Determination of discrete relaxation and retardation time spectra from dynamic mechanical data

A powerful but still easy to use technique is proposed for the processing and analysis of dynamic mechanical data. The experimentally determined dynamic moduli,G() andG(), are converted into a discrete relaxation modulusG(t) and a discrete creep complianceJ(t). The discrete spectra are valid in a time window which corresponds to the frequency window of the input data. A nonlinear regression simultaneously adjust the parametersg _i , _i ,i = 1,2, N, of the discrete spectrum to obtain a best fit ofG, G, and it was found to be essential that bothg _i and _i are freely adjustable. The number of relaxation times,N, adjusts during the iterative calculations depending on the needs for avoiding ill-posedness and for improved fit. The solution is insensitive to the choice of initial valuesg _i,0, _i,0,N ₀. The numerical program was calibrated with the gel equation which gives analytical expressions both in the time and the frequency domain. The sensitivity of the solution was tested with model data which, by definition, are free of experimental error. From the relaxation time spectrum, a corresponding discrete set of parametersJ ₀,, J _d,i and _i of the creep complianceJ(t) can then readily be calculated using the Laplace transform.This paper is dedicated to Professor Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

An automated dynamic viscometer with air bearing and inductive transducers for angular displacement detection

A dynamic viscometer is described, with which the dynamic moduli in the frequency range from 2 × 10^–3 to 10 Hz can be determined for liquids with 10^–3 Pa< |G ^* | <10² Pa. Due to the application of an air bearing and inductive transducers for the detection of the angular displacement of both the drive and the measuring cylinder a sensitive apparatus has been made. Very small strains (₀ 10^–3) can be applied and only a small amount of sample (4 ml) is needed. The operation of the apparatus is fully computer-controlled, thus, long runs at various frequencies and temperatures are possible without operator intervention. The theoretical background, calibration procedure, and operation window are described. A presentation of some measurements on two polyisoprene/polystyrene triblock copolymer solutions concludes the work.     

Messung der dynamischen Eigenschaften verdünnter Polymerlösungen

Zusammenfassung Ein neuartiges Rheometer zur Ermittlung der rheologischen Eigenschaften von Polymerlösungen im linear-viskoelastischen Bereich wird vorgestellt. Durch Anwendung der Laser-Doppler-Anemometrie, die eine im hydro-dynamischen Sinn störungsfreie Meßmethode darstellt, ist es möglich, niedrig-viskose bis hin zu wasserähnlichen Proben zu vermessen.Zur Bewältigung der in großer Zahl anfallenden Daten und Rechenoperationen findet ein Mikrocomputer-System Verwendung. Es erledigt im einzelnen die Aufgaben Versuchsablauf-Steuerung, Meßwerterfassung, Auswertung, Dokumentation und graphische Darstellung der Ergebnisse.Anhand von Messungen an newtonschen Flüssigkeiten werden der Funktionsnachweis erbracht und die Betriebsbereiche der Ringspaltsysteme unterschiedlichen Durchmessers ermittelt.Die viskoelastischen Eigenschaften von wäßrigen Polyethylenoxid- und Polyacrylamid-Lösungen niedriger Viskosität lassen sich über einen Frequenzbereich von 0,1 bis 100 Hz messen. Die Ergebnisse werden mit Hilfe der komplexen Viskositätsfunktion dargestellt. Daraus läßt sich als gleichwertige Darstellung das diskrete Relaxationsspektrum eines konkreten Stoffgesetzes gewinnen.

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A novel rheometer for determining the elastic characteristics of polymeric solutions in the linear-viscoelastic region is described. Laser-Doppler anemometry, which does not introduce hydrodynamic disturbances, is used to enable low viscosity liquids including water-like samples to be measured.A microcomputer system is utilized to control the experiments, store and analyse the measured data, and provide graphical representations of the results. Measurements on Newtonian liquids verified the validity of the method and allowed the operational regions of the annulus systems with different diameters to be determined.The elastic characteristics of low viscosity aqueous solutions of polyethyleneoxide and polyacrylamide can be measured at frequencies from 0.1 to 100 Hz. The results are presented using the complex viscosity function. As a result the discrete relaxation spectrum of a special material law can be obtained as an equivalent representation.

     

Simple linear viscoelastic models of dilute solutions of polymer molecules and emulsion droplets

The complex viscosity of microemulsions shows relaxation processes of which the largest relaxation time is about 10^–5 s or less. This time can be attributed to relaxation of stresses in the surface of emulsion droplets pertaining to interfacial tension. Superimposed on a spherical droplet surface shape fluctuations can occur due to thermal energies. Our aim is to show the influence of thermal shape fluctuations on the complex viscosity of emulsions. The method used in the derivation has also been applied to inflexible rods to demonstrate its feasibility by showing the formal rheological equivalence of in length thermally fluctuating rods and Rouse's simple model of polymers. The emulsion results have been applied to a dilution series of a non-ionic microemulsion.     

Using regularization methods for the determination of relaxation and retardation spectra of polymeric liquids

The application of Tikhonov regularization to the determination of relaxation and retardation spectra of viscoelastic fluids is discussed using simulated and real experimental data. It is thereby shown that with this method consistent results for relaxation and retardation spectra can be obtained from experimental data for different material functions. Furthermore, the differences between the most frequently used variants of Tikhonov regularization and the maximum entropy method are discussed. For most calculations the program FTIKREG has been used, which is an especially fast and reliable implementation of Tikhonov's regularization method.     

A simplified cooperative model of stress relaxation and other consolidation processes in solids

This paper explores the properties of a relaxation function derived from a differential equation mimicking the distribution mechanism of Bose-Einstein statistics in the time domain. Within a significant portion of the process, the relaxation quantity n decreases linearly with log time. The relation between dn/dt and n is an exponential one. In this respect, the present approach produces results largely equivalent to those obtained using the hypothesis of stress-dependent thermal activation or a box-like spectrum of relaxation times, . The spectrum of the model proposed here is discrete, with integer valued fractions of a characteristic -centering the equations.     

Some inverse problems in rheology leading to integral equations

Two inverse problems of the integral type, which are of the general importance to rheology, are discussed. The first of them is the possibility of calculating the molecular weight distribution (MWD) from the flow curve and the second one is the interrelation between creep and relaxation functions. It was shown that the first problem is incorrect and any solution is unstable in respect to minor experimental errors. It means that the general solution of this problem is impossible in principle and only estimations of the width of unimodal MWD can be received from the curvature of the flow curve. The possibility of the correct calculation of the creep function exists in case the relaxation curve being approximated by the sum of exponential members. But the approximation of the relaxation curve within the experimental accuracy is the necessary, but not sufficient condition for the correct solution of this integral inverse problem, because not every mode of approximation leads to the satisfactory prediction of creep function.Delivered as the Courtaulds Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Generalized characteristic of polymer blend melt viscosity

A very simple reduction procedure is suggested for the blend viscosities of different polymer pairs. This procedure is based on the comparison of the blend viscosity, normalized either to the matrix or to the disperse phase viscosity, with the viscosities ratio of the initial polymers ( _m / _d ). We have obtained, for 13 different pairs containing 30% of the second component, the universal linear dependencies, mutual analysis of which allows connection of their special points with the stream morphology. The fibrillous morphology takes place in the range of _m / _d = 0, 1–5. Simultaneous, the thin skin consisting of the disperse phase polymers is formed. These results confirm the predominant role of the viscosities ratio in fibrillar composite material formation in comparison with the interphase tension phenomena.     

The influence of flow-induced non-Newtonian fluid properties on turbulent drag reduction

Friction factors and velocity profiles in turbulent drag reduction can be compared to Newtonian fluid turbulence when the shear viscosity at the wall shear rate is used for the Reynolds number and the local shear viscosity is used for the non-dimensional wall distance. On this basis, an apparent maximum drag reduction asymptote is found which is independent of Reynolds number and type of drag reducing additive. However, no shear viscosity is able to account for the difference between the measured Reynolds stress and the Reynolds stress calculated from the mean velocity profile (the Reynolds stress deficit). If the appropriate local viscosity to use with the velocity fluctuation correlations includes an elongational component, the problem can be resolved. Taking the maximum drag reduction asymptote as a non-Newtonian flow, with this effective viscosity, leads to agreement with the concept of an asymptote only when the solvent viscosity is used in the non-dimensional wall distance.     

Rheological properties of service weathered road bitumens

A study was carried out of the rheological properties of service weathered bitumens and their properties were compared with road performance. Bitumen from 39 test sites was recovered from the uppermost layer of stone particles covering the road surfacing and tested under dynamic and transient loading. Dynamic testing was carried out under forced sinusoidal loading. Testing under transient loading was mainly with a viscosity test conducted at 45°C, but some creep testing in compression was conducted at 0°C. From the results of dynamic testing, master curves of modulus and loss angle were constructed, spanning over ten decades in loading frequency. The hyperbolic expressions of Dickinson and Witt successfully described the frequency dependence of bitumen modulus and loss angle. The Williams, Landel and Ferry (WLF) equation, with newly derived coefficients, described the temperature dependence of the shift factor (or Newtonian viscosity) for the temperature range –10 to 60°C. Attempts were made to compare the measured transient response with that calculated from dynamic results. The result of the viscosity test conducted at 45°C as an indicator of modulus at low temperatures was assessed. Surfacing distress increased as the bitumen viscosity or modulus increased, however service performance of the bitumens was best correlated with the modulus calculated at conditions representative of traffic stressing and lowest site temperature. The changes in the rheological properties of bitumens induced by weathering is discussed.     

Stokes' first problem for linear viscoelastic fluids with finite memory

We consider Stokes' first problem for a viscoelastic fluid. The memory of the fluid is truncated to a finite time interval and discontinuities in the stress relaxation modulus or its derivatives are allowed at the point of truncation. We investigate secondary waves which are generated by the interaction of these singularities in the memory with earlier waves.Dedicated to Prof. Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Bimodal model of slurry viscosity with application to coal-slurries. Part 1. Theory and experiment

The rheological behavior of stable slurries is shown to be characterized by a bimodal model that represents a slurry as made up of a coarse fraction and a colloidal size fine fraction. According to the model, the two fractions behave independently of each other, and the non-Newtonian behavior of the viscosity is solely caused by the colloidal fraction, while the coarse fraction increases the viscosity level through hydrodynamic interactions. Data from experiments run with colloidal coal particles of about 2–3 µm average size dispersed in water show the viscosity of these colloidal suspensions to exhibit a highly shearrate-dependent behavior and, in the high shear limit, to match very closely the viscosity of suspensions of uniform size rigid spheres although the coal volume fraction must be determined semi-empirically. Different amounts of coarse coal particles are added to the colloidal suspension and the viscosity of the truly bimodal slurries measured as a function of shear rate. In agreement with the bimodal model, the measured shear viscosities show the coarse fraction to behave independently of the colloidal fraction and its contribution to the viscosity rise to be independent of the shear rate. It is shown that the shear rate exerted on the colloidal fraction is higher than that applied by the viscometer as a result of hydrodynamic interactions between the coarse particles, and that it is this effective higher shear rate which is necessary to apply in the correlations. For determining the coal volume fraction a relatively simple and quite accurate measurement technique is developed for determining the density and void fraction of coarse porous particles; the technique directly relates volume fraction to mass fraction.     

A slit viscometer for the measurement of the viscosity of aqueous systems at high temperatures

A slit viscometer has been constructed to measure the viscosity of aqueous systems at temperatures up to 140 °C. Liquid is forced backwards and forwards through the slit by the use of varying air pressure. The flow rate is obtained from the time for the liquid to pass conductivity probes located in liquid reservoirs either side of the slit. The pressure difference between two points on the slit wall is determined using a differential pressure transducer. By varying the slit height measurements can be made on liquids with viscosities in the range 10 to 10^–3 Pa s. Shear rates from 10 to 10⁴ s^–1 can be achieved. A simple microcomputer control system enables the shear stress to be automatically increased and decreased stepwise. Representative data on polysaccharide solutions and strach suspensions are presented. The viscometer is particularly well-suited for following temperature-dependent biopolymer transitions and the thermal depolymerisation of water soluble polymers.     

A simple integral constitutive equation for polymeric liquids

An integral constitutive equation describing the strain history in terms of the Cauchy-Green and Finger tensors is presented. Its memory functions, which correspond to a flow-dependent continuous spectrum of relaxation times, depend on the invariants of the Cauchy-Green and Finger rate of strain tensors. Theoretical predictions resulting from this equation are compared with experimental data for some types of flow. Using the Laguerre polynomial expansion, the basic equation is generalized.     

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     

The steady shear viscosity of filled polymeric liquids described by a linear superposition of two relaxation mechanisms

Filled polymeric liquids often exhibit apparent yielding and shear thinning in steady shear flow. Yielding results from non-hydrodynamic particle—particle interactions, while shear thinning results from the non-Newtonian behavior of the polymer melt. A simple equation, based on the linear superposition of two relaxation mechanisms, is proposed to describe the viscosity of filled polymer melts over a wide range of shear rates and filler volume fraction.The viscosity is written as the sum of two generalized Newtonian liquid models. The resulting equation can describe a wide range of shear-thinning viscosity curves, and a hierarchy of equations is obtained by simplifying the general case. Some of the parameters in the equation can be related to the properties of the unfilled liquid and the solid volume fraction. One adjustable parameter, a yield stress, is necessary to describe the viscosity at low rates where non-hydrodynamic particle—particle interaction dominate. At high shear rates, where particle—particle interactions are dominated by interparticle hydrodynamics, no adjustable parameters are necessary. A single equation describes both the high and low shear rate regimes. Predictions of the equation closely fit published viscosity data of filled polymer melts. n power-law index - n ₁,n ₂ power-law index of first (second) term - shear rate - steady shear viscosity - ₀ zero-shear rate viscosity - _{0, 1}, _{0, 2} zero-shear rate viscosity of first (second) term - time constant - ₁, ₂ time constant of first (second) term - µ _r relative viscosity of filled Newtonian liquid - ₀ yield stress - ø solid volume fraction - ø _m maximum solid volume fraction     

Bimodal model of slurry viscosity with application to coal-slurries. Part 2. High shear limit behavior

It is shown that in a truly bimodal coal-water slurry the hydrodynamic interactions between the coarse particles impose on the fine fraction a shear rate higher than that applied externally by the viscometer walls. A semi-empirical function of the coarse volume fraction is obtained for this correction factor to the applied shear rate. The derivation of this shear correction factor is based on lubrication concepts and introduces the maximum packing fraction,ø _m, at which flow can take place.ø _m is obtainable from a simple dry packing experiment. It is shown that the contribution of the coarse particles to the viscosity rise can be successfully described by a viscosity model employing the same concepts used to derive the shear correction factor. The bimodal model is applied in the high shear limit to polymodal coal slurries with a continuous particle size distribution. In the model, the contribution of the coarse particles to the viscosity rise is taken from separate viscosity measurements for the coarse coal particles, while the contribution to the viscosity of the fine coal particles is taken to be that given by the measured viscosity of colloidal suspensions of monomodal rigid spheres. It is shown that there is a ratio of coarse to fine fraction volumes in the continuous size distribution, corresponding to a specific separating particle size, for which the measured viscosities of the polymodal slurries match almost perfectly over the whole solids volume fraction range with the viscosity values obtained using the bimodal approach. The match is found to be relatively insensitive to the precise value of the separating particle size.     

Rheological properties of skim milk gels at various temperatures; interrelation between the dynamic moduli and the relaxation modulus

The rheological properties of rennet-induced skim milk gels were determined by two methods, i.e., via stress relaxation and dynamic tests. The stress relaxation modulusG _c (t) was calculated from the dynamic moduliG andG by using a simple approximation formula and by means of a more complex procedure, via calculation of the relaxation spectrum. Either calculation method gave the same results forG _c (t). The magnitude of the relaxation modulus obtained from the stress relaxation experiments was 10% to 20% lower than that calculated from the dynamic tests.Rennet-induced skim milk gels did not show an equilibrium modulus. An increase in temperature in the range from 20° to 35 °C resulted in lower moduli at a given time scale and faster relaxation. Dynamic measurements were also performed on acid-induced skim milk gels at various temperatures andG _c (t) was calculated. The moduli of the acid-induced gels were higher than those of the rennet-induced gels and a kind of permanent network seemed to exist, also at higher temperatures. G storage shear modulus,N·m^–2; - G loss shear modulus,N·m^–2; - G _c calculated storage shear modulus,N·m^–2; - G _c calculated loss shear modulus,N·m^–2; - G _e equilibrium shear modulus,N·m^–2; - G _ec calculated equilibrium shear modulus,N·m^–2; - G(t) relaxation shear modulus,N·m^–2; - G _c (t) calculated relaxation shear modulus,N·m^–2; - G ^*(t) pseudo relaxation shear modulus,N·m^–2; - H relaxation spectrum,N·m^–2; - t time,s; - relaxation time,s; - angular frequency, rad·s^–1. Partly presented at the Conference on Rheology of Food, Pharmaceutical and Biological Materials, Warwick, UK, September 13–15, 1989 [33].     

Attempts to find a molecular theory for which the high-frequency dynamic viscosity is less than the solvent viscosity

It has been found that for some dilute polymer solutions the dynamic viscosity at very high frequencies is less than the zero-shear-rate solvent viscosity. Such an effect cannot be explained by the usual kinetic theories using bead-spring-rod models. Here we examine several modifications of the kinetic theories that might be expected to explain the experimental facts.This paper is dedicated to Prof. Hanswalter Giesekus on the occasion of his retirement as Editor of Rheologica Acta.     

Generalized Cole-Cole behavior and its rheological relevance

Starting from an analysis of the rheological behavior of the complex modulus predicted by the Cole-Cole formalism, a generalized Cole-Cole ansatz is suggested in order to overcome the related difficulties. The corresponding rheological constitutive equation with fractional derivatives belonging to the generalized Cole-Cole respondance is stated and the characteristic material functions of the linear viscoelasticity theory (like the dynamic modulus and compliance, the relaxation and ratardation functions, the spectra, etc.) are derived. Model predictions of these functions will be compared with experimental results from dynamical measurements and creep data on different polymer systems which show cooperative phenomena (polymeric glasses and gelling systems). One can see that the modified ansatz fits the data very well, in spite of its relative simplicity.