A comparative study of coarse-graining methods for polymeric fluids: Mori-Zwanzigvs. iterative Boltzmann inversion vs. stochastic parametric optimizationZhen Li, Xin Bian, Xiu Yang, and George Em Karniadakis Citation: The Journal of Chemical Physics 145, 044102 (2016); doi: 10.1063/1.4959121 View online: http://dx.doi.org/10.1063/1.4959121 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/145/4?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Incorporation of memory effects in coarse-grained modeling via the Mori-Zwanzig formalism J. Chem. Phys. 143, 243128 (2015); 10.1063/1.4935490 Bayesian parametrization of coarse-grain dissipative dynamics models J. Chem. Phys. 143, 084122 (2015); 10.1063/1.4929557 Derivation of coarse-grained potentials via multistate iterative Boltzmann inversion J. Chem. Phys. 140, 224104 (2014); 10.1063/1.4880555 Coarse-grain molecular dynamics simulations of nanoparticle-polymer melt: Dispersion vs. agglomeration J. Chem. Phys. 138, 144901 (2013); 10.1063/1.4799265 Mixing atoms and coarse-grained beads in modelling polymer melts J. Chem. Phys. 137, 164111 (2012); 10.1063/1.4759504

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THE JOURNAL OF CHEMICAL PHYSICS 145, 044102 (2016)

A comparative study of coarse-graining methods for polymeric fluids:Mori-Zwanzig vs. iterative Boltzmann inversion vs. stochasticparametric optimization

Zhen Li,1 Xin Bian,1 Xiu Yang,2 and George Em Karniadakis1,a)1Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912, USA2Pacific Northwest National Laboratory, Richland, Washington 99352, USA

(Received 11 March 2016; accepted 8 July 2016; published online 25 July 2016)

We construct effective coarse-grained (CG) models for polymeric fluids by employing two coarse-graining strategies. The first one is a forward-coarse-graining procedure by the Mori-Zwanzig (MZ)projection while the other one applies a reverse-coarse-graining procedure, such as the iterativeBoltzmann inversion (IBI) and the stochastic parametric optimization (SPO). More specifically, weperform molecular dynamics (MD) simulations of star polymer melts to provide the atomistic fieldsto be coarse-grained. Each molecule of a star polymer with internal degrees of freedom is coarsenedinto a single CG particle and the effective interactions between CG particles can be either evaluateddirectly from microscopic dynamics based on the MZ formalism, or obtained by the reverse methods,i.e., IBI and SPO. The forward procedure has no free parameters to tune and recovers the MDsystem faithfully. For the reverse procedure, we find that the parameters in CG models cannot beselected arbitrarily. If the free parameters are properly defined, the reverse CG procedure also yieldsan accurate effective potential. Moreover, we explain how an aggressive coarse-graining procedureintroduces the many-body effect, which makes the pairwise potential invalid for the same systemat densities away from the training point. From this work, general guidelines for coarse-graining ofpolymeric fluids can be drawn. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4959121]

I. INTRODUCTION

Although the molecular dynamics (MD) simulation hasbecome a standard computational tool for studying molecularsystems, an all-atom model is computationally prohibitive toaccess systems at large spatial-temporal scales.1 Hence, theatomistic simulation is often unrealistic for many applicationsof biological systems and soft matter physics even at themesoscale. To this end, coarse-grained (CG) approaches havebeen developed to overcome the limitation of scales. The mainidea of coarse-graining is to aggregate atoms or molecules intoclusters and to average out the irrelevant degrees of freedomfor the cluster.2,3 Consequently, the CG model representsa substantial simplification of the microscopic dynamics,but is still able to capture observable dynamics of complexfluids at larger spatial-temporal scales beyond the capabilityof conventional MD simulations.4 Among these CG methods,dissipative particle dynamics (DPD) conserves the momentumof a system and provides the correct hydrodynamic behavior offluids at mesoscale,5,6 which makes DPD one of the currentlymost popular mesoscopic methods.7

The DPD method was invented more than two decadesago for simulating complex fluids at mesoscale.8 Similarto MD, a DPD system consists of interacting particlesand its dynamics is computed by integrating Newton’sequation of motion. But different from MD, it has a softerpotential allowing for a much larger time step. Ever since its

a)Electronic mail: george_karniadakis@brown.edu

inception, the DPD method has found a wide spectrum ofapplications in soft matter including colloidal suspensions,9

smart materials,10 polymer solutions,11 blood rheology,12 andblood coagulation,13 to name but a few. The equations ofmotion for DPD particles are described as14,15

dPI

dt=

J,I

�FCIJ + FD

IJ + FRI J

�, (1a)

FCIJ = aωC(RI J)eI J, (1b)

FDIJ = −γωD(RI J)(eI J · VI J)eI J, (1c)

FRI J = δωR(RI J)θ I JeI J∆t−1/2, (1d)

where I, J are particle indices and P represents the momentum.The relative displacement and relative velocity of twoparticles I and J are defined as RI J = |RI J | = |RI − RJ |and VI J = VI − VJ, respectively; eI J = RI J/RI J is the unitvector along the radial direction of the two. All three forcesare pairwise additive and FC

IJ, FDIJ, and FR

I J are referred toas conservative, dissipative, and random forces, respectively.The total force on particle I is summed over other particleswithin a cutoff radius Rcut. Also, ωC, ωD, and ωR are theweighting functions while the coefficients a, γ, and δ reflectthe individual strength of the three forces. A Gaussian whitenoise θ I J with zero mean and unit variance is generatedfor the random force. In general, the functional form ofFCIJ determines the static properties of the system, such as

pressure, compressibility, and local structure often representedby the radial distribution function (RDF) of particles. Theforms of FD

IJ and FRI J control the dynamic properties,

0021-9606/2016/145(4)/044102/8/$30.00 145, 044102-1 Published by AIP Publishing.

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044102-2 Li et al. J. Chem. Phys. 145, 044102 (2016)

such as viscosity, diffusivity, and time correlation functions.Moreover, the latter two forces together act as a thermostatand satisfy the fluctuation-dissipation theorem (FDT).14 Thisimposes δ2 = 2γkBT and ωD(R) = ω2

R(R), where kB is theBoltzmann constant and T is the temperature of the DPDsystem.

The essence of a valuable CG model is to obtain thefunctional forms of FC

IJ, FDIJ, and FR

I J that describe theeffective interactions between the DPD particles. To constructthe effective DPD force fields from microscopic dynamics,operational strategies generally fall into two classes: a directforward path by evaluating the microscopic dynamics via theMori-Zwanzig (MZ) formalism and a reverse iterative pathby solving an inverse problem targeting certain properties.In the forward-coarse-graining, the CG force fields betweenDPD particles are constructed directly from available MDtrajectories via the Mori-Zwanzig projection.16,17 While thisprocedure may require additional simplifying assumptions forcomputability, in principle there are no free parameters tobe specified. In the reverse-coarse-graining, an initial guessof the CG force fields is posed and a few parameters inthe expression of forces are left undetermined. Subsequently,an (iterative) inverse optimization is carried out to correctthe free parameters so that target mesoscopic properties areobtained. The majority of available CG methods belongto this category, such as the force matching method,18–22

inverse Monte Carlo,23 inverse Boltzmann inversion (IBI),24,25

stochastic parametric optimization (SPO) using Bayesianinference,26,27 and minimization of relative entropy.28 Thefundamental question of coarse-graining is what parametersin the CG model are free to optimize, or are the parametersinterchangeable? In the present work, we shall answer thisquestion by coarse-graining a particular MD simulation asdemonstration via both forward and reverse paths.

In the remainder of this paper, at first we will describea reference MD system in Section II. Subsequently, inSection III we will construct CG models via the Mori-Zwanzigformalism, iterative Boltzmann inversion, and stochasticparametric optimization methods. Section IV presents theresults and discussion. In particular, we examine howclose the values for the pressure, compressibility, radialdistribution function, viscosity, and diffusivity of the CGsystem match the corresponding values of the referenceMD system. Finally, we conclude with a brief summary inSection V.

II. MICROSCOPIC DYNAMICS

Molecular dynamics (MD) simulations of star polymermelts are performed to provide multiscale dynamics forcoarse-graining. More specifically, star polymer molecules inthe MD system are represented as chains of beads connectedby short springs.29,30 Each molecule has ten arms inter-connected with a center bead with two identical monomersper arm, and hence the total number of beads per moleculeis Nc = 21 (see Fig. 1). Excluded volume interaction betweenmonomers is described by Lennard-Jones (LJ) potentialtruncated for only repulsion, also known as the Weeks-Chandler-Andersen (WCA) potential,31

FIG. 1. A typical configuration of a star polymer with 21 monomers/molecule in the MD system. In the coarse-grained (CG) representation, eachmolecule is represented as a single CG particle interacting with other CGparticles by effective CG force fields.

UWCA(r) =

4ϵ(σ

r

)12−

(σ

r

)6+

14

, r < 21/6σ,

0, r ≥ 21/6σ,(2)

where r is the distance between two monomers and ϵ andσ set the energy and length scales, respectively. Moreover,neighboring monomers are connected by a finite extensiblenonlinear elastic (FENE) spring32

UFENE(r) =

−12

kr20 ln

�1 − (r/r0)2� , r < r0,

∞, r ≥ r0,(3)

where k = 30ϵ/σ2 is the spring constant and r0 = 1.5σdetermines its maximum length.29 Such a FENE spring isshort and stiff enough to prevent neighboring bonds fromcrossing each other.30

In the MD simulation, the polymer melt is modeledby 1000 molecules filled in a periodic cubic box of lengthLB = (1000Nc/ρ)1/3 and the number density of monomersis ρ = 0.4σ−3. Unless otherwise indicated, quantities of bothMD and DPD simulations are reported in reduced LJ units, thatis, the length, mass, energy, and time units are set as σ = 1,m = 1, ϵ = 1, and τ = σ(m/ϵ)1/2. The system is coupledwith an external heat bath via the Nose-Hoover thermostat tosample a canonical ensemble.33 In addition, the temperature ofthe system is maintained at kBT = 1.0 and the velocity-Verletintegrator with a time step δt = 0.001τ is adopted.33

III. COARSE-GRAINED MODELS

In this section, we will describe how to constructmesoscopic models using different CG procedures, namelythe Mori-Zwanzig formalism, iterative Boltzmann inversion,and stochastic parametric optimization methods.

A. Mori-Zwanzig formalism

A coarse-grained model can be constructed directly fromthe microscopic dynamics by applying the Mori-Zwanzig(MZ) projection operator.17 Starting from a Hamiltonian ofthe microscopic system and defining proper CG variables,the evolution of the CG variables can be described by ageneralized Langevin equation (GLE) after projecting themicroscopic system to the coarse-grained system.16,17,34–36

Upon a pairwise decomposition, this GLE can be cast into the

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044102-3 Li et al. J. Chem. Phys. 145, 044102 (2016)

DPD equation as37,38

ddt

PI =J,I

FI J(t)

=J,I

⟨FI J⟩ −

t

0KI J(t − s)VI J(s)ds + δFQI J(t)

,

(4)

where FI J is the instantaneous force between CG particles Iand J. The ensemble average ⟨FI J⟩ is taken as the conservativeforce in the CG model. The last term δFQI J is the random forcedue to unresolved degrees of freedom. In general, δFQI J is non-Gaussian and the corresponding dissipative force depends onan integral of the past history of motion weighted by a memorykernel given as KI J(t) = (kBT)−1⟨[δFQI J(t)][δFQI J(0)]T⟩, whichensures that the CG system satisfies the second FDT.39

The momentum of center-of-mass (COM) is often aslow variable due to its inertia, while the random forceis a fast variable due to frequent atomic collisions. Whenthe relaxation time scales of the COM’s momentum andthe random force are clearly separable, the convolutionintegral in the expression of dissipative force can be furthersimplified using a Markovian approximation by replacingthe memory kernel with the Dirac delta function, that is, t

0 KI J(t − s)VI J(s)ds = γI JVI J(t). Furthermore, if only theradial components of CG interactions are considered, Eq. (4)reduces to the classical DPD formulation in Eq. (1).

However, a molecule consists of many discrete monomersin the MD system, therefore, the total force FI J contains notonly the radial components but also the perpendicular compo-nents. Including both radial and perpendicular interactionsin the CG description leads to a full DPD (FDPD) model.37

It is worth noting that the CG interactions are evaluateddirectly from MD simulations and there are no iterativelyoptimized parameters in the MZ-guided DPD models. Forfurther technical details we refer to Refs. 35–38.

B. Iterative Boltzmann inversion

Iterative Boltzmann inversion is primarily used to obtainnon-bonded interactions for reproducing the RDF of areference system.24 Given a RDF gref(R), which is oftenobtained from all-atom MD simulations or from experiments,the potential of mean force (PMF) UPMF(R) between pairs ofCG particles as a function of their distance R can be obtainedthrough the Boltzmann inversion24

UPMF(R) = −kBT ln [gref(R)] , (5)

which is a free energy and cannot be directly used as apairwise potential in a CG model.4 However, UPMF(R) isusually sufficient to serve as an initial guess, UCG

0 (R), forthe pairwise CG potential during an iterative procedure.The first CG simulation performed with UCG

0 (R) yields acorresponding RDF g0(R), which is different from the targetgref(R). Thereafter, a correction term αkBT ln [g0(R)/gref(R)]is added to improve the potential and this procedure isperformed iteratively as40

UCGi+1(R) = UCG

i (R) − αkBT lngi(R)gref(R)

, (6)

where the subscript i denotes the iteration number, andα is a scaling factor that improves the convergence rateand stability of the IBI process.40 Due to a fundamentaltheorem,41 for a given RDF the pairwise potential is uniqueup to a constant. Therefore, the iterative procedure reachesa converged potential that generates the target RDF well.However, in practical implementations the choice of cutoffradius Rcut is arbitrary and as a consequence, there can beas many CG potentials as the choices of Rcut, each of whichreproduces the given target RDF well.

If the pressure is also of concern, only one of thosepotentials with the correct Rcut can generate the correctpressure as that of the reference MD. If a different Rcut isselected initially, a typical strategy is to add a linear termto the potential24,25,42 to correct the pressure at the expenseof the accuracy of the RDF. However, this correction leadsto a significant deviation of the compressibility from thetarget value.25,42 From the discussion above, it is reasonableto hypothesize that for a given MD system the Rcut in the CGpotential is not an arbitrary parameter. Once Rcut is selectedfalsely, the desired thermodynamic properties may not berecovered well by other means.

To evaluate this hypothesis explicitly, we will selectthree different choices of Rcut = 5.23, 7.35, and 10.0, whichare denoted by IBI-1, IBI-2, and IBI-3, respectively. Thefirst value, Rcut = 5.23, is chosen based on the MZ-guidedinteraction range, the second one Rcut = 7.35 takes the valueof the second peak of the given RDF gref(R), and the lastone, Rcut = 10.0, is the longest range of RDF that we considerin the present study. The reference RDF, gref(R), of COMof star polymers is obtained by running MD simulations.Then, the IBI procedure is implemented in the VOTCA43

package while coarse-grained runs are performed usingGROMACS.44

C. Stochastic parametric optimization

Stochastic parametric optimization (SPO) is anotherreverse method to obtain an effective CG potential. The SPOmethod differentiates itself from IBI by assuming empiricalfunctions with undetermined parameters for the interactionpotential. Then, a stochastic parametric optimization isperformed to determine these parameters so that designedtarget properties can be obtained by the optimized CGpotential.19,27,45 For static properties we set the RDF ofCOM of star polymers and the pressure of the referenceMD system as our target properties. Here, we choose twodifferent empirical functions for the CG potential: SPO-1has a form of U1(R) = 5.23a1(s1 + 1)−1(1 − R/5.23)s1+1 withtwo free parameters a1 and s1, while SPO-2 has U2(R)= 0.5a2Rcut(1 − R/Rcut)2 with two free parameters a2 andRcut. The potential U2(R) corresponds to a conservative forcebeing a linear function of distance, which is widely used inclassic DPD simulations.15

Technically, we employ generalized polynomial chaos(gPC)46 to construct a surrogate model for DPD systems usinga linear combination of a set of special basis functions definedin the parameter space.27 The two parameters of DPD modelto be optimized are given by

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044102-4 Li et al. J. Chem. Phys. 145, 044102 (2016)

(a1, s1) = (a1, s1) + (δa1, δs1) · diag(ξ1, ξ2),(a2,Rcut) = (a2, Rcut) + (δa2, δRcut) · diag(ξ3, ξ4), (7)

where (a1, s1) = (150,3.5) with (δa1, δs1) = (30,1.0) forSPO-1, and (a2, Rcut) = (55,4.5) with (δa2, δRcut) = (50,1.0)for SPO-2. Here, ζ1 = (ξ1, ξ2) and ζ2 = (ξ3, ξ4) are i.i.duniform random variables distributed on [−1,1]. To obtainthe optimal parameter set, we first generate 65 samples ofrandom variables based on sparse grid method.47 Then, weconstruct the surrogate model using the 65 samples of DPDsimulations to infer two parameters in the DPD model forachieving the pressure and the RDF of the reference MDsystem.

IV. RESULTS AND DISCUSSIONS

In this section, a quantitative evaluation of the threedifferent CG models by comparing the CG systemswith the reference MD system will be presented anddiscussed, including both static properties (i.e., the radialdistribution function, pressure, and compressibility) anddynamic properties (i.e., diffusivity, viscosity, and velocityautocorrelation function) of the polymeric fluid.

A. Comparison of static properties

The conservative force is responsible for the staticproperties of molecular fluids. Figure 2 shows the distance-dependent pairwise potential from the MZ formalism, whichis defined as the spatial integration of the mean force⟨FI J(R)⟩, that is, UMZ(R) =

∞R ⟨FI J(r)⟩dr . Here, the cutoff

radius Rcut = 5.23 is determined as the distance beyond whichthe pairwise interactions between CG particles vanish.37,38

⟨FI J(R)⟩ has no data available at short distances due to thefact that pairs become improbable at small distance,37 asshown in Fig. 3. In practice, we use a bell-shaped function toextrapolate ⟨FI J(R)⟩ at short distances, which is presented in aglobal view of the MZ-guided pairwise potential in the inset ofFig. 2.

FIG. 2. Coarse-grained pairwise potentials obtained from Mori-Zwanzig for-malism (MZ), iterative Boltzmann inversion (IBI), and stochastic parametricoptimization (SPO). The inset shows a global view of these potentials, wherethe negative values of the potentials of IBI-2 and IBI-3 are displayed by lineswithout symbols.

FIG. 3. Radial distribution functions (RDF) from different CG potentials, incomparison with that of the reference MD system. The vertical dashed lineshows the position of Rcut= 5.23 used for MZ, IBI-1, and SPO-1.

The resultant RDF of the MZ-DPD system as well as theRDF of COM obtained from the reference MD system areplotted in Fig. 3. To quantify the deviation of a RDF from thereference RDF, we define a ℓ2-norm

ℓRDF2 (L) = *

,

L

0 |g(R) − gref(R)|2dR L

0 |gref(R)|2dR+-

1/2

, (8)

where L is a length chosen for comparison of the two RDFs.Here, we take the value of the cutoff distance L = Rcut = 5.23to penalize deviations at small distances. The MZ-guidedpairwise potential results in ℓRDF

2 = 1.20%, which reveals thatthe reference RDF of the MD system has been well reproducedby the MZ-guided DPD model, as shown in Fig. 3. The valuesof ℓRDF

2 and other comparisons for all cases are summarizedin Table I.

In general, the deviation of RDF obtained by IBI from thetarget RDF decreases with the iteration step, but eventuallythe deviation saturates after some number of iterations.43

Specifically, we perform 300 iterations for updating the IBIpotential, and the resultant potentials for different cutoff radiiare shown in Fig. 2. The inset of Fig. 2 gives a global view ofthese potentials, where the negative parts of the potentials ofIBI-2 and IBI-3 are displayed by lines without symbols, whilethe potential of IBI-1 with a cutoff radius of Rcut = 5.23 hasno negative part. The potentials obtained by IBI are assumedexponential functions for Rcut < 2.2 because the RDF becomeszero at short distances.

TABLE I. Quantitative comparison of static and dynamic properties of theMD system and the CG system using different pairwise potentials. Thesymbols P, ℓRDF

2 , κ−1, D, and ν represent pressure, deviation of RDF, dimen-sionless compressibility, diffusivity, and kinematic viscosity, respectively.

System P ℓRDF2 κ−1 ν D

MD 0.198 . . . 17.59 1.413 0.061MZ-FDPD 0.198 0.012 16.34 1.415 0.061SPO-1 0.198 0.023 16.34 1.433 0.065SPO-2 0.208 0.113 17.05 1.456 0.064IBI-1 0.206 0.002 17.07 . . . . . .IBI-2 0.218 0.002 17.58 . . . . . .IBI-3 0.121 0.003 11.39 . . . . . .

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044102-5 Li et al. J. Chem. Phys. 145, 044102 (2016)

FIG. 4. Response surface ofthe relative error function∆= |P−Pref |/Pref+ℓ

RDF2 in the

parameter space for choice of pairwisepotential in the form of (a) U1(R)= 5.23a1(s1+1)−1(1−R/5.23)s1+1 and(b) U2(R)= 0.5a2Rcut(1−R/Rcut)2 totarget the pressure and center-of-massRDF of the reference MD system.

It is shown in Fig. 3 that all the IBI potentials (IBI-1,IBI-2, IBI-3) can reproduce well the RDF of the reference MDsystem with ℓRDF

2 within 0.3% as listed in Table I. However, thepressures from the three CG systems are apparently different.The IBI-1 has a pressure 4.0% higher than that of the referenceMD system, while pressures of the IBI-2 and IBI-3 deviatefrom the reference by +10.1% and −38.9%, respectively. Theerrors on the static properties of different CG systems aresummarized in Table I. These results reveal that the pressureof the reference system can be reproduced in the CG systemwith a correct cutoff radius without deteriorating the qualityof RDF or the compressibility.

For the CG procedure using SPO, we define therelative error of static properties as ∆ = |P − Pref|/Pref + ℓ

RDF2 .

Figure 4(a) shows the response surface of the relative errorfunction ∆(a1, s1) of SPO-1 in the parameter space, while∆(a2,Rcut) of SPO-2 is shown in Fig. 4(b). The optimalparameter set for achieving the target static properties is (a1, s1)= (144.72,3.27) for SPO-1 and (a2,Rcut) = (21.00,4.38) forSPO-2, where the relative error function reaches the minimumin the parameter space, as shown in Fig. 4.

We find that the best result of SPO-2 is ∆min = 16%,which implies significant deviations of pressure and RDF fromthe targets, i.e., P = 0.208 is 5.0% higher than the pressureP = 0.198 of MD system and ℓRDF

2 = 11.3% indicates anobvious deviation of RDF. However, when the cutoff radiusis properly selected, SPO-1 with Rcut = 5.23 yields its bestresult with ∆min = 2% indicating a good reproduction of bothpressure and RDF. The only difference between SPO-1 andSPO-2 is that for SPO-1 we fixed Rcut while for SPO-2 wetook Rcut as a tunable parameter. Results show that the bestCG model of SPO-1 reproduces well the reference MD systemon static properties but the best SPO-2 contains significantdeviations of pressure and RDF from targets, which impliesthat the cutoff radius should not be taken as a free parameterin CG modeling of polymeric fluids, similarly to the findingsin the construction of CG model using IBI.

The dimensionless compressibility is defined by κ−1

= [L]3/ρkBT κT , where κT is the usual isothermal compress-ibility of the fluid, [L] the length scale, and ρ the numberdensity of particles in a volume of [L]3. In particle-basedsystems, the value of κ−1 can be computed by15

κ−1 =1

kBT

(∂P∂ρ∗

)T

=Nc

kBT

(∂P∂ρ

)T

, (9)

where ρ∗ = ρ/Nc is the number density of DPD particles or thenumber density of molecules in the MD system. To obtain theequation of state (EOS) of these systems the monomer densityis varied from ρ = 0.3 to ρ = 0.6 in a step of δρ = 0.025.Figure 5 shows the dependence of pressure P on the monomerdensity ρ for the MD system and the CG systems. We note thatall the CG potentials are obtained based on the MD systemat ρ = 0.4. According to Eq. (9), the compressibility κ−1 ofthese systems can be obtained based on the gradient ∂P/∂ρof the EOS at ρ = 0.4, as listed in Table I.

Figure 5 shows that the CG potentials of MZ,IBI-1, and SPO-1 have similar performance that is consistentwith the equation of state of the reference MD system forρ < 0.45. Although the coarse-graining strategies of MZ, IBI,and SPO are significantly different, the pairwise potentialsobtained by MZ, IBI-1, and SPO-1 converge to the optimalone when the cutoff radius of CG interactions is definedproperly. Here, we note that both IBI-1 and SPO-1 use thecutoff radius Rcut = 5.23, which is obtained from the MZformulation as a guidance for construction of effective CGmodels. Therefore, the CG models of IBI-1 and SPO-1 yieldgood results. However, when IBI-2 and IBI-3 use an arbitrarilyselected Rcut while SPO-2 takes Rcut as a free parameter to beoptimized, none of them can yield a good CG model. To thisend, the MZ-guided CG model has no free parameters and isable to provide accurate information for CG representations,

FIG. 5. Dependence of pressure P on monomer density ρ in the coarse-grained systems using different pairwise potentials, and comparison with theMD system of polymer melt Nc = 21. The inset shows the P-ρ curves forstar polymers of Nc = 11. The coarse-grained pairwise potentials are obtainedfrom MD systems at ρ = 0.4.

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FIG. 6. Probability density functions (PDF) of the gyration radius Rg forMD clusters of the cases Nc = 11 and 21 at two monomer densities ρ = 0.3and 0.6.

which can be taken as necessary guidance for performingreverse CG procedures more effectively.

It is observed in Fig. 5 that all the EOSs of CG modelsusing the pairwise potentials obtained at ρ = 0.4 diverge fromthe MD system as the monomer density increases. The reasonis that if the CG clusters are soft in the MD system and canchange their configurational morphology as the polymer meltbecomes dense, the effective interactions between moleculesdepend on all the neighboring molecules of the star polymer,and hence the many-body interactions should be considered ina CG model. However, when we construct the CG models andcompute the pairwise potentials at ρ = 0.4, we do not includethe many-body effect in the pairwise potential. To quantify thedeformation of CG clusters in the MD system as the monomerdensity changes, we compute the probability density functions(PDFs) of their gyration radius Rg defined by

M R2g =

Nci=1

mir2i =

Nci=1

mi(ri − R)2, (10)

where M is the mass of a CG cluster, m is the mass of amonomer, and ri = ri − R is the relative displacement of amonomer with respect to the COM of the cluster.

Figure 6 illustrates the PDF of Rg for two coarse-graininglevels, Nc = 11 and 21, as the monomer density ρ increasesfrom 0.3 to 0.6. For the star polymer with short armsNc = 11, the cluster has less deformability in the MD system.Therefore, the PDF of Rg does not change as the monomer

density increases, which indicates that the many-body effectis not important for the system of Nc = 11. Consequently,the pairwise potential of Nc = 11 obtained at ρ = 0.4 canbe safely applied to the system at other monomer densitiesand reproduces well the reference MD system, as shown inthe inset of Fig. 5. However, the star polymer with longarms, Nc = 21, is soft and deforms easily as the monomerdensity changes, which corresponds to an obvious shift onPDF of Rg shown in Fig. 6. The deformability of clustermakes the many-body effect non-negligible, and hence allthe CG models of Nc = 21 using pairwise potentials withoutmany-body corrections diverge from the MD system shown inFig. 5. This result suggests that many-body corrections shouldbe included for achieving a transferable CG potential whenCG clusters are soft and can deform significantly. Otherwise,the constructed CG potential is only valid near the trainingpoint and cannot be applied to other systems.

B. Comparison of dynamics properties

In general, the CG potential alone cannot produce the cor-rect dynamic properties, which are irrelevant for the reproduc-ibility of structural correlations.48 Izvekov and Voth20 reportedthat Hamiltonian mechanics on a CG potential surface yieldsfaster diffusion dynamics than its underlying all-atom system.This faster CG dynamics arises from the fact that the effectivefrictional forces, which are induced by the effects of unresolveddegrees of freedom, are not considered in the CG represen-tation. Having obtained the interaction potential, the dynamicproperties of a CG system are determined by the dissipative andrandom forces. In this respect, the IBI method starts from RDFto obtain effective CG potentials that produce correct staticproperties but does not work for dynamic properties, while theSPO method can work for targeting both static and dynamicproperties. Similarly to the optimization of two parametersfor desired pressure and RDF, we can optimize the dissipativeforce in the form of FD

IJ(R) = −γ(1 − R/Rcut)s2 (eI J · VI J)eI Jwith two tunable parameters γ and s2 to target the diffusivityand viscosity of the MD system.

We define the relative error of dynamic propertiesas ∆ = |ν − νref |/νref + |D − Dref |/Dref. Figure 7 shows theresponse surface of ∆(γ, s2) in the parameter space, which isobtained by performing the stochastic parametric optimizationprocess. Let ϕ(R) = γ(1 − R/Rcut)s2, we have the optimalparameter set giving ϕ(R) = 144.0(1 − R/5.23)2.42 for SPO-1

FIG. 7. Response surface ofthe relative error function∆= |ν−νref |/νref+ |D−Dref |/Dref inthe parameter space with a dissipativeforce in the form of FI J(R)=γ(1−R/Rcut)s2(VI JeI J)eI J forcases (a) SPO-1 (Rcut= 5.23), and(b) SPO-2 (Rcut= 4.38) to targetthe viscosity and the center-of-massdiffusivity of the reference MD system.

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044102-7 Li et al. J. Chem. Phys. 145, 044102 (2016)

FIG. 8. Velocity autocorrelation function (VACF) of the coarse-grained sys-tem using different pairwise potentials, and comparison with the referenceMD system. The inset shows the time integral of VACF defined by D(t)= 1

3

t0 ⟨V(τ)V(0)⟩dτ, in which the magnitude of the plateau of D(t) gives

the diffusion constant of each system.

and ϕ(R) = 56.0(1 − R/4.38)1.01 for SPO-2, which result in(D, ν) = (0.065,1.433) of SPO-1 and (D, ν) = (0.064,1.456)of SPO-2 with respect to (D, ν) = (0.061,1.413) of thereference MD system.

According to the Green-Kubo relations,49 the transportproperties of fluids can be expressed in terms of integrals oftime correlation functions. Although the integral propertiessuch as the diffusivity and the viscosity of SPO-1 and SPO-2approximate the values of the MD system, as shown in Table I,Figure 8 reveals that the time correlations in SPO-1 and SPO-2systems are significantly different from that of the MD system.When the inverse methods using optimization are employed toconstruct CG models, even if the target properties are achieved,one should not expect that other behavior besides the targetedones can be correct automatically. In contrast, the MZ-basedCG procedure does not set target properties before construct-ing a CG model as it uses a forward path to compute theeffective CG interactions from MD simulations. Since the CGinteractions are correctly considered in the MZ-guided model,the time correlations and also the integral properties of theCG system agree well with its underlying MD system. ForCG models derived from the reverse CG strategies, only if thetime correction functions were defined as target properties,the dynamics of these CG systems would have correct timeevolution.

V. SUMMARY

We have constructed coarse-grained (CG) models forpolymeric fluids using both forward and reverse coarse-graining strategies. The forward-coarse-graining procedureeliminates irrelevant atomistic variables by using the Mori-Zwanzig (MZ) projection operators and computes the CGinteractions directly from a microscopic dynamics. In contrast,the reverse-coarse-graining procedure obtains the effectiveCG interactions by solving inverse problems, such as theiterative Boltzmann inversion (IBI), starting from a givenradial distribution function (RDF) and iteratively ending withan effective pairwise potential, and the stochastic parametricoptimization (SPO) optimizing undetermined parameters forachieving targeted properties.

In particular, we performed molecular dynamics (MD)simulations of star polymer melts to provide the fields to becoarse-grained. In the CG representation, the internal degreesof freedom of each star polymer are averaged out and theentire molecule of the star polymer is coarsened into a singleCG particle. Then, different CG methods, i.e., MZ, IBI, andSPO, were employed to construct an effective CG modelfor representing the MD system at a cheaper coarse-grainedlevel. Quantitative comparison between these CG modelsindicates that both the forward-coarse-graining and reverse-coarse-graining methods are able to yield an effective CGmodel that recovers well the reference MD system if thefree parameters in a CG model are properly selected. Somequantities, however, such as the cut-off radius, have a stronginfluence and cannot be considered as arbitrarily tunableparameters.

To construct the effective pairwise potential, the three CGtechniques have similar computational efficiency. In particular,the MZ method requires several MD simulations for ensembleaverage of the mean force, while the IBI method needs manyiterations with several DPD simulations in each iteration, andthe SPO method needs many samples of DPD simulationsto construct the surrogate model. For the effective dissipativeand random forces resulting in correct dynamic properties,only the MZ and the SPO techniques can be applied. Ingeneral, the MZ method is more expensive than the SPOmethod. The reason is that we have to run a lot of MDsimulations for ensemble average in the MZ procedure so thatan accurate memory kernel can be obtained. However, despitethe inaccurate time correlation functions, the SPO procedureonly requires many samples of DPD simulations to constructthe surrogate model. Moreover, recently developed methodscan substantially reduce the number of DPD simulationsrequired for constructing the surrogate model in SPO.50–52

We also explained how the coarse-graining procedureintroduces the many-body effect that makes pairwise potentialshighly specific to conditions under which the potentials areobtained. We have demonstrated that coarse-graining rigidCG clusters with less deformability does not introduce theobvious many-body effect, and consequently the CG pairwisepotential can be safely applied to other conditions beyondthe training point. However, in aggressive coarse-graining,the CG clusters are soft and can deform significantly. As aresult, the configurational morphology of soft CG clustersdepends on all neighboring clusters and their configurations,which introduces the many-body effect that cannot be ignoredin CG representations. In order to obtain CG models beingtransferable to a different density, temperature, or composition,the many-body corrections should be incorporated into theCG models at aggressive coarse-graining levels. To this end, itwould be interesting to consider the density-dependence of theeffective CG potentials by using the framework of many-bodyDPD,53,54 where the pairwise interaction depends not only onthe distance but also on the density of neighboring particles.

ACKNOWLEDGMENTS

We acknowledge support from the DOE Center onMathematics for Mesoscopic Modeling of Materials (CM4).

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This work was also sponsored by the U.S. Army ResearchLaboratory and was accomplished under Cooperative Agree-ment No. W911NF-12-2-0023. An award of computer timewas provided by the Innovative and Novel ComputationalImpact on Theory and Experiment (INCITE) program. Thisresearch used resources of the Argonne Leadership ComputingFacility, which is a DOE Office of Science User Facilitysupported under Contract No. DE-AC02-06CH11357. Thisresearch also used resources of the Oak Ridge LeadershipComputing Facility, which is a DOE Office of Science UserFacility supported under Contract No. DE-AC05-00OR22725.Z. Li would like to thank Professor Bruce Caswell and Dr.Xuejin Li for helpful discussions.

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