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Towards online, continuous monitoring for rheometry of complex fluids
Advances in Colloid and Interface Science 206 (2014) 294–302
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Advances in Colloid and Interface Science journal homepage: www.elsevier.com/locate/cis
Towards online, continuous monitoring for rheometry of complex fluids Julia M. Rees ⁎ School of Mathematics and Statistics, Hicks Building, Hounsfield Road, University of Sheffield, S3 7RH, UK
a r t i c l e
i n f o
Available online 13 June 2013 Keywords: Microfluidics Rheology Rheometry Inverse methods Lab on chip Capillary rheometer
a b s t r a c t This paper presents an overview of the developments that have been made towards the design of an inline rheometer that has the capabilities for monitoring in real time the viscous constitutive parameters of nonNewtonian fluids in a pipe flow. This has potential applications for a wide range of fluids, including hydrocolloid solutions and polymer solutions. This is of relevance to many industries, for example the pharmaceutical, lubrication, food and printing industries. The use of mathematical algorithms for inferring rheological parameters from properties of flow field statistics is explored. Particular focus is given to the development of a flow cell rheometer containing a T-junction geometry with the capacity to induce a range of shear rates in the vicinity of the bend, and a distribution of elongational viscosities along the back-wall. Such features create an information-rich flow field that is beneficial for the development of a rheometer with a fast response time that is suitable for commercial purposes. © 2013 Elsevier B.V. All rights reserved.
Contents 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1. Power law fluids . . . . . . . . . . . . . . . . . 1.2. Carreau fluids . . . . . . . . . . . . . . . . . . . 1.3. Viscoelastic fluids . . . . . . . . . . . . . . . . . 2. Applications for rheometry . . . . . . . . . . . . . . . . 2.1. Engineering monitoring and control . . . . . . . . 2.2. Rheometry for medical diagnosis and monitoring . . 3. Capillary methods for rheological characterization . . . . . 4. Inverse methods for determination of rheological parameters 4.1. Power law fluids . . . . . . . . . . . . . . . . . 4.2. Carreau fluids . . . . . . . . . . . . . . . . . . . 4.3. Viscoelastic fluids . . . . . . . . . . . . . . . . . 4.4. T-junction rheometry . . . . . . . . . . . . . . . 5. Discussion and conclusions . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The processing of complex fluids occurs in many engineering applications and in biomedical analysis. Efficiency of such processes can be crucially dependent on the rheology of the fluid. Viscous
⁎ Tel.: +44 114 222 3782. E-mail address: [email protected]. 0001-8686/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cis.2013.05.006
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parameters can affect the texture or appearance of a fluid, and also they can determine the shelf-life stability of a product. At present, the monitoring of viscous parameters requires samples to be taken manually and then analyzed in a cone-and-plate rheometer, or similar device. The ability to perform online continuous rheological characterization in real-time would permit tighter monitoring and control in a wide range of applications that involve the processing of fluids of non-Newtonian fluids, such as those in the food and beverage [1], pharmaceutical [2], oil and lubrication [3], and high-tech printing
J.M. Rees / Advances in Colloid and Interface Science 206 (2014) 294–302
industries [4]. This would lead to a reduction in wastage caused by processes going off spec, and in the level of manual intervention required for product sampling. Currently, no such commercially available device exists that can perform this task efficiently for a broad range of fluids. Quality control is an endpoint test which results in disposal of below spec product. Continuous monitoring would reduce this wastage, potentially completely. Generally the wastage level of a wide range of processes is around 5%. Recently, much attention has been given to the use of flow configurations that yield an information rich data set of measurable flow parameters that can be mapped, through the solution of an inverse problem, to individual constitutive parameters. With the pressure to develop sustainable materials (biomass derived or utilizing waste products) for processing and manufacturing, industry must respond to the greater variability in input feedstock specification [5], as well as material properties varying during processing, with better control systems. Hence there currently exists a strong driver for online, continuous rheometry. The rheological behavior of different types of non-Newtonian fluids is described by different constitutive equations. Some examples are discussed below. 1.1. Power law fluids The viscosity of power law fluids is given by n−1 ; μ ¼ K γ_
ð1Þ
where K is the flow consistency index (SI units Pa sn), γ_ is the shear rate (SI unit s−1), and n b 1 is the flow behavior index. K and n denote the constitutive parameters of the fluid in question and it may be desirable to know what they are in any given situation [6]. For example, the viscosity of bio-oils is highly temperature dependent and follows a power law constitutive equation. The Krohne viscoline inline viscometer manufactured by Delta Instrumentation (http:// www.deltainstrumentation.com/viscosity/viscometer.html) is a novel device that can be used to determine the two rheological parameters, K and n, for power law fluids. The viscoline comprises a section of pipeline that can be inserted into a fluidic process. The pipeline contains two low pressure drop static mixers that are used to generate two different shear regimes. If the flow rate is known then the pipe line viscosity can be calculated from the two pressure drop measurements. The viscoline can operate at shear rates in the range 50 to 500 s−1 and at temperatures of −5 to 200 °C.
For Carreau fluids, models with either three or four parameters are in common usage [7]. In the three parameter version, the viscosity at infinite shear rate is taken to be zero, and the viscosity is given by ð2Þ
where μ0 is the zero-shear-rate viscosity, λ is a parameter with dimensions of time and n is the power-law exponent. The four parameter Carreau model takes the form i 2 ðn−1Þ=2 ; μ ¼ μ ∞ þ ðμ 0 −μ ∞ Þ 1 þ λðγ_ Þ
stress. The Phan–Thein–Tanner (PTT) constitutive law [8] uses the following form for the non-dimensional constitutive equation: Weτt ¼ 2μ 1 D−hτ n
h io þ We τ⋅∇U þ ð∇UÞT ⋅τ−U⋅∇τ þ ζ D⋅∇τ þ ðD⋅τÞT ;
ð4Þ
where h is usually defined using one of the following three forms: Linear model : h ¼ 1 þ
We traceðτÞ; μ1
Quadratic model : h ¼ 1 þ
2 We 1 We traceðτÞ þ traceðτÞ ; μ1 2 μ1
We Exponential model : h ¼ exp traceðτÞ ; μ1
ð5Þ
ð6Þ
ð7Þ
where • • • • • •
τ = T − 2μ2D, T is the extra-stress tensor, D is the rate of deformation tensor, U is the velocity vector, μ1 is the polymeric viscosity, μ2 is the solvent viscosity,
Uλ , where λ is a L relaxation time and U and L are typical velocity and length scales respectively, • and ζ are positive constants used to control the viscoelastic properties.
• We is a Weissenberg number defined as We ¼
The aim of this article is to provide an updated review, with a particular focus on industrial aspects that would potentially benefit from improvements in the ability to perform online rheometric characterization in real-time. Applications of rheometry in engineering, manufacturing and medical processes are outlined in Section 2. Capillary methods for rheological characterization are described in Section 3. In Section 4 some recent research that uses advanced mathematical algorithms to solve inverse problems in order to determine rheological parameters in real time, is outlined. Discussion on the effect of increased variability of feedstocks as a key driver for the development of improved devices for determining rheological parameters online forms the theme for Section 5, where the present situation is summarized. 2. Applications for rheometry
1.2. Carreau fluids
2 ðn−1Þ=2 μ ¼ μ 0 1 þ ðλγ_ Þ
295
ð3Þ
where μ∞ is the viscosity at infinite shear rate. 1.3. Viscoelastic fluids Viscoelastic materials possess both viscous and elastic characteristics when undergoing strain, and therefore exhibit time dependent
2.1. Engineering monitoring and control The monitoring and control of rheological parameters are of importance for a range of engineering and processing systems. Some examples are discussed in this section. Melt flow index (MFI) as a single data point experiment is a wellestablished technique for measuring changes in the average molecular weight of a polymer melt, although its accuracy is not always reliable as it refers to only a single point measurement within a larger spectrum of properties [9]. MFI is a measure of the ease of flow (pliability) of a thermoplastic polymer melt. It is defined as the mass in grams of a polymer passing through a standardized capillary of 2 mm diameter and 8 mm length over a 10 min interval, at a specified temperature, when a standardized load is applied. The MFI only tests a material at a single shear stress and temperature. It provides an indirect measure of molecular weight. A higher MFI is indicative of a lower material viscosity. Many processes use MFI to assess the viscous properties of incoming materials and as an aid to monitoring changes in the process. More accurate rheology tests can be performed using capillary or parallel plate rheometers.
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The three main flow configurations for microfluidic rheology are capillary flow, stagnation and contraction. In capillary devices the shear viscosity, μ, can be determined from the flow rate, Q, and the pressure drop, ΔP, across a straight section of tubing of length L. Thus in order to determine the viscosity for a capillary flow one can either (i) impose a specified pressure gradient and measure the flow rate [10,11] or (ii) impose the flow rate (e.g. using a syringe pump) and measure the associated pressure drop (e.g. using pressure sensors) [12]. Flow rates can be measured by a number of techniques, including micron resolution particle image velocimetry (μPIV) [13]. Capillary rheometers have been used online in polymer and food processing industries for around 40 years [14]. In a conventional online capillary rheometer, if the pressure drop across the system is held constant, the residence time of the material in the rheometer will be dependent upon the properties of the material itself. Online capillary rheometers need to respond quickly to fluctuations in process parameters. They can be used to monitor polymer manufacturing processes or to act as a sensor for closed loop control. The response time behavior for polymer melts of high molecular weight might be a poor indicator of rheological properties due to the associated low flow rates through the capillary. To overcome this, Göttfert® developed a real-time rheometer that incorporates a circulating volumetric flow. This provides a continuous flow of fresh melt through the capillary, even at low shear rates. This circulation transports the polymer melt through transfer lines up to 100 times faster which provides an advantage over conventional capillary online rheometers. Ranganathan et al. [15] present experimental data for the timedependent capillary flow of a high-density polyethylene through a multipass rheometer. They found that the melt compressibility was closely linked to the observed pressure relaxation on cessation of the piston movement. Guillot et al. [11] presented a method for measuring the rheological properties of complex fluids on a microfluidic chip using a methodology that involved using the adjacent parallel flows of two immiscible liquids as a pressure sensor. The ratio of the cross-sectional areas occupied by two immiscible fluids flowing parallel to each other side by side in a rectangular channel is a function of the flow rates and viscosities of the two fluids. Therefore, knowledge of the viscosity of one fluid, together with the flow rates and cross-sectional areas of each fluid, permits the viscosity of the second fluid to be calculated. The viscometric curve of a fluid can be determined by this technique from just 250 μm L of fluid. In a subsequent study, polyethylene oxide solution was used as a test fluid to demonstrate how this technique could be automated to characterize the evolution of the viscosity in a continuous way [16]. Jang and Song [17] studied the rheology of conductive ink flow for printed electronics on a microfluidic chip for two commercial inks containing Ag nanoparticles. Printed electronics offer a promising alternative to conventional semi-conductor fabrication methods. However, the development of devices for printing electronics requires improved knowledge of the flow of ink, and thus the rheology of ink is now a key area for research. Conductive inks typically exhibit non-Newtonian properties due to the metallic particles that they contain [18]. Jang and Song used a microfluidic chip in conjunction with a micro particle image velocimeter for this purpose. They found that the ink flow exhibited a strong-shear thinning behavior as the metallic content, Ag in this case, was increased. The viscosity of many paints is well-fitted by a Carreau model for a wide range of shear rates [19]. Rheological characterization is important in the paint industry as the viscosity needs to be sufficiently low at high shear rates in order to provide a high quality coating to a surface, whilst under conditions of low shear rate the viscosity needs to be high enough to minimize dripping from a brush or roller. Furthermore, during roll coating application of paints on steel sheets, at speeds beyond a critical level, a hydrodynamic instability known as ‘ribbing’ can occur as the paint flows through the applicator. This
generates a waveform patterned interface that is a function of the rheological properties of the paint, as well as of surface tension [20]. Rheological characterization is important for the mixing of nuclear waste sludges, and for its transport to treatment plants [21,22]. These are obviously difficult tasks due to the high costs of handling a radioactive sludge sample for viscometric analysis. These sludges exhibit thixotropic and shear-thinning properties. Due to the dearth of actual data, much recourse is made to simulant sludges. 2.2. Rheometry for medical diagnosis and monitoring Applications for online continuous rheometry in medicine-related disciplines are numerous. Some examples, chosen to highlight the diversity, are described below. Improvements in the rheology of circulating red blood cells may help to reduce the required dosage of erythpoietin which is used for the treatment of urenic anemia. Kobayashi et al. [23] investigated the effect of vitamin E-bonded cellulose membrane dialyzer on carotid atherosclerotic changes and on the viscosity and degree of dysmorphism of red blood cells. Nandi et al. [24] developed a PDMS-based microfluidic system to achieve online coupling of micro dialysis sampling to microchip electrophoresis detection. Samples were injected into a serpentine channel of length 20 cm. The device was used for in vivo analysis of amino acid neurotransmitters in the extracellular fluid of the brain of an anesthetized rat to monitor the permeability of the blood–brain barrier and levels of amino acid neurotransmitters. Their research program has potential applications to brain disorders, including strokes. 3. Capillary methods for rheological characterization Conventional capillary rheometers compute the shear stress as a function of shear rate (rheogram) for different radii. Attempts to adapt capillary rheometers for online situations have proved to be challenging due to their overall (typically) large size, complexity of operation, and cost. Rheological characterization based on velocimetry techniques requires measurement of the velocity profiles across the diameter of a pipe from which the shear rates across the pipe can be deduced. Profiles of shear stress (or viscosity) against shear rate can then be plotted. In capillary rheometry, unless a very long capillary is used (L/D > 100, where D is the capillary diameter), contraction flow at the capillary entrance causes an additional pressure drop due to the stretching of fluid elements which may considerably affect the accuracy of measurements. This entrance effect can be corrected for using the Bagley correction whereby an equivalent length of die is used to represent this extra pressure [25]. Degré et al. [10] used particle image velocimetry (PIV) [13,26] to measure the bulk nonlinear rheology of shear thinning solutions of polyethylene oxide (PEO) in transparent microchannels with rectangular geometry (typically with length, L, of a few centimeters, width O (100 μm), depth O (10 μm)). Different pressures drops, ΔP were applied. PIV involves making measurements of the motion of fluorescent tracer particles that are seeded into a fluid. The PDMS slide was positioned above an inverted fluorescent microscope. By using a cross-correlation technique to analyze a series of images taken with a high speed camera, the velocity field within the channel can be constructed. The velocity profile of a pressure-driven Newtonian fluid in a rectangular channel takes the form of a Poiseuille parabola. Degré et al. [10] found that for the PEO solutions analyzed, a marked departure from this profile was noted. Furthermore, slip at the wall was visible where the velocity profiles failed to extrapolate to zero, which is another feature of nonlinear rheology. Instead of fitting an existing rheological model to their data [10] developed a more general procedure. From consideration of the mechanical equilibrium of the system they showed that the shear stress could be calculated from ΔP ðz−z0 Þ, where x is the flow direction, z is the distance σ xz ¼ − L
J.M. Rees / Advances in Colloid and Interface Science 206 (2014) 294–302
to the wall, and z0 is a chosen plane of the channel that is close to its midplane. The local shear rate can be calculated directly from the du x-component of the velocity, i.e. γ_ ¼ . Plotting σxz against γ_ gave dz the rheological curve. Close agreement was found upon comparison with measurements from a Couette rheometer. This procedure has potential to be used to generate shear rates up to 105 s−1. Its main limitation is that samples must be transparent. Opaque fluids can be analyzed using ultrasonic Doppler velocimetry [27] or by nuclear MRI [28]. Tozzi et al. [29] developed a methodology based on rescaling the velocity profile with the wall shear stress to determine rheological parameters from pressure drop measurements coupled with capillary velocimetry measurements of three fluid classes: glycerol, a crosslinked polyacrylate solution and a micelle surfactant solution. They found that the stress-rescaled velocity profile was a function of the rheological properties of the fluid only and was thus independent of other experimental variables such as flow rate, pressure drop and channel dimensions. Hence, their methodology provides a direct means for rheological characterization. This procedure was also found to be robust in low signal-to-noise conditions. 4. Inverse methods for determination of rheological parameters In many engineering applications, measurements of the desired physical quantities often cannot be made directly, but need to be inferred. This often involves the use of advanced mathematical techniques to solve an ‘inverse problem’ (Fig. 1). The quantities that are directly measurable form a dataset, d. We refer to the set of values that is to be reconstructed as the image, f. The mapping f → d is called the ‘forward problem’. Conventional modeling could be used to solve the forward problem. Inversion algorithms can be used to identify the cause of the measured effect. The mapping d → f comprises the corresponding inverse mapping. Inverse methods can be used to process measurements from a sensing device to obtain estimates of unknown constitutive parameters. To-date it is only rarely that such procedures have been applied to systems where the forward problem involves solving high order spatio-temporal systems of partial differential equations with complex geometries and multiple domains/multiphysics. Advanced computational models with these capabilities have only recently become available. For many applications, the inverse problem draws from advances in algebraic topology, differential geometry and functional analysis. In order for an inverse problem to be well-posed, the following Hadamard conditions must hold [30]: • For any data, d, in the dataset, a solution exists. • The solution is unique in the image space. • The mapping d → f, i.e. the inverse mapping, is continuous.
297
For the purpose of rheological inference, the forward problem could consist of taking a number of observational measurements of the flow field, or it could entail the solution of the governing equations of motion. When tailoring efficient algorithms to determine rheological parameters, it can be useful to run simulations in conjunction with an experimental program as, provided the model is verified as being in close agreement with the experimental data, it can be used to supplement the data set available for solving the inverse problem, thus reducing the number of experiments that need to be performed [31]. Different types of nonNewtonian fluids will be governed by different constitutive equations, containing different parameters. Some examples are now outlined.
4.1. Power law fluids As we saw from Eq. (1), power law fluids are characterized by two parameters, K and n. In order to generalize the procedure, we will denote these two parameters by k1 and k2. Since these parameters cannot be measured directly it is necessary to solve an inverse problem. This requires a minimum of two dynamical measures, Q1 and Q2, say. We therefore need to find a prescription to calculate the inverse problem for the specified data set d = d(k1,k2) for the image f = f(Q1,Q2).
4.2. Carreau fluids As we discussed earlier, both three and four parameter models are in common usage. • For the three parameter model (Eq. (2)) μ0, λ and n denote the constitutive parameters (k1, k2 and k3) of the fluid, thus a minimum of three dynamical measures, Q1, Q2 and Q3, are needed in order to solve the inverse problem. • For the four parameter model (Eq. (3)) μ0, μ∞, λ and n denote the constitutive parameters (k1, k2, k3 and k4) of the fluid, thus a minimum of four dynamical measures, Q1, Q2, Q3 and Q4, are needed in order to solve the inverse problem. Zimmerman et al. [32] simulated the electrokinetic flow of a Carreau fluid in a T-junction microchannel (assuming fixed values for μ0 and μ∞). They evaluated λ and n by computing the map f → d in the form of a look-up table, where Q1 and Q2 were taken to be the standard deviation and skewness of the back wall pressure profile. They demonstrated that this map was globally invertible by a graphical technique.
Fig. 1. Schematic to show methodology of procedure for solving inverse problems.
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4.3. Viscoelastic fluids With reference to Section 1.3, the PTT constitutive law contains five parameters: μ1, μ2, ζ, λ, ", which correspond to constitutive parameters (k1, k2, k3, k4, k5) of the fluid. Therefore, a minimum of five dynamical measures, Q1, Q2, Q3, Q4 and Q5, would be required to solve the inverse problem. 4.4. T-junction rheometry Capillary experiments have conventionally required a separate experiment to be performed for each shear rate under consideration. A step-change to this time-consuming procedure was proposed by Zimmerman et al. [32]. By inducing a flow through a symmetrical channel in the form of a T-junction, a range of shear rates, and hence viscosities, was created as the fluid was forced to turn the corner. In addition, stretching of fluid elements can occur in the vicinity of the back wall of the T-junction, hence, particularly for the case of viscoelastic fluids, a range of elongations can also be induced in a single experiment with this geometrical configuration. A Boger fluid is a dilute polymer solution that is highly elastic but has a constant viscosity [33]. A Boger fluid can be made by dissolving polyacrylamide in a concentrated aqueous sugar solution. By performing experiments with a Boger fluid and then with a Newtonian fluid having the same viscosity it is possible to separate elastic effects from viscous effects. In conventional studies of rheometry of elastic fluids, two opposing jets are used to create a region of the flow that is nearly homogeneous. In contrast, as fluid elements are stretched as they are accelerated along the back wall of a T-junction channel, a rheometer comprising channels with this configuration could be used to study extensional flow effects as the extensional flow region is not homogeneous, and thus the setup is ‘information rich’ in regard to this phenomenon. Zimmerman et al. [34] performed three-dimensional finite element simulations of electrokinetic-driven flow in a microchannel T-junction of a Carreau-type non-Newtonian fluid. They demonstrated the existence of a one-to-one mapping between the Carreau parameters and the back-wall pressure profile for a broad range of operating and physical parameters. Such a map suggested that it might be possible to construct a highly efficient viscometric device based on sensors embedded within a T-junction channel network. In a subsequent computational study, Craven et al. [35] empirically derived a modified slip boundary condition that improves the accuracy of predictions of
Fig. 2. T-junction channel etched onto glass microchip using photolithography and hydrochloric acid etching.
Fig. 3. Schematic of cross-section of channel etched onto PDMS slide by photolithography and hydrofluoric etching.
electroosmotic flow in the vicinity of a sharp corner (such as that in a T-junction microchannel) whilst also delivering an improvement in computational performance as a result of the elimination of a singularity in the velocity field. In a coupled experimental-computational series of investigations by Bandulasena et al. [36–40], the capabilities of the T-junction rheometer were further demonstrated. For the experiments, photolithography and hydrofluoric acid were used to etch the channels onto a PDMS slide (Fig. 2). Hydrofluoric etching produces channels with rounded sides as shown in Fig. 3. The inlet and outlet channels extended about 5 mm from the junction (Fig. 4). The chip was placed in a custom-made chipholder which contained three fluid reservoirs that were used to control a pressure-driven flow through the channels (Fig. 5). A frequency histogram of the shear rate between the corner and the mid-point of the junction for a simulated power-law fluid with n = 0.8 and K = 0.025 Pa sn is shown in Fig. 6 for a pressure drop across the device of 100 Pa. Bandalusena et al. [38] showed that the rheological parameters of a power law fluid could be uniquely determined by measurements of the velocity field or pressure field within the T-junction and the inlet flow rate. However, the range of shear rates produced was restricted by the pressure drop across the device. The next stage in the development was to design an experiment in which the desired shear range could be achieved through precise control of the flow rate. It was then possible to infer K and n by applying the inverse methodology to measurements of the mean pressure at the inlet and at the junction. By performing a creeping flow analysis, i.e. making the assumption that the Reynolds number is sufficiently small that inertial effect can be neglected, the pressure drop, ΔP across the T-junction can be expressed as ΔP = KQnL30 f(a,n), where L0 is a typical length scale (such as the channel depth) and α represents the aspect ratio of the flow domain. Assuming that a Hagen–Poiseuille relationship exists
Fig. 4. Schematic of T-junction channel showing location of hypothetical pressure sensors, P1 and P2.
J.M. Rees / Advances in Colloid and Interface Science 206 (2014) 294–302
299
0.48
P2/P1
0.44
0.4
0.36
0.32
0.4
0.6
0.8
1
n Fig. 7. Plot of P2/P1 plotted against n which is independent of the flow rate Q. Fig. 5. Prototype microscale rheometer suitable for use with nano- to micro-liter sample volumes with moderate viscosity, for instance, rare fluids such as protein mixtures.
between the pressure drop and the volumetric flow rate, since f is dimensionless, an algorithm was derived that permits the inference of K and n from the volumetric flow rate, the pressure drop and the ratio of any two field variables (e.g. the ratio of the mean pressure fields in the vicinity of the inlet and at the junction itself) measured over an averaged region. Under conditions for creeping flow, it is easily demonstrated that the consistency index K, which is dimensional, cannot influence any dimensionless quantity. Therefore, it was deduced that the dimensionless field variables are a function of n only. By performing finite element numerical simulations of the flow of a power-law fluid through the T-junction channels, simulated mean pressure measurements were calculated from hypothetical pressure sensors located at the inlet (P1) and at the center of the junction (P2) Fig. 4. The parametric relationship r = r(n) = P2/P1 was measured, and its inverse, n = r−1(r) was calculated. Linear regression showed that this curve was well-approximated by a quartic fit of the form
∇ ∇ T þ λ1 T ¼ 2μ 0 D þ λ2 D ;
ð9Þ
where • μ0 = μ1 + μ2 is the total fluid viscosity, λ1 is the relaxation time, μ2 is the retardation time, μ0 n o ∂T • T∇ ¼ þ U⋅∇T− ð∇UÞT ⋅T þ T⋅ð∇UÞ is the upper convected time ∂t derivative of the stress tensor (also known as the Oldroyd derivative). • λ2 ¼
0.08
The upper convected derivative is widely used in polymer rheology to describe the behavior of a viscoelastic fluid undergoing large deformation. Figs. 8 to 10 show the results of a finite element simulation of an Oldroyd-B fluid flowing in a T-junction channel for Weissenberg numbers, We = 0, We = 0.8 and We = 1.6. Due to the line of symmetry, only half of the domain is shown, as this saved on computational time. The simulations were all non-dimensionalized, thus the actual scales of the channel width and length are relative. The plots show the velocity field, indicated by scaled arrows, and a greyscale depicts the Frobenius norm of the viscoelastic stress, ‖T‖2. It can be seen that the back wall stress is particularly sensitive to the parameter We. This suggests that statistical measures of the flow field might map to the constitutive parameters in this model. This is a topic of current research. The inverse method based on the T-junction geometry has been found to invert uniquely in all cases that we have studied. We have yet to find a system so nonlinear that it cannot be uniquely inverted. The only question is sensitivity. For power law fluids, sensitivity was found to be within 3% for the flow behavior index which can be attributed to irregularities in the fabrication of the channel geometry. If the wrong constitutive model is chosen, the inversion is poor. However, this is still potentially useful for practical purposes such as control. For an unknown fluid an expert system would test a whole range of models and choose the best performing, which can be assessed from the standard error of fitting several experiments.
0.06
5. Discussion and conclusions
4
3
2
r ¼ 0:3285n −1:1405n þ 1:4808n −1:0067n þ 0:6708:
ð8Þ
This result is plotted in Fig. 7 for a broad range of values of n for which the function is single valued. Since Eq. (8) is quartic, upon algebraic manipulation a closed form solution for n can be found. The Oldroyd-B constitutive model provides a good description for viscoelastic fluids in shear flow [41]. However this model possesses
0.18 0.16 0.14
relative frequency
an unphysical singularity in extensional flow. In the Oldroyd-B model the stress tensor T takes the form
0.12 0.1
0.04 0.02 0
25 75 125 175 225 275 325 375 425 475 525 575 625 675
shear rate (s−1) Fig. 6. Shear rate distribution for a power law fluid with n = 0.8 and K = 0.025 Pa sn from numerical simulations in a T-junction channel network for a pressure drop of 100 Pa and a channel width of 260 μm.
This paper has followed a broadly historical path, beginning with an overview of the melt flow index that is widely used for monitoring polymers despite issues concerning its reliability, and progressing to give an insight into the new methodologies based on inverse techniques that promise much potential for full rheological characterization. The viscosity of a fluid is a measure of its resistance to flow. The more viscous the fluid, the greater is the force that is required to initiate flow. For Newtonian fluids such as water, the shear stress is proportional to the strain rate, the constant of proportionality being the
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Fig. 8. Non-dimensional velocity field (indicated by arrows) and viscoelastic stress tensor (greyscale) for Oldroyd-B fluid with Weissenberg number, We = 0.
Fig. 9. Non-dimensional velocity field (indicated by arrows) and viscoelastic stress tensor (greyscale) for Oldroyd-B fluid with Weissenberg number, We = 0.8.
J.M. Rees / Advances in Colloid and Interface Science 206 (2014) 294–302
301
Fig. 10. Non-dimensional velocity field (indicated by arrows) and viscoelastic stress tensor (greyscale) for Oldroyd-B fluid with Weissenberg number, We = 1.6.
viscosity. Liquids whose behavior deviates from this simple model are called complex, or non-Newtonian, fluids. Common examples include cornstarch, which is used as a thickening agent in liquid-based foods such as soups and sauces, and ketchup, which flows more easily when the bottle is shaken. This complex fluid behavior stems from molecular interactions that fluctuate when the fluid is in motion. The viscosities of complex fluids can depend on a number of parameters via the so-called ‘constitutive equation’. Knowledge of these parameters is known as ‘rheological characterization’ and usually involves complicated calculations and the use of expensive equipment. The equipment is expensive because of multiple moving parts that must be precisely machined and maintained. Rheological parameters can affect the appearance, texture or even the shelf-life stability of a complex fluid. Since most industrial and biological fluids, such as polymer solutions, oils, slurries and blood, exhibit non-Newtonian features, characterization of complex fluids is an essential activity in many science and engineering applications. Knowledge of the rheological characteristics in industrial applications enables optimal process control and hence leads to better quality products. The apparent viscosity of a fluid can be measured by a viscometer. However, a rheometer is needed to obtain the constitutive parameters. There does not currently exist a commercially available inline rheometer that is capable of providing continuous monitoring of rheological parameters in real-time, yet this is a much wished for device for use in many industrial flow processes. At present, in order to perform a rheological characterization, samples need to be taken off-line and analyzed in an expensive cone-and-plate rheometer, or similar device, which is time consuming and costly. Recent developments of rheometry based on the application of advanced mathematical algorithms (inverse methods) hold the potential to be exploited to create the next generation of commercial rheometers that could be operated inline to provide real-time monitoring of the constitutive parameters. This would be of enormous
benefit in a wide range of applications, including the food and beverage industries, oil and lubrication industries, high-tech printing and paints, cements, cosmetics and pharmaceuticals, floor coverings and household commodities (e.g. toothpaste, shampoo). The bio-fuel industry has been highlighted as a particular area of interest due to increased economic and ‘green’ pressures to find sources of energy that are sustainable (biomass derived or utilizing waste products) and that are relatively inexpensive to produce. Newly created technology now enables the production of biofuels more efficiently than was hitherto possible, thus furthering our motivation for exploring the rheology of bio-oils. Industry needs to respond to the greater variability in input feedstock specification, as well as material properties varying during processing, with better control systems. Thus there currently exists a strong driver for online, continuous rheometry. Acknowledgments I am grateful for the support of current and past members of the Rheology Group at The University of Sheffield: HCH Bandulasena, S Brittle, ST Chaffin, TJ Craven, P Fairclough, R Hodgkinson, D Kuvshinov, E Kuvshinova, H Yu and WB Zimmerman. In particular, I would like to thank ST Chaffin and HCH Bandulasena for their assistance with some of the figures. I acknowledge also, Professor Manuel G Velarde, to whom this issue is dedicated, for introducing me to the exciting research area of interfacial dynamics, and its interaction with non-Newtonian behaviors. References [1] Cho H-M, Yoo W, Yoo B. Food Sci Biotechnol 2012;21:1775. [2] Koop HS, Da-Iozzo EJ, de Freitas RA, Franco CRC, Mitchell DA, Silviera JLM. J Pharm Sci 2012;101:2457. [3] Stokes JR, Macakova MR, Chojnicka-Paszun A, de Kruif CG, de Jongh HHJ. Langmuir 2011;27:3474.
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J.M. Rees / Advances in Colloid and Interface Science 206 (2014) 294–302 Vadillo DC, Tuladhar TR, Mulji AC, Mackley MR. J Rheol 2010;54:781. Kenney KL, Smith WA, Gresham GL, Westover TL. Biofuels 2013;4:111. Bachelet C, Dantan Ph, Flaud P. Eur Phys J Appl Phys 2004;25:209. Uddin J, Marston JO, Thoroddsen ST. Phys Fluids 2012;24:073104. Phan-Thein N, Tanner RI. J Non-Newtonian Fluid Mech 1977;93:325. Shenoy AV, Saini DR. Adv Polym Tech 1986;6:1. Degré G, Joseph P, Tabeling P, Lerouge S, Cloitre M, Ajdari A. Appl Phys Lett 2006;89:024104. Guillot P, Panizza P, Joanicot J-B SM, Colin A. Langmuir 2006;22:6438. Pipe CJ, Majmudar TS, McKinley GH. Rheol Acta 2008;47:612. Santiago JG, Wereley ST, Meinhart CD, Beebe DJ, Adrian RJ. Exp Fluids 1998;25:316. Smith JS, Irving NMNH, Simpson RB. Analyst 1970;95:743. Ranganathan M, Mackley MR, Spitteler PH. J Rheol 1999;43:443. Guillot P, Moulin T, Kotitz R, Guirardel M, Dodge A, Joanicot M, et al. Microfluid Nanofluid 2008;5:619. Jang Y-S, Song S. Exp Fluids 2012;53:1. Lee HH, Chou KS, Huang KC. Nanotechnology 2005;16:2434. Bird RB, Curtiss CF, Armstrong RC, Hassager O. Dynamics of Polymeric Liquids. New York: Wiley; 1987. Varela Lopez F, Rosen M. Lat Am Appl Res 2002;32:247. Alderman NJ, Heywood NI. Chem Eng Process 2004;100:27. Chhabra RP. Bubbles, drops and particles in non-Newtonian fluids. 2nd ed. London: Taylor and Francis Group; 2007.
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Kobayashi S, Moriva H, Aso K, Ohtake T. Kidney Int 2003;63:1881. Nandi P, Desai DP, Lunte SM. Electrophoresis 2010;31:1414. Mitsoulis E, Hatzikiriakos SG. Rheol Acta 2003;42:309. Meinhart CD, Wereley ST, Santiago JG. Exp Fluids 1999;27:414. Dogan N, McCarthy MJ, Powell RL. Meas Sci Technol 2005;16:1684. Götz J, Kreibich W, Peciar M, Buggisch H. J Non-Newtonian Fluid Mech 2001;98:117. Tozzi EJ, Bacca LA, Hartt WH, McCarthy KL, McCarthy MJ. J Rheol 2012;56:1449. Hadamard J. Lectures on the Cauchy problem in linear partial differential equations. New Haven: Yale University Press; 1923. Zimmerman WB, Rees JM. Chem Eng Res Des 2009;87:798. Zimmerman WB, Rees JM, Craven TJ. Microfluid Nanofluid 2006;2:481. James DF. Annu Rev Fluid Mech 2009;41:129. Craven TJ, Rees JM, Zimmerman WB. Microfluid Nanofluid 2010;9:559. Craven TJ, Rees JM, Zimmerman WB. Phys Fluid 2008;20:043603. Bandulasena HCH, Zimmerman WB, Rees JM. J Mech Eng Sci Part C 2008;222:761. Bandulasena HCH, Zimmerman WB, Rees JM. Meas Sci Technol 2009;20:114405. Bandalusena HCH, Zimmerman WB, Rees JM. J Non-Newtonian Fluid Mech 2010;165:1302. Bandulasena HCH, Zimmerman WB, Rees JM. Appl Rheol 2010;20:55608. Bandulasena HCH, Zimmerman WB, Rees JM. Meas Sci Technol 2011;22:125402. Oldroyd J. Proc R Soc London A 1950;200:1063.
Contents lists available at ScienceDirect
Advances in Colloid and Interface Science journal homepage: www.elsevier.com/locate/cis
Towards online, continuous monitoring for rheometry of complex fluids Julia M. Rees ⁎ School of Mathematics and Statistics, Hicks Building, Hounsfield Road, University of Sheffield, S3 7RH, UK
a r t i c l e
i n f o
Available online 13 June 2013 Keywords: Microfluidics Rheology Rheometry Inverse methods Lab on chip Capillary rheometer
a b s t r a c t This paper presents an overview of the developments that have been made towards the design of an inline rheometer that has the capabilities for monitoring in real time the viscous constitutive parameters of nonNewtonian fluids in a pipe flow. This has potential applications for a wide range of fluids, including hydrocolloid solutions and polymer solutions. This is of relevance to many industries, for example the pharmaceutical, lubrication, food and printing industries. The use of mathematical algorithms for inferring rheological parameters from properties of flow field statistics is explored. Particular focus is given to the development of a flow cell rheometer containing a T-junction geometry with the capacity to induce a range of shear rates in the vicinity of the bend, and a distribution of elongational viscosities along the back-wall. Such features create an information-rich flow field that is beneficial for the development of a rheometer with a fast response time that is suitable for commercial purposes. © 2013 Elsevier B.V. All rights reserved.
Contents 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.1. Power law fluids . . . . . . . . . . . . . . . . . 1.2. Carreau fluids . . . . . . . . . . . . . . . . . . . 1.3. Viscoelastic fluids . . . . . . . . . . . . . . . . . 2. Applications for rheometry . . . . . . . . . . . . . . . . 2.1. Engineering monitoring and control . . . . . . . . 2.2. Rheometry for medical diagnosis and monitoring . . 3. Capillary methods for rheological characterization . . . . . 4. Inverse methods for determination of rheological parameters 4.1. Power law fluids . . . . . . . . . . . . . . . . . 4.2. Carreau fluids . . . . . . . . . . . . . . . . . . . 4.3. Viscoelastic fluids . . . . . . . . . . . . . . . . . 4.4. T-junction rheometry . . . . . . . . . . . . . . . 5. Discussion and conclusions . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction The processing of complex fluids occurs in many engineering applications and in biomedical analysis. Efficiency of such processes can be crucially dependent on the rheology of the fluid. Viscous
⁎ Tel.: +44 114 222 3782. E-mail address: [email protected]. 0001-8686/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cis.2013.05.006
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parameters can affect the texture or appearance of a fluid, and also they can determine the shelf-life stability of a product. At present, the monitoring of viscous parameters requires samples to be taken manually and then analyzed in a cone-and-plate rheometer, or similar device. The ability to perform online continuous rheological characterization in real-time would permit tighter monitoring and control in a wide range of applications that involve the processing of fluids of non-Newtonian fluids, such as those in the food and beverage [1], pharmaceutical [2], oil and lubrication [3], and high-tech printing
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industries [4]. This would lead to a reduction in wastage caused by processes going off spec, and in the level of manual intervention required for product sampling. Currently, no such commercially available device exists that can perform this task efficiently for a broad range of fluids. Quality control is an endpoint test which results in disposal of below spec product. Continuous monitoring would reduce this wastage, potentially completely. Generally the wastage level of a wide range of processes is around 5%. Recently, much attention has been given to the use of flow configurations that yield an information rich data set of measurable flow parameters that can be mapped, through the solution of an inverse problem, to individual constitutive parameters. With the pressure to develop sustainable materials (biomass derived or utilizing waste products) for processing and manufacturing, industry must respond to the greater variability in input feedstock specification [5], as well as material properties varying during processing, with better control systems. Hence there currently exists a strong driver for online, continuous rheometry. The rheological behavior of different types of non-Newtonian fluids is described by different constitutive equations. Some examples are discussed below. 1.1. Power law fluids The viscosity of power law fluids is given by n−1 ; μ ¼ K γ_
ð1Þ
where K is the flow consistency index (SI units Pa sn), γ_ is the shear rate (SI unit s−1), and n b 1 is the flow behavior index. K and n denote the constitutive parameters of the fluid in question and it may be desirable to know what they are in any given situation [6]. For example, the viscosity of bio-oils is highly temperature dependent and follows a power law constitutive equation. The Krohne viscoline inline viscometer manufactured by Delta Instrumentation (http:// www.deltainstrumentation.com/viscosity/viscometer.html) is a novel device that can be used to determine the two rheological parameters, K and n, for power law fluids. The viscoline comprises a section of pipeline that can be inserted into a fluidic process. The pipeline contains two low pressure drop static mixers that are used to generate two different shear regimes. If the flow rate is known then the pipe line viscosity can be calculated from the two pressure drop measurements. The viscoline can operate at shear rates in the range 50 to 500 s−1 and at temperatures of −5 to 200 °C.
For Carreau fluids, models with either three or four parameters are in common usage [7]. In the three parameter version, the viscosity at infinite shear rate is taken to be zero, and the viscosity is given by ð2Þ
where μ0 is the zero-shear-rate viscosity, λ is a parameter with dimensions of time and n is the power-law exponent. The four parameter Carreau model takes the form i 2 ðn−1Þ=2 ; μ ¼ μ ∞ þ ðμ 0 −μ ∞ Þ 1 þ λðγ_ Þ
stress. The Phan–Thein–Tanner (PTT) constitutive law [8] uses the following form for the non-dimensional constitutive equation: Weτt ¼ 2μ 1 D−hτ n
h io þ We τ⋅∇U þ ð∇UÞT ⋅τ−U⋅∇τ þ ζ D⋅∇τ þ ðD⋅τÞT ;
ð4Þ
where h is usually defined using one of the following three forms: Linear model : h ¼ 1 þ
We traceðτÞ; μ1
Quadratic model : h ¼ 1 þ
2 We 1 We traceðτÞ þ traceðτÞ ; μ1 2 μ1
We Exponential model : h ¼ exp traceðτÞ ; μ1
ð5Þ
ð6Þ
ð7Þ
where • • • • • •
τ = T − 2μ2D, T is the extra-stress tensor, D is the rate of deformation tensor, U is the velocity vector, μ1 is the polymeric viscosity, μ2 is the solvent viscosity,
Uλ , where λ is a L relaxation time and U and L are typical velocity and length scales respectively, • and ζ are positive constants used to control the viscoelastic properties.
• We is a Weissenberg number defined as We ¼
The aim of this article is to provide an updated review, with a particular focus on industrial aspects that would potentially benefit from improvements in the ability to perform online rheometric characterization in real-time. Applications of rheometry in engineering, manufacturing and medical processes are outlined in Section 2. Capillary methods for rheological characterization are described in Section 3. In Section 4 some recent research that uses advanced mathematical algorithms to solve inverse problems in order to determine rheological parameters in real time, is outlined. Discussion on the effect of increased variability of feedstocks as a key driver for the development of improved devices for determining rheological parameters online forms the theme for Section 5, where the present situation is summarized. 2. Applications for rheometry
1.2. Carreau fluids
2 ðn−1Þ=2 μ ¼ μ 0 1 þ ðλγ_ Þ
295
ð3Þ
where μ∞ is the viscosity at infinite shear rate. 1.3. Viscoelastic fluids Viscoelastic materials possess both viscous and elastic characteristics when undergoing strain, and therefore exhibit time dependent
2.1. Engineering monitoring and control The monitoring and control of rheological parameters are of importance for a range of engineering and processing systems. Some examples are discussed in this section. Melt flow index (MFI) as a single data point experiment is a wellestablished technique for measuring changes in the average molecular weight of a polymer melt, although its accuracy is not always reliable as it refers to only a single point measurement within a larger spectrum of properties [9]. MFI is a measure of the ease of flow (pliability) of a thermoplastic polymer melt. It is defined as the mass in grams of a polymer passing through a standardized capillary of 2 mm diameter and 8 mm length over a 10 min interval, at a specified temperature, when a standardized load is applied. The MFI only tests a material at a single shear stress and temperature. It provides an indirect measure of molecular weight. A higher MFI is indicative of a lower material viscosity. Many processes use MFI to assess the viscous properties of incoming materials and as an aid to monitoring changes in the process. More accurate rheology tests can be performed using capillary or parallel plate rheometers.
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The three main flow configurations for microfluidic rheology are capillary flow, stagnation and contraction. In capillary devices the shear viscosity, μ, can be determined from the flow rate, Q, and the pressure drop, ΔP, across a straight section of tubing of length L. Thus in order to determine the viscosity for a capillary flow one can either (i) impose a specified pressure gradient and measure the flow rate [10,11] or (ii) impose the flow rate (e.g. using a syringe pump) and measure the associated pressure drop (e.g. using pressure sensors) [12]. Flow rates can be measured by a number of techniques, including micron resolution particle image velocimetry (μPIV) [13]. Capillary rheometers have been used online in polymer and food processing industries for around 40 years [14]. In a conventional online capillary rheometer, if the pressure drop across the system is held constant, the residence time of the material in the rheometer will be dependent upon the properties of the material itself. Online capillary rheometers need to respond quickly to fluctuations in process parameters. They can be used to monitor polymer manufacturing processes or to act as a sensor for closed loop control. The response time behavior for polymer melts of high molecular weight might be a poor indicator of rheological properties due to the associated low flow rates through the capillary. To overcome this, Göttfert® developed a real-time rheometer that incorporates a circulating volumetric flow. This provides a continuous flow of fresh melt through the capillary, even at low shear rates. This circulation transports the polymer melt through transfer lines up to 100 times faster which provides an advantage over conventional capillary online rheometers. Ranganathan et al. [15] present experimental data for the timedependent capillary flow of a high-density polyethylene through a multipass rheometer. They found that the melt compressibility was closely linked to the observed pressure relaxation on cessation of the piston movement. Guillot et al. [11] presented a method for measuring the rheological properties of complex fluids on a microfluidic chip using a methodology that involved using the adjacent parallel flows of two immiscible liquids as a pressure sensor. The ratio of the cross-sectional areas occupied by two immiscible fluids flowing parallel to each other side by side in a rectangular channel is a function of the flow rates and viscosities of the two fluids. Therefore, knowledge of the viscosity of one fluid, together with the flow rates and cross-sectional areas of each fluid, permits the viscosity of the second fluid to be calculated. The viscometric curve of a fluid can be determined by this technique from just 250 μm L of fluid. In a subsequent study, polyethylene oxide solution was used as a test fluid to demonstrate how this technique could be automated to characterize the evolution of the viscosity in a continuous way [16]. Jang and Song [17] studied the rheology of conductive ink flow for printed electronics on a microfluidic chip for two commercial inks containing Ag nanoparticles. Printed electronics offer a promising alternative to conventional semi-conductor fabrication methods. However, the development of devices for printing electronics requires improved knowledge of the flow of ink, and thus the rheology of ink is now a key area for research. Conductive inks typically exhibit non-Newtonian properties due to the metallic particles that they contain [18]. Jang and Song used a microfluidic chip in conjunction with a micro particle image velocimeter for this purpose. They found that the ink flow exhibited a strong-shear thinning behavior as the metallic content, Ag in this case, was increased. The viscosity of many paints is well-fitted by a Carreau model for a wide range of shear rates [19]. Rheological characterization is important in the paint industry as the viscosity needs to be sufficiently low at high shear rates in order to provide a high quality coating to a surface, whilst under conditions of low shear rate the viscosity needs to be high enough to minimize dripping from a brush or roller. Furthermore, during roll coating application of paints on steel sheets, at speeds beyond a critical level, a hydrodynamic instability known as ‘ribbing’ can occur as the paint flows through the applicator. This
generates a waveform patterned interface that is a function of the rheological properties of the paint, as well as of surface tension [20]. Rheological characterization is important for the mixing of nuclear waste sludges, and for its transport to treatment plants [21,22]. These are obviously difficult tasks due to the high costs of handling a radioactive sludge sample for viscometric analysis. These sludges exhibit thixotropic and shear-thinning properties. Due to the dearth of actual data, much recourse is made to simulant sludges. 2.2. Rheometry for medical diagnosis and monitoring Applications for online continuous rheometry in medicine-related disciplines are numerous. Some examples, chosen to highlight the diversity, are described below. Improvements in the rheology of circulating red blood cells may help to reduce the required dosage of erythpoietin which is used for the treatment of urenic anemia. Kobayashi et al. [23] investigated the effect of vitamin E-bonded cellulose membrane dialyzer on carotid atherosclerotic changes and on the viscosity and degree of dysmorphism of red blood cells. Nandi et al. [24] developed a PDMS-based microfluidic system to achieve online coupling of micro dialysis sampling to microchip electrophoresis detection. Samples were injected into a serpentine channel of length 20 cm. The device was used for in vivo analysis of amino acid neurotransmitters in the extracellular fluid of the brain of an anesthetized rat to monitor the permeability of the blood–brain barrier and levels of amino acid neurotransmitters. Their research program has potential applications to brain disorders, including strokes. 3. Capillary methods for rheological characterization Conventional capillary rheometers compute the shear stress as a function of shear rate (rheogram) for different radii. Attempts to adapt capillary rheometers for online situations have proved to be challenging due to their overall (typically) large size, complexity of operation, and cost. Rheological characterization based on velocimetry techniques requires measurement of the velocity profiles across the diameter of a pipe from which the shear rates across the pipe can be deduced. Profiles of shear stress (or viscosity) against shear rate can then be plotted. In capillary rheometry, unless a very long capillary is used (L/D > 100, where D is the capillary diameter), contraction flow at the capillary entrance causes an additional pressure drop due to the stretching of fluid elements which may considerably affect the accuracy of measurements. This entrance effect can be corrected for using the Bagley correction whereby an equivalent length of die is used to represent this extra pressure [25]. Degré et al. [10] used particle image velocimetry (PIV) [13,26] to measure the bulk nonlinear rheology of shear thinning solutions of polyethylene oxide (PEO) in transparent microchannels with rectangular geometry (typically with length, L, of a few centimeters, width O (100 μm), depth O (10 μm)). Different pressures drops, ΔP were applied. PIV involves making measurements of the motion of fluorescent tracer particles that are seeded into a fluid. The PDMS slide was positioned above an inverted fluorescent microscope. By using a cross-correlation technique to analyze a series of images taken with a high speed camera, the velocity field within the channel can be constructed. The velocity profile of a pressure-driven Newtonian fluid in a rectangular channel takes the form of a Poiseuille parabola. Degré et al. [10] found that for the PEO solutions analyzed, a marked departure from this profile was noted. Furthermore, slip at the wall was visible where the velocity profiles failed to extrapolate to zero, which is another feature of nonlinear rheology. Instead of fitting an existing rheological model to their data [10] developed a more general procedure. From consideration of the mechanical equilibrium of the system they showed that the shear stress could be calculated from ΔP ðz−z0 Þ, where x is the flow direction, z is the distance σ xz ¼ − L
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to the wall, and z0 is a chosen plane of the channel that is close to its midplane. The local shear rate can be calculated directly from the du x-component of the velocity, i.e. γ_ ¼ . Plotting σxz against γ_ gave dz the rheological curve. Close agreement was found upon comparison with measurements from a Couette rheometer. This procedure has potential to be used to generate shear rates up to 105 s−1. Its main limitation is that samples must be transparent. Opaque fluids can be analyzed using ultrasonic Doppler velocimetry [27] or by nuclear MRI [28]. Tozzi et al. [29] developed a methodology based on rescaling the velocity profile with the wall shear stress to determine rheological parameters from pressure drop measurements coupled with capillary velocimetry measurements of three fluid classes: glycerol, a crosslinked polyacrylate solution and a micelle surfactant solution. They found that the stress-rescaled velocity profile was a function of the rheological properties of the fluid only and was thus independent of other experimental variables such as flow rate, pressure drop and channel dimensions. Hence, their methodology provides a direct means for rheological characterization. This procedure was also found to be robust in low signal-to-noise conditions. 4. Inverse methods for determination of rheological parameters In many engineering applications, measurements of the desired physical quantities often cannot be made directly, but need to be inferred. This often involves the use of advanced mathematical techniques to solve an ‘inverse problem’ (Fig. 1). The quantities that are directly measurable form a dataset, d. We refer to the set of values that is to be reconstructed as the image, f. The mapping f → d is called the ‘forward problem’. Conventional modeling could be used to solve the forward problem. Inversion algorithms can be used to identify the cause of the measured effect. The mapping d → f comprises the corresponding inverse mapping. Inverse methods can be used to process measurements from a sensing device to obtain estimates of unknown constitutive parameters. To-date it is only rarely that such procedures have been applied to systems where the forward problem involves solving high order spatio-temporal systems of partial differential equations with complex geometries and multiple domains/multiphysics. Advanced computational models with these capabilities have only recently become available. For many applications, the inverse problem draws from advances in algebraic topology, differential geometry and functional analysis. In order for an inverse problem to be well-posed, the following Hadamard conditions must hold [30]: • For any data, d, in the dataset, a solution exists. • The solution is unique in the image space. • The mapping d → f, i.e. the inverse mapping, is continuous.
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For the purpose of rheological inference, the forward problem could consist of taking a number of observational measurements of the flow field, or it could entail the solution of the governing equations of motion. When tailoring efficient algorithms to determine rheological parameters, it can be useful to run simulations in conjunction with an experimental program as, provided the model is verified as being in close agreement with the experimental data, it can be used to supplement the data set available for solving the inverse problem, thus reducing the number of experiments that need to be performed [31]. Different types of nonNewtonian fluids will be governed by different constitutive equations, containing different parameters. Some examples are now outlined.
4.1. Power law fluids As we saw from Eq. (1), power law fluids are characterized by two parameters, K and n. In order to generalize the procedure, we will denote these two parameters by k1 and k2. Since these parameters cannot be measured directly it is necessary to solve an inverse problem. This requires a minimum of two dynamical measures, Q1 and Q2, say. We therefore need to find a prescription to calculate the inverse problem for the specified data set d = d(k1,k2) for the image f = f(Q1,Q2).
4.2. Carreau fluids As we discussed earlier, both three and four parameter models are in common usage. • For the three parameter model (Eq. (2)) μ0, λ and n denote the constitutive parameters (k1, k2 and k3) of the fluid, thus a minimum of three dynamical measures, Q1, Q2 and Q3, are needed in order to solve the inverse problem. • For the four parameter model (Eq. (3)) μ0, μ∞, λ and n denote the constitutive parameters (k1, k2, k3 and k4) of the fluid, thus a minimum of four dynamical measures, Q1, Q2, Q3 and Q4, are needed in order to solve the inverse problem. Zimmerman et al. [32] simulated the electrokinetic flow of a Carreau fluid in a T-junction microchannel (assuming fixed values for μ0 and μ∞). They evaluated λ and n by computing the map f → d in the form of a look-up table, where Q1 and Q2 were taken to be the standard deviation and skewness of the back wall pressure profile. They demonstrated that this map was globally invertible by a graphical technique.
Fig. 1. Schematic to show methodology of procedure for solving inverse problems.
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4.3. Viscoelastic fluids With reference to Section 1.3, the PTT constitutive law contains five parameters: μ1, μ2, ζ, λ, ", which correspond to constitutive parameters (k1, k2, k3, k4, k5) of the fluid. Therefore, a minimum of five dynamical measures, Q1, Q2, Q3, Q4 and Q5, would be required to solve the inverse problem. 4.4. T-junction rheometry Capillary experiments have conventionally required a separate experiment to be performed for each shear rate under consideration. A step-change to this time-consuming procedure was proposed by Zimmerman et al. [32]. By inducing a flow through a symmetrical channel in the form of a T-junction, a range of shear rates, and hence viscosities, was created as the fluid was forced to turn the corner. In addition, stretching of fluid elements can occur in the vicinity of the back wall of the T-junction, hence, particularly for the case of viscoelastic fluids, a range of elongations can also be induced in a single experiment with this geometrical configuration. A Boger fluid is a dilute polymer solution that is highly elastic but has a constant viscosity [33]. A Boger fluid can be made by dissolving polyacrylamide in a concentrated aqueous sugar solution. By performing experiments with a Boger fluid and then with a Newtonian fluid having the same viscosity it is possible to separate elastic effects from viscous effects. In conventional studies of rheometry of elastic fluids, two opposing jets are used to create a region of the flow that is nearly homogeneous. In contrast, as fluid elements are stretched as they are accelerated along the back wall of a T-junction channel, a rheometer comprising channels with this configuration could be used to study extensional flow effects as the extensional flow region is not homogeneous, and thus the setup is ‘information rich’ in regard to this phenomenon. Zimmerman et al. [34] performed three-dimensional finite element simulations of electrokinetic-driven flow in a microchannel T-junction of a Carreau-type non-Newtonian fluid. They demonstrated the existence of a one-to-one mapping between the Carreau parameters and the back-wall pressure profile for a broad range of operating and physical parameters. Such a map suggested that it might be possible to construct a highly efficient viscometric device based on sensors embedded within a T-junction channel network. In a subsequent computational study, Craven et al. [35] empirically derived a modified slip boundary condition that improves the accuracy of predictions of
Fig. 2. T-junction channel etched onto glass microchip using photolithography and hydrochloric acid etching.
Fig. 3. Schematic of cross-section of channel etched onto PDMS slide by photolithography and hydrofluoric etching.
electroosmotic flow in the vicinity of a sharp corner (such as that in a T-junction microchannel) whilst also delivering an improvement in computational performance as a result of the elimination of a singularity in the velocity field. In a coupled experimental-computational series of investigations by Bandulasena et al. [36–40], the capabilities of the T-junction rheometer were further demonstrated. For the experiments, photolithography and hydrofluoric acid were used to etch the channels onto a PDMS slide (Fig. 2). Hydrofluoric etching produces channels with rounded sides as shown in Fig. 3. The inlet and outlet channels extended about 5 mm from the junction (Fig. 4). The chip was placed in a custom-made chipholder which contained three fluid reservoirs that were used to control a pressure-driven flow through the channels (Fig. 5). A frequency histogram of the shear rate between the corner and the mid-point of the junction for a simulated power-law fluid with n = 0.8 and K = 0.025 Pa sn is shown in Fig. 6 for a pressure drop across the device of 100 Pa. Bandalusena et al. [38] showed that the rheological parameters of a power law fluid could be uniquely determined by measurements of the velocity field or pressure field within the T-junction and the inlet flow rate. However, the range of shear rates produced was restricted by the pressure drop across the device. The next stage in the development was to design an experiment in which the desired shear range could be achieved through precise control of the flow rate. It was then possible to infer K and n by applying the inverse methodology to measurements of the mean pressure at the inlet and at the junction. By performing a creeping flow analysis, i.e. making the assumption that the Reynolds number is sufficiently small that inertial effect can be neglected, the pressure drop, ΔP across the T-junction can be expressed as ΔP = KQnL30 f(a,n), where L0 is a typical length scale (such as the channel depth) and α represents the aspect ratio of the flow domain. Assuming that a Hagen–Poiseuille relationship exists
Fig. 4. Schematic of T-junction channel showing location of hypothetical pressure sensors, P1 and P2.
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0.48
P2/P1
0.44
0.4
0.36
0.32
0.4
0.6
0.8
1
n Fig. 7. Plot of P2/P1 plotted against n which is independent of the flow rate Q. Fig. 5. Prototype microscale rheometer suitable for use with nano- to micro-liter sample volumes with moderate viscosity, for instance, rare fluids such as protein mixtures.
between the pressure drop and the volumetric flow rate, since f is dimensionless, an algorithm was derived that permits the inference of K and n from the volumetric flow rate, the pressure drop and the ratio of any two field variables (e.g. the ratio of the mean pressure fields in the vicinity of the inlet and at the junction itself) measured over an averaged region. Under conditions for creeping flow, it is easily demonstrated that the consistency index K, which is dimensional, cannot influence any dimensionless quantity. Therefore, it was deduced that the dimensionless field variables are a function of n only. By performing finite element numerical simulations of the flow of a power-law fluid through the T-junction channels, simulated mean pressure measurements were calculated from hypothetical pressure sensors located at the inlet (P1) and at the center of the junction (P2) Fig. 4. The parametric relationship r = r(n) = P2/P1 was measured, and its inverse, n = r−1(r) was calculated. Linear regression showed that this curve was well-approximated by a quartic fit of the form
∇ ∇ T þ λ1 T ¼ 2μ 0 D þ λ2 D ;
ð9Þ
where • μ0 = μ1 + μ2 is the total fluid viscosity, λ1 is the relaxation time, μ2 is the retardation time, μ0 n o ∂T • T∇ ¼ þ U⋅∇T− ð∇UÞT ⋅T þ T⋅ð∇UÞ is the upper convected time ∂t derivative of the stress tensor (also known as the Oldroyd derivative). • λ2 ¼
0.08
The upper convected derivative is widely used in polymer rheology to describe the behavior of a viscoelastic fluid undergoing large deformation. Figs. 8 to 10 show the results of a finite element simulation of an Oldroyd-B fluid flowing in a T-junction channel for Weissenberg numbers, We = 0, We = 0.8 and We = 1.6. Due to the line of symmetry, only half of the domain is shown, as this saved on computational time. The simulations were all non-dimensionalized, thus the actual scales of the channel width and length are relative. The plots show the velocity field, indicated by scaled arrows, and a greyscale depicts the Frobenius norm of the viscoelastic stress, ‖T‖2. It can be seen that the back wall stress is particularly sensitive to the parameter We. This suggests that statistical measures of the flow field might map to the constitutive parameters in this model. This is a topic of current research. The inverse method based on the T-junction geometry has been found to invert uniquely in all cases that we have studied. We have yet to find a system so nonlinear that it cannot be uniquely inverted. The only question is sensitivity. For power law fluids, sensitivity was found to be within 3% for the flow behavior index which can be attributed to irregularities in the fabrication of the channel geometry. If the wrong constitutive model is chosen, the inversion is poor. However, this is still potentially useful for practical purposes such as control. For an unknown fluid an expert system would test a whole range of models and choose the best performing, which can be assessed from the standard error of fitting several experiments.
0.06
5. Discussion and conclusions
4
3
2
r ¼ 0:3285n −1:1405n þ 1:4808n −1:0067n þ 0:6708:
ð8Þ
This result is plotted in Fig. 7 for a broad range of values of n for which the function is single valued. Since Eq. (8) is quartic, upon algebraic manipulation a closed form solution for n can be found. The Oldroyd-B constitutive model provides a good description for viscoelastic fluids in shear flow [41]. However this model possesses
0.18 0.16 0.14
relative frequency
an unphysical singularity in extensional flow. In the Oldroyd-B model the stress tensor T takes the form
0.12 0.1
0.04 0.02 0
25 75 125 175 225 275 325 375 425 475 525 575 625 675
shear rate (s−1) Fig. 6. Shear rate distribution for a power law fluid with n = 0.8 and K = 0.025 Pa sn from numerical simulations in a T-junction channel network for a pressure drop of 100 Pa and a channel width of 260 μm.
This paper has followed a broadly historical path, beginning with an overview of the melt flow index that is widely used for monitoring polymers despite issues concerning its reliability, and progressing to give an insight into the new methodologies based on inverse techniques that promise much potential for full rheological characterization. The viscosity of a fluid is a measure of its resistance to flow. The more viscous the fluid, the greater is the force that is required to initiate flow. For Newtonian fluids such as water, the shear stress is proportional to the strain rate, the constant of proportionality being the
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Fig. 8. Non-dimensional velocity field (indicated by arrows) and viscoelastic stress tensor (greyscale) for Oldroyd-B fluid with Weissenberg number, We = 0.
Fig. 9. Non-dimensional velocity field (indicated by arrows) and viscoelastic stress tensor (greyscale) for Oldroyd-B fluid with Weissenberg number, We = 0.8.
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Fig. 10. Non-dimensional velocity field (indicated by arrows) and viscoelastic stress tensor (greyscale) for Oldroyd-B fluid with Weissenberg number, We = 1.6.
viscosity. Liquids whose behavior deviates from this simple model are called complex, or non-Newtonian, fluids. Common examples include cornstarch, which is used as a thickening agent in liquid-based foods such as soups and sauces, and ketchup, which flows more easily when the bottle is shaken. This complex fluid behavior stems from molecular interactions that fluctuate when the fluid is in motion. The viscosities of complex fluids can depend on a number of parameters via the so-called ‘constitutive equation’. Knowledge of these parameters is known as ‘rheological characterization’ and usually involves complicated calculations and the use of expensive equipment. The equipment is expensive because of multiple moving parts that must be precisely machined and maintained. Rheological parameters can affect the appearance, texture or even the shelf-life stability of a complex fluid. Since most industrial and biological fluids, such as polymer solutions, oils, slurries and blood, exhibit non-Newtonian features, characterization of complex fluids is an essential activity in many science and engineering applications. Knowledge of the rheological characteristics in industrial applications enables optimal process control and hence leads to better quality products. The apparent viscosity of a fluid can be measured by a viscometer. However, a rheometer is needed to obtain the constitutive parameters. There does not currently exist a commercially available inline rheometer that is capable of providing continuous monitoring of rheological parameters in real-time, yet this is a much wished for device for use in many industrial flow processes. At present, in order to perform a rheological characterization, samples need to be taken off-line and analyzed in an expensive cone-and-plate rheometer, or similar device, which is time consuming and costly. Recent developments of rheometry based on the application of advanced mathematical algorithms (inverse methods) hold the potential to be exploited to create the next generation of commercial rheometers that could be operated inline to provide real-time monitoring of the constitutive parameters. This would be of enormous
benefit in a wide range of applications, including the food and beverage industries, oil and lubrication industries, high-tech printing and paints, cements, cosmetics and pharmaceuticals, floor coverings and household commodities (e.g. toothpaste, shampoo). The bio-fuel industry has been highlighted as a particular area of interest due to increased economic and ‘green’ pressures to find sources of energy that are sustainable (biomass derived or utilizing waste products) and that are relatively inexpensive to produce. Newly created technology now enables the production of biofuels more efficiently than was hitherto possible, thus furthering our motivation for exploring the rheology of bio-oils. Industry needs to respond to the greater variability in input feedstock specification, as well as material properties varying during processing, with better control systems. Thus there currently exists a strong driver for online, continuous rheometry. Acknowledgments I am grateful for the support of current and past members of the Rheology Group at The University of Sheffield: HCH Bandulasena, S Brittle, ST Chaffin, TJ Craven, P Fairclough, R Hodgkinson, D Kuvshinov, E Kuvshinova, H Yu and WB Zimmerman. In particular, I would like to thank ST Chaffin and HCH Bandulasena for their assistance with some of the figures. I acknowledge also, Professor Manuel G Velarde, to whom this issue is dedicated, for introducing me to the exciting research area of interfacial dynamics, and its interaction with non-Newtonian behaviors. References [1] Cho H-M, Yoo W, Yoo B. Food Sci Biotechnol 2012;21:1775. [2] Koop HS, Da-Iozzo EJ, de Freitas RA, Franco CRC, Mitchell DA, Silviera JLM. J Pharm Sci 2012;101:2457. [3] Stokes JR, Macakova MR, Chojnicka-Paszun A, de Kruif CG, de Jongh HHJ. Langmuir 2011;27:3474.
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