THE JOURNAL OF CHEMICAL PHYSICS 140, 054909 (2014)

Flory-Huggins parameter χ , from binary mixtures of Lennard-Jonesparticles to block copolymer melts

Alexandros Chremos,1,a),b) Arash Nikoubashman,2,b),c)

and Athanassios Z. Panagiotopoulos2

1Department of Chemical Engineering, Centre for Process Systems Engineering, Imperial College London,South Kensington Campus, London SW7 2AZ, United Kingdom2Department of Chemical and Biological Engineering, Princeton University, Princeton,New Jersey 08544, USA

(Received 20 August 2013; accepted 9 January 2014; published online 6 February 2014)

In this contribution, we develop a coarse-graining methodology for mapping specific block copoly-mer systems to bead-spring particle-based models. We map the constituent Kuhn segments toLennard-Jones particles, and establish a semi-empirical correlation between the experimentally de-termined Flory-Huggins parameter χ and the interaction of the model potential. For these purposes,we have performed an extensive set of isobaric–isothermal Monte Carlo simulations of binary mix-tures of Lennard-Jones particles with the same size but with asymmetric energetic parameters. Thephase behavior of these monomeric mixtures is then extended to chains with finite sizes throughtheoretical considerations. Such a top-down coarse-graining approach is important from a computa-tional point of view, since many characteristic features of block copolymer systems are on time andlength scales which are still inaccessible through fully atomistic simulations. We demonstrate theapplicability of our method for generating parameters by reproducing the morphology diagram ofa specific diblock copolymer, namely, poly(styrene-b-methyl methacrylate), which has been exten-sively studied in experiments. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863331]

I. INTRODUCTION

Block copolymers comprise two or more chemically dis-tinct blocks covalently attached to each other. They self-assemble into nanostructured morphologies and have becomea versatile tool for organizing soft matter at the nanometerscale. Numerous applications of block copolymers have beenproposed, given the diverse array of structures that can beachieved. The organized structures are generated by the mi-crophase separation driven by the chemical differences be-tween the components. Synthetic chemistry of the constituentself-assembling molecules can tune the size, shape, and ar-rangement of patterns, which in turn affect the macroscopicmechanical, rheological, and optical properties of the result-ing materials.1

The phase behavior of A-b-B diblock copolymers isdictated by the polymer segment chemical character (quan-tified by the Flory-Huggins interaction parameter χ ), thechain length (the degree of polymerization N), and com-position (monomer volume fraction of the minority blockf). Upon cooling a disordered melt (increasing χ ) belowthe order-disorder temperature, TODT, microphase separationoccurs because polymer chains self-organize into differentmorphologies to minimize the contacts between A and B do-mains under the constrains of connectivity and overall incom-

a)Electronic mail: achremos@imperial.ac.uk.b)A. Chremos and A. Nikoubashman contributed equally to this work.c)Electronic mail: arashn@princeton.edu.

pressibility. Nonetheless, the phase behavior is determinedmainly by the volume fraction f and the combined productχN. Consequently, the phase behavior is often depicted viaa two-dimensional phase diagram showing the equilibriummicrophase as a function of χN and f . Depending on thecomposition, lamellae, perforated lamellae, gyroid, cylindri-cal and spherical morphologies can be produced.2

Computer simulations are a valuable tool for understand-ing the physical properties of polymeric systems, and offerpromising capabilities for designing new materials. However,atomistic simulations are unfeasible for most practical situ-ations, since they can only sample relatively small systemsthat contain only a few microdomains. These system size is-sues become even more problematic for the study of copoly-mers under shear, since the long-range ordering of the mi-crodomains is of central interest in this case (for a recentreview on simulations of shear-induced morphological tran-sitions in block copolymers, see Ref. 3). Furthermore, theuse of realistic shear rates requires the simulation of longtime scales, which are out of reach in such atomistic com-putations. Therefore, the development of accurate coarse-graining methods is essential for the study of block copolymersystems. Typically, one can circumvent these problems bycoarse-graining (CG) the block copolymers, thereby reducingthe degrees of freedom. On the one hand, bottom-up type ap-proaches can be employed for the coarse-graining, where thestructural (and dynamical) characteristics of the system arepreserved from atomistic scales. On the other hand, top-downCG schemes are tailored for preserving macroscopic systemproperties, which are usually more accessible in experimentsthan atomistic properties.

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054909-2 Chremos, Nikoubashman, and Panagiotopoulos J. Chem. Phys. 140, 054909 (2014)

The primary theoretical approach for studying the blockcopolymer phase behavior has been Self-Consistent FieldTheory (SCFT).4–7 In SCFT, the interactions between poly-mer chains are approximated by a spatially inhomogeneousenvironment (represented mathematically by a field) in whichpolymers move independently. The theory provides a wayof predicting the free energies of possible structures, andthus constructs a phase diagram indicating the preferredmorphology for a given set of molecular parameters. Sev-eral additional methods have been developed for the simu-lation of polymeric systems, some inspired by SCFT, suchas the time-dependent Ginzburg-Landau method,8 dynamicdensity functional theory,9 cell dynamics,10 and dissipativeparticle dynamics.11, 12 These methods have been success-ful at describing the structural and dynamical properties ofsimple and complex polymers on a qualitative and semi-quantitative level.7 However, excluded volume effects, whichplay a key role for the static and dynamic properties of poly-meric systems,13, 14 are neglected in these methods, renderingquantitative comparisons to experiments difficult.

On the other hand, bead-spring models with impenetrablemonomer cores have served well for the description of poly-mer models over the last three decades, including polymermelts,15, 16 gels,17, 18 thin films,19–21 nanocomposites,22–24 andbrushes.25–27 Particle-based descriptions preserve the molec-ular structure of the initial polymer to a much higher degreeand can provide insights into the microscopic properties ofthe system. The downside of bead-spring models comparedto SCFT calculations, however, is that the free energy isnot directly accessible, making additional (and potentially)expensive calculations necessary to access such quantities.Although there has been recent progress in merging meanfield and particle-based descriptions of copolymers,28 coarse-grained bead-spring models that take excluded volume effectsinto account have received less attention.

Our model is inspired by the Kremer and Grest bead-spring model,15, 29–31 and consists of Lennard-Jones (LJ) par-ticles that are tethered through massless springs. We keep thesize of the beads and the bonding potential constant for allparticle types, rendering the interaction strengths as the onlytunable parameter. Since it is necessary for any mapping pro-cedure to reproduce the phase behavior of the original sys-tem, one of the key challenges here is to establish a robustconnection between the interaction parameters of the poly-mer model and the experimental Flory-Huggins parameterχ ; in other words, the experimental order-disorder tempera-ture TODT needs to be preserved by the mapping. To this end,we extend the approach brought forward in Refs. 11 and 32(and successfully utilized in Refs. 33 and 34) by developinggeneral semi-empirical relationships between the LJ interac-tions parameters and the χ parameter in the NPT-ensemble.We refine the corrections necessary for chains with finitesizes, and achieve good agreement between predictions andexperiments.

We present the general philosophy behind our blockcopolymer model in Sec. II and highlight the key differencesto previous approaches. We then present a detailed descriptionof the mapping procedure in Subsections II A and II B, wherewe first derive semi-empirical relationships between the LJ in-

teraction parameters and the Flory-Huggins parameter χ for amonomeric binary mixture, and then extend our calculationsto chains of a finite length. In Sec. III, we first demonstrate theapplicability of our approach by studying the structural tran-sitions of a symmetric diblock copolymer system with ad hocinteraction parameters. Then in Subsection III B, we demon-strate how the proposed scheme can be utilized to coarse-grain a specific experimental system, namely, poly(styrene-b-methyl methacrylate) (PS-PMMA). Finally, we draw ourconclusions and present a short outlook in Sec. IV.

II. METHODOLOGY

The general purpose of any theoretical polymer model isto replicate the behavior of the experimental system as ac-curately as computationally feasible. An integral step duringthis procedure is the mapping of the chain with N monomersto an equivalent system with NCG < N CG beads. The result-ing reduction in degrees of freedom is however accompaniedby a loss in (microscopic) detail. One of the key characteris-tics of block copolymer systems is that they phase separateon a microscopic scale when quenched below TODT, form-ing nanoscale domains. Therefore, the faithful reproductionof this phase behavior must be a minimum requirement forany coarse-graining scheme targeted at block copolymer sys-tems, whereas the detailed description of atomic length-scalesis only secondary.

Bead-spring models used to study block copolymerswere initially developed for the NV T -ensemble with theuse of purely repulsive interactions (e.g., Weeks-Chandler-Andersen potential35).29–31 Thermodynamic properties suchas the pressure cannot match the experimental values, dueto the purely repulsive nature of the interactions. More ad-vanced bottom-up CG strategies, such as the Inverse Boltz-mann Method36, 37 or multiscale procedures,38–42 are tailoredfor preserving the structural (and dynamical) characteristicsof the system, but require costly atomistic simulations, whichare often computationally infeasible and not transferable toother systems. Furthermore, preserving the structural prop-erties on the molecular scale does not guarantee the correctthermodynamic behavior in bulk systems.

In this contribution, we follow a different approach,which extends and generalizes the ideas brought forward byKremer et al. in their seminal paper,15 where they mappedspecific polymer chains in a melt onto a bead-spring model.The advantage of such a top-down modeling scheme is that itdoes not rely on any atomistic simulations and only requireseasily accessible information from the original experimen-tal system. The disadvantage is that detailed structural infor-mation is not preserved. We conduct our simulations in theNPT-ensemble, since the majority of experiments are con-ducted at fixed pressure, not constant volume. Additionally,we employ a single interaction potential for all particle typesand include attractive contributions. The pair potential be-tween CG beads is given by:

VLJ(r) ={

4εij

[(σr

)12 − (σr

)6], r ≤ rc

0, r > rc

, (1)

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054909-3 Chremos, Nikoubashman, and Panagiotopoulos J. Chem. Phys. 140, 054909 (2014)

where εij is the interaction strength between two particles iand j, and σ is the diameter of the beads. We chose the LJ po-tential due to its functional simplicity and because it has beenused extensively in prior studies. However, we would like topoint out that our mapping procedure is not restricted to theLJ potential, and can easily be extended to more complex in-teractions, such as the Morse or Mie potentials, that allow fora more realistic description of the system on the microscopicscale.

Each chain consists of two blocks with f N beads of typeA and (1 − f)N beads of type B, where bonds between themare modeled through harmonic potentials:

VH(r) = k(r − l0)2, (2)

where l0 is the equilibrium length of the spring, and k is thespring constant. Here we set l0 = σ and k = 2500 kBT/σ 2

for all bonds, which creates rather rigid links between thebeads. This parameter choice, coupled with the impenetrablecores of the LJ particles, prevents unphysical bond crossings.Bonded interactions are generally different for different parti-cle types. However, in this contribution, we focus on develop-ing a model that realistically reproduces the phase behavior,which is primarily dictated by the non-bonded interaction be-tween A and B beads. Future work will be directed towardsthe study of the impact of the bond length on the conforma-tional and thermodynamical properties of polymer melts.

In our framework, the only free parameters are the in-teraction strengths εij, and we will provide a detailed discus-sion on how the interaction strengths εij have to be chosenfor matching specific systems in what follows. In Sec. III wealso present an extensive example of a specific system (PS-PMMA) to demonstrate the applicability of our approach.

A. Choice of interaction parameters

The first choice for any CG approach is the number ofmonomers that will be mapped on a single CG bead; thecoarser the model is, the softer the effective pair interac-tion between the beads. For very soft potentials, unphysicalbond crossings can occur which violates the essential dynam-ics of the system and renders transport properties invalid.43

Hence, only few monomers should be grouped into a CGbead, if a quantitative description of the dynamic propertiesis of any interest.44, 45 For all polymers, the spatial correla-tions between monomers (e.g., due to restricted bond angles)become negligible after a characteristic length scale known asthe Kuhn length b.46, 47 At length scales above b, the chaincan be treated as a fully flexible polymer consisting of Nb

Kuhn segments instead (where one Kuhn segment representsat least one monomer). In other words, the universal physicalproperties of polymers are independent of the local chemicalstructure.

The Kuhn length provides crucial information about thelength scale and number of monomers that can be representedby a single CG bead, but no information on the interaction pa-rameters εij. The number of repeat units representing a Kuhnsegment is denoted by N

expb and NCG

b , for the experimentaland simulation systems, respectively. In order to establish arelative energy scale between the CG beads, we consider the

critical temperature Tc and volume Vc of the constituent groupof monomers. This information is readily available throughexperiments or calculations within the framework of groupcontribution theory.48 It is noteworthy that the critical temper-ature is not a linear function of the number of monomers.49

We then choose one compound as the reference point andset the number of monomers per CG bead (N exp/N CG) equalto N

expb σ/bCG, where bCG denotes the Kuhn length of a lin-

ear chain comprised of LJ particles.15 For homopolymer sys-tems of LJ chains we obtain for our model parameters bCG

≈ 1.28σ . This is done by calculating the mean-square end-to-end distance of the homopolymer chains as a function ofchain length: 〈R2(N CG)〉 = (bCG)2(N CG − 1). For the othermonomer types, we choose N exp/N CG in such a way so thatthe CG bead has the same (critical) volume as the refer-ence compound. The interaction strengths can then be ex-pressed in units of the reference compound i, and are given byεjj = Tc, j/Tc, i, etc. This procedure can be used to determinethe interaction between two beads of the same species butgives no information on the cross species interaction. Eventhough it is common practice to use the Lorentz-Berthelot(LB) mixing rules εij = √

εiiεjj ,50 copolymer systems arehighly sensitive to the exact value of εij,51, 52 and thereforeit is advisable to use a more sophisticated approach for deter-mining the cross interaction strength. A key ingredient of ourCG scheme is to choose εij so that the experimental TODT isrecovered.

To do so, we need to establish a connection between themodel parameters εij and the experimental Flory-Huggins pa-rameter χ . The latter describes the degree of segregation inpolymer blends and block copolymers, and is typically ex-pressed as function of temperature, T. To establish a rela-tionship between χ and T, we follow the approach of Grootet al.32 Here, the system consists of a two-component mix-ture of A and B particles which are initialized in two blocks,thereby generating a separating interface (see Fig. 1). Aftersome time, the two phases mix to a certain degree, and thefree energy F of the system can be calculated via:

F

kT= φA

NAln φA + φB

NBln φB + χφAφB, (3)

where φA and φB = 1 − φA are the volume fractions, andNA and NB are the number of beads in each A and B block.Minimizing Eq. (3) for monomers, NA = NB = 1, we obtainthat at two-phase coexistence:

χ = ln(1/φA − 1)

1 − 2φA. (4)

FIG. 1. A binary mixture of Lennard-Jones particles. The B-type particlesare rendered transparent for the sake of clarity.

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054909-4 Chremos, Nikoubashman, and Panagiotopoulos J. Chem. Phys. 140, 054909 (2014)

FIG. 2. (a) Concentration profile of a binary mixture of Lennard-Jones par-ticles with asymmetric energetic parameters: εBB/ε = 0.9 and εAB/ε = 0.8.Results for three temperatures are shown. The profile is measured in units ofthe box length. (b) The Flory-Huggins parameter χ as function of the inversesystem temperature for a binary mixture of monomers. The dashed lines arelinear fits to the simulation data.

Based on Eq. (4), we can perform MC simulations to cal-culate φA as a function of T. For a given set of εij there isa temperature range over which the beads segregate into twophases. Once equilibrium is reached, we measure the concen-tration profile along the long dimension of the simulation cell[see Fig. 2(a)]. Repeating the process for different tempera-tures we get φA as a function of T, which can be approximatedby the linear relationship:

χ = α/T + β, (5)

where α and β are parameters for the enthalpic and entropiccontributions to χ , respectively. We found indeed χ∝1/T inthe parameter space of interest, and we were able to fit ourdata very accurately by Eq. (5). Fig. 2(b) shows the simula-tion results of χ (T) for two selected sets of interaction param-eters, demonstrating the good quality of the linear fit. In thesimulation model, the α and β parameters are affected by theinteraction parameters. In other words, for each set of ener-getic parameters the exact values of α and β will change andthe aforementioned procedure has to be repeated. In order to

facilitate this process, we have conducted a systematic param-eter space investigation to derive a semi-empirical expressionof χ (T) as a function of εij.

We set up a simulation of M = 8192 beads in a cuboidalbox with periodic boundary conditions in all directions. Wethen performed NPT Monte Carlo (MC) simulations, whichconsisted of three types of moves, random displacements,particle identity swaps, and random changes in volume. Theparticle identity swaps were necessary to overcome meta-stable/jammed regions for highly dense systems. One MC cy-cle consisted of M attempts to perform one type of movement,where we decided to move a particle with a 60 % probabil-ity, swap its identity with a 20 % probability, or change thesystem volume in the remaining cases. For each state point,40 000 cycles were performed for the system to reach equi-libration and 10 000 cycles for production of the results. Theimposed external pressure was set to zero in all our systems,which corresponds to atmospheric conditions.

Without loss of generality we set εAA = kBT and εAA

≥ εBB ≥ εAB. We incrementally decreased the parameterεBB in steps of 0.05 kBT, and varied εAB with the same stepsize. We then obtained χ (T) [or more precisely α and β inEq. (5)] for each set of parameters, by calculating the concen-tration of the A-rich domain, φA, at 10 different temperatures.In agreement with previously published results,53, 54 we iden-tified two distinct phase regions, which exhibited solid- andliquid-like behavior. Fig. 3(a) shows ρ(T) for εBB = kBT and

-0.12

-0.08

-0.04

0

W4; W

6

0.8

0.9

1.0

ρ σ3

1.2 1.3 1.4 1.5 1.6 1.71 / T [k

B/ε]

3

4

5

6

χ

(a)

(b)

(c)

FIG. 3. (a) The number bead density ρ(T), (b) W4 (squares) and W6 (circles)order parameters, and (c) Flory-Huggins parameter χ as function of inversesystem reduced temperature for a binary mixture with εBB = kBT and εAB= 0.75 kBT.

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054909-5 Chremos, Nikoubashman, and Panagiotopoulos J. Chem. Phys. 140, 054909 (2014)

εAB = 0.75 kBT. Near T ≈ 0.7ε/kB (1/T ≈ 1.3kB/ε) there is asharp change indicating a first order transition. The transitioncan also be identified by plotting the Steinhardt bond orderparameter W4 and W6 against T [see Fig. 3(b)].55 These struc-tural changes significantly affect the Flory-Huggins parameteras well, as shown in Fig. 3(c).

In the solid region of the phase diagram (εAA ≥ εBB

≥ εAB ≥ 0.75 kBT), we can describe α and β by linear re-lationships, which are given by the following equations:

α(εBB, εAB) = 9.8εBB − 18.4εAB + 8.6, (6)

β(εBB, εAB) = −4.6εBB + 6.9εAB − 2.4, (7)

where the uncertainties are <1% for the parameters describ-ing α, and ∼3%–5% for β. Here, α corresponds to the en-thalpic contribution, whereas β reflects the entropic contri-bution. Establishing a similar linear relationship for the liq-uid phase region is a much more difficult undertaking, sinceentropy plays a more significant role in those states, render-ing the standard deviation for the β parameters as too large.Due to these uncertainties, we focus on the parameter-subsetwhere the system exhibits solid-like behavior. This is howevernot a significant restriction, since the energetic parameters ofmost real systems are in this regime (see Sec. III for a typicalexample).

B. Order-disorder transition temperatureand fluctuations

The phase transition from a homogeneous state of di-block copolymers to an ordered state with microphases iscalled an “order-disorder transition” (ODT),56 and the ODTtemperature TODT can be calculated for infinitely long chainsusing mean-field theory.57 However, for chains of finitelength, TODT deviates from mean-field predictions due tofluctuations, and hence a correction term is needed. Thecorrections are controlled by a Ginzburg parameter,58 N̄ ,which is proportional to the copolymer molecular weightN̄ = 63(R3

gρp)2, where ρp is the copolymer number densityin the melt and Rg is the copolymer radius of gyration. ForN̄ → ∞ and fixed χN the Leibler’s mean-field predictionsare asymptotically recovered.56 The mean-field critical pointfor symmetric diblock copolymers is located at

(χN )ODT = 10.495. (8)

For asymmetric diblock copolymers (χN)ODT is larger thanthe value 10.495.57 However, for finite N̄ , the fluctuation cor-rections impose both qualitative and quantitative changes inthe phase diagram, and in particular (χN)ODT is replaced by aweak first order transition at (a lower temperature):59, 60

(χN )ODTeff = (χN )ODT + 41.0N̄−1/3. (9)

More recently, other suggestions have been put forward todescribe more accurately the shift in the ODT due to lowmolecular weight of the chains; for instance, Matsen et al.have proposed the following relationship for symmetric di-block copolymers:6

(χN )ODTeff = (χN )ODT exp (11.35/N) . (10)

In simulations, the chain length is typically NCG

= 10–200 beads, but even at NCG = 100 the shift of TODT

can be significant, since the chain length is not large enoughfor suppressing the fluctuations. Thus in order to obtain aneffective Flory-Huggins parameter (χN)eff, χN is rescaled inthe following way:

(χN )eff = χN(χN )ODT

(χN )ODTeff

. (11)

Using then Eq. (8) in conjunction with Eq. (9) yields:

(χN )eff = χN

1 + 3.9N̄−1/3. (12)

It is a common practice to simplify Eq. (12) by obtaininga relation between N̄ and N (for more details, see Refs. 11and 61). Alternatively, the effective Flory-Huggins parameterbased on Eq. (10) would read:

(χN )eff = χN

exp (11.35/N). (13)

Rearranging Eq. (5) we get for TODT:

TODT = αN

(χN )ODT − βN. (14)

The above relation between TODT and α and β is, however,only valid for systems not affected by fluctuations, e.g., chainswith an infinite length. For a finite chain length affected byfluctuations, one should use Eqs. (5) and (9) instead, yieldingthe final result:

TODT = αN

(χN )ODT + − βN, (15)

where represents the shift of the order-disorder transitiondue to fluctuations, as discussed above. For N → ∞, the cor-rection term converges to zero thus recovering Eq. (14).

As outlined above, the formulation for the prediction ofODT is done within a mean-field framework, where the ex-clude volume effects are not accounted for and thus N repre-sents the number of segments with a Gaussian behavior (oneGaussian segment is equivalent to at least 10 Kuhn segments).In previous simulation studies, the calculation of the (χN)parameter was done by obtaining an effective χ eff and thenmultiplying it by the number of segments N. Although, thisapproach has been used successfully in diblock copolymersystems,29, 62 it has the following drawback: by incorporatingthe effect of molecular architecture into the Flory-Huggins pa-rameter χ eff it becomes difficult to assess the chemical incom-patibility between different monomer types independently ofthe molecular architecture. In other words, each change to themolecular composition necessitates a reevaluation of χ eff, re-sulting in additional and expensive simulations. Nevertheless,the above observations provide insights on how to predict theODT in particle-based simulations. In a particle-based modelthe behavior of the ODT is expected to deviate significantlyfrom the mean-field type of predictions due to excluded vol-ume effects; these excluded volume interactions can affectthe stiffness of a polymer chain, which in turn can signifi-cantly impact the phase behavior of diblock copolymers.63, 64

Therefore, we need to obtain a link between the number of

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054909-6 Chremos, Nikoubashman, and Panagiotopoulos J. Chem. Phys. 140, 054909 (2014)

CG beads with the number of mean-field segments. In gen-eral it is possible to establish such a relationship by mappingthe CG chain on an ideal chain with the use of Kuhn segmentof the LJ chains, NCG

b . However, in block copolymer meltsbelow TODT the chains are stretched due to the phase sepa-ration. Here, not only the individual blocks are stretched intheir respective domains, but also the distance between thecenter of mass of the blocks is increased. For symmetric di-block copolymers (f = 0.5), the mean-square distance of thechain is given by 〈R2〉 = (sbCG)2(NCG/t − 2), where t is a fit-ting parameter and s is the degree of stretching compared to ahomopolymer melt. We find that the stretching increases thesize of the chains by a factor s ≈ 2.6. Thus, by taking intoaccount the additional stretching imposed on the chains, wecan now map the diblock copolymer chain of LJ-chain on itsideal state as follows:

Neff = NCGσ

sbCG+ γ. (16)

The physical significance of γ in Eq. (16) above is to ensurethat the diblock copolymer in its ideal state is always abovethe minimum chain length, i.e., Neff = NA + NB ≥ 2, andtherefore has been set to γ = 2.

We can now calculate the effective combined segregationparameter (χN)eff by combining the χ parameter obtainedfrom the monomeric fluid [Eqs. (5)–(7)] with the effectivenumber of segments, Neff, as derived in Eq. (16):

(χN )eff = (α/T + β) Neff. (17)

Hence, we realize the fluctuation-induced shift of TODT solelyby the mapping of the polymer length. Inserting Eq. (17) thenback into Eq. (14) yields the final expression for TODT:

TODT = α

(χN )ODTsbCG

NCGσ + γ sbCG− β

. (18)

The above equation, Eq. (18), is similar to Eq. (15) andcan be used in conjunction with Eqs. (6) and (7) to adjust thecross species interaction εij (with i �= j) under the imposedconstraint T

expODT = T sim

ODT. Indeed, for conducting the simula-tions, the parameter of interest is the cross energy interactionparameter between the A- and B-type beads, εAB. The param-eter can obtained with the use of Eqs. (6) and (7) from the laststep, which leads to the following relation:

εAB =[

(χN )ODTTODT

NCGσ/(sbCG

) + γ+ εBB (4.6TODT − 9.8)

+ 2.4TODT − 8.6

]/(6.9TODT − 18.4) . (19)

For f < 0.5, diblock copolymers form other structuresthan lamellae, such as spherical, cylindrical, bicontinuous,and perforated lamellar structures, and therefore the indi-vidual copolymer chains are stretched to a different extent.Hence, we need to establish a relationship between f and s.For f = 0 the size of a chain is the same as in a homopoly-mer melt (Neff = NCG), while we found s = 2.6 in the limit f= 0.5. We approximate the intermediate stretching values s(f)through a linear relationship s(f, NCG) ≈ 2.6(ζ f + η), where ζ

FIG. 4. Flow chart of the proposed CG scheme. Input parameters are dis-played in circular nodes, while CG parameters are shown in rectangularnodes. The left and right panels contain all properties of the A and B block,while the middle panel contains the information on the cross-interactionparameter.

and η are adjustable parameters. Taking then into account thetwo constraints at f = 0 and f = 0.5 we get:

s(f,NCG) ≈ 2[2.6 − σ/bCG]f + σ/bCG. (20)

The proposed functional form is an empiric expression, and amore rigorous derivation requires extensive simulations nearthe phase boundaries at different compositions and energeticparameters, which were out of the scope of the current inves-tigation. Nonetheless, the good agreement between our mor-phology diagram and the one obtained by SCFT (cf. Fig. 7)provides an a posteriori justification for our linear approxi-mation of s(f).

To summarize our proposed CG scheme we have con-structed a flow chart, which illustrates how the model param-eters can be obtained for any specific polymer system, seeFig. 4.

III. RESULTS AND DISCUSSION

A. Phase behavior

In order to verify the applicability of our CG scheme,we have studied the phase behavior of symmetric diblockcopolymers (f = 0.5) by performing isobaric-isothermal(NPT) MD simulations. The simulations were performedby the large scale atomic/molecular massively parallel sim-ulator (LAMMPS) package,65 which is a general-purpose,open source molecular dynamics code. Measurements ofthe ODT are computationally and conceptually challenging,since finite size effects need to be avoided and slow dy-namics arises due to large free energy barriers. Therefore,we have employed five different chain lengths, namely, N CG

= 18, 36, 72, 144, and 200. Before extending our approachto a specific experimental system, we first study polymers

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054909-7 Chremos, Nikoubashman, and Panagiotopoulos J. Chem. Phys. 140, 054909 (2014)

FIG. 5. Simulation snapshots of a system consisting of 2000 symmetric(f = 0.5) diblock copolymers with chain length NCG = 144 at different tem-peratures: (a) T = 1.35 ((χN)eff = 16.5); (b) T = 1.65 ((χN)eff = 11.4); and(c) T = 1.75 ((χN)eff = 10.1). Systems are grouped together according to thedegree of segregation. The (χN)eff is calculated with Eq. (17).

with ad hoc interaction parameters εAA = 1.0 kBT, εBB

= 0.921 kBT, and εAB = 0.908 kBT, which correspond to val-ues of α ≈ 0.864 and β ≈ −0.267. At the start of our simula-tions, we removed the attractive contribution of the cross in-teraction by lowering the cutoff to rc = 21/6 σ . This procedureaccelerates the initial microphase separation. Once a lamellaeis formed, the original cross interaction is restored by reset-ting the cutoff to rc = 3.0 σ . Representative snapshots of thesystem are shown in Fig. 5 for different temperatures. Herewe can identify three regimes, namely, the weak (χN ≈ 10),intermediate (χN � 12–50), and strong segregation regimes(χN > 50).57, 60 The strong segregation regime is character-ized by ordered domains that are occupied almost exclusivelyby only one species. In the intermediate segregation regimethe separation becomes less sharp, but the interface betweenblocks is still very narrow. In the weak segregation regime, theinterface between neighboring domains becomes more fuzzy(sinusoidal shape) but lamellae can still be discerned.2, 66

The temperature at which the lamellae (or the appropri-ate morphology for the given volume fraction f) melt gives theTODT for that specific composition. Fig. 6 shows a compari-son between the predictions of TODT and simulation results,and it is clearly visible that the predictions of Eq. (14) overes-timate TODT consistently, due to lack of accounting the effect

0 50 100 150 200N

0

1

2

3

4

T k

Β/ε

TODT

from eq. 14T

ODT from eqns. 12 and 14

TODT

from eqns. 13 and 14T

ODT from eqns. 14 and 16

Simulation

LJ chains - Glass

FIG. 6. Comparison of TODT from predictions and simulations as a functionof chain length. The dashed line represents the glass transition temperatureof LJ-homopolymers.

of fluctuations in the original theory. If we implement a cor-rection according to Eq. (15) then a slightly better agreementwith simulation results can be observed. Nevertheless, devi-ations are still observed for long chain lengths, ODT resultsdo not converge with the mean-field predictions. This is not asurprise since excluded volume effects are not incorporatedin the model. Moreover, for the energetic parameters cho-sen, phase separation would not be possible for chain lengthsN < 10 because TODT is at the same level or lower than theglass transition temperature, Tg. Near the glass transition tem-perature, the polymer chains significantly lose their mobilitydue to the entrapment of the chains by their neighbors. Be-low Tg the system freezes and thus any fluctuations due to thesize of the chains would be suppressed. For more informationregarding the glass transition temperature of (chain) polymersystems we refer the reader to Refs. 67–69. If we use Eq. (18)then it is possible to obtain a significant better agreement withthe simulations results for all chain lengths investigated.

Our focus so far was on symmetric diblock copolymers(f = 0.5), but it is also important to verify our approachfor asymmetric diblock copolymers. To this end, we ex-amined systems composed of 2000 chains with length NCG

= 72, where we kept the interaction parameters the sameas above. By varying the composition of the diblock chain,f = NCG

A /NCG, and (χN)eff, we have produced a morphol-ogy diagram [see Fig. 7(a)]. The good agreement with theresults from SCFT calculations demonstrates the validity ofour modeling technique. In particular, our simulations showthat the combined Flory-Huggins parameter (χN)eff can bedecomposed into the two independent contributions χ andNeff, where χ only needs to be determined once, irrespectiveof copolymer length and composition. The influence of themolecular architecture is then solely captured through Neff:if we ignored the molecular architecture (N = N CG), thenTODT as calculated by Eq. (14) is overestimated; the devia-tion is larger for symmetric diblock copolymers, as shown bythe dotted-dashed line in Fig. 7(b). If a correction is includedthrough Eq. (16) based solely on symmetric diblock copoly-mers, i.e., s ≈ 2.6, then a good description is obtained nearf = 0.5, but TODT is underestimated for asymmetric diblockcopolymers [see dotted line in Fig. 7(b)]. If the chain stretch-ing is incorporated via Eq. (20), then a better description ofTODT is obtained for all values of f, as indicated by the solidline in Fig. 7(b).

Overall, we find that the semi-empirical equations thatprovide the enthalpic and entropic terms [see Eqs. (6) and(7)] in conjunction with Eq. (5) provide a robust frame-work for accurately predicting the Flory-Huggins parameterχ . Moreover, we demonstrate that with the appropriate map-ping scheme, the ODT of our particle-based copolymer modelcan be predicted.

B. Coarse-graining of PS-PMMA

Now that we have successfully demonstrated the practi-cability of our CG scheme for an ad hoc system, we wantto model a specific system. The example is based on thePS-PMMA diblock copolymers, which are commonly usedin experiments70–73 and various applications.74, 75 PS-PMMA

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054909-8 Chremos, Nikoubashman, and Panagiotopoulos J. Chem. Phys. 140, 054909 (2014)

FIG. 7. Comparison between the morphology diagram computed throughSCFT57 and our model. Panel (a): (χN)eff vs. f, where (χN)eff is calculatedthrough Eq. (18) with s calculated by Eq. (20). Symbols correspond to thephases observed in our simulations, while the dashed line corresponds to theglass-transition temperature at TkB/ε = 0.4. Solid lines correspond to thedifferent phases: lamellar (L), gyroid or bicontinuous (G), cylindrical (C), andspherical (S) phases. Panel (b): T vs. f, where the symbols and dashed linesare the same as in (a). The dotted-dashed, dotted, and solid lines correspondto the TODT according to Eq. (14), Eq. (18) with s ≈ 2.6, and Eq. (18) with scalculated by Eq. (20), respectively. The lines for the phase boundaries weretruncated at the glass-transition temperature of the LJ chain model.

chains do not strongly associate with themselves (i.e., chem-ical reactions or formation of hydrogen bonds) and thereforeobey universal polymer laws. As discussed in Sec. II, any flex-ible polymer chain can be mapped on a chain of Kuhn seg-ments. For the constituent compounds of PS-PMMA, namely,styrene (S) and methyl methacrylate (MMA), both the lengthof a Kuhn segment as well as the number of monomers thateach Kuhn segment represents are available in the literature,47

and we have summarized all the necessary information re-quired for the coarse-graining in Table I.

We chose PS as our reference point and map Nexpb /bCG

≈ 5.4 S monomers onto one CG bead. As discussed in de-tail in Sec. II A, we require the critical temperature andvolume of both compounds for determining the interactionstrengths. In this work, we have used the group contribu-tion method of Constantinou and Gani48 and listed the re-sulting values of Tc and Vc in Table I. Also, the number

TABLE I. Physical properties of the constituent monomers styrene (S) andmethyl methacrylate (MMA). Values for b and N

expb have been obtained from

Refs. 47 and 76, while Tc and Vc for a group of N exp/N CG monomers havebeen determined through group contribution theory.48

Monomer bexp (nm) Nexpb Nexp/NCG Tc (K) Vc (cm3/mol)

S 1.8 6.9 5.4 945 1670MMA 1.5 6.0 5.8 884 1670

of MMA monomers mapped to one CG bead, NCGMMA, is set

as NCGMMA ≈ 5.8, which is obtained under the constraint that

the different types of monomers occupy the same volume,Vc,S(NCG

S ) = Vc,MMA(NCGMMA). This is an approximation nec-

essary to take into account the difference in volume betweenthe two monomers, although deviations in the conformationalproperties of the polymers may be expected when the char-acteristic ratios of the two monomers are significantly differ-ent. However, this is not the case for PS-PMMA, where theKuhn lengths are very similar (see Table I), and we antici-pate only a minor impact through this approximation. Finally,Tc provides valuable information on the interaction strengths,yielding εAA = Tc, PS/Tc, PS = 1 and εBB = Tc, PMMA/Tc, PS

= 0.9354. The last missing parameter is the cross interactionstrength εAB, which can be determined through the experi-mental Flory-Huggins parameter χ .

For symmetric PS-PMMA chains with a molecu-lar weight of 30.5 kg/mol the ODT takes place at TODT

= 523 K.77 The cross energy interaction parameter obtainedfrom Eq. (15) is then εAB = 0.9312 kBT, with α = 0.520 andβ = −0.1504. However, if we had used the LB mixing rulesinstead78 then the cross interaction strength would read εAB

= 0.9672 kBT, which gives a negative value for TODT. Thismeans that the system would not microphase separate as thesystem is annealed. The cross energy interaction parameter isessentially the main characteristic to differentiate between thetypes of beads and if it is not significantly different then thesystem would only have one phase for the whole temperaturerange. In fact, TODT should be higher than the glass transitiontemperature (for LJ systems it is approximately at 0.4) be-cause otherwise the system will freeze before the microphaseseparation is initiated. Usage of the Fender-Halsey (FH) mix-ing rules,79 i.e., εAB = 2εAεB/(εA + εB) = 0.9666 kBT wouldlead to the same problem (see Fig. 8). At first glance the dif-ferences in the cross interaction parameter seem to be negligi-ble, but the exact value of εAB has a significant impact on thestructural behavior of the melt. The failure of the LB and FHmixing rules presumably originates from the representation ofseveral monomers as a single bead; indeed, above TODT eachmonomer has a similar probability for having a monomer ofthe same or different type as a neighbor. However, the group-ing of the monomers as a single bead renders this behaviornear TODT impossible. Therefore, a correction is needed to ac-count for this effect and thus the cross interaction parameterwe obtain is smaller than what mixing rules predict. Never-theless, the dependence of TODT on the cross energy interac-tion is significantly affected with the variation of εPS-PMMA,see Fig. 8. Simulations at different values of the cross en-ergy interaction parameter provide further support that our

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054909-9 Chremos, Nikoubashman, and Panagiotopoulos J. Chem. Phys. 140, 054909 (2014)

FIG. 8. Comparison between the prediction of TODT for a PS-PMMA di-block copolymers system with finite chain length (N CG = 100 and f = 0.5)as function of εPS-PMMA. In the first case, TODT is predicted by Eq. (14) withN = N CG, while in the second case TODT is predicted by Eq. (18), whichincludes the chain length correction based on Eq. (16). The dotted lines de-scribe the approximate boundaries of system’s state, i.e., for 0 < TkB/ε < 0.4the system is expected to be in the glassy state. The negative values of TODTrepresent systems that do not phase separate.

methodology can predict the phase behavior (e.g., TODT) ofdiblock copolymers with a particle based model. Finally,we would like to point out that our model is also in goodagreement with the reported experimental behavior even be-low TODT; indeed, for 100 K lower than the TODT of theexample that we have used, our model predicts a value of(χN)eff = 13.5 while experiments report values in the rangeof 11.1–16.5 (for more details on the experimental studies seeRefs. 70–73).

IV. CONCLUSIONS

In this contribution, we present a novel coarse-grainingscheme for block copolymers that aims at replicating themesoscale behavior with an off lattice bead-spring model.We map a group of monomers to Lennard-Jones beads of thesame size for each type of monomers, rendering the cross in-teraction strength as the only free parameter. In order to es-tablish a robust connection between the experimental Flory-Huggins parameter χ and the interaction parameters, we haveperformed extensive isobaric-isothermal Monte Carlo simula-tions of binary mixtures of Lennard-Jones particles, and haveobtained semi-empirical equations that link χ to the modelparameters. We then extended this relationship to chains ofa finite length, by adapting the result of mean-field theorywith an appropriate mapping scheme. In contrast to bottom-up coarse-graining methods, such as the inverse Boltzmannmethod, our model does not require costly atomic simula-tions and only relies on a small set of easily accessible exper-imental parameters. Furthermore, the inclusion of excluded-volume effects potentially allows for a physically morerealistic description of the structural and dynamical propertiescompared to ultrasoft models, which often allow for unphys-ical bond crossing. In order to demonstrate the applicabilityof our copolymer model, we performed molecular dynam-

ics simulations of diblock copolymer melts, which correctlyreproduced the characteristic domain morphologies over abroad range of f and χN values.

ACKNOWLEDGMENTS

This work has been supported by the Princeton Center forComplex Materials (PCCM), a U.S. National Science Foun-dation Materials Research Science and Engineering Center(Grant No. DMR-0819860). A.C. is thankful for the compu-tational resources at Imperial College London.

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