Rheology
Part 1
LMM
Introduction to Rheology
Introduction to Rheology
• Rheology describes the deformation of a body
under the influence of stresses.
• “Bodies” in this context can be either solids,
liquids, or gases.
• Ideal solids deform elastically.
• The energy required for the deformation is
fully recovered when the stresses are
removed.
Introduction to Rheology
• Ideal fluids such as liquids and gases deform
irreversibly -- they flow.
• The energy required for the deformation is
dissipated within the fluid in the form of heat
and cannot be recovered simply by removing
the stresses.
Introduction to Rheology
• The real bodies we encounter are neither
ideal solids nor ideal fluids.
• Real solids can also deform irreversibly under
the influence of forces of sufficient magnitude
• They creep, they flow.
• Example: Steel -- a typical solid -- can be
forced to flow as in the case of sheet steel
when it is pressed into a form, for example for
automobile body parts.
Introduction to Rheology
• Only a few liquids of practical importance come
close to ideal liquids in their behavior.
• The vast majority of liquids show a rheological
behavior that classifies them to a region
somewhere between the liquids and the solids.
• They are in varying extents both elastic and
viscous and may therefore be named “viscoelastic”.
• Solids can be subjected to both tensile and shear
stresses while liquids such as water can only be
sheared.
Ideal solids subjected to shear stresses
react with strain:
Introduction to Rheology
• τ = G ⋅ dL/dy = G ⋅ tan γ ≈ G ⋅ γ
– τ = shear stress = force/area, N/m 2 = Pa
– G = Young’s modulus which relates to the stiffness
of the solid, N/m 2 =Pa
– γ = dL/y = strain (dimensionless)
– y = height of the solid body [m]
– ΔL = deformation of the body as a result of shear
stress [m].
Introduction to Rheology
• The Young’s modulus G in this equation is a
correlating factor indicating stiffness linked
mainly to the chemical-physical nature of the
solid involved.
• It defines the resistance of the solid against
deformation.
Introduction to Rheology
• The resistance of a fluid against any
irreversible positional change of its’ volume
elements is called viscosity.
• To maintain flow in a fluid, energy must be
added continuously.
Introduction to Rheology
• While solids and fluids react very differently
when deformed by stresses, there is no basic
difference rheologically between liquids and
gases.
• Gases are fluids with a much lower viscosity
than liquids.
• For example hydrogen gas at 20°C has a
viscosity a hundredth of the viscosity of water.
Introduction to Rheology
• Instruments which measure the visco-elastic
properties of solids, semi-solids and fluids are
named “rheometers”.
• Instruments which are limited in their use for
the measurement of the viscous flow behavior
of fluids are described as “viscometers”.
Shear induced flow in liquids can occur
in 4 laminar flow model cases:
Flow between two parallel flat plates
• When one plate moves and the other is
stationary.
• This creates a laminar flow of layers which
resembles the displacement of individual
cards in a deck of cards.
Flow in the annular gap between two
concentric cylinders.
• One of the two cylinders is assumed to be
stationary while the other can rotate.
• This flow can be understood as the
displacement of concentric layers situated
inside of each other.
• A flow of this type is realized for example in
rotational rheometers with coaxial cylinder
sensor systems.
Flow through pipes, tubes, or
capillaries.
• A pressure difference between the inlet and
the outlet of a capillary forces a Newtonian
liquid to flow with a parabolic speed
distribution across the diameter.
• This resembles a telescopic displacement of
nesting, tube-like liquid layers sliding over
each other.
Flow through pipes, tubes, or
capillaries.
• A variation of capillary flow is the flow in
channels with a rectangular cross-section such
as slit capillaries.
• If those are used for capillary rheometry the
channel width should be wide in comparison
to the channel depth to minimize the side wall
effects.
Flow between two parallel-plates or
between a cone-and-plate sensor
• Where one of the two is stationary and the
other rotates.
• This model resembles twisting a roll of coins
causing coins to be displaced by a small angle
with respect to adjacent coins.
• This type of flow is caused in rotational
rheometers with the samples placed within
the gap of parallel-plate or cone-and-plate
sensor systems.
Aspects of Rheology
The basic law
• The measurement of the viscosity of liquids
first requires the definition of the parameters
which are involved in flow.
• Then one has to find suitable test conditions
which allow the measurement of flow
properties objectively and reproducibly.
The basic law
• Isaac Newton was the first to express the basic
law of viscometry describing the flow behavior
of an ideal liquid:
   * 
• shear stress = viscosity ⋅ shear rate
The parallel-plate model helps to define
both shear stress and shear rate
Shear stress
• A force F applied tangentially to an area A
being the interface between the upper plate
and the liquid underneath, leads to a flow in
the liquid layer.
• The velocity of flow that can be maintained
for a given force is controlled by the internal
resistance of the liquid, i.e. by it’s viscosity.
Shear stress
• τ =F (force)/A (area)
–N (Newton)/m 2 = Pa [Pascal]
Shear rate
• The shear stress τ causes the liquid to flow in a
special pattern.
• A maximum flow speed Vmax is found at the
upper boundary.
• The speed drops across the gap size y down to
Vmin = 0 at the lower boundary contacting the
stationary plate.
Shear rate
• Laminar flow means that infinitesimally thin
liquid layers slide on top of each other, similar to
the cards in a deck of cards.
• One laminar layer is then displaced with respect
to the adjacent ones by a fraction of the total
displacement encountered in the liquid between
both plates.
• The speed drop across the gap size is named
“shear rate” and in it’s general form it is
mathematically defined by a differential.
Shear rate
In case of the two parallel plates with a linear speed
drop across the gap the differential in the equation
reduces to:
Shear rate
• In the scientific literature shear rate is
denoted as 
• The dot above the γ indicates that shear rate
is the time-derivative of the strain caused by
the shear stress acting on the liquid lamina.
Shear rate
Solids vs Liquids
• Comparing equations [1] and [7] indicates
another basic difference between solids and
liquids:
• Shear stress causes strain in solids but in liquids it
causes the rate of strain.
• This simply means that solids are elastically
deformed while liquids flow.
• The parameters G and η serve the same purpose
of introducing a resistance factor linked mainly to
the nature of the body stressed.
Dynamic viscosity
• Solving equation [2] for the dynamic viscosity
η gives:
Dynamic viscosity
• The unit of dynamic viscosity η is the “Pascal ⋅
second” [Pa⋅s].
• The unit “milli-Pascal ⋅ second” [mPa⋅s] is also
often used.
– 1 Pa ⋅ s = 1000 mPa ⋅ s
• It is worthwhile noting that the previously
used units of “centiPoise” [cP] for the dynamic
viscosity η are interchangeable with [mPa⋅s].
– 1 mPa⋅s = 1 cP
Typical viscosity values at 20°C
[mPa⋅s]
Kinematic viscosity
• When Newtonian liquids are tested by means
of some capillary viscometers, viscosity is
determined in units of kinematic viscosity υ.
• The force of gravity acts as the force driving
the liquid sample through the capillary.
• The density of the sample is one other
additional parameter.
Kinematic viscosity
• Kinematic viscosity υ and dynamic viscosity η
are linked.
Flow and viscosity curves
• The correlation between shear stress and
shear rate defining the flow behavior of a
liquid is graphically displayed in a diagram of τ
on the ordinate and on the abscissa.
• This diagram is called the “Flow Curve”.
• The most simple type of a flow curve is shown
In Figure 4.
• The viscosity in equation(2) is assumed to be
constant and independent of  .
Flow Curve
Viscosity Curve
• Another diagram is very common: η is plotted versus

• This diagram is called the “Viscosity Curve”.
• The viscosity curve shown in Fig. 5 corresponds to the
flow curve of Fig. 4.
• Viscosity measurements always first result in the flow
curve.
• It’s results can then be rearranged mathematically to
allow the plotting of the corresponding viscosity curve.
• The different types of flow curves have their
counterparts in types of viscosity curves.
Viscosity Curve
Viscosity parameters
• Viscosity, which describes the physical
property of a liquid to resist shear-induced
flow, may depend on 6 independent
parameters:
Viscosity Parameters
• “S” - This parameter denotes the physicalchemical nature of a substance being the primary
influence on viscosity, i.e. whether the liquid is
water, oil, honey, or a polymer melt etc.
• “T” - This parameter is linked to the temperature
of the substance. Experience shows that viscosity
is heavily influenced by changes of temperature.
As an example: The viscosity of some mineral oils
drops by 10% for a temperature increase of only
1°C.
Viscosity Parameters
• “p” - This parameter “pressure” is not experienced as
often as the previous ones.
• Pressure compresses fluids and thus increases
intermolecular resistance.
• Liquids are compressible under the influence of very
high pressure-- similar to gases but to a lesser extent.
• Increases of pressure tend to increase the viscosity.
• As an example: Raising the pressure for drilling mud
from ambient to 1000 bar increases it’s viscosity by
some 30%.
Viscosity Parameters
•  -Parameter “shear rate” is a decisive factor
influencing the viscosity of very many liquids.
• Increasing shear rates may decrease or increase
the viscosity.
• “t” Parameter “time” denotes the phenomenon
that the viscosity of some substances, usually
dispersions, depends on the previous shear
history, i.e. on the length of time the substance
was subjected to continuous shear or was
allowed to rest before being tested.
Viscosity Parameters
• “E” - Parameter “electrical field” is related to a family of
suspensions characterized by the phenomenon that their
flow behavior is strongly influenced by the magnitude of
electrical fields acting upon them.
• These suspensions are called either “electro-viscous
fluids” (EVF) or “electro-rheological fluids” (ERF).
• They contain finely dispersed dielectric particles such as
aluminum silicates in electro-conductive liquids such as
water which may be polarized in an electrical field.
• They may have their viscosity changed instantaneously and
reversibly from a low to a high viscosity level, to a doughlike material or even to a solid state as a function of
electrical field changes, caused by voltage changes.
Substances
Types of Fluids
Newtonian Liquids
• Newton assumed that the graphical equivalent of
his equation [2] for an ideal liquid would be a
straight line starting at the origin of the flow
curve and would climb with a slope of an angle α.
• Any point on this line defines pairs of values for τ
and  .
• Dividing one by the other gives a value of η ([8]).
• This value can also be defined as the tangent of
the slope angle α of the flow curve: η = tan α .
Newtonian Liquids
• Because the flow curve for an ideal liquid is
straight, the ratio of all pairs of τ and -values
belonging to this line are constant.
• This means that η is not affected by changes in
shear rate.
• All liquids for which this statement is true are
called “Newtonian liquids” (both curves 1 in Fig.
6).
• Examples: water, mineral oils, bitumen, molasses.
Non-Newtonian Liquids
• All other liquids not exhibiting this “ideal”
flow behavior are called “Non-Newtonian
Liquids”.
• They outnumber the ideal liquids by far.
Pseudo-plastic Liquids
• Liquids which show pseudo-plastic flow
behavior under certain conditions of stress
and shear rates are often just called “pseudoplastic liquids” (both curves 2 in Fig. 6)
• These liquids show drastic viscosity decreases
when the shear rate is increased from low to
high levels.
Pseudo-plastic Liquids
• Technically this can mean that for a given
force or pressure more mass can be made to
flow or the energy can be reduced to sustain a
given flow rate.
• Fluids which become thinner as the shear rate
increases are called “pseudo-plastic”.
• Very many substances such as emulsions,
suspensions, or dispersions of technical and
commercial importance belong to this group.
Pseudo-plastic Liquids
Pseudo-plastic Liquids
• Many liquid products that seem homogeneous
are in fact composed of several ingredients:
particles of irregular shape or droplets of one
liquid are dispersed in another liquid.
• On the other hand there are polymer solutions
with long entangled and looping molecular
chains.
• At rest, all of these materials will maintain an
irregular internal order and correspondingly they
are characterized by a sizable internal resistance
against flow, i.e. a high viscosity.
Pseudo-plastic Liquids
• With increasing shear rates, matchstick-like
particles suspended in the liquid will be
turned lengthwise in the direction of the flow.
• Chain-type molecules in a melt or in a solution
can disentangle, stretch and orient themselves
parallel to the driving force.
• Particle or molecular alignments allow
particles and molecules to slip past each other
more easily.
Pseudo-plastic Liquids
• Shear can also induce irregular lumps of
aggregated primary filler particles to break up
and this can help a material with broken-up filler
aggregates to flow faster at a given shear stress.
• For most liquid materials the shear-thinning
effect is reversible -- often with some time lag -i.e. the liquids regain their original high viscosity
when the shearing is slowed down or terminated:
the chain-type molecules return to their natural
state of non-orientation.
Pseudo-plastic Liquids
• At very low shear rates pseudo-plastic liquids
behave similarly to Newtonian liquids having a
defined viscosity η0 independent of shear rate -often called the “zero shear viscosity”.
• A new phenomenon takes place when the shear
rate increases to such an extent that the shear
induced molecular or particle orientation by far
exceeds the randomizing effect of the Brownian
motion: the viscosity drops drastically.
Pseudo-plastic Liquids
• Reaching extremely high shear rates the
viscosity will approach asymptotically a finite
constant level: η1.
• Going to even higher shear rates cannot
cause further shear-thinning: The optimum of
perfect orientation has been reached.
Pseudo-plastic Liquids
Dilatant Liquids
• There is one other type of material characterized
by a shear rate dependent viscosity: “dilatant”
substances -- or liquids which under certain
conditions of stress or shear rate show increasing
viscosity whenever shear rates increase. (Curves
3 in Fig. 6)
• Dilatancy in liquids is rare.
• This flow behavior most likely complicates
production conditions, it is often wise to
reformulate the recipe in order to reduce
dilatancy.
Plasticity
• It describes pseudo-plastic liquids which
additionally feature a yield point. (both curves 4
in Fig. 6)
• They are mostly dispersions which at rest can
build up an intermolecular/interparticle network
of binding forces (polar forces, van der Waals
forces, etc.).
• These forces restrict positional change of volume
elements and give the substance a solid character
with an infinitely high viscosity.
Plasticity
• Forces acting from outside, if smaller than those
forming the network, will deform the shape of
this solid substance elastically.
• Only when the outside forces are strong enough
to overcome the network forces -- surpass the
threshold shear stress called the “yield point” -does the network collapse.
• Volume elements can now change position
irreversibly: the solid turns into a flowing liquid.
Plasticity
• Typical substances showing yield points
include oil well drilling muds, greases, lipstick
masses, toothpastes and natural rubber
polymers.
• Plastic liquids have flow curves which
intercept the ordinate not at the origin, but at
the yield point level of τ0.
Thixotropy
• For pseudo-plastic liquids, thinning under the
influence of increasing shear depends mainly
on the particle/molecular orientation or
alignment in the direction of flow surpassing
the randomizing effect of the Brownian
movement of molecules.
• This orientation is again lost just as fast as
orientation came about in the first place.
Thixotropy
• Plotting a flow curve of a non-Newtonian
liquid not possessing a yield value with a
uniformly increasing shear rate -- the “upcurve” --, one will find that the “down-curve”
plotted with uniformly decreasing shear rates
will just be superimposed on the “up-curve”:
they are just on top of each other or one sees
one curve only.
Thixotropy
Thixotropy
• It is typical for many dispersions that they not only
show this potential for orientation but additionally for
a time-related particle/molecule-interaction.
• This will lead to bonds creating a three-dimensional
network structure which is often called a “gel”.
• In comparison to the forces within particles or
molecules, these bonds -- they are often hydrogen or
ionic bonds -- are relatively weak: they rupture easily,
when the dispersion is subjected to shear over an
extended period of time (Fig. 9).
Thixotropy
• When the network is disrupted the viscosity drops with
shear time until it asymptotically reaches the lowest
possible level for a given constant shear rate.
• This minimum viscosity level describes the “sol”-status
of the dispersion.
• A thixotropic liquid is defined by it’s potential to have
it’s gel structure reformed, whenever the substance is
allowed to rest for an extended period of time.
• The change of a gel to a sol and of a sol to a gel is
reproducible any number of times.
Thixotropy
Thixotropy
• Fig. 10 describes thixotropy in graphical form.
• In the flow curve the “up-curve” is no longer
directly underneath the “down-curve”.
• The hysteresis now encountered between these
two curves surrounds an area “A” that defines the
magnitude of this property called thixotropy.
• This area has the dimension of “energy” related
to the volume of the sample sheared which
indicates that energy is required to break down
the thixotropic structure
Thixotropy
• For the same shear rate there are now two
different points I and II.
• These two viscosity values are caused by a
shear history at I being much shorter than at
II.
• If it took 3minutes to get to point I and 6
minutes to the maximum shear rate, it will be
9 minutes until point II is reached.
Rheopectic Flow Behavior
• Rheopective liquids are characterized by a
viscosity increase related to the duration of shear.
• When these liquids are allowed to rest they will
recover the original -- i.e. the low -- viscosity
level.
• Rheopective liquids can cycle infinitely between
the shear-time related viscosity increase and the
rest-time related decrease of viscosity.
• Rheopexy and thixotropy are opposite flow
properties.
• Rheopexy is very rare.
Types of Rheometers
Controlled Stress
When to Use
Plate and Cone
Plate and Cone
Plate and Cone
Plate and Cone
Parallel Plate
Parallel Plate
Parallel Plate



Capillary Rheometer
Shear rate calculation for capillary
rheometer
Viscosity calculation for capillary
rheometer