3D modeling of dual wind-up extensional rheometers
Introduction
Extension of macromolecular or polymeric materials is one of the most important rheological characterizations methods. The interest is not merely restricted to the industrial community but also to more theoretical developments. In contrast to shear flow, elongational flow demands material points to separate exponentially with time so that polymer chains experience higher degrees of molecular orientation and stretching. As a matter of fact, many non-linear effects only manifest in extension. Extensional flow measurements are also very sensitive to the molecular structure of the tested polymer.
Macromolecular materials can be liquid or solid like. Examples are dough, gels, elastomers, and plastics. The question of whether any constitutive equation for a macromolecular material is of general validity can only be answered by comparing its predictions with experimental data. Well-defined extensional measurements have shown their importance in the critical evaluation of constitutive theories [3], [4], [5], [6]. In this paper our main concern will be (soft) elastomers and entangled polymeric liquids.
The relevance of elongation flows has motivated the development of different testing platforms intended to impose well-defined elongational flows in the material. Characterizing the extensional behavior of polymeric systems has historically been quite difficult and even though several testing platforms have been presented and widely used their performance is still discussed.
Four rheometers for uni-axial extension are currently commonly used: the Münstedt Tensile Rheometer (MTR) [7], the Meissner elongational rheometer [8] (commercialize as Rheometric Scientific RME), the Sentmanat Extensional Rheometer (SER) [1] and the filament stretch rheometer (FSR) [9]. The MTR operates using translation clamps to extend a long cylindrical shaped sample suspended in an oil bath. In the case of the RME, a film of rectangular shape is held by four belt clamps that perform a counter motion between belt pairs provoking the stretch of the sample. The SER employs a wind-up technique consisting of two cylindrical drums. The sample is attached onto these drums followed by a rotation of both drums in opposite direction. The MTR, RMS as well as the SER are all based on the assumption of an overall homogeneous uni-axial extension. In the FSR the deformation of the sample in the direction of the extension is non-uniform. A short cylindrical polymer sample is held between two plates that are pulled apart to induce the extensional flow. In contrast to the other techniques the FSR relies on the assumption of a homogeneous elongation in the center of the extended sample, where the extension is measured locally using laser microscopy. This subject has resulted in a large amount of numerical studies of the filament stretch initiated by Kolte et al. [10] and Sizaire and Legat [11], and is still of interest [12]. These studies are the main contributions to the understanding of the flow in the FSR.
Numerical studies of the sample deformation in dual wind-up methods as the SER and EVF have according to our knowledge never been presented. It requires a fully three-dimensional, time-dependent as well as free surface viscoelastic flow scheme, with a dynamic contact line at the rotating fixtures. Only the study by Rasmussen et al. [13] on inflation of polymer melts fulfills these requirements. As a case study of dual wind-up methods we will consider the numerical modeling of the stretch of a flat sample in the SER. Please notice that the SER and the Extensional Viscosity Fixture (EVF) [2] rheometer are identical techniques from a modeling perspective. Experimental studies of the MTR, RME and SER techniques with regard to extensional deformation control and uniformity can be found in [1], [14], [15].
The SER (and EVF) employs a wind-up technique designed in such a manner that it can be used for materials as polymer melts and elastomers. Sentmanat [1] experimentally examined the uni-axial polymer melt extension performing series of strain validation experiments for a broad envelope of strain rates. Digital video imaging was used to monitor the evolution of the width of the sample. However, the influence of the material as well as the initial geometry of the sample remains unclear. Here the purpose is to discuss the potential deviations from ideal uni-axial deformation, based solely on numerical computations of theoretically ideal configurations.
Section snippets
Equipment description
The SER can be mounted on any rotational rheometer and consists of two cylindrical drums, as illustrated in Fig. 1. In most cases a polymer sample shaped as a rectangular strip is attached onto these drums that are rotated in opposite directions with the same rotation rate, . Further description of this equipment and operation principles can be found in Sentmanat’s original paper [1]. The drive shaft rotation rate, , is set to be constant, and the set or nominal Hencky strain rate applied to
Elastomers
It is of interest to apply some simple and ideal elasticity models for the strain tensor. These are models as the Neo-Hookean model, which represents a classical network theory and is very strain hardening of nature. Another choice is a model representing an entangled network or an elasticity strain with affine movement of cross-links. Despite widely used, we will not consider empirical elasticity models such as the Mooney–Rivlin, which basically is a mathematical Taylor expansion. All models
Polymer melts and entangled liquids
Considering polymer melts or entangled liquids, integral constitutive models of the K-BKZ type [24], [25] have performed well in describing polymer behavior in complex flow situations [26], [27], [28], [29], [30], [31].
Particularly a K-BKZ version of the MSF (molecular stress function) model [32] has been successful to describe two- and three-dimensional time-dependent complex extensional dominated flows for both linear and branched melts within the experimental error [33], [34], [35], [36],
Conclusion
To summarize, numerical simulations of a dual wind-up drum rheometer have been performed.In the dual wind-up rheometer an ideal uni-axial extension of the sample is assumed at all conditions, whereas our simulations show a more diverse picture.
For (soft) elastomers ideal extension requires a very small initial sample width and thickness of ( mm in the SER) and ( mm in the SER) respectively. This conclusion is based on a Neo-Hookean constitutive equation.
The weighting
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