Entangled polymer melts are usually modelled by considering a test chain that is constrained laterally to its local primitive path [solid line in Figure 3.2.2 (b)]
by a mean field provided by entanglements2 with other polymers. Edwards [48] proposed that this effectively creates a ‘tube’ of confinement around the chain, as shown in Figure 3.2.2. It was further suggested by de Gennes [40] that the polymer chain would undergo one dimensional diffusion along its own length and eventually escape the tube of constraints; unoccupied tube segments are then forgotten as the chain progressively recovers an isotropic distribution of entanglements. This process was termed ‘reptation’ by its analogy with snake-like motion. The most popular ‘tube model’ to result from these initial steps was that of Doi and Edwards [47] (which we term the ‘DE’ theory from here on) whose basic treatment of polymers we briefly outlined in Chapter 2. The theory was an important step in modelling entangled polymers. In particular, it gave good agreement with the experimentally determined ‘damping function’ (as shown in Chapter 2). The theory also predicted the timescale for reptation to be proportional to the cube of the molecular weight of the chains: τd ∝ M3, which is very close to the experimental result τd ∝ M3.4 [47]. However, the theory did not incorporate the relaxation mechanism of convective constraint release (CCR), which describes how constraints forming the tube can be lost when the polymer forming the entanglement itself reptates or retracts through that point; nor the mechanism by which contour length fluctuations (CLF) allow constraints at the ends of the tube to be lost. The theory also did not allow for chain stretch during strongly nonlinear flows with deformation rates approaching the inverse Rouse time, e.g., during shear flow at rates ˙γ > τR−1. Many models based on the original DE theory were developed in an attempt to include such mechanisms [78, 105, 108, 109, 127] including the GLAMM model [67], that provides a stochastic microscopic equation of motion for the chain and the surrounding tube. However, the GLAMM model is computationally intensive; two of its authors therefore derived a single mode approximation to the model for the description of the stress or flow rate response to deformation, by removing the contour dependence from the full GLAMM theory. This single mode approximation is the rolie-poly (ROuse LInear Entangled POLYmers or ‘RP’) model [92], whose constitutive equation for the viscoelastic
(a) X 1) 2) 3) (b)
Figure 3.2.2: (a) Representation of the tube model: points represent the intersection of surrounding polymers with the view plane — the entanglement points surround- ing the test chain effect a ‘tube’ of constraints. (b) Schematic of stress relaxation mechanisms in the RP model: 1) reptation, 2) constraint release, 3) stretch relax- ation.
conformation tensor is given by:
∂tσ = K · σ + σ · KT − 1 τd σ − I − 2(1 − A) τR σ + βA−2δ σ − I , (3.2.3)
where A ≡p3/T , T ≡ tr σ denotes the magnitude of chain stretch in the system, and K is the deformation tensor as defined previously. The timescale τd is the ‘reptation’ time: the timescale on which a test chain escapes its tube of constraints by undergoing 1D diffusion along its own length. τR is the Rouse relaxation time: the timescale on which chain stretch relaxes; these two relaxation times are related by the number of entanglements on a chain: Z = 3τRτd [47]. β is the convective constraint release (CCR) parameter that describes the efficacy of constraint release events3, and has range 0 ≤ β ≤ 1, see Figure 3.2.2 for a schematic of these three mechanisms. δ is a parameter also related to convective constraint release that, following Ref. [92], we set to δ = −12 throughout this thesis. Unfortunately, the CCR parameter is difficult to directly relate to experiment and there is no consensus on its correct value, though a small value β ∼ 0 was used by the authors to best fit the experimental data [92]. In addition, a recent study using a diffusive form of the model found ‘fracture-like’ velocity profiles after a step strain for similarly small values of β, that are qualitatively similar to experiments on entangled polymeric
3More specifically, it is related to the number of constraint release events required to result in
materials [7]. In Chapter 6 we will investigate the effect of varying β on the stress relaxation via the stretch and orientation relaxation mechanisms following a fast ramp-strain, and its resulting effect on linear instability to shear banding. We will also allow β to vary in Chapter 5 where its effect on the overall shear banding properties will be investigated.
Componentwise, in a homogeneous shear flow at rate ˙γ the constitutive equations reduce to4: ∂tT = 2 ˙γσ −τ1 d(T − 3) − 2(1−A) τR [T + βA (T − 3)] , ∂tσ = ˙γσyy −τd1σ −2(1−A)τR (1 + βA) σ, ∂tσyy = −τ1 d (σyy − 1) − 2(1−A) τR [σyy + βA (σyy− 1)] , (3.2.4)
resulting in a system with three dynamical variables.
In the limit of fast stretch relaxation τR→ 0 Eqn 3.2.3 reduces to: [92]:
∂tσ = K · σ − 1 τd
σ − I − 2
3tr K · σ σ + β(σ − I) , (3.2.5) with constant chain stretch T → 3. We denote this the ‘non-stretching’ RP (nRP) model and the former (Eqn 3.2.3) the ‘stretching’ RP (sRP) model in what follows. We also chose units in which τd= 1 and consider small solvent viscosities η Gτd. Componentwise, the non-stretching RP model is given by:
∂tσ = ˙γσyy− 23(1 + β) σ2 −1 τdσ, ∂tσyy = 23˙γ [βσ − (1 + β) σyyσ] −τd1 (σyy− 1) , (3.2.6)
i.e., having only two dynamical variables.
Steady state behaviour
The original DE model results in a negative slope of shear stress in strain ∂γ˙Σ < 0 in the constitutive curve caused by the ‘over-alignment’ of the chains during flow at rates ˙γ > τd−1. Originally this was considered a major failing of the DE model as no evidence of steady state shear banding that would arise from such a constitutive
4Recall that we define σ ≡ σ
10-1 100 101 102 103 γ. 10-1 10 Σ (a) β = 1 10-1 100 101 102 103 γ. 10-1 10 Σ (b) β = 0
Figure 3.2.3: Steady state constitutive curves in the non-stretching RP model for two values of β = 1, 0 in (a), (b), respectively. These are obtained by imposing either a constant shear rate ˙γ and measuring the steady state total shear stress response Σ( ˙γ, t → ∞) (solid line), or a constant total shear stress Σ and measuring the steady state shear rate response Σ[ ˙γ(t → ∞)] (circle symbols). Convergence of the constitutive curves from these two protocols is obtained only for monotonic constitutive curves. Solvent viscosity: η = 10−3, and homogeneity is enforced in the flow gradient direction.
instability existed for entangled polymeric materials at that time. The RP model was then considered a successful step forwards from the DE model, since the action of CCR can increase the total shear stress and remove the negative slope entirely. This is possible because constraint release events cause the tube to make stochastic ‘hops’, so that the tube itself obeys Rouse dynamics. This small deviation from an ‘over-aligned’ state that caused the original constitutive instability can result in a sufficient increase of the total shear stress to result in a monotonic constitutive curve (for sufficiently large values of β; for small values of β a nonmonotonic curve persists — see Figure 3.2.3). As described in Chapter 2, entangled polymeric materials have recently been shown to undergo steady state shear banding for imposed shear rates on the weakest slope of the flow curve, indicating that an underlying nonmonotonic constitutive curve may actually be correct for these materials.
Figure 3.2.3 shows our numerical results for the steady state constitutive curve for the non-stretching RP model obtained by imposing a constant shear rate ˙γ and measuring the total shear stress response Σ( ˙γ, t → ∞). Note that homogeneity is enforced — this is the definition of the ‘constitutive curve’ that we defined earlier. We also plot the constitutive curve obtained by imposing a constant total shear
stress5 and measuring the steady state shear rate response ˙γ(t → ∞). The constitu- tive curves obtained through each protocol differ only when the chosen parameters result in a nonmonotonicity in the constant shear rate protocol. The response under the step stress protocol then results in ‘top-jumping’ of the shear rate response when the imposed stress exceeds that of the underlying maximum Σmax; we investigate this later in Chapter 4.
In the full (sRP) model (Eqn 3.2.4) there are three regimes of the constitutive curve depending on the imposed shear rate: i) ˙γ < τd−1: the linear regime where the total shear stress increases linearly with shear rate; ii) τd−1 < ˙γ < τR−1: a middle regime whose slope depends on CCR (see next); and iii) ˙γ > τR−1: the high shear rate regime in which CCR and chain stretch are the dominant relaxation mechanisms, and the total shear stress again increases with the shear rate6 (note that here we have assumed η GτR). The second (ii) region is negatively sloping if τR τd are sufficiently well separated (i.e., at sufficiently high entanglement number Z) and β and η are sufficiently small, see Figure 3 of [2].