Corotational Maxwell Model Research Articles

The Gordon–Schowalter/Johnson–Segalman model in parallel and orthogonal superposition rheometry and its application in the study of worm-like micellular systems

Parallel and Orthogonal Superposition experiments may be employed to probe a material’s non-linear rheological properties through the rate-dependent parallel and orthogonal superposition moduli, G∥∗(ω,γ̇) and G⊥∗(ω,γ̇), respectively. In a recent series of publications, we have considered the problem of interconversion between parallel and orthogonal superposition moduli as a means of probing flow induced anisotropy. However, as noted by Yamomoto (1971) superposition flows may be used to assess the ability of a particular constitutive model to describe the flow of complex fluids. Herein, we derive expressions for the superposition moduli of the Gordon–Schowalter (or Johnson–Segalman) fluid. This model contains, as special cases, the corotational Maxwell model, the upper (and lower) convected Maxwell models, the corotational Jeffreys model, and the Oldroyd-B model. We also consider the conditions under which the superposition moduli may take negative values before studying a specific, non shear banding, worm like micellular system of cetylpyridinium chloride and sodium salicylate. We find that, using a weakly non-linear analysis (in which the model parameters are rate independent) the Gordon–Schowalter/Johnson–Segalman (GS/JS) model is unable to describe the superposition moduli. However, by permitting strong non-linearity (allowing the GS/JS parameters to become shear rate dependent), the superposition moduli, at all rates studied, are described well by the model. Based on this strongly non-linear analysis, the shear rate dependency of the GS/JS ‘slip parameter’, a, suggests that the onset of shear thinning in the specific worm-like micellular system studied herein is driven by a combination of microstructural modification and a transition from rotation dominated (as in the corotational Jeffreys model) to shear dominated (as in the Oldroyd-B model) deformation of the microstructural elements.

Medium amplitude parallel superposition (MAPS) rheology. Part 2: Experimental protocols and data analysis

An experimental protocol is developed to directly measure the new material functions revealed by medium amplitude parallel superposition (MAPS) rheology. This protocol measures the medium amplitude response of a material to a simple shear deformation composed of three sine waves at different frequencies, revealing a rich dataset consisting of up to 19 measurements of the third-order complex modulus at distinct three-frequency coordinates. We discuss how the choice of input frequencies influences the features of the MAPS domain studied by the experiment. A polynomial interpolation method for reducing the bias of measured values from spectral leakage and reducing variance due to noise is discussed, including a derivation of the optimal range of amplitudes for the input signal. This leads to the conclusion that conducting the experiment in a stress-controlled fashion possesses a distinct advantage to the strain-controlled mode. The experimental protocol is demonstrated through measurements of the MAPS response of a model complex fluid: a surfactant solution of wormlike micelles. The resulting dataset is indeed large and feature-rich, while still acquired in a time comparable to similar medium amplitude oscillatory shear (MAOS) experiments. We demonstrate that the data represent measurements of an intrinsic material function by studying its internal consistency, compatibility with low-frequency predictions for Coleman–Noll simple fluids, and agreement with data obtained via MAOS amplitude sweeps. Finally, the data are compared to predictions from the corotational Maxwell model to demonstrate the power of MAPS rheology in determining whether a constitutive model is consistent with a material’s time-dependent response.

Time-strain separability in medium-amplitude oscillatory shear

We derive and study equations for the weakly nonlinear medium-amplitude oscillatory shear (MAOS) response of materials exhibiting time-strain separability. Results apply to constitutive models with arbitrary linear memory function m(s) and for both viscoelastic liquids and viscoelastic solids. The derived equations serve as a reference to identify which models are time-strain separable (TSS) and which may appear separable but are not, in the weakly nonlinear limit. More importantly, we study how the linear viscoelastic (LVE) relaxation spectrum, H(τ), affects the frequency dependence of the TSS MAOS material functions. Continuous relaxation spectra are considered that are associated with analytical functions (log-normal and asymmetric Lorentzian distributions), fractional mechanical models (Maxwell and Zener), and molecular theories (Rouse and Doi-Edwards). TSS MAOS signatures reveal much more than just the perturbation parameter A in the shear damping function small-strain expansion, h(γ)=1+Aγ2+Oγ4. Specifically, the distribution of terminal relaxation times is significantly more apparent in the TSS MAOS functions than their LVE counterparts. We theoretically show that this occurs because TSS MAOS material functions are sensitive to higher-order moments of the relaxation spectrum, which leads to the definition of MAOS liquids. We also show the first examples of MAOS signatures that differ from the liquid-like terminal MAOS behavior predicted by the fourth-order fluid expansion. This occurs when higher moments of the relaxation spectrum are not finite. The famous corotational Maxwell model is a subset of our results here, for which A = −1/6, and any LVE relaxation spectrum could be used.

Exact analytical solution for large-amplitude oscillatory shear flow from Oldroyd 8-constant framework: Shear stress

The Oldroyd 8-constant model is a continuum framework containing, as special cases, many important constitutive equations for elastic liquids. When polymeric liquids undergo large-amplitude oscillatory shear flow, the shear stress responds as a Fourier series, the higher harmonics of which are caused by the fluid nonlinearity. We choose this continuum framework for its rich diversity of special cases (we tabulate 14 of these). Deepening our understanding of this Oldroyd 8-constant framework thus at once deepens our understanding of every one of these special cases. Previously [C. Saengow et al., Macromol. Theory Simul. 24, 352 (2015)], we arrived at an exact analytical solution for the corotational Maxwell model. Here, we derive the exact analytical expression for the Oldroyd 8-constant framework for the shear stress response in large-amplitude oscillatory shear flow. Our exact solution reduces to our previous solution for the special case of the corotational Maxwell model, as it should. Our worked example uses the special case of the corotational Jeffreys model to explore the role of η∞ on the higher harmonics.

Padé approximants for large-amplitude oscillatory shear flow

Analytical solutions for either the shear stress or the normal stress differences in large-amplitude oscillatory shear flow, both for continuum or molecular models, often take the form of the first few terms of a power series in the shear rate amplitude. Here, we explore improving the accuracy of these truncated series by replacing them with ratios of polynomials. Specifically, we examine replacing the truncated series solution for the corotational Maxwell model with its Pade approximants for the shear stress response and for the normal stress differences. We find these Pade approximants to agree closely with the corresponding exact solution, and we learn that with the right approximants, one can nearly eliminate the inaccuracies of the truncated expansions.

Exact Analytical Solution for Large‐Amplitude Oscillatory Shear Flow

When polymeric liquids undergo large‐amplitude shearing oscillations, the shear stress responds as a Fourier series, the higher harmonics of which are caused by fluid nonlinearity. Previous work on large‐amplitude oscillatory shear flow has produced analytical solutions for the first few harmonics of a Fourier series for the shear stress response (none beyond the fifth) or for the normal stress difference responses (none beyond the fourth) [JNNFM, 166, 1081 (2011)], but this growing subdiscipline of macromolecular physics has yet to produce an exact solution. Here, we derive what we believe to be the first exact analytical solution for the response of the extra stress tensor in large‐amplitude oscillatory shear flow. Our solution, unique and in closed form, includes both the normal stress differences and the shear stress for both startup and alternance. We solve the corotational Maxwell model as a pair of nonlinear‐coupled ordinary differential equations, simultaneously. We choose the corotational Maxwell model because this two‐parameter model (η0 and λ) is the simplest constitutive model relevant to large‐amplitude oscillatory shear flow, and because it has previously been found to be accurate for molten plastics (when multiple relaxation times are used). By relevant we mean that the model predicts higher harmonics. We find good agreement between the first few harmonics of our exact solution, and of our previous approximate expressions (obtained using the Goddard integral transform). Our exact solution agrees closely with the measured behavior for molten plastics, not only at alternance, but also in startup.

Large-amplitude oscillatory shear: comparing parallel-disk with cone-plate flow

We compare the ratio of the amplitudes of the third to the first harmonic of the torque, , measured in rotational parallel-disk flow, with the ratio of the corresponding harmonics of the shear stress, |τ 3|/|τ 1|, that would be observed in sliding-plate or cone-plate flow. In other words, we seek a correction factor with which must be multiplied, to get the quantity |τ 3|/|τ 1|, where |τ 3|/|τ 1| is obtained from any simple shearing flow geometry. In this paper, we explore theoretically, the disagreement between and τ 3/τ 1 using the simplest continuum model relevant to large-amplitude oscillatory shear flow: the single relaxation time co-rotational Maxwell model. We focus on the region where the harmonic amplitudes and thus, their ratios, can be fully described with power laws. This gives the expression for , by integrating the explicit analytical solution for the shear stress. In the power law region, we find that, for low Weissenberg numbers, for the third harmonics , and for the fifth harmonics, . We verify these results experimentally. In other words, the heterogeneous flow field of the parallel-disk geometry significantly attenuates the higher harmonics, when compared with the homogeneous, sliding-plate flow. This is because only the outermost part of the sample is subject to the high shear rate amplitude. Furthermore, our expression for the torque in large-amplitude oscillatory parallel-disk flow is also useful for the simplest design of viscous torsional dampers, that is, those incorporating a viscoelastic liquid between two disks.

Normal stress differences in large-amplitude oscillatory shear flow for dilute rigid dumbbell suspensions

We examine the simplest relevant molecular model for large-amplitude oscillatory shear (LAOS) flow of a polymeric liquid: the suspension of rigid dumbbells in a Newtonian solvent. We find explicit analytical expressions for the shear rate amplitude and frequency dependences of the zeroth, second and fourth harmonics of the first and second normal stress difference responses. We include a detailed comparison of these predictions with the corresponding results for the simplest relevant continuum model: the corotational Maxwell model. We find that the responses of both models are qualitatively alike. The rigid dumbbell model relies entirely on the dumbbell orientation to explain the viscoelastic response of the polymeric liquid, including the higher harmonics in large-amplitude oscillatory shear flow. Our analysis employs the general method of Bird and Armstrong (1972) for analyzing the behavior of the rigid dumbbell model in any unsteady shear flow.We derive the first three terms of the deviation of the orientational distribution function from the equilibrium state. Then, after getting the “paren functions,” we use these for evaluating the normal stress differences for large amplitude oscillatory shear flow. We find the shapes of the first normal stress difference versus shear rate loops predicted to be reasonable [see Fig. 1(a)]. We find that the second normal stress difference is not proportional to the first, and that its shape differs markedly from that of the first [cf. Fig. 1(a) and (b)]. We discover the same remarkable qualitative similarity between the predictions of the rigid dumbbell model and the corotational Maxwell model for the first normal stress difference. We find no qualitative similarities between the dumbbell and the continuum models for any of the predicted coefficients of the second normal stress differences in large-amplitude oscillatory shear flow.

Converging shear rheometer

For highly viscous fluids that slip in parallel sliding plate rheometers, we want to use a slightly converging flow to suppress this wall slip. In this work, we first attack the steady shear flow of a highly viscous Newtonian fluid between two gently converging plates with no slip boundaries using the equation of motion in cylindrical coordinates, which yields no analytical solution. Then we treat the same problem using the lubrication approximation in Cartesian coordinates to yield exact, explicit solutions for dimensionless velocity, pressure and shear stress. This work deepens our understanding of a drag flow through a gently converging slit of arbitrary convergence angle. We also employ the corotational Maxwell model to explore the role of viscoelasticity in this converging shear flow. We then compare these analytical solutions to finite element calculations for both Newtonian and corotational Maxwell cases. A worked example for determining the Newtonian viscosity using a converging shear rheometer is also included. With this work, we provide the framework for exploring other constitutive equations or other boundary conditions in future work. Our results can also be used to design the linear bearings used for the parallel sliding plate rheometer (SPR). This work can also be used to evaluate the error in the shear stress that is caused by bearing misalignment and specify the parallelism tolerance for the linear bearings incorporated into a SPR.

Dilute rigid dumbbell suspensions in large-amplitude oscillatory shear flow: Shear stress response

We examine the simplest relevant molecular model for large-amplitude shear (LAOS) flow of a polymeric liquid: the suspension of rigid dumbbells in a Newtonian solvent. We find explicit analytical expressions for the shear rate amplitude and frequency dependences of the first and third harmonics of the alternating shear stress response. We include a detailed comparison of these predictions with the corresponding results for the simplest relevant continuum model: the corotational Maxwell model. We find that the responses of both models are qualitatively similar. The rigid dumbbell model relies entirely on the dumbbell orientation to explain the viscoelastic response of the polymeric liquid, including the higher harmonics in large-amplitude oscillatory shear flow. Our analysis employs the general method of Bird and Armstrong ["Time-dependent flows of dilute solutions of rodlike macromolecules," J. Chem. Phys. 56, 3680 (1972)] for analyzing the behavior of the rigid dumbbell model in any unsteady shear flow. We derive the first three terms of the deviation of the orientational distribution function from the equilibrium state. Then, after getting the "paren functions," we use these for evaluating the shear stress for LAOS flow. We find the shapes of the shear stress versus shear rate loops predicted to be reasonable.

Temperature Rise in Large-Amplitude Oscillatory Shear Flow from Shear Stress Measurements

Recently, we derived the temperature rise in a viscoelastic fluid undergoing large-amplitude oscillatory shear flow (LAOS) using the corotational Maxwell model [Giacomin et al. Phys. Fluids, 2012, 24, 103101]. The results of this derivation are used to estimate the temperature rise a priori. In the present paper, we show how to calculate the temperature rise in any liquid after measuring the shear stress response to LAOS. Specifically, if the measured response is represented by a Fourier series of odd harmonics, we can then combine this Fourier series with the equation of energy, written in terms of temperature, to calculate the temperature rise. This yields both a time-averaged contribution to the temperature rise, and an oscillating part. We find both contributions to be significant. We estimate the oscillating part with an analytical solution for the worst case, LAOS between adiabatic plates. We can thus use this calculation to see if the ±0.5 °C temperature requirement of the current standard for osci...

Understanding Melt Index and ASTM D1238

Abstract In plastics manufacturing, the melt flow index (MFI) is used as a routine indicator of rheological behavior when more expensive and laborious determinations of well-defined material functions are impractical. The MFI is the mass flow rate in a pressure driven flow through a standardized abrupt cylindrical contraction into a short tube performed under a standardized combination of pressure drop and temperature. In this paper, we use a finite element model to explore the connections between rheological properties and melt index. We explore the role of shear thinning by modeling the flow through the melt indexer using the Bird-Carreau model. We then explore the role of melt viscoelasticity in the MFI using the corotational Maxwell model. We present our results in dimensionless charts designed to help plastics engineers specify the MFI of a plastic for an industrial manufacturing process of known material functions. Worked examples are included to show how to use the results.

Viscous heating in large-amplitude oscillatory shear flow

When measuring rheological properties in oscillatory shear flow, one worries about experimental error due to the temperature rise in the sample that is caused by viscous heating. For polymeric liquids, for example, this temperature rise causes the measured values of the components of the complex viscosity to be systematically low. For such linear viscoelastic property measurements, we use an analytical solution by Ding et al. [J. Non-Newtonian Fluid Mech. 86, 359 (1999)10.1016/S0377-0257(99)00004-X] to estimate the temperature rise. However, for large-amplitude oscillatory shear flow, no such analytical solution is available. Here we derive an analytical solution for the temperature rise in a corotational Maxwell fluid (a model with just two parameters: η0 and λ) subject to large-amplitude oscillatory shear flow. This result can then be generalized to a superposition of corotational Maxwell models for a quantitative estimate of the temperature rise. We chose the corotational Maxwell model because, when generalized for multiple relaxation times, it gives an accurate prediction for molten plastics in large-amplitude oscillatory shear flow. We identify three relevant pairs of thermal boundary conditions: (i) both plates isothermal, (ii) with heat loss by convection from both plates, and (iii) one plate isothermal, the other with heat loss by convection. We find that the time-averaged viscous heating increases as an even power series of the dimensionless shear rate amplitude (Weissenberg number), and that it decreases with the dimensionless imposed frequency (Deborah number). We distinguish between the dimensionless time-averaged temperature rise, $\bar \Theta $Θ¯, and the oscillating part, $\tilde \Theta $Θ̃, where $\Theta \equiv \bar \Theta + \tilde \Theta $Θ≡Θ¯+Θ̃. We solve analytically for the $\bar \Theta $Θ¯ profile through the sample thickness for all three pairs of thermal boundary conditions. For the worst case, two adiabatic walls, we derive an expression for the oscillating part of the temperature rise, $\tilde \Theta $Θ̃. We find this $\tilde \Theta $Θ̃ to be a Fourier series of even harmonics whose contribution to the temperature rise can be as important as $\bar \Theta $Θ¯. If both plates are adiabatic, then the sample temperature rises without bound. Otherwise, it does not.

Viscoelasticity in thermoforming

Abstract This analysis for thermoforming cones from a viscoelastic melt gives the speed of the stresses in the deforming melt. We follow the mechanics of Kershner and Giacomin developed for thermoforming cones from a Newtonian melt, deriving an analytical solution for the corotational Maxwell (CM) model. We have distinguished what happens before and after the melt touches the conical mold (free vs. constrained forming). The CM reduces to the Newtonian solution of Kershner and Giacomin. The upper convected Maxwell (UCM) model is also investigated, but does not yield an analytical solution for conical thermoforming. Our results are confined to forming with a conical mold, the simplest relevant problem in thermoforming, using the simplest relevant constitutive equation, the CM model.

Large-amplitude oscillatory shear flow from the corotational Maxwell model

Using the single relaxation time corotational Maxwell fluid, we derive explicit analytical expressions for the first, third, and fifth harmonics of the alternating shear stress response in large-amplitude oscillatory shear (LAOS). We also derive corresponding expressions for the zeroth, second, and fourth harmonics of both the first and second normal stress differences. These harmonics are found to depend upon just two dimensionless groups: the Deborah and Weissenberg numbers, each of which causes non-Newtonian behavior. The form of the solution for the corotational Maxwell model in LAOS matches the forms of the analytical solutions for two molecular models for dilute solutions and one for concentrated solutions or melts. We also derive an analytical solution for the corotational Maxwell model after startup of LAOS. For this we find that both small and large amplitude cases approach a periodic limit cycle (alternance) at the same rate for both the shear stress response and for the normal stress differences. For molten high density polyethylene that is lightly filled with carbon black, we find good quantitative agreement with measured LAOS behavior when our analytical solution is superposed for multiple relaxation times.

Normal stress differences in large-amplitude oscillatory shear flow for the corotational “ANSR” model

Using the integral form of a nonlinear corotational model, we derive explicit analytical expressions for the zeroth, second, and fourth harmonics of the second normal stress difference in large-amplitude oscillatory shear (LAOS). This model yields an arbitrary normal stress ratio (ANSR) in any simple shearing deformation, including LAOS. This corotational ANSR model adds one parameter to the corotational Maxwell model, a time constant λ 0 controlling the ratio Ψ2/Ψ1 for both the real and imaginary parts of each harmonic of the normal stress difference. The explicit analytical expressions for all harmonics of the alternating shear stress and first normal stress difference responses in LAOS match those obtained previously for the corotational Maxwell model. We evaluate the corotational ANSR model for the case of a single Maxwell relaxation time fluid.

Numerical study around the corotational Maxwell model for the viscoelastic fluid flows

We present numerical results for the FEM (finite element method) presented in [Comput. Methods Appl. Mech. Engrg. 191 (2002) 5045–5065]. This method is devoted to the approximation of fluid flows obeying the Oldroyd model. A particularity of this method, is to take into account the purely viscoelastic case, the so-called Maxwell model, important in practice. Numerical results are given for a fluid flowing in an abrupt plane 4 to 1 contraction. We use the corotational Maxwell model as benchmark in the choice of our computations. Results are also given for the upper convected Maxwell model. Interesting effects appear on the velocity profile: a phenomenon of quasi slip at the downstream wall.

Viscoelastic flows studied by smoothed particle dynamics

A viscoelastic numerical scheme based on smoothed particle dynamics is presented. The concept goes a step beyond smoothed particle hydrodynamics (SPH) which is a grid-free Lagrangian method describing the flow by fluid-pseudo-particles. The relevant properties are interpolated directly on the resulting movable grid. In this work, the effect of viscoelasticity is incorporated into the ordinary conservation laws by a differential constitutive equation supplied for the stress tensor. In order to give confidence in the methodology we explicitly consider the non-stationary simple corotational Maxwell model in a channel geometry. Without further developments the scheme is applicable to ‘realistic’ models relevant for three-dimensional (3D) viscoelastic flows in complex geometries.

Dynamic modelling of melt spinning

Dynamics of melt spinning is reconsidered to determine essential effects in modelling of the process. Basic dynamic equations are reformulated with heat production due to viscous friction in the melt, as well as for non-isochoric deformation. Viscoelastic effects are accounted for by applying co-rotational Maxwell fluid model. Different cooling (heating) zones are taken into account with different cross-blow velocity and temperature of the cooling medium. Crystallization is included in the set of basic equations, and specific effects of crystallization on spinning are calculated. Example computations of velocity, temperature and stress profiles are performed for melt spinning of PET fibres with extreme spinning conditions.

A thermodynamical formulation for heat conducting Jeffreys and Maxwell polymeric fluids

Starting from the basic balance laws and using the Clausius-Duhem inequality, thermodynamically consistent Theological equations of state for a heat conducting polymeric fluid are derived. It is shown that the stress evolution equation is identical to the corotational Jeffreys model. Under certain limiting conditions the model reduces to the well known corotational Maxwell model for viscoelastic materials.