Introduction to Rheology
Notes for the Rheology course – Dr. Berta - September 2011
Despite fact that lessons are in French, the notes for this course are written in English. At a Master degree level a student is supposed toread it and speak it correctly, as it is currently the primary language of Science. However, for didactic purpose, some translations arealready given in this document for the most important terms.
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Contents
1 – Basics of RheologyThe basic terms and definitions: what you need to know to be sure of what you are talking about.
2 - Flow characterisationViscous behaviour, flow curves and associated rheological models.
3 - Creep analysisCreep and basics of viscoelastic behaviour.
4 - Viscoelastic oscillatory characterisationHow to learn about a viscoelastic material with oscillatory tests.
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1. Basics of Rheology
What is Rheology? This is the definition from Wikipedia:
Rheology is the science of deformation and flow.
Ok, now in French so you will have a ready answer when your friends will inevitably ask: “c'estquoi???”
La Rhéologie (du grec “rheo”, couler et “logos”, étude) est l'étude de la déformation et de l'écoulementde la matière sous l'effet d'une contrainte appliquée. Le terme est emprunté à la fameuse expressiond'Héraclite d'Éphèse “panta rei”, “Ça s'écoule toujours”.
Stress and strain. Definitions and measurement in a rheological test.
When a force is applied to a material, the typical curve which describes its behaviour is the stress-strain. It will follow you throughout all the Physics of materials whether polymer, metals, gels etc. Sobefore telling you what is an ideal elastic, a perfect fluid etc. let's make sure these two definitions areunder the belt. When measuring the rheological properties, the force is typically applied by SHEAR(cisaillement).
To define the term STRAIN (déformation) we will consider a cube of material with its base fixed to asurface (See Figure-1).
Figure-1 ideal cube
We apply a constant 'pushing' force, F, to the upper part of the cube. Something like pushing forwardthe top surface of a gelatine cube with the finger. A stress will be applied. The SHEAR STRESS(contrainte de cisaillement) is defined as σ = F/A (A is the area of the upper surface of the cube: l x w)and can be thought as the pressure applied by the force. Since the units of force are Newtons and theunits of area are m2 it follows that the units of Shear Stress are N/m2 . This is referred to as thePASCAL, same unity of pressure indeed (i.e. 1 N/m2 = 1 Pascal). The area is a constant in this case, soif we measure the force applied is easy to work out the stress. The force for a rotating engine isproportional to the torque exerted, which is measurable much more precisely than the force of a fingeron the gelly cube. By measuring the TORQUE (torseur) and knowing the geometry of the material (the
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cube in this case) the applied stress can be estimated.
Figure-2 Shear deformation of a cube
The cube will deform to a new position (Figure-2) This type of deformation (lower fixed, uppermoving) is defined as a SHEAR DEFORMATION.
The displacement δu and h are used to define the SHEAR STRAIN as :
Shear Strain = = δu/h
The shear strain is simply a ratio of two lengths and so has no units. It is important since it enables usto quote pre-defined deformations without having to specify sizes of sample, etc.
Since h is also constant in this case, strain can be estimated by measuring the DISPLACEMENT. Theeesponse to the applied shear will depend on the material, let's consider the two ideal behaviours.
The Hookean response or the ideal elastic solid: simple deformation.
For a purely elastic material Hooke's law states that the stress is proportional to the strain i.e.
Stress = G x Strain
where G is defined as the SHEAR MODULUS (a constant)
Thus doubling the stress would double the strain i.e. the material is behaving with a LINEARRESPONSE. If the stress is removed, the strain returns instantaneously (assuming no inertia) to zero,i.e. the material has undergone a fully recoverable deformation and so NO FLOW HAS OCCURRED.It is the same of a perfect spring. Mechanical energy given by the applied stress is stored by thematerial which recovers to the initial position.
The higher G, the stiffer the material. However stiffness is normally measured by tensile tests ratherthan shear tests, and for polymers is expressed by the Young modulus*1 E. it is in the range ofGigapascal. Good to know, so when you talk about the modulus of an industrial polymer you canreasonably say it is betweeen 1 and 10 GPa without having the slightest idea of the real value. If moreaccuracy is needed, here is a table of Young moduli:
*1 Don't tell me you forgot what Young Modulus is !!!! In such a case run, hide and secretly consult wikipedia.
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It is just for reference (and I left names in French), no need to memorize those values. And if you needeven more details about a specific polymer, you can have a look at
http://www.matbase.com/material/polymers/
The Newtonian response or the ideal viscous fluid: simple flow.
Let us again consider the case of the cube of material as described above but in this case let's assumethat the material behaves as an ideal fluid. When we apply the shear stress (force) the material willdeform as before, but in this case it will not recover the deformation will continually increase at aconstant rate (Figure-3).
The mechanical energy given by the applied stress is dissipated (lost) by the material.
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Polymers and fibers
Caoutchouc 0.001 à 0.1Fibre de carbone haut module 640Fibre de carbone haute résistance 240Kevlar 34.5Nanotubes (Carbone) 1100Nylon 2 à 5Plexiglas (PMMA) 2.38Polyamide 3 à 5Polycarbonate 2.3Polyéthylène (PE) 0.2 à 0.7Polystyrène 3 à 3.4Résines époxy 3.5
Biomaterials
Bec de poussin 50Cartilage 24Cheveux 10Collagène 6Fémur 17,2Humérus 17,2Piquant d'oursin (sea urchin nail) 15 à 65Radius 18,6Soie d'araignée (spider web fiber) 60Tibia 18,1Vertèbre cervicale 0,23Vertèbre lombaire 0,16
Module (GPa)
Figure-3 the viscous behaviour of a deformed ideal material.
The rate of change of strain is referred to as the SHEAR STRAIN RATE (taux de cisaillement) oftenabbreviated to SHEAR RATE and is a function of time. As a consequence, it depends on the speed ofdisplacement.
SHEAR RATE = SHEAR STRAIN / TIME
˙ = / t
The Shear Rate obtained from an applied Shear Stress will be dependant upon the material’s resistanceto flow i.e. its VISCOSITY denoted by the symbol η.Since the flow resistance is the ratio between force and displacement, it follows that
VISCOSITY = SHEAR STRESS / SHEAR RATE
η = σ / ˙
The units of viscosity are [ N x m-2x s ] and are known as Pascal Seconds (Pas). But most used arePoise and centipoise (cP)
1 Pa x s = 10 Poise
If a material has a viscosity which is independent of shear stress, then it is referred to as an ideal orNEWTONIAN fluid.
Again below are given some reference values, the shear rate for some common processes
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and some viscosity values for common liquids and mixtures (room temperature except indicatedotherwise):
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Process
SprayingRubbing (frotter)Curtain coatingMixing / Stirring (melanger)StirringBrushing (brosser)Chewing (mastication)PumpingExtrudingLevellingPaint dripping (Vernis qui coule)Sedimentation
Shear rate [s-1]
104 - 105
104 - 105
102 – 103
102 – 103
101 – 103
101 – 102
101 – 102
100 – 103
100 – 102
10-1 – 10-2
10-1 – 10-2
10-1 – 10-3
Viscosity of fluids with fixed compositions
Liquid
acetone 0.306benzene 0.604blood (37 °C) 3–4castor oil 0.985 985corn syrup 1.3806 1380.6ethanol 1.074ethylene glycol 16.1Glycerol (20 °C) 1.2 1200mercury 1.526methanol 0.544motor oil SAE 10 (20 °C) 0.065 65motor oil SAE 40 (20 °C) 0.319 319nitrobenzene 1.863liquid nitrogen (-196 ºC) 0.158propanol 1.945olive oil 0.081 81quark–gluon plasma 5.00E+011 5.00E+014sulfuric acid 24.2water 0.894
Viscosity[Pa·s]
Viscosity[cP=mPa·s]
3.06×10−4
6.04×10−4
(3–4)×10−3
1.074×10−3
1.61×10−2
1.526×10−3
5.44×10−4
1.863×10−3
1.58×10−4
1.945×10−3
2.42×10−2
8.94×10−4
Models and Greek letters
Here the shear stress is denoted by the symbol σ and the corresponding strain by the symbol In oldertextbooks you may see stress denoted as τ to distinguish it from the tensile stress applied during tensiletests. Strain when applied to a viscous material is often defined by but in some textbooks you findinstead. There may be historical reasons behind for which nobody cares and all this just helpsconfusing students. I can see no reason to change letters when you are measuring essentially the samething. However, to be consistent with the main literature but still logical, in this text I define G andthe modulus and strain measured when a shear stress is applied. E and the same for bending andelongation.
For people lacking imagination there are also the ideal mechanical models: a spring representing anideal elastic behaviour and a dash-pot representing an ideal viscous behaviour. I would prefer an elasticlike those used to be thrown during elementary school lessons (“office consumables” nowadays) and adrop of honey but traditions are traditions so there we go:
Figure-4 Mechanical elements for the Maxwell-Voigt model of material response, guess which is which.
By combining them you can claim that real material will never be totally viscous or totally elastic, but“viscoelastic” (see below). “Those pure behaviours exist only for theoretical Physicists” (sentence forshowing off as an experienced scientist). True, but who ever cares about steel viscosity or oil elasticity?That is why after the understanding the fundamental physics behind elasticity and viscosity, we willlook more in details at the flow behaviour of viscous solutions. Then we will wander in the realm of
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Viscosity of fluids with variable compositions
honey 2–10 2,000–10,000molasses 5–10 5,000–10,000molten glass 10–1,000 10,000–1,000,000chocolate syrup 10–25 10,000–25,000molten chocolate 45–130 45,000–130,000ketchup 50–100 50,000–100,000lard ≈ 100 ≈ 100,000peanut butter ≈ 250 ≈ 250,000
Fluid Viscosity[Pa·s]
Viscosity[cP]
viscoelasticity.
Questions for the mirror, before the exams
What is the difference between tensile and shear stress?
Does shear stress have the same measurement units of pressure?
What is the material modulus G? Can you write the equation which defines it?
The stiffness of a polymer is normally expressed by the Young (E) or by the Shear (G) modulus? Andin which range of GPa does it fall into?
Draw a stress-strain curve for an ideal elastic material
In a pure elastic material the energy given by the applied stress is stored or lost?
Which is the shear strain unit? (sleeping or not question)
What is viscosity? Can you write the equation which defines it?
In a pure viscous material the energy given by the applied stress is stored or lost?
Which one is bigger, cP or Pa*s ? If you want to stay in a range of 1-10 which of these two units youwill use for expressing blood viscosity? And for honey viscosity?
Hookean and Newtonian are limits behaviours. Which one describes simple deformation (elasticity)and which describes simple flow (viscosity)?
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2. Flow characterisation
Flow curves – the non ideal behaviour of viscous materials
Newtonian behaviour is only a limit condition as it was stated before. It assumes that at with a constantstress viscosity is constant when the strain rate increases. Simply a linear relationship. This is notalways the case, depending on conditions of shear and materials properties deviations from linearitymay occur (see below, in Figure-5).
Figure-5 the four most common viscous behaviours of real materials. In French: le fluide est (1) newtonien, (2) deBingham, (3) rhéofluidifiant, (4) rhéoépaississant.
Since it is the relationship of shear stress to shear rate that are strictly related to flow we can directlyshow the flow characteristics of a material by plotting shear stress v shear rate. A graph of this type(Figure-5) is called a “Flow Curve”.
What happens to the measured viscosity in each case? The measured viscosity of a fluid normally canbe seen to behave in one of four ways when sheared, namely :
1 . Viscosity remains constant no matter what the shear rate (Newtonian behaviour) 2 . Viscosity appears to be infinite until a certain shear stress point is achieved (Bingham plastic) 3 . Viscosity decreases as shear rate is increased (Shear thinning behaviour) 4 . Viscosity increases as shear rate is increased (Shear thickening behaviour)
Since the viscosity is not a constant in most cases but a function of shear rate, an “apparent” viscosity isdefined according to the shear conditions. In other words:
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Apparent viscosity = the viscosity of a fluid measured at a given shear rate at a fixed temperature.
That is what the instrument measures, but what we perceive is how viscosity changes with the stresslevel or with the shear strain rate. Imagine spraying and seeing that if you press more on the gas youdon't obtain a good jet anymore. Your mural artistic aspiration will be frustrated and you wish you hada rheometer to test that paint before. So, let's take some time and plot how viscosity varies in the fourcases mentioned earlier. Below (Figure-6), three of them.
Figure-6 Viscosity as a function of shear rate for shear thinning and thickening materials.
In the graph we vary the shear rate but most of the times in real life we apply stress. Those areproportional and but the rheometer may control the displacement during a shear test and measureviscosity as a function of shear rate like in figure-6. In other cases, you may have a stress controlledmachine (rheometer or viscosimeter).
Let's take a closer look to these curves and think what they may represent in real life. Before talkingabout shear thinning and thickening solution, please notice that when the stress/shear rate are lowenough, deviation is not likely to occur but the liquid behaves like a newtonian fluid. This initial steadystate condition is called “lower newtonian plateau”.
A shear thickening material resists deformation more than in linear proportion to the applied force. Forexample, the more effort you put into stirring such a material, the more resistant it becomes to stirring.This is sometimes an indication that the applied force is causing the material to adopt a more orderedstructure. A thick slurry of wet beach sand is often shear thickening: you press it walking on it, squeezeout water and get sand grains closer to each other. At that point, the soft ground becomes harder andyou well feel it with naked feet. Silly putty is another common example of shear thickening behaviour,if you are curious about how it works, have a look at:
http://www.explainthatstuff.com/energy-absorbing-materials.html
Shear thinning behaviour is even more common. Paint is the common example. Its rheologicalbehaviour is due to the polymer chains or 'rods' in a colloidal suspension aligning upon shear andreducing the overall viscosity. Very useful since you want it to be fluid as you apply stress with thebrush but to be not to flow down when sitting on the wall. I don't agree with the fact ketchup can beclassified as shear thinning (as stated in the webpage linked above) but rather like as a “BinghamPlastic fluid”. Fluids obeying Bingham model exhibit a viscous “linear shear-stress to shear-rate”
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˙
behaviour after an initial shear-stress threshold has been reached. This threshold corresponds generallyto the collapse of an initial structure and is called “yield value“ or "yield point". It represents the stressrequired to start fluid movement. Physical causes for the occurrence of yield points in dispersions aregenerally intermolecular particle-particle and particle-dispersing agent interactions. Van Der Waalsforces, Dipole-dipole interactions, Hydrogen bonds, Electrostatic interactions. Or a structure resistantup to a certain stress, like a foam.
Below this threshold the fluid has a virtually infinite viscosity and displays an elastic behaviour. So,you can claim that a “Bingham fluid” can be better defined as viscoelastic rather than viscous.However, for flow measurements the elastic part may be neglected. The reason could be that for itsapplication the material should flow, so the conditions of elasticity at low stresses do not concern us.The Bingham model is widely used in the drilling fluids industry to describe flow characteristics ofmany types of muds. It has been also applied to the laminar flow of fluids into pipes. For BinghamPlastic, it is also worth mentioning that most of the time the initial shear rate is not exactly 0 andviscosity is not infinite but there is some deviation from the model (Figure-7) . And formula may becorrected accordingly if we want to be precise. Deviations in this sense make the fluid resemble ashear-thinning, hence the confusion about ketchup just noted above. And such confusion is also due tothe variety of ketchup brands of course: with some bottled in glass, the amount of shaking required canwell portray the idea of “yield point”, others are squeezed out of a plastic bottle and resemble more aliquid.
Figure-7 Bingham plastic behaviour, some curves show deviation from the model.
To further complicate the picture, over a sufficiently wide range of shears it is often found that thematerial has a more complex characteristic made up of several of the above flow patterns. And if thereare particles in the fluid, their role may not be trivial to explain.
Other definitions for the flow behaviour have been given depending on some memory shape effects(not covered here). More models have been made to explain weird behaviours, relate to microstructuralchanges, new fixtures which better adapt etc. that is why in this section there are on purpose noformulas while in common textbooks you find tons of them for viscosity. Rather than trying to
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remember them, it is much more useful to design a test, see your results and relate to what you knowabout the specific material. How to design a test and select the appropriate instrument will be explainedin a more advanced course. If your material has been investigated before, you can check if thebehaviour you observe relates to a specific model and your curve fits the equation proposed. However,a quick look on the web may satisfy your greed for equations:
http://en.wikipedia.org/wiki/Viscosity
http://physics.info/viscosity/
http://kazmer.uml.edu/Software/ellis-model.html
Those are good if you want to progress into theoretical Colloid Science ad example. Or if you want toimpress friends by explaining the difference between “rheopectic” and “dilatant”.
Temperature-dependent flow behaviour
The viscosity of a material is a function not only of shear rate but also of temperature. Exacttemperature control and accurate indication of the measuring temperature is thus of major importancein 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 phenomenoncan be described with the help of the ARRHENIUS equation:
= ∗exp
where T is the temperature in Kelvin and A and B material constants. The above does not necessarilyhold for polymers since they present a glass transition temperature (so it may just hold above), theymay crosslink etc.
Dependence of viscosity on polymer molecular weight
Molecular weight (Mw) is one of the most important parameters defining polymer performance.Polymer viscosity, or resistance to flow, increases with polymer molecular weight. In other words theability for a polymer to flow decreases with increasing molecular weight. A critical molecular weight,Mc, is defined as the molecular weight above which chain entanglements occur in a polymer melt. In apolymer melt, when molecular weight is lower than Mc, zero shear melt viscosity, η0, is roughlyproportional to molecular weight. At molecular weights above Mc, η0 increases with molecular weightto the power of 3.4 – 3.5.
Mw < Mc η0 = K1M
Mw > Mc η0 = K2M3.4
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Polymer physical properties, such as tensile strength or impact resistance, also increase with increasingmolecular weight. Commercially useful polymers in general have molecular weights higher than M c.By measuring the viscosity you can also work out indirectly the Mw. For who is interested, the fullexplanation and the Mark Houwink equation are given in reference 1.
Questions for the mirror, before the exams
For a newtonian liquid the relationship between stress and shear-rate is linear or exponential?
Other three main behaviours are most commonly observed for fluids, which are they?
Which viscosity behaviour displays the silly putty? Why may be related to walking on the beach?
Which viscosity behaviour is typical of paint?
What is the lower newtonian plateau?
A Bingham fluid displays a viscous behaviour above a certain ... called …. point.
What could be the forces which keep a solution elastic before yielding?
At low enough stress is it correct to say that a Bingham fluid displays an elastic behaviour?
If the initial viscosity is not really infinite for a Bingham fluid and it deviates largely from the model,which kind of behaviour will it resemble?
The temperature dependence of a newtonian liquid is linear or exponential?
When I have a Mw of polymer melt higher than the critical value for entanglement, which is therelationship between viscosity and molecular weight?
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3. Creep analysis
Despite honey is a truly viscous material and, as mentioned earlier, oil elasticity is seldom a concern,elasticity plays an important role for other type of solutions. It strongly affects the flow behaviour andbecomes of great interest. In the case of a Bingham fluid, when the stress level is below the yield point,elasticity is the main characteristic. The creep test provides a method of determining the amount ofelasticity in a sample*2. That is why before talking about viscoelastic characterisation is very importantto understand creep behaviours.
Creep analysis
Creep is defined as 'the slow deformation of a material, usually measured under a constant stress'. If weapply a small stress to a viscoelastic material and hold it constant for a long period of time whilstmeasuring the resultant strain we will see behaviour initially from elastic components followed shortlyby viscoelastic effects. At sufficiently long time scales we will obtain effects only from the viscouscomponents since the resultant strain is large enough to have 'used up' the elastic component.
If we refer back to the previous section on equilibrium flow curves, and a Bingham plastic, we talkedabout solutions behaviour in this later stage when we knew that what we were in a regime of pureviscous flow. In a rheometer we would record data only above a certain stress level (i.e. after the yieldpoint) for such material, if we want to characterise a viscous behaviour. This ignores on pourpose whathappens below that stress level. The creep test instead records the information from the moment weapply the stress and hence gives a measure of elastic, viscoelastic and viscous components. Byapplying small stresses (hence the popularity of stress controlled rheometers for creep measurements) itis also possible to mimic gravitational effects on a sample to assist in predicting effects such assedimentation, and levelling. The shear rates produced under these conditions will typically be of theorder 10-5 to 10-6. Long long times with stress imposed by gravity....
Principle of operation of a creep test
In a creep test a user selected shear stress is 'instantaneously' applied to a sample and the resultantstrain monitored as a function of time. After some predetermined time the stress is removed and thestrain is again monitored. The three typical response curves are shown in figure-8
_________________________________________________________________________________*2 In other cases the amount of elasticity in a liquid is measured by normal forces which develop during a shear test (typicalexample the image of the paste which climbs the rod). You know they exists but we will not discuss about them in thiscourse. More information about is available in Reference 1.
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Figure-8 Creep response for viscous, elastic, and viscoelastic materials.
The third case shows a typical curve produced by a viscoelastic material. The actual shape will bedetermined by the interaction of the viscous and elastic components *3.
Time scales and the Deborah Number
To fully recognise the concept of creep, time factors must be understood with respect to mechanicalbehaviour of the samples. Rapid response times (often fractions of a second) are mainly indicative ofelastic phenomenon whereas viscous phenomenon usually take seconds or even minutes to occur. Thecorrect experiment time is important to enable fast and slow phenomenon to be accurately resolved. Toenable us to put numbers to a materials response characteristics we use a function called the Deborahnumber. Why such a name? The story is given in reference 2.
To understand whether a material will tend to behave more as a fluid or more as a liquid two factorsmust be looked at. These are the time scale of the process/experiment, T and the characteristicrelaxation time of the system,The Deborah number is defined as
De = / T
De < 1 liquid like behaviourDe = 1 viscoelastic behaviourDe > 1 solid like behaviour
_________________________________________________________________________________*3 Since not only the strain level (elastic material) but also the change of strain will be dependant upon the applied stress,the term “compliance” rather than the strain may be used. The compliance is defined simply as the ratio of the strain to theapplied stress and is denoted by the letter J (J=strain/stress). For an elastic J=1/G but for a viscoelastic using this notation,creep curves may be directly compared even if they were not measured under the same applied stress. Just in case youencounter J instead of G (Figure -11), don't be too confused.
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The time τ is infinite for a Hookean elastic solid and zero for a Newtonian viscous liquid and is actuallythe time taken for an applied stress (or strain) to decay to 2/3 its initial value. (The decay isexponential.)
If you think the time of the experiment as an actual process time, the number will determine theoutcome. Let's take water ski ad example. The water relaxation time is very small, but if you speed ishigh enough, to cover a certain distance also the time T will be small. If is constant, you mayincrease the speed to decrease T and get a Deborah number over 1. The water will have an elasticbehaviour on the ski, and you will not sink. Alternatively, you may have lower speed and ski on a fluidwith a higher relaxation time like Nutella for example.A sample may show solid-like behaviour either because it has a relatively long characteristic relaxationtime or it is being subjected to a process of time scales considerably shorter than the materialscharacteristic relaxation time. The creep test takes the sample through short time scale response to longtime scale response as general rule. Then results may be fitted into a model to predict what happens atlonger relaxation times. The viscoelastic behaviour proper of linear amorphous polymers has beenmodelled by Kevin with a spring and a dashpot as a first approximation, and the resulting equationapplying a stress 0 to a material of modulus G is that of an exponential decay, given by the formula:
(t) = ( 0/G)*(1-exp(t/ ))
The relaxation time in Kevin model is defined as time at which the strain is 63.2% of 0/G. But it isimportant to remember that despite the interesting use for predictions, it remains a model and may notreflect the real behaviour of the material. At this point you need another model. And another when thematerial changes so that you will find yourself into a dark forest of springs and dashpots. Since for aChemist macromolecules are probably the most interesting viscoelastic materials, it may be good ideato talk a bit more in depth about their creep behaviours.
Linear amorphous polymers creep: time is money, temperature is time
An example of creep recovery test for a viscoelastic polymer is shown in Figure–9. We apply a stress of20 Pa for 60 seconds. Until 0.1 strain we have an elastic response, then a viscoelastic response. At alonger timescale we will obtain a viscous response. Chains start to disentangle during the elastic regionof the curve, then this process continues but some flow (sliding of chains) starts to occur. When theresponse is purely viscous the chains are disentangled and only slide. As we release the stress, thematerial recovers elastically initially and then viscoelastically for a given time . After the relaxationtime, the response is purely viscous. If we had increased the time of application of our stress or thestress level we would have had a different curve and a shorter relaxation time . The Kevin model witha single exponential decay does not take into account pure elastic and viscous part and a bit moresophisticated equation would be needed. However, if we are just interested in predicting for a verylong time period T at low stress we can measure it at higher stress and fit values into the Kevinequation. And we are ASSUMING an exponential response with stress. If stress is so strong thatpolymer chains break, all this nice model collapses as well. That is why you should make suchpredictions not only with one creep curve but with more at higher stress each time.
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Figure-9 Strain response for a creep-relaxation experiment. The smaller graph represents the ideal behaviour of Kevinmodel. You can see its use for predicting relaxation time but also that does not adapt well to the measured curve.
Figure-10 Stepped temperature creep test. Then superposition of time and temperature to create a mastercurve.
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But let's say we are a bit unlucky and to reach a flow regime it will take too long time to wait. Too longmay be even determined by how long is the course or how long is the research contract, when the bosswants to have answers... all things which have to do with relationship money-time. At this point youmay use the temperature dependence of the creep to compensate your lack of time/money. Supposingthat you have a furnace and have paid electricity bills, increasing the temperature will have a similareffect on the creep curve than increasing stress, ASSUMING that you are in a linear region oftemperature behaviour. That means that you are in a range of temperature where polymers ONLYsoften, less stress is required to move the chains, and at a certain stress level less time is needed tomove them. In such a case you can stay in the linear stress regime and compensate for your littletime/money. Accelerated test methods for long-term creep measurement are usually based on the time-temperature superposition principle. It enables the construction of a master curve, which describes thecreep behaviour over a time range considerably exceeding the measurement time (Figure-10). Theassociated empirical model is expressed by the Williams-Landel-Ferry Equation (WLF) whichspare you for now.
Figure-11 Relaxation modulus and construction of the viscoelastic master curve for PoliyIsoButylene at 25 °C referencetemperature by shifting stress relaxation curves obtained at different temperatures horizontally along the time axis. TheElastic modulus variation determines the regions of mechanical behaviour of the material (upper right corner) .
Remembering Hooke equation, we can do the same for the variation of the modulus G withtemperature, simulating its variation with time. In the early stage (lower Temperature) it will be aconstant (elastic behaviour), then it will vary like in Figure 11. The rubbery plateau (III) denotesviscoelasticity and is reduced with the MW, until a critical value is reached (lower MW) where itdisappears and the polymer behaves like a liquid (example, a paint). The profile of G vs. T alsodepends on glass transition temperature and eventual crystalline regions in the structure. Of course for a
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thermosetting profile changes as cross-linking starts. That is why for the creep-temperature-time curvewe observed a relatively simple case of amorphous linear polymer showing a viscoelastic plateau.When harder materials are tested, it is advisable not only a temperature creep characterisation but also achange of fixture (bending, tensile...). The case of predominantly elastic materials is discussed brieflyin reference 3.
Zero shear viscosity
At sufficiently low shear rates it is found that most materials either have a viscosity that tends toinfinity (yield stress) as in the Bingham model but some other have a viscosity that becomesindependent of shear rate (the low shear Newtonian plateau, Figure-6). It can be calculated with creepexperiments. Let's imagine to perform creep measurements and decrease the stress progressively . InFigure-12 below another common graph of creep experiments is plotted (J instead of G, see page 16)which shows this low shear newtonian plateau. The graph represents the creep response of the materialat several stress levels.
Figure-12 Response for a creep experiment with decreasing stress levels.
We can observe that as the stress decreases the curves get closer to each other. In the viscous regimeattained after enough time the viscosity is given by the slope of the curve. In fact
J=strain /stress
viscosity= 1stress
∗ straintime
= Jtime
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the zero shear viscosity is measured when stress is low enough so that the curves overlap and the slopebelow that point reaches a constant value. A materials zero shear viscosity is useful in predicting suchfactors as storage stability, levelling etc. Again, may estimate the viscosity and time of flow forprocesses which have a very small strain rate. Data fit into models and make again you save time andmoney.
Questions for the mirror, before the exams
Give the definition of creep.
Do you test creep below or above yield point?
You are falling from a plane into a liquid and an Angel in the sky tells you he can change its Deborahnumber according to your will. Assuming that you will be able to get afloat after the impact, would itbe safer to ask a high or low value?
The ideal viscoelastic creep and relaxation behaviour can be described by which model? It implies aexponential or linear relationship between shear and time?
You perform a creep-relaxation experiment for a viscoelastic material. How can you estimate itselasticity from the recovery curve?
You want to estimate the relaxation time for a stiff material. Can you increase stress? Up to whichpoint?
If increasing stress doesn't help, what are you going to do?
Can you draw a curve of G vs. T for a viscoelastic polymer material? The same polymer with higherMW will have a longer or shorter viscoelastic plateau?
If a liquid displays a low shear Newtonian plateau, does it make sense to measure a zero shearviscosity?
How would you calculate it and what would be the typical application for such a measurement?
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4. Viscoelastic oscillatory characterisation
Flow characterisation tests do just that, i.e. tell you how a material is likely to flow under an imposedconstant shear rate or shear stress. They tell only about the materials VISCOUS properties (resistanceto flow). To measure a materials viscoelastic properties we can use creep testing (as described in theprevious section) or alternatively we can use oscillatory techniques. The technique used is to apply astress or strain whose value is changing continuously according to a sine wave equation. Thus theinduced response (strain or stress) will also follow a sine wave.
Thus it can be seen that we can continuously excite the sample but never exceed a certain strain andhence we do not destroy the sample structure. (providing steps are taken to keep the strain smallenough). If we 'over strain' the sample, we will start to destroy the elastic structure of the material andso it is important to keep the strain low. The technique used by the software in controlled stressinstruments is to continuously adjust the applied stress so that the resultant strain is held at a specifiedvalue. If you remember back to section 1 we talked about Hookean deformation (analogous to aspring/elastic being extended) and Newtonian flow (analogous to a viscous dashpot/honey drop) mostmaterials are made up of a combination of these two properties. Hooke's law is a simple linearrelationship, that is if you double the applied stress you double the measured strain. Provided the strainproduced is small enough it is said that you are working in the materials REGION OF LINEARSTRAIN RESPONSE or more simply LINEAR REGION.
Thus, before performing oscillation tests on a material you must verify that the test conditions fall intothis regime. This is easily tested by oscillating at a fixed frequency and slowly increasing the appliedamplitude (strain or stress). The measured values for the viscoelasticity will remain constant. When theapplied stress becomes too great, the induced strain will start to cause the material to 'rupture' i.e. youwill obtain some flow on top of the deformation. This will be seen as the measured value of elasticityfalling whilst the measured viscous component will start to increase. Provided you work at strainsbelow this point you will be working in the materials linear region.
Definition of elastic and viscous components
As stated earlier, if we apply a sinusoidally varying stress to a sample, we will induce a sinusoidallyvarying strain (and vice versa for applied strain) response. If we think back to how the stress effects thesample for a pure solid and a pure liquid you will remember the following :
For a Hookean Solid :
Shear Stress = Shear Strain x G (a constant)
For a Newtonian liquid :
Shear Stress = Shear Strain rate x Viscosity (a constant)
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So for a pure solid the strain is controlled by the absolute value of shear stress, whereas for a liquid it isthe rate of change of strain that is controlled by the stress. If we consider one complete cycle of the sinewave as 360° then we can talk about PHASE ANGLES. This concept is extremely important inRheology so let's brush it up.
Two oscillators that have the same frequency and different phases have a phase difference, and theoscillators are said to be out of phase with each other. The amount by which such oscillators are out ofstep with each other can be expressed in degrees from 0° to 360°, or in radians from 0 to 2π. Thisamount is called phase shift (déphasage) and the corresponding angle will be noted with the Greekletter .
How does the phase shift relate to an angle? see this good animation:
http://www.kwantlen.ca/science/physics/faculty/mcoombes/P2421_Notes/Phasors/doublesine.gif
In the case of a pure solid, since the strain is directly related to the stress, it will be at a maximum whenthe stress is a maximum and zero when the stress is zero. The strain response is said to be totally INPHASE with the applied stress i.e. the PHASE ANGLE = 0° (see Figure-13).
Figure-13 Response to a sinusoidal stress for a solid.
If the material is a pure viscous liquid we find that it will be the strain rate that is exactly following thestress. If you look at the graph of strain rate as a function of time you can see that the strain alternatesbetween a positive and negative extreme accelerating and decelerating between these two values.
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Therefore, when the strain rate is at a maximum the rate of change of strain will be zero, likewise whenthe strain is zero, the rate of change will be a maximum. The resultant strain will therefore be totally(90°) out of phase to the applied stress (see Figure-14).
Figure-14 Response to a sinusoidal stress for a liquid.
Now you may ask yourself: is there any link between creep behaviours, recovery and this phasedifference?
At this point you may imagine that the phase angle since depends on the phase shift, it is directfunction of the Deborah number. It is related indeed, but the phase shift is not the value we talkedabout.
The complex modulus, G' and G”
As stated earlier, most materials are a combination of viscous and elastic components, and are thusviscoelastic. This is particularly important for colloidal solutions and polymers, so in an oscillatoryshear experiment for these the measured phase angle will be somewhere between 0° and 90°. But howto quantify the amount of elasticity and fluidity for the modulus measured? At this point the concept ofcomplex modulus which will follow you throughout all your Rheology career (that is why title isunderlined this time).
Remember Hooke's law? it relates the strain to the stress via a material constant known as theMODULUS, G. (G = stress / strain). In the oscillation test the stress and strain are constantly changingbut we can consider any number of 'instantaneous' values to obtain a value of 'viscoelastic G'. This isreferred to as the materials COMPLEX MODULUS, G* and is obtained from the ratio of the stressamplitude to the strain amplitude.
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This modulus is the 'sum' of the elastic component (referred to as G' often called the STORAGEMODULUS to signify elastic storage of energy since the strain is recoverable in an elastic solid), andthe viscous component (referred to as G'' often called the LOSS MODULUS to describe viscousdissipation [loss] of energy through permanent deformation in flow).
We define the complex modulus as the vector sum of G’ and G”:
G* = G' + iG''
For who studied alternate current this formula is very familiar, with real and imaginary part of thesignal. By measuring the ratio of the stress to the strain (G*) as well as the phase difference betweenthe two (delta, δ) we can define G' and G'' in terms of sine and cosine functions as follows:
G' = G* Cos δ
G'' = G* Sin δ
Since G* is essentially Stress/strain, G' and G'' have units of Pascal (N/m2).
Frequency sweeps and mechanical spectroscopy
Dynamical Mechanical Spectroscopy (DMS) is one fundamental rheological technique used tocharacterize the viscoelastic properties of complex fluids and to investigate their microstructure.During DMS experiments, the system is subjected to a sinusoidal shear deformation, (t):
(t) = 0 sin ( t)
where 0 is the amplitude of the applied strain and ω is the frequency of oscillation. Theshear rate, is:
˙ = 0ω cos ( t)
The material stress response is measured and, by means of some adequate theoretical models, it iscorrelated to the characteristics of the system under investigation.There are two different families of DMS tests, depending on the amplitude, 0 , of the applieddeformation: they are named SAOS (Small Amplitude Oscillatory Shear) and LAOS (Large Amplitude
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Oscillatory Shear). In the former case, 0 is small enough to produce a linear response of the material,in the latter case, the amplitude of 0 becomes large and determines a nonlinear response of thesystems. We stated that at this stage we only look at the linear response and we carefully choose a lowenough value 0 but is good to know this general classification.
For a SAOS
(t) = 0(t) sin ( t + )
Nice but not very informative eh? We can rework the equation and insert the moduli G' and G” wedefined earlier to distinguish the elastic and viscous component of the measured stress.
If we define =G”/ , we can see that for pure elastic and pure viscous material (G' or G” = 0 in eachcase) by substitution we get back to the Hooke and Newton equations. We can thus monitor theviscous or elastic behaviour of a material over a range of frequency assuming it does not changestructure with time and imposing a constant strain. How does all this translate into graphics? Below inFigure-15a a representation of the two ideal limit behaviours in an oscillatory shear test.
Figure-15 Illustrations of frequency-dependent storage and loss G’ and G’’ modulus for a) typical “liquid-like” and “solid-like” and b) viscoelastic materials [reference 4].
For the “liquid-like” fluid, the elastic modulus is much lower than the viscous one and it scales withfrequency as G’∝ ω , the viscous modulus is linear in frequency, G’’∝ . For “solid-like” fluid, G’ >>
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G’’, and G’ is nearly frequency-independent. Real complex fluids often show an intermediateviscoelastic behavior, as can be observed in figure-15b. At low frequencies, the system is “liquid-like”,with G’ << G’’, while more nearly “solid-like” behavior, with G’ > G’’, is found at high frequencies.
At this point after you did all this nice Maths and even got back to the trigonometry classes you may asyourself: what has all this story of moduli to do with spectroscopy and why we do vary frequency ofoscillation? The combination of viscous and elastic components in a material will respond in differentways depending on the 'speed' at which you try to move it. By oscillating over a wide range offrequencies you will obtain the characteristics over a range of time scales as high frequencies relate toshort times whereas low frequencies relate to long time scales. That is how the delay measured relatesto the Deborah number of creep. We impose a strain, smaller molecules relax faster than the longermolecules. If we are fast enough (high frequency) we will measure the response of the small molecule,while at lower frequency we match the relaxation of longer molecules. Imagine to have a “complex stararchitecture” copolymer made by RAFT. Main chain and branches differ in molecular weight. Suchparts which will respond at the shear frequency with different speed. Depending on the frequency, theoverall response will be more affected by one or the other part of the molecule (Figure 16). Is like if themain chain will leave a signature at low frequency (more time taken to relax) and the branches (lowerMw, less time to relax) another at higher frequency.
Figure-16 Frequency sweep graph showing a rheological signature of molecules segment which differ in molecular weight.
In this sense, the frequency sweep can be well compared to spectroscopy, where depending on theenergy (in this case relaxation energy) we will have a peak at a certain frequency. Thinking frequencyas wavelength in the UV-Vis spectroscopy can give an even better idea.
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Time, frequency, stress... where to begin?
As stated earlier, a frequency sweep can give valuable information about the material structure onlywith certain conditions. First of all, the structure should be stable over time. If in doubt, a preliminarytime sweep is advisable. However, for some cases the time sweep can also give valuable information sabout the material examined. In figure 17 below the time sweep for a peptide hydrogel is presented.
Figure-17 Time sweep graph showing a the moduli variation upon time for a peptide hydrogel.
A gel transition is commonly measured as the point where G' crosses G”. In this case due to the verylow initial moduli this is very ill defined. However the curve of G' provides valuable information aboutthe kinetics of gel formation in the first 1-2 hours. A stable structure is obtained only after 5-6 hours.Afterwards a stable gel is obtained and a frequency sweep can be performed assuming that the stress ishigh enough to have signal but not so high to break the structure of the gel. Knowing its characteristicsfrom literature or previous experiments helps to make an educated guess. But if you cannot guess,when time and frequency sweeps have to be made, it would be good idea to make a stress sweepbeforehand to verify that the frequencies have been collected well within the linear viscoelastic regime.A stress sweep can also provide more valuable information. See for example figure 18, where a stressor “amplitude sweep” is shown.
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Figure-18 Amplitude sweep showing the storage and loss moduli are plotted against the shear stress for two colloidalsolutions. A longer viscoelastic linear regime for sample B indicates better stability.
The initial plateau shows that in that stress range the material displays a linear viscoelastic behaviour.For both samples is safe to make frequency sweeps with a stress of 0.5 Pa.However the linear regime for sample B is longer than that of A. Remembering whet we read aboutcreep in section 3, we can also expect a longer stability over time for B than for A since its initialstructure is more resistant (see also reference 6).
Oscillatory temperature sweeps
As for creep measurements, the time-temperature superposition principles can be also applied tooscillatory sweeps. In Figure-18a and 18b, there are shown graphs of G' and G” with curves taken byfrequency sweeps at several temperatures. Moduli get lower with temperature as the material issoftened. Crossover between G' and G” also shifts accordingly. We can calculate the shift and thenapply the WLF model to make a mastercurve (Figure-19c and 19d). As for the case of creep, whensuperposition does not work, it implies that more than one relaxation mechanism is at work in thetemperature range investigated. The system is said to be rheologically complex.In depth knowledge even for a simple system requires a full mathematical description and treatment ofthe WLF model, but the graphs in Figure 19 show how it is done and in practice you may find themastercurve function already in the software provided with the rheometer. You are free of course to dothe superposition manually using Origin or Excel.
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Figure-19 Frequency sweep showing the a) storage and b) loss moduli at several temperatures; then joined into amastercurve d) with a time-temperature superposition.
It is important to note that as for creep, even in oscillatory shear you can use the time-temperaturesuperposition only when the material is in a linear viscoelastic regime. How do you know? Well, sincemost rheometers have a temperature control, at a fixed strain (stress) and frequency you can measure G'and G” at several temperatures. This is called a temperature sweep (Figure-20). In this case, it isinteresting to plot not only G' and G” but also tan . A peak or shoulder of tan curve generallycorresponds to a phase transition of the material. With the glass transition temperature for a polymer adexample, the elasticity is reduced compared to the viscosity and this change of slope is clearly shownby a peak (upper graph of Figure-20). Afterwards the tan is roughly constant. This region correspondsto the viscoelastic plateau (III) in Figure-11. Then the material melts (Tm in the lower graph of Figure-20).
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Figure-20 Temperature sweep for a Polyurethane-Clay nanocomposite elastomer. Below, compared the responses of nano,microcomposite and pure PU. Notice the larger viscoelastic rubbery plateau for the nanocomposite. T' corresponds to a glasstransition of the PU crystalline domains.
In the comparison made in Figure 20 a longer plateau for nanocomposite compared to microcompositeor pure PU, shows that the clay dispersion at that level makes the material more resistant totemperature. Despite there will be always be some small differences due to instrumental features, thetan -T graph can also be conveniently compared to that obtained by DSC to verify the correlationbetween peaks and phase transitions.
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Questions for the mirror, before the exams
• If we want to carry out a test of oscillatory strain in the linear region, can we apply a stress overthe material yield point?
• What is the difference between this test and a creep test? Are we measuring relaxation in bothcases?
• In an oscillatory test the phase angle between stress and strain depends on material response.Which is the angle for a pure elastic response? And for a pure viscous?
• Why are G' and G” called storage and loss modulus? And what is the complex modulus?
• What are LAOS and SAOS tests? Which one of the two are you doing if you are in the linearregime?
• For a liquid-like material is G' higher than G”? and for a solid-like?
• What is the G' and G” dependence upon frequency in each case?
• Imagine to carry out a frequency sweep in linear regime for a branched polymer that looks likea brush. The characteristic for the long backbone is likely to be higher or lower than that ofthe smaller brushes?
• You have a gel which forms over time and you want to characterise it by SAOS. In which orderwould you carry out strain, time, frequency sweeps?
• Which of the 3 tests above mentioned will you use to decide which of two paints has a longershelf time?
• How do we build a mastercurve for frequency sweeps? What are the conditions which allow tomake a time-temperature superposition in this case?
• Imagine to perform a temperature sweep for a polymer and plot tan vs. T. What would a tanpeak normally correspond to?
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References / Further reading
1. http://www.usm.edu/pattonresearchgroup/PSC341/visco.pdf
2. http://iusti.polytech.univ-mrs.fr/~guazzelli/publiperso/Rheo.pdf
3. http://www.perkinelmer.com/CMSResources/Images/44-74546GDE_IntroductionToDMA.pdf
4. Claudia Carotenuto Ph.D Thesis. ADVANCED MECHANICAL SPECTROSCOPY TOINVESTIGATE THE MICROSTRUCTURE OF COMPLEX FLUIDS. Università Federico II.Napoli. 2006
5. http://www.iesmat.com/Lectura recomendada/Productos-MAL/REO-A basic introduction torheology.pdf
6. http://www.rhl.pl/DATA/pliki/files/literatura/v98-156e.pdf
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