Rheology of polydisperse polymers: relationship between intermolecular interactions and molecular weight distribution

An expression of the relaxation function of linear polydisperse polymers is proposed in terms of intermolecular couplings of reptative chains. The relaxation times of each molecular weight are assumed to be shifted according to a tube renewal mechanism accounting for the diffusion of the surrounding chains. The subsequent shift is applied to the relaxation function of each molecular weight obtained from an analytical expression of the complex compliance J ^*(). Therefore the complex shear modulus G ^*() is derived from the overall relaxation function using the probability density accounting for the molecular weight distribution and four species-dependent parameters: a front factor A for zero-shear viscosity, plateau modulus G _N ⁰ , activation energy E and characteristic temperature T . All the main features of the theology of polydisperse polymers are described by the proposed model.     

Constitutive relationships for polymeric materials with power-law distributions of relaxation times

The linear relaxation modulus of polydisperse polymer melts and solutions can often be approximated by a power law,ct ^–m over some range of time,t. If, in addition, the nonlinear rheology is given by a separable integral equation, with a strain-dependent factor typical of those observed experimentally, then some commonly observed empirical rules and equations can be readily derived as approximations, namely the Cox-Merz relationship between complex viscosity and steady-state shear viscosity, Bersted's predictions of steady shear stress and first normal-stress difference from a truncated spectrum of linear relaxation times, and the observation of Koyama and coworkers that the ratio of the nonlinear to the linear time-dependent elongational viscosity is independent of strain rate, over a range of strain rates outside the linear regime.     

A new interpretation for the dynamic behaviour of complex fluids at the sol–gel transition using the fractional calculus

We propose to analyse power law shear stress relaxation modulus observed at the sol–gel transition (SGT) in many gelling systems in terms of fractional calculus. We show that the critical gel (gel at SGT) can be associated to a single fractional element and the gel in the post-SGT state to a fractional Kelvin–Voigt model. In this case, it is possible to give a physical interpretation to the fractional derivative order. It is associated to the power law exponent of the shear modulus related to the fractal dimension of the critical gel. A preliminary experimental application to silica alkoxide-based systems is given.

Molecular dynamics of the glass relaxation process of the soft phase in block copolymers: effect of molecular architecture

To determine the impact of molecular architecture on the molecular dynamics of the glass relaxation processes of soft blocks in different types of block copolymers, model block copolymers with a variation in both molecular architecture and chemical composition were studied. Four block copolymer models, namely, two styrene–butadiene–styrene (S-B-S) block copolymers and two styrene–styrene butadiene–styrene (S-SB-S) were chosen. In each pair of block copolymers, one is linear triblock and the other is star asymmetric. For the sake of comparison, two polybutadiene (PB) homopolymer samples, having similar chain lengths of the PB blocks present in the S-B-S block copolymers, have been investigated. Dynamic mechanical measurements have been carried out for the real and imaginary parts of the complex shear modulus (G′, G”) in the temperature and frequency ranges from −110 to 30 °C and from 10⁻² to 15.9 Hz, respectively. Complete master curves have been constructed for all samples investigated. Moreover, broadband dielectric spectroscopy has been carried out to cover wide temperature and frequency windows, −120 to 0 °C and 10⁻¹ to 10⁷ Hz, respectively. The results showed that the molecular dynamics of the glass relaxation process of the PB or statistical PSB soft phases in the block copolymers is dramatically changed when compared to the PB homopolymer. In addition, the molecular architecture is found to be an important factor in determining the molecular mobility of the soft blocks. The results are discussed in terms of the applied confinement of the counter PS hard phase, block lengths, domain thicknesses and the type of end-to-end junctions between the different polymeric blocks. Paper presented 3rd Annual European Rheology Conference (AERC 2006) April 27–29, 2006, Crete, Greece.     

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].     

The relaxation of concentrated polymer solutions

The focus of this paper is on the viscoelastic properties of concentrated polymer solutions and polymer melts. Dynamic mechanical measurements were performed on various polystyrene/ethylbenzene solutions with polymer concentrations ranging from 40% up to 100% and temperatures from Tg+30°C up to 70°C (230°C for polymer melts). The basis polymers are two commerical grade polystyrenes (BASF) with M _W = 247 kg/mol and 374 kg/mol, respectively. To avoid solvent loss due to evaporating during the measurements, a special sealing technique was used.A phenomenological model which describes quantitatively the relaxation spectrum of concentrated polymer solutions from the flow regime up to the glass transition regime is developed. The relaxation data of the respective polymer melt and the glass transition temperature of the solution are the only input parameters needed. The temperature dependence is described by a universal, concentration invariant WLF-equation. The relaxation spectra are divided into two parts accounting for the entanglement and the segmental relaxation modes, respectively. The relaxation strength related to the flow and entanglement regime scale with c ^2.3, whereas the segmental relaxation strength does not alter with concentration. All relaxation times change with concentration proportional to c ^3.5. Flow curves can be calculated from these relaxation spectra and thus, our results are useful for engineering applications.Roman Symbols a _T Time temperature superposition shift - factor - a _c Time concentration superposition - shift factor in the flow regime - a _c Time concentration superposition - shift factor in the glassy regime - b _T Modulus temperature superposition - shift factor - b _c Modulus concentration shift factor - in the flow regime - b _c Modulus concentration shift factor - in the glassy regime - B Virial coefficients - c Polymer mass fraction kg/kg - c ₁ WLF-parameter - c2 WLF-parameter K - g Relaxation strength of a relaxation Pa mode - G(t) Relaxation modulus Pa - G Storage modulus Pa - G Loss modulus Pa - GN Plateau modulus of linear flexible Pa polymers - (x) Delta function: (0) = 1, - (x<>0)=0 - h() Damping function - H() Relaxation spectrum Pa - J ₀ ^N Recoverable compliance Pa^–1 - m Mass kg - M _c Critical molecular weight kg/mol - M _e Entanglement molecular weight kg/mol - M _w Weight average molecular weight kg/mol - M Number of datapoints - n Scaling exponent - N Number of discrete relaxation modes - T Temperature °C - T _g Glass transition temperature °C - V Volume 1 Greek Symbols Scaling exponent - _f Thermal expansion coefficient K^–1 - Scaling exponent - Shear deformation - Shear rate s^t–1 - Relaxation time s - _c Characteristic relaxation time of thes Cross model - _e Entanglement relaxation time s - Viscosity Pa s - ₀ Zero shear viscosity Pa s - ₀ First normal stress coefficientPa s2 - Segmental friction coefficient - Frequency rad/s Indices f Flow and entanglement regime - g Glass transition regime - i Count parameter - p Polymer - ref Reference state - s Solvent Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland     

Material functions generated by the complex viscosity

Based on the complex viscosity model various steady-state and transient material functions have been completed. The model is investigated in terms of a corotational frame reference. Also, BKZ-type integral constitutive equations have been studied. Some relations between material functions have been derived. C ^–1 Finger tensor - F[], (F ^–1[]) Fourier (inverse) transform - rate of deformation tensor in corotating frame - h(I, II) Wagner's damping function - J (x) Bessel function - m parameter inh (I, II) - m(s) memory function - m _k, n_k integers (powers in complex viscosity model) - P principal value of the integral - parameter in the complex viscosity model - rate of deformation tensor - shear rates - [], [] incomplete gamma function - (a) gamma function - steady-shear viscosity - ^* complex viscosity - , real and imaginary parts of ^* - ₀ zero shear viscosity - ⁺, ₁ ⁺ stress growth functions - ^–, ₁ ^- stress relaxation functions - (s) relaxation modulus - ₁(s) primary normal-stress coefficient - ø(a, b; z) degenerate hypergeometric function - ₁, ₂ time constants (parameters of ^*) - frequency - extra stress tensor     

Determination of the molecular weight distribution of linear polymers by inversion of a blending law on complex viscosities

The method presented in this paper allows to calculate the molecular weight distribution (MWD) of linear homopolymer melts from the complex shear modulus data measured in a wide frequency domain. An empirical blending law on complex viscosities is first developed; as a consequence, the variations of the storage and loss modulus as a function of MWD are presented. This simulation demonstrates also the role of the shape of the MWD itself, and shows that one should not postulate a priori the shape of the MWD. An efficient numerical approach based on a Tikhonov regularization method with constraint is used to solve this ill-posed problem; the MWD is hence derived without any assumption on its shape. This method is first applied on simulated data to prove its numerical efficiency. Then the inversion method is applied on complex moduli data of various commercial polymers (polypropylene, polyethylene and polystyrene) and on an artificial mixture of polystyrene that have been presented in the literature. For amorphous polymers, the coupling of the terminal relaxation domains with the transition region at higher frequency leads to errors in the low molecular weight tail: one way to solve this problem is to cut off the experimental data at the high frequencies. This general method needs only a few physical parameters, namely the scaling law for the Newtonian viscosity η₀=f(M _w ) and the plateau modulus G _N ⁰, and leads to reasonable results with respect to the simplicity of the viscoelastic model used. Received: 27 October 1997 Accepted: 24 February 1998     

Fractional order control of a coupled tank

In this paper, a hybrid system that combines the advantages in terms of robustness of the fractional control and the Sliding Mode Control (SMC) will be proposed. The proposed fractional order SMC is applied to a level control in a nonlinear coupled tank, as a case study. To investigate the capability of the method, a Sliding Mode Controller is alternatively designed. Primarily a sliding surface based on linear compensation networks PD or PID is designed. The work is followed by designation of a fractional form of these networks, PD ^μ or PI ^λ D ^μ . Finally, the performance of the proposed technique is also investigated under disturbance and variation in parameters of system. The simulation results indicate the significance of the fractional order sliding mode controllers.     

Rheological behaviour of asphalts

Summary Stiffness moduli of asphalts of different origins and penetration grades were measured at various temperatures by the sliding plate rheometer. The creep curves obtained were best explained on the basis of Burgers mechanical model. For small loading times and at low temperatures, the stiffness modulus curves converged with the horizontal elastic asymptote and showed maximum values of 5 10⁸ pascals. For longer loading times and at high temperatures the stiffness modulus converged with the viscous asymptote. The stiffness moduli calculated from the combined equations ofSchweyer et al. and ofWilliams-Landell-Ferry for temperatures ranging from –10 to 20°C and those obtained experimentally agreed well only at 20°C. For short loading times (< 50 sec) the experimental stiffiness modulus values agree well with those read from theVan Der Poel nomograph.

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Zusammenfassung Mittels eines Gleit-Platten-Rheometers wurden an einer Reihe von Asphaltproben verschiedener Herkunft und mit verschiedenen Penetrationsgraden bei unterschiedlichen Temperaturen Steifheitsmoduln bestimmt. Die erhaltenen Kriechkurven ließen sich am besten mit Hilfe eines rheologischen Modells nachBurgers darstellen. Bei kurzen Belastungszeiten und niedrigen Temperaturen strebten die Steifheitsmodul-Kurven gegen die das elastische Verhalten beschreibende horizontale Asymptote mit Maximalwerten von 5 10⁸ Pa. Bei längeren Belastungszeiten und höheren Temperaturen strebten diese Kurven jedoch gegen die das viskose Verhalten kennzeichnende Asymptote. Die mittels der kombinierten Gleichungen nachSchweyer et al. und nachWilliams-Landel-Ferry für Temperaturen zwischen –10 und 20°C berechneten Steifheitsmoduln stimmten nur für 20°C gut mit den experimentell ermittelten Werten überein. Für kurze Belastungszeiten (<50 sec.) decken sich die experimentellen Steifheitsmoduln recht genau mit den aus dem Nomogramm nachVan Der Poel abgelesenen Werten.

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With 6 figures and 11 tables     

Estimating the viscoelastic moduli of complex fluids using the generalized Stokes–Einstein equation

We obtain the linear viscoelastic shear moduli of complex fluids from the time-dependent mean square displacement, <Δr ²(t)>, of thermally-driven colloidal spheres suspended in the fluid using a generalized Stokes–Einstein (GSE) equation. Different representations of the GSE equation can be used to obtain the viscoelastic spectrum, G˜(s), in the Laplace frequency domain, the complex shear modulus, G ^*(ω), in the Fourier frequency domain, and the stress relaxation modulus, G _r (t), in the time domain. Because trapezoid integration (s domain) or the Fast Fourier Transform (ω domain) of <Δr ²(t)> known only over a finite temporal interval can lead to errors which result in unphysical behavior of the moduli near the frequency extremes, we estimate the transforms algebraically by describing <Δr ²(t)> as a local power law. If the logarithmic slope of <Δr ²(t)> can be accurately determined, these estimates generally perform well at the frequency extremes. Received: 8 September 2000/Accepted: 9 March 2000     

Finite time stability analysis of PD fractional control of robotic time-delay systems

A finite time stability test procedure is proposed for robotic system where it appears a time delay in PD^α fractional control system. Paper extends some basic results from the area of finite time and practical stability to linear, continuous, fractional order time invariant time-delay systems given in state-space form. Sufficient conditions for this kind of stability, for particular class of fractional time-delay systems are derived.     

Stabilization using fractional-order PI and PID controllers

This paper presents a solution to the problem of stabilizing a given fractional dynamic system using fractional-order PIλ and PI^λD^μ controllers. It is based on plotting the global stability region in the (k _p, k _i)-plane for the PI^λ controller and in the (k _p , k _i , k _d)-space for the PIλDμ controller. Analytical expressions are derived for the purpose of describing the stability domain boundaries which are described by real root boundary, infinite root boundary and complex root boundary. Thus, the complete set of stabilizing parameters of the fractional-order controller is obtained. The algorithm has a simple and reliable result which is illustrated by several examples, and hence is practically useful in the analysis and design of fractional-order control systems.     

Stability analysis of linear fractional differential system with multiple time delays

In this paper, we study the stability of n-dimensional linear fractional differential equation with time delays, where the delay matrix is defined in (R ⁺)^n×n. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. As its an application, we apply our theorem to the delayed system in one spatial dimension studied by Chen and Moore [Nonlinear Dynamics 29, 2002, 191] and determine the asymptotically stable region of the system. We also deal with synchronization between the coupled Duffing oscillators with time delays by the linear feedback control method and the aid of our theorem, where the domain of the control-synchronization parameters is determined.     

Viscoelasticity and diffusion in miscible blends of saturated hydrocarbon polymers

Linear viscoelasticity and tracer diffusion were investigated as functions of temperature, component molecular weight and blend composition for entangled, single-phase blends of nearly monodisperse poly(ethylene-alt-propylene) (PEP) and head-to-head polypropylene (HHPP). Both components are non-polar and, despite evidence for slight differences of component glass temperatures in their blends, the viscoelastic data obey time-temperature superposition rather well. The properties of the blends were compared at constant T-T _g (blend) with predictions of the tube-model theories. The composition dependence of viscosity agrees best with the double-reptation prediction, as had been found earlier for molecular weight blends. The variation in plateau modulus with composition is consistent with reptation, but the changes are too small to provide a definitive test. The tracer diffusion coefficients, D ^* _PEP and D ^* _HHPP are nearly independent of composition, consistent with the reptation prediction and in sharp contrast with tracer diffusion for blends with specific associations. Results for the recoverable compliance depart from this pattern, varying differently and much less strongly with composition than the predictions of either single or double reptation. It thus seems that microstructural blends may behave in significantly more complex ways than molecular weight blends even for components with only weak dispersive interactions and rather modest differences in glass temperature and plateau modulus.Dedicated to Prof. John D. Ferry on the occasion of his 85th birthday.     

Nonlinear viscoelastic properties and change in entanglement structure of linear polymer

Simultaneous measurements of stress relaxation and differential dynamic modulus were made at 268 K over a time scale of 10 to 10⁴⁵ s for nearly monodisperse polybutadiene (M _w =2.2x10⁵, 1,2-structure 70%, M _e =3600) and also one having coarse cross-linking (M _c =29000). Static shear strain ranged from 0.1 to 2.0. In a long-time region (t> _k ), the relaxation modulus G (; t) could be expressed by the product G (0; t) h (y). The observed h() agreed well with the Doi-Edwards theory without use of IA approximation. Both the cured and uncured samples showed initial drop of the differential storage modulus G (), ; t) followed by gradual recovery, but did not attain the value before shearing G (, ; t) for the uncured sample showed smaller values than that for the cured one in the whole measured time scale at the higher strain, confirming the two origins of nonlinear viscoelasticity of well entangled polymer; induced chain anisotropy and induced decrement in entanglement density. G (, ; t) curves for the cured sample agreed well with the BKZ predictions. But the curves for the uncured sample agreed well with the BKZ prediction only at the time scale of t< _k . BKZ prediction showed significant upward deviations at t> _k . Such the differences are discussed in terms of the two origins.Dedicated to Prof. John D. Ferry on the occasion of his 85th birthday.     

On the use of stretched-exponential functions for both linear viscoelastic creep and stress relaxation

The use of the stretched-exponential function to represent both the relaxation function g(t)=(G(t)-G _∞)/(G ₀-G _∞) and the retardation function r(t) = (J _∞+t/η-J(t))/(J _∞-J ₀) of linear viscoelasticity for a given material is investigated. That is, if g(t) is given by exp (?(t/τ)^β), can r(t) be represented as exp (?(t/λ)^µ) for a linear viscoelastic fluid or solid? Here J(t) is the creep compliance, G(t) is the shear modulus, η is the viscosity (η^?1 is finite for a fluid and zero for a solid), G _∞ is the equilibrium modulus G _e for a solid or zero for a fluid, J _∞ is 1/G _e for a solid or the steady-state recoverable compliance for a fluid, G ₀= 1/J ₀ is the instantaneous modulus, and t is the time. It is concluded that g(t) and r(t) cannot both exactly by stretched-exponential functions for a given material. Nevertheless, it is found that both g(t) and r(t) can be approximately represented by stretched-exponential functions for the special case of a fluid with exponents β=µ in the range 0.5 to 0.6, with the correspondence being very close with β=µ=0.5 and λ=2τ. Otherwise, the functions g(t) and r(t) differ, with the deviation being marked for solids. The possible application of a stretched-exponential to represent r(t) for a critical gel is discussed.     

Effect of nanofillers on the phase separation and rheological properties of poly(methyl methacrylate)/poly(styrene-co-acrylonitrile) blends

The viscoelastic characteristics of the blends of poly(methyl methacrylate)/poly(styrene-co-acrylonitrile) (PMMA/SAN) were investigated at various temperatures below, near, and above the phase separation temperature. The investigated polymer system is characterized by a lower critical solution temperature. Rheological behavior of the blends in the region of a phase separation was compared with change of the light scattering intensity. The presence of nanofillers in the blend results in that the phase separation occurs at a higher temperature. At the isothermal conditions, the phase separation begins earlier and proceeds with a higher rate as compared with the same blend without filler. The results of the study show the considerable change of the viscoelastic characteristics of PMMA/SAN when the polymer system passes from the homogeneous state to the heterogeneous one. Such characteristics as the dependence of the storage modulus (G ^′) on the loss modulus (G ^″), the dependence of the loss viscosity (η ^″) on the dynamic viscosity (η ^′), the dependences of the complex viscosity (η*), and the free volume fraction (f) on the blend composition are the most sensitive to the phase separation. The phase separation affects the characteristics G ^′(ω), where ω is the frequency only in a low-frequency range. Temperatures of phase separation were estimated using dependence G ^′(T) at ω, which is the constant in the range of low frequencies.     

On the unsteady rotational flow of fractional Oldroyd-B fluid in cylindrical domains

This paper concerned with the unsteady rotational flow of fractional Oldroyd-B fluid, between two infinite coaxial circular cylinders. To solve the problem we used the finite Hankel and Laplace transforms. The motion is produced by the inner cylinder that, at time t=0⁺, is subject to a time-dependent rotational shear. The solutions that have been obtained, presented under series form in terms of the generalized G functions, satisfy all imposed initial and boundary conditions. The corresponding solutions for ordinary Oldroyd-B, fractional and ordinary Maxwell, fractional and ordinary second grade, and Newtonian fluids, performing the same motion, are obtained as limiting cases of general solutions.     

The Time-Scaled Trapezoidal Integration Rule for Discrete Fractional Order Controllers

In this paper, the time-scaled trapezoidal integration rule for discretizing fractional order controllers is discussed. This interesting proposal is used to interpret discrete fractional order control (FOC) systems as control with scaled sampling time. Based on this time-scaled version of trapezoidal integration rule, discrete FOC can be regarded as some kind of control strategy, in which strong control action is applied to the latest sampled inputs by using scaled sampling time. Namely, there are two time scalers for FOC systems: a normal time scale for ordinary feedback and a scaled one for fractional order controllers. A new realization method is also proposed for discrete fractional order controllers, which is based on the time-scaled trapezoidal integration rule. Finally, a one mass position 1/s^k control system, realized by the proposed method, is introduced to verify discrete FOC systems and their robustness against saturation non-linearity.