Introduction to Rheology

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Introduction to Rheology 
 
 
Basics 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 RheoTec Messtechnik GmbH Phone: ++49 (035205) 5967-0 
 Schutterwaelder Strasse 23 Fax: ++49 (035205) 5967-30 
 D-01458 Ottendorf-Okrilla E-mail: info@rheotec.de 
 Germany Internet: www.rheotec.de 
 
 
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Content 
 
1 Fundamental rheological terms ..................................................................................................2 
1.1 Introduction ..........................................................................................................................................2 
1.2 Definitions ............................................................................................................................................4 
1.2.1 Shear stress..................................................................................................................................4 
1.2.2 Shear rate .....................................................................................................................................4 
1.2.3 Dynamic viscosity .........................................................................................................................6 
1.2.4 Kinematic viscosity .......................................................................................................................7 
1.3 Factors which affect the viscosity ........................................................................................................8 
2 Load-dependent flow behaviour ................................................................................................. 9 
2.1 Newtonian flow behaviour..................................................................................................................10 
2.2 Pseudoplasticity .................................................................................................................................12 
2.3 Dilatancy ............................................................................................................................................13 
2.4 Plasticity and yield point.....................................................................................................................14 
3 Time-dependent flow behaviour ............................................................................................... 16 
3.1 Thixotropy ..........................................................................................................................................16 
3.2 Rheopexy ...........................................................................................................................................18 
4 Temperature-dependent flow behaviour................................................................................... 19 
5 Flow behaviour of viscoelastic materials .................................................................................. 20 
5.1 Viscoelastic liquids.............................................................................................................................20 
5.2 Viscoelastic solids..............................................................................................................................21 
6 Flow behaviour of elastic materials........................................................................................... 22 
6.1 Strain..................................................................................................................................................22 
6.2 Shear modulus ...................................................................................................................................22 
7 Rheometry ................................................................................................................................ 23 
7.1 Tests at controlled shear rate (CSR mode) .......................................................................................24 
7.1.1 Viscosity-time test .......................................................................................................................25 
7.1.2 Viscosity-temperature test ..........................................................................................................26 
7.1.3 Flow and viscosity curves ...........................................................................................................27 
7.2 Tests at controlled shear stress (CSS mode) ....................................................................................30 
7.2.1 Viscosity-time test .......................................................................................................................31 
7.2.2 Viscosity-temperature test ..........................................................................................................32 
7.2.3 Flow and viscosity curves ...........................................................................................................33 
7.2.4 Creep and recovery test .............................................................................................................36 
8 Measuring geometries in a rotation viscometer ........................................................................ 41 
8.1 Coaxial cylinder measuring systems..................................................................................................42 
8.2 Dual-slit measuring systems according to DIN 54453.......................................................................44 
8.3 Cone-plate measuring systems according to ISO 3219 ....................................................................45 
8.4 Plate-plate measuring systems..........................................................................................................47 
 
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1 Fundamental rheological terms 
 
1.1 Introduction 
 
Material scientists have investigated the flow and strain properties of materials since the 17th 
century. The term rheology was first used in physics and chemistry by E.C. BINGHAM and M. 
REINER on 29 April 1929 when the American Society of Rheology was founded in Columbus, Ohio. 
Rheological parameters are mechanical properties. They include physical properties of liquids and 
solids which describe strain and flow behaviour (temporal variation of strain). Strain is observed in 
all materials and substances when exerting external forces. 
 
Rheometry describes measuring methods and devices used to determine rheological properties. 
 
If an external force is exerted on a body, its particles will be displaced relative to each other. This 
displacement of particles is known as strain. Type and extent of strain are characteristic properties 
of a body. 
 
Ideally elastic bodies undergo elastic strain if external anisotropic forces are exerted on them. 
The energy needed for this strain is stored and effects spontaneous full recovery of the original 
form if the external force ceases to act. 
 
Ideally viscous bodies undergo an irreversible strain if external anisotropic forces (e.g. 
gravitational force) are exerted on them. The input energy is transformed. This increasing viscous 
strain is known as flowing. There are only few fluids with practical importance which show (almost) 
ideally viscous behaviour. Most materials are neither ideally viscous nor ideally plastic. They rather 
exhibit different behaviour and are thus called viscoelastic materials. 
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The most simple model to illustrate rheological properties is the parallel plate model. The top 
plate, which has a surface area A [m²], is moved by a force F [N = kgm/s²] at a speed v [m/s].The 
bottom plate remains at rest. The distance between the plates, to which the material adheres, is 
described by h [m]. Now, thinnest elements of the liquid will be displaced between the plates. This 
laminar flow is of fundamental importance for rheological investigations. Turbulent flows increase 
the flow resistance, thus showing false rheological properties. 
 
Fig. 1-1: Parallel plate model 
 
 
Shear rate γ& = v / h in s-1 
 
Shear stress τ = F / A in Pa 
 
Viscosity η = τ / γ& in Pas 
 
Strain γ = dx / h dimensionless 
 
In addition to the expression γ& , the symbol D is also used for the shear rate. 
 
Shear tests are usually conducted using rotation viscometers. In contrast to the parallel plate 
model, the moved surface performs a rotary movement. 
h 
v 
A F 
moved plate 
fix plate 
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( ) Pa
m
NewtonN
AArea
FForcestressShear 2 ===τ
dh
dvrateShear =γ&
1s
m
s/m
hcetanDis
vVelocityrateShear -===γ&
1.2 Definitions 
 
1.2.1 Shear stress 
 
Force F acting on area A to effect a movement in the liquid element between the two plates. The 
velocity of the movement at a given force is controlled by the internal forces of the material. 
 
 
 
 
 
100 Pa = 1 mbar = 1 hPa 
old unit: dyn / cm2 = 0.1 Pa 
 
1.2.2 Shear rate 
 
By applying shear stress a laminar shear flow is generated between the two plates. The uppermost 
layer moves at the maximum velocity vmax, while the lowermost layer remains at rest. The shear 
rate is defined as: 
 
 
 
where dv Velocity differential between adjacent velocity layers 
 dh Thickness differential of the flow layers 
 
In a laminar flow the velocity differential between adjacent layers of like thickness is constant 
(dv = const., dh = const.) The differential can thus be approximated as follows: 
 
 
 
 
 
In addition to γ& , the symbol D is also used for the shear rate in the literature. 
 
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Table 1-1: Typical shear rate ranges 
Process 
 
γ& ranges in s-1 
 
Example of use 
 
Sedimentation of fine particles in 
suspensions 
10-6 ... 10-4 Paints, lacquers, 
pharmaceutical solutions 
Flowing due to surface characteristics 
 
10-4 ... 10-1 Paints, printing inks 
 
Dripping under the effect of the 
gravitational force 
10-2 ... 101 Paints, coatings 
 
Extruding 
 
100 ... 102 Polymers 
 
Chewing/ swallowing 
 
101 ... 102 Food 
 
Spreading butter on a slice of bread 
 
10 ... 50 Food 
 
Mixing, agitating 
 
101 ... 103 Substances in process 
engineering 
Brushing 
 
102 ... 104 Paints, lacquers, pastes 
 
Spraying, spreading 
 
103 ... 106 Paints, lacquers, coatings 
 
Rubbing in 
 
104 ... 105 Creams, lotions 
 
High-speed coating 
 
105 ... 106 Paper coatings 
 
Lubrication of machine parts 
 
103 ... 107 Mineral oils, greases 
 
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Pas
s
Pa
rateShear
stressShearitycosvisDynamic 1 ==γ
τ
=η -&
1.2.3 Dynamic viscosity 
 
Viscosity describes the toughness of a material. 
 
 
 
 
 
The unit Pas (or mPas) is used for the viscosity. 
 
1 Pas = 1000 mPas 
 
old unit: 1 P (Poise) = 100 cP (Centipoise) = 100 mPas (Millipascalsecond) 
 
 
Substance Dyn. viscosity η in mPas 
Acetone 0.32 
Water 1.0 
Ethanol 1.2 
Mercury 1.5 
Grape juice 2...5 
Cream approx. 10 
60 % sugar solution 57 
Olive oil: approx. 100 
Honey approx. 10,000 
Plastic melts 104 ... 108 
Tar approx. 106 
Bitumen approx. 108 
Earth mantle approx. 1024 
 
Table 1-2: Typical viscosities at 20 °C in mPas 
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s
mm
Density
itycosvisDynamicitycosvisKinematic
2
=
ρ
η
=ν
1.2.4 Kinematic viscosity 
 
If ideally viscous materials are tested using a capillary viscometer, such as an UBBELOHDE 
viscometer, the kinematic viscosity ν is determined, not the dynamic viscosity η. The kinematic 
viscosity is related to the dynamic viscosity through the density of the material. 
 
 
 
 
 
Old unit: cSt (Centistokes) = mm² / s 
 
There used to be several device-specific measuring methods to determine the kinematic viscosity, 
e.g. FORD cups, and, accordingly, a large number of units, such as FORD cup seconds, ENGLER 
degrees, REDWOOD or SAYBOLD units. These viscosity-dependent values cannot be converted into 
absolute viscosities η or ν for non-NEWTONian fluids. 
 
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1.3 Factors which affect the viscosity 
 
The flow and strain behaviour of a material may be affected by a number of external factors. The 
five most important parameters are: 
 
Substance 
The viscosity of a material depends on its physical and chemical properties. 
 
Temperature 
Temperature has a major effect on the viscosity. For example, several mineral oils lose about 10 % 
of their viscosity if the temperature is only increased by 1 K. 
 
Shear rate 
The viscosity of most materials depends on the shear rate, i.e. the load. 
 
Time 
The viscosity depends on the strain history of a material, in particular on previous loads. 
 
Pressure 
If great pressure is exerted on a material, its viscosity may increase as particles are organised in a 
more tight structure (resulting in more interaction possibilities). 
 
Other influencing factors are the pH value, magnetic and electric field strength. 
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2 Load-dependent flow behaviour 
 
Flow and viscosity curves 
The flow behaviour of a material is characterised by the relation between shear stress τ and shear 
rate γ& . A γ& -τ diagram is often used for graphic representation. Usually, the shear stress is shown 
on the ordinate and the abscissa the shear rate, irrespective of whether γ& or τ were given for the 
measurement. These diagrams are referred to as flow functions or flow curves. 
 
 
 
 γ& [s-1] 
τ [Pa] 
 
 
Fig. 2-1: Flow curve 
 
Plotting the viscosity over the shear rate γ& or shear stress τ produces the viscosity function or 
viscosity curve. 
 
 
 
η [mPas] 
 γ& [s-1] 
 
Fig. 2-2: Viscosity curve 
 
The measuring result obtained with a viscometer or rheometer is always a flow curve. However, 
the viscosity function can be calculated based on the measured values. 
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2.1 Newtonian flow behaviour 
 
If a NEWTONian material is subjected to a shear stress τ, a shear gradient γ& of viscous flow is 
generated which is proportional to the applied shear stress. The flow function of a NEWTONian 
material is a straight line which runs through the origin of the coordinate system at an angle α. This 
relation between shear stress and shear gradient is described by NEWTON’s law of viscosity. 
 
 
 
 
 
 
η is the material constant of the dynamic shear viscosity. If the viscosity is plotted over the shear 
rate (or shear stress) in a viscosity diagram, a straight line which starts at γ& = 0 s-1 (or τ = 0 Pa) 
and runs parallel to the abscissa is obtained for an ideally viscous material. 
 
 
 
τ 
 γ& 
 
 
η 
 γ& 
 
 
Fig. 2-3: Flow curves of NEWTONian materials Fig. 2-4: Viscosity curves of NEWTONian materials 
 
 
The viscosity of a NEWTONian material or ideally viscous material is independent of the shear 
rate. 
 
Examples of NEWTONian materials: water, mineral oil, sugarsolution, bitumen 
 
.const=
γ
τ
=η &
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According to NEWTON, a viscous body may be represented by the mechanical model of a damper. 
 
 
 
 
 
 
 
Fig. 2-5: Mechanical model of a NEWTONian body 
 
 
It can be shown with the help of this model that the material is continuously deformed in the 
damper as long as a force acts on the piston. If the force ceases to act, the original shape is not 
restored. A viscous strain is characterised in that the energy input to create a flow is fully 
transformed into heat in an irreversible process. Materials which show only little interaction 
between (usually short) molecules exhibit NEWTONian flow behaviour. 
 
Force F
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nK γ⋅=τ &
1nK −γ⋅=
γ
τ
=η &&
2.2 Pseudoplasticity 
 
Many materials exhibit a strong decrease in viscosity if the shear rate grows. This effect is of great 
technical importance. Compared with an ideally viscous material, a pseudoplastic or structurally 
viscous material can be pumped through pipelines with a lower energy input at the same flow 
velocity. 
 
 
 
 
The proportionality factor τ / γ& in the NEWTONian constitutive equation is thus referred to as ηa. ηa is 
the apparent viscosity and denotes the viscosity at a certain shear rate γ&. 
 
Mathematical expression for pseudoplastic materials according to OSTWALD DE WAELE: 
 
 
 
where n < 1 for pseudoplastic materials 
 
Transformed into a viscosity function: 
 
 
 
 
 
τ 
 γ& 
 
 
η 
 γ& 
 
Fig. 2-6: Flow curve of a pseudo- Fig. 2-7: Viscosity curve of a 
 plastic material pseudoplastic material 
 
 
Examples: suspensions, dispersions, paints, lacquers, creams, lotions, gels 
 
Materials are referred to as pseudoplastic if a force acting on the body causes the particle size to 
change, the particles to be oriented in the direction of flow, or an agglomerate to be dissolved. 
τ / γ& = η ≠ const.
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nK γ⋅=τ &
1nK +γ⋅=
γ
τ
=η &&
2.3 Dilatancy 
 
The viscosity of dilatant materials also depends on the shear rate. It increases as the shear rate 
grows. Dilatant behaviour can cause trouble in technological processes. 
 
Mathematical expression according to OSTWALD DE WAELE: 
 
 
 
where n > 1 for dilatant materials 
 
Transformed into a viscosity function: 
 
 
 
 
 
 
τ 
 γ& 
 
 
η 
 γ& 
 
Fig. 2-8 Flow curve of a dilatant material Fig. 2-8 Viscosity curve of a dilatant material 
 
 
Dilatant behaviour is found rather seldom. 
 
Examples: concentrated corn starch dispersions, wet sand, several ceramic suspensions, 
several surfactant solutions 
 
Note: If in case of high shear rates the flow in the measuring gap is no longer laminar, but 
becomes turbulent, this may falsely suggest dilatant behaviour. 
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γ⋅η+=τ &BBf
γ⋅η+=τ &CCf
⋅γ⋅+=τ pH mf &
2.4 Plasticity and yield point 
 
Plasticity describes structurally viscous liquids which have an additional yield point τo. A practical 
example of such a material is toothpaste. At rest, toothpaste establishes an network of inter-
molecular bonding forces. These forces prevent individual volume elements to be displaced when 
the material is at rest. If an external force which is smaller than the internal forces (bonding forces) 
acts on the material, the resulting strain is reversible, as with solids. However, if the external forces 
exceed the internal bonding forces of the network, the material will start flowing, the solid turns into 
a liquid. 
 
Definition of the yield point τo 
Maximum shear stress τ at the shear rate γ& = 0 s-1 
 
Thus, if Fexternal < Finternal, the material does not flow 
if Fexternal > Finternal, the material starts to flow 
 
Examples: Toothpaste, PVC paste, emulsion paint, lipstick, fats, printing ink, butter 
 
 
Flow curves and mathematical expression of materials with a yield point 
Flow curves of plastic liquids do not start in the origin of the coordinate system, but run on the 
ordinate axis until the yield point τo is reached, then they converge from the ordinate. The flow 
curves can be expressed mathematically using a number of equations, depending on the actual 
material. For example, the flow curve of chocolate is typically based on the CASSON model. 
 
Mathematical expression according to BINGHAM: 
 
(where fB = yield point according to 
BINGHAM) 
 
Mathematical expression according to CASSON: 
 
(where fC = yield point according to 
CASSON) 
 
Mathematical expression according to HERSCHEL and BULKLEY: 
 
(where fH = yield point according to 
HERSCHEL and BULKLEY) 
where p < 1 for pseudoplastic and p > 1 for dilatant materials 
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τ 
τ o 
 γ& 
 
 
τ 
τ o
 γ& 
 
Fig. 2-10: Flow curve according to BINGHAM Fig. 2-11: Flow curve according to HERSCHEL 
 and BULKLEY 
 
 
Physical causes for the occurrence of yield points in dispersions are intermolecular particle-particle 
and particle-dispersing agent interactions. 
 
- VAN DER WAALS forces 
- Dipole-dipole interactions 
- Hydrogen bonds 
- Electrostatic interactions 
 
The ST. VENANT model (with static friction) is a mechanical model used to describe plastic 
behaviour. 
 
 
Fig. 2-12: ST. VENANT model for plastic materials 
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3 Time-dependent flow behaviour 
 
3.1 Thixotropy 
 
Thixotropy is a property exhibited by non-NEWTONian liquids, they return to their original viscosity 
only with a delay after the shear force ceased to act. In addition, these materials often also have a 
yield point. 
 
Tomato ketchup is an example of such a material. When stirred or shaken, ketchup becomes 
thinner and only returns to its original viscosity after allowing to rest for a while. Per definition, a 
thixotropic material does not only thin depending on the shear rate, but it additionally returns to its 
original viscosity after a material-specific period of rest. Theses gel-sol and sol-gel changes in 
thixotropic materials are reproducible. 
 
Yoghurt serves as a counter-example: it becomes thinner when stirred, but does not return to its 
original thickness. Yoghurt thus does not exhibit thixotropic flow behaviour. 
 
 
 η 
t 
Shear at 
Sample at rest γ& = const. ( γ& = 0.1 s-1) 
 
Fig. 3-1: Viscosity-time curve of a thixotropic material 
 
 
Two transitional areas can be easily identified in the viscosity-time curve shown above. A gel is 
quickly transformed into a sol at a constant shear force. During the period of rest the material-
specific network structures are re-established, i.e. the sol turns back into a gel. 
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τ 
 γ& 
 
 
η 
 γ& 
 
Fig. 3-2: Flow and viscosity curve of a pseudoplastic and a thixotropic material 
 
 
The flow curve shows that the measured rising and declining curves are not congruent. The area 
between the two curves (hysteresis area) defines the extent of the time-dependent flow behaviour. 
The larger the area the more thixotropic is the material. 
 
Examples: paints, foodstuff, cosmetics, pastes 
UP 
DOWN 
DOWN
UP 
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3.2 Rheopexy 
 
Rheopectic materialsexhibit greater viscosity while they are subject to shear stress. Structures are 
established in the material during the application of mechanical shear forces. The original viscosity 
is only restored with a delay after the shear forces ceased to act on the material, by disintegrating 
this structure. 
 
 
 η 
t 
Shear at 
Sample at rest ( γ& = 0.1 s-1) γ& = const. 
 
Fig. 3-3: Viscosity-time curve of a rheopectic material 
 
 
This process of viscosity increase and decrease can be repeated as often as you wish. 
In contrast to thixotropy, true rheopexy is very rare. 
 
 
τ 
 γ& 
 
η 
 γ& 
Fig. 3-4: Flow and viscosity curve of a pseudoplastic and a rheopectic material 
 
 
Examples: several latex dispersions, several casting slips, several surfactant solutions 
UP UP 
DOWN 
DOWN 
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T
B
eA ⋅=η
4 Temperature-dependent flow behaviour 
 
As mentioned above, the viscosity of a material is a function of temperature. Exact temperature 
control and accurate indication of the measuring temperature is thus of major importance in 
viscosity measurements. 
 
The viscosity-temperature curve of a material is found at a constant shear rate. In most materials, 
the viscosity decreases as the temperature is raised. In ideally viscous materials, this phenomenon 
can be described with the help of the ARRHENIUS equation: 
 
where T .......... temperature in Kelvin 
 A, B ….. material constants 
 
 
T 
η 
 
Fig. 4-1: Viscosity-temperature curve 
 
Viscosity-temperature curves are also often established in order to trace certain reactions. For 
example, the curing temperature can be determined for powder lacquers (see Fig. 4-2, curve a), or 
the chocolate melting process can be followed (see Fig. 4-2, curve b). 
 
a
b 
 
T 
η 
 
Fig. 4-2: Viscosity-temperature curves of certain reactions 
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5 Flow behaviour of viscoelastic materials 
 
Viscoelastic materials exhibit both viscous and elastic properties. Because of physical differences, 
a distinction is made between viscoelastic liquids and viscoelastic solids. The elastic portion of 
viscoelastic liquids is described by HOOKE’S law, which is also known as the spring model, the 
viscous portion by NEWTON’S damper model. 
 
 
5.1 Viscoelastic liquids 
 
Viscoelastic materials and purely viscous materials can only be distinguished if they are stirred, for 
example. Rheological phenomena cannot be observed while they are at rest. In a viscous liquid, a 
rotating agitator causes centrifugal forces, which drive volume elements of the liquid towards the 
wall of the cup. This means that a dip is formed around the agitator shaft. In contrast, in elastic 
liquids the rotating agitator causes normal forces which are so great that they do not only 
compensate the centrifugal forces but exceed them. Consequently, volume elements of the liquid 
are thus dawn up the agitator shaft. This phenomenon of masses “creeping up” a rotating shaft due 
to the acting normal forces is called the WEISSENBERG effect. 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 5-1: Flow behaviour of a viscous and a viscoelastic liquid 
 
 
A force acting on a viscoelastic material (see Fig. 5-2) causes the spring to be deformed 
immediately, but the damper reacts with a delay. If the force ceases to act, the spring will return 
immediately while the damper remains displaced, so that the material partly remains strained. That 
means that there is no full restoration of shape. The amount of the spring return corresponds with 
the elastic portion, the amount of the remaining strain (damper) corresponds with the viscous 
portion. A viscoelastic liquid can thus be described using a model where damper and spring are 
arranged in series. Honouring JAMES C. MAXWELL (1831–1879), this series connection is also 
called the MAXWELL model. 
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Fig. 5-2: Maxwell model 
 
 
Examples of viscoelastic liquids are gels and silicone rubber compounds 
 
 
5.2 Viscoelastic solids 
 
If a force is exerted on a viscoelastic solid a delayed strain will take place, because the 
displacement of the spring is impeded by the damper. If the force ceases to be exerted, the body 
will fully return to its original shape (but again delayed by the damper). That means that there is a 
full restoration of shape. It can thus be said that a viscoelastic solid is characterised in that it has 
the ability of reversible strain. The model to describe such materials is again a combination of 
spring and damper models. In contrast to viscoelastic liquids, however, the two elements are 
connected in parallel, as in the KELVIN-VOIGT model. 
 
 
Fig. 5-3: KELVIN-VOIGT model 
 
 
An example of a viscoelastic solid is hard rubber. 
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Pa
1
Pa
Strain
stressShearGulusmodShear ==
γ
τ
=
γα =
h
s
=tan
6 Flow behaviour of elastic materials 
 
6.1 Strain 
 
A cube with an edge length h shall be investigated here as the volume element in order to illustrate 
the strain γ. The bottom face of the volume element is fixed. A force F acts on its top face, which is 
thereby displaced by the amount s. 
 
 
 
 
 
 
 
 
 
 
 
Fig. 6-1: Unloaded volume element Loaded volume element 
 
 
Mathematical expression: 
 
 
Strain is a dimensionless parameter. A deformation angle α of 45 ° corresponds with a strain γ of 1 
or 100%. 
 
The symbol for the shear rate ( γ& ) can be derived from that for the strain (γ). The shear rate 
describes the strain change dγ during a period of time dt. Consequently, γ& is the derivative of strain 
γ with respect to time t. In other words, the shear rate can be considered to be the strain rate. 
 
 
6.2 Shear modulus 
 
In purely elastic bodies, the ratio of shear stress τ and resulting strain γ is constant. This material-
specific parameter described the stiffness of the material and is known as the shear modulus G. 
 
 
 
 
The shear modulus can be in the MPa (106 Pa) range in very stiff bodies. 
h 
h
s 
F
α 
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7 Rheometry 
 
Viscometers or rheometers used to determine the rheological properties are called absolute 
viscometers if the measured values are based on the basic physical units of force [N], length [m] 
and time [s]. 
 
dynamic viscosity η = [N/m2] · [s] = force / length2 · time = [Pa] · [s] 
 
Viscometers are measuring devices which are used to determine the viscosity depending on the 
rotational speed (= shear rate), time and temperature. 
 
Rheometers are devices which are additionally able to determine the viscous and viscoelastic 
product properties depending on the force (shear stress) exerted in both rotation/ creep test and 
oscillation test. 
 
Absolute viscometers have the advantage that the results of the measurements are independent of 
the device manufacturer or measuring equipment used. The material under investigation is filled 
into the two-part measuring system, where a shear force is applied. 
 
The measurements must be conducted under certain boundary conditions in order to obtain correct 
results. 
 
• The applied shear force must only produce a stratified laminar flow. Vortices and turbulences in 
the measuring system consume much energy so that viscosity values up to 40 % above the true 
viscosity are obtained. 
 
• The material under investigation should be homogeneous. It is not recommended to mix 
substances during the measurement.Due to the different specific weights of the components a 
phase separation of the mix may occur during the measurement. The changed composition of 
the mix in the measuring gap may lead to erroneous viscosity values. 
 
• The material must adhere to the walls. The force exerted on the top plate must be fully 
transmitted on to the sample. This condition is not fulfilled if the boundary layers of the material 
under investigation do not properly adhere to the parts of the measuring system. Problems in 
this respect may be encountered with materials such as fats. 
 
• The elasticity of the sample must not be too high. The shear forces acting on the sample may 
otherwise result in extreme normal forces such that the material creeps out of the measuring 
gap. When handling elastic materials, the shear rate must be chosen such that normal stress 
does not affect the measuring result. 
 
• The shear stress applied according to the laws of viscosity is proportional to the shear rate and 
shall only be as great as is necessary to maintain stationary flow conditions, i.e. a flow at 
constant velocity. The force that is necessary to accelerate or decelerate the flow is not 
considered in this equation. 
 
The NEWTONian relation η = τ / γ& shows that the viscosity η can be determined using either of two 
different measuring methods. 
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7.1 Tests at controlled shear rate (CSR mode) 
 
(CSR … controlled shear rate) 
 
With this measuring method, the rotational speed n is preset at the rheometer, and the shear rate 
is calculated based on the gap h and the rotational speed n (or circumferential speed v of the 
shearing area). The flow resistance moment M (or the shear force F) of the braking, tough material 
under investigation is measured. This torque M is converted into the rheological parameter of 
shear stress using the shear area A of the measuring system. 
 
 
 physical setting rheological setting 
rot. speed n [min-1] shear rate [s-1] 
physical result 
torque M [mNm] 
rheological result 
shear stress τ [Pa] 
τ 
= η 
measurement 
 γ& 
γ&
 
 
The dynamic viscosity η is calculated from the shear stress τ and shear rate γ& (or D). 
 25 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
7.1.1 Viscosity-time test 
 
A constant shear rate is applied for a certain period of time. The shear stress τ is measured as a 
function of time. The viscosity determined according to the viscosity equation is obtained in relation 
to time. This test is used in practice in stability investigations, to study hardening reactions and with 
thixotropic materials. 
 
 
γ& [s-1] 
 
t [s] 
 
Fig. 7-1: Conditions: shear rate = const., temperature = const. 
 
 
3 
1 
2 
 
t [s]
τ [Pa] 
 
3 
1 
2 
 
t [s]
η
 [Pas]
 
 
1 Material with a viscosity which does not change over time (e.g. calibration oil) 
2 Material with a viscosity which decreases over time (e.g. ketchup) 
3 Material with a viscosity which increases over time (e.g. hardening lacquer) 
 
Fig. 7-2: Results of the viscosity-time tests 
 
 26 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
7.1.2 Viscosity-temperature test 
 
At a constant shear rate, a temperature ramp is preset and the viscosity is measured in relation to 
temperature. In practice, this test is conducted when it comes to investigating temperature-
dependent hardening or melting reactions. 
 
 
T 
[°C] 
t [s] 
Fig. 7-3: Conditions: shear rate = const., variable temperature 
 
 
2 
1 
 
T [°C]
τ [Pa] 
 
2 
1 
 
T [°C]
η
 [Pas]
 
 
1 Material with a viscosity which decreases as the temperature rises (e.g. chocolate) 
2 Material with a viscosity which increases as the temperature rises (e.g. hardening lacquer) 
 
Fig. 7-4: Results of the viscosity-temperature tests 
 27 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
7.1.3 Flow and viscosity curves 
 
At a constant temperature, a shear rate - time profile is preset. The shear stress is measured for 
each shear rate value and the corresponding viscosity is calculated from those measuring results. 
 
 
t [s] 
 [s-1] γ&
 
 
Fig. 7-5: Conditions: variable shear rate, temperature = const. 
 
 
A constant viscosity value is found in materials with ideally viscous behaviour (NEWTONian 
materials), such as water. In materials with pseudoplastic behaviour the viscosity decreases as the 
shear rate rises (“shear thinning”). In contrast, in materials with dilatant behaviour the viscosity 
increases as the shear rate rises (“shear thickening”). 
 
2 
4 
3 
1 
 
τ [Pa] 
 γ& [s-1] 
4 
3 
1 
2 
 η
 [Pas]
 γ& [s-1]
 
 
1 NEWTONian material 
2 Pseudoplastic material 
3 Plastic material 
4 Dilatant material 
 
Fig. 7-6: Flow and viscosity curves of materials with different rheological behaviour 
T=const.
 28 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
Flow and viscosity curves are often plotted such that the viscosity values are shown at both rising 
and falling shear rate. Between the two sections of the curve (up ramp, down ramp) there is often a 
section where the shear rate is kept constant. 
 
 
t [s] 
γ& [s-1] 
 
Fig. 7-7: Conditions 
 
 
In addition to load-dependent flow behaviour (pseudoplasticity, dilatancy), the results of the 
measurements then also allow information about time-dependent flow behaviour to be derived. In 
practice, the area between the up and down curves is often used as a measure for time-dependent 
flow properties, i.e. for the thixotropy of the material. Materials which require a long time after 
maximum shear to re-establish their structure show in the diagram a large area between the two 
curves (hysteresis area). 
 
 
τ 
 γ& 
 
 
η 
 γ& 
 
Fig. 7-8: Flow and viscosity curve of a pseudoplastic and a thixotropic material 
 
UP 
UP 
DOWN
DOWN 
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 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
In order to be able to compare measured curves with respect to the hysteresis areas, an identical 
measuring profile must be selected for the individual measurements, i.e. the duration of the 
individual test phases, the total test duration, the shear rate profile and the temperature must be 
identical. 
 
In addition to pseudoplastic behaviour, thixotropic behaviour can also facilitate certain materials to 
be processed. Thanks to the reduced viscosity over time, there is less power required for pumping, 
mixing, spraying or brushing. An advantage of thixotropic coatings is that they spread more 
smoothly after application and bubbles which are possibly created during the application process 
can escape more easily. 
 
Pseudoplasticity and thixotropy are two rheological properties which exist fully independent of 
each other. They should not be mistaken. 
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 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
7.2 Tests at controlled shear stress (CSS mode) 
 
(CSS … controlled shear stress) 
 
In controlled shear stress tests, the torque M is preset at the rheometer, and the shear stress τ is 
calculated from the torque M and the shear area A of the measuring system. The rotational 
speed n achieved by the plunger due to the applied torque is measured. This rotational speed n is 
converted into the rheological parameter of shear rate using the appropriate measuring system 
factor. 
 
 
 physical setting rheological setting 
torque M [mNm] 
shear rate[s-1] 
physical result 
rot. speed n [min-1] 
rheological result 
shear stress τ [Pa] 
τ 
= η 
measurement
γ& γ& 
 
 
The dynamic viscosity η is calculated from the shear stress τ and shear rate γ& . 
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 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
7.2.1 Viscosity-time test 
 
A constant shear stress τ is applied to the material under investigation for a certain period of time. 
The achieved shear rate γ& is measured as a function of time. The viscosity determined according 
to the viscosity equation is obtained in relation to time. This test is applied in practice when 
studying hardening reactions. It boasts an advantage over the controlled shear rate test: during 
hardening the viscosity of the material increases and at a constant shear stress the shear rate falls. 
This gradually reduces the foreign movement which interferes with the hardening process. 
 
 
τ 
[Pa] 
t [s] 
 
Fig. 7-9: Conditions: shear stress = const., temperature = const. 
 
 
 
[s-1] 
2 
3 
1 
 γ& 
t [s] 
3 
2 
1 
 
t [s] 
η
 [Pas]
 
 
1 Material with a viscosity which does not change over time (e.g. calibration oil) 
2 Material with a viscosity which decreases over time (e.g. ketchup) 
3 Material with a viscosity which increases over time (e.g. hardening lacquer) 
 
Fig. 7-10: Results of the viscosity-time tests 
 32 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
τ = const.
7.2.2 Viscosity-temperature test 
 
At a constant shear stress, a temperature ramp is preset and the viscosity is measured in relation 
to temperature. In practice, this test is conducted when it comes to investigating temperature-
dependent hardening or melting reactions. 
 
 
T 
[°C] 
t [s] 
 
Fig. 7-11: Conditions: shear stress = const., variable temperature 
 
[s-1] 
1 
2 
 γ& 
T [°C] 
2 
1 
 
T [°C]
η
 [Pas]
 
 
1 Material with a viscosity which decreases as the temperature rises (e.g. chocolate) 
2 Material with a viscosity which increases as the temperature rises (e.g. jellying starch) 
 
Fig. 7-12: Results of the viscosity-temperature tests 
 
 33 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
T=const.
7.2.3 Flow and viscosity curves 
 
At a constant temperature, a shear stress - time profile is preset. The shear rate is measured for 
each shear stress value and the corresponding viscosity is calculated from those measuring 
results. 
 
 
τ 
[Pa] 
t [s] 
 
Fig. 7-13: Conditions: variable shear stress, temperature = const. 
 
 
A constant viscosity value is found in materials with ideally viscous behaviour (NEWTONian 
materials), such as water. In materials with pseudoplastic behaviour the viscosity decreases as the 
shear stress rises. In contrast, in materials with dilatant behaviour the viscosity increases as the 
shear stress rises. 
 
2 
4 
3 
1 
 
τ [Pa] 
 γ& [s-1] 
4 
3 
1 
2 
 η
 [Pas]
 γ& [s-1] 
1 NEWTONian material 
2 Pseudoplastic material 
3 Plastic material 
4 Dilatant material 
 
Fig. 7-14: Flow and viscosity curves of materials with different rheological behaviour: 
 34 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
T=const. 
In mathematics, the given parameter is usually plotted on the abscissa, and the resulting 
parameter on the ordinate. In contrast, in rheology γ& is plotted on the abscissa and τ on the 
ordinate, irrespective of whether shear rate or shear stress are given for the measurement. 
 
The shear stress test is the only measuring method where the yield point can be determined 
metrologically. Only if the applied shear stress exceeds the network bonding forces in the material 
under investigation the material will start to flow, i.e. a measurable shear rate is obtained. In the 
flow curve shown in Fig. 7-6, material 3 exhibits a yield point. The flow curve does not pass the 
origin of the coordinate system, but shows a certain translation on the Y axis. In plastic materials 
an additional shear stress is required in order to obtain a shear rate in the material. 
 
Also in controlled shear stress tests, flow and viscosity curves are often plotted such that the 
viscosity values are shown at both rising and falling shear stress. Between the two sections of the 
curve (up ramp, down ramp) there is often a section where the shear stress is kept constant. 
 
 
τ 
[Pa] 
t [s] 
 
Fig. 7-15: Conditions 
 
 
In addition to the yield point value (if any), the results of the measurements then also allow 
information about load-dependent flow behaviour (pseudoplasticity, dilatancy), and time-dependent 
flow behaviour (thixotropy, rheopexy) to be derived. 
 
 35 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
 γ& 
 
τ 
 
 γ& 
 
η 
 
Fig. 7-16: Flow and viscosity curve of a pseudoplastic and a thixotropic material 
 
 
In order to be able to compare measured curves with respect to the hysteresis areas, which 
provides information about the thixotropic or rheopectic behaviour of a material, an identical shear 
rate range should be selected for the individual measurements, and the duration of the individual 
test phases, the total test duration and the temperature must be identical. As various shear rates 
can result from a shear stress setting, a shear stress controlled measurement is less suitable for 
comparing hysteresis areas than a shear rate controlled test. The amount of thixotropy and 
rheopexy is determined from the flow curves by finding the difference between the areas under the 
up and down sections of the curve. Thixotropic materials have positive, rheopectic materials 
negative values. 
UP 
UP 
DOWN
DOWN 
 36 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
7.2.4 Creep and recovery test 
 
The creep test forms a simple and quick method used to find the viscoelastic properties of the 
material under investigation (with exact distinction between the viscous and elastic portions). 
 
The mobile part of the measuring arrangement is loaded with a constant shear stress (τx) for a 
certain period of time (t0 bis t2). The sample reacts on this force with a deformation, i.e. the material 
starts to creep. In the second part of the test (t2 bis t4), the material is relieved from the shear 
stress so that it can recover. 
 
τx 
t0 t2 t4
 
τ 
[Pa] 
t [s] 
 
Fig. 7-17: Creep and recovery test 
 
 
7.2.4.1 Ideally elastic materials 
 
A piece of vulcanised rubber will be now be looked at as an example for an ideally elastic material. 
The constant shear stress applied leads to a certain twist in the sample, i.e. it shows the strain γ. 
The angle of such strain is characterised by the spring modulus of elasticity of the purely elastic 
solid. Shear stress and resulting strain γ show a linear relation. If the force is doubled, the strain 
will also double. HOOKE’s spring model serves as a model for ideally elastic materials. The strain is 
maintained as long as the deforming force keeps acting. 100 % of the strain energy is stored in the 
spring. The sample will be 100 % relieved if the force ceases to be exerted. 
 
 37 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
t0 t2 t4
 γ 
t [s] 
 
Fig. 7-18: Creep and recovery curve of an ideally elastic material (according to HOOKE) 
 
 
7.2.4.2 Ideally viscous material 
 
Serving as an example for a material with ideally viscous behaviour, water shows a completely 
different behaviour. The constant shear stress applied leads to a strain γ whichincreases linearly 
over time, i.e. it shows flowing. The input energy is used up for the flowing process. If the sample is 
relieved, the strain γ obtained by this moment of time will be maintained. NEWTON’s damper model 
is used to describe an ideally viscous material. 
 
t0 t2 t4
 γ 
t [s] 
 
Fig. 7-19: Creep and recovery curve of an ideally viscous material (according to NEWTON) 
 
 38 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
7.2.4.3 Viscoelastic liquids and solids 
 
The reaction of viscoelastic liquids on applied shear stress shows characteristics of both elastic 
and viscous strain. A partial recovery by the elastic portion γE can be observed, but the portion of 
viscous strain γV remains. 
 
t0 t2 t4
γE 
γV 
 γ 
t [s] 
γE Elastic portion (recovery) 
γV Viscous portion 
 
Fig. 7-20: Creep and recovery curve of a viscoelastic liquid 
 
 
In a viscoelastic solid, a delayed but complete recovery can be observed, i.e. γV is almost zero. 
 
A material shall now be scrutinised which consists of macromolecules (parallel connection of 
spring 2 and damper 2) linked by springs (spring 1) in a highly viscous oil (damper 3). This model 
is also known as the BURGER model. 
 
 
Fig. 7-21: BURGER model 
 39 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
The applied shear stress initially leads to a spontaneous jump in strain (dilatation of spring 
elements 1 which are situated in the orientation of the strain), the strain rate then drops. During 
that time the macromolecules are oriented, the twisted springs tensioned and the macromolecules 
stretched up to their mechanically maximal possible size (delayed viscoelastic strain of spring 2 
and damper 2). If more force is applied, the strain will again increase linearly if the macromolecules 
are irreversible disentangled and caused to flow with the viscous matrix mass (viscous strain of 
damper 3). 
 
If the test duration is long enough, all dampers and springs finally show maximum dilatation. During 
the relieve phase, two types of recovery take place. Spring 1 returns to its original tension 
immediately (elastic recovery), and the parallel connection of spring 2 and damper 2 recovers with 
a delay (viscoelastic recovery). Damper 3 remains fully displaced, so that a partial strain is 
maintained. If this remaining strain is very small (γV near 0 %), the material is called a viscoelastic 
solid, otherwise it is a viscoelastic liquid. 
 
 
Creep curve (t0 to t2) Recovery curve (t2 to t4) 
 
γ1 ..... Purely elastic strain (spring 1) 
γ2 ..... Viscoelastic strain (parallel connection of spring 2 and damper 2) 
γ3 ..... Purely viscous strain (damper 3) 
γmax … Maximum strain: γ1 + γ2 + γ3 
β ….. Gradient angle of the strain curve on achievement of the stationary flow condition (depending on the 
viscosity of the strained material) 
γE Elastic recovery portion 
γV Viscous recovery portion 
 
Fig. 7-22: Creep and recovery curve with analysed parameters 
 40 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
( )t,f= xτη
( )xf= τη
x)t(J)t( τ⋅=γ
xt/)t(=)t(J γ
β
τ
γ
τη
tan
== x
3
x
0
In the load phase, unstationary flowing occurs in the linearly viscoelastic range between the 
points of time, t0 and t1. The viscosity here depends on the applied shear stress and time. 
 
 
 
Stationary viscous flowing is observed between the points of time, t1 and t2. The viscosity does no 
longer depend on the elapsed loading time. 
 
 
 
The zero viscosity η0 corresponds with the behaviour of damper 3 in the BURGER model (see 
Fig. 7-13). It can be determined as 
 
 
 
 
where the shear rate γ3 = dγ / dt = tan β 
 
If determined at smallest shear rates, the zero viscosity η0 is a material constant, which contains 
information e.g. about the molecular weight of non-networked macromolecules. 
 
In creep tests a constant shear stress is applied and the time-dependent strain is measured. 
Mathematically, the relation between stress and strain can be expressed as follows: 
 
 
 
This equation introduces the time-dependent compliance factor J(t). Like the zero viscosity, it is a 
material-specific quantity. It is a measure of the softness or flexibility of a material. The greater the 
compliance the more can the material be strained under the application of a certain shear stress. 
 
in [Pa] 
 
 41 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
8 Measuring geometries in a rotation viscometer 
 
The measuring systems used in a rotation viscometer usually consist of a rotating and a rigid part. 
The rotating plunger is turned at either a preset speed or a preset torque. 
 
Two measuring principles are distinguished in coaxial measuring systems: 
(1) The SEARLE principle Rotating plunger and resting cup (or bottom plate) 
(2) The COUETTE principle Rotating cylinder (or bottom plate) and resting plunger 
 
Measuring systems which employ the COUETTE principle have drawbacks when it comes to 
temperature control, because it is more difficult technically to seal a rotating face than a fixed one. 
This is why the use of COUETTE systems is often restricted to low speeds. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 8-1: Cylinder measuring arrangement- Cylinder measuring arrangement 
 according to the SEARLE principle according to the COUETTE principle 
 42 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
Ra 
8.1 Coaxial cylinder measuring systems 
 
DIN 53018 describes the coaxial measuring system. Coaxial means that rotating member and 
resting member of the measuring system are disposed on one rotation axis. Such cylinder 
measuring systems are also known as concentric measuring systems. The terms of the parallel 
plate model can be applied to the round cylinder, if its surface area is idealised as many small 
plane faces. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Ri ..... Radius of the plunger (inner cylinder) 
Ra .... Radius of the cup (outer cylinder) 
r ...... Distance of a layer of the liquid from the rotation axis 
v(r) .... Distribution of the circumferential speed in the measuring gap 
 
Fig. 8-2: Cross-section through a Searle cylinder measuring system 
 
 
The shear stress distribution τ(r) and the shear rate distribution γ& (r) are dependent on the radii of 
the measuring system. In order to obtain near-linear distributions γ& (r) and τ(r) , the measuring gap 
must not be too large. DIN 53019 thus specifies a maximum ratio of radii δ = Ra/Ri In other words, 
the DIN standard defines the ratio of radii, but not the absolute radii or gap size. 
 
The ratio of radii δ = Ra/Ri is specified in the DIN standard to be ≤ 1.1, preferably 1.0847. 
Ri
v(r) 
 43 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
3
iR
M0446.0 ⋅
=τ
n291.1 ⋅=γ&
Coaxial cylinder measuring system according to DIN 53019 
 
 
 
 
 
 
 
 
L ….. Length of the plunger (inner cylinder) 
L’’ ….. Immersed skirt length 
L’ ….. Distance between bottom edge of plunger 
and cup base 
Ri ..... Radius of the plunger (inner cylinder) 
Ra ….. Radius of the cup (outer cylinder) 
Rs ..... Radius of the plunger skirt 
α ...... Opening angle of the plunger cone 
 
 
 
Fig. 8-3: Cylinder measuring system according to DIN 53019 / ISO 3219 
 
 
DIN 53019 prescribes the following geometrical arrangement for cylinder measuring systems: 
 
Ratio of radii δ = Ra/Ri ≤ 1.1 (preferably 1.0847) 
90° ≤ α ≤ 150° (preferably 120° ± 1°) 
L / Ri ≥ 3 (preferably 3.00) 
L1 / Ri ≥ 1 (preferably 1.00) 
L2 / Ri ≥ 1 (preferably 1.00)Rs / Ri ≤ 0.3 
 
Calculation of the shear stress τ from the torque M (measuring systems according to DIN 53019) 
 
 
M in [mNm], Ri in [m], τ in [Pa] 
 
Calculation of the shear rate γ& from the rotational speed n (measuring systems according to 
DIN 53019) 
 
 n in [min-1], γ& in [s-1] 
 
α 
 44 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
8.2 Dual-slit measuring systems according to DIN 54453 
 
This special coaxial cylinder measuring system with a particularly large shear area was 
standardised to be able to investigate materials which have a very small viscosity (such as water-
based lacquers, for example). The plunger has the shape of a tube and the cup has a cylindrical 
core section. This dual-slit measuring system thus takes advantage of two shear faces, an inner 
and an outer plunger surface. 
 
 
L
R
R
1
2
R3
R4
 
Fig. 8-4: Dual-slit measuring system according to DIN 54453 
 
 
According to DIN 54453, 
ratio of radii: δ = R4 / R3 = R2 / R1 ≤ 1.15 
immersed length: L ≥ 3 · R3 
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 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
8.3 Cone-plate measuring systems according to ISO 3219 
 
In the cone-plate measuring system, the material to be investigated is disposed between the 
bottom plate and the measuring cone. According to DIN 53018, the cone angle must be rather 
small in order to allow the simplified expression tan β = β (β in rad) to be applied. The cone angle 
of that measuring system is chosen such that for each point on the cone surface the ratio of 
angular speed and distance to the plate is constant. This means that there is a constant shear rate 
across the entire radius of the measuring cone. The tip of the cone must just touch the bottom 
plate in this geometry. In order to prevent the cone tip from wear and at the same time to provide 
the possibility to measure materials which contain fillers, most rheometer manufacturers lift the 
cone tip by a certain amount (30–180 µm). Measuring cones with an angle of 1 ° are most wide-
spread. However, in order to be more flexible when it comes to measuring materials which contain 
fillers, dispersions are often measured using cones with an angle of 4 °. This guarantees a laminar 
flow to be generated in the measuring gap despite the dispersed particles. As a rule of thumb, a 
laminar flow can be assumed as long as the particle diameter is five times smaller than the gap. 
 
In addition to a constant shear rate across the entire measuring gap, the cone-plate measuring 
system has further advantages, such as high shear rates, small sample quantities and easy 
cleaning. 
 
In order to obtain accurate results of the measurements, it is of major importance to conduct them 
very carefully. The measuring cone must be adjusted such that its imaginary tip just touches the 
bottom measuring plate. Another point is that of filling the measuring system. It is filled correctly if 
the material under investigation is visible approx. 1 mm around the entire circumference of the 
cone. The material must not escape from the measuring gap or rest on top of the measuring cone 
during the measurement. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Fig. 8-5: Filling the cone-plate measuring system and detail of a cut-off cone tip 
correct filling 
approx. 1 mm 
flattened 
cone tip 
 46 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
MG
R2
M3
K3 ⋅=
⋅π
⋅
=τ
nk ⋅=β
ω
=γ&
 
 
 
 
 
 
 
 
R ..... Outer radius of the cone 
ß ...... Opening angle of the cone 
 
Fig. 8-6: Cone-plate measuring system according to ISO 3219 / DIN 53018 
 
The area A in the parallel plate model corresponds with the cone area A = π · R². The moving force 
is expressed as F = M / R 
 
Calculation of the shear stress τ from the torque M 
 
 
 
 
Torque M in [mNm], Ri in [m], τ in [Pa] 
 
GK is a constant of the measuring system which depends on the cone radius. The larger the cone 
radius the greater is the sensitivity of the measuring system. 
 
Calculation of the shear rate γ& 
The velocity v in the parallel plate model corresponds with the circumferential speed v = ω · R in 
the rotating system. 
 
ω = 2π · n / 60 where ω is the angular speed in [rad/s] und n the rotational speed in [min-1] 
 
Using the simplified expression tan β = β for small cone angles, the shear rate γ& can be calcul-
ated from the rotational speed n according to the following equation: 
 
 
 
 
where k is a conversion factor which is independent of the cone angle. At a cone angle of 1 ° k = 6, 
at an angle of 2 ° k = 3 and at an angle of 4 ° k = 0.75. This means that at a constant rotational 
speed the shear rate is the higher the smaller the cone angle. Angles are given in [rad] or [degree], 
where 2π rad = 360° i.e. 1 rad = 57.3° and 1° = 0.0175 rad. 
ß 
R 
 47 
 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
( ) MG
R2
M3R P3 ⋅=
⋅π
⋅
=τ
8.4 Plate-plate measuring systems 
 
The plate-plate measuring system consists of two parallel plates. It is characterised by the plate 
radius R and the variable distance h between the plates. The material under investigation is 
disposed between the two plates. DIN 53018 specifies that the plate distance h shall be much 
smaller than the radius of the measuring plate. A gap of between 0.3 mm and 3 mm is 
recommended. Plate-plate measuring devices are used if the material under investigation contains 
large filler particles. The gap should be determined such that it is at least five times as large as the 
largest particles contained in the material. The shear rate is not constant across the entire plate 
radius, like in the cone-plate system, but there is a relatively large shear rate range. The shear rate 
in the centre of the upper measuring plate is zero. The specified shear rate is always that related to 
the outer radius R of the measuring plate, that is the maximum shear gradient. 
 
 
 
 
 
 
 
 
 
 
 
 
 
R ..... Radius of the measuring plate 
H ..... Distance (gap) between the upper and lower measuring plate 
 
Fig. 8-7: Plate-plate measuring system 
 
 
The area A in the parallel plate model corresponds with the rotating area A of the upper measuring 
plate, where A = π·R2. The moving force is expressed as F = M / R. 
 
Calculation of the shear stress τ from the torque M 
 
 
 
 
Torque M in [mNm], R in [m], τ in [Pa] 
 
GP is a constant of the measuring system which depends on the plate radius and the distance h. 
The larger the plate radius the greater is the sensitivity of the measuring system. 
h 
R 
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 RheoTec Messtechnik GmbH, Ottendorf-Okrilla Introduction to rheology V2.1 E.doc 
 
( )
h
R
h
vR ⋅ω==γ&
Calculation of the shear rate γ& 
The plate distance h in the parallel plate model corresponds with the distance h between the upper 
and lower measuring plate; the velocity v in the parallel plate model corresponds with the 
circumferential speed v = ω · R in the rotating system. 
 
ω = 2π · n / 60 where ω is the angular speed in [rad/s] und n the rotational speed in [min-1] 
 
In contrast to the cone-plate measuring system, the shear rate depends on the radius. It is zero in 
the centre of the plate (r = 0), has a maximum at the edge of the plate (r = R) and shows a linear 
gradient. 
 
 
 
 
At a constant angular speed ω or rotational speed n, if the plate distance h is increased the shear 
rate in the measuring gap will fall.