Non-Newtonian fluid mechanics and polymer rheology

2 Non-Newtonian fluid mechanics and polymer rheology K E G E O R G E, Cochin University of Science and Technology, India

Abstract: One of the main reasons for the increased use of polymers in recent decades is the ease of converting them to complex shapes. Extrusion and injection moulding are the most widely used techniques for polymer processing. The success of these techniques depends upon controlling the rheological behaviour of the systems used. The shear flow can be controlled by manipulating the material parameters like molecular weight, molecular weight distribution and chain branching, using modifiers like fillers, plasticizers and other polymers and adjusting the processing variables like temperature, shear and pressure. Elastic behaviour and elongational flow of the system can also become important depending upon the operation. Hence characterization of the system by relevant techniques is necessary. Fundamentals of these rheological concepts are described in this chapter. Key words: polymer processing, polymer melt flow, polymer solutions, polymers as rheology modifiers.

2.1

Introduction

The behaviour of liquids such as polymer melts or polymer solutions is important because they are the base from which polymer products are formed. They are distinguished from ordinary liquids like water or oil by their highly viscous, non-Newtonian and viscoelastic natures. Rheology is the science of deformation of materials or the study of the relationship between the amount of deformation and the force that produces the deformation. By this definition the subject of rheology can apply to both solids and liquids equally well. However, the term rheology generally deals more with the liquid state even though the solid-like behaviour of polymer liquids and the liquid-like behaviour of solid polymers are definitely part of it. Molten polymers or solutions of polymers are very viscous liquids. Further, they are non-Newtonian, which means that their viscosity is dependent on the rate of shearing. Also they exhibit some degree of elasticity like solids. The processing of thermoplastic polymers may involve three distinct thermomechanical stages: 1. The plastication stage in which a polymer goes from solid state (granule, powder, etc.) to a homogeneous liquid state. 13

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Advances in polymer processing

2. The shaping stage where the molten polymer flows under pressure into moulds or dies. 3. The shape stabilization stage where the final shape is stabilized with cooling coupled with drawing, biaxial stretching, blowing, etc. While the liquid state predominates in these three stages, one has to deal with the solid, semi-solid and liquid stages also. The viscosities of molten polymers at usual operating temperatures are in the range of 102 to 104 Pa.s which is 105 to 107 times higher than that of water. This high viscosity has the following practical implications: • • •

Shaping by a die is possible. Heating can be done by viscous dissipation. High pressures are to be applied for making such fluids flow.

The thermal diffusivity of most polymers is in the order of 10–7 m2/s, which is about 1000 times less than the value for copper. So plastication of a polymer using pure conduction from walls of containers will require very long residence times and/or low flow rates, hence it is necessary to split the polymer into thin layers and this accounts for the screw-barrel system being an integral part of extrusion and injection moulding, the most widely used techniques of polymer processing.

2.2

Non-Newtonian behaviour

The simplest constitutive relation between force and deformation for liquids is Newton’s law of viscosity which states that in shear the stress in a liquid is proportional to the rate of shearing:

τ = ηγ˙

2.1

where τ is the applied shear stress, γ˙ is the rate of strain or the shear rate (dγ / dt) which is the local velocity gradient, and η is the constant of proportionality called the Newtonian shear viscosity, which is an intrinsic property of a Newtonian liquid. Generally, low molecular weight liquids exhibit Newtonian behaviour. In contrast, the viscosity of high molecular weight polymer melts and solutions decreases significantly with shear rate (known as shear thinning or pseudoplastic behaviour) (Fig. 2.1). At sufficiently low shear rates the viscosity is independent of shear rate. This viscosity measured at low shear rates, where it is shear rate invariant, is known as the zero shear rate viscosity, η0. In rare cases the curve enters a second linear region of proportionality at high shear rates and the viscosity in this region is called the viscosity at infinite shear rate, η∞. Better names for η0 and η∞ may be initial Newtonian viscosity and final Newtonian viscosity

Non-Newtonian fluid mechanics and polymer rheology η0

15

η0 log η

η

η∞

γ˙ (a)

log γ˙ (b)

2.1 Shear thinning behaviour of polymers: (a) linear scale; (b) logarithmic scale.

respectively. Final Newtonian behaviour is generally not observed due to the extensive breakdown of polymer chains at high shear. The opposite behaviour of pseudoplasticity, i.e. an increase in viscosity with shear rate, known as dilatancy, is less common.

2.2.1

Power law model

The shear rate dependent behaviour can be approximated as a straight line on a log–log plot, suggesting a power law behaviour to describe the dependence on shear rate. This can be given as

τ = kγ˙ n or η = kγ˙ n–1

2.2

(the Ostwald–de Waele equation1–3), where k and n are constants for a given fluid. k is called the consistency index and is a measure of the fluidity of the material, and n, the power-law index, is a measure of the nonNewtonian behaviour. When n = 1, the equation reduces to Newton’s law with k = η. There are some obvious limitations to this simple constitutive model, the most obvious being its invalidity at low shear rates (it predicts infinitely high viscosity) and the non-inclusion of zero shear viscosity as a parameter. Also, negative shear rates do not give a viscosity and the unit of viscosity depends on n. However, the only two parameters which are needed to fit the model can be readily obtained experimentally and for most polymers the power law will suffice over the range of shear normally encountered, even though the model fails at low shear rates.

2.2.2

Polynomial model

This is a modification of the power law to allow for some non-linearity in the flow curve4 and is given by

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Advances in polymer processing

log τ = A0 + A1 log γ˙ + A2 (log γ˙ ) 2

2.3

There are three material parameters: A0, A1 and A2. This model can span a wider range of shear rates and the constants in the model can be evaluated from experimental data by regression analysis.

2.2.3

Other models

There are several other models in use. For example, the Ellis model discussed by Reiner5 allows for a constant viscosity region at low shear rate and is given by α –1 η0 =1+  τ  η τ0 

2.4

There are three constants: η0, τ0 and α. This model has some of the features of the power law since it predicts that the log τ vs log γ˙ relationship is linear at high shear rates and hence α is related to n. The Carreau model6 gives allowance for constant viscosity at both high and low shear rates and is given by

η – η∝ = [1 + ( λγ˙ ) 2 ]( n–1)/2 η0 – η∝

2.5

where η∝, η0, λ and n are constants.

2.2.4

Time-dependent fluids

Another class of non-Newtonian fluids is (shear) time-dependent fluids which are further subdivided into thixotropic and rheopectic. Thixotropic fluids display a reversible decrease in viscosity with time, while rheopectic fluids show a reversible increase in viscosity with time. This means that for thixotropic fluids the viscosity decreases with the time of shearing, while for pseudoplastic fluids the viscosity decreases with the rate of shearing.

2.3

Newtonian shear flow

In polymer processing, flow occurs in well-defined geometry. Two types of flow are encountered commonly: pressure-driven flow and drag flow. Pressuredriven flows occur during melt flow through a die and the filling of compression and injection moulds. Drag flow occurs when a fluid is confined between the walls which are moving relative to each other. This type of flow occurs in the relative movement of screw and barrel.

Non-Newtonian fluid mechanics and polymer rheology

2.3.1

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Hagen–Poiseuille flow

In the case of a pressure-driven steady isothermal flow of an incompressible fluid in a horizontal, circular channel under laminar flow, we can make a simple force balance F2 = F1 + F3 (Fig. 2.2) to get an expression for the shear stress:

τ = r  ∂P  2 ∂Z

2.6

If the pressure drop is uniform, then for a pressure drop ∆P over a length L, the shear stress is given by

τ = ∆P r 2L For a Newtonian fluid

2.7

2.8 τ = ηγ˙ = η ∂v ∂r Substituting this and integrating with the boundary condition v = 0 at r = R (assuming no slip at the wall) we get the velocity profile

( ) 

 v = v 0 1 – r R 

2

2.9

where v0 is the maximum velocity at the centre of the channel. This also shows that the laminar flow profile of a Newtonian fluid through a circular channel is parabolic. The volumetric flow rate can be obtained as P – ∂P dz ∂z

P

Low pressure

High pressure

F3

R

r

F1

F2

τ

z direction

F3 dz

2.2 Force balance for a cylindrical element of fluid under laminar flow to get an expression for the shear stress.

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Q=

π R 4 ∆P 8ηL

2.10

and the wall shear rate ∂v r = R as ∂r

4Q γ˙ w = – π R3

2.11

The negative sign arises because of the chosen coordinate system. That is, although the velocity at the wall is zero, the wall shear stress and wall shear rate are finite. The magnitude of the shear stress and shear rate vary from zero at the central line to maximum values at the wall. This type of flow is known as Hagen–Poiseuille flow.

2.4

Shear flow of a power law fluid

For a power law fluid the corresponding flow equations are

( )

 v = v 0 1 – r R 

  

2.12

( 3nn++11 ) π R v 4Q = – ( 3n + 1 ) 4n π R

Q=

γ˙ w

n+1 n

2

0

2.13

3

2.14

The velocity and shear rate profile of a power law fluid are no longer parabolic and the expression 4Q/πR3 gives only the apparent wall shear rate (γ˙ w app ) . When n = 0.5, the actual wall shear rate is 1.25 γ˙ w app and when n = 0.1 the actual value is 3.25 γ˙ w app . The effect of the power law index n on the velocity profile is shown in Fig. 2.3.7 The velocity profile tends to become plug flow when n → 0. This means that the shear rate is high only in a very narrow region close to the wall and is relatively low over the central portion of the channel. This results in low viscosity near the wall.

2.5

Parameters influencing non-Newtonian behaviour

2.5.1

Temperature dependence of viscosity

When the shear stress vs shear rate curves (flow curves) of polymers are obtained at several temperatures and log viscosity is plotted against log shear rate, the curves converge at increasing shear rates as shown in Fig. 2.4.8 When the natural log of the viscosity of these data is plotted against 1(T in

Non-Newtonian fluid mechanics and polymer rheology

19

3

2.5

n=1

v /

2

n = 1/3 1.5

n = 0.01 1

0.5

0 –1

–0.5

0 r/ R

0.5

1

2.3 Effect of power law index on the reduced velocity profile. 105

Viscosity (Pa.)

150°C

170°C 200°C

104

103

102 10–2

10–1

100 101 Shear rate (s–1)

102

103

2.4 Variation of viscosity with shear rate (log–log plot) for linear low density polyethylene (MI = 1 g/10 min, Mw = 119 600, PI = 3.8) at different temperatures.

kelvin), a straight line with a positive slope is obtained. This is called an Arrhenius plot (Fig. 2.5). From the slope, the activation energy Ea( Eγ˙ or Eτ as the case may be) can be calculated. The flow activation energy is a very important molecular parameter for describing the dependence of viscosity of polymers on temperatures well

Advances in polymer processing

Ea R

logeη

20

1 (K –1) T

2.5 A typical Arrhenius plot.

Table 2.1 Activation energies of some commercial polymers Polymer

Activation energy, kJ/mole

High density polyethylene Linear low density polyethylene Low density polyethylene Polypropylene Polystyrene

25 30 55 40 100

above their glass transition temperature (Tg). A higher activation energy means a higher dependence of viscosity on temperature. Activation energies of some common polymers calculated from their Newtonian viscosities are listed in Table 2.1.9 The figures show that varying the temperature of polymer melts is an efficient method of controlling the viscosity, particularly for high activation energy polymers like polystyrene if thermal degradation can be minimized. The glass transition temperature (Tg) has a great influence on the viscosity. The mobility of chain segments and molecules depends on how far away they are from the Tg and how much free volume is available around them. Williams, Landel and Ferry10 suggested the following equation for the variation of viscosity from the glass transition temperature: log

17.44 ( T – Tg ) ηT = – η Tg 51.6 + ( T – Tg )

2.15

The equation is particularly useful for polymers such as polystyrene whose melt is much closer to its glass transition temperature.

Non-Newtonian fluid mechanics and polymer rheology

2.5.2

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Polymer melts and entanglements

The dependence of zero shear viscosity η0 on the weight average molecular weight Mw is well established for linear polymers of narrow molecular weight distribution. For low molecular weight polymers such as uncured thermosetting resins, the viscosity is directly proportional to the molecular weight when the molecular weight is below a critical molecular weight, Mc, of the polymer (Fig. 2.6):

η0 = K1Mw for Mw < Mc

2.16

The critical molecular weight denotes the point where the polymer chains start to interact or entangle. For most thermoplastics of commercial interest, entanglements are present and hence the viscosity has a much higher dependence on molecular weight:

η 0 = K 2 M w3.4 where M w > M c

2.17

log η0

where K1 and K2 are constants that depend on the molecular structure. Entanglements are an essential feature of polymer chains. They contribute greatly to the properties of polymer melts and solids. For most polymers Mc lies between about 10 000 and 40 000 and increases with increasing chain stiffness. Critical molecular weights of some commercial polymers are listed in Table 2.2.9 For polymer solutions the change in slope occurs at a particular value of c.Mw, where c is the polymer concentration, rather than Mw alone. As the shear rate is increased, the average entanglement density decreases. Since the viscosity is strongly dependent on the number of entanglements, viscosity falls.

3.4

Mc

log Mw

2.6 Dependence of zero shear viscosity on molecular weight.

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Advances in polymer processing Table 2.2 Critical molecular weights of some commercial polymers

2.5.3

Polymer

Mc

Linear polyethylene Polystyrene Polymethyl methacrylate Cis-polyisoprene 1,4-polybutadiene

4000 30000 28000 10000 5000

Molecular weight distribution (MWD)

The transition from the constant zero shear region to the shear thinning region is more abrupt and occurs at a higher shear rate for the narrow molecular weight distribution polymer than for the broadly distributed material. Increasing Mw/Mn also has the effect of decreasing the slope of the shear thinning region. Narrow MWD polymers are known to show melt flow instabilities more easily than their broad MWD counterparts. The viscosity curves of two polymers having the same weight average molecular weight but different MWDs are shown in Fig. 2.7.8

2.5.4

Chain branching

Many industrially important polymers are branched. The branches can be long or short and these in turn can have secondary branching. The branches can be randomly spaced along the backbone chain or several branches can originate from a single point to give a star-shaped molecule. Short branches generally do not affect the viscosity of a molten polymer very much. On the other hand long branches can have a significant effect. Branches that are long but are still shorter than those required for entanglements decrease the zero shear viscosity when compared to a linear polymer of the same molecular weight.11,12 However, if the branches are long enough to participate in entanglements, the branched polymer may have a viscosity much higher than that of a linear polymer of the same molecular weight at low rates of shear.13,14 In general, branched polymers also show more shear thinning than comparable linear polymers.

2.5.5

Effect of additives

Properties of polymers can be modified to a great extent by manipulating their structures. Still, very few polymers are used in their chemically pure form. Additives that are widely used affect rheological properties of the base polymer to which they are added even when they are not added for that purpose.

Number of molecules

Non-Newtonian fluid mechanics and polymer rheology

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Resin-B

Resin-A

100

101

102 103 Molecular weight (a)

104

104

Viscosity (Pa.s)

Resin-B Resin-A 103

102 100

101

102 Shear rate (s–1) (b)

103

104

2.7 Dependence of molecular weight distributions on the flow curve: (a) molecular weight distribution; (b) flow curves (sketch courtesy of D.E. Delaney, Dynisco Polymer Test).

Fillers While fillers generally enhance the viscosity of polymers, the extent of increase depends upon the type of filler and shear. The effect of different types of fillers on the viscosity–shear stress curve of a polymer is shown in Fig. 2.8.15 High aspect ratio fillers such as long glass fibres produce significant improvement in viscosity at low shear, but the effect is not so pronounced at high shear. In the case of non-interacting low aspect ratio fillers, the enhancement is more or less the same at low and high shear. In the case of interacting low aspect ratio fillers, the effect is similar to that of high aspect ratio fillers, i.e. significant improvement in viscosity at low shear, with a

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Advances in polymer processing High aspect ratio filler Agglomerated low aspect ratio filler

Viscosity

Low aspect ratio filler

Base polymer

Shear stress

2.8 Effect of different types of fillers on the viscosity–shear stress behaviour of polymer.

much reduced effect at high shear. The elastic response of filled melts is usually reduced relative to that of the base polymer. However, fillers of high aspect ratio may form a network themselves within the melt and produce anomalous elastic response.16 There may be a dramatic increase in elongational viscosities when fibrous fillers are added to polymer melts.17 The viscosity of filled systems can be easily determined for a Newtonian fluid by using the Einstein equation18,19 for low concentration and low shear:

η = ηs (1 + 2.5φ)

2.18

where ηs is the viscosity of the suspending liquid and φ the volume content of the solids. This equation does not take into account the shape or size of the dispersed particles. At higher filler content, they have to be taken into account. An example is the Krieger–Dougherty equation:20

φ  –[ η ]φm η = ηs  1 + φm  

2.19

where φm is the maximum packing fraction and [η] the intrinsic viscosity. Plasticizers Plasticizers are low molecular weight liquids used for improving the processability of polymers and flexibility of products. Plasticizers act by spacing out the polymer molecules. Their most obvious effect is to reduce

Non-Newtonian fluid mechanics and polymer rheology

25

viscosity, but they also tend to reduce the elastic modulus of the melt, thereby increasing the elastic response at a given stress. The effectiveness of a plasticizer depends on its concentration, compatibility and viscosity.13 Low molecular weight polymers are frequently effective plasticizers.

2.5.6

Effect of pressure

Even though the effect of pressure on viscosity is not as large as the effect of temperature, pressure effects can become significant in processes such as injection moulding where high pressures are employed. Several theories support the idea that viscosity is strongly dependent on the free volume of the system.21–25 Since free volume is directly influenced by pressure, viscosity also depends on pressure.26–28 Viscosity generally increases with increasing pressure and an exponential relation such as

η0 = A.eBP has been found to be useful. polymer.

2.6

2.20 29,30

A and B are constants characteristic of the

Elongational flow

For polymer processes such as fibre spinning, blow moulding, thermoforming, certain extrusion die flows, etc., the major mode of deformation is elongational. The elongational viscosity (µ) of a liquid is much higher than the shear viscosity and for a simple Newtonian fluid Trouton31 has suggested

µ 0 = 3 η0

2.21

This relation holds good for polymer melts in the initial Newtonian region. Thereafter, regions of strain hardening or strain thinning are observed for different materials. The shear and elongational viscosities for two types of polystyrenes are shown in Fig. 2.9. In the region of the Newtonian plateau, the limit of 3 is observed.32 The elongational viscosities as a function of tensile stress for various thermoplastics in common processing conditions are shown in Fig. 2.10.32 The strain-thinning part is due to the reduction in entanglement density by links being slipped off the ends of chains. These are short-lived entanglements near the chain ends. The increase in µ with strain rate is an effect of the long-lived entanglements giving rise to strain hardening.

2.7

Elastic effects in polymer melt flow

Polymeric fluids are also viscoelastic in nature, i.e. they exhibit both viscous and elastic properties. For purely Hookean elastic solids the stress corresponding

26

Advances in polymer processing 5.108

Viscosity, η˙ ,µ (Pa.s)

µ 0 = 1.7·108 Pa-s

Elongational test

µ 0 = 1.6 ·108 Pa-s

108

η0 = 5.5·107 Pa-s

5.107

η0 = 5 ·107 Pa-s

Shear test

107

T = 140°C

5.106

Polystyrene I Polystyrene II

106 102

5.102 103

104 105 Shear/tensile stress τ, σ (Pa)

5.105

2.9 Shear and elongational viscosities of two types of polystyrenes. 6 LDPE

Viscosity, log µ (Pa.s)

5 Ethylene–propylene copolymer

4

PMMA POM

3 PA66

2 3

4 5 Tensile stress, log σ (Pa)

6

2.10 Elongational viscosities of polymers as a function of tensile stress.

to a given strain is time independent, but with viscoelastic materials the stress dissipates over time. Viscoelastic fluids undergo deformation when subjected to stress; however, part of such deformation is gradually recovered when stress is removed.

Non-Newtonian fluid mechanics and polymer rheology

27

The response of a viscoelastic material to an applied stress or strain depends on the rate of application of this stimulus. When the structure of a body is perturbed by external stimuli, the time it takes to reach another equilibrium state or return to its original rest state is characteristic of that structure. A useful parameter to determine the extent of elasticity is the Deborah number De defined by Reiner:33

De = λ tp

2.22

where λ is the relaxation time of the polymer and tp a characteristic process time. In the case of flow through a die, the characteristic process time can be defined by the ratio of die dimension to average speed through the die. A Deborah number of zero represents a viscous fluid and a high Deborah number an elastic solid. As the Deborah number becomes greater than 1, the polymer does not have enough time to relax during the process, resulting in possible extrudate dimension deviation or irregularities such as extrudate swell, shark skin or even melt fracture. Even though the Deborah number provides a reasonable qualitative description of material behaviour, polymers generally cannot be characterized by a single response time. A more realistic description involves the use of a distribution or a continuous spectrum of response times. If a Newtonian fluid is forced through a capillary, the extrudate would swell to a diameter larger than that of the capillary on exit from the die. A viscoelastic material shows a much more pronounced swell on extrusion than a Newtonian material. The reason for this is the generation of forces normal to the shearing direction. Normal stresses are an essential component in polymer processing.

2.8

Polymer blends

Polymer blending is a common technique for manipulating the rheological behaviour. In the case of miscible blends where the constituent polymers mix intimately on a molecular level to form a homogeneous material, the rheological properties, like other properties, are usually a weighted average of the properties of the component polymers. An example of this is provided by blends of poly(phenylene oxide) (PPO) and polystyrene.34 In the case of high viscosity incompatible polymers, the blend may have significantly lower viscosity than either of the constituents as in the case of the blends of polycarbonate and poly(4-methylpentene-1) shown in Fig. 2.11.15 This behaviour is attributed to the weakness of shear planes at the interface between the phases.35 A completely different response may be observed in the blends of low viscosity melts where the blend may have a higher viscosity than either of the constituents as in the case of the blends of polypropylene

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Viscosity(Ns/m2)

Poly(4-methyl pentene-1)

Polycarbonate

103

102

50/50 blend

104

105 Shear stress (N/m2)

106

2.11 Variation of viscosity with shear stress for polycarbonate, poly(4methyl pentene-1) and their 50/50 blend.

and nylon-66 shown in Fig. 2.12.15 This effect is probably due to the large interfacial area produced in a dispersion at a 1 µm level. The flow of such a dispersion requires that the droplet shape be deformed to an ellipsoid, the increase in surface area accounting for the increase in viscosity. Generally, it is not easy to predict the rheology of polymer blends since it depends on so many factors such as compatibility, concentration of the components, viscosity of the components, morphology of the blends, history of blending operations, etc.36 Melt blending is the most widely using technique of blend preparation. Melt blending of two immiscible polymers naturally results in a dispersion of the minor phase component within a matrix of the major component. The dispersed phase droplets tend to minimize their surface area to volume ratio by adopting a spherical or near-spherical shape, and a high shear mixer like a twin screw extruder can generate a fine dispersion with droplet diameters of <1 µm. The deformation of droplets is promoted by the shear stress (τ) exerted by the flow field but resisted by the interfacial stress, σ/R, where σ is the surface tension and R the local radius of the droplet. The ratio of these two stresses is called the capillary number: Ca =

τR σ

2.23

The capillary number will decide whether a droplet will disperse or remain stable within a flow field.

Non-Newtonian fluid mechanics and polymer rheology

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Polypropylene

Viscosity (Ns/m2)

70/30 Nylon/PP blend

102 Nylon 66

101

103

104 Shear stress (N/m2)

105

2.12 Variation of viscosity with shear stress for nylon 66, polypropylene (PP) and 70/30 nylon/PP blend.

2.9

Polymer nanocomposites

Nanofillers or nanomodifiers have revolutionized the area of polymer matrix composites with novel properties and applications. Carbon nanotubes, nanoclay and nanosilica are the nanomodifiers commonly employed to develop polymer nanocomposites. The rheology of these novel materials is of great significance during both the preparation of the composite and the subsequent processing, because proper dispersion of the nanomodifiers is a must for efficient reinforcement. All these nanofillers require modification/coupling for proper reinforcement and such modifications also alter the rheology of the system. Because viscoelastic characterization is highly sensitive to the nanoscale and mesoscale structures of polymeric materials, dynamic rheological measurements are very significant to the study in the rheological behaviour of polymer nanocomposites. Nanoclay is the most widely investigated nanoparticle in a variety of different polymer matrices for a spectrum of applications.37,38 Solution intercalation and melt intercalation are the most widely used techniques for preparation of clay nanocomposites. In the case of solution intercalation the layered silicate is exfoliated into single layers using a solvent in which the polymer is soluble. Such layered silicates can be easily dispersed in an adequate solvent due to the weak forces that stack the layers together. The polymer is then absorbed into the delaminated sheets and when the solvent

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Advances in polymer processing

is evaporated (or the mixture precipitated) the sheet reassembles, sandwiching the polymer to form an ordered, multilayered structure.39 Solution intercalation can also be done from polymers in solution. In the case of melt intercalation, the layered silicate is mixed with the solid polymer matrix in the molten state. Under these conditions and if the layer surfaces are sufficiently compatible with the selected polymer, the polymer can be inserted into the interlayer space or form an intercalated or an exfoliated nanocomposite without the use of any solvent. In the case of clay nanocomposites rheological characterization is found to be a valuable tool to study the degree of exfoliation and dispersion.40,41 The number of particles per unit volume is a key factor determining the characteristic response for clay nanocomposites. Carbon nanotubes (CNTs) possess a very high aspect ratio and have exceptional mechanical properties. Melt compounding is shown to be a suitable way to disperse CNTs homogeneously in a polymer matrix. With very low concentrations of CNTs (0.5 wt%) a significant change in the qualitative rheological behaviour such as a transition from liquid-like to solid-like behaviour is observed.42 The mixing parameters, time, temperature and shear have a significant effect on the nanocomposite structure. The mechanical properties of nanocomposites prepared at optimized conditions are much superior.43 The effect of rotor speed in a torque rheometer during the incorporation of nanosilica in polystyrene on the tensile modulus of polystyrene/silica nanocomposites is shown in Fig. 2.13.44 It may be observed that both the silica content and shear are critical factors in attaining maximum modulus.

2.10

Rheometry

Glass capillary viscometers are widely used to measure the viscosity of low viscosity fluids, and in such instruments it is gravity that provides the pressure 25

Modulus (MPa)

20 15 30 rpm 50 rpm 70 rpm 90 rpm

10 5 0 0

1

2 3 Silica (wt%)

4

2.13 Variation of the modulus of polystyrene/nanosilica composite with silica content prepared at different shear.

Non-Newtonian fluid mechanics and polymer rheology

31

drop for flow. In the case of polymers, glass capillary viscometers can be used at low concentrations. In the case of melts, flow is generated either by forcing a piston to move through a reservoir at constant speed or by applying gas pressure to the melt in the reservoir, thus generating a controlled constant driving pressure.

2.10.1 Melt flow indexer The melt flow indexer is often used in the industry as a simple and quick means to measure the easiness of flow of a polymer. It takes a single point measurement using standard testing conditions. The standard procedure for testing the flow rate of thermoplastics is described in the ASTM D1238 test. The test consists of heating a sample in a barrel and extruding from a short cylindrical die using a piston actuated by a weight. The weight of the polymer in grams extruded during the 10-minute test is the melt flow index (MFI) of the polymer.

2.10.2 Capillary viscometer (rheometer) The most widely used device for generating the shear stress versus shear rate curve (flow curve) is the capillary viscometer (rheometer). For fully developed flow in a circular tube of radius R and length L, the shear stress at the wall of the tube is related to the pressure drop ∆P by the equation

τ w = R. ∆P 2L

2.24

Since it is difficult to measure the pressure at various points in the capillary, it is calculated from the driving pressure (Pd) in the barrel. If the entrance pressure loss is neglected, wall shear stress is calculated from

τw =

RPd 2L

2.25

This equation requires that the capillary be sufficiently long to assure a fully developed flow. However, due to the end effects the actual pressure profile along the length of the capillary exhibits a curvature. This effect is shown schematically in Fig 2.14 and can be corrected by using the procedure adopted by Bagley:45

τw =

Pd L 2 +e R

(

)

2.26

The correction factor e at a particular shear rate can be obtained by plotting pressure drops for various capillary L/D ratios (Fig. 2.15). To obtain the true

32

Advances in polymer processing Barrel

Pd L

Pd

Fully developed flow region

eR Pw Entrance length 0

0

Po L Pa

z

2.14 Pressure distribution in a capillary rheometer.

P 0 –PL

L /D = 0

γ˙ 2

e1

γ˙ 1

L /D e2

2.15 Bagley plot for two shear rates.

shear rate at the wall, dv/dr, the Weissenberg–Rabinowitsch equation46 can be used: d (ln Q )  – dv = γ˙ w = 1 γ˙ app  3 + 4 dr d (ln τ )  

2.27

where γ˙ app is the apparent or Newtonian shear rate given by equation 2.11. The viscosity can be calculated as τ 2.28 η= w γ˙ w The substantial pressure drop at the entrance to the capillary rheometer is associated with the flow of polymer from the barrel to the capillary. This pressure drop arises mainly from the high degree of elongation that occurs in this converging flow. So the apparent extensional viscosity can be estimated from the entrance pressure drop by the method suggested by Cogswell47 and others.

Non-Newtonian fluid mechanics and polymer rheology

33

Extrudate swell Molten polymers tend to swell two to three times in diameter on leaving the die. Extrudate swell can be taken as a measure of the elasticity of the melt. Further, swell is an important phenomenon in certain processing operations such as blow moulding. However, swell depends on factors such as flow rate through the die, die geometry, temperature, elapsed time after the melt leaves the die, etc., in addition to the rheological properties of the melt. Also, until it solidifies, the extrudate will sag under the influence of gravity and will be reduced in diameter. Care should be taken to eliminate the errors due to these defects if extrudate swell is to be obtained as a material property. Extrudate distortion – melt fracture Above a critical shear rate or production rate, two types of extrudate distortion can occur: surface melt fracture and gross melt fracture. The first is a surface defect known as ‘shark skin’. Shark skin is considered a die exit effect while melt fracture is considered a die entry effect. Shark skin is the result of the cracking of the surface of the melt in response to the local concentration of stress that occurs as the melt leaves the die wall. Gross melt fracture arises from the flow instability at the entrance to the die.

2.10.3 Rotational rheometers The cone and plate rheometer The cone and plate rheometer is often used for measuring the viscosity and the first and second normal stress coefficient parameters as a function of shear rate and temperature. A schematic diagram of the cone and plate rheometer is shown in Fig. 2.16. Since angle θ0 is very small, typically less than 5°, the shear rate can be considered to be constant and is given by

γ˙ = Ω θ0

2.29

where Ω is the angular velocity of the cone. The shear stress can also be considered to be constant and can be related to the measured torque, T, as

τ = 3T 3 2 πR The viscosity can be obtained from

τ= τ γ˙

2.30

2.31

If the fluid between the cone and the plate is viscoelastic, the cone and plate

34

Advances in polymer processing Torque

Force Ω

φ

θ

θ0

Fixed plate Pressure transducers

R

2.16 The cone and plate rheometer.

will be pushed apart when the fluid is sheared. The force with which the cone and plate are pushed apart, F, is related to the first normal stress difference N1 in the fluid by the relation F=

π R2 N1 2

2.32

The first normal stress difference is an accurate indicator of the viscoelastic behaviour of a fluid. The Couette rheometer Another rheometer commonly used in industry is the concentric cylinder or Couette flow rheometer schematically shown in Fig. 2.17. The torque T is related to the shear stress that acts on the inner cylinder wall and can be computed as

τi =

T 2 π ri2 L

2.33

If a power law fluid is confined between the outer and inner cylinder walls of a Couette rheometer, the shear rate at the inner wall can be computed using

γ˙ 1 =

2Ω 2/ n   r n 1 –  i    r0   

2.34

Non-Newtonian fluid mechanics and polymer rheology

35

Ω, T

ri

L

ra

Polymer

2.17 The Couette rheometer.

The viscosity can be calculated using the relation τ η= i γ˙ i The power law index n can be calculated as d (log τ i ) n= d (log Ω )

2.35

2.36

2.10.4 Dynamic measurements Vibratory or oscillating deformation is an important mode of mechanical testing. Oscillatory deformation testing represents an efficient method of measuring the elastic and viscous elements of viscoelastic materials. Such tests can be extended to rheological measurements also. In the case of mechanical testing the test measures the complex shear modulus G* = G| + iG||

2.37

The real part of the complex modulus is called the storage modulus G| and the imaginary part the loss modulus G||. The tangent of the phase angle shift is the ratio between the two moduli: tan δ = G||/G|

2.38

Similar to G* a complex viscosity η* can be defined in dynamic rheological measurements by the equation

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Advances in polymer processing

η* = η| – iη||

2.39

where η measures the energy dissipation and η the stored energy, and |

||

tan δ =

η| η ||

2.40

η| is a function of frequency in the same way as steady shear viscosity η is a function of shear rate, and η|| is a measure of elastic response of the material and can be related to the first normal stress difference. Thus the phase relationships in this case are opposite to those of G*.

2.10.5 Extensional rheometry A schematic presentation of two commonly used setups for measuring the elongational viscosity is shown in Fig. 2.18.48 In the Rheotens test,49 the melt is extruded through a round die and the resulting thread is hauled off. The stress on the extrudate is recorded as a function of the melt properties and extension rate. In the Meissner setup50 a long sample (to approach the assumption of an infinite sample) is extended at constant rate. The sample floats in an oil bath which facilitates temperature adjustments.

Fixation

Die

Sample strand

Sample

Oil bath

Takeup rolls and force transducer

2.18 Schematic diagram of the Rheotens setup (left) and the Meissner setup (right) for measuring extensional viscosity.

Non-Newtonian fluid mechanics and polymer rheology

2.11

37

Solution viscosity

Polymer molecules in the dissolved, molten, amorphous and glassy states exist as random coils. This is the result of the relative freedom of rotation associated with the chain bonds of most polymers and the large number of conformations a polymer molecule can adopt. As a consequence of the random coil conformation the volume of a polymer molecule in solution is many times that of its segments alone. The size of the dissolved polymer molecule depends on the degree of polymer–solvent interactions. In a thermodynamically good solvent a high degree of interaction exists between the polymer molecules and the solvent. Consequently the molecular coils are relatively extended. On the other hand in a poor solvent the coils are more contracted. The relationship between the molecular mass of a polymer and the solution viscosity is given by the Mark–Houwink equation51,52 [η] =KMα

2.41

where [η] is the intrinsic viscosity, M the viscosity average molecular weight and α a constant assuming values between 0.5 and 1 depending upon the polymer–solvent interaction. This means that the intrinsic viscosity is much less sensitive to molecular weight than the zero shear viscosity. A value of α = 0.5 corresponds to a system very close to precipitation (known as theta solvent) and α = 1 corresponds to very good solubility. K is a constant characteristic of the polymer independent of its molecular weight and the solvent. The Mark–Houwink relation is valid only for dilute solutions where the interaction between the solute molecules is practically absent. If moderately or highly concentrated solution can be treated as continuum systems, all that has been stated about the melt rheology applies to solutions as well. In the case of dilute polymer solutions the shape of the shear stress–shear rate curves changes with concentration as shown in Fig. 2.19.53 Doolittle has proposed the following relation between viscosity and free volume:54 v η = A ⋅ exp  B 0   vf 

2.42

where η is the viscosity of the solution, v0 is the occupied volume and vf is the free volume; A and B are constants. This equation suggests a general trend in the increase of viscosity with increase in concentration even though the actual variation depends on the type of polymer.

2.12

Molecular rheology

The brief introduction to rheology in this section has been presented treating the materials as a continuum. However, ultimately it is the molecular structure

38

Advances in polymer processing 105

Shear stress (dyne/cm2)

104 103 102 10

9%

1

5%

3%

% 2% 1.0 5% % 0 0.

100 10–1

10–3 10–2 10–1 100 101 102 103 Shear rate (s–1)

104

105

106

2.19 Shear stress versus shear rate curves at different concentrations of polyisobutylene in decalin.

of the material that decides the behaviour of melts or solution. The shape of the polymer chains undergoes constant changes due to natural molecular motions. Therefore only the average size of the molecules may be determined. The most commonly used parameter for describing the molecular size is the root mean square end-to-end distance (r2)1/2 or the root mean square radius of gyration (s2)1/2 which is the root-mean-square distance of elements of the chain from its centre of gravity. These two values of unperturbed coils without any solvent penetration are related to each other as r2 = 6s2

2.43

The average shape of the molecular chain is somewhat elongated, kidney like, rather than spherical. Increase in chain branching results in the coil shape changing to more spherical. With increase in rigidity of chains the chain dimensions increase, eventually approaching the shape of a rod as in the case of polymer liquid crystals.

2.13

Polymer processing

The performance of polymer products is a result of the combined effect of the interaction of their inherent material properties and fabrication variables (Fig. 2.20). Because properties of polymers are drastically affected by processing history, the interaction between material properties and processing variables should be clearly known. The general requirements for polymer products can be classified as: • •

appearance dimensional stability

Non-Newtonian fluid mechanics and polymer rheology

39

Processing window

Intrinsic material properties

Fabrication variables

Product properties

2.20 Interaction of material properties, processing variables and product properties.

• •

properties economics.

Appearance is an indicator of quality and is very important for products like food packages, appliance housings, cosmetic packages, etc. Dimensional stability becomes important for post-forming operations such as thermoforming of sheet, fittings of pipes, etc. Product properties or requirements depend upon the particular use. Since the shear involved in various processing operations is different, the viscosity of the melt at the relevant shear range should be known. The variation of viscosity with the shear rate for polypropylene homopolymer along with the shear rate ranges of various polymer processing techniques are shown in Fig. 2.21. The relevant elongational and elastic behaviours of the fluids also have to be considered.

2.13.1 Unit operations in polymer processing The ease with which polymers may be shaped at relatively low temperatures represents a significant advantage compared to other engineering materials. Extrusion and injection moulding are the most widely used unit operations in polymer processing. Single screw extrusion Extrusion is the most important polymer processing operation. Injection moulding and blow moulding also use extruders. Extrusion involves a series of unit operations such as feeding solid pellets, conveying, melting, pumping

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Advances in polymer processing

1.0E+04

Viscosity (Pa.s)

Roto- Compression moulding mouldng

Pipe/profile extrusion

1.0E+03 Blowmoulding and thermoforming

Film extrusion Injection moulding

1.0E+02 Fibre spining Coating 1.0E+01 1.0E-02

1.0E-01

1.0E+00

1.0E+01 1.0E+02 Shear rate (s–1)

1.0E+03

1.0E+04

2.21 Viscosity–shear rate curve of PP homopolymer (melt flow index (230°C/2.16 kg) 18 g/10 min) at 230°C.

(mixing), forming and cooling. Dimensional tolerances, surface appearances, orientation and internal strains are all related to melt rheology. The basic output from the extruder can be expressed as the difference between the drag flow QD and the pressure flow Qp: Q = QD – Qp

2.44

Drag flow depends on the dimensions of the screw, A, and the screw speed N in revolutions per second: QD = AN

2.45

Pressure flow depends on the pressure drop in the pumping or metering zone as well as on the dimensions of the screw, B: ∆p Qp = B  η L 

2.46

Therefore net flow ∆p Q = AN – B  η L 

2.47

Die shaping generally involves pressure-driven flow through converging channels. Pressure-driven volumetric flow rate (Q) is a function of the pressure drop (∆p), the melt viscosity (η) and the geometry of the flow channel. Thus for Newtonian, isothermal flow through a shaping die,

Non-Newtonian fluid mechanics and polymer rheology

Qdie = C

∆p L

41

2.48

where C is a constant depending upon the geometry of the die. The total pressure drop (∆p) is the sum of the individual pressure losses as the melt flows through the complex flow path of the die. This will include parallel sections where pressure losses are due to shear flow, and converging sections where pressure losses are due to both shear and elongational flows, along with any entrance effects at severe changes in cross-sectional area. If ∆p is very small it will adversely affect output from the die and melt distribution within the die as well as melting and mixing within the extruder. A simplified diagram of a die with a circular cross-section along with the pressure and temperature of the melt as it passes through the flow channel is shown in Fig. 2.22.7 The melt temperature is found to rise due to viscous dissipation. The pressure drop and the magnitude of the temperature rise depend on the thermal and rheological properties of the melt. A uniform average velocity at all points of the die exit is vital if distortion of the extrudate is to be avoided. Melt stresses generated during melt flow along the die channels tend to align the polymer chain molecules axially along the length of the extrudate. Substituting the apparent shear rate in the power law relation, we obtain n 4Q  τ = k  πR 3 

Screw

P

2.49

Breaker plate Taper

Land

T

2.22 Temperature and pressure profile across the die of an extruder.

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Advances in polymer processing

This shows that the die radius R and material properties k and n have significant effects on the shear stress, and hence on the orientation. Higher melt temperatures will tend to reduce orientation due to a reduction in the value of k. This effect is opposed to some extent by the resultant increase in throughput (Q). Similarly any increase in stresses due to higher Q may be opposed by a greater shear thinning of the melt (lower n). Although the degree of molecular orientation created during die shaping is dependent on the melt stresses developed, the level of residual orientation which remains in the final product is dependent on the relative rates of polymer solidification and molecular relaxation and therefore varies with local cooling rates. In addition to shear flow, the taper of the converging die section generates an elongational flow component which increases with taper angle. At low taper angles the shear flow component is higher, but at higher angles the elongational component may predominate. Taper angles greater than that required for minimum pressure generate circulating flow at the entrance to the converging section. These variables broaden the distribution of melt residence times within the die and may give rise to melt stagnation and degradation. The elastic component is also responsible for other effects such as extrudate swell and melt fracture. Flow instability in extrusion Flow instabilities have been observed in extrusion for a long time. It has been noticed that the onset of flow instabilities coincides with the Deborah number approaching 1, which indicates that the instability is related to the viscoelastic nature of the fluid. With increase in flow rate (shear rate) the extrudate passes from smooth to rough surface (shark skin) followed by unstable flow and bulk deformation (melt fracture). Injection moulding The term ‘mouldability’ refers to whether a particular polymer will fill a particular mould within the set limit of temperature and pressure. This is a complex function of the thermal and rheological properties of the polymer melt, processing conditions, flow geometry and heat transfer characteristics of the mould. The complexity of the process has led to the development of software packages for testing mouldability. The moulding area diagram shown in Fig. 2.23 provides a broad boundary for the mouldability. The moulding curve is bonded by four curves. In terms of ram pressure the upper curve indicates the condition at which excessive pressure is generated in the cavity resulting in opening of the mould and the formation of flash. The bottom curve represents the minimum pressure to fill the cavity. In terms of the melt temperature, the left-hand curve indicates the condition at

Non-Newtonian fluid mechanics and polymer rheology

43

Ram pressure

Flashed mouldings

Thermal degradation Short mouldings

Melt temperature

2.23 Moulding area for injection moulding.

which the rate of solidification is too fast and a short moulding results, and the right-hand curve indicates the condition where thermal degradation of the melt starts. The quality of an injection moulding depends on the absence or minimization of internal strains. As the melt is injected into the mould cavity it is subjected to very high shear rate and temperature gradients. The melt is cooled so fast that the highly strained chains do not have enough time to relax. As a result, the chains are frozen into strained conformations. One way to reduce internal stress is to decrease the molecular weight and molecular weight distribution. Mould design has a very important role in improving the quality of the moulding. The shape of the moulding should be designed such that the flow pattern and cooling temperature gradient are as uniform as possible throughout the moulding. Sharp edges and regions where stresses may concentrate should be avoided.

2.14

Applications

2.14.1 Polymers as rheology modifiers Polymers, because of their complex structure, are very effective in controlling solution and dispersion rheology to modify fuels, lubricants, oil field chemicals, water treatment chemicals, coatings and food materials. Polymers modify rheology mostly by virtue of their high molecular weight, chain entanglement and polymer–solvent interaction.55 The power of a polymer to modify fluid rheology arises from its greater volume in solution compared to the volume of its segments. The solution volume swept by the polymer chain is known as the hydrodynamic volume (HDV) and is determined by polymer structural parameters (such as chain length and chain stiffness) and polymer–solvent interactions as well as polymer

44

Advances in polymer processing

associations or repulsions. HDV is also dependent upon temperature, concentration, molecular weight and deformation rate. Polymers as viscosifiers Since polymers have intrinsically large hydrodynamic volume (HDV), only low concentrations of polymers are needed to substantially increase the viscosity of the fluid. When added to lubricating oils, these polymers exist as compact chains in cold oil and expand with increasing temperature because of increased solvation. This effect helps to maintain a uniform viscosity for the oils with temperature changes. The viscosity index is an empirical number that indicates the resistance of a lubricant’s viscosity to changes in temperature Viscosity index improvers for automotive lubricants are non-aqueous rheology modifiers. Similar to temperature stability, rheology modifiers can also provide shear stability. Shear stability is required for oils subjected to high deformation rates between piston and cylinder wall or in gear pumps. The main classes of viscosity index improvers are saturated oil-soluble polymers such as olefin copolymers, hydrogenated styrene–diene copolymers and poly(alkyl methacrylate). In oil or aqueous systems, high viscosities are needed to inhibit sedimentation. This is of importance in oil or water based coatings, food applications, etc. The most cost-efficient way to achieve high viscosities at low shear rates is through the use of clays, although this is not applicable for food application. Use of polymers provides the best performance, but they are often complemented with clay because of the mechanical and thermal degradation limitations of pure polymers. The use of specially designed architecture branched polymers for rheology modification is also widely practised.56–59 Drag reduction The addition of polymers at ppm levels can produce significant improvement in flow. This phenomenon, known as drag reduction, occurs by extension of the laminar flow to very high Reynolds numbers. Hydrocarbon-soluble polymers have been successfully used in crude oil pipelines. Poly(1-octene) has been extensively used for this application.

2.15

References

1. Ostwald W (1925), Kolloid-Z., 36, 99. 2. Ostwald W (1926), Kolloid-Z., 38, 261. 3. De Waele A (1923), ‘Viscometry and plastometry’, J. Oil and Color Chem. Assoc., 6, 33.

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4. Bowers S (1987), ‘Prediction of elastomer flow in multicavity injection moulding’, Plast. Rubb. Proc.& Appl., 7, 101. 5. Reiner M (1949), Deformation and Flow, Lewis, London. 6. Carreau P J (1972), ‘Rheological equation from molecular network theories’, Trans. Soc. Rheol., 16, 19. 7. Wilkinson A N, Ryan A J (1998), Polymer Processing and Structure Development, Kluwer Academic Publishers, Dordrecht. 8. Dealy J M, Saucier P C (2000), Rheology in Plastics Quality Control, Hanser Publishers, Munich. 9. Rohn C L (1995), Analytical Polymer Rheology, Henser Publishers, Munich. 10. Williams M L, Landel R F, Ferry J D (1955), ‘The temperature dependence of relaxation mechanism in amorphous polymers and other glass forming liquids’, J. Am. Chem. Soc., 77, 3701. 11. Mendelson R A (1969), ‘Flow properties of polyethylene melts’, Polym. Eng. Sci., 9, 350. 12. Miltz J, Ram A (1973), ‘Flow behaviours of well characterized polyethylene melts’, Polym. Eng. Sci., 17, 273. 13. Kraus G, Gruver J T (1965), J. Polym. Sci., Part A3, 105. 14. Raju V R, Rachapudy H, Graessley W W (1979), J. Polym. Sci.: Polym. Phys. Ed., 17, 1223. 15. Cogswell F N (1981), Polymer Melt Rheology, George Godwin, London. 16. Mewis J, Spaull A J B (1976), ‘Rheology of concentrated dispersions’, Advances in Colloid and Interface Science, 6(3), 173. 17. Chen Y, White J L, Oyanagi Y (1977), Society of Plastics Engineers Annual Technical Conference, 23, 290. 18. Einstein A (1906), Ann. Physik, 19, 289. 19. Einstein A (1911), Ann. Physik, 34, 591. 20. Krieger I M, Dougherty T J (1959), Trans. Soc. Rheol., 3,137. 21. Litovitz T A, Macedo P B (1965), Physics of Non-crystalline Solids, Interscience, New York, 220. 22. Macedo P B, Litovitz T A (1965), J. Chem. Phys., 42, 245. 23. Hildebrand J H, Lamoreaux R H (1972), Proc. Nat. Acad. Sci., 69, 3428. 24. Sanchez I C (1974), J. Appl. Phys., 45, 4204. 25. Cohen M, Turnbull D (1959), J. Chem. Phys., 31, 1164. 26. Cogswell F N (1973), ‘The influence of pressure in the viscosity of polymer melts’, Plastics and Polymer, 41, 39. 27. Maxwell B, Jiang A (1957), ‘Hydrostatic pressure effect in polymer melt viscosities’, Mod. Plastics, 35, 174. 28. Utracki L A (1943), ‘Pressure dependence of Newtonian viscosity’, Polym. Eng. Sci., 22, 441. 29. Dealy J M, Wissbrum K F (2000), Melt Rheology and its Role in Plastics Processing, Van Nostrand Reinhold, New York. 30. Maeosko C W (1994), Rheology: Principles, Measurements and Applications, VCH Publishers, New York. 31. Trouton F I (1906), Proc. Roy. Soc., 177, 1126. 32. Osswald T A, Menges G (2003), Material Science of Polymers for Engineers, 2nd edition, Hanser Publishers, Munich. 33. Reiner M (1964), Physics Today, 17, 62. 34. Prest W N, Porter R S (1973), ‘Blending laws for high molecular weight polymer melts’, Polymer Journal, 4(2), 163–171.

46

Advances in polymer processing

35. Han C D (1976), Rheology of Polymer Processing, Academic Press, New York. 36. Paul D R, Newman S (eds) (1978), Polymer Blends, Academic Press, New York. 37. Krishnamoorti R, Vaia R A (eds) (2001), Polymer Nanocomposites, Synthesis, Characterization and Modeling, ACS Symposium Series American Chemical Society, 804, Washington, DC. 38. Pinnavaia T J, Beall G W (eds) (2000), Polymer–Clay Nanocomposites, John Wiley & Sons, New York. 39. Koo J H (2006), Polymer Nanocomposites, McGraw-Hill, New York. 40. Wagener R, Reisinger T J G (2008), ‘A rheological method to compare the degree of exfoliation of nanocomposites’, Polymer, 44, 9513. 41. Zhao J, Morgan A B, Harris J D (2005), ‘Rheological characterization of polystyrene– clay nanocomposites to compare the degree of exfoliation and dispersion’, Polymer, 46, 8641–8660. 42. Abdel-Goad M, Potschke P (2005), ‘Rheological characterization of melt processed polycarbonate–multiwalled carbon nanotubes composites’, J. Non-Newtonian Fluid Mech., 128, 2–6. 43. Tillekeratne M, Jullands M, Cser F, Battacharya S N (2006), ‘Role of mixing parameters in the preparation of poly(ethylene vinyl acetate) nanocomposites by melt blending’, J. Appl. Polym. Sci., 100, 2652–2658. 44. Renjanadevi B, George K E, Progress in Rubber Plastics and Recycling Technology, forthcoming issue. 45. Bagley E B (1957), J. Appl. Phys., 28, 621. 46. Rabinowitsch B Z (1929), Phys. Chem., 145, 1. 47. Cogswell F N (1972), ‘Converging flow of polymer melt in extrusion die’, Polym. Eng. Sci., 12, 64. 48. Gahleitner M (2001), ‘Melt rheology of polyolefins’, Prog. Polym. Sci., 26, 895– 944. 49. Wagner M H, Bernnat A, Schulze V J (1998), Rheol., 42, 912. 50. Meissner J (1971), Rheol. Acta., 10, 230. 51. Kuhn W (1933), Kolloid-Z., 62, 269. 52. Houwink R (1940), J. Prakt. Chem., 157, 15. 53. Brodnyan J D, Gaskins F H, Philipoff W (1957), Trans. Soc. Rheol., 1, 109. 54. Doolittle A K (1951), J. Appl. Phys., 22, 1471. 55. Schultz D N, Glass J E (eds) (1991), Polymers as Rheology Modifiers, ACS Symposium Series 462, American Chemical Society, Washington, DC. 56. Mecerreyes D, Jerome R, Dubios P (1999), Adv. Polym. Sci., 147, 1–59. 57. Davis K A, Matyjaszewski K (2002), Adv. Polym. Sci., 159, 153–169. 58. Agarwal U S (2005), ‘Polymer properties’, in Handbook of Polymer Reaction Engineering, Meyer T, Keurentjes J T F (eds), Wiley-VCH, New York, Vol. 2, 679– 720. 59. Rajan M, Velthem P V, Zhang M, Cho D, Chang T, Agarwal U S, Bailly C, George K E, Lemstra P J (2007), ‘Synthesis and characterization of model dumbbell polymers’, Macromolecules, 40, 3080–3089.