Differential viscoelastic constitutive equations for polymer melts in steady shear and elongational flows

J. Non-Newtonian Fluid Mech. 113 (2003) 209–227

Differential viscoelastic constitutive equations for polymer melts in steady shear and elongational flows Martin Zatloukal∗ Polymer Centre, Faculty of Technology, Tomas Bata University in Zl´ın, TGM 275, 762 72 Zl´ın, Czech Republic Received 10 March 2003; received in revised form 22 May 2003

Abstract A slight modification of the dissipation term in the Leonov model using relaxation time-dependent adjustable parameters is proposed to increase the model capability to represent elongational flow behavior. The fitting/predicting capabilities of the proposed modified Leonov model are compared with the eXtended Pom–Pom model and modified White–Metzner model in steady shear and uniaxial extensional flows of LDPE, mLLDPE and PVB melts in a wide range of deformation rates. The input low-shear-rate viscosity and first normal stress coefficient data was measured on the advanced rheometric expansion system (ARES) Rheometrics parallel-plate rheometer, whereas the RH7-2 capillary rheometer was used for the determination of shear viscosity (capillary), first normal stress coefficient (slit die) and uniaxial extensional viscosity (Cogswell method). A newly developed ‘effective entry length correction’ was applied to deal with all uniaxial extensional viscosity data. © 2003 Elsevier B.V. All rights reserved. Keywords: eXtended Pom–Pom model; Modified Leonov model; Modified White–Metzner model; Uniaxial extensional viscosity; Shear viscosity; First normal stress coefficient

1. Introduction Constitutive equations are mathematical relationships that allow computing of the stresses in a liquid for a given flow history. They are often derived from constitutive models, which imply a set of assumptions and idealizations about the molecular or structural forces and motions producing stress. Polymers, characterized by relatively long macromolecules, do not obey simple physical laws because their behavior lies between Newtonian liquids and Hookean solids. As a consequence of this, constitutive equations are not simple for polymers showing specific flow phenomena (die swell, elastic recoil, memory effects, flow instabilities, etc.) [1–4], which complicates polymer processing. ∗

Tel.: +420-57-603-1350; fax: +420-57-603-1444. E-mail address: [email protected] (M. Zatloukal). 0377-0257/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0377-0257(03)00112-5

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These processes can be viewed as transient (time-dependent flows and discontinuous processes as, for example, injection molding, thermoforming, blow molding process) or steady flows (continuous processes—extrusion or coextrusion flows in annular, flat or profile dies, fiber spinning, film casting, film blowing, etc.). A useful tool for understanding the polymer flow in these important industrial processes is modeling. Our attention in this paper will be paid mainly to the steady state flows occurring during continuous polymer processes. The crucial point in modeling is the choice of a suitable constitutive equation which has a capability to correctly represent non-linear behavior of the melts in both elongation and shear. In recent years, significant progress has been made to develop such constitutive equations, for example, the eXtended Pom–Pom model (XPP) [5], a few-parametric modified White–Metzner model (mWM) [6], and the modified Leonov model (mLeonov), whose further improvement is proposed in this work. The aim of the present paper is to evaluate the suitability of these advanced constitutive equations in modeling of steady flows. This will be done through investigation of three polymer melts (LDPE, mLLDPE, PVB) under steady state shear and uniaxial extensional flows, which usually occur in combination in polymer processing.

2. Differential constitutive equations Differential constitutive equations give the present stress as the solution of a particular set of differential equations and seem to be more valuable than integral models, especially for the simulation of general complex flows in both Lagrangian and Eulerian representations. In this study three differential constitutive models (mLeonov, XPP and mWM) are chosen for the investigation; a detailed description is given below. 2.1. Leonov model This constitutive equation is based on heuristic thermodynamic arguments resulting from the theory of rubber elasticity [7–10], which has been found to be mathematically stable [11]. In the model, fading memory of the melts is determined through irreversible dissipation process driven by the dissipation term, b. Mathematically it relates the stress and elastic strain as
 ∂W −1 ∂W τ=2 c , (1) −c ¯¯ ¯¯ ∂I1 ¯¯ ∂I2 where τ is the stress tensor, and the elastic potential, W, depends on the invariants I1 and I2 of the ¯¯ recoverable Finger tensor, c. Although the Leonov model has been found to give an excellent agreement ¯¯ flow experiments for a number of polymer melts and solutions [12], it is to transient and steady shear incapable of representing elongational flows realistically; the steady elongational viscosity is virtually independent of strain rate. Thus, it is not surprising that considerable progress has been made to improve such behavior through generalization of the elastic potential, W [13]:  
  
  3G I1 n+1 I2 n+1 W= (1 − β) −1 +β −1 , (2) 2(n + 1) 3 3

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and dissipation term, b [13–15], in irreversible rate of strain, ep , equation, which is the product of dissi¯¯ pative processes. 
  
  I1 I2 −1 ep = b c − δ −b c − δ . (3) 3 ¯¯ 3 ¯¯ ¯¯ ¯¯ ¯¯ This elastic strain is related to the deformation history as 0

c − c · d − d · c + 2c · ep = 0. (4) ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ ¯¯ Here, G is the linear Hookean elastic modulus, β and n are non-linear model parameters, δ stands for the 0 ¯¯ unit tensor, d is the rate of deformation tensor and c is the Jaumann (corotational) time derivative of the recoverable ¯¯Finger strain tensor. Eq. (2) yields the¯¯Mooney potential for n = 0, and the neo-Hookean potential for n = β = 0. As mentioned above, different functions have been proposed [13,14] for the dissipation term, b, with the aim of improving the capability of the model to predict stresses over the full range of deformation types, especially for elongational ones:
 1 I2 m b= , (5) 4λ I1   −α(I1 + I2 − 6) 1 exp , (6) b= 4λ 2 
−1 2ϑ Ws 1 1+ arctg , (7) b= 4λ π 10G where m, α and ϑ are adjustable parameters, λ is a relaxation time and Ws stands for a symmetrized form of neo-Hookean potential defined as Ws = 21 [W(I1 , I2 ) + W(I2 , I1 )].

(8)

As has been shown by Simhambhatla and Leonov [13], Larson [14] and Tanner [15], these dissipation terms improve either elongational strain hardening or elongational strain softening, but not both simultaneously as needed in the case of LDPE melts. To solve this problem, Leonov recently suggested a formula for the dissipation term, b(I1 ) [16]:

1 sinh[ν(I1 − 3)] exp[−ξ(I1 − 3)] + b(I1 ) = −1 , (9) 4λ ν(I1 − 3) where ξ and ν are adjustable parameters. Unfortunately, the shape of the steady uniaxial viscosity is not smooth in this case due to the occurrence of large unphysical peaks, as demonstrated in Fig. 1 for the case of LDPE IUPAC at 150 ◦ C. The non-linear parameters (ξ = 0.15, ν = 0.03) of the model and relaxation spectrum are taken from [16] and [17], respectively. A significant feature of the Leonov model is the capability to describe basic features of the non-linear viscoelastic behavior of polymer melts, i.e. shear thinning, first and especially second normal stress differences, only with the relaxation spectrum, contrary to other differential models (e.g. Giesekus [18], PTT [19,20], Larson [21], XPP [5]). On the other hand, the Leonov model’s inability to represent elongational

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Uniaxial extensional viscosity (Pa.s)

107

106

105

104 10-5

10-4

10-3

10-2

10-1

1

101

102

103

Extensional rate (1/s) Fig. 1. Steady uniaxial extensional viscosity of LDPE IUPAC melt at T = 150 ◦ C in comparison with the prediction of mLeonov model with the original dissipation term (Eq. (9)).

flows realistically limits the applicability of the model in the area of computational rheology. With the aim of improving the model behavior in elongational flows, a new modification of the Leonov model is proposed and tested in this paper. 2.2. Modified formulation of the Leonov model As mentioned above, the dissipation term (Eq. (9)) proposed in [16] yields unphysical sharp waviness in the steady uniaxial extensional viscosity curve (Fig. 1). This behavior suggests that the exponential dependence of (I1 − 3) with the power of 1 for capturing the hardening phenomenon is too strong. Moreover, dissipation process during the flow can be expected to be more dependent on the type of macromolecules, i.e. on relaxation times. Finally, Eq. (9) is not defined for ν = 0. If these points are taken into account, a new modification of the dissipation term can be proposed in the following form:

 1 sinh[ν(λ)(I1 − 3)] b(I1 ) = exp[−ξ(λ) I1 − 3] + , (10) 4λ ν(λ)(I1 − 3) + 1

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Table 1 Relaxation spectrum and estimated optimal values of the mLeonov model for LDPE IUPAC melt at 150 ◦ C i

1 2 3 4 5 6 7 8

Maxwell parameters

mLeonov model

λi (s)

G0,i (Pa)

ξ

ν

10−4 10−3 10−2 10−1 1 10 102 103

129000 94800 58600 26700 9800 1890 180 1

0 0 0 0 0 0.7 0.43 0.2

0.1 0.1 0.1 0.1 0.1 0.01 0.0015 0.00006

where ξ(λ) and ν(λ) are adjustable parameters which are allowed to vary with the relaxation time, λ. In the proposed Eq. (10), the hardening behavior is captured through the exponential dependence of (I1 −3) with the power of 0.5 rather than 1. It should be mentioned that the modified Leonov model (mLeonov) used in this work is represented by Eqs. (1)–(4) together with the newly proposed dissipation term, Eq. (10) and the neo-Hookean potential, i.e. β = n = 0 in Eq. (2) instead of the generalized one. With the aim of evaluating the mLeonov model capabilities, the proposed modification of the dissipation term is tested for LDPE IUPAC at 150 ◦ C in both steady and transient uniaxial extensional flows as well as in steady shear flow. The relaxation spectrum and measured rheological data of LDPE IUPAC at 150 ◦ C are taken from [17,22,23]. The non-linear parameters, ξ and ν, for the mLeonov model are determined using the steady uniaxial extensional viscosity data only. Both, relaxation spectrum and non-linearity parameters of the mLeonov model are summarized in Table 1. Fig. 2 shows that the fitting capability of the mLeonov model with the proposed dissipation term, Eq. (10), is very good for all flow situations. The small waviness in the strain hardening part of the steady uniaxial extensional viscosity curve reflects the individual relaxation times, which can be improved by choosing a larger number of relaxation times. When the fitting capabilities of the mLeonov model with the unmodified dissipation term (Eq. (9); Fig. 1) and the modified one (Eq. (10); Fig. 2a) are compared, it can be concluded that the proposed dissipation term significantly improves the behavior of the mLeonov model. 2.3. Pom–Pom model The recently introduced Pom–Pom model [24], improved with local branch-point displacement before maximum stretching [25] and adopted in a multimode approach [26], seems to be a breakthrough in the field of viscoelastic constitutive equations due to separation of relaxation times for stretch and orientation. The model is based on the Doi-Edwards reptation tube theory and simplified topology of branched molecules. With this model, correct non-linear behavior in both elongation and shear is accomplished. However, three problems for the differential version of the Pom–Pom model have been detected: first, solutions in steady state elongation show discontinuities; second, the equation for orientation is unbounded for high strain rates because the equation is UCM-like; and finally, the model does not have a second normal stress difference in shear. To overcome these drawbacks, a modified eXtended Pom–Pom model

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Fig. 2. Rheology of LDPE IUPAC melt at T = 150 ◦ C in comparison with newly proposed mLeonov model predictions: (a) steady uniaxial extensional viscosity; (b) steady shear viscosity and first normal stress coefficient; (c) transient uniaxial extensional viscosity. The numbers on the curves indicate extension rates (s−1 ).

has been recently proposed [5]. The key attribute of the XPP model is merging both orientation and stretching mechanisms into a single equation with introduced Giesekus model features for obtaining non-zero second normal stress difference. The XPP does not show the three problems and is easy to implement in finite element packages, because it is written as a single equation for viscoelastic extra-stress tensor. The XPP has the following form:

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Viscoelastic stress: ∇

τ + λ(τ)−1 τ = 2Gd , ¯¯ ¯¯ ¯¯ Relaxation time tensor:

1 α τ + f(τ)−1 δ + G[f(τ)−1 − 1]τ −1 , λ(τ)−1 = λ0b G ¯¯ ¯¯ ¯¯ Extra function: 
  α tr(τ · τ ) 2 1 1 1 −1 ¯¯ ¯¯ , f(τ) = + 1− 1− λ0b λS Λ λ0b Λ2 3G2 Backbone stretch and stretch relaxation time:  tr(τ ) λS = λ0S e−ν(Λ−1) , Λ = 1 + ¯¯ , 3G

ν=

2 , q

(11)

(12)

(13)

(14)

∇

where τ is the stress tensor, τ its upper-convected time derivative, α (level of anisotropy), q (number of arms), λ¯¯ 0S (stretch relaxation¯¯time) are adjustable parameters, which are allowed to vary with the orientation relaxation time, λ0b . The anisotropy parameter, α, is chosen as 0.1/q, as suggested in [5]. It should be pointed out that the Maxwell parameters are G and λ0b = λ. In this model, the stretch relaxation time, λ0S,i , should be physically constrained to lie in the interval λ0b,i−1 < λ0S,i ≤ λ0b,i , and for branched polymers qi should increase with increasing orientation relaxation time λ0b,i , as reported by Inkson et al. [26]. The Leonov as well as the XPP models are, for simplicity, written here in the single mode equations but the calculations are made with the multimode versions. 2.4. Modified White–Metzner model The White–Metzner constitutive equation is a simple Maxwell model for which the viscosity and relaxation time are allowed to vary with the second invariant of the strain rate deformation tensor [6]. It takes the following form: ∇

τ + λ(IId )τ = η(IId )d , (15) ¯¯ ¯¯ ¯¯ where IId is the second invariant of the rate of deformation tensor, λ(IId ) stands for the deformation rate-dependent relaxation time and η(IId ) is the deformation rate-dependent viscosity. Although this modification improves the behavior in steady shear flows, in elongational flows the model predicts unrealistic infinite elongational viscosity. This problem was overcome √by Barnes and Roberts [6], who showed that, for specific functions of λ(IId ) and η(IId ) with (λ0 /K2 ) < ( 3/2) (see Eqs. (16) and (17)), the model does not predict infinite elongational viscosity and can be used for a very good description of elongational viscosity of a wide range of real polymer melts [6], even if the rheology is pressure dependent [27]: η0 η(IId ) = , (16) √ [1 + (K1 2IId )a ](1−n)/a λ(IId ) =

λ0 , 1 + K2 IId

(17)

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where η0 is the Newtonian viscosity, and λ0 , K1 , K2 , n and a are constants. Eqs. (15)–(17) together with the physical constraint for λ0 and K2 mentioned above represent the modified White–Metzner model. It should be pointed out that the use of the phenomenological mWM model is only reasonable in steady flows because in this model the relaxation time depends on IId . Switching on/off the flow causes step-like changes in the relaxation time, which is not very realistic and makes the mWM model a poor candidate for modeling transient industrial flows. The mWM model together with the mLeonov and XPP models were chosen for the investigation of steady state flows because the mWM can reliably represent the steady shear and extensional rheology for different polymer melts [6]. Moreover, contrary to the mLeonov and XPP models it has considerably fewer parameters, and the equations for steady shear and elongational viscosities have analytical expressions. Thus, the fitting procedure for the mWM model is significantly easier compared to Leonov and XPP models. As shown in [28], the mWM model remains stable even for strong extensional flows in mixed numerical scheme with the stresses as unknowns (u–v–p–τ scheme) in FEM analysis of flows through abrupt contractions. These reasons make the mWM model attractive for the investigation of steady shear and elongational rheology of polymer melts. 3. Experimental 3.1. Materials In this work, four materials commonly used in film production have been chosen: • • • •

LDPE Dow LD 150R (Dow Chemicals), ρ = 0.916 g/cm3 ; LDPE Basell Lupolen 1840, ρ = 0.919 g/cm3 ; mLLDPE Engage 8200 (DuPont Dow Elastomers), ρ = 0.870 g/cm3 ; PVB Butacite (DuPont), ρ = 1.08 g/cm3 softening type, commonly used for manufacture of safety car windows.

3.2. Measurements Linear viscoelastic properties (G and G ), low-shear-rate viscosity and first normal stress coefficient data were measured on the advanced rheometric expansion system (ARES) Rheometrics parallel-plate rheometer, whereas the RH7-2 capillary rheometer was used for the determination of shear viscosity (capillary die), first normal stress coefficient (slit die + Han’s methodology [29,30]) and uniaxial extensional viscosity (Cogswell method [31]). The entrance pressure drop measurements were chosen for the uniaxial extensional viscosity determination because there is no other experimental technique enabling to measure at high extensional rates [4,32]. With the aim to obtain reliable steady uniaxial extensional data, a recently proposed ‘effective entry length correction’ [28] has been applied. The detailed information about the techniques is given in Section 3.2.1. 3.2.1. ARES rotational rheometer Rheological characteristics of the chosen materials were measured by advanced rheometric expansion system with parallel-plate geometry using 25 mm diameter plates at selected temperatures. All measurements were performed with a 2 K FRTN1 and 2 K FRTN2 transducers with a lower resolution limit of

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0.02 g cm and 2 g for torque and normal force, respectively. The typical sample thickness ranged from 0.5 to 1.3 mm. The measurements were performed in both oscillatory and transient modes. The oscillatory shear measurements were performed at the frequency range of 0.1–100 rad/s. With the aim of getting the experimental data of the storage, G , and loss, G , moduli for a large range of frequencies, the measurements were performed in a wide range of temperatures (130–320 ◦ C). Possible degradation at very high temperatures was suppressed by inertial nitrogen atmosphere. The well known time–temperature-superposition principle was used to generate master curves for reference temperatures of interest. 3.2.2. Capillary and slit die measurements A twin-bore RH7-2 capillary rheometer produced by Rosand Precision Ltd. was used to determine shear and elongational viscosities. The dimensions of the capillaries and barrel used for tests were as follows. Two orifice dies: L (length) = 0.25 mm and D (diameter) = 1 and 2 mm; two long dies: L = 16 and 32 mm, and D = 1 and 2 mm, respectively; barrel: Db = 15 mm. Capillaries with two different diameters were chosen to enable measurements within a wide range of deformation rates. In order to properly determine the elongational viscosity data from measured entrance pressure drop, a recently proposed strategy [28] has been applied in the following steps. Firstly, apparent ‘entrance’ viscosity, ηENT , was calculated according to Eq. (18) P0 , (18) ηENT = γ˙ a where P0 is the entrance pressure drop, and γ˙ a represents apparent shear-rate. Secondly, the apparent entrance viscosity was fitted by Eq. (19), where the plateau value, ηENT,0 at low shear rates is known from Newtonian viscosity [28]:
  ηENT,0 tanh(α γ˙ a + 1) ξ . (19) log(ηENT ) = log  tanh(1) 1 + (λ γ˙ a )a This is an empirical equation which combines the Carreau–Yasuda model and an additional term that allows a maximum to appear in the entrance viscosity. Parameters α and ξ  control the shape of the entrance viscosity maximum. The predictions of Eq. (19) at very low-shear-rates (at which the measurements are impossible) are taken as experimental data for further calculations. The parameter values for the investigated materials are summarized in Table 2 and the comparison between the measured entrance viscosities and the fitting line are given in Fig. 3a. Finally, Cogswell [31] methods were used to determine elongational viscosity together with a novel ‘effective entry length correction’ [28], which increases the capability of the entrance techniques to evaluate elongational rheological data from capillary measurements. Table 2 Model parameters (Eq. (19)) for LDPE Dow LD 150R melt at T = 200 ◦ C ηENT,0 (Pa s) λ (s) a α (s) ξ

151239 0.530433 0.6457711342 2.670801 0.172863

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LDPE Dow LD 150R at 200 oC

LDPE Dow LD 150R at 200 oC 6.4

6 105

ln(Pexit)

Apparent entrance viscosity (Pa.s)

106

5.6

104 5.2

dln(Pexit)/dln(τxy)=0.7

103 10-2

(a)

10-1

1

101

Apparent shear rate (1/s)

102

4.8 3.2

103

(b)

3.6

4

4.4

4.8

ln(τxy)

Fig. 3. Experimentally determined rheological data from RH7-2 rheometer for LDPE Dow LD 150R melt at 200 ◦ C: (a) apparent entrance viscosity, the line represents the fit by Eq. (19); (b) exit pressure as a function of wall shear stress.

To enable measurement of first normal stress coefficient, ψ1 , at high shear rates with the help of the Han’s method [29,30], a slit die was attached to the barrel of a high pressure capillary viscometer RH7-2. The geometry is presented in Fig. 4, together with pressure transducers details. The width of the slit die was w = 10 mm. The basic idea behind the Han methodology is determination of the pressure at the end of the slit die, Pexit , with the help of the measured pressure profile along the die, and an extrapolation procedure. Then ψ1 can be calculated according to Han’s equation:   d(ln Pexit ) Pexit ψ1 = 2 1 + , (20) γ˙ d(ln τxy ) where τ xy is the wall shear stress, and γ˙ stands for the corrected shear-rate. Both values can be calculated from the following equations:
 ∂p h τxy = − , (21) ∂z 2 
2n + 1 6Q , (22) γ˙ = 3n wh2 where −∂p/∂z is the measured pressure gradient, h means the height of the slit die, Q stands for the volumetric flow rate and n is the slope of the log τ xy versus log(6Q/wh2 ) plot. When measuring on the slit die, it is necessary to keep in mind that the flow disturbance near the die exit must be minimal. As the exit disturbances decrease with increasing shear rate and Weissenberg number, We (i.e. as the amount of energy stored in the fluid increases), the shear measurements were performed

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219

15

31

30 000 PSI

100

60

10 000 PSI

89

10 000 PSI

1 500 PSI 1

Fig. 4. Scheme of the slit die used including the information about pressure transducers Dynisco. All dimensions in mm.

in a constant piston speed mode at the apparent shear-rate range of 5–500 s−1 so that the wall shear stress was higher than 25 kPa in all experiments, as suggested in [30]. This justifies the extrapolation procedure for obtaining Pexit , thus validating the exit pressure method for reliable determination of ψ1 . A typical example of results from slit die measurements is depicted in Fig. 3b, where Pexit is plotted against τ xy in the logarithmic scale for LDPE Dow melt at 200 ◦ C. It can be seen that this dependence is linear, which is also the case of the other investigated materials. 4. Results and discussion The generalized Maxwell model was employed to fit the measured frequency dependent loss and storage moduli for all tested materials to generate discrete relaxation spectra. Fig. 5 clearly shows that for the tested materials this model properly fits all the measured data. Thus, it supports the reliability of the estimated relaxation spectra. The non-linearity parameter, q, and the ratio λb /λS for the XPP model, as well as ξ and ν for the mLeonov model have been determined using the steady uniaxial extensional viscosity data only. For the mWM model, η0 , K1 , a, n, λ0 , K2 have been identified using steady shear and uniaxial viscosity curves. Since the second planar extensional or second normal stress difference data is not available for the tested materials, the anisotropy parameter, α, in the XPP model was chosen as 0.1/q, as suggested in [5]. Discrete relaxation spectra in the terms of relaxation moduli, Gi , and relaxation times, λi , together with the non-linearity parameter for each material are given in Tables 3–7. The use of relaxation time-dependent non-linear parameters in the mLeonov model was necessary for successful fitting of the data by the model.

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Fig. 5. Comparison of Maxwell model with measured complex viscosity, η∗ , storage, G , and loss, G , moduli for tested polymers in oscillatory shear flow at particular temperatures: (a) LDPE Dow LD 150R melt at 200 ◦ C; (b) LDPE Basell Lupolen 1840 melt at 180 ◦ C; (c) mLLDPE Engage 8200 melt at 130 ◦ C; (d) PVB Butacite melt at 130 ◦ C.

The discussion of the results given below concentrates mainly on the comparison of the capabilities of the proposed modification of the Leonov model, as well as XPP and mWM constitutive equations to describe a wide range of measured rheological data for different polymer melts (LDPEs, mLLDPE and PVB), used mainly in film production. Figs. 6–9 show rheological behavior of the three constitutive models and measured steady shear and uniaxial extensional characteristics.

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Table 3 Relaxation spectrum and estimated optimal values of the mLeonov and XPP models for LDPE Basell Lupolen 1840 H melt at 180 ◦ C i

1 2 3 4 5 6 7 8 9

Maxwell parameters

XPP model

mLeonov model

λi (s)

G0,i (Pa)

qi

λ0b,i /λ0S,i

αi

ξ

ν

0.0018 0.0073 0.02955 0.11961 0.48407 1.95915 7.92906 32.0906 129.877

108260 28028.6 23336.4 11290.2 5856.42 2322.13 695.625 128.05 0.56853

1 1 1 1 3 6 6 6 6

3.8 3.8 3.8 3.8 3.8 1.5 3.8 3.8 3.8

0.1 0.1 0.1 0.1 0.0333 0.0167 0.0167 0.0167 0.0167

0 0 0 0.05 0.7 0.2 0 0 0

0.2 0.2 0.2 0.05 0.012 0.01 0.01 0.001 0.001

Table 4 Relaxation spectrum and the XPP and mLeonov models parameters for fitting of LDPE Dow LD 150R melt at 200 ◦ C i

1 2 3 4 5 6 7 8 9

Maxwell parameters

XPP model

mLeonov model

λi (s)

G0,i (Pa)

qi

λ0b,i /λ0S,i

αi

ξ

ν

10−4 10−3 10−2 10−1 1 10 102 103 104

202935.5 110089.3 50474.2 25026.4 10621.6 2779.5 346.3 23.3 1.3

1 1 1 1 1 1 9 9 10

9 9 9 6 6 6 1.7 9 6

0.1 0.1 0.1 0.1 0.1 0.1 0.011 0.011 0.01

0 0 0 0 0 1 0.22 0 0

0.2 0.2 0.2 0.2 0.5 0.6 0.001 0.001 0.001

Table 5 Relaxation spectrum and the XPP and mLeonov models parameters for fitting of mLLDPE Engage 8200 melt at 130 ◦ C i

1 2 3 4 5 6 7

Maxwell parameters

XPP model, physical constraints applied

XPP model, no physical constraints applied

mLeonov model

λi (s)

G0,i (Pa)

qi

λ0b,i /␭0S,i

αi

qi

λ0b,i /λ0S,i

αi

ξ

ν

0.01 0.0379 0.14364 0.54442 2.06338 7.8203 29.6393

244370 33385.8 21737.8 4604.69 1419.21 177.755 16.6116

1 1 1 1 1 1 1

3.78 3.78 3.78 3.78 3.78 3.78 3.78

0.1 0.1 0.1 0.1 0.1 0.1 0.1

1 1 1 1 1 1 1

25 25 15 5 4 3 3

0.1 0.1 0.1 0.1 0.1 0.1 0.1

0 0 0 0 0 0 0

1.2 1.2 1.2 0.3 0.3 0.3 0.3

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Table 6 Relaxation spectrum and the XPP and mLeonov models parameters for fitting of PVB Butacite melt at 130 ◦ C i

1 2 3 4 5 6 7 8 9 10 11 12

Maxwell parameters

XPP model

mLeonov model

λi (s)

G0,i (Pa)

qi

λ0b,i /λ0S,i

αi

ξ

ν

0.01 0.02667 0.07115 0.18979 0.50623 1.35033 3.60187 9.60763 25.6274 68.3584 182.339 486.372

171310 56306.7 54440 42505.8 29254.6 14474.4 6772.14 2577.5 803.37 236.279 15.7930 4.32627

1 1 1 1 1 1 1 1 1 1 1 1

2.66 2.66 2.66 2.66 2.66 2.66 2.66 2.66 2.66 2.66 2.66 2.66

0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

0 0 0 0 0 0 0 0 0 0 0 0

0.9 0.9 0.9 0.9 0.9 0.9 0.1 0.1 0.1 0.1 0.1 0.1

4.1. LDPE In the case of branched LDPEs, Figs. 6a and 7a prove the ability of all three tested models to properly describe and predict shear and elongational viscosities. The other pair of Figs. 6b and 7b reveals that the first normal stress coefficient at low shear rates (up to about 50 s−1 ) is very well predicted by the XPP model but is under-estimated by the mLeonov model. On the other hand, at higher shear rates the XPP tends to overestimate ψ1 , whereas mLeonov model gives a good agreement with the measurements. The prediction of ψ1 by the mWM model is rather poor in this case. 4.2. mLLDPE As can be seen in Figs. 8a and b, the fitting and predicting capability of the tested models is different for mLLDPE than in the previous case. In more detail, the measured rheology of mLLDPE is most precisely fitted and predicted by the mLeonov model. The mWM model describes very well steady shear Table 7 mWM model parameters for fitting of the tested materials at various temperatures Material

η0 (Pa s) K1 (s) a n λ0 (s) K2 (s)

LDPE Basell Lupolen 1840 (180 ◦ C)

LDPE Dow LD 150R (200 ◦ C)

PVB Butacite (130 ◦ C)

mLLDPE Engage 8200 (130 ◦ C)

21000 3.2277 0.59661 0.40678 7.5586 8.988

123247 11.379 0.39073 0.30225 303.98 366.39

165214 0.54309 0.42748 0.01000 3.1283 4.103

15362 0.055849 0.46792 0.094316 0.60585 0.96586

M. Zatloukal / J. Non-Newtonian Fluid Mech. 113 (2003) 209–227

LDPE Dow LD 150R at 200 oC

10

10

6

10

6

105

10

10

First normal stress coefficient (Pa .s2)

Uniaxial extensional and shear viscosities (Pa. s)

LDPE Dow LD 150R at 200 oC 7

223

5

4

ηE,U (measurement) 10

10

ηS (measurement) XPP model mLeonov model mWM model

3

-4

10

-3

(a)

-2

10

10

-1

103

102

ψ 1 (measurement)

101

XPP model mLeonov model mWM model

1

2

10

104

1

10

1

2

10

10

3

(b)

Extensional and shear rates (1/s)

10-1 10-1

101

1

102

103

Shear rate (1/s)

Fig. 6. Experimental data vs. model predictions for LDPE Dow LD 150R melt at 200 ◦ C: (a) steady shear and uniaxial extensional viscosities experimental data vs. model predictions; (b) steady first normal stress coefficient experimental data vs. model predictions.

LDPE Basell Lupolen 1840 at 180 oC

10

10

5

10

4

ηE,U (measurement) 10

3

ηS (measurement)

10

3

10

2

10

1

ψ1 (measurement) XPP model mLeonov model mWM model

1

XPP model mLeonov model mWM model 10

-1

2

10

(a)

First normal stress coefficient (Pa.s 2)

Uniaxial extensional and shear viscosities (Pa. s)

LDPE Basell Lupolen 1840 at 180 oC 6

10 -3

10

-2

10

-1

1

10

1

Extensional and shear rates (1/s)

10

2

10

3

10

(b)

1

10

2

3

10

Shear rate (1/s)

Fig. 7. Experimental data vs. model predictions for LDPE Basell Lupolen 1840 melt at 180 ◦ C: (a) steady shear and uniaxial extensional viscosities experimental data vs. model predictions; (b) steady first normal stress coefficient experimental data vs. model predictions.

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Fig. 8. Experimental data vs. model predictions for mLLDPE Engage melt at 130 ◦ C: (a) steady shear and uniaxial extensional viscosities experimental data vs. model predictions; (b) steady first normal stress coefficient experimental data vs. model predictions; (c and d) comparison between XPP model predictions and experimental data for two fitting model parameter sets. The first set with physical constraints, the other without.

and elongational viscosities, even if small waviness occurs in the fitting line for elongational viscosity, whereas the prediction of ψ1 is poor, similarly to the case of LDPE. The XPP model shows the ability to properly describe steady elongational viscosity up to extensional rate about 10 s−1 ; however, at higher extensional rates the model tends to overestimate the measured elongational viscosity data. In the steady

M. Zatloukal / J. Non-Newtonian Fluid Mech. 113 (2003) 209–227

PVB Butacite at 130 oC

10

10

5

First normal stress coeficient (Pa.s-2)

Extensional and shear viscosities (Pa.s)

PVB Butacite at 130 oC 6

104

ηE,U (measurement) ηS (measurement)

103

XPP model mLeonov model mWM model 102 10-3

(a)

10-2

10-1

1

Shear rate (1/s)

10

5

10

4

10

3

10

10

2

1

ψ1 (measurement) XPP model mLeonov model mWM model

1

101

225

102

10

103

-1

1

(b)

10

1

10

2

3

10

Shear rate (1/s)

Fig. 9. Experimental data vs. model predictions for PVB Butacite melt at 130 ◦ C: (a) steady shear and uniaxial extensional viscosities experimental data vs. model predictions; (b) steady first normal stress coefficient experimental data vs. model predictions.

shear flow the model underestimates both steady shear viscosity and first normal stress coefficient up to the shear rate of 100 s−1 . In the fitting procedure for the XPP model, the q parameter, i.e. the amount of arms attached to the backbone, was considered as a constant (equal to 1) for all relaxation modes, as theoretically expected for linear polymers. The physical constraint is needed in this case because increasing q above the unity causes elongational hardening, which does not occur in the case of the investigated mLLDPE melt. It is interesting that the best fit has been found for λ0b,i /λ0S,i ratio equal to its upper limiting value, i.e. 3.78 for all relaxation modes (the fitting interval was 1 < (λ0b,i /λ0S,i ) < 3.78 for the mLLDPE due to physical constraint for stretch relaxation time [26]). With the aim to find mathematical optimum values of λ0b,i /λ0S,i ratio for all relaxation modes, the physical constraint, λ0b,i−1 < λ0S,i ≤ λ0b,i , for stretch relaxation time was relaxed. In this case, the fitting results of the uniaxial extensional viscosity are much better than in the case when physical constraint was applied, however, the predictions of the shear viscosity and ψ1 fail significantly, as shown in Figs. 8c and d. As a result, the physical constraints should be taken into account during the parameter identification process for the XPP model. 4.3. PVB The steady shear and elongational rheology of highly elastic PVB was taken for testing of the chosen models. As shown in Fig. 9, the mLeonov model shows the best agreement between the measured data and predicted/fitted curves. The mWM model also shows very good fitting and predicting capabilities for steady elongational and shear viscositites, except for the waviness occurring in the fitting line for elongational viscosity.

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Finally, the XPP model tends to overestimate both the steady elongational and shear viscosities at deformation rates higher than 10 s−1 , even if the prediction of ψ1 is very good in this case. Based on the findings given above, it can be concluded that the proposed modification of the Leonov model becomes a good candidate for modeling of complex steady flows of both linear and branched polymers even at deformation rates comparable to those occurring in large-scale extrusion lines, i.e. between 10 and 1000 s−1 . 5. Concluding remarks A modification of the Leonov model has been proposed and tested for the LDPE IUPAC at 150 ◦ C in both steady and transient uniaxial extensional flows as well as steady shear flow. It has been found that the model has a very good capability to describe and predict the behavior of branched LDPE in these flow situations. The fitting and predicting capabilities of the proposed mLeonov model along with the XPP and mWM models were investigated in the steady shear and uniaxial extensional flows for different polymer melts (LDPEs, mLLDPE and PVB). Comparison of the three models has shown the following: • The mWM model is able to reliably describe, with the use of six model parameters only, steady shear and elongational viscosities for both elongational strain hardening and softening melts, but predictions of steady ψ1 are poor. • The XPP model shows an excellent fitting and predicting capability for steady shear and elongational viscosities and ψ1 for elongational hardening melts. The predicting capabilities of the model are excellent especially at lower deformation rates (less than 50 s−1 in this case). On the other hand, for linear melts showing elongational strain softening, the XPP model does not seem to be precise enough. It is believed that a slight adjustment of the evolution of stretch or orientation equation should improve the XPP model capabilities, as it will enable to characterize more properly linear molecular topology of a particular melt. • The newly proposed mLeonov model for the description of steady shear and elongational viscosities and ψ1 has proved to be very good for both strain hardening and softening polymer melts. • The use of relaxation time-dependent non-linear model parameters in the mLeonov model is a substantial modification, and has been found to be the key to the success of the modified model. Acknowledgements The support of the project by the Czech Grant Agency (Grant No. 104/01/P132) and Ministry of Education of the Czech Republic (Grant No. 265200015) are gratefully acknowledged. Special thanks are directed to RETRIM-CZ company for providing PVB material. References [1] R.B. Bird, R.C. Armstrong, O. Hassager, Dynamics of Polymeric Liquids, vol. 1, Wiley, New York, 1987. [2] D.V. Boger, K. Walters, Rheological Phenomena in Focus, Elsevier, Amsterdam, 1993. [3] R.I. Tanner, Engineering Rheology, Oxford University Press, New York, 2000.

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