Elongational viscosity of poly(propylene carbonate) melts: tube-based modelling and primitive chain network simulations

The ERS model is a special form of the molecular stress function (MSF) model, which is a generalized tube segment model with strain-dependent tube diameter (Wagner 1990; Narimissa and Wagner 2019). The extra stress tensor \(\sigma \left(t\right)\) of the MSF model is given by a history integral of the form

\(G\left(t\right)\) is the relaxation modulus, according to Eq. (1); \(t\) is the time of observation when the stress is measured, and t′ indicates the time when a tube segment was created by reptation. The strain measure \({\mathbf{S}}_{DE}^{IA}\) represents the contribution to the extra stress tensor originating from the affine rotation of the tube segments, according to the “independent alignment (IA)” assumption of Doi and Edwards (1978a,b, 1979), and is given bywith \({\varvec{S}}\left(t,{t}^{^{\prime}}\right)\) being the relative- second order orientation tensor. u′u′ is the dyad of a deformed unit vector \({\mathbf{u}}^{\mathrm{^{\prime}}}={\mathbf{u}}^{\mathrm{^{\prime}}}\left(t,{t}^{^{\prime}}\right)\),\({\mathbf{F}}_{\mathrm{t}}^{-1}={\mathbf{F}}_{\mathrm{t}}^{-1}\left(t,{t}^{\mathrm{^{\prime}}}\right)\) is the relative deformation gradient tensor, and \({u}^{^{\prime}}\) is the length of u′. The orientation average is indicated by < … > ₀,i.e. an average over an isotropic distribution of unit vectors \(\mathbf{u}\).

\(f=f\left(t,{t}^{\mathrm{^{\prime}}}\right)\) represents the inverse of the relative tube diameter a/a₀, and at the same time the relative length of a deformed tube segment Wagner and Narimissa (2021),

At time t = t′ the tube segment was created with equilibrium tube diameter a₀ and equilibrium length l₀. Equation (12) is a direct consequence of Eq. (A9) of Doi and Edwards (1978a), who showed that the line density n/l, i.e., the number n of monomer units of length b that are found per length l of the tube, is a well-defined thermodynamic quantity and defines the tube diameter a by the relation n/l = a/b². For \(f\equiv 1\), Eq. (8) reduces to the original Doi-Edwards IA (DEIA) model.

\({\mathbf{S}}_{DE}^{IA}\) is determined directly by the deformation history, according to Eq. (9), while \(f\) is found as solution of an evolution equation, considering affine tube segment deformation balanced by enhanced Rouse relaxation Wagner and Narimissa (2021). With increasing stretch and decreasing tube diameter a, the number of monomers in a control volume of length and diameter a will decrease with the consequence of enhanced relaxation of stretch in this control volume. The incremental increase of the relaxation rate with stretch is proportional to the 4th power of the stretch, and the relaxation rate therefore is proportional to the 5th power of the stretch. The evolution equation of the ERS model is obtained as a balance of extension rate versus relaxation rate Wagner and Narimissa (2021),with \(\mathbf{K}\)  the deformation-rate tensor and initial condition  \(f\left(t,{t}^{^{\prime}}=0\right)=1\). Equations (8) and (13) represent the ERS model and are solved numerically.

In the limit of vanishing chain stretch, Eq. (13) degenerates naturally into the classical stretch evolution equation,with \({\tau }_{R}=\frac{{\varsigma b}^{2}{N}^{2}}{{3\pi }^{2}kT}\) being the Rouse stretch relaxation time. The classical assumption is that \(\varsigma ={\varsigma }_{0}\) is constant i.e. independent of deformation. The evolution Eq. (14) in combination with the stress tensor Eq. (8) leads to a diverging elongational viscosity \({\eta }_{E}\to \infty\) as soon as the Weissenberg number \({Wi}_{R}={\dot{\varepsilon }\tau }_{R}\) approaches a value of \({Wi}_{R}=1\), see e.g. Wagner (2015).

Friction reduction models assume a decreasing monomeric friction coefficient \(\varsigma\) with increasing orientation of chain segments resulting in an effective Rouse time

\({\varsigma }_{0}\) is the friction coefficient at equilibrium. Consequently the stretch evolution Eq. (13) can then be expressed as

Comparison of Eqs. (13) and (16) leads to a normalized friction coefficient,

With \({f}^{5}-1=\left(f-1\right)\left({f}^{4}+{f}^{3}+{f}^{2}+f+1\right)\), this results in (Wagner and Narimissa (2021)

The reduction of the stretch relaxation time with increasing stretch f and increasing tension in the chain can be represented by a monomeric friction coefficient \(\varsigma =\varsigma \left(f\right)\), which reduces with increasing stretch, according to Eq. (18). Thus, the ERS model relates monomeric friction reduction quantitatively and parameter-free to chain segment stretch. In the perspective of the ERS model, monomeric friction reduction is the consequence of faster stretch equilibration of smaller and smaller chain segments of diameter and length \(a={a}_{0}/f\) with increasing stretch. On the other hand, as reviewed by Matsumiya and Watanabe (2021), friction reduction models are based on the assumption of monomeric friction being reduced by flow-induced coalignment of Kuhn segments, quantified by empirical correlations between friction coefficient and a suitable scalar measure of orientation. We remark that in these models, monomeric friction reduction might equally well be correlated with chain segment stretch as in the ERS model, because stretch and orientation are correlated by the orientation tensor S in the evolution Eq. (16) of stretch. From this perspective, we may conclude that there is a common starting point of friction reduction models and the ERS model: It is the importance of faster stretch relaxation of smaller chain segments with increasing stretch.

Figure 2 presents a comparison of the elongational stress growth coefficient \({\eta }_{E}^{+}\left(t\right)\) of the three PPC melts considered with predictions of the ERS model. For PPC69k (Fig. 2a) the relaxation times \({\tau }_{i}\) given in Table 2 had to be time–temperature shifted by a temperature factor a_T = 0.7 in order to obtain agreement with the experimentally observed start-up viscosity. The discrepancy between predictions of the linear-viscoelastic start-up viscosities from SAOS versus VADER measurements was already noted by Masubuchi et al. (2022), as discussed below, and may be due to a small difference between the SAOS reference temperature and the VADER measurement temperature for PPC69. Overall and considering that the ERS model is exclusively based on the LVE characterization of the PPC melts, the agreement of data and model predictions for PPC69k (Fig. 2a) and PPC111k (Fig. 2b) can be rated as fair, for PPC158k (Fig. 2c) even as quantitative.

The comparison of the elongational stress growth coefficient \({\eta }_{\mathrm{E}}^{+}(t)\) of the PS520k melt with predictions of the ERS model is shown in Fig. 3. Again, agreement of data and model prediction is obtained within experimental accuracy.

The experimental data of the steady-state elongational viscosity \({\eta }_{E}(\dot{\varepsilon })\) as a function of strain rate \(\dot{\varepsilon }\) and predictions of the ERS model for the three PPC melts investigated at T = 70 °C and the PS520k melt at T = 130 °C are shown in Fig. 4 a. Within experimental accuracy, the agreement of data and model predictions can be rated as excellent. While the predictive capability of the ERS model has been shown earlier for nearly monodisperse PS melts (Wagner and Narimissa (2021), it is remarkable that the polydispersity of the polymer melts considered here with values of 1.3 to 1.4 does not have a significant effect. The effective “molecular weight average” Rouse time when calculated by Eq. (7) with \(M\equiv {M}_{w}\) seems to be sufficient as the characteristic parameter which determines chain stretch.

At high Weissenberg numbers \({Wi}_{R}\equiv {\dot{\varepsilon }\tau }_{R}\) and large stretch, when the steady state is reached and \(\partial f/\partial t=0\), the evolution Eq. (13) of the ERS model reaches the limit of \({f}^{2}=\sqrt{{5Wi}_{R}}\), and the asymptotic tensile stress from Eq. (8) is given by \(\sigma ={5G}_{N}{f}^{2}={5G}_{N}\sqrt{{5Wi}_{R}}\). As shown in Fig. 4b, the asymptotic reduced elongational viscosity \({\eta }_{E}/\left({G}_{N}{\tau }_{R}\right)=5\sqrt{{5Wi}_{R}^{-1/2}}\) is approached at high \({Wi}_{R}\), while the experimental data feature a power-law slope, which is higher than − 1/2 and for well-entangled polymers closer to a value of − 0.4, as already noted by Wagner et al. (2005). This is due to the broadness of the relaxation spectrum of even monodisperse polymers with the consequence that while the central part of the molecule with long relaxation times is increasingly orientated and stretched with increasing \({Wi}_{R}\), the chain ends with short relaxation times remain nearly isotropic. Figure 4 b is a temperature invariant representation of the data, which also highlights the importance of the Rouse time in the elongational thinning behaviour of polymer melts.

Because model and simulation scheme have been reported earlier (Masubuchi et al. 2001, 2014a, 2016), only a brief description shall be given below. In the primitive chain network (PCN) model, we replace an entangled polymer network with a slip-link network consisting of strands, nodes, and dangling ends. A polymer chain in the system corresponds to a path connecting a pair of dangling ends via strands. At each network node, a slip-link is located to bundle two segments according to the binary assumption of entanglement. The slip-links do not affect the sliding motion of the chain along the path, whereas they suppress the fluctuations in the perpendicular directions. A slip-link is removed when the bundled chain slides off at the chain end. Vice versa, a new slip-link is created at a dangling segment when it protrudes from the connected slip-link to hook another segment randomly chosen from the surrounding ones. The state variables of the system are the positions of slip-links and of dangling ends {R}, the number of Kuhn segments assigned to network strands and dangling segments {n}, and the number of slip-links on each polymer chain {Z}. In the numerical simulation, {R} obeys a Langevin-type equation of motion, in which the force balance is considered between the drag force, the tension acting on the connecting strands, the osmotic force suppressing density fluctuations, and the thermal random force. The time development of {n} is described by a change-rate equation, in which the same force balance is considered as for {R} but in 1-dimension along the chain. We trigger the change of {Z} by monitoring {n} at each dangling segment. When the value of n at an end segment falls below a certain critical value, we remove the connected slip-link and release the bundled segment. On the contrary, when n exceeds another critical value, we create a new slip-link on the subjected dangling segment to randomly hook a segment within a certain distance.

The simulation code employed in this study is the same as that utilized in the earlier studies (Yaoita et al. 2012; Masubuchi et al. 2014a, 2014b, 2021, 2022; Takeda et al., 2015), except that we consider here monomeric friction reduction, according to Eq. (18), as described later. The calculations are made according to non-dimensional quantities normalized by units of length, energy, and time. These units are chosen as the average strand length \({l}_{0}\) under equilibrium conditions, the thermal energy \({k}_{\mathrm{B}}T\), and the diffusion time of the single network node \({\tau }_{0}=2{n}_{0}{\zeta }_{0}{l}_{0}^{2}/6{k}_{\mathrm{B}}T\), where \({\zeta }_{0}\) is the friction of the Kuhn segment at equilibrium, and \({n}_{0}\) is the average number of Kuhn segments on the single strand. The factor of 2 stands for the fact that each network node consists of two strands.

Because the reduction of friction has no effect on the linear-viscoelastic response, the material parameters are the same as in the previous study (Masubuchi et al. 2022). The units of modulus \({G}_{0}\), molecular weight \({M}_{0}\) and equilibration time \({\tau }_{0}\) of a Kuhn segment were chosen as \({G}_{0}=1.9\) (MPa), \({M}_{0}=3.3\) (kg/mol), and \({\tau }_{0}=7.8\times {10}^{-2}\) (s) for PPC melts at \(T=70^\circ{\rm C}\), whereas \({G}_{0}=0.55\) (MPa), \({M}_{0}=11\) (kg/mol), and \({\tau }_{0}=1.3\) (s) were used for the PS melt at \(T=130^\circ{\rm C}\). Note that G₀ and M₀ are different from the plateau modulus G_N and the entanglement molecular weight M_e, as reported earlier (Yang et al. 2021, 2022). The difference is due to fluctuations imposed on network nodes in the PCN model (Masubuchi et al. 2003, 2020; Uneyama and Masubuchi 2021). Finite chain extensibility was considered with the maximum stretch \({\lambda }_{\mathrm{max}}\) defined as \({\lambda }_{\mathrm{max}}=\sqrt{{n}_{K}}\), where \({n}_{K}\) is the number of Kuhn segments on each strand on average. Determining the value of \({n}_{K}\) from \({M}_{0}\), we chose \({\lambda }_{\mathrm{max}}=4.8\) for PPC, and \({\lambda }_{\mathrm{max}}=3.9\) for PS.

Concerning the change of friction, all the dynamics is accelerated when monomeric friction is reduced. Differently from the ERS model described above, not only the chain stretch and contraction, but also the segmental orientation and the entanglement rearrangement are affected by friction reduction. In the previous study (Masubuchi et al. 2022), the friction reduction model proposed by Ianniruberto et al. (2012) was used,

Here, \(\zeta\) is the monomeric friction coefficient, \(S\) is the average strand orientation defined as \(S=\langle {u}_{x}^{2}\rangle -\langle {u}_{y}^{2}\rangle\) with the strand orientation vector \(\mathbf{u}=({u}_{x}, {u}_{y},{u}_{z})\). \({S}_{c}\) and \(\alpha\) are empirical parameters that were chosen as \({S}_{c}=0.1\) and \(\alpha =1.25\) (Masubuchi et al. 2022). In this study, we implemented the ERS friction reduction model of Eq. (18) for comparison by defining the stretch function \(f\) from Eq. (12) as the ratio of the average strand length \(\langle l\rangle\) under flow to the equilibrium strand length \({l}_{0}\),

Here, \(b\) is the Kuhn segment length, and \(\langle \sqrt{n}\rangle\) is the average square root of the number of Kuhn segments on each strand.

The simulations were conducted with periodic boundary conditions. We created 8 independent initial configurations in a flat simulation box with the dimensions of 4 × 45 × 45. After sufficient equilibration, we stretched the system up to 506 × 4 × 4 corresponding to a maximum Hencky strain of 4.8 in order to obtain the elongational stress growth coefficient \({\eta }_{\mathrm{E}}^{+}(t)\). The simulation box used is sufficiently large to accommodate the largest molecule with \(Z=88\) as shown in the previous study (Masubuchi et al. 2022). Due to the nature of stochastic molecular simulations, the stress fluctuates over time and depends on the initial configuration. The magnitude of these inevitable uncertainties is discussed in the Appendix.

To account for the polydispersity of the melts, we assumed a distribution of molecular weights mimicking the polydispersity and the weight average molecular weight of the polymers considered, as presented in Table 3 and used in the previous study (Masubuchi et al. 2022). For comparison with the simulations for PPC158k, we also performed simulations with a monodisperse equivalent Z47 of PPC158k.

Figure 5 shows the simulation results for the PPC melts. The simulation with constant friction (black dotted curves) clearly overestimates the experimental data (symbols), as reported previously (Masubuchi et al. 2022). If we employ the friction model proposed by Ianniruberto et al. (2012), Eq. (19) (red curves), the stress growth coefficient at high strain rates is reduced and approaches the experimental data. The simulation with the friction reduction resulting from the ERS model, Eq. (18) (blue curves), reproduces the data as well, even without any fitting parameter.

Note that the simulation entirely overestimates the data for PPC69k, although the linear viscoelasticity for this specific material has been semi-quantitatively reproduced (Masubuchi et al. 2022). As already mentioned, this discrepancy may be due to the time–temperature shift factor discussed above, and better agreement could be attained if \({\tau }_{0}\) were optimized separately. Nevertheless, for consistency and direct comparison with the previous work (Masubuchi et al. 2022), we decided to use the same parameter set for the PCN simulations of all the PPC melts examined, as employed previously (Masubuchi et al, 2022).

Figure 6 exhibits the simulation results for PS520k. This specific sample was chosen for comparison to PPC158k, and the average number of entanglements per chain and its polydispersity are almost identical to those of PPC158k. Thus, as shown in Table 3, the simulated system is the same as that of PPC158k, except for the maximum stretch ratio. However, the difference of \({\lambda }_{\mathrm{max}}\) does not significantly affect the simulation result, which is essentially the same as that observed in Fig. 5 c.

Figure 7 shows the comparison between experimental data and PCN predictions for the steady-state elongational viscosity \({\eta }_{E}\left(\dot{\varepsilon }\right)\). As reported previously and shown in Figs. 5 and 6, the simulation with constant friction (black dotted curves) overestimates the data showing an upturn at a strain rate close to the contraction rate of the chains. In contrast, the experimental data do not exhibit such an upturn. Even at lower stretch rates than the critical rate for the upturn, the experimental data lie well below the prediction. They monotonically decrease with increasing strain rate when the strain rate is larger than the inverse of the longest relaxation time. The simulations with monomeric friction reduction capture this monotonically decreasing trend. The friction reduction models of Eqs. (18) (blue curves) and (19) (red curves) are in agreement with the experimental data both for the PPC melts (Fig. 7a) and for PS520k (Fig. 7b). Due to saturation of orientation at large strain rates, the simulation results using Eq. (19) show a trend to a certain constant lower-limited value of the viscosity for PPC 158 k, and deviate from the blue curve resulting from use of from Eq. (18), which shows a continued monotonic decrease. However, no experimental data are available at these high elongation rates.

Figure 8 a exhibits the friction change for PPC melts as a function of the strain rate. Equations (18) and (19) prescribe similar magnitudes of friction reduction, as reflected by the general agreement of the \({\eta }_{E}(\dot{\varepsilon })\) predictions. However, the strain rate dependence of the two friction models is qualitatively different from each other: The relative friction reduction resulting from Eq. (18) shows a monotonously decreasing trend, as it is inversely related to the increasing stretch with increasing strain rate. In contrast, Eq. (19) leads to concave curves with a decreasing slope at high elongation rates, in agreement with the increasing saturation of orientation at higher strain rates and the approach to a predicted lower-limited constant viscosity, as shown in Fig. 7. Experimental data at larger strain rates would be needed in order to decide on the importance of either saturation of orientation via the empirical expression of Eq. (19) or increasing stretch as expected by Eq. (18), on friction reduction.

Figure 8 b shows the reduction of the normalized average number of entanglements per chain plotted against the strain rate. For both simulations with friction reduction using Eqs. (18) and (19), the loss of entanglements is smaller, compared to the case with constant friction (shown by dotted black curves), reflecting that the effective strain rate is reduced by friction reduction, as reported earlier (Yaoita et al. 2012). Although the simulated losses of entanglements resulting from Eqs. (18) and (19) are rather similar, the effect of friction reduction on the loss of entanglements is also noticeable by comparison with Fig. 8a: Whenever the friction coefficient is lower, the loss of entanglements is smaller, and vice versa. Note that since the polymer systems examined have molecular weight distributions, the magnitude of the entanglement losses cannot be compared quantitatively to the case of monodisperse systems.

One may argue that the results may depend on the modelling of the molecular weight distribution. To discuss this issue, we conducted simulations for a monodisperse system, for which the molecular weight is assumed to be the same as the weight average molecular weight M_w of PPC158k. The result is shown in Fig. 9. As reported previously, the polydispersity of M_w/M_n = 1.3 does not strongly affect the simulation results, neither for constant friction nor for the friction reduction model, according to Eq. (19). A similar result is obtained here for the ERS friction reduction model of Eq. (18), and we conclude that the molecular weight distribution does not play any significant role in this case.

Another issue of the comparison between predictions of the tube-based ERS constitutive model and the PCN simulation is finite chain extensibility (FENE), which is implemented only in the simulation. Figure 10 shows the PCN simulation result for PPC158k without FENE, yet with the ERS friction reduction model, according to Eq. (18). At large strain rates (\(\dot{\varepsilon }\ge 0.5\) s⁻¹), the simulation with FENE (dotted curve) shows a slight upward deviation from the simulation results obtained by assuming a Gaussian spring (solid curve). This upward trend results from the upper-limited chain stretch in the case of FENE. However, in the experimental window of strain rates investigated, FENE has no significant effect.