Prediction of multiple overshoots in shear stress during fast flows of bidisperse polymer melts

We present a differential constitutive model of stress relaxation in polydisperse linear polymer melts and solutions that contains contributions from reptation, contour-length fluctuations, and chain stretching. The predictions of the model during fast start-up and steady shear flows of polymer melts are in accord with experimental observations. Moreover, in accordance with reported experimental literature (Osaki et al. in J Polym Sci B Polym Phys 38:2043–2050, 2000), the model predicts, for a range of shear rates, two overshoots in shear stress during start-up of steady shear flows of bidisperse polymer melts having components with widely separated molar masses. Two overshoots result only when the stretch or Rouse relaxation time of the higher molar mass component is longer than the terminal relaxation time of the lower molar mass component. The “first overshoot” is the first to appear with increasing shear rate and occurs as a result of the stretching of longer chains. Transient stretching of the short chains is responsible for the early time second overshoot. The model predictions in steady and transitional extensional flows are also remarkable for both monodisperse and bidisperse polymer solutions. The computationally efficient differential model can be used to predict rheology of commercial polydisperse polymer melts and solutions.     

The MSF model: relation of nonlinear parameters to molecular structure of long-chain branched polymer melts

The elongational viscosity data of model PS combs (Hepperle J, Einfluss der Molekularen Struktur auf Rheologische Eigenschaften von Polystyrol- und Polycarbonatschmelzen. Doctoral Thesis, University Erlangen-Nürnberg, 2003) are reconsidered by including the interchain pressure term of Marrucci and Ianniruberto [Macromolecules 37:3934–3942, 2004] in the Molecular Stress Function model [Wagner et al., J Rheol 47(3):779–793, 2003, Wagner et al., J Rheol 49:1317–1327, 2005d]. Two nonlinear model parameters are needed to describe elongational flow, β and . The parameterβ determines the slope of the elongational viscosity after the inception of strain hardening. It is directly related to the molecular structure of the polymer and represents the ratio of the molar mass of the (branched) polymer to the molar mass of the backbone alone. β follows from the hypothesis of Wagner et al. [J Rheol 47(3):779–793, 2003] that side chains are compressed onto the backbone. We consider also the case that side chains are oriented by deformation, but not stretched, and found little difference in the model predictions. The parameter represents the maximum strain energy stored in the polymeric system and determines the steady-state value of the viscosity in extensional flows. The relation of this energy parameter to the molecular structure is discussed. Good correlations between the energy parameter and different coil contraction ratios, as determined either experimentally or calculated theoretically by considering the topology of the macromolecule, are found. The smaller the relative size of the polymer coil, the larger is the energy parameter and the more strain energy can be stored in the polymeric system. Presented at the 3rd Annual European Rheology Conference, AERC2006, Crete, Greece.     

On the Rank 1 Convexity of Stored Energy Functions of Physically Linear Stress-Strain Relations

The rank 1 convexity of stored energy functions corresponding to isotropic and physically linear elastic constitutive relations formulated in terms of generalized stress and strain measures [Hill, R.: J. Mech. Phys. Solids 16, 229–242 (1968)] is analyzed. This class of elastic materials contains as special cases the stress-strain relationships based on Seth strain measures [Seth, B.: Generalized strain measure with application to physical problems. In: Reiner, M., Abir, D. (eds.) Second-order Effects in Elasticity, Plasticity, and Fluid Dynamics, pp. 162–172. Pergamon, Oxford, New York (1964)] such as the St.Venant–Kirchhoff law or the Hencky law. The stored energy function of such materials has the form

Relaxation time spectrum molecular weight distribution relationships

Single exponential decay relationships, which define the molecular weight distribution (MWD) of a polymer as a function of the polymer’s relaxation time spectrum (RTS), have been derived by Wu (Polym Eng Sci 28:538–543, 1988) and Thimm et al. (J Rheol 43:1663–1672, 1999). Experimental validation studies with monodisperse polymers, with quite precisely known MWDs, have been used to test their reliability. It has been established that neither formula is always able to accurately recover the MWDs of monodisperse polymers from their experimentally determined RTS. In this paper, different and more general relationships, based on theoretical results of Anderssen and Loy (Bull Aust Math Soc 65:449–460, 2002a) for decays of the form , where the derivative of θ(t) is a completely monotone function, are derived, analyzed, and applied. It is shown how to transform these general relationships to equivalent single exponential decay relationships for which Laplace transform solutions are derived. In order to illustrate the interrelationship between an RTS and its corresponding MWD, an explicit analytic solution is given. The paper concludes with a discussion of the rheological implications for the BSW model.     

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.     

Oscillatory and simple shear flows of a flour-water dough: a constitutive model

We reported some dynamic and viscometric data on an Australia strong flour-water dough. In oscillatory shear flow experiments, we found the linear viscoelastic strain limit is extremely low, of O(10^–3), consistent with other published data on doughs. The relaxation spectrum derived from the dynamic data is broad, indicating the blend nature of dough. In the start-up of a simple shear flow, we found the shear stress increases nonlinearly with time to a peak value and then decreases rapidly, with no steady-state response. The concept of steady-state viscosity is not very meaningful here, unless the strain at which the measurements are taken is also specified. The stress peaks are strain-rate dependent; but they occur at a strain of O(10), for the strong flour/water dough used, over four decades of strain rates. The experimental data were used to construct a phenomenological model for dough, consisting of an hyperelastic term (representing the elastic gluten network of permanent cross-linked long chain polymers), and a viscoelastic contribution (representing the suspension of starch globules and other long-chain components in dough that are not parts of the permanent cross-linked gluten network). The model predictions compared favourably with experimental data in oscillatory and shear flows.     

A new ALS model for dilute polymer solutions in flows with strong shear components

Ghosh et al. (J Rheol 46:1057–1089, 2002) developed a new model for dilute polymer solutions in flows with strong extensional components. The model based on introducing an adaptive length scale (ALS) as an internal variable was developed to reproduce the fine-scale physics of the Kramers chain. The ALS model describes the polymer molecule as a set of identical segments in which each segment represents a fragment of the polymer that is short enough so that it can sample its entire configuration space on the time scale of an imposed deformation and therefore stretch reversibly. As the molecule unravels, the number of the segments decreases, but the maximum length of each segment increases, so that the constant maximum contour length of the molecule is preserved. Though the single-segment-based ALS model accounts for the orientability of the polymer molecules, it cannot describe the internal motions of the molecules due to the lack of internal nodes. Hence, in this work we consider the more realistic chain (multi-segments) model composed of N _seg springs connected linearly. The model presented in this work is an extension of the ALS model developed by Ghosh et al. (J Rheol 46:1057–1089, 2002). We demonstrate that the ALS varies with the flow strength. Specifically, it is found that as the flow strength increases, the ALS decreases. This implies that as the flow strength increases, the polymer molecule is required to divide into finer and finer segments such that each segment can locally equilibrate with the imposed flow. However, there is a critical number of such subdivisions beyond which further subdivision of the polymer molecule is not required to capture the polymer dynamics for a given flow strength. Both shear viscosity and first normal stress coefficient predictions from ALS model show shear thinning behavior with Weissenberg number. In weak flows, the ALS model and the finitely extensible non-linear elastic model exhibit the same behavior.     

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.     

Diffuse interface model of surfactant adsorption onto flat and droplet interfaces

For applications where droplet breakup and surfactant adsorption are strongly coupled, a diffuse interface model is developed. The model is based on a free energy functional, partly adapted from the sharp interface model of [Diamant and Andelman 34(8):575–580, (1996)]. The model is implemented as a 2D Lattice Boltzmann scheme, similar to existing microemulsion models, which are coupled to hydrodynamics. Contrary to these microemulsion models, we can describe realistic adsorption isotherms, such as the Langmuir isotherm. From the free energy, functional analytical expressions of equilibrium properties are derived, which compare reasonably with numerical results. Interfacial tension lowering scales with the logarithm of the area fraction of the interface unloaded with a surfactant: . Furthermore, we show that adsorption kinetics are close to the classical relations of Ward and Tordai. Prelimary simulations of droplets in shear flow show promising results, with surfactants migrating to interfacial regions with highest curvature. We conclude that our diffuse interface model is very promising for apprehending the above-mentioned applications as membrane emulsification. Paper presented at the Annual European Rheology Conference (AERC) 2005, April 21–23, Grenoble, France.     

On the motion of a gyrostat similar to Lagrange’s gyroscope under the influence of a gyrostatic moment vector

In this paper, the problem of the motion of a gyrostat fixed at one point under the action of a gyrostatic moment vector whose components are ℓ _i (i=1,2,3) about the axes of rotation, similar to a Lagrange gyroscope is investigated. We assume that the center of mass G of this gyrostat is displaced by a small quantity relative to the axis of symmetry, and that quantity is used to obtain the small parameter ε (Elfimov in PMM, 42(2):251–258, [1978]). The equations of motion will be studied under certain initial conditions of motion. The Poincaré small parameter method (Malkin in USAEC, Technical Information Service, ABC. Tr-3766, [1959]; Nayfeh in Perturbation methods, Wiley-Interscience, New York, [1973]) is applied to obtain the periodic solutions of motion. The periodic solutions for the case of irrational frequencies ratio are given. The periodic solutions are analyzed geometrically using Euler’s angles to describe the orientation of the body at any instant t of time. These solutions are performed by our computer programs to get their graphical representations.     

Near-Wake Turbulence Properties in the High Reynolds Number Incompressible Flow Around a Circular Cylinder Measured by Two- and Three-Component PIV

The main objective of the present experimental study is to analyse the turbulence properties in unsteady flows around bluff body wakes and to provide a database for improvement and validation of turbulence models, concerning the present class of non-equilibrium flows. The flow around a circular cylinder with a low aspect ratio () and a high blockage coefficient () is investigated. This confined environment is used in order to allow direct comparisons with realisable 3D Navier–Stokes computations avoiding ‘infinite’ conditions. The flow is investigated in the critical regime at Reynolds number 140,000. A cartography of the velocity fields in the near wake of the cylinder is obtained by PIV and Stereoscopic PIV techniques. Statistical means and phase-averaged quantities are determined. Furthermore, POD analysis is performed on the data set in order to extract coherent structures of the flow and to compare the results with those obtained by the conditional sampling technique. The Reynolds stresses, the strain-rate and vorticity fields as well as the turbulence production terms are determined.     

The interchain pressure effect in shear rheology

Recently, the tube diameter relaxation time in the evolution equation of the molecular stress function (MSF) model (Wagner et al., J Rheol 49: 1317–1327, 2005) with the interchain pressure effect (Marrucci and Ianniruberto, Macromolecules 37:3934–3942, 2004) included was shown to be equal to three times the Rouse time in the limit of small chain stretch. From this result, an advanced version of the MSF model was proposed, allowing modeling of the transient and steady-state elongational viscosity data of monodisperse polystyrene melts without using any nonlinear parameter, i.e., solely based on the linear viscoelastic characterization of the melts (Wagner and Rolón-Garrido 2009a, b). In this work, the same approach is extended to model experimental data in shear flow. The shear viscosity of two polybutadiene solutions (Ravindranath and Wang, J Rheol 52(3):681–695, 2008), of four styrene-butadiene random copolymer melts (Boukany et al., J Rheol 53(3):617–629, 2009), and of four polyisoprene melts (Auhl et al., J Rheol 52(3):801–835, 2008) as well as the shear viscosity and the first and second normal stress differences of a polystyrene melt (Schweizer et al., J Rheol 48(6):1345–1363, 2004), are analyzed. The capability of the MSF model with the interchain pressure effect included in the evolution equation of the chain stretch to model shear rheology on the basis of linear viscoelastic data alone is confirmed.     

Nonlinear rheological properties and stability of dibenzylidene sorbitol fiber networks in poly(propylene oxide)

Dibenzylidene sorbitol (DBS) is known to gel organic liquids and polymers such as poly(propylene oxide) (PPO) by forming long fibers and fiber networks. Potential applications of these networks depend on their ability to withstand large deformations without significant morphological changes. Therefore, we studied the nonlinear rheological properties of the DBS fiber network in PPO for different DBS concentrations. We found that the concentration dependence of critical deformation (transition from linear to nonlinear viscoelastic region) and gel strength (G′ plateau in the linear region) can be explained on the basis of a model for densely cross-linked fiber gels (MacKintosh et al., Phys Rev Lett 75:4425–4428, 1995). Performing periodic strain sweeps, we found that the decrease in gel strength during the deformation cycles can be ascribed to reversible fiber coarsening. Additionally, start-up experiments showed a strong shear thinning behavior, which is in quantitative agreement with the SGM model (Sollich, Phys Rev E 58:738–759, 1998).     

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.     

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) ,

Aging of an unstable w/o gel emulsion with a nonionic surfactant

We study an unstable highly concentrated emulsion of water droplets in oil with a nonionic surfactant. A technique of light diffusion coupled to a rheometer allows simultaneous measurement of average droplet radius and emulsion shear elastic modulus during time. Over the studied range of volume fraction (from 71 to 95%), we show that Princen and Kiss’ (J Colloid Interface Sci 112:427–437, 1986) model does not apply. A dimensional analysis based on the hypothesis of dominant van der Waals forces is proposed for nonionic surfactants, which is in good agreement with experimental data. We also show that the measured average droplet volume increases linearly with time and that the coalescence rate strongly depends on the volume fraction in relation with different topological conformations of droplets.

A mesoscopic tube model of polymer/layered silicate nanocomposites

Rheology of isothermal suspensions of completely exfoliated silicate lamellae in polymer melts is investigated. In order to express more faithfully the physics involved in low shear rates and low frequencies, we model the polymer molecules composing the melt as chains whose motion is confined to a tube formed by surrounding chains and lamellae. In the absence of lamellae, the model reduces to the mesoscopic model of reptating chains developed in Eslami and Grmela (Rheol Acta, 2008). If the chains are seen only as FENE-P dumbbells, the model reduces to the model developed in Eslami et al. (J Rheol 51:1189–1222, 2007). Responses to oscillatory, transient, and steady shear flows are calculated and compared with available experimental data. Particular attention is payed to the region of low shear rate and low frequency.     

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.