Polymer Rheology

P Sunthar

Abstract This chapter concerns the flow behaviour of polymeric liquids. This is ashort introduction to the variety of behaviour observed in these complex fluids. Itis addressed at the level of a graduate student, who has had exposure to basic fluidmechanics. The contents provide only a broad overview and some simple physicalexplanations to explain the phenomena. For detailed study and applications to in-dustrial contexts the reader is referred to more detailed treatises on the subject. Thechapter is organised into three major sections

1 Phenomenology

Interesting behaviour of a material is almost always described by first observingand extracting the qualitative features of the variety of phenomena exhibited. Toa novice, this is the place to start. Some questions to ask would be: What makesthis phenomena different? How to depict it in terms of a simple model? Is there a“law” that can describe the behaviour? Are there other phenomena that obey similarlaws? What role has this played in the state of the universe? Can it be employed forthe betterment of quality of life? What are the consequences of this behaviour toprocesses that manipulate or use the material?

Some of these are simple questions, answers to which may not solve any urgentand present problem. However, it puts it in a larger context.Often, when solvingdifficult problems in rheology using advanced methods and tools, one tends to misssimple laws exhibited by the materials in reality, knowing which may help in gettingan intuitive feel of the solution, and thereby a faster approach to the answer.

Department of Chemical Engineering, Indian Institute of Technology (IIT) Bombay, Mumbai400076 India. e-mail: p.sunthar@gmail.com

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1.1 What are Polymeric Liquids?

Polymeric liquids are precisely what the name states: namely they are like the liquidswe know that flow and have as constituents, some or all being long chain moleculesor polymers. The conventional definition used to definesimple liquids: that they donot support shear stress at rest, cannot be used to define the liquid state of a poly-meric liquid. This is because polymeric liquids, like most other liquids described inthis book arecomplex fluids: They exhibit both liquid and solid like behaviour, andsome of their dynamic properties may not be thermodynamic constants but someeffective constant dependent on the history of forces acting on it. An example is a“viscosity” which is a function of shear rate, or a “viscosity” that changes with time.

1.1.1 Chemical Nature

The common feature of all polymeric liquids considered in this chapter are thatthey have long chain of monomers joined by chemical bonds. They could even beoligomers or very long chain polymers. Most of them have large molecular weightsmore than a 1000 and up to about 109. Many materials we use today are polymersor blends of polymers with other materials. In some stage of their processing mostof them were in one liquid state or other: as solutions or puremolten form.

1.1.2 Physical Nature

The chief physical property of a polymer that distinguishesitself from other fluidsthat exhibit complex behaviour is the linearity of the chain. It is not the high molec-ular weight that leads to the peculiar phenomena but that it is arrangedlinearly : Thelength along the chain is much larger that the other dimensions of the molecule. Forexample, a suspension of polystyrene beads (in solid state)may have high molecu-lar weight per bead. But it may exhibit rheological properties of a suspension ratherthan a polymer solution. This is because for most propertiesthat we measure, suchas the viscosity, it is the beads spherical diameter that matters, and not the molecularweight per bead. There is no linear structure that is exposedto these measurements.On the other hand a solution of polystyrene molecules in cyclohexane is a polymericliquid.

What is the difference between a polymeric liquid to us and a huge bowl of noodlesto a Giant?

A huge bowl of noodles is very much like a several molecules ofpolymers puttogether. Just as we “see” polymers, so the bowl with these noodles would appear toa Giant. The noodles have the necessary linearity, they are flexible like the polymersare, and as a whole can take the shape of the container they areput into. Can we

Polymer Rheology 3

then use a model bowl full of noodles to understand the behaviour of polymericliquids. Not entirely! The key difference between the two systems is temperature orrandom motion. Many of the properties of polymeric liquids we observe, such asthat it flows, are due to therandom linear translating motion of its constituents,similar as in simple liquids. It is the random motion along the length of the chain andthe resulting influence on the other constituents of the material, such as the polymeritself, solvent, or other polymers, that leads to thedefining propertiesof polymericliquids.

“States” of Polymeric Liquids

The simplest definition (though not a theoretically simplermodel to handle) of apolymeric liquid state is themolten state. Take a pure polymer (with no other ad-ditives) to a high enough temperature that it is molten. Thisis like a noodle-state,except that the polymers are in continuous motion. The otherextreme of this stateis when small amounts of a polymer is added as an additive (solute) to a solvent.This state is called thedilute solution. This is very similar to putting one polymerchain in a sea of solvent. It is equivalent to one polymer because the solution isso dilute that the motion of one polymer does not influence theother polymers insolution. Parts of the polymer’s motion influences other parts of the same chain.Between these two states of pure and dilute, we can have a range of proportions ofthe polymer and the other component. Increasing the concentration of the additivepolymer from dilute values, we get to thesemi-diluteregion where the polymers aredistributed in the solution such that they just begin to “touch” each other. Furtherincrease in concentration leads to theconcentrated-solutionwhere there are signif-icant overlaps andentanglements(like in an entangled noodle soup). The moltenstate is like a dry noodle with full of entanglements.

1.2 Who needs to study flow of polymeric liquids?

The primary motivation to study polymeric fluids comes from industrial and com-mercial applications: Polymer processing and consumer products. These fields haveprovided (and continue to provide) with several problems, that are addressed by awide variety of engineers and physicists, to the extent thathas even earned a NobelPrize in Physics for Pierre-Gilles de Gennes in 1991.

The primary question is how is the flow modified in the presenceof these longchain macromolecules. In the polymer processing field one needs to know, as in anyfluid mechanics problem, how to control the flow and stabilityof these liquids asthey pass through various mixers, extruders, moulds, spinnerets, etc. For consumerproducts, on the other hand, we would like to able to control the feel, rates, shapes,etc., of liquids by having additives, which in many cases arepolymers.

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1.3 What is Polymer Rheology?

Clearly industrial flows are complex, not only because the geometries are complex,but also because the constituents are not usually simple. Wehave several compo-nents in a shampoo, performing various actions. Molecular weight distribution of apolymer is another level of complexity, as polymers are rarely synthesised in a sharpmonodisperse population.

This brings in the need to study the behaviour of polymeric liquid in simple flowsand for simple systems, with the hope that the knowledge gained can be appropri-ately used in a complex flow pattern. The wordRheologyis defined as the science ofdeformation and flow, was coined by Prof Bingham in 1920s [1].Rheology involvesmeasurements in controlled flow, mainly the viscometric flowin which the velocitygradients are nearly uniform in space. In these simple flows,there is an applied forcewhere the velocity (or the equivalent shear rate) is measured, or vice versa. They arecalledviscometricas they are used to define an effective shear viscosityη from themeasurments,

η =σxy

γ(1)

whereσxy is the shear stress (measured or applied) andγ is the shear rate (appliedor measured). Viscosity is measured in Pa-s (Pascal second).

Rheology is not just about viscosity, but also about anotherimportant property,namely the elasticity. Complex fluids also exhibit elastic behaviour. Akin to theviscosity defined above being similar to the definition of a Newtonian viscosity, theelasticity of a complex material can be defined similar to itsidealised counter part,the Hookean solid. The modulus of elasticity is defined as

G=σxy

γ(2)

whereγ is called the strain or the angle of the shearing deformation. G is measuredin Pa (Pascal).G is one of the elastic modulii, known as the storage modulus, asit is related to the amount of recoverable energy stored by the deformation.G formost polymeric fluids is in the range 10–104 Pa, which is much smaller than thatof solids (> 1010 Pa). This is why complex fluids, of which polymeric fluids forma major part, are also known assoft matter, i.e. materials that exhibit weak elasticproperties.

Rheological measurements on polymers can reveal the variety of behaviour ex-hibited even in simple flows. Even when a theoretical model ofthe reason for thebehaviour is not known, rheological measurements provide useful insights to prac-tising engineers on how to control the flow (η) and feel (G) polymeric liquids. Inthe following section we sample a few defining behaviours of various properties incommonly encountered flows. In the following sections we present a broad overviewof the variety of phenomena observed in polymeric liquids and how the rheologicalcharacterisation of the liquids can be made.

Polymer Rheology 5

1.4 Visual and Measurable Phenomena

Some of the striking visual phenomena are associated with flow behaviour of poly-meric liquids. These are best seen in recorded videos. We describe a few of themhere and provide links to resources where they may be viewed.There are many morephenomena discussed and illustrated in [2].

1.4.1 Weissenberg Rod Climbing Effect

When a liquid is stirred using a cylindrical rod, the liquid that wets the rod beginsto “climb” up the rod and the interface with the surrounding air assumes a steadyshape dangling to the rod, so long as there is a continuous rotation. In contrast ina Newtonian liquid, there is a dip in the surface of the liquidnear the rod. Rodclimbing is exhibited by liquids that show a normal stress difference. In Newtonianliquids the normal stresses (pressure) are isotropic even in flow, whereas polymericliquids, upon application of shear flow, begin to develop normal stress differencesbetween the flow (τxx) and flow-gradient directions (τyy).

1.4.2 Extrudate or Die Swell

This phenomenon is observed when polymeric melts are extruded through a die. Thediameter of liquid as it exits a circular die can be three times larger that the diameterof the die, whereas in the case of Newtonian fluids it is just about 10% higher inthe low Reynolds number limit. One of the important reasons for this phenomena isagain the normal stress difference induced by the shear flow in the die. As the fluidexits the die to form a free surface with the surrounding air,the accumulated stressdifference tends to push the fluid in the gradient direction.

1.4.3 Contraction Flow

Sudden contraction in the confining geometry leads to very different streamlinepatterns in polymeric liquids. In Newtonian liquids at low Reynolds number, nosecondary flows are observed. Whereas in polymeric liquids, including in dilutepolymer solutions, different patterns of secondary flow areobserved. These includelarge vortices and other instabilities. These flows are undesirable in many situationsin polymer processing as it leads to stagnation and impropermixing of the fluid inthe vortices.

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1.4.4 Tubless Siphon

In a typical syphoning experiment, a tube filled with liquid drains a container con-taining the liquid at a lower pressure, even though the tube goes higher than theliquid surface. When the tube is lifted off the surface of the liquid, the flow imme-diately stops. But this is the case in Newtonian liquids. In the case of polymericliquids, the liquid continues to flow with a free surface withthe air without the tube,as the tube is taken of the surface.

1.4.5 Elastic Recoil

A “sheet” of polymeric liquids pouring down from a vessel canbe literally cut witha pair of scissors. Very similar to a sheet of elastic solid, the top portion of the cutliquid recoils back into the jar.

1.4.6 Turbulent Drag Reduction

In most of phenomena discussed so far the concentration of the polymer was about0.1% or higher, and the viscosity of such systems are usuallylarge that it is not com-mon to encounter large Reynolds number flows that lead to turbulence. In smallerconcentrations of about 0.01% where the solution viscosityis not significantly en-hanced above the solvent’s viscosity, turbulence can be easily observed. The inter-esting feature of such turbulent flows, at least in pipe geometries is that the turbulentfriction on the walls is significantly less, upto nearly five times. This phenomenonhas been used in transportation of liquids and in fire fightingequipment.

1.5 Relaxation time and Dimensionless Numbers

One of the simplest and most important characters of polymeric liquids is the ex-istence of an observable microscopic time scale. For regular liquids the timescalesof molecular motion are in the order of 10−15 seconds, associated with moleculartranslation. In polymeric liquids, apart from this small time scale, there is an impor-tant timescale associated with large scale motions of the whole polymer itself, in theliquid they are suspended in (solutions or melts). This could be from microsecondsto minutes. Since many visually observable and processing time scales are of similarorder, the ratio of these time scales becomes important. Thelarge scale microscopicmotions are usually associated with the elastic character of the polymeric liquids.In the chapter on polymer physics there is a discussion of therelaxation times. Therelaxation time is the time associated with large scale motion (or changes) in thestructure of the polymer, we denote this time scale byλ .

Polymer Rheology 7

The microscopic timescale should be compared with the macroscopic flow timescales. The macroscopic time scales arises from two origins. One is simply thekine-matic local rate of stretching of the fluid packet (strain rate). This is measured bythe local shear rateγ for shearing flows or the local elongation rateε for extensionalflows. The other is adynamic timescale associated with the motion of the fluidpackets themselves. Examples are the time it takes for a fluidpacket to transversea geometry or a section, pulsatile flow, etc. We denote this timescale bytd. Exceptfor viscometric flows, the macroscopic timescales may not beknowna priori, andhave to be determined as part of the solution. For example thenature of the fluid’sviscosity could alter the local shear rateγ, or the time it spends in a particular sec-tion td. To know the this dependence, we need to solve the fluid dynamic equationsin the given geometry.

1.5.1 Weissenberg Number

The ratio of the microscopic time scale to the local strain rate is called as the Weis-senberg Number

Wi = λ γ or λ ε (3)

Note that the strain rate is the inverse of the kinematic timescale. Flows in which theWi are small, Wi� 1, are in which elastic effects are negligible. Most of the floweffects are seen around Wi∼ O (1). For large Wi� 1, the liquid behaves almostlike an elastic solid.

Weissenberg number is used only in situations where there isa homogeneousstretching of the fluid packet in the flow. That is the strain rates are uniform inspace and time. Such a flow is encountered only in viscometricflows and theoreticalanalysis, as it is hardly observed in any practical application.

1.5.2 Deborah Number

In most practical applications the fluid packets undergo a non-uniform stretch his-tory. This means that they could have been subjected to various strain rates at vari-ous times in their motion. Therefore no unique strain rate can be associated with theflow. In these cases it is customary to refer to the Deborah number defined as theratio of the polymeric time scale to they dynamic or flow timescale

De=λtd

(4)

For small De� 1, the polymer relaxes much faster than the fluid packet traversesa characteristic distance, and so the fluid packet is said to have “no memory ofits state” a fewtd back. On the other hand for De∼ O (1), the polymer has notsufficiently relaxed and the state a fewtd back can influence the motion of the packetnow (because this can affect the local viscosity and hence the dynamics).

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Table 1 Scaling of the relaxation timeλ of the polymeric liquid with molecular weightM fordifferent classes of polymeric liquids [3].

Class Scaling

Dilute solution in poor solventλ ∼ M1.0

Dilute solution inθ -conditionsλ ∼ M1.5

Dilute solution in good solventλ ∼ M1.8

Semi dilute solution λchain∼ M2

Entangled Melts λrep∼ M3.4

1.5.3 Relaxation time dependence on Molecular weight

Since the time large scale motion of the molecule (i.e. its relaxation time) dependson the linear size of the molecule, the molecular weight of a polymer has a directbearing on the relaxation times. The dependence on molecular weight isnot abso-lute. That is we cannot say that for a given molecular weight two different polymerswill have the same relaxation time. It is only a scaling dependence for a class ofpolymeric liquids (We can only say that the relaxation time scales with molecu-lar weight power some exponent). This dependence is summarised in Table 1. Thescaling given for semi-dilute and entangled melts is only indicative of the longestrelaxation time; there are several relaxation process in these systems, and the wayexperimental data is interpreted from measurements carried out at various tempera-tures (See section 8.7 of Ref. [3] for details).

1.6 Linear viscoelastic properties

In general the elastic nature of a material is associated with some characteristicequilibrium microstructure in the material. When this microstructure is disturbed(deformed), thermodynamic forces tend to restore the equilibrium. The energy as-sociated with this restoration process is the elastic energy. Polymeric liquids havea microstructure that are like springs representing the linear chain. Restoration ofthese springs to their equilibrium state is through the elastic energy that is “stored”during the deformation process. But polymeric fluids are notideal elastic materials,and they also have a dissipative reaction to deformation, which is the viscous dis-sipation. For small deformations, the response of the system is linear, meaning thatthe response is additive: effect of sum of two small deformations is equal to the sumof the two individual responses. Linear viscoelasticity was introduced in the chapteron Non-Newtonian Fluids.

Linear viscoelastic properties are associated with near equilibrium measurementsof the system, that is the configuration of polymers are not removed far away fromtheir equilibrium structures. Most of the models describedin the “Polymer Physics”chapter deal with such a situation. Study of linear viscoelastic properties can revealinformation about the microscopic structure of polymeric liquids. The term rheology

Polymer Rheology 9

is used by physicists to usually refer to the linear responseand by engineers to referto the large deformation (shear or elongational) or non-linear state. We will firstpresent aspects of the linear response before detailing thenon-linear responses inthe next sections.

An important point to note here is that it is just not polymeric liquids that showelastic behaviour, butall liquids do, at sufficiently small time scales. Typical valuesof relaxation times and the elastic modulus for various liquids is given in Table 2.

Table 2 Linear Viscoelastic properties of common liquids, values are typical order of magnitudeapproximations [4].

Liquid Viscosity Relaxation time Modulusη (Pa.s) λ (s) G (Pa)

Water 10−3 10−12 109

An Oil 0.1 10−9 108

A polymer solution 1 0.1 10A polymer melt 105 10 104

A glass > 1015 105 > 1010

Commonly used tests to study the linear response are

Oscillatory Controlled stress/strain is applied in an small amplitude oscillatorymotion and the response of the strain/stress is measured.

Stress Relaxation A constant strain is applied and the decayof the stresses to theequilibrium value is studied.

Creep A constant stress is applied and the deformation response is measured.

Though all of the above tests can also be carried out in the non-linear regime, as thelimits of linear regime are not knowna priori, sequence of tests are carried out toensure linear response.

1.6.1 Zero-shear rate viscosity

The zero shear rate viscosityη0 is the viscosity of the liquid obtained in the limitof shear rate tending to zero. Though the name suggests that it is a shear viscosity,it is still in the linear response regime because the shear rate is approaching zero.In practise it is not possible to attain very low shear rates for many liquids owingto measurement difficulties. In these cases the viscosity obtained by extrapolatingthe viscosities obtained at accessible shear rates. The zero shear rate viscosity is animportant property to characterise the microstructure of polymeric liquid.

For dilute solutions, since the polymer contribution to thetotal viscosity is usu-ally small, it is useful to define an intrinsic viscosity at zero shear rate as

[η ]0 ≡ limγ→0

[η ]≡ limγ→0

limc→0

η −ηs

cηs(5)

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The intrinsic viscosity scales as

[η ]0 ∼λM

(6)

where the scaling ofλ is given in Table 1 for various types of dilute solutions.In semi-dilute regime, one is more interested in the scalingof the viscosity with

concentration. The usual way to report viscosity is throughthe specific viscosity(and not the intrinsic viscosity)

ηsp0= η0−ηs (7)

The specific viscosity scales linearly with concentration in the dilute regime. In thesemi-dilute regime, underθ -conditions

ηsp0∼ c2 (8)

and in the concentrated (and melt) regime

ηsp0∼ c14/3 (9)

The scaling with respect to concentration in good solvents is weaker to about∼ c1.3

in semi-dilute and∼ c3.7 in the concentrated. More details of this scaling can befound in Ref. [3]. The scaling behaviour is summarised in Figure 1.

1.6.2 Oscillatory Response

The typical response of a polymeric melt to an oscillatory experiment is shown inFigure 2. The symbols used here are the complex modulusG∗: G′ for the storagecomponent (real) andG′′ for the loss component (imaginary), as defined in the Non-Newtonian Fluids chapter.G′ represents the characteristic elastic modulus of thesystem andG′′ measures the viscous response. At high frequencies the response isglassy (which is typically seen at temperatures around glass transition). The domi-nant elastic response is seen in the rubbery region where thestorage modulus showsa plateau. The plateau region is clear and pronounced in higher molecular weightpolymers (with entanglements) in the concentrated solution or the melt states, asshown in Figure 3 [4]. In this region the storage modulus (elastic) is always greaterthan the loss (viscous) modulus. The value ofG′ at the plateau, is known as theplateau modulusG0

N, and is an important property in understanding the dynamicsofthe polymers in the melt state.

The low frequency response (or long time response) is alwaysviscous. The vis-cous regime behaviour is characteristic of all materials (including solids) showingMaxwell behaviour

G′ ≈ Gλ 2ω2 (10)

G′′ ≈ η0 ω (11)

Polymer Rheology 11

Fig. 1 Scaling of the specificviscosityηsp0with concen-tration of polymers. The firsttransition denotes the semi-dilute regime and the secondcorresponds to the entangledregime.

1

2

14/3

Semi−

Dilute

Entangled

Dilute

log c

logη

sp0

c∗

c∗∗

Fig. 2 Typical regimes in thecomplex modulus obtainedusing an oscillatory responseof a polymeric liquid [5].

G’’

G’

Rubbery/PlateauGlassy

Viscous Transition to Flow

log(ω)

log(

G′)

log(

G′′)

∼ λ−1

where,G is the elastic modulus (constant in the Maxwell model), andη0 is the zero-shear rate viscosity. This behaviour is also shown by polymeric liquids in the diluteand semi-dilute regimes. The characteristic relaxation time of the structured liquidcan be obtained from the inverse of the frequency whereG′ andG′′ cross over in theflow transition regime.

λ =G′

G′′ ω(12)

whereG′ andG′′ are measured in the viscous regime with the linear and quadraticscaling respectively. As seen before, increase in molecular weight increases the re-laxation time, so does the increase in concentration. Therefore with increasing con-

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Fig. 3 Increase in the plateauregion with increase in molec-ular weight.

log(ω)

log(

G′)

M

centration or molecular weight of the polymer, the intersection of the two curvesshifts more and more towards the left (low frequencies). Thelow frequency re-sponse of the loss modulus can also be used to obtain the zero-shear rate viscosityη0 from the Equation (11).

The low frequency response, close to the plateau region can also be used to dis-tinguish the type of polymer melt. A schematic diagram of theplateau region be-haviour of storage modulusG′ for various types of polymers is shown in Figure 4.Unentangled melts do not have any plateau region, and directly make the transitionto the Maxwell region. Entangled melts show a plateau region. Cross-linked poly-mers have a wide and predominant plateau region [5]. The transition to the Maxwellbehaviour is almost not seen due to limitations in observingvery cycles, just as insolids. The link between the plateau region and the cross-linking suggests that theentanglement acts like a kind of constraint (like the cross-links) to the motion of thepolymer contour, leading to the plateau region.

1.6.3 Stress Relaxation

The relaxation of the stress modulusG(t) in response to a step strain (for smallstrains) in the linear regime, is equivalent to the oscillatory responseG∗(ω), onebeing the Fourier transform of the other [6]. A schematic diagram of the stress re-laxation is shown in Figures 5 and 6 in linear and logarithmicscale respectively [5].The initial small time response ofG(t) is equivalent to the high frequency responseof G∗(ω), and the long time response is equivalent to the low frequency response.In small times a polymeric substance shows a glassy behaviour, which goes to the

Polymer Rheology 13

Fig. 4 Low frequency re-sponse of various type ofpolymers. Crosslinked Polymer

Entangled Melt

Unentangled Melt

log(ω)

log(

G′)

G0N

Fig. 5 Stress relaxation inresponse to step strain inlinear scale.

t

G(t

)G0

N

Fig. 6 Stress relaxation inresponse to step strain inlogarithmic scale.

log t

log

G(t

)

G0N

plateau region (seen clearly in the logarithmic scale in Figure 6) and finally to theterminal viscous decay. SinceG is related to the elasticity, the response can be un-derstood as being highly elastic at small times, which decays and begins to “flow”at large times: Given sufficiently long time any material flows.

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Fig. 7 Shear thinning ofpolymeric liquids is morepronounced in concentratedsolutions and melts than indilute solutions.

−2

2

−5 −1 3

4

0Dilute Solution

Concentrated solution

Worm−like Micelle

log γ

logη

1.7 Flow Viscosity and Normal Stress

So far we discussed the behaviour of polymeric liquids slightly disturbed from equi-librium, in the linear regime. Here we show the behaviour in flowing systems thatproduce significant deviations from the equilibrium microstructure.

1.7.1 Shear Thinning

Most polymeric liquids have their effective viscosity reduced upon shearing. Theviscosity defined in Equation (1) is a decreasing function ofthe shear rate. Figure 7shows the decrease in the viscosity for two polymeric liquids. The shear thinningis not so much pronounced in dilute solutions as it is in concentrated solutions andmelts [4, 7].

A special class of polymers is known asLiving Polymers, which are long lin-ear structures formed from cylindrical liquid crystallinephases of micelles. Theyare called living polymers because they form and break alongtheir length owingto thermodynamic and flow considerations. They are also known asworm like mi-celles. At equilibrium, their behaviour is very similar to a high molecular weightconcentrated solution of polymers: large viscosity and elastic modulus. Howeverupon shearing they can break leading to the behaviour exhibited by low molecu-lar weight counterparts, aligning in the shearing direction. The viscosity reductionupon shearing is therefore very significant and sharp in these systems, as depictedschematically in Figure 7. Examples of every day use are shampoo and shower gelswhich are very viscous at rest, but can be easily flown out of the container by grav-itational forces (which are sufficient to overcome the viscosity after they begin toflow).

Polymer Rheology 15

1.7.2 Normal stresses

The normal stress difference is zero for a liquid that is isotropic. Polymeric liquidshaving microstructure can develop anisotropy in the orientation of the constituentpolymers in flow, there by leading to normal stress differences. The normal stressbehaviour shows a similar behaviour to that of shear stress.The normal stress dif-ference has two components

N1 = τxx− τyy (13)

N2 = τyy− τzz (14)

where for planar Couette flow,x is the direction of flow,y is the direction of thegradient andz is the vorticity direction. Similar to the viscosity, whichis a coefficientof the shear stress, we can define two coefficients for the normal stresses: the onlydifference is the denominator which isγ2, because it is the lowest power of the shearrate that the normal stresses depend on. The coefficients arecalled as First normalstress coefficients,Ψ1 andΨ2 defined as:

Ψ1 =N1

γ2 (15)

Ψ2 =N2

γ2 (16)

The normal stress coefficients show shear thinning very similar to the viscosity,however for very large shear rates, the absolute value for the normal stress differencecan become larger than the shear stress as shown in Figure 8. Such a behaviour isseen in concentrated solutions and in melts [4]. The second normal stressN2 isusually zero for polymeric liquids.

1.7.3 Elongational Flow Viscosity

Extensional or elongational flow where the local kinematicsdictates that the fluidelement is stretched in one or more directions and compressed in others. Liquids be-ing practically incompressible, the elongational stretchconserves the volume. Elon-gational flows are encountered many situations. Though there are very few systemswhere it is purely extensional, there are several cases where there is significant elon-gation along with rotation of fluid elements, the two basic forms of fluid kinematicsdue to flow. Sudden changes in flow geometry such as contraction or expansion,spinning of fibres, stagnation point flows, breakup of jets ordrops, blow moulding,etc.

The simplest elongation is called as the uniaxial elongation the velocity gradienttensor for which is given by

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Fig. 8 Comparison of theabsolute values of normalstresses with thta of the shearstresses as a function of shearrate for concentrated solutionsand melts.

log γ

logσ,

N1

N1

σ

∇v = ε

1 0 00 −1

2 00 0 −1

2

(17)

which corresponds to stretching along thex direction and equal compression alongy andz directions.ε is called the elongation or extension rate. The elongational (ortensile) viscosity, is defined as

ηE =σxx−σyy

ε=

σxx−σzz

ε(18)

The elongational viscosity, like the shear viscosity is a function of the shear rate.However, in the case of elongational flows, it is difficult to measure the steady statevalue. In experiments it is only possible to access a time dependent valueη+

E (t, ε)which is a transient elongational viscosity or more precisely the tensile stress growthcoefficient [8]. The elongational viscosity is defined as theasymptotic value of thiscoefficient for large timest → ∞. The behaviour of the transient growth coefficientsfor various elongation rates is shown in Figure 9. The elongational viscosity abruptlyincreases to a high value in short times [1]. This phenomenonis called as strainhardening. The elongational strain is measured by theHenky strain defined as

ε = ε t = log

(

L(t)L0

)

(19)

Polymer Rheology 17

Fig. 9 Transient growth co-efficient (trbansient elonga-tional viscosity) for increasingstrain ratesε .

log t

logη+ E ε

Fig. 10 Behaviour of theelongational viscosity withelongation rate for variouspolymeric liquids. The elon-gational viscosity is not nec-essarily the steady state value,and it could be the maximumor the value at the terminalHencky strain.

SolutionsBranched Melts

Linear Melts

log ε, log γ

logη

logη

E

η

ηE

×3

whereL0 is the initial length of an element along the stretch direction andL(t) isthe deformed length that grows asL(t) = L0eε t The elongational viscosity attains amaximum and then falls down.

It is not possible to easily measure the terminal or asymptotic value of the vis-cosity. This is because for a Hencky strain ofε = 7, the elongation required is about1100 times the initial value. It is a convention therefore toreport the maximumvalue of the transient tensile growth coefficientη+

E as a function of the strain ratefor practical applications. The “Steady values” of the viscosity reported are eitherthe maximum values at a given strain rate, or the value of the coefficient at a givenexperimentally accessible Hencky strain. The behaviour ofthis viscosity is shownin Figure 10 for various polymeric liquids [4].

Another useful way to report the elongational viscosity is the Trouton ratio [1],which is defined as

TR =ηE(ε)

η(√

3ε)(20)

where the shear viscosity is measured at a shear rateγ =√

3ε (see further detailsin the chapter on Non-Newtonian Fluid Mechanics). For Newtonian (or inelasticliquids) the elongational viscosity is three times the shear viscosity. For polymeric

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Fig. 11 Trouton ratios forvarious polymeric liquids.Dilute solutions show themost dramatic increase in theelongational viscosity.

3

100

1000

Melts

Inelastic liquid

Dilute Solution

log ε, log γ

log

TR

1/2λ

liquids, at low shear (and elongation) rates, the Trouton ratio TR is alwaysTR ≈ 3.The Trouton ratio plots, such as the one shown in Figure 11 provide an indicationof the extent of elongational viscosity effects in relationto the shear viscosity [4].The most dramatic effects are in dilute solutions of polymers where, dependingon the molecular weight of the polymer the Trouton ratio can be several orders ofmagnitude higher than unity.

2 Modelling and Physical Interpretation

Modelling in polymer rheology is mainly of two types: Phenomenological andmolecular modelling. Some general phenomenological models have been discussedin the chapter on Non-Newtonian Fluid Mechanics. The basic molecular modelshave been discussed in the chapter on Polymer Physics: the Rouse and Zimm mod-els [9]. Here we present some simple physical interpretations of the polymer rheo-logical behaviour.

2.1 Polymer solution as a suspension

The simplest explanation of the polymeric liquid viscosityis that of a dilute solution.A dilute solution is like a suspension of colloidal particles, except that the particlesare not spherical. The polymers in equilibrium have a coil like structure (see chapteron Polymer Physics). Considering the polymer coils to represent colloidal particles,the viscosity can be related to the volume fraction. For dilute colloidal suspensionthe viscosity of the suspension is more than the solvent viscosity, and is given bythe Einstein expression

η = ηs

(

1+52

φ)

(21)

Polymer Rheology 19

whereφ is the volume fraction of the colloidal particle. This approximation is validin the limit φ → 0, for spherical particles. Polymer coils are not rigid spheres, butbehave more like a porous sphere (on the average, because theshape of the coilchanges continuously due to thermal Brownian forces from solvent). Because ofthis the increase in viscosity is reduced. For dilute polymers the viscosity can bewritten as

η = ηs(1+UηRφ) (22)

whereUηR is a constant< 5/2. This constant is an universal constant for all poly-mers and solvents chemistry and depends only on the solvent quality (which couldbe poor, theta, or good solvents). The Zimm theory predicts this constant to be 1.66,whereas experiments and simulations incorporating fluctuating hydrodynamic inter-actions (without the approximations made in Zimm theory) observe the value to beUηR = 1.5.

2.2 Shear Thinning

As noted before, the extent of shear thinning is much more significant in concen-trated solutions and melts in comparison with dilute solutions (see Figure 7). Theexplanation for viscosity decrease in dilute solutions is subtle and involves severalfactors [9, 7]. We will restrict to simple explanations in entangled systems (i.e. inconcentrated solutions and melts). In the entangled state,the microstructure of thesolution is like a network: the entanglements act like nodesof a covalently bondednetwork junctions which restricts the motion of the polymers and therefore of thesolution. This is the reason for the high viscosity of the solution. However, uponshearing the chains begin to de-entangle and align along theflow. This reduces theviscosity.

2.3 Normal Stresses

The existence of normal stresses in polymeric systems is dueto the anisotropy in-duced in the microstructure because of flow. This can be easily understood in thecase of dilute polymer solutions. In the absence of flow, the coil like structure ofthe molecules assume a spherical pervaded volume, on the average. The flow causesthe molecules to stretch towards the direction of flow and tumble. This results in anpervaded volume that is ellipsoidal and oriented towards the direction of flow. Therestoring force is different in the two planesxxandyy. This results in the anisotropicnormal forces.

20 P Sunthar

Fig. 12 Theoretical pre-diction of the elongationalviscosity in the “standardmolecular theory” or the tubemodel, for linear polymermelts.

Reptation Orientation Stretching

Ful

ly S

tret

ched

log ε

logη

E

τd τR

2.4 Elongational Viscosity

As shown in Figure 11 dilute solutions have a dramatic effecton the elongationalviscosity. The main reason for this is in the dilute state at equilibrium, the polymersare in a coil like state in a suspension whose viscosity is only slightly more than thesolvent viscosity. Whereas upon elongation the chains are oriented in the direction ofelongation. After a critical elongation rate given by the Weissenberg number Wi=0.5, the thermal fluctuations can no longer hold against the forces generated on theends of the polymer due to flow. The chains then begin to stretch. This leads toanisotropy and stress differences between the elongation and compression planes,and is observed as an increase in the elongational viscosityin Equation (18). Sincethe polymers are not infinitely extensible, the stress difference between the planessaturates to the value given by that in the fully stretched state (which is like a slenderrod).

The behaviour of elongational viscosity in concentrated solutions and melts isdifferent. The zero shear rate viscosity is itself much higher than the solvent vis-cosity. A semi-quantitative model for the behaviour in linear polymer melts can beinterpreted in terms of the tube model or also called as the “standard moleculartheory” [5]. At strain rates small compared to the inverse reptation time (or dis-entanglement time)τ−1

d , thermal fluctuations are stronger than the hydrodynamicdrag forces induced by the flow, and the chain remain in close to equilibrium mi-crostructure. The viscosity is nearly same as the zero-shear rate viscosity, as shownschematically in Figure 12. At higher strain ratesε � τ−1

d , the tubes (the pervadedvolume of the polymers), are disentangled and begin to orient along the extensionaldirection, which decreases the viscosity. This continues till the strain rate becomescomparable to the inverse Rouse relaxation timeτ−1

R , when chain stretching begins(similar to the case in dilute polymer solutions). This causes an increase in the vis-cosity. The terminal region of constant viscosity is occursdue to the chains attainingtheir maximum stretch. Most of these theoretical predictions have not been verifiedby experiments, mainly due to the difficulty in consistentlymeasuring the elonga-tional viscosity at very large strain rates and strains.

Polymer Rheology 21

Other reading

An historical account of rheology in general is discussed inRef. [10]. Ref [7] con-tains an extensive literature survey of various aspects of polymer rheology, and gen-eral rheology. A compilation of recent works in melt rheology is given in Ref. [11].

Figures in this chapter

The figures in this chapter have been schematically redrawn for educational pur-poses. They may not bear direct comparisons with absolute values in reality. Wherepossible, references to the sources containing details or other original referenceshave been provided.

References

1. H.A. Barnes, J.F. Hutton, K. Walters,An Introduction to Rheology(Elsevier, Amsterdam,1989)

2. R.B. Bird, R.C. Armstrong, O. Hassager,Dynamics of Polymeric Liquids - Volume 1: FluidMechanics, 2nd edn. (John Wiley, New York, 1987)

3. M. Rubinstein, R.H. Colby,Polymer Physics(Oxford University Press, 2003)4. H.A. Barnes,Handbook of Elementary Rheology(University of Wales, Institute of Non-

Newtonian Fluid Mechanics, Aberystwyth, 2000)5. J.M. Dealy, R.G. Larson,Structure and Rheology of Molten Polymers: From Structure to Flow

Behaviour and Back Again(Hanser Publishers, Munich, 2006)6. C.W. Macosko,Rheology: Principles, Measurements, and Applications(Wiley-VCH, 1994)7. R.G. Larson,The Structure and Rheology of Complex Fluids(Oxford University Press, New

York, 1999)8. J.M. Dealy, J. Rheol.39, 253 (1995)9. R.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager,Dynamics of Polymeric Liquids - Vol-

ume 2: Kinetic Theory, 2nd edn. (John Wiley, New York, 1987)10. R.I. Tanner, K. Walters,Rheology: An Historical Perspective(Elsevier, 1998)11. J.M. Piau, J.F. Agassant,Rheology for Polymer Melt Processing(Elsevier, Amsterdam, 1996)