DOI: 10.1007/s13367-019-0024-3
Multi-chain slip-spring simulations for polyisoprene melts
Yuichi Masubuchi1,2,* and Takashi Uneyama1,2
1Department of Materials Physics, Nagoya University, Nagoya 4648603, Japan
2Center of Computational Science, Nagoya University, Nagoya 4648603, Japan
(Received June 27, 2019; final revision received October 3, 2019; accepted October 5, 2019) The multi-chain slip-spring (MCSS) model is a coarse-grained molecular model developed for efficient sim- ulations of the dynamics of entangled polymers. In this study, we examined the model for the viscoelasticity of polyisoprene (PI) melts, for which the data are available in the literature. We determined the conversion factor for the molecular weight from the fitting of the molecular weight dependence of zero-shear viscosity.
According to the obtained value, we calculated the linear viscoelasticity of several linear PI melts to deter- mine the units of time and modulus. Based on the conversion factors thus determined, we predicted linear viscoelasticity of 6-arm star PI melts, and viscosity growth under high shear for linear PI melts. The pre- dictions were in good agreement with the data, demonstrating the validity of the method. The conversion factors determined were consistent with those reported for polystyrene melts earlier, whereas the relations between the conversion factors are still unknown.
Keywords: molecular simulations, rheology, entanglement
1. Introduction
The multi-chain slip-spring (MCSS) model is one of the coarse-grained multi-chain models for simulations of entangled polymers at the molecular level (Masubuchi 2014; 2016a). Due to the slow relaxation nature, molec- ular dynamics simulations with atomistic details are prac- tically difficult for entangled polymers, yet attempted (Baig et al., 2010a; 2010b; Stephanou et al., 2010). The widely used and established approach is the bead-spring type simulations (Kremer and Grest, 1990). Recent studies with highly optimized computation codes have reported some simulation results for moderately entangled polymer melts, in which the bead number per chain was increased up to several hundred (Xu et al., 2015). For this approach, the computation costs are still high due to the inter-bead interaction, which is short-range and repulsive like the Lennard-Jones interaction. Such a short-range and strong repulsion limits the size of the time step for the numerical integration rather small. In this respect, the simulations with softer interactions that allow interpenetration of the particles are favorable for the computation. However, the simulations with such an interaction cannot reproduce the entangled polymer dynamics unless ad-hoc uncrossability between the chains is implemented (Pan and Manke, 2003). Attempts have been made for the realization of uncrossability via the inter-bond interaction (rather than
the inter-bead interaction) (Kumar and Larson, 2001) and via the geometrical consideration (Padding and Briels, 2001). However, such rigorous treatments for uncrossabil- ity demand computation comparable to calculations with the conventional bead-spring simulations. An alternative approach is the slip-spring model, in which the entangle- ment is replaced by a virtual spring that connects bead- spring chains without short-range repulsive interactions.
To mimic the entanglement, the anchoring points of the virtual spring slide along the connected chains. The virtual spring is annihilated by a certain probability when one of the anchoring points reaches the chain end. At each chain end, a new virtual spring is created to maintain the aver- age number of springs in the system. Although the com- putational efficiency is less than the multi-chain slip-link model (Masubuchi, 2016b; Masubuchi et al., 2001; 2004) owing to the soft interactions the simulations for moder- ately entangled systems are quite efficient.
Inspired by the single-chain version (Likhtman, 2005;
Uneyama, 2011), the multi-chain slip-spring models have been developed by a few groups in parallel (Chappa et al., 2012; Langeloth et al., 2013; Ramírez-Hernández et al., 2013; Uneyama and Masubuchi, 2012; Vogiatzis et al., 2017). The critical issue for multi-chain modeling is the inter-bead interaction. For the single-chain version, one of the ends for the virtual spring is anchored at a certain point in space. Thus, the virtual spring does not connect different chains, and the conformational distribution of the chain reduces to that of an ideal chain when we integrate- out the degrees of freedom for the anchoring points. This construction ensures that the ideal chain statistics are straightforwardly retained.
In contrast, for the multi-chain models, because both
# This paper is based on an invited lecture presented by the corresponding author at the 30th Anniversary Symposium of the Korean Society of Rheology (The 18th International Symposium on Applied Rheology (ISAR)), held on May 21-24, 2019, Seoul.
*Corresponding author; E-mail: mas@mp.pse.nagoya-u.ac.jp
ends of the virtual spring connect the mobile chains, the conformational distribution functions for the chains are affected by the inclusion of virtual springs. Indeed, they play a role as an attractive interaction between the seg- ments to compress the chains. Because the entanglement is purely kinetic and does not affect the chain statistics, we must compensate for the artificial effect of the virtual spring on the chain statistics. The rigorous treatment by the inter-bead repulsive, yet soft interaction, has been pro- posed (Chappa et al., 2012; Uneyama and Masubuchi, 2012). The intermolecular interaction with finite com- pressibility (Ramírez-Hernández et al., 2013) and the weak repulsive interaction employed in dissipative parti- cle dynamics simulations (Langeloth et al., 2013; Masu- buchi et al., 2016) are effective as well.
Nevertheless, the models with adequate treatments for the artifacts of the inclusion of virtual springs are capable to simulate a variety of polymeric systems including block- copolymers (Ramírez-Hernández et al., 2013; 2018), mix- tures of long and short chains (Langeloth et al., 2014), polymer solutions (Masubuchi et al., 2016), branch poly- mers (Masubuchi, 2018), network polymers (Masubuchi and Uneyama, 2019; Megariotis et al., 2018), etc. The sim- ulations under fast shear flows have also been attempted (Masubuchi, 2015; Masubuchi and Uneyama, 2018a;
Ramírez-Hernández et al., 2013; 2015). The other import- ant direction is the linkage to the atomistic models (Sgou- ros et al., 2017; Vogiatzis et al., 2017) and the other coarse-grained models (Masubuchi and Uneyama, 2018a).
In this specific study, we shall report the consistency of the MCSS simulations with the literature data for poly- isoprene (PI) melts (Auhl et al., 2008; Matsumiya et al., 2014). In the earlier studies, we have evaluated the model by the comparison of rheological data for polystyrene (PS) (Masubuchi, 2018), and by the comparison to the generic bead-spring model (Masubuchi and Uneyama, 2018a;
2018b). Because the polymer dynamics is virtually uni- versal and not strongly dependent on the chemistry (Ferry, 1980), the evaluation already performed would be suffi- cient. However, parameter determination for specific chemistries is also necessary for practical use. Besides, the parameters must be evaluated for consistency to the other materials and the theories. We have performed the MCSS simulations for several linear polymers and a few star- branched polymers. For the linear polymers, we also per- formed the simulations under fast shear flows. We have confirmed that the simulation results are in semi-quanti- tative agreement with the experimental data. The param- eters used are consistent with those for PS. Details are shown below.
2. Model and Simulations
We used the MCSS model proposed earlier (Uneyama
and Masubuchi, 2012) with the modifications for shear flows (Masubuchi, 2015) and the branch point dynamics (Masubuchi, 2018). In this model, a number of Rouse chains are dispersed in a simulation box with a periodic boundary condition. The Rouse chains are connected by virtual springs, for which both ends hop between the Rouse beads along the chain. The virtual spring is anni- hilated with a certain probability when one of the ends comes to the chain end, and vice versa, new virtual springs are introduced at the chain ends.
The free energy is given by
(1)
The first and second terms on the right-hand side are the contributions from the Rouse springs and the virtual springs, respectively. b is the average bond length between the Rouse segments, Ri,k is the position of the Rouse beads k on the chain i, and Ns is the parameter that determines the intensity of the virtual spring. S,j is the connectivity matrix for the virtual spring . The third term is the soft- core repulsive interaction between the Rouse beads to eliminate the artificial effect on the chain statistics from the virtual springs. e/kT is the activity of the virtual springs to control the density of virtual springs in the system by a grand canonical manner according to the chemical potential .
It must be noted that both of the Rouse spring and the virtual spring are Gaussian and the effect of finite chain extensibility is not considered, as seen in the free energy given in Eq. (1). We use the linear springs owing to our previous study (Masubuchi, 2015), in which the molecular motion under moderate shear is reasonably reproduced with the adopted linear springs. Meanwhile, similar earlier studies considered the finite chain extensibility (Chappa et al., 2012; Sgouros et al., 2017).
We derived the governing equations and algorithms for the dynamics from the total free energy of the system to fulfill the detailed balance. The time development of Ri,k
obeys the Langevin equation of motion, and the SLLOD method was implemented for the application of shear flows (Masubuchi, 2015). The Glauber dynamics manage the hopping of the virtual springs. For the branch poly- mers, we also consider the hopping across the branch point (the so-called SHAB), yet such a mechanism is not essential for symmetric star polymers (Masubuchi, 2018).
The annihilation and creation of the virtual springs are in balance under equilibrium by the master equation. In addi- tion to the parameters that appear in the free-energy (e/kT and Ns), we introduce the friction coefficients for the bead motion and that for the kinetics of virtual spring s for
F kBT --- = 3
2b2 ---
i k
Ri,k+1–Ri k 2 + 3 2Nsb2 ---
RS,1,S,2–RS,3,S,42 + e /kTi,k,j,l
exp 32Nsb2 ---
– Ri,k–Rj l2
these dynamical equations. For simplicity, we assume =
s. See the previous publication for further details and the numerical implementation (Uneyama and Masubuchi, 2012).
We performed the simulations for linear, and star-branched polymer melts with dimensionless units, for which the units of length, energy, and time are the bond length b, the thermal energy kBT, and the diffusion time of a single bead . For convenience, to compare the simulated results with experimental data, we employ the unit of molecular weight (i.e., corresponding molecular weight for the single bead) M0 and the unit of modulus G0 ~ b3/ kBT, instead of the units of length and energy. For each case, 200 molecules are placed in the simulation box.
Owing to the previous studies, e/kT and Ns were fixed at 0.036 and 0.5, respectively. The bead number density b
was 4. These values attain a particular density of slip- springs, which are distributed along the chain with the average number of Rouse beads between consecutive anchoring points of slip-springs, = 3.5.
We employed periodic boundary conditions with the Lees-Edwards boundary under shear. We sufficiently equili- brated the systems before data acquisition. We calculated the linear relaxation modulus G(t) from the stress auto- correlation obtained from equilibrium simulations for an extended period that is at least ten times larger than the longest relaxation time. From G(t), storage and loss mod- uli, and , were obtained by the fitting of G(t) to a multi-mode Maxwell function. The zero-shear vis- cosity 0 was obtained from G(t). For statistics, eight inde- pendent simulation runs starting from different initial configurations were performed for each condition, and the results reported below are the averaged values.
3. Results
Figure 1 shows the molecular weight M dependence of
0 for linear polymers in comparison to the data for PI melts reported experimentally (Abdel-Goad et al., 2004;
Auhl et al., 2008; Gotro and Graessley, 1984). The sim- ulation result indicated by the solid red curve reasonably captures the data, including the onset of entanglement.
Namely, the viscosity increases with an increase of M with a power-law manner, for which the exponent is unity in the low-M regime, whereas it is around 3.5 in the high-M regime. The transition between unentangled to entangled regimes is shown in a further clarity in the bottom panel, in which the viscosity is reduced by M3 according to is the scaling prediction given by the classical tube model (Doi and Edwards, 1986). Indeed, the minimum of 0/M3 indi- cates the critical molecular weight for the onset of entan- glement. From this comparison, the conversion factor for the molecular weight of PI melt is determined as M0 = 400 (g/mol).
Figure 2 shows the linear viscoelastic response for linear
and star polymers. The literature data of PI melts (Auhl et al., 2008; Matsumiya et al., 2014) are shown for compar- ison for the samples that have the molecular weights com- parable to the simulated chains according to the conversion factor M0 ob tained ab ove. For linear PI (seen in the top panel), the conversion factors for time and modulus are chosen as 0= 5 × 107 s and G0= 2 × 106 Pa at 25oC.
These values give the unit viscosity 0G0= 1 Pa·s, which is consistent with the result shown in Fig. 1. With the same set of the conversion factors, the data for several dif- ferent molecular weights are reasonably reproduced, except the high- behavior for the shortest chain, due to an unavoidable cut-off of the model.
For branch polymers, the conversion factors must be shared with those for linear polymers. Because the exper- imental data for star PI employed here (Matsumiya et al., 2014) were taken at 40oC, which is different from that for the linear polymers, we utilized the shift factors for PI
0 b2/kBT
Ne SS
G G
Fig. 1. (Color online) Molecular weight dependence of zero- shear viscosity for linear polymers (top) and the viscosity nor- malized by the classical reptation prediction (bottom). The data for PI melts at 25oC (Abdel-Goad et al., 2004; Auhl et al., 2008;
Gotro and Graessley, 1984) are shown for comparison by sym- bols, and the solid curve indicates the simulation result.
melts reported experimentally (Auhl et al., 2008) to obtain the conversion factors. The determined values are 0= 1.5
× 106 s and G0= 2.1 × 106 Pa (at 40oC), and the compar- ison based on the conversion is shown in the bottom panel. The prediction is not excellent, although it works to some extent. We wish to note that better reproducibility has been reported for linear and branch polystyrenes (Masubuchi, 2018), for which the reference temperature is identical and the data are reported from the same group.
Figure 3 shows the viscosity growth under start-up shear for a few linear polymers, for which the linear viscoelas- ticity is shown in Fig. 2. For the nonlinear viscoelasticity, the conversion factors must be identical to those for linear viscoelasticity. We accommodated the temperature differ- ence employing the shift factors reported (Auhl et al., 2008) and obtained the conversion factors at 35oC as
0= 2.8 × 103 s and G0 = 1.7 × 106 Pa. The simulation
results according to these conversion factors are in excel- lent agreement for the short PI (Mw = 13.5 k shown in the top panel). In contrast, the simulations are not entirely consistent with the data for the long PI (Mw = 33.6 k shown in the bottom panel). In particular, the predictions for high shear rates are underestimated. This inconsistency might be due to the model construction, in which we assume that the system is not far from equilibrium. Mean- while, we wish to note that the MCSS prediction under high shear is in good agreement with the results obtained from the bead-spring simulations (Masubuchi and Uneyama, 2018b). Note also that we could attain better Fig. 2. (Color online) Linear viscoelasticity of linear (top) and 6-
arm star-branched (bottom) polymer melts compared to the experimental data for PI with corresponding molecular weights indicated in the figures (Auhl et al., 2008; Matsumiya et al., 2014). For the star polymers, the molecular weight and the bead number of each arm are indicated. The experimental data (shown by symbols) were taken at 25oC for linear PI and 40oC for star PI, respectively. The simulation results are shown by solid and dot- ted curves. Filled symbols and solid curves are for G', and unfilled symbols and dotted curves are for G'', respectively.
Fig. 3. (Color online) Viscosity growth under start-up shear for linear polymers compared with the experimental data for PI melts taken at 35oC (Auhl et al., 2008). The bead number per chain and the corresponding molecular weight of PI are N = 34 and Mw = 13.5 k (top), and N = 84 and Mw = 33.6 k (bottom).
The simulation results and the experimental data are shown by the red curve and triangle, respectively. The blue dotted curve shows the linear viscoelastic envelope. The shear rate is indicated in ter ms of the Weissenber g number for the longest r elaxation time.
agreement if the conversion factors could be tuned sepa- rately.
4. Discussion
Let us discuss the unit of molecular weight M0, which has been determined as 400 for PI in this study. This value is consistent with the previous research for PS, for which M0= 1250 (Masubuchi, 2018) if we consider the entan- glement molecular weight. Because M0 is the molecular weight being carried by the single bead, and = 3.5, the molecular weight between two anchoring points of slip-springs is 1400 for PI and 4375 for PS. Both of these values are around four times smaller than the tabu- lated values for the entanglement molecular weight Me. For instance, Me values for PI and PS have been reported as 4820 (Auhl et al., 2008) and 14470 (Likhtman and McLeish, 2002) based on the tube model. A possible rea- son for this difference is the model dependence of the rela- tion b etween the plateau modulus and the density of entanglement. As we have discussed in the previous pub- lications (Masubuchi et al., 2003; Masubuchi and Uneyama, 2018a), the relation between the plateau modulus and the number density of entanglements is model dependent as written by GN= AnkBT. Here, A is the model-dependent factor that depends on the fluctuations considered, and n is the number density of entanglement segments. For most of the cases, the value of Me experimentally reported is obtained for A = 1, which is for the affine network theory without any fluctuation at entanglement. A = 0.8 for the tube model due to the chain sliding at the entanglement, and A = 0.5 for the cross-link network models with Brown- ian motion of the entanglement. In the previous study (Masubuchi and Uneyama, 2018a), we have reported that our MCSS simulation gives A = 0.17. This value seems reasonable because of the fluctuations that include the hopping of the slip-spring and the Brownian motion of the beads. Based on A = 0.17, can be calculated from Me
mentioned above as 1024 and 3074 for PI and PS, respec- tively. These values are even smaller than the deter- mined from the fitting. It would be due to the difference in the segment density.
The unit of modulus G0 obtained for PI is also consistent with that obtained for PS earlier (Masubuchi, 2018). For PI, G0= 2 × 106 Pa (at 25oC), whereas for PS, G0= 9.1 × 105 Pa (at 169.5oC). Both of these values are around three times larger than the plateau modulus reported, implying the consistency between the simulations for PI and PS.
However, the relation between G0 and M0 seems non-triv- ial. From the dimensional analysis, we obtain a relation- ship G0= BRT/M0. Here, B is a constant, is the density, and R is the gas constant. Note that this relation is similar to, yet different from that discussed above for GN and Me. For the tube model proposed by Likhtman and McLeish
(Likhtman and McLeish, 2002), it has been reported that
= 4820 and = 6 × 105 Pa for the PI melt exam- ined in this study. With the density of = 0.913 g/cm3, these values give BLM= 1.3. For our case, B = 0.35. We have no explanation for this value of B.
Let us turn our attention to the unit of time determined as 0= 5 × 107 s. According to the study by Auhl et al.
(2008), the values for the tube model are = 1.3 × 105 s. The unit of molecular weight can explain the difference for the unit of time. For the tube model, = 4820 has been reported, whereas our simulation gives M0= 400.
Because the unit of time corresponds to the Rouse relax- ation time of the segment, the ratio for the unit of time would be the square of the ratio for the unit of molecular weight. This estimation is consistent with the values men- tioned above. Similar consistency with the tube model can be confirmed for PS.
It is fair to note that Ramírez-Hernández et al. (2015) have reported a similar simulation study for the rheology of linear PI melts examined above. Their model, so-called TIEPOS (theoretically informed entangled polymer simu- lations), is the same with our MCSS model in the funda- mental construction, in which virtual springs connect bead-spring chains with weakly repulsive interactions.
The main difference is that they consider the inter-bead interaction based on the compressibility, whereas we introduce the interaction to retain the ideal chain statistics.
The other difference is that they allow only one slip-spring connected to a single bead to avoid passing between slip- springs along the chain. Probably due to these differences, the effect of slip-spring on the chain dynamics seems not the same with each other. To reproduce the viscoelasticity of PI melt with Mw = 33.6 k, we need the bead number per chain of N = 84, whereas they only have N = 32. The average number of slip-springs on one chain Z is 25 for our case, whereas 8.6 for them. This comparison demon- strates that their model has an advantage in the level of coarse-graining. However, the efficiency of their entan- glement is inexplicable from the discussion for men- tioned above. According to the difference of Z, a naïve estimation of A value for TIEPOS would be ca. 0.5, which is close to the cross-link networks. This estimation implies that the incompressibility significantly suppresses the fluc- tuations around their entanglement. Besides, they com- pared the statistical distributions of their model with predictions by a mean-field single-chain model and reported that the simulation data agree well with the theoretical predictions. Their result implies that the fluctuations in the TIEPOS model are close to those in the mean-field single- chain model. However, in multi-chain models, in general, the environment around a chain fluctuates (because the environment itself consists of fluctuating chains) and the statistical distributions deviate from the mean-field statis- tics without fluctuations. The friction for the slip-spring Me SS
Me SS
Me SS
Me SS
M0LM G0 LM
0 LM M0 LM
Me SS
kinetics is the other important parameter, but the setting in TIEPOS seems the same with our simulations. Neverthe- less, further assessment is necessary for the nature of the multi-chain slip-spring models.
5. Conclusions
We performed MCSS simulations with the model parameters fixed at e/kT = 0.036, Ns= 0.5, = s, and b = 4 to compare the results with the experimental data for PI melts. We determined the conversion factor for the molec- ular weight (i.e., the molecular weight carried by the sin- gle Rouse bead) as M0= 400 (g/mol) from the molecular weight dependence of the zero-shear viscosity for linear polymers. According to the M0 value, we calculated the linear viscoelasticity for several linear and 6-arm star- branch polymers, for which the experimental data are available for PI with the corresponding molecular weight.
For the linear polymers, the simulation results agree with the data for the conversion factors obtained by the fitting as 0= 5 × 107 s and G0= 2 × 106 Pa at 25oC. Based on these values, we determined the conversion factors for dif- ferent temperatures according to the shift factors reported experimentally. With these conversion factors, the simu- lation results for linear viscoelasticity of star PI taken at 40oC, and for viscosity growth under high shear of linear PI obtained at 35oC are in reasonable agreement with the experimental data. The conversion factors thus determined are consistent with those for PS. However, the relations between the conversion factors are still unknown, and fur- ther evaluation and systematic studies are necessary. For example, the dependence of conversion factors on the model parameters is worth investigating. The other import- ant direction is the effect of chemistry as made in the pres- ent study. Although adequate dataset is rather rare, the systematic rheological measurements for polystyrene derivatives with various sidechain lengths (Matsushima et al., 2017) is ideal, for example. Nevertheless, studies in such directions are ongoing, and the results will be pub- lished elsewhere.
Acknowledgments
This study was supported in part by Grant-in-Aid for Scientific Research (A) (17H01152), (B) (19H01861) and for Scientific Research on Innovative Areas (18H04483) from JSPS.
Appendix
In this section, let us exhibit a few characteristic features of the model in relation to our earlier studies.
For most of the multi-chain models, the density of entanglements decreases with an increase in the flow rate
(Chappa et al., 2012; Masubuchi, 2015; Sgouros et al., 2017; Uneyama et al., 2009). Figure 4 shows such behav- ior for the case of N = 84 (Mw = 33.6 k) under the steady- Fig. 4. (Color online) Normalized entanglement density under steady-state as a function of the Weissenberg number. The red square shows the result of this study for N = 84. The black circle and triangle show the results from the atomistic simulation (Baig et al., 2010b) and the multi-chain slip-link simulation (Uneyama et al., 2009).
Fig. 5. (Color online) Time development of normalized viscosity (top) and for the rotation angle of the end-to-end vec- tor (bottom) for N = 84 under the start-up shear flow with Wid = 34.1.
sin2
state. For comparison, the data from the atomistic molec- ular simulations (Baig et al., 2010b) and the multi-chain slip-link simulations (Uneyama et al., 2009) are also shown.
As reported earlier (Sgouros et al., 2017), the flow-rate dependence of the entanglement density is virtually uni- versal when the molecular weight is sufficiently large.
This universality is non-trivial because the definition of entanglement is model-dependent. Nevertheless, physics is unknown.
For polymers under fast-shear, Nafar-Sefiddashti et al.
(2015) have observed a tumbling motion by the atomistic simulations. Costanzo et al. (2016) proposed a constitutive equation that accounts for the tumbling. Masubuchi et al.
(2018) have demonstrated that the tumbling occurs in the simulations by a multi-chain slip-link model. We observed similar tumbling behavior for our simulations. Figure 5 bottom panel shows the time development of for N = 84 at Wid = 34.1, where is the rotation angle formed by the shear direction and the end-to-end vector projected onto the vorticity plane. Staring from = 0.5, which is the value for the isotropic state, decreases as time goes due to the orientation. Before reaching the steady-state, shows an undershoot, indicating a coherent rotation of the molecules. The position of the undershoot roughly coincides with that for the viscosity shown in the top panel. As reported for the other molec- ular model, this tumbling behavior is in harmony with the theory proposed by Costanzo et al. (2016).
References
Abdel-Goad, M., W. Pyckhout-Hintzen, S. Kahle, J. Allgaier, D.
Richter, and L.J. Fetters, 2004, Rheological properties of 1,4- polyisoprene over a large molecular weight range, Macromol- ecules 37, 8135-8144.
Auhl, D., J. Ramirez, A.E. Likhtman, P. Chambon, and C. Ferny- hough, 2008, Linear and nonlinear shear flow behavior of monodisperse polyisoprene melts with a large range of molec- ular weights, J. Rheol. 52, 801-835.
Baig, C., P.S. Stephanou, G. Tsolou, V.G. Mavrantzas, and M.
Kröger, 2010a, Understanding dynamics in binary mixtures of entangled cis-1,4-polybutadiene melts at the level of primitive path segments by mapping atomistic simulation data onto the tube model, Macromolecules 43, 8239-8250.
Baig, C., V.G. Mavrantzas, and M. Kröger, 2010b, Flow effects on melt structure and entanglement network of linear poly- mers: Results from a nonequilibrium molecular dynamics sim- ulation study of a polyethylene melt in steady shear, Macromolecules 43, 6886-6902.
Chappa, V.C., D.C. Morse, A. Zippelius, and M. Müller, 2012, Translationally invariant slip-spring model for entangled poly- mer dynamics, Phys. Rev. Lett. 109, 148302.
Costanzo, S., Q. Huang, G. Ianniruberto, G. Marrucci, O. Has- sager, and D. Vlassopoulos, 2016, Shear and extensional rhe- ology of polystyrene melts and solutions with the same number
of entanglements, Macromolecules 49, 3925-3935.
Doi, M. and S.F. Edwards, 1986, The Theory of Polymer Dynam- ics, Clarendon press, Oxford.
Ferry, J.D., 1980, Viscoelastic Properties of Polymers, 3rd ed., John Wiley & Sons, Inc, New York.
Gotro, J.T. and W.W. Graessley, 1984, Model hydrocarbon poly- mers: Rheological properties of linear polyisoprenes and hydrogenated polyisoprenes, Macromolecules 17, 2767-2775.
Kremer, K. and G.S. Grest, 1990, Dynamics of entangled linear polymer melts: A molecular-dynamics simulation, J. Chem.
Phys. 92, 5057-5086.
Kumar, S. and R.G. Larson, 2001, Brownian dynamics simula- tions of flexible polymers with spring-spring repulsions, J.
Chem. Phys. 114, 6937-6941.
Langeloth, M., Y. Masubuchi, M.C. Böhm, and F. Müller-plathe, 2013, Recovering the reptation dynamics of polymer melts in dissipative particle dynamics simulations via slip-springs, J.
Chem. Phys. 138, 104907.
Langeloth, M., Y. Masubuchi, M.C. Böhm, and F. Müller-Plathe, 2014, Reptation and constraint release dynamics in bidisperse polymer melts, J. Chem. Phys. 141, 194904.
Likhtman, A.E., 2005, Single-chain slip-link model of entangled polymers: Simultaneous description of neutron spin-echo, rhe- ology, and diffusion, Macromolecules 38, 6128-6139.
Likhtman, A.E. and T.C.B. McLeish, 2002, Quantitative theory for linear dynamics of linear entangled polymers, Macromol- ecules 35, 6332-6343.
Masubuchi, Y., 2014, Simulating the flow of entangled polymers, Annu. Rev. Chem. Biomol. Eng. 5, 11-33.
Masubuchi, Y., 2015, Effects of degree of freedom below entan- glement segment on relaxation of polymer configuration under fast shear in multi-chain slip-spring simulations, J. Chem.
Phys. 143, 224905.
Masubuchi, Y., 2016a, Molecular Modeling for Polymer Rheol- ogy, In: Reference Module in Materials Science and Materials Engineering, Elsevier Inc., 1-7.
Masubuchi, Y., 2016b, PASTA and NAPLES: Rheology Simu- lator, In: Computer Simulation of Polymeric Materials, Springer Singapore, Singapore, 101-127.
Masubuchi, Y., 2018, Multichain slip-spring simulations for branch polymers, Macromolecules 51, 10184-10193.
Masubuchi, Y., G. Ianniruberto, F. Greco, and G. Marrucci, 2003, Entanglement molecular weight and frequency response of sliplink networks, J. Chem. Phys. 119, 6925-6930.
Masubuchi, Y., G. Ianniruberto, F. Greco, and G. Marrucci, 2004, Molecular simulations of the long-time behaviour of entangled polymeric liquids by the primitive chain network model, Model. Simul. Mater. Sci. Eng. 12, S91-S100.
Masubuchi, Y., G. Ianniruberto, and G. Marrucci, 2018, Stress undershoot of entangled polymers under fast startup shear flows in primitive chain network simulations, Nihon. Reoroji.
Gakk. 46, 23-28.
Masubuchi, Y., J.-I. Takimoto, K. Koyama, G. Iannir uber to, G.
Marrucci, and F. Greco, 2001, Brownian simulations of a net- work of reptating primitive chains, J. Chem. Phys. 115, 4387- 4394.
Masubuchi, Y., M. Langeloth, M.C. Böhm, T. Inoue, and F.
sin2
sin2
sin2
sin2
Müller-Plathe, 2016, A multichain slip-spring dissipative par- ticle dynamics simulation method for entangled polymer solu- tions, Macromolecules 49, 9186-9191.
Masubuchi, Y. and T. Uneyama, 2018a, Comparison among multi-chain models for entangled polymer dynamics, Soft Mat- ter 14, 5986-5994.
Masubuchi, Y. and T. Uneyama, 2018b, Comparison among multi-chain simulations for entangled polymers under fast shear, ECS Trans. 88, 161-167.
Masubuchi, Y. and T. Uneyama, 2019, Retardation of the reaction kinetics of polymers due to entanglement in the post-gel stage in multi-chain slip-spring simulations, Soft Matter 15, 5109- 5115.
Matsumiya, Y., Y. Masubuchi, T. Inoue, O. Urakawa, C.-Y. Liu, E. van Ruymbeke, and H. Watanabe, 2014, Dielectric and vis- coelastic behavior of star-branched polyisoprene: Two coarse- grained length scales in dynamic tube dilation, Macromole- cules 47, 7637-7652.
Matsushima, S., A. Takano, Y. Takahashi, and Y. Matsushita, 2017, Dynamic viscoelasticity of a series of poly(4-n-alkylsty- rene)s and their alkyl chain length dependence, Polymer 133, 137-142.
Megariotis, G., G.G. Vogiatzis, A.P. Sgouros, and D.N. Theo- dorou, 2018, Slip spring-based mesoscopic simulations of polymer networks: Methodology and the corresponding com- putational code, Polymers 10, 1156.
Nafar Sefiddashti, M.H., B.J. Edwards, and B. Khomami, 2015, Individual chain dynamics of a polyethylene melt undergoing steady shear flow, J. Rheol. 59, 119-153.
Padding, J.T. and W.J. Briels, 2001, Uncrossability constraints in mesoscopic polymer melt simulations: Non-Rouse behavior of C120H242, J. Chem. Phys. 115, 2846-2859.
Pan, G. and C.W. Manke, 2003, Developments toward simulation of entangled polymer melts by dissipative particle dynamics (DPD), Int. J. Mod. Phys. B 17, 231-235.
Ramírez-Hernández, A., B.L. Peters, L. Schneider, M. Andreev, J.D. Schieber, M. Müller, M. Kröger, and J.J. de Pablo, 2018,
A detailed examination of the topological constraints of lamel- lae-forming block copolymers, Macromolecules 51, 2110- 2124.
Ramírez-Hernández, A., B.L. Peters, M. Andreev, J.D. Schieber, and J.J. de Pablo, 2015, A multichain polymer slip-spring model with fluctuating number of entanglements for linear and nonlinear rheology, J. Chem. Phys. 143, 243147.
Ramírez-Hernández, A., F.A. Detcheverry, B.L. Peters, V.C.
Chappa, K.S. Schweizer, M. Müller, and J.J. de Pablo, 2013, Dynamical simulations of coarse grain polymeric systems:
Rouse and Entangled dynamics, Macromolecules 46, 6287- 6299.
Sgouros, A.P., G. Megariotis, and D.N. Theodorou, 2017, Slip- Spring Model for the linear and nonlinear viscoelastic proper- ties of molten polyethylene derived from atomistic simulations, Macromolecules 50, 4524-4541.
Stephanou, P.S., C. Baig, G. Tsolou, V.G. Mavrantzas, and M.
Kröger, 2010, Quantifying chain reptation in entangled poly- mer melts: Topological and dynamical mapping of atomistic simulation results onto the tube model, J. Chem. Phys. 132, 124904.
Uneyama, T., 2011, Single chain slip-spring model for fast rhe- ology simulations of entangled polymers on GPU, Nihon.
Reoroji. Gakk. 39, 135-152.
Uneyama, T. and Y. Masubuchi, 2012, Multi-chain slip-spring model for entangled polymer dynamics, J. Chem. Phys. 137, 154902.
Uneyama, T., Y. Masubuchi, K. Horio, Y. Matsumiya, H. Wata- nabe, J.A.A. Pathak, C.M. Roland, and C.M. Roland, 2009, A theoretical analysis of rheodielectric response of type-A poly- mer chains, J. Polym. Sci. Pt. B-Polym. Phys. 47, 1039-1057.
Vogiatzis, G.G., G. Megariotis, and D.N. Theodorou, 2017, Equa- tion of state based slip spring model for entangled polymer dynamics, Macromolecules 50, 3004-3029.
Xu, X., J. Chen, and L. An, 2015, Simulation studies on archi- tecture dependence of unentangled polymer melts, J. Chem.
Phys. 142, 074903.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
