Numerical investigation of local defectiveness control of diblock copolymer patterns

We numerically investigate local defectiveness control of self-assembled diblock copolymer patterns through appropriate substrate design. We use a nonlocal Cahn-Hilliard (CH) equation for the phase separation dynamics of diblock copolymers. We discretize the nonlocal CH equation by an unconditiona...

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Дата:	2016
Автори:	Jeong, D., Choi, Y., Kim, J.
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Опубліковано:	Інститут фізики конденсованих систем НАН України 2016
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Цитувати:	Numerical investigation of local defectiveness control of diblock copolymer patterns / D. Jeong, Y. Choi, J. Kim// Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33001: 1–10. — Бібліогр.: 35 назв. — англ.

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spelling	irk-123456789-1542272019-06-16T01:25:22Z Numerical investigation of local defectiveness control of diblock copolymer patterns Jeong, D. Choi, Y. Kim, J. We numerically investigate local defectiveness control of self-assembled diblock copolymer patterns through appropriate substrate design. We use a nonlocal Cahn-Hilliard (CH) equation for the phase separation dynamics of diblock copolymers. We discretize the nonlocal CH equation by an unconditionally stable finite difference scheme on a tapered trench design and, in particular, we use Dirichlet, Neumann, and periodic boundary conditions. The value at the Dirichlet boundary comes from an energy-minimizing equilibrium lamellar profile. We solve the resulting discrete equations using a Gauss-Seidel iterative method. We perform various numerical experiments such as effects of channel width, channel length, and angle on the phase separation dynamics. The simulation results are consistent with the previous experimental observations. Проведено числове дослiдження керування локальною дефектнiстю самоорганiзованих структур дiблоккополiмерiв за допомогою вiдповiдної конструкцiї субстрату. Використовується нелокальне рiвняння Кана-Хiлларда для динамiки фазового роздiлення дiблок-кополiмерiв. Здiйснено дискретизацiю нелокального рiвняння з використанням безумовно стiйкої схеми скiнченної рiзницi на звуженiй канавцi зразка i, зокрема, використано крайовi умови Дiрiхле, Ньюмана i перiодичнi граничнi умови. Значення при крайових умовах Дiрiхле отримано згiдно з рiвноважним ламеларним профiлем, що вiдповiдає енергетичному мiнiмуму. Ми розв’язуємо отриманi дискретнi рiвняння, використовуючи iтеративний метод ГауссаЗейделя. Проведено рiзнi числовi експерименти, такi як вплив ширини каналу, довжини каналу та кута на динамiку фазового роздiлення. Результати симуляцiй вiдповiдають попереднiм експериментальним спостереженням. 2016 Article Numerical investigation of local defectiveness control of diblock copolymer patterns / D. Jeong, Y. Choi, J. Kim// Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33001: 1–10. — Бібліогр.: 35 назв. — англ. 1607-324X DOI: 10.5488/CMP.19.33001 PACS: 02.60.Cb, 02.60.Lj, 02.70.Bf, 02.70.Pt arXiv:1609.06974 http://dspace.nbuv.gov.ua/handle/123456789/154227 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We numerically investigate local defectiveness control of self-assembled diblock copolymer patterns through appropriate substrate design. We use a nonlocal Cahn-Hilliard (CH) equation for the phase separation dynamics of diblock copolymers. We discretize the nonlocal CH equation by an unconditionally stable finite difference scheme on a tapered trench design and, in particular, we use Dirichlet, Neumann, and periodic boundary conditions. The value at the Dirichlet boundary comes from an energy-minimizing equilibrium lamellar profile. We solve the resulting discrete equations using a Gauss-Seidel iterative method. We perform various numerical experiments such as effects of channel width, channel length, and angle on the phase separation dynamics. The simulation results are consistent with the previous experimental observations.
format	Article
author	Jeong, D. Choi, Y. Kim, J.
spellingShingle	Jeong, D. Choi, Y. Kim, J. Numerical investigation of local defectiveness control of diblock copolymer patterns Condensed Matter Physics
author_facet	Jeong, D. Choi, Y. Kim, J.
author_sort	Jeong, D.
title	Numerical investigation of local defectiveness control of diblock copolymer patterns
title_short	Numerical investigation of local defectiveness control of diblock copolymer patterns
title_full	Numerical investigation of local defectiveness control of diblock copolymer patterns
title_fullStr	Numerical investigation of local defectiveness control of diblock copolymer patterns
title_full_unstemmed	Numerical investigation of local defectiveness control of diblock copolymer patterns
title_sort	numerical investigation of local defectiveness control of diblock copolymer patterns
publisher	Інститут фізики конденсованих систем НАН України
publishDate	2016
url	http://dspace.nbuv.gov.ua/handle/123456789/154227
citation_txt	Numerical investigation of local defectiveness control of diblock copolymer patterns / D. Jeong, Y. Choi, J. Kim// Condensed Matter Physics. — 2016. — Т. 19, № 3. — С. 33001: 1–10. — Бібліогр.: 35 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT jeongd numericalinvestigationoflocaldefectivenesscontrolofdiblockcopolymerpatterns AT choiy numericalinvestigationoflocaldefectivenesscontrolofdiblockcopolymerpatterns AT kimj numericalinvestigationoflocaldefectivenesscontrolofdiblockcopolymerpatterns
first_indexed	2025-07-14T04:40:49Z
last_indexed	2025-07-14T04:40:49Z
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fulltext	Condensed Matter Physics, 2016, Vol. 19, No 3, 33001: 1–10 DOI: 10.5488/CMP.19.33001 http://www.icmp.lviv.ua/journal Numerical investigation of local defectiveness control of diblock copolymer patterns D. Jeong, Y. Choi, J. Kim∗ Department of Mathematics, Korea University, Seoul 136-713, Republic of Korea Received October 21, 2015, in final form December 22, 2015 We numerically investigate local defectiveness control of self-assembled diblock copolymer patterns through appropriate substrate design. We use a nonlocal Cahn-Hilliard (CH) equation for the phase separation dynamics of diblock copolymers. We discretize the nonlocal CH equation by an unconditionally stable finite difference scheme on a tapered trench design and, in particular, we use Dirichlet, Neumann, and periodic boundary con- ditions. The value at the Dirichlet boundary comes from an energy-minimizing equilibrium lamellar profile. We solve the resulting discrete equations using a Gauss-Seidel iterative method. We perform various numerical ex- periments such as effects of channel width, channel length, and angle on the phase separation dynamics. The simulation results are consistent with the previous experimental observations. Key words: diblock copolymer, nonlocal Cahn-Hilliard equation, local defectivity control PACS: 02.60.Cb, 02.60.Lj, 02.70.Bf, 02.70.Pt 1. Introduction A diblock copolymer is a linear chain consisting of two blocks of different types of monomers bonded covalently to each other. The two blocks are mixed above the critical temperature; however, the copoly- mer melt undergoes phase separation below the critical temperature due to the incompatibility of dif- ferent blocks [1]. As a result of phase separation, periodic structures including lamellae [2–7], spheres [2, 3, 8–12], cylinders [2, 3, 6, 10, 13, 14], hexagons [2, 3, 7, 10, 13–17], and gyroids [2, 3, 10] are observed in a mesoscopic-scale domain. Figure 1. Examples of local defects. In recent years, self-assembly of block copolymer has come out as a promising patterning tool to over- come the scaling limits in nano-lithography and generate suboptical lithographic patterns [18]. However, one of the problems is the lack of complete pattern orientation due to a high density of defects [19]. In figure 1, we can observe various examples of local defect in the block copolymer. Therefore, it is very important to control the local defects of self-assembled polymer patterns with the application of these materials. As the efforts to rectify this, many researches and techniques such as electric fields [20], flow ∗Corresponding author, E-mail: cfdkim@korea.ac.kr. © D. Jeong, Y. Choi, J. Kim, 2016 33001-1 http://dx.doi.org/10.5488/CMP.19.33001 http://www.icmp.lviv.ua/journal D. Jeong, Y. Choi, J. Kim [21], shear application [22–24], thermal treatment [25], chemically pre-patterned surface (chemoepitaxy) [26, 27], and topographical confinement (graphoepitaxy) [28] have been carried out to reduce the defect density in specific pattern-forming block copolymer thin films. Among the controlling method, authors in [19] proposed an appropriate substrate design and achieved a defect-free pattern formation. In this paper, we focus on numerically realizing the situation presented in [19] and we describe in detail the numerical method which is used in the numerical simulations. We use the mathematical model proposed by Ohta and Kawasaki [29]. Let φ be the difference of the local volume fraction of A and B monomers. Then, the nonlocal Cahn-Hilliard (CH) equation in a two- dimensional domain is ∂φ(x, t ) ∂t = ∆µ(x, t )−α[ φ(x, t )− φ̄] , (1) µ(x, t ) = F ′(φ(x, t ) )−ε2∆φ(x, t ), (2) where x = (x, y) and t are the spatial and temporal variables, respectively. F (φ) = 0.25(φ2 − 1)2 is the Helmholtz free energy, ε is the gradient energy coefficient, α is inversely proportional to the square of the total chain length of the copolymer, and φ̄ = ∫ Ωφ(x,0)dx/\|Ω\| is the average concentration over the domain Ω [30]. In equation (1), α[φ(x, t )− φ̄] term indicates the long-range interaction and plays an important part in pattern formation. If α= 0, then equations (1) and (2) describe the process of the reduction in the total interfacial energy of a microstructure as the classical CH equation. The total system energy is given as E (φ) = ∫ Ω [ F (φ)+ ε2 2 ∣∣∇φ∣∣2 ] dx+ α 2 ∫ Ω ∫ Ω G(x−y) [ φ(x)− φ̄][ φ(y)− φ̄] dydx , (3) whereG is the Green’s function of −∆ in Ω with periodic boundary conditions, i.e., −∆G(x) = δ(x). Then, the evolving equations (1) and (2) can be derived using the H−1 gradient flow for the free energy (3), and equation (3) can be rewritten as E (φ) = ∫ Ω [ F (φ)+ ε2 2 ∣∣∇φ∣∣2 ] dx+ α 2 ∫ Ω ∣∣∇ψ∣∣2 dx , where ψ satisfies −∆ψ=φ− φ̄ with periodic boundary conditions [2]. Now, we will solve equations (1) and (2) on a trench domain. Figure 2 represents the physical do- main (Ω) and boundaries (Γ1, Γ2). On Γ1, Dirichlet boundary condition for φ and homogeneous Neumannboundary condition for µ are used. On Γ2, the periodic boundary condition for both φ and µ is used. The rest of this paper is organized as follows. In section 2, we describe the numerical method and so- lution. In section 3, we present several numerical experiments. Conclusions are summarized in section 4. . Γ1 Γ1 Γ2Γ2 Ω Γ 1 [φ : Dirichlet boundary, µ : Neumann boundary] Γ 2 [φ, µ : Periodic boundary] Figure 2. Illustration of the physical domain (Ω) with boundaries Γ1 and Γ2. 33001-2 Numerical investigation of local defectiveness control of diblock copolymer patterns 2. Numerical method 2.1. Discretization of domain First, assume that we have a domain Ω as shown in figure 2. The domain Ω is defined by the angle θ, reference values a and b for the trench wall as represented in figure 3. Here, the trench walls are determined with symmetric points (−a,b), (a,b), (−a,−b), and (a,−b). Then, we cover the domain Ω by a rectangular domain ΩR = (−Lx ,Lx )× (−Ly ,Ly ) with a Cartesian grid of mesh size h. Now, we discretize the rectangular domain ΩR with the uniform mesh size h = 2Lx /Nx = 2Ly /Ny inboth x- and y -directions. Here, Nx and Ny are the number of grid points in x- and y -directions, respec- tively. We denote cell-corner points as (xi , y j ) = (hi ,h j ) for i = 0, . . . , Nx and j = 0, . . . , Ny . Let φn i j and µn i jbe approximations of φ(xi , y j , tn) and µ(xi , y j , tn), respectively, where tn = n∆t and ∆t is the temporal step size. Lx−Lx Ly −Ly x y 0 (a, b)(−a, b) (a,−b)(−a,−b) Γ1 Γ1 Γ2Γ2 θ ΩR Γ 1 [φ : Dirichlet boundary, µ : Neumann boundary] Γ 2 [φ, µ : Periodic boundary] Figure 3. Illustration of the parameters over the whole domain ΩR = (−Lx ,Lx )× (−Ly ,Ly ). Γ1 and Γ2are boundary of the computational domain which is determined from θ. Trench walls are defined with symmetric points (−a,b), (a,b), (−a,−b), and (a,−b). 2.2. Numerical solution In this paper, we apply a non-linearly stabilized splitting scheme [31] to the nonlocal CH equations (1) and (2) as follows: φn+1 i j −φn i j ∆t = ∆hµ n+1 i j −α ( φn+1 i j − φ̄ ) , (4) µn+1 i j = ( φn+1 i j )3 −φn i j −ε2∆hφ n+1 i j , (5) where φ̄=∑ xi j ∈Ωh φ0 i j /∑ xi j ∈Ωh 1. Here,Ωh is the computational domain which is represented bymarkedcircle in figure 4. To solve equations (4) and (5), we use the Gauss–Seidel iterative method. Given solution φn i j , let φn+1,0 i j =φn i j be an initial guess. For eachm Ê 0, we generate the updated solution φn+1,m+1 i j and µn+1,m+1 i j from φn+1,m i j and µn+1,m i j by ( 1 ∆t +α ) φn+1,m+1 i j + 4 h2µ n+1,m+1 i j = φn i j ∆t +αφ̄+ µn+1,m+1 i−1, j +µn+1,m i+1, j +µn+1,m+1 i , j−1 +µn+1,m i , j+1 h2 , (6) 33001-3 D. Jeong, Y. Choi, J. Kim φ : Dirichlet boundary, µ : Neumann boundary Interior domain Ω h Periodic boundary Figure 4. Inner grid points (•) which are on the computational domainΩh , Dirichlet (φ) and homogeneousNeumann (µ) boundary points (◦), and periodic boundary points (�). [ −4ε2 h2 −3 ( φn+1,m i j )2 ] φn+1,m+1 i j +µn+1,m+1 i j =−φn i j −2 ( φn+1,m i j )3 −ε2 φn+1,m+1 i−1, j +φn+1,m i+1, j +φn+1,m+1 i , j−1 +φn+1,m i , j+1 h2 . (7) We continue the above iterations until l2-norm error between two successive approximations of φ is lessthan a given tolerance tol, that is, ∥∥φn+1,m+1 −φn+1,m∥∥ 2 < tol. 2.3. Boundary conditions For a numerical solution, we consider three different conditions at each boundary as follows: • φi j = ‖φeq‖∞ for xi j ∈ Γ1. • ∇hµi j = 0 for xi j ∈ Γ1. • φ0 j =φNx+1, j and µ0 j =µNx+1, j for j = 1, . . . , Ny +1. Here, ‖φeq‖∞ represents the maximum value of numerical solution at equilibrium state. In subsection2.4, we will describe more details for ‖φeq‖∞.Near the boundaries, we should use some special formulae. For example, let us consider the position (xi , y j ) in figure 5. By the Dirichlet boundary condition, we already know the value at A and B . We define φA φB βh αh φij φi+1,j φi,j−1 θ (a, b) (a) µA µB ph qh µ ij µ i+1,j µ i,j−1 µp µq θ (a, b) (b) Figure 5. (a) Dirichlet condition and (b) Neumann condition on curved boundary. 33001-4 Numerical investigation of local defectiveness control of diblock copolymer patterns ∆Dxx and ∆Dy y as the discrete second derivatives near the boundary as follows: ∆Dxxφi j = ( φi+1, j −φi j h − φi j −φA αh )( αh +h 2 )−1 , (8) ∆Dy yφi j = ( φB −φi j βh − φi j −φi , j−1 h )( βh +h 2 )−1 , (9) where 0 < α, β < 1, and φA = φB = ‖φeq‖∞. Therefore, the discrete Laplacian operator near the bound-ary with Dirichlet condition is defined as ∆Dhφn+1 i j = ∆Dxxφ n+1 i j +∆Dy yφ n+1 i j . For other points, the discreteLaplacian is similarly defined. We also define the discrete Laplacian operator near the boundary with Neumann boundary condition as ∆Nhµn+1 i j =∆Nxxµ n+1 i j +∆Ny yµ n+1 i j . Here, ∆Nxxµi j = ( µi+1, j −µi j h − µi j −µq αh )( αh +h 2 )−1 , (10) ∆Ny yµi j = ( µp −µi j βh − µi j −µi , j−1 h )( βh +h 2 )−1 , (11) where α and β are defined as in figure 5 (a). µp and µq are obtained by using a linear interpolation, µp = pµi+1, j + (1−p)µi j and µq = qµi j + (1−q)µi , j−1 [see figure 5 (b)]. 2.4. Optimal wavelength having minimum discrete total energy We describe an algorithm for finding the total energy-minimizing wavelength [1, 4]. We define the optimal wavelength L∗ as the period of the hexagonal lattice that has the lowest energy. In other words, L∗ means the smallest length having the global minimum of the domain-scaled discrete total energy. To calculate L∗, we solve equations (1) and (2) until a numerical equilibrium state is reached with the given values of hx , ∆t , ε, and α. The initial condition is φ(x,0) = 0.1cos(2πx/Lx ) in Ω = (0,Lx ), where Lx starts at 2hx and increases in steps of 2hx . Let M be the smallest even integer such that the domain- scaled total energy E d/Lx is minimized. Construct the quadratic polynomial passing the three points( (M −2)hx , E d/[(M −2)hx ] ), (Mhx , E d/(Mhx ) ), and ( (M +2)hx , E d/[(M +2)hx ] ); then, define the op- timal length L∗ as the critical point of the polynomial [see figure 6 (a)]. For more details, see refer- ences [1, 4]. We define the numerical equilibrium state as that in which the consecutive error is not larger than the prescribed tolerance, that is,max1ÉiÉNx (\|φk+1 i −φk i \|)/∆t É 1.0×10−6. The maximum value of equilibrium wave is defined as ‖φeq‖∞ = max1ÉiÉNx \|φeqi \| in figure 6 (b). We replace the Dirichlet problem solution with ‖φeq‖∞ in this paper. Ed Lx Lx oprimal length L ∗ LM−2 LM LM+2 0 ‖φeq‖∞ L∗ x φ (a) (b) Figure 6. (a) Schematic of algorithm to search for the optimal length L∗. Here, LM−2 = (M −2)hx , LM = Mhx , and LM+2 = (M +2)hx . (b) Illustration of maximum value ‖φeq‖∞ of equilibrium wave. 33001-5 D. Jeong, Y. Choi, J. Kim 3. Numerical experiments In this section, we perform a number of numerical tests. Throughout the numerical experiments, unless otherwise specified, we use ε = 1/(20 p 2), α = 100, L∗ = 0.375, h = L∗/10, ∆t = 0.1h, ‖φeq‖∞ = 0.6134, and θ =π/4. We examine the evolution of a randomperturbation about the average concentration φ̄= 0 on simple rectangle domainΩR = (−25L∗,25L∗)×(−15L∗,15L∗)withNx = 500,Ny = 300. The initial condition is set to φ(x, y,0) = φ̄+0.01 rand(x, y). Here, rand(x, y) is a random number between −1 and 1. Also, we use tol = 10−4 for stopping criterion of the Gauss-Seidel iteration. 3.1. Discrete total energy We first define the discrete total energy as E d(φn) = Nx∑ i=1 Ny∑ j=1 { h2F (φn i j )+ ε2 2 [( φn i+1, j −φn i j )2 + ( φn i , j+1 −φn i j )2 ] + α 2 [( ψn i+1, j −ψn i j )2 + ( ψn i , j+1 −ψn i j )2 ]} . Note that ψ satisfies −∆ψ=φ− φ̄ with periodic boundary conditions [2]. Figure 7 shows the temporal evolution of the normalized discrete total energy E d(φn)/E d(φ0). In figure 7, we can see that the normalized discrete total energy (which is denoted by the solid line) is nonincreasing as time proceeds. Moreover, the four small figures represent the numerical solution at times t = 30∆t , 100∆t , 700∆t , 2000∆t , respectively. 0 1 2 3 4 5 6 7 0.7 0.8 0.9 1.0 t Ed(φn) Ed(φ0) Figure 7. Time evolution of the normalized discrete total energy E d(φn )/E d(φ0). Here, the small figures indicate the concentration field φ at times t = 30∆t , 100∆t , 700∆t , 2000∆t , respectively. 3.2. The effect of channel width To investigate the effect of the channel width, we fix a = 5L∗ with b = 2L∗ and b = 5L∗. Figures 8 (a) and (b) show the temporal evolution of φ with the trench widths 2b = 4L∗ and 2b = 10L∗, respectively. We can observe that the self-assembled pattern is completely defect-free and is aligned parallel to the trench walls within the narrow trench area; all the defects reside in the wider regions on either side, which is consistent with the experimental results [19]. 33001-6 Numerical investigation of local defectiveness control of diblock copolymer patterns (a) (b) t = 30∆t t = 100∆t t = 2000∆t Figure 8. The effect of different trench width: (a) 4L∗ and (b) 10L∗. Evolution times are given below each figure. 3.3. The effect of channel length In this section, we simulate two cases with respect to a narrow channel length. For this test, we use two different values a = 4.5L∗ and a = 9L∗ when we fix b = 4.5L∗. The numerical results can be seen in figure 9. Similarly to the previous tests, we can see that the numerical solution in the narrow channel has the defect-free lamella pattern. (a) (b) t = 30∆t t = 100∆t t = 2000∆t Figure 9. The effect of different trench length: (a) 9L∗ and (b) 18L∗. Evolution times are given below each figure. 3.4. The effect of angle To see the dynamics of the angle, we only change the angle as θ =π/3, π/4, and π/6with a = b = 5L∗. Figure 10 represents the temporal evolution of pattern formation in channels with respect to the angle. In all three cases, we observe that the numerical solution in the narrow channel has aligned lamella patterns parallel to the trench walls. Also, within the narrow trench region, the self-assembled pattern is defect-free unlike the side region where all the defects are located. Figure 11 shows the profiles of φ at equilibrium state for each ε= 0.02, 0.03, and 0.04. From the result in figure 11, as ε value is increasing, we observe that the amplitude of φ is smaller and the wavelength is wider. 33001-7 D. Jeong, Y. Choi, J. Kim (a) (b) (c) t = 40∆t t = 100∆t t = 2000∆t Figure 10. The effect of the angle: (a) θ = π/3, (b) π/4, and (c) π/6. Evolution times are given below each figure. 3.5. Comparison of Dirichlet and Neumann boundary conditions In this section, we compare numerical results by the Dirichlet and Neumann boundary conditions. We have the comparison test on the same geometry shown in figure 10 (c). Figure 12 (a) shows the tem- poral evolution of φ when applying Dirichlet and homogeneous Neumann conditions for φ and µ on the boundary Γ1, respectively. Figure 12 (b) represents the temporal evolution of φ when applying homoge-neous Neumann condition forφ and µ on the boundary Γ1. As we expected, we obtain the lamella patternin the narrow channel when we apply the Dirichlet boundary condition on Γ1. However, the numericalsolution with the zero homogeneous boundary condition for φ has many defects in the narrow channel and a contact angle of 90◦ on all boundaries. 0 0.1 0.2 0.286 0.346 0.396 −1 −0.5 0 0.5 1 x φ ǫ = 0.02 ǫ = 0.03 ǫ = 0.04 Figure 11. Profiles of φ at the equilibrium state when ε = 0.02, 0.03, and 0.04. We reprinted from [35], with permission from the Current Applied Physics. 33001-8 Numerical investigation of local defectiveness control of diblock copolymer patterns (a) (b) t = 40∆t t = 100∆t t = 2000∆t Figure 12. Time evolution ofφwhen applying (a) the Dirichlet and (b) the homogeneous Neumann bound- ary condition for φ on Γ1. The other boundary conditions are used to be equal to the previous examples.Here, we denote the simulation time on the bottom of figures columns line. 4. Conclusions In this paper, we numerically investigated the local defectiveness control of self-assembled diblock copolymer patterns through appropriate substrate design. We used a nonlocal Cahn-Hilliard equation for the phase separation dynamics of diblock copolymers. We discretized the nonlocal CH equation by an unconditionally stable finite difference scheme on a tapered trench design and, in particular, we used Dirichlet, Neumann, and periodic boundary conditions. The value at the Dirichlet boundary is obtained from energy-minimizing wavelength. We solved the resulting discrete equations using the Gauss-Seidel iterative method. We performed various numerical experiments to know the effect of the channel width, length, and angle. Our simulation results were consistent with real experimental observations. Acknowledgement The first author (D. Jeong) was supported by a Korea University Grant. The corresponding author (J.S. Kim) was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (NRF-2014R1A2A2A01003683). References 1. Jeong D., Lee S., Choi Y., Kim J., Curr. Appl. Phys., 2015, 15, 799; doi:10.1016/j.cap.2015.04.033. 2. Choksi R., Peletier M.A., Williams J.F., SIAM J. Appl. Math., 2009, 69, 1712; doi:10.1137/080728809. 3. Fink Y., Urbas A.M., Bawendi M.G., Joannopoulos J.D., Thomas E.L., J. Lightwave Technol., 1999, 17, 1963; doi:10.1109/50.802981. 4. 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Theory Simul., 2000, 9, 363; doi:10.1002/1521-3919(20000801)9:7<363::AID-MATS363>3.0.CO;2-7. 33. Choksi R., Ren X., J. Stat. Phys., 2003, 113, 151; doi:10.1023/A:1025722804873. 34. Choksi R., Peletier M.A., Williams J.F., SIAM J. Appl. Math., 2009, 69, 1712; doi:10.1137/080728809. 35. Jeong D., Shin J., Li Y., Choi Y., Jung J.H., Lee S., Curr. Appl. Phys., 2014, 14, 1263; doi:10.1016/j.cap.2014.06.016. Числове дослiдження керування локальною дефектнiстю структур дiблок-кополiмерiв Д. Йонг, Й. Чоi,Ю. Кiм Факультет математики, Корейський унiверситет, Сеул 136-713, Республiка Корея Проведено числове дослiдження керування локальною дефектнiстю самоорганiзованих структур дiблок- кополiмерiв за допомогою вiдповiдної конструкцiї субстрату. Використовується нелокальне рiвняння Кана-Хiлларда для динамiки фазового роздiлення дiблок-кополiмерiв. Здiйснено дискретизацiюнелокаль- ного рiвняння з використанням безумовно стiйкої схеми скiнченної рiзницi на звуженiй канавцi зразка i, зокрема, використано крайовi умови Дiрiхле,Ньюмана i перiодичнi граничнi умови. Значення при крайо- вих умовах Дiрiхле отримано згiдно з рiвноважним ламеларним профiлем, що вiдповiдає енергетично- му мiнiмуму. Ми розв’язуємо отриманi дискретнi рiвняння, використовуючи iтеративний метод Гаусса- Зейделя. Проведено рiзнi числовi експерименти, такi як вплив ширини каналу, довжини каналу та кута на динамiку фазового роздiлення. Результати симуляцiй вiдповiдають попереднiм експериментальним спостереженням. Ключовi слова: дiблок-кополiмери, нелокальне рiвняння Кана-Хiлларда, керування локальною дефектнiстю 33001-10 http://dx.doi.org/10.1126/science.1168108 http://dx.doi.org/10.1039/c4sm01884b http://dx.doi.org/10.1088/0953-8984/27/19/194101 http://dx.doi.org/10.1016/j.colsurfa.2008.08.015 http://dx.doi.org/10.1002/app.40790 http://dx.doi.org/10.1002/adma.200602470 http://dx.doi.org/10.1126/science.273.5277.931 http://dx.doi.org/10.1063/1.2723673 http://dx.doi.org/10.1002/adma.200400643 http://dx.doi.org/10.1088/0953-8984/27/10/103102 http://dx.doi.org/10.5488/CMP.18.23801 http://dx.doi.org/10.1088/0953-8984/23/25/254213 http://dx.doi.org/10.1038/nature01775 http://dx.doi.org/10.1021/ma302464n http://dx.doi.org/10.1002/1521-4095(200108)13:15%3C1152::AID-ADMA1152%3E3.0.CO;2-5 http://dx.doi.org/10.1021/ma00164a028 http://dx.doi.org/10.1016/0167-2789(95)00005-O http://dx.doi.org/10.1557/PROC-529-39 http://dx.doi.org/10.1002/1521-3919(20000801)9:7%3C363::AID-MATS363%3E3.0.CO;2-7 http://dx.doi.org/10.1023/A:1025722804873 http://dx.doi.org/10.1137/080728809 http://dx.doi.org/10.1016/j.cap.2014.06.016 Introduction Numerical method Discretization of domain Numerical solution Boundary conditions Optimal wavelength having minimum discrete total energy Numerical experiments Discrete total energy The effect of channel width The effect of channel length The effect of angle Comparison of Dirichlet and Neumann boundary conditions Conclusions

Numerical investigation of local defectiveness control of diblock copolymer patterns

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