A composition transferable and time-scale consistent coarse-grained model for cis-polyisoprene and vinyl-polybutadiene oligomeric blends

We study a coarse-grained model for a binary blend system composed of cis-polyisoprene and vinyl-polybutadiene. Since the slow relaxation dynamics of polymers may require very long simulation times, coarse-grained descriptions are regularly used in order to reduce computational cost while keeping the essential physics. Relaxation dynamics of a coarse-grained model is sometimes accelerated by the smooth coarse-grained potentials. However, the magnitude of the acceleration may be different in different components in a multi-component system. In order to simulate a time-scale consistent dynamics, the acceleration effects should be the same across the different components. Here, we investigate a time-scale consistent coarse-grained model for a binary polymer blend. For the coarse-grained equation of motion, we adopt the Langevin equation and adjust the friction coefficients by focusing on the relaxation times of the first normal mode of the polymers. A united-atom model is used as a reference system of the coarse-graining. Since it is found that the solubility parameter of the atomistic model is much larger than the experimental result, our simulation model is not applicable for the quantitative predictions, but we utilize it as a example system to study a time scale mismatch of a coarse-grained model. We find that the coarse-grained potentials and the friction coefficients derived for one blend composition captures different compositions of the blend. Furthermore, it is found that the magnitude of the acceleration effects of the blend rarely depends on the composition ratio. This implies that our coarse-grained model can be used for inhomogeneous systems.


Introduction
Polymeric materials such as plastics, rubbers, and gels, are ubiquitous in our surroundings. They are soft and have long relaxation time which is compatible with the time scale of our daily life. The mechanical and rheological properties are crucially important in their practical use [1]. To understand relationships between chemical structure and the physical properties is a central subject of material design [2]. Molecular dynamics simulations have been often utilized to study the structure-property relationship. However, since relaxation modes of long polymers are widely distributed in time and can display a very long relaxation time, require a very long time to study mechanical and rheological properties in atomistic simulations. Therefore, many coarse-grained descriptions [3][4][5][6][7][8][9][10][11][12][13][14] such as bead-spring models and slip-spring models have been proposed to perform the molecular dynamics simulations. While phenomenological models can capture essential physics of the polymeric dynamics, significantefforts are often required for the parametrization to yield quantitative predictions of a real material.
In a structure-based coarse-graining method [15][16][17][18], a coarse-grained model is derived based on the chemical structure of a target material. It has been reported that several physical properties such as a stress relaxation function of polymer melts [19][20][21], entanglement molecular weight [19,22], and a chemical potential of additives [23] can be quantitatively reproduced in the coarse-grained models.
According to the structure-based coarse-graining, coarse-grained potentials are derived based on a mapping rule to specify coarse-grained variables. In an iterative Boltzmann inversion method [18], the local structure represented by radial distribution functions of coarse-grained beads is reproduced as a target property of the coarse-grained model. The coarse-grained potentials often accelerate the relaxation dynamics compared to the corresponding microscopic model when we simply use the potentials with the Newtonian equation of motion. A single numerical factor is often introduced to rescale the simulation time for the simplicity although the numerical factor may depend on the thermodynamic state point of the coarse-graining. It has been demonstrated that the time-recaling approach can be used to obtain a long time scale properties such as a diffusion property in homogeneous polymer melt systems [15,17,20,[24][25][26][27][28][29]. The acceleration effect can be advantageous for the coarse-graining from the perspective ofreducing the computational cost.
In industrial applications, polymeric materials are typically made from many ingredients by blending different polymers and nano-fillers [30]. The acceleration of relaxation dynamics of different species upon coarse-graining must be consistent in the blend systems so as to study the dynamical properties. In fact, different acceleration effects of different components have been already reported in multi-component systems of ionic liquids, lipids, polymer melts with small additives, and polymer nanocomposites [23,24,[31][32][33][34][35]. For example, it is shown in [24] that in a system of polystyrene polymers with ethylbenzene molecules the acceleration factor of the ethylbenzene is smaller than that of the polystyrene beads.
Since a quantitative prediction of the time rescaling factor based on a mapping rule is not trivial, the effects on activation energies and time scale dependent scaling factors have been studied [21,[36][37][38][39]. Furthermore, applying a processing in which polymers are mixed or demixed at certain temperatures and/or under shear flows, one needs a coarse-graining model that is transferable to different temperatures, compositions, and rheological situations. Although there are some studies for polymer blends using the structure-based coarse-grained method in which the morphologies and the radial distribution functions are mainly concerned [40][41][42][43][44][45], the dynamical properties have not been well investigated. The recent progress and challenges of coarse-graining methods with consistent dynamics properties are reviewed in [46].
A natural way to resolve the time scale mismatch is to derive appropriate equations of motion for coarse-grained variables. One approach is to derive a generalized Langevin equation based on the Mori-Zwanzig projection operator method [47,48]. The Mori-Zwanzig equation derived from a microscopic autonomous equation of motion of an isolated Hamilton system is exact and keeps the time-reversal symmetry of the system, since the dynamics of the fast variables is also involved in the equation. To close the equation of motion, with respect to the coarse-grained variables defined by the projection, some approximations and idealizations are applied to the terms of the dissipation kernel and the random forces. They may introduce a violation of the time-reversal symmetry into the coarse-grained equation of motion. For instance, a Markovian approximation is often applied by assuming a clear time scale separation between the coarse-grained variables and the degenerated degrees of freedom. Dissipative forces and random forces which reflect the degenerated degrees of freedom satisfy the fluctuation dissipation relation of second kind [49]. There are several studies on this method [50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65]. The formulation of the dissipative particle dynamics (DPD) [66,67] is often investigated as a coarse-grained equation of motion in which the dissipation kernel depends on relative positions between two coarse-grained beads and can be evaluated from the corresponding microscopic simulations.
The issue of the time scale mismatch in multi-component systems hasrecently beenstudied in polymer solutions and in a polymer matrix with penetrant molecules [65]. Deichmann and van der Vegt showed that the diffusion properties in the polymer solutions can be well reproduced by the Mori-Zwanzig DPD-type coarse-grained equation of motion. It is also reported that, although the penetrant molecules in the coarse-grained model display slightly faster diffusion dynamics than that in the atomistic simulation, the coarse-grained model reproduces better diffusion coefficients compared to those in another coarse-grained model which is described by the Newtonian equation of motion with a Nose-Hoover thermostat.
There are also some studies in which, by assuming stochastic equation of motion for coarse-grained beads, the parameters of the equation of motion are determined through a Bayesian optimization with the corresponding microscopic trajectories [68][69][70]. Dissipation parameters can be also evaluated as the parameters in the assumed stochastic equation. Another example is a Markov state model based approach by defining a probability distribution of states of interest [71][72][73]. The thermodynamic states evolve according to a transition matrix which may include a violation of time-reversal symmetry of the dynamics, that is, a dissipation effect. The transition matrix can be derived and optimized from the corresponding microscopic trajectories.
However, deriving dissipative friction kernel usually requires extensive statistics. Several theoretical and numerical approaches has been studied for the dissipation kernel [60,61,[74][75][76]. It is also not trivial to define proper states of a Markov state model in advance while keeping the transition matrix reasonably small. If the dissipation kernel is properly obtained as well as the coarse-grained potential, it is expected that the coarse-grained model recovers the original time scale as examined for velocity auto-correlation function and diffusion coefficients in [55,64], and then the acceleration effect of relaxation dynamics is lost. Practically, major molecular dynamics packages, such as LAMMPS [77], Gromacs [78], and ESPResSo++ [79], do not support these kinds of equation of motion such as a non-Markovian DPD equation by default or require some programming efforts.
In this study, we consider another simple approach to adjust time scales of different components. We adopt theLangevin equation [80] and attempt to adjust the friction coefficients of the coarse-grained beads. An application of the Langevin equation has been studied in [50] for liquid methanol with the multi-scale coarse-graining method [81]. They showed that the time dependence of the velocity auto-correlation function can be well reproduced if a sufficient resolution is kept in the coarse-grained model. The friction coefficients have been evaluated from the corresponding microscopic simulation.
When a friction coefficient is sufficiently large, Lanvegin dynamics undergoes an overdamped regime where the inertial degrees of freedom are degenerated as fast variables and the relaxation time can be changed by the strength of the friction coefficient [82,83]. This approach has been studied in a water system in a context of an adaptive resolution simulation scheme to make the dynamics of the atomistic water and the coarse-grained water consistent [84]. It is demonstrated that the diffusion behavior can be adjusted by increasing the friction coefficient while keeping the static properties. Furthermore, Salerno et al also showed in polyethylene melts [20] that the stress relaxation time can be adjusted by varying a friction strength of the Langevin equation of motion. Although short time scale behaviors such as a velocity auto-correlation may be changed by modifying the friction coefficients, it is expected that the coarse-grained model can approximately describe slow dynamics of interests if the dynamics is well represented by the overdamped equation. The aim of this paper is to investigate the applicability of this method to a polymer blend system.
As an example, we investigate a mixture of cis-polyisoprene and atactic vinyl-polybutadiene. This is experimentally known as a miscible polymer blend [85]. The procedure to derive the coarse-grained model is divided into 3 steps. In the first step, we derive two coarse-grained models of the homopolymer melts of cis-polyisoprene and vinyl-polybutadiene, respectively. Then, we obtain two sets of coarse-grained force fields of the cis-polyisoprene and the vinyl-polybutadiene, respectively. By keeping these force fields, we derive interaction potentials between the beads of the different polymers in the second step. In the last step, we adopt the Langevin equation of motion for the polymers and adjust the friction coefficients to reproduce the ratio of the characteristic relaxation times of the two polymers. After the derivation, we also study a composition transferability of the coarse-grained model whose coarse-grained potentials and friction coefficients are fixed to those derived in a certain composition.
This manuscript is organized as follows. After introducing a united-atom model of cis-polyisoprene and vinyl-polybutadiene in section 2.1. We derive two coarse-grained models to describe the homopolymer melts of cis-polyisoprene in section 2.2.1 and of vinyl-polybutadiene in section 2.2.2, respectively. The cross interactions between the two polymers and the friction coefficients of the Langevin equation are derived referring a 50/50 blend system in the volume fractions in section 2.2.3 and section 2.3. In section 3, we examine the coarse-grained model in the different compositions, and then summarize the results in section 4.

Atomistic model
Atomistic molecular dynamics simulations are performed as microscopic descriptions in this study with the united-atom models developed in [86][87][88] for cis-polyisoprene and in [89,90] for atactic vinyl-polybutadiene. These united-atom models are derivatives of one united-atom model for polybutadiene [91]. The same force field parameters are assigned to the same carbon atom types. Therefore, we can naturally extend the interaction parameters to the blend systems. The force field parameters used in this study is summarized in A. The reference temperature and pressure of our simulation are chosen as 393 K (k B T ∼ 3.264 kJ/mol) and 1.0 bar as a rubbery state. All molecular dynamics simulations are performed by Gromacs 5.1.4 [78].
As a homopolymer melt of cis-polyisoprene, we have prepared 512 chains composed of 24 monomers each in a periodic boundary box as studied in [19]. The initial configuration of the system is prepared from a dilute gas state and compressed by NPT simulations for around 90 ns to the target pressure. After the compression, the system is further equilibrated by a 60 ns simulation with a Nose-Hoover thermostat [92,93] and a Parinello-Rahmann barostat [94] of the damping factors 0.5 ps and 2.0 ps, respectively, which is almost four times longer simulation than the chain end-to-end vector relaxation time. The data for coarse-graining are sampled from a 30 ns trajectory of a NVT simulation after the equilibration. The mass density and the Kuhn length at 393 K and 1.0 bar are around ρ cIP m = 0.85 g/cm 3 and l cIP K = 0.84 nm, respectively, which agree well with the evaluated value 0.86 g/cm 3 from an experiment [95], and 0.83 nm for polyisoprene of cis-content 93% [96]. The relaxation time of the end-to-end vectors of the chains is about 15.4 ns.
As a homopolymer melt of vinyl-polybutadiene, we have also prepared 512 chains of 24 monomers in another periodic boundary box. The system is equilibrated by the same way as done for the cis-polyisoprene melt. The density and the Kuhn length obtained by the simulation at 393 K and 1.0 bar are around ρ vBD m =0.91 g/cm 3 and l vBD K =1.4 nm. The mass density in the simulation is considerably larger than the value 0.83 g/cm 3 from experiments [97,98]. Note that in the two experiments the vinyl content and the poly-dispersity are high and quite narrow although the molecular weight is much larger than that of the simulation. This density mismatch is also reported in a simulation [90] and presumed due to the force field parameters of the model. On the other hand, the Kuhn length is fairly in a good agreement with an experimental result 1.37 nm [96] (temperature of the experiment is not exactly the same as our case). The relaxation time of the end-to-end vectors of the chains is around 3.46 ns in our simulation. Here we use this model as a model intended to describe vinyl-polybutadiene with keeping the discrepancy of the density in mind. It is noted that any systematical modifications of the force field parameters have not been examined in this study. Since the parameters for butadiene are originally derived from the quantum chemistry calculations and experiments for the polymers whose main contents are cis and trans connection [89,91], a modification for vinly-butadiene remains as a matter to be studied further.
Let us define the volume fraction of cis-polyisoprene in a blend of the two polymers as where w cIP and w vBD are the weight fraction of cis-polyisoprene and vinyl-polybutadiene in the blend, respectively. ρ cIP m and ρ vBD m are the mass densities in the cases of the homopolymer melts, respectively. The volume fraction of vinyl-polybutadiene ϕ vBD is also defined by the same way and ϕ vBD = 1 − ϕ cIP .
Besides the two homopolymer melts, we consider three blend systems of different compositions, ϕ cIP = 0. 25 The 50/50 blend system is also prepared from a low density gas state to a condensed state by a long NPT simulation. After this preparation, we have further performed a NPT simulation for 100 ns with the Nose-Hoover thermostat and the Parinello-Rahmann barostat for the equilibration. For coarse-graining, a 40 ns trajectory is sampled in a NVT simulation of the 50/50 blend system after the equilibration. For the other 25/75 (ϕ cIP = 0.25) and 75/25 (ϕ cIP = 0.75) blends, we have removed some chains from the 50/50 blend to change the volume fraction. Then, the systems are equilibrated by NPT simulations.
A blend of cis-polyisoprene and vinyl-polybutadiene is known as a miscible polymer blend in experiments [85] (references are therein). They are miscible even in high molecular weights. The solubility parameter χ evaluated by a scattering experiment [99,100] is extremely small and negative around −8.5 × 10 −4 at 393 K. In order to estimate the solubility parameter χ in our simulation model, scattering functions are calculated in the 50/50 blend. The random phase approximation in a mean-field approach for a polymer blend [101,102] leads to a simple relationship among the scattering functions and the solubility parameter as in the incompressible limit. Here, S comp (q) is the composition scattering function as defined below, and P(q) represents the form factor of a single polymer chain in the unperturbed state. v 0 is the reference volume of the system, and v cIP and v vBD are the average volume of the scatters of the cis-polyisoprene and vinyl-polybutadiene, respectively. Let us use the coarse-grained beads defined in figure 1 as the scatters of the scattering functions. The circles shown in figure 1 represent coarse-grained beads. The position of a bead is defined as the center-of-mass position of the inside atoms of the bead. The number of coarse-grained beads along the chain, N CG , is 48 for both polymers in our system. We set the reference volume v 0 = N tot /V where N tot is the total number of the coarse-grained beads in the system and V is the total volume of the system. v cIP and v vBD can be evaluated as the average volume of the coarse-grained beads of the polymers.
for |q| > 0 and δϕ cIP = δϕ vBD = 0 at q = 0. Thus, the composition scattering function S comp (q) in the isotropic system is described as for q > 0 where j and k stand for all scatters. f j = v cIP /v 0 for a scatter of cis-polyisoprene, and f j = −v vBD /v 0 for a scatter of vinyl-polybutadiene. The ensemble average ⟨·⟩ is evaluated by the time-average of the simulation trajectory.
In equation (2), P cIP and P vBD are approximately evaluated by the intra-chain scattering functions of the respective polymers in the blend. They are normalized as P(q → 0) = N CG . Since the chains obey Gaussian statistics in a large scale that satisfies ql K > 1, the form factor P(q) can be described by the Debye function [101,102].
In figure 2, the contribution from the first and second terms of the right hand side of equation (2) is plotted in red, and the third term is plotted in blue. The gyration radii in the simulation are around 1.4 nm for cis-polyisoprene and 1.1 nm for vinyl-polybutadiene. In the inset of figure 2, the whole contribution of the right hand side of equation (2) is drawn. Although it does not seem to be a constant, by averaging data of the two smallest q we obtain a rough estimate χ ≈ 0.035 in the scale of v 0 . Thus, χN CG , which is expected as a scale invariant parameter, is roughly evaluated as about 1.7 in the simulation.
As described previously, the experimentally expected χ is negative and the absolute value is extremely smaller than our simulation result. The larger bulk density of vinyl-polybutadiene in the simulation could cause the discrepancy of the miscibility of the blend. Although more larger systems are needed to evaluate more precisely the deviation of the miscibility, it is implied that the simulation model cannot quantitatively reproduce the blend properties. This indicates that our simulation model in this study should be understood as a toy model to investigate a relaxation time consistent coarse-grained model of the blend.

Coarse-grained model 2.2.1. Cis-polyisoprene melt
A coarse-grained model of the cis-polyisoprene homopolymer melt is derived in the same way as studied in [19]. We have here chosen the cut-off length of the non-bonded interactions as r cut = 1.5 nm to reduce  (2) and (b) the third term are plotted, respectively. The coarse-grained beads represented in figure 1 are used as the scatters to compute the scattering functions. Inset, χ evaluated from equation (2) as a function of q 2 is plotted.
further the computational cost although it is 2.0 nm in [19]. Two types of coarse-grained beads A and B are defined in the monomers as illustrated in figure 1(a). The bead A contains two carbons and the bead B contains three carbons. For the bonded interactions, we define two bond potentials, two angle potentials, and two dihedral potentials. Note that since the positions of the coarse-grained beads are defined at the center-of-mass positions of the carbons inside the beads, the equilibrium bond lengths of AB and BA are different. The bonded potentials are derived through the Boltzmann inversion of the probability distribution functions of those variables obtained in the single chain simulation in vacuum [103]. We derive pressure-corrected non-bonded potentials in order to minimize the pressure difference of the blend polymers. The pressure-correction is preformed iteratively as [104] with the iterative Boltzmann inversion (IBI) method [18] where X,Y , respectively represent the ith and (i + 1)-th candidates of the coarse-grained potential between X and Y beads, ∆P i = P i − P target is the deviation from the target pressure, and g (i) X,Y is the radial distribution function under the potential of V (i) X,Y . The symbols X and Y represent the coarse-grained beads A or B. The pressure correction and IBI are performed alternately after the convergence of IBI steps. We have used the VOTCA package [104] for the iteration. The parameters f and α 1 are the coefficients that we can adjust to optimize the convergence of the iteration. Here, we have set empirically f = 0.003 so that f ∆P i ≈ 1 at the first step, and α 1 = 1.0. g As studied in [19], the coarse-grained model with this mapping rule can well reproduce the Kuhn length and the linear viscoelaticity. The mean squared internal distance, ⟨R(s) 2 ⟩/s, is plotted with respect to s in figure 5(a) where R(s) is the end-to-end distance of a sub-chain composed of s + 1 beads. It is verified that not only for the end-to-end distance of the chains but bead-bead correlation in the small s region agree with those obtained from the corresponding atomistic simulation.

Vinyl-polybutadiene melt
A coarse-grained model of the vinyl-polybutadiene homopolymer melt is derived with the mapping rule represented in figure 1(b). The size of the coarse-grained beads are chosen to be almost the same as the beads for cis-polyisoprene. The bead C is defined from three carbons but the carbons connecting the vinyl group  are shared by the two sequent C beads in equal half weight. The beads C and S are connected alternately along the chain as studied for polystyrene in [105]. 1-5 bonded interactions between C-C and S-S along the  chain are also explicitly defined as done in [105] in order to hold the correct distance distribution between the pairs. The non-bonded interactions along the chain act from the 1-6 bonded pairs.
The bonded and non-bonded interactions are derived by the same way as for the cis-polyisoprene melt. The bonded interactions are derived by the Boltzmann inversion of the probability distribution functions obtained in the single chain simulation. The non-bonded interactions are also derived from IBI and the pressure correction as discussed for cis-polyisoprene. The resultant potentials are shown in figure 6. The distributions of the bonded variables are well reproduced as shown in figure 7(a), (b), and (c). The agreement of the radial distribution functions with those of the atomistic simulation are also verified as shown in figure 7(d).
The mean square internal distance of the coarse-grained model also agrees well with the result of the atomistic model as shown in figure 5(b). The Kuhn length of the coarse-grained model is about l CGvBD K = 1.37 nm which agrees with l vBD K = 1.4 nm of the atomistic model. We conclude that the two coarse-grained models for cis-polyisoprene and vinyl-polybutadiene have similar qualitiesas the representability of the polymers.

The 50/50 Blend
To derive the coarse-grained potentials between two polymers, let us assume that 1) the bonded interactions obtained in the single chain simulations can be used in the blend systems as well as in the homopolymer melts, and 2) we can also keep the non-bonded interactions derived in the homopolymer melts. Namely, the bonded interactions and the six non-bonded potentials between A-A, A-B, B-B, C-C, C-S, and S-S beads are assumed to be composition transferable. Thus, we here derive the cross interactions between A-C, A-S, B-C and B-S beads in the 50/50 blend system. To derive the cross interaction potentials, we have used IBI described in equation (8) and the coordination iterative Boltzmann inversion (C-IBI) which is described as [ where C X,Y (r ≈ r o ). Therefore, it is expected that IBI and C-IBI can correct the coarse-grained potentials in a different way.
We start from IBI with initial guesses of V (0) X,Y (r) = [u X,X (r) + u Y,Y (r)]/2 where u X,X (r) represents the interaction potential between X-X beads. The initial guesses are chosen because the potentials obtained by Boltzmann inversion from the target radial distribution functions cause a phase separation in the first step of IBI. Since the solubility and the radial distribution functions of the different polymer's beads are sensitive to a small difference of the interaction potentials, we have set α 1 = α 2 = 0.01 empirically. In the iteration, we have performed bathes of IBI and C-IBI alternately. In each step, 4.5 ns simulation is performed which is longer than the first normal mode relaxation time of the coarse-grained chains. During the iteration, the Nose-Hoover thermostat with a coupling factor 1.0 ps is applied to each type of the polymers. In the next subsection, we consider Langevin thermostats instead of the Nose-Hoover thermostats to study the relaxation dynamics of the polymers.
After the iteration, we obtain the coarse-grained potentials as represented in figure 8. The radial distribution functions obtained in the coarse-grained system show good agreement with those in the atomistic simulation as shown in figure 9. It is verified that the interaction potentials obtained from the homopolymer melts work well for the blend.
The mean square internal distance curves of the respective polymers are represented in figure 10. The good agreement at small s implies that the bonded potentials give accurate local configurations along the chains. For large s, the end-to-end distances of both polymers are about 7% shrink by blending in the atomistic simulation. Although the coarse-grained cis-polyisoprene reproduces the shrinkage effect, the coarse-grained vinyl-polybutadiene remains in almost the same end-to-end distance as that in the homopolymer melt. This implies that the coarse-grained model is slightly more compatible compared with the atomistic model. The composition transferability of the coarse-grained potentials is verified in section 3.

Relaxation times of the two polymers
As a characteristic relaxation time of the chains, we focus on the auto-correlation function of the first normal mode defined as [3] with the number of coarse-grained beads N CG along the chain, and the coordinates of coarse-grained beads R j . The relaxation time τ R is evaluated by fitting C(t) to a single exponential function e −t/τR . A time rescaling factor p is introduced so as to fit C(pt) of the coarse-grained model to C(t) of the corresponding atomistic model. In the coarse-grained 50/50 blend with the Nose-Hoover thermostats, the two polymers show similar relaxation times of the first normal modes. The ratio of the relaxation time of cis-polyisoprene to that of vinyl-polybutadiene, ζ = τ cIP R /τ vBD R , is obtained as approximately 1.3 although the ratio in the atomistic simulation is~2.8. The relaxation times of the coarse-grained polymers weakly depend on the damping factors of the thermostats. This discrepancy results from the different magnitude of the time scale acceleration effect of the two polymers.
Therefore, the ratio of the relaxation times of the two polymers needs to be corrected keeping as large an acceleration of the coarse-grained model as much as possible. Since cis-polyisoprene displays too fast relaxation compared to vinyl-polybutadiene, we attempt to make cis-polyisoprene slower while keeping the relaxation dynamics of vinyl-polybutadiene as fast as possible. To perform this, we consider a simple Langevin equation described as for the coarse-grained beads R j with the mass m, a friction coefficient γ, and Gaussian white noise ξ j (t) which satisfies the fluctuation dissipation relation U represents the total interaction energy of the coarse-grained model. If the friction coefficient is sufficiently large, the relaxation time is inversely proportional to γ. The values of m, γ, and then ξ may depend on the types of the coarse-grained beads. As the friction coefficients, we use the same value of γ CGcIP for the two bead types of the cis-polyisoprene and the same value of γ CGvBD for the two bead types of the vinyl-polybutadiene.
Since the friction coefficient does not change the equilibrium distribution because of the fluctuation dissipation relation, it can be used to vary the relaxation time while keeping the static properties unchanged. We first fix the friction coefficient of the beads of vinyl-polybutadiene, γ CGvBD , to be sufficiently small. Then, we increase γ CGcIP which is the friction coefficient of the beads of cis-polyisoprene until the ratio becomes Figure 11. Dependency of the friction coefficients on the relaxation times. In (a), the first normal mode relaxation time of vinyl-polybutadiene is represented with respect to the friction coefficient for C and S beads. The ratio of the relaxation times, ζ, is plotted against the friction coefficient for A and B beads where γ CGvBD is fixed to 0.5 ps −1 . The dash lines is an eye guide. In the Langevin simulations, we find that γ CGvBD = 0.5 ps −1 is small enough as shown in figure 11(a) and smaller than the intrinsic segmental friction of the chains as confirmed below. While keeping the value of γ CGvBD , we vary γ CGcIP . The ratio of the relaxation times is represented in figure 11(b) with respect to γ CGcIP . With approaching the overdamped regime, the relaxation time of cis-polyisoprene is increased and the target ratio is reproduced around γ CGcIP = 6.67 ps −1 . As shown in figure 12, the auto-correlation functions of the first normal modes in the coarse-grained model with γ CGvBD = 0.5 ps −1 and γ CGcIP = 6.67 ps −1 agree well with those of the atomistic simulation where we have used a single time rescaling factor p = 1.7 for both polymers.
It should be noted that the damping time of inertia with the Langevin thermostat is typically estimated as the order of γ −1 . Therefore, changing the value of γ may change the relaxation dynamics in this short time scale. According to the renormalization group method [107], by expanding the underdamped equation described as equation (12) in the limit of γ −1 → 0, the overdamped equation described as can be derived theoretically. Therefore, when dynamics of interests changes in a much longer time scale than γ −1 , the overdamped picture becomes a good approximation although we cannot expect that the overdamped coarse-grained model well reproduce the short time behaviors around γ −1 . In the overdamped Langevin equation, the friction coefficient controls the time scale as equation (14) is invariant under the transformation of t → t ′ = pt and γ → γ ′ = pγ where p is a constant and independent of j. We adjust the balance of γ CGcIP and γ CGvBD in the overdamped picture. To confirm the segmental relaxation time τ s as a basic time scale of the chain dynamics and the effective segmental friction coefficient ς/m K where m K is the mass of a segment. We apply the Rouse model [108], which is a bead-spring model of a single ideal chain in a medium, in the blend system as an approximation. The Rouse relaxation time of the first normal mode is described with the segmental friction ς, the Kuhn length l K , and the number of Kuhn segments N K along the chain as As summarized in table 1, for both types of the polymers that the segmental relaxation time is much larger than the effective damping time m K /ς of the segment. This implies that the polymeric relaxation dynamics can be well described in the overdamped picture in which the potential interactions play an important role compared with the inertial relaxation.

Composition transferability
As a verification of composition transferability of the coarse-grained model, we consider the blends of ϕ cIP = 0, 0.25, 0.75, and 1.0. ϕ cIP = 0 and 1.0 corresponds to the single homopolymer melts of vinyl-polybutadiene and cis-polyisoprene, respectively. The coarse-grained potentials and the friction coefficients derived in the 50/50 blend system are used for the simulations of the different compositions.
As the static properties, the radial distribution functions and the mean square internal distance curves are represented in figure 13 and figure 14. The systems ϕ cIP = 0 and 1.0 are omitted because they are the same as shown in figures. 4, 5, and 7. Although the beads of the minor component are slightly more structured in the atomistic model compared with the coarse-grained model, the coarse-grained model captures quite well the changes of the radial distribution functions. Moreover, the tendency of the weak shrinkage in the end-to-end distance caused of the minor component is also well reproduced as seen in figure 14.
For the relaxation dynamics, we verify the ratio of the relaxation times as well as the time rescaling factor. The auto-correlation functions of the first normal modes excellently agree with the results of the atomistic simulation as shown in figure 15. The time rescaling factor p is evaluated in each composition by comparing results from the atomistic model and the coarse-grained model. Although p in the pure vinyl-polybutadiene system is around 20 % larger than the others, it is found that p does not much depend on composition.
In figure 16, the relaxation times of the two polymers with respect to the blend composition are represented where p = 1.7 is used in the all cases of the coarse-grained simulations. The agreement in figure  16 implies that the change of the relaxation times are mainly caused by the interaction potentials and the effect of the friction coefficients are minor compared with the composition difference in the present model.

Summary
We have derived a coarse-grained model of the oligomeric polymer blend of cis-polyisoprene and vinyl-polybutadiene based on a united-atom model. Since the atomistic model of the blends studied shows considerably less miscibility compared with the experimental results, it is considered as a model system to study a time-scale consistent coarse-grained model. The discrepancy presumably comes from the force field parameters for vinyl-polybutadiene because the homopolymer melt displays a higher mass density in the simulation compared to the experimental result.
The non-bonded interaction potentials between the coarse-grained beads of the different polymers are derived in the 50/50 blend system while keeping the bonded and the non-bonded interactions of the beads within the identical polymers which are derived in their homopolymer melt systems. As characteristic    coarse-grained beads. It is found that the present coarse-grained model reproduces very well the radial distribution functions of the systems for different blend compositions. In addition to the static property, the ratios of the relaxation times of the two polymers are also well reproduced by the coarse-grained model derived in the 50/50 blend. Moreover, it is also found that the time rescaling factors of the different blend compositions are almost the same although they deviate maximally around 20 %.
The adjusted friction coefficients in the Langevin equation are smaller than the effective segmental friction coefficients evaluated based on the Rouse model, respectively. Although the short time scale behaviors such as the inertial relaxation may be changed by the modification of the friction coefficients, it is confirmed that the segmental relaxation time is much longer than the damping time of the segmental friction in the Rouse picture. Therefore, in the coarse-grained model, it is expected that Rouse-like dynamics is well described by the overdamped equation of motion where the potential interactions play an essential role. CH2b, C = , CH = , CH2 160.00 0.000 CH, C = , CH = , CH2 160.00 180.0 The composition transferability implies that the coarse-grained model can be applicable in inhomogeneous states such as a mixing process at least near equilibrium conditions when hydrodynamic interactions are not important. To what extent the coarse-grained model works far from equilibrium states still has to be tested because all coarse-grained parameters are derived and verified in the equilibrium state. It should be also noted that the coarse-grained potentials and the friction coefficients depend on the thermodynamics state point in which they are derived. Further studies are necessary to obtain thermodynamics transferability such as temperature transferability.
The examinations in the systems of different molecular weights and different solubility situations are also left for the future studies. In general, a polymer blend undergoes phase separation when χ is positive and molecular weights of the polymers are sufficiently large. If the coarse-grained potentials are transferable not only for composition but for molecular weight, it is expected that the coarse-grained model can capture morphological differences by the blend composition. To verify static and dynamical properties of the coarse-grained model, interfacial energy and composition fluctuation dynamics near phase boundaries also need to be investigated. Since the method to derive the time scale consistent model studied here is not specific for the polymers, it can be examined for other blend systems.

Appendix A. Parameters of the united-atom model
The force filed parameters for cis-polyisoprene and vinyl-butadiene are taken from the literatures [86][87][88][89][90]. The parameter used in our simulations is summarized in table A1 for the bonded parameters and in table A2 for the non-bonded parameters. The corresponding symbols of the united carbons atoms are shown in figure  A1. The bond lengths are fixed by LINCS algorithm. Harmonic functions described as (1/2)k θ (θ − θ 0 ) 2 are used for the angluar potentials. For the proper and improper dihedral angles, Ryckaert-Bellemans function described as ∑ 5 n=0 (−1) n c n cos n ϕ and a harmonic function as (1/2)k ϕ (ϕ − ϕ 0 ) 2 are used, respectively.  Lennard-Jones potentials described as 4ϵ[(σ/r) 12 − (σ/r) 6 ] with a cut-off length 1.1 nm are assumed for the non-bond interactions in the simulation.