A constitutive model with moderate chain stretch for linear polymer melts

J. Non-Newtonian Fluid Mech. 123 (2004) 185–199

A constitutive model with moderate chain stretch for linear polymer melts M.A. Tchesnokova,∗ , J. Molenaara,b , J.J.M. Slotc,d , R. Stepanyanc b

a Department of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands c Department of Applied Physics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands d Material Science Centre, DSM Research, P.O. Box 18, 6160 MD Geleen, The Netherlands

Received 26 February 2004; received in revised form 26 August 2004; accepted 30 August 2004

Abstract In our previous publication, we presented a molecular model to describe the dynamics of the interfacial layer between a flowing polymer melt and a die wall. We showed that the ensemble-averaged behavior of polymer molecules adsorbed on the wall could be successfully described in terms of the so-called bond vector probability distribution function (BVPDF). The BVPDF couples the chain orientation and chain stretch on the level of single segment, and thus is an extension of the orientation distribution function of Doi and Edwards introduced for inextensible chains. In this paper, the developed formalism is extended to molecules in the polymer bulk. We show how the well-known Doi and Edwards theory (DE) for inextensible chains based on the orientation distribution function can be naturally extended to include chain stretch and (convective) constraint release (CCR). The final constitutive equation accounts for such mechanisms on polymer chains as reptation, retraction, convection, contour length fluctuations, and (convective) constraint release. It is valid for both linear and non-linear flow regimes. The proposed theory is quantitative, and contains the same input parameters as the original DE model. As an application of the full theory, a simple equation of motion for the stress tensor is derived. Despite the simplicity, its predictions are found to be in good agreement with available experimental data over a wide range of flow regimes and histories. © 2004 Elsevier B.V. All rights reserved. Keywords: Reptation; Polymer extrusion; (Convective) constraint release; Bond vector; Constitutive equation; Bond vector probability distribution function

1. Introduction The flow behavior of entangled polymer melts has attracted a lot of attention and has been a recurring topic over the past decades. A number of theories have been proposed to describe the dynamics of such systems. One of the most successful microscopic models was developed by Doi and Edwards [1] who applied the tube concept by de Gennes [2] to the case of entangled, monodisperse, linear polymer melts. Their model combines reptation with instantaneous and complete chain retraction within the mesh of moving constraints created by surrounding chains. Predictions of the Doi and Edwards theory (DE) for large-step shear strains are known ∗

Corresponding author. Fax: +31 53 4894833. E-mail address: [email protected] (M.A. Tchesnokov).

0377-0257/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnnfm.2004.08.007

to be in an excellent agreement with experimental data. The DE model, however, failed to predict other non-linear shear properties, such as the steady-state viscosity or the relaxation of stress after cessation of steady shearing. A refinement of the DE model by Marrucci and Grizutti, referred to as the DEMG model [3,4], allows retraction to be gradual and incomplete so that chains can be stretched by the flow. The inclusion of chain stretch leads to improved predictions for transient startup behavior, such as overshoots in the first normal stress difference and in the shear stress. However, it did not remove all the flaws of the DE model. In particular, both the DE and DEMG models predict a maximum in the shear stress σxy as a function of the shear rate γ˙ (at γ˙ approximately equal to the inverse reptation time) followed by a region where σxy decreases asymptotically as γ˙ −0.5 . This has never been observed experimentally. Instead,

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the shear stress was found to be a monotonic function of the shear rate. The flaws of the DE model and the more sophisticated DEMG model led to a conclusion that an important relaxation mechanism was missing in the theory. This mechanism becomes especially important if the shear rate exceeds the inverse reptation time, that is, where both the DE and DEMG models predict a significant chain alignment with the flow. Recently, Marrucci [5] and Ianniruberto and Marrucci [6] have described a mechanism for stress relaxation in entangled polymers under flow which they called convective constraint release (CCR). It stems from the fact that in the presence of flow, chains are continually being stretched and relaxing back. As they do so, entanglements in which the ends of a chain are involved are also continually being relaxed. It was recognized that constraint release is a slow, and probably not important, process when a monodisperse melt is at rest or in slow flow. However, when there is a flow that is fast compared to the inverse reptation time, constraints surrounding a chain may be swept away rapidly. In this regime, CCR may play a crucial role in the dynamics of polymer chains. Following these ideas, Mead et al. [7] and Ianniruberto and Marrucci [8,9] have developed non-linear constitutive models that account for both chain stretch and CCR. These theories model the effect of CCR on stress by modifying the overall chain relaxation time. However, the assumption that CCR acts in a global manner “hides” any information about the local chain conformation on length scales less than the chain length. An essential refinement of the description of constraint release was made by Viovy et al. [10] who proposed to treat (C)CR as a hopping (Rouse-like) motion of the tube itself with the time scale set by the inverse frequency of constraint release events, and with a hopping distance of the order of the tube diameter. This approach is based on the microscopic consideration of constraint release, and therefore allows a detailed description of the local influence of constraint release on the chain conformation. Recently, Milner and coworkers [11,12] proposed a microscopic non-linear constitutive model which combined the sophisticated treatment of CCR through a Rouse-like motion of the tube together with chain stretch. Its further refinement by Graham et al. [13] allows chain stretch to be non-uniform along the chain contour. In parallel, a number of models were proposed which are based on the microscopic consideration of the melt dynamics, and formulated in terms of a set of stochastic differential equations suitable for numerical sim¨ ulations, e.g. Ottinger and Beris [14] and Hua and Schieber [15]. The general conclusion of all these models is that incorporation of CCR and chain stretch leads to improved predictions for both linear and non-linear flow regimes compared to those of the DE (DEMG) model. Moreover, it was recognized that inclusion of chain length fluctuations (CLF) is important to predict the correct scaling laws for the molecular weight dependence of the longest relaxation time and steadystate viscosity [1]. This implies that a successful constitutive

model should account for CCR, CLF, and chain stretch in a consistent way. In this paper, an attempt is taken to construct a new microscopic non-linear model for linear entangled polymer melts which incorporates state-of-the-art ideas of the existing models, such as treatment of CCR as the local Rouse-like motion of the tube, CLF, gradual and incomplete non-uniform chain stretch. The resulting model is quantitative and contains no adjustable parameters, except for those that can be extracted from independent molecular measurements. Its numerical evaluation is quite simple, yet it still provides a reasonable accuracy over a wide range of flow regimes and histories.

2. The parameterized chain In order to describe the behavior of a melt under a flow, we need a reliable mathematical description of the single chain dynamics in a mesh of moving constraints imposed by surrounding molecules. According to de Gennes [2], these constraints build a tube in which the chain is confined. In the absence of constraint release, motion of the chain in its tube can be presented as one-dimensional “reptative” motion, so that the description of the single chain dynamics can be significantly “eased”. However, in the presence of constraint release, the tube itself undergoes a three-dimensional Rouselike motion. In this case, motion of the primitive path of the chain is similar to the Brownian motion of a Rouse chain found in dilute solutions [1]. As known from the Rouse theory, every spatial conformation of a chain can be described by a set of the position vectors {Rn }n=N n=1 , where Rn is the position vector of the nth monomer, and N is the number of monomers per chain. Similarly, the configuration of the primitive path of a chain will ˆ 0 , t), where the parambe specified by the space curve R(s eter s0 runs over a certain fixed interval, the same for all chains (see Fig. 1). This parameter ‘labels’ the same physiˆ 0 , t) cal segment of the primitive path at all times, so that R(s gives the position vector of the segment s0 at time t. Note ˆ 0 , t) determines the position of the primithat the curve R(s tive path, and thus only captures time averaged behavior of a physical chain, in contrast to the position vectors Rn of a Rouse chain. Since in a tube-based model the motion of a physical chain is only represented via the motion of its primitive path, the primitive path will, for short, be referred to as ‘chain’ throughout the paper. Moreover, all stochastic (i.e. pertaining to a single chain) variables will carry a hat sign. Since we have freedom in selecting the interval for s0 , let us choose it as −L0 L0 ≤ s0 ≤ , 2 2

(1)

where L0 is the equilibrium chain length. Due to symmetry arguments, the ensemble-averaged values are even functions of s0 . The points s0 = 0 and s0 = ±L0 /2 correspond to the

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187

3. Equation of motion for the bond vectors

Fig. 1. The parameterized bulk molecule: sˆ (s0 , t) is the physical position of the segment s0 along the primitive path at time t; L0 is the equilibrium length ˆ 0 , t) is the position vector of the segment s0 ; λˆ is the local of the chain; R(s stretching of the primitive path at s0 and time t; uˆ is the unit tangent vector.

center and free ends of a chain, respectively. A similar description of the single chain spatial conformation was used in [11–13]. The physical position of a segment along the primitive path is defined by the curvilinear coordinate sˆ (s0 , t), the corresponding arclength from the origin to the segment (see Fig. 1). Attention must be paid that sˆ (s0 , t) is a function of s0 and time. Contrary to s0 , sˆ (s0 , t) is chain dependent which is indicated by its hat sign. However, in the absence of flow, when chains are not stretched, sˆ (s0 , t) ≡ s0 . Taking into acˆ 0 , t) of the primitive count Eq. (1), the local stretching λ(s path at position s0 and time t is given by ˆ 0 , t) = ∂ˆs(s0 , t) . λ(s ∂s0

(2)

Clearly, it describes local stretch of the primitive path by the ˆ 0 , t) ≡ 1 at any point flow. Note that in the absence of flow, λ(s along the chain. Alternatively to the description in terms of the position ˆ 0 , t), every spatial conformation of a chain can vectors R(s also be described by a set of the so-called bond vectors (see Fig. 1) defined as ˆ ˆ 0 , t) = ∂R(s0 , t) . b(s ∂s0

(3)

Its orientation and modulus coincide with the tangent and local stretching of the primitive path, respectively. The bond vector ‘couples’ the local orientation and stretch on the level of a single segment. Note that in the case of inextensible ˆ 0 , t) coincides with the correchains or absence of flow, b(s ˆ 0 , t) to the primitive path. sponding unit tangent vector u(s ˆ 0 , t) for every Given the parameterization function R(s chain in the melt, the dynamics of the whole system is completely defined. Unfortunately, this requires a solution of an enormous system of coupled equations, and therefore is hardly possible. The description of the system with the help of the bond vectors contains less information than that based on the position vectors. However, as will be shown later on, it includes all the necessary information we need to know to calculate different macroscopic parameters of practical interest, such as stress and viscosity. Moreover, we will show that the formalism based on the bond vectors is simple, and has a clear physical interpretation.

ˆ 0 , t) Given an equation of motion for each bond vector b(s of a chain in the melt, the time evolution of its primitive path is defined. However, Eq. (3) makes it clear that, in order to derive this equation, we must first study the dynamics of the ˆ 0 , t). In the absence of constraint release position vectors R(s ˆ 0 , t) obeys the following equation of motion and reptation, R(s [16]: ¯ ˆ 0 + υˆ 0 (s0 , t)t, t). ˆ 0 , t + t) = [I¯ + t K(t)] R(s R(s

(4)

It describes motion of a chain in a moving mesh of constraints produced by the flow. The motion of the mesh deforms the primitive path (this is usually referred to as convection) and ¯ is characterized by the so-called velocity gradient tensor K [1] which specifies how fast the velocity of the mesh changes ¯ is a function of position and time. We in space. In general, K assume, however, convection to be homogeneous on the scale of a single chain, and hence drop the spatial dependence in ¯ In the case of shear flow, K ¯ has only one non-zero comK. ponent Kxy equal to the shear rate. In Eq. (4), the expression in square brackets specifies convection of the primitive path over the time interval between t and t + t (t is small). Since the flow stretches the chain, convection is always accompanied by chain retraction. Due to retraction, the segment s0 “slides” along the primitive path. Its displacement along the s0 -axis over the time interval between t and t + t is given by υˆ 0 (s0 , t)t, where υˆ 0 is the retraction velocity [16] given by (for definiteness we assume s0 ≥ 0) 1 υˆ 0 (s0 , t) = ˆλ(s0 , t)




s0

ˆ t), dx ξ(x,

0

∂λˆ ˆ − K␣␤ uˆ ␣ uˆ ␤ λ. ξˆ = ∂t

(5)

Here, summation is assumed over repeating indices. In Eq. ˆ t) is the retraction rate of the segment x at time t, so (5), ξ(x, that υˆ 0 (s0 , t) actually gives the retraction rate of a part of a ˆ 0 , t) chain between x = 0 and x = s0 . The local stretching λ(s of the segment s0 at time t was defined earlier in Eq. (2). Milner et al. [11] derived equations similar to Eqs. (4) and (5). However, the retraction rate ξˆ was assumed to be position independent. Graham et al. [13] extended the results by Milner to account for non-uniform chain stretch. The expression for υˆ 0 (s0 , t) found in [13] can be obtained from Eq. (5) if we assume that a chain inside its tube can be represented as a bead-spring system. Then, ξˆ is proportional to the tensile ˆ  · u) ˆ where κ is force along the chain contour given by κ (R the spring constant, u is the unit tangent vector to the tube, ˆ  is the second derivative of the position vector R ˆ with and R respect to s0 . Differentiating both sides of Eq. (4) with respect to s0 and taking the limit t → 0 yield the following equation of ˆ 0 , t) in the absence of reptation motion for the bond vector b(s

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and constraint release: ˆ 0 , t) ∂b(s ¯ · b(s ˆ 0 , t) + ∂ (υˆ 0 (s0 , t) b(s ˆ 0 , t)). = K(t) ∂t ∂s0

(6)

The RHS of Eq. (6) contains two contributions. The first term corresponds to convection, and therefore describes affine ‘rotation’ and ‘stretch’ of the bond vector by the flow. The second term arises from retraction, and describes ‘sliding’ and ‘shrinking’ of the bond vector along the chain contour. Note ˆ and vanishes at that υˆ 0 also depends on the flow rate (via λ) rest. The next step towards a complete equation of motion for the bond vectors is to incorporate reptation into Eq. (6). According to Doi and Edwards [1], reptation, i.e. curvilinear diffusion of the entire chain along its own contour, can be modelled as an one-dimensional Rouse motion. Let ζˆ (s0 , t) be the physical shift of the segment s0 along the primitive path (over a small time interval t) due to reptation. Due to the Brownian nature of reptation, ζˆ (s0 , t) is stochastic and can be modelled by a Wiener process. Hence, ζˆ  = 0,

2

ζˆ  = 2Dc t,

(7)

where Dc is the diffusion coefficient of the Rouse motion [1] given by Dc =

kB T . Nς

(8)

Here, kB , N, ς, and t are the Boltzmann constant, the number of monomers per chain, the monomeric friction coefficient, and the absolute temperature, respectively. The vari√ ance of ζˆ is proportional to t, as is typical for Wiener processes. In the case of pure reptation, the equation of moˆ 0 , t) can be written as [1] tion for R(s ˆ 0 + ζˆ 0 , t), ˆ 0 , t + t) = R(s R(s

(9)

where ζˆ 0 is a displacement along the s0 -axis of the segment s0 over the time interval from t to t + t. It is related to the corresponding physical shift ζˆ along the chain contour via ζˆ 0 (s0 , t) =

ζˆ (s0 , t) . ˆ 0 , t) λ(s

(10)

Since reptation is independent of retraction and convection, combination of Eqs. (4) and (10) together with differentiation with respect to s0 leads to the following equation of motion for the bond vectors: ¯ ˆ 0 + υˆ 0 t + ζˆ 0 , t) ˆ 0 , t + t) = [I¯ + t K(t)] b(s b(s   ∂υˆ 0 ∂ζˆ 0 ˆ b(s0 , t), + t + ∂s0 ∂s0

(11)

which now accounts for convection, reptation, and retraction. As is seen, the contribution of reptation has a similar form to that of retraction. However, according to Eq. (5), the retraction velocity υˆ 0 of the segment s0 depends on the stretch and orientation of other segments with the coordinates x ≤ s0 .

This implies that Eq. (5) is non-local with respect to s0 , and equations of motion for all the segments of a chain are coupled. In the absence of chain stretch, i.e. λˆ ≡ 1, the retraction velocity υˆ 0 (5) reads as
s0 υˆ 0 (s0 , t) = − dx K␣␤ uˆ ␣ uˆ ␤ , (12) 0

ˆ 0 , t) where use was made of the fact that in this regime, b(s ˆ 0 , t) coincides with the corresponding unit tangent vector u(s to the primitive path. One may ascertain that Eq. (11) then boils down to the equation of motion for uˆ derived by Doi and Edwards [1] for inextensible chains.

4. The bond vector probability distribution function of bulk chains In the previous section, we derived the equation of motion for the bond vectors of a chain which accounts for reptation, convection, and retraction. In general, the motions of neighboring chains in the melt are highly correlated via constraint release. This implies that its solution in the presence of constraint release would require consideration of macroscopically large system of coupled equations. In reality, however, we are only interested in macroscopic parameters of the melt which are actually expressed via averages over the ensemble of chains. For example, the local stress σ␣␤ in the melt is given by [1]
G0 L0 σ␣␤ (t) = ds0 bˆ ␣ (s0 , t)bˆ ␤ (s0 , t), L0 0 G0 =

3kB Tc , Ne

(13)

where bˆ ␣ , G0 , Ne , and c are the ␣-component of the bond ˆ the elastic modulus, the number of monomers per vector b, entanglement segment, and the chain density, respectively. In Eq. (13), the brackets · · · denote averaging over the ensemble of chains. If Nall is the total number of chains in the ensemble and N(b, s0 , t) is the number of chains whose bond vector at position s0 and time t is equal to b, then the average can be written as
ˆ ˆ b␣ (s0 , t)b␤ (s0 , t) = d3 b bˆ ␣ bˆ ␤ f (b, s0 , t), (14) R3

where f (b, s0 , t) is the bond vector probability distribution function (BVPDF) defined as f (b, s0 , t) =

N(b, s0 , t) . Nall

(15)

Clearly, f (b, s0 , t) is the fraction of chains whose bond vector at s0 and time t is equal to b. Eq. (15) makes it explicit that the BVPDF is normalized
d3 b f (b, s0 , t) = 1. (16) R3

M.A. Tchesnokov et al. / J. Non-Newtonian Fluid Mech. 123 (2004) 185–199

Here, the integral is taken over all possible values of the bond vector. From Eq. (14), it follows that the BVPDF can be also formally written as −L0 L0 ≤ s0 ≤ . (17) 2 2 Both definitions of the BVPDF (Eqs. (15) and (17)) are equivalent. Note that the latter (see Eq. (17)) shows explicitly that the BVPDF involves ensemble averaging, and consequently is an even function of s0

ˆ 0 , t)], f (b, s0 , t) = δ[b − b(s

f (b, s0 , t) = f (b, −s0 , t). As follows from Eq. (17), given the equation of motion for the bond vectors (see Eq. (11)), one can derive that for f (b, s0 , t). To this end, let us calculate f (b, s0 , t + t) for small t. Substituting Eq. (11) into Eq. (17), we arrive at ˆ 0 + υˆ 0 t + ζˆ 0 , t) f (b, s0 , t + t) = δ[b − b(s ˆ 0 , t), s0 , t)], − gˆ (b(s

(18)

where ¯ · b(s ˆ 0 , t), s0 , t) = t K(t) ˆ 0 , t) + t ∂υˆ 0 b(s ˆ 0 , t) gˆ (b(s ∂s0 +

∂ζˆ 0 ˆ b(s0 , t). ∂s0

(19)

As seen, gˆ contains contributions of convection, retraction, and reptation. The convection and retraction terms are√ proportional to t, whereas the last term is proportional to t (see Eq. (7)). The characteristic time scale of retraction is usually much smaller than that of reptation [1]. Therefore, in the presence of retraction, reptation can be considered as a simultaneous and coordinated motion of all the segments of a chain, so that the chain moves in its tube as a whole. This implies that on the time scale of reptation, ζˆ 0 is independent of s0 , and the last term on the RHS of Eq. (19) can be neglected. Then, gˆ is simply proportional to t. For small t, gˆ is small compared ˆ So, expanding the RHS of Eq. (18) in powers of gˆ and to b. discarding second-order terms, we have ˆ 0 + υˆ 0 t + ζˆ 0 , t), t] f (b, s0 , t + t) = δ[b − b(s ∂ ˆ 0 , t)]. (20) · ˆg(b, s0 , t)δ[b − b(s ∂b The last term on the RHS can be evaluated as follows: −

ˆ 0 , t)] ˆg(b, s0 , t) δ[b − b(s   ¯ · b − ξ(b) b f (b, s , t) = t K(t) 0 b


s0 b 3   3   − t 2 d b ξ(b ) d b |b | b R3 R3 0   ∂ (3)  × dx f (b , x|b , y|b, s0 , t) , ∂y y=s0

(21)

189

where the notation b = |b| is used and the third-order BVPDF f (3) (b , x|b , y|b, s0 , t) is the fraction of chains whose bond vectors at x, y, and s0 are equal to b , b , and b, respectively. ˆ 0 , t) only depends In Eq. (21), use was made of the fact that ξ(s ˆ 0 , t) (see on the position s0 and time t via the bond vector b(s ˆ 0 , t) = ξ(b(s ˆ 0 , t)). Note that the last term Eq. (5)), so that ξ(s ¯ λ/∂s ¯ 0 ), which is on the RHS of Eq. (21) is proportional to ξ(∂ of the order of (λˆ − 1)2 , and can be neglected if the chain stretch is small. Next, expanding the first term on the RHS of Eq. 20 in powers of t and neglecting second-order contributions, we have ˆ 0 + υˆ 0 t + ζˆ 0 , t), t] δ[b − b(s   ∂ ∂ 1 ∂2 2 = 1 + υˆ 0 t + ζˆ 0 + (ζˆ 0 ) 2 ∂s0 ∂s0 2 ∂s0 ˆ 0 , t)] , × δ[b − b(s

(22)

where use was made of the fact that ζˆ 0 is proportional to √ t (see Eq. (7)). The second term on the RHS was already studied in [16]. It describes relaxation of the BVPDF due to retraction and can be evaluated as  ˆ 0 , t)] ∂δ[b − b(s υˆ 0 ∂s0
s0
1 ∂ (2)  = d3 b ξ(b ) dx f (b , x|b, s0 , t) b R3 ∂s0 0


s0 1 3  3    + 2 d b d b ξ(b ) |b | b R3 R3 0   ∂ (3)  × dx , (23) f (b , x|b , y|b, s0 , t) ∂y y=s0 where f (2) (b , x|b, s0 , t) is the BVPDF of the second order. The last two terms on the RHS of Eq. (22) describe relaxation of the BVPDF due to reptation. Since ζˆ 0 is a zero-mean noise, and is thus uncorrelated with the corresponding bond vector at the same point, the linear (with respect to ζˆ 0 ) term vanishes. Reptation is the dominant relaxation mechanism for stress in the absence of flow. In flow regimes where chain stretch becomes important, it gives only a minor contribution compared to fast convection and retraction, and can be neglected. On the other hand, for flow regimes in which reptation is relevant, chains are hardly stretched, and the following approximation holds  2 δ[b − b(s ˆ 0 , t)] ∂ ∂2 f (b, s0 , t) (ζˆ 0 )2 ≈ 2D t. (24) c ∂s02 ∂s02 This approximation actually means that displacements due to reptation along the s0 -axis and along the chain contour coincide. Finally, summing up all the contributions in Eqs. (21)– (24), the full equation of motion for the BVPDF f (b, s0 , t)

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reads as ∂f (b, s0 , t) = Dc ∂t

∂2 f (b, s ∂s02

0 , t)

+

1 b







s0

dx 0

R3

d3 b ξ(b )

∂f (2) (b , x|b, s0 , t) ∂ − · [· · ·] ∂s0 ∂b
s0

1 3   + 2 dx d b ξ(b ) d3 b |b | b 0 R3 R3 ×

×

∂f (3) (b , x|b , y|b, s0 , t)

y=s0 , ∂y

(25)

where

  b ¯ [· · ·] = K(t) · b − ξ(b) f (b, s0 , t) b

b − 2 d3 b d3 b ξ(b )|b | 3 b R3 R
s0

∂ (3)  × dx f (b , x|b , y|b, s0 , t) y=s0 . ∂y 0

(26)

Eq. (25) describes the ensemble-averaged dynamics of the spatial configuration of a chain in the melt in the presence of flow. It accounts for reptation, convection, and retraction. The first term on the RHS pertains to relaxation of the BVPDF due to reptation. It has the form of a diffusion process with the coefficient Dc defined in Eq. (8). This makes it clear that after the time TD = L20 /π2 Dc , the chain will escape its initial tube, thereby “renewing” its spatial configuration. The second and last terms on the RHS of Eq. (25) stem from retraction. As seen, they contain higher order BVPDFs f (2) and f (3) which include information about correlations between separate chain segments along the chain contour. So, in general, the equation of motion for the BVPDF f (b, s0 , t) is not closed, and requires consideration of higher order functions. However, in a flow regime where only correlations between neighboring segments are important, a closure approximation can be used. For example, the second-order BVPDF f (2) can be approximated as follows: f (2) (b , s0 |b, s0 , t) 

≈ (1 − e−(|s0 −s0 |/lc ) )f (b, s0 , t)f (b , s0 , t)  s0 + s0  + e−(|s0 −s0 |)/lc f b, , t δ(b − b ). 2

(27)

A similar approximation can be written for f (3) . Here, lc is the characteristic correlation length (lc  L0 ) which specifies the range of interactions between different segments along the chain contour. Eq. (27) makes it explicit that if the distance between two segments is larger than lc , any correlations between their orientations and stretch can be ignored. Note that lc is a function of the flow rate. In the absence of flow, a chain in the melt can be considered as a random walk so that lc = 0. If the flow is fast enough, then lc ≈ L0 , and the closure approximation becomes poor. In this regime, the so-

lution of full Eq. (25) is required. However, even for a flow whose rate is of the order of TR−1 , i.e. the inverse Rouse time [1], chain stretch is small [7]. This implies that on the length scale of one segment, the flow can still be considered as a perturbation, and lc is small compared to L0 . Therefore, it is reasonable to expect that the closure approximation in Eq. (27) is applicable even in the non-linear flow regime with flow rates larger than TR−1 . Incorporation of CCR which destroys any correlations between different chain segments may postpone it to even higher flow rates. The third term on the RHS of Eq. (25) has the form of a divergence and contains both a contribution of convection and retraction. It depends on the direction of the vector b, and then determines deviation of the BVPDF from that at rest which is isotropic. Eq. (25) shows that in general, the BVPDF cannot be decomposed in a pure ‘orientational’ and ‘stretch’ part. This means that the local orientation and local stretch are coupled, so the pre-averaged approximation ˆ ≈ λ ˆ uˆ ␣ uˆ ␤ , uˆ ␣ uˆ ␤ λ which is widely used in the literature for all flow regimes, can only be applied for regimes of small chain stretch where λˆ is close to unity. Finally, in order to solve Eq. (25), one must also specify the boundary conditions for f (b, s0 , t). Since the ends of a chain are free to ‘choose’ their direction and all the directions are equally probable, f (b, s0 , t) at s0 = ± L0 /2 is isotropic. Besides that, due to fast retraction equilibration processes active at the free ends [7], stretching at both ends can be neglected, and so  L0 1 f b, ± ,t = δ(|b| − 1). (28) 2 4π

5. Convective constraint release The formalism developed above can be made fully quantitative by specifying the frequency of convective constraint release ν as a function of the molecular and flow parameters. First, note that constraints on a chain can be released by either reptation or retraction of surrounding chains. The former is referred to as thermal constraint release (TCR), the latter as convective constraint release. To find ν, let us point out a single molecule in the melt and follow its evolution in time. The flow “stretches” the chain. This is followed by its retraction which results in a movement of the test chain inside its tube. This process can be imagined as if the chain were pulled by one of its ends through the melt. The tube in which the test chain is moving is built out of neighboring chains. In turn, the test chain also imposes constraints on surrounding molecules. So, if one of the ends of the test chain moves (inside its tube) a distance equal to the mean entanglement spacing a¯ , it will release one (if entanglements are pair-wise contacts) or more entanglements.

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(in the presence of retraction)   L0 L0 ∗ ∗ ¯ ˆ ˆ +∆ ,t ,0 . E(t , 0) · R 2 2

Fig. 2. Time evolution of the test chain under imposed flow: (a) the test chain at time t = 0; (b) the test chain in the absence of retraction at time t = t ∗ ; (c) the test chain in the presence of retraction at time t = t ∗ .

Let t ∗ be the time necessary for an end of the test chain to move the distance a¯ . An expression for t ∗ can be found by carrying out the following thought experiment. Imagine that at time t = 0, we “switch off” retraction so that chains can be infinitely stretched. Then, at time t ∗ , we take a snapshot of the test chain. Next, we “switch on” retraction and repeat the experiment. The obtained pictures are presented in Fig. 2. Comparison of the t = 0 and t = t ∗ (no retraction) snapshots yields that the position vectors of the test chain taken at time t = 0 and t = t ∗ satisfy ¯ ∗ , 0) · R(s ˆ 0 , 0), ˆ 0 , t ∗ ) = E(t R(s

−L0 L0 ≤ s0 ≤ , 2 2

(29)

¯ ∗ , 0) is the deformation tensor [1] which specifies where E(t how the tube of the test chain is deformed by the flow over the time interval from t = 0 to t = t ∗ . From Eq. (29), it follows that at time t = t ∗ , the segment s0 = L0 /2 − a0 has the position vector  ¯ ∗ , 0) · R ˆ L0 − a 0 , 0 . (30) E(t 2 Now we compare the t = 0 and t = t ∗ (with retraction) snapˆ 0 , t ∗ ) be the distance (along the s0 -axis) that is shots. Let ∆(s passed by the segment s0 due to retraction over the time interval from t = 0 to t = t ∗ . Therefore, ¯ ∗ , 0) · R(s ˆ 0 + ∆(s ˆ 0 , t ∗ ) = E(t ˆ 0 , t ∗ ), 0), R(s

0

=

t∗

dt 0

1 ˆλ(s0 , t)




s0

ˆ 0 , t), dx ξ(s

(33)

By definition, t ∗ is the time necessary for the segment s0 = L0 /2 to move a distance equal to the mean entanglement spacing. In terms of the position vectors, this implies that after time t ∗ , this segment arrives at the same point at which the position vector of the segment s0 = L0 /2 − a0 of the ‘nonretractable’ chain is located (see Fig. 2). So the following relation holds:  ¯ ∗ , 0) · R ˆ L0 − a 0 , 0 E(t 2   L0 L0 ∗ ∗ ¯ ˆ ˆ = E(t , 0) · R (34) +∆ ,t ,0 , 2 2 which is in fact an implicit equation for t ∗ . Substitution of Eq. (32) into Eq. (34) yields 
∗
t L0 /2 ˆ (35) dt dx ξ(x, t) = −a0 . 0

0

Here the averaging over the ensemble has been introduced since every chain has in general its own t ∗ . In Eq. (35), use was made of the boundary conditions for the local stretchˆ t), Eq. ˆ L0 /2, t) = 1 (see Eq. (28)). Given ξ(x, ing, that is, λ(± ∗ (35) becomes an integral equation with respect to t . Since a0 is the equilibrium entanglement spacing and therefore constant, t ∗ does not depend on time explicitly. However, t ∗ may ˆ t). Note that the retraction rate of depend on time via ξ(x, chain segments close to the free ends is of the order of 1/τe (τe = TR /ZB2 is the Rouse time of one segment) whereas for those in the middle, the retraction rate is given 1/TR . Therefore, Eq. (35) shows that it is mainly retraction of segments near chain ends that causes constraint release. For a flow regime in which t ∗  τe , on the time scale TR , the chain ends can be considered to be at local equilibrium. Then, from Eqs. (44) and (35), we find an explicit expression for t¯∗ , the ensemble averaged t ∗ 
−1 L0 /2 ∗ ¯ ¯ t ≈ a0 dx |ξ(x, t)| , (36) 0

−L0 L0 ≤ s0 ≤ , (31) 2 2 If υˆ 0 (x, t) is the retraction velocity (along the s0 -axis) of the ˆ 0 , t ∗ ) can be segment s0 at time t, then the displacement ∆(s written as
t∗ ∗ ˆ ∆(s0 , t ) = dt υˆ 0 (s0 , t)


191

(32)

0

where use has been made of Eq. (5). According to Eq. (30), at time t = t ∗ , the chain end s0 = L0 /2 has the position vector

¯ t) is the mean retraction rate of the segment x where ξ(x, which is also averaged over the time scale much larger than τe , but smaller than TR . Note that in the steady-state, Eq. (36) becomes exact. Given ¯t ∗ , one can readily find the CCR frequency ν which is equal to the inverse mean life-time of a constraint. Let us assume that all entanglements in the melt are pair-wise contacts between separate chains, so that movement of a chain over the distance a¯ will destroy only one entanglement. Since entanglements are built out of different parts of chains, not all of them will be released after time ¯t ∗ . Let us point out a unit volume in the melt which contains, say, N chains. If ZB is the mean number of constraints per chain, then the total number of entanglements (at time

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t = 0) in the chosen volume is Nent = ZB N/2. The number of entanglements in this volume at time t = ¯t ∗ is Nent (t = t¯∗ ) =

ZB N − N. 2

(37)

Here, since we are only interested in the mean life-time of an entanglement, the entanglement creation mechanism was ignored. Let τCCR be the mean life-time of an entanglement due to CCR. Therefore, Nent (t = ¯t ∗ ) − Nent (t = 0) 2 t¯∗ =− . =− Nent (t = 0) τCCR ZB From Eqs. (36) and (38), we finally have 
 L0 /2 2 1 ¯ t)| , ≈2 ν= dx |ξ(x, τCCR L0 0

(38) t∗ = (39)

where we used that L0 = a0 ZB [1]. The extra-prefactor 2 on the RHS stems from the fact that both ends of a chain contribute to CCR. In the single relaxation time approximation, ¯ t) ∝ [λ(x, ¯ t) − 1] (see Eq. (44)). Therefore, in the linξ(x, ear flow regime at flow rates γ˙ < TD−1 (TD is the reptation time), where chains are hardly stretched, ν < TD−1 so that CCR plays only a minor role in the chain dynamics and the DE (DEMG) model can be used. However, at higher flow rates γ˙ > TD−1 , τCCR < TD , and CCR becomes an important relaxation mechanism on polymer chains. Note that at very high flow rates γ˙ > TR−1 , the single relaxation time approximation becomes poor, and one has to use the exact expression ¯ t) [16] for the retraction rate ξ(x, ξ¯ = 3ZB Dc

∂2 λ¯ ∂x2

,

rate ξ¯ in Eq. (39) is given by Eq. (40) whereas in ([18]) by Eq. (44). Until now, we have ignored the presence of thermal constraint release. This is only correct if the polymer chains are long enough. In a real experiment, however, the mean number of entanglement segments per chain is sometimes rather small, and TCR may also play a role in the chain dynamics. An explicit form of the frequency of TCR can be readily found from Eq. (7). If t ∗ is the mean time needed for a chain to pass the distance a0 due to reptation, then

(40)

where Dc is the reptation diffusion coefficient defined in Eq. (8). Taking into account Eq. (5), Eq. (39) can be rewritten as follows:  
L0 /2 2 ∂λ¯ ν=2 . (41) dx K␣␤ < uˆ ␣ uˆ ␤ λˆ > − L0 0 ∂t Clearly, ν contains two contributions. In order to explain their origin, let us imagine a chain which is pulled by one of its ends through a melt. If the chain is inextensible, then the average velocity between the chain and the melt is proportional to K␣␤ < uˆ ␣ uˆ ␤ > (the first term), the projection of the mesh velocity on the chain contour. In the presence of stretch, this motion is also accompanied with retraction which provides an additional velocity between the chain end and the melt (the second term). Graham et al. [13] and Likhtman and Graham [18] also derived an explicit expression for the frequency ν of CCR in terms of the local retraction rate. Eq. (39) can be written in the form similar to that found in [13] if we further assume that λ¯ 2 ≈ λ2 and change the integral over the s0 -axis to that along the chain contour. Note, however, that Eq. (39) agrees with that found in [18] only in the case of small chain stretch. The difference is that at high shear rates, the retraction

a02 . 2Dc

(42)

Here, chain stretch is neglected. Therefore, from Eqs. (38) and (42), the frequency νTCR of TCR can be written as νTCR

  C 2 4Dc √ 1+ ≈ , ZB ZB a02

(43)

where we also took into account the effect of CLF on the frequency of constraint release (see, for example, Doi and Edwards [1]). The coefficient C ≈ 1.69 was found by numerical simulations in [19]. According to Eq. (51), the diffusion coefficient due to TCR is ZB times less than that of reptation.

6. The equation of motion for the BVPDF ˆ 0 , t) In general, the retraction rate per single segment ξ(s (40) is represented by a set of relaxation times Tp = TR /p2 (p = 1, 3, 5, . . .), where the Rouse time TR is the longest relaxation time needed for a chain to restore its equilibrium length. The characteristic relaxation time of the second mode is about 10 times smaller than that of the first mode. This implies that for the flow rates less than 10TR−1 , higher order modes with Tp (p = 3, 5, . . .) can be considered instantaˆ 0 , t) can be approximated as neous so that ξ(s ˆ ˆ 0 , t) ≈ − λ(s0 , t) − 1 . ξ(s TR

(44)

This is the single relaxation time approximation in which all the segments are assumed to have the same relaxation time equal to the Rouse time TR . It is, however, not applicable for segments near the chain ends, since their relaxation time is given by τe , the Rouse time of single segment. Note that a similar approximation was used in [7]. For a flow regime in which the approximation (44) is valid, chain stretch is still small [7], and a remarkable reduction of the full theory comes out. Taking into account Eq. (44), from Eqs. (25) and (26), the equation of motion for the BVPDF reads as (terms of the order of lc2 , lc (λ − 1), and (λ − 1)2  are

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neglected)

193

equilibrium BVPDF)

3νa02 ∂f (b, s0 , t) = Dc + ∂t 4



∂2 f (b, s0 , t) ∂s02

−

¯ 0 , t) − b 1 λ(s − f (b, s0 , t) TR b 
s0  1 ∂f (b, s0 , t) ¯ + dx ξ(x, t) b ∂s0 0   ∂ ¯ · b + 1 b − 1 b f (b, s , t) , − · K(t) 0 ∂b TR b (45) where use was made of the closure approximation for f (2) (see Eq. (27)). Moreover, as was argued before, the terms with f (3) are of the order of (λ − 1)2 , and therefore can be neglected (compared to terms of the order of (λ − 1)). ¯ 0 , t) and λ(s ¯ 0 , t) are the ensemble-averaged In Eq. (45), ξ(s retraction rate and local stretching of the segment s0 at time ¯ 0 , t) can be in turn expressed via t, respectively. Note that λ(s the BVPDF as
¯ 0 , t) = λ(s d3 b bf (b, s0 , t), (46) R3

where the integral is taken over all possible values of the bond vector. So, Eq. (45) is a second-order non-linear integropartial differential equation. Note that a contribution due to CR was added in Eq. (45). As shown in [16], it has the form of a diffusion process with the coefficient proportional to the frequency of constraint release ν and the equilibrium tube diameter a0 squared. The explicit expression for ν was derived in the previous section (see Eq. (41)). Eq. (45) accounts for such mechanisms on polymer chains in the melt as reptation, retraction, convection, and (convective) constraint release, but ignores contour length fluctuations (CLF) [1]. However, the present formalism readily admits inclusion of CLF. These fluctuations involve motion of the chain ends into the tube, thereby temporarily creating a higher than the average density of monomers near the chain ends. When the chain ends move outward again, they are free to choose their direction so that the initial orientation of the tube relaxes. According to Milner and McLeish [17], the mean relaxation time of chain segments due to CLF increases exponentially with the distance from the chain ends and can be written as τ(s0 ) ≈ τ0 e0.75ZB (1−2s0 /L0 ) , 2

(47)

where ZB is the mean number of constraints per chain. Here, use was made of the fact that chain stretch near the ends is small so that s(s0 , t) ≈ s0 . For the time constant τ0 , we use TR /2. A more accurate prefactor, which depends on s0 , was derived in [17]. The contribution of CLF to the equation of motion for the BVPDF (45) can be then written as (feq is the

 1  f (b, s0 , t) − feq , τ(s0 )

feq (b) =

1 δ(b − 1). (48) 4π

The final equation of motion for the BVPDF (see Eqs. (45) and (48)) is at the heart of the present model. It describes both evolution of the local chain orientation and the local chain stretch in flow, and includes all major mechanisms on bulk chains. It is a natural extension of the well-known theory of Doi and Edwards [1] developed for inextensible chains. In contrast to the DE model, the present formalism self-consistently includes chain stretch and CCR, and is valid for both linear and non-linear flow regimes. Let us show now that in the limit of inextensible chains, Eq. (45) can be reduced to that of Doi and Edwards. In the absence of chain stretch, the BVPDF can be presented as f (b, s0 , t) = δ(b − 1)φ(u, s0 , t),

(49)

where φ(u, s0 , t) is the orientation distribution of unit tangent vectors introduced in the DE theory. Noting that for inextensible chains, λ¯ ≡ 1 along the chain contour, from Eq. (5), it follows that the retraction rate per segment ξ(b) is equal to −K␣␤ u␣ u␤ , and the retraction velocity υˆ 0 is given by Eq. (12). Substituting Eq. (49) into Eq. 45 and integrating over b, one may ascertain that Eq. (45) boils down (without the CR term) to the well-known equation for φ(u, s0 , t) [1], derived without independent alignment approximation.

7. The bond vector correlation function As an application of the developed theory, a simple constitutive equation will be derived in this section. To this end, let us introduce the bond vector correlation function S␣␤ (s0 , t) defined as the second moment of the BVPDF
ˆ ˆ d3 b b␣ b␤ f (b, s0 , t). S␣␤ (s0 , t) = b␤ (s0 , t)b␣ (s0 , t) = R3

(50) According to Eq. (13), the local stress in the melt is proportional to the averaged along the chain contour value of S␣␤ (s0 , t). The equation of motion for S␣␤ follows directly from Eq. (45), namely multiplying both sides by b␣ b␤ and then integrating over all possible values of the bond vector, we find ∂S␣␤ (s0 , t) ∂t = K␣γ S␤γ (s0 , t) + K␤γ S␣γ (s0 , t)

 ¯ 0 , t) − 1 3νa02 ∂2 S␣␤ (s0 , t) λ(s + Dc + − 2 S␣␤ (s0 , t) TR 2 ∂s02
s0 ¯ t) ∂S␣␤ (s0 , t) − 1 (S␣␤ − S eq ), (51) + dx ξ(x, ␣␤ ∂s0 τ(s0 ) 0

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where terms of the order of (λ − 1)2  are neglected. Here eq S␣␤ = δ␣␤ /3 is the equilibrium (no-flow) value of S␣␤ . Note √ that the mean local stretching λ¯ ≈ S␣␣ , so that Eq. (51) is a non-linear integro-partial-differential equation. The first two terms on the RHS correspond to convection, and describe local rotation and stretch of a chain. They show that in the presence of flow S␣␤ may contain non-diagonal elements, proportional to the corresponding shear stresses. The third term contains contributions due to reptation and constraint release. Both have the form of a diffusion process, and thus can be treated as diffusion along and perpendicular the chain contour, respectively. The boundary conditions for S␣␤ can be readily obtained from those for the BVPDF (see Eq. (28)). Note that S␣␤ ‘couples’ both the local orientation and stretch. Therefore, Eq. (51) does not only capture the dynamics of the local orientation in flow, but also the dynamics of the local stretch. In fact, the equation of motion for the ¯ 0 , t) of the segment s0 directly folmean local stretching λ(s lows from Eq. (51) by taking the trace of both sides. Eq. (51) shows that apart from retraction, a chain may also relax its stretch via CR. This agrees with the proposal of Mead et al. [7], who showed that CR may also lead to relaxation of the local stretch via removal of constraints on a ‘tout’ piece of chain. They argued that at high flow rates, when chains are stretched, removal of a constraint will more likely result in relaxation of the local stretch than in relaxation of the local orientation. However when chain stretch is small, both mechanisms occur at the same frequency. This is also described by Eq. (51), namely given an explicit expression for the CR frequency ν, one may directly find the total relaxation time Ttot of chain stretch in the presence of CCR. Taking the trace of both sides of Eq. (51) and regrouping terms proportional to (λ¯ − 1), in the single relaxation time approximation, we find that 1 1 1 ≈ + , Ttot TR TCCR

TCCR =

2 ZB2 1 , 3 π2 ν

(52)

where uniform chain stretch was assumed along the contour, so that TCCR is the overall relaxation time of chain stretch due to CCR, in accord with the Verdier–Stockmayer model [1]. A similar expression for Ttot was found in [7]. In the absence of chain stretch, the correlation function S␣␤ turns into the orientation tensor uˆ ␣ uˆ ␤  of the DE (DEMG) model which is usually calculated with the help of the famous Doi–Edwards tensor Q␣␤ [1]. Note that the present approach does not include Q␣␤ , and the equation of motion for uˆ ␣ uˆ ␤  is written out directly. Ianniruberto and Marrucci [8,9] proposed a simple constitutive model which is based on the equation of motion for the averaged (along the chain contour) orientation tensor uˆ ␣ uˆ ␤ . Note that Eq. (51) can be written in the form similar to that found by Ianniruberto and Marrucci if we further assume that the mean local stretching is independent of the local orientation and position along the chain contour, and neglect high-order relaxation modes for CR and reptation.

The molecular model developed by Milner and coworkers [11,12] and Graham et al. [13] provides an alternative approach to constitutive modelling which is based on the so-called tangent correlation function f (s, s , t) = bˆ ␣ (s , t)bˆ ␤ (s, t). Note that f (s, s , t) can be expressed via the second-order BVPDF f (2) , and therefore can be considered as an extension of the “one-point” bond vector correlation function S␣␤ (50) introduced in this paper. The tangent correlation function f (s, s , t) satisfies a very sophisticated non-linear equation of motion which is also based on the assumption of small correlation length between separate chain segments. Its solution demands intensive numerical calculations. In contrast, the equation of motion for S␣␤ (s0 , t) (51) is only two-dimensional and can be easily solved numerically using conventional computation methods for non-linear equations. Later on, we will show that despite the simple structure of the constitutive equation (51), the present model provides a good agreement with experimental data over a wide range of flow regimes and histories. Recently, Likhtman and Graham [18] proposed a simplified version of the ‘full’ model [13]. They derived an equation of motion (the Rolie–Poly equation) for the stress tensor which accounts for convection, retraction, and CCR. Note that Eq. (51) can be written in the form similar to that of the Rolie–Poly equation if we assume that the chain stretch is position independent, and neglect CLF and high-order modes of reptation and constraint release.

8. Results and discussion The constitutive equation (Eq. (51)) was solved numerically using the conventional Newton method for non-linear equations. As the original DE model, the theory contains only four parameters: the reptation time TD , the Rouse time TR , the mean number of entanglement segments ZB , and the elastic modulus G0 . All these parameters may be extracted from independent rheological measurements, and therefore cannot be considered as adjustable. So the present model does not involve fitting parameters. In the absence of fluctuations, one has TD = 3ZB TR and the number of independent parameters reduces to three. In Figs. 3–6, the steady-shear predictions of the model are presented. In Fig. 4, we plot the shear stress σxy and the first normal stress difference N1 = σxx − σyy as functions of the ˙ As is seen, both σxy and N1 increase monotonishear rate γ. cally with γ˙ and have three different regimes, in accordance with experimental observations [20,21]. As expected from the DE model, in each case, there is a linear viscoelastic regime at low shear rates γ˙ ≤ TD−1 for which σxy ∝ γ˙ and N1 ∝ γ˙ 2 . In the non-linear regime at flow rates TD−1 < γ˙ < TR−1 , the ˙ whereas N1 conshear stress increases only slightly with γ, tinues to grow rapidly. In this regime, CCR plays a dominant role in the stress relaxation. Since the DE model does not account for CCR, it leads to the unrealistic behavior in

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Fig. 3. Shear stress and first normal stress difference vs. shear rate for ZB = 30. The dotted lines are predictions for inextensible chains.

which σxy ∝ γ˙ −0.5 and N1 approaches a constant value. In the vicinity of the shear rate γ˙ ≈ TR−1 , the slopes of both curves again become steep due to significant tube stretch. The system enters the third regime in which these stretch

Fig. 4. Shear stress vs. shear rate for various ZB (τe is the Rouse time of single segment).

195

Fig. 5. Mean chain lengthening vs. shear rate for different ZB .

effects are very important. Fig. 3 shows that in this regime, the theory of inextensible chains with CCR (Eq. (51) without stretch) substantially underestimates the actual values of σxy and N1 . Fig. 4 shows the shear stress σxy versus the shear rate γ˙ for different molecular weights of polymer chains (ZB is proportional to the molecular weight). As is seen, longer chains reach “the plateau regime” earlier than shorter ones, as expected from the DE model. At high shear rates, all the curves corresponding to different molecular weights nearly merge into a single one, in accord with the behavior observed in experiments [22,23]. ˆ Fig. 5 shows the mean chain lengthening λ = L/L 0 versus the shear rate γ˙ for different molecular weights of polymer chains. Clearly, in the presence of flow, chains are always stretched. However, chain stretch becomes especially important as γ˙ approaches TR−1 (at γ˙ ≈ TR−1 , chains are found to have the same amount stretch around 15%). Written as a ˙ R , all the curves function of the dimensionless shear rate γT corresponding to different molecular weights nearly superimpose to a single one. As TR ∝ ZB2 [1], this implies that ˙ longer chains are more stretched, λ ∝ ZB2 , and for a given γ, as expected. Note also that λ increases linearly with γ˙ for all tested molecular weights. In Fig. 6, the steady-state extinction angle χ is shown as a function of the shear rate. It is seen that χ first drops rapidly with increasing shear rate, and then more slowly, similar to the behavior predicted by the CV model of Mead et al. [7] and observed experimentally. Both the DE and the DEMG models predict an excessively steep decrease in χ with increasing γ˙ which means a significant tube alignment with flow at shear

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Fig. 6. Extinction angle (in degrees) vs. shear rate. ZB = 50.

TD−1 .

rates above Fig. 7 shows early time relaxation of the shear stress after cessation of steady shear flow. As found, the curves corresponding to shear rates γ˙ much smaller than TR−1 nearly merge. This implies that the stress relaxation is independent of the steady-state shear rate. Clearly, these curves show relaxation of the initially aligned with the flow chain configurations due to reptation. At high shear rates, the rate of re˙ similar to the behavior observed laxation increases with γ, in experiments [24]. Apparently, this is a consequence of relaxation of chain stretch and its effect on the rate of CCR. According to Eq. (41), the CCR frequency is proportional to the mean chain lengthening, so that more stretched chains are expected to relax faster. Note that in the case of inextensible chains with CCR, one would expect that the stress relaxation ˙ rate is independent of γ. In Figs. 8 and 9, the model predictions are shown for the case of step shear deformation. Fig. 8 shows the relaxation of the shear modulus G(γ, t) after step shear of various magnitudes. As is seen, after a small step strain, chains are hardly stretched, so that stress relaxes via reptation. In contrast, after a large step strain, chains are stretched substantially, so that first stress relaxes via relaxation of stretch. However, at larger times, the stress relaxation is again governed by reptation, so that all the curves can be superposed into a single one by dividing G(γ, t) by the corresponding value of the damping function h(γ) [1], in accordance with experimental data by Osaki et al. [25]. In Fig. 9, we plot the step shear damping function h(γ) as a function of the shear strain. The result is compared with that of the DE model. Note that the DE model is known

Fig. 7. Early time relaxation of shear stress, normalized by its initial value at steady state vs. time after cessation of steady-state shear flow for different prior shear rates. ZB = 30.

Fig. 8. Relaxation of dimensionless shear modulus G = 3σxy /(γG0 ) vs. time after step shear strains of different magnitudes γ. ZB = 30.

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Fig. 9. Step shear damping function vs. shear strain. Dots represent the result of the DE model without IAA [1].

to be in an excellent agreement with experimental data on the step–strain response of monodisperse melts and solutions [1]. Therefore, it is important that the present theory should preserve the ability to predict accurately the response for step strain deformations. As is seen, despite the different

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form of the constitutive equation, the agreement with the DE damping function is very good over a wide range of strain amplitudes. In the rest of this paper, the predictions of the model will be compared with the available experimental data for two flow histories: steady-state shear flow and transient startup of simple shear. In Fig. 10, the shear stress σxy and the first normal stress difference N1 as functions of the shear rate are compared with data by Mead et al. [7] for a 3 wt% solution of 8.4 × 106 molecular weight polystyrene in tricresyl phosphate at room temperature. The Rouse time TR , the disentanglement time TD , and the mean number of entanglement segments ZB were reported to be around 0.3 s, 2.0 s, and 20, respectively. The elastic modulus was found to be G0 ≈ 5900 Pa by a best fit to the data. It is seen that the model predictions are in good agreement with the experimental curves over a wide range of flow rates from the linear to the non-linear regime. In Fig. 11, model predictions for the viscosity and first normal stress difference as functions of time after startup of simple shear flow are compared with the data by Osaki et al. [25] for the f128-10 solution of polystyrene in tricresyl phosphate at 40 ◦ C. The Rouse time TR and mean number of entanglement segments ZB per chain were estimated by Graham et al. [13] to be around 3 s and 8, respectively. The elastic modulus was found to be G0 ≈ 4000 Pa by a best fit to the data, which agrees with the value estimated in [13]. As is seen, the model predictions are in good agreement with the experimental data over a wide range of flow rates. Fig. 11 also shows that inclusion of chain stretch improves agreement between the experimental data and model predictions.

Fig. 10. Shear stress and first normal stress difference vs. shear rate for a 3 wt% solution of 8.4 × 106 molecular weight polystyrene in tricresyl phosphate at room temperature [7]. The solid lines are the model predictions.

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Fig. 11. Shear viscosity and first normal stress difference vs. time after startup of shear flow of various rates for f128-10 solution of polystyrene in tricresyl phosphate at 40 ◦ C [25]. The solid lines are the model predictions. The dotted lines are the model predictions for inextensible chains for γ˙ = 1.74 s−1 .

9. Conclusion and remarks In this paper, a quantitative molecular model is proposed to describe the dynamics of polymer chains in flow. The formalism is based on the so-called bond vector probability distribution function, found to be a successful mathematical ‘tool’ in constitutive modelling. It is shown that the original DE theory based on the orientation distribution function can be naturally extended to incorporate chain stretch and constraint release. Basing on the equation of motion for the BVPDF, a simple constitutive equation is derived for the stress tensor which accounts for chain stretch, reptation, convection, retraction, CLF, and (C)CR. It contains no adjustable parameters, except for those that can be extracted from independent rheological measurements.

Acknowledgements This research is supported by the Technology Foundation STW, applied science division of NWO, and the technology programme of the Ministry of Economic Affairs of The Netherlands.

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