David M. Hoyle

Large amplitude oscillatory shear and Fourier transform rheology analysis of branched polymer melts

In this paper, the predictions of the Pompom constitutive model in medium and large amplitude oscillatory shear (LAOS) are examined using Fourier transform rheology (FTR). FTR is commonly used in combination with small amplitude oscillatory shear to fit linear Maxwell parameters to dynamic moduli, and in this paper, this process is expanded to larger strain amplitudes and to further terms in the Fourier series. For both small and large amplitudes, these higher harmonics are dependent on the nonlinear Pompom parameters and the Pompom parameter space is explored to see how experimental oscillatory shear data can infer molecular detail. In the regime of small and medium strain amplitude, there exists an asymptotic solution to the Pompom equations which depends only on the ratio of the orientation and stretch relaxation times, τb and τs. This asymptotic solution is found to be accurate up to strains of order unity and the branching priority, q, only affects the stress response at larger strains. The Pompom pa...

Cross-slot extensional rheometry and the steady-state extensional response of long chain branched polymer melts

Stress-optical measurements at a flow stagnation point in confined geometries such as the cross-slot provide an elegant way to perform extensional testing for polymer melts. This technique is especially useful for samples which have a steady-state that cannot be reached (easily) in standard elongational rheometry, for example, highly branched polymers which show a non-homogeneous deformation that occurs in stretching experiments for Hencky strains above 4. In contrast to filament stretching, the cross-slot provides one point at which steady-state extensional flow may be sustained indefinitely. In this study, a Cambridge multi-pass rheometer [Coventry, K. D., and M. R. Mackley, J. Rheol. 52, 401–415 (2008)] is used to generate planar elongational flow in a cross-slot geometry for different polyethylene melts. The experimental results are compared to finite element flow simulations using the multi-mode Pompom constitutive equations. The steady-state elongational viscosity at the stagnation point is computed...

Transient overshoot extensional rheology of long-chain branched polyethylenes: Experimental and numerical comparisons between filament stretching and cross-slot flow

This work analyses the high-strain extensional behavior of long-chain branched polyethylenes, employing two novel extensional rheometer devices, the filament stretching rheometer and the cross-slot extensional rheometer. The filament stretching rheometer uses an active feedback loop to control the imposed strain rate on a filament, allowing Hencky strains of around 7 to be reached. The cross-slot extensional rheometer uses optical birefringence patterns to determine the steady-state extensional viscosity from planar stagnation point flow. The two methods probe different strain-rate regimes and in this paper we demonstrate the agreement when the operating regimes overlap and explore the steady-state extensional viscosity in the full strain-rate regime that these two complimentary techniques offer. For long-chain branched materials, the cross-slot birefringence images show a double cusp pattern around the outflow centre line (named W-cusps). Using constitutive modeling of the observed transient overshoot in extension seen in the filament stretching rheometer and using finite element simulations we show that the overshoot explains the W-cusps seen in the cross-slot extensional rheometer, further confirming the agreement between the two experimental techniques.

The long-chain dynamics in a model homopolymer blend under strong flow: small-angle neutron scattering and theory

We use small-angle neutron scattering (SANS) measurements to provide a detailed picture of the non-linear dynamics of the long chains in a model polystyrene blend. By a weighted subtraction of SANS measurements from two otherwise identical blends with different deuteration fractions, we isolate the single-chain form factor of the long-chain component of a model blend flowing through a 4 : 1 contraction–expansion flow. Complementary flow-birefringence also provides a measure of chain deformation on finer length-scales. In addition, higher flow Weissenberg numbers than in previous studies on monodisperse melts were achieved, leading to greater anisotropy in the measured single-chain structure factor. The short residence time inside the slit means that the chains are still oriented in the flow direction as they enter the contraction exit, leading to a rapid reversing flow. We compare these data to a simple generalisation of a non-linear tube model. Our model predictions are entirely ab initio, with all model parameters being determined from independent equilibrium measurements. The model shows very good agreement with the experimental data across the full range of length-scales for the contraction entrance and subsequent relaxation within the slit. However, there is conspicuous disagreement between theory and experiments at the contraction exit, in both the SANS and birefringence predictions, which we attribute to the reversing flow that occurs in this region.

Criteria for extensional necking instability in complex fluids and soft solids. Part I: Imposed Hencky strain rate protocol

We study theoretically the necking dynamics of a filament of complex fluid or soft solid in uniaxial tensile stretching at constant imposed Hencky strain rate ϵ , by means of linear stability analysis and nonlinear (slender filament) simulations. We demonstrate necking to be an intrinsic flow instability that arises as an inevitable consequence of the constitutive behavior of essentially any material (with a possible rare exception, which we outline), however carefully controlled the experimental conditions. We derive criteria for the onset of necking that are reportable simply in terms of characteristic signatures in the shapes of the experimentally measured rheological response functions, and should therefore apply universally to all materials. As evidence of their generality, we show them to hold numerically in six popular constitutive models: The Oldroyd B, Giesekus, FENE-CR, Rolie-Poly, and Pom-pom models of polymeric fluids, and a fluidity model of soft glassy materials. Two distinct modes of neck...

Age-dependent modes of extensional necking instability in soft glassy materials.

We study the instability to necking of an initially cylindrical filament of soft glassy material subject to extensional stretching. By numerical simulation of the soft glassy rheology model and a simplified fluidity model, and by analytical predictions within a highly generic toy description, we show that the mode of instability is set by the age of the sample relative to the inverse of the applied extensional strain rate. Young samples neck gradually via a liquidlike mode, the onset of which is determined by both the elastic loading and plastic relaxation terms in the stress constitutive equation. Older samples fail at smaller draw ratios via a more rapid mode, the onset of which is determined only by the solidlike elastic loading terms (though plastic effects arise later, once appreciable necking develops). We show this solidlike mode to be the counterpart, for elastoplastic materials, of the Considère mode of necking in strain-rate-independent solids.

Non-linear step strain of branched polymer melts

Long-chain branched polymer melts such as low density polyethylene (LDPE) and branched metallocene polyethylenes show strong time-strain separability in step strain. Constitutive models of the multi-mode Pom-pom form are highly successful in modeling the stress generated by general flow histories for these materials. However, a single Pom-pom mode is not time-strain separable and reconciling this to the step-strain phenomenon has been a challenge. We investigate multi-mode integral Pom-pom models and a differential approximation to compare time-strain separation, with respect to mode density. Here we show that for a wide class of branched distributions, a family of damping functions can be derived with a response that is very close to separable. We evaluate the family for both LDPE and branched high density polyethylene melts and show that a damping function derived from the multi-mode Pom-pom model gives an accurate prediction of the damping behavior in step-strain experiments.

Hydrolytic degradation of ROMP thermosetting materials catalysed by bio-derived acids and enzymes: from networks to linear materials

This paper reports the first example of degradable ROMP thermosetting materials catalysed by bio-derived acids and cutinase from Thermobifida cellulosilytica (Thc_Cut1). The ROMP thermosetting materials are based on norbornene dicarboximides containing acetal ester groups only in the crosslinking moiety. The insoluble cross-linked materials were subjected to acid-catalysed hydrolysis using bio-derived acetic and citric acids as well as enzymatic degradation using Thc_Cut1, resulting in the materials becoming completely soluble in dichloromethane. 1H NMR and rheological analysis performed on materials after acid-catalysed hydrolysis showed characteristics indistinguishable to those of the linear polymer analogues. These analyses confirmed the cleavage of the crosslinking moiety upon degradation with the main backbone chains remaining intact. The glass transition temperatures of the polymer materials after acid-catalysed hydrolysis were the same as those observed for the linear polymer analogues. TGA showed that the cross-linked polymers were thermally stable to 150 °C, beyond which they showed weight losses due to the thermal cleavage of the acetal ester linkages.

Necking after extensional filament stretching of complex fluids and soft solids.

Abstract We perform linear stability analysis and nonlinear slender filament simulations of extensional necking in complex fluids and soft solids, during the stress relaxation process following an interrupted strain ramp. We start by deriving analytical criteria for necking within a highly simplified and generalised scalar constitutive model. Within this, we find two different possible modes of necking: one associated with an upward curvature in the stress relaxation function on a log-linear plot, and another related to a carefully defined ‘elastic’ derivative of the tensile force with respect to an imagined sudden strain increment. We showed these two criteria to agree fully with simulations of the Oldroyd B and Giesekus models of polymeric solutions, and with the Rolie-Poly model of more concentrated polymeric solutions and melts, without polymer chain stretch. With chain stretch included, we find a slightly more complicated analytical criterion for necking during the stress relaxation, although with key ingredients that closely mirror counterpart ingredients of the simpler criteria obtained within the scalar model. We show this criterion to agree fully with slender filament simulations of the Rolie-Poly model with chain stretch, and with the scenario discussed by the Copenhagen group in [1,2] . In particular, we see delayed necking after strain ramps with an accumulated strain exceeding ϵ ¯ ≈ 0.7 , for ramp rates exceeding the inverse chain stretch relaxation timescale. We discuss finally an analogy between this delayed necking following an interrupted extensional strain ramp and delayed shear banding following an interrupted shear strain ramp [3]. This work provides the counterpart, for interrupted extensional strain ramps, to earlier papers giving criteria for necking in the protocols of constant imposed Hencky strain rate [4]  and of constant imposed tensile stress or constant imposed tensile force [5] .

Non‐linear Step Strain of Branched Polymer Melts

The Pom‐pom model by McLeish and Larson (Journal of Rheology 42(81–110), 1998) is a highly successful molecular theory for describing the rheology of long chain branched melts. However, there is a long‐standing puzzle in step strain: how can a model that is intrinsically non‐separable recover empirical strain‐time separation? We investigate the Pom‐pom model in step‐strain, comparing the qualitatively different behaviour of the single mode integral and differential orientation. Despite this difference when both models are used in a multi‐mode form, their behaviour is shown to be comparable. Although neither integral nor differential model can predict exact time‐strain separability, both can create a region in which the approximation is a very good one before the longest stretch time has been reached.By transforming to a continuous spectrum we find under certain assumptions, a parameter sub‐space where an analytic damping function can be derived. We survey a range of materials produced by two different syn...