lable at ScienceDirect

Polymer 78 (2015) 69e80

Contents lists avai

Polymer

journal homepage: www.elsevier .com/locate/polymer

Supramolecular assembly of diblock copolymer blends withhydrogen-bonding interactions modeled by Yukawa potentials

Xu Zhang a, Jingyuan Lin b, Liquan Wang a, **, Liangshun Zhang a, Jiaping Lin a, *,Liang Gao a

a Shanghai Key Laboratory of Advanced Polymeric Materials, State Key Laboratory of Bioreactor Engineering, Key Laboratory for Ultrafine Materials ofMinistry of Education, School of Materials Science and Engineering, East China University of Science and Technology, Shanghai 200237, Chinab School of Marine Science, Shanghai Ocean University, Shanghai 201306, China

a r t i c l e i n f o

Article history:Received 11 June 2015Received in revised form4 September 2015Accepted 26 September 2015Available online 30 September 2015

Keywords:AssemblyHydrogen bondSelf-consistent field theory

* Corresponding author.** Corresponding author.

E-mail addresses: lq_wang@ecust.edu.cn (L. Wang

http://dx.doi.org/10.1016/j.polymer.2015.09.0650032-3861/© 2015 Elsevier Ltd. All rights reserved.

a b s t r a c t

We extended self-consistent field theory to study the phase behavior of supramolecular blends of diblockcopolymers with hydrogen-bonding interactions. The hydrogen-bonding interactions are described byYukawa potentials. The hydrogen-bonding donors and acceptors were modeled as two blocks smearedwith opposite screened charges. Hierarchical microstructures such as parallel and perpendicularlamellae-in-lamellae were observed. The appearance of parallel/perpendicular lamellae-in-lamellaedepends on the strength of hydrogen-bonding interactions related to the density of hydrogen bondsand the characteristic lengths of the Yukawa potentials. Phase diagrams were correspondingly mappedout, and the domain size, interfacial width and free energies were examined to gain insight into thephase transitions. It was also found that the revealed mechanism can account for some experimentalphenomena that are not well explained yet. The present method can be readily further extended to morecomplicated supramolecular systems with hydrogen-bonding interactions.

© 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Supramolecular polymers are kinds of materials which utilizenon-covalent bonding interactions such as hydrogen-bonding in-teractions and electrostatic interactions to connect chemicallydissimilar polymers into a dynamic molecular architecture [1e5].The introduction of non-covalent bonding interactions into poly-mer blend systems would effectively suppress the macrophaseseparation. In the supramolecular polymers, themixing blocks withvarying ratios offer a straightforward and efficient way to generatea series of microstructures, without having to synthesize compli-cated copolymers. Moreover, the supramolecular polymers wouldoffer a great platform for fabricating materials with highly tunableproperties, such as environmental adaptation and self-repair ca-pacity, owning to the temperature/pH-dependence of the non-covalent interactions [6,7].

Inspired by the ability of supramolecular polymers to adjust the

), jlin@ecust.edu.cn (J. Lin).

versatilities and response to environments, plenty of efforts havebeen made to understand the phase behaviors of supramolecularpolymers [8e16]. A widely-studied system is the blends of AB- andB'C-diblock copolymers with B- and B'-blocks miscible byhydrogen-bonding interactions. Matsushita et al. prepared blendsof polyisoprene-b-poly(2-vinylpyridine) (PI-b-P2VP) and poly(4-hydroxystyrene)-b-polystyrene (P4HS-b-PS) diblock copolymers,where the hydrogen bonds can be formed between the P2VP- andP4HS-blocks [8]. The change of themolar ratio and the symmetry ofdiblock copolymers can lead to a variety of hierarchical micro-structures such as perpendicular lamellae-in-lamellae and spheres-in-lamellae. Fredrickson et al. designed a mixture of poly(ethyleneoxide)-b-poly(styrene-r-4-hydroxystyrene) (PEO-b-PS4HS) andpoly(styrene-r-4-vinylpyridine)-b-poly(methyl methacrylate)(PS4VP-b-PMMA) with various fractions of hydrogen-bondedphenolic and pyridyl units [9,10]. The effect of the hydrogen-bonding strengths on the hierarchical microstructures was exam-ined by controlling over the relative densities of hydrogen-bondingdonors and acceptors. It was found that hierarchical square ar-ranged nanostructures and hexagonally packedmicrostructures areformed at different ratios of donors to acceptors.

In addition to the experiments, theory and simulations were

X. Zhang et al. / Polymer 78 (2015) 69e8070

utilized to study the phase behavior of the supramolecular poly-mers [17e20]. One powerful theory successful in accounting forgeneral phase behaviors of complex polymeric systems is self-consistent field theory (SCFT) [21e36]. It has been applied to su-pramolecular polymers with hydrogen-bonding interactions[37e41]. Recently, there are two approaches for tackling thehydrogen-bonding interactions in the framework of SCFT. One ofthe methods is to use a negative FloryeHuggins parameter to ac-count for the hydrogen-bonding interactions. Fredrickson et al.developed this method to examine the self-assembly of AB/B'Cdiblock copolymer blends. An agreement with the experimentalobservations proves that this method is sufficient to understandqualitative aspects of the self-assembly of supramolecular systems[9,10]. However, it is unphysical to treat the hydrogen-bondinginteractions with a negative FloryeHuggins parameter. This isbecause that the hydrogen-bonding donors and acceptors arechemically incompatible (cN> 0) although they are attracted eachother. Another method is to treat hydrogen-bonding donors andacceptors as chemically reactive polymers with hetero-complementary bonding groups [37e41]. In this method, theimmiscibility between chemically different polymers are stilldescribed by FloryeHuggins parameters. The drawback of thismethod is that it suffers from difficulties in studying the systemswith multiple hydrogen bonds. This is due to the fact that thecomputational cost dramatically increases with increasing thenumber of hydrogen bonds. (Note that the supramolecular systemsgenerally contain a large number of hydrogen bonds in each block.)To overcome the drawbacks of these two methods, new SCFTmethod should be developed for the supramolecular copolymers.

In this work, we assumed the hydrogen-bonding interactions aselectrostatic dipoleedipole interactions. The dipoleedipole in-teractions are implicitly described by Yukawa potentials which isalso named screened Coulomb potentials [42,43]. Based on thisassumption, a SCFTcoupledwith the screened Poisson equation canbe derived. This method was first developed by Nogovitsin et al. forinvestigating the behavior of polyelectrolyte chains in aqueous saltsolutions [44]. We extended this method to a different polymersystem, namely, supramolecular blends of diblock copolymers withhydrogen-bonding interactions. The hydrogen-bonding donors andacceptors are modeled as two charged blocks smeared withopposite screened charges. Such an approach can overcome thedeficiencies of two commonly used methods d may rationallydescribe the immiscibility between different polymers and couldbe suit for multiple-hydrogen-bond supramolecular systems[42,43]. Furthermore, the method can bridge a gap between thehydrogen-bonding interactions and electrostatic interactions byvarying the parameters in Yukawa potential. Then, we applied thismethod to study the supramolecular blends of AB- and B'C-diblockcopolymers where B- and B'-blocks can be associated with thehydrogen bonds. The effect of the density of hydrogen bonds andtwo characteristic lengths of the Yukawa potential on the phasebehaviors were examined. It was found that the parallel-to-perpendicular lamellae-in-lamella transition occurs by varyingthese parameters. Based on the obtained data, phase diagramswere mapped out. This work provides reasonable informationabout the phase behavior of supramolecular copolymers, and theSCFT method can be readily extended to more complicated supra-molecular systems.

2. Theoretical method

A system of volume V consisting of nAB AB-diblock copolymersand nB'C B'C-diblock copolymers was considered. The incompres-sibility condition constrains the bulk density of segment number toa constant value r0 at each point in the space. The block copolymers

are assumed to be monodisperse with the same statistical segmentlength a. The total statistical segment of P-diblock copolymers(P ¼ AB and B'C) is NP, where NAB¼N and NB'C ¼ aB'CN for AB- andB'C-diblock copolymers, respectively. Here, aP ¼ NP/N is the lengthratio of P-diblock copolymer to AB-diblock copolymer. For AB-diblock copolymers, the volume fraction of A- and B-blocks is fAand fB, respectively. Similarly, for B'C-diblock copolymers, the vol-ume fraction of B'- and C-block is fB0 and fC, respectively. Thus,fA þ fB ¼ 1 and fB'þ fC ¼ 1. In the supramolecular system, one blockcan be connected with another block through hydrogen bonds. Thetwo kinds of blocks attracted by hydrogen bonds were smearedwith opposite screened charges. The valence number of theseblocks is denoted as zI (I ¼ A, B, B0, and C), and the correspondingcharge is zIe (e is the elementary charge).

The partition function of this supramolecular system is given by

Z ¼YP

znPPnP!

YnP

k¼1

ZDRk;P

!�Z

DjYP

YnP

k

�Xqk;PðsÞ

exp�� HkBT

�d

� ZdrbfeðrÞ

�Yrd

"XI

bfIðrÞ � 1

#(1)

Here, zP is the partition function of a single chain due to thekinetic energy; nP is the number of P-diblock copolymer; Rk,P is thespace curve of the kth P-polymer chain; j is the electrostatic po-tential at r position. qk,P(s) denotes the charge density of the sthsegment of kth chain of P-diblock copolymer.

Pqk;PðsÞ

is a summation

over the charge distributions of the kth chain of P-diblock copol-ymer. The Hamiltonian H of the system consists of three parts

H ¼ H0 þ H1 þ H2 (2)

where

H0 ¼ 3kBT2a2

XP

XnP

k¼1

ZNP

0

ds����dRkðsÞ

ds

���� (3)

is the elastic energy of Gaussian chains. The kB and T is the Boltz-mann constant and temperature, respectively.

H1 ¼ r0kBT2

XI; JIsJ

ZdrcIJbfIðrÞbfJðrÞ (4)

describes the short-range interaction energy of the system, wherecIJ is the FloryeHuggins interaction parameter between I- and J-species (in unit of kBT). The local microscopic densities of differentspecies at the position r are defined by

bfIðrÞ ¼1r0

XP

XnP

k¼1

ZNP

0

dssI;PðsÞd�r� Rk;PðsÞ

�(5)

Here, sI,P(s) ¼ 1 (or 0) represents that the sth segment of P-diblock polymers is of type I (or not). The hydrogen bond is usuallydescribed as an electrostatic dipoleedipole interaction. To mimiccorrelation effect between dipoles in a way compatible with theself-consistent field theory, we add a field j(r) to the energy of adipole (donor/acceptor) at position r [44,45]. The Yukawa potentialhas been used to describe the interaction between dipoles (e.g.water) implicitly, and found that it can successfully account for thecorrelation effects between dipoles [42,43]. The electrostatic en-ergy of the system is given by

X. Zhang et al. / Polymer 78 (2015) 69e80 71

H2 ¼Z

drdr0bfeðrÞjðr� r0Þbfeðr0Þ (6)

where the microscopic densities of the screened charges are

bfeðrÞ ¼XP

XnP

k¼1

ZNP

0

dsqk;PðsÞzPðsÞed�r� Rk;PðsÞ

�(7)

where j(r) is derived from a Yukawa potential. The Yukawa po-tential is

jðrÞ ¼ �he�ljrj

jrj ¼ �h

Z4p

k2 þ l2eikrdk (8)

QP ¼

ZDRexp

8<:�Z NP

0ds�

32a2

����dRðsÞds

����2 þ uðRðsÞÞ þ qPðsÞzPðsÞe4ðRðsÞÞ�9=;

ZDRexp

8<:� 32a2

Z NP

0ds����dRðsÞds

����29=;

(14)

where h and l are the two characteristic lengths of the Yukawapotential. The h is the amplitude of potential related to thedielectric permittivity and the l is the screening length that de-termines the range of the potential. The Yukawa potential is set tobe attractive in order to account for hydrogen-bonding interactionsbetween various blocks. This type of potential was previously usedto account for the interaction of hyaluronic acid chains in salt so-lutions [44]. To decouple the interaction in Eq. (6), the identity (8)and HubbardeStratonovich transformation were used. The elec-trostatic energy then becomes

H2 ¼Z

drbfeðrÞjðrÞ �Z

dr1

8ph

jVjðrÞj2 þ l2jðrÞ2

(9)

Subsequently integral representations for the delta functionsare inserted into the partition function (Eq. (1)). We use thefollowing integral representations,

d�fIðrÞ � bfIðrÞ

� ¼ Z duI exp�Z

drr0uIðrÞ�fIðrÞ � bfIðrÞ

��(10)

d

"XI

fIðrÞ � 1

#¼Z

dx exp

(Zdrr0xðrÞ

"XI

fIðrÞ � 1

#)(11)

it introduces a density field fI(r) constrained to bfIðrÞ and aneffective chemical potential field uI(r) conjugated to the densityfields. The Lagrange multiplier x(r) is introduced to enforce theincompressibility constraint (

PfI(r) ¼ 1). The partition function

(eq. (1)) becomes

Z ¼Z Y

I

ðDfIDuIÞDjDx exp�� FkBT

�(12)

where F is the free energy of the system, written as

NFr0VkBT

¼ NV

Zdr

(12

XI; JIsJ

cIJfIðrÞfJðrÞ �XI

uIðrÞfIðrÞ

� kBT8pr0h

hjV4ðrÞj2 þ l24ðrÞ2

i� xðrÞ

"1�

XI

fIðrÞ#)

�XP

cPaP

ln�QP

V

�(13)

Here, cP ¼ nPN/r0V is the volume-averaged fraction of P-diblockcopolymers and 4(r) ¼ j(r)/kBT. QP is the single chain partitionfunction for the P-diblock copolymers, given by

The partition function, QP, can be expressed in terms of propa-gators, qP(r,t), which are written as QP ¼ !drqP(r,1). The contourlength t starts from one end of the polymer chain (t ¼ 0) to theother (t ¼ 1). The spatial coordinate r is rescaled by Rg, whereRg2 ¼ Na2/6. The propagator qP(r,t) represents the probability of

finding segments t at position r, which satisfies the followingmodified diffusion equations

vqPðr; tÞvt

¼ V2qPðr; tÞ � NXI

½uIðrÞ þ qIzI4ðrÞ�sI;PðtaPNÞ�qPðr; tÞ

(15)

subject to the initial condition qP(r,0) ¼ 1. The backward propaga-tors qP*(r,t) satisfy the following modified diffusion equations

vq*Pðr; tÞvt

¼ V2q*Pðr; tÞ � NXI

½uIðrÞ þ qIzI4ðrÞ�sI;PðtaPNÞ�q*Pðr; tÞ

(16)

subject to the initial condition qP*(r,0) ¼ 1.

The mean-field equations are obtained by the saddle-pointapproximation, setting dF/dfI ¼ 0, dF/duI ¼ 0, and dF/dx ¼ 0. TheSCFT equations can be written as

uIðrÞ ¼X

I; J ¼ A;B;B0;CIsJ

cIJfJðrÞ þ xðrÞ (17)

fIðrÞ ¼X

P¼AB;B0C

cPVaPNQP

Z10

qPðr; tÞq*Pðr;1� tÞsI;PðtaPNÞdt (18)

XI¼A;B;B0;C

fIðrÞ ¼ 1 (19)

X. Zhang et al. / Polymer 78 (2015) 69e8072

The electrostatic potential is determined by a screened Poissonequation, which is obtained by minimizing the free energy withrespect to the electrostatic potential, i.e. dF/d4¼ 0, which is given asV2 � l2

4ðrÞ ¼ �feðrÞ

g(20)

where the constant g¼ 1/8ph. fe(r) is a summation over the chargedensities at position r, which is written as

feðrÞ ¼XI

zIqIfIðrÞ (21)

One of the main tasks is to solve the screened Poisson equation.This equation was solved with spectral method, whose computa-tional cost is much smaller than that of solving diffusion equations(15) and (16). Therefore, the computational cost does not increasetoo much. Moreover, the cost is independent of the number ofhydrogen bonds, and thereby the method suits for multiple-hydrogen-bond systems. The detailed spectral method is given as

4ðrÞ ¼ F�1�� 1k2 þ l2

,F�� feðrÞ

g

��(22)

where F and F�1 represent forward and inverse Fourier transformoperations, respectively.

The free energy can be split into the contributions of enthalpy Uand entropy S, i.e., F ¼ U � TS, where U ¼ Uc þ Ue. These termspossess the following expressions:

NUc

r0VkBT¼ N

2V

ZdrXI; JIsJ

cIJfIðrÞfJðrÞ (23)

NUe

r0VkBT¼ N

V

Zdr�4ðrÞfeðrÞ �

kBT8pr0h

hjV4ðrÞj2 þ l24ðrÞ2

i�(24)

NSr0VkB

¼ NV

Zdr

(XI

uIðrÞfIðrÞ þ 4ðrÞfeðrÞ)

þXP

cPaP

ln�QP

V

�(25)

The SCFT equations were solved by using a variant of the algo-rithm developed by Fredrickson and co-workers [45e48]. Initialfield configurations started from a general random state. Thediffusion equations were solved via the pseudo-spectral methodand operator splitting formula scheme, where the fast Fouriertransforms were performed by the software package developed atMIT [49]. All of the simulations in the present work were carriedout with periodic boundary conditions. Each calculation utilized astep size of Dt ¼ 0.01 and the spatial resolution Dx< 0.1Rg. Thenumerical simulations proceeded until the relative accuracy in the

fields (measured byffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPI

Rdr½unew

I ðrÞ � uoldI ðrÞ�2=P

I

Rdr

r) is smaller

than 10�10 and the incompressibility condition was achieved[50,51]. The free energy was minimized with respect to the size ofsimulation box, as suggested by Bonbot-Raviv and Wang [52].

Fig. 1. Sketch of supramolecular blends of AB- and B'C-diblock copolymers. The h and l

are the two characteristic lengths of the Yukawa potential j(r). The h is the amplitudeof potential related to the dielectric permittivity and the l is the screening length thatdetermines the range of the potential.

3. Results and discussion

The SCFT was applied to examine the phase behavior of supra-molecular blends of diblock copolymers (AB/B'C) with hydrogen-bonding interactions. Such interactions are implicitly modeled byYukawa potentials. In this work, we focused on the case that the

hydrogen bonds can only occur between B- and B'-blocks, as widelystudied in experiments [8e10]. As shown in Fig. 1, the B- and B'-blocks were smeared with hydrogen bonds. The interactions be-tween them are through Yukawa potentials. Thus, we set zA¼ zC¼ 0and qA ¼ qC ¼ 0. The valence number of the charges on the block iszB ¼ �zB' ¼ z. The volume fraction of the charges on the block isqB ¼ qB' ¼ q. For the simplicity of investigations, the AB- and B'C-diblock copolymers were treated to have the same volume-averaged fraction and equal length, i.e. cAB ¼ cB'C ¼ 0.5 andaAB ¼ aB'C¼ 1 (or NAB ¼ NB'C ¼ N). The interaction strength betweendifferent blocks was set as cABN ¼ cAB'N ¼ cBCN ¼ cB'CN ¼ 20 andcACN ¼ 10, leaving cBB'N as the single alterable interaction strength.Additionally, to ensure the charge neutrality, the volume fraction ofthe charged B-blocks is identical to that of charged B'-blocks,parameterized by fA ¼ fC ¼ f and fB ¼ fB' ¼ 1-f. Therefore, the vari-ables in the studies are q, l, g, and cBB'N.

3.1. Effect of hydrogen-bonding strength on hierarchicalmicrostructures

We first studied the microphase separation of symmetricaldiblock copolymers with f ¼ 0.5. Because of the symmetry of suchdiblock copolymers, the blends are able to form lamellar micro-structures. Fig. 2 shows the hierarchical microstructures obtainedfrom the symmetrical diblock copolymer blends with cBB'N ¼ 5,g ¼ 0.01, and l ¼ 0.1 at various values of q. The blue, olive, green,and red colors are assigned to A-, B-, B'-, and C-blocks, respectively.Hierarchical microstructures such as parallel lamellae-in-lamellae(Ljj, see Fig. 2a) and perpendicular lamellae-in-lamellae (L⊥, seeFig. 2b) were observed at q ¼ 0.25 and q ¼ 0.35, respectively. Thesemorphologies of parallel and perpendicular lamellae-in-lamellaeself-assembled from supramolecular blends of diblock co-polymers have also been observed in experiments [8,53]. FromFig. 2a and b, we can see that the A-domains (blue color) and C-domains (red color) are surrounded by B-domains (olive color) andB'-domains (green color), respectively. This is due to the restrictionof the covalence connected diblock copolymer architecture. Moreremarkable is that the B- and B'-blocks are contacted and inter-mixed to form BB'-domains since the attractive interactionsthrough attractive Yukawa potentials. In addition, the two-dimensional density profile of parallel (Ljj) and perpendicular (L⊥)hierarchical lamellae was respectively presented in Fig. 2c and d toillustrate the hierarchy of the lamellae-in-lamellae, where the blackdot lines are used to emphasize the underlying order of large-length-scale and small-length-scale lamellae. As can be seen fromFig. 2c, the small-length-scale lamellae (marked by yellow arrows)are parallel to the large-length-scale lamellae (marked by light bluearrows). From Fig. 2d, it can be seen that the small-length-scalelamellae are perpendicular to the large-length-scale lamellae.

To examine the domains formed by B- and B'-blocks and getmore deep insight into the hierarchical lamellae-in-lamellae, weanalyzed the variation of block densities in hierarchical lamellae-in-lamellae. The one-dimensional density profiles of each block

Fig. 2. (a) Parallel lamellae-in-lamellae Ljj (q ¼ 0.25) and (b) perpendicular lamellae-in-lamellae L⊥ (q ¼ 0.35) self-assembled from the supramolecular blends of AB- and B'C-diblockcopolymers with cBB'N ¼ 5, g ¼ 0.01, and l ¼ 0.1. The (c) and (d) show the corresponding cross sections of (a) and (b), respectively. The blue, olive, green, and red colors are assignedto A-, B-, B'-, and C-blocks, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

X. Zhang et al. / Polymer 78 (2015) 69e80 73

for parallel lamellae-in-lamellae Ljj and perpendicular lamellae-in-lamellae L⊥ along the chosen x arrows are shown in Fig. 3. It can beseen from Fig. 3a and b that, for both the Ljj and L⊥, the hydrogen-bonding donors and acceptors (B- and B'-blocks) are not homoge-neously mixed but microphase-separated. The B- and B'-blockstend to localize at the two sides of A-domains and C-domains,respectively. As shown in Fig. 3c, the A- and C-blocks contact witheach other to form A/C interfaces, which further demonstrates thatthe microstructures are perpendicular lamellae-in-lamellae. It isdue to the fact that the A/C contacts are energetically favored interms of interfacial energy. From Fig. 3, it is noted that the degree ofpeak separation between fB and fB0 in Ljj (Fig. 3a) are stronger thanthat in L⊥ (Fig. 3b). This indicates that the degree of microphaseseparation between B- and B'-blocks in Ljj are stronger than that inL⊥; namely, the B- and B'-blocks are more homogeneously mixed inL⊥ than that in Ljj. This would be discussed in more detail in thefollowing sections.

We further analyzed the variation of free energies upon theformation of the observed parallel lamellae-in-lamellae Ljj andperpendicular lamellae-in-lamellae L⊥. Fig. 4 shows the changes offree energy for Ljj and L⊥ as a function of charge density q, screeninglength l, and inverse amplitude g of Yukawa potential related todielectric permittivity, respectively. As can be seen from Fig. 4a, theLjj is shown to be more stable than the L⊥ at the smaller value of q.

This indicates that the lamellae-in-lamellae can be transformedfrom Ljj to L⊥ as the charge density increases. It can also be seenfrom Fig. 4a that the phase transition from Ljj to L⊥ is roughly atq¼ 0.308 which is the intersection of the two free energy curves. Incontrast to q, the screening length l and inverse amplitude g ofYukawa potential play the opposite roles in determining the sta-bility of the hierarchical lamellae-in-lamellae (see Fig. 4b and c). Asthe l and g increases, the L⊥ microstructures are transformed intothe Ljj microstructures. The transition points are l ¼ 0.45 andg ¼ 0.013, respectively. Because the increase in q or decrease in l

and g leads to an increase in the attraction between B- and B'-blocks, it implies that the perpendicular lamellae-in-lamellae L⊥

are preferred for the supramolecular blends with strong hydrogen-bonding interactions. Since the increase in q or decrease in l and g

play similar effect on the hydrogen-bonding interactions, we onlyconcentrated on the influence of charge density q in the followingwork.

To further clarify the dependence of the hierarchical lamellae-in-lamellae on the strength of hydrogen bonds, we calculated thedomain size, the order parameter profiles, and the interfacial widthfor supramolecular blends with various charge densities (q). Fig. 5shows the lamellar period D/Rg of the hierarchical lamellae-in-lamellae as a function of q for the supramolecular blends of AB-and B'C-diblock copolymers with cBB'N¼ 5, g ¼ 0.01, and l¼ 0.1. As

Fig. 3. One-dimensional density profiles of the blocks along the x direction of thehierarchical lamellae-in-lamellae of the supramolecular blends of AB- and B'C-diblockcopolymers with cBB'N ¼ 5, g ¼ 0.01, and l ¼ 0.1 at (a) q ¼ 0.25 and (b, c) q ¼ 0.35. D isthe lamellar period along the x direction which is indicated by white arrows in theinserts. The inserts show the cross sections, where the colors appear as same as thosein Fig. 2. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

Fig. 4. Plots of free energies as a function of (a) q at g ¼ 0.01 and l ¼ 0.1, (b) l atg ¼ 0.01 and q ¼ 0.35, and (c) l at g ¼ 0.01 and q ¼ 0.35 for the supramolecular blendsof AB- and B'C-diblock copolymers with cBB'N ¼ 5 (n ¼ r0V/N). The transition pointbetween Ljj and L⊥ is indicated by an arrow. The inserts show two-dimensional densityprofiles of parallel lamellae-in-lamellae Ljj and perpendicular lamellae-in-lamellae L⊥.The colors appear as same as those in Fig. 2. (For interpretation of the references tocolor in this figure legend, the reader is referred to the web version of this article.)

X. Zhang et al. / Polymer 78 (2015) 69e8074

can be seen, the increase in the charge density q produces adecrease in lamellar period. This can be attributed to the fact thatthe B- and B'-blocks are strongly associated at larger charge density.As the phase transition from Ljj to L⊥ occurs, the lamellar period oflarge-length-scale lamellae shows an increase. This discontinuousvariation of lamellar periods implies that the phase transition isfirst-order. For the Ljj, both the periods of large-length-scalelamellae and B/B'-domains decrease. While for the L⊥, the periodof large-length-scale lamellae shows a slight change and the periodof small-length-scale lamellae dramatically decreases. This in-dicates that the A- and C-blocks become restricted to a narrower

space along the normal direction as the association of B- and B'-blocks increases.

Fig. 6 illustrates the effect of charge density q on the orderparameter profiles fB(x)-fB0(x) of the parallel lamellae-in-lamellaeLjj and perpendicular lamellae-in-lamellae L⊥. (Note that jfB(x)-fB0(x)j ¼ 0 and 1 represents that the B- and B'-blocks are disorderand complete separation, respectively.) As can be seen from Fig. 6,the values of q have a pronounced influence on the order parameterprofiles. As shown in Fig. 6a and b, with increasing the q values, theorder parameter profiles for both the hierarchical lamellae-in-lamellae show a shift to zero. This suggests that the increase in q

Fig. 5. Lamellar period D/Rg of Ljj and L⊥ as a function of q for the supramolecularblends of AB- and B'C-diblock copolymers with cBB'N ¼ 5, g ¼ 0.01, and l ¼ 0.1. Thelarge-length-scale lamellar period DL of Ljj and L⊥, the period of BB'-domains DBB0 of Ljj

and the small-length-scale lamellar period DS of L⊥ are schematically presented in theinsets, where the DS of Ljj is equal to 2DBB'. The colors appear as same as those in Fig. 2.(For interpretation of the references to color in this figure legend, the reader is referredto the web version of this article.)

X. Zhang et al. / Polymer 78 (2015) 69e80 75

leads to a weaker segregation between B- and B'-blocks due to theincreased association of B- and B'-blocks by hydrogen bonds. Tounderstand such an effect, the interfacial width (w¼ (dfB/dx)�1 atthe interface [54]) was also calculated. The results for the Ljj and L⊥

were provided in the inserts of Fig. 6a and b, respectively. It can be

Fig. 6. Order parameter profiles fB(x) � fB0(x) along the x direction of (a) Ljj and (b) L⊥

of the supramolecular blends of AB- and B'C-diblock copolymers with various values ofq. The other parameters are cBB'N ¼ 5, g ¼ 0.01, and l ¼ 0.1. D is the period of the large-length-scale lamellae. The inserts show the interfacial width and the correspondingtwo-dimensional lamellae where the colors appear as same as those in Fig. 2. (Forinterpretation of the references to color in this figure legend, the reader is referred tothe web version of this article.)

Fig. 7. Phase diagram in (a) qel space at g ¼ 0.01 and (b) qeg space at l ¼ 0.1 for thesupramolecular blends of AB- and B'C-diblock copolymers with various values of cBB'N.The inserts show the two-dimensional parallel lamellae-in-lamellae Ljj and perpen-dicular lamellae-in-lamellae L⊥. The colors appear as same as those in Fig. 2. (Forinterpretation of the references to color in this figure legend, the reader is referred tothe web version of this article.)

seen that the interfaces become broad as the q increases. This alsoimplies that the segregation between B- and B'-blocks is weaker inthe blends of AB/B'C diblock copolymers with a relatively largercharge density.

Based on the obtained results, we plotted the phase diagrams inqel and qeg spaces for the hierarchical lamellae-in-lamellae. Thediagrams are shown in Fig. 7. The phase boundaries are obtained bycomparing the free energies of the Ljj and L⊥. In qel space, as shownin Fig. 7a, the phase boundaries from Ljj to L⊥ shift towards highervalue of q as the l increases. In qeg space, as shown in Fig. 7b, thephase boundaries also show a shift to higher value of q as the g

increases. This effect can be rationalized by considering associationof B- and B'-blocks by hydrogen bonds. The increase in l and g

corresponds to the increase of the screening effect and the decreaseof the dielectric permittivity, respectively, weakening thehydrogen-bonding interaction between B- and B'-blocks. This re-quires more hydrogen bonds on the B- and B'-blocks to associatethem together (larger q), leading to a shift of boundaries to higher qwith increasing l and g.

In addition, the effect of interaction strength cBB'N betweenhydrogen-bonding donors and acceptors on the phase behavior ofAB/B'C diblock copolymer blends was also presented in Fig. 7. Inboth the phase diagrams, the phase boundaries shift upwards tohigher q as cBB'N increases. This indicates that stronger hydrogen-bonding interactions are needed for the supramolecular blendswith higher cBB'N to form the perpendicular lamellae-in-lamellaeL⊥. An increase in the value of cBB'N (�0) can weaken the associa-tion of B- and B'-blocks. To compensate this effect, the hydrogen-

X. Zhang et al. / Polymer 78 (2015) 69e8076

bonding interactions have to be increased, for example, increasingthe density q of hydrogen bonds on the B- and B'-blocks. This is thereason that the phase boundaries shows a shift upwards to higher qwith increasing cBB'N.

3.2. Energy analysis of parallel-to-perpendicular lamellae-in-lamellae transition

To further understand the effect of hydrogen-bonding interac-tion on the structures, we calculated the interfacial energy Uc/nkBT,electrostatic energy Ue/nkBT, and entropic loss �S/nkB for the su-pramolecular AB- and B'C-diblock copolymer blends with variouscharge densities q. Fig. 8 shows the variations of these energies as afunction of q, where the black dot line at q ¼ 0.308 indicates theboundary of parallel-to-perpendicular lamellae transition. As the q

increases, i.e., as the hydrogen-bonding interaction increases, boththe interfacial energyUc/nkBT and entropy loss -S/nkB increases, andthe electrostatic energy Ue/nkBT decreases slightly. Upon phasetransition from the Ljj to L⊥, the interfacial energy and entropic lossshow a dramatic increase, and the electrostatic energy shows anabrupt decrease.

Fig. 8. Plots of (a) interfacial energy Uc/nkBT, (b) electrostatic energy Ue/nkBT, and (c)entropic loss �S/nkB as a function of q for the supramolecular AB- and B'C-diblockcopolymer blends with cBB'N ¼ 5, g ¼ 0.01, and l ¼ 0.1 (n ¼ r0V/N). The inserts showtwo-dimensional density profiles of hierarchical parallel lamellae-in-lamellae Ljj andhierarchical perpendicular lamellae-in-lamellae L⊥. The colors appear as same as thosein Fig. 2. (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article.)

In according to the above free energy analyses, we provided ascheme to illustrate the phase transitions. The scheme is presentedin Fig. 9, in which possible conformations of the supramolecularcopolymers in various structures are illustrated. Combining thescheme with the energy variations (see Fig. 8), the phenomenonthat the Ljj transforms to L⊥ with increasing the hydrogen-bondinginteractions can be explained as follows. At lower hydrogen-bonding interactions, the supramolecular blends are phase-separated into Ljj with large period. In this way, the interfacial en-ergy and entropic loss is minimized (Fig. 8a and c), since the specificarea of A/B(B0) and C/B(B0) interfaces can be effectively reduced.With increasing the hydrogen-bonding interactions, the electro-static energy favors the mixing of B- and B'-blocks, and the B- andB'-blocks becomemore associated (Fig. 9a and b). The association ofB- and B'-blocks weakens the segregations of B- and B'-blocks,narrowing the lamellar period. The specific area of A/B(B0) and C/B(B0) interfaces increases, and the interfacial energy and entropicloss also increases (Fig. 8a and c). Thus, this is a process balanced byincreasing electrostatic attraction and sacrificing the conforma-tional entropy and interfacial energy.

When the interfacial energy increases to the critical point, thefree energy cannot be minimized in Ljj, and then the L⊥ is favored.Compared with Fig. 9b and c, we can see that there are more B/B0

interfaces for forming hydrogen bonds in the L⊥, as indicated by thewhite regions. (Note that in the Ljj the hydrogen bonds are formedat the B/B0 interfaces parallel to large-length-scale lamellae, whilein the L⊥ the hydrogen bonds can be formed at the B/B0 interfacesnot only parallel but also perpendicular to large-length-scalelamellae). As a result, the electrostatic energy decreases (Fig. 8b).However, the additional A/C interfaces is produced in the L⊥,leading to an increase in the interfacial energy (Fig. 8a). For the L⊥

at relatively lower hydrogen-bonding interactions, the period ofsmall-length-scale lamellae is larger. This is favored by conforma-tional entropy due to broad distribution of the blocks (Figs. 8c and9c). As hydrogen-bonding interactions increases, the chain distri-bution is restricted to a narrower space owing to the increasedattraction between B- and B'-blocks (Fig. 9d), and the conforma-tional entropic loss increases (Fig. 8c). Consequently, the period ofsmall-length-scale lamellae decreases as the hydrogen-bondinginteractions increases (Fig. 5). Overall, the transition from Ljj to L⊥

and the decrease in their respective period are attributed to theoptimization of electrostatic energy with sacrificing the entropyand interfacial energy.

3.3. Comparison with experimental observations

Some experimental findings are available in the literature forsupporting our simulation results. Supramolecular blends of AB-diblock copolymers and B'-homopolymers where the B and B0 canform hydrogen bonds are the special AB/B'C supramolecular blendswhere the length of C-blocks vanishes. There are lots of studies onAB/B0 blend systems. For example, Chen et al. investigated thephase behavior of poly(4-vinylphenol-b-styrene) (PVPh-b-PS)blended with poly(4-vinylpyridine) (P4VP), poly(methyl methac-rylate) (PMMA), and PVPh, respectively [16]. (Note that thehydrogen-bonding strength of the blend systems is in a sequence ofPVPh-b-PS/P4VP > PVPh-b-PS/PMMA > PVPh-b-PS/PVPh.) Theyfound a nearly linear growth of the lamellar period with the con-centration of PVPh-homopolymers in the PVPh-b-PS/PVPh blendsystems. While in the PVPh-b-PS/PMMA and PVPh-b-PS/P4VPblend systems, the lamellar spacing shows a decrease as the ho-mopolymer concentration increases.

To reproduce such behaviors, we carried out additional calcu-lations for the lamellar spacing of AB/B0 supramolecular blends.Fig. 10 shows the obtained normalized lamellar spacing for the

Fig. 9. Sketch of possible conformation of supramolecular copolymers in various structures, where the hydrogen-bonding interactions increase along the arrows. White regionsindicate the possible interfaces for forming hydrogen bonds.

X. Zhang et al. / Polymer 78 (2015) 69e80 77

symmetric AB/B0 blend systems. The parameters were set as thosein the work of Dehghan and Shi [37]. The block lengths areNA¼NB¼NB0, and the interaction strengths are cABN ¼ 12,cAB'N ¼ 15, and cBB'N ¼ 2. In addition, g ¼ 0.01 and l ¼ 0.1. Thedensities of the hydrogen-bonding accepters and donors are fixedas q ¼ qB ¼ qB'. As shown in Fig. 10, at the weak hydrogen-bondingregime (q ¼ 0.05), the lamellar spacing D/D0 increases as the con-centration of B'-homopolymers increases. While at the stronghydrogen-bonding regimes (q ¼ 0.25 and q ¼ 0.5), the increase inhomopolymer concentration results in a decrease in the lamellarspacing. These results are consistent with the experimental find-ings and those predicted by the interpolymer-complexationmodels[16,37]. This further verifies that our method is an appropriateapproach to study the phase behaviors of the polymer blends withhydrogen-bonding interactions.

In addition, the revealed mechanism for the Ljj-to-L⊥ transitioncould account for some experimental phenomena that need to befurther clarified. Fredrickson et al. prepared blends of PEO-b-PS4HS(AB) and PS4VP-b-PMMA (B'C) with various fractions of hydrogen-bonded phenolic and pyridyl units [9,10]. As the number ofhydrogen-bonding donors in PEO-b-PS4HS (AB) copolymers

Fig. 10. Normalized lamellar spacing D/D0 for the symmetric AB/B0 blend systems atvarious densities q ¼ qB ¼ qB0 of hydrogen-bonding acceptors and donors. The otherparameters are NA¼NB¼NB0 , cABN ¼ 12, cAB'N ¼ 15, cBB'N ¼ 2, g ¼ 0.01, and l ¼ 0.1. D0

is the lamellar spacing for the AB-copolymers.

increases (the number of hydrogen-bonding acceptors in PS4VP-b-PMMA (B'C) copolymers are fixed), the structures formed by theblends are transformed from tetragonal cylinders (Fig. 11a) tohexagonal cylinders (Fig. 11b) (both Fig. 11a and b are reproducedfrom ref. 10 with permission of ACS). In tetragonal cylinders(Fig. 11a), the A-cylinders are packed in a tetragonal lattice with C4symmetry and distributed uniformly around the C-cylinders. Inhexagonal cylinders (Fig. 11b), A- and C-blocks are intermixed toform cylinders with C6 symmetry. Fredrickson et al. provided anindefinite explanation for the phase transitions with donor-acceptor stoichiometry, which is as follows. Because of the exces-sive phenolic groups (hydrogen-bonding donors) on B-blocks, theB-blocks can interact with A- and C-blocks by hydrogen bonds. Theincreased interaction between B-blocks and A/C-blocks drives thephase transition from tetragonal to hexagonal cylinders.

Our work could provide an explanation for the tetragonal-to-hexagonal cylinder transition in another way. In the work carriedout by Fredrickson et al., the increase in the number of hydrogen-bonding donors leads to the increase of hydrogen-bonding in-teractions between B- and B0 blocks. (Note that although thehydrogen bonding involves functional groups at localized sites thatsaturate upon bonding, the hydrogen bonding cannot reach 100%due to the entropic reason of the copolymers.) According to thisexperimental evidence and the mechanism revealed for the Ljj-to-L⊥ transition, it can be deduced that the phase transition fromtetragonal cylinders to hexagonal cylinders is mainly caused by theincrease in hydrogen-bonding interactions. The deduction is basedupon following consideration. In the hexagonal structures, the B-and B'-blocks are uniformly mixed since A- and C-blocks are mixedin the cylinders. Compared with the hexagonal structures, the B-and B'-blocks cannot be effectively mixed in the tetragonal struc-tures with separated A- and C-cylinders. This is because the B- andB'-blocks are respectively chemically bonded to A and C-blocks, theB- and B'-blocks have to stretch themselves to mix themselveshomogeneously as the A- and C-cylinders are separated. Due toenhanced mixing of B- and B'-blocks, as the tetragonal-to-hexagonal cylindrical transition occurs, both the interfacial en-ergy and entropic loss increase. To compensate these two unfa-vorable energies and lower total free energy, the electrostaticenergy has to decrease dramatically. As a result, only the supra-molecular block copolymer systems with stronger hydrogen-

Fig. 11. SFM phase images of solvent annealed films from supramolecular block copolymer blends of PS4VP-b-PMMA/PEO-b-PS4HS (reproduced from ref. 10 with permission ofACS) for (a) tetragonal packing with C4 symmetry and (b) hexagonal packing with PEO and PMMA mixed. Hierarchical cylindrical microstructures self-assembled from the su-pramolecular blends of AB- and B'C-diblock copolymers with fA ¼ fC ¼ 0.3, fB ¼ fB' ¼ 0.7, cABN ¼ cAB'N ¼ cBCN ¼ cB'CN ¼ 20, cACN ¼ 10, cBB'N ¼ 5, g ¼ 0.01, and l ¼ 0.1: (c) tetragonalpacking with C4 symmetry (q ¼ 0.05), (d) hexagonal packing with A- and C-blocks mixed (q ¼ 0.30), and (e) hexagonal packing with separated A- and C-cylinders (q¼0.10). Thecolors in the cross sections in (c) and (e) appear as same as those in Fig. 2. In (d), the polymers are schematically shown, and the purple and dark green colors are assigned to mixedAC- and BB'-domains, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

X. Zhang et al. / Polymer 78 (2015) 69e8078

bonding interactions can meet the demand.Based on the above analysis, we can predict that an increase in

the hydrogen-bonding interactions (increase q or decrease l/g)could lead to a phase transition from tetragonal to hexagonal cyl-inders. To test the prediction, we carried out an additional calcu-lation for the supramolecular blends of asymmetric diblockcopolymers. The lengths of AB and B'C-block copolymers were setequal and the compositions of A- and C-blocks were set asfA ¼ fC¼ 0.3, according to the work of Fredrickson et al. Without theloss of generality, the other parameters are set ascABN ¼ cAB'N ¼ cBCN ¼ cB'CN ¼ 20, cACN ¼ 10, cBB'N ¼ 5, g ¼ 0.01,and l ¼ 0.1. The obtained results are shown in Fig. 10cee. As shownin the Fig. 10c, the supramolecular blends self-assemble intotetragonal structures with separated A- and C-cylinders at q ¼ 0.05.At higher value of q, for example, q ¼ 0.30, the hexagonal structureswithmixed AC-cylinders is formed (Fig. 11d). The result well provesthe prediction, and therefore our explanation for the phase tran-sition can support the experimental phenomena. In addition, wefound a new structure d hexagonal packing structure with sepa-rated A- and C-cylinders (Fig. 11e)d that has not been observed yetin the experiments and other SCFT calculations [10]. The formationof this type of structures is also driven by the hydrogen-bondinginteractions.

The agreement with experimental observations demonstratesthat the present method is capable of studying the supramolecularblends with hydrogen-bonding interactions. Although this methodwas originally developed for the polyelectrolytes in salt solutions[44], it also shows great power in studying polymer systems withhydrogen-bonding interactions. In contrast with the presentmethod, the commonly used methods may not well reproduce theexperimental observations of Fredrickson et al. [9,10,37e41] Thedifficulty in reproducing these experimental findings by thecommonly used methods may be as follows. Since our workrevealed that the interfacial energy exhibit marked influence on thetetragonal-to-hexagonal cylindrical transition, the treatment of

hydrogen-bonding interaction with negative FloryeHuggins pa-rameters may underestimate this important interaction [10]. As aresults, the method that treats the hydrogen-bonding interactionsas negative FloryeHuggins parameters could not produce theseexperimental observations. In addition, the present method canreproduce some other behaviors that the method with negativeFloryeHuggins parameters cannot well address. For example, themethod with negative FloryeHuggins parameters cannot properlydescribe the change of lamellar spacing induced by the addition ofthe B'-homopolymers to AB-diblock copolymers [37], while ourmethod can do it (see Fig. 10). On the other hand, the hydrogen-bonding interaction, associated with the density and strength ofhydrogen bonds, plays a decisive role in determining the phasetransitions. Therefore, it is crucial to model the hydrogen-bondinginteractions appropriately (in particular the multiple hydrogenbonds on each block). However, for the method that treatshydrogen-bonding donors/acceptors as reactive polymers [37e41],it is a rather hard task to tackle the blocks with multiple hydrogenbonds. The present work provides a new SCFT method that over-comes these two drawbacks and can be further extended othersupramolecular systems with complicated hydrogen-bondinginteractions.

Finally, it should be noted that saturation effect is an importantproperty of the hydrogen bonding. However, this effect is lacking inthis method because the Yukawa potential is a pair-wise potential.Some other features of the hydrogen bonds such as directional arealso lacking due to the technical limitation of the SCFT. Neverthe-less, it is demonstrated by comparing with available experimentaland theoretical results that the hydrogen bonding of polymerscould be simulated well by the present method with Yukawapotential.

4. Conclusions

A self-consistent field theory was used for studying the

X. Zhang et al. / Polymer 78 (2015) 69e80 79

supramolecular self-assembly of AB- and B'C-diblock copolymerblends with hydrogen-bonding interactions between B- and B'-blocks. The hydrogen-bonding interactions were described by theYukawa potential, and the hydrogen-bonding donors and acceptorsweremodelled as B- and B'-blocks smeared with opposite screenedcharges. Parallel lamellae-in-lamellae and perpendicular lamellae-in-lamellae were observed. Through comparing with the free en-ergies, phase diagrams were mapped out. The perpendicularlamellae-in-lamellae were found to be stable for the supramolec-ular blends with strong hydrogen-bonding interactions that areassociated with higher charge density, smaller screening length, orsmaller inverse amplitude of Yukawa potentials. The analyses ofenthalpy and entropy reveal that the formation of perpendicularlamellae-in-lamellae at strong hydrogen-bonding interactions isfavored by electrostatic energy. The obtained results can provide areasonable explanation for the tetragonal-to-hexagonal cylindertransitions observed in the experiments for which the mechanismis not well-understood yet. The present SCFTmethod can be readilyapplied to other supramolecular blends with more complicatedhydrogen-bonding interactions.

Acknowledgments

This work was supported by National Natural Science Founda-tion of China (21304035, 21234002, 21474029), Key Grant Project ofMinistry of Education (313020), National Basic Research Program ofChina (No. 2012CB933600), Fundamental Research Funds for theCentral Universities (222201314024), and Research Fund for theDoctoral Program of Higher Education of China (20120074120002).Support from project of Shanghai Municipality (13JC1402001) isalso appreciated.

References

[1] F. Zeng, Y. Han, Z. Yan, C. Liu, C. Chen, Supramolecular polymer gel with multistimuli responsive, self-healing and erasable properties generated by hoste-guest interactions, Polymer 54 (2013) 6929e6935.

[2] M. Hetzer, C. Fleischmann, B.V.K.J. Schmidt, C. Barner-Kowollik, H. Ritter, Vi-sual recognition of supramolecular graft polymer formation via phenolph-thaleinecyclodextrin association, Polymer 54 (2013) 5141e5147.

[3] A. Noro, K. Ishihara, Y. Matsushita, Nanophase-separated supramolecular as-semblies of two functionalized polymers via acid-base complexation, Mac-romolecules 44 (2011) 6241e6244.

[4] K. Dobrosielska, S. Wakao, A. Takano, Y. Matsushita, Nanophase-separatedstructures of AB block copolymer/C homopolymer blends with complemen-tary hydrogen-bonding interactions, Macromolecules 41 (2008) 7695e7698.

[5] N. Houbenov, A. Nyk€anen, H. Iatrou, N. Hadjichristidis, J. Ruokolainen, C. Faul,O. Ikkala, Fibrillar constructs from multilevel hierarchical self-assembly ofdiscotic and calamitic supramolecular motifs, Adv. Funct. Mater. 18 (2008)2041e2047.

[6] T. Aida, E.W. Meijer, S.I. Stupp, Functional supramolecular polymers, Science335 (2012) 813e817.

[7] J. Ruokolainen, R. M€akinen, M. Torkkeli, T. M€akel€a, T. Serimaa, G. ten Brinke,O. Ikkala, Switching supramolecular polymeric materials with multiple lengthscales, Science 280 (1998) 557e560.

[8] T. Asari, S. Matsuo, A. Takano, Y. Matsushita, Three-phase hierarchical struc-tures from AB/CD diblock copolymer blends with complemental hydrogenbonding interaction, Macromolecules 38 (2005) 8811e8815.

[9] C. Tang, E.M. Lennon, G.H. Fredrickson, E.J. Kramer, C.J. Hawker, Evolution ofblock copolymer lithography to highly ordered square arrays, Science 322(2008) 429e432.

[10] C. Tang, S. Hur, B.C. Stahl, K. Sivanandan, M. Dimitriou, E. Pressly,G.H. Fredrickson, E.J. Kramer, C.J. Hawker, Thin film morphology of blockcopolymer blends with tunable supramolecular interactions for lithographicapplications, Macromolecules 43 (2010) 2880e2889.

[11] S. Shinde, S.K. Asha, Self-assembly directed template photopolymerization ofperylenebisimide-poly(4-vinylpyridine): nano organization, Polymer 65(2015) 115e123.

[12] S. Sami, E. Yildirim, M. Yurtsever, E. Yurtsever, E. Yilgor, I. Yilgor, G.L. Wilkes,Understanding the influence of hydrogen bonding and diisocyanate symme-try on the morphology and properties of segmented polyurethanes and pol-yureas: computational and experimental study, Polymer 55 (2014)4563e4576.

[13] S. Ma, X. Qi, Y. Cao, S. Yang, J. Xu, Hydrogen bond detachment in polymer

complexes, Polymer 54 (2013) 5382e5390.[14] G. Song, Y. Zhang, D. Wang, C. Chen, H. Zhou, X. Zhao, G. Dang, Intermolecular

interactions of polyimides containing benzimidazole and benzoxazole moi-eties, Polymer 54 (2013) 2335e2340.

[15] D. Tsiourvas, M. Arkas, Columnar and smectic self-assembly deriving from nonionic amphiphilic hyperbranched polyethylene imine polymers and inducedby hydrogen bonding and segregation into polar and non polar parts, Polymer54 (2013) 1114e1122.

[16] S.-C. Chen, S.-W. Kuo, U.-S. Jeng, C.-J. Su, F.-C. Chang, On modulating the phasebehavior of block copolymer/homopolymer blends via hydrogen bonding,Macromolecules 43 (2010) 1083e1092.

[17] F. Tanaka, M. Ishida, A. Matsuyama, Theory of microphase formation inreversibly associating block copolymer blends, Macromolecules 24 (1991)5582e5589.

[18] H.J. Angerman, G. ten Brinke, Weak segregation theory of microphase sepa-ration in associating binary homopolymer blends, Macromolecules 32 (1999)6813e6820.

[19] J. Huh, W.H. Jo, Theory on phase behavior of triblocklike supramoleculesformed from reversibly associating end-functionalized polymer blends,Macromolecules 37 (2004) 3037e3048.

[20] Y. Han, W. Jiang, Self-assembly of the AB/BC diblock copolymer mixture basedon hydrogen bonding in a selective solvent: a Monte Carlo study, J. Phys.Chem. B 115 (2011) 2167e2172.

[21] G. Quan, M. Wang, C. Tong, A numerical study of spherical polyelectrolytebrushes by the self-consistent field theory, Polymer 55 (2014) 6604e6613.

[22] C.L. Ting, Z.-G. Wang, Minimum free energy paths for a nanoparticle crossingthe lipid membrane, Soft Matter 8 (2012) 12066e12071.

[23] X. Ye, T. Shi, Z. Lu, C. Zhang, Z. Sun, L. An, Study of morphology and phasediagram of p-shaped ABC block copolymers using self-consistent-field theory,Macromolecules 38 (2005) 8853e8857.

[24] G. Quan, Y. Zhu, C. Tong, The numerical study of the adsorption of bi-disperseflexible polyelectrolytes onto the surface of two charged objects, Polymer 54(2013) 6834e6842.

[25] X. Wang, M. Goswami, R. Kumar, B.G. Sumpter, J. Mays, Morphologies of blockcopolymers composed of charged and neutral blocks, Soft Matter 8 (2012)3036e3051.

[26] Y. Xia, Z. Sun, T. Shi, J. Chen, L. An, Y. Jia, Self-assembly of rod-terminallytethered three-armed star-shaped coil block copolymer: investigation of thepresence of the branching in the coil to the self-assembled behavior, Polymer49 (2008) 5596e5601.

[27] Y. Xia, J. Chen, Z. Sun, T. Shi, L. An, Y. Jia, Self-assembly of linear ABCcoilecoilerod triblock copolymers, Polymer 51 (2010) 3315e3319.

[28] L. Wang, J. Lin, Discovering multicore micelles: insights into the self-assemblyof linear ABC terpolymers in midblock-selective solvents, Soft Matter 7 (2011)3383e3391.

[29] Y. Zhuang, L. Wang, J. Lin, L. Zhang, Phase behavior of graft copolymers inconcentrated solution, Soft Matter 7 (2011) 137e146.

[30] X. Ye, B.J. Edwards, B. Khomami, Block copolymer morphology formation ontopographically complex surfaces: a self-consistent field theoretical study,Macromol. Rapid Commun. 35 (2014) 702e707.

[31] X. Ye, X. Yu, T. Shi, Z. Sun, L. An, Z. Tong, A self-consistent field theory study onthe morphologies of linear ABCBA and H-shaped (AB)2C(BA)2 block co-polymers, J. Phys. Chem. B 110 (2006) 23578e23582.

[32] D. Sun, Z. Sun, H. Li, L. An, Study of morphology and phase diagram of the H-shaped (AC)B(CA) ternary block copolymers using self-consistent field theory,Polymer 50 (2009) 4270e4280.

[33] D. Sun, Z. Sun, H. Li, L. An, Effects of asymmetric interaction energies on themicro-phase separation behavior of H-shaped (AC)B(CA) ternary blockcopolymer systems: a real space self-consistent field theory study, Macromol.Theory Simul. 19 (2010) 100e112.

[34] X. Cao, L. Zhang, L. Wang, J. Lin, Insights into ordered microstructures andordering mechanisms of ABC star terpolymers by integrating dynamic self-consistent field theory and variable cell shape methods, Soft Matter 10(2014) 5916e5927.

[35] R. Wang, Z. Jiang, H. Yang, G. Xue, Side chain effect on the self-assembly ofcoil-comb copolymer by self-consistent field theory in two dimensions,Polymer 54 (2013) 7080e7087.

[36] Y.-C. Hsu, C.-I. Huang, W. Li, F. Qiu, A.-C. Shi, Micellization of Linear A-b-(B-alt-C)n multiblock terpolymers in A-selective solvents, Polymer 54 (2013)431e439.

[37] A. Dehghan, A.-C. Shi, Modeling hydrogen bonding in diblock copolymer/homopolymer blends, Macromolecules 46 (2013) 5796e5805.

[38] X. Zhang, L. Wang, T. Jiang, J. Lin, Phase behaviors of supramolecular graftcopolymers with reversible bonding, J. Chem. Phys. 139 (2013)184901e184912.

[39] Y. Zhuang, L. Wang, J. Lin, Hierarchical nanostructures self-assembled fromdiblock copolymer/homopolymer blends with supramolecular interactions,J. Phys. Chem. B 115 (2011) 7550e7560.

[40] E.H. Feng, W.B. Lee, G.H. Fredrickson, Supramolecular diblock copolymers: afield-theoretic model and mean-field solution, Macromolecules 40 (2007)693e702.

[41] W.B. Lee, R. Elliott, K. Katsov, G.H. Fredrickson, Phase morphologies inreversibly bonding supramolecular triblock copolymer blends, Macromole-cules 40 (2007) 8445e8454.

[42] P.U. Kenkare, C.K. Hall, Modeling of phase separation in PEG-salt aqueous

X. Zhang et al. / Polymer 78 (2015) 69e8080

two-phase systems, AIChE J. 42 (1996) 3508e3522.[43] P. Koehl, H. Orland, M. Delarue, Beyond the PoissoneBoltzmann model:

modeling biomoleculeewater and waterewater Interactions, Phys. Rev. Lett.102 (2009) 087801e087804.

[44] E.A. Nogovitsin, Y.A. Budkov, Self-consistent field theory investigation of thebehavior of hyaluronic acid chains in aqueous salt solutions, Phys. A 391(2012) 2507e2517.

[45] G.H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers, Ox-ford University Press, Oxford, U.K., 2006.

[46] F. Drolet, G.H. Fredrickson, Optimizing chain bridging in complex block co-polymers, Macromolecules 34 (2001) 5317e5324.

[47] F. Drolet, G.H. Fredrickson, Combinatorial screening of complex block copol-ymer assembly with self-consistent field theory, Phys. Rev. Lett. 83 (1999)4317e4320.

[48] V. Ganesan, G.H. Fredrickson, Field-theoretic polymer simulations, Europhys.

Lett. 55 (2001) 814e820.[49] FFTW, http//www/fftw.org/.[50] J.U. Kim, M.W. Matsen, Droplets of structured fluid on a flat substrate, Soft

Matter 5 (2009) 2889e2895.[51] L. Wang, J. Lin, Q. Zhang, Self-consistent field theory study of the solvation

effect in polyelectrolyte solutions: beyond the PoissoneBoltzmann model,Soft Matter 9 (2013) 4015e4025.

[52] Y. Bohbot-Raviv, Z.-G. Wang, Discovering new ordered phases of block co-polymers, Phys. Rev. Lett. 85 (2000) 3428e3431.

[53] W.-C. Chen, S.-W. Kuo, F.-C. Chang, Self-assembly of an AeB diblock copol-ymer blended with a C homopolymer and a CeD diblock copolymer throughhydrogen bonding interaction, Polymer 51 (2010) 4176e4184.

[54] M.W. Matsen, R.B. Thompson, Equilibrium behavior of symmetric ABA triblockcopolymer melts, J. Chem. Phys. 111 (1999) 7139e7146.