distributions of freely quenched and injection molded specimens of two engineering thermoplastics presented in Part I of this paper were compared with simulation results. The effect of the thermal history on the density development in the freely quenched samples was investigated by employing the fictive temperature concept in conjunction with the experimentally determined material characteristics. The density distribution in freely quenched specimens of poly(ether imide) could be readily predicted. The combined effects of the cooling rate and the pressure history on the density distributions of injection molded samples were also investigated. A phenomenological model was developed to combine the effects of thermal and pressure history on the density distribution. Overall, the predicted results for injection molded specimens agreed well with the experimental results.

Short time structural relaxation processes in liquids: Comparison of experimental and computer simulation glass transitions on picosecond time scales

Recent Brillouin scattering studies of relaxation in the simple ion system Ca++–K+/NO−3 (two particles of argon structure and a small anion) provide the experimental basis for a direct comparison of glass transformation phenomenology in computer simulated LJ argon ‘‘glass,’’ with that in laboratory substances studied on the same time scale. Most of the attenuated glass transition characteristics observed for LJ argon are found in the ionic system, and the same relation of the dispersion midpoint to the so-called ‘‘ideal’’ glass transition temperature is observed. Analysis of the real and imaginary parts of the complex longitudinal modulus shows that at high temperatures the relaxation function for Ca++–K+–NO−3, in strong contrast with that at normal low temperature behavior, closely approaches a simple exponential decay with Arrhenius form for the relaxation time. Furthermore, the high temperature Arrhenius plot extrapolates naturally to the reciprocal quasilattice vibration frequency determined by far infrared absorption studies on the glassy solid. Because of the low activation energy in the high temperature regime the glass ‘‘transition’’ observed with decreasing temperature is smeared out almost beyond recognition. It is argued that this will be a rather general phenomenon for hyperquenched glasses and that for such cases, the fictive temperature concept which associates the glass structure with an internally equilibrated liquid structure (that at Tfictive) must collapse. Parallels between LJ argon and Ca+–K+–NO3–glasses are discussed.

Photostructural change in the Urbach tail in chalcogenide glasses

Absorption properties in the Urbach tail range were studied in relation to the photostructural change (PSC) or the photodarkening effect in amorphous-chalcogenide semiconductors. A reversible absorption change in the Urbach tail of amorphous-chalcogenide films due to photoirradiation and heat treatment is first reported. The Urbach tail is relaxed by photoirradiation and recovered by heat treatment, an effect which is associated with the effect of the photoenhanced far-infrared band broadening in amorphous-chalcogenide films. Temperature dependence of the absorption is found to be closely associated with the frequency of the dominant far-infrared absorption of the vibration mode. It is indicated that optically excited randomness, due to the photostructural change, contributes to the photoenhanced relaxation of the exponential tail slope. Analysis of the tail slope parameter reveals strong electron-phonon couplings at the Ge-Se (S) or As-Se (S) bond. The photostructural change physical mechanism is discussed, and the frozen-in fictive temperature concept is found adequate.

Experimental study of stress relaxation during annealing

In order to calculate the residual stresses in a flat sheet of glass during annealing on the basis of its thermal history, the viscosity and coefficient of thermal expansion must be determined throughout the transformation range. The structural modifications of the material are taken into account by using fictive temperature concept. Three constitutive assumptions for the rate of the fictive temperature are discussed. A very good agreement is obtained between measured and calculated residual stresses.

Nonlinear Viscoelastic Behavior of Glass at Its Transformation Range

A mechanical model is employed to simulate the deformation behavior of glass rods subjected to tensile loads which varied linearly with time. Assumptions are made about the nonlinearity of viscosity and modulus, and about the distribution function of viscosity coefficients as a function of retardation time. A modification of the model is introduced to apply the fictive temperature concept to viscosity data and is also applied to the case of stress-release under constant strain.

Effect of Thermal History on Glass Expansion Characteristics

Thermal expansion data are presented for two glasses for both heating and cooling at constant rates. Heating data were taken on samples with varying thermal histories brought about by either constant‐rate cooling or by constant‐temperature stabilization before the expansion test. The fictive temperature concept is discussed in regard to these data. Expansion data are correlated with density and refractive index at room temperature for a variety of heat‐treatments.

Limitations of the Fictive Temperature Concept

The fictive temperature formalism, in which the structural state of the glass is characterized by the fictive temperature T, is very useful in the qualitative description of the effects of thermal treatment in the transformation range. Experimental data are presented which show, however, that such a one‐parameter description has only a limited quantitative significance. When samples of a borosilicate crown, glass, annealed at various constant cooling rates, are compared with samples annealed at constant temperature, it is found that a single parameter does not specify all physical properties simultaneously and that the two types of samples differ markedly in transformation‐range kinetic phenomena. Properties studied include density, refractive index, low‐temperature thermal expansion, approach to equilibrium at constant temperature, and thermal expansion during heating in the transformation range.