Concentration Of Multivalent Cations Research Articles

Influence of Zr-doping on the structure and transport properties of rare earth high-entropy oxides

Fluorite-type ceria-based ceramics are well established as oxygen ion conductors due to their high conductivity, superseding state-of-the-art electrolytes such as yttria-stabilized zirconia. However, at a specific temperature and oxygen partial pressure they occasionally exhibit electronic conduction attributed to polaron hopping via multivalent cations (e.g. Pr and Ce). (Ce, La, Pr, Sm, Y)O2−δ is a high-entropy oxide with a fluorite-type structure, featuring low concentrations of multivalent cations that could potentially mitigate polaron hopping. However, (Ce, La, Pr, Sm, Y)O2−δ undergoes a structural transition to the bixbyite-type structure above 1000 °C. In this study, we introduce Zr doping into (Ce, La, Pr, Sm, Y)O2−δ to hinder the structural transition at elevated temperatures. Indeed, the fluorite structure at elevated temperatures is stabilized at approximately 10 at.% Zr doping. The total conductivity initially increases with doping, peaking at 5 at.% Zr doping, and subsequently decreases with further doping. Interestingly, electronic conductivity in (Ce, La, Pr, Sm, Y)1−x Zr x O2−δ under oxidizing atmospheres is not significant and is lowest at 8 at.% Zr. These results suggest that ceria-based high-entropy oxides can serve as oxygen ion conductors with a significantly reduced electronic contribution. This work paves the way for new compositionally complex electrolytes as well as protective coatings for solid oxide fuel cells.

Abundant phosphorus expected for possible life in Enceladus’s ocean

Saturn's moon Enceladus has a potentially habitable subsurface water ocean that contains canonical building blocks of life (organic and inorganic carbon, ammonia, possibly hydrogen sulfide) and chemical energy (disequilibria for methanogenesis). However, its habitability could be strongly affected by the unknown availability of phosphorus (P). Here, we perform thermodynamic and kinetic modeling that simulates P geochemistry based on recent insights into the geochemistry of the ocean-seafloor system on Enceladus. We find that aqueous P should predominantly exist as orthophosphate (e.g., HPO42-), and total dissolved inorganic P could reach 10-7 to 10-2 mol/kg H2O, generally increasing with lower pH and higher dissolved CO2, but also depending upon dissolved ammonia and silica. Levels are much higher than <10-10 mol/kg H2O from previous estimates and close to or higher than ∼10-6 mol/kg H2O in modern Earth seawater. The high P concentration is primarily ascribed to a high (bi)carbonate concentration, which decreases the concentrations of multivalent cations via carbonate mineral formation, allowing phosphate to accumulate. Kinetic modeling of phosphate mineral dissolution suggests that geologically rapid release of P from seafloor weathering of a chondritic rocky core could supply millimoles of total dissolved P per kilogram of H2O within 105 y, much less than the likely age of Enceladus's ocean (108 to 109 y). These results provide further evidence of habitable ocean conditions and show that any oceanic life would not be inhibited by low P availability.

Improved extraction efficiency of natural nanomaterials in soils to facilitate their characterization using a multimethod approach

Characterization of natural nanomaterial (NNM) physicochemical properties – such as size, size distribution, elemental composition and elemental ratios - is often hindered by lack of methods to disperse NNMs from environmental samples. This study evaluates the effect of extractant composition, pH, and ionic strength on soil NNM extraction in term of recovery and release of primary particles/small aggregate sizes (i.e., <200 nm). The extracted NNMs were characterized for hydrodynamic diameter and zeta potential by dynamic light scattering and laser Doppler electrophoresis, natural organic matter desorption by UV–Vis spectroscopy, element composition by inductively coupled plasma-mass spectroscopy (ICP-MS), size based elemental distribution by field flow fractionation coupled to ICP-MS, and morphology by transmission electron microscopy. The extracted NNM concentrations increased following the order of NaOH ≤ Na2CO3 < Na2C2O4 < Na4P2O7. Na4P2O7 was the most efficient extractant and results in 2–12 folds higher NNM extraction than other extractants. The Na4P2O7 extracted NNMs exhibited narrower size distribution with smaller modal size relative to NaOH, Na2CO3, Na2C2O4 extracted NNMs. Thus, Na4P2O7 enhances the extraction of primary NNMs and/or smaller NNM aggregates (i.e., size <200 nm). Na4P2O7 promote soil microaggregates breakup and release of NNMs by reducing free multivalent cation concentration in soil pore water by forming metal-phosphate complexes and by enhancing NNM surface charge via phosphate sorption on NNM surfaces. Additionally, the extracted NNM concentrations increased with the increase in extractant concentration and pH, except at 100 mM where the high ionic strength might have induced NNM aggregation. The improved NNM-extraction will improve the overall understanding of the physicochemical properties of NNMs in environmental systems. This study presents the key properties of NNMs that can be used as background information to differentiate engineered nanomaterials (ENMs) from NNMs in complex environmental media.

Multivalent cation induced attraction of anionic polymers by like-charged pores.

The efficiency of nanopore-based polymer sensing devices depends on the fast capture of anionic polyelectrolytes by negatively charged pores. This requires the cancellation of the electrostatic barrier associated with repulsive polymer-pore interactions. We develop a correlation-corrected theory to show that the barrier experienced by the polymer can be efficiently overcome by the addition of multivalent cations into the electrolyte solution. Cation adsorption into the pore enhances the screening ability of the pore medium with respect to the bulk reservoir which translates into an attractive force on the polymer. Beyond a critical multivalent cation concentration, this correlation-induced attraction overcomes the electrostatic barrier and triggers the adsorption of the polymer by the like-charged pore. It is shown that like-charge polymer-pore attraction is suppressed by monovalent salt but enhanced by the membrane charge strength and the pore confinement. Our predictions may provide enhanced control over polymer motion in translocation experiments.

Chemical Analysis of Underground Drinking Water Quality in Ghaziabad

Underground water contamination is a kind of water pollution or water destruction, in which various contaminants unify together to degrade quality and nature of water. Present scenario most probably support the water contamination by the humans and up to less extent by the natural disasters. The major ions which are responsible to maintain the quality of ground water are carbonate (CO 3 2- ) and bicarbonate (HCO 3 - ), Chloride (Cl - ), Sulphate (SO 4 -- ), Nitrate (NO 3 -- ), Phosphate (PO 4 3- ), and Fluoride (F - ). The above ions are present in the form of anions. These anions are in general maintain by the cations as Ca ++ ,Cu ++ , Mg ++ , Na + , and K + .Among the above briefed ions Ca ++ and Mg ++ are the ones which are responsible for the hardness of the water. These are found abundantly in the area, where manufacturing of chalk and limestone works take place. Water's hardness is determined by the concentration of multivalent cations in the water. This hard water if consumed by animals or humans lead to the terrific pathological conditions. Three water samples are collected from different locations of Pilkhuwa, District Hapur near to the location of the industry with limestone works, during summer. The ionic concentrations are expressed in mg/L.

Transport of soil-aged silver nanoparticles in unsaturated sand

Engineered nanoparticles released into soils may be coated with humic substances, potentially modifying their surface properties. Due to their amphiphilic nature, humic coating is expected to affect interaction of nanoparticle at the air-water interface. In this study, we explored the roles of the air-water interface and solid-water interface as potential sites for nanoparticle attachment and the importance of hydrophobic interactions for nanoparticle attachment at the air-water interface. By exposing Ag nanoparticles to soil solution extracted from the upper soil horizon of a floodplain soil, the mobility of the resulting “soil-aged” Ag nanoparticles was investigated and compared with the mobility of citrate-coated Ag nanoparticles as investigated in an earlier study. The mobility was determined as a function of hydrologic conditions and solution chemistry using column breakthrough curves and numerical modeling. Specifically, we compared the mobility of both types of nanoparticles for different unsaturated flow conditions and for pH=5 and pH=9. The soil-aged Ag NP were less mobile at pH=5 than at pH=9 due to lower electrostatic repulsion at pH=5 for both types of interfaces. Moreover, the physical flow field at different water contents modified the impact of chemical forces at the solid-water interface. An extended Derjaguin-Landau-Verwey-Overbeek (eDLVO) model did not provide satisfactory explanation of the observed transport phenomena unlike for the citrate-coated case. For instance, the eDLVO model assuming sphere-plate geometry predicts a high energy barrier (>90 kT) for the solid-water interface, indicating that nanoparticle attachment is less likely. Furthermore, retardation through reversible sorption at the air-water interface was probably less relevant for soil-aged nanoparticles than for citrate-coated nanoparticles. An additional cation bridging mechanism and straining within the flow field may have enhanced nanoparticle retention at the solid-water interface. The results indicate that the mobility of engineered Ag nanoparticles is sensitive to solution chemistry, especially pH and the concentration of multivalent cations, and to the unsaturated flow conditions influencing particle interaction at biogeochemical interfaces.

Evaluation of the impact of multivalent metal ions on the permeation behavior of Dolutegravir sodium

Interactions between active pharmaceutical ingredients (APIs) and polyvalent cations are an important factor within drug absorption in the gastrointestinal tract. Dolutegravir sodium, as a second-generation integrase stand transfer inhibitor for the treatment of HIV was investigated regarding chelation with Al3+, Ca2+, Fe3+, Mg2+ and Zn2+ ions at three different molar ratios. Furthermore, the influence of drug–ion chelates on the permeability of the drug across two intestinal membrane models was analyzed. For this purpose, Caco-2 monolayer model and Ussing chamber technique utilizing freshly excited rat intestinal mucosa were chosen and a buffer system without additional Mg2+ and Ca2+ ions was tested regarding cell detachment. The addition of polyvalent cations in an equal molar ratio to the drug solution decreased the dissolved drug by at least 11%. An increased multivalent cation concentration in a ratio of 1:10 afforded an API drop in the solution of at least 88% with the exception of Mg2+. In particular, Dolutegravir sodium was chelated with iron ions to nearly 100%. Overall, the higher the amount of metal ions in the solution, the lower was the detected amount of the drug. The permeation experiments across the Caco-2 monolayer and the rat intestinal mucosa pointed out that the addition of AlCl3, CaCl2 and ZnCl2 in a molar ratio of 10:1 to the drug led to significantly decreased drug permeation. According to these results the co-administration of Al3+, Ca2+ or Zn2+ as well as of supplementary medications containing these polyvalent ions is in case of oral Dolutegravir delivery not recommended.

The influence of multivalent cations on the flocculation of activated sludge with different sludge retention times

The mechanism governing the flocculation of activated sludge (AS) with different sludge retention times (SRTs) was studied in this paper. AS samples were cultivated in 8 lab-scale reactors with SRTs of 5 d, 7.5 d, 10 d, 12.5 d, 15 d, 20 d, 30 d, and 40 d. The bulk solution, loosely bound extracellular polymeric substances (LB-EPS), tightly bound EPS (TB-EPS), and pellet were extracted for all 8 AS samples. There was a clear trend that the effluent turbidity decreased as the SRT increased, and we deduced that this is because AS samples with longer SRTs have lower interaction energy barriers and lower LB-EPS content. Furthermore, the concentrations of multivalent cations (especially trivalent cations) in the pellets were found to be closely correlated to the AS flocculability, total interaction energy (Wtot), and LB-EPS content. The multivalent (especially trivalent) cations possess greater binding ability, and this ability to bind tightly to AS in large quantities is responsible for the superior flocculability of AS samples with longer SRTs. Hence, the concentrations of multivalent cations in the pellets are an important indicator of AS flocculability. We deduced that variations in the quantities of multivalent cations that tightly bind with the AS rather than remaining in the influent are the core reason behind observed fluctuations in the AS flocculability with different SRTs.

Fouling mitigation in membrane bioreactors using multivalent cations

Several cations have been used to study the effect of mineral coagulants on activated sludge properties and membrane fouling in submerged membrane bioreactors (MBRs). The flocculability and settling properties of activated sludge were studied in various concentrations of sodium, potassium, magnesium, calcium, ferrum (in Fe3+ form), and aluminum. Significant effect of cations and their concentrations on different parameters have been analyzed by 2-way ANOVA. Results showed that multivalent cations induce flocculation in bioreactors while monovalent cations have a detrimental effect on flocculability of activated sludge. The ratio of tight bound extracellular polymeric substances (TB-EPSs) to loosely bound (LB-EPSs) increased with concentration of multivalent cations that was in accordance with enhancement of biosorption in high concentration of multivalent cations. Moreover, these cations improved the settleability and dewatering properties of activated sludge. They also mitigated membrane fouling in the MBRs so that calcium and aluminum reduced membrane fouling to nearly 30%. Simple but useful correlations were developed for description of activated sludge properties based on easy measurable parameters that could be used also for estimation of membrane fouling. It was found that the ratio of TB-EPS to LB-EPS and also flocs size distribution are the main parameters affecting membrane fouling.

Selectivity of polyamide nanofiltration membranes for cations and phosphoric acid

The results shown in this paper are part of a study to investigate the use of nanofiltration membranes for the recycling of phosphorus from sewage sludge. It especially contains the results of an experimental study about the influencing factors pressure, pH and feed composition on the selectivity of nanofiltration membranes for phosphoric acid and multivalent cations. For the use of pretreated sewage sludge an effective pressure of 11 bar could be identified as an effective operational point. The selectivity turned out to increase with decreasing pH which was studied in a pH-interval of pH 1–4. An increasing concentration of multivalent cations in the feed solution had a negative effect on the permeability of phosphoric acid. Accordingly it could be observed that the nanofiltration worked significantly more effective with sewage sludge ash eluates of the concentration 10 g/l than eluates of the concentration 100 g/l.

Interplay of ion binding and attraction in DNA condensed by multivalent cations

We have measured forces generated by multivalent cation-induced DNA condensation using single-molecule magnetic tweezers. In the presence of cobalt hexammine, spermidine, or spermine, stretched DNA exhibits an abrupt configurational change from extended to condensed. This occurs at a well-defined condensation force that is nearly equal to the condensation free energy per unit length. The multivalent cation concentration dependence for this condensation force gives the apparent number of multivalent cations that bind DNA upon condensation. The measurements show that the lower critical concentration for cobalt hexammine as compared to spermidine is due to a difference in ion binding, not a difference in the electrostatic energy of the condensed state as previously thought. We also show that the resolubilization of condensed DNA can be described using a traditional Manning–Oosawa cation adsorption model, provided that cation–anion pairing at high electrolyte concentrations is taken into account. Neither overcharging nor significant alterations in the condensed state are required to describe the resolubilization of condensed DNA. The same model also describes the spermidine3+/Na+ phase diagram measured previously.

Long-range electrostatic interactions influence the orientation of fos-jun binding at AP-1 sites

Heterodimeric transcription regulatory proteins that bind palindromic DNA sequences can potentially bind their recognition sites in two opposite orientations. The orientation of transcription factor binding can control transcriptional activity by altering interactions with proteins that bind to adjacent regulatory elements. Fos-Jun heterodimers bind to AP-1 sites with different flanking sequences in opposite orientations. A gel-based fluorescence resonance energy transfer assay, gelFRET, was used to define the mechanism whereby amino acid residues and nucleotide base-pairs outside the Fos-Jun-AP-1 contact interface determine the orientation of heterodimer binding. Exchange of three amino acid residues adjacent to the basic DNA contact regions between Fos and Jun reversed the binding orientation. The effects of these amino acid residues on the orientation of heterodimer binding depended on base-pairs flanking the core AP-1 recognition sequence. Single amino acid and base-pair substitutions had parallel effects on DNA bending by Fos-Jun-AP-1 complexes and on heterodimer orientation. The binding orientation exhibited a close correspondence with both the difference in bending propensities of opposite sides of the AP-1 site as well as the difference in bending potentials of the Fos and Jun subunits of the heterodimer. The influence of flanking DNA sequences on heterodimer orientation was attenuated in the presence of high concentrations of multivalent cations. Base substitutions up to one helical turn from the center of the AP-1 site affected the binding orientation. Modification of flanking base-pairs with positively or negatively charged functional groups had opposite effects on the orientation of heterodimer binding. These changes in DNA charge had converse effects on the orientation preferences of heterodimers in which charged amino acid residues adjacent to the basic regions were exchanged between Fos and Jun. These results indicate that the orientation of heterodimer binding is determined primarily by minimization of the electrostatic free energy of the Fos-Jun-AP-1 complex. Consequently, long-range electrostatic interactions influence the architecture of nucleoprotein complexes.

Reentrant condensation of DNA induced by multivalent counterions

A theory of condensation and resolubilization of a dilute DNA solution with growing concentration of multivalent cations, N, is suggested. It is based on a new theory of screening of a macroion by multivalent cations, which shows that due to strong cation correlations at the surface of DNA the net charge of DNA changes sign at some small concentration of cations N0. DNA condensation takes place in the vicinity of N0, where absolute value of the DNA net charge is small and the correlation induced short range attraction dominates the Coulomb repulsion. At N&amp;gt;N0 positive DNA should move in the opposite direction in an electrophoresis experiment. From comparison of our theory with experimental values of condensation and resolubilization thresholds for DNA solution containing Spe4+, we obtain that N0=3.2 mM and that the energy of DNA condensation per nucleotide is 0.07 kBT.

Effect of aluminum on the growth and nutrition of tulip-poplar seedlings

Atmospheric deposition of nitrogen and sulfate is acidifying some ecosystems, potentially increasing the availability of aluminum in the soil solution. The effects of Al on tulip-poplar (Liriodendron tulipifera L.) were investigated with seedlings grown in sand culture. Seedlings were exposed to Al at four concentrations ranging from 0 to 800 µM. Reductions of 77% in shoot biomass and 71% in root biomass were observed at 200 µM Al. Aluminum reduced the concentrations of multivalent cations in leaves and roots. The Ca concentration of leaves was reduced at 400 µM Al, and the Mg concentration of roots was significantly reduced at 200 µM Al. Relationships between tissue Ca, Mg, and Al indicated that Al alters the translocation of Ca and Mg to the shoots and the accumulation of Mg in the roots. If mature trees respond in a similar fashion as seedlings noted here, then tulip-poplar has a low threshold for A1 toxicity, which may lead to declines in the health of tulip-poplar populations in ecosystems exposed to elevated levels of acidic deposition.

Effect of experimental parameters on the ESI FT-ICR mass spectrum of fulvic acid.

Fulvic acid (FA) is a heterogeneous mixture of organic macromolecules found in the waters, soils, and sediments of the earth's surface. The ability of electrospray ionization (ESI) to effectively transfer large ions from the solution phase to the gas phase and the coupling of ESI to the high-mass-resolution capabilities of Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR MS) provide a potential method for the mass spectrometric analysis of FA. Positive- and negative-ion ESI FT-ICR MS analyses of four reference International Humic Substances Society FAs were performed. The spray solution composition was found to have a dramatic effect on the ion distributions, with high-mass aggregates (m/z approximately 2000-4000) being formed in less polar spray solutions. Positive-ion spectra for each FA obtained under optimum conditions resulted in number-average molecular weights ranging from 1700 to 1900. The mass spectra were extremely complex, with ion distributions on the order of m/z approximately 500-3000. The presence of more than one ion at each nominal mass was routinely observed. Negative-ion ESI analysis of the FA samples resulted in the observation of multiply charged ions whose distributions could be affected by the acidification of the spray solution. Solution parameters which have been reported to affect molecular weight distributions of FA such as pH, ionic strength, and concentration of multivalent cations were found to have little or no effect on the observed m/z distributions.

Shifting the Equilibrium Mixture of Gramicidin Double Helices Toward a Single Conformation with Multivalent Cationic Salts

The conformation of the polypeptide antibiotic gramicidin is greatly influenced by its environment. In methanol, it exists as an equilibrium mixture of four interwound double-helical conformers that differ in their handedness, chain orientation, and alignment. Upon the addition of multivalent cationic salts, there is a shift in the equilibrium to a single conformer, which was monitored in this study by circular dichroism spectroscopy. With increasing concentrations of multivalent cations, both the magnitude of the entire spectrum and the ratio of the 229-nm to the 210-nm peak were increased. The spectral change is not related to the charge on the cation, but appears to be related to the cationic radius, with the maximum change in ellipticity occurring for cations with a radius of ∼1 Å. The effect requires the presence of an anion whose radius is greater than that of a fluoride ion, but is otherwise not a function of anion type. It is postulated that multivalent cations interact with a binding site in one of the conformers, known as species 1 (a left-handed, parallel, no stagger double helix), stabilizing a modified form of this type of structure.

Self-association of G-rich oligodeoxyribonucleotides under conditions promoting purine-motif triplex formation.

Efficient purine-motif triple-helix formation with guanosine/thymidine-rich oligodeoxyribonucleotides requires the presence of divalent cations (e.g., Mg2+) or polyamines at physiologic concentrations. However, under such conditions, we found that G-rich oligonucleotides were capable of self-association. Mixing experiments indicated a stoichiometry of two G-rich oligonucleotide strands in each complex. Dimerization was proportional to the oligonucleotide length, facilitated by increasing concentrations of multivalent cations, and inhibited by monovalent cations that promote G-quartet formation (e.g., K+, Rb+ NH4+). Although dimer formation was relatively slow (t(1/2) approximately 20 minutes), these species were quite stable, with dissociation rates on the order of days. Methylation protection experiments indicated that these dimers exhibited protected N7 position on most all guanines consistent with Hoogsteen base pairing, although this pattern differed from that observed under conditions favoring intramolecular quadruplex formation. Most important, G-rich oligonucleotide dimers were less capable of purine-motif triplex formation than were their denatured counterparts. Thus, these data indicated that G-rich oligodeoxyribonucleotides can form alternate self-associated structures under conditions that do not favor standard quadruplex formation and that these species can have altered properties with regard to their recognition of biologic targets.

Standard Enthalpies of Adsorption of Di- and Trivalent Cations on Alumina

Standard enthalpies of Ni(II), Gd(III), and Y(III) adsorption on alumina can be estimated from the temperature effect on apparent equilibrium constants of surface reactions. The results are relatively insensitive to the choice of the adsorption model, although the calculated apparent equilibrium constants of surface reactions at a given temperature are model dependent and they often differ by several orders of magnitude from one model to another. The conditions of the experiment—solid-to-liquid ratio, initial concentration of multivalent cations in solution, presence of various 1:1 and 2:1 electrolytes at various concentrations—do not significantly affect the calculated values of standard enthalpies of adsorption. Assuming that the total number of surface sites and the number of protons released per one adsorbed heavy metal cation do not depend on temperature, the standard enthalpies of adsorption on alumina at constant pH are 65 kJ/mol for Ni, 90 kJ/mol for Gd, and 90 kJ/mol for Y. The presence of alkaline earth metal cations (up to 0.05 mol dm−3) does not significantly influence the adsorption of nickel and gadolinium on alumina. Thus, barium and magnesium do not compete with heavy metal cations for the surface sites.

The effect of multivalent cations on adhesion time for cellular adhesion to solid surfaces

The dynamic behavior of the adhesion of a charge-regulated cell to a solid surface of constant potential is investigated. In particular, the effect of the presence of multivalent cations in the suspension medium on adhesion time is discussed. By neglecting the effect of hydrodynamic retardation and assuming that the bulk liquid phase is stagnant, we show that the presence of multivalent cations has the effect of retarding cell adhesion. At a fixed level of ionic strength, the adhesion time increases with the increase of the concentration of multivalent cations in the suspension medium, and decreases with the increase in magnitude of the Hamaker constant. For a fixed concentration of cations, the adhesion time decreases with the increase of ionic strength. The effect of the magnitude of Hamaker constant on adhesion time is appreciable if both the ionic strength and the concentration of cations are high.

Prediction of Viscosity for Partially Hydrolyzed Polyacrylamide Solutions in the Presence of Calcium and Magnesium Ions

Abstract Loss of solution viscosity in water of increasing ionic strength is a major problem encountered in the use of the partially hydrolyzed polyacrylamide polymers for improved oil recovery. It is recognized widely that the viscosity loss is more drastic in the presence of multivalent cations than is observed for sodium ions. There is, however, little information available on the relationships between total ionic strength, concentrations of multivalent cations, and solution viscosities. The purpose of this study is to establish relationships between total ionic strength, concentration of calcium or magnesium ions, polymer concentration, and the resulting viscosity for partially hydrolyzed polyacrylamides with varying degrees of hydrolysis. Solutions at constant ionic strength with varying ratios of calcium or magnesium to sodium ions are compared, and the loss of viscosity as a function of the fraction of divalent cations in the system is determined. For shear rates in the power-law region, the fractional loss in viscosity is a function of the fraction of multivalent cations and, in the range studied, is independent of the total ionic strength. A more complicated relationship is found at lower shear rates where the fractional viscosity loss does vary with total ionic strength. The relationship in the power-law region should prove valuable in predicting viscosities on the basis of the dependence of viscosity on ionic strength and on multivalent cation concentration at a single ionic strength, eliminating the need for many individual measurements of viscosity. More work is needed before useful predictions will be possible at lower shear rates.