An appraisal of rheological models as applied to polymer melt flow

Summary An attempt is made at giving an appraisal of some representative rheological models of both differential and integral type, using the standard rheological measurements of six polymer melts. Experimental data obtained were the steady shear viscosity and the first normal stress difference by means of aWeissenberg rheogoniometer over the range of shear rates: 10^–2 ~ 10 sec^–1, and by means of aHan slit/capillary rheometer over the range of shear rates: 10 ~ 10³ sec^–1. Also measured by means of theWeissenberg rheogoniometer were the dynamic viscosity and dynamic elastic modulus over the range of frequencies: 0.3 × 10^–2 ~ 3 × 10² sec^–1. Rheological models chosen for an appraisal are theSpriggs 4-constant model, theMeister model, and theBogue model.It is found that the capability of the three models considered is about the same in their prediction of the rheological behavior of polymer melts in simple shearing flow. It is pointed out however that, due to the ensuing mathematical complexities, the usefulness of these models is limited to the study of flow problems associated with simple flow situations. Therefore, in analysing the complex flow situations often encountered with various polymer processings, the authors suggest use of the empirical models of the power-law type for both the viscosity and normal stress functions.With 11 figures, 4 schemas and 1 table     

Investigation of the linear flow regime of commercial polymers by numerical conversion of MVM creep measurements

A new experimental and numerical method has been developed to characterize the terminal flow behavior of polydisperse, commercial grade polymer melts over a wide dynamic range of time/frequency scales. Experimentally, an MVM rheometer specifically designed for long time scale (t 10⁴ s) creep measurements is used to measure the creep compliance of three commercial polymers: two high density polyethylenes and one polystyrene. The long time scale MVM creep data are complemented in the short time scale regime by creep data from an industrial plate-plate rheometer. The time-dependent creep data is combined and converted to a discrete retardation spectra using a nonlinear regularization algorithm to address the ill-posed nature of the interconversion. The retardation spectrum is analytically converted to dynamic moduli and compared with independently measured dynamic moduli. In the overlapping frequency region, calculations and measurements show excellent agreement and the combined data span a much larger dynamic range than either independent data set. The calculated and measured dynamic moduli data are combined and a retardation spectrum with a vastly expanded dynamic range is generated. Combining long time scale MVM creep compliance data and dynamic moduli data exploits the intrinsic sensitivities of controlled strain and controlled stress rheological experiments and is a powerful means to greatly expand the experimentally accessible dynamic range of time/frequency. This approach is particularly useful for commercial polymers with broad molecular weight distributions and commensurately large distributions of relaxation times.     

Polymer motion as detected via dielectric spectra of 1,4-cis-polyisoprene under large amplitude oscillatory shear (LAOS)

In order to investigate the global polymer chain motion under large amplitude oscillatory shear (LAOS), the dielectric properties under LAOS are measured by a new rheo-dielectric combination. The design of the rheo-dielectric setup including a new fixture and modified oven is explained in detail. For 1,4-cis-polyisoprene, having type-A dipoles parallel to the backbone, the dielectric dipoles can detect the global polymer chain motion via the end-to-end vector. Thus rheological and dielectric (rheo-dielectric) properties reflect the dynamics of the polymer chain motion under LAOS. In this study, we investigate the rheo-dielectric properties under LAOS with 1,4-cis-polyisoprene as model component. As the strain amplitude was increased under LAOS, the relaxation strength from dielectric properties decreased for the whole spectra without changing the shape of the dielectric spectra. These results are analyzed on the basis of the molecular model of dynamic tube dilation (DTD) induced by the convective constraint release (CCR). It is found that the global chain motion under LAOS flow is affected by both rheological frequency and strain amplitude. It is also observed that segmental motion is affected via the oscillatory frequency under LAOS. This result differs from experiments under steady shear.     

Relaxation patterns of long,linear, flexible,monodisperse polymers: BSW spectrum revisited

Theoretical predictions for the dynamic moduli of long, linear, flexible, monodisperse polymers are summarized and compared with experimental observations. Surprisingly, the predicted 1/2 power scaling of the long-time modes of the relaxation spectrum is not found in the experiments. Instead, scaling with a power of about 1/4 extends all the way up to the longest relaxation times near τ/τ _max = 1. This is expressed in the empirical relaxation time spectrum of Baumgaertel-Schausberger-Winter, denoted as “BSW spectrum,” and justifies a closer look at the properties of the BSW spectrum. Working with the BSW spectrum, however, is made difficult by the fact that hypergeometric functions occur naturally in BSW-based rheological material functions. BSW provides no explicit solutions for the dynamic moduli, G ^′(ω), G ^″(ω), or the relaxation modulus G(t). To overcome this problem, close approximations of simple analytical form are shown for these moduli. With these approximations, analysis of linear viscoelastic data allows the direct determination of BSW parameters.     

Dynamic viscosity of various samples of polypropene and polyvinylchloride

Summary Complex dynamic viscosity experiments carried out on a balance rheometer CONTRAVES (Dr. Képès' system) for the polypropene samples and on a RHEOMETRICS viscometer apparatus for the polyvinylchloride formulations have allowed to determine the rheological behaviour of these materials. In the case of polypropene, the Maxwell-Wiechert model seems to be in good agreement with the rheological behaviour of this material. For the polyvinylchloride, a dynamic power-law has been proposed. This law correlates the complex dynamic viscosity modulus | ^*| with the pulsation of the frequency. It shows a certain analogy with that determined in the static mode.With 7 figures and 8 tables     

Rheological study of segmented polyurethane and polycaprolactone blends

The dynamic rheological properties of segmented polyurethane based on polycaprolactone diol (PU-PCL diol) and poly (ɛ-caprolactone) (PCL) blends were experimentally examined and theoretically analyzed using Palierne model. PU–PCL diol was melt blended with PCL in three different ratios of 20, 30, and 40%. Initial modeling attempts failed to fit the experimental data of these blends, as the model overpredicted their rheological data. This failure is believed to be due to partial dissolution of PCL in PU. According to our obtained results from differential scanning calorimetry and dynamic thermal mechanical analysis, pure PU–PCL diol is diluted by its homopolymer. By calculating the complex modulus of this diluted matrix using a simple mixing rule and its incorporation into the model, good fitting of the Palierne equation to the experimental data was obtained.     

Comparative rheological studies of polyamide-6 and its low loaded nanocomposite based on layered silicates

We study some rheological properties for polyamide-6 (PA-6) and a low concentrated clay nanocomposite melt based on polyamide-6 and montmorillonite. Simple shear experiments, carried out for both the neat system and nanocomposite at two different temperatures, include start up shear flows, stress relaxation after cessation of steady flow and oscillatory shear. The dynamic data for the neat PA-6 matrix differ markedly from that of the nanocomposite system, even if it has very low nanofiller concentration. Thermal stability of the PA-6 matrix imposed many restrictions on rheological studies of our systems. Therefore an experimental window was established via rheological and thermal characterization of the materials, wherein the polymer matrix was confirmed to be thermally stable. The relaxation spectra for both polymer systems were determined from linear dynamic experiments using the Pade-Laplace procedure. A rough estimation of nanocomposite volume fraction at percolation allowed us to attribute the occurrence of extra (relative to the neat polymer) Maxwell modes observed for the nanocomposite to the formation of a particulate network above the percolation threshold.     

Rheological characterization and constitutive modeling of bread dough

Bread dough (a flour–water system) has been rheologically characterized using a parallel-plate, an extensional, and a capillary rheometer at room temperature. Based on the linear and nonlinear viscoelastic and viscoplastic data, two constitutive equations have been applied, namely a viscoplastic Herschel–Bulkley model and a viscoelastoplastic K–BKZ model with a yield stress. For cases where time effects are unimportant, the viscoplastic Herschel–Bulkley model can be used. For cases where transient effects are important, it is more appropriate to use the K-BKZ model with the addition of a yield stress. Finally, the wall slip behavior of dough was studied in capillary flow, and an appropriate slip law was formulated. These models characterize the rheological behavior of bread dough and constitute the basic ingredients for flow simulation of dough processing, such as extrusion, calendering, and rolling.     

A class of linear viscoelastic models based on Bessel functions

In this paper we investigate a general class of linear viscoelastic models whose creep and relaxation memory functions are expressed in Laplace domain by suitable ratios of modified Bessel functions of contiguous order. In time domain these functions are shown to be expressed by Dirichlet series (that is infinite Prony series). It follows that the corresponding creep compliance and relaxation modulus turn out to be characterized by infinite discrete spectra of retardation and relaxation time respectively. As a matter of fact, we get a class of viscoelastic models depending on a real parameter \(\nu > -1\). Such models exhibit rheological properties akin to those of a fractional Maxwell model (of order 1/2) for short times and of a standard Maxwell model for long times.     

RHEOLOGICAL FRACTURE ANALYSIS OF CABLE BY CONSIDERING THE SECOND ORDER EFFECTS OF RANDOM SLACK

In this paper,we consider the cable as a bundle consisting of n sub-bundles,with mparallel tension members per sub-bundle,and the tension members themselves are polymeric yarns im-pregnated with a resin matrix.The nonfailed members at any instant must share an applied system loadaccording to some rule,since there is a clearly expressed dependence of the fracture on the durationand character of the loading.So then,the fracture of cable is a process of nonlinear dynamic evolu-tion,which accommodates to the non-equilibrium thermodynamics of irreversible processes by itself.Let us assume that the polymeric yarns are as viscoelastic solid,under certain probabilistic assump-tions,according to the principles of theology of bodies with defects,the relationship between the singlemember loading and failure and the bundle loading are investigated.It can be shown that the bundlefailure time is asymptotically normally distributed as the number of members grows large.After astudy of the second order effects of random slack,it is known that the asymptotic mean and varianceare functions of the parameters of loading and single member rhcological behavior.Hence the loss inthe asymptotic bundle strength mean brought about by random member and sub-bundle slack,L.,andthe loss in the asymptotic bundle strength variance caused by random member slack,Δ_(?),are deter-mined.And finally,it is known that the asymptotic time of failure can make up a considerable part ofthe fracture of cable,and the fracture of cable is a time-dependent process of rheological fracture.     

Memory Across Eye-Movements: 1/f Dynamic in Visual Search

The ubiquity of apparently random behavior in visual search (e.g., Horowitz & Wolfe, 1998) has led to our proposal that the human oculomotor system has subtle deterministic properties that underlie its complex behavior. We report the results of one subject's performance in a challenging search task in which 10,215 fixations were accumulated. A number of statistical and spectral tests revealed both fractal and 1/f structure. First, scaling properties emerged in differences across eye positions and their relative dispersion (SD/M)—both decreasing over time. Fractal microstructure also emerged in an iterated function systems test and delay plot. Power spectra obtained from the Fourier analysis of fixations produced brown (1/f ²) noise and the spectra of differences across eye positions showed 1/f (pink) noise. Thus, while the sequence of absolute eye positions resembles a random walk, the differences in fixations reflect a longer-term dynamic of 1/f pink noise. These results suggest that memory across eye-movements may serve to facilitate our ability to select out useful information from the environment. The 1/f patterns in relative eye positions together with models of complex systems (e.g., Bak, Tang & Wiesenfeld, 1987) suggest that our oculomotor system may produce a complex and self-organizing search pattern providing maximum coverage with minimal effort.     

Rheological characterization and modeling of end-functionalized polybutadienes

The rheological characterization and modeling of a series of polybutadienes obtained by anionic solution polymerization is presented in this work. The polybutadienes are synthesized using two different initiators: R,R′,R′′-silyloxyalkyllithium (F1) and R,R′,R′′-silylalkyllithium (F3). In addition, a polybutadiene obtained with a conventional alkyllithium initiator (n-butyllithium) is used as a reference. The rheological characterization is carried out under small amplitude oscillatory shear in the stress-controlled mode. Microstructure, molecular weight, and molecular weight distribution are determined by FTIR and GPC. The vinyl content of the polybutadienes synthesized using the functionalized initiators is similar to that obtained with n-butyllithium (8–11%). Materials obtained with F1 show a relatively low polydispersity within a narrow molecular weight range (250,000–300,000 g/mol), while samples obtained with F3 cover a wider range of molecular weights (65,000–670,000 g/mol) and display higher values of polydispersity. In all cases, a parallel reaction using propylene oxide in the termination step is done to place a functional group at the chain ends. The effect of this group on the rheological behavior appears to be negligible. Three rheological models are used and their predictions of the experimental data are compared. The models include the Doi and Edwards reptation model, expressions using a discrete spectrum of relaxation times based in the rubber-like liquid constitutive equation and the fractional Maxwell equation in which a given analytical relaxation-spectrum is used. Relevant relations are obtained between the models' parameters and the molecular properties of these systems, which in turn are related to the presence of functional groups at the polymer chain ends.     

含缺陷粘弹性材料的序参量本构方程

根据缺陷演化期形成的局域温度场的实验成果，用该内部温度场定义缺陷场。然后对统计相关细观缺陷的演化过程结合这些缺陷的生成速率和概率进行分析，认为含缺陷粘弹性材料的动态破坏过程可以用热波传播的程序按通常方法定量描述。     

Time- and space-resolved spectroscopy of dynamic hohlraum interiors

A dynamic hohlraum is created when an annular z-pinch plasma implodes onto a cylindrical 0.014 g/cc 6-mm-diameter CH₂ foam. The impact launches a radiating shock that propagates toward the axis at 350 μm/ns. The radiation trapped by the tungsten z-pinch plasma forms a 200 eV hohlraum that provides X-rays for indirect drive inertial confinement fusion capsule implosion experiments. We are developing the ability to diagnose the hohlraum interior using emission and absorption spectroscopy of Si atoms added as a tracer to the central portion of the foam. Time- and space-resolved Si spectra are recorded with an elliptical crystal spectrometer viewing the cylindrical hohlraum end-on. A rectangular aperture at the end of the hohlraum restricts the field of view so that the 1D spectrometer resolution corresponds approximately to the hohlraum radial direction. This enables distinguishing between spectra from the unshocked radiation-heated foam and from the shocked foam. Typical spectral lines observed include the Si Lyα with its He-like satellites and the He-like resonance sequence including Heα, Heβ, and Heγ, along with some of their associated Li-like satellites. Work is in progress to infer the hohlraum conditions using collisional–radiative modeling that accounts for the radiation environment and includes both opacity effects and detailed Stark broadening calculations. These 6-mm-scale radiation-heated plasmas might eventually also prove suitable for testing Stark broadening line profile calculations or for opacity measurements.     

A modified fractional model to describe the entire viscoelastic behavior of polybutadienes from flow to glassy regime

The dynamic mechanical behavior of a series of monodisperse polybutadienes has been investigated from the flow regime to the glassy state. Assuming a linear superposition of the entanglement and glass behavior a mathematical model of the spectrum of relaxation times is developed. It consists in a self-similar spectra for the entanglement contribution and a Fractional Maxwell Fluid (FMF) for the glassy contribution. The model closely represents the master curves of dynamic moduli over 15 logarithmic decades of frequency with three parameters for the flow regime (GN ⁰ _{N 0} and a cut-off parameter _max) and four parameters for the FMF. It is shown that one of the parameter of the FMF is similar to the power-law exponent of a self-similar spectra previously proposed in the literature to model the transition to glass.     

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.     

Rheological scaling and modeling of shear-enhanced crystallization rate of polypropylene

Shear-induced isothermal crystallization of a commercial isotactic polypropylene (iPP) has been investigated by using a rotational rheometer at the steady shear rates ranging from 0.00012 s⁻¹ to 1 s⁻¹, and the temperatures from 135 to 145 °C. Two time scales can be utilized to characterize the crystallization rates: one is the level-upturn onset time of the viscosity; another is that of the normal force. Plotting the onset times against the corresponding onset strain, a common critical value for all the undercooling temperatures can be identified, below which the shear flows have no significant effect on the crystallization rates. Furthermore, we propose a concept of dimensionless onset work; this parameter can make the normalized onset times approximately temperature-invariant in the range of our experiment. Our modeling of the quiescent crystallization is based on the nucleation theory of Ziabicki; the results indicate two-dimensional crystallite growth on pre-existing nuclei. The shear enhanced crystallization is modeled by estimating the excess free energy induced by the flow, and using the rheological model recently proposed by Marrucci, in which the required relaxation times are derived from our rheological measurements. The results imply that the crystallization under the present low shear rates is still two-dimensional crystallite growth on pre-existing nuclei, thus supporting the athermal nucleation theory proposed by Janeschitz-Kriegl. Compared with the experimental data, the modeling is only partially successful. Further improvements encompassing the effects of shear flows on the non-linear increase of the number density of athermal nuclei and on the acceleration of polymer chain disentanglement are needed.     

Generalized Cole-Cole behavior and its rheological relevance

Starting from an analysis of the rheological behavior of the complex modulus predicted by the Cole-Cole formalism, a generalized Cole-Cole ansatz is suggested in order to overcome the related difficulties. The corresponding rheological constitutive equation with fractional derivatives belonging to the generalized Cole-Cole respondance is stated and the characteristic material functions of the linear viscoelasticity theory (like the dynamic modulus and compliance, the relaxation and ratardation functions, the spectra, etc.) are derived. Model predictions of these functions will be compared with experimental results from dynamical measurements and creep data on different polymer systems which show cooperative phenomena (polymeric glasses and gelling systems). One can see that the modified ansatz fits the data very well, in spite of its relative simplicity.     

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.