Defectivity in Laterally Confined Lamella-Forming Diblock

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Defectivity in Laterally Confined Lamella-Forming Diblock Copolymers: Thermodynamic and Kinetic Aspects Hassei Takahashi,†,‡ Nabil Laachi,† Kris T. Delaney,† Su-Mi Hur,†,‡ Corey J. Weinheimer,§ David Shykind,§ and Glenn H. Fredrickson†,‡,* †

Materials Research Laboratory, University of California, Santa Barbara, California 93106, United States Department of Chemical Engineering, University of California, Santa Barbara, California 93106, United States § Intel Corporation, Hillsboro, Oregon 97124, United States ‡

ABSTRACT: We use self-consistent field theory (SCFT) to study the directed self-assembly of laterally confined diblock copolymers. In this study, we focus on systems in which the self-assembled lamellae are oriented parallel to selective sidewalls in a channel. While well-ordered, perfect lamellae are observed in narrow channels both experimentally and numerically, undesirable defective structures also emerge. We therefore investigate the energetics of two categories of isolated defects (dislocations and disclinations) for various segregation strengths and channel dimensions, and establish conditions that favor the formation of defects. We also determine the energy barrier and the transition path between defective and perfect states using the string method. We find that only a few kT of energy are necessary to overcome the kinetic barrier and remove a defect, sharply contrasting with the large gain in free energy (many tens of kT) that is necessary for the formation of a defect from the pristine state.

I. INTRODUCTION With optical lithography rapidly approaching its scaling limits, progress in nanofabrication is increasingly relying on the development of alternative patterning routes. Particular interest lies in nanopatterning tools that yield features below the ≈40 nm (half-pitch) limit of conventional line-space lithography.1 In the search for alternative patterning techniques, block copolymers are excellent candidates as self-assembled templates for lithography. Because of their ≈10 nm scale of microdomain ordering and the variety of shapes they produce, block copolymers have the potential to define high resolution features at a fraction of the cost of today’s lithography processes.1−3 It has been shown, however, that thermally excited long wavelength phonon modes are sufficient to disrupt the long-range order in unbound hexagonal lattices and smectics, systems that are closely related to self-assembled block copolymers.4,5 As a result, thin films of block copolymers will organize into randomly oriented grains exhibiting poor long-range order without careful control or guidance of the assembly.2,3 Directed self-assembly (DSA) can be achieved in many ways, and among them, chemoepitaxy6 and graphoepitaxy (or lateral confinement) 7,8 in particular offer potentially inexpensive routes to dramatically reducing defect populations. Many successful examples have demonstrated the versatility of these approaches for producing a library of wellordered, periodic structures for surface patterning applications (see the review by Bang et al.9). For block copolymer lithography to advance into large-scale manufacturing and represent a viable alternative to photolithography, perhaps the most difficult obstacle that must be overcome is achieving sufficiently low defect levels. To date, typical defect densities remain orders of magnitude away from the target concentration of 0.01 defects/cm2 set by the © 2012 American Chemical Society

International Technology Roadmap for Semiconductors (ITRS).1,10 A successful approach for suppressing defects to such low levels has proved elusive, and without improved understanding of the fundamental causes for the prevalence of defects, this obstacle could very well limit the potential applications for directed self-assembly going forward. Several groups have studied defectivity in various graphoepitaxial systems.11−13 In layered smectic systems, such as lamellar and cylindrical (lyingdown monolayer) block copolymers, it was found both theoretically14 and experimentally15 that, at thermal equilibrium, isolated defects occur at a density nd ≈ ac −2 exp( −Ed /kT )

(1)

where ac is the size of the defect core, typically of order 10 nm. The key quantity Ed in the expression above is the free energy cost of creating a single defect in an otherwise defect-free film; numerical determination of this thermodynamic defect formation energy allows us to make predictions of the equilibrium defect concentration nd. In a recent study, Mishra et al. further examined the effect of microdomain spacing on defect densities in systems arranged in monolayers and bilayers of cylinder-forming block copolymers.16 They showed lower microdomain spacing, while desirable for nanopatterning applications, leads to higher defect densities by lowering the energetic cost of defect formation.16 These studies show that analyzing the factors that control the defect formation energy Ed is crucial to achieving defect density targets. Received: May 16, 2012 Revised: June 19, 2012 Published: July 18, 2012 6253

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closely approximating those of the incompressible model.26 Upon transforming the particle-based theory into a field-based model, we obtain a Hamiltonian that can be expressed as a functional of a pressure field W+(x) and an exchange potential field W−(x)23,24

The metastable defects we consider here are thermodynamic in nature, and hence their concentrations follow the above scaling. Nonequilibrium defects that are kinetically frozen during the annealing or quenching processes are beyond the scope of our study and will not be treated. We will address, however, kinetic pathways or mechanisms that connect metastable defects to defect-free structures. In this work, we employ self-consistent field theory (SCFT) to study the self-assembly of laterally confined lamella-forming block copolymer systems. With this computational approach, we determine factors that influence defect densities under equilibrium conditions through an investigation of the energetics of two types of isolated defects under confinement. While several theoretical and computational studies have investigated the confinement of symmetric lamella-forming block copolymer films between hard walls,17−19 we are not aware of any study focusing on a numerical determination of defect formation energies in these systems. We have further examined kinetic pathways by using the recently developed string method20 to compute minimum energy paths (MEP) for the transition from the defective state to the ordered state. This method not only reveals the mechanism by which the transition occurs, but also provides the kinetic barrier for escaping the defective state, which can be used to estimate anneal times required.

H[W+ , W −] = C +

⎡

∫ dx ⎢⎣ χ1N W −(x)2 2χw̅ Nϕw(x) − 2ζNϕ(x)iW+(x) + W+(x)2

χN + 2ζN ⎤ χ N + 2 w ϕw(x)W −(x)⎥ − Cϕ ̅ V ⎦ χN ln Q [WA , WB]

(2)

where χw̅ ≡ (χwA + χwB)/2, C = ρ0Rg3/N, ϕw(x) = ρw(x)/ρ0 is the normalized wall density, ϕ(x) = ρp(x)/ρ0 is the normalized total polymer density, ϕ̅ is its spatial average over the entire volume V, and Q ≡ Q[WA, WB] is the single chain partition function, a functional of the conjugate potential fields, WA(x) ≡ iW+(x) − W−(x) and WB(x) ≡ iW+(x) + W−(x), acting on Aand B-segments, respectively. The single-chain partition function can be calculated from Q = V−1∫ dx q(x, 1), where the propagator q ≡ q(x, s) is the solution to the modified diffusion equation26

II. ENERGETICS OF ISOLATED DEFECTS A. Self-Consistent Field Theory Model. We model a system of symmetric AB diblock copolymers confined between selective sidewalls in a narrow trench of width w and volume V. We describe the polymer as a standard continuous Gaussian chain composed of N segments, a fraction fA = 0.5 of which are of type A. Interactions between A and B monomers are mediated through the Flory parameter χ. In the case of poly(styrene-b-methyl methacrylate) (PS-b-PMMA) for example, the value of χ is slowly varying with temperature and a reference value of χN = 25 roughly corresponds to a polymer with an overall molecular weight of Mw ≈ 72 g/mol and a radius of gyration of approximately 7.2 nm.21 Lateral confinement is modeled using the masking method, first introduced by Matsen,22 where the impenetrable sidewalls are described by a mask with a prescribed wall density, ρw. The total density ρ0 is therefore given (in an incompressible model) by ρ0 = ρp(x) + ρw(x), where ρp(x) is the total monomer density associated with the confined block copolymers in the system. A hyperbolic tangent functional form for the wall density is used to exclude the polymers from the mask region where the polymer density is vanishingly small.22−24 The width of the wall-polymer interface is chosen to be 0.25 Rg, unless otherwise specified, where Rg is the unperturbed radius of gyration of the block copolymer. The walls of the mask also preferentially interact with the A-blocks to induce and favor parallel alignment of the lamellae in the longitudinal direction of the channel where periodic boundary conditions are imposed. These interactions are modeled, by analogy to monomer−monomer interactions, via a Flory-like parameter χw = (χwA − χwB)/2 where χwK(K = A, B) describes interactions between the wall and K-type monomers.23,24 We further opted for a compressible model in our study, which applies a harmonic penalty for local deviations from the total density ρ0, with magnitude controlled by a parameter ζ. We take advantage of the improved convergence characteristics of the compressible model25 and with a high penalty (in this study, ζN = 1000), we obtain values

∂q = ∇2 q − W (x , s)q ∂s

(3)

in which W = WA(x) for 0 ≤ s < fA and W = WB(x) for fA ≤ s ≤ 1. We note that all lengths have been expressed in units of Rg. The initial condition for eq 3 is q(x , 0) = 1

(4)

In the mean-field approximation, the thermodynamic properties of the confined melt are obtained from saddle-point configurations of the Hamiltonian in eq 2, i.e., solutions of δH[W+ , W −] δH[W+ , W −] = = 0. δW+(x) δW −(x)

(5)

Finally, in our implementation, the modified diffusion equation was solved using an operator-splitting, pseudospectral method,27 while field configurations were relaxed to their saddle points using an explicit Euler scheme.26 A detailed formulation of the model and the SCFT implementation can be found elsewhere.23,24 We note that all the calculations presented in section II.B and section II.C as well as the MEP calculations using the string method in section IV rely on two key assumptions: (i) The melts are perfectly monodisperse and, (ii) the bottom substrate of the channel and the top surface of the film are completely neutral (nonselective) to both polymer blocks. The latter assumption ensures that the self-assembled morphologies are homogeneous in the direction normal to the substrate and can therefore be fully described using two-dimensional (2D) simulations (see Figure 1a). The defect free energies and energy barriers that we report were nevertheless computed by extrapolating our 2D results to a homogeneous 3D channel with a depth of 4Rg, about one lamellar period. (Note that the extensive free energy of a structure homogeneous in the film normal direction is linearly proportional to the film thickness.) In doing so, the number of chains in the channel was estimated 6254

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Figure 1. (a) A-type density profile in a well-ordered lamella-forming diblock copolymer between two confining walls with A-favorable wetting conditions. Bright regions are A-rich microdomains resulting form 3D (left) and (2D) simulations. (b) Examples of density profiles of dislocations (left) and disclinations (right) we consider in this study. In all these profiles, χN = 25, χwN = −5 and w = 18Rg. For the 3D profile, the height of the channel is 4Rg.

Figure 2. Plot of a dimensionless excess free energy (due to confinement and wall interactions) as a function of channel width w. The excess free energy is reduced using the number of chains nc in the trench and the width w. The inset shows domains of stability of an assembly with n lamellar periods. The χN and χwN parameters are the same as in Figure 1.

As shown in the figure, perfectly ordered structures with a different number of periods will arise for various channel widths, consistent with previous studies on symmetric block copolymers between selective hard walls.19,22 The domains of stability, in which assembly with n periods has lower excess free energy than either n + 1 or n − 1 periods, are also evident from the inset of Figure 2. Within the domain of stability for n periods, the excess free energy is minimized at an optimal (commensurability) width, wn, for which the assembly is most stable. At wn, residual elastic strains due to confinement are nearly suppressed and the excess free energy is mainly dominated by the width-independent surface energy. The period, wc, of the block copolymer under confinement can be obtained from the sequence of optimal channel widths (or from the density profiles). In the case of χN = 25, we find wn = (n − 1)wc + 2b, where wc ≈ 4.15Rg, about 3% smaller than in bulk (wbulk ≈ 4.29Rg), and the parameter b ≈ 1.99Rg is the size of a c brush of the A-component near the walls that arises from the imposed mask. As the width of the channel increases or decreases away from the optimal width for n periods, tensile or compressive strains build up. At some critical value, their relaxation requires a rearrangement of the lamellae resulting in the addition or removal of a period. As can be seen from Figure 2, the stability domain of n periods is centered around the optimal width wn and extends approximately half a period for increasing and decreasing values away from wn. Introducing a dimensionless strain εn ≡ (w − wn)/wn, and using the approximation 2b ≈ wc, the window of stability of n periods is given by values of εn inside the interval [−1/2n, 1/2n]. This indicates a narrowing of the stability windows for larger channels.18,19,22 C. Defect Free Energies: Commensurability and Polymer Size Effects. In addition to well-ordered systems, SCFT simulations produce a rich variety of defective metastable structures. The characterization of such defects poses several challenges, both conceptually and numerically. Because of the high compliance of melts of block copolymers, defective morphologies are numerous and often span large domains, contrasting with discrete, well localized defects in crystals. On numerical grounds, defects can be hard to stabilize inside the

using a monomer density of 1 g/cm3, reasonably close to experimental values for PS and PMMA. Additional refinements through the investigation of the effects of polydispersity and bottom substrate interactions are the subject of section III. B. Domains of Stability of Defect-Free Structures. As expressed in eq 1, the overall concentration of isolated defects is an exponential function of their formation energy Ed, itself computed from Ed ≡ ΔF = Fd − Fp

(6)

where ΔF is the excess free energy due to the defective state with a free energy Fd, relative to a perfect state with a free energy Fp. In this section, we focus on the perfect lamellar state and defer the discussion of defects in laterally confined block copolymers to section II.C. A representative density profile of a perfectly ordered system with four periods (three internal A lamellae and two A brushes at the mask walls) is shown in Figure 1 where a single period refers to a repeat unit that consists of a pair of one A- and one B-lamella. The goal of this section is to determine the optimal circumstances that favor the emergence and the stability of well-ordered structures. Note that due to selective interactions at the side walls of the trench, we are only interested in “integer” arrangements of the lamellae and situations where an assembly involves half-periods19,22 will not be considered in the following. The total free energy Fp of the perfectly ordered system can be decomposed as Fp = Fb + Fex = Fb + Fel + Fw

(7)

where Fb is the free energy of the polymer in bulk and Fex is the total excess free energy. The latter can be further decomposed into an excess free energy, Fel, due to confinement (of elastic origin) and an excess free energy, Fw, due to wall interactions (surface energy). We note that, by definition, Fw depends only on local interactions near the walls and therefore, the variation of the excess free energy with respect to the channel dimension mainly results from the elastic component. A plot of the nondimensionalized excess free energy as a function of channel width is shown in Figure 2. 6255

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channel as they are continuously evolving both in nature and in size, and often are associated with a slow convergence rate compared to defect-free structures. We therefore chose to focus our study on two types of isolated defects, dislocations and disclinations, as they are in abundance experimentally and were more robust to changes in polymer parameters over the relatively narrow channel widths corresponding to n = 3 and n = 4 that were the primary focus of our investigations.28 In a recent experimental study, Mishra et al.16 showed that in laterally confined thin films of lying-down cylinders in wide channels, defects first appeared as paired dislocations (although widely separated across the film). When the annealing temperature was increased, the concentration of the paired dislocations rose until the first disclinations emerged. While SCFT neglects the thermal fluctuations necessary to support equilibrium populations of such defects, SCFT simulations conducted from random initial conditions in wide channels do lead to a preponderance of metastable defect structures that resemble those thermally populated in experimental systems. In the present study, however, we focus on relatively narrow channels where defects are reduced in number, size, and complexity by strong preferential wetting of the sidewalls. This is the regime of current technological interest; under these conditions, the defects that can be obtained in SCFT simulations tend to be isolated defects of an elementary nature. In particular, we limit our discussion to the elementary “dislocation” and “disclination” type defects represented in Figure 1b). We note that these are not true elementary defects, but rather reflect a tight dislocation pair and a disclination bound to a large Burgers vector dislocation. Our SCFT simulations in narrow channels with selective sidewalls have not turned up more “elementary” structures than these. 1. Wall Conditions. In the laterally confined channel geometry, wall conditions (commensurability and selectivity) are evidently key factors in promoting the self-assembly of defect-free lamellae. Park et al.29 observed that a well-ordered PS-b-PMMA lamellar structure with up to 8 periods was readily produced, with wider channels resulting in defects in the middle of the channel. Using a smaller molecular weight of PSb-PMMA, Ruiz et al.30 reported perfect lamellae in channels accommodating 10 periods. In this limit of a large number of periods n, i.e., when the channel width is large compared to the length of a single period, commensurability effects are negligible and defects arise, as in 2D smectics, from entropic effects driven by a thermal bath of phonon modes.13,14 Here we are concerned with understanding defectivity in the low n regime (n < 5) where channel confinement plays an essential role in the orientation of the chains. In addition to geometrical factors, the selectivity or wetting conditions at the sidewalls are equally important for inducing orientational order in the trench. For symmetric PS-b-PMMA systems, selective wetting can be achieved for example with Au or photoresist sidewalls.9,29,30 While control over the strength of this interaction is limited by chemistry, it is instructive to predict what benefits might be gained from varying the χw parameter. Using our 2D model, we can better understand how proximity to, and the strength of the selective interactions at the walls affect defectivity. We performed SCFT simulations in channels accommodating 3 to 7 periods and containing one of the two defects introduced above. By varying w and χwN, we examined both the effects of channel width and sidewall interactions on defect energies. The results of our simulations, summarized in Figure 3 for a fixed value of χwN = −5.0, indicate that many tens of kT gain in

Figure 3. Plots of the defect formation energy for dislocations (bottom) and disclinations (top) for various channel widths and different number of periods, n. Plots are shown for χN = 25 and χwN = −5 and the film thickness is assumed to be 4Rg.

free energy are necessary for the formation of defects from the pristine state. The energy cost is even larger for disclinations than for dislocations because they are accompanied by a larger distortion of the lamellae and stronger strain fields. Furthermore, the plots in Figure 3 also indicate that for a given number of periods, the free-energy cost of a defect is highest at a width slightly larger than the commensurability width for perfect lamellae in the same channel conditions. Deviations away from this optimal width lead to several tens of kT decrease in the formation energy and hence, to a strong increase in the concentration of defects. We also note that as the number of periods increases, the maximum formation energy also increases until it saturates to a period-independent, bulk value for very wide channels. While the intensive elastic energy of an unconfined system (defective or defect-free) is reduced when confinement constraints are present, the defect formation energies reported here are extensive properties and are computed from free energy differences between two different states. Another feature uncovered from the plots of Figure 3 is the asymmetric behavior of the defect formation energy with regard to deviations from the optimal width. This asymmetry is further highlighted in Figure 4 where the excess free energy due to the defect is symmetric for small values around zero strain. For larger strains, tension leads to lower values of the defect formation energy compared to compression of the same magnitude, suggesting that defect concentrations should be more sensitive to tensile strains. This is reminiscent of the buckling effect first predicted by Wang31 and confirmed experimentally later by Cohen et al.,32 where tensile strains introduce wavy (defective) patterns, ultimately leading to a buckling instability, in what is otherwise a perfect lamellar structure in the absence of strain. In addition to width variations, we also investigated the effect of changing the strength of the interactions at the walls. While increasing the strength of wall interactions does affect the absolute free energy of a defective or a defect-free configuration, we found that the differencethe defect formation energyis not very sensitive to the magnitude of the selectivity. The main effect of the presence of the selective walls is to nucleate a preferential alignment of the lamellae longitudinally, which then propagates to the center of the 6256

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Figure 4. Plots of a dislocation formation energy in a channel accommodating 3 lamellar periods (n = 3, solid line) and a parabola representing a perfectly symmetric fit of ΔF under compressive strains (dashed line). The two curves highlight the asymmetry of the defect formation energy when tensile or compressive strains are applied. At small values of the strain, the formation energy is Hookean and becomes more sensitive to tensile strains than compressive ones when the magnitude of the strain increases. ΔF is obtained for χN = 25 and χwN = −5 in a film that is 4Rg in thickness.

Figure 5. Plots of the defect formation energy of dislocations in channels with three lamellar periods for various χN parameters. Plots are shown for χwN = −5 and the film thickness is fixed at 4Rg.

larger size of the defect at stronger segregation and the associated increase in interfacial contact energy and chain stretching, and suggests that higher molecular weight block copolymers should exhibit much lower equilibrium defect densities. However, as the molecular weight increases, particularly above the entanglement lengths for the blocks, defects may be difficult to remove by annealing due to the kinetic constraints presented by entanglements, but also because reptative dynamics of lamellar block copolymers require surmounting energy barriers even for motion parallel to the layers.33 3. Comments on Defect Formation Energy. We presented in the previous section defect formation energies of block copolymers in narrow channels computed from SCFT simulations and found typical values in the range of many kT for dislocations in channels hosting n = 3 and n = 4 lamellar periods. In this section, we compare our results to estimates of the formation energy obtained from an analytical treatement inspired by an early work of de Gennes34 and Pershan35 in the case of smectics and extended later by Mishra et al.16 to systems of laterally confined lying-down cylinders in wide channels. Using analytical calculations of the strain field of an isolated dislocation by de Gennes34 and Peshran,35 Chandrasekhar and Ranganath36 arrived at the following expression for the dislocation formation energy per unit height of the film

channel. Because the magnitude of the interactions only manifests as a local pinning effect near the walls, as long as the core of the defect is far away from the walls, there is not a substantial difference between this contribution and that of a defect-free structure. In other words, what seems to matter in reducing defect concentrations is the promotion of lateral ordering through a preferential wetting at the wall, not the strength of the interaction leading to the ordering, at least above a χw threshold sufficient to pin lamellae at both walls. 2. χN Effects. Let us turn our attention now to the effect of χN on the propensity of defects in confining channels. As mentioned above, Ruiz et al. successfully aligned perfect lamellae in very wide channels using small molecular weight polymers.30 In the case of PS-b-PMMA, the Flory parameter χ is weakly dependent on temperature21 and variations of χN predominantly result from variations of the molecular weight of the block copolymer. To understand the sensitivity of defect energetics to variations in molecular weight, we performed simulations for χN = 21, 25 and 30, which, at an anneal temperature of 190 °C for symmetric PS-b-PMMA, approximately correspond to molecular weights for each block of 29 kg/mol, 36 kg/mol, and 42 kg/mol for the copolymer, respectively. The resulting defect formation energies due to isolated dislocations in channels accommodating 3 lamellar periods are shown in Figure 5. For each χN, the defect formation energy is maximal at some optimal channel width, determined by commensurability between the channel dimension and the natural period of the polymer. Since higher χN values produce larger domain spacing, we observe a shift of the peak defect formation free energy to wider channels. Experimentally, this allows for control of feature sizes by tuning the molecular weight of the chains in the melt. We also found that for a given period, increasing χN results in significantly higher defect formation energies; defect formation energies at χN = 30 are approximately twice those at χN = 21. This increase in the free-energy cost of producing a defect is evidently caused by the

εb = ε0 +

λb 2 B 2ac

(8)

where b is the length of the Burgers vector associated with a defect of core size ac and a core energy ε0, λ is the penetration depth, estimated by Amundson and Helfand to be approximately 0.25Rg for lamella-forming block copolymers,37 and B is the compression modulus normal to the layers. Ignoring the core energy for wide channels and setting ac ≈ b ≈ wc (wc being the size of a single period), a theoretical estimate of the formation energy of a dislocation inside the trench of height d ≈ wc is given by Ed ≈ 0.25R gwc 2B

(9)

A factor of 2 was introduced in the above expression to account for the tight pair of dislocations we focus on in this study 6257

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to take place and underestimating the formation energy of the tight pair of dislocations we consider in this study.

although we acknowledge that this is an overestimate, since some elastic energy is released by binding the pair. In the case of PS-b-PMMA with χN = 25, the commensurate width was calculated above as wc ≈ 4.15Rg with Rg ≈ 7.2 nm,21 and eq 9 reduces to Ed ≈ 1600B where B is expressed in units of kT/ nm3. Adopting a similar approach to Hammond et al.,15 the compression modulus B can be estimated from the elastic part of the free energy of perfect lamella, plotted in Figure 2. Using the dimensionless strain introduced in section II.B, the change in energy per chain δf when the layer dimensions are changed by a small strain amount δε is calculated in the linear regime from δf =

BM w εδε ρ 5AkT

III. POLYDISPERSITY AND SUBSTRATE INTERACTIONS We have shown that the canonical SCFT model described above is a valuable tool for investigating the effect of a number of factors on equilibrium defect energies. In this section, the assumptions of monodispersity and depth homogeneity are relaxed and the model is further refined to account for polydispersity and substrate interaction effects. A. Effect of Polydispersity. In order to investigate the effect of polydispersity on defect free energies, we have investigated a binary symmetric diblock copolymer blend under narrow confinement. This model is an extension of the twodimensional SCFT model outlined above, with two additional parameters describing the composition and (discrete) polydispersity of the blend: (i) The molar fraction φ1 of chains of type 1 and, (ii) the ratio γ = N2/N1 of chain lengths for the two polymer types in the blend. We note that the two block copolymers have the same symmetric composition and differ only in total molecular weight. For this binary blend, the polydispersity index PDI ≡ Nw/Nn is calculated using

(10)

where Mw and ρ are the molecular weight and mass density of the block copolymer and 5 A is Avogadro’s number. Inserting experimental values of PS-b-PMMA, i.e., ρ ≈ 1 g/cm3 and Mw ≈ 72 kg/mol when χN = 25, we obtain the following expression of B in units of kT/nm3 B≈

2 0.0083 ∂ f kT ∂ε 2

Nn = N1[φ1 + γ(1 − φ1)] (11)

Nw = N1

Using data from the plot in Figure 2 to evaluate ∂2f/∂ε2, the compression modulus is approximately 0.03 kT/nm3. The defect formation energy for a dislocation is therefore, Ed ≈ 50kT, in good agreement with the range 50−90kT for dislocations under similar conditions obtained from SCFT and plotted in Figure 3. A better agreement is expected if the core energy, ε0, is also included in the formation energy in eq 9, especially in the case of narrow channels where the strain field is limited by the confinement. Experimental studies on monolayers of lying-down cylinders of polystyrene-b-poly(2-vinylpyridine) (PS−PVP) by Mishra et al. reveal that defect densities are extremely sensitive to variations as small as 2 nm in the microdomain spacing.16 Using eq 1 and arguments similar to those leading to eq 9 and eq 11, they estimated defect formation energies for dislocations at 140 °C in the range 12−19 kT, much smaller than values we computed using SCFT in the case of lamella-forming diblocks. We outline here a couple of reasons for the observed discrepancies. The PS−PVP polymers used in the experimental study have a molecular weight ≈25 kg/mol, a minority block volume fraction ≈0.23 and a Flory parameter χ ≈ 0.1, translating to a segregation strength, χN ≈ 25. While this value is similar to that we used in our simulations, it is closer to the order−disorder transition in the case of cylinders with comparable minority block fractions than for lamellae. A PS− PVP system with χN > 35 is therfore more appropriate for a comparison with our lamella-forming diblocks. Since an increase in χN for the cylinder-forming system increases both the microdomain spacing and the compression modulus,16 eq 9 indicates that much larger defect formation energies are expected in cylinder-forming systems when χN > 35. Using Figure 12 from ref16, we should expect a 2-fold increase in Ed if χN is increased from 25 to 40 for cylinder-forming diblocks with a fixed composition. In addition to discrepancies due to χN values, the channels used to confine the PS−PVP diblocks in ref 16 are wide enough to contain many dozens of cylindrical periods, thus allowing dislocations of different Burgers vectors

(12)

φ1 + γ 2(1 − φ1) φ1 + γ(1 − φ1)

(13)

where Nn and Nw are the number- and weight-average chain lengths of the binary distribution, respectively. The interaction terms of the Hamiltonian in eq 2 remain unchanged, but to account for the two polymer types in the film, the entropic term In Q is replaced by −Cϕ ̅ V

N1 {φ ln Q 1[WA , WB] + (1 − φ1) ln Q 2[WA , WB]} Nn 1 (14)

where C is defined using the reference chain length N1. A number of theoretical and experimental studies have examined the effect of polydispersity, both continuous and discrete, on film morphology in unconfined systems.38 Broadening of the molecular weight distribution (MWD) leads to shifting phase boundaries and larger lattice periods,39 which we have demonstrated using the binary blend model. In Figure 6, we plotted the commensurability width, expressed in units of Rg,n, the radius of gyration of an unperturbed chain of number-average molecular weight Nn, as a function of PDI for binary blends with equivalent Nn. The curves show an approximately linear increase in domain spacing at low PDI, with the slope decreasing at higher PDI. We note that results for the three values of Nn/N1 we studied did not collapse onto one curve, but instead produced three curves for a given PDI. This is because PDI is a poor descriptor of the breadth of the molecular weight distribution,40 particularly for a binary blend. We chose to express the average molecular weight in terms of Nn, rather than weight-averaged molecular weight Nw, because the volume fractions of each block can be expressed more naturally with Nn. In some cases it has been shown Nw is the better choice,41 but we observed no evidence that is the case here. Prior to this work, the effect of polydispersity on defectivity has largely been overlooked, primarily because it is difficult to isolate its contribution to defect densities experimentally. We 6258

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Figure 8. Density profiles of A-type monomers in a polydisperse blend. (Left) Total density of A-monomers. (Right) Density of Amonomers belonging to the shorter chains. Short chains mostly occupy interfacial areas while long chains are more densely distributed around the defect (areas within white circles). Density profiles obtained for PDI = 1.4, Nn/N1 = 1.3, χNn = 25, and χwNn = −5.

Figure 6. Plots of the commensurability width as a function of the polydispersity index, PDI, for various binary blends with equivalent number-average molecular weight, Nn. Plots are shown for χNn = 25 and χwNn = −5.

began our study of equilibrium defectivity in polydisperse blends by computing defect free energies of dislocations in binary blends with equivalent N n at their respective commensurability widths. In Figure 7, we observe that defect

Figure 9. Plots of the formation energy of dislocations as a function of channel width for binary blends with a polydispersity index, PDI = 1.2 (top) and PDI = 1.4 (bottom). In all these runs, χNn = 25 and χwNn = −5. Same legend as in Figure 7.

energies of dislocations at a range of channel widths for various binary blends for PDI = 1.2 and 1.4. The curves for blends with PDI = 1.2 take on a steeper shape about the commensurability width compared to that of the monodisperse limit, and a further increase in the MWD to PDI = 1.4 produces even narrower curves. This result indicates commensurability is of greater significance on defect populations in polydisperse blends. While our results show that a monodisperse system most effectively suppresses equilibrium defects, with monotonically decreasing defect formation energies with increasing PDI, we note the decrease in defect energy is quite modest, particularly at PDI < 1.1 or within the range of values typical of anionically synthesized polymers. This suggests that deviation from the monodisperse condition is likely not the limiting factor in achieving defect density targets. More critical is the batch-tobatch reproducibility of the MWD. Channel dimensions should be tailored to the properties of the blend used to produce lamellae with consistent natural domain spacing, so the MWD should be as consistent as possible. This is an even more significant consideration for blends with broad MWD, given the increased sensitivity of defect formation energies to strain at high PDI. It is also possible to fine-tune feature sizes by blending block copolymers of varying molecular weight, as shown in a recent

Figure 7. Plots of the formation energy of dislocations as a function of the polydispersity index, PDI, in various binary blends with equivalent number-average molecular weight, Nn. For all of these data, channel widths are commensurate with 3 lamellar periods of the corresponding defect-free polydisperse blend. Also, χNn = 25, χwNn = −5, and the film thickness was taken to be 4Rg,n.

free energies decrease for broader MWD. This trend is intuitive given the rationale for the larger domain spacing observed in polydisperse blends; just as the presence of longer chains diminishes the stretching energy to allow for wider lamellae, a polydisperse blend can arrange its polymers such that the stress caused by the defect is minimized, translating to a lower energetic cost of the defect core and surrounding strain field. This is illustrated in the density profiles of A monomers in Figure 8. As shown there, monomers of the shorter chains are most concentrated at the interface of the A-rich and B-rich domains, whereas segments from the longer chains are preferentially located at the middle of the domains and in the area surrounding the defect. We further sought to investigate how commensurability effects are influenced by polydispersity in the blend. In Figure 9, we plotted the defect formation 6259

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study by Zhang et al.42 In addition to demonstrating the ability to produce domains of any size between those of the individual components blended, the authors found binary blends produced lower defect densities compared to either of the components individually. As was noted in their discussion, this is most likely due to kinetic factors; with longer chains demonstrating limited mobility, the blending of shorter chains act as plasticizers to improve ordering. The importance of such nonequilibrium considerations aside, the lower values of Ed that we have observed in polydisperse systems unequivocally implies larger defect densities for fully equilibrated systems. B. Effect of Non-Neutral Substrates. The 2D SCFT model simulates systems for which ideal sidewall and substrate conditions are achieved, with preferential wetting at the wall and neutral interactions at both top and bottom surfaces to promote ordering only in the lateral direction. A neutral substrate is typically achieved using a mixed polymer brush or random copolymer, but it is difficult to confirm neutrality of a substrate in a graphoepitaxial trench. If the substrate deviates significantly from neutrality, any inhomogeneities that develop in the direction normal to the substrate make the defect free energies calculated by the 2D model invalid. Therefore, it is of interest to study defect formation energies as a function of substrate interaction strength to understand how tolerant the system is to weakly interacting substrates. The 3D SCFT model is similar to the model described in section II.A, but requires resolution of the third spatial dimension. There is an additional Flory-type parameter, χw,sub, to describe the substrate−polymer interaction, while the top surface is assumed to be nonselective to either species. We performed 3D SCFT calculations for substrate-segment interactions ranging from χw,subN = 5 (B-wetting) to χw,subN = −5 (A-wetting), with a neutral top surface and A-wetting sidewalls. For simplicity, we only consider channels at commensurate widths and we set χN = 25 and χwN = −5. In the interest of studying defects analogous to those studied in our 2D simulations, we initialized with field configurations of defects homogeneous in depth (taken to be 8Rg, or ≈2wc) and allowed the fields to relax to nearby saddle points. In Figure 10, we plot the resulting defect formation energies as a function of the substrate selectivity. The curve is approximately parabolic, and as we deviate from the substrate neutrality condition, an increase in the defect formation energy is observed. While this trend is counterintuitive at first glance, a closer look at the density profiles shown in the inset of Figure 10 reveals the cause of such behavior. At χw,subN = 5, the block copolymers in the vicinity of the polymer−substrate interface rearrange so as to minimize the contact area between the A-rich domain and the substrate. This results in the slightly tapered domains shown. For A-wetting substrates, the polymers strain to produce tapered B-rich domains. While the tapering effect is observed for both defect-free and defective structures, polymers in the defective configurations are already strained, making it more difficult to compensate for the unfavorable polymer− substrate interfacial contacts. The significance of this result is that thermodynamically, neutral substrates do not suppress defects. The problem with non-neutral substrates is an effect not captured in our SCFT simulations; with high enough selectivity, competing lyingdown lamellae nucleate more readily from the substrate, which would increase the likelihood of creating defects of types that have not been studied here. Han et al. found that in systems with weakly preferential substrates, the lateral orientation (i.e.,

Figure 10. Plots of the formation energy of dislocations as a function of substrate selectivity, χw,subN, at a width commensurate with 3 and 4 lamellar periods, a thickness = 8Rg(≈2wc) and χN = 25. For a system with 3 lamellar periods, the inset shows cross sections of density profiles in the longitudinal direction of a channel of width w = 12.5Rg. In these runs, χw,subN = −5 (top), χw,subN = 0 (middle), and χw,subN = 5 (bottom).

standing-up lamellae) can be promoted by controlling film thickness such that it is a noninteger multiple of the lamellar period.43 Therefore, although we have shown that laterally confined systems can tolerate deviations from the neutral substrate condition, it is advisible when the selectivity becomes comparable to the sidewall selectivity, to control film thickness in addition to channel width to promote the standing-lamella morphology. In Figure 11, we illustrate the effect of the various competing morphologies that arise from different wetting conditions at side and bottom walls of the trench. To eliminate any orientational bias, all of these morphologies are converged saddle points of the Hamiltonian quenched from random initial configurations of the pressure and exchange fields. With only a weakly interacting substrate near the commensurability width, perpendicular lamellae are produced (Figure 11a). However, at channel widths for which the lamellae must strain to produce perpendicular lamellae (see Figure 11, parts b and d), or with substrates which are strongly selective (Figure 11c), we observe structures resembling lamellae which are parallel to the substrate. If the sidewalls are not selective, there is no aligning force promoting the standing-up lamellae morphology, and lying-down lamellae (Figure 11e) and lamellae perpendicular to the sidewalls (Figure 11f) form readily. These structures emphasize the importance of neutral substrates in the promotion of well-ordered lamellae.

IV. DEFECT KINETICS: THE STRING METHOD AND MINIMUM ENERGY PATHS Predicting defect concentrations from defect formation energies extracted from simulation rests on the assumption that the system is in equilibrium. An example of such a prediction in the case of defects with a core size ac ≈ 30 nm and a formation energy ΔF ≈ 50kT yields a concentration in eq 1 of nd ≈ 10−11 cm−2, much lower that the ITRS target of 10−2 cm−2. As a result, it is likely that kinetic effects would account for many of the defects observed experimentally and an examination of defect kinetics in the narrow channel geometry is of great interest. Specifically, we seek to identify minimum energy paths 6260

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pressure field, W+, can be relaxed to its saddle-point value consistent with a particular constrained value of W−. As a consequence, in the mean field approximation, the free energy landscape F ≈ H[W *+ [W −], W −] = H[W −]

is a functional of the single chemical potential field W−. In eq 16, the pressure field W*+[W−] is the solution to δH[W+, W−]/ δW+ = 0 and is therefore, also a functional of the exchange field W−(x). The choice of a partial saddle-pont approximation for F guarantees a convex SCFT free energy and that all intermediate string states are consistent with the compressible model we used in our study. The computation begins by discretizing the string into a chain of states parametrized by a reaction coordinate α, such that α = 0 and α = 1 correspond to the defective and perfect states, respectively. After initializing the string, typically with a linear interpolation of the W− field between the two local minima, the string is relaxed toward the MEP using a two-step iterative process. In the first step, the string is updated by evolving each of the sampled states in the direction of the local force. In the second step, a reparameterization of the string by interpolation is performed to evenly redistribute the new configurations along the path. This step is necessary to prevent the collapse of sampled states into the local minima at the string end points after the first step. Successive iterations are performed until a stationary solution is obtained. The steady state trajectory is not only informative about the two local minima of the free energy landscape, but also about the height of the free energy barrier that needs to be crossed for transition between the two minima. Because of the use of a reasonably small number of points/states along the string, the discretized string solution only approximates the exact transition state. We consequently incorporated the climbing image technique44 to calculate the kinetic barrier to high precision. In this method, the transition state is approached from the nearest points on the discretized string using a steepest ascent relaxation scheme, slightly different from the evolution equation in eq 15, but one that yields higher accuracy in identifying the transition state and barrier.44 B. Minimum Energy Paths and Kinetic Barriers: Dislocations. The string method outlined above was implemented for the same 2D block copolymer system to study transition pathways from defective to well-ordered configurations in confining channels. An example of such a calculation is shown in Figure 12 for the transition of a dislocation (tight pair) to defect-free lamellae. For the case of dislocation defects, all of our calculations result in a single melting mechanism, similar to that depicted in Figure 13. Starting at the defective state (α = 0), the polymers in the defect core region must rearrange to reconnect the broken upper lamella. The additional distortion and excess interfacial area required to achieve the transition state (α = 0.16) produces the kinetic barrier (a local maximum along the MEP) shown in Figure 12. Once the microdomains have merged, the system has overcome the kinetic barrier, and the link connecting the lamellae breaks to form the defect-free configuration (α = 1). Since minimal mixing of the blocks is observed in this mechanism, the kinetic barrier is relatively small and only a few kT are sufficient for the transition between the two local minima. Beyond uncovering the melting mechanism of dislocations, we have also investigated commensurability effects on the

Figure 11. Three-dimensional density profiles of lamella-forming block copolymers in trenches with various selectivity conditions at side and bottom walls illustrating competing morphologies during the selfassembly process. The relevant parameters are as follows: (a) χw,subN = 2.5, χwN = −2.5, w = 12.5Rg, (b) χw,subN = 2.5, χwN = −2.5, w = 15Rg, (c) χw,subN = 5, χwN = −5, w = 12.5Rg, (d) χw,subN = 5, χwN = −5, w = 15Rg, (e) χw,subN = −2.5, χwN = 0, w = 12.5Rg, and (f) χw,subN = 0, χwN = 0, w = 15Rg. A width w = 12.5Rg is closely commensurate with three lamellar periods while w = 15Rg is close to the limit of stability of a 3period assembly. In all of these runs, χN = 25 and the thickness is 8Rg(≈2wc).

(MEPs) on the free energy landscape that connect the elementary defect states to the pure state. Field configurations that are local maxima along an MEP are saddle points that represent possible transition states. The associated free energy barriers to be surmounted in crossing these transition states from the metastable defect can consequently be used in a kinetic model to estimate the kinetic rate of escape from a defect configuration. A. Outline of the String Method. A powerful method for computing MEP transition pathways in complex, high dimensionality systems is the so-called “string method”, recently developed by E et al.20 In this method, two local minima of the free energy landscape are connected by a string of intermediate states in configuration space and the MEP corresponds to the energetically most probable trajectory for the kinetic transition between the two states. If the string is described by some continuous curve ψ in configuration space, then the MEP is the stationary solution to the evolution equation20,44 ψ̇n = −(∇F )⊥

(16)

(15)

where ψ̇ n denotes the velocity normal to the curve and (∇F)⊥ is the force component acting in the direction normal to the string. A detailed discussion of the theory can be found elsewhere, and in the following we briefly describe the method as applied to a field-based theory.44,45 We seek a string describing the minimum-energy transition path between the two local minima corresponding to the metastable defective state and the defect-free state. In the case of a field-based theory for block copolymers, Cheng et al.45 argued that an appropriate configuration space for the string is restricted to the exchange potential field, W−, whereas the 6261

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Figure 12. Example of a pathway for the transition between the defective (α = 0) and the defect-free state (α = 1) obtained using the string method. The free energies are calculated relative to the defective state. The kinetic barrier to the transition occurs at an intermediate state where α ≈ 0.16. In this example, χN = 25, χwN = −5, and w = 12.5Rg (n = 3). The energies were scaled to films with a thickness of 4Rg.

Figure 14. Plot of the barrier height to escape a dislocation defect as a function of the width for channels with 3 (left) and 4 (right) lamellar periods. In these runs, χN = 25 and χwN = −5. The energies were scaled to films with a thickness of 4Rg.

In order to further examine the effect of the wall on kinetic barriers, we have also varied the number of lamellae in the film, as well as the polymer-wall interaction strength. In Table 1, we Table 1. Kinetic Barriers for Films with Various Number of Lamellar Periods at Commensurate Widthsa

a

number of periods, n

commensurate width [Rg]

kinetic barrier [kT]

3 5 7

12.8 21.4 29.9

3.25 4.01 4.40

Values reported for χN = 25, χwN = −5, and films of thickness 4Rg.

show our results for the kinetic barriers at commensurate widths for n = 3, 5, and 7 lamellar periods. Previously we found that pinning the defect between narrow channels decreased the formation energy cost of defects. In the case of defect melting, however, the distortion required to reach the transition state is accompanied by higher kinetic barriers when the number of lamellae is increased, hence, unfavorably impacting the annihilation rates of defects in wider channels. Beyond commensurability effects, the pinning effect of the polymer-wall interactions, shown in Figure 15, indicates that an increase in the absolute value of χw is accompanied by a lowering of the transition barrier. This may be somewhat surprising considering that the wall interaction strength did not change defect formation energies appreciably. However, the kinetic barriers (≈ 3kT) are small relative to defect formation energies (≈ 100kT) and it is reasonable that the aligning force imposed by the pinning of polymers at the wall affects only the kinetic barriers significantly. We also note that the relatively small change in the barrier height with varying wall interactions might simply be an artifact of the commensurate witdh depending slightly on χw. We now focus on the effect of block segregation strength on transition barriers. In Figure 16, we have plotted the kinetic barriers for disclocation defects for χN = 21, 25, and 30, at widths commensurate with the domain spacing at these segregation strengths. Kinetic barriers clearly demonstrate a strong dependence on χN. The excess interfacial area and chain stretching at the transition state contribute to the height of the

Figure 13. Snapshots of the melting mechanism of the dislocation in Figure 12. The two arms of the dislocation first break and then rearrange to form perfect lamellae. The transition state occurs at α = 0.16, shown in the large bottom image.

transition barrier. In Figure 14, kinetic barriers are plotted as a function of channel width for dislocations with n = 3 and n = 4 lamellar periods, at χN = 25, χwN = −5 and a thickness of 4Rg. Similar to defect formation energies in Figure 3, kinetic barriers are nonmonotonic functions of channel widths with a minimum value where defect annihilation is fastest. However, for a given stability domain, the barrier is minimal at a width slightly smaller than the commensurate width. For channels slightly narrower than the commensurate width, chains at the defect core are more compressed than their counterparts in the commensurate case. As a result, the stretching of polymers that occurs at the transition state requires a smaller strain energy and hence, a lower barrier when the channels are slightly narrower. 6262

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Figure 17. Transition pathways for the symmetric (orange) and the asymmetric (blue) melting of disclinations obtained using different string initializations with χN = 25, χwN = −5, w = 12.5Rg (n = 3), and a thickness of 4Rg. The initial kinetic barriers of both transitions are 5.2 kT and 2.1 kT, respectively. Snapshots of initial, intermediate, and final density profiles for both paths are also shown.

Figure 15. Plot of the barrier height to escape a dislocation as a function of χwN in channels with widths commensurate with three lamellar periods. In these runs, χN = 25 and w = 12.5Rg. The energies were scaled to films with a thickness of 4Rg.

transition state barrier of 5.2kT. The asymmetric mechanism, also pictured in Figure 17, has one terminated lamella meet the bend and produces an intermediate metastable state α = 0.34, found to be a dislocation. The relative rates at which the system escapes the disclination are determined by the initial kinetic barrier for these competing mechanisms. For all the parameter sets we studied, we found that the one-step symmetric melting of disclinations has a higher barrier and should proceed much slower than escape via the asymmetric pathway. Disclinations should thus transform into dislocations at a faster rate than directly into defect-free lamellae, which helps to explain the prevalence of dislocation defects in confined channels.16 Beyond the 2D melting mechanisms described above, more complex pathways can emerge from full 3D calculations when inhomogeneities in depth are present. For example, the domains may meet at the bottom of the film and “zip up” to the top of the film. Because we cannot measure the kinetic barriers associated with such pathways without performing costly 3D string calculations, our results can be considered an upper bound of kinetic barriers. D. Kinetic Barriers and Defect Annealing Rates. The kinetic barriers computed in the previous section can be used to assess relative annealing times for defect annihilation. Using a Kramers-like approach, the average time, τ, necessary to cross a barrier height, Eb, and reach a defect-free configuration is given by44,46

Figure 16. Plot of the barrier height as a function of χN for dislocations in channels with widths commensurate with 3 lamellar periods (i.e., w = 12.2Rg, 12.8Rg, and 13.4Rg for χN = 21, 25, and 30, respectively). In these runs, χwN = −5 and the energies were scaled to films with a thickness of 4Rg.

kinetic barrier, and the cost of both increases with the segregation strength. This suggests an increased difficulty in producing well-ordered lamellae at higher molecular weight, consistent with comparable experimental studies.29 We note that other kinetic factors, such as entanglement effects, can also dramatically affect transition rates and lock the long polymers in long-lived nonequilibrium defective configurations. C. Disclinations and Uniqueness of Kinetic Pathways. While we found a single mechanism for the dislocation-toperfect lamellae transition for every parameter set we studied, the uniqueness of MEP trajectories is not guaranteed by the SCFT free-energy functional and different initializations of the string may therefore result in multiple string solutions. One such example is reported by Cheng et al. where two individual MEPs for the bulk lamella (L) → gyroid (G) transition were found.45 Similarly, our string calculations show that disclinations annihilate to perfect states via two distinct pathways. In the symmetric mechanism, pictured in Figure 17, both terminated lamellae simultaneously meet the disclination bend. This corresponds to the MEP in Figure 17 with a single

τ ≈ τ0 exp[Eb /kT ]

(17)

where τ0 is a prefactor that can be approximated by τ0 ≈

ξ2 D0

(18)

since the transition occurs primarily by parallel diffusion along microdomains. In eq 18, D0 is the diffusion coefficient of a block copolymer chain parallel to the lamellae and ξ is the length scale of the diffusion. In the PS-b-PMMA system, the diffusion coefficient is limited by the PMMA block which has a much greater monomeric friction factor relative to the PS block. Therefore, we use D0 ≈ 10−12 cm2/s, the parallel diffusion coefficient for PMMA in the case of symmetric PS-b-PMMA 6263

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Macromolecules with a number-average molecular weight of ≈51 kg/mol at 195 °C.30,33 The relevant length scale, ξ, is the distance between the broken lamellae, measured at approximately one lamellar period wc ≈ 30 nm. Using these values, we estimate the anneal times for a dislocation melting shown in Table 2. While the times

anneal time

3 5 7 10

200 s 25 min 3h 60 h

predicted are short for commensurate widths (barrier ≈3kT), they represent the time required to remove a single dislocation defect. To anneal many defects, some of which are more complex in nature, would undoubtedly involve longer times. The significance of commensurability effects are again demonstrated in these results; slight deviations from the commensurate width resulting in an increase in the kinetic barrier from 3kT to 5kT requires an ≈8-fold increase in the anneal time. Similarly, because kinetic barriers increase approximately linearly with χN for the three values studied, anneal times increase exponentially with increased segregation strength (or molecular weight) within this range. These two factors are particularly critical for anneal rates, and careful consideration should be given to ensure optimal conditions for defect removal.

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REFERENCES

(1) Herr, D. J. C. J. Mater. Res. 2011, 26, 122. (2) Park, C.; Yoon, J.; Thomas, E. L. Polymer 2003, 44, 6725. (3) Segalman, R. A. Mater. Sci. Eng. R: Rep. 2005, 48, 191. (4) Mermin, N. D. Phys. Rev. 1968, 176, 250. (5) Caillé, A. CR Acad. Sci. Paris Ser. B 1972, 274, 891. (6) Kim, S. O.; Solak, H. H.; Stoykovich, M. P.; Ferrier, N.; De Pablo, J. J.; Nealey, P. F.; et al. Nature 2003, 424, 411. (7) Segalman, R. A.; Yokoyama, H.; Kramer, E. J. Adv. Mater. 2001, 13, 1152. (8) Sundrani, D.; Sibener, S. J. Macromolecules 2002, 35, 8531. (9) Bang, J.; Jeong, U.; Ryu, D. Y.; Russell, T. P.; Hawker, C. J. Adv. Mater. 2009, 21, 4769. (10) www.itrs.net (11) Tiron, R.; Chevalier, X.; Couderc, C.; Pradelles, J.; Bustos, J.; Pain, L.; Navarro, C.; Magnet, S.; Fleury, G.; Hadziioannou, G. J. Vacuum Sci. Technol. B: Microelectron. Nanometer Struct. 2011, 29, 06F206. (12) Bosse, A. W.; Sides, S. W.; Katsov, K.; García-Cervera, C. J.; Fredrickson, G. H. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 2495. (13) Cheng, J. Y.; Mayes, A. M.; Ross, C. A. Nature Mater. 2004, 3, 823. (14) Toner, J.; Nelson, D. R. Phys. Rev. B 1981, 23, 316. (15) Hammond, M. R.; Cochran, E.; Fredrickson, G. H.; Kramer, E. J. Macromolecules 2005, 38, 6575. (16) Mishra, V.; Fredrickson, G. H.; Kramer, E. J. ACS Nano 2012, 6, 2629. (17) Kellogg, G. J.; Walton, D. G.; Mayes, A. M.; Lambooy, P.; Russell, T. P.; Gallagher, P. D.; Satija, S. K. Phys. Rev. Lett. 1996, 76, 2503. (18) Kikuchi, M.; Binder, K. J. Chem. Phys. 1994, 101, 3367. (19) Geisinger, T.; Mueller, M.; Binder, K. J. Chem. Phys. 1999, 111, 5241. (20) E, W.; Ren, W.; Vanden-Eijnden, E. Phys. Rev. B 2002, 66, 052301. (21) Russell, T. P.; Hjelm, R. P., Jr; Seeger, P. A. Macromolecules 1990, 23, 890. (22) Matsen, M. W. J. Chem. Phys. 1997, 106, 7781. (23) Bosse, A. W.; García-Cervera, C. J.; Fredrickson, G. H. Macromolecules 2007, 40, 9570. (24) Hur, S.-M.; García-Cervera, C. J.; Kramer, E. J.; Fredrickson, G. H. Macromolecules 2009, 42, 5861. (25) Chantawansri, T. L.; Hur, S.-M.; García-Cervera, C. J.; Ceniceros, H. D.; Fredrickson, G. H. J. Chem. Phys. 2011, 134, 244905. (26) Fredrickson, G. H. The equilibrium theory of inhomogeneous polymers; Oxford University Press: New York, 2006, Vol. 134. (27) Tzeremes, G.; Rasmussen, K.; Lookman, T.; Saxena, A.; et al. Phys. Rev. E 2002, 65, 33. (28) The two defects were captured from initial exploratory runs that yielded highly defective morphologies. The isolated defects were then

V. CONCLUSIONS The success of block copolymer lithography hinges, among other criteria, on the ability to control defectivity. In order to achieve low defect levels consistent with ITRS targets, a systematic approach is required to identify the conditions that most effectively suppress defect formation. Simulation fills this need, and we have demonstrated that SCFT and the string method together allow for investigation of equilibrium defect densities, as well as estimation of kinetic rates for defect annihilation. Using SCFT, we showed that equilibrium densities of isolated dislocations and disclinations in narrowly confined symmetric block copolymers with strong preferential wall wetting conditions are well below target densities. In contrast, we found the melting of elementary defects into a perfectly ordered state can be impeded by kinetic barriers. In both cases, the effects of the trench (commensurability and selectivity) and polymer characteristics (χN parameter, polydispersity) were investigated in detail. The computational framework used here can similarly be applied to the study of defects in other grapho- or chemoepitaxially ordered systems such as lying-down or standing-up cylinders. Our results should ultimately provide useful guidelines for the design of optimized patterning processes where defect concentrations are minimized.

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ACKNOWLEDGMENTS

This work was funded by Intel Corporation and the Focus Center Research Program (FCRP) - Center on Functional Engineered Nano Architectonics (FENA). The calculations presented in this work were largely conducted using computational facilities at Intel. Partial calculations of defect energetics were conducted using the computational resources of the California NanoSystems Institute (CNSI) and Materials Research Laboratory (MRL) at the University of CaliforniaSanta Barbara. The MRL Central Facilities are supported by the MRSEC Program of the NSF under Award No. DMR 1121053; a member of the NSF-funded Materials Research Facilities Network (www.mrfn.org). We finally wish to thank Robert Bristol, Intel Components Research, for experimental and theoretical conversations about DSA and lithography.

Table 2. Anneal Times Calculated from Kinetic Barriers and Constants for the PS−PMMA System kinetic barrier [kT]

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest. 6264

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embedded in a nondefective structure and the resulting configuration was used as a seed to study the energetics of stable defects. (29) Park, S. M.; Stoykovich, M. P.; Ruiz, R.; Zhang, Y.; Black, C. T.; Nealey, P. F. Adv. Mater. 2007, 19, 607. (30) Ruiz, R.; Sandstrom, R. L.; Black, C. T. Adv. Mater. 2007, 19, 587. (31) Wang, Z.-G. J. Chem. Phys. 1994, 100, 2298. (32) Cohen, Y.; Albalak, R. J.; Dair, B. J.; Capel, M. S.; Thomas, E. L. Macromolecules 2000, 33, 6502. (33) Ferry, J. D., Viscoelastic properties of polymers; John Wiley & Sons Inc: New York, 1980) (34) de Gennes, P.-G. Solid State Commun. 1972, 10, 753. (35) Pershan, P. S. J. Appl. Phys. 1974, 45, 1590. (36) Chandrasekhar, S.; Ranganath, G. S. Adv. Phys. 1986, 35, 507. (37) Amundson, K.; Helfand, E. Macromolecules 1993, 26, 1324. (38) Lynd, N. A.; Meuler, A. J.; Hillmyer, M. A. Prog. Polym. Sci. 2008, 33, 875. (39) Cooke, D. M.; Shi, A. C. Macromolecules 2006, 39, 6661. (40) Rane, S. S.; Choi, P. Chem. Mater. 2005, 17, 926. (41) Sides, S. W.; Fredrickson, G. H. J. Chem. Phys. 2004, 121, 4974. (42) Zhang, X.; Murphy, J. N.; Wu, N. L. Y.; Harris, K. D.; Buriak, J. M. Macromolecules 2011, 44, 9752. (43) Han, E.; Kang, H.; Liu, C. C.; Nealey, P. F.; Gopalan, P. Adv. Mater. 2010, 22, 4325. (44) E, W.; Ren, W.; Vanden-Eijnden, E. J. Chem. Phys. 2007, 126, 164103. (45) Cheng, X.; Lin, L.; Zhang, P.; Shi, A. C. Phys. Rev. Lett. 2010, 104, 148301. (46) Gardiner, C. W., Stochastic Methods; Springer: Berlin, 1985.

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