PHYSICAL REVIEW E 88, 042602 (2013)

Microphase separation in comblike liquid-crystalline diblock copolymers

S. K. Mkhonta,1,2 K. R. Elder,3 Zhi-Feng Huang,1 and Martin Grant41Department of Physics and Astronomy, Wayne State University, Detroit, Michigan 48201, USA

2Department of Physics, University of Swaziland, Private Bag 4, Kwaluseni M201, Swaziland3Department of Physics, Oakland University, Rochester, Michigan 48309, USA

4Department of Physics, McGill University, Montreal, Quebec, Canada H3A 2T8(Received 14 January 2013; revised manuscript received 30 July 2013; published 10 October 2013)

The interplay between liquid crystallinity and microphase separation in comblike liquid-crystalline diblockcopolymers is examined via a Brazovskii-type phenomenological model using both analytical and numericalcalculations. For symmetric diblock copolymers we determine a critical electric field that is required to tilt theorientation of the constituent liquid crystals of the polymer side chains in the microphase-separated lamellar state.Such electrically induced reorientation of the liquid-crystal molecules can lead to substantially large changes oflamellar periodicity. Our numerical results show that highly aligned polymer lamellar domains can self-assemblewhen the liquid-crystal ordering precedes microphase separation, and that weak electric fields can be used todirect the self-assembly process due to the dielectric anisotropy of the liquid-crystal side chains. We also findthat phase separation of asymmetric diblock copolymers can coexist with a network of liquid-crystal nematicorientations, with domain morphology depending on the details of copolymer and liquid-crystal coupling.

DOI: 10.1103/PhysRevE.88.042602 PACS number(s): 83.80.Uv, 81.16.Rf, 82.35.Jk, 89.75.Kd

I. INTRODUCTION

Liquid-crystalline diblock copolymers (LC BCPs) forman increasingly important class of self-assembling macro-molecules [1–3] and are of fundamental importance foremerging technological applications such as flexible organicsemiconductors [4], biomaterial engineering [5], and pho-tonic crystal displays [6]. The liquid-crystalline behavior isincorporated into block copolymers (BCPs) usually via twoways. Rodlike homopolymers can be used as one of thebuilding blocks to form a rod-coil LC BCP. Small liquid-crystal(LC) molecules can be linked as side chains or as part ofthe polymeric backbone to form a side-chain LC BCP ora main-chain LC BCP, respectively. The interplay betweenliquid crystallinity and microphase separation in LC BCPsleads to rich behavior of macromolecular self-assembly.

Recent experimental and theoretical studies of LC BCPshave identified some novel microseparated phases that arecompletely different from those observed in regular BCPs.Rod-coil BCPs have been found to assemble into arrowheads,wavy and broken lamellar, zigzag, and plucked phases [7–9].In side-chain LC BCPs the length of the LC segments ison the order of ∼1 nm and the gyration radius of thepolymeric backbone is on the order of ∼10 nm. The LCsegments can order inside the polymer domains, leading tolamellar-within-lamellar and columnar-hexagonal-in-lamellarphases [10–12]. The coupling between liquid crystallinityand copolymer microphase separation can be utilized for thecontrol and manipulation of self-assembly procedure leadingto highly aligned domains [13–15], a process desired in manyblock copolymer technological applications. The amount ofthe liquid-crystal content can also be used to control themorphology and size of the BCP domains [16,17].

The dynamic behavior of LC BCPs is much less studiedand understood as compared to their equilibrium properties[2,3]. A Brazovskii-type free energy functional [18] has beenapplied to address mechanical properties of comblike LCBCPs (see Fig. 1), as will also be adopted in this study.

In addition to self-consistent field theory and Monte Carloor molecular dynamics simulations, Landau-Brazovskii-typephenomenological models have been extensively used todescribe molecular self-assembly in regular BCPs [19–24].Predictions from these models have been found consistent withexperimental observations, especially in the weak segregationlimit [20]. Computer simulations based on the Landau-Brazovskii approach are less computational intensive com-pared to other simulation techniques [25,26]. Being simplerand much faster, this approach often serves as a quick link toexperimental results [21,27].

In this work we utilize the Brazovskii-type model for LCBCPs to examine the coupling between liquid-crystal andcopolymer phases, particularly its effects on the alignmentand morphology of the polymeric domains. In comblike LCBCPs, the side-chain LCs usually adopt a preferred orientationat the domain interface of the phase-separated system [1,28].In the lamellar polymeric state, this interfacial anchoringbehavior favors a uniform nematic direction relative to thelamellar normal. Such LC ordering can direct microphaseseparation, leading to well-aligned domains of a lamellarpattern in symmetric diblock copolymers, and various domainmorphologies in asymmetric LC BCPs. Our simulations alsodemonstrate that the lamellar ordering and alignment processcan be accelerated via applying a modest electric field, due tothe dielectric anisotropy of the side-chain LC molecules. Inaddition, our analytic calculations identify another importanteffect of electric field and liquid crystallinity: i.e., tilting the LCmolecules via an electric field can lead to substantial changesof the lamellar periodicity.

The rest of the article is organized as follows: In Sec. IIwe introduce our phenomenological model. In Sec. III wepresent analytical results that predict the magnitude of electricfields that can switch the direction of LC molecules withinthe bulk of a phase separated system. In Sec. IV we presentnumerical results showing domain evolution after a thermalquench. Last, in Sec. V we conclude by summarizing our

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MKHONTA, ELDER, HUANG, AND GRANT PHYSICAL REVIEW E 88, 042602 (2013)

FIG. 1. (Color online) A schematic illustration of a comblike LCBCP with the LC side chains in the (a) randomly oriented side chainsand (b) uniformly oriented side chains. The shaded disk or ellipseillustrates the approximate shape of the polymeric backbone.

results and suggesting possible problems that can be tackledby the model.

II. FREE ENERGY FUNCTIONAL

The description of comblike LC BCPs combines twowell-established fields: microphase separation in BCPs [20]and liquid-crystal ordering [29,30]. The Landau-Brazovskiifree energy functional for block copolymer ordering consistsof a single order parameter φ(�r,t) that defines the level ofmicrophase separation [31]. On the other hand the Landau-deGennes free energy functional for liquid-crystal orderingconsists of at least two order parameters: a tensorial orderparameter Q(�r,t) that defines the orientational ordering of themesogens and a complex order parameter ρ(�r,t) that describestheir spatial distribution.

The tensorial order parameter for a uniaxial nematic systemis defined by [29]

Qij = dS

d − 1

(ni nj − 1

dδij

), (1)

where d is the spatial dimensionality, S is the magnitude ofthe order parameter, ni is the ith component of n, and δij isthe Kronecker delta function. By definition, the tensor Qij issymmetric and traceless. Its eigenvector n(�r,t) describes thelocal average orientation of the LC molecules at a given po-sition �r at time t and |n(�r,t)| = 1, with orientation invariancen(�r,t) → −n(�r,t). The eigenvalue S(�r,t) describes the extentof fluctuations about n(�r,t). For a uniform alignment, S = 1.In a region with completely random fluctuations, where thereis no preferred orientation, S = 0 (i.e., isotropic phase). Fora system with a slow spatial variation of the nematic directorfield S(r) � const.

Depending on the architecture of the LC BCP chain, theattached LCs may exhibit a smectic ordering where theyorganize themselves in layers. This mesophase is characterized

by a complex order parameter [32,33]

ρ(�r,t) = ρ0eiω, (2)

where modulus ρ0 defines the amplitude of the one-dimensional ordering of the center-of-mass of the moleculeswith a phase ω. The smectic spacing is given by d0 = 2π/qs

with qs = | �∇ω|.In this work we consider a melt of nematic LC BCP

chains where the mesogens are attached with flexible spacesof random lengths (as in Fig. 1). In the nematic state theattached nematogens are homogeneous in space and thereforeare characterized by a single order parameter Q, since ρ = 0.The Landau-de Gennes free energy functional for this systemis then given as [29,34,35]

1

kBTFLdG =

∫d�r

[e2

2QijQji − e3

3QijQjkQki

+ e4

4(QijQji)

2 + L0

2(∂kQij )2

], (3)

where the summation over repeated subscripts is implied. HereT is temperature, kB is the Boltzmann constant, e2 ∼ (1 −TNI /T ) where TNI is the critical temperature for a hypotheticalsecond-order transition to the nematic state, e3 and e4 arematerial dependent parameters that are assumed to be constantnear the transition point, and L0 is the elastic constantfor the nematic phase. In practice, a nematic material hasthree independent elastic constants; hence this approximationdepends on the material details and is usually valid for systemscomposed of small-aspect-ratio nematogens [29].

Substituting Eq. (1) into Eq. (3), the Landau-de Gennes freeenergy functional can be rewritten in terms of two fields S(�r,t)and n(�r,t). In three dimensions this gives [34]

1

kBTFLdG =

∫d�r

(3

4e2S

2 + 1

4e3S

3 + 9

16e4S

4

+ 3

4L0| �∇S|2 + K

2[( �∇ · n)2 + | �∇ × n|2]

), (4)

where K = 4L0S2 is the Frank-Oseen elastic constant and we

have neglected coupling terms including spatial derivatives inboth S and n. For a system where the strength of nematic orderis spatially homogeneous Eq. (4) reduces to

1

kBTFLdG =

∫d�r

{K

2[( �∇ · n)2 + | �∇ × n|2]

}+ const. (5)

In the following we examine this limit to study problems thatare associated with the coupling of the nematic director orien-tation to the polymer domain interfaces in an inhomogeneousstate.

Microphase separation in BCP melts is usually char-acterized by an order parameter φi(�r) = ci(�r) − fi , whichrepresents the deviation of the local volume fraction ci(�r)of ith-type monomers at a position �r from its globalaverage 〈ci(�r)〉 = fi . In A/B diblock copolymers wherethere are two different monomer species, the condition ofincompressibility requires that cA(�r) + cB(�r) = 1. Thus, thissystem can be described by a single order parameter φA(�r) =cA(�r) − fA, as used in the seminal work of Leibler [31]for the Landau-Brazovskii free energy functional expansionof BCPs. Here we utilize the local concentration difference

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MICROPHASE SEPARATION IN COMBLIKE LIQUID- . . . PHYSICAL REVIEW E 88, 042602 (2013)

φ(�r) = cB(�r) − cA(�r) = φB(�r) − φA(�r) + (1 − 2fA), to char-acterize microphase separation. Note that the spatial aver-age 〈φ(�r)〉 = 1 − 2fA, and 〈φ(�r)〉 = 0 represents a symmet-ric BCP melt with equal concentration between the twomonomers.

In a nematic LC BCP melt, the nematic director of the side-chain LCs is naturally coupled to the gradients of the polymerconcentration field. We thus modified the Landau-Brazovskiifree energy functional for regular BCPs [19–23] to include thecoupling terms between �∇φ and n. The modified free energyfunctional for nematic LC BCPs is then given by

FLB

kBT=

∫d�r

{ξ 2

2

(q2

0φ + ∇2aφ

)2 + τ

2φ2 + w

4φ4

+ α

2(n · �∇φ)2 + K

2[( �∇ · n)2 + | �∇ × n|2]

}, (6)

where

∇2a = ∇2 + a1(n · �∇)2 + a2|n × �∇|2, (7)

and the gradient operators are defined as (n · �∇)2 = ni nj ∂i∂j

and |n × �∇|2 = εijkεilmnj nl∂k∂m, where the subscripts referto components in a Cartesian coordinate system and εijk is theLevi-Civita symbol.

For parameters given in Eq. (6), those for regular symmetricBCP melts are known as [36]

q0∼= 1.95/Rg, τ = 2Nρm(χc − χ ), (8)

χc = 10.49/N, ξ 2 = 1.5ρmc2R2g/q

20 , (9)

where N is the polymerization index and χ ∼ 1/T is theFlory-Huggins interaction parameter. Assuming the monomersize is b, the radius for a Gaussian chain is R2

g∼= 1/6Nb2, and

the chain density in the incompressible melt is ρm = 1/(Nb3).The dimensionless ratios w/ρm and parameter c are of orderunity, and in this work we adopt the approximation of w/ρm =1 and c = 1 [21,36]. Note that for nematic LC BCPs, τ andw are also dependent on the strength of the nematic orderS. The contribution of S on the value of τ is essential inunderstanding the effects of nematic order on the transitionpoint of microphase separation. Such studies are available inthe literature for the case of liquid-crystal/polymer mixtures[37] and rod-coil block copolymers [38–40].

The rest of the parameters in Eq. (6), i.e., α, a1, anda2, determine the coupling strength between n and thegradients of φ(�r) and reflect the amount of the LC content.The positive dimensionless parameters a1 and a2 define theanisotropy of the chain conformation as a result of isotropic-to-nematic transition. The stretching of polymer chains in anematic environment has been observed experimentally fornematic liquid-crystalline polymers [41]. The stretched chainis experimentally characterized by a molecular anisotropyaspect ratio κ , and its relation to a1 and a2 will be shownbelow. Note that the coupling due to a1 and a2 does notfix the relative orientation between the nematic director n

and �∇φ, but changes the molecular length scales of thepolymers [42]. The parameter α breaks this internal rotationalsymmetry and describes the tendency of the mesogens toadopt a preferred orientation at the domain interface of the

microphase-separated polymer chains. This tendency has beenobserved experimentally [1].

A dimensional analysis of Eq. (6) gives

α � hξ 2q20 = 1.5hρmc2R2

g, (10)

where h is a dimensionless proportionality constant. This resultis similar to the derivations of Carton-Leibler [43] for theanchoring energy in polymer-polymer interfaces. The signof α determines whether the LCs will favor a parallel orperpendicular orientation relative to the polymer interface. Inexperiments, the precise anchoring condition depends on thelength of the flexible spacers that link the nematogens to thepolymer chain [1,44]. Note that more details of coupling canbe obtained via self-consistent field equations describing therandom-walk type statistics of chain configurations [45].

In the nematic state with the nematic direction along the x

axis, n = x, Eq. (6) reduces to

F

kBT=

∫d�r

[ξ 2

2

(q2

0φ + ∂xxφ + κ2∇2⊥φ

)2

+ α

2(∂xφ)2 + τ

2φ2 + w

4φ4

], (11)

where we have rescaled ∇2a → ∇2

a/(1 + a1), ∇2⊥ = ∂yy + ∂zz

and the anisotropic ratio

κ =√

1 + a2

1 + a1, (12)

where a1 and a2 are related to the graft density of the side-chainLCs [46]. Since in regular BCPs Rg ∼ 1/q0, the first termin Eq. (11) implies that the free energy favors the gyrationradius R

‖g ∼ 1/q0 and R⊥

g ∼ κ/q0 along and perpendicularto the nematic direction, respectively. Thus κ represents themolecular aspect ratio of the polymeric blocks in the nematicstate. In this work we consider κ > 1, for which the polymericbackbone is stretched perpendicular to the nematic direction.This is the usual conformation of LC BCPs with side-chainLC molecules [10,11].

In two dimensions where n = x cos θ + y sin θ , Eq. (6)reduces to

F (φ,θ )

kBT=

∫d2r

[α

2(cos θ∂xφ + sin θ∂yφ)2

+ ξ 2

2

(q2

0φ + A∂xxφ + B∂yyφ + C∂xyφ)2

+ τ

2φ2 + w

4φ4 + K

2| �∇θ |2

⎤⎦, (13)

where A = 1 + (κ2 − 1) sin2 θ , B = 1 + (κ2 − 1) cos2 θ , andC = (1 − κ2) sin 2θ . The corresponding model dynamics isassumed to be dissipative, driven to minimize the free energy,and nonconservative for θ and conservative for φ [47], i.e.,

∂θ

∂t= −Mθ

δF

δθ,

∂φ

∂t= Mφ∇2 δF

δφ, (14)

where Mθ and Mφ are the mobilities of θ (�r,t) and φ(�r,t) fields,respectively.

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MKHONTA, ELDER, HUANG, AND GRANT PHYSICAL REVIEW E 88, 042602 (2013)

III. ANALYTICAL CALCULATIONS

It is of theoretical and experimental interest to characterizethe elastic interaction between the BCP concentration fieldφ(�r) and the nematic director n(�r). The conventional approachis to control one of the fields using an anisotropic externalforce. Here we use a uniform electric field to manipulate thedirection of n(�r) in the lamellar ordered state. In this way wecan also calculate the magnitude of the electric field requiredto change the lamellar spacing.

A. Electric fields in LC-BCPs

The coupling of an electrostatic field �E to both the polymercomposition field and the nematic director has been addressedin previous studies [21,29,30,48]. When an electrostatic fieldcouples to φ(�r) via a dielectric contrast mechanism, the relatedfree energy contribution can be expressed as [21,48]:

1

kBTFdielec = β

2

∫d �q( �E · q)2φ�qφ−�q, (15)

where φ�q is the Fourier transform of the order parameterφ(�r), β = (εA − εB)2/{4kBT [εAfA + εB(1 − fA)]}, and εA

and εB are the static dielectric constants for blocks A andB, respectively. Fdielec is derived by expanding the dielectricconstant of the inhomogeneous melt and solving Maxwell’sequations [48].

The coupling of the nematic director to the electrostaticfield can be derived in a similar way [49]. This leads to thefree energy contribution [29,30,49]

1

kBTFani = −�

2

∫d�r(n · �E)2, (16)

where � = ε0�ε/(kBT ), ε0 is the permittivity of free space,and �ε = ε‖ − ε⊥, where ε‖ and ε⊥ are the dielectric constantsmeasured parallel and perpendicular to �E, respectively. Notethat the dielectric contrast mechanism operates at a lengthscale of 10–100 nm (polymer microdomain size) while thedielectric anisotropy mechanism operates at a length scale of5–10 A (dimension of the LC molecules). This means that theLCs can enhance the electric response of the copolymer melt.

B. In-plane electric field

We consider an electric field, �E = Ex, that is appliedto the lamellar domains of an ordered LC BCP melt withn = x cos θ + y sin θ (as depicted in Fig. 2). In the weaksegregation limit, the monomer concentration variation canbe approximated by a one-mode approximation as

φ(y) = A cos(qy), (17)

where 2π/q corresponds to the lamellar thickness and A is theamplitude of the concentration wave.

Substituting Eq. (17) into Eq. (6) and including the electricfield contribution, we obtain the free energy per unit volume(in units of kBT ):

f = [τ + αq2 sin2 θ + ξ 2

(Bq2 − q2

0

)2]A2/4

+ 3wA4/32 − �E2 cos2 θ/2. (18)

FIG. 2. (Color online) A schematic illustration of liquid-crystalalignment by an external electric field �E. (a) A nematic system ofLC molecules with the direction of the electric dipole moment ofeach molecule indicated by an arrow. Left: LC molecules with adipole moment parallel to their molecular axis (�ε > 0) tend toorient parallel to �E. Right: LC molecules with a dipole momentperpendicular to their molecular axis (�ε < 0) tend to orientperpendicular to �E. (b) A system of LC BCPs in the lamellarphase with the side-chain LC molecules having negative dielectricanisotropy �ε < 0. An external electric field applied parallel to thelamellar normal will tend to rotate the LC molecules and thus changethe stretch direction of the copolymer chains.

Minimizing the above free energy density with respect to A

and q, we have

A = 2√[ − τ − αq2 sin2 θ − ξ 2

(q2

0 − Bq2)2]

/3w (19)

and

q =√

q20

B − α sin2 θ

2ξ 2B2. (20)

Substituting Eqs. (19) and (20) into Eq. (18), we can determinethe free energy density as

f (θ ) = − 1

6w

(τ + q2

0α sin2 θ

B − α2 sin4 θ

4ξ 2B2

)2

− 1

2�E2 cos2 θ. (21)

When E = 0 the free energy minimum corresponds to θ =0 [18]. However, when E �= 0 the orientation of the LCmolecules that minimizes the free energy depends on E and is

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MICROPHASE SEPARATION IN COMBLIKE LIQUID- . . . PHYSICAL REVIEW E 88, 042602 (2013)

determined by the equation

1

B

[τ + α sin2 θ

B

(q2

0 − α sin2 θ

4ξ 2B

)]

×(

1 + κ2 − 1

B sin2 θ

)(q2

0 − α sin2 θ

2ξ 2B

)= 3w

2α�E2.

(22)

We can obtain a simple expression of θ (E) in the limit of weakor strong electric field coupling, as detailed below.

1. Small electric field limit

In the weak electric field limit the LC molecules are slightlytilted from their zero electric field orientation, and thus we canexpect θ (and sin2 θ ) to be small. The free energy density f (θ )can be expanded as

f = − τ 2

6w− �E2

2+

(�E2

2− τν

3

)sin2 θ

+ ν

6

[2τ (κ−2 − 1) + wν

(τ

2ξ 2q40

− 1

)]sin4 θ

+O(sin6 θ ), (23)

where ν = αq20/(wκ2). The competition between the effects

of the external electric field and the LC interfacial anchoring isincorporated in the sin2 θ term of Eq. (23), leading to a criticalvalue of electric field

Es =√

2ντ

3�. (24)

Note that τ < 0 and thus the tilting of the side-chain mesogensby the electric field will be effective only when � < 0. For E

slightly above Es the free energy is minimized when

sin2 θ = (E/Es)2 − 1

2(1 − κ−2) + αq20κ−2

[1/τ − 1/

(2ξ 2q4

0

)] . (25)

Thus, when E > Es the side-chain LC molecules with anegative dielectric anisotropy will be tilted towards a directionperpendicular to the applied electric field. Note that Eqs. (24)and (25) can also be obtained from the general result (22) viaan expansion up to first order of sin2 θ .

2. Large electric field limit

In the large electric field limit the electrostatic contributionin Eq. (18) dominates, and the LC molecules with � < 0 areexpected to tilt towards θ = π/2. Thus cos2 θ would be small,and the system free energy density can be expanded as

f = − τ 2l

6w+

(Hτl

3w− �E2

2

)cos2 θ

+ 1

6w

{[(−3κ2 − 2)H + αq2

0κ4]τl − H 2

}cos4 θ

+O(cos6 θ ), (26)

where τl = τ + αq20 − (α/2ξ )2 and H = αq2

0κ2[1 −α/(2ξ 2q2

0 )]. In this case the free energy is minimized byθ = π/2 (cos θ = 0) if the coefficient of the cos2 θ termbecomes negative. This leads to another critical value of

electric field

El =√

2Hτl

3w�. (27)

It is interesting to note that in the small α limit, El � κ2Es .For field strength just below El , f is minimized when

cos2 θ = 3w�(E2

l − E2)/

2

τl

[(3κ2 − 2)H − αq2

0κ4] − H 2

, (28)

which can also be derived via directly expanding Eq. (22) upto O(cos2 θ ). This implies that the liquid crystal moleculesbegin to rotate away from the direction perpendicular to thefield when E < El .

C. Electric field effect on lamellar spacing

The magnitude of critical electric fields Es and El canbe estimated for a typical system as follows. Representativevalues of model parameters τ , κ , and �ε are available in theliterature [21,30,41,50–52]. To get an estimate of α = hξ 2q2

0[see Eq. (10)] we consider a system (as illustrated in Fig. 2)where E is tuned from values slightly less than Es to E > El ,so that the side-chain LC molecules can be tilted from θ = 0to θ = π/2. The corresponding change of the lamellar spacingis given by

δm = λ0 − λπ/2

λ0= 1 − κ−1

(1 − h

2

)−1/2

, (29)

where λθ=0,π/2 = 2π/q as given in Eq. (20). For h → 0, δm →1 − κ−1, i.e., the copolymer chain anisotropy determines themaximum change of the lamellar spacing. As h (α) increases,δm decreases and becomes zero at a critical value of h = 2(1 −1/κ2). This means the allowed range of h is [0, 2 − 2/κ2]. Thisconstraint is consistent to the fact that for a Gaussian polymerchain where κ = 1, there is no effective microscopic couplingbetween LCs’ director and block copolymer interfaces.

For typical experimental values for a BCP melt: copoly-mer chain volume vp = 500 nm3, T = 400 K, Nχ = 11[21,50,51], chain anisotropy of the side-chain LC polymerκ = 1.6 [41], LC dielectric anisotropy �ε = −10 [53] andh ∼ 0.1 � 2(1 − 1/κ2), we obtain that Es = 1.8 V/μm andEl = 4.4 V/μm. Figure 3 shows the electrically inducedreorientation of the LC molecules and the associated changeof the lamellar spacing. As can be seen in this figure, a fullrotation of the LC side chains due to the applied field leadsto a decrease of the lamellar spacing by about 40%. Thus inthe small α limit the domain size of the LC BCPs systemis highly tunable. LC BCPs exhibiting this tunable behaviorwould be of large importance for applications in photoniccrystal technologies, as has been explored recently in variousresearch that showed very promising results [54–56].

IV. NUMERICAL RESULTS

For convenience it is useful to rewrite the total free energyfunctional Eq. (6) in dimensionless units, i.e.,

�r ′ = q0�r, τ = τ

ξ 2q40

, ψ =√

w

ξ 2q40

φ. (30)

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MKHONTA, ELDER, HUANG, AND GRANT PHYSICAL REVIEW E 88, 042602 (2013)

FIG. 3. Orientation of the liquid crystal molecules and the lamellar wave vector as a function of applied electric field. The solid line is from thefull one mode solutions determined numerically from Eqs. (18)–(20). The dashed lines are the approximate solutions given in Eqs. (25) and (28).

Equation (13) then becomes

F0 = 1

2

∫ddr ′

{(ψ + ∇2

aψ)2 + τψ2 + ψ4

2

+h(n · �∇ψ)2 + K[( �∇ · n)2 + | �∇ × n|2]

}, (31)

where h = α/(ξ 2q20 ), K = wK/(ξ 4q6

0 ), and F0 =wξ−4qd−8

0 F/kBT is the dimensionless system free energyin the absence of an external field. Similarly the contributionof an applied electrostatic field [Eqs. (15) and (16)] can beexpressed in the following dimensionless form:

Fdielec = β

2E2

0

∫d �q ′[x · q]2ψ�q ′ψ−�q ′ (32)

and

Fani = − �

2E2

0

∫d2r ′(n · x)2, (33)

where E0 = | �E|/Es , x is the direction of the electric field,β = βE2

s /(ξ 2q20 ), � = w�E2

s /(ξ 4q80 ), and ψ�q ′ is the Fourier

transform of ψ(�r ′).We also express time in dimensionless unit, i.e., t ′ =

Mφξ 2q60 t , and thus the dynamic equation (14), including the

electric field contribution, becomes

∂θ

∂t ′= −δF0

δθ− �

2E2

0 sin(2θ ),(34)

∂ψ

∂t ′= ∇2 δF0

δψ+ βE2

0∂2xψ.

Here we have assumed that the time scale for θ and ψ

are the same, i.e. Mφ = ξ 2q20w−1Mθ . For convenience the

dimensionless time and length scales are simply referredwithout the primes in the text below.

We numerically solve Eq. (34) in two dimensions by usingan explicit Euler scheme, with details given in the Appendix.The steps in the discretized space and time are chosen as�x = �y = 1 and �t = 0.001. The initial conditions for anisotropic homogeneous state are defined by the concentrationfield ψ(�r) having small random fluctuations superposed on itsglobal average ψ0 and the local nematic director completelyrandomly oriented, i.e., θ (�r,0) ∈ [−π/2,π/2]. For a nematichomogeneous state, the nematic director is assumed to beuniform, that is, θ (�r,0) ∼= const. We performed the numericalsimulations on a square grid with 2562 points, using periodicboundary conditions on both directions.

A. Phase separation in symmetric LC BCPs

We first consider microphase separation in nematic LCBCPs where fA = 1/2 (〈ψ(�r)〉 = 0). This limit represents asymmetric LC BPCs melt where the A/B blocks have thesame volume factions. We set the nematic elastic constantK = 2, h = 0.5, and κ = √

2. Note that here h is sufficientlylarge but less than its physical limit 2(1 − 1/κ2) as discussedin the previous section. For |τ | � 1 our calculations remainwithin the weak segregation limit [19], and we found consistentresults for τ ∈ [−0.05,−0.5]. The results given below forthe symmetric LC BCPs corresponds to τ = −0.15. Figure 4shows the dynamics of microphase separation in a melt aftera thermal quench from an isotropic homogeneous state [seeFig. 4(a)–4(c)] and from a nematic homogeneous state [seeFig 4(d)–4(f)]. Note that the equilibrium state for both systemsis a perfectly aligned lamellar phase. Comparing the twocases, we observe that when nematic order is imposed prior tomicrophase separation, the emerging domains are well alignedalong the nematic direction. The alignment process can be

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MICROPHASE SEPARATION IN COMBLIKE LIQUID- . . . PHYSICAL REVIEW E 88, 042602 (2013)

FIG. 4. (Color online) Time evolution of the concentration field ψ in symmetric LC-BCPs. The ordering process is initiated from anisotropic homogeneous state in (a)–(c) and a nematic homogeneous state in (d)–(f). Simulation parameters are set as h = 0.5, κ = √

2, andK = 2.0. Only a quarter of the simulation grid is shown here.

understood by examining the orientation of the LCs around aregion where the polymer domains are curved.

Figure 5 shows that the nematic director is nonuniformaround a lamellar defect. This deformation is penalized by thenematic elastic energy, and thus in the nematic state curveddomains are disfavored. Our numerical results also show thatdomain evolution from an isotropic homogeneous state to analigned lamellar state is quite slow. This suggests that an

external field is required to accelerate the domain alignmentprocess. Similar results were observed in the dynamics ofrod-coil BCPs [39].

B. Domain alignment in an electric field

Anisotropic external fields such as electric fields can beapplied to BCP melts to produce highly aligned microstructure.

FIG. 5. (Color online) Coupled defects obtained from a self-assembling LC BCP system. The image on the left shows the concentrationfield ψ(�r,t), and the right panel gives the corresponding profile of the LC director n(�r,t).

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MKHONTA, ELDER, HUANG, AND GRANT PHYSICAL REVIEW E 88, 042602 (2013)

FIG. 6. (Color online) Simulated morphologies of the concentration field ψ(�r) in the presence of an electric field �E = E0x at time t = 300after a thermal quench from an isotropic homogeneous state. (a) and (b) show the systems of regular BCPs for (a) E0 = 0.03 and (b) E0 = 1.Systems of LC BCPs with side-chain LCs having a positive dielectric anisotropy � = 10 are given in (c) for E0 = 0.03 and (d) for E0 = 0.06.The cases of LC BCPs with side-chain LCs having a negative dielectric anisotropy (� = −10) are shown in (e) for E0 = 0.03 and (f) forE0 = 0.06. The direction of �E is indicated in the figures. Only a quarter of the simulated sample size is shown.

There are several recent theoretical and experimental studieson the effect of electric fields in regular BCPs [21,24,48,57,58],in rod-coil BCPs [39,40], and in semiflexible LC BCPs [59].Here we consider the two contributions of the electric fieldto the system free energy as given in Eqs. (32)and (33). Thedielectric contrast mechanism will favor �∇ψ to be perpendic-ular to the electric field direction. The dielectric anisotropymechanism will dictate the direction of the nematic directorwith respect to �E. As realized in the previous section, thelater mechanism will indirectly specify the orientation of the

polymer domains. Hence the two electric field contributionscompete during the alignment process depending on the sign ofthe dielectric anisotropy. We explore this behavior in numericalsimulations.

We study microphase separation in three different systems:(1) LC BCPs with � = 10, (2) LC BCPs with � = −10,and (3) regular BCPs (where h = 0 and κ = 1). We assumethat these three systems have similar A/B monomers witha dielectric contrast β = 0.1. We also assume that the LCBCP chains have the same molecular architecture: κ = √

2

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MICROPHASE SEPARATION IN COMBLIKE LIQUID- . . . PHYSICAL REVIEW E 88, 042602 (2013)

-1

-0.5

0

0.5

1

0 100 200 300

Ave

rage

dom

ain

alig

nmen

t P(t

)

Time t

FIG. 7. (Color online) Time dependence of average microdomainalignment P (t) during microphase separation of BCPs in the presenceof an electric field �E = E0x. The curves shown from top to bottomcorrespond respectively to regular BCPs at E0 = 1.0 (open triangles),LC BPCs with � = 10 at E0 = 0.06 (solid squares) and at E0 = 0.03(open squares), regular BCPs at E0 = 0.03 (solid triangles), and LCBPCs with � = −10 at E0 = 0.03 (open circles) and at E0 = 0.06(solid circles).

and h = 0.5. Simulation results given in Fig. 6 show that theLC BCP melt can be aligned by modest electric fields that aremuch smaller than that for regular BCPs of same dielectriccontrast. An electric field with E0 = 0.06 produces highlyaligned domains in the LC BCP melts that are comparableto those of regular BCPs at E0 = 1. Similar results can be

obtained by monitoring the average domain alignment:

P (t) =∫

d2r[(∂yψ)2 − (∂xψ)2]∫d2r| �∇ψ |2 , (35)

as shown in Fig. 7.These results indicate that the electric field favors a

perpendicular lamellar alignment in the system where theside-chain LCs have a negative dielectric anisotropy � [seeFigs. 6(e), 6(f), and 8(c)]. This state is unusual since thealternating lamellae are perpendicular to �E, and thus there is anenergy cost due to the dielectric contrast mechanism [Eq. (33)].Note that since � < 0, the LCs are also oriented perpendicularto the electric field, and therefore this state minimizes theLC anchoring energy α|n · �∇ψ |2 for a sufficiently largeh. However, in the small α limit, the dielectric contrastmechanism dominates, and the lamellar domains align alongthe electric field direction as shown in Fig. 8(a).

Figure 8 also shows the circularly average structure factorS(q) for the two types of alignment states. In the perpendicularalignment state the characteristic wavenumber of the systemis qc ≈ 1/κ . In the parallel state the lamellar domains arecompressed and qc ≈ 1. Note that this behavior is consistentwith our analytical calculations, i.e., Eq. (29) which showsthat the lamellar wavenumber can change up to a factorof (1 − κ−1) when the relative orientation between the LCsdirector and the lamellar interface varies by 90◦. The tran-sition between parallel-to-perpendicular states, as illustrated

FIG. 8. (Color online) Evolution of LC BCPs with β = 0.1, � = −10, and κ = √2 in the presence of an electric field of magnitude

E0 = 0.1 (direction shown by an arrow), after a thermal quench from an isotropic homogeneous state. The concentration profiles ψ(�r,t) takenat time t = 300 are shown for systems with (a) h = 0.01, (b) h = 0.07, and (c) h = 0.5. (d) Structure factor for systems of different values ofh taken at time t = 300. (e) Time dependence of the average microdomain alignment P (t) for different values of h.

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MKHONTA, ELDER, HUANG, AND GRANT PHYSICAL REVIEW E 88, 042602 (2013)

in Fig. 8, depends on three material parameters: �, β, andh. The nematic director in all the results shown in Fig. 8is perpendicular to �E since the side-chain LCs are stronglycoupled to the electric field (|�| is large). Thus the parallelstate is only possible in the small h limit. For larger h theA/B monomer interface orientation is strongly coupled to the

nematic director, and thus the system prefers a perpendicularalignment.

C. Phase separation of asymmetric LC BCP

An asymmetric melt is obtained when fA �= 1/2 (ψ0 =1 − 2fA �= 0). For ψ0 < 0 the B monomers are of minority

FIG. 9. (Color online) Morphologies of a phase separating asymmetric LC BCP system taken at time t = 300. In (a) K = 0.1, h = 0.5, andκ = 1.4, in (b) K = 2, h = 0.5, and κ = 1.4, and in (c) K = 2, h = 0.1, and κ = 2. Left panels give the distribution of the concentration fieldψ(�r), while right panels show the corresponding profile of the nematic director n(�r). These figures show only a small portion of the simulationgrid.

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MICROPHASE SEPARATION IN COMBLIKE LIQUID- . . . PHYSICAL REVIEW E 88, 042602 (2013)

phase and usually segregate into circular domains below theorder-disorder transition point in regular BCPs. In LC BCPsthe behavior would be very different since curved interfacesare disfavored due to the competition between the nematicelastic energy and domain interfacial energy. In rod-coil BCPsit has been observed that the competition leads to ellipsoidaldomains or to the stabilization of the lamellar phase in theasymmetric quench [39,40].

Here we explore the different morphologies that can occurin nematic LC BCPs system with τ = −0.25 and ψ0 = −0.2in the absence of electric fields. In Fig. 9 we demonstratethe effect of the increasing nematic constant on the LCBCP morphologies. In the small K limit circular domainsof BCPs are obtained, coexisting with a network of nematicdefects as shown in Fig. 9(a). In this limit the minimizationof the domain interfacial energy dominates the microphaseseparation process. On the other hand, when the value ofK increases the domains merge to form stripes parallel tothe nematic orientation field. In this case the microphaseseparation process is dominated by the minimization of thenematic deformation energy. Similar results have been foundin rod-coil BCPs [39]. We also find that in the small h limitellipse-shaped domains can coexist with perfect nematic order(uniformly aligned nematic director), as shown in Fig. 9(c).This shows that the relative orientation between the LCmolecules and the polymer domain interfaces can changewhen the Frank-Oseen energy dominates over the contributionof LC anchoring energy. In Figs. 9(a) and 9(b) the nematicdirectors tend to be parallel to the domain interface, but theperpendicular orientation occurs for the case of Fig. 9(c) withsmall enough LC anchoring energy.

V. CONCLUSIONS

Microphase separation in comblike LC BCPs exhibits newfascinating behavior due to the strong influence of side-chainLC molecules. Our study demonstrates that one can controlthe morphology, size, and orientation of the microphase-separated domains via the LC molecules. In the lamellar phaseelectrically induced tilting of the LC molecules relative to thelamellar plane can lead to large changes in the lamellar spacing.The critical electric field required to tilt the LCs within thelamellar domains is also determined. The control of lamellardomain size using electric fields has been achieved in regularBCPs [60]. However one needs strong electric fields, on theorder of ∼10 kV/cm. We anticipate that comblike LC BCPswould be ideal in these experiments because of the dielectricanisotropy of the LC molecules and the possible tilting of theanisotropic polymer chains as demonstrated in this study (see,e.g., Fig. 3).

We have also studied the microstructure formation inself-assembling LC BCP systems that are thermally quenchedfrom a disordered state, and demonstrated that relatively weakelectric fields can well align the polymer microstructure. Inour numerical simulations, when both LC and BCP orderingprocesses are initiated simultaneously from the homogeneousisotropic state, a slow process of lamellar domain alignmentis observed. On the other hand, in LC BCP systems where theLC molecules have a large dielectric anisotropy, a modestelectric field can direct the LC ordering; as a result the

emerging microphase-separated copolymer domains will bewell aligned along the nematic direction. We have alsoexamined microphase separation in asymmetric LC BCPs,for which the competition between domain interfacial energyand nematic elasticity leads to the creation of various domainmorphologies of the minority polymer phase, and also anetwork of liquid-crystal nematic defects. All these resultshave indicated the important role played by the couplingbetween liquid-crystalline and copolymer microphases, inparticular on achieving the structural control and long-rangeordering of microstructure self-assembly in macromolecules.

ACKNOWLEDGMENTS

K.R.E. acknowledges support from NSF under Grant No.DMR-0906676. Z.-F.H. acknowledges support from NSFunder Grant No. DMR-0845264. M.G. was supported by theNatural Sciences and Engineering Research Council of Canadaand by le Fonds quebecois de la recherche sur la nature et lestechnologies.

APPENDIX: NUMERICAL METHOD

The dynamics of the model is governed by Eq. (34), asdetermined by the minimization of the free energy functionalEq. (31). The functional derivatives of the two fields are givenas follows:

δF0

δψ= τψ + [L1 + ∂xx(AL1) + ∂yy(BL1) + ∂xy(CL1)]

−h[∂x(L2 cos θ ) + ∂y(L2 sin θ )] (A1)

and

−δF0

δθ= K∇2θ + L1(κ2 − 1)[(∂xxφ − ∂yyφ) sin 2θ

− 2 cos 2θ∂xyψ] + αL2[sin θ∂xφ − cos θ∂yψ],

(A2)

where L1 = (ψ + A∂xxψ + B∂yyψ + C∂xyψ) and L2 =cos θ∂xψ + sin θ∂yψ .

In our numerical algorithm for solving the spatiotemporalevolution of the system, a simple Euler method is adopted,where we replace ∂tφ(�r,t) in Eq. (14) by (ψm+1

i,j − ψmi,j )/�t ,

∂xxψ(�r,t) by

∂xxφmi,j = 1

(�x)2

[ψm

i+1,j + ψmi−1,j − 2ψm

i,j

](A3)

and ∇2ψ(r,t) by

∇2ψmi,j = 1

(�x)2

(2

3

∑nn

+ 1

6

∑nnn

−10

3

)ψm

i,j , (A4)

which include contributions from both nearest neighbors (nn)and next-nearest neighbors (nnn) [61,62]. The time index m

refers to the discretized time tm = m�t . Likewise the spatialindex (i,j ) corresponds to position (i�x,j�x).

For the dynamics of the nematic director angle θ , in ournumerical calculation of Eq. (14) we replace ∂tθ (�r,t) by

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(θm+1i,j − θm

i,j )/�t , and ∇2θ (r,t) by

∇2θmi,j = 1

2�x2

[2

3

∑nn

sin 2(θmi,j − θm

nn

)

+ 1

6

∑nnn

sin 2(θmi,j − θm

nnn

)], (A5)

where θmi,j is the angle between the liquid-crystal director

vector and the x-axis at site (i,j ) and time index m. TheLaplacian of the orientational field is expressed as a sinu-soidal function to ensure the degeneracy of angular values;for example sites with θi,j = pπ for p = 0,±1,±2, . . . areequivalent.

[1] C. O. Osuji, J. T. Chen, G. Mao, C. K. Ober, and E. L. Thomas,Polymer 41, 8897 (2000).

[2] B. D. Olsen and R. A. Segalman, Mater. Sci. Eng. R 62, 37(2008).

[3] Q. Wang, Soft Matter 7, 3711 (2011).[4] I. Mcculloch, M. Heeney, C. Bailey, K. Genevicius, I. Mcdonald,

M. Shkunov, D. Sparrowe, S. Tierney, R. Wagner, W. Zhang,M. L. Chabinyc, R. J. Kline, M. D. Mcgenee, and M. F. Toney,Nat. Mater. 5, 328 (2006).

[5] Y.-B. Lim, K.-S. Moon, and M. Lee, J. Mater. Chem. 18, 2909(2008).

[6] C. Osuji, C.-Y. Chao, I. Bita, C. K. Ober, and E. L. Thomas,Adv. Funct. Mater. 12, 753 (2002).

[7] J. T. Chen, E. L. Thomas, C. K. Ober, and G. Mao, Science 273,343 (1996).

[8] V. Pryamitsyn and V. Ganesan, J. Chem. Phys. 120, 5824 (2004).[9] J. Gao, W. Song, P. Tang, and Y. Yang, Soft Matter 7, 5208

(2011).[10] J. Ruokolainen, M. Saariaho, O. Ikkala, G. ten Brinke, E. L.

Thomas, M. Torkeli, and R. Serimaa, Macromolecules 32, 1152(1999).

[11] A. J. Soininen, I. Tanionou, N. ten Brummelhuis, H. Schlaad,N. Hadjichristidis, O. Ikkala, J. Raula, R. Mezzenga, andJ. Ruokolainen, Macromolecules 45, 7091 (2012).

[12] K. K. Tenneti, X. Chen, C. Y. Li, X. Wan, X. Fan, Q. Zhou,L. Rong, and B. S. Hsiao, Soft Matter 4, 458 (2008).

[13] Y. Tao, H. Zohar, B. D. Olsen, and R. Segalman, Nano Lett. 7,2742 (2007).

[14] M. Gopinadhan, P. W. Majewski, and C. O. Osuji,Macromolecules 43, 3286 (2010).

[15] B. O. Olsen, X. Gu, A. Hexemer, E. Gann, and R. A. Segalman,Macromolecules 43, 6531 (2010).

[16] Y. Tao, B. D. Olsen, V. Ganesan, and R. A. Segalman,Macromolecules 40, 3320 (2007).

[17] E. Verploegen, T. Zhang, Y. S. Jung, C. Ross, and P. T.Hammond, Nano Lett. 8, 3434 (2008).

[18] S. K. Mkhonta, K. R. Elder, and M. Grant, Eur. Phys. J. E 32,349 (2010).

[19] I. W. Hamley and V. E. Podneks, Macromolecules 30, 3701(1997).

[20] I. W. Hamley, Development in Block Copolymer Science andTechnology (John Wiley and Sons, Chichester, 2004).

[21] Y. Tsori, F. Tournilhac, D. Andelman, and L. Leibler, Phys. Rev.Lett. 90, 145504 (2003).

[22] Z.-F. Huang, F. Drolet, and J. Vinals, Macromolecules 36, 9622(2003).

[23] Z.-F. Huang and J. Vinals, Phys. Rev. E 69, 041504 (2004).[24] X.-F. Wu and Y. A. Dzenis, Phys. Rev. E 77, 031807 (2008).

[25] M. Pinna, A. V. Zvelindovsky, S. Todd, and G. Goldbeck-Wood,J. Chem. Phys. 125, 154905 (2006).

[26] M. Pinna and A. V. Zvelindovsky, Soft Matter 4, 316 (2008).[27] M. Pinna, L. Schreier, and A. V. Zvelindosky, Soft Matter 5, 970

(2009).[28] A. Anthamatten and P. T. Hammond, J. Polym. Sci. B: Polym.

Phys. 39, 2671 (2001).[29] P. G. de Gennes and J. Prost, Physics of Liquid Crystals (Oxford

University Press, Oxford, 1995).[30] M. Kleman and O. D. Lavretovich, Soft Matter Physics:

An Introduction (Springer-Verlag, New York, 2003).[31] L. Leibler, Macromolecules 13, 1602 (1980).[32] P. Biscari, M. C. Calderer, and E. M. Terentjev, Phys. Rev. E 75,

051707 (2007).[33] P. K. Mukherjee, H. Pleiner, and H. R. Brand, Eur. Phys. J. E 4,

293 (2001).[34] M. Ravnik and S. Zumer, Liq. Cryst. 36, 1201 (2009).[35] Z. Bradac, S. Kralj, and S. Zumer, J. Chem. Phys. 135, 024506

(2011).[36] Y. Tsori and D. Andelman, Macromolecules 35, 5161 (2002).[37] A. M. Lapena, S. C. Glotzer, S. A. Langer, and A. J. Liu, Phys.

Rev. E 60, R29 (1999).[38] M. Motayama, N. Yamazaki, M. Nonomura, and T. Ohta,

J. Phys. Soc. Jpn. 72, 991 (2003).[39] N. Yamazaki, M. Motayama, M. Nonomura, and T. Ohta,

J. Chem. Phys. 120, 3949 (2004).[40] T. Ohta, N. Yamazaki, M. Motoyama, K. Yamada, and

M. Nonomura, J. Phys.: Condens. Matter 17, S2833 (2005).[41] M. D. Kempe, N. R. Scruggs, R. Verduzo, J. Lal, and J. A.

Kornfield, Nat. Mater. 3, 177 (2004).[42] S. K. Mkhonta, D. Vernon, K. R. Elder, and M. Grant, Europhys.

Lett. 101, 56004 (2013).[43] J.-P. Carton and L. Leibler, J. Phys. France 51, 1683 (1990).[44] H. Yu, T. Kobayashi, and H. Yang, Adv. Mater. 23, 3337 (2011).[45] W. Zhao, T. P. Russell, and G. M. Grason, J. Chem. Phys. 137,

104911 (2014).[46] R. Wang and Z.-G. Wang, Macromolecules 43, 10096 (2010).[47] N. Provatas and K. R. Elder, Phase Field Methods in Material

Science and Engineering (Wiley-VCH, Weinheim, 2010).[48] A. V. Kyrylyuk, A. V. Zvelindovsky, G. J. A. Sevink, and J. G.

E. M. Fraaije, Macromolecules 35, 1473 (2002).[49] A. Onuki, J. Phys. Soc. Jpn. 73, 511 (2004).[50] K. Almdal, J. H. Rosedale, F. S. Bates, G. D. Wignall, and

G. H. Fredrickson, Phys. Rev. Lett. 65, 1112 (1990).[51] F. S. Bates, W. Maurer, T. P. Lodge, M. F. Schulz, M. W. Matsen,

K. Almdal, and K. Mortensen, Phys. Rev. Lett. 75, 4429 (1995).[52] B. D. Olsen, M. Shah, and V. Ganesan, Macromolecules 41,

6809 (2008).

042602-12

MICROPHASE SEPARATION IN COMBLIKE LIQUID- . . . PHYSICAL REVIEW E 88, 042602 (2013)

[53] V. Reiffenrath, J. Krause, H. J. Plach, and G. Weber, Liq. Crys.5, 159 (1989).

[54] H. Kosonen, J. Ruokolainen, T. Haatainen, M. Torkkeli,R. Serimaa, G. Ten Brinke, and O. Ikkala, Nat. Mater. 3, 872(2004).

[55] Y. Kang, J. J. Walish, T. Gorishyy, and E. L. Thomas, Nat. Mater.6, 957 (2007).

[56] Y. Lu, H. Xia, G. Zhang, and C. Wu, J. Mater. Chem. 19, 5952(2009).

[57] T. Xu, A. V. Zvelindovsky, G. J. A. Sevink, K. S. Lyakhova,H. Jinnai, and T. P. Russell, Macromolecules 38, 10788 (2005).

[58] V. Olszowka, M. Hund, V. Kuntermann, S. Scherdel,L. Tsarkova, A. Boker, and G. Krausch, Soft Matter 2, 1089(2006).

[59] R. R. Netz and M. Schick, Phys. Rev. Lett. 77, 302 (1996).[60] K. Schmidt, H. G. Schoberth, M. Ruppel, H. Zettl, H. Hansel,

T. W. Weiss, V. Urban, G. Krausch, and A. Boker, Nat. Mater.7, 142 (2007).

[61] J. J. Christensen and A. J. Bray, Phys. Rev. E 58, 5364(1998).

[62] H. Qian and G. F. Mazenko, Phys. Rev. E 69, 011104(2004).

042602-13