30 Chapter 3. Conventional Rheometry: State-Of-The

CHAPTER 3. CONVENTIONAL RHEOMETRY: STATE-OF-THE-ART

This chapter reviews literature on conventional rheometries. Section 3.1 briefly introduces conventional rheometers. In sections 3.2 and 3.3, viscometers commonly used for the viscosity measurements of fluids, which have been used for hemorheology studies, are demonstrated. Section 3.4 provides conventional methods of measuring yield stresses of fluids. Section 3.5 presents the drawbacks of conventional viscometers for clinical applications.

3.1. Introduction

Numerous types of rheometers have been used to measure the viscosity and yield stress of materials [Tanner, 1985; Ferguson and Kemblowski, 1991; Macosko,

1994]. In the present study, rheometer refers to a device that can measure both viscosity and yield stress of a material, whereas viscometer can measure only the viscosity of the material. In addition, only shear viscometers will be discussed in the study since the other type, extensional viscometers, are not very applicable to relatively low viscous fluids, such as water and whole blood.

Typically, shear viscometers can be divided into two groups [Macosko, 1994]: drag flows, in which shear is generated between a moving and a stationary solid surface, and pressure-driven flows, in which shear is generated by a pressure difference over a capillary tube. The commonly utilized members of these groups are 31 shown in Fig. 3-1. Numerous techniques have been developed for determining the yield stress of fluids both directly and indirectly.

Most of these viscometers can produce viscosity measurements at a specified, constant shear rate. Therefore, in order to measure the viscosity over a range of shear rates, one needs to repeat the measurement by varying either the pressure in the reservoir tank of capillary tube viscometers, the rotating speed of the cone or cup in rotating viscometers, or the density of the falling objects. Such operations make viscosity measurements difficult and labor intensive. In addition, these viscometers require anticoagulants in blood to prevent blood clotting. Hence, the viscosity results include the effects of anticoagulants, which may increase or decrease blood viscosity depending on the type of anticoagulant [Rosenblum, 1968; Crouch et al., 1986;

Reinhart et al., 1990; Kamaneva et al., 1994].

Drag-flow type of viscometers includes a falling object (ball or cylinder) viscometer and a rotational viscometer. However, the falling object viscometer is not very convenient to use for clinical applications. In the case of the falling object viscometer, the relatively large amount of a test fluid is required for the viscosity measurement. In addition, since the testing fluid is at a stationary state initially, the type of viscometer is not very applicable to a thixotropic fluid like whole blood. The principle of the falling object viscometer is provided in Appendix B.

For the yield measurement of blood, most researchers have used indirect methods rather than direct methods for practical reasons [Nguyen and Boger, 1983; de Kee et al., 1986; Magnin and Piau, 1990]. Thus, the details of direct methods will 32 not be discussed in this chapter. As indirect methods, data extrapolation and extrapolation using constitutive models are introduced and discussed in this chapter.

Rheometers

Viscosity Yield Stress Measurements Measurements

Drag Pressure- Indirect Direct Flows Driven Flows Methods Methods

Capillary- Data Extrapolation Tube Extrapolation using Viscometer Constitutive Models

Falling/ Rotational Rolling Viscometer Object Viscometer

Fig. 3-1. Rheometers.

3.2. Rotational Viscometer

In a rotational viscometer, the fluid sample is sheared as a result of the rotation of a cylinder or cone. The shearing occurs in a narrow gap between two surfaces, usually one rotating and the other stationary. Two frequently used geometries are Couette (Fig. 3-2) and cone-and-plate (Fig. 3-3).

3.2.1. Rotational Coaxial-Cylinder (Couette Type)

In a coaxial-cylinder system, the inner cylinder is often referred to as bob, and the external one as cup. The shear rate is determined by geometrical dimensions and the speed of rotation. The shear stress is calculated from the torque and the geometrical dimensions. By changing the speed of the rotating element, one is able to collect different torques, which are used for the determination of the shear stress- shear rate curve. Figure 3-2 shows a typical coaxial-cylinder system that has a fluid

R confined within a narrow gap ( i ≥ 0.99 ) between the inner cylinder rotating at Ω Ro and the stationary outer cylinder.

Once the torque exerting on either inner or outer cylinder is measured, the shear stress and shear rate can be calculated as follow [Macosko, 1994]:

M i M o τ (Ri ) = 2 or τ (Ro ) = 2 (3-1) 2πRi H 2πRo H 35

ΩR Ri γ&(Ri ) ≅ γ&(Ro ) = when 1 > ≥ 0.99 (3-2) Ro − Ri Ro where

Ri and Ro = radii of inner and outer cylinders, respectively

R + R R = i o 2

M i and M o = torques exerting on inner and outer cylinders, respectively

H = height of inner cylinder

Ω = angular velocity.

3.2.2. Cone-and-Plate

The common feature of a cone-and-plate viscometer is that the fluid is sheared between a flat plate and a cone with a low angle; see Fig. 3-3. The cone-and-plate system produces a flow in which the shear rate is very nearly uniform. Let’s consider a fluid, which is contained in the gap between a plate and a cone with an angle of β .

Typically, the gap angle, β , is very small ( ≤ 4o ). The shear rate of the fluid depends on the gap angle, β , and the linear speed of the plate. Assuming that the cone is stationary and the plate rotates with a constant angular velocity of Ω , the shear stress and shear rate can be calculated from experimentally measured torque, M , and given geometric dimensions (see Fig. 3-3) as follows [Macosko, 1994]:

3M Ω τ = and γ& = . (3-3) 2πR 3 β 36

Ω

H Ro

Fig. 3-2. Schematic diagram of a concentric cylinder viscometer.

Torque measurement device

Fluid R Cone

Plate

Ω

Fig. 3-3. Schematic diagram of a cone-and-plate viscometer.

3.3. Capillary-Tube Viscometer

The principle of a capillary tube viscometer is based on the Hagen-Poiseuille

Equation which is valid for Newtonian fluids. Basically, one needs to measure both pressure drop and flow rate independently in order to measure the viscosity with the capillary tube viscometer. Since the viscosity of a Newtonian fluid does not vary with flow or shear, one needs to have one measurement at any flow velocity.

However, for non-Newtonian fluids, it is more complicated because the viscosity varies with flow velocity (or shear rate).

In a capillary-tube viscometer, the fluid is forced through a cylindrical capillary tube with a smooth inner surface. The flow parameters have to be chosen in such a way that the flow may be regarded as steady-state, isothermal, and laminar.

Knowing the dimensions of the capillary tube (i.e., its inner diameter and length), one can determine the functional dependence between the volumetric flow rate and the pressure drop due to friction. If the measurements are carried out so that it is possible to establish this dependence for various values of pressure drop or flow rate, then one is able to determine the flow curve of the fluid.

For non-Newtonian fluids, since the viscosity varies with shear rate, one needs to vary the pressure in the reservoir in order to change the shear rate, a procedure that is highly time-consuming. After each run, the reservoir pressure should be reset to a new value to obtain the relation between flow rate and pressure drop. In order to determine the flow curve of a non-Newtonian fluid, one needs to establish the functional dependence of shear stress on shear rate in a wide range of these variables. 39

Figure 3-4 shows the schematic diagram of a typical capillary-tube viscometer,

which has the capillary tube with an inner radius of Rc and a length of Lc . It is assumed that the ratio of the capillary length to its inner radius is so large that one may neglect the so-called end effects occurring in the entrance and exit regions of the capillary tube. Then, the shear stress at the tube wall can be obtained as follows:

r∆P τ = c (3-4) 2Lc

Rc ∆Pc τ w = (3-5) 2Lc where

τ and τ w = shear stresses at distance r and at tube wall, respectively

r = distance from the capillary axis

∆Pc = pressure drop across a capillary tube.

It is of note that the shear stress distribution is valid for fluids of any rheological properties.

In the case of a Newtonian fluid, the shear rate at tube wall can be expressed by taking advantage of the well-known Hagen-Poiseuille Equation as:

4Q 4V γ&w = 3 = (3-6) πRc Rc where

γ&w = wall shear rate

4 π Rc ∆Pc 2 Q = = πRc ⋅V = volumetric flow rate (Hagen-Poiseuille Equation) 8µ Lc

V = mean velocity. 40

Compressed air

Air

Test fluid

Reservoir tank Capillary tube

Lc 2Rc

Collected test fluid

Balance

Fig. 3-4. Schematic diagram of a capillary-tube viscometer.

3.4. Yield Stress Measurement

Whether yield stress is a true material property or not is still a controversial issue [Barnes and Walters, 1985]. However, there is generally an acceptance of its practical usefulness in engineering design and operation of processes where handling and transport of industrial suspensions are involved. The minimum pump pressure required to start a slurry pipeline, the leveling and holding ability of paint, and the entrapment of air in thick pastes are typical problems where the knowledge of the yield stress is essential.

Numerous techniques have been developed for determining the yield stress both directly and indirectly based on the general definition of the yield stress as the stress limit between flow and non-flow conditions. Indirect methods simply involve the extrapolation of shear stress-shear rate data to zero shear rate with or without the help of a rheological model. Direct measurements generally rely on some independent assessment of yield stress as the critical shear stress at which the fluid yields or starts to flow.

The value obtained by the extrapolation of a flow curve is known as

“extrapolated” or “apparent” yield stress, whereas yield stress measured directly, usually under a near static condition, is termed “static” or “true” yield value.

3.4.1. Indirect Method

Indirect determination of the yield stress simply involves the extrapolation of experimental shear stress-shear rate data at zero shear rate (see Fig. 3-5). The extrapolation may be performed graphically or numerically, or can be fitted to a suitable rheological model representing the fluid and the yield stress parameter in the model is determined.

3.4.1.1. Direct Data Extrapolation

One of most common procedures is to extend the flow curve at low shear rates to zero shear rate, and take the shear stress intercept as the yield stress value. The technique is relatively straightforward only if the shear stress-shear rate data are linear. With nonlinear flow curves, as shown in Fig. 3-5, the data may have to be fitted to a polynomial equation followed by the extrapolation of the resulting curve fit to zero shear rate. The yield stress value obtained obviously depends on the lowest shear rate data available and used in the extrapolation. This shear rate dependence of the extrapolated yield stress has been demonstrated by Barnes and Walters (1985) with a well-known yield stress fluid, Carbopol (carboxylpolymethylene). They concluded that this fluid would have no detectable yield stress even if measurement was made at very low shear rates of 10-5 s-1 or less. This finding should be viewed with caution, however, since virtually all viscometric instruments suffer wall slip and 43 other defects which tend to be more pronounced at low shear rates especially with yield stress fluids and particulate systems [Wildermuth and Williams, 1985; Magnin and Piau, 1990]. Thus, it is imperative that some checking procedure should be carried out to ascertain the reliability of the low shear rate data before extrapolation is made.

3.4.1.2. Extrapolation Using Constitutive Models

A more convenient extrapolation technique is to approximate the experimental data with one of the viscoplastic flow models. Many workers appear to prefer the

Bingham model which postulates a linear relationship between shear stress and shear rate. However, since a large number of yield stress fluids including suspensions are not Bingham plastic except at very high shear rates, the use of the Bingham plastic model can lead to unnecessary overprediction of the yield stress as shown in Fig. 3-5

[Nguyen and Boger, 1983; de Kee et al., 1986]. Extrapolation by means of nonlinear

1 1 Casson model can be used from a linear plot of τ 2 versus γ& 2 . The application of

Herschel-Bulkley model is less certain although systematic procedures for determining the yield stress value and the other model parameters are available

[Heywood and Cheng, 1984].

Even with the most suitable model and appropriate technique, the yield stress value obtained cannot be regarded as an absolute material property because its accuracy depends on the model used and the range and reliability of the experimental 44 data available. Several studies have shown that a given fluid can be described equally well by more than one model and hence can have different yield stress values

[Keentok, 1982; Nguyen and Boger, 1983; Uhlherr, 1986].

3.4.2. Direct Method

Various techniques have been introduced for measuring the yield stress directly and independently of shear stress-shear rate data. Although the general principle of the yield stress as the stress limit between flow and non-flow conditions is often used, the specific criterion employed for defining the yield stress seems to vary among these techniques. Furthermore, each technique appears to have its own limitations and sensitivity so that no single technique can be considered versatile or accurate enough to cover the whole range of yield stress and fluid characteristics.

Usually, the direct methods are used for fluids having yield stresses of greater than approximately 10 Pa [Nguyen and Boger, 1983]. Therefore, as mentioned earlier, the direct method is not very convenient to use for the yield stress measurement of blood since the yield stress of human blood is approximately 1 to 30 mPa [Picart et al.,

1998].

Fig. 3-5. Determination of yield stress by extrapolation [Nguyen and Boger, 1983].

3.5. Problems with Conventional Viscometers for Clinical Applications

3.5.1. Problems with Rotational Viscometers

Over the years, rotational viscometers have been the standard in clinical studies investigating rheological properties of blood and other body fluids. Despite their popularity, rotational viscometers have some drawbacks that limit their clinical applicability in measuring whole blood viscosity. They include the need to calibrate a torque-measuring sensor, handling of blood, surface tensions effects, and the range of reliability.

The torque-measuring sensor can be a conventional spring or a more sophisticated electronic transducer. In either case, the sensor requires a periodic calibration because repeated use of the sensor can alter its spring constant. The calibration procedure is often carried out at manufacturer’s laboratory because it requires an extremely careful and elaborate protocol, requiring the viscometer unit to be returned for service.

Another concern is the need to work with contaminated blood specimens.

After each measurement, the blood sample must be removed from the test section, and the test section must be cleaned manually. Not only is this procedure time- consuming, but also it poses a potential risk for contact with contaminated blood.

Surface tension effects arise in the use of the coaxial-cylinder viscometer because surface tension is relatively high for blood and macromolecular solutions.

The contact area between the blood and an inner cylinder is not uniform along the 47 periphery. The bob (inner cylinder) is pulled in different directions and revealed in fluctuating torque readings, introducing serious errors in viscosity measurement.

Another inherent difficulty in measuring whole blood viscosity using rotational viscometers is the limited shear rate range. In the extremes of the reputed range (whether high shear or low shear, depending on the instrument), the detected torque values do not have sufficient accuracy. Usually, manufacturers recommend discarding viscosity data if the torque is less than 10% of the maximum value of the sensor. This restriction is a major concern. For example, in the case of Brookfield rotational viscometer, the minimum shear rate is often limited at approximately 30-50 s-1 due to the 10% restriction.

There are other clinical, practical considerations in using the rotational viscometer. For example, it is usually necessary to treat the blood sample with a measurable amount of anticoagulant, such as ethylenediaminetetraacetic acid (EDTA) or heparin, to prevent coagulation during viscosity measurements. The reason for this is that the contact area among blood, rotational viscometer component, and air is relatively large for the size of the blood sample, and it usually takes a relatively long time to complete viscosity measurements over a range of shear rates. Treating blood with such anticoagulants results in an altered sample, and subsequent viscosity measurements do not reflect the intrinsic values of unadulterated blood.

3.5.2. Problems with Capillary-Tube Viscometers

There are some drawbacks in the use of conventional capillary-tube viscometers for clinical applications. The range of shear rate is limited to high shears over 100 s-1. Although one can produce viscosity data at lower shear rates below 100 s-1 with a sophisticated vacuum system, the capillary tube system is basically designed and operated to obtain viscosity at the high shear range. Since it is essential to obtain blood viscosity at low shear rates below 10 s-1, the traditional capillary tube viscometer is not suitable for measuring the viscosity at low shear rates. However, capillary-tube viscometer is simple in its design and uses gravity field to drive test fluid such that there is no need for calibration.

It takes a relatively long time to complete viscosity measurements over a range of shear rates because at each shear rate, a sufficient quantity of a fluid sample must be collected for an accurate measurement of flow velocity. After the measurement at one shear rate, the pressure at the reservoir tank must be readjusted to either increase or decrease shear rate. Then, the next shear rate case resumes. Thus, anticoagulants must be added to whole blood for the viscosity measurement over a range of shear rates.