Systematic and simulation-free coarse graining of multi-component polymeric systems: Structure-based coarse graining of binary polymer blends

Accepted Manuscript Systematic and simulation-free coarse graining of multi-component polymeric systems: Structure-based coarse graining of binary polymer blends Qiang Wang PII:

S0032-3861(17)30376-2

DOI:

10.1016/j.polymer.2017.04.008

Reference:

JPOL 19588

To appear in:

Polymer

Received Date: 9 January 2017 Revised Date:

26 March 2017

Accepted Date: 4 April 2017

Please cite this article as: Wang Q, Systematic and simulation-free coarse graining of multi-component polymeric systems: Structure-based coarse graining of binary polymer blends, Polymer (2017), doi: 10.1016/j.polymer.2017.04.008. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Systematic and Simulation-Free Coarse Graining of

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Multi-Component Polymeric Systems: Structure-based

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Coarse Graining of Binary Polymer Blends Qiang Wang∗

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Department of Chemical and Biological Engineering

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Colorado State University

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Fort Collins, CO 80523-1370, USA April 10, 2017

Abstract

We proposed a systematic and simulation-free strategy for the structure-based

coarse graining of multi-component polymeric systems using the polymer reference interaction site model theory, and as an example applied it to a simple model system of binary polymer blends. Our structure-based coarse graining ensures that the original and coarse-grained (CG) systems have the same intra- and intermolecular pair

∗

E-mail: [email protected].

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correlation functions involving the CG segments (each representing the center-ofmass of a group of consecutive monomers on the same chain in the original system). When applied to binary polymer blends, it does not change the spinodal curve (thus also the critical point), regardless of the original model system and closure used.

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We also quantitatively examined how the effective non-bonded pair potentials between CG segments and the thermodynamic properties of CG systems vary with the

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coarse-graining level. As expected, the structure-based coarse graining cannot give thermodynamic consistency (i.e., the same interchain internal energy per chain and

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virial pressure) between the original and CG systems at any coarse-graining level.

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Introduction

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1

We recently proposed a systematic and simulation-free strategy for coarse graining of homopolymer melts, where we use integral-equation theories (including the reference

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interaction site model (RISM),1 the polymer reference interaction site model (PRISM),2 and the multi-block PRISM3 theories), instead of many-chain molecular simulations

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(MCMS), to obtain the structural and thermodynamic properties of both the original and coarse-grained (CG) systems, and quantitatively examine how the effective pair potentials between CG segments and the thermodynamic properties of CG systems vary with the coarse-graining level.4 Our strategy is quite general and versatile. It is much faster than those using MCMS, thus effectively solving the transferability problem in coarse graining,5 and further avoids the problems caused by finite-size effects and

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statistical uncertainties in MCMS. We have demonstrated these advantages by applying

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our strategy to both the structure-based4 and the relative-entropy-based6 coarse graining of homopolymer melts. Here, we extend our previous work4 and propose a systematic

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and simulation-free strategy for the structure-based coarse graining of multi-component

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polymeric systems, and as an example apply it to binary polymer blends (BPB).

Polymer blends have been investigated in numerous experimental, simulation, and theoretical studies due to their both fundamental and practical importance in polymer science and engineering.7 In contrast to the case of homopolymer melts, however, coarse graining of polymer blends has been performed only by a few groups: Murat and Kremer coarse grained each chain in symmetric BPB as one (purely repulsive) ellipsoid that reproduces the chain mean-square radius of gyration in the original system.8

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Their

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approach was later improved by Eurich and Maass.9

Sun and Faller reported the first coarse graining of BPB using the iterative Boltzmann inversion, where each monomer is coarse grained as one segment.10 They performed

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molecular dynamics simulations of an original system containing 24 atactic polystyrene (PS) chains and 36 cis-polyisoprene (PI) chains, each having 15 monomers with atomistic

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details, to provide the radial distribution functions between PI-PI, PS-PS, and PI-PS CG segments to be closely matched by adjusting the non-bonded pair potentials between these CG segments.

The CG bonding and bending potentials were directly trans-

ferred from their previous coarse graining of the corresponding homopolymers,11, 12 and

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attractive linear tail functions were used to correct the pressure of the CG model of BPB.10

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Recently, Wu developed a scheme to coarse grain each monomer in stereoregular polymers as one segment, and applied it to two BPB composed of either isotactic or

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syndiotactic poly(methyl methacrylate) chains each having 15 monomers and isotactic poly(vinyl chloride) chains each having 24 monomers.13 In this scheme, the CG non-

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bonded pair potentials are first obtained using the conditional reversible work method from the all-atom constraint distance stochastic dynamics simulations of trimer/trimer pairs in vacuum, and then used to obtain the CG bonding, bending and torsion potentials using the iterative Boltzmann inversion from the all-atom stochastic dynamics simulations of the corresponding single chain in vacuum.13 In a subsequent study, to achieve better representability Wu obtained the CG bonding, bending and torsion potentials from the all-atom simulations of the corresponding homopolymer melts instead.14

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Finally, Guenza and co-workers performed structure-based coarse graining of BPB using the PRISM theory,2 where the original model system is taken as continuous Gaussian chains with the Dirac δ-function interactions and each chain is coarse grained as only one (spherical) particle;15, 16 their analytical approach requires several approximations,

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as given in detail later. With the hypernetted-chain closure17 for the CG system, they further obtained the effective pair potentials between CG particles.18

As shown in

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Sec. 3.3 below, however, this closure qualitatively fails in reproducing the thermodynamic properties of the original systems used in this work and in our previous work4 when the number of CG segments on each chain becomes large.

Since proposed in 1987 by Schweizer and Curro,2 PRISM theory has been widely

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applied to many polymeric systems with great success;19–21 this supports the application

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of our systematic and simulation-free coarse-graining strategy to multi-component polymeric systems such as polymer blends, block copolymers, polymer solutions,

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nanocomposites, etc. In particular, the self-consistent PRISM theory22 has been applied to united-atom models of isotactic/syndiotactic polypropylene blends and isotactic polypropylene / polyethylene blends, and achieved good agreement with experiments and

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molecular dynamics simulations.23, 24 To illustrate our general strategy for structure-based coarse graining of multi-component polymeric systems using the PRISM theory, in this work we apply it to a simple model system of BPB.

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2.1

Models and Methods

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A general strategy for structure-based coarse graining using PRISM theory

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Let us consider an M-component original system consisting of MP > 0 polymeric species (for which coarse graining is to be performed) and MS ≡ M − MP ≥ 0 small-molecule

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species (e.g., solvent molecules or small particles, for which no coarse graining is to be performed) in volume V at thermodynamic temperature T . To simplify our notation, here we consider each chain as a hompolymer and each small molecule as a single interaction site; it is straightforward to generalize our methods to more complex systems of heteropolymers and multi-site small molecules. Each molecule of polymeric species I

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(= 1, . . . , MP ) in the system has NI monomers (interaction sites), the number density

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of which is denoted by ρI ≡ nI NI /V with nI being the number of molecules of species I; for simplicity, we call each molecule of small-molecule species K (= MP + 1, . . . , M) a

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“monomer” and ρK ≡ nK /V .

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As explained in our previous work,4 there are three steps in our simulation-free strategy for structure-based coarse graining using the polymer reference interaction site model (PRISM) theory. First, PRISM theory for the above original system gives19–21          mm mm  ˆ mm ˆ mm H ˆ mm ˆ mm 0 ˆ mm H ˆ mm  ˆ ˆ H Ω c c Ω 0 H  PP PS  =  PP   PP PS   PP  +  PP PS  . (1) mm mm  ˆ mm H ˆ mm ˆ ˆ ˆ mm H ˆ mm  H 0 Ω cˆmm cmm 0 Ω H SS SS SP SS SP ˆ SS SP SS ˆ mm is a symmetric MP × MP matrix with its (I,J) element given by ρI ρJ h ˆ mm In Eq. (1), H PP IJ

(for J = 1, . . . , MP ) and hmm IJ (r) being the intermolecular monomer-monomer (denoted by 6

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ˆ mm the superscript “mm”) total pair correlation function (PCF) between species I and J, H PS ˆ mm , H ˆ mm = (H ˆ mm )T , and is an MP ×MS matrix with its (I,K−MP ) element given by ρI ρK h IK SP PS ˆ mm ˆ mm is a symmetric MS × MS matrix with its (K−MP ,L−MP ) element given by ρK ρL h H SS KL

(for L = MP + 1, . . . , M); cˆmm PP is a symmetric MP × MP matrix with its (I,J) element given

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by cˆmm and cmm IJ IJ (r) being the intermolecular monomer-monomer direct PCF between species I and J, cˆmm ˆmm PS is an MP × MS matrix with its (I,K−MP ) element given by c IK ,

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T cˆmm cmm cmm SP = (ˆ PS ) , and ˆ SS is a symmetric MS ×MS matrix with its (K−MP ,L−MP ) element

ˆ mm given by cˆmm ˆ Imm , KL ; ΩPP is a diagonal MP ×MP matrix with its (I,I) element given by ρI NI ω P I PNI mm 2 mm mm ω ˆ Imm ≡ N ˆ I,ij /NI , and ωI,ij (r) being the normalized (i.e., ω ˆ I,ij (q = 0) = 1) i=1 j=1 ω

intramolecular monomer-monomer PCF between the monomers i and j of species I, and

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ˆ mm is a diagonal MS × MS matrix with its (K−MP ,K−MP ) element given by ρK ; and Ω SS R∞ throughout this work we use the short-hand notation fˆ = (4π/q) 0 drf (r)r sin(qr)

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to denote the 3D Fourier transform of a radial function f (r) with q being the wavenumber.

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Eq. (1) needs to be solved together with an approximate relation between hmm IJ (r) and cmm IJ (r) (i.e., a closure) for each IJ pair (here {I,J} = 1, . . . , M). For example, the Percus-Yevick (PY),25 hypernetted-chain (HNC),17 and random-phase approximation

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(RPA)26 closures give, respectively,   [1 − exp (βuIJ (r))] (hmm  IJ (r) + 1)   mm cmm −βuIJ (r) + hmm IJ (r) = IJ (r) − ln (hIJ (r) + 1)     −βuIJ (r)

PY HNC ,

(2)

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where β ≡ 1/kB T with kB being the Boltzmann constant, and uIJ (r) is the non-bonded isotropic pair potential between two monomers of species I and J, respectively, separated at distance r. 7

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Next, for the purpose of coarse graining, we consider the center-of-mass of lI consecutive monomers on each chain of species I (= 1, . . . , MP ) in the system as a

(non-interacting) sites, Eq. (1) is expanded as 

ˆ mm H ˆ mm H ˆ ms H PP PS PP





ˆ mm 0 Ω ˆ ms Ω PP PP



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coarse-grained (CG) segment of I. With these centers-of-mass introduced as auxiliary

cˆmm PP

ˆcmm PS

cˆms PP



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      ˆ mm ˆ mm ˆ ms    mm mm ms  ˆ mm 0   HSP HSS HSP  =  0 Ω    ˆ ˆ ˆ c c c SS      SP SS SP  ˆ sm H ˆ sm H ˆ ss ˆ sm 0 Ω ˆ ss ˆcsm ˆsm ˆss H Ω PP PS PP PP PP PP c PS c PP  mm    ms mm ˆ mm ˆ ms   ˆ ˆ ˆ Ω 0 ΩPP H H H      PP PS PP    PP   ˆ mm ˆ mm ˆ ms  ˆ mm 0  . ×  0 Ω +    H H HSP SS     SP SS       ˆ sm ˆ ss ˆ sm H ˆ sm H ˆ ss  Ω 0 Ω H PP

PP

PS

PP

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PP

(3)

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ˆ ms /lJ (for ˆ ms is an MP × MP matrix with its (I,J) element given by ρI ρJ h In Eq. (3), H PP IJ J = 1, . . . , MP ) and hms IJ (r) being the intermolecular monomer-segment (denoted by the superscript “ms”) total PCF between a monomer of species I and a segment of

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ˆ sm = (H ˆ ms )T , H ˆ ms is an MS × MP matrix with J in the expanded original system, H PP PP SP ˆ ms /lJ , H ˆ sm = (H ˆ ms )T , and H ˆ ss is a symmetric its (K−MP ,J) element given by ρK ρJ h KJ PS SP PP

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ˆ ss /lI lJ and hss (r) being the MP × MP matrix with its (I,J) element given by ρI ρJ h IJ IJ intermolecular segment-segment (denoted by the superscript “ss”) total PCF between ms species I and J; cˆms ˆms PP is an MP × MP matrix with its (I,J) element given by c IJ and cIJ (r)

being the intermolecular monomer-segment direct PCF between a monomer of species I T ˆsm ˆms and a segment of J in the expanded original system, c cms PP = (ˆ PP ) , c SP is an MS × MP T ˆsm ˆss matrix with its (K−MP ,J) element given by cˆms cms KJ , c PS = (ˆ SP ) , and c PP is a symmetric ss MP × MP matrix with its (I,J) element given by cˆss IJ and cIJ (r) being the intermolecular

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ˆ ms is a diagonal MP × MP segment-segment direct PCF between species I and J; and Ω PP PNI PNI /lI ms ms matrix with its (I,I) element given by ρI i=1 α=1 ω ˆ I,iα /NI and ωI,iα (r) being the

normalized intramolecular monomer-segment PCF between the monomer i and the

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ˆ sm = Ω ˆ ms , and Ω ˆ ss is a diagonal MP × MP matrix with its segment α of species I, Ω PP PP PP P I /lI PNI /lI ss ss 2 ss ss (I,I) element given by ρI NI ω ˆ I /lI , ω ˆ I ≡ (lI /NI )2 N ˆ I,αγ , and ωI,αγ (r) being α=1 γ=1 ω

the normalized intramolecular segment-segment PCF between the segments α and γ of

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species I.

Because introducing the auxiliary sites does not change the intermolecular total ˆ mm in Eqs. (1) and (3) gives PCFs in the original system, when MS > 0, equating H SP  sm  mm ms ˆ ˆc ˆ ˆ sm = 0, thus ˆcms = 0, which also ensures the equality of H ˆ mm in Ω SS SP ΩPP + HPP SP SS

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ˆ mm cˆms H ˆ sm = 0). Furthermore, equating H ˆ mm in both both equations (which requires Ω SS SP PS PP

cˆss PP

=

−ˆcsm PP

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equations gives 

ˆ mm + H ˆ mm Ω PP PP

  sm −1  ms −1 mm sm ˆ ˆ ˆ ˆ ˆcms , ΩPP + HPP − Ω Ω PP PP PP

(4)

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ˆ mm finally gives and equating H PS 

  mm   sm −1 mm mm sm sm ˆ ˆ ˆ ˆ ˆ ˆ HPS − ΩPP + HPP ΩPP + HPP HPS = 0.

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ˆcsm PP

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ˆss ˆsm If MS ≥ MP , these two equations give ˆcsm PP = c PP = 0; otherwise, setting c PP = 0 (which is ˆ mm ˆ mm an assumption27 ) gives ˆcss PP = 0 and the equality of HPP , as well as HPS (when MS > 0), in Eqs. (1) and (3). With these intermolecular direct PCFs, we obtain after some algebra  mm −1  mm −1 ms ˆ ss = Ω ˆ sm Ω ˆ ˆ mm Ω ˆ ˆ , H H Ω PP PP PP PP PP PP   −1 ˆ mm . ˆ sm = Ω ˆ sm Ω ˆ mm H H PS PS PP PP 9

(6) (7)

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Finally, PRISM theory for the CG system gives    mm    mm    mm ˆ ms mm ˆ ms  mm  ˆ ˆ ˆ ˆ ˆ ˆ H HSP Ω 0 c c Ω 0 H HSP  +  SS  ,  SS  =  SS   SS SP   SS ss ss   sm ss sm ss ˆ ˆ ˆ ˆ ˆ ˆ ˆcPS cˆPP 0 Ω H H H H 0 Ω PP

PP

PP

PS

PP

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PS

(8)

where ˆcSP is an MS × MP matrix with its (K−MP ,J) element given by cˆKJ and cKJ (r)

being the intermolecular monomer-segment direct PCF between a monomer of species K

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and a segment of species J in the CG system (which is different from cms KJ (r) = 0 in the expanded original system), ˆcPS = (ˆcSP )T , and ˆcPP is a symmetric MP × MP matrix with its (I,J) element given by cˆIJ and cIJ (r) being the intermolecular segment-segment direct PCF between species I and J in the CG system (which is different from css IJ (r) = 0 in the expanded original system). Note that the structure-based coarse graining is achieved by

find cˆPS =

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ˆ (given by Eqs. (6) and (7)) and Ω ˆ in Eqs. (3) and (8). From Eq. (8) we using the same H



ˆ ss Ω PP

+

ˆ ss H PP

−1

ˆ sm H PS



−1 ˆ mm Ω SS

−

cˆmm SS



,

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   −1 ss −1 ss ss ms ss ˆ ˆ ˆ ˆ ˆ cˆPP = ΩPP HPP − ˆcPS HSP ΩPP + HPP .

(9) (10)

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The effective non-bonded CG pair potentials v = {vIJ (r)} (here I = 1, . . . , MP and J = 1, . . . , M) are then obtained from a closure given by   ln [1 − cIJ (r)/ (hss  IJ (r) + 1)]   ss βvIJ (r) = hss IJ (r) − cIJ (r) − ln (hIJ (r) + 1)     −cIJ (r) which is the same as Eq. (2) but for the CG system.

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PY HNC , RPA

(11)

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To illustrate the above general strategy for structure-based coarse graining using the PRISM theory, in this work we apply it to binary polymer blends as described in detail below.

Structure-based coarse graining of binary polymer blends

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2.2

Here we consider the original system of binary polymer blends A/B (i.e., MP = 2 and

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MS = 0). First of all, we note that our structure-based coarse graining, given by Eq. (6) in this case, does not change the spinodal curve (thus also the critical point), which occurs ˆ mm (q = 0) (thus H ˆ ss (q = 0)) diverge. This is regardless of when all the elements of H PP PP the original model system and closure used, and is one of the major conclusions of our work.

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For simplicity, hereafter we consider symmetric blends where the only difference

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between A and B is their labeling; that is, we have NA = NB = Nm , ρA = ρB = ρ/2 ˆ mm = h ˆ mm , cˆmm = cˆmm , and the with ρ being the total number density of monomers, h BB AA BB AA

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normalized single-chain structure factor ω ˆ Amm = ω ˆ Bmm = ω ˆ mm . PRISM theory for the original system, Eq. (1), can then be simplified as

and

2 2 2 Nm [ˆ cmm cmm cmm ˆ mm ] (ˆ ω mm )2 AA − (ρ/2)Nm ((ˆ AA ) − (ˆ AB ) ) ω 2 ((ˆ 2 2 ω mm )2 1 − ρNm cˆmm ˆ mm + (ρ2 /4) Nm cmm cmm AA ω AA ) − (ˆ AB ) ) (ˆ

(12)

2 mm Nm cˆAB (ˆ ω mm )2 . 2 ((ˆ 2 2 ω mm )2 1 − ρNm cˆmm ˆ mm + (ρ2 /4) Nm cmm cmm AA ω AA ) − (ˆ AB ) ) (ˆ

(13)

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ˆ mm = h AA

ˆ mm = h AB

To fully specify our original system here, we use the continuous Gaussian chain (CGC) model for the chain connectivity, where Nm → ∞ but each chain has a finite root-mean-square end-to-end distance Re,0 in the ideal-chain state. We also use 11

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2 2 βuAA (r) = βuBB (r) = (¯ κ/ρc Nm ) δ(r) and βuAB (r) = [(¯ κ + χ)/ρ ¯ c Nm ] δ(r) for the non-

bonded pair potentials between monomers, where κ ¯ ≥ 0 and χ¯ ≥ 0 control, respectively, the system compressibility and A-B incompatibility, ρc ≡ ρ/Nm is the total chain number density, and δ(r) denotes the Dirac δ-function. When both κ ¯ and χ¯ are finite, we refer

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to this model as the soft-core CGC-δ (or Gaussian thread) model, which is equivalent to

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that used by Helfand and Tagami.28

To solve Eqs. (12) and (13), in this work we use the PY closure for the original mm system. Linearizing it for the AA interaction gives cmm AA (r) = − (hAA (r) + 1) βuAA (r) = 2 2 mm 3 − (hmm κ/ρc Nm ) δ(r), thus a constant CAA ≡ Nm cˆAA /Re,0 and AA (r = 0) + 1) (¯

(14)

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p hmm (r = 0) = − N¯ CAA /¯ κ − 1, AA

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6 where N¯ ≡ ρ2c Re,0 is the invariant degree of polymerization29 controlling the system fluc-

tuations; as shown in Sec. IIA of our previous work,4 since βuAA (r) → 0 in the soft-core

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CGC-δ model, this linearization is exact. Similarly, linearizing the PY closure for the AB

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2 mm 3 interaction gives a constant CAB ≡ Nm cˆAB /Re,0 and

p N¯ CAB /(¯ κ + χ) ¯ − 1. hmm (r = 0) = − AB

Eqs. (12) and (13) can then be re-written as h √  i 2 2 mm ¯ mm C − N /2 (C − C ) ω ˆ (ˆ ω mm )2 ˆ AA AA AB h AA √ =
2 3 2 Re,0 1 − N¯ CAA ω ˆ mm + N¯ /4 (CAA − CAB ) (ˆ ω mm )2 and

ˆ mm h CAB (ˆ ω mm )2 AB √ = ,
2 3 2 Re,0 1 − N¯ CAA ω ˆ mm + N¯ /4 (CAA − CAB ) (ˆ ω mm )2 12

(15)

(16)

(17)

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respectively. We further assume ideal-chain conformations; that is, ω ˆ mm for the CGC-δ model is given by the well-known Debye function PD (˜ q ) = 2[exp(−x) + x − 1]/x2 with x ≡ q˜2 /6 and q˜ ≡ qRe,0 . Substituting the inverse Fourier transform of Eqs. (16) and (17),

AA

1 2π 2

Z

∞ 0

AB

(18)

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and

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respectively, into Eqs. (14) and (15), we finally have  i h √ √ 2 2 Z ∞ q˜2 CAA − ¯ N /2 (CAA − CAB ) PD (˜ q ) PD2 (˜ q) 1 CAA N¯ √ d˜ q =− −1
2π 2 0 κ ¯ 1 − N¯ CAA PD (˜ q ) + N¯ /4 (C 2 − C 2 ) P 2 (˜ q)

√ CAB N¯ q˜2 CAB PD2 (˜ q) √ d˜ =− q − 1,
2 2 κ ¯ + χ¯ 1 − N¯ CAA PD (˜ q) + N¯ /4 (CAA − CAB ) PD2 (˜ q)

(19)

from which we solve CAA and CAB for given N¯ , κ ¯ and χ ¯ using the globally convergent Newton-Raphson method30 with a convergence criterion of the maximum absolute residual

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error of Eqs. (18) and (19) being less than 10−10 . To calculate the improper integrals, we

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note that, at large q˜, PD (˜ q ) = 12˜ q −2 − 72˜ q −4 + O(˜ q −6 ) and the integrand g(˜ q) in Eq. (19), √ for example, approaches 0 as g(˜ q) = 144CAB q˜−2 + 1728CAB ( N¯ CAA − 1)˜ q −4 + O(˜ q −6). R∞ R q˜ R 1/˜q We therefore split the integral into two parts: 0 d˜ q g(˜ q) = 0 1 d˜ q g(˜ q) + 0 1 dtt−2 g(1/t),

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and calculate each part using Romberg integration31 with an accuracy of 10−10 . To q √ ensure that g(˜ q) begins to approach 0 asymptotically for q˜ > q˜1 , we take q˜1 > 12 N¯ /ǫr

(note that |CAA | < 10) with ǫr = 0.1 being the required ratio of the absolute value of the O(˜ q −4)-term to that of the O(˜ q −2)-term in g(˜ q); the improper integral in Eq. (18) is p calculated in the same way. In the case of κ ¯ = 0 (where CAA = 0), we take q˜1 > 12/ǫr with ǫr = 0.01 instead.

Next, coarse graining each chain as N segments and setting cˆms ˆms AA = c AB = 0 (thus cˆms ˆms BB = c BA = 0 for symmetric blends), we obtain from the structure-based coarse graining 13

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ˆ ss = (ˆ ˆ mm h ω ms /ˆ ω mm )2 h IJ IJ

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(i.e., Eq. (6))

for IJ=AA and AB, where ω ˆ ms is given by Eq. (A8) in our previous work.4

(20) This

result can be compared with the first equality of Eq. (19) in our previous work4 for the

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structure-based coarse graining of homopolymer melts. We also note that a similar result

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was obtained by Guenza and co-workers, who considered only the case of N = 1 and used
2 the approximation ω ˆ ms (N = 1) ≈ exp −q 2 Re,0 /36 .15, 16, 18 Finally, Eq. (10) becomes

(21)

D

cˆAA 3 Re,0

√  . ss 3 2 ss 3 2 ˆhss /R3 + ˆ ˆ ¯ N /2 (hAA /Re,0) − (hAB /Re,0) ω ˆ ss AA e,0 h i = √
 ss ˆ ss /R3 + N¯ /4 (h ˆ /R3 )2 − (h ˆ ss /R3 )2 N 2 (ˆ ω ss)2 + N¯ ω ˆ ss h e,0 e,0 e,0 AA AA AB

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and

(22)

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ˆ ss /R3 h cˆAB e,0 AB h i , = √
 ss 3 Re,0 ˆ ss /R3 + N¯ /4 (h ˆ /R3 )2 − (h ˆ ss /R3 )2 N 2 (ˆ ω ss )2 + N¯ ω ˆ ssh e,0 e,0 e,0 AA AA AB

where ω ˆ ss is given by Eq. (A12) in our previous work.4 With Eqs. (20), (16), and (17),

cˆAA 3 Re,0

AC C

we have after some algebra

h √  i 2 2 (ˆ ω ms /ˆ ω ss)2 CAA + N¯ /2 (CAA − CAB ) ((ˆ ω ms )2 /ˆ ω ss − ω ˆ mm ) h i = √
2 2 2 ms 2 ss mm 2 ms 2 ss mm ¯ ¯ N 1 + N CAA ((ˆ ω ) /ˆ ω −ω ˆ ) + N /4 (CAA − CAB ) ((ˆ ω ) /ˆ ω −ω ˆ ) (23)

and cˆAB (ˆ ω ms /ˆ ω ss )2 CAB h i. = √
2 3 2 Re,0 2 2 ms 2 ss mm ms 2 ss mm ¯ ¯ N 1 + N CAA ((ˆ ω ) /ˆ ω −ω ˆ ) + N /4 (CAA − CAB ) ((ˆ ω ) /ˆ ω −ω ˆ ) (24) 14

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The effective CG pair potentials βvAA (r) = βvBB (r) and βvAB (r) are then obtained from Eq. (11).

To calculate the inverse Fourier transform f (˜ r) ≡ (1/2π 2)

R∞ 0

ˆ 3 )˜ d˜ q (f/R q r˜)/˜ r, e,0 q sin(˜

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ss where r˜ ≡ r/Re,0 and f is hss ˜, AA , hAB , cAA , or cAB , we note that, at large q p ω ˆ ms ≈ 6π/N exp(−˜ q 2 /72N)/˜ q and therefore approaches 0 exponentially. With

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limq˜→∞ ω ˆ ss = 1/N, we can then use an upper cut-off q˜c > 0, where the absolute value of the integrand in the inverse Fourier transform |g(˜ qc )| is required to be smaller than the relative accuracy of double-precision numbers (i.e., the machine epsilon) ǫ ≈ 10−16 . This R∞ is justified by the following: For I = q˜c d˜ q g(˜ q) with g(˜ q) = exp(−A˜ q 2 ) and A > 0, we

D

have I/g(˜ qc) = q˜c−1 /2A + O(˜ qc−3) at large q˜c ; while this result is for f (˜ r = 0), we expect √ it to be valid also for f (˜ r > 0). Assuming |CAA | = |CAB | = 10, we choose q˜c = 100 N

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to ensure that the above requirement is met for both r˜ = 0 and r˜ > 0 (even in the case

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of κ ¯ = 0). We can then calculate the above inverse Fourier transforms for r˜ > 0 using P −1 ˆ s the inverse discrete sine transform defined as F (˜ rj ) ≡ 2 M qk ) sin(πjk/M), where k=1 F (˜

r˜j = j˜ rc /M (j = 0, . . . , M), q˜k = πk/˜ rc (k = 0, . . . , M), M + 1 is the number of equally

AC C

spaced points spanning [0, q˜c ], and Fˆ s (˜ qk ) is the corresponding discrete sine transform (i.e., DST-I). This introduces a cut-off r˜c = πM/˜ qc in r˜; we take M = 4000 to ensure that r˜c is large enough for our subsequent calculation of the thermodynamic properties of the CG system (see below), and perform the inverse discrete sine transform using FFTW.32 In addition, we calculate the above inverse Fourier transforms for r˜ = 0 using Romberg integration31 with an accuracy of 10−10 .

We also calculate the following thermodynamic properties: For the original system 15

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of symmetric blends described by the soft-core CGC-δ model, its interchain internal energy per chain due to the IJ interaction is √ √ 2 ¯ N2 Z ∞ π N r N¯ CIJ m mm βum = dr (h (r) + 1) βu (r) = − , IJ c,IJ IJ 3 1 + δIJ 0 Re,0 4(1 + δIJ )

(25)

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where δIJ = 1 for IJ = AA and 0 for IJ = AB, and its interchain virial pressure due to the IJ interaction is 2 π N¯ Nm =− 3(1 + δIJ )

Z

∞

0

dr

r2 dβuIJ (r) N¯ CIJ (hmm =− , IJ (r) + 1) 3 Re,0 d ln r 4(1 + δIJ )

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3 βRe,0 PIJm

(26)

where we have used dδ(r)/d ln r = −3δ(r) in 3D obtained from ∇·δ(r)r = 0. Note that the linearized PY closure is used to obtain the last equality in Eqs. (25) and (26). For the

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D

CG system, its interchain internal energy per chain due to the IJ interaction is √ Z π N¯ N 2 ∞ βuc,IJ = d˜ r r˜2 (hss r) + 1) βvIJ (˜ r ), IJ (˜ 1 + δIJ 0

(27)

and its interchain virial pressure due to the IJ interaction is π N¯ N 2 =− 3(1 + δIJ )

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3 βRe,0 PIJ

Z

0

∞

d˜ r r˜3 (hss r ) + 1) IJ (˜

dβvIJ (˜ r) . d˜ r

(28)

AC C

We use Romberg integration31 with the above cut-off r˜c to calculate the improper integrals in Eqs. (27) and (28); this is justified by the exponential decay of |βvIJ (˜ r )| towards 0 at large r˜ found in our numerical results shown below. Finally, we calculate the derivative in Eq. (28) using the following sixth-order-accurate finite-difference formulae: With the Taylor expansion it is easy to show that, given equally-spaced data points (xi , fi ) (i = 1, . . . , m ≥ 7) with xi+1 − xi = ∆ (i = 1, . . . , m − 1), we can estimate the first-order

16

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derivative fi′ at xi as    (−147fi + 360fi+1 − 450fi+2 + 400fi+3 − 225fi+4 + 72fi+5 − 10fi+6 )/60∆ for i = 1       (−10fi−1 − 77fi + 150fi+1 − 100fi+2 + 50fi+3 − 15fi+4 + 2fi+5 )/60∆ for i = 2 ′ fi = ;   (2fi−2 − 24fi−1 − 35fi + 80fi+1 − 30fi+2 + 8fi+3 − fi+4 )/60∆ for i = 3       (−fi−3 + 9fi−2 − 45fi−1 + 45fi+1 − 9fi+2 + fi+3 )/60∆ for i = 4, . . . , m − 3 (29)

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for i = m, m−1, and m−2, a formula similar to the above for i = 1, 2, and 3, respectively, can be used. These formulae have an error of O(∆6 f (7) ).

3

Results and Discussion

D

Here we first present in Secs. 3.1 and 3.2 results of the original and coarse-grained (CG)

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systems of symmetric binary polymer blends (BPB), respectively, then compare in Sec. 3.3 their thermodynamic properties (i.e., the interchain internal energy per chain and virial pressure). The latter clearly shows that the structure-based coarse graining cannot give

3.1

AC C

work.4, 33

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thermodynamic consistency at any coarse-graining level, consistent with our previous

Original system

Our original system of symmetric BPB described by the soft-core CGC-δ model, which is equivalent to that used by Helfand and Tagami28 and numerically solved here using the polymer reference interaction site model (PRISM) theory2 and the Percus-Yevick closure25 with ideal-chain conformations, has three parameters: κ ¯ controlling the system compressibility, χ¯ controlling the A-B incompatibility, and the invariant degree of 17

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polymerization29 N¯ controlling the system fluctuations. In this work, we focus on the effects of κ ¯ and χ, ¯ and set N¯ = 104 ; as shown in our previous work,4 other values of N¯ do not change our results qualitatively.

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In the limit of κ ¯ → ∞, the value of χ¯ becomes irrelevant and we have CAA = CAB ; that is, the system reduces to the hard-core CGC-δ model for homopolymer melts. This model

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system has only one parameter (i.e., N¯ ), and was first studied by Schweizer and Curro34 and also examined in Appendix B of our previous work.4 On the other hand, at χ ¯=0 and finite κ ¯ , we also have CAA = CAB ; that is, the system reduces to the soft-core CGC-δ model for hompolymer melts. This model system has two parameters (i.e., κ ¯ and N¯ ), is equivalent to the well-known Edwards model,35 and was also examined in Appendix B

D

of our previous work.4 For completeness, Fig. 1 shows that CAA = CAB in this case is

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proportional to κ ¯ for small κ ¯ and monotonically approaches the corresponding value of

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the hard-core CGC-δ model C0∞ at a rate of κ ¯ −1 (i.e., CAA − C0∞ ∝ κ ¯ −1 ) for large κ ¯. At finite χ ¯ > 0 and κ ¯ ≥ 0, the properties of our original system of symmetric BPB

AC C

are completely specified by CAA and CAB (which depend on κ ¯ , χ, ¯ and N¯ ), as shown in Sec. 2.2 above. First of all, the critical point occurs when the denominator in Eqs. (16) h ih i √ √ and (17) at q = 0 vanishes, i.e., 1 − (CAA + CAB ) N¯ /2 1 − (CAA − CAB ) N¯ /2 = 0;

since CAA ≤ 0 and CAB ≤ 0 as required by Eqs. (14) and (15), respectively, we have √ ∆C ≡ CAA − CAB = 2/ N¯ at the critical point. The mean-field critical point is obtained √ √ RPA RPA if one applies the RPA closure, which gives CAA = −¯ κ/ N¯ and CAB = −(¯ κ + χ)/ ¯ N¯ , thus χ¯RPA = 2. In contrast, the PY closure used in this work accounts for, in an c approximate way, the system fluctuations/correlations neglected at the mean-field level. 18

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10

1

C

AA

C

(

C

)

AB

C

AA

0

0.01 0

10

1

10

2

10

3

10

4

10

5

10

6

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10

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0.1

Figure 1: Logarithmic plot of CAA = CAB obtained numerically for the original system of soft-core CGC-δ model for homopolymer melts solved using the PRISM theory and the PY closure with ideal-chain conformations, where C0∞ denotes the corresponding value

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D

(≈ −9.6223) for the hard-core CGC-δ model (i.e., κ ¯ → ∞). N¯ = 104 .

Fig. 2(a) shows that ∆C > 0 is proportional to χ ¯ at small χ ¯ (i.e., away from the

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critical point), exhibits negative deviation from this linear relation at χ ¯ close to the critical point, and decreases with increasing κ ¯.

It also shows that, at κ ¯ = 100,

AC C

CAA (χ¯ = 0) − CAA (χ) ¯ is proportional to χ¯ at small χ; ¯ the same is found for other values of κ ¯ > 0 (data not shown). Finally, Fig. 2(b) shows that, at χ ¯ = 2, ∆C decreases towards 0 at a rate of κ ¯ −2 at large κ ¯ (−CAA and CAA − C0∞ in this case virtually overlap with the corresponding curve at χ¯ = 0 shown in Fig. 1); the same scaling is found for other values of χ¯ > 0 (data not shown). This supports our above statement that the value of χ¯ is irrelevant in the limit of κ ¯ → ∞. Comparing with the simple RPA (mean-field) √ predictions (e.g., ∆C RPA = χ/ ¯ N¯ ), we see that these results are due to the system

19

0.04

0.02

0.01

1E-3

1E-5

C

0.02

1E-3

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C (

0) 100) 0.019

C

(

AA

0.1

0) C

AA

2

2.1

2.13

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1E-7 C (

1E-4

1E-9

1

10 15

(b)

0

10

1

10

2

10

3

10

4

10

5

10

6

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(a)

10

Figure 2: Logarithmic plot of ∆C ≡ CAA − CAB obtained numerically for the original √ system of symmetric BPB. In part (a), the horizontal line marks its value of 2/ N¯ at the critical point, CAA (χ¯ = 0) − CAA (χ) ¯ at κ ¯ = 100 is also plotted, and the inset shows the enlarged ∆C at κ ¯ = 0 close to the critical point, where dots are the actual calculated

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D

values. In part (b), χ¯ = 2. N¯ = 104 in both parts.

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fluctuations/correlations treated approximately by the PY closure.

Guenza and co-workers also used the CGC-δ model as their original system of

AC C

BPB,15, 16, 18 following Tang and Schweizer.36 In addition to using the Pad´e approxi
2 mation PP (q) = 1/ 1 + q 2 Re,0 /12 for ω ˆ mm , they further assumed cˆIJ = c0 − ǫIJ /ρ for

{I,J}={A,B} instead of using a closure for the original system, where c0 < 0, ǫIJ < 0, and χ ≡ ǫAB − (ǫAA − ǫBB ) /2 is an input parameter equivalent to our χ/N ¯ m . Their assumption is therefore equivalent to the RPA closure leading to the mean-field results, as indicated by their χ-value of 2/Nm at the spinodal temperature16, 18, 36 of symmetric BPB (i.e., the critical point). We finally note that, even with the above approximations,

20

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mm their analytical results of hIJ (r), given by Eq. (12) in Ref. [16] or Eq. (7) in Ref. [18],

do not strictly satisfy the PRISM equations due to the additional approximations used in Eqs. (4.15)∼(4.18) in Ref. [36], which keep only the leading-order terms in ρ.

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Since the systematic and simulation-free structure-based coarse graining of homopolymer melts described by the hard-core CGC-δ model was studied in detail in our previous

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work,4 in the following we take two original systems of symmetric BPB close to their critical point as examples: Case I is at κ ¯ = 100 and χ ¯ = 12, and Case II is at κ ¯ = 0 and χ¯ = 2; note that CAA = 0 in Case II, which leads to some qualitative differences from Case I.

Coarse-grained systems

D

3.2

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Here we present the interchain total and direct PCFs in the CG systems of various N, as well as the corresponding CG potentials obtained with various closures. To gain more

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insights into some of our numerical results, however, we first present an approximate

3.2.1

AC C

analytical solution for the interchain total PCFs in the CG system. Pad´ e approximation for interchain total PCFs

With the assumed ideal-chain conformations in the original system, we can obtain from Eq. (20) the [0/2]-order Pad´e approximant at small q˜ as ˆ ss h AA 3 Re,0

 √ 2 36N 2CAA − d N¯   ≈ , √  √ 18N 2CAA − d N¯ 4 − 4CAA N¯ + dN¯ − a˜ q2

21

(30)

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√ 2 2 2 2 where d ≡ CAA −CAB ≤ 0 and a ≡ 8CAA (1−2N)+4 N¯ [(5N − 3)CAA − (3N − 1)CAB ]− 2(4N − 3)dCAA N¯ + (N − 1)d2N¯ 3/2 , and

ˆ ss h 72NCAB AB   , ≈ √ 3 Re,0 18N 4 − 4CAA N¯ + dN¯ − b˜ q2

(31)

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transform of the above, we then have   √ 2 9N 2CAA − d N¯ exp − q hss r) ≈ − AA (˜ πa˜ r and

3˜ r

√

¯ 4−2CAA √ N ¯ ¯ 4−4CAA N +dN



18NCAB exp − q πb˜ r

D

hss r) ≈ − AB (˜

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√ where b ≡ 4 − 8N + 4(N − 1)CAA N¯ + dN¯ < 0, respectively. Taking the inverse Fourier

−

1 2N

3˜ r

√ ¯ 4−2CAA √ N ¯ +dN ¯ 4−4CAA N

 √ . √ it is easy to show that 4 − 2CAA N¯ 4 − 4CAA N¯ + dN¯

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+

1 2

− 

1 2N

−

CAA √ ¯ 2CAA −d N



 (32)



;

(33)

> 1/2. As shown below,

Eqs. (32) and (33) work well when the original system is close to the critical point. Case I: κ ¯ = 100 and χ¯ = 12

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3.2.2

AC C

In this case, we have CAA ≈ −0.7666 and CAB ≈ −0.7856; that is, the original system solved with the PY closure and ideal-chain conformations is still in the mixed phase (which is stabilized by the system fluctuations neglected at the mean-field level). Figs. 3(a) and 3(b) show that hss r ) < 0 for small r˜ . 0.3, which is due to the repulsion AA (˜ between A monomers in the original system, and that hss r ) > 0 for larger r˜ and AA (˜ hss r ) < 0 for all r˜, both of which are mainly due to the stronger repulsion between A AB (˜ and B monomers than that between A monomers. We also see that hss r) and hss r) AA (˜ AB (˜ for various N approximately collapse, respectively, when r˜ & 1. The insets of Figs. 3(a) 22

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0.34

0.00

0.30

0

5

15

20

r

N 10 N 30

N 100

(r)

AB

1E-11

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(a)

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0.00

N 100 (Pade)

N 4

1E-9

N 10 N 100

1E-11 0

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r

4

5

-0.75 0.0

1.5

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1.0

(d)

r

AA

(r)|

-0.60

2

0.5

-0.40

h

ss

1E-7

AA

(r)

1E-5

ss

N 2

D

AA

r|h

AA

ss

(r)

(r)|

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h

2

r

0.02 N 100 (Pade)

1E-3

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N 1

r|h

1E-3

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0

(b)

r

ss

N 4

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(c)

ss

0.10

N 100

1E-11

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-0.20

N 30

N 1

1E-5

1E-9

h

N 10

1E-9

0.20

rh

(r)

N 4

AB

1E-7

ss

(r)

AA

ss

N 1

1E-5

rh

(r)

AA

h

ss

-0.10

N 1 (Pade)

1E-3

N 1 (Pade)

1E-3

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0.02

0.02

N 2

1E-5

1E-7 N 4

1E-9

N 10 N 100

1E-11 0

1

2

3

4

5

r

0.5

1.0

1.5

r

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Figure 3: The interchain total PCFs between the CG segments, (a) hss r ) and (b) hss r ), AA (˜ AB (˜ for symmetric BPB at κ ¯ = 100 and χ¯ = 12. Parts (c) and (d) show hss r) for symmetric AA (˜

AC C

BPB at κ ¯ = 100 and χ ¯ = 0 and in the limit of κ ¯ → ∞, respectively. The inset in each part shows the corresponding semi-logarithmic plot, where “Pade” denotes the results given by the Pad´e approximant, Eqs. (32) and (33). N¯ = 104 . and 3(b) further show that both r˜hss r) and r˜hss r ) decay exponentially towards 0 for AA (˜ AB (˜ r˜ & 1; this is well explained by the Pad´e approximant, Eqs. (32) and (33). Note that the deviation (quick fall-off) of the N = 100 result from the exponential decay shown in the insets of Figs. 3(a) and 3(b) is due to r˜c ≈ 12.57 used in this case; our use of the inverse 23

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sine transform enforces hss rc ) = hss rc ) = 0. As shown below, however, this r˜c -value is AA (˜ AB (˜ large enough to ensure the accuracy of our calculated thermodynamic properties.

We also note that the Pad´e approximation (thus the exponential decay) does not

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work well for systems far from the critical point. For example, Figs. 3(c) and 3(d) show hss r ) for symmetric BPB at κ ¯ = 100 and χ¯ = 0 (where CAA = CAB ≈ −0.7407) and in AA (˜

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the limit of κ ¯ → ∞ (where CAA = CAB ≈ −9.6223), respectively; we find that hss r) AA (˜ oscillates around 0 for small N . 3 (as indicated by the cusps shown in their inset), and that the Pad´e approximation (which gives hss r ) < 0 for all r˜) clearly deviates from our AA (˜ numerical results for r˜ & 1.5. We further see that r˜hss r) for (effectively) homopolymers AA (˜

D

is nearly independent of N for N & 10.

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On the other hand, Figs. 4(a) and 4(b) show that cˆIJ < 0 (IJ=AA and AB) monotonically √ 3 approaches 0 with increasing qs ≡ q˜/ N. We also see that N 2 cˆIJ /Re,0 for various N

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approximately collapse when plotted as a function of qs ; its behavior at small qs is well 3 explained by the Taylor expansion N 2 cˆIJ /Re,0 ≈ CIJ (1 − qs2 /18), which supports the

AC C

collapse for various N. Figs. 4(c) and 4(d) further show that, compared to hss r) shown IJ (˜ in Figs. 3(a) and 3(b), cIJ (˜ r) decays to 0 much faster (especially for large N) and then oscillates around 0. Finally, the normalization of r˜ by N k with k = −0.644 for the horizontal axis in Figs. 4(c) and 4(d) is explained at the end of this subsection.

The CG potentials βvIJ (˜ r) are calculated from hss r) and cIJ (˜ r ) according to Eq. (11). IJ (˜ Fig. 5 shows that βvIJ (˜ r = 0) > 0, which measures the CG potential strength, exhibits a maximum at small N for all the three closures used for the CG system; the monotonic 24

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(r)|

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(r)

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1

0.8

/R

3

1

0.1

0.5

1.0

N

1.5 0.644

2.0

2.5

r

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Figure 4: Logarithmic plots of the Fourier transform of the interchain direct PCFs in the CG system, (a) cˆAA and (b) cˆAB , where the insets show the corresponding linear plots,

AC C

and the linear plots of (c) cAA (˜ r ) and (d) cAB (˜ r), where the insets show the corresponding semi-logarithmic plots, for symmetric BPB at κ ¯ = 100 and χ ¯ = 12. N¯ = 104 . decrease of βvIJ (˜ r = 0) with increasing N after the maximum is due to the soft-core CGC-δ model used for the original system here, which gives βuIJ (˜ r = 0) = 0 as shown in our previous work.4 We also see that βvAB (˜ r = 0) > βvAA (˜ r = 0) for each closure, due to the stronger repulsion between A and B monomers than that between A monomers in HNC RPA the original system. We further find that βvIJ (˜ r = 0) is larger than both βvIJ (˜ r = 0)

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v(r

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AA

AB

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HNC PY RPA

0.16 1

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0.4

Figure 5: Logarithmic plot of the strength of the effective CG pair potentials between AA and AB segments obtained with various closures for the CG system, for symmetric BPB at κ ¯ = 100 and χ¯ = 12. N¯ = 104 .

D

PY RPA PY and βvIJ (˜ r = 0) for all N, and that βvIJ (˜ r = 0) > βvIJ (˜ r = 0) for small N . 10 but

TE

the opposite occurs for larger N; these are consistent with the values of hss r = 0) shown IJ (˜

EP

in Figs. 3(a) and 3(b), as well as those of cIJ (˜ r = 0) shown in Figs. 4(c) and 4(d).

PY PY Figs. 6(a) and 6(b) show vAA (˜ r ) and vAB (˜ r ), respectively, normalized by their strength.

AC C

PY PY We see that vIJ (˜ r )/vIJ (˜ r = 0) for various N approximately collapse, when r˜ is RPA RPA normalized by N k with k = −0.644; similar results are found for vIJ (˜ r)/vIJ (˜ r = 0) HNC HNC (data not shown) and for vIJ (˜ r )/vIJ (˜ r = 0) as shown in Figs. 6(c) and 6(d).

The normalization of r˜ by N k for the horizontal axis in Fig. 6 is explained at the end of this subsection.

Furthermore, the insets of Figs. 6(c) and 6(d) show that

HNC βvIJ (˜ r ) ≈ −cIJ (˜ r) + (hss r))2 /2 (when |hss r )| is small) is dominated by cIJ (˜ r) for small IJ (˜ IJ (˜ HNC r˜ or N and by hss r ) for large r˜ or N. In particular, we find that βvAB (˜ r) > 0 for N ≥ 17, IJ (˜

26

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1.0

1.0

0)

RI PT

1

N 4

1

N 100

2

(a)

N

0.644

AB

PY

AB

PY

|v

10

0.644

15

r

N 1 N 4

0

(b)

r

N 30

1

N

N 100

2 0.644

3

4

r

1.0

0.1

0

5

10

D

v

N 1 N 2 N 4

0

N 10

1

2

3

N

0.644

(r)| AB

HNC

2

1E-8

0

5

10

15

r

N 2

0

(d)

r

1E-6

1E-10

N 4

4

1E-4

N 1

0.0

N 100

TE

0.0

15

HNC

AA

r

r | v

0)

1E-10

AB

1E-8

0.5

N 100 (Pade)

0.01

(r)/v

HNC

(r

1E-6

AB

2

AA

(r)/v

1E-4

v

(r)| AA

r | v

HNC

0) 0.5

0.1

N 100 (Pade)

0.01

(r

4

5

N

N 10

3

1.0

HNC

0

0.0 N 30

1E-9

1E-12

M AN U

0

(r)|/v

0) (r v

N 1

N 10

HNC

15

r

1E-6

SC

AA

0.644

AB

PY

10

N

0.0

(c)

(r)/v

5

PY

0

0.5

AB

AA

PY PY

1E-9

|v

AA

(r)|/v

0) (r

AA

1E-6

1E-12

v

PY

(r)/v

PY

0.5

1E-3

(r

1E-3

(r

0)

1

N 10

1

N 100

2

N

0.644

3

4

r

EP

Figure 6: Effective CG pair potentials between AA and AB segments obtained with various closures for the CG system for symmetric BPB at κ ¯ = 100 and χ¯ = 12. The insets are

AC C

semi-logarithmic plots, where “Pade” denotes the results given by the Pad´e approximant (see main text for details). N¯ = 104 . and that (˜ rhss r ))2 /2 with hss r ) given by the Pad´e approximant, Eqs. (32) and (33), well IJ (˜ IJ (˜ describes the exponential decay at large r˜ and N shown in the insets of Figs. 6(c) and 6(d).

Finally, the range of CG potentials can be measured by their second moment,

27

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3

3 r

AA RPA

LS fit

AA HNC

1

r

1

0.1 10

(a)

100

1

(b)

M AN U

N

RPA AB

AB

0.1 1

AB

HNC

1/sqrt(3N)

AA

HNC

RI PT

r

HNC

SC

r

10

LS fit

1/sqrt(3N)

100

N

Figure 7: Logarithmic plot of the range of the CG potentials between (a) AA and (b) AB segments obtained with various closures for the CG system, for symmetric BPB at κ ¯ = 100 and χ ¯ = 12. “LS” denotes “least-squares”. N¯ = 104 . d˜ r r˜4 vIJ (˜ r )/


RPA 2

σIJ

R∞ 0

d˜ r r˜2 vIJ (˜ r ). As in our previous work,4 we find

D

∞ 0

R   ∇2q˜ cˆIJ q˜=0 d˜rr˜2 cIJ (˜ r) 1 2 dˆ cIJ d2 cˆIJ =− =− + = R cˆIJ (˜ q = 0) cˆIJ (˜ q = 0) q˜ d˜ q d˜ q 2 q˜=0 d˜rcIJ (˜ r)

=

TE

qR

2CIJ /9N 3 + CIJ /9N 3 1 = . 2 CIJ /N 3N

EP

σIJ ≡

(34)

AC C

In our numerical evaluation of the improper integrals in σIJ , we again use the upper cut-off r˜c given above. We also measure the range of CG potentials by the location of their first root, r˜IJ , which is the smallest r˜ where vIJ (˜ rIJ ) = 0. To accurately locate r˜IJ , we use the HNC HNC cubic polynomial interpolation for βvIJ (˜ r ). Since βvAB (˜ r ) > 0 for N ≥ 17, however, r˜IJ PY RPA is not calculated in such cases. We also note that r˜IJ = r˜IJ .

√ RPA Fig. 7 shows the range of CG potentials as a function of N; both σIJ = 1/ 3N and √ PY σIJ ≈ 1/ 3N are supported by our numerical results (data not shown). We see that 28

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RI PT

√ HNC HNC σIJ ≈ 1/ 3N for small N . 4, indicating that σIJ is dominated by cIJ (˜ r ) in such HNC cases. For larger N, however, σIJ exhibits a minimum (at N = 8 for AA and 7 for AB)

and then monotonically increases with increasing N. On the other hand, we find that RPA r˜IJ ∝ N −0.644±0.003 using the least-squares fitting of the linearly transformed data; as

SC

in our previous work,4 we normalize r˜ by N k with k = −0.644 for the horizontal axis in Figs. 4(c), 4(d), and 6 above to approximately collapse the CG potentials for various N.

M AN U

HNC RPA We also see that r˜IJ ≈ r˜IJ with a maximum deviation of < 0.02 (< 20%) occurring at

N = 100 for IJ=AA and of < 0.04 (< 12%) at N = 16 for IJ=AB; in all cases, we find HNC RPA r˜IJ > r˜IJ .

Case II: κ ¯ = 0 and χ ¯=2

3.2.3

D

In this case, we have CAA = 0 and CAB ≈ −0.01905 (that is, the original system here

TE

solved with the PY closure and ideal-chain conformations is still in the mixed phase), and only highlight the qualitative differences from Case I. Fig. 8 shows hss r ) and hss r ), AA (˜ AB (˜

EP

which can be compared with Figs. 3(a) and 3(b); the only qualitative difference is that hss r ) > 0 for all r˜ here, which is due to the repulsion between A and B monomers, AA (˜

AC C

although there are no interaction between monomers of the same type, in the original system.

On the other hand, Figs. 9(a) and 9(b) show cˆAA and cˆAB , respectively, which can be compared with Figs. 4(a) and 4(b).

We see that cˆAA here exhibits qualita-

tively different behavior; that is, cˆAA > 0 and has a maximum around qs = 4.5. 3 We also see that N 3 cˆAA /Re,0 for various N approximately collapse; its behavior at √ 2 3 N¯ CAB qs4 /3240, small qs is well explained by the Taylor expansion N 3 cˆAA /Re,0 ≈

29

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0.03

0.06 0.02

RI PT

0.02

N 1 (Pade)

N 100 5

10

15

(r)

N 2

1E-7

N 4

1E-9

0.02

20

1E-5

N 10

N 100

1E-11

0

r

0.00

SC

0

rh

ss

ss

N 10

AB

(r)

N 4

h

ss

rh

(r)

0.04

N 2

1E-7

1E-11

0.01

N 1

1E-5

1E-9

AA

ss

h

AA

(r)

N 1

0.02

N 1 (Pade)

1E-3

AB

1E-3

5

10

15

20

r

0.00 0

1

2

(a)

3

0

(b)

M AN U

r

1

2

3

r

Figure 8: The interchain total PCFs between the CG segments, (a) hss r ) and (b) hss r ), AA (˜ AB (˜ for symmetric BPB at κ ¯ = 0 and χ ¯ = 2. The insets show the corresponding semilogarithmic plots, where “Pade” denotes the results given by the Pad´e approximant,

D

Eqs. (32) and (33). N¯ = 104 .

3 We further note that N 3 cˆAA /Re,0

TE

which supports the collapse for different N.

EP

3 and N 2 cˆAB /Re,0 for various N here collapse, respectively, better than in Figs. 4(a)
R∞ 3 and 4(b). Since N 3/2 cAA (rs ) = dqs N 3 cˆAA /Re,0 qs sin(qs rs )/2π 2 rs and 0 R∞
3 N 1/2 cAB (rs ) = 0 dqs N 2 cˆAB /Re,0 qs sin(qs rs )/2π 2 rs , where rs ≡ N 1/2 r˜, the ap-

AC C

proximate collapse in Figs. 9(a) and 9(b), respectively, provides the basis for Figs. 9(c) and 9(d), which can be compared with Figs. 4(c) and 4(d).

Figs. 10(a) and 10(b) show, respectively, the effective CG pair potentials between AA and AB segments at r˜ = 0 obtained with various closures for the CG system, which can be RPA compared with Fig. 5. In particular, the RPA closure gives βvIJ (˜ r ) = −cIJ (˜ r). We see RPA that −βvAA (˜ r = 0) > 0 and is approximately proportional to N −1.476±0.002 as obtained

30

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1E-3

N 10 N 4

0.0004

e,0

3

0.0006

/R

N 100

AB

0.0007

N 2

0.015 1E-7

1E-9

5

10

1

(a)

13

10

q

1E-13

30

(b)

N 100

13 10

q

30

s

AB

3/2

1E-9

1/2

c

1E-11 1E-13 0

0.001

2

4

N

1.0

1.5

N

1/2

(r)|

0.02

N 10 N 4

1E-5

N 2

1E-7

N 1

1E-9 1E-11 1E-13 0

2

0.01

0.0

(d)

r

4

N

1/2

6

8

r

0.00

2.0

TE

0.5

r

8

D

0.000

1/2

6

0.03

AB

(r)

AA

|c

N 1

|c

N 2

1/2

N 4

1E-7

N 100

1E-3

N

(r)|

1E-5

N

(r)

AA

c

3/2

N

10

0.1

0.04

N 10

(c)

5 1

0.046

0.01

1E-3

-0.0004 0.0

0

0.1

s

0.004

0.002

N 1

SC 0.000

0

0.1

0.003

N 2

1E-11

M AN U

0.0000

N 4

0.005

0.0002

1E-13

N 10

0.010

N 1

1E-11

N 100

1E-5

N

1E-9

0.020

2 ^ N c

3 ^ N c

AA

/R

3

e,0

1E-5

1E-7

RI PT

0.1

1E-3

0.5

1.0

N

1/2

1.5

2.0

r

EP

Figure 9: Logarithmic plots of the Fourier transform of the interchain direct PCFs in the CG system, (a) cˆAA and (b) cˆAB , where the insets show the corresponding linear plots,

AC C

and the linear plots of (c) cAA (˜ r ) and (d) cAB (˜ r), where the insets show the corresponding semi-logarithmic plots, for symmetric BPB at κ ¯ = 0 and χ¯ = 2. N¯ = 104 . RPA from the least-squares fitting of the linearly transformed data, and that βvAB (˜ r = 0) > 0

and is approximately proportional to N −0.491±0.001 ; as expected, these exponents obtained at r˜ = 0 are slightly different from −3/2 and −1/2 above, respectively, obtained at small qs . The same results are found for the CG potentials given by the PY closure, PY βvIJ (˜ r ) ≈ −cIJ (˜ r ) + hss r )cIJ (˜ r ) − (cIJ (˜ r ))2 /2, due to the smallness of cIJ (˜ r) and hss r) IJ (˜ IJ (˜

31

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0.004

0.05

RI PT

0.000 1E-3

-0.001

v

1E-4

v

1E-5

AA

(r

AA

3E-6

HNC AA

v

RPA

(r

AA

0)

0)

(r

1

-0.0028 100

10

(a)

HNC

(r

0)

v

RPA

LS fit for

v

AB

(r

HNC

(r

AB

0) 0)

10

100

N

0

N 10 N 1

-0.8 4

6

N

1/2

N 10

AB

HNC

v

N 50 N 100

2

1E-6

0.4

N 100 (Pade)

1E-8 0

0.2

1

2

3

8

5

0.0

10

0

(d)

r

4

r

N 4

TE

2

N 50

D

v

N 100

5

N 4

1E-4

r

4

0.6

AB

3

HNC

2

(r)/v

1

AB

0

HNC

2

(r

1E-7

r

0

N 1

(r)

0)

1E-6

1E-8

4E-3

0.8

v

(r)| AA

HNC

r | v

0)

1

AA

(r

AB

AB

0)

0)

1

N 100 (Pade)

1E-5

(r)/v

HNC

(r

LS fit for

M AN U

1E-4

AA

(r

PY

1.0

2

HNC

AB

0.004

(b)

N 2.4

(c)

v

-0.002

0)

LS fit for v

0.01

0)

(r

PY

v

SC

v

RPA

RPA

2

4

6

N

1/2

8

10

r

EP

Figure 10: Effective CG pair potentials between AA and AB segments obtained with various closures for the CG system for symmetric BPB at κ ¯ = 0 and χ ¯ = 2. Parts (a) and

AC C

HNC (b) are logarithmic plots, except that βvAA (˜ r = 0) in (a) is plotted on the linear scale

(the right axis), where “LS” denotes “least-squares”; and the insets in parts (c) and (d) are semi-logarithmic plots, where “Pade” denotes the results given by the Pad´e approximant. N¯ = 104 .

PY PY (compared to 1) in this case; that is, βvAA (˜ r ) and βvAB (˜ r) are closely represented by

Figs. 9(c) and 9(d), respectively, which can be compared with Figs. 6(a) and 6(b).

32

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RI PT

RPA PY In contrast, qualitatively different results from those of βvIJ (˜ r ) and βvIJ (˜ r ) are HNC found for βvIJ (˜ r ) ≈ −cIJ (˜ r) + (hss r))2 /2 (when |hss r )| is small). In particular, we see IJ (˜ IJ (˜ HNC in Fig. 10(a) that βvAA (˜ r = 0), which is dominated by cAA (˜ r = 0) for small N and by

hss r = 0) for large N, monotonically increases from about −2.7×10−3 at N = 1 to about AA (˜

SC

1.7 × 10−4 at N = 100; it is < 0 for N ≤ 7 and > 0 for N ≥ 8, and levels off at large N. HNC We also see in Fig. 10(b) that βvAB (˜ r = 0), which is mainly determined by cAB (˜ r = 0) for

M AN U

small N but by both cAB (˜ r = 0) and hss r = 0) for large N, is approximately proportional AB (˜ to N −0.433±0.001 as obtained from the least-squares fitting of the linearly transformed HNC HNC data. On the other hand, Figs. 10(c) and 10(d) show vAA (˜ r ) and vAB (˜ r ), respectively,

normalized by their value at r˜ = 0, which can be compared with Figs. 6(c) and 6(d);

D

HNC HNC the qualitative difference is that here βvAA (˜ r ) > 0 for N ≥ 8 and βvAB (˜ r) > 0 for all N.

TE

Finally, Fig. 11 shows the range of CG potentials as a function of N, which can be compared with Fig. 7.

RPA Note that, for Case II (where CAA = 0), σAA cannot

EP

be calculated numerically as the denominator in Eq. (34) is 0; a similar problem is PY PY RPA HNC encountered for σAA as vAA (˜ r ) ≈ vAA (˜ r ) here. Also, r˜AA is not calculated for N ≥ 8

AC C

HNC HNC HNC and r˜AB is not calculated here. We see that both σAA and σAB level off around 1 at √ RPA large N. On the other hand, we find that r˜AA deviates from 1/ 3N by a maximum RPA HNC of only 0.004 (at N = 2), and that r˜AB ∝ N −1/2 for N & 10. We also see that r˜AA

quickly decreases towards 0 with increasing N ≤ 7, consistent with our results shown in Figs. 10(a) and 10(c).

33

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1 HNC AA HNC AB

0.1

r

RPA AB RPA AA

SC

r

RI PT

2

1/sqrt(3N) r 0.01 1

10

AA

100

M AN U

N

HNC

Figure 11: Logarithmic plot of the range of the CG potentials between AA and AB segments obtained with various closures for the CG system for symmetric BPB at κ ¯=0 and χ ¯ = 2. N¯ = 104 .

Comparison of thermodynamic properties between the orig-

D

3.3

Case I: κ ¯ = 100 and χ¯ = 12

EP

3.3.1

TE

inal and CG systems

3 m m 3 m In this case, we have βum c,AA = βRe,0 PAA /100 ≈ 9.583 and βuc,AB = βRe,0 PAB /100 ≈

AC C

19.64. Figs. 12(a) and 12(b) show the differences in these thermodynamic properties between the original and CG systems; the RPA and PY closures are used for the CG system to obtain its properties. We find that the RPA closure underestimates all these quantities, 3 m except βum c,AA for N ≤ 3 and βRe,0 PAA for N ≤ 7. More problematically, all of its devia-

tions monotonically increase with increasing N at large N; the original system therefore cannot be recovered even in the limit of N → Nm (i.e., without coarse graining), as expected. In contrast, while the PY closure also underestimates all the quantities except 3 m 3 m βRe,0 PAA for N ≥ 49 and βRe,0 PAB for N ≥ 43, all of its deviations are much smaller

34

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100

4

400

RI PT

2 1

0.01

0.2

100

1

(b)

N

u

HNC c

u

RPA c

HNC

P

RPA

P

AA

0.003

TE

1

D

0.01

AB 10

AB

m

AB e,0

|P

3

)

RPA

P

HNC

(P

e,0

10

0.1

3

M AN U

c

RPA

u

100

1

c

HNC

0.2 100

N

1000

)

10

(u

PY

10000

100

(c)

10

RPA

R

(a)

10

1

AB

SC

PY

0.01

P

c,AB

R

c,AB

u m

(u

u

AA

RPA

1

10

c,AB

AA e,0

P m 3

|P

0.1

P

c,AA

R

u

1

P

)

|

AA

|

c,AA

u c,AA

|u

m

1 0.1

|

100

10

1 0.3

100

N

EP

Figure 12: Logarithmic plot of the interchain internal energy per chain uc and virial pressure P of the CG system due to (a) the AA and (b) the AB interactions obtained with the RPA and PY closures, for symmetric BPB at κ ¯ = 100 and χ¯ = 12. Part (c) is a

AC C

logarithmic plot of the difference in these quantities of the CG system obtained between the HNC and RPA closures. N¯ = 104 . than those of the RPA closure at large N. In the limit of N → Nm , only the PY closure is expected to recover the original system.

Fig. 12(c) shows the differences between the thermodynamic properties of the CG sys-

35

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RI PT

tem obtained with the HNC closure and those with the RPA closure. We find that

RPA 3 β uHNC ≈ βRe,0 PIJHNC − PIJRPA /100 > 0 scales approximately with N 2 at large c,IJ − uc,IJ

N for IJ=AA and AB. This indicates the qualitative failure of the HNC closure in repro-

β

uHNC c,IJ

−

uRPA c,IJ




√ Z π N¯ N 2 ∞ = d˜ rr˜2 (hss r) + 1) [hss r ) − ln (hss r ) + 1)] ; IJ (˜ IJ (˜ IJ (˜ 1 + δIJ 0

(35)

M AN U

and (27), we obtain

SC

ducing the thermodynamic properties of the original system at large N. From Eqs. (11)

that hss r) is nearly independent of N at large N as shown in Fig. 3 (also in Fig. 8) then IJ (˜
RPA gives the approximate scaling of β uHNC − u with N 2 in such cases. In addition, c,IJ c,IJ from Eqs. (11) and (28), we obtain PIJHNC

−

PIJRPA




Z ∞ π N¯ N 2 d ss =− [h (˜ r) − ln (hss r) + 1)] d˜ r r˜3 (hss r) + 1) IJ (˜ IJ (˜ 3(1 + δIJ ) 0 d˜ r IJ  r˜=∞ π N¯ N 2 =− r˜3 (hss r) + 1) [hss r ) − ln (hss r ) + 1)] r˜=0 IJ (˜ IJ (˜ IJ (˜ 3(1 + δIJ )  Z r˜=∞  3 ss  ss ss − [hIJ (˜ r) − ln (hIJ (˜ r) + 1)] d r˜ (hIJ (˜ r ) + 1) r˜=0 Z π N¯ N 2 r˜=∞ ss = [hIJ (˜ r ) − ln (hss r ) + 1)] IJ (˜ 1 + δIJ r˜=0   r˜3 ss 2 ss × (hIJ (˜ r) + 1) r˜ d˜ r + d (hIJ (˜ r ) + 1) , (36) 3

AC C

EP

TE

D

3 βRe,0

where the second equality is obtained via integration by parts, and the third one is obtained when hss r ) approaches 0 faster than r˜−3/2 at large r˜ (which is supported by Figs. 3 and 8); IJ (˜ neglecting the last hss r) in the last line of Eq. (36) and comparing it to Eq. (35), we then IJ (˜

√ RPA 3 find β uHNC = βRe,0 PIJHNC − PIJRPA / N¯ , which is supported by Fig. 12(c). c,IJ − uc,IJ Finally, we note that similar results are also found for homopolymer melts (i.e., symmetric BPB at χ¯ = 0). Fig. 13(a) and its inset compare the interchain internal energy 36

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10

0.1

0.01

10

0.006

100

1

(b)

N

10

HNC

P

RPA

)

1000

100

1

N

1

RPA

P

(P

100

HNC

P

e,0

) u 10

c

1 0.6 100

N

M AN U

(a)

1

RPA

3

40

HNC

3200

c

RPA

e,0

4000

c

50

(u

u

c

R

3

P

Orig

5000

u

100

10

RPA

c

SC

PY

HNC

RI PT

u

HNC

6000

60

30

6000

60

7000

R

70

Figure 13: (a) Semi-logarithmic plot comparing the interchain internal energy per chain uc and virial pressure P (inset) of the CG system obtained with various closures to the corresponding quantity of the original system (denoted by “Orig”) for homopolymer melts at κ ¯ = 100. (b) Logarithmic plot of the difference in uc , as well as in P , of the CG system

TE

D

obtained between the HNC and RPA closures for homopolymer melts. N¯ = 104 . per chain uc = 4uc,AA and virial pressure P = 4PAA , respectively, of the CG system

EP

obtained with the three closures to the corresponding quantity of the original system 3 m for the case of κ ¯ = 100, where we have βum c = βRe,0 P /100 ≈ 37.04. We see that

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the RPA closure underestimates both quantities and that the deviations monotonically increase with increasing N. While the PY closure also underestimates both quantities 3 except βRe,0 P m for N ≥ 44, it has much smaller deviations than those of the RPA closure

at large N. Fig. 13(a) and its inset are similar to our numerical results for the case of κ ¯ → ∞ shown, respectively, in Figs. 6(b) and 6(a) of our previous work.4 Fig. 13(b)

RPA 3 HNC RPA shows that β uHNC − u ≈ βR P − P /100 > 0 scales approximately with c c e,0

N 2 at large N for both κ ¯ = 100 and → ∞. This again indicates the qualitative failure of the HNC closure in reproducing the thermodynamic properties of the original system at 37

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large N, and directly contradicts the full thermodynamic consistency recently claimed by Guenza and co-workers.37 As clearly shown in our previous work, both numerically4 and analytically,33 their claim is simply due to the (incorrect) approximations used in their

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analytical derivation. Case II: κ ¯ = 0 and χ ¯=2

3.3.2

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3 m m 3 m In this case, we have βum c,AA = βRe,0 PAA = 0 and βuc,AB = βRe,0 PAB /100 ≈ 0.4762.

Figs. 14(a) and 14(b) show the differences in these thermodynamic properties between the original and CG systems obtained with the RPA and PY closures, which can be compared with Figs. 12(a) and 12(b), respectively.

We find that the RPA closure

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3 m underestimates all these quantities, except βRe,0 PAA for N > 5. More problematically,

RPA 3 m RPA both β um monotonically increase with increasing c,AB − uc,AB and βRe,0 PAB − PAB

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N; the original system therefore cannot be recovered even in the limit of N → Nm , as in

Case I. In contrast, while the PY closure also underestimates all the quantities except

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3 m 3 m βRe,0 PAA for N > 1 and βRe,0 PAB for all N, all the deviations from the original system

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approach 0 at large N.

Finally,

Fig. 14(c) shows the differences between the thermodynamic proper-

ties of the CG system obtained with the HNC closure and those with the RPA closure, which can be compared with Fig. 12(c). We once again find that

RPA 3 β uHNC ≈ βRe,0 PIJHNC − PIJRPA /100 > 0 scales approximately with N 2 c,IJ − uc,IJ at large N for IJ=AA and AB, which indicates the qualitative failure of the HNC closure

in reproducing the thermodynamic properties of the original system at large N.

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Figure 14: Logarithmic plot of the interchain internal energy per chain uc and virial pressure P of the CG system due to (a) the AA and (b) the AB interactions obtained with the RPA and PY closures, for symmetric BPB at κ ¯ = 0 and χ¯ = 2. Part (c) is a

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logarithmic plot of the difference in these quantities of the CG system obtained between the HNC and RPA closures. N¯ = 104 . That the structure-based coarse graining cannot give thermodynamic consistency at any coarse-graining level is actually expected, due to the fact that the CG pair potentials do not have the many-body nature of the potential of mean force in coarse graining. This is consistent with the finding in our previous work4, 33 and by several other

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groups.5, 38

Conclusions

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We have extended our previous work4 and proposed a systematic and simulation-free strategy for the structure-based coarse graining of multi-component polymeric systems

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using the polymer reference interaction site model (PRISM) theory,2 and as an example applied it to a simple model system of binary polymer blends (BPB). Our structure-based coarse graining, given by Eqs. (6) and (7), ensures that the original and coarse-grained (CG) systems have the same intra- and intermolecular pair correlation functions (PCFs) involving the CG segments (each representing the center-of-mass of a group of consecutive

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monomers on the same chain in the original system). When applied to BPB, it does not

system and closure used.

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change the spinodal curve (thus also the critical point), regardless of the original model

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For simplicity, here we have chosen symmetric BPB A/B described by the softcore CGC-δ (or Gaussian thread) model as the original system, where continuous

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Gaussian chains each of Nm → ∞ monomers interact with the Dirac-δ function potential; this model is equivalent to that used by Helfand and Tagami.28 Numerically solved using the PRISM theory with the Percus-Yevick (PY) closure25 and ideal-chain conformations, our original model system has three parameters: κ ¯ controlling the system compressibility, χ¯ controlling the A-B incompatibility, and the invariant degree of polymerization29 N¯ controlling the system fluctuations. In the limit of κ ¯ → ∞, the value of χ ¯ becomes irrelevant and our original system reduces to the hard-core Gaussian thread model for 40

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homopolymer melts.34 On the other hand, at χ ¯ = 0 and finite κ ¯ , it reduces to the soft-core CGC-δ model for hompolymer melts, which is equivalent to the well-known Edwards model.35 Here we have taken two original systems of symmetric BPB close to their critical point (but still in the mixed phase) as examples: Case I is at κ ¯ = 100 and

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χ¯ = 12, and Case II is at κ ¯ = 0 and χ ¯ = 2; N¯ = 104 in both cases. Since the interchain direct PCF between A monomers is 0 in Case II, it leads to some qualitative differences

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from Case I as given below.

We have examined in detail the interchain total and direct PCFs between IJ(=AA and AB) segments in the CG systems of various N (the number of CG segments on each chain), hss r ) and cIJ (˜ r ), respectively, where r˜ ≡ r/Re,0 with Re,0 being the IJ (˜

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root-mean-square end-to-end distance of an ideal chain in the original system. We have

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found that hss r ) > 0 (except that hss r ) < 0 for r˜ . 0.3 in Case I) and hss r ) < 0. AA (˜ AA (˜ AB (˜ For r˜ & 1, hss r) and hss r ) for various N approximately collapse, respectively, and AA (˜ AB (˜

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both r˜hss r ) and r˜hss r) decay exponentially towards 0 with increasing r˜. We have AA (˜ AB (˜ also presented an approximate analytical solution for hss r ), given by Eqs. (32) and (33), IJ (˜ which works well at r˜ & 1 for systems close to their critical point, but not for systems

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far away from their critical point. Compared to hss r ), cIJ (˜ r) decays to 0 much faster IJ (˜ (especially for large N) and then oscillates around 0. Its 3D Fourier transform cˆIJ < 0 √ monotonically approaches 0 with increasing qs ≡ qRe,0 / N (except that cˆAA > 0 exhibits 3 a maximum around qs = 4.5 in Case II), where q is the wavenumber, and N 2 cˆIJ /Re,0 3 3 (N 3 cˆAA /Re,0 instead of N 2 cˆAA /Re,0 in Case II) for various N approximately collapse

when plotted as a function of qs .

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The effective non-bonded pair potentials between CG segments vIJ (˜ r) are obtained from hss r ) and cIJ (˜ r) with either the PY, the hypernetted-chain (HNC),17 or the IJ (˜ random-phase approximation (RPA)26 closure for the CG system, as given in Eq. (11). The variation of vIJ (˜ r = 0) with N is shown in Fig. 5 for Case I and in Figs. 10(a) and

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10(b) for Case II. We have measured the range of these CG potentials by their second

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moment σIJ and the location of their first root r˜IJ , which are shown in Fig. 7 for Case I √ √ RPA PY and Fig. 11 for Case II. In particular, we have found that σIJ = 1/ 3N , σIJ ≈ 1/ 3N PY RPA PY RPA (σAA cannot be calculated numerically in Case II), r˜IJ = r˜IJ , r˜IJ ∝ N −0.644±0.003 in √ RPA RPA Case I, and r˜AA ≈ 1/ 3N and r˜AB ∝ N −1/2 (for N & 10) in Case II; the scaling of r˜IJ

with N can be used to approximately collapse vIJ (˜ r)/vIJ (˜ r = 0) for various N.

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Finally, for 1 ≤ N ≤ 100 examined in this work, both the RPA and PY closures

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for the CG system underestimate the interchain internal energy per chain um c,IJ and virial pressure PIJm due to the IJ interaction of the original system in most cases. At large N,

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all the deviations given by the RPA closure increase with increasing N (except those for IJ=AA in Case II); it therefore cannot recover the original system even in the limit of N → Nm (i.e., without coarse graining), as expected. Only the PY closure is expected

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to recover the original system in this limit. On the other hand, at large N, that hss r) IJ (˜ RPA 2 is nearly independent of N gives uHNC c,IJ − uc,IJ ∝ N for the CG system; we have also
√ RPA HNC found that uHNC − PIJRPA / N > 0. This indicates the qualitative c,IJ − uc,IJ ≈ PIJ

failure of the HNC closure in reproducing the thermodynamic properties of the original system at large N.

Similar results are also found for homopolymer melts, directly

contradicting the full thermodynamic consistency recently claimed by Guenza and co-workers.37 The structure-based coarse graining therefore cannot give thermodynamic 42

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consistency at any coarse-graining level, which is expected because the CG pair potentials do not have the many-body nature of the potential of mean force in coarse graining. This is consistent with the finding in our previous work4, 33 and by several other groups.5, 38

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While applied to symmetric BPB described by the simple soft-core CGC-δ model as an example in this work, our systematic and simulation-free strategy for the

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structure-based coarse graining of multi-component polymeric systems proposed here is quite general and versatile, and can certainly be applied to more complicated chain models and complex systems. It is much faster than those using many-chain molecular simulations, thus effectively solving the transferability problem in coarse graining,5 and further avoids the problems caused by finite-size effects and statistical uncertainties in

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many-chain molecular simulations. As explained in our previous work,4 our strategy here

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combined with fine graining serves well for the ultimate goal of multi-scale modeling of

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multi-component polymeric systems.

Acknowledgement

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Financial support for this work was provided by Colorado State University, which is gratefully acknowledged.

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[2] Schweizer, K. S.; Curro, J. G. Integral-Equation Theory of the Structure of Polymer Melts. Phys. Rev. Lett. 1987, 58, 246.

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[34] Schweizer, K. S.; Curro, J. G. RISM theory of polymer liquids: analytical results for continuum models of melts and alloys. Chem. Phys. 1990, 149, 105.

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A systematic and simulation-free strategy for coarse graining polymeric systems; Ensures structural consistency of coarse-grained (CG) segments; Does not change the spinodal curve (thus also the critical point) of polymer blends; Cannot give thermodynamic consistency between the original and CG systems.

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• • • •