Linear Viscoelastic Data Research Articles

Laun's rule for predicting the first normal stress coefficient in complex fluids: A comprehensive investigation using fractional calculus

Laun's rule [H. M. Laun, “Prediction of elastic strains of polymer melts in shear and elongation,” J. Rheol. 30, 459–501 (1986).] is commonly used for evaluating the rate-dependent first normal stress coefficient from the frequency dependence of the complex modulus. We investigate the mathematical conditions underlying the validity of Laun's relationship by employing the time-strain–separable Wagner constitutive formulation to develop an integral expression for the first normal stress coefficient of a complex fluid in steady shear flow. We utilize the fractional Maxwell liquid model to describe the linear relaxation dynamics compactly and accurately and incorporate material nonlinearities using a generalized damping function of Soskey–Winter form. We evaluate this integral representation of the first normal stress coefficient numerically and compare the predictions with Laun's empirical expression. For materials with a broad relaxation spectrum and sufficiently strong strain softening, Laun's relationship enables measurements of linear viscoelastic data to predict the general functional form of the first normal stress coefficient but often with a noticeable quantitative offset. Its predictive power can be enhanced by augmenting the original expression with an adjustable power-law index that is based on the linear viscoelastic characteristics of the specific material being considered. We develop an analytical expression enabling the calculation of the optimal power-law index from the frequency dependence of the viscoelastic spectrum and the strain-softening characteristics of the material. To illustrate this new framework, we analyze published data for an entangled polymer melt and for a semiflexible polymer solution; in both cases our new approach shows significantly improved prediction of the experimentally measured first normal stress coefficient.

Characterization of Nonlinear Viscoelastic Properties of Enhanced Oil Recovery Polymer Systems Using Steady-Shear Rheometry

SummaryEnhanced oil recovery (EOR) polymer systems such as hydrolyzed polyacrylamide (HPAM) solutions flowing in porous media at high fluxes were reported to cause shear-thickening, a nonlinear viscoelastic (NLVE) phenomenon. Bulk rheological studies are usually performed to characterize the viscous and viscoelastic behaviors of polymer solutions in porous media. To characterize the viscoelastic properties, oscillatory shear rheology based on linear viscoelastic (LVE) data and extensional rheology based on NLVE data have been used. Although both extensional flow and high-speed shear flow can stretch and thicken polymer solutions in a nonlinear fashion, steady-shear rheometry has been used to measure only the viscous and thinning behavior of HPAM systems. In this paper, (a) the thickening ability of HPAM polymer systems formulated with contrasting salinity and molecular weight (MW) concentration is characterized in bulk shear rheometry to analyze if the thickening in the pure shear field correlates with the thickening behavior reported in porous media and other rheometry and (b) the thickening intensity of the polymer solutions of varying salinity concentration in the bulk shear field is compared with the reported mechanical degradation effects in the literature.The shear-thickening index obtained by fitting a power-law model to thickening data in steady-shear rheograms is higher for the high-saline, high-MW low-concentration HPAM systems that have inherently higher nonlinear viscoelasticity. Furthermore, the thickening behavior observed due to salinity variation in bulk steady shear contrasts with linear oscillatory shear behavior but conforms to the thickening behavior observed in porous media and in the extensional field. This signifies that polymer-EOR researchers performing comparative studies on low-salinity and high-salinity polymer floods, and having a shear rheometer at their disposal, must look beyond linear oscillatory rheology and make use of nonlinear steady-shear rheology. There is a direct relation between the shear-thickening index of saline HPAM solutions and their mechanical degradation intensity. This paper shows that the high-speed data in steady-shear rheometry that are usually ignored in EOR literature have useful information and the notion that HPAM solutions are purely thinning in the bulk shear field needs to be reconsidered.

A single-chain model for the linear viscoelasticity of unentangled melts of associating polymers

Existing single-chain models for unentangled associating polymers account for the association by assigning the sticky junctions a large value of the monomeric friction coefficient, which prevents them from moving in space unless stickers effectively dissociate. With such an assumption, comparison of model predictions with linear viscoelastic data is not fully successful in the intermediate range of frequency. In this work, we improve agreement with data by developing a single-chain model where sticky junctions are allowed to quickly move in space without dissociating. We also account for a random distribution of the stickers but differently from the recent model by Jiang et al. [Macromolecules 53, 3438–3451 (2020)]. Predictions of the model are successfully compared with unentangled melt data for two different copolymer chemistries and different sticker concentrations. Particularly significant are the data by Cui et al. [J. Rheol., 62, 1155–1174 (2018)] of melts of polymers with only two stickers per chain, revealing that sticky junctions are in fact also endowed with fast mobility.

Kramers–Kronig relations for nonlinear rheology. Part II: Validation of medium amplitude oscillatory shear (MAOS) measurements

The frequency dependence of third-harmonic medium amplitude oscillatory shear (MAOS) modulus G33∗(ω) provides insight into material behavior and microstructure in the asymptotically nonlinear regime. Motivated by the difficulty in the measurement of MAOS moduli, we propose a test for data validation based on nonlinear Kramers–Kronig relations. We extend the approach used to assess the consistency of linear viscoelastic data by expressing the real and imaginary parts of G33∗(ω) as a linear combination of Maxwell elements: the functional form for the MAOS kernels is inspired by time-strain separability (TSS). We propose a statistical test based on fitting a sum of Maxwell elements using LASSO (least absolute shrinkage and selection operator) regression, and call it the SMEL test. It works well on a broad range of materials and models including those that do not obey TSS. It successfully copes with experimental data that are noisy or confined to a limited frequency range. When Maxwell modes obtained from the SMEL test are used to predict the first-harmonic MAOS modulus G31∗, it is possible to identify the range of time scales over which a material exhibits TSS.

Processing of viscoelastic data via a generalized fractional model

Data fitting and interconversion remain to be a difficult task in linear viscoelasticity. On the basis of transfer function, a generalized fractional model is proposed to fit linear viscoelastic data. The new model is modified from the fractional Maxwell model, and it is abbreviated as the MFM model in this paper. Common fractional viscoelastic models, including the fractional Kelvin and fractional Maxwell models, can be considered as special cases of the MFM model. Apart from inheriting the properties of the fractional Maxwell model, the MFM model is capable of adjusting the transient region between the power-law and plateau regions. Case studies are provided to apply the MFM model in data fitting of various kinds of viscoelastic data, and the results generally demonstrate improved fitting quality with fewer fitting modes compared with the generalized Maxwell model. In addition, the MFM model is used as a venue to convert time-domain data (including creep compliance data, creep data with ringing, and realistic relaxation data) to dynamic moduli data. Compared with classical numerical methods for data interconversion, this method appears to be more direct and convenient.

Calculation of molecular weight distribution using extended Cole-Cole model and quadratic mixing rule

We suggest a numerical method to calculate molecular weight distribution from linear viscoelastic data. The calculation method consists of three components: (1) a viscoelastic model of a monodisperse polymer as a function of molecular weight; (2) the mixing rule connecting viscoelastic data of monodisperse and polydisperse polymers through molecular weight distribution; (3) an algorithm which calculates the molecular weight distribution from the chosen mixing rule. Since we cannot measure the relaxation modulus of all monodisperse samples, we need an accurate monodisperse model for any molecular weight. It is known that a dynamic test is more reliable than a relaxation test, while the mixing rule needs relaxation modulus. Hence, we should have a smart numerical method that can convert dynamic data to relaxation modulus with the minimum conversion error. If we use the numerical method, then we have to generate numerical data from the model. Then it takes quite a long time. On the other hand, if we have a monodisperse model with the analytical relaxation spectrum, then calculation time can be reduced dramatically. Since the conversion from relaxation modulus to dynamic modulus suffers from smaller errors than the reverse conversion because of ill-posedness of the interconversion, the analytical conversion can be implemented more quickly at an acceptable level of errors. This paper proposes a new method satisfying the requirements.

Rethinking the Analysis of the Linear Viscoelastic Behavior of an Epoxy Polymer near and above the Glass Transition

In this communication, we propose a different approach for analyzing linear viscoelastic relaxation data that is more faithful to the underlying physics and naturally accommodates the thermorheological complexity that is observed in glass-forming polymers. Specifically, the linear viscoelastic behavior was evaluated for a diglycidyl ether of bisphenol-A epoxy cured with 4,4′-methylenedianaline with a glass transition temperature (Tg) of 101.5 °C. The dynamic storage and loss moduli were measured from 10–2 to 101.7 Hz for 19 temperatures between 90 and 180 °C. The experimental window was extended by two orders of magnitude using stress relaxation experiments for temperatures between 90 and 112.5 °C. The G′ and G″responses for this single-phase polymer are thermorheologically complex, thus precluding the construction of master curves via time–temperature superposition. The traditional method of determining the relaxation spectrum implicitly assumes a constant spectral density where the spectral strength cha...

Reversible Gelation of Entangled Ionomers

Dynamics of entangled associative polymers near the sol-to-gel transition is examined both theoretically and experimentally. Explicit expressions are provided for the elastic modulus and the relaxation time at varying degrees of gelation e. Linear viscoelasticity data are presented for samples derived from polystyrene containing about 5 entanglements per chain, which are sulfonated to different levels and neutralized with sodium hydroxide. The characteristic features of the sol-to-gel transition are observed, which can be categorized into three regimes according to the value of e: sol (−1 < e ≤ 0), mixture of sol and gel (0 < e ≤ 1), and gel (1 ≤ e). The sol samples exhibit a steep relaxation with no well-defined power-law scaling, suggesting a strong dynamic dilution effect. The mixtures of sol and gel fail to obey the time–temperature superposition at high T where the ionic association is activated, and thus, the relaxation dynamics of the sol chains and the gel network vary with temperature differently...

Rheological mucoadhesion and cytotoxicity of montmorillonite clay mineral/hybrid microparticles biocomposite

A biocomposite based on montmorillonite clay mineral (MMT) and hybrid biopolymer microparticles (MP) carrying atorvastatin (AC), these were designed for oral drug carrier was characterized by scanning and transmission electron microscopy (SEM &TEM), differential scanning calorimetry (DSC) and in vitro biocompatibility by cell viability on fibroblast cells. The rheological synergism relation (by linear oscillatory tests at different pH and frequency) with the mucoadhesive properties revealed interactions between isolated mucin and the MP/MMT biocomposite and those between the mucin and the individual components. Results of these tests showed that the MP/MMT biocomposite provides an improvement in the thermal stability and hence good biocompatibility exhibiting anti-proliferation effects at high concentrations. To our knowledge, no previous reports exist on the determination of the rheological synergism in biocomposites with MMT clay either alone. Moreover, we found a dependence of the rheological synergism on pH and frequency provided by linear viscoelastic data at low frequencies. The MP/MMT biocomposite possesses the highest rheological synergism related to the strongest interaction with mucin at pH 1.6.

On fitting data for parameter estimates: residual weighting and data representation

We describe a fundamental problem in fitting rheological data to infer parameters and thus structural information about a material. The results may depend greatly on subjective choices in the fitting method making such inferences non-unique and therefore highly uncertain or wrong. We study experimental data and demonstrate that the most commonly used fitting scheme for oscillatory linear viscoelastic data is deceptively wrong for the purpose of inference, i.e., viscoelastic moduli (G′,G″) weighted by the experimental data. Our results establish best practices for fitting rheological data, linear viscoelastic or otherwise, with strong implications for inverse problems of structural inference.

Enhanced Foamability with Shrinking Microfibers in Linear Polymer.

Strain hardening has important roles in understanding material structures and polymer processing methods, such as foaming, film forming, and fiber extruding. A common method to improve strain hardening behavior is to chemically branch polymer structures, which is costly, thus preventing users from controlling the degree of behavior. A smart microfiber blending technology, however, would allow cost-efficient tuning of the degree of strain hardening. In this study, we investigated the effects of compounding polymers with microfibers for both shear and extensional rheological behaviors and characteristics and thus for the final foam morphologies formed by batch physical foaming with carbon dioxide. Extensional rheometry showed that compounding of in situ shrinking microfibers significantly enhanced strain hardening compared to compounding of nonshrinking microfibers. Shear rheometry with linear viscoelastic data showed a greater increase in both the loss and storage modulus in composites with shrinking microfibers than in those with nonshrinking microfibers at low frequencies. The batch physical foaming results demonstrated a greater increase in the cell population density and expansion ratio with in situ shrinking microfibers than with nonshrinking microfibers. The enhancement due to the shrinkage of compounded microfibers decreasing with temperature implies that the strain hardening can be tailored by changing processing conditions.

Effects of particle stiffness on the extensional rheology of model rod-like nanoparticle suspensions.

The linear and nonlinear rheological behavior of two rod-like particle suspensions as a function of concentration is studied using small amplitude oscillatory shear, steady shear and capillary breakup extensional rheometry. The rod-like suspensions are composed of fd virus and its mutant fdY21M, which are perfectly monodisperse, with a length on the order of 900 nm. The particles are semiflexible yet differ in their persistence length. The effect of stiffness on the rheological behavior in both, shear and extensional flow, is investigated experimentally. The linear viscoelastic shear data is compared in detail with theoretical predictions for worm-like chains. The extensional properties are compared to Batchelor's theory, generalized for the shear thinning nature of the suspensions. Theoretical predictions agree well with the measured complex moduli at low concentrations as well as the nonlinear shear and elongational viscosities at high flow rates. The results in this work provide guidelines for enhancing the elongational viscosity based on purely frictional effects in the absence of strong normal forces which are characteristic for high molecular weight polymers.

Molecular Design of Highly Stretchable Ionomers

Application of ionomers is often disturbed by their brittleness originating from limited stretchability of the network strands physically cross-linked by the ionic sites therein. Thus, an effective method of improving the ductility is to increase the length of network strands (and/or entanglements). Considering this point, this study examined linear viscoelasticity (LVE) and nonlinear elongational rheology of unentangled copolymers of hexyl methacrylate (HMA) and the ionic monomer sodium 4-vinylbenzenesulfonate hydrate (SSNa). The ionized SSNa monomer, being randomly distributed along the chain backbone at a concentration ranging from <1 to ∼4 monomers per chain, served as the physical cross-link (or physical branching point). The LVE data showed a sol-to-gel transition, and the ductility of the sample turns out to be strongly related to the degree of gelation. Analysis of those data gave an average length of the network strands, and the ductility of the ionomer samples detected in the nonlinear elongatio...

Linear Viscoelasticity and Cation Conduction in Polyurethane Sulfonate Ionomers with Ions in the Soft Segment–Single Phase Systems

PEO-based polyurethane sulfonate ionomers with various PEO chain lengths and sodium counterions were synthesized and characterized by both linear viscoelasticity (LVE) and dielectric relaxation spectroscopy (DRS). Since p-phenylene diisocyanate is a small hard segment, the ionomers were found to be single phase with only one DSC Tg which increases with both ion content and hard segment content. An electrode polarization model was used to simultaneously determine the temperature dependence of conducting ion concentration and their mobility, which show Arrhenius and Vogel–Fulcher–Tammann (VFT) temperature dependences, respectively. The polymer dipole relaxation probed by DRS was found to be between 103 and 106 times faster than the mechanical segmental relaxation observed in LVE data near Tg, suggesting that most polymer modes need to wait for the ions to rearrange. The dielectric relaxation, mechanical relaxation, and ionic conductivity all show good correlation with DSC Tg and are related to ion rearrange...

Linear and Extensional Rheology of Model Branched Polystyrenes: From Loosely Grafted Combs to Bottlebrushes

Monodisperse comb polystyrenes (comb-PS) with loosely to densely grafted architectures up to loosely grafted bottlebrush structures were synthesized via anionic polymerization. This comb-PS series, named PS290-Nbr-44, had the same entangled backbone, Mw,bb = 290 kg/mol, corresponding to a number of entanglements along the backbone Zbb ≅ 20, and similar branch length, Mw,br ≅ 44 kg/mol or Zbr ≅ 3, but varied in the number of branches per molecule, Nbr, from 3 to 190 branches. Consequently, the average number of entanglements between two consecutive branch points along the backbone (branch point spacing), Zs, ranged from well entangled, Zs ≅ 5, to values that were far less than one entanglement, Zs ≅ 0.1. Linear viscoelastic data including the zero-shear rate viscosity, η0, diluted modulus, GN,s0, and a new diluted modulus extracted from the van Gurp–Palmen plot, |G*| at δ = 60°, were analyzed as a function of the Mw of the combs. Scaling of η0 versus Mw revealed three different regions for increasing Nbr o...

Non-terminal behavior as a finger print to follow droplet deformation

It is well-established that rheology can trace structural dynamics and conjectures the forming/formed morphology. The objective of this work is to trace the droplet deformation via conducting fiber spinning process. In this regard, two types of polymer blends Polypropylene (PP)/Polyamide6 (PA6) and Polypropylene/Polytrimethylene terephthalate (PTT) with various ratios (99/1, 94/6, 90/10, and 80/20) were melt-blended on co-rotating twin screw extruder and thereafter, were spun. Scanning electron microscopy (SEM) images represent a droplet and fibrillar morphologies for as-extruded samples and spun blend fibers respectively. Unlike PA6 as dispersed phase, PTT droplets in PP/PTT blend system shows improved spinnability. Linear melt viscoelastic data were gathered by performing rheological approaches in linear region. Conducting frequency sweep at 240°C is revealed a terminal behavior for Gʹ and a Newtonian plateau for η* at low frequencies. However, at 195°C dynamic rheological data pertinent to fibers represent a non-terminal trend for Gʹ and viscosity upturn at low frequencies respectively. These distinguishable characteristics can be considered as spinnability of droplets which reflect fibrillation mechanism, and fibrillar growth can be followed by scanning the viscoelastic responses at low frequency region which enhanced upon fibril growth. According to SEM and rheological test results, PTT, as an engineering plastics, manifests higher capability of forming nano fibrils.

On the use of continuous relaxation spectra to characterize model polymers

There have been many more papers published on methods for inferring a continuous relaxation spectrum from linear viscoelastic data than have been published on the use of such methods for polymer characterization. But a continuous spectrum can reveal key details of the relaxation process that are not evident from plots of the storage and loss moduli, particularly in the case of model polymers prepared for research. In particular there are sharp peaks in the spectrum where Rouse relaxation is complete and further relaxation is inhibited by entanglements in the plateau zone and at the entry to the terminal zone where molecules have escaped their tubes. Examples of continuous spectra that reveal these relaxation phenomena are presented.

Structure preservation of Aloe vera (barbadensis Miller) mucilage in a spray drying process

Aloe vera (barbadensis Miller) mucilage in powder form was obtained by spray-drying following by suspension in aqueous solution, to enable microstructure recovery. The rheological behavior of the reconstituted mucilage was evaluated as a function of mucilage concentration, temperature, pH and ionic-strength. Mucilage solutions exhibited shear-thinning non-Newtonian behavior. The viscosity was found dependent on ionic-strength. This dependence is more evident when divalent cations are used, although a strong rise in viscosity upon increasing pH is observed. Linear viscoelastic data show a predominant viscous behavior, but with a crossover point (storage module G′ = loss module G″) suggesting a change in molecular conformation to a random-coil arrangement of the mucilage microstructure. The spray-dried powders were compared with fresh mucilage, with regard to chemical composition and mechanical flow behavior. Results reveal a small structure modification during the spray-drying process, evidencing preservation of the mucilage microstructure when optimum spray-drying conditions are used, i.e., 1.5 L/h inlet flow, temperature of 150 °C and atomization rate of 27,500 rpm.

Semianalytical methods for the determination of the nonlinear parameter of nonlinear viscoelastic constitutive equations from LAOS data

Various viscoelastic constitutive equations have been developed to describe nonlinear viscoelastic flows. Most equations contain two kinds of parameters: Nonlinear and linear ones. The linear parameters correspond to relaxation time spectrum and can be determined from linear viscoelastic data. Meanwhile, the nonlinear parameters cannot be determined by linear viscoelastic data. The determination of the nonlinear parameters requires both reliable nonlinear data and complex procedures for fitting numerical solution of differential equations to the nonlinear data. If an analytical solution of viscoelastic model is available then dramatic reduction of difficulty is expected in the determination of the nonlinear parameters. Previous studies on analytical solution of large amplitude oscillatory shear (LAOS) are based on series expansion which is effective up to only third harmonic. Since it is practically impossible to obtain the analytical solution of higher order than fifth, we suggest a new method which extracts semianalytical solutions (SAS) for some relevant quantities of LAOS from the numerical solutions of nonlinear viscoelastic constitutive equations: The Giesekus model and the Phan-Thien/Tanner model. The SAS includes the effects of higher harmonics which cannot be achieved by low-order series expansion technique. The series expansion is applicable to limited Wi ≪ 1 but any De, while the SAS are applicable to De &amp;lt; 1 but any Wi. The methods developed here are helpful not only for the estimation of nonlinear parameters of viscoelastic models but also for the investigation of the origin of strain-frequency superposition in LAOS.

Melt flow instabilities in polyethylene resins

The melt flow instabilities of four polyethylene (PE) pipe resins were investigated and critically compared. By comparing the flow curve obtained by capillary rheology measurements with the linear viscoelastic (LVE) data, the apparent slip was seen in the four polyethylenes. It was proposed that the high content of small chains could cause the significant wall slip, but the slip was not the origin mechanism of sharkskin melt fracture. The bimodal resins that had higher content of low molecular weight component first showed a matte surface at lower stresses. The matte surface remained until the extrudate converted to a wavy appearance as shear rates were increased. It was observed that the onset stress of sharkskin melt fracture was almost proportional to the plateau modulus. Furthermore, the chain relaxation time of the resins also influences the surface melt fracture behavior of the resin.