Extensional flow oscillatory rheometry

Abstract

The measurement of extensional viscosity, particularly for low viscosity complex fluids, has long been recognised as both an essential and challenging rheological task. Many of the techniques currently available have drawbacks that either preclude the use of low viscosity fluids or provide varying levels of applied fluid strain. By combining oscillatory flow with a stagnation point extensional flow field, conditions of steady-state stretching using only tiny volume displacements can be achieved. This extensional flow oscillatory rheometer has four electronically controlled micro-pumps positioned at the end of each channel of a cross-slots flow cell, creating planar extension, a differential pressure transducer records flow resistance measurements. The geometry permits the shear and extensional rheological components to be separately determined. An optical probe records simultaneous flow field stability, microstructure and molecular orientation data. Results are presented for dilute (10–100 ppm) polystyrene and hyaluronan polymer solutions, showing the capability of the technique for measuring the extensional viscosity and non-ideal Trouton ratios, controlling the fluid strain, following the evolution of molecular strain, assessing molecular flexibility and deriving the molecular weight distribution of high polymers.

Introduction

It has long been recognised that extensional viscosity is a material parameter of fundamental importance, necessary to complement data from conventional shear rheometers and to fully characterise fluid properties. The measurement of extensional viscosity, particularly for low viscosity fluids (of order 1 mPa s to 1 Pa s), is not straightforward and still remains one of the most challenging aspects of rheometry. For complex fluids, e.g. containing polymeric or surfactant additives, which have the ability to build up changes in local structure with time, the mechanical response to an irrotational extensional flow field can be dramatically different to that produced in shear flow. The widespread and routine use of these fluids in many industrial and technological processes, in which extensional flow components are present, e.g. extrusion, coatings, moulding, film blowing, fibre spinning, lubrication, drag reduction and porous media flow (enhanced oil recovery and filtration), is a driving force behind the need to assess extensional properties.

Polymer solutions have been studied widely since it has been empirically established that solutions of parts per million of high molecular weight polymers can exhibit remarkable non-Newtonian behaviour in extensional flows. The most striking examples being the observation of turbulent drag reduction (Fabula et al. [1]) and dramatic flow thickening beyond a critical flow rate in porous media flows (Dauben and Menzie [2]). Such phenomena have been attributed to, perhaps the most fundamental physical property of high polymers, that of extensibility. In stretching flows especially, a transition from an ambient coil state to a stretched out conformation might be expected to dramatically modify flow behaviour. In a landmark paper, de Gennes [3] predicted that the coil–stretch phenomenon in dilute solutions in stretching flow fields could be critical, with a sudden transition from a coil to a close to fully extended state as the strain rate is increased. This he envisaged to be due to an increase in frictional interaction of the moving solvent with the coil as it begins to extend and become more free-draining—resulting in a run-away process. Hysteresis is predicted in the coil–stretch–coil cycle. Similar ideas were advanced in the same year by Hinch [4]. An enormous increase in extensional viscosity would be anticipated for the stretched out molecules. A dual condition must be realized if a polymer molecule is to achieve full chain extension, in a stretching flow field. The flow field must provide a sufficient strain rate and maintain it for a long enough time for a given fluid element to attain the required strain. A strain of 100× or more may be necessary to stretch a long, highly flexible polymer.

A variety of techniques have been developed for measuring the extensional properties of fluids, e.g. capillary entrance flows, thread-line rheometers, opposed jets and two- and four-roll mills, and a comparison of them using a single polymeric test fluid was attempted in the M1 project (Sridhar [5]). The outcome of this study was a three-decade wide variation in the measured extensional viscosity (η_E) as a function of applied strain rate ($\dot{\epsilon }$) (James and Walters [6]). The conclusion drawn was that the measured η_E values were transient and strongly dependent upon variables other than strain rate. The authors reported that differences between the techniques could be rationalised on the basis of the enormous variation in available fluid strain across the techniques.

Capillary entrance or orifice flows, which measure the pressure drop across a contraction, have been widely used to estimate the extensional properties of polymer melts (Gibson [7] and Kim and Dealy [8]). They can also be used to study low viscosity fluids (James and Saringer [9]) and this technique can potentially provide very large velocity gradients using simple, easy to operate apparatus. The problems associated with this technique are that interpretation of the data is not straightforward even for Newtonian fluids, limited strain and residence time in the flow, turbulence at high flow rates, the presence of recirculating vortices, extreme strain rates around re-entrant corners and flow instabilities in viscoelastic fluids (Binding and Walters [10] and Boger [11]) which leads to a varied and unknown experimental strain rate.

Filament stretching devices have been used recently to measure the extension of dilute polymer fluids (Anna et al. [12], Stelter et al. [13]). Fluid is contained between two end-plates which move apart under controlled separation and subject the fluid to nearly ideal uniaxial extension. Sensors allow the measurement of the tensile force in the filament and the mid-filament diameter as the plates move apart. Such techniques are normally limited to relatively high viscosity or elastic fluids which can support a thread and minimise ‘gravitational sagging’ (McKinley and Sridhar [14]). For ‘dilute’ polymer solutions, high viscosity oligomers are generally used as a solvent, such that the zero-shear-rate shear viscosity of the fluid is around 1 Pa s, with 20–30 Pa s being more typical (McKinley and Sridhar [14]). Progress continues to be made and recently results have been obtained with relatively dilute water based solutions of highly elastic polymers such as poly(acrylamide) leading to estimates of polymer conformational relaxation times [13]. The elongation of the thread to arbitrarily large Hencky strains is generally not possible due to ‘necking’ or capillary instabilities or end-plate instabilities that lead to break-up of the filament. The maximum controllable extension rate is limited by the slowest response of the components in the velocity–position feedback system, and to access the steady-state extensional viscosity of polymeric systems, fluid relaxation times of the order of 1–100 s are necessary (Anna et al. [12]). The scale of the apparatus is such that both solvent volatility and temperature control become important and non-trivial issues, e.g. it is difficult to envisage filament stretching as a technique which can address ‘in vivo’ problems in the rapid growth field of bio-rheology.

Opposed jets devices are classed as ‘stagnation point’ devices for measuring extensional viscosity, along with the four-roll mill and the cross-slots geometries. The opposed jets system consists of two cylindrical jets, which are immersed facing each other in fluid. The fluid is then sucked simultaneously through both jets (Frank et al. [15]). A stagnation point exists at the centre of symmetry, where the velocity is zero but the extension rate is finite. Along the axial region of the opposed jets, a fluid element will be accelerated from the central zone towards the jet entrance. The flow field created is simple uniaxial extension to a first approximation, high strain rates can be realised and the pressure drop across the jets provides a measure of the effective extensional viscosity of the fluid (Carrington et al. [16], [17]). Those streamlines which pass close enough to the stagnation point provide fluid elements with sufficient residence time to accumulate very high strain (Crowley et al. [18]). Fluid elements on other streamlines further from the centre of the system have a lower residence time and hence experience a lower strain. As a consequence the available strain in the opposed jets system ranges from low to infinite in principle, depending on the streamline followed. Extension of polymer molecules in stagnation point flows has been shown to approach highly stretched states by direct observation of fluorescent labelled DNA (Smith and Chu [19]). Hysteresis of the coil–stretch, stretch–coil behaviour of DNA has also been observed (Schroeder et al. [20]) as predicted by de Gennes [3]. Whilst the central region between the jets is subject to an irrotational flow, the presence of the jet walls and capillaries also gives rise to regions of shear flow. The stagnation point can be lost due to elastic instabilities and therefore this region needs to be monitored with an optical probe to ensure the extensional flow is maintained (Müller et al. [21]).

A commercial rheometer based on the opposed jets, the Rheometrics RFX (Cathey and Fuller [22]) was available until recently, but the lack of flow field visualisation was seen as a shortcoming. The RFX used a pivoting jet and a torque rebalancing transducer to maintain a constant jet separation and provide a measure of fluid stress, rather than measuring the pressure drop. This approach minimised the contribution from the shear flow in the pipe work, but the measured torque contained an additional contribution from fluid inertia (Hermansky and Boger [23]). Furthermore, the RFX had no facility for observation of the flow field, leading to considerable uncertainty in the assumption of a hyperbolic flow field. Fluid volumes required in opposed jets systems are relatively high, typically 200–500 ml are required, which may inhibit the use of some biological systems or may be hazardous in the case of some organic solvents.

It would be attractive to use oscillatory flow in an extensional rheometer, as has been done with shear rheometers for many years; this would dramatically reduce the volume of fluid required. The essential problem for polymer solutions is that extensional flows unravel molecules, this requires large strains and so long exposure to the stretching flow field, which is difficult to achieve in conventional extensional flow devices (capillaries, fibre drawing, etc.). The key to the instrument described in this paper is to combine oscillatory flow with a stagnation point in a cross-slots apparatus, to create an extensional flow oscillatory rheometer. This geometry is similar to the opposed jets, but by making the jets slit shaped with the slit directions parallel, the flow field corresponds to pure shear (planar extension) (Scrivener et al. [24] and Kwan et al. [25]). Pure shear flow is realised along the outgoing direction within the central crossover region. The exact centre is again a locality of zero velocity, which provides a site of infinite residence time so that stretching (to steady-state along certain streamlines due to unlimited strain) will occur, even for tiny volume displacements. This means experimentally that extensional viscosities can be measured using extremely small sample volumes, due to the small scale of the cross-slots cell (channel widths are typically in the size range 100–750 μm), and the fact that a fluid element needs only to be oscillated within the stretching region.

The oscillation is achieved by using four micro-pumps situated at the end of each slot channel. By using micro-pumps which are driven electronically by computer synthesized waveforms, there is the potential to provide any repetitive or one-off flow profile required, e.g. the variation of ramp rate, waveform and frequency could be set to model porous media flows and to show structural development in a Lagrangian frame. The pressure drops, to provide extensional viscosity information, are measured with pressure transducers in two limbs of the cross-slots. The pumping rates of each limb of the cross-slots can also be controlled separately to give asymmetric flows, so flow with and without the stagnation point can be investigated and the effects of shear flow at the walls subtracted, to give a measure of the fluid response to pure extension only. The fixed geometry also overcomes the problem of fluid inertia that affected the pivoting-jet approach. The sample is contained in the cell only, which provides a clean inert environment for biological and sensitive samples, and allows for ease of temperature control. Continuous extensional cycling at high strain rates permitted by this geometry can be used to promote and investigate thermo-mechanical degradation, as for instance previously studied in steady extensional flow of DNA and synthetic polymers [26].

Macromolecular strain in response to the flow field is assessed by an optical probe to measure birefringence [27], [28], [29], [30], [31], [32], [33], [34]. The generation of a molecular strain profile with distance from the centre of the cross-slots allows the resolution of strain along the streamlines and provides a level of information which takes the technique beyond being just a viscosity ‘indexer’. The optical probe also provides information on the stability of the flow field, and coupled with the control given by the micro-pumps, reduced perturbation of the flow field is possible by controlling the molecular strain, so enabling work on semi-dilute solutions.

A plot of birefringent intensity against strain rate is essentially a cumulative molecular weight distribution. This technique for assessing molecular weight becomes easier the longer the polymer molecule is and therefore permits assessment of ultra high molecular weight tails and their rheological effects.