Advanced rheological analysis methods

As mentioned in Chapter 1, people who adhere to the academic school of thought for viscosity measurements have more complex needs than those who adhere to the pragmatic and theoretical schools. They need viscometric data defined in rheological terms. This generally requires a complete mathematical description of the measurement parameters of the viscometer and an analysis of the rheological behavior of the fluid studied.

The previous chapters have described the different types of fluid behavior and their relationship to measurements made with Brookfield viscometers and their accessories. The appendices describe the significant measurement parameters of this equipment and propose simplified formulas for calculating the shear rates and shear stress.

However, for many, this information is still inadequate to achieve the type of analysis they want to perform. After identifying a particular type of flow and defining it mathematically, these users need more information to understand how the fluid reacts in a given situation and how to control that reaction. It is for these people that this chapter was written.

Here you will find the formulas from which derive the simplified information on shear rates and shear stress provided in the appendices. Different methods of analyzing Newtonian and non-Newtonian fluids are also presented. The information provided here represents the junction point of many of the useful methods developed by Brookfield Engineering Laboratories or others. Other specific methods, generally adapted to a particular rheological problem, are sometimes applicable. Do not hesitate to consult Labomat Essor if you wish to obtain more information.

Define the measurement parameters for the various mobile geometries

This chapter presents the equations making it possible to define the operating parameters concerning the geometries of moving parts encountered with Brookfield viscometers and their accessories. Definitions and values that are not presented can be found in Appendix A.

Brookfield cylindrical mobiles

The following equations apply to cylindrical mobiles only when used without a caliper and are valid for all models of Brookfield viscometers.

Note: Rc should not be greater than 2xRb to obtain well defined shear rates.

Brookfield Coaxial Cylinders

The “coaxial cylinder” geometry can be found with the ULA and SSA accessories, the Thermosel system, the DIN adapter, the spiral adapter and the PVS rheometer.

Cone / plane geometry

These equations can be used with all models of Wells / Brookfield cone / plane viscometers and CAP viscometers

Disc-shaped and T-shaped needles

The standard disc-shaped mobiles supplied with most viscometer models and the T-shaped mobiles of the helipath accessory, as well as the mobiles with special shapes other than cylindrical or conical shapes, do not have equations allowing to define them sometimes that the speed of rotation of the mobile is assimilated to the shear rate, particularly when T-shaped mobiles are used. This is incorrect because mathematical models do not exist to represent the viscosimetric functions of T-mobiles. However, computational models are available for disc-shaped mobiles. Refer to technical document AR-82 available from Brookfield Engineering Laboratories.

Spiral needle adapter

The spiral adapter is composed of a screw-shaped mobile surrounded by a concentric cylinder. This combination allows the product to be pumped continuously through the adapter. The fluid reaches a stabilized flow state during which viscosity can be measured. The approximate shear rate is 0.667xN (in s-1), where N represents the rotational speed of the mobile (RPM).

"Pale" mobiles for pasty products

The Brookfield KU-1 + viscometer uses a blade-shaped spindle to measure product-induced spring torsion at a spindle rotational speed of 200 RPM. Unlike conventional mobile viscometers, the resulting viscosity is expressed in Krebs units (KU) or in grams (g). Due to the particular shape of the mobile, no calculation of the shear rate is possible. A mobile for pasty product is available as an option. Of the same shape, it consists of two side bars of cylindrical shape. The resulting viscosity is expressed in grams. It is applicable for materials of very high consistency

Other special mobiles

Brookfield can manufacture special mobiles on request. This activity is organized with the cooperation of the sales department of Labomat Essor. Please contact our services

Analysis of non-Newtonian fluids not dependent on time

The equations we presented above provide perfectly defined viscosity data for both Newtonian and non-Newtonian fluids. For Newtonian fluids, this simple analysis is sufficient because variations in shear rate will have no effect on the viscosity of the fluid. However, when the material is non-Newtonian, the situation is more complicated. While the equations give a precise definition of the measurement made with a given mobile at a given speed, the figures obtained with another mobile and / or another speed will probably be different. What will be the right values? All and none. These different figures form part of the rheological description of the fluid and must therefore be considered for its analysis. In this section, we describe several methods of analyzing products whose viscosity is not time dependent.

Ratio methods

A common method to characterize and quantify non-Newtonian flow is to define the viscosity ratio of the fluid measured at two different speeds (with the same spindle). These measurements are generally carried out at speeds which differ by a factor of 10 (for example, 2 and 20 RPM, 10 and 100 RPM, etc.) but any other factor can be considered. When making the report, the viscosity measured at the lowest speed should be placed at the numerator and that measured at the highest speed at the denominator. Thus, for shear-thinning fluids, the ratio will be greater than 1 and will be all the more so as the degree of shear fluidity increases. Conversely, for shearthickening fluids, the ratio will be less than 1 and will be all the more so the greater the dilating behavior. This ratio is often referred to as the "thixotropy index". This name is misleading because this ratio quantifies the non-Newtonian behavior not dependent on time and not the thixotropy which is a phenomenon dependent on time. The analysis of time dependent properties is detailed in section 5.4. A similar method eliminates the calculation of viscosity and simply uses the percentage twist measurements from the internal spring to define what is called the "viscosity ratio":

Graphic methods

The most basic graphical method for the analysis of non-Newtonian fluids consists in constructing a curve of viscosity measurements as a function of the speed of rotation of the mobile (using the same mobile for all the measurements). Typically, viscosity is positioned on the Y axis and velocity (RPM) on the X axis. The slope and shape of the resulting curve indicate the type and degree of flow behavior. To see examples of this type of curve, see the illustrations in section 4.4 dealing with the different types of non-Newtonian behavior. Another method is to make a graph by positioning the viscosity on the X axis and the velocity on the Y axis. If the graph is drawn on log-log paper, the result is often a straight line. When this is the case, the slope of the curve (which indicates the type and degree of non-Newtonian behavior) and the point of intersection of the line with the x-axis (indicating the yield point, s 'there is) can be used as empirical constants. When the shear rates and stresses are known, as is the case for cylindrical spindles and for coaxial geometry, these values can substitute for speed and viscosity measurement in the methods described above. Thus, the predictions of viscosity values at other shear rates can be made by interpolation, or by extrapolation for values not accessible with the particular geometry of the mobile. When using these methods with disc-shaped mobiles, it is best to plot the speed on the Y axis and perform all measurements with the same mobile. It is perfectly plausible to assume that for a given moving body, the shear rate is proportional to the speed of rotation. Thus, the shear rate at 30 RPM for example is ten times greater than the shear rate at 3 RPM.

Template method

A more sophisticated technique for the analysis of non-Newtonian fluids involves the use of "jigs". Its use is limited to fluids which meet the Power Law, i.e. those which exhibit only one non-Newtonian behavior and not those which change from one behavior to another when the range of shear varies. For example, a material which changes from shear thinning behavior to shear thickening behavior as soon as a certain shear rate threshold is exceeded does not follow the power law over the full range of shear rate measured.The template method can only be used for measurements carried out using cylindrical mobiles or coaxial cylinders. The data is positioned on a template to determine a constant called STI. The STI is a convenient way to characterize non-Newtonian behavior, as is the viscosity index. Some parameters of the used viscometer and the STI are then positioned on a second jig which is then used to predict the viscosity of the fluid at other shear rate values. It is a useful method for predicting viscosity at shear rates not accessible with Brookfield viscometers and for characterizing the behavior of fluids under particular conditions. A complete description of the template method, including the two templates, is described in technical bulletin AR-49 available from Labomat Essor

Determination of the yield point

Some fluids behave like solids when the shear is zero. They will not flow out until some force is applied to them. This force is called the “yield point” and measuring it is very often very useful. Threshold values can be used to determine whether a pump has sufficient power to move a fluid and often correlate with other properties for suspensions and emulsions. The difficulty in pouring a liquid is directly related to its flow threshold. A simple method of determining the yield point is to calculate the "Brookfield cutoff value" using the ratio:

With this method, Newtonian fluids will exhibit a threshold value of zero, while plastic fluids will exhibit a threshold value all the greater as the predictable viscosity at zero shear is important. A more accurate method of determining yield point involves plotting a viscosity (x-axis), / speed (Y-axis) curve. The line thus obtained is extrapolated to zero speed. The corresponding viscosity measurement value represents the threshold value. If cylindrical mobiles are used to make the measurement, the threshold value can be calculated using the following equation:

Extrapolating the line down to 0 RPM is easy when the line is linear. We call this behavior a Bingham flow. If the line is curved, as with shear thinning or shear thickening behavior, an estimate of x1 must be made by continuing the curve until the intersection of the X axis (0 on the Y axis). This estimated value of x1 is then subtracted from all other readings that make up the graph. These new values are positioned on a log-log paper, viscosity as a function of speed. This graph will very often be a straight line for fluids meeting the power law if the value of x1 has been accurately evaluated. A curved line indicates that a new estimate of x1 should be made. Once a straight line is obtained, the angle that this line forms with the Y axis (RPM) is measured. The power law index of this fluid can then be calculated from the following equation:

Another method of determining the threshold value and the plastic viscosity when the graph of a series of viscosity measurements as a function of the speed of rotation does not give a straight line, consists in positioning on a graph the square root of the shear stress as a function of the square root of the shear rate. This very often straightens the line and facilitates extrapolation to 0 RPM. This method is more particularly applicable to shear-thinning fluids having a threshold value in accordance with the behavior model known under the name of Casson behavior. You will find more information in technical bulletins AR-77 and AR-79 available from Labomat Essor.

Time-dependent analysis of non-Newtonian fluids

In most cases, the analysis of thixotropic and / or anti-thixotropic fluids consists in plotting the curve of change of the viscosity values as a function of time. The simplest method is to select a moving body and a speed (preferably a low speed) and to let the viscometer run for a while, noting the viscosity values at regular intervals.

It is important to carefully control the temperature of the fluid so that no temperature variation will affect the results. A change in material viscosity over time indicates time-dependent behavior: a decrease in viscosity means thixotropy, an increase, anti-thixotropy (or sometimes drying out of the material).

A second method consists in plotting the curve for measuring viscosity as a function of speed using a single mobile. Starting with the lowest speed, the viscosity measurement is carried out successively for each higher speed until the measurement scale is exceeded (100%). The curve of these measurements corresponds to the ascending curve. Without stopping the viscometer, the speed is successively reduced to the starting value and the viscosity is measured at each speed. These values correspond to the falling curve. It is best to leave enough time between each gear change. If the fluid is independent of time, the ascending curve and the descending curve will be confused. If they are not, the product is time dependent. The positioning of the two curves indicates the flow behavior. If the ascending curve indicates higher viscosity values than the descending curve, the fluid is thixotropic, otherwise it is anti-thixotropic.

An indication of the viscosity recovery time (return to the initial viscosity after shearing) can be obtained by stopping the viscometer at the end of the falling curve, waiting for a given period of time, then restarting the viscometer and taking a measurement. immediate. A more sophisticated approach is to calculate the "thixotropic fracture coefficient". This number quantifies the degree of thixotropy (or anti-thixotropy) that a material exhibits. Plot the viscosity versus logarithm of time curve with a unique mobile / speed combination taking measurements at regular intervals. This generally produces a straight line. Then apply the following equation:

The thixotropic behavior plot can sometimes be used to predict the freezing point of a fluid. One method is to plot the logarithm of viscosity as a function of time using a single moving / speed combination. If the resulting line shows a steep slope, gelation will be likely to occur. If the line curves and flattens, gelation is highly unlikely. Another technique is to plot the time as a function of the inverse of the viscosity reading (in %). In this method, the freezing point can be read from the curve where the curve reaches the viscosity reading of 100 %. Fluids that do not gel show an asymptotic curve at the Y axis.

Dependence of viscosity on temperature

The viscosity of most fluids decreases with increasing temperature. By measuring the viscosity at two temperatures (using the same mobile and the same speed), it is possible to predict the curve representing the temperature dependence of the viscosity of a product by applying the following equations:

Various methods

There are many other possible techniques for analyzing the rheological behaviors of fluids under a variety of conditions. We don't have enough space here for a more detailed study, but you can get more information by contacting Labomat Essor on this subject:

• Evaluation of tension and shear rate values with the use of disc-shaped mobiles (AR - 82).
• Techniques for determining viscosity at very low shear rates using spring relaxation techniques (AR - 84).
• Computerized analysis of certain rheological characteristics

Mathematical models

The analysis of viscosity data can be improved by the use of mathematical models. Non-Newtonian behaviors can be expressed simply through an equation, and in some cases the parameters of the mathematical model can be used to infer the performance of a fluid under use. Newtonian flows are defined by a proportional response in shear stress for a change in shear rate (the relationship is linear). Non-Newtonian fluids exhibit a non-linear voltage / rate relationship. Newton's equation has been modified many times in an attempt to characterize non-Newtonian behavior.

The best known equations are the Bingham, Casson, NCA / CMA Casson equations and the power law equation.

as well as Herschel Buckley's law

The chocolate industry uses the NCA / CMA version of the Casson equation to rate chocolate before the final process. This equation closely approximates the plastic behavior of chocolate. Moreover, experience shows that the slope of the square root of the viscosity indicates the response of the chocolate during the movements of the manufacturing process (mixing, pumping). Likewise, the interception of the curve with the Y axis 2 square root of the yield point indicates the force necessary to initiate the flow (molding, coating). A particular batch of chocolate can be modified to achieve specific performance required for the next step in the manufacturing process. The drilling oil industry in the United States uses the power law equation to evaluate the performance of drilling muds and fracture fluid. The latter is a material that is injected into wells which are no longer operational in order to allow more oil to be recovered. The power law equation provides a very good approximation of the shear thinning behavior of this fluid. The consistency index k indicates the low shear flow behavior of the sludge when it is at the far end of the well. Fracture fluid can be modified in its storage container to achieve the proper flow characteristics for pumping into the well.

In the two cases described above, the efficient use of the mathematical model will make it possible to avoid the use of the wrong fluid and, in the end, to use unsuitable equipment or to refuse the product. The mathematical model should be used as a tool to better understand and interpret viscosity measurements. The use of mathematical models normally requires the collection of viscosity measurements under defined conditions of shear rate and stress. Many rover geometries providing accurate shear rate and stress data are available for your Brookfield viscometer.

Brookfield offers several software, some devices even have the intrinsic capabilities of data analysis according to a variety of mathematical models.