Molecular pathways for defect annihilation indirected self-assemblySu-Mi Hura,b,c, Vikram Thapard, Abelardo Ramírez-Hernándeza,b, Gurdaman Khairab, Tamar Segal-Peretza,Paulina A. Rincon-Delgadillob, Weihua Lie,f, Marcus Müllere, Paul F. Nealeya,b, and Juan J. de Pabloa,b,1

aMaterials Science Division, Argonne National Laboratory, Lemont, IL 60439; bInstitute for Molecular Engineering, The University of Chicago, Chicago,IL 60637; cSchool of Polymer Science and Engineering, Chonnam National University, Gwangju 500757, Korea; dSchool of Chemical and BiomolecularEngineering, Cornell University, Ithaca, NY 14853; eInstitut für Theoretische Physik, Georg-August-Universität, 37077 Göttingen, Germany; and fDepartmentof Macromolecular Science, Fudan University, 200433 Shanghai, China

Edited by Caroline Ross, Massachusetts Institute of Technology, Cambridge, MA, and accepted by the Editorial Board September 28, 2015 (received for reviewApril 29, 2015)

Over the last few years, the directed self-assembly of blockcopolymers by surface patterns has transitioned from academiccuriosity to viable contender for commercial fabrication of next-generation nanocircuits by lithography. Recently, it has becomeapparent that kinetics, and not only thermodynamics, plays a keyrole for the ability of a polymeric material to self-assemble into aperfect, defect-free ordered state. Perfection, in this context, impliesnot more than one defect, with characteristic dimensions on theorder of 5 nm, over a sample area as large as 100 cm2. In this work,we identify the key pathways and the corresponding free energybarriers for eliminating defects, and we demonstrate that an ex-traordinarily large thermodynamic driving force is not necessarilysufficient for their removal. By adopting a concerted computationaland experimental approach, we explain the molecular origins ofthese barriers and how they depend on material characteristics, andwe propose strategies designed to overcome them. The validity of ourconclusions for industrially relevant patterning processes is estab-lished by relying on instruments and assembly lines that are onlyavailable at state-of-the-art fabrication facilities, and, through thisconfluence of fundamental and applied research, we are able todiscern the evolution of morphology at the smallest relevant lengthscales—a handful of nanometers—and present a view of defect anni-hilation in directed self-assembly at an unprecedented level of detail.

directed self-assembly | copolymer | defect | minimum free energy path |string method

Over the last decade, the directed self-assembly (DSA) ofblock copolymers has rapidly evolved from mere intellec-

tual curiosity (1–4) to a potentially crucial step in the commercialfabrication of next-generation electronic circuits. Indeed, thecharacteristic length scale of ordered self-assembled copolymerdomains is in the range of 5–50 nm. Furthermore, their size andshape can be manipulated through simple processing steps, therebymaking them attractive for the production of semiconductor de-vices, nanofluidic devices, or high-density storage media (5, 6). Thegeneral idea behind copolymer DSA is that a surface pattern—chemical or topographic—can be used to guide the assembly of apolymeric material into an ordered, device-like structure that is freeof defects. In so-called “density multiplication” patterning strategies(7, 8), the spacing or pitch of the surface features can be muchlarger than the characteristic dimensions of the copolymer ofinterest. One can thus prepare coarse surface patterns, which areeasier to create, and rely on the copolymer to self-assemble intofeatures whose density is considerably larger. Fig. 1 shows aschematic representation of the process for obtaining a lamellarmorphology on a stripe-patterned substrate under a one-to-three(or 3X) density multiplication strategy. Patterned stripes interactpreferentially with one of the blocks and guide the assembly ofthin copolymer films into ordered lamellae that are perpen-dicular to the substrate. Because the glass transition temperatureof polymers is often well above room temperature, DSA isgenerally enabled by elevated temperatures (thermal annealing),

by addition of solvents (solvent annealing), or through combinedstrategies (9).Because of its simplicity, thermal annealing remains the more

widely studied means for achieving perfect assembly, and, forthat reason, it is also the approach considered here. Also notethat recent studies indicate that results for solvent annealing canbe mapped onto those of thermal annealing (10) through theproper renormalization of material parameters (11). Fig. 1 showsan experimental scanning electron microscopy (SEM) imagewhere, after thermal annealing, one can appreciate large areas ofordered lamellae that are interdispersed with rare, isolated defects.The original literature on copolymer lithography implicitly

assumed that the final morphologies that emerge in DSA rep-resent equilibrium states of the material. Recent studies havechallenged that view and suggest that DSA structures often rep-resent metastable states of matter, whose characteristics depend onthe process of assembly. For applications in the semiconductorindustry, defects can only be tolerated at the level of approximatelyone per 100 cm2 area. Experimentally observed defect densities,however, can be much higher than those predicted from the cor-responding defect free energies, which are on the order of hundredsof thermal energy units, kBT (12, 13). Experiments also indicatethat, for long annealing times, defect density decays with temper-ature (14). Taken together, such observations lead us to believe thatthe defects that arise in DSA represent kinetically trapped,

Significance

A molecular model is used to calculate the free energy of for-mation of ordered and disordered copolymer morphologies.We rely on advanced methodologies to identify the minimumfree energy pathways that connect such states of thematerial. Our predictions for defect formation and annealingare compared with experimental observations. Our results pro-vide a detailed molecular view of isolated block copolymer de-fects, which measure approximately 5 nm and represent isolatedevents in large areas. They are true “needles in the hay stack”that can only be studied by concerted molecular simulations anddedicated access to production-level fabrication tools. We showthat defect annealing is an activated process, where defects areeliminated by operating near the order−disorder transition.

Author contributions: S.-M.H., M.M., P.F.N., and J.J.d.P. designed research; S.-M.H., V.T.,A.R.-H., G.K., T.S.-P., P.A.R.-D., W.L., M.M., P.F.N., and J.J.d.P. performed research; S.-M.H.,V.T., A.R.-H., M.M., P.F.N., and J.J.d.P. analyzed data; and S.-M.H. and J.J.d.P. wrotethe paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. C.R. is a guest editor invited by the EditorialBoard.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. Email: depablo@uchicago.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1508225112/-/DCSupplemental.

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nonequilibrium structures that cannot be explained by equilib-rium thermodynamics alone. For emerging applications in nano-technology, it is essential that their stability and, more importantly,their pathways for removal be understood.In this study, we address three fundamental questions that are

central to deployment of DSA in commercial technologies: (i) Whatis the free energy of defects in DSA during density multiplication?(ii) Is defect removal an activated process and, if so, (iii) how do thekinetic pathways for defect removal depend on material and patterncharacteristics? To examine these issues, we rely on experimentsand a 3D model of the thermodynamics and kinetics of the co-polymer. We evaluate the free energy landscape as a function ofthe local composition, and we identify the minimum free energypaths (MFEP) between defective and ordered states, along withunderlying transition states or saddle points. The results of simu-lations are validated through comparison with detailed imagesfrom fully 3D, transmission-electron-microscopy (TEM) tomog-raphy in thin copolymer films. We conclude this introductory sec-tion by noting that although our focus is thermal annealing, a futurestudy will consider the more specialized problem of how thermalpathways can be altered (or not) through controlled addition ofsolvents (11, 15, 16).

Model and MethodsThe model considered here relies on a particle-based representation of thecopolymer molecules but adopts the same Hamiltonian H as that generallyused in self-consistent field theory (17, 18). Such a model has been describedin the literature and need not be recounted here. For completeness, how-ever, a more extensive discussion is provided in SI Text. Importantly, themodel has been shown to provide a representation of copolymer thin filmsthat is in good agreement with available experimental data (19, 20). We relyon the string method to identify the MFEP between two metastable stateson a free energy landscape (21–27). An initial string is constructed by linearinterpolation between the two end states. In this work, we connect a de-fective morphology, α= 0, and a defect-free lamellar morphology, α= 1. Asexplained in SI Text, at each iteration of the string method, the mean forceat discretized nodes along the string is numerically estimated by conductingfield-theoretic umbrella sampling simulations. After each iteration, thestring is updated according to the potential of mean force perpendicular tothe string.

ResultsMFEP for Edge Dislocation Dipole. As can be appreciated in Fig. 1,the most commonly observed defects in block copolymer thinfilms are edge dislocations that have either an A or a B blockcore (13, 28). They consist of a half-domain of A or B materialterminated in the middle of the regular lamellar domain, withdistorted nearby planes of the internal AB interfaces. A single

dislocation is a topological defect that cannot disappear by itself.However, pairs of dislocations with opposing Burgers vectors(dislocation dipoles) can cancel each other and disappear by trans-ferring a defect core from one lamellar domain to a neighboringstripe of the smectic structure (gliding motion). In this study, wefocus on dislocation dipoles with cores of the same-species block.Fig. 2A shows the free energy difference, ΔF, along the MFEP

between a defective morphology, α= 0, and a defect-free, lamellarmorphology, α= 1, in units of kBT. The MFEP exhibits two bar-riers and one shoulder at α = t1, t2, and t3, respectively. Themorphologies for these three points along the MFEP are shownin Fig. 2B. The red contours correspond to the internal AB in-terfaces where A and B species having a vanishing local densitydifference.A first transition state is observed at t1 ≈ 14=127; it has the

highest free energy barrier along the path. A commitor proba-bility analysis, according to procedures outlined in SI Text, indi-cates that this first transition state does correspond to a commitorprobability of 0.5 (Fig. S1). The first transition state correspondsto the formation of a partial connection (a “bridge”) between oneedge dislocation and the neighboring bent A layer (marked as 1 inFig. 2A, Inset). A full connection will then break the branched Bdomain that surrounds the A core and transfer the edge disloca-tion by half a lamellar period, hereby switching into a B core edgedislocation (gliding motion). We emphasize here that the positionof the peak in the MFEP corresponds to the incipient state con-necting the A domains from two different regions, where a mo-lecular bridge is built between them; the “transition” bridgetypically comprises a handful of molecules, and a more detailedmolecular-level analysis of its characteristics is provided in SI Textand in Fig. S2. The downhill descent after that first peak corre-sponds to the growth of the bridge. Once it attains a certain

Fig. 1. (A) Schematic representation of 3X density multiplication of blockcopolymers on chemical patterns and (B) experimental SEM image of DSAstructure after thermal annealing showing an isolated defect.

Fig. 2. (A) MFEP between a defective (apposing pair of dislocations) and adefect-free lamellar structure for χN= 25. The abscissa represents the re-action coordinate along the pathway, α∈ ½0,1�. The ordinate corresponds tothe free energy difference from the starting defective morphology, in unitsof kBT. Inset shows defect morphology where darker shaded regions cor-respond to areas where the chemical guiding pattern acts. Numbers indicatepositions of sequential morphological changes, discussed in MFEP for EdgeDislocation Dipole. (B) Morphologies at α= 0, t1, t2, t3, and 1.

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size—but before it detaches from the B domain—it enters along-lived metastable state.Bridge formation incurs an enthalpic penalty, associated with

A blocks straddled over a chemically incompatible B-rich do-main. The formation of the excess interfacial area during theannihilation process contributes to the free energy barrier.However, the precise position of the local maximum (plateau) ofthe interfacial area is located slightly after the barrier of theMFEP. The corresponding changes in the total bonded energy(chain stretching) and interfacial energy along the reaction pathare shown in Fig. S3. Those energies reveal that, in addition tothe energetic contributions incurred by a growing interface, thenarrow connection of the A domains that is necessary to initiatethe bridge alters the molecules’ configurations, which becomestretched. This behavior is reminiscent of what is seen whenpolymer chains transfer between a double potential well (29). Ascan be seen in Fig. S4, the interfacial area increases after defectannihilation, but the decrease in the magnitude of the bondedenergy is larger than the increase in the interfacial energy, leadingto an overall decrease of the free energy along the pathway. Notethat the magnitude of the difference in the sum of bonded andinterfacial energy between the defective (α= 0) and defect-free(α= 1) states matches the defect formation energy estimated fromthe MFEP, serving to provide a consistency check on the analysispresented here. Also note that the transition state for the incipientconnection between the A domains can occur at either the top (airsurface) or the bottom (nonpreferential substrate) of the lamellae.Bridge formation in the middle of the polymer film is less ad-vantageous than near the surface; this is due to the neutrality ofthe bottom and top surfaces, and the higher probability of findinga chain end near the surface (for entropic reasons). Moreover, thedepletion layer that arises near a hard wall leads to fewer polymersegments, thereby facilitating bridge formation in a region thatexhibits fewer AB contacts.Once a bridge is formed, it can grow more easily via the

parallel motion of polymer chains along the AB interface; chainscan migrate into the intermediate B domain by moving along thenewly formed interface, without incurring unfavorable contactsbetween A and B segments. The possibility of having chains crossthe domain through the bridge is presumably higher than thatof crossing without a bridge, by an amount proportional toexpð−χN=2Þ (27).For completeness, we also compare the results of our string

calculation to those of unrestrained, dynamic Monte Carlo (MC)simulations. Note that MC simulations with single-bead dis-placements have been demonstrated to capture the time evolu-tion of morphological changes in block copolymer melts (30,31); some of our diffusive MC simulation results for defect

annihilation are shown in Fig. 3A. One can see that the sequenceof morphologies identified here for different values of α alongthe MFEP are qualitatively consistent with those observed in MCsimulations. This agreement serves to indicate that the orderparameter adopted here for MFEP calculations is appropriate,and that there are no additional bottlenecks in the kineticsthat arise from the single-chain dynamics in the complexmorphology of the transition state. Also note that, when χN issmall, ΔFb is comparable to the thermal energy, and the con-cept of the MFEP as the typical, well-defined transformationpath breaks down.The MC simulations can also be used to examine single-chain,

center-of-mass trajectories during the transition period. Fig. 3 Band C shows center-of-mass trajectories for chains located nearthe bridge area during a fixed time interval. Fig. 3B depictstrajectories before the bridge is formed, and Fig. 3C presentstrajectories for the same chains after the bridge has formed. Thesimulations confirm that the chains cross the domain after thebridge has been formed; otherwise, they move primarily alongthe interfaces. This view is supported by simulations and exper-iments on diffusion of block copolymer chains in bulk lamellarmorphologies; diffusion is anisotropic, and is slower along thedirection perpendicular to the interface than along the directionparallel to the interface (32, 33).Once the A block wets the surface completely (either from the

bottom or the top), but before it completely fills region 1 (compareFig. 2A) with A segments across the entire film, the MFEP exhibitsa second free energy barrier, t2, corresponding to the formation of abridge on the other core of the A dislocation. This second bridge isformed in a manner analogous to that of the first transition state.Bridge formation leads to depletion of A blocks in the bent region(indicated by number “3”), thereby facilitating the connection ofthe B domain (bright region) in the middle; the third peak cor-responds to B bridge formation in region 3, but has a much smallerheight (or is a shoulder).Note that the MFEP follows the 3D kinetic pathway not only

for the transition states but also for long-lasting metastable statesthat may exist in between. Traditional experimental SEM imagesof the top of the film are unable to reveal the morphologicalchanges that occur across the film. These changes, however, can

Fig. 3. (A) Defect annihilation in diffusive MC simulations. (B and C) Center-of-mass trajectories of chains near the first transition state. (B) Trajectoriesbefore the bridge is formed, and (C) trajectories for the same chains afterthe bridge has formed.

Fig. 4. The 3D structure of DSA dislocation defects obtained from TEMtomography (Top), along with predicted metastable state along the MFEP(Bottom). For clarity, the region corresponding to the evolving defect ishighlighted with a yellow contour in both simulations and experiments. For3D TEM tomography, alumina was grown in three cycles of sequential in-filtration synthesis (SIS); the polymer was then etched using oxygen plasma,leaving behind an alumina nanostructure that follows the polymethylmethacrylate (PMMA) domains.

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be extracted from a recent experimental technique, namely 3DTEM tomography (34). Experimental details are provided in SIText. Our measurements are shown in Fig. 4, along with thecorresponding simulated morphologies, and confirm the existenceof the 3D metastable states identified by the MFEP.

Effect of Segregation Strength. Having established the validity ofour pathway predictions by comparison with experiment anddynamic simulation, we now examine in a systematic manner theeffect of material characteristics, in the form of the segregationstrength, χN, on the barriers, ΔFb, along the MFEP and theexcess free energy, ΔFd, of defects. The periodicity and width ofthe guiding stripes are properly rescaled according to the naturaldomain spacing of lammellar, L0, corresponding to different χN.The effects of additional chain compression and stretching ofchains (12, 35) are therefore excluded here. Fig. S5 shows theMFEPs for χN = 20, 25, 30, and 37. For all χN considered here,the MFEPs between a dislocation dipole and a defect-free la-mellar morphology exhibit three peaks (or two peaks and oneshoulder at smaller χN), showing that, regardless of χN, the systemfollows a unique topological pathway for defect removal, but withdifferent ΔFb and ΔFd. Fig. 5 presents ΔFb (black) from the firstpeak and ΔFd (blue) as a function of χN. The solid line corre-sponds to results for 3X guiding stripes of W =L0=2 and ΛN = 0.5(see Fig. 2), whereas the dashed lines correspond to assemblywithout chemical patterns, on homogeneous neutral substrates. Asdiscussed in MFEP for Edge Dislocation Dipole, the free energybarrier is related to the penalty associated with A blocks bridgingthe B domain. Fig. 5 shows that the free energy barriers decreaseas the segregation strength decreases, because the free energypenalty is proportional to χN. Moreover, as χN becomes smallerand approaches the order–disorder transition (ODT), the widthof AB interfaces becomes wider and the magnitude of fluctua-tions also increases, making it much easier for chains to makeconnections with neighboring domains. At the same time, thethermodynamic driving force ΔFd decreases monotonically as χNdecreases, because the excess free energy due to the distortedinternal AB interface decreases. Note that past self-consistentsimulations of graphoepitaxy in two dimensions (12) and che-moepitaxy in three dimensions (27) predicted that ΔFb and ΔFddepend almost linearly on χN. Also note that such self-consistentfield calculations predict a vanishing ΔFb before χN reaches theODT (by extrapolation of the ΔFb vs. χN curve for a neutralsubstrate) (27). Similarly, our particle-based simulations show

that free energy barrier heights decrease linearly with χN forboth chemically patterned (solid black line in Fig. 5) and neutralsubstrates (dashed black line). In agreement with these self-consistent field calculations and computer simulations (13), ΔFdis found to be large—on the order of 100 kBT—even for smallestχN where defects exist; again, this result emphasizes that theprobability of forming defects at equilibrium is extremely small.In experiments, defects arise in the process of structure forma-tion (via spin casting, for example); defect removal is an activatedprocess, and free energy barriers proportional to χN helpstabilize them.One important practical consideration is to identify optimal

annealing processes for defect removal. The segregation strength,χN, can be manipulated through temperature or solvent concen-tration, and these two variables can be controlled in experimentsto achieve optimal results, i.e., low defectivity in short processingtimes. Ideally, one would seek to find conditions having small ΔFband large ΔFd. Li et al. suggest that the region around χN⋆ = 18for neutral substrates (or larger values for dense guiding patterns)provides the best condition for defect annihilation on nonpreferentialsubstrates or dense guiding patterns (27) because the excess freeenergy ΔFd of defects significantly exceeds the thermal energykBT, yet the free energy barrier ΔFb, which separates the meta-stable defect from the perfectly ordered structure, is small com-pared with kBT, i.e., there exists a certain range of temperature—or solvent concentration or molecular weight—where thermalfluctuations remove defects but do not create new ones. In thefollowing, we explore the defect removal mechanisms for guidingpatterns with density multiplication and more complex defectstructures by computer simulation.Fig. 5 shows that a chemical pattern with 3X density multi-

plication helps decrease ΔFb and increase ΔFd for all χN, therebyfacilitating defect annihilation. Results for ΔFb and ΔFd fordifferent substrate conditions at χN = 25 are summarized in TableS1. At this point, it is important to emphasize that the strain fieldthat accompanies a dislocation decays with the distance from thecore. These strain field-mediated forces result in an attractionbetween dislocations with opposite Burgers vectors, and themetastable dislocation dipoles correspond to situations wherethese forces disappear. Previous studies of the free energy as-sociated with isolated dislocations relied on thermodynamicintegration (13) and examined finite-size effects. Such studiesconfirmed that the defect formation free energy, ΔFd, reaches itsasymptotic value for sample sizes comparable to those consideredhere. For a dislocation pair having opposite Burgers vectors, ascreening of the strain fields is expected in the far field. Strain fieldeffects are therefore smaller for dislocation pairs, and the excessfree energy and the barrier should not exhibit significant finite-sizeeffects. These predictions are confirmed by results from simula-tions of adjacent dislocation dipoles in simulation boxes of varyingsizes. When the simulation box size is small, the interaction anddistortion by neighboring dislocation dipoles (via the periodicboundary conditions) increases the defect formation energy, ΔFd,and destabilizes the defects (the free energy barrier height is smaller).However, when the system size is sufficiently large, at approximatelyLx ≈ 6L0, our calculations show that both ΔFb and ΔFd reach theirasymptotic limits, thereby leading us to place the characteristicdimension of the strain field at around 6L0. All results reportedhere are from simulations with Lx = 6L0.In contrast to the case with a homogeneous substrate, 3X

guiding stripes that attract the A block with ΛN = 0.5 reduce ΔFbby about 20%. The magnitude of expðΔFb=kBTÞ, which is pro-portional to the time scale for crossing the first transition stateand reaching the second metastable state, decreases by a factorof five with respect to the transition time observed on the neutralsubstrate. This result shows that guiding stripes on a chemicallypatterned substrate facilitate defect annihilation significantly, evenwhen those defects are not directly on top of the guiding stripe.

Fig. 5. Free energy difference between defective and defect-free lamellarstructures, ΔFd (blue), and highest free energy barrier height (first transitionstate), ΔFb (black), in units of kBT, as a function of χN. Simulation data for 3XDSA (circles) and neutral substrates (triangles) are fitted with solid anddashed lines, respectively.

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Importantly, calculations for guiding stripes that deviate fromperfect commensurability reveal that the thermodynamic equi-librium morphology is not affected, but that the kinetics of DSA(i.e., the time required for defect removal) can vary consider-ably. As explained in SI Text, we also considered the magnitudeof barrier heights as a function of the relative position of adefect to the guiding stripes. Surprisingly, we found that whenone of the edge dislocations lies on the guiding stripe, it isharder to remove the associated defect than when it is ona neutral background (Fig. S6 and Table S2).

Dislocation Dipoles Separated by Multiple Lamellae. At this point, itis instructive to note that, although the kinetic barriers for ad-jacent dislocation dipoles are predicted to be small for low valuesof χN, experimentally observed defect densities can, in somecases, be large. The answer to this apparent contradiction is thatprevious studies were limited to cases where the two edge dis-locations were located next to each other. Generally, the initialstages of structure formation lead to dislocation dipoles that areapart in both the perpendicular and parallel directions to thestripes. The strain energy of dislocations leads to an attractiveforce between them, causing them to gradually approach eachother by a “climbing” or a “gliding” motion (36). Here, climbingrefers to an edge dislocation displacement along the stripes, andperpendicular to the Burgers vector; gliding describes a defectmovement perpendicular to the internal AB interfaces. Unlikethe defects in solid crystals, in which climbing by the emission orabsorption of vacancies is more difficult, defects in block co-polymer thin films are expected to exhibit a slower gliding mo-tion, because gliding necessarily involves interdiffusion acrossdomain boundaries and interface destruction, a highly unfavorableprocess due to the enthalpic penalty for block mixing mentionedearlier. Climbing motion, on the other hand, can be achieved viarelatively inexpensive chain displacements parallel to the inter-faces. These views are confirmed by recent experiments by Tongand Sibener, who monitored the dynamics of layered structuresof cylinder-forming block copolymers confined in a channel (36).In response to the strain field-mediated interactions, disloca-

tion cores with opposite Burgers vectors attract each other andclimb along the stripes until they reach a metastable, force-freeconfiguration. Here we therefore focus on gliding motion out ofthis metastable configuration, and examine how barrier heightsvary for each of the peaks as the perpendicular distance betweenedge dislocations varies.Fig. 6 presents the MFEP of dislocation dipoles separated by

two lamellar periods at χN = 20. As shown in the representativeconfigurations of the initial defect configuration, two edge dis-locations are located right next to the A-attractive guiding stripes,and intermediate curved domains are located on top of the neutraldomain. Along the defect annihilation path, there is a small firstbarrier that corresponds to bridge formation of the bright B blocks,followed by a downhill descent corresponding to the growth of thebridge (increasing the wetting layer of B). Another small barrierand a downhill section follow as the bridge of the B block on theother side forms and grows. Approximately at the reaction co-ordinate α = 0.3, the system reaches a 3D metastable structure.Beyond that structure, the free energy rises again. This uphillprocess is associated with the formation of the B bridge, and isfollowed by transition states with a large barrier height. This firsttransition state has a slight connection of the red A domain,which occurs after the B bridge increases on one side, as shownin the fifth morphology of Fig. 6B. When the connected A do-main vanishes completely, the B core edge dislocation is transferredinto an A core edge dislocation. A similar transition occurs on theother side, and eventually results in a dislocation dipole with a re-duced distance of L0. The morphology of this dislocation dipole isthe same as that of the initial defect structure (tight dislocationdipole) studied in MFEP for Edge Dislocation Dipole, and the

subsequent portion of the MFEP is the same as the black curve inFig. S4 for χN= 20. We would like to emphasize that, although thebarrier height for annihilation of dislocation dipoles that are ad-jacent to each other is predicted to be very small, if the two edgesof a dislocation pair are initially apart, the system must first crosstransition states having much larger barrier heights, on the orderof 15  kBT, even for materials having a small χN = 20. This ismore than 3 times larger than the barrier height for dislocationdipoles that are adjacent to each other.Note, additionally, that the kinetic pathway for defect removal

consists of multiple barriers and metastable states and that theprobability of a metastable state going back in the direction towardthe original defect state is not negligible, i.e., the system performsa stochastic jump process between the metastable states.These observations help explain the higher defect density that

is sometimes observed in experiments, and emphasizes that, forDSA, it is important to know how barrier height ΔFb varies as afunction of the distance between dislocations. Previous experi-mental studies (36, 37) that attempted to fit data using a uniformdiffusion, regardless of the distance between dislocation dipoles,were unlikely to capture the diffusion of moving defects.

ConclusionsWe have examined the MFEP for defect annihilation of dislo-cation pairs in lamellar structures of block copolymer thin films,with an emphasis on industrially relevant DSA strategies thatrely on a chemically patterned substrate and density multiplica-tion. Multiple metastable states, separated by free energy bar-riers, arise along the MFEP. The largest barrier height, ΔFb, fora tight dislocation dipole corresponds to the formation of a firsttransition state in which a molecular bridge connecting the edgeand the nearest curved domain is initiated. From a Kramers-likeapproach, the transition time is expected to be exponentiallyproportional to ΔFb. This will therefore be the most time-con-suming step along the path. MFEP and diffusive MC simulations,which can predict the evolution of morphologies, show that oncea bridge is formed, it can grow rapidly as additional molecules

Fig. 6. (A) MFEP between a dislocation dipole separated by two periods anddefect-free lamellar structures . The x axis represents the reaction coordinatealong the pathway, α∈ ½0,1�. The y axis is free energy difference with thestarting morphology (defective) in units of kBT. (B) Morphologies along theMFEP in A. All of the images correspond to bottom views.

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travel through the growing bridge. The proposed mechanism ofbridge formation and growth is confirmed by experimental mea-surements using 3D TEM tomography. As the incompatibilitybetween blocks, χN, increases, the defect formation energy ΔFdalso increases. At the same time, however, the barrier height ΔFbincreases. Thus, we anticipate that even for the large thermo-dynamic driving forces that arise in high-χN materials, defectannihilation is inhibited by the presence of large kinetic barriers.We have also shown that a chemical pattern helps reduce barrierheights significantly, even for defects located on the neutralbackground. The annihilation time can therefore be reduced by afactor of five with respect to the case of a homogeneous surfacewithout any guiding patterns. Finally, we showed that kineticbarriers are very sensitive to the distance between dislocations.The barrier height of a dislocation dipole separated by two layers

requires more than 4 times the activation energy for them toapproach each other and annihilate. This finding suggests thatthe density of defects separated by multiple lamellae should bemuch higher than that of adjacent dislocations.

ACKNOWLEDGMENTS. This work is supported by the US Department ofEnergy, Office of Science, Office of Basic Energy Sciences, Materials Scienceand Engineering. W.L. and M.M. received financial support from theEuropean Union Seventh Framework Programme under Grant Agreement619793 CoLiSA.MMP. We are grateful for valuable computing resourcesprovided on Blues, a high-performance computing cluster operated by theLaboratory Computing Resource Center at Argonne National Laboratory, andfor resources provided by the Midway Research Computing Center at theUniversity of Chicago. An award of computer time was provided by theInnovative and Novel Computational Impact on Theory and Experiment(INCITE) program of the Argonne Leadership Computing Facility at ArgonneNational Laboratory.

1. Park M, Harrison C, Chaikin PM, Register RA, Adamson DH (1997) Block copolymerlithography: Periodic arrays of ∼1011 holes in 1 square centimeter. Science 276(5317):1401–1404.

2. Rockford L, et al. (1999) Polymers on nanoperiodic, heterogeneous surfaces. Phys RevLett 82(12):2602–2605.

3. Kim SO, et al. (2003) Epitaxial self-assembly of block copolymers on lithographicallydefined nanopatterned substrates. Nature 424(6947):411–414.

4. Segalman RA, Yokoyama H, Kramer EJ (2001) Graphoepitaxy of spherical domainblock copolymer films. Adv Mater 13(15):1152–1155.

5. Albert JN, Epps TH, III (2010) Self-assembly of block copolymer thin films.Mater Today13(6):24–33.

6. Kim HC, Park SM, Hinsberg WD (2010) Block copolymer based nanostructures: Ma-terials, processes, and applications to electronics. Chem Rev 110(1):146–177.

7. Ruiz R, et al. (2008) Density multiplication and improved lithography by directedblock copolymer assembly. Science 321(5891):936–939.

8. Cheng JY, Rettner CT, Sanders DP, Kim HC, Hinsberg WD (2008) Dense self-assemblyon sparse chemical patterns: Rectifying and multiplying lithographic patterns usingblock copolymers. Adv Mater 20(16):3155–3158.

9. Gotrik KW, Ross CA (2013) Solvothermal annealing of block copolymer thin films.Nano Lett 13(11):5117–5122.

10. Nealey PF, Wan L (2012) US patent publication US 20120202017 A1 (August 9, 2012).11. Hur SM, et al. (2015) Simulation of defect reduction in block copolymer thin films by

solvent annealing. ACS Macro Lett 4(1):11–15.12. Takahashi H, et al. (2012) Defectivity in laterally confined lamella-forming diblock

copolymers: Thermodynamic and kinetic aspects. Macromolecules 45(15):6253–6265.13. Nagpal U, Müller M, Nealey PF, de Pablo JJ (2012) Free energy of defects in ordered

assemblies of block copolymer domains. ACS Macro Lett 1(3):418–422.14. Welander AM, et al. (2008) Rapid directed assembly of block copolymer films at el-

evated temperatures. Macromolecules 41(8):2759–2761.15. Jung YS, Ross CA (2009) Solvent-vapor-induced tunability of self-assembled block

copolymer patterns. Adv Mater 21(24):2540–2545.16. Gotrik KW, et al. (2012) Morphology control in block copolymer films using mixed

solvent vapors. ACS Nano 6(9):8052–8059.17. Matsen MW, Schick M (1994) Stable and unstable phases of a diblock copolymer melt.

Phys Rev Lett 72(16):2660–2663.18. Fredrickson G (2006) The Equilibrium Theory of Inhomogeneous Polymers (Oxford

Univ Press, Oxford).19. Detcheverry FA, Pike DQ, Nealey PF, Müller M, de Pablo JJ (2009) Monte Carlo sim-

ulation of coarse grain polymeric systems. Phys Rev Lett 102(19):197801.20. Detcheverry FA, Liu G, Nealey PF, de Pablo JJ (2010) Interpolation in the directed

assembly of block copolymers on nanopatterned substrates: Simulation and experi-ments. Macromolecules 43(7):3446–3454.

21. E W, Ren W, Vanden-Eijnden E (2002) String method for the study of rare events. PhysRev B 66(5):052301.

22. Miller TF, 3rd, Vanden-Eijnden E, Chandler D (2007) Solvent coarse-graining and thestring method applied to the hydrophobic collapse of a hydrated chain. Proc NatlAcad Sci USA 104(37):14559–14564.

23. E W, Ren W, Vanden-Eijnden E (2007) Simplified and improved string method forcomputing the minimum energy paths in barrier-crossing events. J Chem Phys126(16):164103.

24. Cheng X, Lin L, e W, Zhang P, Shi AC (2010) Nucleation of ordered phases in block

copolymers. Phys Rev Lett 104(14):148301.25. Ting CL, Appelö D, Wang ZG (2011) Minimum energy path to membrane pore for-

mation and rupture. Phys Rev Lett 106(16):168101.26. Müller M, Smirnova YG, Marelli G, Fuhrmans M, Shi AC (2012) Transition path from

two apposed membranes to a stalk obtained by a combination of particle simulations

and string method. Phys Rev Lett 108(22):228103.27. Li W, Nealey PF, de Pablo JJ, Müller M (2014) Defect removal in the course of directed

self-assembly is facilitated in the vicinity of the order-disorder transition. Phys Rev

Lett 113(16):168301.28. Kim SO, et al. (2006) Defect structure in thin films of a lamellar block copolymer self-

assembled on neutral homogeneous and chemically nanopatterned surfaces.Macromolecules

39(16):5466–5470.29. Sebastian KL, Paul AKR (2000) Kramers problem for a polymer in a double well. Phys

Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 62(1 Pt B):927–939.30. Edwards EW, et al. (2005) Mechanism and kinetics of ordering in diblock copolymer

thin films on chemically nanopatterned substrates. J Polym Sci Part B Polym Phys

43(23):3444–3459.31. Detcheverry FA, Nealey PF, de Pablo JJ (2010) Directed assembly of a cylinder-forming

diblock copolymer: Topographic and chemical patterns. Macromolecules 43(15):

6495–6504.32. Ramírez-Hernández A, et al. (2013) Dynamical simulations of coarse grain polymeric

systems: Rouse and entangled dynamics. Macromolecules 46(15):6287–6299.33. Lodge TP, Dalvi MC (1995) Mechanisms of chain diffusion in lamellar block copoly-

mers. Phys Rev Lett 75(4):657–660.34. Segal-Peretz T, et al. (2015) Characterizing the three-dimensional structure of block

copolymers via sequential infiltration synthesis and scanning transmission electron

tomography. ACS Nano 9(5):5333–5347.35. Kim B, et al. (2014) Thermodynamic and kinetic aspects of defectivity in directed self-

assembly of cylinder-forming diblock copolymers in laterally confining thin channels.

J Appl Polym Sci 131(24):40790.36. Tong Q, Sibener SJ (2013) Visualization of individual defect mobility and annihilation

within cylinder-forming diblock copolymer thin films on nanopatterned substrates.

Macromolecules 46(21):8538–8544.37. Ruiz R, Bosworth JK, Black CT (2008) Effect of structural anisotropy on the coarsening

kinetics of diblock copolymer striped patterns. Phys Rev B 77(5):054204.38. Liu G, Thomas CS, Craig GSW, Nealey PF (2010) Integration of density multiplication in

the formation of device-oriented structures by directed assembly of block copolymer–

homopolymer blends. Adv Funct Mater 20(8):1251–1257.39. Branduardi D, Faraldo-Gómez JD (2013) String method for calculation of minimum

free-energy paths in Cartesian space in freely-tumbling systems. J Chem Theory

Comput 9(9):4140–4154.40. Helfand E, Tagami Y (1972) Theory of the interface between immiscible polymers. II.

J Chem Phys 56(7):3592–3601.41. Liu CC, et al. (2013) Chemical patterns for directed self-assembly of lamellae-

forming block copolymers with density multiplication of features. Macromolecules

46(4):1415–1424.

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