defects and their removal in block copolymer thin film simulationskatsov/pubs/pub023.pdf ·...

Defects and Their Removal in Block CopolymerThin Film Simulations

AUGUST W. BOSSE,1 SCOTT W. SIDES,2 KIRILL KATSOV,3 CARLOS J. GARCÍA-CERVERA,4 GLENN H. FREDRICKSON5

1Department of Physics, University of California, Santa Barbara (UCSB), California

2Tech-X Corporation, Boulder, Colorado

3Materials Research Laboratory, UCSB, California

4Department of Mathematics, UCSB, California

5Departments of Chemical Engineering and Materials, UCSB, California

Received 15 January 2006; revised 28 March 2006; accepted 29 March 2006DOI: 10.1002/polb.20905Published online in Wiley InterScience (www.interscience.wiley.com).

ABSTRACT: In recent years, there has been increased interest in using microphase-separated block copolymer thin films as submicrometer/suboptical masks in nextgeneration semiconductor and magnetic media fabrication. With the goals of removingmetastable defects in block copolymer thin film simulations and potentially examiningequilibrium defect populations, we report on two new numerical techniques that canbe used in field-theoretic computer simulations: (1) a spectral amplitude filter (SF) thatencourages the simulation to relax into high symmetry states (representing zero defectstates), and (2) different variants of force-biased, partial saddle point Monte Carlo algo-rithms that allow for barrier crossing toward lower energy defect-free states. Beyondtheir use for removing defects, the force-biased Monte Carlo algorithms will be seento provide a promising tool for studying equilibrium defect populations. © 2006 WileyPeriodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2495–2511, 2006Keywords: diblock copolymers; mean-field theory; microdomain defects; Monte Carlo;self-consistent field theory; simulations; thin films

INTRODUCTION

Microphase-separated block copolymer thin filmshave emerged as a promising new vehicle forthe creation of submicrometer/suboptical masksin next generation semiconductor and magneticmedia.1–4 Depending on the specific architec-ture and composition of the block copolymer, themicrophase-separated system can yield a varietyof ordered microdomains. Microphase separation

Correspondence to: A. W. Bosse (E-mail: [email protected])Journal of Polymer Science: Part B: Polymer Physics, Vol. 44, 2495–2511 (2006)© 2006 Wiley Periodicals, Inc.

occurs on the 10-nm scale, yet the resultingmicrodomains often exhibit defect structures thatextend to the micrometer scale.

Not surprisingly, large ordered arrays of nano-meter-scale microdomains represent a promisinglithographic tool. For example, in the case of mag-netic media fabrication (flash memory), one mightdesire a large, regular array of magnetic dots. Aphase-separated block copolymer thin film couldact as a lithographic mask for the creation of sucha well-ordered magnetic array. However, if blockcopolymer masking techniques are to evolve intoa realistic and commercially viable fabricationscheme, one must have control over, or at least anunderstanding of, defect structures and the factors

2495

2496 BOSSE ET AL.

that control long-range lateral order in copolymerthin films.

The problem of controlling and understand-ing microdomain ordering in block copolymer thinfilms has garnered much attention from polymerexperimentalists.5–7 For example, Segalman et al.have shown that a polystyrene-b-(2-vinylpyridine)(PS-PVP) block copolymer thin film (with fPVP

= 0.129 and N = 670), which consists of a singlelayer of spherical PVP-domains in a PS matrix,behaves as a 2D crystal–hexatic–liquid system,consistent with the predictions of the Kosterlitz–Thouless–Halperin–Nelson–Young (KTHNY) the-ory of 2D melting.5 According to this theory,melting of a 2D crystal of microdomains proceedsthrough two continuous phase transitions, first toa hexatic phase and then into a liquid. Each phaseis characterized by a well defined equilibriumdistribution of microdomain defects.

In addition to equilibrium defect populations,it is common to observe nonequilibrium distrib-utions of metastable defects. Qualitatively, it isuseful to think of metastable states as correspond-ing to local minima in a free energy landscape andbeing separated from each other by substantial(at least a few kBT) activation barriers. In exper-iments, one often handles nonequilibrium defectsvia thermal annealing. In computer simulations,both particle-based8 and field-theoretic,9 it is typ-ically very difficult to relax toward a defect-freestate or an equilibrium defect population becauseof the time limitations that are inherent to themethods.

There are various field-based simulationmethodologies that are successful at yieldingdefect-free states, including unit-cell calculationsby means of finite differences, pseudo-spectraltechniques, or the spectral method of Matsen andSchick.9,10 These methods require either seedingthe simulation cell with a specific microdomainmorphology or imposing a specific symmetry sothatonlycertainsymmetry-constrainedstructurescan form. However, if one is primarily interested inequilibrium defect populations, or if one wishes toallow symmetries to set in in an unbiased manner(for example, if the symmetries of the microphasesare unknown), then one needs to apply large-cellsimulations. In such simulations, periodic bound-ary conditions are typically imposed, but the sizeof the computational cell is taken to be much largerthan the microdomain period.

In the present paper we address two issues thatrelate to large-cell field theoretic simulations ofblock copolymers:

• We develop techniques for achieving defect-free mean-field solutions. This is especiallyimportant when conducting simulations ina discovery mode, where the microdomainsymmetry is not known in advance. Thehighly defective metastable states obtainedin such simulations often do not possess suf-ficient long-range order to clearly identify theunderlying defect-free periodic ground state.

• We develop simulation techniques that cangenerate equilibrium defect populations,such as those observed by Segalman et al.5

in block copolymer thin films.

Consequently, the purpose of this study is todevelop computational methods for removing andevolving microdomain defects in unbiased large-cell block copolymer simulations. For the purposeof removing defects in block copolymer simula-tions, we develop two computational methods: (1) aspectral amplitude filtering (SF) algorithm, and(2) several variants of force-biased Monte Carlo(MC) algorithms. The MC methods also appear tobe useful in generating equilibrium defect popula-tions, and so address our second objective.

In Background section we provide some back-ground relevant to the problem of microdomaindefects in block copolymer thin films. Model andSCFT section details our polymer model, the mean-field approximation leading to self-consistent fieldtheory (SCFT), and our pseudo-spectral numer-ical methods, including the semi-implicit Seidel(SIS) technique for relaxing mean-field solutions.In Spectral Amplitude Filtering section we intro-duce and describe an SF algorithm for the elim-ination of defects. Partial Saddle Point MonteCarlo section describes a family of force-biased MCalgorithms (PSP-MC), based on a partial saddle-point (PSP) approximation to the underlying fieldtheory model. Results section provides a compar-ison of the ability of these techniques to elimi-nate microdomain defects in two-dimensional (2D)diblock copolymer thin films. Finally, Discussionand Conclusions section summarizes our resultsand provides some concluding remarks.

BACKGROUND

As mentioned earlier, KTHNY theory predictsequilibrium populations of lattice defects in 2Dsystems (here “lattice” refers to the characteris-tic lattice formed by the phase-separated micro-domains). A useful way to identify lattice defects

Journal of Polymer Science: Part B: Polymer PhysicsDOI 10.1002/polb

DEFECTS REMOVAL IN BLOCK COPOLYMER SIMULATIONS 2497

is via a Voronoi diagram.5 The diagram is con-structed from the polyhedra formed by perpendic-ular bisection of the bonds between all neighboringlattice points. The number of sides of the result-ing polyhedra indicates the number of nearestneighbors; thus, the Voronoi diagram allows oneto quickly identify the coordination of each latticepoint (here we define “lattice point” as the centroidof a microdomain).

For the system of interest, an asymmetric ABdiblock copolymer with composition f > 0.5 (Ablock volume fraction) in 2D, the copolymer meltcan microphase separate into an ordered hexago-nal array of B cylinders, depending on the specificvalues of f and the Flory parameter χAB.10,11 In2D, the only nonlamellar microphase is a hexag-onally ordered cylindrical phase. For a perfecthexagonal lattice, each B cylinder will have sixnearest neighbors. Therefore, topological latticedefects are characterized by microdomains withless than or more than six nearest neighbors; inother words, Voronoi polygons with other than sixsides. It is worth noting that the three phases con-sidered in the KTHNY theory can be distinguishedby the number and type of nonhexagonal Voronoipolygons.5

The standard field theory model of an incom-pressible block copolymer melt on which oursimulations are based9,10,12 applies Hubbard–Stratonovich (HS) fields to decouple the variouspolymers in the system. The mean-field approx-imation (SCFT) corresponds to a saddle-pointapproximation for the functional integrals overthese fields. To compute SCFT solutions, i.e. to con-duct SCFT simulations, it is necessary to have anumerical procedure for solving a Fokker–Planckequation describing the statistical mechanics of asingle copolymer in the HS “self-consistent” fields,and a second numerical algorithm for relaxing thefields to approach a saddle point. For the firstpurpose, we use a second-order accurate, pseudo-spectral algorithm based on operator splitting.13,14

In relaxing the fields, we use a variation of a con-tinuous steepest descent search.9 A fictitious timevariable is introduced, and at each moment in timethe system is moved along the gradient of theenergy functional to approach a saddle point.

Such SCFT simulations ultimately terminateat a stable or metastable saddle point solution,corresponding to a mean-field solution of theblock copolymer partition function. The continu-ous steepest descent scheme lacks a mechanismfor moving between metastable states (i.e., barriercrossing between saddle points). Although there

has been recent work on improving the speedand stability of the steepest descent search,15 bar-rier crossing and the avoidance of metastable,defect-laden states still present a problem.

One possible method of reducing defect popu-lations is to extend the steepest descent searchbeyond the mean-field approximation (SCFT) andincorporate thermal fluctuations. Such fluctua-tions may be sufficient to enable barrier cross-ing and annealing of defect structures. Thereare various ways to conduct such “field-theoreticsimulations,” the two most popular of which areMC and complex Langevin (CL) methods.9,16,17

Although these methods successfully extend thetheory beyond the mean-field approximation, theyhave problems of their own. First, and perhapsmost important, any given configuration obtainedby the stochastic methods are not solutions per se,but are snapshots suitable for calculating statis-tical averages. Here we are primarily concernedwith “ground-state” SCFT solutions. Second, fullMC/CL sampling requires a doubling in the num-ber of field variables used when compared withSCFT. Third, the CL approach is subject to accu-racy and stability limitations that can makethe simulations prohibitively slow for our pur-poses (although there has been recent progressin this issue18). Finally, full MC simulations offield-theoretic models of polymers exhibit a “signproblem”9,17 that can slow or even prevent conver-gence of this approach.

With the primary goal of removing metastabledefects in SCFT simulations of block copolymerthin films, we examine two algorithms: (1) anSF algorithm that dynamically amplifies symme-tries developing in the system and encourages thesteepest descent search to move in the appropri-ate direction, and (2) a family of force-biased MCalgorithms,which introduce controlled noise to thesteepest descent search, thus potentially facilitat-ing barrier crossing. As for our secondary goal,the MC algorithms developed here satisfy detailedbalance, and so they may prove useful in simulat-ing equilibrium defect populations as predicted byKTHNY theory and observed by Segalman et al.5

We note that Knoll et al. have recently reportedon the use of dynamic density functional theoryto examine the kinetics of the phase transitionbetween parallel cylinders and perforated lamellain block copolymer thin films.19 The results areinteresting and shed light on how defects moveand annihilate under diffusive dynamics duringthe course of such phase transitions. Nonethe-less, the Knoll et al. simulations focused on


2498 BOSSE ET AL.

microdomain kinetics during a block copolymerphase transition. Here we are concerned withrapidly locating mean-field solutions through theuse of fictitious, nonconservative dynamics, withstrategies for defect removal,and with the efficientsampling of equilibrium defect populations.

MODEL AND SCFT

Both of our microdomain defect removal algo-rithms are built on top of a standard field theorymodel for an incompressible AB diblock copoly-mer melt.10,17 In this section we provide a shortreview of the basic polymer model, the SCFT (sad-dle point) approximation, and the SIS algorithmfor obtaining saddle point solutions.

Block Copolymer Model

We consider n monodisperse AB diblock copoly-mers in a volume V . The index of polymerizationis denoted by N, and the fraction of A monomer isdenoted by f . We assume the statistical segmentlengths of the two polymers are equal bA = bB = b;therefore, in three spatial dimensions, the unper-turbed radius of gyration is given by R2

g0 = b2N/6.We restrict our attention to incompressible poly-mer melts, and thus the fixed, average monomerdensity is given by ρ0 = nN/V . The block copoly-mers are modeled using the continuous Gauss-ian chain model with space curves rα(s). Here α

= 1, 2, . . . , n is the polymer index, and s ∈ [0, 1] isa polymer contour length variable that runs froms = 0 at the beginning of the A side to s = 1 at theend of the B side. The canonical partition functionis given by (here and throughout we set kBT = 1)

Z =∫ (

n∏α=1

Drα

)δ[ρA + ρB − ρ0] exp(−H0 − HI),

(1)

where H0 is the harmonic chain stretching penalty,

H0 = 14R2

g0

n∑α=1

∫ 1

0ds

∣∣∣∣drα(s)ds

∣∣∣∣2

, (2)

and HI is the Flory monomer–monomer interac-tion,

HI = χ

ρ0

∫V

dr ρA(r)ρB(r). (3)

Here χ = χAB is the Flory interaction parame-ter for interactions between A and B monomers.

The microscopic A and B monomer densities aredefined as follows:

ρA(r) = Nn∑

α=1

∫ f

0ds δ(r − rα(s)), (4)

ρB(r) = Nn∑

α=1

∫ 1

fds δ(r − rα(s)). (5)

In the partition function eq 1,∫ (∏n

α=1 Drα

)repre-

sents a functional integral over all chain configu-rations of all of the polymers, and δ[ρA + ρB − ρ0]is a delta functional that enforces the incompress-ibility constraint of the melt, ρA(r) + ρB(r) = ρ0 atall points.

The next step is to perform a Hubbard-Stratonovich transformation on the partition func-tion. This replaces monomer–monomer interac-tions (contained in HI) with a single polymerinteracting with an external field. First we intro-duce a change of variables

ρ±(r) = ρA(r) ± ρB(r). (6)

The change of variables allows us to write

HI = −14

χ

ρ0

∫V

dr [ρ−(r) − (2f − 1)ρ0]2, (7)

up to a constant shift in energy. We can nowconvert eq 1 into a field theory by performing aHubbard-Stratonovich transformation

exp[− 1

4a

∫V

dr b2(r)

]=

∫Dw− exp

[∫V

dr[−aw2

−(r) + b(r)w−(r)]]

,

(8)

and using the exponential representation of thedelta functional,

δ[c] =∫

Dw+ exp[−i

∫dr w+(r)c(r)

]. (9)

Here b(r) = ρ−(r) − (2f − 1)ρ0, a = ρ0/χ , andc(r) = ρA(r)+ρB(r)−ρ0. After some simple algebraand rescalings, we obtain

Z =∫

DW+DW− exp (−H[W+, W−]) (10)



where

H[W+, W−] = C∫

Vdx [−iW+(x) + (2f − 1)W−(x)

+ W2−(x)/χN] − CV log Q[iW+ − W−, iW+ + W−].

(11)

Here, x is the dimensionless spatial coordinatex = r/Rg0 and C = (ρ0/N)R3

g0 is a dimensionlesschain concentration. From here on, all lengths areexpressed in units of Rg0 (e.g., V/R3

g0 → V ). Theparameter C can be viewed as an effective coor-dination number (the average number of chainsthat penetrate the coil of a given copolymer) andwill be seen to control the strength of fluctuationsin the system. The conjugate fields w±, defined ineqs 8 and 9, are also given a new normalization:W± = Nw±.

In the Hamiltonian eq 11, Q represents the par-tition function for a single AB diblock copolymerin an external field. The A block interacts with thefield WA = iW+ − W− and the B block interactswith the field WB = iW+ + W−. We can calculate Qusing the propagator, q(x, 1; [WA, WB]):

Q[WA, WB] = 1V

∫V

dx q(x, 1; [WA, WB]). (12)

The propagator q(x, 1; [WA, WB]) gives the proba-bility density of finding a polymer whose chain endis at position x; it satisfies the following modifieddiffusion equation:

∂

∂sq(x, s) = ∇2q(x, s) − ψ(x, s)q(x, s), (13)

subject to the initial condition q(x, 0) = 1 and with

ψ(x, s) ={

iW+(x) − W−(x), 0 < s < fiW+(x) + W−(x), f < s < 1.

(14)

The microscopic monomer densities are computedusing the propagator as follows:

φA(x; [WA, WB]) = 1Q

∫ f

0ds q(x, s)q†(x, 1 − s), (15)

φB(x; [WA, WB]) = 1Q

∫ 1

fds q(x, s)q†(x, 1 − s). (16)

Here, φJ(x; [WA, WB]) = ρJ(x; [WA, WB])/ρ0, J= A, B is the reduced local monomer density (i.e.,the local volume fraction of segments of type J),and q†(x, s) is a backwards propagator. The back-wards propagator is needed because the copolymer

chain is not symmetric; it satisfies the followingequation:

∂

∂sq†(x, s) = ∇2q†(x, s) − ψ†(x, s)q†(x, s) (17)

subject to the initial condition q†(x, 0) = 1 andwith

ψ†(x, s) ={

iW+(x) + W−(x), 0 < s < 1 − fiW+(x) − W−(x), 1 − f < s < 1.

(18)

Self-Consistent Field Theory

Up to this point we have performed a series ofexact transformations on the original polymer par-tition function eq 1 to yield a new, formally equiv-alent field theory with partition function given byeq 10. We now shift our attention to mean fieldapproximations to the partition function given byeq 10.

The factor C appears in front of all terms ineq 11; therefore, for C → ∞ we can use themethod of steepest descent to validate saddle pointsolutions of the above theory. The saddle pointsolutions represent mean-field approximations toeq 10.17 The saddle point equations are given by

δH[W+, W−]δW±

∣∣∣∣W±

= 0, (19)

where W± are the saddle-point configurations ofthe fields W±.

Equation 19 represents four equations, one foreach component of the complex fields W±; however,it is known that the saddle-point configuration ofW+ is strictly imaginary and that of W− is strictlyreal.17 Therefore, we introduce the real valuedpressure field � = iW+ = −Im[W+] and the realvalued exchange field W = W− = Re[W−].

A continuous steepest descent search is the sim-plest and one of the most efficient ways to solvethe saddle point equations. As mentioned earlier,we introduce a fictitious time variable t, and ateach time we move the field values in the directionof the gradient of the Hamiltonian. By combin-ing eqs 11 and 19, the saddle point equations aregiven by

δH[�, W]δ�(x)

= C[φA(x) + φB(x) − 1] = 0, (20)


2500 BOSSE ET AL.

and

δH[�, W]δW(x)

= C[(2f − 1) + 2W(x)/χN

− φA(x) + φB(x)] = 0. (21)

Since the saddle point value W+ = −i� is purelyimaginary, the saddle point search is actually a“steepest ascent” in �. We write,

∂

∂t�(x, t) = δH[�, W]

δ�(x, t), (22)

∂

∂tW(x, t) = −δH[�, W]

δW(x, t). (23)

Clearly, when eqs 22 and 23 are stationary, thesaddle point equations are satisfied.

This completes the framework for SCFT. Weare able to search for mean-field solutions of thepolymer theory by iterating the following steps:

1. Initialize �(x, 0) and W(x, 0).2. Solve the modified diffusion equations: eqs

13 and 17.3. Calculate Q, φA, and φA using eqs 12, 15,

and 16.4. Update �(x, t) and W(x, t) by integrating

eqs 22 and 23 forward over a time intervalt.

5. Repeat Steps 2–5 until a convergence crite-rion has been met.

Numerical Methods

In the aforementioned Step 2, we solve the mod-ified diffusion equations using the pseudo-spectral operator splitting method developed byRasmussen and Kalosakas.9,13 This is a stable,fast, O(s2) accurate algorithm for solving themodified diffusion equation. In eq 13 we identifythe linear operator L = ∇2 − ψ(x, s). Formally,we can find the solution at a set of discrete con-tour locations s by propagating along the polymerchain,

q(x, s + s) = esLq(x, s), (24)

starting from the initial condition q(x, 0) = 1.Motivated by the Baker–Campbell–Hausdorff

identity,20 we perform an O(s2) splitting of esL:

esL = e−sψ(x,s)/2 es∇2e−sψ(x,s)/2 + O(s3). (25)

The potential ψ(r, s) is diagonal in real space andthe Laplacian is diagonal in Fourier space; thus,

explicit multiplication by e−sψ(x,s)/2 is evaluatedin real space, and the action of es∇2 is evaluatedas an explicit multiplication in Fourier space. Wemove between real and Fourier space using thefast Fourier transform (FFT).21

In executing Step 4 of the SCFT algorithm men-tioned earlier, our primary goal is to quickly solvethe saddle point equations in as stable a manneras possible. The accuracy of the time integrationis not important because we seek only station-ary states. Furthermore, we wish to avoid beingtrapped in metastable states that contain defects.In other words, we wish to find the lowest energy,or “ground state” mean-field solution.

It is convenient to discretize the system spa-tially on a 2D square lattice with periodic bound-ary conditions, and we assume that the system isuniform and finite in the third direction. That is,we set

xi = ix, i = 0, . . . , nx − 1,

yj = jy, j = 0, . . . , ny − 1,(26)

where nx and ny are the number of lattice pointsin the x and y directions respectively. In order tosimplify notation, we subsequently use a single,bold face index i to represent the ordered pair(i, j). The lattice parameters are defined as follows:x = Lx/nx and y = Ly/ny, where Lx and Ly arethe dimensions of the system in units of Rg0. Thedimension of the system in the uniform z-directionis denoted by Lz. Accordingly, the volume of thesystem is V = LxLyLz, the total number of latticepoints is M = nxny, and the area per grid point is2x = xy. This particular discretization repre-sents a uniform collocation grid, which allows us torapidly transform between real and Fourier spacevia the FFT, as mentioned earlier.

In the z-direction we assume that the systemis uniform, which roughly corresponds to the thinfilm morphology that is of interest here. How-ever, under this assumption of uniformity, the filmthickness Lz always appears in combination withC, the dimensionless polymer concentration, andcan simply be adsorbed into its definition.

Finally, we also discretize the chain contourvariable s:

sl = ls, l = 0, . . . , ns, (27)

where ns is the number of steps along the poly-mer backbone, and s = 1/ns is the contour stepsize. The fictitious time variable t is also sampled



at discrete intervals:

tm = mt, m = 0, . . . , nt. (28)

The value of t chosen depends on the methodused to solve the relaxation equations, and nt

defines the number of iterations used to relax theSCFT equations.

Semi-implicit Seidel

Until recently, only simple forward Euler algo-rithms and related explicit schemes had beenemployed for solving the relaxation equations,eqs 22 and 23:

�m+1i = �m

i + t∂H[�m, Wm]

∂�mi

, (29)

Wm+1i = Wm

i − t∂H[�m, Wm]

∂Wmi

, (30)

where superscripts m represent discrete stepsin the time variable tm. Although the forwardEuler algorithm has been successfully applied,it has a restrictive stability threshold; thus, oneis often limited to rather small time steps t.It is not uncommon that forward Euler steppingcan require many tens-of-thousands of field itera-tions to obtain a reasonably accurate saddle pointsolution.

A more stable algorithm for solving the relax-ation equations has been recently proposed byCeniceros and Fredrickson.15 There are variousways to interpret this time stepping algorithm (seeRefs. 9 and 15 for detailed discussions), but it isimplemented by simply adding and subtractinglinearized expressions (in � or W) for the deriv-atives of the Hamiltonian to the right hand sideof eqs 29 and 30. The subtracted term is evalu-ated at the current time m, and the added termis evaluated at the future time m + 1. The resultis an SIS update scheme that is implementedpseudospectrally in both real and Fourier space.For the present purposes, it is important to real-ize that the SIS algorithm is significantly morestable than the forward Euler algorithm. As aresult, a much larger time step can be used. It isnot uncommon for the SIS algorithm to outper-form the forward Euler algorithm by one or twoorders of magnitude in the number nt of time stepsrequired to achieve an SCFT solution of prescribedaccuracy.

SPECTRAL AMPLITUDE FILTERING

Although the SIS relaxation method is well suitedfor quickly finding a stationary saddle point solu-tion, it does not allow barrier crossing. As aresult, it is rather susceptible to getting caughtin metastable mean-field solutions. In the Resultssection we provide specific examples of metastableconfigurations that were reached via an SIS relax-ation.

The SF algorithm is designed to be used inconjunction with another relaxation method. TheSF algorithm is relatively intuitive. Consider arelaxed metastable solution of the saddle pointequations for a symmetric diblock copolymer withχN above the order–disorder transition (ODT).The density profile of the lowest-energy solutionfor such a system is a lamellar phase with per-fect one-dimensional compositional order. How-ever, metastable solutions typically exhibit mor-phologies that are locally lamellar, but whoselong-range order is destroyed by defects such asdislocations and kink grain boundaries. Even withthese defects, the Fourier transform of such com-position patterns retains some of the symmetry ofthe lowest energy ordered state. In Fourier space,the defects produce a spreading of the dominantpeaks and a roughened landscape in which thesepeaks are embedded. With this in mind, if we areable to sharpen the large amplitude peaks of thepower spectrum and remove the lower amplitudeFourier components, the resulting real space den-sities will exhibit increased long-range order. Asimple “spectral amplitude filter” that zeroes outall Fourier coefficients whose amplitude is lessthan some fraction of the largest peak amplitudeshould be effective in this regard.

Figure 1 (left) represents the modulus of thediscrete Fourier transform of W (exchange poten-tial), |Wk|, corresponding to a perfect hexagonalarray of microdomains (i.e., a composition profileexhibiting no microdomain defects) for a f = 0.7and χN = 17 AB diblock melt. This defect-freestate was reached by applying the spectral ampli-tude filter described later. Note that the Fourieramplitude spectrum consists of an array of sharppeaks. The location of these peaks correspondsto the hexagonal symmetry of the ordered micro-domains: the six largest peaks are equally spacedaround and equidistant from the (0, 0) mode.

Figure 1 (right) represents the Fourier ampli-tude spectrum of W corresponding to a disorderedarray of microdomains (i.e., a composition pro-file exhibiting multiple microdomain defects) for


2502 BOSSE ET AL.

Figure 1. (left) |Wk| for a defect-free ordered hexagonal lattice of microdomains. Thisstructure was obtained using the SF algorithm combined with SIS relaxation. (right)|Wk| for a defective array of microdomains. Although the array of microdomains lackslong-range order, short-ranged six-fold symmetry is apparent in the Fourier amplitudespectrum. Although hard to identify, there are six first-order peaks in |Wk|. In bothfigures, the x and y axes represent discrete modes in Fourier space.

a f = 0.7 and χN = 17 AB diblock melt. Notethat although there are still six first-order peaks inthe Fourier amplitude spectrum corresponding tohexagonal symmetry, the peaks are dramaticallyspread out. The SF algorithm is used to isolateand sharpen the largest peaks, and thus producea Fourier amplitude spectra resembling Figure 1(left).

To implement the SF algorithm, we identify thehighest amplitude peaks in the amplitude spec-trum of the � and W fields, then we zero out allwavenumbers that have an amplitude that is lessthan some fraction fmax of the maximum ampli-tude. The SF algorithm is summarized below:

1. �k = FFT[�i] and Wk = FFT[Wi].2. Filter �k: For all k, if (|�k| < fmax max[|�|]),

then �k = 0.3. Filter Wk:For all k, if (|Wk| < fmax max[|W|]),

then Wk = 0.4. �i = FFT−1[�k] and Wi = FFT−1[Wk].

This filtering algorithm is applied by alter-nating with a series of field relaxation steps, sothat the system is allowed to relax toward asaddle point, is then filtered, rerelaxed, filtered,

etc. Although in principle any suitable relax-ation scheme can be paired with the SF, we havefound that the SIS algorithm performs partic-ularly well in this role. The number of relax-ation steps between applications of the filter andthe amplitude threshold fmax of the filter canbe tuned to achieve optimal performance of thescheme.

As we shall see, the SF combined with SISrelaxation is a very powerful tool for elimina-tion of defects from 2D hexagonally ordered blockcopolymer mesophases. However, if the Fourierspace representation of � and/or W is particu-larly feature rich, then the SF can be too aggres-sive. This has been observed in nontrivial 3Dmicrophases such as the Ia3d cubic gyroid phase.The gyroid phase has a particularly complicatedFourier amplitude spectrum, with multiple peaksof various amplitudes, each playing a defining rolein the gyroid symmetry. When the SF is appliedwith too tight a threshold fmax in such cases, it isnot uncommon to see the filter direct the systeminto a nongyroid, metastable microphase (e.g., thelamellar phase).22 Thus, one must exercise cautionwhen applying this technique to complex blockcopolymer phases.



The SF algorithm is easily extended to morecomplicated polymer systems; however, its use-fulness will depend largely on the complexity ofthe Fourier amplitude spectrum. One can imag-ine extending the SF to an M-block copolymersystem (e.g., M = 3 corresponding to an ABCcopolymer). In such a system, one would apply thespectral filter to each conjugate potential field Wi,∀i = 1, 2, . . . , M, and to the pressure field W+. ForM > 2, we expect a large number of candidatemesophases. As mentioned earlier, however, thespectral filter does not perform well when appliedto complex block copolymer phases whose Fourierspectrum contains many secondary peaks of vary-ing intensity that are important to establish thesymmetry of the phase. Further study is neces-sary under such circumstances in order to explorethe usefulness of the SF.

PARTIAL SADDLE POINT MONTE CARLO

In this section we detail another algorithmdesigned to remove microdomain defects fromblock copolymer simulations, but that also has thepotential for achieving equilibrium defect popu-lations. As outlined earlier, our model of an ABblock copolymer melt includes two potential fields,W+ and W−. The pressure field W+ effectivelyenforces the local incompressibility constraintφA(r) + φB(r) = 1, and the exchange field W−couples to the A and B composition of the system.

Microdomain defect structures represent localchanges in the melt composition. Accordingly, itseems reasonable that by introducing controlledfluctuations to the exchange field relaxation, thesystem may relax to lower-energy, defect-free con-figurations. The PSP approximation allows oneto add fluctuations in W− while instantaneouslyrelaxing W+ to its mean-field value at the fixednew W− configuration. In the following, we definea family of MC methods to introduce fluctuationsin W−, hence the name partial saddle point MonteCarlo (PSP-MC).

The PSP approximation has been successfullyimplemented as an approximate way of sam-pling the equilibrium properties of various fluc-tuating polymer systems23,24 beyond the mean-field approximation. Although not exact, the PSPapproximation renders the effective Hamiltonianstrictly real and thus side-steps the sign problempresent in the full field theory. The PSP approxi-mation is reasonably accurate for the incompress-ible AB diblock melt model at large to moderate

values of C,23 but its general validity in other con-texts is questionable.9 Here, we simply use thePSP-MC method to add targeted, controlled noiseinto the system with the hope that this will facili-tate defect motion and annihilation. To the extentthat the PSP approximation is valid, the schemepotentially also provides a route to simulating andstudying equilibrium populations of defects.

The implementation of PSP-MC is straightfor-ward. The functional integrals associated withthe pressure field are evaluated using the saddle-point approximation, while the functional inte-grals associated with the exchange field are leftto be evaluated by MC sampling. A PSP-MC algo-rithm consists of the following steps, beginning attime tm and field configuration Wm:

1. Apply an MC trial move of choice to advanceWm to obtain a trial configuration WT.

2. Use the SIS relaxation method to relax �m

to its local saddle point value �T consistentwith the trial configuration WT.

3. Using the trial configurations WT and �T,and with a probability weight that satis-fies detailed balance, determine whether toaccept the combined trial move Wm → WT

and �m → �T. If the move is accepted, setWm+1 = WT and �m+1 = �T; otherwise, setWm+1 = Wm and �m+1 = �m.

We note that, in this PSP-MC scheme, trial movesinvolve global, rather than single site, changes inthe field configuration. Single site trial moves infield-theoretic polymer simulations are grosslyinefficient because the modified diffusion equa-tions must be re-solved to update the energy,regardless of the number of grid points on whichWT differs from Wm.

As seen earlier, restricting the pressure field W+to its saddle-point configuration � = iW+ yieldeda real-valued effective Hamiltonian. Therefore wewere able to implement traditional MC algorithmsrequiring a positive definite probability weightP ∝ e−H . This idea can be extended to morecomplicated polymer systems.

In general, one will need to restrict all pressure-like potential fields to their saddle-point configu-rations in order to ensure a real-valued Hamilton-ian. Here, pressure-like refers to potential fieldswhose saddle-point configuration is purely imag-inary. Potential fields with a real-valued saddle-point configuration are called exchange-like.

We can further examine the requirements of ageneralized PSP-MC algorithm by introducing a


2504 BOSSE ET AL.

monomer–monomer interaction matrix. Considera polymer system with M different monomer types(e.g., a M-block copolymer), each with a differentmonomer–monomer interaction parameter χij, fori, j = 1, 2, . . . , M. For such a system, the general-ized monomer–monomer interaction energy can begiven by

HI = 12ρ0

∫V

dr ρTU ρ, (31)

where ρ is an M-dimensional microscopic monomerdensity vector with components ρi(r), for i= 1, 2, . . . , M (each defined by an expression anal-ogous to eqs 4 or 5), ρT is the transpose of ρ, andU is the M × M monomer–monomer interactionmatrix with components

Uij = ζ + χij. (32)

This matrix captures compressibility with ζ andthe Flory monomer–monomer interactions withχij. The ζ → ∞ limit yields strict melt incompress-ibility. The Flory parameter is symmetric, χij = χji,and has zero-valued diagonal elements χii = 0,∀i = 1, 2, . . . , M.

The matrix U is symmetric, Uij = Uji, and thuspossesses real-valued eigenvalues, λj ∈ R,

Uuj = λjuj, ∀i = 1, 2, . . . , M, (33)

where uj are the eigenvectors of U. The sign of theeigenvalues determines whether the correspond-ing conjugate potential fields ωj have real-valuedor purely imaginary saddle-point configurations.∗

Specifically, ωj fields with λj > 0 are pressure-like,while ωj fields with λj < 0 are exchange-like. Oncewe have identified and classified the potentialfields, we can implement a generalized PSP-MCroutine: the functional integrals associated withthe pressure-like fields are evaluated using thesaddle-point approximation, while the functionalintegrals associated with the exchange-like fieldsare left to be evaluated by MC sampling.

At present, we have not examined the useful-ness of PSP-MC algorithms in the context of morecomplicated polymer systems. In this paper, werestrict our attention to AB diblock copolymermelts in 2D.

∗The eigenvectors are used to define a new density vector with components j(r) = 〈ρ(r), uj〉 (analogous to the ρ± vari-ables from Model and SCFT section). Here 〈·, ·〉 is the innerproduct.We then perform a Hubbard-Stratonovich transforma-tion, which yields the corresponding conjugate potentials ωj .

In this work, we apply two MC methods inthe PSP approximation: the force-biased “smartMonte Carlo” (SMC) method of Rossky, Doll, andFriedman,25 and a colored-noise (i.e., Fourier-accelerated) variation of the smart MC method.The primary advantage of force-biased MCmethod is decreased equilibration time in com-parison with conventional Metropolis MC. Eachforce-biased MC kick involves a move along thegradient of the Hamiltonian (in the same way thatthe steepest descent relaxations move in the direc-tion of the gradient), followed by a random kick. Invarious tests, we have observed that force-biasedMC methods produce a 1–2 order-of-magnitudereduction in equilibration time for field-based sim-ulations, when compared with that by MetropolisMC. Here we detail two variations of the force-biased smart MC routine: a white noise and acolored noise version.

In the colored noise implementation, we choosea Fourier space noise distribution that haswavenumbers peaked around q∗, the peak inthe diblock structure factor.12 This wavenumberroughly corresponds to the fundamental diblocklength scale Rg0. By carefully selecting the dis-tribution of the noise, it is our hope to not onlydecrease equilibration time but also allow barriercrossings that correspond to large scale rearrange-ments of the microdomains toward less defectiveconfigurations.

Beginning at an arbitrary time tm and field con-figurations �m and Wm, the PSP SMC update pro-cedure is implemented as follows (for more explicitdetails, particularly with respect to detailed bal-ance and the trial-move acceptance probabilities,see Ref. 25):

1. Evaluate the Hamiltonian Hm = H[�m, Wm]and the “W-force” Fm

i = −∂Hm/∂Wmi .

2. Generate a trial configuration WT:

WTi = Wm

i + Fmi + Ri, (34)

where the random M-vector Ri is selectedfrom the distribution

P({Ri}) = 1(4π)M/2

exp

[−1

2

M∑i

R2i

2

](35)

3. Relax � to its instantaneous, WT depen-dent, saddle-point trial configuration, �T,using SIS updates until the relative errorin H[�T, WT] is below a specific, predeter-mined value ε. Note that the force required



to implement SIS relaxation for the pres-sure field � is calculated at a fixed value ofthe exchange field W = WT.

4. Evaluate the Hamiltonian HT = H[�T, WT],the “W-force” FT

i = −∂HT/∂WTi , and the

expression

τ = P({Wmi − WT

i − FTi })

P({WTi − Wm

i − Fmi })e−(HT−Hm). (36)

5. Select a random number r ∈ [0, 1]. If r ≤ τ ,then accept the trial move and set �m+1

= �T and Wm+1 = WT, else reject the trialmove and set �m+1 = �m and Wm+1 = Wm.

There are a few technical points worth men-tioning. First, the dimensionless polymer concen-tration (coordination number) C multiplies everyterm in the Hamiltonian and, thus, controls theextent to which fluctuation corrections to thesaddle-point approximation (i.e.,SCFT) are impor-tant. Simply stated, for large C, fluctuations areless important, and the MC procedure will pro-duce results that resemble those of an SCFTsimulation. For smaller C, fluctuations are moreimportant, and we can expect them to have anappreciable effect on defect populations and defectmobility in the system. Second, the parameter

sets the magnitude of the force and (by detailedbalance) the magnitude of the random kick. Foroptimal performance, is adjusted to ensure thetrial-move acceptance ratio is ≈0.5.

The colored-noise variation of the SMC algo-rithm represents only a small change to theabove-mentioned algorithm. Beginning at an arbi-trary time tm and field configurations �m and Wm,the PSP colored-noise SMC update procedure isimplemented as follows:

1. Evaluate the Hamiltonian Hm = H[�m, Wm]and the “W-force” Fm

i = −∂Hm/∂Wmi .

2. Generate a trial configuration WT:

WTi = Wm

i +M∑i′

|i−i′|Fmi′ + Ri (37)

where the random M-vector Ri is selectedfrom the distribution

PCN({Ri})

∝exp

[−1

2

M∑i

M∑i′

RiRi′(−1)|i−i′|/2

]. (38)

3. Relax � to its instantaneous, WT depen-dent, saddle-point trial configuration, �T,using the SIS scheme. Continue relaxinguntil the relative error in H[�T, WT] isbelow a specific, predetermined value ε.Note that the required “�–force” is evalu-ated at fixed exchange potential W = WT.

4. Evaluate the Hamiltonian HT = H[�T, WT],the “W-force” FT

i = −∂HT/∂WTi , and the

expression:

τ = PCN({

Wmi − WT

i − ∑Mi′ |i−i′|FT

i′})

PCN({

WTi − Wm

i − ∑Mi′ |i−i′|Fm

i′})

× e−(HT−Hm). (39)

5. Select a random number r ∈ [0, 1]. If r ≤ τ ,then accept the trial move and set �m+1

= �T and Wm+1 = WT, else reject the trialmove and set �m+1 = �m and Wm+1 = Wm.

Again, there are a few technical points worthmentioning. The M × M matrix |i−i′|, whichdefines the distribution of (spatially) colored noise,is assumed to be translationally invariant andsymmetric. Because the discrete Fourier trans-form of such a matrix is diagonal, the procedurefor generating random numbers Ri and evaluat-ing the trial configuration WT is greatly simplified.Thus, it is natural to perform colored-noise SMCupdates in Fourier space. For a function fj definedon the simulation lattice, we define the Fouriertransform

fk =M∑j

fj e−ikj, (40)

and the inverse Fourier transform

fj = 1M

M∑k

fk eikj. (41)

The distribution of Rk is given by

PCN({Rk}) ∝ exp

[−1

2

M∑k

R−kRk

2Mk

]

=M∏k

exp

[−1

2R−kRk

2Mk

]. (42)

Note that −1|i−i′| is diagonal in Fourier space: −1

k

= 1/k. Also note that since Rj is real, R−k = R∗k.


2506 BOSSE ET AL.

Equation 42 tells us that the distributions ofboth Re(Rk) and Im(Rk) are Gaussian, each with(k dependent) variance 2Mk; hence the term“colored noise.”

In Fourier space, eq 37 becomes

WTk = Wm

k + kFmk + Rk, (43)

where we have used the cyclic convolution theo-rem. Similarly, eq 39 can be expressed as

τ = PCN({

Wmk − WT

k − kFTk

})PCN({WT

k − Wmk − kFm

k }) e−(HT−Hm). (44)

The variance k is selected to be a function peakedat the wavenumber k0 = 2 ≈ q∗ roughly corre-sponding to Rg0. For example,

k = [e−(k−k0)2/2σ2 + δ], (45)

or

k = Sk = k2

k40 + k4

. (46)

Here k0, σ , and δ remain constant, while isadjusted to ensure the acceptance ratio is ≈0.5.Equation 45 allows us to adjust the breadth ofharmonics of the kicks by changing σ . We canintroduce a uniform “white noise” background dis-tribution by setting δ = 0. Alternatively, eq 46,while also peaked at k0, captures the observedlow-k and high-k behavior of the diblock structurefactor Sk.12

In the next section we present results that illus-trate the effectiveness of the SIS, SF, and PSP-MCalgorithms at yielding defect-free states. Further-more, we also address the possibility that thePSP-MC algorithms may represent a useful toolfor examining equilibrium defect populations.

RESULTS

In the simulations reported here we applied peri-odic boundary conditions in both lateral directions,and we set Lx = Ly = 48Rg0, nx = ny = 192,ns = 70, f = 0.7, χN = 17, and C = 20 or 100.

For the PSP-MC algorithms, we relax � (usingthe SIS scheme) until the relative error in H[�, W]is below ε = 0.001. On average, approximately fiveSIS steps in � are needed for each trial MC update.In the SIS update scheme, C always appears as a

multiplicative factor next to t, and thus repre-sents a modification to the SIS time step. In allSIS updates, the overall time step tC is set totC = 100.

As mentioned earlier, all colored-noise SMCruns have a Fourier space noise distribution thatis peaked at k0 = 2 ≈ q∗. For the Gaussiancolored-noise distributions, we set δ = 0 (i.e., nobackground white noise).

For the SIS runs, we relax the system fornt = 20,000 total time steps (this ensures a wellrelaxed saddle point solution). For the PSP-MCruns we also run for nt = 20,000 total timesteps (attempted MC moves); however, the PSP-MC runs are split into two distinct sections: a10,000 iteration equilibration run, and a 10,000iteration production run (the value of or isfixed over the production run).

For the SF runs, we apply the spectral ampli-tude filter at tm = 5000 with fmax = 0.75, thenagain at tm = 6000 with fmax = 0.50, and finally attm = 7000 with fmax = 0.25. Before, between, andafter SF applications, we relax the system towarda saddle point solution with the SIS algorithm.After the final spectral filter application, we con-tinue to relax the system using the SIS algorithmuntil tm = nt = 20,000.

For each scheme, we performed two indepen-dent runs, each with a different random initialcondition, and at the end of the run performed aVoronoi analysis and computed the average defectdensity reported.

For all PSP-MC runs, we began with a ran-dom initial condition, followed by exactly five SISsteps, and then started the actual MC run. Thefive SIS steps are necessary because the first fewinitial moves away from a random initial statewere observed to always increase the value ofH[�, W], usually by such a large amount thatthe probability of accepting such an MC move isnegligible.

The above-mentioned set of parameters islargely arbitrary; however, there were a few guide-lines that we adhered to when selecting the systemparameters: (1) defects vs. run time—we wanteda simulation cell large enough to exhibit signif-icant numbers of defects (SCFT simulations onsmall systems often do not exhibit defects), butsmall enough to keep run times manageable. Withthis size, our runs (20,000 field iterations) lasted≈20 h for SIS/SF and ≈60 h for PSP-MC on asingle-processor workstation. (2) We set dx = dy= 0.25Rg0 in order to ensure adequate resolutionto resolve the domain interfacial regions, and we



selected the number of contour points (ns = 70)to be the smallest value that does not appreciablyaffect the solutions obtained. (3) The most detailedexperimental studies of defects in thin film blockcopolymer systems are performed near the ODT.5–7

While we elected not to use the exact values of fand χN corresponding to the experimental stud-ies, we selected parameters f = 0.7 and χN = 17that not only place the system near the ODT atboth C = 100 and C = 20, but also at a compositionwhere small χN variations sustain the cylindricalphase. This allowed us the opportunity to studyχN dependent effects without the requirement ofchanging f .

The procedure for obtaining defect densitiesinvolves four steps: (1) run the simulations,(2) generate the Voronoi diagrams, (3) count thenumber of pentagons n5 (or heptagons n7) and thetotal number of polygons ntot, and (4) calculatethe defect densities:

ρD = n5

ntot= n7

ntot. (47)

For well-relaxed systems in the hexatic or crys-tal phase, n5 + n6 + n7 = ntot, as observed in theexperiment.5

In the present section we compare five differ-ent defect annealing schemes: SCFT implementedwith SIS relaxation (SIS), spectral amplitude fil-tering in tandem with SIS relaxation (SF), PSPwhite-noise SMC, PSP colored-noise SMC with aσ = 1 Gaussian noise distribution (SMC1), andPSP colored-noise SMC with k given by eq 46(SMCS).

In Figure 2 we provide a representative com-position pattern for a well-relaxed SIS simula-tion (φA in white). This system clearly exhibits asignificant concentration of microdomain defects.The lowest energy mean-field density consists ofa hexagonally ordered lattice of B-rich micro-domains (i.e., each microdomain has 6 nearestneighbors). Here, there are various microdomainswith five or seven nearest neighbors. As we shallsee, the defects are more easily identified (andenumerated) with Voronoi diagrams.

In Figures 3 and 4, we provide representa-tive Voronoi diagrams resulting from simulationruns using SIS, SF, SMC, SMC1, and SMCS.Recall that the number of sides of the polygoncentered on a cylinder represents the number ofnearest-neighbor cylindrical domains. Thus, hexa-gons (in white) represent perfect 6-coordinatedmicrodomains, while pentagons (in light gray) andheptagons (in dark gray) represent defects (in this

Figure 2. Composition profile,φA (in white), for a well-relaxed SCFT simulation (using SIS relaxation). Notethat there are numerous microdomain defects, charac-terized by a breakdown of perfect sixfold coordination.The x and y axes are in units of Rg0.

case disclinations). It is important to note thatthe relaxed SIS density has a nonzero defect den-sity. This is a common occurrence and the primarymotivation for this work.

In Figure 5 we present average pentagon den-sity versus number of iterations (tm). For each ofthe PSP-MC relaxations, we ran with C = 100 andC = 20. We see from Figure 5 that the SF relax-ation was the most successful of the five schemesat removing defects. In fact, for our system, SFrelaxation always produced defect-free states. Thesecond most successful algorithm was the SISrelaxation, followed by the various PSP-MC algo-rithms.

It is not surprising that the SF was the mostsuccessful at removing microdomain defects. TheFourier landscape of hexagonally ordered cylin-ders is rather simple (the first-order structure con-sists of six equally spaced peaks, all of which areequidistant from the (0,0) mode). Consequently,the spectral amplitude filter is consistently suc-cessful at filtering out “noise” associated withmicrodomain defects.

It is interesting that the SIS relaxation schemeconsistently outperformed the various PSP-MCalgorithms in the elimination of defects. There area few possible explanations for this observation.

First, the amplitude of the thermal fluctuationsmay be too large or too small. We examined thePSP-MC at two values of C. For C = 100 we noticed


2508 BOSSE ET AL.

Figure 3. Representative Voronoi diagrams for SF (left) and SIS (right) relaxations at20,000 iterations. In both figures, the x and y axes are in units of Rg0.

very little microdomain movement. C = 100 is arelatively high value (recall C → ∞ gives SCFT),and thus the energy provided by the MC fluc-tuations may be below the energy required to“jump” out of the local minima corresponding tometastable defects. C = 20, on the other hand,represents a rather low value where fluctuationsare significant.26 In this case, the fluctuations arelikely large enough that the system is continu-ously kicked in to and out of both local and globalminima. Our observations are consistent with con-siderable defect motion at C = 20. One couldimagine that there exists a “sweet spot” in C, inter-mediate between C = 20 and C = 100, where oneor all of the PSP-MC algorithms are capable ofachieving states with low defect densities. Morelikely, however, it would be necessary to gradu-ally ramp C from a low value to a high valuein a type of “simulated annealing” procedure toachieve and maintain low defect densities in suchMC simulations.

Second, it is possible that we have actually real-ized our secondary goal of developing methodsof sampling equilibrium defect populations. Foreach of the PSP-MC algorithms, the defect frac-tion appears to level off to a fixed value. Thisis not desirable from a defect removal stand-point; however, it is exactly what one would expectwhen observing equilibrium populations of defectscharacteristic of a 2D crystalline or hexatic phase.

Finally, it is also possible that the time scalefor defect removal is far longer than the 20,000iterations we have selected here. In order to par-tially examine this issue, we set up two of oursimulations to focus intensely on moving defects.The SMC1 relaxations have a noise distributionin Fourier space that is sharply peaked at q∗. Thepeak is so sharp that the energies and densitiesyielded by the SMC1 update differ significantlyfrom the other PSP-MC updates. It is our beliefthat the SMC1 updates are nonergodic, and thusunsuitable for statistical sampling of the diblockpartition function. However, the SMC1 algorithmis particularly well suited for defect removal. TheSMC1 algorithm was not the most successful atremoving defects on the time scale examined; how-ever, longer test runs of the SMC1 scheme (beyond20,000 iterations) suggest that it continues toslowly remove defects (in fact, the SMC1 defectfractions approached those of the SIS relaxation).Although this may represent a partial success,the time scales are too long to be a practicalmethod of removing defects. In addition, since theSMC1 relaxation is nonergodic, it does not repre-sent a useful tool for examining equilibrium defectpopulations.

We have yet to discuss the results of theSMCS algorithm. Even though the SMCS Fourierspace noise distribution is sharply peaked atk = 2 ≈ q∗, the method does appear to be ergodic



Figure 4. Representative Voronoi diagrams for SMC (C = 100) (top left), SMCS(C = 100) (middle left), SMC1 (C = 100) (bottom left), SMC (C = 20) (top right), SMCS(C = 20) (middle right), and SMC1 (C = 20) (bottom right) at 20,000 iterations. In allfigures, the x and y axes are in units of Rg0.


2510 BOSSE ET AL.

Figure 5. Fraction of pentagons vs. simulation time. The SF relaxation is the mostsuccessful at removing defects while the SMC scheme is the least successful. For the PSP-MC schemes, the defect fractions appear to level off, and thus may represent equilibriumpopulations.

(i.e., it yielded similar density profiles and ener-gies to other ergodic PSP-MC methods such asMetropolis MC and the white noise SMC). Ofall the PSP-MC algorithms at C = 100, theSMCS scheme was the most successful at remov-ing defects over the time scale examined. Sincethis update scheme does appear to fully samplephase space, and it appears to equilibrate morerapidly than the other PSP-MC algorithms, itmay represent a useful tool for examining equi-librium defect populations and possibly studyingthe crystal–hexatic transition in 2D block copoly-mer thin films. Further study will be necessaryto determine the computational feasibility of suchinvestigations.

DISCUSSION AND CONCLUSIONS

We have developed a family of field updatealgorithms primarily intended to systematicallyremove metastable defects encountered in sim-ulations of block copolymer fluids. In our trialson 2D melt films of AB diblock copolymers with

hexagonal mesophase order, the SF algorithm wasthe most effective at achieving defect-free, mean-field (SCFT) solutions. Our MC algorithms basedon the PSP approximation (PSP-MC) incorporatethermal fluctuations, and while less effective atannealing defects, may prove useful for samplingequilibrium defect populations.

Our numerical simulations illustrate that theSF algorithm is very efficient at removing defectsin the 2D AB block copolymer systems studied.However, preliminary tests on AB systems in 3Dshow that the method can run into difficulty forcomplex block copolymer mesophases (such asgyroid), where lower amplitude Fourier peaks thatare crucial for specifying symmetry can becomeburied in a rough landscape created by defects. Itremains to be seen how useful the SF algorithmwill be when applied to simulations of ABC blockcopolymer systems that possess large numbers ofcomplex phases.

Experimental studies of defect populations inblock copolymer thin films indicate that a singlelayer of spherical microdomains exhibit order-ing characteristics consistent with KTHNY theory,



including equilibrium defect densities.5 Theergodic PSP-MC algorithms developed here (SMCand SMCS) appear to be useful algorithms incomputational studies of equilibrium defect popu-lations and numerical investigations of the applic-ability of KTHNY theory to block copolymer thinfilms. Of the PSP-MC variants, the SMCS methodappears the most promising for such studies, as itis not only ergodic but also injects noise concen-trated near wavenumbers k = 2 ≈ q∗ that areeffective at producing defect motion. We hope toreport on further investigations along these linesin the near future.

G. H. Fredrickson and A. W. Bosse derived partialsupport from NSF grant DMR-0312097. The work ofC. J. García-Cervera was funded by NSF grant DMS-0411504. A. W. Bosse received partial supported fromThe Frank H. and Eva B. Buck Foundation. Thiswork made use of MRL Central Facilities supported bythe MRSEC Program of the National Science Founda-tion under Award No. DMR05-20415. In addition, weacknowledge use of the Hewlett-Packard cluster in theCNSI Computer Facilities. We are also greatful to GilaStein and Edward Kramer for providing the MATLABcode used to generate the Voronoi diagrams.

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