# Introduction to Rheology

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A Basic Introduction to Rheology

RHEOLOGY AND VISCOSITY

Introduction
Rheometry refers to the experimental technique used to determine the rheological properties of materials; rheology being defined as the study of the flow and deformation of matter which describes the interrelation between force, deformation and time. The term rheology originates from the Greek words ‘rheo’ translating as ‘flow’ and ‘logia’ meaning ‘the study of’, although as from the definition above, rheology is as much about the deformation of solid-like materials as it is about the flow of liquid-like materials and in particular deals with the behavior of complex viscoelastic materials that show properties of both solids and liquids in response to force, deformation and time.
There are a number of rheometric tests that can be performed on a rheometer to determine flow properties and viscoelastic properties of a material and it is often useful to deal with them separately. Hence for the first part of this introduction the focus will be on flow and viscosity and the tests that can be used to measure and describe the flow behavior of both simple and complex fluids. In the second part deformation and viscoelasticity will be discussed.
Viscosity
There are two basic types of flow, these being shear flow and extensional flow. In shear flow fluid components shear past one another while in extensional flow fluid component flowing away or towards from one other. The most common flow behavior and one that is most easily measured on a rotational rheometer or viscometer is shear flow and this viscosity introduction will focus on this behavior and how to measure it.

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Shear Flow
Shear flow can be depicted as layers of fluid sliding over one another with each layer moving faster than the one beneath it. The uppermost layer has maximum velocity while the bottom layer is stationary. For shear flow to take place a shear force must act on the fluid. This external force takes the form of a shear stress (σ) which is defined as the force (F) acting over a unit area (A) as shown in Figure 1. In response to this force the upper layer will move a given distance x, while the bottom layer remains stationary. Hence we have a displacement gradient across the sample (x/h) termed the shear strain (γ). For a solid which behaves like a single block of material, the strain will be finite for an applied stress – no flow is possible. However, for a fluid where the constituent components can move relative to one another, the shear strain will continue to increase for the period of applied stress. This creates a velocity gradient termed the shear rate or strain rate ( ) which is the rate of change of strain with time (dγ/dt).

Figure 1 – Quantification of shear rate and shear stress for layers of fluid sliding over one another
When we apply a shear stress to a fluid we are transferring momentum, indeed the shear stress is equivalent to the momentum flux or rate of momentum transfer to the upper layer of fluid. That momentum is transferred through the layers of fluid by collisions and interactions with other fluid components giving a reduction in fluid velocity and kinetic energy. The coefficient of proportionality between the shear stress and shear rate is defined as the shear viscosity or dynamic viscosity (η), which is a quantitative measure of the internal fluid friction and associated with damping or loss of kinetic energy in the system.

Newtonian fluids are fluids in which the shear stress is linearly related to the shear rate and hence the viscosity is invariable with shear rate or shear stress. Typical Newtonian fluids include water, simple hydrocarbons and dilute colloidal dispersions. Non-Newtonian fluids are those where the viscosity varies as a function of the applied shear rate or shear stress. It should be noted that fluid viscosity is both pressure and temperature dependent, with viscosity generally increasing with increased pressure and decreasing temperature. Temperature is more critical than pressure in this regard with higher viscosity fluids such as asphalt or bitumen much more temperature dependent than low viscosity fluids such as water.

To measure shear viscosity using a single head (stress controlled) rotational rheometer with parallel plate measuring systems, the sample is loaded between the plates at a known gap (h) as shown in Figure 2. Single head rheometers are capable of working in controlled stress or controlled rate mode which means it is

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possible to apply a torque and measure the rotational speed or alternatively apply a rotational speed and measure the torque required to maintain that speed. In controlled stress mode a torque is requested from the motor which translates to a force (F) acting over the surface area of the plate (A) to give a shear stress (F/A). In response to an applied shear stress a liquid like sample will flow with a shear rate dependent on its viscosity. If the measurement gap (h) is accurately known then the shear rate (V/h) can be determined from the measured angular velocity (ω) of the upper plate, which is determined by high precision position sensors, and its radius (r), since V = r ω. Other measuring systems including cone-plate and concentric cylinders are commonly used for measuring viscosity with coneplate often preferred since shear rate is constant across the sample. The type of measuring system used and its dimensions is dependent on the sample type and its viscosity. For example, when working with large particle suspensions a coneplate system is often not suitable.

Figure 2 – Illustration showing a sample loaded between parallel plates and shear profile generated across the gap
Shear thinning
The most common type of non-Newtonian behavior is shear thinning or pseudoplastic flow, in which the fluid viscosity decreases with increasing shear. At low enough shear rates, shear thinning fluids will show a constant viscosity value, η0, termed the zero shear viscosity or zero shear viscosity plateau. At a critical shear rate or shear stress, a large drop in viscosity is observed, which signifies the beginning of the shear thinning region. This shear thinning region can be mathematically described by a power law relationship which appears as a linear section when viewed on a double logarithmic scale (Figure 5), which is how rheological flow curves are often presented. At very high shear rates a second constant viscosity plateau is observed, called the infinite shear viscosity plateau. This is given the symbol η∞ and can be several orders of magnitude lower than η0 depending on the degree of shear thinning.
Some highly shear-thinning fluids also appear to have what is termed a yield stress, where below some critical stress the viscosity becomes infinite and hence characteristic of a solid. This type of flow response is known as plastic flow and is characterized by an ever increasing viscosity as the shear rate approaches zero (no visible plateau). Many prefer the description ‘apparent yield stress’ since some materials which appear to demonstrate yield stress behavior over a limited shear rate range may show a viscosity plateau at very low shear rates.

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Figure 3 - Typical flow curves for shear thinning fluids with a zero shear viscosity and an apparent yield stress
Why does shear thinning occur? Shear thinning is the result of micro-structural rearrangements occurring in the plane of applied shear and is commonly observed for dispersions, including emulsions and suspensions, as well as polymer solutions and melts. An illustration of the types of shear induced orientation which can occur for various shear thinning materials is shown in Figure 4.

Figure 4 - Illustration showing how different microstructures might respond to the application of
shear
At low shear rates materials tend to maintain an irregular order with a high zero shear viscosity (η0) resulting from particle/molecular interactions and the restorative effects of Brownian motion. In the case of yield stress materials such interactions result in network formation or jamming of dispersed elements which must be broken or unjammed for the material to flow. At shear rates or stresses high enough to overcome these effects, particles can rearrange or reorganize in to string-like layers, polymers can stretch out and align with the flow, aggregated structures can be broken down and droplets deformed from their spherical shape. A consequence of these rearrangements is a decrease in molecular/particle interaction and an increase in free space between dispersed components, which both contribute to the large drop in viscosity. η∞ is associated with the maximum degree of orientation achievable and hence the minimum

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attainable viscosity and is influenced largely by the solvent viscosity and related hydrodynamic forces.
Model fitting
The features of the flow curves shown in Figure 3 can be adequately modeled using some relatively straight forward equations. The benefits of such an approach are that it is possible to describe the shape and curvature of a flow curve through a relatively small number of fitting parameters and to predict behavior at unmeasured shear rates (although caution is needed when using extrapolated data). Three of the most common models for fitting flow curves are the Cross, Power law and Sisko models. The most applicable model largely depends on the range of the measured data or the region of the curve you would like to model (Figure 5). There are a number of other models available such as the Carreau-Yasuda model and Ellis models for example. Other models accommodate the presence of a yield stress, these include Casson, Bingham, and HerschelBulkley models.

η0 is the zero shear viscosity; η∞ is the infinite shear viscosity; K is the cross constant, which is indicative of the onset of shear thinning; m is the shear thinning index, which ranges from 0 (Newtonian) to 1 (Infinitely shear thinning); n is the power law index which is equal to (1 – m), and similarly related to the extent of shear thinning, but with n → 1 indicating a more Newtonian response; k is the consistency index which is numerically equal to the viscosity at 1 s-1.

Figure 5 – Illustration of a flow curve and the relevant models for describing its shape

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Shear thickening
While most suspensions and polymer structured materials are shear thinning, some materials can also show shear thickening behavior where viscosity increases with increasing shear rate or shear stress. This phenomenon is often called dilatancy, and although this refers to a specific mechanism for shear thickening associated with a volume increase, the terms are often used interchangeably.
In most cases, shear thickening occurs over a decade of shear rates and there can be a region of shear thinning at lower and higher shear rates. Usually dispersions or particulate suspensions with high concentration of solid particles exhibit shear thickening. Materials exhibiting shear thickening are much less common in industrial applications than shear thinning materials. They do have some useful applications such as in shock absorbers and high impact protective equipment but for the most part shear thickening is an unwanted effect which can lead to major processing issues.
For suspensions, shear thickening generally occurs in materials that show shear thinning at lower shear rates and stresses. At a critical shear stress or shear rate the organized flow regime responsible for shear thinning is disrupted and so called ‘hydro-cluster’ formation or ‘jamming’ can occur. This gives a transient solid-like response and an increase in the observed viscosity. Shear thickening can also occur in polymers, in particular amphiphilic polymers, which at high shear rates may open-up and stretch, exposing parts of the chain capable of forming transient intermolecular associations.
Thixotropy
For most liquids shear thinning is reversible and the liquids will eventually gain their original viscosity when the shearing force is removed. When this recovery process is sufficiently time dependent the fluid is considered to be thixotropic. Thixotropy is related to the time dependent microstructural rearrangements occurring in a shear thinning fluid following a step change in applied shear (Figure 6). A shear thinning material may be thixotropic but a thixotropic material will always be shear thinning. A good practical example of a thixotropic material is paint. A paint should be thick in the can when stored for long periods to prevent separation, but should thin down easily when stirred for a period time – hence it is shear thinning. Most often its structure does not rebuild instantaneously on ceasing stirring – it takes time for the structure and hence viscosity to rebuild to give sufficient working time.
Thixotropy is also critical for leveling of paint once it is applied to a substrate. Here the paint should have low enough viscosity at application shear rates to be evenly distributed with a roller or brush but once applied should recover its viscosity in a controlled manner. The recovery time should be short enough to prevent sagging but long enough for brush marks to dissipate and a level film to be formed. Thixotropy also affects how thick a material will appear after it has been processed at a given shear rate, which may influence customer perception, or whether a dispersion is prone to separation and/or sedimentation after high shear mixing for example.

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Figure 6 - Illustration showing microstructural changes occurring in a dispersion of irregularly
shaped particles in response to variable shear
The best way to evaluate and quantity thixotropy is using a three step shear test as shown in Figure 7. A low shear rate is employed in stage one which is meant to replicate the samples at near rest behavior. In stage two a high shear rate is applied for a given time to replicate the breakdown of the sample's structure and can be matched to the process of interest. In the third stage the shear rate is again dropped to a value generally equivalent to that employed in stage one and viscosity recovery followed as a function of time. To compare thixotropic behavior between samples the time required to recover 90% (or a defined amount) of the initial viscosity can be used. This time can therefore be viewed as a relative measure of thixotropy - a small rebuild time indicates that the sample is less thixotropic than a sample with a long rebuild time.

Figure 7 - Illustration showing a step shear rate test for evaluating thixotropy and expected response for non-thixotropic and thixotropic fluids
As well as monitoring viscosity recovery following application of high shear, it is also possible to work in oscillatory mode either side of an applied shear rate step and therefore directly monitor changes in G’ (elastic structure) with time. See the section on viscoelasticity for more details on this test mode.

Yield Stress
Many shear thinning fluids can be considered to possess both liquid and solid like properties. At rest these fluids are able to form intermolecular or interparticle networks as a result of polymer entanglements, particle association, or some other interaction. The presence of a network structure gives the material predominantly solid like characteristics associated with elasticity, the strength of which is directly related to the intermolecular or interparticle forces (binding force) holding the network together, which is associated with the yield stress.
If an external stress is applied which is less than the yield stress the material will deform elastically. However, when the external stress exceeds the yield stress the network structure will collapse and the material will begin to flow as if it is a liquid. Despite yield stress clearly being apparent in a range of daily activities

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such as squeezing toothpaste from a tube or dispensing ketchup from a bottle, the concept of a true yield stress is still a topic of much debate. While a glassy liquid and an entangled polymer system will behave like a solid when deformed rapidly, at longer deformation times these materials show properties of a liquid and hence do not possess a true yield stress. For this reason the term 'apparent yield stress' is widely used. Figure 8 shows a plot of shear stress against shear rate for various fluid types. Materials which behave like fluids at rest will have curves that meet at the origin since any applied stress will induce a shear rate. For yield stress fluids the curves will intercept the stress axis at a non-zero value indicating that a shear rate can only be induced when the yield stress stress has been exceeded. A Bingham plastic is one that has a yield stress but shows Newtonian behavior after yielding. This idealized behavior is rarely seen and most materials with an apparent yield stress show non-Newtonian behavior after yielding which is generalized as plastic behavior.

Figure 8 – Shear stress/shear rate plots depicting various types of flow behavior
There are a number of experimental tests for determining yield stress, including multiple creep testing, oscillation amplitude sweep testing and also steady shear testing; the latter usually with the application of appropriate models such as the Bingham, Casson and Herschel-Bulkley models.

Where σY is the yield stress and ηB the Bingham viscosity, represented by the slope of shear stress versus shear rate in the Newtonian region, post yield. The Herschel-Bulkley model is just a power law model with a yield stress term and hence represents shear thinning post yield, with K the consistency and n the power law index. All of the various tests for measuring yield stress are discussed in [5].

One of the quickest and easiest methods for measuring the yield stress is to perform a shear stress ramp and determine the stress at which a viscosity peak is observed (Figure 9). Prior to this viscosity peak the material is undergoing elastic deformation where the sample is simply stretching. The peak in viscosity

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represents the point at which this elastic structure breaks down (yields) and the material starts to flow. If there is no peak this indicates that the material does not have a yield stress under the conditions of the test.
Yield stress can be related to the stand-up properties (slump) of a material, the stability of a suspension, or sagging of a film on a vertical surface, as well as many other applications.

Figure 9 – Linear shear stress ramp and shear strain response (left) and corresponding viscosity against shear stress for materials with and without a yield stress
Viscoelasticity
As the name suggests, viscoelastic behavior describes materials which show behavior somewhere between that of an ideal liquid (viscous) and ideal solid (elastic). There are a number of rheological techniques for probing the viscoelastic behavior of materials, including creep testing, stress relaxation and oscillatory testing. Since oscillatory shear rheometry is the primary technique that is used to measure viscoelasticity on a rotational rheometer this will be discussed in greatest detail, although creep testing will be also introduced.
Elastic behaviour
Structured fluids have a minimum (equilibrium) energy state associated with their ‘at rest’ microstructure. This state may relate to inter-entangled chains in a polymer solution, randomly ordered particles in a suspension, or jammed droplets in an emulsion. Applying a force or deformation to a structured fluid will shift the equilibrium away from this minimum energy state, creating an elastic force that tries to restore the microstructure to its initial state. This is analogous to a stretched spring trying to return to its undeformed state.

Figure 10 – The response of an ideal solid (spring) to the application and subsequent removal of a strain inducing force

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A spring is representative of a linear elastic solid that obeys Hooke’s law, in that the applied stress is proportional to the resultant strain as long as the elastic limit is not exceeded, and will return to its initial shape when the stress is removed as shown in Figure 10. If the elastic limit is surpassed the spring will be permanently distorted. These same principles can also be applied to simple shear deformation as illustrated in Figure 11.

Figure 11 – Quantification of stress, and strain for an ideal solid deforming elastically in shear
For simple shear elastic deformation the constant of proportionality is the elastic modulus (G). The elastic modulus is a measure of stiffness or resistance to deformation just as viscosity is a measure of the resistance to flow. For a purely elastic material there is no time dependence so when a stress is applied an immediate strain is observed and when the stress is removed the strain immediately disappears. This can be expressed as:
Viscous Behaviour
Just as a spring is considered representative of a linear elastic solid that obeys Hooke’s law, a viscous material can be modeled using a dashpot which obeys Newton’s law. A dashpot is mechanical device consisting of a plunger moving through a viscous Newtonian fluid.

Figure 12 – Response of an ideal liquid (dashpot) to the application and subsequent removal of a strain inducing force
When a stress (or force) is applied to a dashpot, the dashpot immediately starts to deform and goes on deforming at a constant rate (strain rate) until the stress is removed (Figure 12). The energy required for deformation or displacement is

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