  # The A-Z of Rheology & Viscosity - from Angles to Zero Shear Viscosity

## Viscosity & Rheology Glossary

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Angular Frequency

Oscillation frequency measured in radians per second.  Angular frequency is often used in preference to frequency in Hertz.  One particular benefit of this approach is that the result of an oscillatory frequency sweep, when plotted as complex viscosity v angular frequency, can be equated to a shear viscosity v shear rate profile for some materials.  This convenient equivalence is known as the Cox-Merz relationship.

Angle: Loss or Phase Angle

Apparent Viscosity

Unlike Newtonian liquids, which have a constant viscosity at a defined temperature, the viscosity of a non-Newtonian fluid is dependent upon the shear conditions at the time,  i.e. during the measurement or in some in-use or processing flow condition. at a given imposed shear rate or shear stress.  This viscosity is known as the apparent viscosity.

Bingham Model or Bingham Equation

A simple straight-line rheological model that relates shear stress and shear rate and quantifies yield stress and high-shear viscosity.

Brookfield Viscosity

A simple viscosity value, usually reported in centipoise (cP), obtained with a Brookfield Viscometer, usually fitted with a dip-in spindle but may also be fitted with some specialist accessories such as Cone/Plate, Helipath stand and T-Bar spindles or a concentric cylinder measuring system such as a Small Sample Adaptor or Ultra-Low Accessory. Learn More...

Carreau Model or Carreau Equation

A relative of the Cross Model that is often fitted to viscosity vs shear rate profiles.

Casson Model or Casson Equation

A commonly used rheological model that quantifies yield stress and high shear viscosity, typically used for inks and molten chocolate.

Complex Modulus

The overall resistance to deformation of a material, regardless of whether that deformation is recoverable (elastic) or non-recoverable (viscous). Symbol G*.  Complex modulus is a useful property to quantify as it is a direct measure of the rigidity of a material's soft solid structure when exposed to stresses below the yield stress.  For that reason it is a good indicator of visible attributes such the flexibility or stiffness of a material.  Along with the other viscoelastic moduli, complex modulus is obtained by performing oscillation rheology techniques.

Complex Viscosity

Complex Modulus divided by Angular Frequency.  Symbol: η, units: typically Pa.s

There are two particular situations where complex viscosity comes in useful:

In oscillatory temperature sweeps for profiling the cure behavior of a resin complex viscosity is often observed to identify the minimum viscosity the resin achieves at elevated temperature prior to the cure commencing.

Complex viscosity, when plotted as a function of angular frequency, can, for some materials, be correlated to shear viscosity as a function of shear rate.  This useful correlation, known as the Cox-Merz relationship, enables the generation of effective viscosity/shear rate profiles from oscillation frequency sweeps for some materials that are impossible to test under shear using traditional techniques.

Compliance

The ratio strain/stress. Symbol: J, units: 1/Pa. Compliance is often plotted as a function of time in a creep test. This enables the direct visual comparison of results gained using differing applied stresses.

Creep Test

Creep testing entails applying a small constant stress to a sample and monitoring its deformation over time.  When a viscoelastic material is subjected to a creep test the initial stage of the test is dominated by elastic, recoverable deformation.  As the test progresses the sample reaches an elastic equilibrium and only residual viscous non-recoverable flow persists.  From the gradient of the strain/time plot in the later viscous-flow stage of the test zero-shear viscosity can be calculated.  By extrapolating the straight-line regression from this part of the curve to an intercept on the strain axis it is possible to obtain the equilibrium elastic strain obtained from the sample – the maximum elastic recoverable strain under the specific imposed stress.  Strain values can be divided by the applied stresses to obtain compliance, (symbol: J(t) ), useful for where differing stresses are employed and the results are to be overlaid.

Critical (or Yield) Strain

The strain at which the yielding (i.e. disruption) of elastic, energy-storing structure commences.

Cross Model or Cross Equation

The Cross model is often fitted to profiles of viscosity against shear rate, where the “full” flow curve - commencing at the zero-shear viscosity plateau into the shear thinning region and ending in the infinite shear viscosity plateau - is available. The zero-shear viscosity and infinite shear viscosity plus a rate constant and time constant are then quantified by the model.

Crossover Frequency

In an oscillatory (or dynamic) frequency sweep, the frequency at which the elastic and viscous moduli cross, usually marking the transition from the terminal (viscous) region to the rubbery plateau (elastic) region.

Dilatancy

Shear thickening – non-Newtonian behaviour where viscosity increases with increasing shear rate.

Elastic Modulus

See Storage Modulus

Herschel Bulkley Model

A rheological model that combines the Power law model with a yield stress variable.

Instantaneous Elastic Compliance

In a creep test, the compliance achieved instantaneously upon the impostion of stress.

Kinematic Viscosity

The dynamic viscosity divided by density.  Typically reported in Stokes or centiStokes.

Loss Angle

Another name for Phase Angle

Loss Modulus

Loss modulus is a measure of the energy dissipated in a material in which a deformation (for example sinusoidal oscillatory shear) has been imposed.  Loss modulus can be thought of that proportion of the total rigidity (the complex modulus) of a material that is attributable to viscous flow, rather than elastic deformation.  Symbol G”, typically reported in Pascals (Pa).

Loss Tangent

Another name for Tan Delta, the ratio of loss modulus to storage modulus.

Newtonian Fluid

A fluid which exhibits a viscosity that is independent of the current shear conditions.

Non-Newtonian Fluid

A fluid which exhibits a viscosity that is dependent upon the shear conditions.

Ostwald (or Ostwald de Waele) Model

Same as Power law Model

Phase Angle

The phase difference between the stress and strain in an oscillatory test.  A measure of the presence and extent of elastic behaviour in a fluid. Symbol δ.

Power Law Model

A useful rheological model that describes the relationship between viscosity or shear stress and shear rate over the range of shear rates where shear thinning occurs in a Non-Newtonian fluid. Quantifies overall viscosity range and degree of deviation from Newtonian behaviour.  More about the Power Law Model...

Pseudoplastic

Same as shear-thinning

Relaxation Time

A time constant describing the rate of relaxation of stresses in a material (eg a viscoelastic fluid) that has been deformed to a defined strain.

Rheopexy

Time dependent viscosity increase at constant shear rate – often known as anti-thixotropy.

Shear Rate

The velocity gradient perpendicular to the direction of shear flow (dv/dx).  Units 1/s or s-1.  Shear rate is the fundamental quantification of the speed of a shear flow resulting from the application of a shear stress to a liquid.  When a non-Newtonian liquid is to be measured it is crucial to understand the importance of defining the shear rate at which the measurement is to be performed as a test at low shear will result in a different viscosity reported to that measured at high shear.

Shear Stress

The shear force per unit area.  Symbol s or t.  Units of Pascals (Pa)

Shear Strain

A unit-less quantity, the relative displacement of the faces of a sheared body (for example a layer of fluid) divided by the distance between them.

Shear Thinning

Viscosity decrease with increasing shear rate.  Shear-thinning is the most common form of non-Newtonian behaviour and is seen in suspensions, emulsions, polymer solutions and gels.  The change in viscosity with shear rate for some materials can span several orders of magnitude, thus emphasising the adage that for most products “viscosity is a plot, not a dot”.

Storage Modulus

Storage modulus is a measure of the energy stored in a material in which a deformation (for example sinusoidal oscillatory shear) has been imposed.  Put simply, storage modulus can be thought of that proportion of the total rigidity (the complex modulus) of a material that is attributable to elastic deformation.  Symbol G', typically reported in Pascals (Pa).

Sisko Model

A viscosity model that combines the Power Law model with an infinite-shear viscosity parameter. The Sisko model is often a useful model to fit to viscosity data gained across a mid-to-high range of shear rates.

Tan Delta

The tangent of the phase angle – the ratio of viscous modulus (G") to elastic modulus (G') and a useful quantifier of the presence and extent of elasticity in a fluid.  Tan delta values of less than unity indicate elastic-dominant (i.e. solid-like) behaviour and values greater than unity indicate viscous-dominant (i.e. liquid-like) behaviour.

Thixotropy

Time dependent viscosity change at constant shear rate.  A thixotropic fluid is a shear-thinning fluid that takes a finite time to reach an equilibrium viscosity following a step change in the applied shear rate.  Put simply, a highly thixotropic material is one that takes a significant time to either thin to a low viscosity on commencement of shearing or recover its viscosity on cessation of shearing.    Thixotropy measurements

Vane Spindle

A four or six bladed paddle measuring system that can be employed instead of cones, plates, cylinders or spindles on a rheometer or viscometer.  The vane is often used to eliminate slippage and sample damage upon loading.

Viscoelasticity

The phenomenon of exhibiting both elastic (solid-like or energy storing) and viscous (liquid-like or energy dissipating) properties.

Viscosity

The resistance of a fluid to flow.  In shear deformation viscosity is the ratio of applied shear stress to resulting shear rate. Typically reported in units of Poise (P) and centiPoise (cP) or Pascal seconds (Pa.s) or milliPascal seconds (mPa.s).  Symbol, typically η, units: typically Pa.s

Viscous Modulus

See Loss Modulus

Wall slip

A phenomenon seen in dispersions where a low-viscosity slip layer forms at a smooth surface due to localised depletion (wall depletion) of the dispersed phase.  Wall slip leads to poor viscosity data at low shear rates but can be eliminated by employing crosshatched (or serrated), roughened or grooved measuring systems.  Learn more about wall slip and its elimination...

Yield Stress

A structured fluid, such as an emulsion or suspension, possesses a yield stress.  This is the stress that must be applied to disrupt internal structure (due to colloidal or other interactions) and elicit a significant decrease in viscosity.  The presence of a yield stress can imbue products with desirable handling, appearance and storage properties.  Yield stress can be measured using both oscillatory and viscometric sweep methods.

Yield stress of very delicate structures

Yield stress testing of topical products

Quantifying yield using viscosity / stress profiles

Zero-shear Viscosity

The viscosity at the limit of low shear rate. In other words, the maximum plateau value attained as shear stress or shear rate is reduced.  Zero-shear viscosity is effectively the viscosity of a product whilst at rest.  For that reason, in the case of a dispersion, an elevated continuous phase zero-shear viscosity can play a vital role in inhibiting ongoing sedimentation or creaming processes.

Zero-shear viscosity assessments

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