First Normal Stress Difference Research Articles

Structural-Phenomenological Rheological Model for Engineering Calculations of Polymeric Media Flows

Studying the behavior of flows of solutions and polymer melts in the field of nonlinear viscoelasticity allows describing the rheological properties in more details and more accurately assess the adequacy of rheological models. A new structural-phenomenological model is proposed to describe the rheological behavior of melts of branched polymers. This model can be recommended for engineering calculations of flows of polymeric media. The model is obtained from the modified Vinogradov-Pokrovsky model which is based on the microstructur-al approach and describes the dynamics of a polymer fluid. Stationary viscometric functions for simple shear and uniaxial tension, as well as stationary shear viscosity, the first-difference coefficient of normal stresses, stationary viscosity at uniaxial tension, are calculated using the obtained model. Also, the influence of model parameters on the form of the functions has been studied. It is shown that the model describes with good accuracy the nonlinear viscoelastic behavior of flowing polymer systems: an anomaly in viscosity, a drop in the coefficient of the first difference of normal stresses, and a nonmonotonic nature of the dependence of the steady-state elongation viscosity on the tensile rate. The viscometric functions data are compared with the experimental data for an industrial polyethylene melt sample.

An assessment of models for flow-enhanced nucleation in an n-alkane melt by molecular simulation

Flow-enhanced nucleation of the crystal phase under shear and uniaxial extension for a monodisperse melt of n-pentacontahectane (C150H302 or C150) chains was studied by nonequilibrium molecular dynamics simulation. The resulting acceleration in the crystal nucleation rate was correlated with macroscopically measurable properties of the flow field and with microscopic conformational statistics. Based on the fidelity of the observed correlations, several empirical models reported in the literature were evaluated for their abilities to account for the observed enhancement of the nucleation rate due to flow, and new models are proposed for data that do not comport with existing models. In agreement with prior reports, the nucleation rate was found to correlate well with first-normal stress difference, the second invariant of the deviatoric conformation tensor, and the stretch ratio, albeit with some differences from the existing models. New models based on conformational invariants for Kuhn segments are proposed and shown to describe the simulation data more accurately than those based on conformational behavior of entire chains. Within the applicability of the stress-optical rule, related models are proposed based on invariants of the extra stress tensor.Flow-enhanced nucleation of the crystal phase under shear and uniaxial extension for a monodisperse melt of n-pentacontahectane (C150H302 or C150) chains was studied by nonequilibrium molecular dynamics simulation. The resulting acceleration in the crystal nucleation rate was correlated with macroscopically measurable properties of the flow field and with microscopic conformational statistics. Based on the fidelity of the observed correlations, several empirical models reported in the literature were evaluated for their abilities to account for the observed enhancement of the nucleation rate due to flow, and new models are proposed for data that do not comport with existing models. In agreement with prior reports, the nucleation rate was found to correlate well with first-normal stress difference, the second invariant of the deviatoric conformation tensor, and the stretch ratio, albeit with some differences from the existing models. New models based on conformational invariants for Kuhn segments are propo...

Shear-Induced Migration of Rigid Particles near an Interface between a Newtonian and a Viscoelastic Fluid.

Simulations of rigid particles suspended in two-phase shear flow are presented, where one of the suspending fluids is viscoelastic, whereas the other is Newtonian. The Cahn–Hilliard diffuse-interface model is employed for the fluid–fluid interface, which can naturally describe the interactions between the particle and the interface (e.g., particle adsorption). It is shown that particles can migrate across streamlines of the base flow, which is due to the surface tension of the fluid–fluid interface and a difference in normal stresses between the two fluids. The particle is initially located in the viscoelastic fluid, and its migration is investigated in terms of the Weissenberg number Wi (shear rate times relaxation time) and capillary number Ca (viscous stress over capillary stress). Four regimes of particle migration are observed, which can roughly be described by migration away from the interface for Wi = 0, halted migration toward the interface for low Wi and low Ca, particle adsorption at the interface for high Wi and low Ca, and penetration into the Newtonian fluid for high Wi and high Ca. It is found that the angular velocity of the particle plays a large role in determining the final location of the particle, especially for high Wi. From morphology plots, it is deduced that the different dynamics can be described well by considering a balance in the first-normal stress difference and Laplace pressure. However, it is shown that other parameters, such as the equilibrium contact angle and diffusion of the fluid, are also important in determining the final location of the particle.

Study on the Viscoelastic Fluid Behavior in Hollow Fiber Membrane Fabrication Process Using Giesekus’ Model

In the present study, attempts are made to study the behavior of a viscoelastic fluid in hollow fiber membrane production process. For this purpose, the fluid is considered to obey the Giesekus’ model and flow is taken as steady, two-dimensional and isothermal flow. The flow field is obtained using DEVSS/SUPG scheme combined with a free surface tracking method. The obtained results are compared to the case of Newtonian fluid. It is shown that the extrudate swell is 20% more in the case of viscoelastic fluid. Moreover, the elastic behavior of the viscoelastic fluid significantly affects the velocity field in the extrudate outside the die. It causes the fluid flow to become reversed in the middle part of the extrudate as the direct result of the stress relaxation process, whereas such a behavior is not seen for Newtonian fluid. Also, the first normal stress difference (FNSD) distribution in the extrudate is obtained and its variation is discussed. Furthermore, the effect of operational parameters including die back pressure and take-up velocity are investigated. Finally, the predicted extrudate swell for various operation conditions is compared to experimental results showing a good agreement between experimental and simulated data.

Optical Rheology of a Polydimethylsiloxane Fluid in Jeffrey-Hamel Type Flow

The rheo-optical behavior of a viscoelastic polydimethylsiloxane (PDMS) fluid was examined at room temperature for various pressure drops (flow rates) across a Jeffrey Hamel type converging wedge flow cell. The strain-rates were computed from local velocity measurements using laser Doppler anemometry (LDA), and the planar extensional flow behavior of the polymer melt was studied via birefringence measurements along the centerline of the flow cell. The linear relation between the stress and polarizability tensors were confirmed over a range of strain rates that extended well into the non-Newtonian region. The first normal stress difference (FNSD) was calculated from the local velocity measurements using a two-term Goddard-Miller model with a single Maxwell-type relaxation time constant of 0.0174 s computed from Rouse model and a zero-shear viscosity of 300 Pa.s. A linear stress-optical coefficient of 1.41 x 10-10 Pa-1 was obtained for PDMS in planar extensional flow at room temperature from the flow birefringence measurements and the first normal stress difference (FNSD) computed using the Goddard-Miller model. This compares well with values for PDMS in the range of 0.909 – 1.84 x 10-10 Pa-1 at room temperature as reported by various researchers.

Semi-implicit stress formulation for viscoelastic models: Application to three-dimensional contraction flows

In this work we propose a new formulation for the divergence of the viscoelastic stress for the collocated (cell-centered) finite-volume method. The reformulation allows for a semi-implicit handling of the constitutive equation, which promotes the numerical stability. Simulations of a three-dimensional planar and a square–square contraction show the robustness of this technique. The new formulation is completely devoid of unphysical checkerboard patterns of the velocity, which are present when using standard approximations for the divergence in conjunction with non-staggered grid methods. The consistency is ensured by giving results, which are independent of the time-step Δt for steady-state problems. The results for the planar contraction are generally in good agreement with experimental data for velocity, shear stress and first-normal stress difference. Stable simulations for the square–square contraction could be performed over a wide range of Deborah numbers. The vortex length is in agreement with the experimental results in the Newtonian-like and vortex-enhancement flow regime, however, the results deviate from the experiments in the diverging streamline regime.

Diffuse interface modeling of the morphology and rheology of immiscible polymer blends

A diffuse interface model is used to simulate a step-shear experiment of a binary immiscible polymer blend. The gradient theory used in diffuse interface modeling makes it possible to incorporate interfacial tension and governs the process of coalescence and breakup without any additional decision criteria. The interface tensor q, a direct outcome of the model, is used to relate microstructural information to the first-normal stress difference N1. The results obtained are in qualitative agreement with experiments reported in the literature.

The non-Newtonian rheology of dilute colloidal suspensions

The non-Newtonian rheology is calculated numerically to second order in the volume fraction in steady simple shear flows for Brownian hard spheres in the presence of hydrodynamic and excluded volume interactions. Previous analytical and numerical results for the low-shear structure and rheology are confirmed, demonstrating that the viscosity shear thins proportional to Pe2, where Pe is the dimensionless shear rate or Péclet number, owing to the decreasing contribution of Brownian forces to the viscosity. In the large Pe limit, remnants of Brownian diffusion balance convection in a boundary-layer in the compressive region of the flow. In consequence, the viscosity shear thickens when this boundary-layer coincides with the near-contact lubrication regime of the hydrodynamic interaction. Wakes are formed at large Pe in the extensional zone downstream from the reference particle, leading to broken symmetry in the pair correlation function. As a result of this asymmetry and that in the boundary-layer, finite normal stress differences are obtained as well as positive departures in the generalized osmotic pressure from its equilibrium value. The first normal stress difference changes from positive to negative values as Pe is increased when the hard-sphere limit is approached. This unusual effect is caused by the hydrodynamic lubrication forces that maintain particles in close proximity well into the extensional quadrant of the flow. The study demonstrates that many of the non-Newtonian effects observed in concentrated suspensions by experiments and by Stokesian dynamics simulations are present also in dilute suspensions.

Second normal stress difference relaxation in a linear polymer melt following step-strain

ABSTRACT: Flow birefringence is used to study stress relaxation following step straindeformations of a well entangled polyisoprene melt. The optical method employs mul-tiple light paths to fully sample the three-dimensional stress tensor, and hence providesmeasurements of all three independent shear material functions (shear stress and bothfirst and second normal stress differences). Experiments are complicated by multipleorders in retardation. However, data show that the ratio of the second to the firstnormal stress difference is a strain thinning function, with magnitude intermediatebetween the predictions of the Doi-Edwards model with and without independentalignment. © 1998 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 36: 2671–2675, 1998 Keywords: second normal stress difference; melt rheology; Doi-Edwards model INTRODUCTION Understanding the nonlinear rheology of polymers isimportant for the rational development of polymer pro-cessing technology, and at the same time can providedeep insights into the nature of relaxation processes inchain molecules. A prerequisite for the development ofmolecular models of chain dynamics is an extensivebase of reliable experimental data which characterizethe response of polymers to well-defined deformationhistories. Development of rheometric techniques to ac-quire such data has been an active, but demanding fieldof research. Even in simple shearing flows, completecharacterization of the stress tensor is rarely at-tempted. In shear, the stress tensor for an incompress-ible material may be represented in terms of threestress components: the shear stress, s, the first normalstress difference,

Rheological Characterization of Polyisobutylene Solutions from Rod-Climbing Experimentation

Abstract Rod-climbing constants and first-normal stress differences of polyisobutylene (PIB) solutions are obtained from rod-climbing experiments. For a low deformation rate, the polymer solution can be regarded as a second-order fluid, and the rod-climbing constant for second-order fluids is correlated with the rheological properties of the polymer solutions. The rod-climbing constants for solutions of PIB in polybutene decrease with temperature, and increase with rod size, polymer concentration, molecular weight of the polymer, and solvent viscosity. The rheological properties were measured using RMS-800, and compared with those obtained from rod-climbing experimentation.

Viscoelasticity of melts of polyethylene blends. 2

1. The activation energies of viscoelastic relaxation and viscous flow of melts of LDPE/HDPE blends of the same composition are the same. 2. The viscosity of melts of blends of linear and branched polyethylenes with low shear stresses is higher and corresponds to the value predicted by the rule of logarithmic additivity for high shear stresses. 3. The shear rate corresponding to the appearance of elastic turbulence increases exponentially with an increase in the proportion of HDPE in the polyethylene blend. 4. The equilibrium shear modulus of the melt can be regulated within broader limits by mixing HDPE and LDPE than by altering the molecular characteristics of the polymer. 5. The Havriliak-Negami equation in principle cannot describe the ratio of the rates of accumulation and dissipation of energy in steady-state shear flow whenPw/2ġgη0 > 0.5 (not the experimentally observed viscosity superanomaly is predicted, i.e.,dσ12/dġg < 0). 6. The first difference of normal stresses in shear flow, calculated with the die swell with the Bagley equation, is close to the experimental results for HDPE melts and HDPE/LDPE blends if the concentration of HDPE > 60%; for LDPE melts and blends containing HDPE < 60%, the best agreement of the predicted and experimental values is attained in calculations with the Tanner dependence.

Orthogonal superposition of small and large amplitude oscillations upon steady shear flow of polymer fluids

The orthogonal superposition of small and large amplitude oscillations upon steady shear flow of elastic fluids has been considered. Theoretical results, obtained by numerical methods, are based on the Leonov viscoelastic constitutive equation. Steady-state components, amplitudes and phase angles of the oscillatory components of the shear stress, the first and second normal stress differences as functions of shear rate, deformation amplitude and frequency have been calculated. These oscillatory components include the first and third harmonic of the shear stresses and the second harmonic of the normal stresses. In the case of small amplitude superposition, the effect of the steady shear flow upon the frequency-dependent storage modulus and dynamic viscosity has been determined and compared with experimental data available in literature for polymeric solutions. The predicted results have been found to be in fair agreement with the experimental data at low shear rates and only in qualitative agreement at high shear rates and low frequencies. A comparison of the present theoretical results has also been made with the predictions of other theories. In the case of large amplitude superposition, the effect of oscillations upon the steady shear flow characteristics has been determined, indicating that the orthogonal superposition has less influence on the steady state shear stresses and the first difference of normal stresses than the parallel superposition. However, in the orthogonal superposition a more pronounced influence has been observed for the second difference of normal stresses.

Influence of the rigidity of the dynamometer, specimen, and equipment on the measurement of the first difference of normal shear stresses

1. In the measurement of the first difference of normal stresses σ1 in nonstationary flow regimes of viscoelastic liquids (e.g., in the case of periodical deformation and in the case of transition processes of stationary regimes) on rheogoniometers using cone-plate working units, the rigidity of the material specimen in the direction of the rotation axis of the cone K*n is of the same order of magnitude as the rigidity of the dynamometer; consequently, K*n must be taken into account in the calculation of σ1. 2. An analytical relationship has been established for the calculation of the rigidity of the specimen using a cone-plate geometry in the direction of the rotation axis of the cone, based on a known modulus of the material. 3. Errors have been pointed out that may occur in the measurement of the phase angle, depending on the changes in the temperature of the equipment frame during the measurement.

Effect of fibre formation on elastic properties of melts prepared from polymer mixtures

Elastic properties (high-elastic deformation, first difference of normal stresses, shear modulus) of melts prepared from polyoxymethylene-copolyamide, polyoxymethylene-copolyamide-plasticizer mixtures were evaluated according to the extent of swelling of extruded annealed products. It was shown that polyoxymethylene and copolyamide of weak elasticity form mixtures which markedly differ in elastic properties from initial components. Thus, high elastic deformation γ e for melts of mixtures reached 45–100 rel. units, which is over one order of magnitude higher than γ e for thermoplastic materials such as polyethylene (or even for typical elastomers such as polybutadienes). The addition of a plasticizer increases elasticity of the polyoxymethylene-copolyamide melt. Such high elastic deformations are due to formation of ultra-thin polyoxymethylene fibres in the copolyamide during the flow of melt prepared from mixtures through the capillary. The applicability of ratios of evaluating parameters of high-elasticity for melts of polymer mixtures via swelling is explained.

Polarization‐optical investigation of normal and shear stresses in flow of polymers

AbstractA method has been suggested for calculating the first difference of normal stresses characterizing the flow of polymers at high shear stresses. The calculations are based on the results of rheooptical measurements in a slit of rectangular cross section. It has been found, for several samples of high molecular weight polybutadienes and polyisoprenes, that the flow behavior of the representatives of the given polymer homologous series having different molecular weights is characterized by a general relationship between the first normal stress differences and the shear stresses in those cases where the polymers are characterized by narrow molecular weight distributions. It has also been established that the first normal stress difference sharply increases in the region of shear stresses which immediately precedes the spurt—a jumpwise increase of the flow rate at a certain critical value of shear stress; while for polymers of wide molecular weight distribution the increase of the normal stress difference in the region of high values of shear stresses is retarded. Equilibrium swell of the extrudate has been measured and the first normal stress difference determined by the rheo‐optical method has been found to agree satisfactorily with the values calculated from the swelling ratios according to theoretical models.

Viscous and high-elasticity properties of solutions of monodispersed polybutadienes in the region of critical concentrations

It is shown that there is a critical concentration corresponding to the formation of a compact, fluctuating network, which results in a sharp change in the concentration dependence of the viscosity, first difference of normal stresses and high-elasticity deformations. It is found that at concentrations below the critical concentration high-elasticity deformations do not develop, although normal stresses exist. In consequence, the relationship whereby the modulus of high elasticity is equivalent to the ratio between the square of the viscosity and the coefficient of normal stress is fulfilled only for concentrations higher than the critical.

Experimental investigation of the rheological properties of concentrated polymer solutions

MOST investigations of the rheological properties of concentrated polymer solutions have beech restricted to the study of viscosity of solutions [1], although during the flow of polymer systems numerous specific effects arise, which can only be understood and explained when the solutions are regarded as viscoelastic liquids. The problem therefore is how to generalize the experimental data from investigations of various characteristics of solutions and the relations between them so that an indiection of the role of concentration and mode of deformation in the parameters measured could be achieved. This study, which was carried out using solutions of polystyrene in decalin at different concentrations, reports experimental determinations of three main characteristics of :heological behaviour of flowing polymer systems: shear stress z, first difference of normal stresses a and high elastic deformation ~e. Generalizations of these characteristics are proposed in order to plot eoncentration-invariant characteristics of the rheological properties of solutions.

Viscoelastic behavior of an amorphous polymer under oscillations of large amplitude

AbstractThe influence of periodic shear deformation and steady flow on a typical amorphous polymer is discussed. Forced sinusoidal vibrations were applied and the complex viscosity was determined. The action of a vibration of finite amplitude is equivalent to steady flow with a definite finite shear rate. Both processes cause truncation of the long‐time part of the relaxation specturm. It may be accepted to a first approximation that the long‐time boundary of the remaining part of the relaxation spectrum conforms to the long‐time part of the initial spectrum, even if the plateau region of the spectrum is truncated. The concept of limiting truncation of the short‐time part of the spectrum is introduced, this corresponding to the minimum absolute value of the complex viscosity versus reduced frequency and the lowest values of the dynamic and apparent viscosities. With an approximate representation of the relaxation spectrum, calculations were made of the maximum values of the viscosity and the coefficient relating the first difference of normal stresses to the square of the shear rate, and also of the apparent viscosity and normal stresses as functions of the shear rate. The calculated values are compared with experimental measurements, and it is shown that the correlation of the apparent viscosity and the absolute value of the complex viscosity is distributed at high frequencies, being superseded by a correlation between the apparent and dynamic viscosities.