J. Chem. Phys. 139, 184901 (2013); https://doi.org/10.1063/1.4828941 139, 184901

© 2013 AIP Publishing LLC.

Phase behaviors of supramolecular graftcopolymers with reversible bondingCite as: J. Chem. Phys. 139, 184901 (2013); https://doi.org/10.1063/1.4828941Submitted: 19 August 2013 . Accepted: 22 October 2013 . Published Online: 08 November 2013

Xu Zhang, Liquan Wang, Tao Jiang, and Jiaping Lin

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THE JOURNAL OF CHEMICAL PHYSICS 139, 184901 (2013)

Phase behaviors of supramolecular graft copolymerswith reversible bonding

Xu Zhang, Liquan Wang,a) Tao Jiang, and Jiaping Lina)

Shanghai Key Laboratory of Advanced Polymeric Materials and Key Laboratory for Ultrafine Materials ofMinistry of Education, School of Materials Science and Engineering, East China University of Science andTechnology, Shanghai 200237, China

(Received 19 August 2013; accepted 22 October 2013; published online 8 November 2013)

Phase behaviors of supramolecular graft copolymers with reversible bonding interactions were ex-amined by the random-phase approximation and real-space implemented self-consistent field the-ory. The studied supramolecular graft copolymers consist of two different types of mutually in-compatible yet reactive homopolymers, where one homopolymer (backbone) possesses multifunc-tional groups that allow second homopolymers (grafts) to be placed on. The calculations carried outshow that the bonding strength exerts a pronounced effect on the phase behaviors of supramolecu-lar graft copolymers. The length ratio of backbone to graft and the positions of functional groupsalong the backbone are also of importance to determine the phase behaviors. Phase diagrams wereconstructed at high bonding strength to illustrate this architectural dependence. It was found thatthe excess unbounded homopolymers swell the phase domains and shift the phase boundaries. Theresults were finally compared with the available experimental observations, and a well agreementis shown. The present work could, in principle, provide a general understanding of the phase be-haviors of supramolecular graft copolymers with reversible bonding. © 2013 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4828941]

I. INTRODUCTION

Supramolecular polymers employ non-covalent bondinginteractions, such as hydrogen bonding, host-guest modula-tion, metal-ligand coordination, electrostatic interaction, andπ -π stacking, to link chemically dissimilar functional poly-mers together into larger macromolecules varying in size,architecture, and/or composition. To date, although the tech-nologies of synthesizing polymers with sophisticated topolo-gies (e.g., block, graft, and dendrimer copolymers) havebeen successfully burgeoned,1–3 it is still a challenge forthe molecular chemistry field to improve the ability to ad-just the versatilities and responsibilities through chemical de-sign. Therefore, increasing attention has been paid to thesupramolecular polymers, which allows various microstruc-tures to be fabricated without having to synthesize com-plicated copolymers.4 In addition, the supramolecular poly-mers offer a great platform for fabricating materials withhighly tunable properties, such as the thermal manipulation ofviscosity,5 temperature-dependent conductivity,6 controlleddrug delivery,7 thermo-, chemo-, or mechano-responses, andluminance.8

Owing to the ongoing interests in supramolecular poly-mers, a large number of polymer blends with non-covalentbonding interactions have been studied experimentally.9–15

The introduction of non-covalent bonding interactions intoblend systems would effectively suppress the macrophaseseparation and generate various microstructures. Forexample, Matsushita et al. reported that the poly(styrene-

a)Authors to whom correspondence should be addressed. Electronic ad-dresses: jlin@ecust.edu.cn and lq_wang@ecust.edu.cn.

b-2-vinylpyridine)/poly(4-hydroxystyrene) blends can formmicrophase-separated structures due to the non-covalentbonding interactions between poly(2-vinylpyridine) andpoly(4-hydroxystyrene).9 Fredrickson et al. designedsupramolecular structures from mixtures of poly(ethyleneoxide)-b-poly(styrene-r-4-hydroxystyrene) and poly(styrene-r-4-vinylpyridine)-b-poly(methyl methacrylate) with dif-ferent fractions of hydrogen-bonded phenolic and pyridylunits.12 Square arranged nanostructures and hexagonallypacked structures were discovered by controlling the ratio ofdonors to acceptors.

In addition to these experimental studies, some the-oretical studies began to emerge. Inspired by an earlyrandom-phase approximation (RPA) theory about systemswith reversible bonds exploited by Tanaka, Matsuyama,and co-workers,16–18 a complete self-consistent field theory(SCFT) of supramolecular block copolymers implemented ingrand canonical ensemble has been developed by Fredrick-son et al.19, 20 Quantities of phase behaviors, including mi-crophase separation into ordered mesophases and coexistenceof homogeneous phases and mesophases, were discoveredfrom supramolecular diblock and triblock copolymers. Basedon the studies of Fredrickson et al., our group has carriedout a study on the phase behaviors of supramolecular tri-block copolymers composed of AB-diblock copolymers andC-homopolymers, in which the reversible bonds exist be-tween B free end and one of C free ends.21 A series of hierar-chical nanostructures, such as tetragonal cylinders surroundedby hexagonal cylinders, alternating hexagonal cylinders, andtetragonal cylinders surrounded by octagonal spheres, wereobserved.

0021-9606/2013/139(18)/184901/12/$30.00 © 2013 AIP Publishing LLC139, 184901-1

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Among the studies of supramolecular polymers, most ofthem are focused on linear supramolecular polymers. Limitedattention has been paid to nonlinear supramolecular polymerswith complex architectures.6, 22, 23 These available studies re-vealed that the molecular architecture is an important factorin determining the self-assembled structures, phase behaviors,and material properties.24–32 Matsushita et al. have prepared ablend sample of carboxyl-terminated poly(dimethylsiloxane)(PDMS-COOH) and polyethylenimine (PEI), where acid-base complexes can occur between the carboxylic acidof poly(dimethylsiloxane) and various amino groups onpolyethylenimine.33 The lamellar structures were always at-tained regardless of the compositions. In addition, the non-linear supramolecular graft copolymer systems were alsoreported to be able to form aggregates such as micellesin solutions.34–39 In comparison to the situation of linearsupramolecular polymers, theoretical studies regarding thephase behaviors of nonlinear supramolecular polymers suchas supramolecular graft copolymers are still limited, and alsomany mechanisms behind the phase behaviors remain unclear.Thus, a systematical theoretical study on the phase behaviorsof supramolecular polymers is desired.

In the present work, we undertook an investigation onthe phase behaviors of supramolecular graft copolymers withreversible bonding interactions. The RPA method40–47 wasfirst applied to determine the scattering function in the dis-ordered state and the spinodal which is the proximity of theorder-disorder transitions (ODTs). Since the phase diagramof supramolecular graft copolymers is much more complex, astudy of spinodal can help us understand the contour of dia-gram and facilitates the construction of phase diagrams. Withthe help of the information obtained from RPA calculations,we then utilized real-space implemented SCFT to study the

phase behaviors of supramolecular graft copolymers, such asphase diagrams, domain size, and interfacial width. The SCFTphase diagrams were constructed at high bonding strength.The phase behaviors at high bonding strength were furtherexplained with a mean-field analysis of the copolymer densi-ties. The results were finally compared with the existing ex-perimental observations.

II. RESULTS AND DISCUSSION

In this work, we studied the phase behaviors ofsupramolecular graft copolymer systems with reversiblebonding interactions. The systems are composed of twochemically distinct yet reactive homopolymer species. Asshown in Fig. 1, the reversible bonds can occur betweenthe chemically distinct homopolymers, one of which (speciesB) is mono-functional at one of its free ends and the other(species A) is multi-functional (m functional points) along thechains. As a consequence, the equilibrium system consists ofa melt blend of A- and B-homopolymer “reactants” with graftcopolymer “products.” The flexible A- and B-homopolymers,denoted by Ah and Bh, are modeled as monodisperse con-tinuous Gaussian chains with the degree of polymerizationNAh and NBh. The graft copolymer products with one B-graftand two B-grafts are defined as mono-graft copolymers (de-noted by AgB) and di-graft copolymers (denoted by AgBB),respectively. The jth functional point is located at τ j and givenby τ j = τ 1 + (j − 1)(1 − 2τ 1)/(m − 1), where 1 ≤ j ≤ m.Both A- and B-segments have the same statistical length a.The total statistical segment number of the largest referencesupramolecules is expressed as N = NAh + mNBh and set asN = 300. The length ratio of A-homopolymers to largest ref-erence supramolecular graft copolymers is parameterized by

FIG. 1. Schematic diagram for the reversibly bonding reaction in the supramolecular graft copolymer blends with (a) m = 1 and (b) m = 2. The zi representsthe activity of i species. The subscripts of Ah, Bh, AgB, and AgBB represent A-homopolymer, B-homopolymer, graft copolymer with one B-graft, and graftcopolymer with two B-grafts, respectively.

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αAh = NAh/N, whereas the other length ratios are given asαBh = (1 − αAh)/m, αAgB = αAh + αBh, and αAgBB = αAh

+ 2αBh. The interaction strength between dissimilar compo-nents is characterized by Flory-Huggins parameter χ . Thefree energy decrease for bonding reaction in units of the ther-mal energy kBT is denoted by h, and the ratio h/χN is used todescribe the supramolecular bonding strength.19, 20

This section includes Subsections II A–II C: “RPA studyof spinodals,” “SCFT study of phase behaviors at high bond-ing strength,” and “comparison with available experimentalobservations.” The results in Subsection II A are obtainedfrom RPA calculations. Details regarding the RPA method aredescribed in Appendix A. The results presented in Subsec-tion II B are based on the SCFT (for the details of SCFT, seeAppendix B). In the SCFT calculations, the phase diagramsare evaluated by comparing the free energies of the disor-dered phase (Dis) and the ordered phases obtained from thecalculations started from a deterministic initial condition thathas symmetry of lamellae (L), hexagonally packed cylinders(C), body-centered cubic spheres (S), and bicontinuous gyroidphases (G).

A. RPA study of spinodals

We first examined the effect of bonding strength h/χNon the spinodals of supramolecular graft copolymers withNAh = mNBh, as shown in Fig. 2. In these diagrams, thevertical axis is the interaction strength between A- and B-segments with single functional point obtained by normal-izing χN with respect to m, i.e., χN/m, which is chosen toaccount properly for the effect of changing m, not simplyincreasing the degree of polymerization.28, 43 The horizon-tal axis is the total volume fraction of A-segments, φA,tot.Figure 2(a) shows the spinodals for the m = 1 case. It wasfound that the spinodal curves are almost symmetric with re-spect to φA,tot ≈ 0.5 and shift upwards as h/χN increases. Ath/χN = 0, a parabolic spinodal curve was observed, simi-lar to the parabolic immiscible region of two incompatiblehomopolymers above the critical point for phase separation.As h/χN increases, the parabolic region is contracted intoan hourglass shape. With further increasing h/χN, the hour-glass shape is retracted into an upper double-concave shapewith a lower closed loop. The phase separation may takeplace both in the closed loop and above the double-concavecurve. At stronger h/χN, the closed loop diminishes and even-tually disappears, while the double-concave curve shifts up-wards to higher χN/m. For the double-concave spinodals, thetwo concaves are separated by a low temperature melting“eutectic” point, which has been observed in various poly-meric systems.18, 20, 48 This point is roughly located at φA,tot

≈ 0.5, which is consistent with the stoichiometric composi-tion of mono-graft copolymers (supramolecular graft copoly-mers with one B-graft, AgB).

The spinodals for the m = 2 case are shown inFig. 2(b). With increasing h/χN, the spinodals shift up-wards to higher χN/m and their shapes undergo a transforma-tion from parabola to deformed hourglass, then to an upperdouble-concave curve with a lower closed loop, and finally to

FIG. 2. RPA spinodal curves in χN/m − φA,tot space for supramoleculargraft copolymers with various h/χN at (a) m = 1, αAh = 0.5, and τ 1 = 0.5and (b) m = 2, αAh = 0.5, and τ 1 = 0.25. Critical points for the homoge-nous phases are indicated with red squares, and the Lifshitz point, LP, is la-beled. (c) RPA spinodal surface for the supramolecular graft copolymers withm = 2, αAh = 0.5, and τ 1 = 0.25 corresponding to the case of (b). The redline with squares along the top of the surface is the homogeneous critical line,which terminates at Lifshitz point. The critical line can go ahead beyond thispoint but is not indicated here.

double-concave shape. This behavior is similar to that of the m= 1 case. However, in contrast to the m = 1 case, the spinodalsfor the m = 2 case tend to be highly asymmetric with regard toφA,tot ≈ 0.5 and slant towards smaller φA,tot. For the double-concave spinodals, there is a prominent low-φA,tot concave.Note that the low temperature melting “eutectic” point is stilllocated at nearly φA,tot ≈ 0.5, which agrees with the stoichio-metric composition of di-graft copolymers (supramoleculargraft copolymers with two B-grafts, AgBB). In addition, thevalley of high-φA,tot concaves is located at φA,tot ≈ 2/3, whichis a stoichiometric composition of A-blocks in the mono-graftcopolymers.

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From Figs. 2(a) and 2(b), we learned that the supramolec-ular blends with higher h/χN are more difficult to undergophase separation because of the upward shift of the spin-odals. Usually, the phase separation takes place above thecritical point at φA,tot

c = αBh1/2/(αAh

1/2 + αBh1/2) and χNc

= (αAh−1/2 + αBh

−1/2)2/2 in the homopolymer blends with-out non-covalent interactions. The critical points for differenth/χN were indicated by red squares in Figs. 2(a) and 2(b). Theonset of mesophases, when coinciding with a liquid-liquidcritical point, is a special type of tricritical point known asan isotropic Lifshitz point (LP).19, 20 To illustrate the Lifshitzpoint more clearly, we plotted the RPA spinodal surface forthe supramolecular graft copolymers with m = 2, αAh = 0.5,and τ 1 = 0.25, which is shown in Fig. 2(c). This correspondsto the case of Fig. 2(b). The red line with squares along the topof the surface is the homogeneous critical line, which termi-nates at Lifshitz point. The critical line can go ahead beyondthis point but is not indicated here. For all the points along thecritical line at lower bonding strength, i.e., h/χN < (h/χN)LP,there is only a simple liquid-liquid critical behavior. Higherbonding strength than (h/χN)LP leads to mesophase criticalpoints.19, 20

In addition to h/χN, the influence of molecular architec-ture on the spinodals was also examined. Figure 3 shows thespinodals at various αAh (the length ratio of A-homopolymersto the largest reference supramolecular graft copolymers). Ascan be seen, the spinodals assume typical double-concaveshapes, where the two concaves are separated by a low tem-perature melting “eutectic” point. Note that the lower closedloops do not exist due to the high h/χN. At lower αAh, the

FIG. 3. RPA spinodal curves in χN/m − φA,tot space for supramoleculargraft copolymers with various αAh at (a) m = 1, τ 1 = 0.5, and h/χN = 1.0and (b) m = 2, τ 1 = 0.25, and h/χN = 0.6. The insets show the χN/m valueof the low temperature melting “eutectic” point plotted as a function of φA,tot(solid line) and αAh (scattered squares).

high-φA,tot concaves are more prominent than the low-φA,tot

ones. As αAh increases, the low-φA,tot concaves are gradu-ally enlarged and eventually become dominant. Moreover, theχN/m of “eutectic” point first decreases and then increases asαAh increases, as shown in the inset of the figures (scatteredsquares). When plotted as a function of φA,tot (solid line), thecurve was found to be overlapped with that as a function ofαAh. This implies that the φA,tot and αAh have similar effecton the “eutectic” point. The main difference between m = 1(Fig. 3(a)) case and m = 2 (Fig. 3(b)) case is the locationwhere the minimum χN/m of “eutectic” point appears. Theminimum χN/m of “eutectic” point for m = 1 is located atabout αAh = 0.5, while that for m = 2 is located at aboutαAh = 0.4686. At αAh = 0.5 for the m = 1 case (or αAh

= 0.4686 for the m = 2 case), the two concaves of the spin-odal are nearly equal.

Besides αAh, the functional point position τ j is alsoan important parameter affecting the phase behaviors.Figures 4(a) and 4(b) present the results for the m = 1 andm = 2 cases, respectively. For the m = 1 case with αAh = 0.5and h/χN = 1.0, the spinodal curves are almost symmetricwith respect to φA,tot ≈ 0.5 and shift upwards as τ 1 increases.This implies that the supramolecular graft copolymers withlarger τ 1 are more difficult to undergo phase separation. Inaddition, the low temperature melting “eutectic” points ap-pear at the stoichiometry of the reaction products (i.e., φA,tot

≈ αAh) and shift upwards to higher χN/m as τ 1 increases (seethe inset of Fig. 4(a)). For the m = 2 case with αAh = 0.5and h/χN = 0.6, as shown in Fig. 4(b), with increasing τ 1

from 0.2 to 0.4, the spinodals first shift upwards to higherχN/m (τ 1 < 1/3) and then shift downwards to lower χN/m (τ 1

FIG. 4. RPA spinodal curves in χN/m − φA,tot space for supramoleculargraft copolymers with various τ 1 at (a) m = 1, αAh = 0.5, and h/χN = 1.0and (b) m = 2, αAh = 0.5, and h/χN = 0.6. The insets show the χN/m valueof the low temperature melting “eutectic” point plotted as a function of τ 1.

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> 1/3). In order to demonstrate this effect more clearly, likethe m = 1 case, the χN/m of the low temperature melting “eu-tectic” points was plotted as a function of τ 1, which is shownin the inset of Fig. 4(b). As can be seen, the maximum χN/mvalue occurs at τ 1 ≈ 1/3, implying that the phase separationof supramolecular graft copolymers with nearly symmetricalfunctional point position requires the lowest temperature, i.e.,highest χN/m. It is noted that the spinodals are nearly inde-pendent of τ 1 when φA,tot is significantly large or small forboth the m = 1 and m = 2 cases.

In this subsection, we have employed the RPA method toinvestigate some general features of the order-disorder enve-lope of supramolecular graft copolymer systems. The RPAmethod is particularly suited for elucidating the effects ofcopolymer architectures such as the number of functionalpoints on the spinodals in a systematic way. Although a pre-view of the envelopes of phase diagrams can be achievedby the RPA calculations, more accurate numerical solutionsare still desired to get a complete picture of the phase be-haviors of supramolecular graft copolymers. The SCFT hasemerged as a powerful tool to explore the phase behaviorsof copolymers49–59 and has been extended to supramolecularpolymer systems.19, 20 Thus, it is feasible to study the phasebehaviors of supramolecular graft copolymers by utilizing theSCFT.

B. SCFT study of phase behaviors at high bondingstrength

The above RPA calculations of spinodals have pro-vided a general understanding of the order-disorder transitionof supramolecular graft copolymers. In this subsection, weturned to the detailed calculations of the phase diagrams usingthe SCFT, guided by the RPA results. The phase boundarieswere obtained by comparing the free energies of given struc-tures including disordered phases (Dis), lamellae (L), hexag-onally packed cylinders (C), body-centered cubic spheres (S),and bicontinuous gyroid phases (G).60, 61 In addition, two-phase coexistence regions (2φ) were also found, which areindicated by gray shadows in the SCFT phase diagrams. Thesubscripts A and B in C, S, and G denote that the minor-ity domains of ordered structures are formed by A and Bblocks, respectively. In the phase diagrams, a portion of thephase boundaries was determined by extrapolation due to theprohibitive resolutions in calculating the phases at very lowφA,tot.20 For a comparison with the RPA results, the spinodals(red solid lines) were also mapped in the phase diagrams.

Figure 5(a) displays the SCFT phase diagram in χN/m− φA,tot space for the supramolecular graft copolymers withm = 1, αAh = 0.5, and τ 1 = 0.5 at h/χN = 1.0. The mostdistinctive characteristic of the phase diagram is the nearlysymmetric ODT boundary with double-concave shape. Here,we only focused on the case of high h/χN, and therefore,the closed-loop-shaped domain does not exist. This case cor-responds to various experimental systems containing low-molecular-weight polymers (low N) with high density of non-covalent bonds (high h).33 The ODT boundary was found tobe nearly coincident with the spinodal obtained from the RPAcalculations at the range of φA,tot ≈ 0.40–0.60. However, be-

FIG. 5. Phase diagrams in χN/m − φA,tot space for supramolecular graftcopolymers with (a) m = 1, αAh = 0.5, τ 1 = 0.5, and h/χN = 1.0 and (b)m = 2, αAh = 0.5, τ 1 = 0.25, and h/χN = 0.6. Dis labels the disorderedregions. The ordered regions are denoted as S (body-centered cubic phases),C (hexagonally packed cylinders), G (bicontinuous gyroid), and L (lamella).The subscripts A and B in C, S, and G denote that the minority domains ofordered structures are formed by A and B blocks, respectively. The blacksolid lines with circles, black dotted lines, and red solid lines are the phaseboundaries calculated by SCFT, phase boundaries obtained by extrapolation,and spinodals calculated by RPA, respectively. The gray regions indicate thetwo-phase coexistence regions.

yond this range, the ODT boundary is external to the spin-odal. The ODT envelope is also separated by a low tempera-ture melting “eutectic” point. This point is located in the CB

phase at about φA,tot ≈ 0.5, corresponding to the stoichiom-etry of mono-graft copolymers. Although the ODT bound-ary shows symmetry about φA,tot ≈ 0.50, the phase regionsare asymmetrical. The large stable region of hexagonal (CA)and spherical (SB) phases dominates the most interior of thelow- and high-φA,tot concaves, respectively. The lamellar re-gion is almost located in the interior of the low-φA,tot con-cave. In addition, the two-phase coexistence regions (2φ) ap-pear between two neighboring ordered phases, indicated bygray shadows. Note that some two-phase coexistence regionsare so narrow that they merge into single lines in the phasediagrams. However, they really exist. For example, at χN/m= 13, the 2φ between L and GB is in the range of φA,tot

= 0.46643–0.46669, whereas the 2φ between GB and CB is inthe range of φA,tot = 0.46694–0.46719. The multiphase point,which is at the intersection of CA, GA, L, GB, CB, 2φ, and Dis,appears at φA,tot ≈ 0.40 and χN/m ≈ 12 (indicated by a greencircle in Fig. 5(a)).

The SCFT phase diagram in χN/m − φA,tot space for thesupramolecular graft copolymers with m = 2, αAh = 0.5,and τ 1 = 0.25 at h/χN = 0.6 is shown in Fig. 5(b). The

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ODT boundary takes on an asymmetric double-concave shapewith a prominent low-φA,tot concave. And it was found to beexternal to (at higher and lower φA,tot) or nearly coincidentwith (at moderate φA,tot) the spinodal, which is similar to them = 1 case. The lamellar phase is still located in the interiorof the low-φA,tot concave, and the multiphase point is roughlyat φA,tot ≈ 0.35 and χN/m ≈ 9.95 (indicated by a green circlein Fig. 5(b)). In the phase diagram, similar to the m = 1 case,the large stability region of hexagonal (CA) and spherical (SB)phases dominates the most interior of the low- and high-φA,tot

concaves, respectively.One notable feature of these phase diagrams (Figs. 5(a)

and 5(b)), in contrast to block copolymers, is that they arenot symmetrical about φA,tot = 0.50, but show a shift towardslower φA,tot except the phase boundary between SB and Dis,which is similar to the neat graft copolymers.45 This behav-ior can be rationalized by considering the stretching energy ofthe graft copolymer chains. The graft copolymers experiencea lateral crowding of the backbone due to the fact that moreblocks of backbone are accumulated on the same side of theinterface. This lateral crowding can be partially alleviated byallowing the interface to curve away from the backbone do-mains. These result in an enhanced preference for backboneto remain on the convex side of the interface, which causesthe shifts of phase diagrams towards lower volume fractionof the backbone (i.e., φA,tot). For the more detailed explana-tion of this shifting behavior, it can be referred to our previousstudy.28, 45

Comparing with the phase diagram of neat graft copoly-mers, however, some differences were also discovered. First,the ODTs manifest a double-concave shape, rather than aparabola in the phase diagram of neat graft copolymers.28, 62

Second, the order-order transition (OOT) boundaries showsome shifts as compared to the phase diagram of neat graftcopolymers. For the m = 1 case, the OOTs show a shift to-wards lower φA,tot in the region of φA,tot < 0.50 and higherφA,tot in the region of φA,tot > 0.50, relative to the phase di-agram of neat mono-graft copolymers.62 For the m = 2 case,as φA,tot < 0.50, the OOTs also show a shift towards lowerφA,tot.28 When φA,tot > 0.50, the shift behavior is much morecomplicated because of the diverse constituent polymers. Theappearance of these behaviors can be mainly ascribed to theexcess unbounded homopolymers.

To have an insight into how the excess unbounded ho-mopolymers influence the phase behaviors, we presentedthe volume fractions of different constituent polymers.Figure 6(a) shows the volume fractions φi (i = Ah, Bh, andAgB) as a function of φA,tot for m = 1 at χN/m = 13.5, i.e.,along the level line at χN/m = 13.5 in Fig. 5(a). At φA,tot

= 0.5, the volume fraction of mono-graft copolymers (φAgB)is nearly 100%, and the volume fraction of A-homopolymersand B-homopolymers (φAh and φBh) is nearly zero. This rep-resents a neat mono-graft copolymer system, which prefersto form the CB phase as referring to the Grason’s work.62 InGrason’s work, the symmetric mono-graft copolymers, calledA2B miktoarm star copolymers, prefer the CB phase with thevolume fraction of A-blocks fA = 0.5 at χN/m ≈ 13.5. AsφA,tot decreases from 0.5 to 0, the φAgB decreases and the φBh

increases linearly. To satisfy the strong bonding reaction, the

FIG. 6. Volume fractions of the constituent polymers as a function of φA,totfor the supramolecular graft copolymers corresponding to Figs. 5(a) and 5(b).The species are labeled in the legends. Diagram (a) depicts the species vol-ume fractions at χN/m = 13.5 for the system shown in Fig. 5(a). Diagram (b)displays the species volume fractions for a cut at χN/m = 11 in Fig. 5(b). Allother parameters and notations for (a) and (b) are the same as in Figs. 5(a)and 5(b), respectively. The gray regions represent the two phase coexistenceregions (2φ).

φAh ≈ 0 is maintained due to the dearth of A-homopolymers.As a result, in the φA,tot < 0.5 range, the B-domains areswollen by the excess unbounded B-homopolymers. Theswelling effect becomes more predominated with decreasingthe φA,tot, leading to a shift of OOTs to lower φA,tot. During thephase transitions, the B-domains become the major domainsgradually and B-homopolymers dominate the Dis phase atvery low φA,tot. On the other hand, with increasing φA,tot from0.5 to 1, the φAgB decreases and the φAh increases. A shift ofOOTs to higher φA,tot occurs because of the swelling effect ofexcess unbounded A-homopolymers. The major A-domainsare swollen by the excess unbounded A-homopolymers.

Figure 6(b) shows the volume fractions φi (i = Ah,Bh, AgB, and AgBB) versus φA,tot at χN/m = 11 for them = 2 case. As φA,tot increases, the volume fraction ofdi-graft copolymers (φAgBB) increases linearly until φA,tot

≈ 0.5 and then decreases. For the B-homopolymers, theφBh decreases linearly from the maximal value to zero. Atlower φA,tot (<0.5), the φAgB and φAh are very close tozero, indicating that the blends consist of B-homopolymersand di-graft copolymers almost without unbounded A-homopolymers and mono-graft copolymers. The excess un-bounded B-homopolymers swell the B-domains, causing ashift of OOTs to lower φA,tot as compared to the phase dia-grams of neat graft copolymers.28 By contrast, at higher φA,tot

(>0.5), the φBh becomes very close to zero, but the φAh andφAgB possess nonzero value. The φAgB has a peak at aboutφA,tot ≈ 0.71, which is larger than the stoichiometric point of

184901-7 Zhang et al. J. Chem. Phys. 139, 184901 (2013)

FIG. 7. One-dimensional density profiles along the x direction of all blocksof the supramolecular graft copolymers with m = 1 and χN/m = 13.5 forthe (a) lamellar phase at φA,tot = 0.337259 and the (b) cylindrical phaseat φA,tot = 0.605580. The coordinate x is scaled by the spacing D betweentwo adjacent centers of B-domains. All other parameters are the same as inFig. 5(a). The insets show the two-dimensional nanostructures, where the redand blue colors represent A- and B-block rich domains, respectively.

mono-graft copolymers (φA,tot = αAh/αAgB = 2/3). This maybe due to the presence of di-graft copolymers, causing themaximal φAgB to move to higher φA,tot. Hence, in addition tothe excess unbounded A-homopolymers, there are mono-graftcopolymers remaining in the system. This results in a muchmore complicated shift behavior of OOTs.

From above results, we learned that the excess un-bounded homopolymers play a role in swelling the domains.To further understand this behavior, the one-dimensionaldensity profiles of each block in lamellar and cylindri-cal phases were given in Fig. 7. In the lamellar phase,as shown in Fig. 7(a), it can be seen that the unboundedB-homopolymers are mainly distributed at the center ofB-domains, and thus swell the B-domains. Due to the swellingeffect of excess unbounded B-homopolymers and the restric-tion of supramolecular graft copolymers, the bonded B-blocksare prone to being located close to the interface. To counter-balance the deficiency of A-blocks, the bonded A-blocks arepushed into the center of A-domains to stabilize the lamellarphase. In the cylindrical phase (Fig. 7(b)), the unbounded A-homopolymers are rich at the center of A-domains and swellA-domains, while the content of unbounded B-homopolymersis nearly zero (can also be seen from Fig. 6(a)). The bondedA-blocks and B-blocks of mono-graft copolymers tend to belocated at the center of A-domains and B-domains, respec-tively. From Fig. 7, we knew that the unbounded homopoly-mers prefer the center of domains and thus swell the corre-sponding domains.

FIG. 8. (a) Domain size D/Rg and (b) interfacial width w/Rg of CB phase asa function of φA,tot for supramolecular graft copolymers in the two cases ofm = 1 at χN/m = 13.5 and m = 2 at χN/m = 12.5. All other parameters areidentical to those in Fig. 5.

The effect of excess unbounded homopolymers onswelling the phase domains was also reflected by a variationof the domain size and interfacial width. Figure 8(a) showsthe domain size (or period) D/Rg of CB phase as a function ofφA,tot for the supramolecular graft copolymers in both the m= 1 and m = 2 cases. Notably, an increase in φA,tot producesa first decrease and then increase in cylindrical period, wherethe minimum D/Rg value occurs at φA,tot ≈ 0.5. With decreas-ing (or increasing) φA,tot from 0.5, the excess unbounded B-homopolymers (or A-homopolymers) are generated, causingthe graft (or backbone) domains to be swollen. This swellingeffect leads to an increase in domain size D/Rg. Figure 8(b)shows the interfacial width w/Rg as a function of φA,tot for theCB phase. Here, we evaluated the width as w ≡ (dφA/dz)−1

at the interface.63 The broadest interfacial width appears atφA,tot ≈ 0.5, indicating that the segregation of CB phase be-comes weaker as φA,tot approaches to 0.5. As can be seen, theinterfacial width becomes narrower rapidly as φA,tot decreasesfrom 0.5 to the low-φA,tot boundary of CB phase. Moreover,as φA,tot increases from 0.5 to the high-φA,tot boundary of CB

phase, the width first narrows rapidly at relatively low φA,tot

and then slightly at relatively high φA,tot.To clarify the effect of variation in φA,tot on the interfacial

width, we then calculated the junction distributions, where thejunction is the grafted functional point of A-homopolymers.45

Figure 9 plots the junction distributions for the cylindri-cal phase of supramolecular graft copolymers with variousφA,tot. The coordinate x = 0 corresponds to the middle ofA-domains. When φA,tot ≈ 0.5, the local volume fraction ofjunctions has a maximum value at the center of A-domains.Thus, the junctions are mainly distributed at the center of

184901-8 Zhang et al. J. Chem. Phys. 139, 184901 (2013)

FIG. 9. Local junction distribution ρ(x) for mono-graft copolymers with m= 1 at χN/m = 13.5 in various φA,tot. The coordinate x normal to the interfaceis scaled by the spacing D between two adjacent centers of B-domains, andthe middle of A-domain occurs at x = 0. All other parameters are the sameas in Fig. 5(a). The inset shows the two-dimensional structure, where the redand blue colors represent A- and B-block rich domains, respectively.

A-domains. As φA,tot increases, the volume fraction of junc-tions has a decrease at the center of A-domains but an increaseat the interface. This suggests that the junctions move fromA-backbone domains into interfaces and accumulate moredensely. The dense junction distribution narrows the interfa-cial width. This can be explained as follows. The ODT of graftcopolymers is higher than that of homopolymer blends, andtherefore the segregation becomes strong as the fraction ofgraft copolymers decreases. The strong segregation betweendifferent polymers loads to a narrow interfacial width, whichis associated with the junction distributions. Near φA,tot = 0.5,the volume fraction of graft copolymer is maximum and thusthe interfacial width is broadest.

C. Comparison with available experimentalobservations

Some existing experimental results are available in litera-tures for comparing with our theoretical predictions. Recently,Matsushita and co-workers have prepared a blend sample ofPDMS-COOH and PEI, where acid-base complexes can oc-cur between the carboxylic acid of PDMS-COOH (graft) andvarious amino groups on PEI (backbone).33 The weight ratioof PDMS-COOH: PEI were varied as 50: X (X = 1–10). Itwas found that the lamellar structures are always formed re-gardless of X when X > 3. To make a comparison with theexperiments, we did a further study on the phase behaviors ofsupramolecular graft copolymers with m = 2, αAh = 0.5, τ 1

= 0.25, and h/χN = 0.6. In accord with the fact that PDMS-COOH and PEI are strongly separated in the experiments, theχN/m was chosen to be a high value of 50. Through the calcu-lations, we found that the lamellar phase is stable in the rangeof 0.45 < φA,tot < 0.8, where the experimental data of X from3 to 6 are included. It reveals that the lamellar phase couldbe stable for a wide range of φA,tot value at the high value ofχN/m, similar to the observations in the experiments reportedby Matsushita et al.33

Figure 10 shows the comparison of the lamellar periodbetween theoretical calculations and experimental observa-tions. The experimental data (inset a) were calculated from theexperiments carried out by Matsushita et al.33 As can be seen,

FIG. 10. Domain size D/Rg of L phase as a function of φA,tot for supramolec-ular graft copolymers with m = 2, αAh = 0.5, τ 1 = 0.25, and h/χN = 0.6 atstrong segregation χN/m = 50. The inset (a) shows the experimental datawhich are calculated from the experiments carried out by Matsushita et al.33

The total volume fraction of PEI for the experiments is estimated as φA,tot= X/(X + 1.6), where X = 3–10. The inset (b) illustrates the D of the Lphase.

both the lamellar periods obtained from the theoretical calcu-lations and experiments show similar increase trends as φA,tot

increases from 0.5. Paying attention to stoichiometry betweencarboxylic acid and amine in the experiments, the stoichio-metric balance should be attained at X = 1.6 (i.e., roughlyφA,tot = 0.5). As a result, there exists an excess amount ofunbounded PEI (backbone). This is consistent with our cal-culations. In our calculations, there is also an excess amountof unbounded backbone as φA,tot is higher than the stoichio-metric composition. Moreover, we found that the spacing in-creases as φA,tot decreases from the stoichiometric composi-tion due to the swelling effect of excess unbounded grafts.However, this phenomenon has not been observed in this ex-periment, and thus is expected to be demonstrated in futureexperimental studies. Overall, the SCFT calculations well re-produced the experimental results and also predicted some un-known phase behaviors. The present results could provide ageneral understanding of the phase behaviors in supramolec-ular graft copolymer melts and also offer guidance for furtherinvestigations.

III. CONCLUSIONS

In this study, we explored the phase behaviors ofsupramolecular graft copolymers with reversible bonding in-teractions using the RPA and real-space implemented SCFT.The RPA calculations show that the spinodals undergo atransformation of parabola → hourglass → an upper dou-ble concaves with a lower closed loop → double concavesas h/χN increases. Variations in αAh at high bonding strengthcan change the spinodal from double-concave shape with aprominent high-φA,tot concave to double-concave shape witha prominent low-φA,tot concave. As τ 1 increases, the spinodalsshift towards higher χN/m for the m = 1 case, and the spin-odals first shift upwards to higher χN/m and then shift down-wards to lower χN/m for the m = 2 case. When φA,tot is largeor small enough, both the αAh and τ 1 show less marked effecton the spinodals. By using the SCFT calculations, the SCFTphase diagrams in χN/m − φA,tot space for supramoleculargraft copolymers with high bonding strength were mapped

184901-9 Zhang et al. J. Chem. Phys. 139, 184901 (2013)

out. In contrast to the phase diagrams of neat graft copoly-mers, the OOTs shift towards lower φA,tot when φA,tot < 0.50and higher φA,tot when φA,tot > 0.50. This is due to the factthat the excess unbounded homopolymers swell the phase do-mains and shift the phase boundaries. With increasing thecontent of excess unbounded homopolymers, the domain sizeincreases and the interfacial width decreases. The present re-sults provide a general understanding of the phase behaviorsof supramolecular graft copolymers with reversible bonding.

ACKNOWLEDGMENTS

This work was supported by National Natural ScienceFoundation of China (Grant Nos. 21304035 and 21234002),Key Grant Project of Ministry of Education (Grant No.313020), National Basic Research Program of China (GrantNo. 2012CB933600), Fundamental Research Funds for theCentral Universities (Grant No. 222201314024), and Re-search Fund for the Doctoral Program of Higher Educationof China (Grant No. 20120074120002). Support from projectof Shanghai municipality (Grant No. 10GG15) is also appre-ciated.

APPENDIX A: RANDOM-PHASE APPROXIMATION

The RPA theory is suited for weaker segregation cases.Leibler worked out RPA expansion of block copolymer meltto fourth order in the density fluctuations.64 Since we payour attention only to the spinodal instability, the second-orderterm of the free energy expansion is enough. The density fluc-tuations in disordered state are described by density-densitycorrelation function S(q), which is the Fourier transform ofS(r). The RPA scattering function S−1(q) is a function of sin-gle chain correlation functions and is written as

S−1(q) ≡ 1

N(F (q) − 2χN )

= SAA(q) + 2SAB(q) + SBB(q)

SAA(q)SBB(q) − S2AA(q)

− 2χ, (A1)

where SAA(q), SAB(q), and SBB(q) are the second-order corre-lation functions. The SAA(q), SAB(q), and SBB(q) are based onthe integration over a subset of segment-segment correlationfunctions.43 For the m = 1 case, the correlation functions are

SAA(q)/N = φAhαAhg(1, αAhx) + φAgBg(αAh, x), (A2)

SBB(q)/N = φBhαBhg(1, αBhx) + φAgBg(αBh, x), (A3)

SAB(q)/N = 2mφAgB v(αBh, x)v(τ1αAh, x), (A4)

where

g(f, x) = 2[f x + exp(−f x) − 1]/x2, (A5)

v(f, x) = [1 − exp(−f x)]/x, (A6)

u(f, x) = 2[v(f, x) − v(f,mx)]/v2(f, x). (A7)

In above equations, x = q2Rg2, and Rg

2 = a2N/6 is theideal radius of gyration for the largest reference graft copoly-mer. The φAh, φBh, and φAgB are the volume fraction ofA-homopolymers, B-homopolymers, and mono-graft copoly-mers, respectively. For the m = 2 case, the correlation func-tions are

SAA(q)/N = φAhαAh g(1, αAhx)

+φAgBαAgB g(αAh/αAgB, αAgBx)

+φAgBB g(αAh, x), (A8)

SBB(q)/N = φBhαBh g(1, αBhx)

+φAgBαAgB g(αBh/αAgB, αAgBx)

+mφAgBB[g(αBh, x) + v2(αBh, x)

+ exp(−fAtx)v2(αBh, x)u(fAt, x)/x], (A9)

SAB(q)/N = φAgBαAgB v(αBh/αAgB, αAgBx)

× [v(τ1αAh/αAgB, x) + v(τ2αAh/αAgB, x)]

+ 2mφAgBBv(αBh, x)[1 − exp(−τ1αAhx)

× v(fAt,mx)/v(fAt, x)]/x, (A10)

where fAt = αAh(1 − 2τ 1)/(m − 1) and the φAgBB is thevolume fraction of di-graft copolymers. Here, we chose φBh

as a single independent density and the bonding constraintsas well as incompressible conditions determine the otherdensities.

For the m = 1 case, the volume fractions ofA-homopolymers and mono-graft copolymers are φAh

= αAhαBh(1 − φBh)/d1 and φAgB = (1 − φBh)φBheh-lnN/d1,where d1 = αAhαBh + φBheh-lnN. The total volumefractions of A- and B-segments, denoted by φA,tot andφB,tot in the present work, are φA,tot = φAh + αAhφAgB

and φB,tot = φBh + αBhφAgB, respectively. For the m= 2 case, the volume fractions of A-homopolymers,mono-graft copolymers, and di-graft copolymers are φAh

= αAhαBh2(1 − φBh)/d2, φAgB = 2αAgBφAhφBheh-lnN/αAhαBh,

and φAgBB = φBhφAgBeh-lnN/2αBhαAgB, where d2 = αAhαBh2

+ 2αBhαAgBφBheh-lnN + (φBheh-lnN)2. The total volume frac-tions of A- and B-segments are φA,tot = φAh + αAhφAgB/αAgB

+ αAhφAgBB and φB,tot = φBh + αBhφAgB/αAgB + 2αBhφAgBB,respectively.

Ultimately, we carried out the numerical procedures forRPA method by iterating over the initial guessed bondingstrength h/χN. For a given h/χN and also a specified φBh

sp,the function F(q) of Eq. (A1) is minimized with respect to qat q = q∗, and the magnitude of S−1(q∗) converges to zeroat the spinodal point, which yields the spinodal point χNsp

= F(q∗)/2. The density φBh was adjusted from 0 to 1, thedivergence over q∗ was reevaluated and the process was re-peated until the desired χNsp is obtained. For a given bondingstrength h/χN, the φBh

sp and χNsp can be determined throughthis process. The values, χNsp/m and φA,tot

sp, establish a sin-gle point on the spinodal curve for a given bonding strengthh/χN.

184901-10 Zhang et al. J. Chem. Phys. 139, 184901 (2013)

APPENDIX B: SELF-CONSISTENT FIELD THEORY

For a grand canonical ensemble, the incompressible bi-nary homopolymer blends are in a system with volume V.The incompressibility condition constrains the bulk segmentnumber density to a constant value ρ0 at each point in thespace. In grand canonical ensemble, it is convenient to formu-late the model by utilizing the species chemical potentials μi.In this binary homopolymer blends, a single chemical poten-tial is sufficient to control the relative volume fractions of A-and B-homopolymers since the incompressibility sustains thetotal volume fraction and the additional graft copolymer prod-ucts have their volume fractions constrained by the conditionsfor the chemical reaction equilibriums (one for each reaction).Therefore, the relative volume fractions of all species in thesupramolecular polymer blends are tuned by only one chemi-cal potential μBh.

As mentioned above, the incompressibility also permitsus to assign zAh = 1 and retain zBh as the single indepen-dent activity. The other activities of additional graft copoly-mer products are determined from the mass-action laws forthe chemical reaction equilibriums,19, 20

zAgB

zAhzBh= zAgB

zBh= Keq/N = eh−ln N ⇒ zAgB = zBhe

h/N,

(B1)zAgBB

zAgBzBh= Keq/N = eh−ln N ⇒ zAgBB = z2

Bh e2h/N2,

where Keq = exp(h) is an equilibrium constant. The zi, repre-senting the activity of i species, is z0exp(−μi/kBT).

Within the mean-field theory, the configuration of a sin-gle polymer is determined by a set of effective chemical po-tential fields, ωA(r) and ωB(r), replacing actual interactionsin the melt blend. The potential fields are conjugated to thedensity fields, φA(r) and φB(r). We invoke an incompressibil-ity (φA(r) + φB(r) = 1) by introducing a Lagrange multi-plier ξ (r) that enforces the incompressibility constraint. Thegrand canonical Hamiltonian for the supramolecular blendHG, scaled by a reference number of polymer chains ρ0V/N,can be written as19, 20

fG = βNHG

ρ0V= −

∑i

κiziQi + 1

V

∫dr{χNφA(r)φB(r)

−ωA(r)φA(r) − ωB(r)φB(r)

+ ξ (r)[φA(r) + φB(r) − 1]}, (B2)

where the Qi = ∫drqi(r,1) are the partition functions of a sin-

gle non-interacting chain subject to the fields ωA(r) and ωB(r)in terms of the backbone propagator qi(r,s). The index i be-longs to {Ah, Bh, AgB} for the m = 1 case and {Ah, Bh,AgB, AgBB} for the m = 2 case. The quantities κ i, represent-ing the combinatorial factors, are equal to {1, 1, 1} for the m= 1 case of i = {Ah, Bh, AgB} and {1, 1, 2, 1} for the m = 2case of i = {Ah, Bh, AgB, AgBB}. The contour length s startsfrom one end of the homopolymer chain (s = 0) to the other(s = 1). The spatial coordinate r is rescaled by RA, where RA

2

= a2NAh/6.In the SCFT model, the backbone propagator is divided

into m + 1 segments based on reactive functional points and

is given by27–29, 45

qA,i(r, s) = q(j )A,i(r, s). (B3)

Here, q(j )A,i(r, s) is the backbone propagator of the jth segment

of i species between τ j and τ j+1. The contour length s is sub-ject to τ j ≤ s < τ j+1 for j = 0, 1, . . . , m. In particular, τ 0 andτm+1 are the positions of the two free ends of the backbone(i.e., τ 0 ≡ 0 and τm+1 ≡ 1). Each segment of the backbonepropagator satisfies the modified diffusion equation

∂q(j )A,i(r, s)

∂s= R2

A∇2q(j )A,i(r, s) − αAhωA(r)q(j )

A,i(r, s), (B4)

subject to the following initial condition

q(j )A,i(r, τj ) =

{q

(j−1)A,i (r, τj )qB(r, 1) σj = 1,

q(j−1)A,i (r, τj ) σj = 0,

(B5)

where j = 1, 2, . . . , m and q(0)A,i(r, 0) = 1. σ j = 1 (or 0) rep-

resents that the jth functional point along A-homopolymerbackbone is (or not) grafted by B-homopolymers. Here,qB(r,s) is a propagator for B-graft that satisfies the followingmodified diffusion equation

∂qB(r, s)

∂s= αBh

αAhR2

A∇2qB(r, s) − αBhωB(r)qB(r, s), (B6)

and is subject to the initial condition qB(r,0) = 1 for the freeend of the graft at s = 0. The backward propagator of the A-backbone is also subdivided into m + 1 segments accordingto functional points and is given by

q̄A,i(r, s) = q̄(j )A,i(r, s), (B7)

where q̄(j )A,i(r, s) is the backward propagator for the jth seg-

ment between 1 − τm+1-j and 1 − τm-j. The contour length sis subject to (1 − τm+1-j) < s ≤ (1 − τm-j) for j = 0, 1, . . . , m.Each segment obeys Eq. (B4) and is subject to the followinginitial conditions

q̄(j )A,i(r, 1−τm+1−j )=

{q̄

(j−1)A,i (r, 1 − τm+1−j )qB(r, 1) σj =1,

q̄(j−1)A,i (r, 1 − τm+1−j ) σj =0,

(B8)where j = 1, 2, . . . , m and q̄

(0)A,i(r, 0) = 1. The backward

propagator q̄Bj,i(r, s) of each B-branch attached to the jthfunctional point satisfies Eq. (B6) and starts on the end ofthe B-chain tethered to the backbone. The initial conditionq̄Bj,i(r, 0) is the product of the propagator and backward prop-agator of the backbone approaching the jth functional pointand satisfies27–29, 45

q̄Bj,i(r, 0) = qA,i(r, τj )q̄A,i(r, 1 − τj )

q2B(r, 1)

. (B9)

For the m = 1 case, the segment densities φA(r) and φB(r) aregiven as follows:

φA(r) = αAh

{∫ 1

0ds qAh(r, s)qAh(r, 1 − s) + zBh

eh

N

×∫ 1

0ds qA,AgB(r, s)q̄A,AgB(r, 1 − s)

}, (B10)

184901-11 Zhang et al. J. Chem. Phys. 139, 184901 (2013)

φB(r) = αBh

{zBh

∫ 1

0ds qBh(r, s)qBh(r, 1 − s) + zBh

eh

N

×∫ 1

0ds qB(r, s) q̄B,AgB(r, 1 − s)

}. (B11)

For the m = 2 case, the segment densities φA(r) and φB(r) aregiven as follows:

φA(r) = αAh

{∫ 1

0ds qAh(r, s)qAh(r, 1 − s) + 2zBh

eh

N

×∫ 1

0ds qA,AgB(r, s) q̄A,AgB(r, 1 − s)

+ z2Bh

e2h

N2

∫ 1

0ds qA,AgBB(r, s) q̄A,AgBB(r, 1 − s)

},

(B12)

φB(r) = αBh

{zBh

∫ 1

0ds qBh(r, s)qBh(r, 1 − s) + 2zBh

eh

N

×∫ 1

0ds qB(r, s) q̄B,AgB(r, 1 − s)

+m∑

j=1

z2Bh

e2h

N2

∫ 1

0ds qB(r, s) q̄Bj,AgBB(r, 1 − s)

}.

(B13)

Finally, the minimization of free energy with respect to φA(r),φB(r), and ξ (r) is achieved by satisfying the mean-fieldequations

ωA(r) = χNφB(r) + ξ (r), (B14)

ωB(r) = χNφA(r) + ξ (r), (B15)

φA(r) + φB(r) = 1. (B16)

The numerical solution of the mean-field equations wasstarted from deterministic initial states constructed from a su-perposition of the leading harmonics for the microstructures.The modified diffusion equations were solved via the pseudo-spectral method and operator splitting formula scheme.65, 66

The stable structures were obtained by exhaustively compar-ing the free energy of the computed structure with the freeenergies of other structures which could exist in the melt. Allof the simulations in the present work were carried out withperiodic boundary conditions. Each calculation utilized a stepsize of �s = 0.01 along the chain contour for both the A-and B-blocks. In the calculations, we chose the spatial resolu-tion �x < 0.14RA (�x < 0.1Rg). The numerical simulationsproceeded until the relative accuracy in the fields (measured

by√∑

I

∫dr[ωnew

I (r) − ωoldI (r)]2/

∑I

∫dr) is smaller than

10−6 and the incompressibility condition was achieved. Thefree energy was minimized with respect to the size of simula-tion box, as suggested by Bohbot-Raviv and Wang.67

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