Mathematics and Rheology

Summary The great advances made in Rheology during the last forty years owe much to the application of mathematics. But in some cases, there have been misunderstandings. As a result, rheologists have sometimes been unnecessarily restricted. Mathematics is limited only by self-consistency: Rheology deals with the physical world.In particular, three terms are discussed: infinity, zero and negative, all of which have at least two meanings which have been confused. Rheological phenomena cannot be explained merely by mathematical formulation. Unnecessary mathematics should be avoided but professional rheologists must not shirk mathematics.This paper was read in an abbreviated form, at aConference of the British Society of Rheology in April 1971 The Proceedings were not published.     

Rheology of molten polymers

Summary TheCross equation describes the flow of pseudoplastic liquids in terms of an upper and a lower Newtonian viscosity corresponding to infinite and zero shear, and ₀, and of a third material constant related to the mechanism of rupture of linkages between particles in the intermediate, non-Newtonian flow regime, Calculation of of bulk polymers is important, since it cannot be determined experimentally. The equation was applied to the melt flow data of two low density polyethylenes at three temperatures.Using data in the non-Newtonian region covering 3 decades of shear rate to extrapolate to the zero-shear viscosity resulted in errors amounting to about onethird of the measured ₀ values. The extrapolated upper Newtonian viscosity was found to be independent of temperature within the precision of the data, indicating that it has a small activation energy.The ₀ values were from 100 to 1,400 times larger than the values at the corresponding temperatures.The values of were large compared to the values found for colloidal dispersions and polymer solutions, but decreased with increasing temperature. This shows that shear is the main factor in reducing chain entanglements, but that the contribution of Brownian motion becomes greater at higher temperatures.

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Zusammenfassung Die Gleichung vonCross beschreibt das Fließverhalten von pseudoplastischen Flüssigkeiten durch drei Konstante: Die obereNewtonsche Viskosität (bei sehr hohen Schergeschwindigkeiten), die untereNewtonsche Viskosität ₀ (bei Scherspannung Null), und eine Materialkonstante, die vom Brechen der Bindungen zwischen Partikeln im nicht-Newtonschen Fließbereich abhängt. Die Berechnung von ist wichtig für unverdünnte Polymere, wo man sie nicht messen kann.Die Gleichung wurde auf das Fließverhalten der Schmelzen von zwei handelsüblichen Hochdruckpolyäthylenen bei drei Temperaturen angewandt. Die Werte von ₀, durch Extrapolation von gemessenen scheinbaren Viskositäten im Schergeschwindigkeitsbereich von 10 bis 4000 sec^–1 errechnet, wichen bis 30% von den gemessenen ₀-Werten ab. Die Aktivierungsenergie der war so klein, daß die-Werte bei den drei Temperaturen innerhalb der Genauigkeit der Extrapolation anscheinend gleich waren. Die ₀-Werte waren 100 bis 1400 mal größer als die-Werte.Im Verhältnis zu kolloidalen Dispersionen und verdünnten Polymerlösungen war das der Schmelzen groß, nahm aber mit steigender Temperatur ab. Deshalb wird die Verhakung der Molekülketten hauptsächlich durch Scherbeanspruchung vermindert, aber der Beitrag derBrownschen Bewegung nimmt mit steigender Temperatur zu.

     

Rheology of plastic fats

Summary A plastic material is obtained by rapid cooling of a solution of glyceryl tristearate in paraffin oil. The fat crystals produced form a three-dimensional network structure, which is mainly responsible for the rheological properties of the material.The properties have been investigated by means of creep measurements at deformations not exceeding a few percent, and in a concentric cylinder viscometer at very low shear rates.Results of measurements are discussed in terms of the properties of the three-dimensional network of fat crystals.

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Zusammenfassung Eine plastische Substanz wird durch rasches Abkühlen einer Lösung von Glyceryltristearat in Paraffinöl erhalten. Es entstehen dabei zusammenhängende Fettkristalle, welche eine feste Struktur bilden. Die Theologischen Eigenschaften des Materials werden hauptsächlich durch diese vernetzte Struktur der festen Teilchen bestimmt.Die Messung der Eigenschaften erfolgte mittels Kriechversuchen bei sehr kleinen Deformationen und im Rotationsviskosimeter bei sehr niedrigem Geschwindigkeitsgefälle. Die Ergebnisse dieser Messungen werden an Hand von den Eigenschaften der vernetzten Struktur der Fettkristalle diskutiert.

     

Rheology of oil-in-water emulsions

The effect of interfacial tension on the steady-flow and dynamic viscoelastic behavior of emulsions are studied experimentally. At very low inter-facial tensions and low volume fractions, the viscosity decreases with increasing shear rate and becomes constant at high shear rates. The high-shear-rate Newtonian viscosity is not affected by interfacial tension, but the transition from pseudoplastic to Newtonian flow shifts to lower shear rates as the interfacial tension decreases. At an interfacial tension of 5 × 10^–3 Nm^–1, the viscosity decreases, passes through a minimum, and then increases as the shear rate is increased. The dilatant behavior may be attributed to elastic responses of interfaces during collision of drops. At high volume fractions, the emulsions show remarkable elasticity resulting from the interfacial energy associated with deformation of liquid films. The modulus and viscosity are proportional to interfacial tension and inversely proportional to drop size.     

Rheology of coal suspensions

Proper design of operations encountered in preparation, transport and employment of suspensions like coal slurries and coal-oil mixtures require an accurate knowledge of their rheological behaviour.Such concentrated suspensions generally exhibit non-Newtonian behaviour (shearthinning) which is more pronounced at higher coal concentrations. The nature of the dispersing medium influences the aggregation state of the disperse phase and, consequently, affects the stability and the rheology of the systems. In the present paper coal suspensions prepared with different dispersing media and covering a wide range in solid phase concentration are studied, by using a rotational coaxial cylinders viscometer.Different models have been taken into consideration for correlating experimental data. In particular, in order to describe the dependence of viscosity on shear rate and solid phase concentration, the suitability of the model suggested by Smith and Bruce is evaluated. Accordingly, the aggregation state of the disperse phase as well as its dependence on shear rate and dispersing medium can be estimated.     

Rheology of columnar-grained ice

Creep of columnar-grained ice, under uniaxial compressive force normal to the columns, is shown to be composed of an instantaneous elastic response followed by a delayed elastic and viscous deformation. Both the delayed elastic and viscous strains are shown to have equal activation energies. Thus, this ice can be considered as a thermorheologically simple material with a nonlinear stress dependence. A simple phenomenological relationship has been developed that can be used for further analysis of the creep compliance function presented in a normalized form.     

Rheology of polymer blends

Rheological properties of blends of amorphous and crystalline polymers were studied for a broad range of compositions and temperatures. It was established that below the melting pointT _m the viscoelastic properties of blends of crystalline polymers are similar to those of polymers filled with mineral fillers. In both cases these properties are influenced by the existence, in such systems, of a temporary structural network formed by mineral or polymeric particles and its subsequent breakdown under the action of shear stresses. It was found that an anomalous decrease in the melt viscosity of the main component on addition of a small amount of a second polymer depended on deformation conditions. The comparison of data on viscoelastic properties and thermodynamic interaction between the components in the melt, estimated from the parameter ₂₃ of a new Flory theory, shows that the sharp drop of viscosity takes place in the region of microphase separation due to the appearance of an excess free volume in the interphase region. Calculation of the relaxation spectra for various blends also revealed marked changes when various amounts of a second component were added to the main polymer.     

Rheology of zirconium adipate sol

Summary Employing aStormer viscosimeter, cup and rotor type, the rheological behaviour of gel yielding zirconium adipate sol has been investigated. Application of Staudinger's viscosity relation _sp =K _m Mc, gives the extent of aggregation during sol-gel transformation. In the gelling region a sharp rise in specific viscosity, nearly forty fold is observed. Rheological data is used in determining the gelling time and compares favourably well with values obtained by other methods.

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Zusammenfassung Mit Hilfe einesStormer-Viskosimeters vom koaxialen Zylindertyp wurde das rheologische Verhalten von Zirkonium-Adipat-Sol, das gelieren kann, untersucht. Die Anwendung der Staudinger Beziehung _spez=K _m M. c gibt das Ausmaß der Aggregierung während der SolGel-Umwandlung wieder. Im Gelierungsgebiet wird eine starke Zunahme in der spezifischen Viskosität — beinahe auf das 40fache — beobachtet. Rheologische Daten werden verwendet, um die Gelierzeit zu bestimmen, und sie können mit den Resultaten anderer Methoden gut verglichen werden.

     

Rheology of carbon nanofiber-reinforced polypropylene

The rheological properties of two different nanocomposite systems consisting in the dispersion of carbon nanofibers (CNFs) in polypropylene are investigated. The nanoreinforced systems were identically prepared with two CNFs that differ only in the length of the fibers being otherwise identical to analyze the effect of fiber aspect ratio. Linear dynamic viscoelasticity and the steady-state rheology of the two different nanocomposites are presented. The system reinforced with CNFs with larger aspect ratio shows several rheological features that resemble peculiarities of rodlike polymers in the nematic liquid crystalline phase.     

Rheology of zirconium adipate sol

Summary Rheological study of gel yielding zirconium adipate sol shows that it exhibits plasticity with a definite yield value. Influence of ultrasonic radiations, frequency 1 Mc/s, on the plasticity has been investigated. Ultrasonic agitation decreases the plasticity leading to a significant increase in the gelling time. An increase in the conductance and a decrease in p_H is simultaneously observed. These ultrasonic effects have been discussed and explained.     

Rheology of inhomogeneous immiscible blends

Spatial inhomogeneities of blends are taken into account by letting the free energy depend on gradients of the quantities chosen to characterize the interface. The governing equations as well as new stresses that arise due to the inhomogeneity are derived from a requirement of compatibility with thermodynamics.     

Rheology of concentrated microgel solutions

Viscosity, modulus, and yield stress for 0–6 wt% aqueous solutions of Carbopol 941 were investigated using constant shear rate, constant shear stress, and dynamic oscillatory experiments. The microgel character of the polymer was evident from the solid-like behavior of the solutions above 1 wt%. Yield stress increased with concentration, but yield occurred at a critical shear strain of 40%, independent of concentration. The static stress-strain relationship became non-linear at ~ 25% strain, in fair agreement with the onset of non-linear response in the storage modulus at ~ 10% strain. Small strain moduli from static and low frequency measurements agreed rather well; modulus values obtained from the recoverable strain after yielding were 30–40% smaller. Solutions flowed at near-constant stress in the low shear rate regime; at higher rates the stress increases with shear rate more rapidly. The viscosity did not obey the Cox-Merz rule. Steady-state viscosity scaled with polymer concentration to the 3/4 power. Results were interpreted using a cellular, deformable sphere model for the polymer, in analogy to emulsions and foams.     

Rheology of network forming systems

Summary The curing reactions of an epoxy-anhydride and a rubber vulcanization system were followed by rheological means. Dynamic shear measurements were used to follow the reaction. The rheological quantities were related to the reaction kinetics of the systems. From these relations, overall activation energies, reaction orders and rate constants were interpreted from the data. These compare favorably with results reported in the literature. The cures of several diallyl phthalate compounds were also examined with dynamic shear measurements. The results correlated well with thermoset molding experience.With 7 figures     

Rheology of brown coal-water suspensions

Techniques for measuring the fundamental flow properties of as-mined Victorian brown coal suspensions are developed. Flow properties are presented for Morwell, Yallourn and Loy Yang coals as a function of concentration for fixed particle size distribution. Even at the relatively low solids concentration of 20 to 30 percent by weight, the suspensions exhibit complex non-Newtonian characteristics. Generally at high concentration, the suspensions are thixotropic with a shear rate dependent viscosity and exhibit a yield stress. The inherent thixotropy of the coals can be exploited and it is technically but perhaps not economically feasible to transport the as-mined coals in a pipeline. The power requirement to pump the coals in a pipeline is found to be 10 to 30 times that required to pump black coal in the Black Mesa pipeline in the U.S.A. The differences in the rheological properties of the three coals is related to the surface properties of the coal particle — expressed in terms of carboxylate and inorganic cation content, and to differences in the pore volume of the coals. The flow characteristics of one coal can be converted to that of another simply by varying either the carboxylate or ionic content of the suspension.