parametrization of chain molecules in dissipative particle...

Parametrization of Chain Molecules in Dissipative Particle DynamicsMing-Tsung Lee, Runfang Mao, Aleksey Vishnyakov, and Alexander V. Neimark*

Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, NewJersey 08854-8058, United States

*S Supporting Information

ABSTRACT: This paper presents a consistent strategy for para-metrization of coarse-grained models of chain molecules in dissipativeparticle dynamics (DPD), where the soft-core DPD interactionparameters are fitted to the activities in solutions of reference compoundsthat represent different fragments of target molecules. The intercompo-nent parameters are matched either to the infinite dilution activitycoefficients in binary solutions or to the solvent activity in polymersolutions. The respective calibration relationships between activity andintercomponent interaction parameter are constructed from the results ofMonte Carlo simulation of the coarse-grained solutions of referencecompounds. The chain conformation is controlled by the near neighborand second neighbor bond potentials, which are parametrized by fitting the intramolecular radial distribution functions of thecoarse-grained chains to the respective atomistic molecular dynamics simulations. The consistency, accuracy, and transferabilityof the proposed parametrization strategy is demonstrated drawing on the example of nonionic surfactants of the poly(ethyleneoxide) alkyl ether (CnEm) family. The lengths of tail and head sequences are varied (n = 8−12 and m = 3−9), so that the criticalmicelle concentration ranges from 10 to 0.1 mM. The surfactants are modeled at different coarse-graining levels using DPDbeads of different diameters. We found consistent agreement with experimental data for the critical micelle concentration andaggregation number, especially for surfactants with relatively long hydrophilic segments. Depending on the system, we observedsurfactant aggregation into spheroidal, elongated, or core−shell micelles, as well as into irregular agglomerates. Using the modelsat different coarse-graining levels for the same molecules, we found that the smaller the bead size the better is agreement withexperimental data.

1. INTRODUCTION

Dissipative particle dynamics (DPD) has become a commontool for mesoscale modeling of self-assembly in surfactant andpolymeric systems. In DPD, the system components aremodeled by beads, each of which represents several atoms thatare lumped together. The beads interact via short-range softrepulsion potentials that affords superb computationalefficiency. Despite their simplicity, DPD models providequalitative and sometimes quantitative description of thestructure and thermodynamics of quite complex systems.Examples include micellization in surfactant solutions,1−3

morphology of cell membranes and lipid bilayers,4,5 andphase segregation in polyelectrolyte membranes,6−8 to name afew. However, quantitative predictions with DPD are hinderedby apparent problems with customization and parametrizationof coarse-grained models. Commonly, parametrization of DPDmodels is based on a top-down approach; the parameters ofinterbead potentials are fitted to reproduce certain physico-chemical properties, like solubility or interfacial tension, whichare known either from experiments or from establishedtheoretical models, like Flory−Huggins (FH) theory9 orCOSMO-RS method.10 A rigorous bottom-up approach toupscaling of atomistic potentials is hardy practical, because itinvolves expensive and case-specific atomistic simulations, and

yet the coarse-grained models obtained often do not predictcorrectly the macroscopic properties.11,12

The basic principles of top-down parametrization of DPDmodels were summarized and illustrated on several case studysystems by Groot and Warren (GW).13 Detailed analysis of theGW correlation and its applicability to parametrization of DPDmodels at different coarse-graining levels is given in SupportingInformation, section S2. Due to its simplicity and computa-tional efficiency, the GW model is widely used in the literaturefor modeling various systems.1,2,4−6,8 Therefore, parametriza-tion approaches, which are briefly reviewed in the followingparagraph, are mostly developed with the GW model in mind.However, substantial attention is now being paid tomodifications of GW model in order to widen the limit theapplicability of DPD to quantitative description of self-assembly. In particular, the efforts have been directed to (1)using different effective bead diameters and force cutoff fordifferent bead types, (2) introducing dependence of theintracomponent repulsion parameters on the bead type orcoarse graining level, and (3) applying additional (that isbeyond weak harmonic bonds between neighboring beads)

Received: January 3, 2016Revised: April 29, 2016Published: May 11, 2016

Article

pubs.acs.org/JPCB

© 2016 American Chemical Society 4980 DOI: 10.1021/acs.jpcb.6b00031J. Phys. Chem. B 2016, 120, 4980−4991

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http://pubs.acs.org/doi/suppl/10.1021/acs.jpcb.6b00031/suppl_file/jp6b00031_si_001.pdf


pubs.acs.org/JPCB

http://dx.doi.org/10.1021/acs.jpcb.6b00031

restraints in order to account for the actual conformations ofthe molecules.Groot and Rabone4 scaled the intracomponent parameters

based on the physical unit of bead diameter and linkedintercomponent parameters to physical properties of surfactantand lipid molecules. The phospholipid model was parametrizedto match the density and elasticity because these properties arecritically influenced by the conformations of alkane tails14

whose rigidity is ignored in the standard GW model. For thisreason, the intracomponent repulsion between the lipid headheads in ref 4 was made stronger than that for other bead types.Travis et al.15 suggested determining both intra- andintercomponent parameters from the cohesive energy densityof individual components. Maiti and McGrother16 rederivedGW theory by taking into account the bead size. Theysuggested that the intercomponent parameters should be scaledwith coarse grained level, yet the intracomponent parametershould remain unaffected. Their DPD parameters are fitted tosurface tension between liquid phases and the cohesive energydensities. Spaeth et al.17 introduced beads of different effectivediameters. The mixtures of soft DPD spheres of different sizeswere studied in depth by Kacar et al.18 The parameters forDPD models of individual components were designed toproduce the same pressure in pure liquids; the bead densities indifferent phases were allowed to differ. The authors obtained anequation of state for mixtures of single-bead components andrelated the intercomponent DPD parameters to the FHparameters. Extending the Monte Carlo study for soft potentialby Wijmans et al.,19 Liyana-Arachchi et al.20 presented adetailed MC study of thermodynamic properties and liquidstructure of single-component DPD fluids, as well as the phasediagrams of binary mixtures whose intracomponent is unified orvarious by bead type. They suggested an equation of state forsingle-bead DPD mixtures that allowed facile fitting of theparameters to experimental LLE data. The other targetproperty for fitting DPD parameters is the surface tensionbetween liquid phases. The correlations between the surfacetension and DPD parameter first obtained by Groot andWarren are often used in DPD simulations of polymers.21−23

Maiti and McGrother16 extended this approach to beads ofdifferent diameters. Extension of this parametrization to othersystems is hindered by a limited availability of experimentaldata on the surface tension.Here, we present a systematic parametrization methodology,

some fragments of which were discussed in our recentpapers.3,24−26 In particular, we suggested determining theDPD repulsion parameters by matching the infinite dilutionactivity coefficients (IDAC) of reference solutions.3 Thecalibration relationships between intercomponent parameterand IDAC were obtained by MC simulation of respectivebinary mixtures of reference DPD fluids. Alternatively, therepulsion parameters were obtained from the concentrationdependence of the solvent activity in respective polymersolutions. The proposed approach is systematically testeddrawing on the example of a series of poly(ethylene oxide) alkylether CnEm surfactants of different composition with the spreadof CMC over 3 decimal orders. A molecule of CnEm surfactantconsists of n hydrophobic CH3 or CH2 groups and mhydrophilic oxyethylene units (−CH2OCH2−). Quantitativeprediction of micellar properties is a benchmark problem fortesting parametrization algorithms in DPD and other coarse-grained simulation methods.3,27,28 By varying the chaincomposition and comparing the simulation results with

experimental data on critical micelle concentration (CMC)and aggregation number (Nag), we demonstrate and verify therobustness of the proposed methodology and its consistencythrough different coarse grained levels. The method is outlinedas follows: (1) dissection of system compounds into equal sizefragments, which correspond to reference compounds, andchoice of the bead size that determines the level of coarse-graining, (2) determination of the intracomponent repulsionparameter from the density and compressibility of the solvent atgiven coarse-graining level, (3) determination of the bondpotentials between the same type beads from the fitting toatomistic MD simulations of conformations of respectivehomo-oligomers, (4) determination of intercomponent (mis-match) parameters from the fitting of the infinite dilutionactivity coefficient (IDAC) in binary solutions of referencecompounds to the results of MC simulations of respective DPDfluids, (5) if IDAC reference data is not available, determinationof intercomponent (mismatch) parameters from the fitting ofthe activity in aqueous solution of respective referencecompound oligomers, and (6) determination of head−tailbond parameters and verification of the overall set ofparameters against atomistic MD simulations of the con-formations of short molecules composed of a few head and tailgroups. In this work, the parametrization strategy is applied tothe simplest situation, where all bead types have the same Rcand same aII in any particular simulation, but these restrictionsare not inherent, and the same strategy can be extended tosystems with differently sized beads.The rest of the paper is structured as follows: Section 2

describes DPD models of CnEm surfactants with differentcoarse-graining levels and the force field used in DPDsimulations. Section 3 discusses the choice of intracomponentrepulsion parameters. In section 4, we describe how to definethe rigidity on the head and tail segments on the basis ofatomistic MD modeling of conformations of alkanes and PEOmolecules in bulk phases. We fit the bonded potentials of theDPD model to reproduce the conformations. In section 5, wechoose reference compounds for the tail and head blocks atdifferent coarse-graining levels; we also describe in detail theprotocol of matching the intracomponent repulsion parametersto the activities in solution of reference compounds, includingIDAC and the solvent activities in polymer solutions. Insections 6 and 7, the DPD models obtained in sections 2−5 areapplied to modeling self-assembly and calculating the criticalmicelle concentration (CMC) and the aggregation number(Nag) in aqueous solutions of CnEm surfactants. In addition, wecompare the performance of different methods of bond rigiditycalibration, using FENE and harmonic potentials parametrizedeither from the fitting to atomistic simulations or using anempirical correlation between the stiffness and the number offlexible torsions suggested in ref 7. Conclusions are summarizesin section 8.

2. SURFACTANT MODELSA CnEm surfactant molecule consists of n hydrophobic CH3 orCH2 groups and m hydrophilic oxyethylene units(−CH2OCH2−). We model several systems with alkyl taillength from n = 8 to n = 12 and PEO head from m = 3 to m =9. In coarse-grained models, the surfactant molecule is dissectedinto fragments represented by beads. The first DPD model ofCnEm was introduced by Groot and Rabone, who studied theeffect of surfactants on the properties and rupture of lipidbilayers.4 Parametrization was based on FH model parameters.

The Journal of Physical Chemistry B Article

DOI: 10.1021/acs.jpcb.6b00031J. Phys. Chem. B 2016, 120, 4980−4991

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Other coarse-grained simulation techniques were also appliedto CE surfactants; for example, Jusufi et al.29 derived hard-corepotentials from atomistic simulation of short PEO oligomers inthe water. CMC and aggregation number for CnEm (n = 6−12,m = 3−9) near-quantitatively reproduced the experimentalresults.Coarse-grained representations of CnEm surfactants are

shown in Figure 1. The molecule is dissected into hydrophobic

and hydrophilic fragments represented by tail (T) and head(H) beads. The hydroxyl group terminating the Em block is alsoincluded in the end H bead to reduce the number ofparameters.3,4,25 The coarse-grained model containing x tailbeads and y head beads is denoted as TxHy. The tail and headfragments are to be of approximately equal volume to fulfill therequirement of the equal bead size in the basic DPD model.13

The size of the water (W) bead is chosen to match the volumesof T and H beads. Based on the neutron reflection data30

interpreted by Groot and Rabone,4 two ethyl groups areassumed to occupy the same volume as one oxyethylene or twowater molecules. For example, the T bead representing

(CH2CH2CH2CH2) fragment of the hydrophobic alkyl blockand H bead representing (CH2OCH2CH2OCH2) oxyethylenefragment have approximately the same volume as a W beadrepresenting four water molecules. Noteworthy, the coarse-graining level and the bead size, Rc, is determined by the chosendissection of surfactant molecules. In order to explore therobustness of the proposed parametrization method, weexplored several dissection schemes at different coarse-graininglevels quantified by the number, Nw, of water molecules in a Wbead. From the liquid densities of octane (chosen as a referencealkane to estimate the tail bead volumes) and water at ambientconditions, we estimated using a standard procedure13 that atthe bead density of ρ = 3, Rc equals to 0.65, 0.71, 0.77, and 0.81nm when W bead contains Nw = 3, 4, 5, and 6 water molecules,respectively. Details of coarse graining are listed in the Table 1.The important condition for the choice of the chain

dissection is the availability of the data on activities formixtures of the reference compounds, which may berepresented by either single T and H beads or their dimersor oligomers. The activity data for reference compounds areused for parametrization of interaction potentials. Thereference compounds must be liquid at given conditions,because standard DPD models cannot adequately representvapor−liquid and solid−liquid equilibria. For example, at thecoarse-graining level of Nw = 4, n-octane is chosen as thereference compound and modeled as a dimer of T beads; at thecoarse-graining level of Nw = 6, the reference compound is n-hexane modeled as a single T bead. The reference compoundsfor the models considered are given in Table. 1.

3. INTRACOMPONENT REPULSION PARAMETERSNonbond interaction between coarse grained beads i and j inDPD is described by a linear, repulsive force,

= − <⎛⎝⎜

⎞⎠⎟a

r

R rr RF r

r( ) 1 forij ij

ij ij

ijij

(C)IJ

cc

(1)

This force decays to zero at the effective bead diameter, Rc, andis characterized by the repulsion parameter aIJ. The intra-component repulsion parameter, aII, is set equal for all beadtypes and fitted to the isothermal compressibility of water.13

This approximation is reasonable for dilute surfactant solutions

Figure 1. Examples of coarse-grained models of CnEm surfactants withdifferent bead diameter Rc. T beads (shown in red) denotehydrophobic tail, and H beads (shown in green) denote hydrophilichead. Water beads (blue) with corresponding bead size are illustratedby NW, the number of water molecules in a W bead.

Table 1. CnEm Surfactant Molecules and the Corresponding DPD Models TxHy at Different Coarse-Graining Levels Determinedby the Bead Size, Rc (diameter in nm)a

CnEm TxHy

n m x y Rc (nm) T bead fragment H bead fragment box size (Rc) reference compound for tail model for reference compounds

8 4 2 2 0.71 (CH2)4 (CH2OCH2)2 30 octane dimer8 6 2 3 0.71 (CH2)4 (CH2OCH2)2 30 octane dimer8 8 2 4 0.71 (CH2)4 (CH2OCH2)2 30 octane dimer10 5 2 2 0.77 (CH2)5 (CH2OCH2)2.5 45 decane dimer12 3 4 2 0.65 (CH2)3 (CH2OCH2)1.5 60 hexane dimer12 6 4 4 0.65 (CH2)3 (CH2OCH2)1.5 60 hexane dimer12 9 4 6 0.65 (CH2)3 (CH2OCH2)1.5 60 hexane dimer12 8 3 4 0.71 (CH2)4 (CH2OCH2)2 64 octane dimer12 3 2 1 0.81 (CH2)6 (CH2OCH2)3 60 hexane monomer12 6 2 2 0.81 (CH2)6 (CH2OCH2)3 60 hexane monomer12 9 2 3 0.81 (CH2)6 (CH2OCH2)3 60 hexane monomer

aSubscript n and m are the numbers of methylene and ethyleneoxide groups, and x and y are the numbers of tail and head beads. System size(simulation box length in Rc) is chosen with respect to the experimental CMC (the smaller CMC, the bigger the box). Last two columns show thereference compounds and their DPD models used for the calibration for intercomponent parameters (details discussed in section 5).



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(volume fraction of surfactant in this work is around 2% to 4%),where water occupies most of the volume. Because of adifference in opinions in the literature (for example, differentintracomponent parameters were used in similar simulations ofNafion8,31) on the scaling of the intracomponent repulsionparameter with the bead size, we performed MC simulations onthe dependence of the compressibility κ on aII for a given Rc.The simulation results are shown in Supporting Information,section S2. The dependence of the inversed reduced isothermalcompressibility, κsim

−1, on aII can be approximated by acorrelation, according to which aII increases linearly withRc

3.13 The intracomponent parameters for different Rc areshown in Table 2. They are in reasonable agreement with thevalues used in the literature.4,13

4. PARAMETRIZATION OF BOND POTENTIALSTo maintain the chain integrity, the neighboring beads areconnected by nearest neighbor bonds (1−2 bond). Additionalbonds are applied between the second neighbor beads (1−3bond) in order to describe the chain rigidity, which significantlyinfluences the mechanism of surfactant micellization.25 We useboth harmonic (eq 2) and FENE (eq 3) potential:

= − −r

K r rF rr

( ) ( )ij ijij

ijij

(B)B 0

(2)

=− −

−< +

= ∞ ≥ +

−⎜ ⎟⎛⎝

⎞⎠

K r rr r r

r r r

F r

F

( )( )

1for

for

ij ijr ij

r r

r

ij m

ij ij m

r

(B)B 0

( ) 0

(B)0

ij

ij

ij

m

02

2

(3)

where r0 is the equilibrium bond length, rm is the maximumbond length (if applicable), and KB is the bond stiffness.For comparison purposes, we follow our original para-

metrization algorithm, where the harmonic potential is used forboth 1−2 and 1−3 bonds of the surfactant model. Bondparameters are fitted by comparing the conformation of unitedatom MD simulation for a fluorocarbon melt with its DPDrepresentative at coarse grained size Rc = 0.71 nm.3 We adaptthe same approximation of this simplistic model that theequilibrium bond length and stiffness are scaled by the numberof flexible torsions, ntors, of the connected DPD bead centers:KB(rij − r0) = 320/ntors and r0 = 0.2ntors. The simplistic modelwas not fitted to the atomistic MD conformations. Theparameters at all coarse grained levels are given in Table 2c.Compared with harmonic potential, FENE potential was

found26 to be better in describing the atomistic conformation ofthe surfactant molecule. In particular, the finite possible bondextension in FENE model allowed reproduction of theasymmetry of 1−4 and 1−5 distance distributions, whiledistributions obtained with the harmonic potential were moreor less symmetric.7 The parameters of 1−2 and 1−3 bonds areobtained by a bottom-up approach, where the conformations ofcoarse-grained chains are fitted to the results of atomistic MDsimulation for the reference compounds. The referencecompound for the tail of CnEm surfactant is hexadecane, andit is poly(ethylene oxide) (PEO, MW = 400) for surfactanthead. Based on the molecule dissection rule described insection 2, a hexadecane molecule was dissected into equalfragments of 3, 4, 5, or 6 carbons at Rc equals 0.65, 0.71, 0.77,and 0.81 nm. We calculated intramolecular distance distribu-tions between the centers of mass of the fragments. Thesedistributions were matched to the distributions of the distancesbetween corresponding beads of the coarse-grained model ofhexadecane melt by adjusting the 1−2 and 1−3 FENE bondparameters (coarse-grained C15H32 was modeled for Rc = 0.65nm and Rc = 0.77 nm, C16H34 was modeled for Rc = 0.71 nmand Rc = 0.77 nm, C18H38 was modeled for Rc = 0.81 nm).Figure 2a shows the beads and corresponding fragments for

Rc = 0.71 nm. The obtained bead-to-bead distance distributionsare presented in Figure 2b. The comparison of atomistic MDand DPD results shows that by using 1−2 and 1−3 FENEbonds, we are able to mimic the flexibility of the alkane tail inDPD simulations fairly well. The FENE bonds turn out to besuperior to the harmonic bonds in reproducing the rigidity ofsurfactant molecules. The detailed description of thesimulations and fitting of DPD bonded parameters to theatomistic MD conformations is given in Supporting Informa-tion, section S3. In order to obtain bond parameters for thehydrophilic PEO head beads, we simulated PEO fragments ofnine monomers using a similar procedure. The shapes ofintramolecular bead−bead distance distributions in PEO meltsare much more complex compared with alkanes. In general,PEO chains are more flexible with a shorter persistent length.

Table 2. Intracomponent Repulsion Parametersa and 1−2and 1−3 Bond Parametersb of DPD Surfactant Model atDifferent Coarse Grained Bead Size, Rc

c

(a) Surfactant Tail, FENE Potential

Rc (nm) aII bond type KB r0 rm

0.65 78.5 1−2 280.0 0.50 21−3 80.0 1.11 4

0.71 106.1 1−2 280.0 0.61 21−3 20.0 1.50 4

0.77 133.7 1−2 180.0 0.70 21−3 8.0 1.85 4

0.81 161.2 1−2 190.0 0.81 21−3 8.0 1.75 4

(b) Surfactant Head, FENE Potential

Rc (nm) aII bond type KB r0 rm

0.65 78.5 1−2 100.0 0.56 21−3 15.0 1.30 4

0.71 106.1 1−2 64.0 0.66 21−3 5.0 1.42 4

0.77 133.7 1−2 30.0 0.75 21−3 2.0 1.55 4

0.81 161.2 1−2 30.0 0.85 21−3 2.2 1.90 4

(c) Harmonic Potentials, Simplistic Model

tail head

Rc (nm) bond type KB r0 KB r0

0.65 1−2 106.7 0.6 71.1 0.91−3 53.3 1.2 35.6 1.8

0.71 1−2 80.0 0.8 53.3 1.21−3 40.0 1.6 26.7 2.4

0.81 1−2 53.3 1.2 35.6 1.81−3 26.7 2.4 17.8 3.6

aDiscussed in section 3. bDiscussed in section 4. cSections a and bcontain bond parameters for surfactant tail and head, when FENEpotential is used. Section c contains bond parameters when harmonicpotential is used. The unit of aII is kBT/Rc, the unit of KB is kBT/Rc

2,and the unit of r0 and rm is Rc.



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Small peaks on MD distributions correspond to particularpreferential conformations of PEO chains. However, the overallrigidity of the PEO chains is reproduced within reasonableaccuracy as shown in Figure 2c. The bonding parametersobtained with this procedure are given in Table 2 a,b.It is worth mentioning that the equilibrium distances for the

second neighbor bonds are in several systems longer than twicethe equilibrium length of the corresponding nearest neighborbonds. In this paper, 1−2 bonds are short (re < Rc), nearestneighbors experience conservative repulsion from each other,and the actual distances between them exceed re. The best fit toMD simulations is obtained with 1−3 bonds with equilibriumlength close to the sum of average nearest neighbor distances(see Supporting Information, section S7).

5. INTERCOMPONENT REPULSION PARAMETERS

Intercomponent repulsion parameters (aIJ in eq 1) for tail (T),head (H), and water (W) beads are obtained from the activitiesin reference binary solutions. For hydrophobic beads T, wechoose different linear alkanes as reference compounds; thechoice depends on the bead size Rc. Since the referencecompound has to be a liquid well below its boiling temperatureat the simulation conditions, the smallest suitable alkane ishexane. Hexane serves as a natural reference compound for Tbead with the coarsest dissection of Rc = 0.81 nm, where one Tbead effectively represents six methyl/methylene groups.Correspondingly, hexane in this case is modeled as a singlebead T molecule. At finer coarse-graining levels, referencealkanes are presented as dimers of tail beads TT. For Rc = 0.65nm, T bead comprised of three CHx groups, hexane is modeledas a TT dimer. The reference compounds for Rc = 0.71 nm andRc = 0.77 nm are octane and decane, respectively. The referencecompounds and models are listed in Table 1.The intercomponent parameters between T and W beads are

fitted to the experimental IDAC for reference compounds.3

Since the reference compounds are represented either by singlebeads or by bead dimers, we considered the respective binarysystems and constructed the calibration relationships for theIDAC ln γ∞(ΔaIJ) as functions of the mismatch parameter ΔaIJ= aIJ − aII. The chemical potential of solute was calculated byequilibrating the bath of pure reference compound ofcomponent I (modeled by monomer I or dimer I−I) in aseries of canonical Monte Carlo simulations augmented by

Widom insertion of a test molecule of the reference compoundof component J (monomer J or dimer J−J).For a given aIJ, γ

∞ is calculated using eq 4 (see derivation inSupporting Information section 5):

γρρ

+

= ⟨ − ⟩ − ⟨ − ⟩

∞⎛⎝⎜⎜

⎞⎠⎟⎟

b

b

E kT E kT

ln ln

ln exp( /( )) ln exp( /( ))

IJJ

I

I

J

JJins

IJins

(4)

γ∞ values obtained experimentally or by thermodynamicmodeling are interpolated onto the reference curves. For typeJ, bJ is the number of physical molecules that a bead of type Jrepresents. For example, if the octane molecule is presented asa dimer with Rc = 0.71 nm, one T bead is 1/2 of the molecule.The W bead in this case represents four water molecules; thusbW = 4. It should be noted that the pressure of a DPD fluid ofdimers is different from that of monomers. In order to maintainthe equal pressure in the fluids of monomers and fluids ofdimer, one has to adjust the dimer fluid density to match thepressure of the monomer fluid. Such adjustment is similar tothat of Kacar et al.18 and contributes to the correction term ρI/ρJ in eq 4 compared with the original formula.3 ρI is the beaddensity for component I at the pressure of solvent bulk. Asillustrated by the schematic in Figure 3, the density ρI fordimers or polymers differs from that of the solvent, which is setto 3.Although the term ln(ρI/ρJ) is rather small (ρ = 3 for

monomer solvent, ρ = 3.11, 3.05, and 3.01 for TT dimer at Rc =0.65, 0.71, and 0.81 nm), the density difference between thesingle-bead fluid and the fluid of dimers substantially affects theinsertion energies and therefore dramatically changes thecalibration curves γ∞(ΔaIJ). This is because the short andstiff bonds, which reproduce the conformations of themolecules (Table 2), affect the structure of the dimer fluid,which becomes quite different from the structure of the single-bead fluid with the same Rc. In the simulation of surfactantsolution, water constitutes the bulk (96% in volume) of thesystem and alone determines the pressure in the NVTensemble. The hydrophobic cores of the micelles are effectivelyunder constant pressure conditions created by the solvent, eventhough the DPD simulations are carried out in the canonicalensemble. For this reason, the reference γ∞(ΔaIJ) curves for thesystems with dimers were calculated at the pressure

Figure 2. Examples of fitting FENE bond parameters in the DPD model (at Rc = 0.71 nm) to the molecular conformations of liquid alkanes and apoly(ethylene oxide) to the results of atomistic MD simulations at T = 298 K. Panel a shows the schematics of mapping the corresponding moleculefragment to the center of mass of coarse grained bead; r12, r13, and r14 are distances between the centers of beads separated by 0, 1, and 2 beads.Panels b and c show the distribution of r12, r13, and r14 from MD (solid lines) and from DPD simulations (dashed lines) of the reference system n-alkane (b) and PEO (c).



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corresponding to the solvent (a single-bead fluid) at thereduced density of ρ = 3. Note that the issue is not essentialwhen the dimer has relatively soft bond4,13 or the 1−2 bondequilibrium length equals 0.8Rc.

3 In such cases, the structureand the pressure of a dimer fluid differ insignificantly from amonomer solvent.With the aforementioned procedure, we obtained the

reference correlations between γ∞ and ΔaIJ for differentcombinations of monomer and dimer fluids. The correlationsare given in Table 3. All reference relationships are monotonic,

but we have to note that the slope depends on the bead size, asshown in Figure 4. Even for the same physical solution of twomolecules of similar sizes (say, hexane and water), differentvalues of Δa are needed to achieve the same solubility fordifferent coarse-grained levels, especially when these liquids arevery dissimilar and strongly separate. This is related to thedependence of the DPD fluid structure on aII and should betaken into account when Δa is calculated from thethermodynamic properties of reference solutions, includingthe calculations of Δa from FH parameters χ. For alkane−watermixtures, γ∞ are estimated from the experimental solubilities

and interpolated onto the corresponding reference curves.Following ref 3, we averaged the aTW parameters obtained fromwater solubilities in alkanes and alkane solubilities in water. Wehave to mention that the experimental errors for such insolublecompounds as octane and decane are very substantial, butseveral sources give values that are reasonably consistent, andthey are used in the present work. We took the valuesrecommended by the IUPAC series,32−34 which collect andexamine different resources. The resulting parameters are givenin Table 4.

5.1. Tail−Water Repulsion Parameter. The tail−watermismatch parameters depend on the bead size. Generally, thelarger the bead size, the higher the ΔaTW. The reason thatΔaTW for Rc = 0.65 nm is higher than that for Rc = 0.71 nm isdue to the shorter bond enforced on the tail−tail link, and itneeds to be compensated somehow to reach the correct overallhydrophobicity. Such effects are not considered if Δa ismapped to the FH parameters, for the FH theory assumesflexible molecules. In the simplistic model, equilibrium bondlength is scaled with Rc; therefore aTW monotonically increaseswith Rc. It is difficult to tell to what extent the linearity (Δa ∝Rc) suggested in ref 16 holds, but the increase of Δa with Rc issignificant.

5.2. Head−Water and Head−Tail Repulsion Parame-ter. For the H beads, ethers (such as CH3OCH2CH2OCH3)appear to be the most suitable reference compounds. However,we face a shortage of reliable data on activities for theappropriate reference solutions. Groot and Rabone,4,27 fittedH−W parameters to LLE data for water−PEO solutions athigher temperatures, where the PEO−water system undergoesa liquid−liquid separation. They obtained χ = 0.3 and ΔaHW =1.3 kT/Rc. Here we opted to take short PEO oligomers withmolecular weight of 400 Da for reference compounds, becausethe data on water activity in PEO−water solutions was reportedfor a wide range of concentrations.35,36 The procedure of aHWparameter fitting is essentially the same as that in ref 24. Weperform MC simulations with DPD models for PEO−watermixtures, where the overall bead densities are adjusted to yieldthe bulk water pressures similarly to the dimer simulationsdescribed above.

Figure 3. Example of the MC simulations for calculations ofcalibration curves (eq 4). The calibration relations (eq 4) arecalculated (a) by trial insertion of a monomer J in the bath of I−Idimer and (b) by insertion of a J−J dimer into the bath of I monomer.Thus, for each ΔaIJ two MC simulations are performed. Additionally,we perform simulations to obtain insertion energies in the referencesystems, which is the bath of the inserted particle (monomer J in case aand dimer J−J in case b. In order to maintain equal pressures in thefluid of monomers and fluid of dimers, the density of the fluid ofdimers is adjusted.

Table 3. Parameters of Linear Fits to CalibrationRelationships ln γA

∞ − ln(bJ ρI/(bI ρJ)) = pΔaTW + q

solute = monomer,solvent = monomer

solute = monomer,solvent = dimer

solute = dimer, solvent =monomer

Rc(nm) p q

Rc(nm) p q

Rc(nm) p q

0.65 0.31 0 0.65 0.29 0.33 0.65 0.63 −2.280.71 0.32 0 0.71 0.33 0.69 0.71 0.74 −1.230.77 0.33 0 0.77 0.34 −0.12 0.77 0.81 −1.800.81 0.35 0

Figure 4. Calibration relationship between Δa (ΔaIJ = aIJ − aII) andIDAC for two single-bead components of types I and J. The density-dependent term in eq 4 is independent of aIJ.



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Figure 5 shows the activity of water in the PEO−watersolution as a function of PEO mass fraction at 298.15 K for Rc =

0.65 nm. The activity of water, aW(x), is calculated for differentvalues of ΔaHW and compared with experimental data.36 TheDPD model cannot describe the water activity in PEO over theentire range of concentrations (which is not surprising taking inaccount the DPD phase diagrams for polymers19). The bestoverall agreement for Rc = 0.65 and 0.71 nm is achieved withΔa = 1.5kT/Rc, although activities of water in polymer-richsolutions are underestimated with this parameter. The resultingΔa agrees well with the estimates of Groot and Rabone.4 For Rc

= 0.77 and 0.81 nm, MC statistics was very poor, and we usedthe same ΔaHW for all coarse-grained levels. For soluble fluids,γ(Δa) reference correlations generally show very weakdependence on the coarse-graining size. For tail and headinteraction, the only available data also requires theextrapolation from a higher temperature.37 We employed inthis case the mixing rule proposed by Groot and Rabone basedon the chemical structure of ethylene oxide, and set ΔaTHequaled to one-third of ΔaTW (given in Table 4). Head−tailinteractions for Rc = 0.71 nm were estimated from the IDACv a l u ed f o r d i g l yme−o c t a n e s y s t em . D i g l yme ,CH3OCH2CH2OCH3, and octane are presented as symmetricdimers HH and TT, and a calibration curve γ(aHT) wasconstructed in ref 3. Because aHW showed only a weakdependence on Rc, H−T mismatch is relatively low, and this

Table 4. Mismatch Parameters for T−W Interactions Obtained with Eq 4a

Rc (nm) compound model solubility [ref] Δa (MD-fitted) Δa (simplistic) model

0.65 hexane dimer 2.4 × 10−6 [32] ΔaTW 21.2 22.33H2O monomer 6.1 × 10−6 [32] ΔaWT 30.5 15.2

avg. ΔaTW 25.9 18.80.71 octane dimer 1.1 × 10−7 [34] ΔaTW 20.4 24.6

4H2O monomer 6.1 × 10−4 [34] ΔaWT 26.4 13.6avg. ΔaTW 23.4 19.1

0.77 decane dimer 3.3 × 10−9 [33] ΔaTW 23.65H2O monomer 6.3 × 10−4 [33] ΔaWT 28.7

avg. ΔaTW 26.10.81 hexane monomer 2.4 × 10−6 [32] ΔaTW 31.5 41.6

6H2O monomer 6.1 × 10−4 [32] ΔaWT 26.0 15.8avg. ΔaTW 28.8 28.7

aMD-fitted model uses the calibration relationships in Table 3. The simplistic model uses the approximate bond length from Table 2c. Solubilitiesare given in mole fractions.

Figure 5. Water activity versus mass fraction of PEO in water−PEOmixture at 300 K. Experimental data (empty squares) is from Malcolmand Rowlinson.36 Cyan squares are simulation results by using ΔaHW =1.5, which produces the best overall agreement with experiment at theentire range of composition. The red circles are simulation resultsusing ΔaHW = 4.5 (red circles), which complies with experimental lowwater concentrations.

Table 5. Experimental and Simulation CMC (in log10 scale) for CnEm Surfactant at Different Tail (n) and Head Length (m) atDifferent Coarse Grained Level, Rc

a

CnEm surfactant MD-fitted

n m Rc [nm] exptl38 log10[CMC] simplistic log10[CMC] log10[CMC] relative error Nag aggregation morphology

8 4 0.71 −2.1 −2.3 −2.0 0.1 905 irregular8 6 0.71 −2.0 −2.2 −1.8 0.1 539 irregular8 8 0.71 −2.0 −2.2 −1.7 0.1 83 irregular and spherical10 5 0.77 −3.1 −2.8 0.2 251 irregular12 3 0.65 −4.0 −4.4 0.2 728 globular12 6 0.65 −4.2 −4.2 0.5 258 vesicular12 9 0.65 −4.1 −3.8 −4.0 0.1 111 spherical and cylindrical12 8 0.71 −4.0 −3.8 −3.2 0.1 311 vesicular12 3 0.81 −4.0 −4.0 0.6 471 globular12 6 0.81 −4.2 −3.7 0.4 392 globular12 9 0.81 −4.1 −4.3 −3.6 0.4 200 spherical and cylindrical

aSimulation CMCs are calculated by two methods in determining bond parameter. Only a few characteristic systems are calculated by simplisticmodel for the purpose of comparison. Nag is the average aggregation number, and the aggregation pattern is exemplified in Figure 8.



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parameter is not crucial for the system behavior, we used thesame mismatch ΔaHT = 6.5kT for all bead sizes.

6. CALCULATIONS OF CMC OF CnEm SURFACTANTWith the models and parameters described above, we calculatethe CMC of select CnEm surfactants. The DPD simulations arepretty standard and described in Supporting Informationsection S1. CMC is estimated from the average number offree surfactant molecules in the solution. Two surfactantmolecules were assumed to belong to the same aggregate if anytwo of their tail or middle beads overlapped. The concentrationof “monomeric” surfactant (that is, surfactant belonging toaggregates smaller than a certain critical size nmono) in water wastreated as the CMC. Reference 25 confirms that for nonionicsurfactants this procedure is reasonably accurate, and CMCshowed no significant dependence on the total surfactantloading or box size. The relative error is calculated from themean square deviations of the free monomer concentrationsfrom the average. The average aggregation numbers of allaggregates is reported as the aggregation number for thesystem, Nag. The details of CMC and Nag calculations may befound in Supporting Information, section S4.Table 5 summarizes the micellar properties obtained from

the simulations compared with the available experimental data.CMC is mostly determined by the alkyl tail chain length n.Addition of one ethylene monomer decreases CMC by about 1order of magnitude. The influence of m on CMC is muchweaker. Generally, CMC slowly increases with m. For thepurpose of evaluation of the accuracy of the proposedparametrization method, we compared the results of the seriesof simulations for the chains with given tail length n anddifferent head lengths m, using, when possible, the models withdifferent bead sizes. The results are shown in Figure 6. Theupper panel presents data for n = 8, which was the shortesthydrophobic block considered; we modeled C8E4, C8E6, andC8E8 surfactants with the bead size Rc = 0.71 nm. SimulatedCMC increases as the hydrophilic block becomes longer, whichqualitatively agrees with experimental observation. The slight

overestimation is visible but comparable to the statisticaluncertainty of both simulations and experiments. Surprisingly,the simplistic model gives similar results but slightly under-estimates the CMC. For Rc = 0.77 nm, decane is chosen as areference compound for the tails and presented as a dimer. TheDPD results for n = 10 showed the same tendencies as for n =8: MD-fitted coarse-grained model slightly overestimated andthe simplistic model slightly underestimated the CMC.Nevertheless, overall the decrease of CMC with n is reproducedvery reasonably with both models, confirming the consistencyof our parametrization scheme.For surfactants with longer hydrophobic tail, n = 12, the

bottom panel in Figure 6 presents CMCs using different coarse-graining levels. The statistical errors for these surfactants aremuch higher both in simulations and experiments, because oflower free surfactant concentrations and large micelle sizefluctuations. The finest coarse-graining with Rc = 0.65 nm givesvery good agreement with the experiments. The Rc = 0.81 nmmodel slightly overestimates CMC, but the difference withexperiment is still comparable to the statistical uncertainty ofboth simulations and experiments (almost exact agreement wasobtained for C12E3). Noteworthy, the reference compound forthe hydrophobic tail is hexane in both cases (Table 1), but it ispresented as a monomer for Rc = 0.81 nm and as a dimer for Rc= 0.65 nm. The only system with a serious discrepancy with theexperiment is C12E8; the Rc = 0.71 nm model overestimatesCMC by the factor of 6.5. Overall, we find semiquantitative toquantitative agreement with the experiment. A selected systemC12E9 modeled using the simplistic model at Rc = 0.65 and 0.81nm also has good agreement with the experimental CMC. Thisindicates that although the representation of the moleculestructure in the simplistic model is not as accurate as the MD-fitted model, it can still be applied to surfactants with alkyl tailsand yield a reasonable CMC at low overall concentrations.3

It should be stressed that the presented data confirm therobustness of the proposed parametrization strategy. Themodels at different coarse-graining levels, parametrized usingexperimental data on different reference compounds andatomistic MD simulations provide consistent results andpredict CMC for the surfactants considered with reasonableaccuracy.

7. MICELLE SIZES AND SHAPES ANDAGGLOMERATION OF MICELLES

To analyze the role of the coarse-grained level on the structureof micellar aggregates beyond CMC, we used the asphericityfactor, which quantifies the shape of the micelle, namely, thedeparture from spherical symmetry.39 The asphericity factor(A) is calculated as

= − + − + −AR

R R R R R R1

2[( ) ( ) ( )]

g4 1

22

21

23

23

22

2

(5)

where R12, R2

2, and R32 are eigenvalues of the gyration tensor

= ∑ − −=S N S S S S1/ ( )( )ij lN

il i jl j1CM CM . Si

CM stands for thecenter of mass of the micelle in coordinate i (i denotes x, y, orz). The asphericity factor is 0 for perfectly spherical micellesand reaches 1 for perfect cylinders.Figure 7 shows the distribution of asphericity factors of

micelles in 2 vol % solution of C12E9 generated with the T4H6model (Rc = 0.65 nm) and T2H3 model (Rc = 0.8 nm). Thefiner T4H6 model (that is, with smaller Rc) predicts two

Figure 6. Experimental and simulation CMC data for CnEm surfactantat different n (categorized by panels) and m (x-axis). Experimentalvalues are shown as crosses, and the trend lines are only forvisualization purposes. Two sets of simulation CMC are reported byusing different methods in calculating bond parameters: values for theMD-fitted model are shown as green open symbols, and values for thesimplistic model are shown as red filled symbols. Shapes of thesymbols indicate the coarse grained level: triangle −0.65 nm, square−0.71 nm, diamond −0.77 nm, and circle −0.81 nm.



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pronounced peaks on the asphericity distribution correspond-ing to spherical and elongated micelle, see also Figure 8a forvisualization. The average aggregation number is Nag = 111,which agrees well with the experimental value reported byZulauf et al. (about 100 for C12E8 at 303 K).40 The coarserT2H3 model produces mostly elongated micelles with nocharacteristic values for A. The average aggregation number isabout Nag = 200, exceeding the experimental value. This effectis related to the packing of the beads in the hydrophobic core:the bead size of the coarser model is comparable to thepersistent length, and the chain therefore has no rigidity. Itappears that despite the “softness” of DPD potentials, which areunable to reproduce the atomistic details, the finer T4H6 modelwith smaller and more tightly spaced beads gives a betterrepresentation of the hydrophilic tail packing in the micellecore compared with the coarser T2H3 model.The tendency of micelles to agglomerate into bigger

aggregates is overestimated by our models, especially forsurfactants with shorter head segment, and prediction capabilitygenerally worsens as Rc increases. In experiments, theaggregation number generally decreases as the hydrophilicblock becomes longer, due to the entropic repulsion betweenthe hydrophilic chains. For the same hydrophobic segmentlength n = 12, this tendency is qualitatively reproduced in

simulations, but it is much more pronounced compared withthe experiments (Table 5). Experimental Nag increases from Nag= 100 for C12E8 to Nag = 150 for C12E6; in simulations with theT4H6 model (Rc = 0.65 nm), Nag increases from 111 and 258.The analysis of DPD configurations for C12E6 reveals a mixtureof regular spherical micelles, elongated micelles, and core−shellmicelles (Figure 8b). Core−shell micelles are especiallycommon in solutions of C12E3 (Figure 8c). Aggregation ofmicelles into bigger agglomerates is also observed for n = 8 withRc = 0.71 nm. Although the micelles in the C8E8 system are notof a well-defined spherical shape, they have a singlehydrophobic core, and the aggregation number (Nag = 83)agrees very reasonably with the experimental value (Nag =72).41 For the shorter molecule C8E4, the volumetricconcentration ϕ = 0.04 at which the DPD simulations areperformed is above the aggregation transition. The micelles areagglomerated via core-to-core contacts into larger agglomeratesof complex irregular shapes with the formal Nag reaching 905(sic!). In experiment, ϕ = 0.04 is below aggregation transition,with Nag = 147.Figure 9 shows the size distribution of the aggregates in the

C8E8 system. To show the importance of carefully assigning the

Figure 7. Normalized asphericity distribution of the micelles for modelC12E9 surfactant. The finer model, T4H6 at Rc = 0.65 nm (black line),produces well-defined spherical micelles; the coarser model, T2H3 atRc = 0.81 nm (red line), produces micelles of more irregular shape andagglomeration between small micelles.

Figure 8. Characteristic aggregation patterns in solutions of 4 vol % CnEm surfactants simulated by the T4H6 model with Rc = 0.65 nm bead size.Surfactant tail beads are shown in red, and head beads are shown in cyan. Water beads are not shown. Zoomed snapshots present the internalstructure of the aggregates: (a) C12E9, mixture of spherical and elongated micelles; (b) C12E6, mixture of spherical, elongated, and core−shellmicelles; (c) C12E3, core−shell micelles.

Figure 9. Normalized distribution of the aggregate sizes for C8E8. TheMD-fitted bond model produces well-defined mostly sphericalmicelles. The simplistic bond model leads to larger and irregularaggregates.



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bond potentials, we present for comparison the results obtainedwith the simplistic bond model. The MD-fitted model predictsmostly spherical micelles with a well-defined peak and Nag = 83.Some elongated micelles containing up to 350 molecules arealso detected, but overall a very reasonable agreement with theexperimental Nag is observed. The simplistic model produceslarger and irregular aggregates. The reason is that the simplisticmodel overestimates the end-to-end distances and the overallrigidity of CE surfactants compared with the MD-fitted model(Supporting Information S6). The rigidity decreases the chainentropy and provokes the formation of larger aggregates asdiscussed in our earlier work,25 resulting in the presence ofseveral peaks in higher values of micelle size in Figure 9. TheNag is overestimated as 128.Noteworthy, there were few attempts in the literature to

quantitatively reproduce micellization in particular experimentalsystems in DPD simulations and to study the transformationfrom spherical micelles to ordered mesophases with increasingsurfactant concentration. Therefore, it is not clear whether theobserved disagreements between experimental and simulatedmorphologies indicate deficiencies of the proposed para-metrization or the limitations of the DPD model in general.It is desirable yet perhaps impossible within the constraints ofthe DPD model to reproduce both the onset of micellization(CMC) and the consecutive morphological transformation in awide concentration ranges above CMC with a single set ofparameters. Observed underestimation of the concentration ofthe onset of micelle agglomeration may be related to the use ofsoft-core models for relatively small surfactant molecules likethose considered in this work.

8. CONCLUSIONWe presented a systematic strategy for parametrization of chainmolecules in DPD simulations, which is based on acombination of top-down and bottom-up approaches. Thekey components of the proposed approach are the choice of thecoarse-graining level (size of beads) and the dissection ofmolecules into head and tail beads with regard to the givenmolecular chemical structure, determination of interbeadrepulsion interaction parameters from the fitting of theexperimental infinite dilution activity coefficients or solubilitiesof reference compounds to the results of the MC simulation ofthe respective coarse-grained solutions, and control of chainrigidity by introduction of the first (1−2) and second (1−3)neighbor bond potentials with parameters fitted to the resultsof atomistic MD simulations. The proposed approach wasillustrated and its robustness discussed drawing on the exampleof poly(ethylene oxide) alkyl ether CnEm surfactants of differentcomposition with the DPD models constructed at differentcoarse-graining levels and parametrized based on the data fordifferent reference compounds.In this work, we have thoroughly examined the techniques of

parametrization of DPD models based on fitting the mismatchparameters to the infinite dilution activity coefficients of thereference compounds. A reference compound is chosen toclosely resemble a fragment of a target molecule represented asa bead or a dimer of beads. A reliable parametrization techniqueshould, in general, produce similar results independent of thechoice of coarse-graining level (that is the bead diameter) orthe choice of reference compounds. In order to check theconsistency of our parametrization strategy, we modeledaqueous solutions of CnEm surfactants, which consist ofhydrophobic alkane segment and hydrophilic PEO segment.

We constructed DPD models of the CnEm surfactants fordifferent length of each segment. The parameters for thenearest neighbor and second neighbor bonds, which providedesired connectivity and rigidity, were matched to the results ofatomistic simulations. We also employed a simplistic model forthe bonds, with the bond length and stiffness calculated fromthe number of flexible torsions between the bead centers.3 Thelatter approach is not universally applicable, but it gave goodagreement with experiments on CMC and Nag of severalchemically different surfactants. We parametrized the repulsionparameters for DPD models of different coarse-graining levelfrom the IDAC of n-alkanes (from C6 to C10) in water and ofwater in alkanes at room temperature. Water interactions withthe hydrophilic segment were parametrized from the bestmatch to experimental water activities in water−PEO mixturesof low molecular weight.The models of CnEm surfactant with different Rc para-

metrized with different reference alkanes show consistentresults for CMC. We correctly reproduced the increase ofCMC with the hydrophilic block length and most of thesimulated CMC values agree (not without an exception) withthe experimental data, for both MD-fitted and simplistic bondmodels. For CnEm surfactants with longer hydrophilic segment,we also obtained reasonable agreement for the micelle size. Thedownside of the models is an apparent overprediction of atendency to agglomeration of micelles into bigger aggregates. Inseveral systems, we obtained various complex aggregates (suchas cylindrical micelles, core−shell micelles, or irregularagglomerates), with the Nag values larger than those reportedexperimentally. The prediction of Nag generally worsens forshort hydrophilic segments and larger bead sizes. The observeddiscrepancy in the onset of aggregation is most likely attributedto the limitations of the DPD model itself. Noteworthy, theCMC in the systems considered here is varied by 3 decimaldecades, and the fact that we were able to reproduce theexperimental data with reasonable (in most cases quantitative,accounting for the spread of experimental data) agreementtestifies toward the robustness of the proposed approach.Overall, the parametrization of DPD models from the

activities appears to be a relatively straightforward and reliabletechnique, and the accuracy of the models derived with thisprocedure is very reasonable, considering inherent uncertaintiesin micellization experiments and approximations of the DPDmethod itself. The parametrization of bond potentials requiresstandard and inexpensive atomistic MD simulations ofsurfactant fragments. The calculation of the activity calibrationrelationships, γ∞(ΔaIJ), are obtained very easily for smallerbead sizes using NVT simulations with Widom insertions. Theaccuracy of calculations of γ∞(ΔaIJ) steeply declines with a, andadvanced MC techniques may be needed when fluids of lowcompressibility are modeled with relatively large beads (∼Rc >0.7 nm). Noteworthy, while the γ∞(ΔaIJ) calibration for singlebead reference compounds is universal for given bead size, inthe case of dimer reference compounds, these correlations arespecific to the bond parameters obtained by fitting to MDconfigurations.The proposed parametrization strategy can be applied to

other more complex aqueous and organic soft matter systems,like polymer−surfactant gels, polymer brushes, and polyelec-trolytes. It should be noted that the self-assembled morphologyof such irregular systems might be sensitive to the coarse-grained level, as shown above with example of CnEm, since theentropic repulsion between the hydrophilic blocks is difficult to



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reproduce in soft-core models with large beads. Therefore, finercoarse-graining is recommended for chain molecules with smallhydrophilic and hydrophobic blocks to provide at least 3−4beads in each block, at the expense of increased computationalcosts. The latter could be lowered by extension of the proposedparametrization strategy to DPD models with beads of differentdiameters16 and using larger coarse-grained level for thesolvent.

■ ASSOCIATED CONTENT*S Supporting InformationThe Supporting Information is available free of charge on theACS Publications website at DOI: 10.1021/acs.jpcb.6b00031.

Section S1, DPD force field and simulation details,section S2, review of GW parametrization and thecalculation of intracomponent repulsion parameters fromsolvent compressibility for different Rc, section S3, detailsof MD simulations employed in section 4, section S4,details of calculations of CMC and Nag, section S5,derivation of eq 4, section S6, analysis of surfactantconformations in solution, and section S7, influence of1−3 bond length on chain conformation (PDF)

■ AUTHOR INFORMATIONCorresponding Author* Phone: +1-848-445-0834. E-mail: [email protected] authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work is supported by NSF Grant DMR-1207239 andDTRA Grant HDTRA1-14-1-0015.

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parametrization of chain molecules in dissipative particle...

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