Thermodynamics of relaxation processes in the glass transition region

Relaxation processes in the glass transition region, especially the recovery of the volume and the physical ageing of polymers, do not follow the common (linear) theory of relaxation. On the contrary, they show a development which depends on the previous history, may be non-monotonous and requires a relaxation time that may have negative values and a pole. These phenomena can be explained if the single relaxation time is replaced by a spectrum of relaxation times and the relaxation times are supposed to be subjected to a feedback via certain structure- and temperature-parameters (as, for instance, in the KAHR-theory).However, the feedback and a pole of the relaxation time arise already for a single internal degree of freedom by themselves, if, in the non-equilibrium thermodynamics, a dynamic and a static temperature are strictly differentiated. In the case of the relaxation of the diffusive translational motion of the molecules in the glass transition region the dynamic temperature is identical with the socalled fictive temperature introduced by Tool.With regard to the relaxation of the volume three different temperature regions must be distinguished: A fluid region at high temperatures where the relaxation is controlled by the free volume and complies with the linear theory at least approximately; a glass-like region at low temperatures where the relaxation is controlled by the thermal expansivity of the free volume and where, under certain conditions, the statements set up by Davies and Jones are valid; an intermediate region (the glass transition region) where the free volume as well as its coefficient of expansivity are decisive. In that region the effective relaxation time of the volume may have a pole and the dynamic temperature may approach its equilibrium value by discontinuous jumps or in a chaotic manner.Dedicated to Professor Dr. J. Meissner (ETH Zürich) on the occasion of his 60th birthday     

Testing of a constitutive equation with free volume dependent relaxation spectrum

Summary A model of non-linear viscoelasticity with relaxation times dependent upon free volume is here proposed. The free volume is related to the isotropic part of the stress tensor by means of a simple differential equation.The model predictions are compared with a large amount of experimental results taken on polymeric melts or concentrated solutions and reported in the literature. The single parameter of the model is determined, within each set of data, by fitting of the viscosity curve. A satisfactory agreement is obtained with data taken under both elongation and shear for which also the relaxation behavior after single and double strain steps is considered.

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Zusammenfassung Es wird das Modell einer nicht-linear viskoelastischen Flüssigkeit vorgeschlagen, bei dem die Relaxationszeiten vom freien Volumen abhängen. Das freie Volumen wird dabei durch eine einfache Differentialgleichung mit dem isotropen Teil des Spannungstensors verknüpft.Die Voraussagen des Modells werden mit einer großen Anzahl von experimentellen Ergebnissen an Polymerschmelzen und konzentrierten Lösungen verglichen, die in der Literatur mitgeteilt sind. Der einzige Parameter des Modells wird für jeden Datensatz durch Anpassung der Viskositätskurve bestimmt. Man erhält eine befriedigende Übereinstimmung für die Meßwerte sowohl von Dehn- als auch von Scherversuchen, wobei ebenfalls das Relaxationsverhalten nach dem Aufprägen einfacher und doppelter Dehnungsstufen betrachtet wird.

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With 16 figures and 1 table     

Stochastic models of relaxation phenomena

Summary The present work is concerned with simple stochastic models for the representation of the linear viscoelastic material behaviour. It is shown that such models can be employed in the same manner as the usual mechanical models. However in contrast to the mechanical models, the stochastic models are related to the microstructure of the medium by means of a material functional. In the present work the latter is shown only in a very reduced form.Creep and relaxation functions of such materials are considered first in terms of microscopic quantities, which are random variables. By introducing the concept of a mesoscopic domain within the material sample the transition from the microscopic to the macroscopic or phenomenological representation can be achieved.

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Zusammenfassung In der vorliegenden Arbeit werden einfache stochastische Modelle beschrieben. Es wird gezeigt, daß solche Modelle ebenso wie die einfachen Anordnungen von Federn und Dämpfern zur Veranschaulichung des linearen viskoelastischen Verhaltens eines Materials benutzt werden können.Im Gegensatz zu den letzteren sind jedoch die stochastischen Modelle mit der Materialstruktur mittels einer Materialfunktion verbunden. Diese wird in einer äußerst vereinfachten Form dargestellt. Kriech- und Relaxations-Funktionen werden mittels dieser Modelle vorerst für die eingeführten mikroskopischen Variablen abgeleitet und dann mittels Einführung des Begriffes einer mesoskopischen Region im. Material wird der Übergang zur makroskopischen Theorie vollzogen.

     

Heat capacities and volumetric changes in the glass transition range: a constitutive approach based on the standard linear solid

A novel approach to represent the glass transition is proposed. It is based on a physically motivated extension of the linear viscoelastic Poynting–Thomson model. In addition to a temperature-dependent damping element and two linear springs, two thermal strain elements are introduced. In order to take the process dependence of the specific heat into account and to model its characteristic behaviour below and above the glass transition, the Helmholtz free energy contains an additional contribution which depends on the temperature history and on the current temperature. The model describes the process-dependent volumetric and caloric behaviour of glass-forming materials, and defines a functional relationship between pressure, volumetric strain, and temperature. If a model for the isochoric part of the material behaviour is already available, for example a model of finite viscoelasticity, the caloric and volumetric behaviour can be represented with the current approach. The proposed model allows computing the isobaric and isochoric heat capacities in closed form. The difference \(c_\mathrm{p} -c_\mathrm{v} \) is process-dependent and tends towards the classical expression in the glassy and equilibrium ranges. Simulations and theoretical studies demonstrate the physical significance of the model.     

The time temperature position of the glass-rubber transition of amorphous polymers and the free volume

Summary Accurate measurements of the shear creep compliance in the glass rubber transition region have been performed by means of a combined torsional pendulum and torsional creep apparatus over a broad experimental window for commercial PS, PMMA and PVC and for two slightly crosslinked rubbers, NR and PU. For the three plastic materials, specific volume vs. temperature curves were obtained by volume dilatometry under different rates of cooling.Shear creep compliance was found to obey the time-temperature superposition principle. In the temperature region above the glass transition temperature, the time-temperature shift function could be accurately described by a W.L.F. equation. However, the W.L.F. constantsc ₁ andc ₂ differed for the different polymers and were different from the universal W.L.F. values. No choice for a reference temperature could be found to make the W.L.F. constants of the different polymers to coincide within experimental scatter.Specific volumes in the rubbery state and the thermal expansion coefficients in the glassy state were found to be independent of rate of cooling. Specific volumes in the glassy state and glass transition temperature both increased with rate of cooling.A simple hypothesis could be used to discuss the results, in which free volume is defined by the excess specific volume above glass transition on the one hand, and by its relation to mobility (the Doolittle equation), on the other. This hypothesis contains three independent parameter values, viz. the thermodynamic glass transition temperatureT , the thermal expansion coefficient of the free volume fraction, _f, andB of the Doolittle equation. By combining the results of both measurements, all three parameters can be separately determined. Values forB are found between 0.2 and 0.4, values of the fractional free volume frozen at the glass transition are found between 0.8 and 1.1%.

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Zusammenfassung Es wurden genaue Messungen der Scher-Kriechfunktion im Glas-Kautschuk-Übergangsgebiet in einem ausgedehnten logarithmischen Zeitintervall mit Hilfe eines Torsionspendel- und Kriechautomaten durchgeführt. Untersucht wurden kommerzielles PS, PMMA und PVC sowie zwei leicht vernetzte Kautschuke, NR und PU. Für die drei Kunststoffe wurde das spezifische Volumen als Funktion der Temperatur bei verschiedenen Kühlgeschwindigkeiten bestimmt.Die Scher-Nachgiebigkeit erfüllte das Zeit-Temperatur-Verschiebungsprinzip. Im Temperaturbereich über der Glastemperatur kann die Zeit-Temperatur-Verschiebungsfunktion mit großer Genauigkeit durch die W.L.F.-Gleichung beschrieben werden. Die W.L.F.-Konstantenc ₁ undc ₂ sind jedoch für die verschiedenen Polymere verschieden und stimmen nicht mit den Universalwerten dieser Konstanten überein.Spezifisches Volumen im gummielastischen Zustand und thermischer Ausdehnungskoeffizient im Glaszustand hängen nicht von der Kühlgeschwindigkeit ab. Spezifisches Volumen im Glaszustand und die Glastemperatur nehmen mit steigender Kühlgeschwindigkeit zu.Die Ergebnisse konnten mit Hilfe einer einfachen Hypothese beschrieben werden, in der das freie Volumen einerseits aus dem spezifischen Volumen definiert wird, das oberhalb der Glastemperatur zusätzlich auftaut, andererseits durch seinen Zusammenhang mit der molekularen Beweglichkeit eingeführt wird (Doolittle-Gleichung). Diese Hypothese enthält drei unabhängige Parameter, nämlich die thermodynamische GlastemperaturT , den Ausdehnungskoeffizienten des relativen freien Volumens _f und den ParameterB der Doolittle-Gleichung. Durch Kombination der Meßresultate können alle drei Parameter bestimmt werden. Wir finden fürB Werte zwischen 0,24 und 0,42 und für das beim Glasübergang eingefrorene relative freie Volumenf _g Werte zwischen 0,8 und 1,1%.

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With 16 figures and 4 tables     

Study of low-frequency relaxation phenomena in thin-film capacitors

Experiments are described to determine the dispersion characteristics at low frequencies of thin-film capacitances. It is supposed that mobile positive ions are responsible for the observed relaxation phenomena. The number of mobile ions and their relaxation time are investigated in function of the applied voltage for thin films of SiO, Al₂O₃, Ta₂O₅ and polytetrafluoraethylene. Theoretical models to explain the experimental data are also given.     

Investigation on nonequlibrium radiation and relaxation phenomena in shock tubes

The experimental results for the excited time of the nonequlibrium radiation and the ionization behind strong shock waves are presented. Using an optical multichannel analyzer, InSb infrared detectors and near-free-molecular Langmuir probes, the infrared radiation, the electron density of air and the nonequlibrium radiation spectra at different moments of the relaxation process in nitrogen test gas behind normal shock waves were obtained, respectively, in hydrogen oxygen combustion driven shock tubes. The project supported by the National Natural Science Foundation of China (19982005 and 10032050), and the National Defense Science Foundation of China (95JBA4.2ZK0402)     

A thermo-mechanically-coupled large-deformation theory for amorphous polymers in a temperature range which spans their glass transition

Amorphous thermoplastic polymers are important engineering materials; however, their non-linear, strongly temperature- and rate-dependent elastic-viscoplastic behavior is still not very well understood, and is modeled by existing constitutive theories with varying degrees of success. There is no generally agreed upon theory to model the large-deformation, thermo-mechanically-coupled, elastic-viscoplastic response of these materials in a temperature range which spans their glass transition temperature. Such a theory is crucial for the development of a numerical capability for the simulation and design of important polymer processing operations, and also for predicting the relationship between processing methods and the subsequent mechanical properties of polymeric products. In this paper we extend our recently published theory [Anand, L., Ames, N. M., Srivastava, V., Chester, S. A., 2009. A thermo-mechanically-coupled theory for large deformations of amorphous polymers. Part I: formulation. International Journal Plasticity 25, 1474–1494; Ames, N. M., Srivastava, V., Chester, S. A., Anand, L., 2009. A thermo-mechanically coupled theory for large deformations of amorphous polymers. Part II: applications. International Journal of Plasticity 25, 1495–1539] to fill this need.     

The effect of volume relaxation on elastohydrodynamic lubrication

In elastohydrodynamic lubrication (EHD) three important non-Newtonian effects arise. These are volume viscoelasticity, shear viscoelasticity, and the variation of viscosity with shear rate. All these effects tend to decrease the shear stress or traction.In this paper the effect of volume relaxation of EHD is examined using experimental viscosity data obtained in a simple viscometric flow. It is shown that the viscosity of a fluid during EHD is unlikely to reach its equilibrium value. Approximations to the viscosity as a function of time lead to the conclusion that volume and shear viscoelasticity have effects which are of the same order of magnitude and will be difficult to separate except by an exact knowledge of the shear rate and pressure profiles.     

A constitutive model in linear thermoviscoelasticity of polymers based on the concept of cooperative relaxation

New constitutive relations are derived for amorphous glassy polymers based on the concept of cooperative relaxation. A polymer is treated as a system of rearranging regions (flow units) embedded into a homogeneous elastic matrix. The viscoelastic (time-dependent) response of a medium is explained by rearrangements of segments of long chains in relaxing regions which occur at random instants. The kinetics of rearrangement is described in the framework of the Eyring concept of thermally activated processes, whereas the energy of any flow unit is assumed to randomly change at the instant of its reformation. Based on experimental data, phenomenological formulas are proposed for material functions. Adjustable parameters are found by fitting observations for mixtures of nylon with lithium halides in isothermal tensile relaxation tests. The thermoviscoelastic response in other tests is studied numerically. It is demonstrated that the material behavior predicted by the constitutive model in non-isothermal tests substantially differs from that predicted by conventional models whose adjustable parameters are determined by using the same experimental data. Received September 30, 1998     

Pressure dependence of glass transition temperature in polymers

The pressure dependencedT _g /dP of glass transition temperatureT _g has received considerable interest due to its connection with solid state thermodynamic properties and theories of glass transition. Free volume considerations (1, 2) led to an estimate of the pressure effect onT _g , showing thatdT _g /dP had to depend on thermal expansion and compressibility changes atT _g through the equation: [1] $$\frac{{dT_g }}{{dP}} = \frac{{\Delta \beta }}{{\Delta \alpha }}$$ whereΔβ=β _e ?β _g andΔα=α _e ?α _g Later work (3, 4, 5, 6) has shown that eq. [1] is not verified by experimental facts, the ratioΔβ/Δα being much larger than (dT _g /dP) exp. Recent analysis of the properties of glasses obtained under different pressures have complicated the situation, showing that the experimental value ofdT _g /dP depends, of course, on the polymer usedbut also on the experimental procedure used in its determination. Since it is obvious that in order to measure anyΔT _g -value we need to operate on at leasttwo glasses, these should be identical in all properties which could influenceT _g except pressure. Any difference in morphology,which could lead to a change in T _g at constant pressure, should therefore be avoided in order to get a sound value for the pure pressure effectdT _g /dP. To reveal this effect, we have performed (7)dT _g /dP determinations on two polymers, polyvinylacetate (PVAC) and polyvinylchloride (PVC), following three different procedures:

1. Measurement of the changeΔT _g induced by application of a pressure incrementΔP on the liquid polymer (T>T _g ). This is the procedure normally used; the liquid is cooled down at a fixed rate of temperature change (～5 °C/day) andT _g is dilatometrically recorded at 1 atmosphere. Then the polymer is taken again to the liquid state, pressure ΔP is applied and, at the same rate, the system is cooled down isobarically; the newT _g is recorded anddT _g /dP calculated.
2. Measurement of the change ΔT_g induced by application of a pressure increment ΔP on the glassy polymer (T _g ). Once determinedT _g at 1 atmosphere, pressureΔP is applied on the glass, time is given to the system to equilibrate; then the glass is heated isobarically. Intersection of the glassy line to the liquid line in a volume/temperature plot gives the newT _g and therefore allows the calculation ofdT _g /dP.
3. Measurement ofΔT _g during the heating of a glass along an isochor (5, 8). Here the polymer glass is heated at constant volume, by application of an increasing pressure at increasing temperatures given by(?P/?T) _v . By repeating this procedure two times, starting from two different specific volumes of the glass, two values ofT _g at different pressures can be recorded anddT _g /dP calculated.

   Table 1 shows the result of this work     

   16.

   Phase volume and size effects on the terminal relaxation of ABS melts     

   Dr. L. Castellani  P. Lomellini 《Rheologica Acta》1994,33(5):446-453

   The dynamical viscoelasticity of ABS melts with different particle size was investigated at various levels of rubbery phase contents. The effects of the rubber are more pronounced in the terminal zone: a transition from viscoelastic liquid to viscoelastic solid behavior was observed which can be interpreted as a physical gelation occurring at a critical rubbery phase content. This critical content resulted in being smaller in the case of smaller particles. A quantitative explanation of the experimental findings was proposed in terms of the average interparticle distance and overlapping of the chains grafted onto the neighboring rubber particles. The gel-like transition appeared to correspond to an approximately constant level of grafted chains overlapping.Presented in part at the Symposium Recent Developments in Structured Continua, Montreal (Canada), 26–28 May 1993.     

   17.

   The time-dependent stress-optical behavior of polycarbonate in the glass transition region     

   Dipl. Ing. R. Wimberger-Friedl  Dipl. Ing. J. G. De Bruin 《Rheologica Acta》1991,30(5):419-429

   The mechanical and stress-optical behavior of Bisphenol-A polycarbonate was investigated in the glass-transition region. For this purpose, optical creep experiments were carried out in shear and elongation on a tensile tester specially designed for use on a microscope state. A Kohlrausch Williams Watts equation (KWW) with a temperature-independent parameter could successfully be applied to the curves describing the time-dependent values of the stress-optical coefficient for several temperatures. The temperature dependence of the corresponding retardation time could be established and described by the WLF equation. For variable stresses the time-dependent birefringence is obtained from a generalized linear stress-optical rule as modeled according to linear superposition. The time-temperature superposition principle was applied to all measurements. With the dynamic moduli some deviations were observed at the transition from the rubbery plateau to the relaxation. The strain-optical coefficient was found to decrease with increasing time and strain. The strain dependence was found to be independent of temperature at constant stress.     

   18.

   The glass transition in carbon black reinforced rubber     

   P. P. A. Smit 《Rheologica Acta》1966,5(4):277-283

   Summary It can be stated that the influence of the presence of carbon black in a rubber vulcanisate with regard to the dynamic properties can be explained by assuming physical adsorption of rubber on the black surface. The effect can be represented by assuming an adsorbed layer having different properties from the bulk rubber taking the layer thickness to be at least 20 Å. Adsorption — desorption from this layer causes non-linearity and may contribute to the losses observed in the rubber-glass transition. This adsorption is analogous to the adsorption of simple liquids in microporous systems such as silicagel and activated carbon.     

   19.

   Investigation of laminar to turbulent transition phenomena effects on impingement heat transfer     

   Mustafa Kemal Isman  Philip J. Morris  Muhiddin Can 《Heat and Mass Transfer》2016,52(10):2027-2036

        

   20.

   Thermal contraction and volume relaxation of amorphous polymers     

   R. Greiner  F. R. Schwarzl 《Rheologica Acta》1984,23(4):378-395

   Two experimental techniques are described for the determination of the change of specific volume of polymers with temperature and aging time, which allow measurements between – 160 °C and + 200 °C. Four technical amorphous polymers, PS, PVC, PMMA and PC have been investigated. Volume-temperature curves under constant rate of cooling are presented and interpreted with respect to relaxation processes known from other physical investigations. The rate dependence of dilatometric glass transition temperatures is compared with the time dependence of rheometric glass transition temperatures from shear creep data. Volume relaxation data at constant aging temperature are presented. Aging is found to proceed until very low temperatures in the glassy state for e.g. PMMA.For polystyrene, a comparison is made between the predictions of a very simple theory of volume relaxation due to Kovacs with experimental data, using additional information from volume temperature curves and the time temperature shift of the shear creep transition. The theory predicts a rate of volume relaxation which is much lower than that found by experiment.     

Phase volume and size effects on the terminal relaxation of ABS melts

The dynamical viscoelasticity of ABS melts with different particle size was investigated at various levels of rubbery phase contents. The effects of the rubber are more pronounced in the terminal zone: a transition from viscoelastic liquid to viscoelastic solid behavior was observed which can be interpreted as a physical gelation occurring at a critical rubbery phase content. This critical content resulted in being smaller in the case of smaller particles. A quantitative explanation of the experimental findings was proposed in terms of the average interparticle distance and overlapping of the chains grafted onto the neighboring rubber particles. The gel-like transition appeared to correspond to an approximately constant level of grafted chains overlapping.Presented in part at the Symposium Recent Developments in Structured Continua, Montreal (Canada), 26–28 May 1993.     

The time-dependent stress-optical behavior of polycarbonate in the glass transition region

The mechanical and stress-optical behavior of Bisphenol-A polycarbonate was investigated in the glass-transition region. For this purpose, optical creep experiments were carried out in shear and elongation on a tensile tester specially designed for use on a microscope state. A Kohlrausch Williams Watts equation (KWW) with a temperature-independent parameter could successfully be applied to the curves describing the time-dependent values of the stress-optical coefficient for several temperatures. The temperature dependence of the corresponding retardation time could be established and described by the WLF equation. For variable stresses the time-dependent birefringence is obtained from a generalized linear stress-optical rule as modeled according to linear superposition. The time-temperature superposition principle was applied to all measurements. With the dynamic moduli some deviations were observed at the transition from the rubbery plateau to the relaxation. The strain-optical coefficient was found to decrease with increasing time and strain. The strain dependence was found to be independent of temperature at constant stress.     

The glass transition in carbon black reinforced rubber

Summary It can be stated that the influence of the presence of carbon black in a rubber vulcanisate with regard to the dynamic properties can be explained by assuming physical adsorption of rubber on the black surface. The effect can be represented by assuming an adsorbed layer having different properties from the bulk rubber taking the layer thickness to be at least 20 Å. Adsorption — desorption from this layer causes non-linearity and may contribute to the losses observed in the rubber-glass transition. This adsorption is analogous to the adsorption of simple liquids in microporous systems such as silicagel and activated carbon.     

Thermal contraction and volume relaxation of amorphous polymers

Two experimental techniques are described for the determination of the change of specific volume of polymers with temperature and aging time, which allow measurements between – 160 °C and + 200 °C. Four technical amorphous polymers, PS, PVC, PMMA and PC have been investigated. Volume-temperature curves under constant rate of cooling are presented and interpreted with respect to relaxation processes known from other physical investigations. The rate dependence of dilatometric glass transition temperatures is compared with the time dependence of rheometric glass transition temperatures from shear creep data. Volume relaxation data at constant aging temperature are presented. Aging is found to proceed until very low temperatures in the glassy state for e.g. PMMA.For polystyrene, a comparison is made between the predictions of a very simple theory of volume relaxation due to Kovacs with experimental data, using additional information from volume temperature curves and the time temperature shift of the shear creep transition. The theory predicts a rate of volume relaxation which is much lower than that found by experiment.