Dynamics of coaxial torsional resonators consisting or segments with different radii,material properties and surrounding media

High-Q torsional resonators constitute the most sensitive transducers for high frequency dynamic viscoelastic measurements of dilute polymer solutions. Most such resonators described in the literature are segmented. Because of the need for torque and torsional displacement transducers the Q-value of the individual segments most often differ, but normally all segments have the same radius.A detailed analysis of the dynamics of such resonators when both the radii, material properties and surrounding media may be different for each segment, is presented. For resonators where all segment lengths equal an integer multiple of a quarter of the torsional wavelength, we find that the Q-value of the resonator as a whole is mainly determined by the Q-value of the segment with the smallest radius. We further find that reduction of the radius of the segment surrounded by polymer solution results in a stronger mechanical coupling between the resonator as a whole and the polymer solution. These findings suggest that the segment radii are important optimization parameters of segmented torsional resonators used to measure the high frequency dynamic viscoelastic properties of e.g. polymer solutions.     

Driven torsion pendulum for measuring the complex shear modulus in a steady shear flow

To investigate the viscoelastic behavior of fluid dispersions under steady shear flow conditions, an apparatus for parallel superimposed oscillations has been constructed which consists of a rotating cup containing the liquid under investigation in which a torsional pendulum is immersed. By measuring the resonance frequency and bandwidth of the resonator in both liquid and in air, the frequency and steady-shear-rate-dependent complex shear modulus can be obtained. By exchange of the resonator lumps it is possible to use the instrument at four different frequencies: 85, 284, 740, and 2440 Hz while the steady shear rate can be varied from 1 to 55 s^–1. After treatment of the theoretical background, design, and measuring procedure, the calibration with a number of Newtonian liquids is described and the accuracy of the instrument is discussed.Notation a radius of the lump - A geometrical constant - b inner radius of the sample holder - c constant - C ₁, C ₂ apparatus constants - D damping of the pendulum - e _x , e _y , e _z Cartesian basis - e _r , e , e _z orthonormal cylindrical basis - E geometrical constant - E _t , ⁰ E _t , _t relative strain tensor - f function of shear rate - F _t relative deformation tensor - G (t) memory function - G ^* complex shear modulus - G Re(G ^* ) - G Im(G ^* ) - h distance between plates - H ^* transfer function - , functional - i imaginary unit: i ²= – 1 - I moment of inertia - J _exc excitation current - J ₀ amplitude of J _exc - k ^* = k – ik complex wave number - K torsional constant - K fourth order tensor - l length of the lump - L mutual inductance - M _dr driving torque - M _liq torque exerted by the liquid - ⁰ M _liq, _liq steady state and dynamic part of Mliq - n power of the shear rate - p isotropic pressure - Q quality factor - r radial position - R,R ₀, R _c Re(Z ^*, Z ₀ ^* , Z _c ^* ) - s time - t, t time - T temperature - T, ⁰ T, stress tensor - u velocity - U lock-in output - ₀ velocity - V _det detector output voltage - V _sig, V _cr signal and cross-talk part of V _det - x Cartesian coordinate - X , X ₀, X _c Im(Z ^*, Z ₀ ^* , Z _c ^* ) - y Cartesian coordinate - z Cartesian coordinate, axial position     

Methode zum Messen des Langzeitverhaltens von Thermoplasten

Zusammenfassung Es wird eine neue berührungslose optische Dehnungsmeßeinrichtung für biaxiale Langzeit-Zug- bzw. Torsionsversuche vorgestellt. Die Ergebnisse der Versuche an kugel- und kurzglasfasergefülltem Polyamid 6 (PA6) zeigen, daß dieses System genügend genau arbeitet. Das untersuchte Material weist sowohl im Zugversuch als auch im Torsionsversuch nichtlineares Verhalten auf. Bei überlagerten Zug/Torsions-Belastungen verschieben sich die Dehnungen, verglichen mit denen bei gleicher einaxialer Last zu höheren Werten. Durch die Füllstoffzugabe ergeben sich unterschiedliche Verstärkungseffekte. Die faserverstärkten Proben sind erwartungsgemäß eher in Richtung der Rohrachse verstärkt, während das kugelgefüllte System eine höhere Verstärkungswirkung bei Torsionsbelastung erkennen läßt.

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A new optical device for strain measurement in long-term tension and torsion tests is presented. The new system has been compared with others and the results of the experiments with glass bead and short, glas-fiber-filled Polyamid 6 show that it works well. With a dead-weight loading machine used it is possible to perform combined tension/torsion creep experiments or simple tension and simple torsion tests. The materials under consideration here exhibit nonlinear behavior in tension as well as in torsion tests. Under combined loading, which is tensile stress with a superimposed torsional stress, or vice versa, the amounts of the strains compared with those of the one-dimensional stress state are increasing. The fillers produce different reinforcement effects. The bead-filled system is stiffer when subjected to a torsional load, which may be explained by different load carrying mechanism in tension and torsion. The glassfiber-filled system is stiffer in axis direction, because the fibers are aligned to a considerable extent by the flow of the melt during molding.

Vortrag gehalten auf der Jahrestagung der Deutschen Rheologischen Gesellschaft, TH Darmstadt, 19.–21. 4. 1989.     

Methode zum Messen des Langzeitverhaltens von Thermoplasten

Zusammenfassung Es wird eine neue berührungslose optische Dehnungsmeßeinrichtung für biaxiale Langzeit-Zug- bzw. Torsionsversuche vorgestellt. Die Ergebnisse der Versuche an kugel- und kurzglasfasergefülltem Polyamid 6 (PA 6) zeigen, daß dieses System genügend genau arbeitet. Das untersuchte Material weist sowohl im Zugversuch als auch im Torsionsversuch nichtlineares Verhalten auf. Bei überlagerten Zug/Torsions-Belastungen verschieben sich die Dehnungen, verglichen mit denen bei gleicher einaxialer Last zu höheren Werten. Durch die Füllstoffzugabe ergeben sich unterschiedliche Verstärkungseffekte. Die faserverstärkten Proben sind erwartungsgemäß eher in Richtung der Rohrachse verstärkt, während das kugelgefüllte System eine höhere Verstärkungswirkung bei Torsionsbelastung erkennen läßt.

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A new optical device for strain measurement in long-term tension and torsion tests is presented. The new system has been compared with others and the results of the experiments with glass bead and short, glas-fiber-filled Polyamid 6 show that it works well. With a dead-weight loading machine used it is possible to perform combined tension/torsion creep experiments or simple tension and simple torsion tests. The materials under consideration here exhibit nonlinear behavior in tension as well as in torsion tests. Under combined loading, which is tensile stress with a superimposed torsional stress, or vice versa, the amounts of the strains compared with those of the one-dimensional stress state are increasing. The fillers produce different reinforcement effects. The bead-filled system is stiffer when subjected to a torsional load, which may be explained by different load carrying mechanism in tension and torsion. The glass-fiber-filled system is stiffer in axis direction, because the fibers are aligned to a considerable extent by the flow of the melt during molding.

Vortrag gehalten auf der Jahrestagung der Deutschen Rheologischen Gesellschaft, TH Darmstadt, 19.–21. 4. 1989.     

Shear-induced structures in macroscopic dispersions

Observations are reported on structures induced in an oscillatorily sheared dispersion of glass spheres in a silicon oil and in a highly viscoelastic fluid (mixture of polyisobutylenes). The shear flow used in these experiments is oscillatory with a frequency 200 Hz) and has displacement amplitudes of the order of magnitude as the sphere diameter (40 µm).The interpretation of the structures observed is based on non-linear effects which, even when small, in comparison to the viscous effects, may lead to the irreversible formation of structures over a long time, i.e. a time much larger than the period of the shear. A comparison between similar experiments performed with two different suspending fluids (silicon oil, in which the spheres line up perpendicular to the velocity direction; polyisobutylene mixture, in which the spheres line up in the velocity direction) confirm our interpretation.     

Direct measurement of static yield properties of cohesive suspensions

A technique of yield stress investigation based upon the combined use of two devices (an applied stress rheometer and an instrument for measuring the propagation velocity of small amplitude, torsional shear waves) is described. Investigations into the low shear rate rheological properties of illitic suspensions are reported for shear rates, typically, in the range 10^–4— 10^–1 s^–1 under applied stresses in the range 0.01 — 10 Nm^–2 and involving shear strains between 10^–1 and 10^–4. Results are presented which demonstrate that the technique does not invoke the excessive structural disruption of material associated with applied shear rate based methods (direct and otherwise) and the widely encountered problem of wall slip at the surface of rotational measuring devices is avoided using miniature vane geometries. Results are compared with those obtained using smooth-walled cyclindrical measuring devices in both applied stress and applied shear rate instruments.Yield measurements are considered in relation to the structural properties of the undisturbed material state and shear moduli obtained by studying the propagation of small amplitude (10^–5 rad), high frequency (~ 300 Hz) torsional shear waves through the test materials are reported. Experimental techniques and instrument modifications to permit these measurements are described.     

A novel geometry for rheological characterization of viscoelastic materials

A novel geometry for generating a viscometric flow presents advantages of both cone and plate and parallel plate geometries, regarding uniform shear field and adjustable range of measurement. Kinematics and dynamics of the generated flow have been described mathematically utilizing an orthogonal curvilinear coordinate system based on the shapes of the shearing surfaces which are similar to the surface that generates the flow. Simple equations that allow the calculation of quantities of experimental interest in the rheological characterization of liquid materials, namely, shear rate, shear stress and two normal stress differences, have also been derived.The geometry, called pseudosphere, was tested with two types of fluids (Newtonian and pseudoplastic). Results show that the geometry can be used with low viscosity liquids (Newtonian liquids) by only adjusting the gapH. The behavior of pseudoplastic fluids for both low and moderately high viscosity could also be studied with this geometry. Very reproducible results were obtained when compared with those obtained with cone and plate geometry. Regions of lower shear rate could be studied using only the pseudosphere geometry.     

Viscoplasticity of agglomerated suspensions

This work is a theoretical study on the effects of agglomeration on the fluidity and plasticity of a suspension of neutrally buoyant particles in a Newtonian fluid. The dynamics of a cluster of permanently attached spherical particles in a simple shear field is analyzed. The viscous and plastic components of the drag force acting on each of the agglomerated particles is then calculated and found to depend on the size of the individual particle unit, its location being relative to the center of the cluster and the material properties of the engulfing fluid. This information in conjunction with the knowledge of the interparticle cohesive forces is used to establish criteria for the agglomerate size reduction during dispersive mixing. From the kinematics of the cluster movement and the forces acting on its particulate components the rate of energy dissipation is calculated and utilized to evaluate the viscosity and yield stress of the suspension. These rheological parameters depend on the volume fraction and architecture of the agglomerate, the number of fused particles per cluster, and the viscosity of the suspending fluid. The analysis is also extended to include the case of polydispersity of agglomerate sizes.     

Lubricated and nonlubricated squeezing flow of a double layered array of two power law liquids

Force-time relationships of a double-layered array of two power law liquids in squeezing flow at a constant displacement rate were generated with a computer. As in the case of a single layer, lubrication, or lack of it, has the strongest influence on the magnitude of the forces and the flow pattern. Transient flow regimes that were prominent in the behavior of Newtonian liquid arrays in lubricated squeezing flow were also found in the behavior of the power law liquids. Their prominence was influenced by the liquid's flow index and it was drastically magnified as the differences in the liquid's consistency increased.     

Voltage–amplitude response of alternating current near half natural frequency electrostatically actuated MEMS resonators

This paper uses the Reduced Order Model (ROM) method to investigate the nonlinear-parametric dynamics of electrostatically actuated MEMS cantilever resonators under soft Alternating Current (AC) voltage of frequency near half natural frequency of the resonator. The voltage is between the resonator and a ground plate, and provides a nonlinear parametric actuation for the resonator. Fringe effect and damping forces are included. The resonator is modeled as an Euler–Bernoulli cantilever. Two methods of investigations are compared, Method of Multiple Scales (MMS), and Reduced Order Model. Moreover, the instabilities (bifurcation points) are predicted for both cases, when the voltage is swept up, and when the voltage is swept down. Although MMS and ROM are in good agreement for small amplitudes, MMS fails to accurately predict the behavior of the MEMS resonator for greater amplitudes. Only ROM captures the behavior of the system for large amplitudes. ROM convergence shows that five terms model accurately predicts the steady-states of the resonator for both small and large amplitudes.     

Shear creep and recovery of a technical polystyrene

The torsional creep and recoverable bahaviour of a technical polystyrene is reported over seven orders of magnitude of the value of the compliance from 10^–8 to 10^–1 Pa^–1 and over more than seven decades in time. The results for the recoverable compliance J_R (t) reveal a dispersion region seen between the glass transition and the steady-state recoverable compliance J_e. The limiting value of the final dispersion J_e = 4.7 · 10^–4 Pa^–1 indicates a broad molecular-weight distribution. The steady-state recoverable compliance J_e is independent of the temperature. The temperature dependence of the final dispersion was found to be indistinguishable from that of viscous flow. However, this temperature dependence differs significantly from that of the glass-rubber transition. A proposal has been made for the construction of creep compliance and recoverable compliance over an extended time scale.     

The peculiarities of the rheological behaviour of coal tars

The results are discussed of rheological studies of coal tars with different concentrations of substances insoluble in toluene at periodical, steady-state and combined periodical-steady shear deformations in a wide range of deformation frequencies, rates and amplitudes in the temperature region from 223 to 333 K. The temperatures of structural and mechanical glassingT _g , the activation energies of viscous flow and initial viscoelastic constants of these systems have been determined. Temperature and temperature-frequency dependencies of dynamical parameters have been obtained, the pre-steady-state and the steady-state flow modes of permanent deformation have been studied and thixotropic parameters have been evaluated at the combined action of vibration and permanent deformation.     

Orthogonal superposition of small and large amplitude oscillations upon steady shear flow of polymer fluids

The orthogonal superposition of small and large amplitude oscillations upon steady shear flow of elastic fluids has been considered. Theoretical results, obtained by numerical methods, are based on the Leonov viscoelastic constitutive equation. Steady-state components, amplitudes and phase angles of the oscillatory components of the shear stress, the first and second normal stress differences as functions of shear rate, deformation amplitude and frequency have been calculated. These oscillatory components include the first and third harmonic of the shear stresses and the second harmonic of the normal stresses. In the case of small amplitude superposition, the effect of the steady shear flow upon the frequency-dependent storage modulus and dynamic viscosity has been determined and compared with experimental data available in literature for polymeric solutions. The predicted results have been found to be in fair agreement with the experimental data at low shear rates and only in qualitative agreement at high shear rates and low frequencies. A comparison of the present theoretical results has also been made with the predictions of other theories.In the case of large amplitude superposition, the effect of oscillations upon the steady shear flow characteristics has been determined, indicating that the orthogonal superposition has less influence on the steady state shear stresses and the first difference of normal stresses than the parallel superposition. However, in the orthogonal superposition a more pronounced influence has been observed for the second difference of normal stresses.     

Rubber-like elasticity by boundary element method: Finite deformation of a circular elastic slice

We briefly review the phenomenological theory of rubber-like elasticity and report a microstructural model that leads us to eventually adopt a particular constitutive equation, which includes the Neo-Hookean and the Mooney materials. A numerical implementation of the Boundary Element method for solving a general two-dimensional or axisymmetric finite deformation problem is described and tested with some simple deformations. The resulting program is used to analyse the finite deformation of a circular elastic slice perfectly bonded to two parallel rigid end plates; the bottom plate is stationary and the top plate is given a constant displacement. The problem has a free surface which must be found as part of the solution. The results indicate that the Boundary Element method can be an efficient tool for stress-strain analyses with rubber-like materials.     

On the cone and plate deformation of a rubber-like material: Instability

In this report it is shown that the place boundary-value problem, for a small deformation superimposed on the large cone-and-plate deformation of a Mooney-Rivlin material, has no unique solution at some displacement angles of the cone. The kinematics of the small deformation are chosen such that the problem is reduced to a set of ordinary differential equations. With respect to this restricted class of kinematics the cone-and-plate deformation is unstable.     

Finite rise time step strain modeling of nearly monodisperse polymer melts and solutions

The step shear strain experiment is one of the fundamental transient tests used to characterize the rheology of viscoelastic polymer melts and solutions. Many melts and solutions exhibit homogeneous deformation and stress relaxation; in these cases the transient dynamics can be modeled by completely ignoring momentum effects and imposing singular kinematics. Recently, however, it has been observed that there are certain classes of nearly monodisperse melts and solutions that exhibit anomalous nonhomogeneous deformation and stress relaxation (Morrison and Larson (1990), Larson, Khan, and Raju (1988), Vrentas and Graessley (1982), and Osaki and Kurata (1980)). We demonstrate that, for these classes, a finite rise time must be incorporated, some source of inhomogeneity must be present, and a small amount of added Newtonian viscosity is necessary. We examine five nonlinear and quasilinear models; the Johnson-Segalman, Phan Thien Tanner, Giesekus, White-Metzner, and Larson models. We determine which mathematical features of the models are necessary and/or sufficient to describe the observed experimental behavior.     

Viscoelastic measurement of complex fluids using forced oscillating torsion resonator with continuously varying frequency capability

Traditional torsional resonators, often obtaining the viscoelastic moduli of complex fluids only at one or several given discrete frequencies, lack the continuously varying frequency capability. This is an obvious disadvantage of the traditional torsional resonator technique. This paper presents an improved strategy, based on our previous discrete-frequency-measuring method (Wang et al., J Rheol 52:999–1011, 2008), to overcome such restriction and thus accomplish the continuously varying frequency capability of the traditional torsional resonator for measuring the viscoelastic properties of complex fluids. The feasibility of this strategy is demonstrated with the Newtonian fluids (several water–glycerol solutions) of viscosities varying from 10 to 1,400 cp by using our homemade torsion resonator apparatus in the 10 ~ 2,500 rad/s frequency range (continuous frequencies). Some results for typical viscoelastic polymers (two polyethylene oxide (PEO) aqueous solutions) are also given. Additionally, a comparison of the PEO results is made with the common rheometer technique. It is demonstrated that this improved strategy could enable the traditional torsional resonators, with one oscillating resonance mode, to work as the microrheological technique and the common rheometer technique in the continuous frequency range.     

Relaxation dynamics of selected polymer chain segments and comparison with theoretical models

Simultaneous measurement of infrared dichroism and birefringence is used to study selected polymer segment dynamics in isotopically labeled block copolymers. Two different polymers were studied: polybutadiene and poly (ethylene propylene). The first type consisted of a triblock with a short middle block labeled and a diblock with a short end block labeled, while the second type consisted of a triblock with three equal blocks and the end blocks labeled. Results of step strain experiments at –10°C for polybutadiene and at room temperature for poly(ethylene propylene) indicated that segments located at chain ends relax faster than segments located at chain centers. These experimental data were compared to the predictions of two molecular models: the bead-spring model of Rouse and the tube model of Doi and Edwards, and it was found that both models correctly predict the qualitative features of segmental relaxation. However, the tube-model predictions were closer to the experimental results. In addition, when the effects of orientational coupling interactions between segments in the melt were incorporated into this model, its predictions quantitatively agreed with the experimental results. The orientational coupling coefficient for poly(ethylene propylene) was 0.45 as measured from previous work, and for polybutadiene it was found to be 0.4.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.Dedicated to Prof. R.S. Stein, University of Massachussets at Amherst, USA, on the occasion of his 65th birthday.