Modelling elongational and shear rheology of two LDPE melts

Experimental data of two low-density polyethylene (LDPE) melts at 200°C for both shear flow (transient and steady shear viscosity as well as transient and steady first normal stress coefficient) and elongational flow (transient and steady-state elongational viscosity) as published by Pivokonsky et al. (J Non-Newtonian Fluid Mech 135:58–67, 2006) were analysed using the molecular stress function model for broadly distributed, randomly branched molecular structures. For quantitative modelling of melt rheology in both types of flow and in a very wide range of deformation rates, only three nonlinear viscoelastic material parameters are needed: Whilst the rotational parameter, a ₂, and the structural parameter, β, are found to be equal for the two melts considered, the melts differ in the parameter describing maximum stretch of the polymer chains.     

Large amplitude oscillatory elongation flow

A filament stretching rheometer (FSR) was used for measuring the elongation flow with a large amplitude oscillative elongation imposed upon the flow. The large amplitude oscillation imposed upon the elongational flow as a function of the time t was defined as where ε is the Hencky strain, is a constant elongational rate for the base elongational flow, Λ the strain amplitude (Λ ≥ 0), and Ω the strain frequency. A narrow molecular mass distribution linear polystyrene with a molecular weight of 145 kg/mol was subjected to the oscillative flow. The onset of the steady periodic regime is reached at the same Hencky strain as the onset of the steady elongational viscosity ( Λ = 0). The integral molecular stress function formulation within the ‘interchain pressure’ concept agrees qualitatively with the experiments.     

Viscoelastic properties of POSS–styrene nanocomposite blended with polystyrene

Polyhedral oligomeric silsesquioxane (POSS) are hybrid nanostructures of about 1.5 nm in size. These silicon (Si)-based polyhedral nanostructures are attached to a polystyrene (PS) backbone to produce a polymer nanocomposite (POSS–styrene). We have solution blended POSS–styrene of with commercial polystyrene (PS), , and studied the rheological behavior and thermal properties of the neat polymeric components and their blends. The concentration of POSS–styrene was varied from 3 up to 20 wt.%. Thermal analysis studies suggest phase miscibility between POSS–styrene and the PS matrix. The blends displayed linear viscoelastic regime and the time–temperature superposition principle applied to all blends. The flow activation energy of the blends decreased gradually with respect to the matrix as the POSS–styrene concentration increased. Strikingly, it was found that POSS–styrene promoted a monotonic decrease of zero-shear rate viscosity, η ₀, as the concentration increased. Rheological data analyses showed that the POSS–styrene increased the fractional free volume and decreased the entanglement molecular weight in the blends. In contrast, blending the commercial PS with a PS of did not show the same lubrication effect as POSS–styrene. Therefore, it is suggested that POSS particles are responsible for the monotonic reduction of zero-shear rate viscosity in the blends.     

Molecular orientation in sheared molten thermotropic random copolyester

The macromolecular alignment and texture orientation in sheared thermotropic copolyester were investigated using in situ wide-angle X-ray scattering (WAXS) and polarizing optical microscopy (POM). The molecular behavior was correlated with viscoelastic properties. The polymer is a random copolyester based on 60 mol% 1,4-hydroxybenzoic acid (B) and 40 mol% ethylene terephthalate (ET) units. X-ray scattering showed that the molecular chains were aligned along the flow direction. The degree of molecular orientation, , is an increasing function of the applied shear rate. However, rheo-optics showed that shear flow could not orient the polydomain texture, i.e., neither defect stretching nor elimination of defects was observed. Instead, shear compressed the microdomains and gave rise to long-range orientation correlations. Rheology showed that the nematic melt is viscoelastic, the loss modulus G″ dominates the elastic modulus G′, and the dynamic viscosity η* is shear thinning. Moreover, the steady shear viscosity, η, also behaved shear thinning, while the first normal stress difference N ₁ remained positive. The empirical Cox–Merz rule did not hold, , within the shear rate range studied. The microscopic and rheological properties suggest that B–ET is a flow-aligning nematic polymer.     

Coarse-grained model of entangled polymer melts in non-equilibrium

A coarse-grained model developed for entangled polymeric systems and calibrated to represent melts in equilibrium (Rakshit, Picu, J Chem Phys 125:164907(1)–(10), 2006) is used to model shear flows. The model is a hybrid between multimode and mean-field representations: chain inner blobs are constrained to move along the chain backbone and the end blobs are free to move in 3D and continuously redefine the diffusion path for the inner blobs. Therefore, contour length fluctuations and reptation are captured. Constraint release is implemented by tracing the position of chain ends and performing a local relaxation of the chain backbones once end retraction is detected. This algorithm takes advantage of the multi-body nature of the model and requires no phenomenological parameters other than the length of an entanglement segment. The model is used to study start-up and step strain shear flows and reproduces features observed experimentally such as the overshoot during start-up shear flow, the Lodge–Meissner law, the monotonicity of the steady state shear stress with the strain rate, and shear thinning at large . These simulations are performed in conditions in which using a fully refined model of the same system would have been extremely computationally demanding or simply impossible with the current methods.     

Local Bifurcation Analysis of a Second Gradient Model for Deformations of a Rectangular Slab

In this paper we carry out a derivation of the equilibrium equations of nonlinear elasticity with an added second-gradient term proportional to a small parameter . These equations are given by a fourth order semilinear system of pdes. We discuss different types of possible boundary conditions for these equations. We then specialize the equations to a rectangular slab and study the linearized problem about a homogenous deformation. We show that these equations admit solutions representable as Fourier series in one of the independent variables. Furthermore, we obtain the characteristic equation for the eigenvalues (possible bifurcation points) for the linear problem and derive asymptotic representations for this equation for small . We used these expressions to show that in the limit as the characteristic equation for converges uniformly (in certain regions of the parameter space) to the corresponding characteristic equation for . When the base material () is that of a Blatz–Ko type, we get conditions for the existence of eigenvalues of the linear problem with and small. Our numerical results in this case indicate that the number of bifurcation points is finite when and that this number monotonically increases as . For the problem with we get conditions for the existence of local branches of non-trivial solutions.      

Zur analytischen Beschreibung des Füllstoffeinflusses auf das Fließverhalten von Kunststoff-Schmelzen

From literature some representative equations have been compiled describing the influence of filler on the viscosity of polymer melts. By application of these on the experimental results obtained from GF-SAN it was found that the relative viscosity _R , i.e. the ratio of the viscosities of the filled and unfilled melt, shows a pronounced dependence on the shear rate but not on the shear stress. Defining _R with constant and not with constant (as it is usually done), an analytical approach is possible independent of Further the influence of pressure, temperature and filler content on the zero-shear viscosity of filled polymer melts may be expressed by a modified Arrhenius equation.

     

Modeling elongational viscosity of blends of linear and long-chain branched polypropylenes

To enhance the melt strength of a conventional linear polypropylene (L-PP), blends with a long-chain branched polypropylene (LCB-PP) were produced by adding 2, 5, 10, 25, 50, and 75 wt% of LCB-PP to L-PP and mixing in a twin screw extruder. It was found that, already, an addition of 10% or less of LCB-PP to L-PP leads to significant strain hardening. Elongational viscosity data of L-PP and LCB-PP and those of their blends were analyzed by the use of the molecular stress function (MSF) theory. While L-PP is characterized by the MSF parameter, β=1 (typical for linear melts), and shows very little chain stretch (), melt elongational behavior of LCB-PP is characterized by the MSF parameters, β=2 (typical of LCB melts), and (which corresponds to a maximum stretch of molecular chains by a factor of 15). By extruding LCB-PP, a refining effect is observed similar to the refining effects seen in low density polyethylene (LDPE), which reduces the steady-state elongational viscosity and reduces to 121. A second-order mixing rule for the fractional relaxation moduli, g _i , was found to show good agreement with the linear-viscoelastic data of the blends. To simulate the elongational viscosities of the L-PP/LCB-PP blends, a similar second-order mixing rule was used for the MSF parameter, β, while a first-order mixing rule was found to be appropriate for . This allows for a quantitative prediction of the time-dependent elongational viscosities of all L-PP/LCB-PP blends on the basis of the linear and nonlinear parameters of the mixing components L-PP and LCB-PP only. Comparison between the steady-state elongational viscosities as obtained from creep experiments shows good agreement with predictions.     

Rheological behavior of poly(methyl methacrylate)/polystyrene (PMMA/PS) blends with the addition of PMMA-<Emphasis Type="BoldItalic">ran</Emphasis>-PS

In this work, the dynamic behavior of poly(methyl methacrylate)/polystyrene blend to which P(S_0.5-ran-MMA_0.5) was added was studied. Several blend (ranging from 5 to 20 wt% of dispersed phase) and copolymer (up to 20 wt% with respect to dispersed phase) concentrations were studied. The rheological behavior of the blends was compared to Bousmina’s (Rheol Acta 38:73–83, 1999) and Palierne’s (Rheol Acta 29:204–214, 1990) generalized models. The relaxation spectra of the blends were also inferred, and the results were analyzed in light of the analysis of Jacobs et al. [J Rheol 43:1495–1509, 1999]. The relaxation spectra of the blends with smaller dispersed phase (below 10 wt%) and larger copolymer concentrations (above 0.4 wt%) showed the presence of four relaxation times, two corresponding to the blend phases, τ _F , corresponding to the relaxation of the shape of the dispersed phase of the blend and that can be attributed to the relaxation of Marangoni stresses tangential to the interface between the dispersed phase and matrix. The experimental values of and were used to infer the interfacial tension (Γ) and the interfacial complex shear modulus (β) for the different blends, Γ decreased with increasing copolymer concentration. β decreased with increasing blend dispersed phase concentration and decreasing copolymer concentration. The predictions of Palierne’s generalized model were found to corroborate the experimental data once the values of Γ and β, found analyzing the relaxation spectra, were used in the calculations. Bousmina’s model was found to corroborate the data only for larger dispersed phase concentration. Paper was presented at the 3rd Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Oscillating Solutions and Large ω-limit Sets in a Degenerate Parabolic Equation

The paper deals with positive solutions of the initial-boundary value problem for with zero Dirichlet data in a smoothly bounded domain . Here is positive on (0,∞) with f(0) = 0, and λ₁ is exactly the first Dirichlet eigenvalue of −Δ in Ω. In this setting, (*) may possess oscillating solutions in presence of a sufficiently strong degeneracy. More precisely, writing , it is shown that if then there exist global classical solutions of (*) satisfying and . Under the additional structural assumption , s > 0, this result can be sharpened: If then (*) has a global solution with its ω-limit set being the ordered arc that consists of all nonnegative multiples of the principal Laplacian eigenfunction. On the other hand, under the above additional assumption the opposite condition ensures that all solutions of (*) will stabilize to a single equilibrium.      

Vector potential–vorticity relationship for the Stokes flows: application to the Stokes eigenmodes in 2D/3D closed domain

The unsteady dynamics of the Stokes flows, where , is shown to verify the vector potential–vorticity ( ) correlation , where the field is the pressure-gradient vector potential defined by . This correlation is analyzed for the Stokes eigenmodes, , subjected to no-slip boundary conditions on any two-dimensional (2D) closed contour or three-dimensional (3D) surface. It is established that an asymptotic linear relationship appears, verified in the core part of the domain, between the vector potential and vorticity, , where is a constant offset field, possibly zero.     

Uniqueness and Nonuniqueness of Viscosity Solutions to Aronsson’s Equation

For a bounded domain and , assume that is convex and coercive, and that has no interior points. Then we establish the uniqueness of viscosity solutions to the Dirichlet problem of Aronsson’s equation:

A Jordan Curve Spanned by a Complete Minimal Surface

In this paper we construct complete (conformal) minimal immersions which admit continuous extensions to the closed disk, . Moreover, is a homeomorphism and is a (non-rectifiable) Jordan curve with Hausdorff dimension 1. It turns out that the set of Jordan curves constructed by the above procedure is dense in the space of Jordan curves of with the Hausdorff metric.     

Explicit Effective Constants for an Inhomogeneous Porothermoelastic Medium

An arbitrary anisotropic micro-inhomogeneous (composite) poroelastic medium is considered, containing a random set of ellipsoidal inhomogeneities with different poroelastic characteristics. The properties of these constituents are described by the linear porothermoelastic theory of Biot. One of the self-consistent schemes named effective field method is used to develop explicit expressions for the effective porothermoelastic constants (tensor of the frame elastic compliances , tensor of the generalized Skempton’s coefficients , tensor of thermal expansion coefficients , Biot’s constants , and the heat capacity at constant stress for the static porothermoelastic theory. It is shown that for two components composite porous material these expressions are interconnected and can be expressed only via the components of tensor . Some special cases are considered for the isotropic main material (matrix).     

Asymptotic Behaviour of a Semilinear Elliptic System with a Large Exponent

Consider the problem where Ω is a bounded convex domain in , N > 2, with smooth boundary . We study the asymptotic behaviour of the least energy solutions of this system as . We show that the solution remain bounded for p large. In the limit, we find that the solution develops one or two peaks away from the boundary, and when a single peak occurs, we have a characterization of its location.This research was supported by FONDECYT 1061110 and 3040059.     

Uniform Exponential Dichotomy and Continuity of Attractors for Singularly Perturbed Damped Wave Equations

For , we consider a family of damped wave equations , where − Λ denotes the Laplacian with zero Dirichlet boundary condition in L ²(Ω). For a dissipative nonlinearity f satisfying a suitable growth restrictions these equations define on the phase space semigroups which have global attractors A _η, . We show that the family , behaves upper and lower semicontinuously as the parameter η tends to 0⁺.     

On Dual Configurational Forces

The dual conservation laws of elasticity are systematically re-examined by using both Noether's variational approach and Coleman–Noll–Gurtin's thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy–momentum tensor. Some previously unknown and yet interesting results in elasticity theory have been discovered. As an example, we note the following duality condition between the configuration force (energy–momentum tensor) and the dual configuration force (dual energy–momentum tensor) ,

Guidelines for Modeling a 2D Rough Wall Channel Flow

The effects of 2-D roughness elements on the Reynolds stress anisotropy tensor, and of the energy dissipation rate anisotropy tensor of a turbulent channel flow are investigated using data obtained from direct numerical simulations (DNSs). The roughness elements consist of transverse square rods of size , placed on one wall of the channel only. While is kept constant (, is the half-width of the channel), the spacing between the rods is varied from to . The results show that the variation in can dramatically change the structure of the wall region flow. The modeling of the near-wall region needs to reflect the structural changes caused by the variation in . On the basis of the Reynolds stress budgets, attention needs to be given to the turbulent energy and pressure diffusion terms while local isotropy may be a reasonable approximation for the energy dissipation rate, especially over a range of for which the drag is near its maximum.     

A Non-Newtonian Fluid with Navier Boundary Conditions

We consider in this paper the equations of motion of third grade fluids on a bounded domain of or with Navier boundary conditions. Under the assumption that the initial data belong to the Sobolev space H ², we prove the existence of a global weak solution. In dimension two, the uniqueness of such solutions is proven. Additional regularity of bidimensional initial data is shown to imply the same additional regularity for the solution. No smallness condition on the data is assumed.     

A Helmholtz/Weyl Decomposition in Energy Space

For a bounded region in a Helmholtz/Weyl decomposition of the Sobolev space is given,with orthogonality with respect to the strain-energy inner product of elasticity (anisotropic or isotropic).