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A 2D model for tube orientation and tube squeezing in fast flows of polymer melts
J. Non-Newtonian Fluid Mech. 128 (2005) 42–49
A 2D model for tube orientation and tube squeezing in fast flows of polymer melts Giuseppe Marrucci ∗ , Giovanni Ianniruberto Universit`a Federico II, Piazzale Tecchio 80, I-80125 Napoli, Italy Received 15 October 2004; received in revised form 17 January 2005; accepted 21 January 2005
Abstract Motivated by recent data of Hassager and coworkers [A. Bach, K. Almdal, H.K. Rasmussen, O. Hassager, Elongational viscosity of narrow molar mass distribution polystyrene, Macromolecules, 36 (2003) 5174–5179], we develop a new tube model describing the non-linear behaviour of entangled monodisperse linear polymers. Within the context of well established tube theories, the model accounts for the effect of flow on tube diameter, thus somehow picking up a long standing suggestion by Wagner and Schaeffer [M.H. Wagner, J. Schaeffer, Non-linear strain measures for general biaxial extension of polymer melts, J. Rheol., 32 (1992) 1–26] among others. Since tubes with a deformation dependent cross section are rather difficult to deal with, we here limit model development to an artificial two-dimensional situation. The simple 2D model has the advantage of illustrating the new physical assumptions more transparently, and it already proves to correctly predict most qualitative features typically shown by shear and elongational data in the non-linear range. © 2005 Elsevier B.V. All rights reserved. Keywords: Non-linear rheology; Entangled polymers; Tube squeezing; Chain stretching
1. Introduction The rheological behaviour of polymeric liquids in the entangled state does not seem to be fully described, especially for what concerns the non-linear response in fast flows. We have recently shown [1] that new, accurate data of elongational viscosity on monodisperse polystyrene melts by Hassager and coworkers [2] cannot be interpreted within the available models [3–5]. Further improvement of the theory for entangled polymers appears to be required. Dynamic theories for entangled polymers typically neglect the repulsive, excluded-volume interactions among chain segments. Of course, entanglements resulting from chain uncrossability are accounted for, but here we refer to the direct dynamic effect of interchain interactions. For example, in the expression for the stress tensor, the classical theory only considers traction along the “primitive path” (or “tube” axis) of the entangled chains [6], i.e., it only accounts for intrachain forces. Interchain repulsive interactions arise ∗
Corresponding author. E-mail address: [email protected] (G. Marrucci).
0377-0257/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2005.01.007
from local, very fast bumping events, and the usual assumption is that they only contribute an irrelevant isotropic term to the stress tensor. Conversely, we tend to believe that the interchain repulsions may well contribute to the stress tensor and, more generally, to the long-time dynamics insofar as the tube conformation is long lived, and hence those bumping events remain self-correlated in time. Evidence for the relevance of interchain repulsive interactions can be found in atomistic and coarse-grained simulations of polymer melts. Indeed, the atomistic simulations of C100 H202 by Moore et al. [7] show that interchain and intrachain contributions to viscosity are comparable in magnitude. Also in the recent coarse-grained simulations of fully entangled polyethylene melts (up to C800 H1602 ) by Padding and Briels [8], the collision events between chains attempting (in vain) to cross each other are found to contribute significantly to the stress tensor. (The reader should not be mislead by the sentence at the bottom of p. 10278 of [8] stating that “the stress is dominated by interactions between bonded blobs” since, as explained by the authors immediately thereafter, the interchain repulsive collisions are accounted for in the model by introducing an extra-term in the intrachain tension.)
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A simple way of accounting for interchain repulsive interactions is through the concept of “pressure” against the “box” of surrounding chains, i.e., against the tube “wall”. The formula for such pressure is a classical one, and is in fact reported in [6]. In our previous work [1], we have shown that by considering the dynamical effect of the lateral pressure exerted by the chains on their surroundings, the data reported in [2] can indeed be interpreted. However, the model developed in [1] only applies to the special situation of steady elongational flows, and it appears urgent, therefore, to extend the model to the general case so as to verify its predictions against the important case of steady shear flows, as well as of transient situations. In this work, although we formulate the model for arbitrary flows, we also introduce a non-trivial limitation. This is dictated by the following complication. In the non-linear situation, i.e., when anisotropy sets in, also the interchain interactions are expected to become anisotropic. The cage surrounding a chain segment looses the axial symmetry around the primitive path or, in other words, the tube cross section is no longer circular. Now, dealing with the dynamics of chains constrained by tubes with elliptical cross sections is no simple matter; hence we simplify the problem in this work by restricting our model to a fictitious two-dimensional world. For such a simplified problem, the variables describing the tube constraint reduce to two only: an angle, θ, specifying the local tube orientation, and a tube diameter, a, measuring the lateral constraint. Notice that we also ignore the influence of the coordinate along the chain, i.e., our model belongs to the so-called “single segment” theories. Reduction to two dimensions temporarily avoids the complication of tubes with non-circular cross sections. It is worth recalling that elliptical cross sections have already been examined in [9], but only with regard to the expression for the stress tensor, i.e., without considering the full dynamic evolution. In that work, among other things, it was noted that the interchain repulsive interactions (i.e., the pressure against the tube wall) also affect the numerical front factor of the plateau modulus, which then increases over the value obtained by neglecting them. The prediction that the modulus increases due to interchain interactions can perhaps help explaining recent results by Likhtman on polyethylene and polyethylene phthalate [10,11]. The paper is organized as follows. In the next section, we summarize our previous results [1], which were aimed at interpreting Hassager’s steady-state data of extensional flows, fast enough to reach full orientation of the primitive chains. In the same section, we further clarify the relevant aspects of entangled polymer dynamics introduced in [1]. In Section 3, we write the equations of the 2D dynamics for the general case when tubes change in time their orientation as well as their size. In Section 4, numerical results for shear and extensional flows are reported, which appear to show all relevant qualitative features typically observed in the experiments. Final comments and perspectives of future work conclude the paper.
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2. Model ingredients The main novelties introduced in our previous paper [1] are: (i) the pressure effect due to interchain repulsive interactions, (ii) the resulting dynamics of tube diameter under flow, and (iii) the characteristic time of tube diameter relaxation. In this section, these items are briefly reviewed, and further commented upon for better clarity. 2.1. Pressure on tube wall Within the tube model for entangled polymers, it appears natural to represent interchain repulsive interactions through the thermal pressure p on the tube wall. Classical kinetic theory [6] then provides the result [1]: p ≈ kT
Nb2 a4 L
(1)
where kT is the thermal energy, N and b the number and length of monomers (actually Kuhn segments), respectively, a the tube diameter, and L is the length of the tube segment (assumed to be straight) containing those monomers. The near-equality sign in Eq. (1) means that the numerical coefficient of order unity is ignored. From the pressure of Eq. (1), the force F pushing against the wall of the tube segment is obtained as: F ≈ kT
Nb2 a3
(2)
Notice the sensitivity of this force to the tube diameter a, which appears as a3 . It is understood that the kinetic pressure of Eq. (1) (or the corresponding force of Eq. (2)) is balanced at equilibrium, i.e., when the tube diameter is a0 . Conversely, if flow modifies the tube diameter, a driving force towards equilibrium arises, which we assume to be given by: 1 1 F ≈ kTNb2 (3) − 3 a3 a0 2.2. Dynamics of tube diameter in fast elongational flows In our previous work [1], we examined the limiting case, depicted in Fig. 1, where tubes are fully aligned to the elongation direction of the flow. As shown in the figure, tubes are
Fig. 1. A tube aligned to the extension direction of an uniaxial elongational flow. Flow from the side effectively squeezes the tube.
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G. Marrucci, G. Ianniruberto / J. Non-Newtonian Fluid Mech. 128 (2005) 42–49
Fig. 2. The chain in the cage of topological constraints (tube). Fluctuations of blob size occur over all time scales. The tube geometry fluctuations considered here occur over the Rouse time of the surrounding chains. Over such a time, the neighboring chains permanently accommodate the disturbance rather than elastically pushing the blob back.
systematically squeezed by the incoming flow. By assuming affine deformation, flow generates a rate of diameter reduction given by:
in proportion with ε˙ 1/2 , in full agreement with Hassager’s data [2].
a˙ affine ≈ −˙εa
2.3. The new relaxation time
(4)
where ε˙ is the extension rate of the flow. To find the steady state situation for any given ε˙ , we equate the force of Eq. (3) to the friction force ζ a˙ affine , i.e., 1 1 kTNb2 (5) − 3 ≈ ζ ε˙ a a3 a0 Eq. (5) predicts that a decreases with increasing ε˙ , soon approaching the asymptotic power law a ∝ ε˙ −1/4 . To calculate the elongational stress in the well aligned case considered in Fig. 1, one should note that, while a decreases, the tube length L correspondingly increases so as to maintain equilibrium in the monomer density along the tube, the equilibrium relationship between a and L being aL = Nb2 [6]. Assuming monomer density equilibrium along the tube implies that the longitudinal friction is ineffective in stretching the chain, i.e., that ε˙ (though large enough to orient the tubes) remains smaller than the reciprocal Rouse time τ R of the chain. Under these conditions, the elongational stress σ is given by [1]: σ ≈ νkT
Nb2 a2
(6)
where ν is number of chains per unit volume. In view of the scaling law for a, the elongational stress is predicted to scale
The force balance of Eq. (5) describes a steady state. If, conversely, a transient situation is considered, a new time constant τ p naturally emerges from the dynamics of the tube diameter. The new time is strictly related to the lateral friction coefficient ζ of Eq. (5) in the following way [1]: ζa04 a02 ≈ τR (7) kTNb2 b2 The last equality in Eq. (7) comes out by arguing on the meaning of the lateral friction coefficient ζ. Such a friction is related to the process schematically depicted in Fig. 2, where a chain blob fluctuates in size while the surrounding chains relax the corresponding disturbance through a longitudinal retraction within their own tubes. It is through such a mechanism that the Rouse time τ R of the chains appears in the picture. Eq. (7) thus shows that the relaxation time τ p of the tube diameter scales with the polymer molecular mass M like a Rouse time, and yet is much larger than τ R because of the factor a02 /b2 . The predicted proportionality between τ p and M2 also is in good agreement with Hassager’s data. Notice that the relaxation mechanism of the tube diameter depicted in Fig. 2 is not expected to affect the tube orientation. The latter remains determined by the entanglement topology, which can only change through the well known mechanisms of reptation, constraint release, and tube length fluctuations.
τp =
Fig. 3. An elongational step strain along a tube segment. After the affine deformation, relaxation starts. For the case τ d /τ p > 1, the tube geometry relaxes first (up to time τ p ). Next, the orientational relaxation takes place (in the disengagement time τ d ).
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In other words, the relaxation process occurring over the τ p timescale (i.e., with the lateral friction coefficient ζ/N) affects the “geometry” of the tube segments, not the topology of the chain network. To better explain our view of the relevant physics, let us first define subchains and tube segments on the basis of a fixed number of monomers (specifically, those contained at equilibrium between consecutive entanglements). With this choice the number of tube segments remains constant also under non-equilibrium conditions, while only the tube geometry changes. The evolution of the geometry of a tube segment after a step strain is then schematically illustrated in Fig. 3. It is there emphasized that after the affine deformation (which makes diameter and length to change in opposite directions) the aspect ratio (or geometry) of the tube segment relaxes in the course of time because of two mechanisms: (i) a fast Rouse-like retraction (within τ R ) and (ii) a much slower effect of the interchain pressure (within τ p ). Finally, orientation relaxes within τ d . Of course, in this example we have assumed that τ d > τ p . Should the opposite be true, orientation and geometry would relax simultaneously within τ d (except for the initial Rouse process). Note finally that constraint release induced by the Rouse retraction process (ignored in Fig. 3) further complicates the picture.
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However, with a frequency equal to the inverse orientational relaxation time τ (to be specified below) θ is randomized. In other words, at each time step t, the orientation θ is given the probability t/τ to randomly renew. The orientational relaxation time τ accounts for reptation (through the disengagement time τ d ) as well as for constraint release, both thermal and convective, in the following way: 1 da −1 −1 −1 τ = τd + β τd + k : uu + (9) a dt where the first term in bracket accounts for thermal constraint release, and the last two terms for CCR. The unknown numerical coefficient β weighs the constraint release importance. As for the last two terms, the scalar product k: uu describes in the usual way the average advective lengthening of the tube (with k the velocity gradient of the flow, and u a unit vector indicating local tube orientation). The last term, usually written as −1/L(dL/dt) (with L the average tube length), is justified by the equilibrium relationship aL = constant (already mentioned in Section 2.2), holding true in view of the assumption that flow is slower than τR−1 . The term 1/a(da/dt) in Eq. (9) couples the θ-dynamics to the variable a. We now move on to the dynamics of the tube diameter a, which also involves several contributions. We write the rate of change of a (made non-dimensional by taking the ratio to a0 ) in the following way:
3. The 2D model
a˙ = a˙ affine + τp−1 (a−3 − 1) + fa
The model summarized in the previous section is limited to the particular situation of tubes fully oriented in the elongational direction [1]. We here extend the model to arbitrary situations, in which tube segments are variously oriented, and variously deformed. In the three dimensions of the real world, the variables describing the tube orientation and geometry are numerous; therefore, in the present work, we start approaching this problem by concentrating on a simplified 2D analysis. Further simplifications here adopted are as follows: (i) we assume that the Rouse time τ R is effectively zero, though τ p (which is a large multiple of τ R ) is not, and is in fact very important. In other words, we neglect the possibility that the chain is stretched because of the longitudinal friction along the tube, hence limiting our analysis to flows slower than τR−1 . However, chains do not generally remain unstretched, because in fact tube squeezing will occur, and tube length will correspondingly increase. (ii) We neglect any dependence on position along the chain; hence we ignore chain end fluctuations, and higher modes of reptation (so-called single segment, or toy, model). As anticipated in the Introduction, with these simplifications the model only requires two state variables: the tubesegment orientation θ and diameter a. Consistently with the classical Doi–Edwards theory, the orientation θ is made to change affinely:
where the first two terms on the right hand side correspond to those appearing in Eq. (5) when accounting for the definition of τ p as in Eq. (7). The last term, fa , is a random Gaussian term, with the following moments:
θ˙ = θ˙ affine
(8)
fa = 0;
fa (t)fa (t ) = 2τp−1 δ(t − t )
(10)
(11)
where δ (. . .) is the Dirac delta function and, in view of the non-dimensionality of a, the reciprocal time τp−1 plays the role of a diffusion coefficient. Close to equilibrium, the relaxation process described in the Langevin form, Eqs. (10) and (11), can equivalently be written in the form of a Smoluchowski equation for the distribution function ψ(a) ∂ψ ∂ψ ∂ − − + (a−3 − 1)ψ (12) = τp−1 ∂a ∂a ∂t leading to the following equilibrium distribution 1 ψ0 (a) = C exp − 2 − a 2a
(13)
with C the normalization constant also shown in Fig. 4. Notice that a = 1 (i.e., a = a0 ) is the most probable value, while the average value of a is somewhat larger than unity. While the equilibrium a distribution is independent of θ, under non-equilibrium conditions the tube diameter is coupled to orientation through the affine term of Eq. (10), which is θ-dependent (as specified below). Furthermore, relaxation
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G. Marrucci, G. Ianniruberto / J. Non-Newtonian Fluid Mech. 128 (2005) 42–49
Here, θ = 0 defines the shear or the extension direction, respectively. We conclude this section by indicating the relevant average for stress calculations. Before doing so, we recall that in the conventional approach [3–6], and in view of the fact that the velocity gradient of the flow is assumed to be smaller than the reciprocal Rouse time, no chain stretching is expected, i.e., the stress is purely orientational. Hence, the relevant average would simply be uu. In the present model, however, chain stretch does occur as a consequence of tube squeezing, and therefore the stress is given by an average involving both tube orientation and tube diameter. For the (non-dimensional) stress T, we will then use the form: uu (16) T= 2 a Fig. 4. Equilibrium distribution of tube diameters.
which generalizes Eq. (6) in a simple way.
of the tube diameter towards equilibrium occurs in two ways. One is through the “lateral” dynamics described by Eqs. (10) and (11). The second is through the “longitudinal” dynamics due to reptation. Indeed, whenever a segment randomizes its orientation θ because of reptation (an event occurring at each time step with probability t/τ d ), it also randomizes its diameter a by picking a new value according to the equilibrium distribution ψ0 (a). Notice that, conversely, when an “internal” tube segment randomizes its orientation because of constraint release (an event occurring with probability t/τ –t/τ d ), the tube diameter is assumed not to change. The fact that the tube diameter can go back to equilibrium in two ways, i.e., either in a time τ p by pushing back the surrounding constraints or in a time τ d by reptating out, has important consequences. Indeed, depending on the ratio τ d /τ p , the transition from the linear behaviour to the asymptotic power law mentioned in Section 2.2 occurs in a qualitatively different way. Such a prediction, anticipated in our previous work [1] and shown in the next section, has been recently confirmed by new data of Hassager on polystyrene melts of smaller molecular mass [12]. In the following section, we will present numerical results obtained with the above described dynamical rules for the case of shear and elongational flows, both at steady state and during start up. The flow dependent quantities appearing in the dynamical equations are given by:
4. Numerical results The non-dimensional constitutive parameters of the model are only two, namely the ratio of relaxation times τ d /τ p , and the constraint release effectiveness β. Consistently with the double reptation picture, the latter parameter is fixed at the value β = 1, unless specified otherwise. The ratio τ d /τ p is expected to increase roughly as M1.3/1.6 . For the case of polystyrene melts, the analysis reported in [1] indicates that for M = 200,000 the ratio τ d /τ p is nearly unity. We will therefore examine values of that ratio both above and below unity. Concerning numerical aspects, we used Euler integration in time, and in most cases a population of 30,000 tube segments proved sufficient to obtain good statistics. In slow flows, noise was reduced by using a simple variance reduction technique. In all cases, error bars are within a few percent. In line with our previous paper [1], Fig. 5 shows results for steady elongational flows in the form of (non-dimensional)
Shear flows: θ˙ affine = −γ˙ sin2 θ
˙ sin θ cos θ a˙ affine = −γa
˙ k : uu = γsin θ cos θ
(14)
Elongational flows: θ˙ affine = −˙ε sin(2θ) k : uu = ε˙ cos(2θ)
a˙ affine = −˙εa cos(2θ) (15)
Fig. 5. Non-dimensional elongational stress vs. ε˙ τp for several values of the time ratio τ d /τ p , corresponding to different M. From right to left τ d /τ p = 0.03, 0.1, 1, 10, 100.
G. Marrucci, G. Ianniruberto / J. Non-Newtonian Fluid Mech. 128 (2005) 42–49
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Fig. 6. Shear stress and normal stress difference in steady shear flows for τ d /τ p = 1.
Fig. 8. Tangential and normal components of tensor uu in steady shear flows for τ d /τ p = 1.
tensile stress σ versus ε˙ τp for several values of τ d /τ p , i.e., for various M. These results show the same qualitative features anticipated in Fig. 4 of our previous work [1], and confirmed by the more recent data of Hassager [12] (see his Fig. 2). Notice in particular the two rightmost curves in Fig. 5 for low M’s, which exhibit an elongational viscosity higher than the Trouton value in the transition from the linear range to the asymptotic power law σ ≈ (˙ετp )1/2 . Fig. 6 reports steady shear results for tangential and normal stresses for the case τ d /τ p = 1. Notice that a quasi-plateau for the tangential component is found before yielding towards the asymptotic 1/2 power law. It is noteworthy that such a plateau-like region is found in shear, while (for the same values of parameters) in elongational flow the transition from linear to asymptotic behaviour occurs rather abruptly (middle curve in Fig. 5). This difference between the two flows can be understood by examining what happens to the tube diameter a in the two cases. Fig. 7 indeed shows that the elongational flow is more effective than shear in squeezing the tubes. In its turn, this difference in behaviour is related to the well known fact that elongational flow, differently from shear, is
irrotational. In a shear flow, because of rotation, tubes get squeezed for 0 < θ < π/2 to enlarge again for −π/2 < θ < 0, while in elongational flow tubes permanently align around θ = 0 and get continuously squeezed. Going back to Fig. 6, it is also worth noting the difference between normal and shear stresses in the non-linear range. Differently from the tangential component, N1 does not show ˙ any curvature change, and keeps rising with a power-law in γ. To understand this behaviour, it is useful to consider the orientational average uu, the tangential and normal components of which are reported in Fig. 8. The tangential component goes though a maximum and then decreases towards its CCR plateau, while the normal component keeps increasing before reaching saturation. The difference in Fig. 6 then reflects that in Fig. 8, as modified by the effect of a2 in the denominator of the average uu/a2 : the sloping down of the tangential component in Fig. 8 becomes the quasi-plateau of the flow curve in Fig. 6, and the moderate increase of the normal component in Fig. 8 becomes the sustained growth of N1 in Fig. 6. It is noteworthy that experiments systematically indicate the behaviour shown in Fig. 6, rather than that of Fig. 8 pre-
Fig. 7. Average tube diameter in steady shear and elongational flows for τ d /τ p = 1.
Fig. 9. Same as Fig. 6 but for a larger value of the time ratio τ d /τ p , i.e., for higher M.
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˙ d = 1, 3, 10, 30, 100, from top to Fig. 10. Curves of transient viscosity (a) and normal stress coefficient (b) in shear start-up for τ d /τ p = 10. Shear rates are γτ bottom.
dicted by orientation alone. In other words, data seem to support the concept that chain stretch is significant throughout the non-linear range, and not merely when the reciprocal Rouse time is approached. Tube squeezing might then be the explanation for premature chain stretching, as suggested by Wagner and coworkers [13,14]. Fig. 9 is the analogous of Fig. 6 for τ d /τ p = 10, i.e., for a larger value of M. The plateau-like region for the shear stress is here more pronounced, and a slight inflection also appears in the normal stress curve. Altogether, however, the qualitative appearance is not very different at higher M. Moving now to transient flows, Fig. 10 reports typical results of tangential and normal stresses in start up of shear. The tangential component exhibits overshoots when the shear ˙ d∼ rate reaches the value γτ = 1, as typically shown by data. As regards the normal component, overshoots in Fig. 10 are completely suppressed by CCR. They appear in the model only when the CCR effectiveness is reduced by choosing β values smaller than unity. Fig. 11 shows that with β = 0.5 (instead of β = 1 as considered so far) overshoots start appearing also in the normal stress component. Even smaller
Fig. 12. Curves of transient viscosity in elongational start-up for τ d /τ p = 1. Elongation rates are ε˙ τd = 0.1 (linear regime), 10, 100, 1000, from top to bottom.
values of β, however, while generating more pronounced overshoots, would also produce unpleasant instabilities. We postpone further analysis of these aspects to a more complete model. Finally, Fig. 12 shows the transient viscosity in start up of elongational flows for several values of ε˙ . The qualitative features shown in Fig. 12 are in good agreement with Hassager’s data [2]. Particularly noteworthy is the strain hardening effect. Such a behaviour is predicted (and shown by the data) when the steady elongational viscosity falls in the asymptotic power-law regime of Fig. 5.
5. Conclusions The model presented in this paper appears promising in describing the non-linear behaviour of entangled polymer melts. Let us summarize in the following all the positive features: Fig. 11. Curves of transient normal stress coefficient in shear start-up for τ d /τ p = 10 and β = 0.5. Shear rates as in Fig. 10.
◦ In steady uniaxial elongational flows at high stretching rates, the model predicts the power law observed by Has-
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◦
◦
◦
◦ ◦
sager and coworkers [2]. The M-dependence of the asymptotic power law is also interpreted by the model [1]. Still in steady elongational flows, the transition from the linear Trouton regime to the asymptotic power law is predicted to show different qualitative features depending on M (see Fig. 5). In particular, for moderately entangled melts the model predicts that the elongational viscosity first rises above the Trouton value, and then decreases towards the asymptotic power law. Such an unusual behaviour, anticipated in [1] and confirmed here, has been experimentally found long ago by M¨unstedt [15] as well as very recently by Hassager [12]. In steady shear flows, the model predicts a plateau-like region of the flow curve, which extends well into the nonlinear region, before yielding to the asymptotic power law. The quasi-plateau is a characteristic feature of the experimental flow curves of monodisperse polymers. Still in steady shear flows, the normal stress difference does not show a plateau, and keeps growing with increasing shear rate with a significant slope. Also this feature is in line with observations. Notice in this regard that CCR theories without chain stretch predict that both tangential and normal stresses approach a plateau. Predictions of the model in shear and elongational start up appear to show the correct qualitative features. It is fair to recall, however, that the model remains limited to qualitative aspects, both because of its 2D formulation, and for the nature of “single segment” toy model. Work is in progress to remove both simplifications.
Acknowledgments One of the authors (G.I.) acknowledges financial support from the Campania regional government (Regional Bill no. 5 dated 28.05.2002). The authors are very grateful to the anonymous referees for their stimulating comments leading to significant improvements of the paper.
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References [1] G. Marrucci, G. Ianniruberto, Interchain pressure effect in extensional flows of entangled polymers, Macromolecules 36 (2004) 3934–3942. [2] A. Bach, K. Almdal, H.K. Rasmussen, O. Hassager, Elongational viscosity of narrow molar mass distribution polystyrene, Macromolecules 36 (2003) 5174–5179. [3] D.W. Mead, R.G. Larson, M. Doi, A molecular theory for fast flows of entangled polymers, Macromolecules 31 (1998) 7895–7914. [4] G. Marrucci, G. Ianniruberto, Flow-induced orientation and stretching of entangled polymers, Phil. Trans. R. Soc. A 361 (2003) 677–688. [5] R.S. Graham, A.E. Likhtman, T.C.B. McLeish, S.T. Milner, Microscopic theory of linear, entangled polymer chains under rapid deformation including chain stretch and convective constraint release, J. Rheol. 47 (2003) 1171–1200. [6] M. Doi, S.F. Edwards, The Theory of Polymer Dynamics, Clarendon Press, Oxford, 1986. [7] J.D. Moore, S.T. Cui, H.D. Cochran, P.T. Cummings, Transient rheology of a polyethylene melt under shear, Phys. Rev. E 60 (1999) 6956–6959. [8] J.T. Padding, W.J. Briels, Coarse-grained molecular dynamics simulations of polymer melts in transient and steady shear flow, J. Chem. Phys. 118 (2003) 10276–10286. [9] G. Ianniruberto, G. Marrucci, Stress tensor and stress-optical law in entangled polymers, J. Non-Newtonian Fluid Mech. 79 (1998) 225–234. [10] A.E. Likhtman, Problems of the standard tube theory and new sliplink model, in: Proceedings of Yonezawa Rheology Workshop 2004Reconstruction of Polymer Rheology, Yonezawa (Japan), August 20–21, 2004, p. 135. [11] A.E. Likhtman, Neutron spin-echo leads to the new model of entanglements, in: Proceedings of the XIVth International Congress on Rheology, Seoul (Korea), August 22–27, 2004, p. MS04-1. [12] O. Hassager, Polymer fluid mechanics: molecular orientation and stretching, in: Proceedings of XIVth International Congress on Rheology, Seoul (Korea), August 22–27, 2004, p. NF01-1. [13] M.H. Wagner, J. Schaeffer, Nonlinear strain measures for general biaxial extension of polymer melts, J. Rheol. 32 (1992) 1–26. [14] M.H. Wagner, P. Rubio, H. Bastian, The molecular stress function model for polydisperse polymer melts with dissipative convective constraint release, J. Rheol. 45 (2001) 1387–1412. [15] H. M¨unstedt, Dependence of the elongational behaviour of polystyrene melts on molecular weight and molecular weight distribution, J. Rheol. 24 (1980) 847–867.
A 2D model for tube orientation and tube squeezing in fast flows of polymer melts Giuseppe Marrucci ∗ , Giovanni Ianniruberto Universit`a Federico II, Piazzale Tecchio 80, I-80125 Napoli, Italy Received 15 October 2004; received in revised form 17 January 2005; accepted 21 January 2005
Abstract Motivated by recent data of Hassager and coworkers [A. Bach, K. Almdal, H.K. Rasmussen, O. Hassager, Elongational viscosity of narrow molar mass distribution polystyrene, Macromolecules, 36 (2003) 5174–5179], we develop a new tube model describing the non-linear behaviour of entangled monodisperse linear polymers. Within the context of well established tube theories, the model accounts for the effect of flow on tube diameter, thus somehow picking up a long standing suggestion by Wagner and Schaeffer [M.H. Wagner, J. Schaeffer, Non-linear strain measures for general biaxial extension of polymer melts, J. Rheol., 32 (1992) 1–26] among others. Since tubes with a deformation dependent cross section are rather difficult to deal with, we here limit model development to an artificial two-dimensional situation. The simple 2D model has the advantage of illustrating the new physical assumptions more transparently, and it already proves to correctly predict most qualitative features typically shown by shear and elongational data in the non-linear range. © 2005 Elsevier B.V. All rights reserved. Keywords: Non-linear rheology; Entangled polymers; Tube squeezing; Chain stretching
1. Introduction The rheological behaviour of polymeric liquids in the entangled state does not seem to be fully described, especially for what concerns the non-linear response in fast flows. We have recently shown [1] that new, accurate data of elongational viscosity on monodisperse polystyrene melts by Hassager and coworkers [2] cannot be interpreted within the available models [3–5]. Further improvement of the theory for entangled polymers appears to be required. Dynamic theories for entangled polymers typically neglect the repulsive, excluded-volume interactions among chain segments. Of course, entanglements resulting from chain uncrossability are accounted for, but here we refer to the direct dynamic effect of interchain interactions. For example, in the expression for the stress tensor, the classical theory only considers traction along the “primitive path” (or “tube” axis) of the entangled chains [6], i.e., it only accounts for intrachain forces. Interchain repulsive interactions arise ∗
Corresponding author. E-mail address: [email protected] (G. Marrucci).
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from local, very fast bumping events, and the usual assumption is that they only contribute an irrelevant isotropic term to the stress tensor. Conversely, we tend to believe that the interchain repulsions may well contribute to the stress tensor and, more generally, to the long-time dynamics insofar as the tube conformation is long lived, and hence those bumping events remain self-correlated in time. Evidence for the relevance of interchain repulsive interactions can be found in atomistic and coarse-grained simulations of polymer melts. Indeed, the atomistic simulations of C100 H202 by Moore et al. [7] show that interchain and intrachain contributions to viscosity are comparable in magnitude. Also in the recent coarse-grained simulations of fully entangled polyethylene melts (up to C800 H1602 ) by Padding and Briels [8], the collision events between chains attempting (in vain) to cross each other are found to contribute significantly to the stress tensor. (The reader should not be mislead by the sentence at the bottom of p. 10278 of [8] stating that “the stress is dominated by interactions between bonded blobs” since, as explained by the authors immediately thereafter, the interchain repulsive collisions are accounted for in the model by introducing an extra-term in the intrachain tension.)
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A simple way of accounting for interchain repulsive interactions is through the concept of “pressure” against the “box” of surrounding chains, i.e., against the tube “wall”. The formula for such pressure is a classical one, and is in fact reported in [6]. In our previous work [1], we have shown that by considering the dynamical effect of the lateral pressure exerted by the chains on their surroundings, the data reported in [2] can indeed be interpreted. However, the model developed in [1] only applies to the special situation of steady elongational flows, and it appears urgent, therefore, to extend the model to the general case so as to verify its predictions against the important case of steady shear flows, as well as of transient situations. In this work, although we formulate the model for arbitrary flows, we also introduce a non-trivial limitation. This is dictated by the following complication. In the non-linear situation, i.e., when anisotropy sets in, also the interchain interactions are expected to become anisotropic. The cage surrounding a chain segment looses the axial symmetry around the primitive path or, in other words, the tube cross section is no longer circular. Now, dealing with the dynamics of chains constrained by tubes with elliptical cross sections is no simple matter; hence we simplify the problem in this work by restricting our model to a fictitious two-dimensional world. For such a simplified problem, the variables describing the tube constraint reduce to two only: an angle, θ, specifying the local tube orientation, and a tube diameter, a, measuring the lateral constraint. Notice that we also ignore the influence of the coordinate along the chain, i.e., our model belongs to the so-called “single segment” theories. Reduction to two dimensions temporarily avoids the complication of tubes with non-circular cross sections. It is worth recalling that elliptical cross sections have already been examined in [9], but only with regard to the expression for the stress tensor, i.e., without considering the full dynamic evolution. In that work, among other things, it was noted that the interchain repulsive interactions (i.e., the pressure against the tube wall) also affect the numerical front factor of the plateau modulus, which then increases over the value obtained by neglecting them. The prediction that the modulus increases due to interchain interactions can perhaps help explaining recent results by Likhtman on polyethylene and polyethylene phthalate [10,11]. The paper is organized as follows. In the next section, we summarize our previous results [1], which were aimed at interpreting Hassager’s steady-state data of extensional flows, fast enough to reach full orientation of the primitive chains. In the same section, we further clarify the relevant aspects of entangled polymer dynamics introduced in [1]. In Section 3, we write the equations of the 2D dynamics for the general case when tubes change in time their orientation as well as their size. In Section 4, numerical results for shear and extensional flows are reported, which appear to show all relevant qualitative features typically observed in the experiments. Final comments and perspectives of future work conclude the paper.
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2. Model ingredients The main novelties introduced in our previous paper [1] are: (i) the pressure effect due to interchain repulsive interactions, (ii) the resulting dynamics of tube diameter under flow, and (iii) the characteristic time of tube diameter relaxation. In this section, these items are briefly reviewed, and further commented upon for better clarity. 2.1. Pressure on tube wall Within the tube model for entangled polymers, it appears natural to represent interchain repulsive interactions through the thermal pressure p on the tube wall. Classical kinetic theory [6] then provides the result [1]: p ≈ kT
Nb2 a4 L
(1)
where kT is the thermal energy, N and b the number and length of monomers (actually Kuhn segments), respectively, a the tube diameter, and L is the length of the tube segment (assumed to be straight) containing those monomers. The near-equality sign in Eq. (1) means that the numerical coefficient of order unity is ignored. From the pressure of Eq. (1), the force F pushing against the wall of the tube segment is obtained as: F ≈ kT
Nb2 a3
(2)
Notice the sensitivity of this force to the tube diameter a, which appears as a3 . It is understood that the kinetic pressure of Eq. (1) (or the corresponding force of Eq. (2)) is balanced at equilibrium, i.e., when the tube diameter is a0 . Conversely, if flow modifies the tube diameter, a driving force towards equilibrium arises, which we assume to be given by: 1 1 F ≈ kTNb2 (3) − 3 a3 a0 2.2. Dynamics of tube diameter in fast elongational flows In our previous work [1], we examined the limiting case, depicted in Fig. 1, where tubes are fully aligned to the elongation direction of the flow. As shown in the figure, tubes are
Fig. 1. A tube aligned to the extension direction of an uniaxial elongational flow. Flow from the side effectively squeezes the tube.
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Fig. 2. The chain in the cage of topological constraints (tube). Fluctuations of blob size occur over all time scales. The tube geometry fluctuations considered here occur over the Rouse time of the surrounding chains. Over such a time, the neighboring chains permanently accommodate the disturbance rather than elastically pushing the blob back.
systematically squeezed by the incoming flow. By assuming affine deformation, flow generates a rate of diameter reduction given by:
in proportion with ε˙ 1/2 , in full agreement with Hassager’s data [2].
a˙ affine ≈ −˙εa
2.3. The new relaxation time
(4)
where ε˙ is the extension rate of the flow. To find the steady state situation for any given ε˙ , we equate the force of Eq. (3) to the friction force ζ a˙ affine , i.e., 1 1 kTNb2 (5) − 3 ≈ ζ ε˙ a a3 a0 Eq. (5) predicts that a decreases with increasing ε˙ , soon approaching the asymptotic power law a ∝ ε˙ −1/4 . To calculate the elongational stress in the well aligned case considered in Fig. 1, one should note that, while a decreases, the tube length L correspondingly increases so as to maintain equilibrium in the monomer density along the tube, the equilibrium relationship between a and L being aL = Nb2 [6]. Assuming monomer density equilibrium along the tube implies that the longitudinal friction is ineffective in stretching the chain, i.e., that ε˙ (though large enough to orient the tubes) remains smaller than the reciprocal Rouse time τ R of the chain. Under these conditions, the elongational stress σ is given by [1]: σ ≈ νkT
Nb2 a2
(6)
where ν is number of chains per unit volume. In view of the scaling law for a, the elongational stress is predicted to scale
The force balance of Eq. (5) describes a steady state. If, conversely, a transient situation is considered, a new time constant τ p naturally emerges from the dynamics of the tube diameter. The new time is strictly related to the lateral friction coefficient ζ of Eq. (5) in the following way [1]: ζa04 a02 ≈ τR (7) kTNb2 b2 The last equality in Eq. (7) comes out by arguing on the meaning of the lateral friction coefficient ζ. Such a friction is related to the process schematically depicted in Fig. 2, where a chain blob fluctuates in size while the surrounding chains relax the corresponding disturbance through a longitudinal retraction within their own tubes. It is through such a mechanism that the Rouse time τ R of the chains appears in the picture. Eq. (7) thus shows that the relaxation time τ p of the tube diameter scales with the polymer molecular mass M like a Rouse time, and yet is much larger than τ R because of the factor a02 /b2 . The predicted proportionality between τ p and M2 also is in good agreement with Hassager’s data. Notice that the relaxation mechanism of the tube diameter depicted in Fig. 2 is not expected to affect the tube orientation. The latter remains determined by the entanglement topology, which can only change through the well known mechanisms of reptation, constraint release, and tube length fluctuations.
τp =
Fig. 3. An elongational step strain along a tube segment. After the affine deformation, relaxation starts. For the case τ d /τ p > 1, the tube geometry relaxes first (up to time τ p ). Next, the orientational relaxation takes place (in the disengagement time τ d ).
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In other words, the relaxation process occurring over the τ p timescale (i.e., with the lateral friction coefficient ζ/N) affects the “geometry” of the tube segments, not the topology of the chain network. To better explain our view of the relevant physics, let us first define subchains and tube segments on the basis of a fixed number of monomers (specifically, those contained at equilibrium between consecutive entanglements). With this choice the number of tube segments remains constant also under non-equilibrium conditions, while only the tube geometry changes. The evolution of the geometry of a tube segment after a step strain is then schematically illustrated in Fig. 3. It is there emphasized that after the affine deformation (which makes diameter and length to change in opposite directions) the aspect ratio (or geometry) of the tube segment relaxes in the course of time because of two mechanisms: (i) a fast Rouse-like retraction (within τ R ) and (ii) a much slower effect of the interchain pressure (within τ p ). Finally, orientation relaxes within τ d . Of course, in this example we have assumed that τ d > τ p . Should the opposite be true, orientation and geometry would relax simultaneously within τ d (except for the initial Rouse process). Note finally that constraint release induced by the Rouse retraction process (ignored in Fig. 3) further complicates the picture.
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However, with a frequency equal to the inverse orientational relaxation time τ (to be specified below) θ is randomized. In other words, at each time step t, the orientation θ is given the probability t/τ to randomly renew. The orientational relaxation time τ accounts for reptation (through the disengagement time τ d ) as well as for constraint release, both thermal and convective, in the following way: 1 da −1 −1 −1 τ = τd + β τd + k : uu + (9) a dt where the first term in bracket accounts for thermal constraint release, and the last two terms for CCR. The unknown numerical coefficient β weighs the constraint release importance. As for the last two terms, the scalar product k: uu describes in the usual way the average advective lengthening of the tube (with k the velocity gradient of the flow, and u a unit vector indicating local tube orientation). The last term, usually written as −1/L(dL/dt) (with L the average tube length), is justified by the equilibrium relationship aL = constant (already mentioned in Section 2.2), holding true in view of the assumption that flow is slower than τR−1 . The term 1/a(da/dt) in Eq. (9) couples the θ-dynamics to the variable a. We now move on to the dynamics of the tube diameter a, which also involves several contributions. We write the rate of change of a (made non-dimensional by taking the ratio to a0 ) in the following way:
3. The 2D model
a˙ = a˙ affine + τp−1 (a−3 − 1) + fa
The model summarized in the previous section is limited to the particular situation of tubes fully oriented in the elongational direction [1]. We here extend the model to arbitrary situations, in which tube segments are variously oriented, and variously deformed. In the three dimensions of the real world, the variables describing the tube orientation and geometry are numerous; therefore, in the present work, we start approaching this problem by concentrating on a simplified 2D analysis. Further simplifications here adopted are as follows: (i) we assume that the Rouse time τ R is effectively zero, though τ p (which is a large multiple of τ R ) is not, and is in fact very important. In other words, we neglect the possibility that the chain is stretched because of the longitudinal friction along the tube, hence limiting our analysis to flows slower than τR−1 . However, chains do not generally remain unstretched, because in fact tube squeezing will occur, and tube length will correspondingly increase. (ii) We neglect any dependence on position along the chain; hence we ignore chain end fluctuations, and higher modes of reptation (so-called single segment, or toy, model). As anticipated in the Introduction, with these simplifications the model only requires two state variables: the tubesegment orientation θ and diameter a. Consistently with the classical Doi–Edwards theory, the orientation θ is made to change affinely:
where the first two terms on the right hand side correspond to those appearing in Eq. (5) when accounting for the definition of τ p as in Eq. (7). The last term, fa , is a random Gaussian term, with the following moments:
θ˙ = θ˙ affine
(8)
fa = 0;
fa (t)fa (t ) = 2τp−1 δ(t − t )
(10)
(11)
where δ (. . .) is the Dirac delta function and, in view of the non-dimensionality of a, the reciprocal time τp−1 plays the role of a diffusion coefficient. Close to equilibrium, the relaxation process described in the Langevin form, Eqs. (10) and (11), can equivalently be written in the form of a Smoluchowski equation for the distribution function ψ(a) ∂ψ ∂ψ ∂ − − + (a−3 − 1)ψ (12) = τp−1 ∂a ∂a ∂t leading to the following equilibrium distribution 1 ψ0 (a) = C exp − 2 − a 2a
(13)
with C the normalization constant also shown in Fig. 4. Notice that a = 1 (i.e., a = a0 ) is the most probable value, while the average value of a is somewhat larger than unity. While the equilibrium a distribution is independent of θ, under non-equilibrium conditions the tube diameter is coupled to orientation through the affine term of Eq. (10), which is θ-dependent (as specified below). Furthermore, relaxation
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Here, θ = 0 defines the shear or the extension direction, respectively. We conclude this section by indicating the relevant average for stress calculations. Before doing so, we recall that in the conventional approach [3–6], and in view of the fact that the velocity gradient of the flow is assumed to be smaller than the reciprocal Rouse time, no chain stretching is expected, i.e., the stress is purely orientational. Hence, the relevant average would simply be uu. In the present model, however, chain stretch does occur as a consequence of tube squeezing, and therefore the stress is given by an average involving both tube orientation and tube diameter. For the (non-dimensional) stress T, we will then use the form: uu (16) T= 2 a Fig. 4. Equilibrium distribution of tube diameters.
which generalizes Eq. (6) in a simple way.
of the tube diameter towards equilibrium occurs in two ways. One is through the “lateral” dynamics described by Eqs. (10) and (11). The second is through the “longitudinal” dynamics due to reptation. Indeed, whenever a segment randomizes its orientation θ because of reptation (an event occurring at each time step with probability t/τ d ), it also randomizes its diameter a by picking a new value according to the equilibrium distribution ψ0 (a). Notice that, conversely, when an “internal” tube segment randomizes its orientation because of constraint release (an event occurring with probability t/τ –t/τ d ), the tube diameter is assumed not to change. The fact that the tube diameter can go back to equilibrium in two ways, i.e., either in a time τ p by pushing back the surrounding constraints or in a time τ d by reptating out, has important consequences. Indeed, depending on the ratio τ d /τ p , the transition from the linear behaviour to the asymptotic power law mentioned in Section 2.2 occurs in a qualitatively different way. Such a prediction, anticipated in our previous work [1] and shown in the next section, has been recently confirmed by new data of Hassager on polystyrene melts of smaller molecular mass [12]. In the following section, we will present numerical results obtained with the above described dynamical rules for the case of shear and elongational flows, both at steady state and during start up. The flow dependent quantities appearing in the dynamical equations are given by:
4. Numerical results The non-dimensional constitutive parameters of the model are only two, namely the ratio of relaxation times τ d /τ p , and the constraint release effectiveness β. Consistently with the double reptation picture, the latter parameter is fixed at the value β = 1, unless specified otherwise. The ratio τ d /τ p is expected to increase roughly as M1.3/1.6 . For the case of polystyrene melts, the analysis reported in [1] indicates that for M = 200,000 the ratio τ d /τ p is nearly unity. We will therefore examine values of that ratio both above and below unity. Concerning numerical aspects, we used Euler integration in time, and in most cases a population of 30,000 tube segments proved sufficient to obtain good statistics. In slow flows, noise was reduced by using a simple variance reduction technique. In all cases, error bars are within a few percent. In line with our previous paper [1], Fig. 5 shows results for steady elongational flows in the form of (non-dimensional)
Shear flows: θ˙ affine = −γ˙ sin2 θ
˙ sin θ cos θ a˙ affine = −γa
˙ k : uu = γsin θ cos θ
(14)
Elongational flows: θ˙ affine = −˙ε sin(2θ) k : uu = ε˙ cos(2θ)
a˙ affine = −˙εa cos(2θ) (15)
Fig. 5. Non-dimensional elongational stress vs. ε˙ τp for several values of the time ratio τ d /τ p , corresponding to different M. From right to left τ d /τ p = 0.03, 0.1, 1, 10, 100.
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Fig. 6. Shear stress and normal stress difference in steady shear flows for τ d /τ p = 1.
Fig. 8. Tangential and normal components of tensor uu in steady shear flows for τ d /τ p = 1.
tensile stress σ versus ε˙ τp for several values of τ d /τ p , i.e., for various M. These results show the same qualitative features anticipated in Fig. 4 of our previous work [1], and confirmed by the more recent data of Hassager [12] (see his Fig. 2). Notice in particular the two rightmost curves in Fig. 5 for low M’s, which exhibit an elongational viscosity higher than the Trouton value in the transition from the linear range to the asymptotic power law σ ≈ (˙ετp )1/2 . Fig. 6 reports steady shear results for tangential and normal stresses for the case τ d /τ p = 1. Notice that a quasi-plateau for the tangential component is found before yielding towards the asymptotic 1/2 power law. It is noteworthy that such a plateau-like region is found in shear, while (for the same values of parameters) in elongational flow the transition from linear to asymptotic behaviour occurs rather abruptly (middle curve in Fig. 5). This difference between the two flows can be understood by examining what happens to the tube diameter a in the two cases. Fig. 7 indeed shows that the elongational flow is more effective than shear in squeezing the tubes. In its turn, this difference in behaviour is related to the well known fact that elongational flow, differently from shear, is
irrotational. In a shear flow, because of rotation, tubes get squeezed for 0 < θ < π/2 to enlarge again for −π/2 < θ < 0, while in elongational flow tubes permanently align around θ = 0 and get continuously squeezed. Going back to Fig. 6, it is also worth noting the difference between normal and shear stresses in the non-linear range. Differently from the tangential component, N1 does not show ˙ any curvature change, and keeps rising with a power-law in γ. To understand this behaviour, it is useful to consider the orientational average uu, the tangential and normal components of which are reported in Fig. 8. The tangential component goes though a maximum and then decreases towards its CCR plateau, while the normal component keeps increasing before reaching saturation. The difference in Fig. 6 then reflects that in Fig. 8, as modified by the effect of a2 in the denominator of the average uu/a2 : the sloping down of the tangential component in Fig. 8 becomes the quasi-plateau of the flow curve in Fig. 6, and the moderate increase of the normal component in Fig. 8 becomes the sustained growth of N1 in Fig. 6. It is noteworthy that experiments systematically indicate the behaviour shown in Fig. 6, rather than that of Fig. 8 pre-
Fig. 7. Average tube diameter in steady shear and elongational flows for τ d /τ p = 1.
Fig. 9. Same as Fig. 6 but for a larger value of the time ratio τ d /τ p , i.e., for higher M.
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˙ d = 1, 3, 10, 30, 100, from top to Fig. 10. Curves of transient viscosity (a) and normal stress coefficient (b) in shear start-up for τ d /τ p = 10. Shear rates are γτ bottom.
dicted by orientation alone. In other words, data seem to support the concept that chain stretch is significant throughout the non-linear range, and not merely when the reciprocal Rouse time is approached. Tube squeezing might then be the explanation for premature chain stretching, as suggested by Wagner and coworkers [13,14]. Fig. 9 is the analogous of Fig. 6 for τ d /τ p = 10, i.e., for a larger value of M. The plateau-like region for the shear stress is here more pronounced, and a slight inflection also appears in the normal stress curve. Altogether, however, the qualitative appearance is not very different at higher M. Moving now to transient flows, Fig. 10 reports typical results of tangential and normal stresses in start up of shear. The tangential component exhibits overshoots when the shear ˙ d∼ rate reaches the value γτ = 1, as typically shown by data. As regards the normal component, overshoots in Fig. 10 are completely suppressed by CCR. They appear in the model only when the CCR effectiveness is reduced by choosing β values smaller than unity. Fig. 11 shows that with β = 0.5 (instead of β = 1 as considered so far) overshoots start appearing also in the normal stress component. Even smaller
Fig. 12. Curves of transient viscosity in elongational start-up for τ d /τ p = 1. Elongation rates are ε˙ τd = 0.1 (linear regime), 10, 100, 1000, from top to bottom.
values of β, however, while generating more pronounced overshoots, would also produce unpleasant instabilities. We postpone further analysis of these aspects to a more complete model. Finally, Fig. 12 shows the transient viscosity in start up of elongational flows for several values of ε˙ . The qualitative features shown in Fig. 12 are in good agreement with Hassager’s data [2]. Particularly noteworthy is the strain hardening effect. Such a behaviour is predicted (and shown by the data) when the steady elongational viscosity falls in the asymptotic power-law regime of Fig. 5.
5. Conclusions The model presented in this paper appears promising in describing the non-linear behaviour of entangled polymer melts. Let us summarize in the following all the positive features: Fig. 11. Curves of transient normal stress coefficient in shear start-up for τ d /τ p = 10 and β = 0.5. Shear rates as in Fig. 10.
◦ In steady uniaxial elongational flows at high stretching rates, the model predicts the power law observed by Has-
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◦
◦
◦
◦ ◦
sager and coworkers [2]. The M-dependence of the asymptotic power law is also interpreted by the model [1]. Still in steady elongational flows, the transition from the linear Trouton regime to the asymptotic power law is predicted to show different qualitative features depending on M (see Fig. 5). In particular, for moderately entangled melts the model predicts that the elongational viscosity first rises above the Trouton value, and then decreases towards the asymptotic power law. Such an unusual behaviour, anticipated in [1] and confirmed here, has been experimentally found long ago by M¨unstedt [15] as well as very recently by Hassager [12]. In steady shear flows, the model predicts a plateau-like region of the flow curve, which extends well into the nonlinear region, before yielding to the asymptotic power law. The quasi-plateau is a characteristic feature of the experimental flow curves of monodisperse polymers. Still in steady shear flows, the normal stress difference does not show a plateau, and keeps growing with increasing shear rate with a significant slope. Also this feature is in line with observations. Notice in this regard that CCR theories without chain stretch predict that both tangential and normal stresses approach a plateau. Predictions of the model in shear and elongational start up appear to show the correct qualitative features. It is fair to recall, however, that the model remains limited to qualitative aspects, both because of its 2D formulation, and for the nature of “single segment” toy model. Work is in progress to remove both simplifications.
Acknowledgments One of the authors (G.I.) acknowledges financial support from the Campania regional government (Regional Bill no. 5 dated 28.05.2002). The authors are very grateful to the anonymous referees for their stimulating comments leading to significant improvements of the paper.
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