RheologyPart 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 “visco-elastic”.

• 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 physical-chemical 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 dough-like 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 “pseudo-plastic 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 “up-curve” --, 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