### Linear viscoelasticity

Figure 1 shows the linear viscoelasticity of the examined bidisperse blends. With the parameters shown in Table 1 the linear viscoelasticity is reasonably reproduced as earlier reported for the other bidisperse polystyrene melts [21]. In particular, for Blend1 the second plateau (at *ω* ∼ 2 × 10^{−3} rad/s) is correctly captured. On the other hand, for the other two samples the simulation shows the faster relaxation than experiment so that the longest relaxation time is somewhat underestimated.

### Transient elongational viscosity

Figure 2 shows the transient viscosity \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \). The simulation nicely captures \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \) at \( \dot{\varepsilon } > 1/\tau_{\text{RL}} = 0.016\;{\text{s}}^{ - 1} \) (where *τ*
_{RL} is the viscoelastic Rouse relaxation time of the long chain calculated from the relation *τ*
_{R} = *Z*
^{2}_{
L
}
*τ*
_{0}/2*π*
^{2} as derived earlier [22]). For example, for \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \) of Blend1 the simulation result (red curve) at \( \dot{\varepsilon } = 0.3\,\;{\text{s}}^{ - 1} \) (the leftmost curve) almost coincides with the data (symbol). Fair coincidence can be seen for all the blends at \( \dot{\varepsilon } = 0. 3, \, 0. 1 {\text{ and }}0.0 3 \,\;{\text{s}}^{ - 1} \). (Note that for Blend2 \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \) at \( \dot{\varepsilon } = 0.3\,\,{\text{s}}^{ - 1} \) is not available in the literature.) On the other hand, the simulation underestimates the data at \( \dot{\varepsilon } \le 1/\tau_{\text{RL}}. \) In particular, at \( \dot{\varepsilon } = 0.00 3 {\text{ and }}0.00 1\, {\text{s}}^{ - 1} \), \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \) from the simulation is close to the linear viscoelastic envelope (black dotted curve, obtained from the simulated *G*′ and *G*″) but the data show clear deviations from the linear viscoelastic response. This discrepancy in \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \) between the data and simulation results is partly attributable to the discrepancy in *G*′ and *G*″, in which the simulation predicts faster relaxation than experiment for Blend2 and Blend3 as mentioned for Figure 1. However, the discrepancy in \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \) is also seen for Blend1, for which *G*′ and *G*″ are quantitatively reproduced. The mechanism of this discrepancy is unknown.

Concerning the effect of SORF, the simulations with and without SORF exhibited no difference in \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \), except for Blend3 at \( \dot{\varepsilon } = 0.3\,\,{\text{s}}^{ - 1} \) (the leftmost curve). At this condition, the simulated \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \) without SORF (blue curve) is significantly higher than the result with SORF (red curve). In comparison to the monodisperse melts, in which SORF appears to reduce the viscosity at \( \dot{\varepsilon } > 1/\tau_{\text{R}} \), in the bidisperse blends the activation of SORF is suppressed by the short chains, as discussed later. Nevertheless, the simulation with SORF reproduces the data better than that without SORF.

### Steady state elongational viscosity

Figure 3 shows the steady state viscosity \( \eta_{\text{E}} (\dot{\varepsilon }) \). As mentioned for \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \), the simulation reasonably captures the data at \( \dot{\varepsilon } > 1/\tau_{\text{RL}} \) whereas it underestimates the data at \( \dot{\varepsilon } \le 1/\tau_{\text{RL}} \). The simulated \( \eta_{\text{E}} (\dot{\varepsilon }) \) (red and blue curves) is consistent with the linear viscoelasticity at low \( \dot{\varepsilon } \); the simulated \( \eta_{\text{E}} (\dot{\varepsilon }) \) (blue and red curves) converges to 3*η*
_{0} (dotted line) around \( \dot{\varepsilon } = 1/\tau_{1} \). On the other hand, the experimental \( \eta_{\text{E}} (\dot{\varepsilon }) \) is higher than 3*η*
_{0} suggesting some slow relaxation modes for which the relaxation time is longer than the simulated *τ*
_{1}. The simulation failed to predict these behaviors by unknown reasons. The effect of SORF (difference between red and blue curves) appears only at \( \dot{\varepsilon } \) which is well-beyond 1/*τ*
_{RL}, as mentioned for \( \eta_{\text{E}}^{ + } (\dot{\varepsilon },t) \).

### Decoupling analysis on the role of the long chain

In this section, we attempt molecular level analysis on the steady state viscosity for Blend1 to reveal the significance of the long chain contribution on the basis of the decoupling approximation [18] in which the stress is decomposed as

$$ \sigma \approx \left( {\frac{Z}{{Z_{0} }}} \right)f_{\text{FENE}} \lambda^{2} S $$

(2)

Here, *Z* is the average number of subchains and *Z*
_{0} is its equilibrium value, *f*
_{FENE} is the FENE spring constant, *λ* is the stretch normalized with respect to the maximum stretch, and *S* is the subchain orientation defined as *S* = *u*
^{2}_{
x
}
− *u*
^{2}_{
y
}
(where **u** is the subchain orientation vector and *x* and *y* are the stretch and normal directions, respectively). The numerical prefactor is neglected here. Figure 4 shows the decoupled measures (*Z*/*Z*
_{0}, *S*, *λ*
^{2} and *f*
_{FENE}) plotted against \( \dot{\varepsilon } \). In the top panel, as shown in the earlier studies, the total viscosity (black curve) is dominated by the minor long chain component (red solid curve), and the short chain contribution (red dotted curve) is relatively small. In the bottom panel, the reduced friction with respect to the equilibrium value, \( \zeta /\zeta (0) \), is shown for comparison.

The behavior of molecular measures shown in Figure 4 can be divided into five regimes with respect to \( \dot{\varepsilon } \). In the low strain rate limit (\( \dot{\varepsilon } < 1/\tau_{1} \): Region I) *S* linearly increases with increasing \( \dot{\varepsilon } \) with no stretch both for long and short chains. This linearity of *S* appears in the viscosity that maintains the Trouton’s viscosity. At the strain rates located in the range \( 1/\tau_{1} < \dot{\varepsilon } < 1/\tau_{\text{RL}} \) (Region II), the stretch of the long chain occurs in addition to the linear growth of *S*. This simultaneous growths of *S* and *λ* induce the increase of \( \eta_{\text{E}} (\dot{\varepsilon }) \), and consequently \( \eta_{\text{E}} (\dot{\varepsilon }) \) is lifted up beyond 3*η*
_{0} to show the strain hardening. (One may argue that the long chain stretch should be observed at \( \dot{\varepsilon } > 1/\tau_{\text{RL}} \), and not at the lower strain rates. However, the stretch transition is not sharp and the stretch gradually starts from lower \( \dot{\varepsilon } \), as discussed for elastic dumbbells [6]. It is also noted that the *τ*
_{RL} value discussed here is the viscoelastic Rouse relaxation time that is 1/2 of the end-to-end relaxation time. For the monodisperse melt of long chain exhibits similar stretching behavior as shown later.) Interestingly, the short chain viscosity also increases with increasing \( \dot{\varepsilon } \) even without stretch. This increase of short chain viscosity is due to the coupling between long and short chains through entanglement. Nevertheless, the short chain contributes much less than the long chain.

In the higher strain rate range (\( 1/\tau_{\text{RL}} < \dot{\varepsilon } \)), there exist two critical strain rates. One of the critical rates is located around 10^{−1} s^{−1} beyond that the short chain stretches (see dotted line in the 2nd panel). The other one is around 3 × 10^{−1} s^{−1} beyond that the friction decreases due to SORF (see bottom panel). These critical strain rates are indicated as dash-dotted vertical lines in Figure 4 to classify the regions III–V. At the strain rates in the range \( 1/\tau_{\text{RL}} < \dot{\varepsilon } < \) 10^{−1} s^{−1} (Region III), the growth rate of *S* for the long chain is declined as *S* approaches to the maximum value (*S* ≤ 1 by definition). The growths of *λ* and *f*
_{FENE} for the long chain retain the viscosity constant. The linear growth of the short chain viscosity is maintained, even though the short chain stretch does not occur in this region, but the contribution is still small. On the other hand, in the Region IV (\( 10^{ - 1} {\text{s}}^{ - 1} < \dot{\varepsilon } < 3 \times 10^{ - 1} {\text{s}}^{ - 1} \)), as the short chain stretch grows, the short chain contribution in the viscosity also grows to be comparable to the long chain contribution. The long chain viscosity increases mainly owing to the increase of *f*
_{FENE}. In the highest strain rate region (Region V), SORF occurs due to the stretch/orientation of the short chains that dominates the stretch/orientation order parameter of the system as a whole.

### Comparison of the long chain behavior to the monodisperse system

To illuminate the difference of the long chain behavior between monodisperse and bidisperse systems, the decoupled measures for both cases are shown in Figure 5 as functions of the Weissenberg number *Wi*
_{RL} defined with respect to *τ*
_{RL}. The subchain number is omitted for simplicity.

In the slow flow regime (*Wi*
_{RL} < 1), the remarkable difference is observed in the growth of *S* (3rd panel). Although it is common for both cases that the *S* growth starts at \( \dot{\varepsilon } \approx 1/\tau_{1} \), the shortened *τ*
_{1} of the bidisperse system due to the short chains retards the *S* growth (red solid curve) in comparison to that of the monodisperse case (black dotted curve). Interestingly, *S* for the bidisperse system steeply increases to reach a similar value to the monodisperse system at *Wi*
_{RL} = 1. As a consequence, the bidisperse system exhibits the simultaneous growth of *S* and *λ*, which causes the strain hardening (*η*
_{E} > 3*η*
_{0}). On the other hand, for the monodisperse system the *λ* growth separately occurs after the *S* growth. Since the growth rates for *λ* and *S* against \( \dot{\varepsilon } \) are less than the linear relationship, strain softening occurs (*η*
_{E} < 3*η*
_{0}). Note that the critical *Wi*
_{RL} value at which *λ* starts growing is around 2 × 10^{−1} and it is common for the monodisperse and bidisperse systems. (As mentioned in the previous section, it is noted that the stretch observed at the strain rates lower than *Wi*
_{RL} = 1 is just due to the nature of the Rouse chain [6].) The growth rate of *λ* is also similar to each other in *Wi*
_{RL} < 1.

It is fair to mention that the simultaneous growth of *S* and *λ* for bidisperse systems has been suggested earlier (but rather implicitly). Wagner [13] analyzed the dataset of Nielsen et al. [5] by the molecular stress function theory to report that the strain hardening is induced by the dynamic tube dilation through the reduction of the inter-chain tube pressure. Although Wagner did not discuss the effect of long chain orientation, his picture is essentially common with ours in a sense that the reduced motional constraint for the long chain induces the strain hardening.

In the fast flow region (*Wi*
_{RL} ≥ 1), the *S* growth saturates and *λ* growth dominates the *η*
_{E} behavior. In this respect, the remarkable difference between the monodisperse and bidisperse systems is the appearance of SORF. See bottom panel, in which the friction decreases with increasing \( \dot{\varepsilon } \) for the monodisperse system (black dotted curve) whereas it retains the equilibrium value for the bidisperse system (red solid curve) up to the much higher value of *Wi*
_{RL} (∼20). The difference in friction is induced by the total stretch/orientation of the system. For the monodisperse system, the high stretch/orientation of the system as a whole induces SORF. This occurrence of SORF suppressed *λ* and *f*
_{FENE}, and then the monotonic decrease of *η*
_{E} is attained. On the contrary, for the bidisperse system, the average stretch/orientation are maintained at low values due to the short chains (that is not stretched/oriented at this flow rate). Then the growth rate of *λ* against \( \dot{\varepsilon } \) is close to the linear relationship, which compensates the declined growth rate of *S* and maintains *η*
_{E} > 3*η*
_{0}.