Modeling and Simulation in Polymers.

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4 Computational Polymer Processing Evan Mitsoulis 4.1 Introduction 4.1.1 Polymer Processing

Polymer processing is defined as the “engineering activity concerned with operations carried out on polymeric materials or systems to increase their utility” [1]. Primarily, it deals with the conversion of raw polymeric materials into finished products, involving not only shaping but also compounding and chemical reactions leading to macromolecular modifications and morphology stabilization, and thus to “valueadded” structures [2]. The subject matter of polymer processing in general has been superbly laid out in the 1979 monograph by Tadmor and Gogos and has been updated significantly in the 2006 second edition [2]. Notable additions to the subject are the two very recent books by C.D. Han on rheology and processing of polymers [3]. They provide a wealth of information on rheological modeling of various polymer processes and an extensive list of relevant references. An excellent overview article has also appeared [4], where important up-to-date information on the polymerprocessing industry is given, together with the most important developments in various polymer processes, and a most informative list of references and textbooks on the subject for further reading. Synthetic polymers can be classified into two categories. Thermoplastics (by far the largest volume) can be melted by heating, solidified by cooling, and remelted repeatedly. Major types are polyethylene (PE), polypropylene (PP), polystyrene (PS), polyvinyl chloride (PVC), polycarbonate (PC), polymethyl methacrylate (PMMA), polyethylene terephthalate (PET), and polyamide (PA, nylon). Thermosets are hardened by the application of heat and pressure, owing to cross-linking, that is, the creation of permanent three-dimensional networks. They cannot be softened by heating for reprocessing. Bakelite, epoxies, and most polyurethanes are thermosets.

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This chapter is devoted to the processing of thermoplastics and its computations. Thermoplastics are usually processed in the molten state. Molten polymers have very high viscosity values and exhibit shear-thinning (pseudoplastic) behavior. As the rate of shearing increases, the viscosity decreases, owing to alignments and disentanglements of the long molecular chains. The viscosity also decreases with increasing temperature. In addition to the viscous behavior, molten polymers exhibit elasticity. Elasticity is responsible for a number of unusual rheological phenomena [5–7], including stress relaxation and normal stresses. Slow stress relaxation causes frozenin stresses in injection-molded and extruded products. Normal stress differences are responsible for some flow instabilities during processing and also extrudate swelling, that is, the significant increase in cross-sectional area when a molten material is extruded out of a die. The most important polymer processing operations are extrusion and injection molding (Vlachopoulos and Strutt, 2003). Extrusion is material-intensive and injection molding is labor-intensive. Both these processes involve the following sequence of steps: (a) heating and melting the polymer, (b) pumping the polymer to the shaping unit, (c) forming the melt into the required shape and dimensions, and (d) cooling and solidification. Other processing methods include calendering, roll coating, wire coating, fiber spinning, film casting, film blowing, blow molding, thermoforming, compression molding, and rotational molding (Figure 4.1). 4.1.2 Historical Notes on Computations

All these processes have been analyzed computationally in varying degrees of complexity. The mathematical analysis of the processes has followed the development of high-speed digital computers. Computational polymer processing was first tackled in the 1950s using analytical solutions for a few tractable problems, such as flows in channels with the power-law model of pseudoplasticity [8]. Then, the advent of digital computers saw numerical solution of simple problems in simple geometries, using mainly the finite difference method (FDM) due to its simplicity. With an increase in computer power, the 1970s saw the utilization of the more involved finite element method (FEM), which proved more capable in handling complicated geometries and boundary conditions (BCs) [9]. Two-dimensional (2D) flow problems in polymer processing were then handled for the first time [10]. The 1980s saw an overwhelming majority of computational works dealing with viscoelasticity (see book by Crochet et al. [11], and a review article by Crochet and Walters [12]). The efforts were directed toward overcoming the high Weissenberg number (Wi) problem (HWNP) [13], which did not allow solution of the viscoelastic models above a critical Wi number on the order of 1 (see below), where most phenomena were not very different from their inelastic counterparts. As expected, and due to a gigantic effort by many researchers around the world, this problem was eventually resolved successfully by using numerical schemes best suited for hyperbolic equations [14] in the late 1980s and early 1990s. Crochet’s group and others then managed to reach highly viscoelastic numerical solutions in the range 1 < Wi < 100 [15].

4.1 Introduction

Figure 4.1 Photographs of various polymer processes. (From various issues of Plastics Engineering.)

From these significant efforts on viscoelasticity, the first 2D and then 3D FEM codes developed into commercial packages, with varying degrees of success, in the 1990s. Among these, some software particularly suited for polymer processing were POLYFLOW [16] and POLYCAD [17]. More recently, polymer processing software were made available from www.compuplast.com [18] and www.polyXtrue.com [19]. Other software were more generic in nature for viscous fluids, such as FIDAP, FLUENT, NEKTON, PHOENICS, and so on [20]. All these software use sophisticated mesh generation schemes and solvers for solving the governing differential equations along with appropriate boundary conditions. The FEM is the numerical method of choice for most packages, while inroads have also been made with the finite volume method (FVM) and the boundary element method (BEM). This chapter reviews some major contributions regarding polymer melt flows that appear in polymer processing and discusses several issues (usually still unresolved) and their influence in polymer processing. Due to the vastness of the subject matter, the outlay of what follows is rather personal and subjective, but it is hoped that it will add some focus to the present state of affairs in polymer processing computations.

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4.2 Mathematical Modeling 4.2.1 Governing Conservation Equations

In order to study polymer flows in processing equipment, it is essential to consider first the governing flow equations. The flow of incompressible fluids (such as polymer solutions and melts, at least in situations where they are considered as incompressible for pressures below 100 MPa) is governed by the conservation equations of mass, momentum, and energy [2, 6, 21], that is, r  v ¼ 0;

ð4:1Þ

rv  rv ¼ rp þ r  t;

ð4:2Þ

rcp v  rT ¼ k r2 T þ t : rv;

ð4:3Þ

where v is the velocity vector, p is the scalar pressure, t is the extra stress tensor, r is the density, cp is the heat capacity, k is the thermal conductivity, and T is the temperature. The above system of conservation equations is usually called the Navier–Stokes equations in fluid mechanics. 4.2.2 Constitutive Equations

The above system of conservation equations is not closed for non-Newtonian fluids due to the presence of the stress tensor t. The required relationship between the stress tensor t and the kinematics (velocities and velocity gradients) must be given by appropriate rheological constitutive equations, and this is an eminent subject in theoretical rheology [5, 6, 22]. A cartoon showing the importance of stresses for polymers has been put forward in Figure 4.2, where a zebra is losing its stripes under stress. The implicit message is that polymers under stress exhibit unusual, unexpected, and counterintuitive behavior or that the wrong constitutive equation may give stresses totally inappropriate for a polymer undergoing deformation and flow. For purely viscous fluids, the rheological constitutive equation that relates the stresses t to the velocity gradients is the generalized Newtonian model [5, 6, 21] and is written as _ c; _ t ¼ gðcÞ

ð4:4Þ

_ is the apparent viscosity where c_ ¼ rv þ rvT is the rate-of-strain tensor and gðcÞ given in its simplest form by the power-law model [5] _ ¼ K c_ n1 ; gðcÞ

ð4:5Þ

4.2 Mathematical Modeling

Figure 4.2 The stress state of a polymer melt is essential for any computation. A cartoon showing a zebra losing its stripes is a good analogue for the computation of a polymer melt with the wrong model.

where K is the consistency index and n is the power-law index (usually 0 < n < 1, representing a degree of shear-thinning). Another popular model for viscosity computations, among others, is the Carreau model [5] given by _ 2 _ ¼ g1 þ g0 ½1 þ ðlC cÞ gðcÞ

n1 2

ð4:6Þ

and the Cross model [23] given by _ ¼ g1 þ gðcÞ

g0 _ 1n 1 þ ðlC cÞ

:

ð4:7Þ

In the above equations, g0 is the zero shear rate viscosity, g1 is the infinite shear rate viscosity, lC is a time constant, and n is again the power-law index. The magnitude c_ of the rate-of-strain tensor is given by rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 _ ; c_ ¼ IIc_ ¼ ðc_ : cÞ ð4:8Þ 2 2 where IIc_ is the second invariant of the rate-of-strain tensor. The Carreau model describes well the shear-thinning behavior of polymer solutions and melts for all shear rates and exhibits two plateaus for low and for high shear rates, while for intermediate to high shear rates it represents well the power law. For example, for a low-density polyethylene (LDPE) melt, experimental data at different temperatures are fitted well with the Carreau model, as evidenced in Figure 4.3. The effect of temperature on the viscosity is of primordial importance in polymer processing, where tight control of temperatures is required for a successful

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Figure 4.3 Non-Newtonian viscosity of an LDPE melt at several temperatures [5].

operation. The viscosity as a function of temperature is given by an exponential relationship, according to gT ¼ g0 exp ½bT ðTT0 Þ;

ð4:9Þ

where bT is a temperature-shift factor in the expression that relates viscosity to temperature and g0 is the viscosity at a reference temperature, T0. The values of bT for polymers are usually in the range of 0.01–0.04/ C, but occasionally they may reach 0.1/ C or more for some polymers. Another expression for the temperature-dependence of the viscosity is the Arrhenius law [21]:    Ea 1 1 ; ð4:10Þ gT ¼ g0 exp  Rg T T0 where Rg is the ideal gas constant (¼ 8.13 J/(K mol)), Ea is the activation energy (J/mol), T is the absolute temperature (K), and T0 is the absolute reference temperature (K). Then combining Eqs. (4.8) and (4.9) yields bT ¼

Ea : Rg T T0

ð4:11Þ

Non-Newtonian fluids (polymer solutions and melts) are rheologically complex materials, which exhibit both viscous and elastic effects, and are therefore called viscoelastic [6]. Regarding viscoelasticity, a plethora of constitutive equations exist with varying degrees of success and popularity. Standard textbooks on the

4.2 Mathematical Modeling

subject [5, 6, 22, 24–26] list categories of these equations and their predictions in several types of flow and deformation. There are constitutive equations of differential type, of integral type, molecular models, and so on. From the differential models, eminent in the early 1980s were the upper convected Maxwell (UCM) and the Oldroyd-B models. Then, in the 1990s the Phan-Thien/Tanner (PTT) and the Giesekus models were among the most popular, while in the 2000s the “pom-pom” model has been the model of choice for computations. From the integral models, the KBKZ model (from the initials of Kaye, Bernstein, Kearsley and Zapas) has been by far the most popular. This subject matter is more fully explored in Chapter 1. As an example of a popular viscoelastic constitutive equation used in the past 25 years, which possesses enough degree of complexity so as to capture as accurately as possible the complex nature of polymeric liquids, we present here the K-BKZ integral constitutive equation with multiple relaxation times proposed by Papanastasiou et al. [27] and further modified by Luo and Tanner [28]. This is often referred to in the literature as K-BKZ/PSM model (from the initials of Papanastasiou, Scriven, Macosko) and is written as t¼

1 1q

  ðt X N   Gk tt0 HðIC1 ; IIC1 Þ Ct1 ðt0 Þ þ qCt ðt0 Þ dt0 ; exp  l l k k k¼1

ð4:12Þ

1

where t is the stress tensor for the polymer, lk and Gk are the relaxation times and relaxation moduli, respectively, N is the number of relaxation modes, q is a material constant, Ct is the Cauchy–Green tensor, Ct1 is the Finger strain tensor, and IC1 , IIC1 are its first and second invariants. The function H is a strain-memory (or damping) function, and the following formula was proposed by Papanastasiou et al. [27]: HðIC1 ; IIC1 Þ ¼

a ; ða3Þ þ bIC1 þ ð1bÞIIC1

ð4:13Þ

where a and b are nonlinear model constants to be determined from shear and elongational flow data, respectively. The q-parameter (a negative number) relates the second normal stress difference N2 ¼ t22–t33 to the first N1 according to N2 q ¼ : N1 1q

ð4:14Þ

The linear viscoelastic storage and loss moduli, G0 and G00 , can be expressed as a function of frequency v as follows: G0 ðvÞ ¼

N X

Gk

k¼1

00

G ðvÞ ¼

N X k¼1

Gk

ðvlk Þ2 1 þ ðvlk Þ2 ðvlk Þ 1 þ ðvlk Þ2

;

ð4:15aÞ

:

ð4:15bÞ

These functions are independent of the strain-memory function, and only lk and Gk can be determined from dynamic data of the viscoelastic moduli. As an example,

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we give in Figure 4.4a and Table 4.1 the fitting of experimental G0 and G00 data for another popular benchmark LDPE melt (the IUPAC-LDPE, melt A) [29], where it is shown that a spectrum of eight relaxation times, ranging from 104 to 103 s is able to capture well the data at all frequencies. The strain-memory function is derived from the first and second invariants of the Finger strain tensor. For simple shear flow, the strain-memory function is given as HðIC1 ; IIC1 Þ ¼

a ; a þ c2

ð4:16Þ

where c is the shear strain. The strain-memory function in simple shear flow depends on a but not on b. This is expected since a is viewed as a shear parameter, while b is viewed as an elongational parameter. The above constitutive equation has been successfully used for fitting the data for many polymer solutions and melts [27, 30]. Apart from fitting well the G0 and G00 , the model can give a good fit for other rheological data, as shown in Figure 4.4b, with the values of the model for parameters a, b, and q given in Table 4.1. The full fitting involves the determination of the relaxation spectrum (parameters N, lk, and Gk) from experimental data on the storage and loss moduli, G0 and G00 . Then, the nonlinear parameters, a and b, are determined from shear and elongational data, in this case from shear viscosity, gS, first normal stress difference, N1, and uniaxial elongational viscosity, gE. The value of q is usually set to be a small negative number (around 0.1) according to experimental evidence [28]. Other extensional viscosities in planar extension, gP, and in biaxial extension, gB, are predicted by the model. These predictions can also be extended to transient effects for all rheological functions at different times [27]. This integral model has been used in numerical flow simulations for a number of flow problems more or less successfully (see Refs [28, 31–34]). A recent review [35] on the subject gives a list of problems solved with this model through numerical simulation, including many flows from polymer-processing operations. Other flows solved with a number of different constitutive equations can be found in a recent book on computational rheology [36]. 4.2.3 Dimensionless Groups

Before proceeding with the boundary conditions, it is interesting to examine the relevant dimensionless numbers in polymer processing. The dimensionless groups are calculated at a reference temperature, here taken as the temperature of the process, T0. As a characteristic length, it is usually assumed the smallest dimension, for example, in a capillary tube its radius, R. As a characteristic speed, it is usually assumed the average velocity defined by V ¼ Q=pR2 :

ð4:17Þ

4.2 Mathematical Modeling

106

106

104

104

103

103 G"

102

102 G'

101

101

100

100

10-1 10-4

10-3

10-1

10-1 100 101 Frequency, w (s-1)

102

Shear (Extensional) Viscosity, hS, (hE, , hP, hB) (Pa.s)

107

Loss Modulus, G" (Pa)

105

10-1 103

107

(b) hE 106

106

hP 105

105

hB hS

104

104

N1

102

100 10-4

10-3

10-1

10-1

102

100

101

102

First Normal Stress Difference, N1 (Pa)

Storage Modulus, G' (Pa)

(a) 105

100 103

Shear (Extensional) Rate, g (e) (s-1) Figure 4.4 Rheological data and their best fit for the IUPAC-LDPE melt-A using the K-BKZ/PSM integral constitutive equation with eight relaxation modes and the data of Table 4.1 [32]. Symbols correspond to experimental data [29], solid lines correspond to their best fit.

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Table 4.1 Material parameter values used in Eq. (4.12) for fitting the data of the IUPAC-LDPE (sample A) melt at 150  C (a ¼ 14.38, q ¼ 1/9) [28].

k

lk (s)

Gk (Pa)

bk ()

1 2 3 4 5 6 7 8

104 103 102 101 100 101 102 103

1.29  105 9.48  104 5.86  104 2.67  104 9.80  103 1.89  103 1.80  102 1.00  100

0.018 0.018 0.08 0.12 0.12 0.16 0.03 0.002

A characteristic apparent shear rate is then defined according to c_ a ¼ V=R

ð4:18Þ

and a characteristic viscosity is given as a function of apparent shear rate and reference temperature, that is,  ¼ gðc_ a ; T0 Þ: g

ð4:19Þ

For a power-law model, the characteristic viscosity can be found as ðT0 Þ ¼ g

t ¼ K c_ n1 a ; c_ a

ð4:20Þ

where the material parameters K and n are calculated at T0. The relative importance of inertia forces in the equation of momentum is assessed by the Reynolds number, defined for Newtonian fluids by Re ¼

rVD ; m

where D is the characteristic diameter (¼ 2R). For power-law fluids, Boger and Walters [7] gives a generalized Re  n rV 2n Dn 4n Re ¼ n1 : 3n þ 1 8 K

ð4:21Þ

ð4:22Þ

It is noted that for most polymer melt flows the Re number is usually small, in the range of 0.0001–0.01. Therefore, these flows are inertialess or creeping. For viscoelastic fluids with a relaxation time l, several dimensionless groups can be defined, but these can be seen as being equivalent [21]. For example, the Deborah number (De) is defined as De ¼

l _ ¼ lc; tp

ð4:23Þ

4.2 Mathematical Modeling

where l is a material relaxation time, tp is a process relaxation time usually taken to be _ and c_ is a shear rate usually evaluated at the channel wall. The equal to 1/c, Weissenberg number (Wi, also written as We or Ws) is defined as Wi ¼ l

V : R

ð4:24Þ

The recoverable shear or stress ratio (SR) is defined as SR ¼

N1;w ; 2tw

ð4:25Þ

where N1,w ¼ t11–t22 is the first normal stress difference and tw is the shear stress, both evaluated at the channel wall. The equivalence is evident when we take c_ ¼ V=R;

N1;w ¼ Y1 c_ 2 ;

_ tw ¼ gc;

l ¼ Y1 =2g;

ð4:26Þ

where Y1 is the first normal stress difference coefficient and g is the shear viscosity. The case of De ¼ Wi ¼ SR ¼ 0 corresponds to inelastic fluids (l ¼ Y1 ¼ 0), while it is understood that De ¼ Wi ¼ SR ¼ 1 corresponds to the elastic effects being as important as the viscous effects, and for De ¼ Wi ¼ SR > 1 the elastic effects dominate the flow over the viscous effects. The relative importance of surface tension effects (usually for polymer solutions) is assessed by the capillary number defined by Ca ¼

mV ; c

ð4:27Þ

where c is the surface tension. For very viscous fluids, such as polymer melts, the surface tension effects are negligible (Ca ! 1), and the boundary terms, including a force balance with the capillary forces, can be set to zero. The relative importance of each term in the energy equation is assessed through a variety of dimensionless groups [37, 38]. The Peclet number is defined by Pe ¼

rCp VD : k

ð4:28Þ

The Peclet number is a measure of convective heat transfer with regard to conductive heat transfer. High Pe values indicate a flow dominated by convection. From a numerical point of view, these flows are notorious because of instabilities that manifest themselves in the form of spurious oscillations in the temperature field. Special upwinding techniques must then be used to remedy the oscillations [37]. Another group related to Pe is the Graetz number defined by Gz ¼

rCp VD2 D ¼ Pe ; kL L

ð4:29Þ

where L is the axial length of the die. The Graetz number can be understood as the ratio of the time required for heat conduction from the center of the capillary to the

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wall and the average residence time in the capillary. As with Pe, a large value of Gz means that heat convection in the flow direction is more important than conduction toward the walls. The Nahme number is defined by Na ¼

V 2 bT g : k

ð4:30Þ

The Nahme number is a measure of viscous dissipation effects compared to conduction; hence, it is an indicator of coupling of the energy and momentum equations. For values of Na > 0.1–0.5 (depending on geometry and thermal boundary conditions), the viscous dissipation leads to considerable coupling of the conservation equations, and a nonisothermal analysis is necessary. The relative importance of heat transfer mode at the boundaries is expressed in terms of the dimensionless Biot number defined by   qT ri ; ð4:31Þ Biot ¼ qr w ðTs Tw Þ where ri is the local radius (gap), Ts is some temperature of the surroundings, and Tw is the local boundary wall temperature. A high value of Biot (>100) approaches isothermal conditions (Biot ¼ 1), while a low value of Biot (