Temperature and concentration dependent viscosity and gelation temperature of ABA triblock copolymer solutions

In solutions of ABA-triblock copolymers in a poor solvent for A thermoreversible gelation can occur. A three-dimensional dynamic network may form and, given the polymer and the solvent, its structure will depend on temperature and polymer mass fraction. The zero-shear rate viscosity of solutions of the triblock-copolymer polystyrene-polyisoprene-polystyrene in n-tetradecane was measured as a function of temperature and polymer mass fraction, and analyzed; the polystyrene blocks contained about 100 monomers, the polyisoprene blocks about 2000 monomers. Empirically, in the viscosity at constant mass fraction plotted versus inverse temperature, two contributions could be discerned; one contribution dominating at high and the other one dominating at low temperatures. In a comparison with theory, the contribution dominating at low temperatures was identified with the Lodge transient network viscosity; some questions remain to be answered, however. An earlier proposal for defining the gelation temperature T _gel is specified for the systems considered, and leads to a gelation curve; T _gel as a function of polymer mass fraction.Mathematical symbols {} functional dependence; e.g., f{x} means f is a function of x - p _log logarithm to the base number p; e.g., ¹⁰log is the common logarithm - exp exponential function with base number e - sin trigonometric sine function - lim limit operation - – in integral sign: Cauchy Principal Value of integral, e.g., - derivative to x - partial derivative to x Latin symbols dimensionless constant - b constant with dimension of absolute temperature - constant with dimension of absolute temperature - B dimensionless constant - c mass fraction - dimensionless constant - constant with dimension of absolute temperature - d ^* dimensionless constant - D{0} constant with dimension of absolute temperature - e base number of natural (or Naperian) logarithm - g distribution function of inverse relaxation times - G relaxation strength relaxation function - h distribution function of relaxation times reaction constant enthalpy of a molecule - H Heaviside unit step function - i complex number defined by i ² = –1 - j{0} constant with dimension of viscosity - j index number - k Boltzmann's constant - k _H Huggins' coefficient - m mass of a molecule - n number - N number - p index number - s entropy of a molecule - t time - T absolute temperature Greek symbols as index: type of polymer molecule - as index: type of polymer molecule - shear as index: type of polymer molecule - shear rate - small variation; e.g. T is a small variation in T relative deviation Dirac delta distribution as index: type of polymer molecule - difference; e.g. is a difference in chemical potential - constant with dimension of absolute temperature - (complex) viscosity - constant with dimension of viscosity - [] intrinsic viscosity number - inverse of relaxation time - chemical potential - number pi; circle circumference divided by its diameter - mass per unit volume - relaxation time shear stress - angular frequency     

Evaluation of the dynamic-mechanical properties of solids from a cooperative model for stress relaxation

For many solid materials the stress relaxation process obeys the universal relationF = – (d/d lnt)_max = (0.1 ± 0.01) ( ₀ — _i ), regardless of the structure of the material. Here denotes the stress,t the time, ₀ the initial stress of the experiment and _i the internal stress. A cooperative model accounting for the similarity in relaxation behaviour between different materials was developed earlier. Since this model has a spectral character, the concepts of linear viscoelasticity are used here to evaluate the corresponding prediction of the dynamic mechanical properties, i.e. the frequency dependence of the storageE () and lossE () moduli. Useful numerical approximations ofE () andE () are also evaluated. It is noted that the universal relation in stress relaxation had a counterpart in the frequency dependence ofE (). The theoretical prediction of the loss factor for high-density polyethylene is compared with experimental results. The agreement is good.     

Thermal expansivity of particulate composites: Interlayer versus molecular model

We continue the comparison of the results of an interlayer model, based on the theory of elastic continua, and a molecular model, derived from a theory of mixtures, previously presented in terms of bulk moduli K. We now derive expressions for the dependence of the thermal expansivity _c on the volume fraction _f of the filler, at low and elevated values of _f . Correspondencies between the characteristic parameters, viz. adhesion and repulsion ratios on the one hand, and interlayer content and thermal properties of matrix, filler, and layer, on the other, are examined. Since in the molecular theory both andK are derived from an equation of state, the identical set of parameters determines both functions and suggests correlations between them.     

Influence of the formation of associations of macromolecules on the rheo-optical properties of their dilute solutions. I. Theory

Assuming the formation of doublets in the flow according to a mass action law, the shear rate and the concentration dependence of the extinction angle, of the birefringence, and of the average coil expansion are calculated for dilute solutions of flexible macromolecules. It is shown that this reversible association process has a strong influence on the measurable parameters in a flow birefringence experiment. c concentration (g/cm³) - h ² mean square end-to-end distance at shear rate - h ₀ ² mean-square end-to-end distance at zero-shear rate - n refractive index of the solution (not very different from the solvent for a very dilute solution) - E mean coil expansion - K ₀,K constant of the mass action law - M molecular weight - R _G gas constant - T absolute temperature - ₁ – ₂ optical anisotropy of the segment - ₀ Deborah number: - Deborah number: - shear rate - ₀, reduced concentration - _s viscosity of the solvent - [] ₀ intrinsic viscosity at zero-shear rate - [] intrinsic viscosity at shear rate - extinction angle - N _a Avodagro's number - _n magnitude of the birefringence     

The evolution of linear viscoelasticity during the vulcanization of polyethylene

The evolution of linear viscoelasticity during the vulcanization of polyethylene is studied through the gel point. The material in the vicinity of the gel point is described by two scaling laws: one characterizes the viscoelasticity at the critical point and a second characterizes the evolution of viscoelasticity near the gel point. Time Resolved Mechanical Spectroscopy is used to observe both scaling phenomena. The material at the gel point displays power law relaxation over five decades of time with a power-law relaxation exponent equal to 0.32. This study conforms with previous findings that this exponent is composition-dependent. The longest relaxation time diverges in the vicinity of the gel point as _max |p _c - p| ^–1/, and we find = 0.2. This result conforms with previous reports that this exponent may be independent of composition. The Arrhenius flow activation energy for this material undergoes three-fold changes during crosslinking up to the gel point. A single-adjustable-parameter stretched-exponential-power law relaxation function is an inadequate representation of crosslinked materials over any significant range of extent of the reaction up to the gel point.     

Fragile networks and rheology of concentrated suspensions

Suspensions consisting of particles of colloidal dimensions have been reported to form connected structures. When attractive forces act between particles in suspension they may flocculate and, depending on particle concentration, shear history and other parameters, flocs may build-up in a three-dimensional network which spans the suspension sample. In this paper a floc network model is introduced to interpret the elastic behavior of flocculated suspensions at small deformations. Elastic percolation concepts are used to explain the variation of the elastic modulus with concentration. Data taken from the suspension rheology literature, and new results with suspensions of magnetic -Fe₂O₃ and non-magnetic -Fe₂O₃ particles in mineral oil are interpreted with the model proposed.Non-zero elastic modulus appeared at threshold particle concentrations of about 0.7 vol.% and 0.4 vol.% of the magnetic and non-magnetic suspensions, respectively. The difference is attributed to the denser flocs formed by magnetic suspensions. The volume fraction of particles in the flocs was estimated from the threshold particle concentration by transforming this concentration into a critical volume concentration of flocs, and identifying this critical concentration with the theoretical percolation threshold of three-dimensional networks of different coordination numbers. The results obtained indicate that the flocs are low-density structures, in agreement with cryo-scanning electron micrographs. Above the critical concentration the dynamic elastic modulus G was found to follow a scaling law of the type G ( _f - _f ^c ) ^f , where _f is the volume fraction of flocs in suspension, and _f ^c is its threshold value. For magnetic suspensions the exponent f was found to rise from a low value of about 1.0 to a value of 2.26 as particle concentration was increased. For the non-magnetic a similar change in f was observed; f changed from 0.95 to 3.6. Two other flocculated suspension systems taken from the literature showed a similar change in exponent. This suggests the possibility of a change in the mechanism of stress transport in the suspension as concentration increases, i.e., from a floc-floc bond-bending force mechanism to a rigidity percolation mechanism.     

Zum Mischen rheologisch inhomogener Stoffsysteme auf Einschneckenmaschinen

Zusammenfassung Das Mischen von Stoffen mit unterschiedlichen rheologischen Eigenschaften in Schneckenmaschinen ist in der Kunststoffauf- und -verarbeitung eine Standardaufgabe. Trotzdem gibt es hierfür kein zufriedenstellendes mathematisch-physikalisches Modell. Daher werden zunächst einfache Mischmodelle diskutiert. Auf der Basis dieser Modelle wird dann unter Berücksichtigung der Besonderheiten des Plastifizierextruderprozesses eine Mischgütebeziehung mathematisch formuliert. Die experimentelle Überprüfung erfolgt mit Hilfe der Grauwertanalyse extrudierter Zweistoffsysteme, bei denen ein Stoff mit Ruß eingefärbt war. Da der Mischprozeß hochgradig stochastisch ist, streuen die Meßergebnisse. Unter Berücksichtigung dieses Tatbestandes ist der theoretische Ansatz zufriedenstellend.

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Mixing of polymer resins with different rheological properties is a usual demand in plastics processing using screw extruders. A mathematical model describing this processing problem sufficiently is not known, however. Therefore, simple mixing models will be discussed. Based on these, a concept for the calculation of mixing homogeneity will be presented, including the particular requirement of the plasticating screw process. An experimental investigation utilizes the grey-value analysis of extruded two-component materials, which in one phase is carbon-black filled. Considering the fact that the mixing process is highly random, the theoretical model leads to a good level of aggreement with the scattering measurement data.

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b Schneckenkanalbreite - B Bandbreite der Grauwerte - c Konstante - mittlere Konzentration, bezogen auf die Grauwertbandbreite - h Höhe, Gangtiefe, Schneckenkanalhöhe - h ₀ Gangtiefe der Einzugszone - h ₁ Gangtiefe der Ausstoßzone - L Länge - gemittelte Schmelzebettlänge - n Exponent des Potenzfließgesetzes - s Standardabweichung der Grauwerte bezogen auf die Grauwertbandbreite - S Standardabweichung der Grauwerte - t Verweilzeit - t ₁ kürzeste Verweilzeit - mittlere Verweilzeit - ₀ Umfangsgeschwindigkeit - mittlere Geschwindigkeit - V Volumenstrom - w Dicke eines Kontrollelements - w Ausstreichdicke eines Kontrollelements - x Koordinate - Mittelwert der Grauwerte - y Koordinate - Scherdeformationswinkel - Scherdeformation - mittlere Scherdeformation - Schergeschwindigkeit - Viskosität - ₁ dimensionslose kürzeste Verweilzeit - dimensionsloser Volumenstrom - _LSM laminarer Schermischgrad - _{LSM, the} theoretischer laminarer Schermischgrad - _{LSM, exp} experimenteller laminarer Schermischgrad - ² Varianz der Verweilzeit im Schmelzebett - Schubspannung - Gangsteigungswinkel der Schnecke - ø Volumenanteil - dimensionslose Kennzahl     

On some nearly viscometric flows of viscoelastic liquids

Superposition of oscillatory shear imposed from the boundary and through pressure gradient oscillations and simple shear is investigated. The integral fluid with fading memory shows flow enhancement effects due to the nonlinear structure. Closed-form expressions for the change in the mass transport rate are given at the lowest significant order in the perturbation algorithm. The elasticity of the liquid plays as important a role in determining the enhancement as does the shear dependent viscosity. Coupling of shear thinning and elasticity may produce sharp increases in the flow rate. The interaction of oscillatory shear components may generate a steady flow, either longitudinal or orthogonal, resulting in increases in flow rates akin to resonance, and due to frequency cancellation, even in the absence of a mean gradient. An algorithm to determine the constitutive functions of the integral fluid of order three is outlined.Nomenclature A _n Rivlin-Ericksen tensor of order . - A _k Non-oscillatory component of the first order linear viscoelastic oscillatory velocity field induced by the kth wave in the pressure gradient - d Half the gap between the plates - e _x, e _z Unit vectors in the longitudinal and orthogonal directions, respectively - G(s) Relaxation modulus - G History of the deformation - Stress response functional - I() Enhancement defined as the ratio of the frequency dependent part of the discharge to the frequencyindependent part of it at the third order - I ^*() Enhancement defined as the ratio of the increase in discharge due to oscillations to the total discharge without the oscillations - k Power index in the relaxation modulus G(s) - k _i ^–1 Relaxation times in the Maxwell representation of the quadratic shear relaxation modulus (s ₁, s ₂) - m _i ^–1, n _i ^–1 Relaxation times in the Maxwell representations of the constitutive functions ₁(s ₁,s ₂,s ₃) and ₄ (s ₁, s ₂,s ₃), respectively - P Constant longitudinal pressure gradient - p Pressure field - _mx ,⁽³⁾ _nz ,⁽³⁾ Mean volume transport rates at the third order in the longitudinal and orthogonal directions, respectively - ₀,⁽³⁾, ₁,⁽³⁾ Frequency independent and dependent volume transport rates, respectively, at the third order - s = t- Difference between present and past times t and      

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].     

Pulsatile blood flow through arterioles

An analytical study was made to examine the effect of vascular deformability on the pulsatile blood flow in arterioles through the use of a suitable mathematical model. The blood in arterioles is assumed to consist of two layers — both Newtonian but with differing coefficients of viscosity. The flow characteristics of blood as well as the resistance to flow have been determined using the numerical computations of the resulting expressions. The applicability of the model is illustrated using numerical results based on the existing experimental data. r, z coordinate system - u, axial/longitudinal velocity component of blood - p pressure exerted by blood - _b density of blood - µ viscosity of blood - t time - , displacement components of the vessel wall - T _t0,T ₀ known initial stresses - density of the wall material - h thickness of the vessel wall - T _t,T stress components of the vessel - K _l,K _r components of the spring coefficient - C _l,C _r components of the friction coefficient - M _a additional mass of the mechanical model - r ₁ outer radius of the vessel - thickness of the plasma layer - r ₁ inner radius of the vessel - circular frequency of the forced oscillation - k wave number - E ₀,E _t, , _t material parameters for the arterial segment - µ _p viscosity of the plasma layer - Q total flux - Q _p flux across the plasma zone - Q _h flux across the core region - Q mean flow rate - resistance to flow - P pressure difference - l length of the segment of the vessel     

Rheological characteristics of glass-fibre-filled polypropylene melts

The rheological properties of glass fibre-filled polypropylene melts have been investigated. A high pressure capillary rheometer has been used for the experimental study. The effect of shear rate, temperature, and fibre concentration on the melt viscosity and viscoelastic properties have been studied. An equation has been proposed to correlate the melt viscosity with shear rate, temperature and fibre content. A master curve relation on this basis has been brought out using the shift factora _T . a _T shift factor (=/ _r ) - A _i coefficients of the polynomical of eq. (1) (i = 0, 1, 2, ,n) - B constant in the AFE equation (eq. (2)) (Pa s) - B constant in eq. (3) - D extrudate diameter - d capillary diameter - activation energy at constant shear rate (kcal/mole) - E activation energy at constant shear stress (kcal/mole) - T melt temperature (K) - X fraction glass fibre by weight - shear rate (s^–1) - shear viscosity (Pa s) - normal stress coefficient (Pa s²) - ₁ – ₂ first normal-stress difference (Pa) - shear stress (Pa) - r at reference temperature     

Free convection from a horizontal surface with mass diffusion

The flow due to free convection and mass diffusion of the steady laminar flow of an incompressible fluid on upper side of a hot horizontal surface has been studied. Similar solutions have been obtained for the boundary layer equations under certain restricting assumptions. The effects of mass diffusion on the free convection flow field have been analyzed. Numerical solutions have been obtained for some particular cases of gases with Prandtl number and Schmidt number close to unity.

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Zusammenfassung Es wird die freie Konvektionsströmung und die Diffusion in der inkompressiblen laminaren und stationären Grenzschicht auf der Oberseite einer geheizten Platte behandelt. Die Grenzschichtgleichungen für dieses Problem wurden für diejenigen Fälle gelöst, die zu ähnlichen Lösungen führen. Der Einfluß der Diffusion auf die Strömung wurde untersucht. Numerische Beispiele für einige Fälle mit Prandtl-Zahl und Schmidt-Zahl nahe Eins sind berechnet und angegeben worden.

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Nomenclature b Width of plate perpendicular to the plane of paper - C Concentration of the diffusing species in the main gas - C ₁, C₂ Equation constants, defined in text - C _p Specific heat at constant pressure - D Coefficient of mass diffusion - f _w Flow parameter, dimensionless - h Heat transfer coefficient - l Length of plate from which heat transfer occurs - m ₁ Molecular weight of the ambient gas - m ₂ Molecular weight of the diffusing gas - p Fluid pressure defined in text - u, v Fluid velocity components inx andy-directions respectively. - T Absolute temperature at any point in the free convection field - T ₀ Absolute temperature of the ambient atmosphere - Gr Grashof number,(l ³g T/v²) - Gr_m Diffusion Grashof number, - Gr_TM Gr + Gr_M - Nu Average nusselt number, - Pr Prandtl number,/ - Sc Schmidt number,/D h - Thermal diffusivity, - Coefficient of cubical expansion - Dynamic viscosity - Fluid density - Kinematic viscosity - ()_w Value of quantity at the horizontal surface - ()(x) Local value of the quantity - ()_t Refers to diffusing fluid     

A model for the thermorheological behavior of viscoelastic fluids

Using a power-law ansatz for the temperature dependence of the shear modulus on the level of internal variables, the thermorheological behavior is modeled for viscoelastic fluids of a special group of rheological constitutive equations (rate-type models). The model parameter introduced characterizes thermoelastic contributions. The relation between the model parameter and the physical quantities appearing in deformation processes is discussed. Based on the chosen temperature dependence of the shear modulus, thermodynamically consistent equations like the nonlinear rheological constitutive equation and the temperature equation are derived. The special cases of entirely entropy and energy elastic fluids are also considered. The thermorheological behavior (exo-, - or endothermal processes) of a viscoelastic fluid in a stress-growth experiment followed by relaxation is analyzed with respect to the model parameter.     

Primary normal-stress coefficient prediction at high shear rates

On the basis of a brief analysis of well known normal-stress calculation methods, the necessity of improved models of prediction is elaborated. A modified form of the so-called mirror relation which meets these requirements is presented. In combination with the Carreau viscosity equation, an analytical solution is given which leads to a Carreau normal-stress coefficient equation and, thus, to a simple method of calculation. The comparison between measured normal stresses and those determined by experiments shows that the values calculated in accordance with the presented method agree well with the measured values, especially within the range of high shear rates. The parameters andK to be selected for this purpose are determined in dependence on the slope of the viscosity function ₁ at high shear rates for each polymer individually, using empirical relations so that the global selection of parameters, which is common practice with other methods, is obviated. In an appendix a method for deriving the relations between material functions on the basis of operator calculation is given.Extended version of a paper read at the 2^nd Symposium on Rheology of the GDR in Tabarz/Thuringia, December 7–11, 1987     

Measurements of slip at the wall during flow of high-density polyethylene through a rectangular conduit

A hot-film probe has been used to measure slip of high-density polyethylene flowing through a conduit with a rectangular cross section. A transition from no slip to a stick-slip condition has been observed and associated with irregular extrudate shape. Appreciable extrudate roughness was initiated at the same flow rate as that at which the relationship between Nusselt number and Péclet number for heat transfer from the probe departed from the behavior expected for a no-slip condition at the conduit wall. A ₁ constant defined by eq. (A3) - C dimensionless group used in eq. (7) - C _p heat capacity - D constant in eq. (13) - f u _s/u - f _lin defined by eq. (A6) - G storage modulus - G loss modulus - k thermal conductivity - L length of hot film in thex-direction - L _eff effective length of large probe found from eq. (A3) - Nu _L Nusselt number, defined for a lengthL by eq. (2) - (Nu _L)₀ value ofNu _L atPe = 0 (eq. (A 1)) - Pe Péclet number,uL/ - Pe ₀ Péclet number in slip flow, eq. (6) - Pe ₁ Péclet number in shear flow, eq. (4) - q _L average heat flux over hot film of lengthL - R _i resistances defined by figure 8 - R _AB correlation coefficient defined by eq. (14) for signalsA andB - T temperature - T _s temperature of probe surface - T ambient temperature - T T _s –T - u average velocity - u _s slip velocity - V _b voltage indicated in figure 8 - W probe dimension (figure 6) - x distance in flow direction (figure 1) - y distance perpendicular to flow direction (figure 1) - thermal diffusivity,k/C _p - wall shear rate - _5% thickness of lubricating layer during probe calibration for introduction of an error no greater than 5%; see Appendix I - ^* complex viscosity - density - time - _c critical shear stress, eq. (13) - _w wall shear stress - frequency characterizing extrudate distortion (figures 12 and 13), or frequency of oscillation during rheometric characterization (figures 18–20) - * quantity obtained from normalized Nusselt number, eq. (A1), or complex viscosity ^* - A actual (small) probe (see Appendix I) - M model (large) probe (see Appendix I)     

Comparison between uniaxial and biaxial elongational flow behavior of viscoelastic fluids as predicted by differential constitutive equations

Behavior of polymer melts in biaxial as well as uniaxial elongational flow is studied based on the predictions of three constitutive models (Leonov, Giesekus, and Larson) with single relaxation mode. Transient elongational viscosities in both flows are calculated for three constitutive models, and steady-state elongational viscosities are obtained as functions of strain rates for the Giesekus and the Larson models.Change of elongational flow behavior with adjustable parameter is investigated in each model. Steady-state viscosities _E and _B are obtained for the Leonov model only when the strain-hardening parameter is smaller than the critical value _cr determined in each flow. In this model, uniaxial elongational viscosity _E increases with increasing strain rate , while biaxial elongational viscosity _B decreases with increasing biaxial strain rate _B . The Giesekus model predictions depend on the anisotropy parameter . _E and _B increase with strain rates for small _B while they decrease for large . When is 0.5, _E in increasing, but _B is decreasing. The Larson model predicts strain-softening behavior for both flows when the chain-contraction parameter > 0.5. On the other hand, when is small, the steady-state viscosities of this model show distinct maximum around = _B = 1.0 with relaxation time . The maximum is more prominent in _E than in _B .     

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     

An experimental study on effects of solid concentration and ζ-potential on pseudoplastic flow of suspensions

The pseudoplastic flow of suspensions, alumina or styrene-acrylamide copolymer particles in water or an aqueous solution of glycerin has been studied by the step-shear-rate method. The relation between the shear rate,D, and the shear stress,, in the step-shear-rate measurements, where the state of dispersion was considered to be constant, was expressed as = AD ^1/2 +CD. The effective solid volume fraction,ø _F, andA were dependent on the shear rate and expressed byø _F =aD ^b andA = D . Combining the above relations, the steady flow curve was expressed by = D ^{1/2 +} + ₀ (1 – a D ^b/0.74)^–1.85 D, where ₀ is the viscosity of the medium.With an increase in solid volume fraction and a decreases in the absolute value of the-potential, the flow behavior of the suspensions changed from Newtonian ( = = b = 0), slightly pseudoplastic ( = b = 0), pseudoplastic ( = 0) to a Bingham-like behavior.The change in viscosity of the medium had an effect on the change in the effective volume fraction.     

On the relaxation of shear and normal stresses of viscoelastic fluids following constant shear rate experiments

By means of a cone and plate rheometer the relaxation of the shear stress and the first normal stress difference in polymer liquids upon cessation of a constant shear rate were examined. The experiments were conducted mostly in a high shear rate region of relevance for the processing of these materials. The relaxation behavior at these shear rates can only be measured accurately under extremely precise specifications of the rheometer. To determine under which conditions the integral normal thrust is a convenient measure for the relaxing local first normal stress difference the radial distribution of the pressure in the shear gap was measured. The shape of relaxation of both the shear stress and the first normal stress difference could be closely approximated for the entire measured shear rate and time range by a two parameter statistical function. In the range of measured shear rates, one of the parameters, the standard deviationS, is equal for the shear and the normal stress, and is independent of the shear rate within the limit of experimental error. The second parameter, the mean relaxation timet _50, of the shear stress andt _50, of the first normal stress difference, can be calculated approximately from the viscosity function and only a single relaxation experiment.     

Representation of the rheological properties of polymer melts in terms of complex fluidity

In dynamic rheological experiments melt behavior is usually expressed in terms of complex viscosity ^* () or complex modulusG ^* (). In contrast, we attempted to use the complex fluidity ^* () = 1/µ ^* () to represent this behavior. The main interest is to simplify the complex-plane diagram and to simplify the determination of fundamental parameters such as the Newtonian viscosity or the parameter of relaxation-time distribution when a Cole-Cole type distribution can be applied. ^* () complex shear viscosity - () real part of the complex viscosity - () imaginary part of the complex viscosity - G ^* () complex shear modulus - G() storage modulus in shear - G() loss modulus in shear - J ^* () complex shear compliance - J() storage compliance in shear - J() loss compliance in shear - shear strain - rate of strain - angular frequency (rad/s) - shear stress - loss angle - ^* () complex shear fluidity - () real part of the complex fluidity - () imaginary part of the complex fluidity - ₀ zero-viscosity - ₀ average relaxation time - h parameter of relaxation-time distribution