lable at ScienceDirect

Current Applied Physics 14 (2014) 1263e1272

Contents lists avai

Current Applied Physics

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

Numerical analysis of energy-minimizing wavelengths of equilibriumstates for diblock copolymers

Darae Jeong a, Jaemin Shin b, Yibao Li c, Yongho Choi a, Jae-Hun Jung d, Seunggyu Lee a,Junseok Kim a, *

a Department of Mathematics, Korea University, Seoul 136-713, Republic of Koreab Institute of Mathematical Sciences, Ewha W. University, Seoul 120-750, Republic of Koreac Department of Computational Science and Engineering, Yonsei University, Seoul, 120-749, Republic of Koread Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260-2900, United States

a r t i c l e i n f o

Article history:Received 23 April 2014Received in revised form13 June 2014Accepted 17 June 2014Available online 8 July 2014

Keywords:Nonlocal CahneHilliard equationLamellar phaseWavelengthPhase separationDiblock copolymer

* Corresponding author. Tel.: þ82 2 3290 3077; faxE-mail address: cfdkim@korea.ac.kr (J. Kim).URL: http://math.korea.ac.kr/%4ecfdkim

http://dx.doi.org/10.1016/j.cap.2014.06.0161567-1739/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

We present a robust and accurate numerical algorithm for calculating energy-minimizing wavelengths ofequilibrium states for diblock copolymers. The phase-field model for diblock copolymers is based on thenonlocal CahneHilliard equation. The model consists of local and nonlocal terms associated with short-and long-range interactions, respectively. To solve the phase-field model efficiently and accurately, weuse a linearly stabilized splitting-type scheme with a semi-implicit Fourier spectral method. To findenergy-minimizing wavelengths of equilibrium states, we take two approaches. One is to obtain anequilibrium state from a long time simulation of the time-dependent partial differential equation withvarying periodicity and choosing the energy-minimizing wavelength. The other is to directly solve theordinary differential equation for the steady state. The results from these two methods are identical,which confirms the accuracy of the proposed algorithm. We also propose a simple and powerful formula:h ¼ L*/m, where h is the space grid size, L* is the energy-minimizing wavelength, and m is the number ofthe numerical grid steps in one period of a wave. Two- and three-dimensional numerical results arepresented validating the usefulness of the formula without trial and error or ad hoc processes.

© 2014 Elsevier B.V. All rights reserved.

1. Introduction

A diblock copolymer is a polymer consisting of a block of onetype of monomer, A, joined chemically to a block of another type ofmonomer, B [1]. Below a critical temperature, the two sequencesare incompatible and the copolymer melt undergoes phase sepa-ration [2]. This occurs on the mesoscopic scale and the observedmesoscopic domains show periodic structures including lamellae,spheres, cylinders, and gyroids [3]. These phase separations havebeen observed experimentally [4e16], studied mathematically[3,17e26], and simulated numerically [27e39]. For the mathe-matical approach to phase separation, dynamic mean field theory,based on the self-consistent field theory [40,41], has been widelystudied. In this work, we use the density functional theory pro-posed by Ohta and Kawasaki [24]. This is dependent on themonomer density [42], which is the minimization of a nonlocal

: þ82 2 929 8562.

CahneHilliard (CH) free energy defined by an order parameter.Here, a diblock copolymer is described by the order parameter f,which represents the relative local density difference between twomonomers constituting the copolymer. The standard CH free en-ergy, generally used for the modeling of phase separation [43], issupplemented with a nonlocal term [44].

Numerical experiments are useful to predict self-assemblystructures of the diblock copolymer, containing lamellae, spheres,and cylindrical tubes. However, because of the non-convexity of theOhtaeKawasaki functional, the numerical results can draw themetastable state and local minimizer. We would like to refer [45]for the study on assessing the lowest energy state and escapingfrom certain metastable states by using a spectral weighting.Although there have been a number of numerical studies on diblockcopolymer phase separation, there has been little investigation ofthe accurate calculation of the energy-minimizing wavelengths ofequilibrium states. Therefore, the main purpose of this paper is todevelop a robust and accurate numerical algorithm for calculatingthe energy-minimizing wavelengths of equilibrium states indiblock copolymers. We also propose a simple and powerful

D. Jeong et al. / Current Applied Physics 14 (2014) 1263e12721264

formula for the space grid size. This controllability of the space stepsize is essential in the computational simulations for the physicalexperiments.

This paper is organized as follows. In Section 2, we brieflydescribe a phase field model of diblock copolymer. In Section 3, wepresent a linearly stabilized splitting-type scheme with a semi-implicit Fourier spectral method to solve the phase-field modelefficiently and accurately. In Section 4, we perform a series of nu-merical experiments. Finally, our conclusion and discussion aregiven in Section 5.

2. Mathematical model

We consider a diblock copolymer consisting of two homopoly-mer blocks A and B. The governing equation is the OhtaeKawasakimodel [24]. Let the order parameter f(x) ¼ rA(x) � rB(x) be definedas the difference between the local volume fractions of A and B atthe point x. The governing equation is the following partial differ-ential equation (PDE),

vfðx; tÞvt

¼ D�F 0ðfðx; tÞÞ � ε

2Dfðx; tÞ�� a

�fðx; tÞ � f

�; (1)

where x2U3Rd (d¼ 1,2,3) is the domain, F(f) ¼ 0.25(f2�1)2 is theHelmholtz free energy, ε is the gradient energy coefficient, a isinversely proportional to the square of the total chain length of thecopolymer [46], and f ¼ RUfðx;0Þdx=RUdx is the total mass. Wenow briefly review the derivation of Eq. (1). The free energy func-tional is represented as the sum of two parts as εtotal ¼ εshort þ εlong.First, εshort(f) denotes the short-range part of the free energyfunctional,

3shortðfÞ :¼ZU

�FðfÞ þ ε

2

2jVfj2

�dx:

Second, εlong(f) denotes the long-range part of the free energyfunctional,

3longðfÞ :¼a

2

ZU

ZU

Gðx � yÞ�fðxÞ � f��fðyÞ � f

�dydx;

where G is the Green's function having the following property:DG(x�y) ¼ �d(x�y). Here, periodic boundary conditions areassumed and d is Dirac delta function. By taking a variational de-rivative and letting f ¼ F0, we have,

d 3shortðfÞdf

¼ f ðfÞ � ε2Df; (2)

d 3longðfÞdf

¼ a

ZU

Gðx � yÞ�fðyÞ � f�dy: (3)

Next, we deduce Eq. (1) by substituting Eqs. (2) and (3) into theevolution equation.

ft ¼ D

�d 3shortðfÞ

dfþ d 3longðfÞ

df

�¼ D

�f ðfÞ � ε

2Df�� a

�f� f

�:

Note that if j satisfies �Dj ¼ f� f with the periodic boundarycondition [3], then we can represent εlong(f) as,

3longðfÞ ¼aZ

DxjðxÞ�Z

DyGðx � yÞjðyÞdy

2U U

dx ¼ a

2

ZU

jVjðxÞj2dx:

Therefore, an alternative form of the total system energy is givenas,

3totalðfÞ ¼ZU

�FðfÞ þ ε

2

2jVfj2

�dx þ a

2

ZU

jVjj2dx;

which we will use for the numerical evaluation of the total energy.Differentiate the energy εtotal and the total mass

RUfdxwith respect

to time, and using the periodic boundary condition, we have,

ddt

3totalðtÞ ¼ �ZU

jVðmþ ajÞj2dx � 0 andddt

ZU

fdx ¼ 0;

where m¼ f(f)� ε2Df. Therefore, the total energy is non-increasing

and the total mass is conserved in time.

3. Numerical solution

In this section, we present a fully discrete scheme for thefollowing one-dimensional nonlocal CH equation with the periodicboundary condition on U ¼ (0,L):

vfðx; tÞvt

¼ v2f ðfðx; tÞÞvx2

� ε2v

4fðx; tÞvx4

� a�fðx; tÞ � f

�: (4)

We use the equidistant grid xm ¼ (m � 1)h form ¼ 1,…,MwhereM is an even number and h ¼ L/M. Let fn

m be an approximation off(xm,nDt), where Dt is the temporal step size. We define the gridfunction fn ¼ (f1

n,…,fMn ) as a vector of grid point values. In partic-

ular, we denote the constant vector as 1 ¼ (1,…,1). The discreteFourier transform and its inverse transform are defined by,

bfnp ¼

XMm¼1

fnme

�ixmxp ; (5)

fnm ¼ 1

M

XM=2

p¼1�M=2

bfnpe

ixmxp ; (6)

where xp ¼ 2p(p�1)/L. The second- and fourth-order partial de-rivatives are,

v2f

vx2¼ � 1

M

XM=2

p¼1�M=2

x2pbfnpe

ixxp ;v4f

vx4¼ 1

M

XM=2

p¼1�M=2

x4pbfnpe

ixxp :

To solve Eq. (4), we use the linearly stabilized splitting-typescheme [47] with a semi-implicit Fourier spectral method[33,48e50]. The linear terms are treated implicitly and thenonlinear term is treated explicitly,

fnþ1m � fn

mDt

¼ 2v2fnþ1

m

vx2� ε

2v4fnþ1

m

vx4þ v2gnm

vx2� a

�fnþ1m � f

�; (7)

where gnm ¼ f ðfnmÞ � 2fn

m and f ¼PMm¼1f

0m=M. If a ¼ 0, then Eq. (7)

becomes the linear stabilized scheme for the CH equation [47]. Forthe higher temporal order spectral scheme of the nonlocal CH

Fig. 1. (a) Maximum error of the numerical solution, measured at t ¼ 10�2. (b) Pointwise approximation error for increasing resolution.

Fig. 2. Temporal evolutions of (a) the normalized discrete total energy 3htotalðtÞ= 3htotalð0Þand (b) the total mass

PMm¼1f

nm=M.

D. Jeong et al. / Current Applied Physics 14 (2014) 1263e1272 1265

equation, please refer to [51]. Using Eq. (6), Eq. (7) can be trans-formed into the discrete Fourier space as follows:

bfnþ1p ¼

bfnp � Dtx2pbgnp þ aDtfb1p

1þ aDt þ 2Dtx2p þ ε2Dtx4p

: (8)

After updating bfnþ1p with bfn

p , we recover fnþ1m from bfnþ1

p by theinverse discrete Fourier transform, Eq. (6). To satisfy the mass

conservation property, we should havePM

m¼1fnþ1m ¼PM

m¼1fnm. By

using Eq. (5), we only need to prove bfnþ11 ¼ bfn

1. From Eq. (8), we getbf11 ¼ ðbf0

1 þ aDtMfÞ=ð1þ aDtÞ ¼ ðbf01 þ aDtbf0

1Þ=ð1þ aDtÞ ¼ bf01.

Here, we have used the definitions of f and Fourier transform. Ifbfn1 ¼ bf0

1, we have bfnþ11 ¼ ðbfn

1 þ aDtMfÞ=ð1þ aDtÞ ¼ ðbf01þ

aDtbf01Þ=ð1þ aDtÞ ¼ bf0

1. Therefore, we can get bfn1 ¼ bf0

1 for all n.Thus, our proposed scheme inherits the mass conservation. Let the

Fourier interpolation of fn be IMfnðxÞ ¼ 1MPM=2

p¼1�M=2bfnpe

ixxp : We

define the discrete energy as 3htotalðfnÞ ¼ 3hshortðfnÞ þ 3hlongðfnÞ,where,

3hshortðfnÞ ¼ h

XMm¼1

"F�fnm�þ ε

2

2

�ddx

IMfnðxmÞ�2#;

3hlongðfnÞ ¼ h

XMm¼1

a

2

�ddx

IMjnðxmÞ�2

:

Here, we use the Fourier spectral method to solve the Poisson

equation �v2jnm

vx2 ¼ fnm � f. Note that we set bjn

1 ¼ 0 to guarantee aunique solution. We also perform the numerical simulation for thenonlocal CH equation on two- and three-dimensional domainsU ¼ (0,L)d, d ¼ 2,3, using a multigrid method for the CH equation[52,53].

4. Numerical experiments

In this section, we describe a series of numerical experiments.We perform a convergence test and a linear stability analysis,calculate the discrete total energy dissipation and distribution atequilibrium states, compute the optimal wavelength with theminimum discrete total energy, study the effect of a, ε, and f, andcompare the PDE solver with an ordinary differential equation(ODE) solver. We implement two- and three-dimensional numer-ical experiments using the new formula for an optimal grid spacing.Throughout the experiments, unless otherwise specified, we useε ¼ 1=ð20

ffiffiffi2

pÞ, a ¼ 100, h ¼ 0.00125, and Dt ¼ 0.0625.

4.1. Numerical convergence test

We examine the convergence of our proposed scheme. Theinitial state is f(x,0) ¼ 0.1cos(4px) in U ¼ (0,1). To calculate theaccuracy of the proposed method, the numerical solutions fn arecomputed on the grids M ¼ 2s for s ¼ 3,4,…,7. For each case, thecalculation is run until T ¼ 10�2 with the time step Dt ¼ 10�6 andε¼ 0.05.We define the error of a grid to be themaximumpointwiseerror, given by L∞ ¼ max

��IM~f� IMfn��. We define ~f as the reference

solution, which is the numerical solution usingM ¼ 210 grid points.Fig. 1(a and b) show the maximum errors and pointwise error ofapproximation for increasing resolution, respectively. As can beseen, the convergence of the results under mesh refinements isevident.

4.2. Discrete total energy dissipation

Fig. 2 illustrates the temporal evolutions of the normalizeddiscrete total energy 3htotalðtÞ= 3htotalð0Þ and the total mass until T¼ 20with M ¼ 800. The initial condition is f(x,0) ¼ 0.1r and(x) with thezero average f ¼ 0 in U ¼ (0,1), where rand(x) is a random valuebetween �1 and 1. Fig. 2(a) shows that the discrete total energy

Fig. 3. Growth rate given by linear stability theory and numerical solution as a func-tion of k with a ¼ 0 and a ¼ 100.

Table 1Optimal length L* with varying h and fixed Dt ¼ 0.1.

h L*

0.010000 0.3751113926052600.005000 0.3747218016357600.002500 0.3747116722567410.001250 0.3746876097050820.000625 0.374681591974566

Table 2Optimal length L* with varying Dt and fixed h ¼ 0.00125.

Dt L*

1.0000 0.3746876097050250.5000 0.3746876097050550.2500 0.3746876097050650.1250 0.3746876097050540.0625 0.374687609705047

D. Jeong et al. / Current Applied Physics 14 (2014) 1263e12721266

decreases and dots are associated with profiles in the insertedchart. Fig. 2(b) confirms that the total mass is preserved.

4.3. Linear stability analysis

We examine the linear stability of the trivial stationary solutionf ¼ f on the one-dimensional space. Under a linear stability anal-ysis [2,44,54], the solution of Eq. (1) is assumed to have the form,

fðx; tÞ ¼ fþXk¼1

∞bkðtÞcosðkxÞ; (9)

where jbkðtÞj≪1. Substituting Eq. (9) into Eq. (1), we have,

dbkdt

¼ �k2h3f2 � 1þ ε

2k2ibk þ abk; (10)

up to first order. The solution of Eq. (10) is bk(t) ¼ bk(0)exp(hkt),

where hk ¼ �k2½3f2 � 1þ ε2k2� � a is the growth rate. Therefore,

we have the maximal growth rate if we take the wavenumber as

kmax ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 3f2

q=ð

ffiffiffi2

pεÞ. Note that this is independent of the value

of a. Next, the theoretical growth rate hk is compared to the nu-merical growth rate given by our proposed scheme for differentvalues of k and a. The numerical growth rate is defined by~hk ¼ logð max

1�m�M

��fnm

��= max1�m�M

��f0m

��Þ=T: For the numerical test, we

take the initial condition f(x,0) ¼ 0.01cos(kx) on U ¼ (0,2p). With

Fig. 4. Domain-size-scaled discrete total energy 3htotal=L as a function of L.

Dt ¼ 10�6, the numerical simulations are run up to T ¼ nDt ¼ 10�4.Fig. 3 suggests that the theoretical values from the linear analysis(solid (a ¼ 0) and dashed (a ¼ 100) lines) and the numerical re-sults (circle (a ¼ 0) and triangle (a ¼ 100) symbols) are in goodagreement. Moreover, the numerical result shows that the growthrate is maximum at k ¼ 20, which is consistent with the theo-retical result.

4.4. Discrete total energy at equilibrium states

We define the numerical solution to be in the steady state, whenthe consecutive error is less than a given tolerance, tol,max

1�m�M

��fnþ1m � fn

m

��=Dt < tol: From now on, we will use tol ¼ 10�12.We calculate the domain-size-scaled discrete total energy 3htotal=Lwith an equilibrium solution in (0,L). The numerical result in Fig. 4shows 3htotal=L as a function of L. The initial condition is taken asf(x,0) ¼ 0.1 rand (x) and shown in the jagged lines. The inscribedfigures correspond to equilibrium state profiles at the indicateddomain sizes. We can observe that the domain-size-scaled discretetotal energy is minimized when Lz 0.38 (one wave) and 0.76 (twowaves). This doubling length is due to the periodic boundarycondition.

4.5. Optimal length L* having minimum discrete total energy3htotal=L

�

We describe an algorithm for finding the total energy-minimizing wavelength. We define the optimal length L* as the

Fig. 5. Local minimum of 3htotal=L regardless of the initial wavenumbers.

Fig. 6. (a) Optimal length L* as a function of the value of a. (b) Equilibrium states with varying a. (c) Comparison between linear stability theory and numerical simulation results.(d) Various equilibrium states with a ¼ 50 and L ¼ 5L*.

D. Jeong et al. / Current Applied Physics 14 (2014) 1263e1272 1267

smallest length having the global minimum of scaled discrete totalenergy. To calculate L*, we solve Eq. (7) until it reaches a numericalequilibrium state with given values of h, Dt, ε, and a. The initialcondition is f(x,0) ¼ 0.01cos(2px/L) in U ¼ (0,L), where L starts at2 h and increases in steps of 2 h. LetM be the smallest even integersuch that 3htotal=ðMhÞ is minimized in U ¼ (0,Mh). Construct thequadratic polynomial passing the three points ððM � 2Þh; 3htotal=

ððM � 2ÞhÞÞ, ðMh; 3htotal=ðMhÞÞ, and ððM þ 2Þh; 3htotal= ððM þ 2ÞhÞÞ,then define the optimal length L* as the critical point of thepolynomial.

Table 1 shows the optimal length L* for varying h with Dt ¼ 0.1.The result indicates that the solution with h ¼ 0.00125 is

Fig. 7. (a) Optimal length L* against ε. (b) Profiles of f at

sufficiently accurate. We also tested the effect of the time step Dt onthe optimal length L* (see Table 2). We will use h ¼ 0.00125 andDt ¼ 0.0625 in the subsequent sections.

4.6. Initial wavenumber-independent local minimum of 3htotal=L

We show that the domain-size-scaled discrete total energy3htotal=L has the same energy-minimizing wavelength which is in-dependent on the initial wavenumbers. For this numerical simu-lation, we use the two different initial conditions asf(x,0) ¼ 0.5cos(2px/L) and f(x,0) ¼ 0.5cos(4px/L) in (0,L). Fig. 5shows 3htotal=L as a function of L with two different initial

the equilibrium state when ε ¼ 0.02,0.03, and 0.04.

Fig. 8. (a) Effect of f on the optimal length L*. (b) Steady state phase-field profiles for different f.

Fig. 9. Steady state solutions from PDE and ODE with two different initialconfigurations.

D. Jeong et al. / Current Applied Physics 14 (2014) 1263e12721268

wavenumbers which are indicated by the star line (2p/L wave-number) and circled line (4p/Lwavenumber). In Fig. 5, the domain-size-scaled discrete total energy is minimized at the same wave-lengths L z 0.37 and 0.75 regardless of the initial wavenumbers.The shaded areas containing L ¼ 0.37 and L ¼ 0.75 represent oneand two period of waves, respectively.

4.7. Effect of a on L*

Fig. 6(a) shows the optimal length L* as a function of the value ofthe long-range interaction parameter a. Here, we use the initialcondition f(x,0) ¼ 0.01cos(2px/L). The energy-minimizing wave-length becomes shorter as a increases. Fig. 6(b) shows equilibriumprofiles with a ¼ 10,50, and 150. It also shows that the waveamplitude decreases as a increase. In Fig. 6(c), the shaded region isthe positive growth phase plane of a and wavenumber k. Thedashed line is the maximum growth wavenumber of the linearstability analysis, and the thick solid line is the numerical simula-tion result, 2p/L*, with varying a. As a approaches 200, 2p/L* con-verges to 20. Fig. 6(d) represents three cases of differentequilibrium states with different random initial conditions. Here,a ¼ 50, L ¼ 5L*, and h ¼ 0.00125 are used. When L ¼ 5L*, we expectthe energy minimizing steady solution is a five-periodic wave.However, as shown in Fig. 6(d), the steady state solutions are four-,five-, or six-periodic waves. These phenomena can be explainedfrom Fig. 6(c). There are many growing modes other than the en-ergy minimizing mode. This implies that we could have differentequilibrium solutions form different initial conditions.

4.8. Effect of ε on L*

Fig. 7(a) shows the optimal length L* as a function of the value ofε. Here, we use the initial condition f(x,0)¼ 0.01cos(2px/L). We canobserve that the energy minimizing wavelength L* also increases asε increases. Fig. 7(b) shows profiles of f at equilibrium state whenε ¼ 0.02, 0.03, and 0.04. We also observe that the amplitude of thephase-field is smaller as ε increases.

4.9. Effect of f on L*

Fig. 8(a) illustrates the effect of f on the optimal length L*. Here,we use the initial condition fðx;0Þ ¼ 0:01 cosð2px=LÞ þ f. Asshown in Fig. 8(a), the values of L* increase from f ¼ 0 to f ¼ 0:26,and then decrease when the total mass f is larger than 0.26.Fig. 8(b) shows the steady state phase-field profiles for different f.

4.10. Comparison of results from PDE and ODE

We compare the steady state solutions given by two methods toshow that the results are reliable. The first method solves thesteady state equation of Eq. (1) using a MATLAB (ver. 7.14, R2012b,The Mathworks, Inc., MA) solver bvp5c [55e57]. The equilibriumsolution for Eq. (1) should satisfy,�f3 � f� ε

2fxx

�xx

� af ¼ 0; x2ð0; LÞ;

which is subject to the periodic boundary conditions,

fð0Þ¼fðLÞ¼0; fxð0Þ¼fxðLÞ; fxxð0Þ¼fxxðLÞ; fxxxð0Þ¼fxxxðLÞ:

The secondmethod is to evolve Eq. (1) until it reaches the steadystate. Fig. 9 shows the steady state solutions for different initialconditions. The initial states are 0.5sin(2px/L) and 0.9sin(2px/L),where L ¼ 0.38. For the evolution, we use M ¼ 64. The solid line isthe solution obtained by the ODE solver, and the circles are thosegiven by the Fourier spectral method. From this result, we canconfirm that the steady solution from the PDE does not becomestuck in a metastable state that has a larger total energy than theglobal minimizers.

We have an equilibrium solution fðxÞ ¼ tanh½x=ðffiffiffi2

pεÞ� on the

infinite domain when a ¼ 0. Therefore, f varies from �0.9 to 0.9over a distance of about 2

ffiffiffi2

pε tanh�1ð0:9Þ. If ε ¼ hm=2

ffiffiffi2

ptanh�1

ð0:9Þ, then we have approximately m grid points across the

Fig. 10. (a) and (b) are linear fittings of log(L*) versus the values of log(a) and log(ε), respectively.

D. Jeong et al. / Current Applied Physics 14 (2014) 1263e1272 1269

interfacial transition layer [58]. However, there is no non-trivialclosed-form solution of the nonlocal CH equation (a s 0). Wepropose the following simple and powerful formula: if we want tohave m numerical grid steps in one period of a wave, we defineh ¼ L*/m, where L* is the energy minimizing optimal length. Thiscontrollability of the numerical grid steps is very important in thecomputational simulations for the physical experiments. Forexample, if h is too large, it is difficult to get sufficiently resolvednumerical results. On the other hand, if h is too small, large scalecomputations are extremely costly for higher-dimensional (2D or3D) simulations. Therefore, the proposed formula for the choice ofthe spatial step size h will be useful to do accurate and efficientnumerical simulations without trial and error or ad hoc processes.

4.11. Scaling law for L* in terms of a and ε

In this section, we numerically investigate the scaling law for L*

in terms of a and ε. In Ref. [44], the authors presented a formula forL* with f ¼ 0 as,

Fig. 11. Lamellar pattern with (a) h ¼ 0.375/10, (b) h ¼ 0.375/20, and (c) h ¼ 0.375/40. Dt ¼ 0second rows represent the numerical results at t ¼ 5 and t ¼ 1000, respectively.

L� ¼ 64

ffiffiffi2

pε

a

!1=3

þ O�ε2=3�:

First, we fix the value of ε as 1=ð20ffiffiffi2

pÞ and change a from 0 to

200. As shown in Fig. 10(a), we find the following linear fitting:log(L*) ¼ �0.3017log(a) þ 0.3983, which is close to the theoreticalprediction. Next, we fix a ¼ 100 and change the value of ε from0.001 to 0.0106. Fig. 10(b) shows the numerical result with thelinear fitting: log(L*) ¼ 0.3506log(ε)� 1.2793. Here, we have a goodagreement with the theoretical scaling factor, 1/3.

4.12. Two-dimensional experiment

To validate the proposed formula h ¼ L*/m, we perform thesimulation of phase separation on the two-dimensional domainU ¼ (0,L)2. Fig. 11(a)e(c) show the lamellar pattern with differentspatial step sizes h¼ 0.375/10, 0.375/20, and 0.375/40, respectively.For this numerical test, we use Dt ¼ 0.1, a ¼ 100, ε ¼ 1=ð20

ffiffiffi2

pÞ,

.1, a ¼ 100, ε ¼ 1=ð20ffiffiffi2

pÞ, 512 � 512 mesh grid, L ¼ 512 h, and f ¼ 0 are used. First and

Fig. 12. Hexagonal pattern with (a) h ¼ 0.31/10, (b) h ¼ 0.31/20, and (c) h ¼ 0.31/40. Parameters are used as Dt ¼ 0.1, a ¼ 100, ε ¼ 1=ð30ffiffiffi2

pÞ, 512 � 512 mesh grid, L ¼ 512 h, and

f ¼ �0:3. First and second rows represent the numerical results at t ¼ 5 and t ¼ 1000, respectively.

D. Jeong et al. / Current Applied Physics 14 (2014) 1263e12721270

512 � 512 mesh grid, and L ¼ 512 h with an initial conditionf(x,y,0) ¼ 0.15 rand(x,y). In Fig. 11, first and second rows representthe numerical results at t ¼ 5 and t ¼ 1000, respectively. By theproposed formula h ¼ L*/m, we can expect approximately 512/mwaves along the one-axis. As can be seen in the second row ofFig. 11(a)e(c), have approximately 50, 25, 12 periodic waves alongthe one-axis, respectively.

Fig. 13. Lamellar pattern with (a) h ¼ 0.375/10, (b) h ¼ 0.375/20, and (c) h ¼ 0.375/40. Paraf ¼ 0. First and second rows represent the numerical results at t ¼ 5 and t ¼ 1000, respec

Now, we consider the hexagonal pattern with different spatialstep sizes. For the numerical test, we use ε ¼ 1=ð30

ffiffiffi2

pÞ, a ¼ 100,

512 � 512 mesh grid, and L ¼ 512 h. The initial condition is set asf(x,y,0) ¼ �0.3þ0.15 rand (x,y). Fig. 12(a)e(c) represent the tem-poral evolutionwith h ¼ 0.31/10, 0.31/20, and 0.31/40, respectively.In Fig. 12(a)e(c), results at t ¼ 1000 have approximately 50, 25, 12periodic waves along the one-axis, respectively.

meters are Dt ¼ 0.1, a ¼ 100, ε ¼ 1=ð20ffiffiffi2

pÞ, 128 � 128 � 128 mesh grid, L ¼ 128 h, and

tively.

Fig. 14. Spherical pattern with (a) h ¼ 0.258/10, (b) h ¼ 0.258/20, and (c) h ¼ 0.258/40. Parameters are Dt ¼ 0.1, a ¼ 200, ε ¼ 1=ð30ffiffiffi2

pÞ, 128 � 128 � 128 mesh grid, and f ¼ �0:3.

First and second rows represent the numerical results at t ¼ 5 and t ¼ 1000, respectively.

D. Jeong et al. / Current Applied Physics 14 (2014) 1263e1272 1271

4.13. Three-dimensional experiment

In this section, we perform the simulation of phase separationon the three-dimensional domain U ¼ (0,L)3. We simulate lamellarpattern with different spatial step sizes h ¼ 0.375/m, wherem ¼ 10,20, and 40. Parameters are Dt ¼ 0.1, a ¼ 100, ε ¼ 1=ð20

ffiffiffi2

pÞ,

and L ¼ 128 h. And the initial condition is f(x,y,z,0) ¼ 0.15 rand(-x,y,z) on 128 � 128 � 128 mesh grid. We observe that (a), (b), and(c) in the second row of Fig. 13 have approximately 12, 6, and 3periodic waves along the one-axis, respectively.

Fig. 14(a)e(c) show the spherical patterns with h ¼ 0.258/m,m ¼ 10, 20, and 40, respectively. Here, we use a ¼ 200,ε ¼ 1=ð30

ffiffiffi2

pÞ, and f ¼ �0:3. In the second row of Fig. 14, we see

that (a), (b), and (c) have approximately 12, 6, and 3 periodic wavesalong the one-axis, respectively.

5. Conclusions

We proposed a robust and accurate numerical algorithm forcalculating the energy-minimizing wavelengths of equilibriumstates in diblock copolymers. The mathematical model for diblockcopolymers is the nonlocal CahneHilliard equation, which consistsof local and nonlocal terms associated with short- and long-rangeinteractions, respectively. We used the linearly stabilizedsplitting-type schemewith a semi-implicit Fourier spectral methodto solve the governing equation. To find energy-minimizingwavelengths of the equilibrium states, we considered both PDEand ODE approaches. The results from these methods were iden-tical, which confirms the accuracy of the proposed algorithm. Thenumerical results indicate that (1) the wavelength becomes shorteras the value of the long-range interaction parameter increases; (2)the wavelength increases with the interfacial energy coefficientvalue; (3) the most unstable wavelength given by the linear sta-bility analysis is different from the energy-minimizing wavelengthas shown in Fig. 6(c); (4) numerical equilibrium states can havenon-energy minimizing wavelengths (Fig. 6(d)). We also proposed

a simple and powerful formula: h¼ L*/m, which is very useful in thecomputational simulations for the physical experiments. Forexample, it could be used in setting an initial domain size for thealgorithm of simulated annealing in order to escape from certainmetastable states [45]. Two- and three-dimensional numerical re-sults were presented validating the usefulness of the formulawithout trial and error or ad hoc processes.

Acknowledgments

The first author (D. Jeong) was supported by a Korea UniversityGrant. The corresponding author (J.S. Kim) acknowledges the sup-port of the MSIP (Ministry of Science, Ict & future Planning).

References

[1] C. Singh, M. Goulian, A.J. Liu, G.H. Fredrickson, Phase behavior of semiflexiblediblock copolymers, Macromol 27 (1994) 2974e2986.

[2] F. Liu, N. Goldenfeld, Dynamics of phase separation in block copolymer melts,Phys. Rev. A 39 (1989) 4805e4810.

[3] R. Choksi, M.A. Peletier, J.F. Williams, On the phase diagram for microphaseseparation of diblock copolymers: an approach via a nonlocal CahneHilliardfunctional, SIAM J. Appl. Math. 69 (2009) 1712e1738.

[4] F.S. Bates, G.H. Fredrickson, Block copolymers-designer soft materials, Phys.Today 52 (1999) 32e38.

[5] I.W. Hamley, The Physics of Block Copolymers, Oxford University Press, NewYork, 1998, p. 24.

[6] M. Imai, K. Sakai, M. Kikuchi, K. Nakaya, A. Saeki, T. Teramoto, Kinetic pathwayto double-gyroid structure, J. Chem. Phys. 122 (2005) 214906.

[7] M. Seul, D. Andelman, Domain shapes and patterns: the phenomenology ofmodulated phases, Science 267 (1995) 476e483.

[8] Y.S. Joseph, S. Kim, Q. Zhang, R. Hoogenboom, J.D. Hong, Multilayer filmscomposed of a thermoresponsive cationic diblock copolymer and a photo-responsive dye, Polymer 54 (2013) 4894e4901.

[9] A.T. Lorenzo, M.L. Arnal, A.J. Müller, A. Boschetti de Fierro, V. Abetz,Confinement effects on the crystallization and SSA thermal fractionation ofthe PE block within PE-b-PS diblock copolymers, Eur. Polym. J. 42 (2006)516e533.

[10] P.A. George, J.J. Cooper-White, Kinetically constrained block copolymer self-assembly a simple method to control domain size, Eur. Polym. J. 45 (2009)1065e1071.

D. Jeong et al. / Current Applied Physics 14 (2014) 1263e12721272

[11] S. Akasaka, T. Okamoto, T. Osaka, T. Matsushita, H. Hasegawa, 3D analysis ofthe lattice defects in the gyroid network structure of a block copolymer/ho-mopolymer blend e the line-like defects in the gyroid network structure, Eur.Polym. J. 479 (2011) 651e661.

[12] B. Nandan, B.K. Kuila, M. Stamm, Supramolecular assemblies of block co-polymers as templates for fabrication of nanomaterials, Eur. Polym. J. 47(2011) 584e599.

[13] D. Borah, C.D. Simao, R. Senthamaraikannan, S. Rasappa, A. Francone, O. Lorret,M. Salaun, B. Kosmala, N. Kehagias, M. Zelsmann, C.M. Sotomayor-Torres,M.A. Morris, Soft-graphoepitaxy using nanoimprinted polyhedral oligomericsilsesquioxane substrates for the directed self-assembly of PS-b-PDMS, Eur.Polym. J. 49 (2013) 3512e3521.

[14] B.M.D. O'Driscoll, R.A. Kelly, M. Shaw, P. Mokarian-Tabari, G. Liontos,K. Ntetsikas, A. Avgeropoulos, N. Petkov, M.A. Morris, Achieving structuralcontrol with thin polystyrene-b-polydimethylsiloxane block copolymer films:the complex relationship of interface chemistry, annealing methodology andprocess conditions, Eur. Polym. J. 49 (2013) 3445e3454.

[15] O. Karagollu, M. Gorur, A. Turkan, B. Sengez, F. Gode, F. Yilmaz, A novelamperometric glucose biosensor based on poly(glycidylmethacrylate-co-(3-thienylmethylmethacrylate)), Curr. Appl. Phys. 13 (2013) 725e730.

[16] M.-S. Jeong, J.H. Kim, J.-H. Ko, Y.H. Ko, K.J. Kim, Pressure dependence ofacoustic behaviors and refractive index of amorphous Kel F-800 copolymerstudied by Brillouin spectroscopy, Curr. Appl. Phys. 13 (2013) 1774e1777.

[17] A. Aksimentiev, M. Fialkowski, R. Holyst, Ultrafast dynamics and spectroscopyof bacterial photosynthetic reaction centers, Adv. Chem. Phys. 121 (2002)1e88.

[18] M. Bahiana, Y. Oono, Cell dynamical system approach to block copolymers,Phys. Rev. A 41 (1990) 6763e6771.

[19] J.W. Cahn, J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial freeenergy, J. Chem. Phys. 28 (1958) 258.

[20] P.C. Fife, Models for phase separation and their mathematics, Electron. J. Diff.Equ. 48 (2000) 1e26.

[21] M.E. Gurtin, On a theory of phase transitions with interfacial energy, Arch.Ration. Mech. Anal. 87 (1985) 187e212.

[22] O.V. Borisov, E.B. Zhulina, Theory of self-assembly of triblock ter-polymers inselective solvent towards corona-compartmentalized (Janus) micelles, Poly-mer 54 (2013) 2043e2048.

[23] L. Leibler, Theory of microphase separation in block copolymers, Macromol 13(1980) 1602e1617.

[24] T. Ohta, K. Kawasaki, Equilibrium morphology of block copolymer melts,Macromol 19 (1986) 2621e2632.

[25] L. Modica, The gradient theory of phase transitions and the minimal interfacecriterion, Arch. Ration. Mech. Anal. 98 (1987) 123e142.

[26] T. Teramoto, Y. Nishiura, Morphological characterization of the diblockcopolymer problem with topological computation, Jpn. J. Ind. Appl. Math. 27(2010) 175e190.

[27] F.A. Detcheverry, D.Q. Pike, U. Nagpal, P.F. Nealey, J.J. de Pablo, Theoreticallyinformed coarse grain simulations of block copolymer melts: method andapplications, Soft Matter 5 (2009) 4858e4865.

[28] T.B. Martin, A. Seifpour, A. Jayaraman, Assembly of copolymer functionalizednanoparticles: a Monte Carlo simulation study, Soft Matter 7 (2011)5952e5964.

[29] C.A. Mujica-Martinez, J.C. Arce, Mini-bandstructure tailoring in pi-conjugatedperiodic block copolymers using the envelope crystalline-orbital method, Int.J. Quantum Chem. 110 (2010) 2532e2540.

[30] K.Ø. Rasmussen, G. Kalosakas, Improved numerical algorithm for exploringblock copolymer mesophases, J. Polym. Sci. Part B Polym. Phys. 40 (2002)1777e1783.

[31] L.T. Yan, E. Maresov, G.A. Buxton, A.C. Balazs, Self-assembly of mixtures ofnanorods in binary, phase-separating blends, Soft Matter 7 (2011) 595e607.

[32] M. Pinna, A.V. Zvelindovsky, Large scale simulation of block copolymers withcell dynamics, Eur. Phys. J. B 85 (2012) 1e18.

[33] A.W. Bosse, Phase-field simulation of long-wavelength line edge roughness indiblock copolymer resists, Macromol. Theory Simul. 19 (2010) 399e406.

[34] J.G.E.M. Fraaije, S.K. Nath, K. Remerie, J. Groenewold, Phase evolution theoryfor polymer blends with extreme chemical dispersity: parameterization of

DDFT simulations and application to poly (propylene) impact copolymers,Macromol. Theory Simul. 20 (2011) 133e145.

[35] M. Pinna, I. Pagonabarraga, A.V. Zvelindovsky, Modeling of block copolymer/colloid hybrid composite materials, Macromol. Theory Simul. 20 (2011)769e779.

[36] J.S. Meena, P.P. Thankachan, Theoretical studies of the ring opening metath-esis reaction of 3,3-dimethyl cyclopropene with molybdenum catalyst, Comp.Theor. Chem. 1024 (2013) 1e8.

[37] E. Yildirim, M. Yurtsever, The role of diisocyanate and soft segment on theintersegmental interactions in urethane and urea based segmented co-polymers: a DFT study, Comp. Ther. Chem. 1035 (2014) 28e38.

[38] D. Wang, X. Zhang, W. Ding, X. Zhao, Z. Geng, Density functional theory designand characterization of D-A-A type electron donors with narrow band gap forsmall-molecule organic solar cells, Comp. Theor. Chem. 1029 (2014) 68e78.

[39] J. Qin, G.S. Khaira, Y. Su, G.P. Garner, M. Miskin, H.M. Jaeger, J.J. de Pablo,Evolutionary pattern design for copolymer directed self-assembly, Soft Matter9 (2013) 11467e11472.

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

[41] J.A. Pople, D.L. Beveridge, N.S. Ostlund, Recent progress in approximate sel-consistent-field theory, Int. J. Quantum Chem. 1 (1967) 293e305.

[42] R. Choksi, X. Ren, On the derivation of a density functional theory formicrophase separation of diblock copolymers, J. Stat. Phys. 113 (2003)151e176.

[43] A. Segatti, On the hyperbolic relaxation of the CahneHilliard equation in 3D:approximation and long time behaviour, Math. Mod. Meth. Appl. Sci. 17(2007) 411e437.

[44] I. Ohnishi, Y. Nishiura, M. Imai, Y. Matsushita, Analytical solutions describingthe phase separation driven by a free energy functional containing a long-range interaction term, Chaos 9 (1999) 329e341.

[45] R. Choksi, M. Maras, J.F. Williams, 2D phase diagram for minimizers of aCahneHilliard functional with long-range interactions, SIAM J. Appl. Dyn.Syst. 10 (2011) 1344e1362.

[46] Y. Nishiura, I. Ohnishi, Some mathematical aspects of the micro-phase sepa-ration in diblock copolymers, Phys. D. 84 (1995) 31e39.

[47] D.J. Eyre, An Unconditionally Stable One-step Scheme for Gradient Systems,1998. Unpublished article, www.math.utah.edu/~eyre/research/methods/stable.ps.

[48] L.Q. Chen, J. Shen, Applications of semi-implicit Fourier-spectral method tophase field equations, Comput. Phys. Commun. 108 (1998) 147e158.

[49] J. Zhu, L.Q. Chen, J. Shen, V. Tikare, Coarsening kinetics from a variable-mobility CahneHilliard equation: application of a semi-implicit Fourierspectral method, Phys. Rev. E 60 (1999) 3564.

[50] X.F. Wu, Y.A. Dzenis, Guided self-assembly of diblock copolymer thin films onchemically patterned substrates, J. Chem. Phys. 125 (2006) 174707.

[51] B. Bene�sov�a, C. Melcher, E. Süli, An implicit midpoint spectral approximationof nonlocal CahneHilliard equations, SIAM J. Numer. Anal. 52 (2014)1466e1496.

[52] J. Kim, A numerical method for the CahneHilliard equation with a variablemobility, Commun. Nonlinear Sci. Numer. Simul. 12 (2007) 1560e1571.

[53] D. Lee, J. Huh, D. Jeong, J. Shin, A. Yun, J. Kim, Physical, mathematical, andnumerical derivations of the CahneHilliard equation, Comput. Mater. Sci. 18(2014) 216e225.

[54] J.F. Blowey, M.I.M. Copetti, C.M. Elliott, Numerical analysis of a model forphase separation of a multicomponent alloy, IMA J. Numer. Anal. 16 (1996)111e139.

[55] J.D. Evans, V.A. Galaktionov, J.R. King, Unstable sixth-order thin film equation:II. Global similarity patterns, Nonlinear 20 (2007) 1843.

[56] J. Kierzenka, J.F. Shampine, A BVP solver based on residual control and theMaltab PSE, ACM Trans. Math. Softw. 27 (2001) 299e316.

[57] J. Kierzenka, L.F. Shampine, A BVP solver that controls residual and error,JNAIAM J. Numer. Anal. Ind. Appl. Math. 3 (2008) 27e41.

[58] J. Kim, Phase-field models for multi-component fluid flows, Commun. Com-put. Phys. 12 (2012) 613e661.