Elongational flow and rheology of monodisperse polymers in solution

We describe the utilization of idealized stagnation point extensional flows, produced by opposed jets, for birefringence visualization of induced molecular strain and flow resistance measurements. We identify rheological changes associated with the coil---stretch transition which occurs beyond a critical strain-rate in elongational flow-fields. In dilute solutions of monodisperse atactic polystyrene, increases in extensional viscosity are observed as isolated molecules become stretched. The largest increases in extensional viscosity, however, are found only beyond a critical concentration and strain rate, and are associated with the stretching of transient networks of interacting molecules. These results parallel similar effects seen in porous media flow and capillary entrance experiments. We determine the molecular weight dependence of the critical concentration which scales as M^−0.55 in agreement with pairwise interaction of coils, but is much lower than conventional values of the critical polymer concentration, c^*. We believed that polydispersity may play an important role in the development of such transient networks, and in controlling the degradation behaviour during flow.     

An experimental evaluation of the behaviour of mono and polydisperse polystyrenes in Cross-Slot flow

The behaviour of a number of mono and polydisperse polystyrenes are probed experimentally in complex extensional flow within a Cross-Slot geometry using flow-induced birefringence. Polystyrenes with similar molecular weight (M _w) and increasing polydispersity (PD) illustrated the effect of PD on the principal stress difference (PSD) pattern in extensional flow. Monodisperse materials exhibited only slight asymmetry at moderate flowrates, although increased asymmetry and cusping was observed at high flowrates. The response of monodisperse materials of different M _w at various flowrates is presented and characterised by Weissenberg numbers for both chain stretch and orientation using a theory for linear entangled polymers. The comparison of stress profiles against Weissenberg number for each process is used to determine whether the PSD pattern observed is independent of M _w and elucidate which relaxation mechanism is dominant in the flow regimes probed. For monodisperse materials, at equivalent chain orientation Weissenberg number (We _τd), different molecular weight materials were seen to exhibit similar steady state PSD patterns independent of We _τR (chain stretch We). Whilst no obvious critical Weissenberg number (We) was found for the onset of increased asymmetry and cusping, it was found to occur in the “orientating flow without chain stretch” regime.     

Piezomagnetic and piezoelectric poling effects on mode I and II crack initiation behavior of magnetoelectroelastic materials

The ferrite and ferroelectric phase of magnetoelectroelastic (MEE) material can be selected and processed to control the macroscopic behavior of electron devices using continuum mechanics models. Once macro- and/or microdefects appear, the highly intensified magnetic and electric energy localization could alter the response significantly to change the design performance. Alignment of poling directions of piezomagnetic and piezoelectric materials can add to the complexity of the MEE material behavior to which this study will be concerned with.Appropriate balance of distortional and dilatational energy density is no longer obvious when a material possesses anisotropy and/or nonhomogeneity. An excess of the former could result in unwanted geometric change while the latter may lead to unexpected fracture initiation. Such information can be evaluated quantitatively from the stationary values of the energy density function dW/dV. The maxima and minima have been known to coincide, respectively, with possible locations of permanent shape change and crack initiation regardless of material and loading type. The direction of poling with respect to a line crack and the material microstructure described by the constitutive coefficients will be specified explicitly with reference to the applied magnetic field, electric field and mechanical stress, both normal and shear. The crack initiation load and direction could be predicted by finding the direction for which the volume change is the largest. In contrast to intuition, change in poling directions can influence the cracking behavior of MEE dramatically. This will be demonstrated by the numerical results for the BaTiO₃–CoFe₂O₄ composite having different volume fractions where BaTiO₃ and CoFe₂O₄ are, respectively, the inclusion and matrix.To be emphasized is that mode I and II crack behavior will not have the same definition as that in classical fracture mechanics where load and crack extension symmetry would coincide. A striking result is found for a mode II crack. By keeping the magnetic poling fixed, a reversal of electric poling changed the crack initiation angle from θ₀=+80° to θ₀=−80° using the line extending ahead of the crack as the reference. This effect is also sensitive to the distance from the crack tip. Displayed and discussed are results for r/a=10⁻⁴ and 10⁻¹. Because the theory of magnetoelectroelasticity used in the analysis is based on the assumption of equilibrium where the influence of material microstructure is homogenized, the local space and temporal effects must be interpreted accordingly. Among them are the maximum values of (dW/dV)_max and (dW/dV)_min which refer to as possible sites of yielding and fracture. Since time and size are homogenized, it is implicitly understood that there is more time for yielding as compared to fracture being a more sudden process. This renders a higher dW/dV in contrast to that for fracture. Put it differently, a lower dW/dV with a shorter time for release could be more detrimental.     

Effects of middle plane curvature on vibrations of a thickness-shear mode crystal resonator

We study the effects of a small curvature of the middle plane of a thickness-shear mode crystal plate resonator on its vibration frequencies, modes and acceleration sensitivity. Two-dimensional equations for coupled thickness-shear, flexural and extensional vibrations of a shallow shell are used. The equations are simplified to a single equation for thickness-shear, and two equations for coupled thickness-shear and extension. Equations with different levels of coupling are used to study vibrations of rotated Y-cut quartz and langasite resonators. The influence of the middle plane curvature and coupling to extension is examined. The effect of middle plane curvature on normal acceleration sensitivity is also studied. It is shown that the middle plane curvature causes a frequency shift as large as 10⁻⁸ g⁻¹ under a normal acceleration. These results have practical implications for the design of concave–convex and plano-convex resonators.     

Fiber spinning of very dilute solutions of polyacrylamide in water

The extensional viscometer developed earlier by the authors was refined and used to extend very dilute (50 ppm) solutions of polyacrylamide in distilled water. A slender liquid filament was stretched by the use of a suction device, and this resulted in the spinning of the fiber. By varying the volumetric flow rate and the filament length, stretch rates in the 100–1000 s⁻¹ range were easily obtained. The corresponding tensile stresses were very large, and these gave apparent extensional viscosities of the order of 200 P (20 Pa s). In contrast to this, the material functions in shear were difficult to measure, except for the shear viscosity which showed pronounced shear thinning. It was found that all the measurements, in shear as well as extension, could be explained based on the four constant Johnson-Segalman constitutive equation.     

Body wave propagation in rotating elastic media

We investigate the propagation of elastic waves through an elastic medium submitted to an angular rotation Ω. Wave propagation is shown to be directly related to the Kibel number Ki=ω/Ω, where ω is the wave frequency. Two dispersive waves W₁ and W₂ are obtained which tend to the classical dilatational and shear waves, respectively, when Ki tends to infinity. Wave W₁ shows a cutoff frequency ω_c=Ω below which it does not propagate. The case of small angular rotation Ω is also studied. The corrections to be introduced to dilatational and shear waves are then shown to be of order O(Ki⁻¹).     

Characterization of material inhomogeneity by stationary values of strain energy density

The strain energy density theory is applied to analyze the fracture instability of a mechanical system whose behavior is governed by the interaction of geometry, load and material inhomogeneity. This is accomplished by obtaining the location of the global and local relative minima of the strain energy density function dW/dV denoted, respectively, by [(dW/dV)_min]_g and [(dW/dV)_min]_¢l. The former refers to a fixed global coordinate system for the entire solid while the latter corresponds to local coordinate systems referred to each material point. An unique length parameter “ℓ” representing the distance between [(dW/dV)_min^max]_g and [(dW/dV)_min^max]_ℓ can thus be found and serves as a measure of the degree of system instability tending toward failure by fracture.Numerical results are obtained and displayed graphically for the case of a solid containing an inclusion of dissimilar material. The changes that take place in material inhomogeneity, loading type and physical dimensions of the solid and inclusion are reflected through ℓ. The method suggests the compatibility of ℓ for each member of a multi-component structure in order to avoid premature failure of a single member.     

Flow birefringence of dilute polymer solutions in two-dimensional flows

Summary Flow birefringence measurements have been obtained on three molecular weight samples (2–8 × 10⁶ M _W ,M _W /M _N = 1.14–1.3) of polystyrene in dilute solution (50–100 ppm) in a viscous polychorinated biphenyl solvent. The flows were generated using a four roll mill which could simulate a wide range of two dimensional flows in which the flow type (i.e. the ratio of the rate of rotation to the rate of strain) could be varied independently of the velocity gradient. The normalized birefringence, corrected for concentration, (n/nc), was observed to approach a saturation value at high velocity gradients in purely extensional flow. This saturation value was independent of both the molecular weight and the concentrationc, in agreement with theory. In addition, the magnitude of the saturation value is consistent with nearly fully extended chains and suggests extensions in the range of 20–50 times the rest state size. The data for the birefringence over a wide range of flows was found to be well correlated against the eigenvalue of the velocity gradient tensor in agreement with the results of the strong/weak flow theories ofTanner (1976) andOlbricht et al. (1980).The experiments are compared with a simple dumbbell model which incorporates the effects of a nonlinear spring variable hydrodynamic friction, and internal viscosity. It is shown that this simple model can simulate the experimental results surprisingly well if the effects of molecular weight distribution and finite transit times in the flow device are taken into account.

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Zusammenfassung Es werden Doppelbrechungsmessungen an verdünnten Lösungen (50–100 ppm) von drei Polystyrolproben mit verschiedenem Molgewicht (M _W = 2–8 10⁶,M _W /M _N = 1,14–1,3) in einem polychlorierten Diphenyl-Lösungsmittel (Pyralene 4000) durchgeführt. Die Strömung wird in einer Vier-Walzen-Apparatur erzeugt, die zweidimensionale Strömungen in einem weiten Typenbereich (d. h. mit verschiedenen Verhältnissen von Rotations- zu Deformationsgeschwindigkeit) unabhängig vom Geschwindigkeitsgradienten herzustellen gestattet. Die bezüglich der Konzentration korrigierte normierte Doppelbrechung (n/nc) strebt bei reiner Dehnströmung mit hohem Geschwindigkeitsgradienten gegen einen Sättigungswert. Dieser ist in Übereinstimmung mit der Theorie sowohl vom Molgewicht als auch von der Konzentrationc unabhängig. Weiterhin entspricht die Größe dieses Sättigungswerts der Annahme nahezu vollständig gestreckter Ketten mit Ausdehnungen vom 20–50fachen der Ausdehnung im Ruhezustand. Die Doppelbrechungswerte sind in einem weiten Strömungsbereich gut mit dem Eigenwert des Tensors des Geschwindigkeitsgradienten korreliert, was mit den Ergebnissen der Theorien starker und schwacher Strömungen vonTanner (1976), sowieOlbricht und Mitarbeitern (1980) übereinstimmt.Die Experimente werden mit den Voraussagen der Theorie des einfachen Hantelmodells verglichen, allerdings unter Einschluß eines nichtlinearen Federgesetzes, einer variablen hydrodynamischen Reibung und einer inneren Viskosität. Man findet, daß dieses einfache Modell die experimentellen Ergebnisse überraschend genau wiederzugeben vermag, wenn die Einflüsse der Molgewichtsverteilung sowie einer endlichen Einstellzeit in der Strömungsanordnung mitberücksichtigt werden.

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With 17 figures and 1 table     

Viscoelastic properties of liquid crystals of aqueous biopolymer solutions

The formation of aqueous, lyotropic phases of the biopolymers xanthan (M = 1.6·10⁶) and schizophyllan (M _W = 335 000) is investigated with stationary shear flow and oscillatory measurements, as well as with the aid of polarization microscopy, because these polymers show very different viscoelastic properties from coiled vinyl polymers. Xanthan and schizophyllan exhibit the same typical behavior observed in anisotropic solutions when the viscosity is plotted as a function of concentration and of shear rate. It has also been observed that the first normal stress difference for concentrated xanthan solutions shows a saturation effect at increasing shear rate. In oscillatory measurements only schizophyllan exhibits an maximum for the storage modulus. The absence of a such an elasticity maximum in the case of the xanthan solution may be attributed to the significantly higher flexibility of the xanthan helix. A comparison of the critical concentrations calculated according to Flory's theory and the experimentally determined values shows that the two-phase region is distinctly broader than the theory predicts. This deviation cannot be attributed to the flexibility of the polymer, but can, however, be explained by intermolecular interactions. In contrast to the non-charged schizophyllan the polyelectrolyte xanthan is affected in addition to the attractive interactions (H-bonds) by electrostatic repulsion forces.This paper was partly presented at the 198th ACS National Meeting in Miami Beach, Florida, Sept. 10–15, 1989 .     

Nonequilibrium stretching dynamics of dilute and entangled linear polymers in extensional flow

We propose an extension of the FENE-CR model for dilute polymer solutions [M.D. Chilcott, J.M. Rallison, Creeping flow of dilute polymer solutions past cylinders and spheres, J. Non-Newtonian Fluid Mech. 29 (1988) 382–432] and the Rouse-CCR tube model for linear entangled polymers [A.E. Likhtman, R.S. Graham, Simple constitutive equation for linear polymer melts derived from molecular theory: Rolie–Poly equation, J. Non-Newtonian Fluid Mech. 114 (2003) 1–12], to describe the nonequilibrium stretching dynamics of polymer chains in strong extensional flows. The resulting models, designed to capture the progressive changes in the average internal structure (kinked state) of the polymer chain, include an ‘effective’ maximum contour length that depends on local flow dynamics. The rheological behavior of the modified models is compared with various results already published in the literature for entangled polystyrene solutions, and for the Kramers chain model (dilute polymer solutions). It is shown that the FENE-CR model with an ‘effective’ maximum contour length is able to describe correctly the hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation also observed and computed by Doyle et al. [P.S. Doyle, E.S.G. Shaqfeh, G.H. McKinley, S.H. Spiegelberg, Relaxation of dilute polymer solutions following extensional flow, J. Non-Newtonian Fluid Mech. 76 (1998) 79–110] and Li and Larson [L. Li, R.G. Larson, Excluded volume effects on the birefringence and stress of dilute polymer solutions in extensional flow, Rheol. Acta 39 (2000) 419–427] using Brownian dynamics simulations of bead–spring model. The Rolie–Poly model with an ‘effective’ maximum contour length exhibits a less pronounced hysteretic behavior in stress versus birefringence in start-up of uniaxial extensional flow and subsequent relaxation.     

Der Zusammenhang zwischen der Dehnspannung von Kunststoffschmelzen im Düseneinlauf und im Schmelzbruch

Zusammenfassung In der vorliegenden Arbeit wurde der Schmelzbruch darauf zurückgeführt, daß im Düseneinlauf eine kritische Dehnspannung der Kunststoffschmelze überschritten wird. In der Regel ist die Einlaufströmung von einem Zirkularwirbel umgeben und kann als annähernd einachsige Dehnströmung betrachtet werden, wobei die Scherung vernachlässigbar ist. Die maximale Dehnspannung wird dann nahezu gleich dem Einlaufdruckverlust. Auf diese Weise läßt sich der Schmelzbruch auch bei extrem kurzen Düsen (sog. Lochdüsen), wo er besonders augenfällig in Erscheinung tritt, deuten.Die mit Hochdruck-Polyäthylen bei einer Temperatur von 150 °C durchgeführten Experimente führten auf eine kritische Dehnspannung von 11,5±0,5 kp × cm^–2, die unabhängig von der Dehnung oder der Dehngeschwindigkeit war.Das Strömungsfeld im Düseneinlauf kann man leicht sichtbar machen, wenn man im Füllraum eines Kapillarviskosimeters verschieden eingefärbte Scheiben aufeinander schichtet, auf Extrusionstemperatur erwärmt und nach nur teilweiser Extrusion unter Druck wieder erstarren läßt. Beim Aufschneiden des zurückgebliebenen Füllrauminhalts ergeben sich charakteristische Bilder, die quantitative Schlüsse auf die Dehnung und die Dehngeschwindigkeit zulassen. Daraus und aus dem oben erwähnten Einlaufdruckverlust lassen sich die Dehnspannungen berechnen.

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Summary In this paper it will be shown that melt fracture starts, when the extensional stress in the die inlet exceeds a critical value. If the flow in the die inlet is surrounded, as usual, by an annular vortex, we have a nearly unidirectional extension of the fluid, so that the maximum extensional stress has almost the value of the inlet pressure loss. With very short dies the extrudate distortions are especially well observable, and the onset of melt fracture occurs at a critical extrusion pressure.Experiments with a high pressure polyethylene of 150 °C temperature resulted in a critical extensional stress of 11,5 + 0,5 kp · cm^–2, this value being independent from extension or rate of extension.The flow pattern in the die inlet space becomes observable, when discs of two different colors are put in alternating order into the cylinder of a capillary rheometer and are then melted and partially extruded. The remaining melt in the cylinder is then cooled and sliced in the solid state. The characteristic flow patterns produced in this way can be used to calculate the extension and rate of extension vs. the flow coordinate.

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Auszugsweise als Vortrag gehalten auf der Jahrestagung der Deutschen Rheologen inDarmstadt am8. Juni 1971 bzw. auf dem2. Darmstädter Kunststoff-Kolloquium am15. Oktober 1971.

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Wir danken dem Land Hassen für die finanzielle Förderung dieser Arbeit.     

Rheology and morphology of starch/synthetic polymer blends

Corn starch and maleic anhydride functionalized synthetic polymers were melt blended in a Haake twin-screw extruder. The amount of starch in the blends was 60 and 70% by weight. The synthetic polymer used was either styrene maleic anhydride (SMA) or ethylene propylene maleic anhydride copolymer (EPMA). The blends did not exhibit normal thermoplastic behavior; and hence, rheological data was obtained by extrusion feeding the material through a slit die or cylindrical tube viscometer. The starch/SMA blends were extruded through a slit viscometer with a 45% half entry angle, while the starch/EPMA blends were extruded through a cylindrical tube viscometer with a half entry angle of 37.5°. For the blends, data could be obtained at low to moderate shear rates (10< _app<200s^–1). At higher shear rates, blends exhibited slip and/or degradation of starch. The viscosity of the blends exhibited shear-thinning properties. Regrinding and re-extruding through the viscometer a second time showed a significant reduction in shear viscosity for starch/SMA blends. Gel permeation chromatography data indicated that starch macromolecules degraded upon successive extrusion. Extensional viscosity, as estimated from entrance pressure drop method for starch/EPMA blends showed stretch thinning properties. Regrinding and re-extruding showed that the samples were more sensitive to changes in extensional viscosity as observed from the Trouton ratio versus extension rate plot. Optical microscopy showed the presence of starch granules after melt blending, the size of which was related to the torque (or stress) generated during extrusion. The higher the torque, the smaller the size of the starch granules. Successive extrusion runs reduced the number of unmelted granules.Nomenclature A,B Constants associated with power law fluids (Pa s^{m or n}) - e Entrance correction - H Height of slit die (m) - m, n Flow behavior index in shear and extension flow respectively - P _s Shear component of the entrance pressure drop (Pa) - P _e Extensional component of the entrance pressure drop (Pa) - Q volumetric flow rate (m³S^–1) - R _o radius of barrel exit (m) - R ₁ radius of cylindrical die (m) - T _r Trouton ratio - w width of slit die (m) - pressure gradient (Pam^–1) - half die entry angle - P _en Entrance Pressure Drop (Pa) - apparent extension rate (s^–1) - apparent shear rate (s^–1) - _w wall shear stress (Pa) - first normal stress difference in uniaxial extension (Pa)     

Differential constitutive equation for entangled polymers with partial strand extension

Beginning with a formal statement of the conservation of probability, we derive a new differential constitutive equation for entangled polymers under flow. The constitutive equation is termed the Partial Strand Extension (PSE) equation because it accounts for partial extension of polymer strands in flow. Partial extensibility is included in the equation by considering the effect of a step strain with amplitude E on the primitive chain contour length. Specifically, by a simple scaling argument we show that the mean primitive chain contour length after retraction is L=L ₀ E ^1/2, not the equilibrium length L ₀ as previously thought. The equilibrium contour length is infact recovered only after a characteristic stretch relaxation time λ _s that is bounded by the reptation time and longest Rouse relaxation time for the primitive chain. The PSE model predictions of polymer rheology in various shear and extensional flows are found to be in good to excellent agreement with experimental results from several groups. Received: 16 July 1997 Accepted: 22 January 1998     

On the number of droplets in aerosols

The number of droplets which may be formed with a supersaturated vapor in presence of a gas cannot exceed a number proportional to (p_v−p_v0)⁴ where p_v and p_v0 denote at the same temperature the pressure of the supersaturated vapor–gas mixture and the pressure of the saturated vapor–gas mixture. The energy necessary to the droplet formation is also bounded by a number proportional to (p_v−p_v0)².     

Simultaneous determination of shear and extensional viscosities and wall slip by on-line rheometry with parameter fitting: an application to EPDM rubber

Shear and extensional viscosities and wall slip are determined simultaneously under extrusion processing conditions using an on-line rheometer. Because it is not possible to independently control flow rate and temperature, classical methods for interpretation of capillary data cannot be used with on-line rheometry. This limitation is overcome using computational optimization to fit parameters in a flow model. This consists of three parts, representing shear viscosity, extensional viscosity, and wall slip. Three-parameter, power law forms, based on local instantaneous deformation rates and including temperature dependence, are used for each, and analytic solutions applied for entry flow and flow in the capillary. For entry flow, the Cogswell–Binding approach is used, and for developed flow in the capillary a solution incorporating wall slip is derived. The rheometer, with interchangeable capillaries, is mounted in place of the die on a rubber profile extrusion line. Pressure drops and temperatures for extrusion of an EPDM rubber through 2 mm diameter capillaries of length 0, 2, 3, 4, and 5 mm are logged and flow rates determined for a range of extruder speeds (5 to 20 rpm). Pressures ranged from 60 to 75 bar and temperatures from 86 to 116 °C. Mean flow velocity in the capillaries was between 5 × 10⁻³ and 5 × 10⁻¹ m s⁻¹. The nine material parameters are optimized for best fit of the analytic pressure drops to experimental data, using about 100 data points, with the Levenberg–Marquardt method. It is concluded that flow is dominated by extension and wall slip. Shear flow appears to play little part. The slip model indicates that slip velocity increases much more rapidly than the wall shear stress (in the range 0.5–1 MPa) and decreases with temperature for a given stress level. Results for the (uniaxial) extensional viscosity represent an engineering approximation to this complex phenomenon at the high strains (approximately 200) and high extension rates (up to 800 s⁻¹) applying in the extrusion. Results indicate a slight extension hardening and a decrease with temperature. Results are put into the context of the available studies in the literature, which, particularly with regard to wall-slip and extensional flow, consider conditions far removed from those applying in industrial extrusion. The present methods provide a powerful means for flow characterization under processing conditions, providing data suitable for use in computer simulations of extrusion and optimization of die design.     

Mechanics of adhesive contact on a power-law graded elastic half-space

We consider adhesive contact between a rigid sphere of radius R and a graded elastic half-space with Young's modulus varying with depth according to a power law E=E₀(z/c₀)^k (0<k<1) while Poisson's ratio ν remaining a constant. Closed-form analytical solutions are established for the critical force, the critical radius of contact area and the critical interfacial stress at pull-off. We highlight that the pull-off force has a simple solution of P_cr=−(k+3)πRΔγ/2 where Δγ is the work of adhesion and make further discussions with respect to three interesting limits: the classical JKR solution when k=0, the Gibson solid when k→1 and ν=0.5, and the strength limit in which the interfacial stress reaches the theoretical strength of adhesion at pull-off.     

Comment on “Correlation of entrainment for annular flow in horizontal pipes”, by Pan,L., Hanratty,T.J., Int. J. Multiphase flow, 28(3), (2002), pp. 385–408

The above referenced paper, published in International Journal of Multiphase Flow (Pan and Hanratty, 2002), proposed an entrainment fraction correlation for annular flow in horizontal pipes. The entrainment fraction in annular flow is defined as the ratio of the mass flow rate of the liquid droplets in the gas to the total mass flow of liquid, F_E = W_LE/W_L. The proposed correlation was verified with experimental data for liquids with viscosities close to that of water. The proposed entrainment fraction correlation includes another correlation for the critical film flow rate, W_F,cr to estimate a maximum entrainment fraction F_E,max. It is shown that the critical film flow rate correlation can result in negative maximum entrainment fraction values, for low liquid flow rates.     

Mesoscopic dynamics of polymer chains in high strain rate extensional flows

We take a step towards accessing the physics of viscoelastic liquid breakup in high speed, high strain rate flows by performing Brownian dynamics computations of dilute uniaxial, equibiaxial, and ellipsoidal polymeric extensional flows. Our computational implementation of the bead-spring model, when tailored to the DNA molecule, consistently with recent works of Larson and co-workers, is shown: (a) to predict a coil-stretch transition at Deborah number De≈0.5, and (b) to reproduce the experimental longest relaxation time. Furthermore, after adapting the model parameters to represent the polyethylene oxide (PEO) chain (for M=10⁶ Da), we find it possible to reproduce our own experimental data of the longest relaxation time, the transient extensional viscosity of dilute solutions at small Deborah numbers, and a coil-stretch transition at Deborah number De≈0.5. Extended to large Deborah numbers, the model predicts that polymer stretching is controlled by: (a) the randomness of the initial conditions that, in combination with rapid kinematically imposed compression, lead to the formation of initially frozen chain-folds, and (b) the speed with which thermo-kinematic processes relax these folds. The slowest fold relaxation occurs during uniaxial extension. As expected, the introduction of stretching along a second direction enhances the efficiency of fold relaxation mechanisms. Even for Deborah numbers (based on the chain longest relaxation time) of the order of one thousand, there is a large variation in the time a polymer needs in order to extend fully, and the effects of Brownian motion cannot be ignored. The computed Trouton ratios and polymer contributions to the total stress as functions of Hencky strain provide information about the relative importance of elastic effects during polymeric liquid stretching. At high strain rates, the steady state elastic stresses increase linearly with the Deborah number, resembling at that stage an anisotropic Newtonian fluid (constant extensional viscosity).     

Rheological properties of wheat flour doughs

Summary A comparative study was made of the large deformation and rupture properties of doughs from a medium strength and a weak wheat flour. Experiments were made by stretching, at a uniform rate, dough rings immersed in a liquid of matching density to prevent the rings from deforming under their own weight. Data were obtained on doughs differing in water content at temperatures from 5 to 45 °C and extension rates from 0.132 to 52.6 inches per minute.Essentially, the tensile properties of each dough could be represented by four characteristic functions, each dependent on only one of the variables: strain, time, temperature, and water content. The strain function (), equaled (In)/, where is the extension ratio, for extensions up to about 90% and, in some instances, up to nearly 200%. Thus, over extended ranges of strain, isochronal stress-strain data (representing comparable states of stress relaxation) gave a direct proportionality between true stress and theHencky strain, _H=In; the proportionality constant is the constant strain rate modulus,F (t^*), evaluated at the (isochronal) timet ^*. The modulusF(t,T,W — a function of timet, temperatureT, and water absorptionW-equals (T/T ₀)F (t ^*,T ₀,W ₀) (t/t ^* a _T a _W)ⁿ, wheren is a negative constant characteristic of the flour,F (t ^*,T ₀,W ₀) is the modulus at timet ^* at the reference temperatureT ₀ for a dough having the reference water absorptionW ₀;a _T anda _W are shift factors that account for the change of relaxation times with temperature and water content. The factor ay obeyed theArrhenius equation and gave activation energies of about 7.7 and 22.8 kcal for doughs from the medium strength and weak flour, respectively. Rupture data obtained at different temperatures and extension rates were superposed by usinga _T data and also were represented by failure envelopes. The shift factora _W appears to depend somewhat on temperature, especially for the weaker flour.Differences in the rheological behavior of doughs from the two flours were evident in: (1) the range over which the isochronal stress-strain behavior could be linearized; (2) the magnitude of the characteristic exponentn; (3) the magnitude and the temperature dependence of the moduli; (4) the activation energies; (5) the effect of temperature ona _W; and (6) several characterizing plots prepared to represent rupture data.

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Zusammenfassung Es wurde eine Vergleichsuntersuchung betreffend große Verformungen und Brucheigenschaften von Teigen aus zwei Weizenmehltypen durchgeführt. Die Versuche wurden in der Weise ausgeführt, daß man Teigringe, die in eine Flüssigkeit von entsprechender Dichte eingetaucht waren, um zu verhindern, daß sich die Ringe unter der Wirkung ihres eigenen Gewichtes verformten, mit gleichbleibender Geschwindigkeit streckte. Es wurden Werte für verschiedene Teige ermittelt, die sich durch ihren Wassergehalt unterschieden, und zwar bei Temperaturen von 5–45 °C und Dehnungsgeschwindigkeiten von 0,132–52,6 Zoll pro Minute.Die Zugeigenschaften eines jeden Teiges konnten im wesentlichen durch vier charakteristische Funktionen dargestellt werden, von denen jede lediglich von einer der Veränderlichen: Verformung, Zeit, Temperatur und Wassergehalt abhing. Die Verformungsfunktion () war gleich (In)/, worin das Dehnungsverhältnis bedeutet, und zwar für Dehnungen bis zu ungefähr 90%, in einigen Fällen sogar bis zu fast 200%. Somit ergab sich über einem ausgedehnten Verformungsbereich für die isochronen Kraft-Dehnungs-Werte (die vergleichbare Zustände der Spannungsrelaxation darstellen) eine direkte Proportionalität zwischen wahrer Spannung undHencky- Verformung, _H=In; die Proportionalitätskonstante ist der ModulF (t^*), genommen bei konstanter Verformungsgeschwindigkeit und der (isochronen) Zeit t^*. Der ModulF (t, T, W)eine Funktion der Zeitt, der TemperaturT und der WasserabsorptionW — läßt sich darstellen durch (T/T ₀)F (t ^*,T ₀,W ₀ ) (t/t ^* a _T a _W)ⁿ, worinn eine für das Mehl charakteristische negative Konstante undF (t ^*,T ₀,W ₀) der Modul zur Zeitt ^* bei der BezugstemperaturT ₀ ist für einen Teig, der die Bezugs-WasserabsorptionW ₀ hat;a _T unda _W sind Verschiebungsfaktoren, die der Veränderung der Relaxationszeiten mit der Temperatur und dem Wassergehalt Rechnung tragen. Der Faktor ay gehorcht derArrhenius-Gleichung und ergibt Aktivierungsenergien von ungefähr 7,7 bzw. 22,8 kcal für die Teige der beiden Mehltypen. Bruchwerte, die bei verschiedenen Temperaturen und verschiedenen Dehnungsgeschwindigkeiten erhalten wurden, ließen sich durch Verwendung vona _T-Daten überlagern und durch eine Brucheinhüllende (failure envelope) darstellen. Der Verschiebungsfaktora _W scheint etwas von der Temperatur abzuhängen, und zwar besonders bei dem schwächeren Mehl.Unterschiede im rheologischen Verhalten von Teigen aus den beiden Mehlsorten bestanden offensichtlich in Bezug auf: (1) den Bereich, über den das isochrone Kraft-Dehnungs-Verhalten linearisiert werden konnte, (2) die Größe des charakteristischen Exponentenn, (3) die Größe und die Temperaturabhängigkeit der Moduln, (4) die Aktivierungsenergien, (5) die Wirkung der Temperatur aufa _W und (6) einige graphische Darstellungen zur Charakterisierung der Bruchwerte.