Effect of pressure in capillary flow of polystyrene

Several corrections possibly required for capillary flow are based on the existence of a linear relationship between the pressure drop along the capillary and the length-to-diameter ratio at a given temperature and shear rate. Recently, the appearance of nonlinearities in this relationship has created some concern as to the cause of this behavior. The occurrence and an explanation of the nonlinearities for polystyrene form the basis of this study. A narrow-distribution, low molecular weight (20,400) polystyrene was tested in eight capillaries at temperatures of 140 and 160°C to initiate the discussion of the nonlinearity in a ΔP (pressure) versus L/D (length/diameter of capillary) plot. The sample exhibits negligible extrudate swelling at all pressures which reinforces the idea that pressure is influencing the flow. The pressure dependence of viscosity is determined using the equivalent expression of the WLF equation derived from free volume theory. Justification for its use is presented. A pressure correction, representing the increased shear stress necessary for flow of the higher viscosity material, is found to linearize the ΔP versus L/D data. A narrow-distribution, high molecular weight polystyrene (670,000) is subjected to a similar analysis at 165°C by using nine capillaries. The situation is quite different, as the high molecular weight sample is not nearly as ideal as the low molecular weight polystyrene.     

Experimental study of the immiscible displacement of shear-thinning fluids in pore networks

The pore scale mechanisms and network scale transient pattern of the immiscible displacement of a shear-thinning nonwetting oil phase (NWP) by a Newtonian wetting aqueous phase (WP) are investigated. Visualization imbibition experiments are performed on transparent glass-etched pore networks at a constant unfavorable viscosity ratio and varying values of the capillary number (Ca), and equilibrium contact angle (theta(e)). Dispersions of ozokerite in paraffin oil are used as the shear-thinning NWP, and aqueous solutions of PEG colored with methylene blue are used as the Newtonian WP. At high Ca values, the tip splitting and lateral spreading of WP viscous fingers are suppressed; at intermediate Ca values, the primary viscous fingers expand laterally with the growth of smaller capillary fingers; at low Ca values, network spanning clusters of capillary fingers separated by hydraulically conductive noninvaded zones of NWP arise. The spatial distribution of the mobility of shear-thinning NWP over the pore network is very broad. Pore network regions of low NWP mobility are invaded through a precursor advancement/swelling mechanism even at relatively high Ca and theta(e) values; this mechanism leads to irregular interfacial configurations and retention of a substantial amount of NWP along pore walls; it becomes the dominant mechanism in displacements performed at low Ca and theta(e) values. The residual NWP saturation increases and the end WP relative permeability decreases as Ca increases and both become more sensitive to this parameter as the shear-thinning behavior strengthens. The shear-thinning NWP is primarily entrapped in individual pores of the network rather than in clusters of pores bypassed by the WP. At relatively high flow rates, the amplitude of the variations of pressure drop, caused by fluid redistribution in the pore network, increase with shear-thinning strengthening, whereas at low flow rates, the motion of stable and unstable menisci in pores is reflected in strong pressure drop fluctuations.     

Statistical-mechanical theory of rheology: Lennard-Jones fluids

The generalized Boltzmann equation for simple dense fluids gives rise to the stress tensor evolution equation as a constitutive equation of generalized hydrodynamics for fluids far removed from equilibrium. It is possible to derive a formula for the non-Newtonian shear viscosity of the simple fluid from the stress tensor evolution equation in a suitable flow configuration. The non-Newtonian viscosity formula derived is applied to calculate the non-Newtonian viscosity as a function of the shear rate by means of statistical mechanics in the case of the Lennard-Jones fluid. For that purpose we have used the density-fluctuation theory for the Newtonian viscosity, the modified free volume theory for the self-diffusion coefficient, and the generic van der Waals equation of state to compute the mean free volume appearing in the modified free volume theory. Monte Carlo simulations are used to calculate the pair-correlation function appearing in the generic van der Waals equation of state and shear viscosity formula. To validate the Newtonian viscosity formula obtained we first have examined the density and temperature dependences of the shear viscosity in both subcritical and supercritical regions and compared them with molecular-dynamic simulation results. With the Newtonian shear viscosity and thermodynamic quantities so computed we then have calculated the shear rate dependence of the non-Newtonian shear viscosity and compared it with molecular-dynamics simulation results. The non-Newtonian viscosity formula is a universal function of the product of reduced shear rate (gamma*) times reduced relaxation time (taue*) that is independent of the material parameters, suggesting a possibility of the existence of rheological corresponding states of reduced density, temperature, and shear rate. When the simulation data are reduced appropriately and plotted against taue*gamma* they are found clustered around the reduced (universal) non-Newtonian viscosity formula. Thus we now have a molecular theory of non-Newtonian shear viscosity for the Lennard-Jones fluid, which can be implemented with a Monte Carlo simulation method for the pair-correlation function.     

Rheology of polydimethylsiloxane swollen with supercritical carbon dioxide

Viscosity curves were measured for polydimethyl siloxane (PDMS) melts swollen with dissolved carbon dioxide at 50 and 80°C for shear rates ranging from 40 to 2300 s⁻¹, and for carbon dioxide contents ranging from 0 to 21 wt %. The measurements were performed with a capillary extrusion rheometer modified for sealed, high-pressure operation to prevent degassing of the melt during extrusion. The concentration-dependent viscosity curves for these systems are self-similar in shape, exhibiting low-shear rate Newtonian plateau regions followed by shear-thinning “power-law” regions. Considerable reduction of viscosity is observed as the carbon dioxide content is increased. Classical viscoelastic scaling methods, employing a composition-dependent shift factor to scale both viscosity and shear rate, were used to reduce the viscosity data to a master curve at each temperature. The dependence of the shift factors on polymer chain density and free volume were investigated by comparing the shift factors for PDMS-CO₂ systems to those obtained by iso-free volume dilutions of high molecular weight PDMS. This comparison suggests that the free volume added to PDMS upon swelling with dissolved carbon dioxide is the predominant mechanism for viscosity reduction in those systems. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys, 35: 523–534, 1997     

Rheology of Water-in-Oil Emulsions with Different Drop Sizes

Systemic experiments have been conducted to investigate the effect of drop sizes on the rheology of water-in-oil (W/O) emulsions. Three sets of emulsions with different average drop sizes were first prepared and then the corresponding rheologies were determined using a concentric viscometer. Results indicated that the flow behavior of concentrated emulsions changes qualitatively from Newtonian flow to non-Newtonian flow with shear rates. In Newtonian flow regime, a smaller drop size leads to a higher viscosity, and the increments are more pronounced at high dispersed phase volume fractions. Two local remarkable increases of the emulsion viscosity with dispersed phase volume fractions correspond to the percolation and glass-transition, respectively. In non-Newtonian flow regime, emulsions show shear-thinning behavior and can be fitted well by the power law model. For emulsions with volume fractions between 0.132 and 0.325, the flow index and consistency constant show power law relationship with the water content. Furthermore, the shear-thinning effect becomes stronger in the emulsions with smaller drop sizes. A correlation has been successfully developed for determining the clusters’ sizes in W/O emulsions and shows excellent agreement with the experimental data. As a consequence, a microscopic understanding (cluster level) was presented for the shear-thinning behavior of the emulsions in this study.     

A predictive approach based on the Simha–Somcynsky free‐volume theory for the effect of dissolved gas on viscosity and glass transition temperature of polymeric mixtures

The gas concentration and pressure effects on the shear viscosity of molten polymers were modeled by using a unified approach based on a free volume theory. A concentration and pressure dependent “shift factor,” which accounts for free volume changes associated with polymer‐gas mixing and with variation of absolute pressure as well as for dilution effects, has been herein used to scale the pure polymer viscosity, as evaluated at the same temperature and atmospheric pressure. The expression of the free volume of the polymer/gas mixture was obtained by using the Simha and Somcynsky equation of state for multicomponent fluids. Experimental shear viscosity data, obtained for poly(ε‐caprolactone) with nitrogen and carbon dioxide were successfully predicted by using this approach. Good agreement with predictions was also found in the case of viscosity data reported in the literature for polystyrene and poly(dimethylsiloxane) with carbon dioxide. Free volume arguments have also been used to predict the T_g depression for polystyrene/carbon dioxide and for poly(methyl methacrylate)/carbon dioxide mixtures, based on calculations performed, again, with the Simha and Somcynsky theory. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 1863–1873, 2006     

Statistical theory of multiple-site linear wall-adsorption capillary Chromatography

Based on the mass-balance principle, a particular diffusion equation to describe the movement of solute molecules in the stagnant layer of multiple-site solid surfaces is constructed. From the equation, the moments of residence time in a step on multiple-site surfaces are derived. Similarly, the moments in a step in the mobile phase are also derived from a diffusion-drift equation. According to the probability theory, there exists a general relationship between the moments of an elution curve and the moments in a step. Through this relationship, the expressions of the elution-curve moments are derived from the step moments. In this paper, the details related to multiple-site linear wall-adsorption capillary chromatography are described and added in the equations to determine the step moments. The resultant expressions of the elution-curve moments involve various factors, such as adsorption–desorption rate constants, equilibrium constants, axial and radial dispersions in the mobile phase. Afterwards, the moment expressions are used to analyze the peak tailing. The results show that a small quantity of sites with a slow desorption rate will lead to a large peak asymmetry.     

Electroosmosis through a Cation-Exchange Membrane: Effect of an ac Perturbation on the Electroosmotic Flow

The viscous behavior of oil-in-water (O/W) emulsions is studied over a broad range of dispersed-phase concentrations (φ) using a controlled-stress rheometer. At low-to-moderate values of φ (φ<0.60), emulsions exhibit Newtonian behavior. The droplet size does not exert any influence on the viscosity of Newtonian emulsions. However, at higher values of φ, emulsions exhibit shear-thinning behavior. The viscosity of shear-thinning emulsions is strongly influenced by the droplet size; a significant increase in the viscosity occurs when the droplet size is reduced. With the decrease in droplet size, the degree of shear thinning in concentrated emulsions is also enhanced. The viscosity data of Newtonian emulsions are described reasonably well by the cell model of Yaron and Gal-Or (Rheol. Acta 11, 241 (1972)), which takes into account the effects of the dispersed-phase concentration as well as the viscosity ratio of the dispersed phase to continuous phase. The relative viscosities of non-Newtonian emulsions having different droplet sizes but the same dispersed-phase concentration are scaled with the particle Reynolds number. The high shear viscosities of non-Newtonian emulsions can be predicted fairly well by the cell model of Yaron and Gal-Or (Rheol. Acta 11, 241 (1972)). Copyright 2000 Academic Press.     

Effect of pH on deagglomeration and rheology/morphology of aqueous suspensions of goethite nanopowder

The kinetics of deagglomeration in diluted suspensions of goethite nanopowder, as well as the rheology and morphology of the resulting suspensions, strongly depends on pH. At pH 3, nanopowder can be dispersed as separate nanoparticles, and the resulting suspension is Newtonian, with the viscosity only marginally higher than the viscosity of water. At pH between 5 and 12, nanoparticles tend to reaggregate and form weak aggregates/flocs. Morphology changes from a Newtonian suspension of primary nanoparticles to a non-Newtonian, shear-thinning suspension of large, porous, interconnected flocs with the yield stress reaching a maximum at an isoelectric point. The effect of pH on morphology and rheology is reversible, and as pH is reduced to 3, the suspension becomes Newtonian, with viscosity marginally higher than the viscosity of water. The rheological models based on DLVO theory do not allow prediction of the effect of pH on viscosity and yield stress, but the flow curves of goethite suspensions can be described by a fractal model with five adjustable parameters.     

An experimental study on the rheological properties of aqueous ceria dispersions

The rheological properties of aqueous ceria dispersions are studied experimentally. In particular, the effects of particle concentration, temperature, pH, and ionic strength are discussed. If the volume fraction is below 2%, ceria slurry exhibits Newtonian behavior, and for higher volume fractions, shear-thinning behavior is observed. The effect of temperature on the behavior of ceria slurry is found to be pH-dependent. If pHIEP, the viscosity slightly increases with increased temperature. A shift of IEP to a higher value of pH was observed for ionic strength, even for indifferent electrolytes. The influence of pH on the rheological properties of ceria slurry decreases if the ionic strength is high. The pH at which viscosity and yield stress are maximum coincide with IEP only for low ionic strengths. The slopes of acidic and basic branches of viscosity against pH and yield stress against pH curves are not symmetrical at high ionic strength, and the alkaline branch deviates significantly from Hunter's theory.     

High‐pressure rheology of polystyrene melts plasticized with CO2: Experimental measurement and predictive scaling relationships

A high‐pressure extrusion slit die rheometer was constructed to measure the viscosity of polymer melts plasticized by liquid and supercritical CO₂. A novel gas injection system was devised to accurately meter the follow of CO₂ into the extruder barrel. Measurements of pressure drop, within the die, confirm the presence of a one‐phase mixture and a fully developed flow during viscosity measurements. Experimental measurements of viscosity as a function of shear rate, pressure, temperature, and CO₂ concentration were conducted for three commercial polystyrene melts. The CO₂ was shown to be an effective plasticizer for polystyrene, lowering the viscosity of the polymer melt by as much as 80%, depending of the process conditions and CO₂ concentration. Existing theories for viscoelastic scaling of polymer melts and the prediction of T_g depression by a diluent were used to develop a free volume model for predicting the effects of CO₂ concentration and pressure on polymer melt rheology. The free volume model, dependent only on material parameters of the polymer melt and pure CO₂, was shown to accurately collapse the experimental data onto a single master curve independent of pressure and CO₂ concentration for each of the three polystyrene samples. This model constitutes a simple predictive set of equations to quantify the effects of gas‐induced plasticization on molten polymer systems. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 3168–3180, 2000     

Effect of Particle Size Distribution on the Rheology of Dispersed Systems

The rheological properties of aqueous polystyrene latex dispersions from three synthetic batches, with nearly the same z-average particle sizes, 400 nm, but varying degrees of polydispersity, 0.085, 0.301, and 0.485, respectively, were systematically investigated using steady-state shear and oscillatory shear measurements. The particles were sized with photon correlation spectroscopy and transmission electron microscopy and were stabilized sterically with PEO–PPO–PEO triblock copolymer (Synperonic F127). Results from steady-state shear measurements show that the viscosities of the systems exhibit shear-thinning behavior at high solid fractions. However, the degree of shear thinning depends on the breadth of particle size distribution, with the narrowest distribution suspension exhibiting the highest degree of shear thinning. The Herschel–Bulkley relationship best describes the flow curves. The relative viscosities as a function of volume fraction data were compared, and it was found that the broadest distribution suspension had the lowest viscosity for a given volume fraction. In addition, the data were fitted to the Krieger–Dougherty equation for hard spheres. A reasonable agreement of theory with experiment is observed, particularly and surprisingly for the very broad distribution. However, when the contribution to the volume due to the adsorbed polymer layer is considered, the agreement between experiment and theory becomes closer for all the suspensions, although the agreement for the broad distribution suspension is now worse. Fitting the Dougherty–Krieger theory to the experimental data based on our experimental maximum packing fractions gives very good agreement for all the systems studied. From oscillatory shear measurements, the moduli were obtained as a function of frequency at various latex volume fractions. The results show general change of the dispersions from viscous (G" > G′) at low volume fractions (0.25–0.30) to moderately elastic (G′ > G") at moderately high volume fractions (0.41–0.45). The change at this concentration level is likely due to some compression and interpenetration of the stabilizing polymer chain at the periphery, indicating the dominance of the interparticle forces. Overall, the very broad distribution was found to have the lowest elastic modulus for a given volume fraction.     

Correlation of the viscosity of pure liquids at high pressures based on an equation of state

A novel model has been presented for correlating the dynamic viscosity of Newtonian liquids at high pressures. The proposed model was started with the activation volume, which could simultaneously be influenced by temperature and pressure. The core of the model was based on Hu and Liu's work to calculate the compressibility factor. The final expression may contain two adjustable parameters, namely κ₁ and κ₂, which had been determined by fitting literature viscosity data. The results show that the agreement between experimental data of viscosity and the calculated ones with the proposed model was reasonably good for the selected systems. It was found that the logarithm of parameter κ₁ was a linear function of the reciprocal of the temperature, and κ₁ was approximately equal to viscosity of liquid at 0.1 MPa. Besides, for linear chain hydrocarbons, the logarithm of the parameter κ₁ was completely a linear function of the number of the carbon atoms under certain temperature. A comparison between model with two-parameter and one with one-parameter had been given to show their applicability. The results calculated by not only two-parameter but also one-parameter were superior to corresponding ones by previous pressure equation for estimating viscosity.     

Continuous Viscometric Detection in Size Exclusion Chromatography

Abstract

Continuous viscometric detection is based on the measurement of pressure drop in an on-line small capillary tube in which chromatographic eluents flow at constant flow rate. This detector is always coupled with a concentration detector (usually refractometer) and located before it to avoid back pressure in the refractometer. In order to obtain reliable information for polymer samples, it is generally necessary to connect these two detectors to a computer which performs data acquisition and treatment.

First, we discuss the problem of shape, geometry and dimensions of the viscometer. The typical characteristics are the result of a compromise between contradictory targets, mainly small internal volume, low shear rate and low pressure drop. It is shown that Poiseuille's laminar flow is only obtained when coiling radius of the measurement tube is greater than 6 cm, which is not the case inside the refractometer. Accordingly, two pressure transducers are necessary to eliminate pressure drop data coming from refractometer.

In a second part, we show how to extract information from pressure variation data. By using concentration data, pure solvent pressure and sample pressure it is possible to calculate intrinsic viscosity extrapolated to zero concentration at each point of the chromatogram. By comparison with intrinsic viscosity of the polymer used for calibration, a correction of hydrodynamic volume according to Benoit's universal calibration leads to absolute molecular weights.

In addition, for a linear polymer, the knowledge of log [η] versus log M leads to the determination of Mark-Houwink relationship coefficients. For branched polymers, viscosity laws are curved and the comparison between the linear law corresponding to the linear equivalent polymer and the experimental law allows the determination of the g' branching parameter distribution.     

Rheology of molten polystyrene with dissolved supercritical and near-critical gases

The viscosities of polystyrene melts containing three different dissolved gases, carbon dioxide, and the refrigerants R134a (1,1,1,2-tetrafluoroethane) and R152a (1,1-difluoroethane) are investigated at pressures up to 20 MPa. These pressures reach near-critical and supercritical conditions for the three gas components, and produce polymer–gas solutions containing up to 10 wt % gas. The measurements are performed in a sealed high-pressure capillary rheometer at 150 and 175°C, and at shear rates ranging from 1–2,000 s⁻¹. Very large reductions in melt viscosity are observed at high gas loading; at 150°C, 10 wt % R152a reduces the Newtonian viscosity by nearly three orders of magnitude relative to pure polystyrene. The viscosity data for all three polystyrene–gas systems follows ideal viscoelastic scaling, whereby the set of viscosity curves for a polymer-gas system can be scaled to a master curve of reduced viscosity vs. reduced shear rate identical to the viscosity curve for the pure polymer. The viscoelastic scaling factors representing the effect of dissolved gas content on rheological behavior are found to follow roughly the same variation with composition for all three polystyrene gas systems. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 2771–2781, 1999     

Strain-Dependent Rheological Model and Pressure Wave Prediction for Shut in and Restart of Waxy Oil Pipelines

Waxy oil gelation and rheology is investigated and modeled using strain-dependent viscosity correlations. Rotational rheometry shows a sharp viscosity increase upon gel formation. High creeping flow viscosities are observed at small deformation conditions prior to yielding. A new strain-dependent rheological model, following analogous formulation to the Carreau–Yasuda shear rate-dependent model, captures viscosity reduction associated with yielding. In addition, shear viscosity and extensional viscosity are investigated using a capillary rheometry method. Distinct shear-thinning behavior is observed in the shear mode of deformation, while distinct tension-thinning behavior is observed in the extensional mode of deformation for the model fluid systems. High Trouton ratios are obtained for the gelled model fluid systems, confirming strongly non-Newtonian fluid rheology. Finally, axial pressure wave profiles are computed at real pipeline dimensions for idealized moderate yield stress fluids using a computationally efficient 1D pipeline simulator. The Rønningsen time-dependent gel degradation model is used to emulate the fluid rheology in the simulator. Axial stress localization phenomena are shown to depend on the overall magnitude of gel degradation as established by the reduction in yield value. A high degree of gel degradation serves to afford flow commencement in a timely manner.     

Spreading dynamics and dynamic contact angle of non-Newtonian fluids

The spreading dynamics of power-law fluids, both shear-thinning and shear-thickening fluids, that completely or partially wet solid substrate was investigated theoretically and experimentally. An evolution equation for liquid-film thickness was derived using a lubrication approximation, from which the dynamic contact angle versus the contact line moving velocity relationship was evaluated. In the capillary spreading regime, film thickness h is proportional to xi3/(n+2) (xi is the distance from the contact line), whereas in the gravitational regime, h is proportional to xi1/(n+2), relating to the rheological power exponent n. The derived model fit the experimental data well for a shear-thinning fluid (0.2% w/w xanthan solution) or a shear-thickening fluid (7.5% w/w 10 nm silica in polypropylene glycol) on a completely wetted substrate. The derived model was extended using Hoffmann's proposal for partially wetting fluids. Good agreement was also attained between model predictions and the shear-thinning fluid (1% w/w cmc solution) and shear-thickening fluid (10% w/w 15 nm silica) on partially wetted surfaces.     

The links-nodes-blobs model for shear-thinning-yield-stress fluids

A new model based on fractal and percolation concepts is proposed to explain the rheological behavior of shear-thinning yield-stress fluids. Suspension particles of the fluids are described in terms of the links-nodes-blobs (L-N-B) model. The complex suspension rheology can be interpreted via the similarity of the L-N-B model to the Rouse chain model. Consequently, the empirically universal relationship between the dimensionless shear stress, T, and the dimensionless shear rate, Γ, which was recently suggested by Coussot as T = 1+KΓ ⁿ at Γ<0.3 and approaches Newtonian behavior at Γ>50, can be derived in terms of microscopic properties of a suspension of the force-free particles, fractal dimensions of the percolation system, and the critical lengths of the percolation system. According to our study, a more precise and more general universal relationship, which fits experimental data well over a wide range from Γ = 10⁻⁷10³, is proposed as T = 1+Γ+KΓ ⁿ . The parameter K in the universal equation can be expressed as a function of the dimensionless cross-section of the blobs, the distribution of links, and fractal dimensions of the percolation system, while the exponent n in the universal equation is a function of the fractal dimensions only. The transition point of a shear-thinning yield-stress fluid from shear-thinning to Newtonian behavior was explicitly interpreted. Received: 22 March 1999 Accepted in revised form: 1 June 1999