Network constitutive equation with internal viscosity: application to stress jump prediction

J. Non-Newtonian Fluid Mech. 95 (2000) 135–146

Network constitutive equation with internal viscosity: application to stress jump prediction N. Sun, C.F. Chan Man Fong, D. De Kee∗ Department of Chemical Engineering, Tulane University, New Orleans, LA 70118, USA Received 15 February 2000; received in revised form 20 July 2000

Abstract A stress jump, defined as the instantaneous gain or loss of stress on startup or cessation of a deformation, has been predicted by various models and has relatively recently been experimentally observed. In this paper, the internal viscosity idea is incorporated into the transient network model. Via appropriate approximations, we obtain a closed constitutive equation where the total stress equals the sum of an elastic contribution and a viscous contribution. As the latter is rate dependent, the model predicts a stress jump and we consider data on shear flow in this contribution. We successfully compare the model predictions with the stress jump measurements of Liang and Mackay [C.H. Liang, M.E. Mackay, J. Rheol. 37 (1993) 149]. The model yields good quantitative predictions of the steady, transient and dynamic material functions. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Constitutive equation; Internal viscosity; Stress jump

1. Introduction The determination of a constitutive equation which adequately describes the rheological properties of a material under all flow conditions is one of the central problems in rheology. The problem can be approached via continuum mechanics or via molecular theory. The continuum approach identifies admissible variables for rheological models but takes no explicit consideration into account of the molecular structure of the material. Generally, the equations based on the continuum mechanics yield a better fit of the experimental data. On the other hand, the molecular approach yields specific information regarding the relationship between rheological parameters and the molecular structure (properties). The network model was originally developed for solid rubber-like materials, where chemical crosslinks form permanent junctions, possibly undergoing affine motion. This was later extended to liquid polymeric materials by Lodge [1] and Yamamoto [2–4] who considered the junctions to be temporary and liable to be destroyed and created. Many successful rheological models have evolved from the network concept. In the Lodge network model, the creation and destruction rates of the junctions are assumed to be constant. ∗

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This model is able to describe many of the phenomena associated with linear viscoelasticity [5] but predicts a constant shear viscosity and normal stress coefficient, which is not usually observed with polymeric materials. One way of overcoming this difficulty is to allow those two rates to be functions of a macroscopic variable. Based on the different choices of the macroscopic variable, the modified network models can be labeled as rate-dependent, stress-dependent or strain-dependent. More details can be found in [5,6]. In this contribution, we propose a further improvement by including an internal viscosity in the network model. In the 1990s, there has been an increasing interest in the stress jump phenomenon, which is defined as the instantaneous finite change in stress due to an instantaneous change in deformation rate. Such discontinuities were predicted by De Kee and Carreau [7] and measured in shear flow by Liang and Mackay [8] and in extensional flow by Orr and Sridhar [9], and by Spiegelberg and McKinley [10]. There are two possible mechanisms to account for a stress jump; a viscous intermolecular force and a viscous intramolecular force. The former force arises from the interaction between solvent molecules and polymer molecules and was used in the rigid dumbbell model [11] and in the De Kee–Carreau model [7], to name just two. The latter force relies on the idea of an internal viscosity (IV) [12,13]. The internal viscosity was first suggested by Kuhn and Kuhn [14], and models generated were found to be able to describe high-frequency dynamic behavior, dielectric relaxation behavior and non-linear properties [15]. In the 1980s, the IV idea began to be associated with stress jump in the bead–rod–spring molecular model. As far as we know, this IV idea has been used only with the bead–rod–spring (dumbbell) model. Here we introduce the intramolecular force into the network theory.

2. Theory The polymer molecules are considered to form a network of segments of various lengths linked together by junctions or entanglements where the polymer–polymer interactions are localized [11]. For polymeric liquids, they are temporary entanglements of various lifetimes with a distribution function Ψ (Q, t) such that: Ψ (Q) dQ is number of segments per unit volume at time t that have end-to-end vector in the range dQ about Q. Here we consider only one type of junctions. The equation of continuity for the distribution function is given by [11]: ! ∂Ψ ∂ ˙ ] + L¯ − Ψ , · [QΨ =− (1) ∂t ∂Q λ where L¯ is the creation rate for segments of length Q and the probability per unit time that a segment will be destroyed is λ−1 . For non-affine motion, the equation of motion is given by ˙ = [κ · Q] − 1 ξ [γ˙ · Q] = κˆ · Q, Q 2

(2a)

where κ = ∇v + , γ˙ is the deformation rate tensor, ξ is a slip parameter and κˆ = κ − 21 ξ γ˙ .

(2b)

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The combination of Eqs. (1) and (2) gives the equation of change for the conformation tensor QQ:   䊐 QQ hQQi = Lδ − , (3a) λ Z (3b) hQQi = QQΨ dQ, 䊐

where hQQi stands for the Gordon–Schowalter derivative of hQQi. The total stress tensor is written as π = −hQF c i.

(4)

Here we represent the segment by a spring and a dashpot connected in parallel. Thus, the tension is given by Fc = HQ + ϕ

˙ ·Q Q Q2

Q,

(5)

where H is the spring constant. The second term in Eq. (5) is due to the internal viscosity which has been widely adopted in the dumbbell model [13,16–18]. ϕ is the internal viscosity coefficient. Combining Eqs. (2), (4) and (5) gives   QQQQ π = −H hQQi − ϕ κˆ : , (6a) Q2 π = π e + π v.

(6b)

We identify π e as the elastic contribution, equal to the second moment term in Eq. (6a), and π v as the viscous contribution, given by the fourth-order tensor contribution in Eq. (6a). The expression of π e , when combined with Eq. (3), turns out to be the non-affine Lodge model: 䊐

π e + λπ e = −

ηm δ. λ(1 − ξ )

(7)

Using Wick’s theorem with Gaussian distribution assumption [19] combined with a Peterlin [20] approximation, we can decompose the fourth-order tensor into second moment terms and thus obtain the expression for π v . π =ϕ v

κˆ : π e π e + (1 − ξ )π e · γ˙ · π e H tr π e

.

(8)

Eqs. (6b), (7) and (8) form the new constitutive equation for the non-affine network model with internal viscosity. This new equation is given by π =π +ϕ e

κˆ : π e π e + (1 − ξ )π e · γ˙ · π e H tr π e

where π e is obtained via Eq. (7).

,

(9)

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3. Results and discussion 3.1. Steady shear In steady shear flow, the model predicts the following shear viscosity η, primary normal stress coefficient ψ 1 and secondary normal stress coefficient ψ 2 : εηm ηm π21 + (10a) = fη (γ˙ ), η(γ˙ ) = − 2 2 γ˙ 1 + ξ(2 − ξ )λ γ˙ λ ψ1 (γ˙ ) = −

π11 − π22 2ληm = g1 (γ˙ ), 2 γ˙ 1 + ξ(2 − ξ )λ2 γ˙ 2

(10b)

ψ2 (γ˙ ) = −

π22 − π33 ξ ληm =− + ε(1 − ξ )ηm g2 (γ˙ ), 2 γ˙ 1 + ξ(2 − ξ )λ2 γ˙ 2

(10c)

where 1 + 2(2 − 2ξ + ξ 2 )λ2 γ˙ 2 − ξ(2 − ξ )λ4 γ˙ 4 , 3 + 2(1 + 2ξ − ξ 2 )λ2 γ˙ 2 + ξ(2 − ξ )(2 + 2ξ − ξ 2 )λ4 γ˙ 4   (1 − ξ )2 λγ˙ 2 , g1 (γ˙ ) = 1 + 3ε 3 + (2 + 2ξ − ξ 2 )λ2 γ˙ 2   2 + ξ(ξ + 1)λ2 γ˙ 2 . g2 (γ˙ ) = 3 + 2(1 + 2ξ − ξ 2 )λ2 γ˙ 2 + ξ(2 − ξ )(2 + 2ξ − ξ 2 )λ4 γ˙ 4 fη (γ˙ ) =

(10d) (10e) (10f)

Following Chan Man Fong and De Kee [21], we assume the creation and loss of network junctions to be rational functions of the second invariant of the deformation rate tensor, i.e. the shear rate γ˙ in simple shear flow. This implies that we may write η0 , (11a) ηm = 1 + bγ˙ n λ = λ(γ˙ ) =

λ0 , 1 + cγ˙ m

(11b)

where m and n are related to the slopes of the shear viscosity and primary normal stress coefficient in the power law region. These slopes are equal to (2 − 2m + n) and (2 − m + n), respectively, and in general m > n. We also assume that, the internal parameter ε = ϕ/H , has the same characteristics. That is, it is also a function of a macroscopic variable. The microscopic properties are connected to the macroscopic properties and in order to limit the number of model parameters, we assume the internal parameter to be given by ε(γ˙ ) = ε0

λ(γ˙ ) . ηm (γ˙ )

(11c)

That is to say, the internal viscosity is also ‘shear-thinning’. More discussion is given in the stress jump section. It ensures the existence of an infinite shear viscosity (η∞ ) at high shear rate. Thus, a model with eight parameters: ε 0 , ξ , η0 , λ0 , b, c, m and n, is obtained, that in addition to the stress and primary normal

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Fig. 1. Viscosity η and primary normal stress coefficient ψ 1 vs. shear rate. (䊉) Viscosity data of a LDPE I melt (data from [43], adapted from [23]); (䊏) normal stress coefficient data of the LDPE I melt (data from [43], adapted from [23]); (䉲) viscosity data of a 2 wt.% polyacrylamide solution; (䉱) primary normal stress coefficient data of the 2 wt.% polyacrylamide solution; (—) predictions of Eqs. (10a) and (10b). The model parameters for LDPE I melt are η0 = 4.41 × 104 Pa s, b = 1.84, m = 0.85, λ0 = 53.3 s, c = 43.4, n = 0.54, ε0 = 4.77 and ξ = 0.126, and for the 2 wt.% polyacrylamide solution are η0 = 1.41 × 103 Pa s, b = 12.7, m = 0.76, λ0 = 163 s, c = 55.0, n = 0.33, ε0 = 0.723 and ξ = 0.203; (· · · ) prediction of the EWM model [23]; (– ·· –) prediction of the EWM model with parameters ensuring satisfactory viscosity prediction [23].

stress jump phenomenon, successfully describes a variety of material functions such as: η, ψ 1 , η+ , ψ1+ , η0 and G0 . We note that the recent dumbbell model used to describe stress jump predictions are formulated in terms of at least four parameters. Recent versions using the dumbbell model assume a constant IV coefficient. However, other than the 0.010 wt.% xanthan solution, modeled by Hua and Schieber [22] and Schieber [16], no comparisons of model predictions with experimental data have been reported for this dumbbell type model. Fig. 1 successfully compares viscosity and primary normal stress coefficient data of a LDPE I melt and of a 2 wt.% solution of polyacrylamide in a 50 wt.% mixture of water and glycerin to model predictions. In 1992, Souvaliotis and Beris [23] proposed an extended White–Metzner (EWM) model involving more than ten parameters and compared their model predictions with the Phan-Thien–Tanner (PTT) model predictions on the LDPE I melt data. Both the EWM and PTT single-mode models failed to adequately predict the steady shear viscosity data. Multi-mode versions of these models required different parameters to satisfactorily predict the viscosity and primary normal stress coefficient data. Fig. 1 also shows the predictions of the multi-mode EWM model for the ψ 1 data (· · · ) and for the ψ 1 data using parameters that yield good viscosity predictions (– ·· –). In this case, the ψ 1 data are severely underpredicted. The multi-mode PTT model exhibits behavior similar to that of the multi-mode EWM model. We stress that our eight parameter model outperforms the multi-mode EWM and PTT models in terms of η and ψ 1 predictions and that in addition, our model yields excellent stress jump predictions. Furthermore, by setting b = 0 (i.e. ηm = η0 ), we can reduce the number of the model parameters to six and still achieve reasonable predictions.

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3.2. Stress jump A stress jump may occur at the startup of flow or may be associated with stress relaxation. In the latter case, Liang and Mackay [8] defined and measured a stress jump ratio given by R − (t = 0, γ˙0 ) =

η− (t = 0, γ˙0 ) . η(γ˙0 ) − ηs

(12)

They also listed some of the constitutive equations that can predict a stress jump. Gerhardt and Manke [24] have shown that for a stress jump to occur, the memory function of an integral constitutive equation must have a discontinuity. The model proposed by De Kee and Carreau [7], has been successfully compared [25] with stress jump experimental data. Although the stress jump could be due to the Newtonian solvent as represented by the Oldroyd-B model, more scientists tend to relate it to the intramolecular forces, i.e. the IV of the molecule. The exact meaning of the IV is not clear yet. Some [16,18] attribute the IV to the internal friction of gauche/trans rotations along the polymer backbone that have energy barriers [26], and friction between two segments of the polymer chain which are far away along the backbone but near in space [27]. The latter force may diminish with shear since polymer chains will stretch along the flow field. This adds to the complexity in formulating the IV. Zimmerman and Williams [28] summarized four versions of the IV models with respect to their ability to fit experimental data. Those four models evolved mainly from the formulas of the IV force proposed by Cerf [29–31], which was proportional to the deformational velocity of the beads. The evaluation of the deformational velocity depended on an angular rotation rate and it was controversial. They would either lead to an asymmetric stress tensor, or after imposing a ‘symmetrized function’, yield a limited range of viscosity predictions. These models are no longer used. Schieber [32] examined several dumbbell models with the IV force and he proved that the model with the formulation given by Eq. (5) satisfies an explicitly stated interdependence of the fluctuating and dissipative forces, known as the fluctuation–dissipation theorem. Eq. (8) allows us to conclude that, π v dissipates immediately upon cessation of flow as it is shear rate dependent. π e on the other hand is related to the structure configuration and is time dependent. Thus, the stress jump ratio can be written as R − (t = 0, γ˙0 ) =

e (t = 0, γ˙0 ) π12 . π12 (γ˙0 ) − ηs γ˙0

(13)

Combining Eqs. (10) and (13) one can obtain the expression of the stress jump ratio as a function of γ˙0 . Fig. 2 compares the model predictions with the stress jump data of three xanthan gum solutions (0.010, 0.020 and 0.050 wt.%) in a solvent of 75 wt.% fructose and 25 wt.% water. The model parameters are calculated from the steady shear viscosity curves as shown in Fig. 3. We note that those are the only shear stress jump data on polymer solutions available in the literature [33]. As noted earlier, Schieber [16] and Hua and Schieber [22] also showed a well comparison of the 0.010 wt.% xanthan stress jump data with the predictions of a modified dumbbell model. Some similar measurements have been reported on liquid crystalline polymers [34] and on suspensions [35,36]. In the former case, Smyth and Mackay [34] observed a local maximum in the stress jump ratio curves and they attributed this to the sliding of domains formed by defects in the structure, which gives rise to the dissipation of hydrodynamic energy at the interdomain boundary (the defect regions). We also note that in the case of extensional flow, the viscous contribution on the right side of Eq. (9) vanishes instantly upon cessation of elongational flow. Thus, the model also

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Fig. 2. Stress jump ratio R− vs. shear rate for three xanthan gum solutions: (䊉) 0.010 wt.%; (䊊) 0.020 wt.%; (䉲) 0.050 wt.% (data from [8]). Model predictions: (—) 0.010 wt.%; (· · · ) 0.020 wt.%; (– – –) 0.050 wt.%.

predicts a stress jump in extensional flow. In addition, we wish to point out that systems, obeying the stress-optical rule cannot exhibit substantial stress jump. Smyth et al. [37] examined two xanthan gum solutions and compared the mechanical viscosity measurements to rheo-optical measurements. They observed shear stress jumps as well as the breakdown of the stress-optical rule. However, they noted that, if only the elastic stress was used, the stress-optical rule still held true. The elastic stress is proportional to the birefringence index [38,39]. Polymer melts seem to be dominated by the elastic stress. Liang and Mackay [8] did not observe a stress jump for a variety of polymer melts in their laboratory.

Fig. 3. Viscosity vs. shear rate for three xanthan gum solutions: (䊉) 0.010 wt.%; (䉱) 0.020 wt.%; (䊏) 0.050 wt.% (data from [8]); (—) predictions of Eq. (10a).

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3.3. Startup flow + In startup shear flow, a finite shear rate γ˙∞ is applied at t = 0. The stress π12 (t, γ˙0 ) is measured and the stress growth and primary normal stress growth functions are defined as

η+ (t, γ˙∞ ) = −

+ (t, γ˙∞ ) π12 , γ˙∞

(14a)

+ + − π22 π11 . 2 γ˙∞

(14b)

ψ1+ (t, γ˙∞ ) = −

To solve Eq. (9), we separate all quantities into time-dependent and time-independent parts. The latter corresponds to the steady-state. The solutions for the non-affine Lodge model are +

ηe = ηe (γ˙∞ )(1 − e−t/λ [cos(pt) − pλ sin(pt)]),    1 e+ e −t/λ ψ1 = ψ1 (γ˙∞ ) 1 − e sin(pt) , cos(pt) + pλ

(15a) (15b)

√ + + where ηe and ψ1e refer to the elastic contribution and p = γ˙∞ ξ(2 − ξ ). Via Eqs. (9) and (15), one obtains the stress growth and primary normal stress growth functions η+ (t) and + ψ1 (t). Because the elastic component yields the major contribution to the stress growth evolutions, we can analyze the model behavior through Eq. (15). The predicted behavior is in agreement with experimental results [11]. That is, at low shear rate, the functions increase monotonously and at higher values of γ˙ , stress overshoots appear. The overshoot occurs earlier and with a larger magnitude at higher shear rate. The quantitative comparisons between the model predictions and the LDPE I melt data are shown in Fig. 4. The model parameters are those obtained by fitting the steady shear data in Fig. 1. The model correctly predicts the small overshoot and the η+ data at large t but underestimates the data at small t (see

Fig. 4. Stress growth (η+ ) and primary normal stress growth (ψ1+ ) vs. time for the LDPE I melt data: (䊉) η+ ; (䊏) ψ1+ (data from [43] adapted from [39]); (—) model predictions.

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Fig. 5. Stress growth function η+ vs. time for the 2 wt.% polyacrylamide solution: (—) η+ data; (– – –) model predictions.

Fig. 4). We believe this to be due to the fact that for the shear rate of the test (1 s−1 ) our model draws on one value of λ(γ˙ ) and on one value of ηm (γ˙ ). That is to say: if the behavior at large t is quantitatively correct, we can expect deviations at small t, since the model parameters are those of the steady-state measurements of Fig. 1. Nevertheless, the model predictions are satisfactory and do not warrant an increase in the number of model parameters, which would result by considering, say a two-mode version of the model. Acceptable predictions are obtained for the ψ1+ LDPE data. Fig. 5 shows the model predictions on the stress growth function for the 2 wt.% polyacrylamide solution. Again, the model parameters are those of Fig. 1, and yield reasonable stress growth predictions. Oscillations at high shear rate values are predicted due to the trigonometric functions in Eq. (15a), i.e. the exponential term does not dampen fast enough. Similar behavior is also predicted via modified dumbbell models [16–18]. Note that, the theoretical curves of η+ (t) do not start from the origin due to the internal viscosity. There is a stress jump at the startup of the flow. This discontinuity is much more difficult to measure than the one at cessation of flow and no data are available yet. On cessation of steady simple shear flow, the Gordon–Schowalter derivative reduces to the partial derivative and the model yields an exponential decay function with a relaxation time λ(γ˙0 ). However, we may postulate that, once the flow stops, λ(γ˙0 ) will be recovering to the value λ0 via Brownian motion. That is to say, the value of λ on cessation of flow is time dependent. This can yield improved predictions of the stress relaxation. 3.4. Small amplitude oscillatory flow In the linear limit, Eq. (9) yield the following for the dynamic viscosity (η0 ) and the storage modulus (G0 ): ηm , (16a) η0 = 1 + λ2 ω2

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Fig. 6. Viscosity and primary normal stress coefficient vs. shear rate and dynamic viscosity and storage modulus vs. frequency for a 3 wt.% polyacrylamide solution: (䊉) η; (䊏) ψ 1 (䉱) η0 ; (䉲) G0 . Model predictions (—) of Eqs. (10) and (16) are η0 = 2.82 × 103 Pa s, b = 19.4, m = 0.75, λ0 = 210 s, c = 94.2, n = 0.47, ε0 = 0.859 and ξ = 0.128.

G0 =

ηm λω2 . 1 + λ2 ω2

(16b)

These equations calculated from our model are identical in form to the Maxwell model predictions except that, λ and ηm in Eq. (16) are functions of an independent macroscopic variable. The Maxwell model contains only one relaxation time and it is quantitatively inadequate to describe the linear viscoelasticity of polymeric materials. A relaxation spectrum is desired when comparing with experimental results. Inspired by the Cox–Merz relation, we propose a heuristic extension by replacing γ˙ with ω in the formulas of λ and ηm as in Eq. (11). Fig. 6 shows η0 and G0 data for a 3 wt.% solution of polyacrylamide in a 50 wt.% mixture of water and glycerin, as well as the data of the shear viscosity and primary normal stress coefficient. Eqs. (10) and (16) are shown to provide excellent predictions of these material functions, without the need to introduce multiple modes. That is, without introducing extra model parameters.

4. Conclusions The new constitutive equation developed from a modified network model with internal viscosity (IV) is able to predict a variety of material functions such as the non-Newtonian viscosity (η), primary normal stress coefficient (ψ 1 ), dynamic viscosity (η0 ), storage modulus (G0 ), stress growth (η+ ), normal stress growth (ψ1+ ), stress relaxation (η− ) and normal stress relaxation (ψ1− ), as well as the stress jump phenomenon, in terms of a reasonable number of model parameters. The internal viscosity coefficient is assumed to decrease with increasing shear rate in that the friction of the segments of a polymer chain is less likely to occur at high shear. The ad hoc choice of the IV formula in this paper ensures a second Newtonian region at high shear rate. The model predictions are successfully compared with experimental data on a LDPE I melt, on xanthan gum solutions and on polyacrylamide solutions.

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The shear rate dependent formulation of the internal parameter gives rise to some interesting results. At low shear rate, the elastic component dominates the viscous component. The reverse is predicted at high shear rate. If ε varies with γ˙ as ε ∝ γ˙ −i with i < m − n, the model predicts shear-thickening following the power law shear-thinning region. That is to say, the dependence of the intramolecular friction on the macroscopic flow determines the material behavior at high shear rate. This could be interpreted either as an increase in the friction of segments and/or as an entrapment of dangling segments into the active network [6], increasing the number of junctions (or segments), with a corresponding increase in viscosity. The choice of the rate of deformation as the dependent variable was discussed by De Kee and Wissbrun [40]. Rajagopal and Srinivasa [41] provide a thermodynamic framework for such rate type models. We do not claim that our equations are valid for all flows. That is, we seek a local equilibrium rather than a global equilibrium. Leonov [42] lists a variety of equations that are dissipative and/or Hadamard unstable. Nature is not always globally stable. For example, the Oldroyd-B model may not be stable but is nonetheless useful. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

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