Thermodynamics and Kinetics of Defect Motion and Annihilation in the

Article pubs.acs.org/Macromolecules

Thermodynamics and Kinetics of Defect Motion and Annihilation in the Self-Assembly of Lamellar Diblock Copolymers Weihua Li†,‡ and Marcus Müller*,† †

Institute for Theoretical Physics, Georg-August University, 37077 Göttingen, Germany State Key Laboratory of Molecular Engineering of Polymers, Department of Macromolecular Science, Fudan University, Shanghai 200433, China

‡

ABSTRACT: The thermodynamics of dislocations in thin films of lamella-forming diblock copolymers and their climb and glide motions are investigated using single-chain-in-mean-field (SCMF) simulations and self-consistent field theory (SCFT) in conjunction with the string method. The glide motion of a defect perpendicular to the stripe pattern is characterized by large free energy barriers. The barriers not only stem from altering the domain topology; an additional barrier arises from a small-amplitude but long-range domain displacement. In contrast, the climb motion along the stripes does not involve a free energy barrier in accord with the continuous translational invariance along the stripe. Thus, the perpendicular distance (“impact parameter”) between a pair of defects is approximately conserved. Dislocation pairs with opposite Burgers vectors attract each other and move toward each other (“collide”) via climb motion. We find that the forces between apposing defects significantly depend on system size, and the Peach−Koehler force in smectic structures only becomes accurate for extremely large system sizes. Moreover, we observe in SCMF simulations that the defect annihilation time qualitatively and nonmonotonously depends on the defects’ perpendicular distance and rationalize this finding by the collective kinetics along the minimum free energy path (MFEP) and the single-chain dynamics in an inhomogeneous environment.

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studied by self-consistent field theory (SCFT).9,18,23,24 Imperfections due to stripe misalignment or disconnections result in the occurrence of topological defects in the quasi-twodimensional patterns similar to what is observed in smectic liquid crystals.9 One prototypical defect type are dislocations. These defects not only appear in two-dimensional lamellar morphologies in thin films without patterned substrate, but they are also observed in single-layer lying cylinders guided by topographical trenches25 or free-standing lamellae guided by periodic chemical patterns with imperfect guiding conditions.26 Thermodynamically, the (meta)stability of a defect is characterized by its excess free energy, ΔFd, i.e., the free energy difference with respect to the corresponding defect-free morphology.18 The free energy of a defect dictates the probability that a defect is created by thermal fluctuations in a defect-free morphology. Except for the ultimate vicinity of the transition to the disordered phase, ΔF ∼ kBT 5̅ with kB denoting Boltzmann’s constant and T temperature. The large values of the invariant degree of polymerization, 5̅ ∼ 104 , prevent the spontaneous formation of defects. This finding implies that experimentally observed defects cannot be conceived as rare equilibrium fluctuations in a defect-free

INTRODUCTION Block copolymer self-assembly provides a useful platform for the fabrication of various ordered nanostructures.1−4 Varying molecular architectures including chain topology and number of blocks or species, one can fabricate a vast diversity of equilibrium structures for a wide spectrum of potential applications.5−7 For applications in microelectronics, extremely small defect densities on the order of 1 defect in 100 cm2 are required, and chemically or topographical substrate patterns are employed to guide the structure formation. This directed selfassembly (DSA) of block copolymers offers a promising bottom-up patterning technique that is currently regarded as one of the most appealing next-generation lithography techniques.3,8,9 On one hand, DSA aims to generate largescale, defect-free, geometrically simple and dense structures.10−13 On the other hand, DSA targets the design of irregular, device-oriented structures, of which some structural units resemble the geometry of defects.14−17 For both application aspects it is critical to understand and control the thermodynamics and kinetics of defect formation and annihilation.18,19 One of the most widely studied patterns in DSA are lines and spaces (L/S) that are formed by the self-assembly of AB block copolymers in thin films yielding perpendicularly standing lamellae10 or single-layer lying cylinders.20−22 These L/S structures in the context of DSA have also been intensively © XXXX American Chemical Society

Received: May 22, 2016 Revised: July 26, 2016

A

DOI: 10.1021/acs.macromol.6b01088 Macromolecules XXXX, XXX, XXX−XXX

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annihilation of pairs of dislocations with opposite Burgers vectors formed in single-layer lying cylinders using in situ visualization techniques.25 They observed the climb motion along the stripe pattern and glide motion perpendicular to the stripes. The latter of the two modes, dislocation glide, was dramatically slower.25 From the time evolution of the distance, L(t), between defects, L2 ∼ t, they concluded that the attractive force between defects is inversely proportional to their distance, in accord with the Peach−Koehler force in nematic phases.40−44 Here we study dislocation glide and climb in lamella-forming block copolymers by particle-based simulations45,46 and selfconsistent field theory (SCFT). In particular, we focus on the role of the distance of the dislocation pair perpendicular to the stripe direction and the forces between defects. First, the kinetics of structure evolution is investigated by Monte Carlo (MC) simulations employing the single-chain-in-mean-field (SCMF) algorithm.45,46 These simulations yield direct insights into structure formation including inter alia intermediate morphologies during defect motion and annihilation as well as estimates of the defect-annihilation rate. Subsequently, these simulation results are complemented by SCFT calculations using the pseudospectral method of SCFT.47,48 These SCFT calculations provide information about the MFEPs, the free energy of defect structures, and interaction between defects.

structure, but they represent metastable states, into which the kinetics of structure formation became trapped.27 From the kinetics point of view, defect (meta)stability is dictated by the complex, rugged free energy landscape surrounding the local minimum that represents the defect. There are a variety of paths on the free energy landscape connecting the minimum of the considered metastable defect and its neighboring minima that representing other metastable defects or the stable, defect-free structure. The time evolution will closely mimic the minimum free energy path (MFEP) on the free energy landscape because fluctuation effects are suppressed for high molecular weights, 5̅ . There may be multiple MFEPs connecting a metastable defect to the defectfree structure representing the global free energy minimum. The probability of a specific path or defect-annihilation mechanism is dictated by the largest free energy barrier along the path; i.e., the mechanism with the smallest free energy barrier occurs more likely and determines the defectannihilation rate. Given a suitable initial condition for the transformation pathway, the string method28 allows to efficiently compute a MFEP. This technique has been implemented in SCFT calculations29 and particle-based simulations,30 and it has found fruitful applications for computing pathways of collective structure changes in self-assembling systems.18,29,31−34 Using SCFT calculations, Li et al. have studied the metastability and annihilation mechanism of a prototypical tight dislocation pair in lamella-forming AB diblock copolymers on uniform and chemically patterned substrates.18 These SCFT calculations reveal that the free energy barrier, ΔFb, linearly varies with χN,18,32,35 where χ denotes the Flory−Huggins interaction parameter characterizing the immiscibility of A and B species and N is the number of segments. Importantly, ΔFb vanishes at a critical value of χN* ≈ 18 for nonpatterned substrates, which is significantly higher than the order−disorder transition (ODT), χNODT ≈ 10.5; i.e., there exists an interval of incompatibilities, χNODT < χN < χN*, where ΔFb vanishes and defects are not even metastable but spontaneously annihilate whereas ΔFd is too large for new defects to form by fluctuations.18 This critical value χN* can be further increased by using an external guiding field identifying optimal conditions for DSA. These conclusions have been corroborated by particle simulations.34 At weak and intermediate incompatibilities, χN < χN*, the width of the internal AB interfaces and the thermal fluctuations of their local positionline edge roughness (LER)will be enhanced in the block copolymer system.36−39 These lowsegregation features may appear detrimental for DSA applications, but (i) we will show that one can anneal defects at low χN and then quench the system to larger segregation in order to reduces these fluctuation effects and (ii), importantly, in typical DSA applications one component of the copolymer is removed in an etching step (in the glassy state), thereby replacing the internal AB interfaces with a low interface tension by a polymer−vapor surface with a significantly larger surface tension, and the concomitant surface flow will smoothen highfrequency LER. In the present study we investigate how dislocation pairs interact and move with respect to each other in simple geometries. This study provides information about how metastable tight dislocations pairs are potentially formed in the course of structure formation. In recent elegant experiments, Tong and Sibener have tracked the motion and

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MODEL AND TECHNIQUES Single-Chain-in-Mean-Field (SCMF) Simulation. In single-chain-in-mean-field (SCMF) simulation we employ a soft, coarse-grained particle model. Polymer molecules are represented by a bead−spring model with harmonic bonds.45,46 /b[{r}] 3(N − 1) = kBT 2R e0 2

n

N−1

∑ ∑ [ri(s + 1) − ri(s)]2 i=1 s=1

(1)

where ri(s) denotes the position of bead number s on polymer i = 1, ..., n. The n macromolecules are each composed of N = 32 coarse-grained beads. This discretized Edwards Hamiltonian of Gaussian macromolecules is complemented by nonbonded interactions that take the form of a local density functional /nb[{r}] kBT N̅

=

∫ Rdr 3 [ κ2N (ϕÂ + ϕB̂ − 1)2 − e0

χN ̂ (ϕ − ϕB̂ )2 ] 4 A (2)

where the density of A beads is given by ϕÂ (r|{r}) =

1 ρ

n

N

∑ ∑ i = 1 s = fA N

δ(r − ri(s)) (3)

The first half of the copolymer consists of A beads, f = 1/2, and similar expressions hold for the density of B beads. Here, ρ = nN/V denotes the segment number density, and 5̅ = ρR e0 3/N = 128. The Flory−Huggins parameter χN = 30 quantifies the repulsion between A and B blocks. We set the inverse isothermal compressibility to κN = 50. The nonbonded interactions are computed via the segment densities on a collocation grid with typical grid spacing, ΔL = Re0/6, using a linear assignment. Additional details about the computational model and techniques are discussed in ref 49. We study thin films with thickness Lz = 1.26Re0 ≈ (3/4)d0, where d0 denotes the equilibrium lamellar period because it maximally frustrates lying structures.50 Both confining film B


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Macromolecules surfaces are hard and nonpreferential, whereas periodic boundary conditions are applied in the two lateral directions. In order to exploit the difference between the strong bonded and weak, slowly varying nonbonded forces, we use the SCMF simulation algorithm; i.e., we replace the weak, slowly varying nonbonded forces by external fields that mimic the interaction of a bead with its instantaneous surrounding while updating the bead positions by Smart Monte Carlo moves.51 After each Monte Carlo sweep (SMC) the fluctuating external fields are recalculated from the instantaneous densities. This quasiinstantaneous field approximation provides an accurate description of fluctuations.45 The Smart Monte Carlo moves use the strong bonded interactions to propose a trail displacement of the beads, resulting in an overdamped dynamics. The soft, nonbonded interactions do not prevent chain contours from crossing each other, and after less than 10 SMC, the single-chain dynamics is accurately described by the Rouse model, appropriate for nonentangled polymer systems.51 Importantly, the particlebased simulations capture the relation between the single-chain dynamics and the kinetics of the collective densities, which is difficult in dynamic SCFT or continuum models because the Onsager coefficient of the incompressible system is both nonlocal and density dependent. We use the time, τ = Re02/D = 8900 SMC moves with D being the self-diffusion coefficient, that a macromolecule in the disordered phase, χN = 0, needs to diffuse its size, to identify the time scale.51,52 Self-Consistent Field Theory and String Method. In the SCFT calculations we study a similar system as in the particle-based simulations, f = 1/2, χN = 30, d0 = 1.825Re0, and Lz = 0.8d0 except that the melt is strictly incompressible and the chain contour is finely discretized, Δs = 0.01. The two confining, nonpreferential surfaces in the xy-plane are modeled by reflective boundaries. Within the standard framework of SCFT of Gaussian chain model,53,54 the free energy can be expressed as F 5̅ kBTV /R e0 3

= − ln 8 +

∫

partition function 8 = (1/V ) ∫ dr q(r, s)q†(r, s) (for any 0 ≤ s ≤ 1). For the equilibrium calculation of the defect morphologies, we implement the pseudospectral method47,48 to solve the modified diffusion equations coupled with an initialization scheme that targets the desired defects.18,55 Minimization of the free energy functional leads to the selfconsistent equations for the saddle-point values

(8)

ϕA (r) =

1 Q

∫0

ϕB(r) =

1 Q

∫f

f

1

G[W, U[W ]] 5̅ kBTV /R e0 3

ds q(r, s)q†(r, s)

(9)

ds q(r, s)q†(r, s)

(10) (11)

=

∫

⎧ dr ⎪ [W (r)]2 ⎨ − U (r) V⎪ ⎩ χN

+(2f − 1)W (r)} − ln 8

(12)

where only the pressure-like field, U(r) = (wA + wB)/2, has to be adjusted so that the saddle-point densities, eqs 9 and 10, obey the incompressibility condition for a given exchange potential, W(r).54,56,57 Thus, the EPD functional solely depends on the order parameter, W(r). In the following we study the MFEP of the EPD free energy functional, G[W]. The string is defined as a sequence of orderparameter configurations, Wα(r), where 0 ≤ α ≤ 1 denotes the reaction coordinate along the string. It is discretized into Δα = 1/40 and 1/32 for the climb and glide paths, respectively. Explicit configurations and intermediates, Wα(r), are obtained by cubic spline interpolation along α for fixed position, r. The difference of reaction coordinates, Δα = α1 − α2, between two morphologies is proportional to Δα2 ∝ ∫ dr [Wα1(r) − Wα2(r)]2, and the constant of proportionality is chosen so that the two fixed starting and ending morphologies, W0(r) and W1(r), correspond to α = 0 and 1. The MFEP is defined by the condition that the component of δG/δW(r) perpendicular to the direction of the MFEP vanishes. This string is found via an iterative procedure.28 Because of the complexity of the free energy landscape, there may be multiple MFEPs connecting two given starting and ending morphologies, W0(r) and W1(r). These can be explored by providing initial guesses for intermediate morphologies, Wα(r), along the path that, for instance, could be provided by the time evolution observed in particle-based simulations.

(4)

(5)

with initial condition q(r, s = 0) = 1

wB(r) = χNϕA (r) + ξ(r)

In order to characterize the morphology of the incompressible system, one can either use the local density difference, ϕA − ϕB, or the thermodynamically conjugated variable, the exchange chemical potential, W ≡ (wA − wB)/2. In the following we choose the latter variable as order parameter.18,29,32 An equivalent formalismexternal potential dynamics (EPD)can be derived, with the same physical content, that does not have explicit density fields.54,56,57 This auxiliary field SCFT with implicit densities employs an additional partialsaddle-point approximation for the auxiliary field, U, to make the free energy functional convex. The EPD form of the free energy functional is given by

where ϕA and ϕB denote the volume fraction of A and B segments, and wA and wB denote the mean fields acting on the respective segment species. ξ(r) is the Lagrange multiplier that enforces incompressibility, ϕA(r) + ϕB(r) = 1. 8 quantifies the configurational partition function of a single chain subjected to the mean fields. The configurations of a copolymer can be described by two conjugate propagator functions, q(r, s) and q†(r, s), which indicate the probability of finding the segment s at the spatial position r starting from the two distinct ends, respectively. The propagator function, q(r, s), satisfies the modified diffusion equation ∂q(r, s) 1 = ∇2 q(r, s) − w(r, s)q(r, s) ∂s 6

(7)

1 = ϕA (r) + ϕB(r)

dr {χNϕA (r)ϕB(r) − wA(r)ϕA (r) V

− wB(r)ϕB(r)− ξ(r)[1 − ϕA (r) − ϕB(r)]}

wA(r) = χNϕB(r) + ξ(r)

(6)

where w(r,s) = wA(r) for s < f, and otherwise w(r, s) = wB(r). The propagator, q†(r,s), obeys the same differential equation (eq 5) but with a different initial condition, q†(r,s=1) = 1. The propagator functions can be used to compute the single-chain C


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Figure 1. Snapshots of the SCMF simulations for three dislocation pairs with impact parameters, Δ, in the direction perpendicular to the stripes. The snapshots depict a fraction of the simulation box of extension ∼7d0 along the stripe direction. The box sizes in the three cases are (a) 12.59Re0 × 25.19Re0 × 1.26Re0 for the evaporation climb with Δ = 0, (b) 22.67Re0 × 10.07 Re0 × 1.26Re0 for the unconstrained climb with Δ = d0, and (c) 15.11Re0 × 10.07Re0 × 1.26Re0, for the stagnation climb with Δ = 2d0. Times are indicated in the figures in units of the relaxation time, τ = Re02/D, in the disordered melt.

During the calculation of MFEP, the most time-consuming step is the determination of the saddle-point field U(r) for a given W(r). To this end we apply a highly efficient, semi-implicit algorithm.58

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RESULTS AND DISCUSSION We study the time evolution of two apposing edge dislocations with opposite Burgers vectors in a lamellar structure. These dislocations attract each other via the long-range distortion of the stripe structure. As we shall explicitly quantify by SCFT in the Glide Motion section, the motion of defects perpendicular to the stripesglide motionis hampered by a sizable free energy barrier. Thus, the distance between the dislocation defects perpendicular to the stripesthe impact parameter, Δapproximately remains conserved as the two dislocations collide. In the following we study the collision process for three impact parameters, Δ/d0 = 0, 1, and 2. As shown in Figures 1 and 2 the time evolution of these different colliding defect pairs qualitatively differs, and the time scale of defect annihilation varies nonmonotonously with the impact parameter, Δ. This behavior depends on the answer to three questions: (i) How large is the distortion-mediated, attractive force between defects with opposite Burgers vectors? (ii) What is the mobility of a defect; i.e., how fast do the defects approach each other in response to the attractive force? (iii) Does the collision of defects result in defect annihilation? Kinetic Evolution of Defects Observed by SCMF Simulations. To examine the mechanism of defect motion and annihilation, we present in Figure 1 a series of snapshots of SCMF simulations characterizing the time evolution of the

Figure 2. Time evolution of the distance L between the defect cores along the direction of the stripes for the evaporation climb, Δ = 0, and the unconstrained climb, Δ = d0. The dashed line corresponds to a linear time dependence of the distance L(t)/Re0 = 6.2 − 0.24t/τ whereas the solid line depicts an exponential time dependence, L(t)/ Re0 = 5.7 exp(−e−5.64t/τ).

three defect morphologies. Note that the system size perpendicular to the stripes, Ly ≈ 15d0 (a) or 6d0 (b, c), is commensurate with the equilibrium lamellar spacing, d0. The system size in the SCMF simulations and SCFT calculations does not vary during the course of defect motion and annihilation; only the defect-free lamellar structure is stressfree. This is the typical setting of directed self-assembly (DSA) D


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Figure 3. Time evolution of a tight dislocation pair formed after the collision of two edge dislocation with impact parameter Δ = 2d0 at time t = 337τ and χN = 30, after a quench to χN = 20. At the lower incompatibility the defect spontaneously annihilates within 5.4τ.

dislocations have different cores (A- and B-type cores) instead of the same core.18 This stagnation of motion in the metastable tight dislocation pair indicates that the force between the close, apposed defects vanishes. Therefore, we denote this collision with Δ = 2d0 as “stagnation climb”. In this second stage, the morphology only evolves extremely slowly in order to reduce the strain imposed by the periodic boundary conditions, but this tightly bound defect pair remains metastable over the entire simulation that has been extended to t = 1000τ. Thus, there must be a sizable free-energy barrier, ΔFb ≫ kBT , for the defect to overcome in order to transform into the defect-free morphology and a concomitant protracted annihilation time scale, τ exp(ΔFb/kBT). The formation of tightly bound, metastable states upon collision is also characteristic for larger impact parameters, Δ ≥ 2d0, as observed in previous simulations.34 The value of this barrier critically depends on the incompatibility, χN.18 We predicted by SCFT that the defectannihilation free energy barrier, ΔFb, vanishes around χN* ≈ 18. In order to verify this prediction, we altered at t = 337τ, well inside the second phase, χN = 30 to χN = 20. At this incompatibility the segregation between domains is weaker, and the preferred lamellar spacing is slightly smaller that at the larger χN. Indeed, we observe in Figure 3 that the structure spontaneously converts into a defect-free stripe pattern within 5.4τ. Thus, decreasing the incompatibility, χN, toward χN* but remaining above χNODT, we alter the qualitative properties of the free energy landscape in the vicinity of a tight dislocation pair from a local minimum to an unstable structure. Thermodynamics and Free Energy Landscape as Obtained by the Minimum Free Energy Path (MFEP). In order to obtain quantitative insights into the free energy landscape and estimate the thermodynamic forces between the apposing defects, we have calculated the MFEPs for the three dislocation climbs, Δ/d0 = 0, 1, and 2, by coupling SCFT calculations with the string method. The results are presented in Figures 4, 5, and 6, respectively. The free energy difference, Δf = R e0ΔF /LzkBT 5̅ , along the path is measured with respect to ending morphology, which is a defect-free state for Δ/d0 = 0 and 1 or the tight, metastable dislocation pair for Δ = 2d0. Our calculations employ rather large lateral system sizes, Lx × Ly = 9d0 × 8d0. The system dimension perpendicular to the stripes, Ly, is comparable to or larger than the typical distance of guiding patterns in chemo- or graphoepitaxy. Again, we consider the case where Ly is commensurate with d0. In the first case with Δ = 0 (see Figure 4), initially, the free energy Δf linearly decreases with the reaction coordinate, α. In

where chemical or topographical guiding patterns are employed to direct the structure formation of the diblock copolymer materials into defect-free, registered structures. Typically, the periodicity of the guiding pattern is a small integer multiple, n ≤ 6, of the equilibrium lamellar spacing; here the role of the periodic boundary conditions is analogue to that of the chemical or topographical guiding pattern in DSA. In the first case, Δ = 0, the two defects attract each other, continuously approach each other, collide, and annihilate. The entire process requires a protracted time, 500τ. As we will discuss below, this intrinsic slowness stems from an evaporation-like mechanism, and therefore we denote this case as “evaporation climb”. Figure 2 presents the time dependence of the distance L(t) of the defect cores in the direction parallel to the stripes. In the time, during which this parallel distance decreases by ΔL = 4Re0, individual chains diffuse a distance of the order 2Dt = R e0 2t /τ ≈ 25R e0 because the diffusion of nonentangled chains along the stripes closely resembles the Rouse behavior in the disordered state.51,59 In the second case, Δ = d0, the sequence of morphologies is qualitatively similar to the case Δ = 0: the defects continuously move toward each other, collide, and annihilate. At the collision stage, t = 19.44τ, the defect pair has formed a disconnected stripe. Such a structure has been predicted to be unstable by Li et al.,18 and indeed, it immediately evolves into the defect-free morphology in the simulations. The time scale of defect annihilation at Δ = d0 is a factor 25 faster than at Δ = 0 because the approach of the defect core only requires single-chain diffusion along the internal AB interfaces. Since the single-chain dynamics does not constrain the climb motion, we denote this case as “unconstrained climb”. The distance, L, between the defect cores approximately decreases linearly in time (cf. Figure 2). Thus, on large distances the collective change of the structure, i.e., defect motion ΔL ∼ −t, is faster than single chains diffuse along the stripes Δr ∼ √t, which is allowed because defect motion does only necessitate short-range, singlechain motion. In the third case with impact parameter Δ = 2d0, the annihilation process is composed of two phases: In the first phase, t < 28τ, the two defects approach each other, and their collision velocity is comparable to the previous, unconstrained climb with Δ = d0. From the second-to-last snapshot we observe that the climb motion during the first phase lasts until the two dislocations pass by each other and form a tight dislocation pair that is similar to the metastable defect investigated in our previous work except that the two E


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Figure 4. (a) MFEPs for the removal of two apposing dislocations with impact parameter, Δ = 0, for two different boundary conditions: Ly = 8d0; i.e., the 8 outer lamellae adopt their equilibrium spacing whereas the 9 inner lamellae are compressed (black cross symbols), and Ly = 8.458d0, i.e., inner and outer lamellae are equally frustrated (blue circle symbols). The red dashed line indicates a linear fit for the linear part, α < 0.7, of the data in the first case. As the reaction coordinate linearly correlates with the distance between the defect cores, L (L = 8.212Re0, 4.398Re0, and 2.464Re0 for α = 0, 0,4, and 0.6), a distance-independent attractive force, −dΔf/dL ≈ 0.498, is extracted. (b1−b3) Three intermediate morphologies located at α = 0.725, α = 0.825, and α = 0.850, respectively. The inset shows the enlarged MFEP in the second case.

Figure 6. (a) MFEP for the evolution of a pair of dislocations with impact parameter Δ = 2d0. The constant force is −Re0 dΔf/dL ≈ 0.490. (b1−b3) Three intermediate morphologies located at α = 0.500, α = 0.850, and α = 0.925, respectively.

distance-independent force, K ∼ −(dΔF/dα)(dα/dL), between the two apposing dislocations with opposite Burgers vectors. When the defect cores approach each other, L ∼ d0, there is a shoulder in the free energy profile but no barrier. Morphological snapshots in the vicinity of the shoulder of the free energy profile are depicted in Figure 4. The sequence of structures and the details of the defect-annihilation process nicely agree with the simulation snapshots of Figure 1, suggesting that the MFEP of the EPD free energy functional, G[W], accurately captures the defect-annihilation path. In the second case with Δ = d0 (cf. Figure 5), the free energy Δf also decreases linearly with the reaction coordinate for L0 − L(t) ∼ α < 0.725, indicating a distance-independent attractive force between the defects. Deviations from this linear behavior occur for L ∼ d0, indicating a strong attraction and thermodynamic driving force toward defect annihilation. Again the sequence of morphologies along the string and the absence of a free energy barrier nicely match the configuration snapshots of the simulations. Since the third case with Δ = 2d0, presented in Figure 6, kinetically evolves into a metastable tight dislocation pair, we choose this tight dislocation pair as the ending morphology of the string. Note that this ending morphology at α = 1 with two identical cores is not exactly the same as that of the last simulation snapshot with two different cores (cf. Figure 1). The former defect can be formed by connecting the A-end core with the upper A-lamella in the latter morphology. The evolution from this metastable tight dislocation pair into a defect-free structure has been studied by SCFT18 and simulations.34 Similar to the previous case, the excess free energy linearly decreases along the MFEP until the two dislocations meet at α ≈ 0.8. When the two dislocations pass by each other, there are deviations from the linear relation between Δf and α that stem from the distortion of the domains as the metastable state is formed. The initial, linear behavior again indicates a constant attractive force, and the MFEP successfully predicts the

Figure 5. (a) MFEP for the removal of a pair of dislocations with impact parameter Δ = d0. The constant force extracted from the linear part of the free energy profile is −Re0 dΔf/dL ≈ 0.495. Three typical points, located at α = 0.725, α = 0.850, and α = 0.900 are indicated by numbered arrows, and their isosurfaces of A density are plotted in panels b1−b3.

this regime, α < 0.7, the reaction coordinate correlates well with the distance, L(t), between the defect cores in the direction parallel to the stripes. Thus, a linear dependence indicates a F


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overdamped motion with velocity dL/dt ∼ −K. Indeed, this linear time dependence of L(t) is observed in the simulations (cf. Figure 2). Thus, the minuscule Onsager coefficient Λ for changes of the collective density or the free energy barrier of the motion of individual B blocks across the encircling A domain explains the intrinsic slowness of the climb motion with impact parameter Δ = 0 compared to the unconstrained climb with Δ = d0. We note that (i) this difference will decrease if we reduce the incompatibility, χN, and (ii) this difference will not occur for lying cylinders that also form a smectic stripe structure but allow material transport above or below the cylinders. Indeed, experiments observed that the ordering kinetics and defect annihilation is faster in monolayers of lying cylinders than in standing lamellae.61 Interaction of Two Apposing Dislocations. The MFEPs of climb motion of defects in Figures 4−6 indicate that the attractive force between defects along the stripe direction is largely independent from their distance, L, parallel to the stripes for L > 2d0 and the distances accessible in our calculations. Furthermore, it is approximately independent from the impact parameter, Δ. The L independence of the force, K = −dΔF/dL, is corroborated by the constant velocity with which defects approach each other in the simulations during the unconstrained climb, Δ = d0. This finding of SCFT and simulation differs from the classical, distortion-mediated Peach−Koehler (PK) force acting on pairs of dislocations in smectics that is expected to decay like K ∼ 1/L.40−44 This Peach−Koehler force has been derived by a far-field approximation of the strain-field-mediated interaction between dislocations in an infinitely extended, continuous, elastic medium that is characterized by its compression and bending moduli, B0 and K0, respectively. Wang62 explicitly calculated these elastic constants for lamella-forming block copolymers. Order of magnitude estimates are given by B0 ∼ 5̅ kBT /d03 and K 0 ∼ 5̅ kBT /d0. These elastic constants give rise to a characteristic length scale, λ = K 0/B0 , that scales like the domain spacing, λ ∼ d0. The Peach−Koehler force, in turn, is proportional to the film thickness Lz and scales like KPK ∼ − 5̅ kBTLz /(d0L). From the simulation snapshots in Figure 1 we observe that the main distortion of the lamellar structure in our finite-sized system consists in a uniform compression of the lamellae between the dislocations because of the presence of one additional lamellar domain; i.e., the central, inner lamellae between the dislocation cores have a smaller periodicity than the outer lamellae. In our simulations, the Np lamellae in the outer region adopt their equilibrium spacing, d0 = Ly/Np, whereas the lamellar period of the lamellae between the defect cores is d′0 = Np/(Np + 1) d0 because of the finite number of lamellae Np. Assuming a uniform compression of these inner lamellae (cf. Figure 7), we obtain an estimate for the excess free energy

formation of a metastable state where the force vanishes and the defect motion stagnates. Evaporation vs Unconstrained Climb: Onsager Coefficient or Single-Chain Barrier. Quantitatively, the forces between defects with impact parameters Δ = 0 and d0 (as well as 2d0) are very similarin all cases the force is approximately independent from their mutual distance L along the stripes (for L > 2d0) and has the nearly equal magnitude, 0.490 ≤ −Re0 dΔf/dL ≤ 0.498. Thus, the free energy landscape alone cannot explain the qualitative difference in time scales of a factor 25 between dislocation collisions with Δ = 0 and Δ = d0 as observed in the simulations. In order to understand the collective kinetics of structure formation not only the thermodynamic driving force but also the Onsager coefficient, Λ, is important. Λ establishes a relation between the collective dynamics of the densities and the underlying motion of the individual macromolecules. It quantifies how efficient a gradient in the chemical potential, μ, can be translated into a composition current, j = Λ∇μ. In an incompressible system, Λ ∼ ϕA(r)ϕB(r). Thus, a thermodynamic driving force has great difficulties to generate a composition current across a strongly segregated domain because ϕA(r)ϕB(r) ≈ e−χN/2 ≪ 1 for a symmetric copolymer f = 1/2. In the first case with Δ = 0, the two opposing dislocations form an encircled droplet (B domain). A composition current through the encircling A domain is required for the reduction of the distance, L, between the dislocation cores. This “evaporation” of B blocks through the strongly segregated A domain is an intrinsically slow collective process60 with a rate proportional to Λ ∼ exp(−χN/2). One can complement this description of the kinetics of the collective density by a consideration of the single-chain dynamics. In the case of the evaporation climb, Δ = 0, the constant thermodynamic force, K, gives rise to an excess pressure on the B blocks of the encircled domain. The encircled B blocks individually “evaporate” through the surrounding A domain with a rate, ν, that is proportional to the free energy barrier of the B-block motion crossing an A lamella. Schematically, at most all N/2 segments of the B block are exposed to the A environment at the saddle point of this thermally activated, single-chain process. Thus, ν ∼ exp(−χN/ 2)/τ, where we have assumed that the attempt frequency of tunneling scales like the single-chain relaxation time, τ. We note that this ν is proportional to the ratio of the perpendicular selfdiffusion coefficient D⊥ to the parallel self-diffusion coefficient D along the stripes.59 The individual “evaporation” processes of B blocks lead to an exponential decrease of the number, nB, of B block in the encircled domain resulting in L0 − L(t ) ∼ nB(0) − nB(t ) ∼ exp( −νt ) with ντ ∼ e−χN /2

(13)

Figure 2 demonstrates that this predicted exponential approach of the dislocation cores is observed in the simulations. In contrast to the first case, no transfer of material across a domain is required in the case with Δ = d0, where the B domain between the two ending stripes is continuous. Thus, the absence of the encircling domain, which constraints material transfer, gives rise to a significantly faster defect annihilation. In the absence of any further constraint, the L-independent attractive force, K, between the defects gives rise to an

1 B0 LLyLz[(d0′ − d0)/d0]2 + FC 2 1 + FC 5̅ kBT /d0 3LNpd0 × Lz (Np + 1)2

ΔFBD = ∼ ∼ G

5̅ kBT

LLz 2

Np

d0 (Np + 1)2

+ FC DOI: 10.1021/acs.macromol.6b01088 Macromolecules XXXX, XXX, XXX−XXX

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calculations, the comparison highlights that the Peach−Koehler force is negligible in comparison to the boundary-induced force if Ly is a small integer multiple of d0, which is the typical size in DSA applications. We also note that the boundary-induced force in the commensurate case, Ly = 8d0, eliminates the barrier of defect annihilation, whereas in the special limit that we eliminate the boundary-induced force by fine-tuning Ly, the two opposing dislocation cores have to overcome a barrier in order to annihilate. Thus, the boundary-induced force facilitates defect annihilation in the typical DSA setting. (ii) In the second case, the reduced dimensionality allows us to systematically vary the system size Ly perpendicular to the stripes and consider system sizes as large as Lx × Ly = 24d0 × 48d0 discretized into a lattice of 576 × 1152 grid points. We consider two apposing dislocations with opposite Burgers vectors (cf. Figure 7) and calculate the excess free energy ΔFd as a function of the dislocation distance, L, by SCFT calculations. Because of the attraction between the dislocations, the state of two dislocations with a distance L is not metastable, and we have to impose a constraint to stabilize the structure. The free energy, ΔFd(L), will depend on the specific choice of the constraint, and we use a very gentle integral criterion to constraint L. To this end we define

Figure 7. Typical two-dimensional morphology of two apposing dislocations with Δ = 0 and a distance L apart. The two outer regions are composed of Np = 24 equilibrium lamellae with period d0, whereas the central region between the defects is composed of Np = 25 compressed lamellae with period Np/(Np + 1)d0. Additionally, the domain of integration for the constraint 3 in eq 14 is indicated by black solid curves.

3[ϕB] =

1 d0′

∫droplet drϕB(r)

(14)

where the integration is extended over the additional B domain that the two apposing dislocations generate. Since this additional, droplet-shaped B domain is encircled by an A domain, 3 is insensitive to the definition of domain of integration. We impose the constraint 3[ϕB] = L′ via a Lagrange multiplier λ > 0, which corresponds to an increased exchange chemical potential of the B species inside the droplet. With this constraint, the two apposing defects become metastable and thus the SCFT calculations can converge. In practice, a suitable initial guess for a system with L′ < L can be generated from a converging solution at L by SCFT iterations of the unconstrained system because the two dislocations approach each other in the course of the SCFT iteration. From the converged morphology, we geometrically determine the distance, L, between the defect cores. Assuming that the interaction of the dislocation pair consists of a linear superposition of the compression force and the Peach−Koehler force, we write the excess free energy in the form

where FC is the L-independent excess free energy of the defect cores and the distortion at the boundary between the inner and outer lamellae. As the two defect cores move toward each other, the Np + 1 compressed lamellae are replaced by Np relaxed outer lamellae but FC remains unaltered. Thus, this uniform compression results in a boundary-induced force KBD = −dΔFcomp/dL ∼ − 5̅ kBTLz/(d0Ly), i.e., a force between the two dislocation cores that is independent from their distance L along the stripe direction. This boundary-induced force, KBD, due to the compression of inner lamellae competes with the Peach−Koehler force, KPK: For large distances, L > LBD ∼ Ly the boundary-induced force is important, whereas for shorter distances the Peach−Koehler force dominates the interaction. In order to test these considerations and quantify the relative strength of Peach−Koehler and boundary-induced forces, we have performed (i) MFEP calculations for the case that Ly does not match an integer multiple of the equilibrium domain spacing and (ii) two-dimensional SCFT calculations of large systems. (i) Specifically, we consider the evaporation climb, Δ = 0, and choose the system size, Ly = 8.458d0, perpendicular to the stripe direction such that the free energy per chain in the Np outer lamellae is exactly identical to the value of the Np + 1 inner lamellae; i.e., inner and outer lamellae are equally frustrated by the periodic boundary conditions. The corresponding MFEP is presented in Figure 4. In accord with the expectation that this specific choice of Ly eliminates the boundary-induced force, we do not observe a compressioninduced, linear dependence of the free energy profile. Although the residual interactions between the dislocation cores, originating from the distortion of the morphology, cannot be accurately resolved within the accuracy of our SCFT

Δfd (L) = c BD

Np

L L + c PK ln + cC R e0 (Np + 1) R e0 2

(15)

where the three constants cBD, cPK, and cC quantify the amplitudes of the boundary-induced free energy, the Peach− Koehler free energy, and the core energy, respectively. Differentiation yields the normalized interaction force KR e0 2 5̅ kBTLz

= −R e0

dΔfd dL

= −c BD

Np 2

(Np + 1)

+ c PK

R e0 L (16)

In Figure 8 we present the excess free energy, Δfd(L), as a function of the distance, L, between the apposing dislocations for various system sizes, Ly/d0 = 12, 16, 24, and 48, perpendicular to the stripe direction. Similar to the previous H


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Figure 8. Excess free energy Δfd, of the defect morphology for various system sizes, Ly, as a function of the dislocation distance, L. Ly = 12d0, 16d0, 24d0, and 48d0, varies from top to bottom. The solid lines indicate a linear fit to the data at large L.

Figure 10. Interaction, Δf, between four dislocation cores which have pairwise condensed into tight dislocation dipoles for Ly = 8d0. The distance, L, is indicated in one of the morphologies obtained by twodimensional SCFT calculations.

3D results for smaller systems, Δfd linearly depends on the distance, L, corresponding to a constant, attractive force between the defects. Thus, even for these large systems the boundary-induced force dominates the behavior. In accord with the phenomenological considerations, we find that the force decreases as we increase the perpendicular system extension, Ly. In Figure 9, we plot the magnitude of the force,

the boundary-induced forces result in a pairwise condensation into tight, metastable dislocation dipoles (via stagnation climbs). Figure 10 demonstrates that the strain-field-mediated and boundary-induced interactions between these tight dislocation dipoles are very small; i.e., the structure is almost completely stress-free. Only when the mutual distance L becomes comparable to 2d0 there is a small attraction, and at that distance, the deviation of the morphology from that of two undisturbed, tight dislocation dipoles is clearly visible. Glide Motion of a Defect. In the previous section we have considered the motion of a defect along the stripe direction. By virtue of the translational invariance along the stripe, the motion of an isolated defect along the stripe can occur in infinitesimally small steps that do not incur a free energy change. The situation is dramatically different for the motion of a defect perpendicular to the stripe pattern because there is no continuous translational symmetry in that direction but the motion occurs in discrete jumps of the lamellar spacing. Therefore, glide motion is a thermally activated process that in soft matter systems like liquid crystals or copolymer materials is strongly suppressed.43 This behavior is in accord with recent experiments25 and the simulation data in Figure 1, where the impact parameter, Δ, is conserved during defect motion. Here we explicitly calculate the free energy barrier involved in the glide of a dislocation perpendicular to the stripe. This barrier of dislocation glide is of interest for two reasons: (i) At the late stages of ordering, when the defect density is small, glide motion is involved in defect annihilation. For instance, the annihilation of a dislocation pair with opposite Burgers vectors and Δ > 2d034 requires a sequence of rearrangements that resembles the glide motion of one of the dislocations. (ii) One could expect that free energy barriers of a dislocation glide behave similar as the free energy barriers of annihilating a metastable dislocation pair because both processes involve the breaking and reconnecting of lamellar domains. Such an agreement of the free energy barriers would suggest that the breaking and reconnecting of lamellar domains is the elementary process of defect annihilation and ordering kinetics. To facilitate the determination of the MFEP of the glide process by three-dimensional SCFT calculations, we consider an apposing dislocation pair with Δ = 0 (similar to Figure 4) as starting morphology, α = 0, of the string. The distance between the apposing dislocations along the stripe direction is L = Lx/2, and we impose a mirror symmetry along the stripe direction, x

Figure 9. Magnitude of the force (i.e., the slope of the linear portion of the excess free energy in Figure 8), acting on two dislocations, as a function of Np/(Np + 1)2. The blue solid line indicates the simple linear extrapolation.

i.e., the slope of the excess free energy indicated by the straight lines in Figure 8, as a function of Np/(Np + 1)2, as suggested by the phenomenological considerations. Importantly, a linear extrapolation toward Np → ∞, indicated by the blue solid line, reveals a finite abscissa offset that provides an order-ofmagnitude estimate of the Peach−Koehler force. We observe that the data for the largest perpendicular size, Ly = 48d0, slightly deviates from the suggested linear behavior, indicating that simple assumptions may become inaccurate for large Ly. Up to now we have only considered the limit of vanishingly low defect density by studying a pair of dislocations and the interactions between themselves and with the periodic boundaries. At earlier times during the structure formation, the defect density is larger and the long-range strain fields of multiple defects will interfere. In order to explore the interaction between multiple defects, we consider four dislocation (see Figure 10). Dislocations with opposite Burgers vector are characterized by the impact parameter, Δ = 2d0, and I


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one of the two confining, nonpreferential surfaces of the film. The excess free energy arises from the creation of a wedgeshaped A domain, where the unfavorably thin domain thickness at the top of the film frustrates the packing of the macromolecules and increases the area of the internal AB interfaces (cf. Figure 11b1). We note that the corresponding free energy barrier, Δf b ≈ 0.06, is slightly larger than the free energy barrier, 0.05, encountered in breaking an A domain in the course of annihilating a tight dislocation pair.18 This finding indicates that the barriers for breaking connections are influenced by the local geometry and the concomitant local distortion of the domains giving rise to stretched or compressed domains and a local increase of the internal AB interface area. As the breaking of the A domain progresses from the top of the film toward the substrate, the free energy is reduced and a new metastable state is formed at α ≈ 0.1094. In this metastable state, only the top half of the A domain is broken, but the bottom portion of the Y-shaped connection remains intact. This metastable structure has a lower free energy than the isolated, quasi-two-dimensional dislocation, α = 0. Indeed, such a partially broken A domain is also observed as long-lived structure in the simulation of the stagnation climb for t < 28τ; cf. bottom row of snapshots in Figure 1. The portion 0.3594 ≤ α ≤ 0.5 of the MFEP path corresponds to breaking the bottom half of the A connection, completing the transformation from B-core dislocation to an Acore dislocation. By virtue of the compositional symmetry of the copolymer and geometric symmetry of the intermediate defect morphologies, the MFEP path is symmetric with respect to α = 1/4. The configurations along the MFEP at α = 0.1094, 1/4, and 0.3594 demonstrate that the large free energy barrier, Δf b ≈ 0.3, at α = 0.25 is not associated with altering the domain connectivity. In fact, the local morphologies at α = 0.1094 and 0.25 are very similar. Instead, the barrier at α = 1/4 arises from a subtle but long-range distortion of the morphology. In the initial B-core dislocation, α = 0, the system exhibits a mirror symmetry with respect to the center of the isolated inner B domain that forms the defect core. Likewise, the final A-core dislocation, α = 1/2, is symmetric with respect to the center of the isolated, inner A domain. Since the domain spacings, din and dout, of the inner and outer lamellae differ, such a shift of the symmetry axis by din/2 also requires that the relative positioning of the inner and outer lamellae is altered. As sketched in Figure 12, the upward shift of the symmetry axis by din/2 between α = 0 and α = 1/2 is accompanied by a

with respect to x = Lx/2. The dislocation pair corresponds to an extra lamella domain for Lx/4 < x < 3Lx/4. In the initial stage, α = 0, the dislocation cores are formed by an isolated B droplet (i.e., B-core dislocation), whereas in the final stage, α = 1/2, there are two apposing A-core dislocations. The ending morphology, α = 1/2, of the string is obtained from the initial one, α = 0, by breaking the upper misconnections of the two dislocations as shown in Figure 11. Obviously, starting and ending morphologies have equal free energies in a symmetric block copolymer, and the interval 0 ≤ α ≤ 1/2 corresponds to half a glide process.

Figure 11. (a) MFEP for the glide motion of a pair of apposing dislocations. The system size in the normal direction is Ly = 8/(1 − 1/ 18)d0. (b1−b3) Isosurface plots of A density of the portion around the right dislocation inside the red box for three intermediate morphologies located at α ≈ 0.0625, α ≈ 0.1094, and α = 0.2500 and their cleaved portion at the left-top showing the cross section of the broken A domain.

The inner domain spacing, Lx/4 < x < 3Lx/4, is din = Ly/(Np + 1), and the outer periodicity is dout = Ly/Np with Np = 8. We choose the perpendicular system size Ly = Np(Np + 1) d0/(Np + 1/2), asserting that the inner Np + 1 lamellae are compressed and the outer Np lamellae are stretched so that their free energies per chain in the stretched and compressed domains are approximately equal. This choice results in a cancellation of the boundary-induced forces (discussed in the section Interaction of Two Apposing Dislocations). The MFEP of the glide motion of the apposing dislocations is presented in Figure 11. Its multiple free energy extrema are indicated by the numbered arrows. The concomitant intermediate morphologies along the MFEP are depicted in panels b1−b3 of Figure 11, where we only show the lateral area around the right dislocation core marked by the red frame in the panel of the morphology at α = 0 and the top-left portion of this area inside the blue box is further enlarged to show the cross section of the broken A domain. The first maximum along the MFEP, at α = 0.0625, corresponds to triggering the rupture of the misconnected A domain that participates in the Y-shaped junction. Similar to the MFEP of annihilating a tight dislocation pair, this rupture of the A domain is a truly three-dimensional process that starts at

Figure 12. Sketch of the symmetry axis of the B-core dislocation at α = 0 and the A-core dislocation at α = 1/2. The upward shift of the symmetry axis by din/2 also imposes a shift of the inner and outer domains with respect to each other by Δy = (dout − din)/2. J


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Kawasaki model63 by using a composition-dependent Onsager coefficient. For the Ohta−Kawasaki model, however, such a composition dependence is often ignored because it results in a significant increase in computational complexity. Upon collision at intermediate segregation, χN = 30, the opposing defects may either annihilate, Δ/d0 = 0 and 1, or form a metastable, tight dislocation pair, Δ = 2d0. In the latter case, the annihilation process involves a free energy barrier associated with breaking and reconnecting lamellar domains (in accord with previous studies18,34). Breaking and reconnecting events are also involved in the motion of a dislocation perpendicular to the stripe pattern glide motion. Our MFEP calculations reveal, however, that the free energy barriers associated with altering the domain topology are not the only and may not even be the dominant free energy barriers to be overcome in the course of dislocation glide in a finite-sized system. Instead, our MFEP calculations indicate that the main barrier arises from a small-amplitude but long-range shift of lamellar domains with respect to each other. Thus, our study emphasizes that the ordering kinetics and defect annihilation is not only dictated by the free energy barriers of locally breaking and rejoining domains, but instead, long-range strain fields and compression may give rise to qualitatively different, significant gradients and barriers in the free energy landscape. By virtue of the long-range character, these strain-mediated effects are relevant for system sizes in DSA applications and may open opportunities to control defect motion and annihilation by tailoring the geometry of confinement.

downward shift of the outer domains with respect to the inner ones by Δy = (dout − din)/2. This shift is minuscule Δy ≈ d0/ 2Np but it is applied to all Np lamellae. Therefore, the change of the symmetry axis between the two metastable states α = 0.1094 and 0.3594 results in a long-range strain field that gives rise to an important free energy barrier. The entire MFEP is composed of (i) the initial breaking to the top half of the Y-shaped connection, α = 0 → 0.1094, (ii) a shift of the inner versus the outer lamellae by Δy, α = 0.1094 → 0.3594, and (iii) the final severance of the bottom half of the Yshaped connection, α = 0.3594 → 1/2. For a symmetric block copolymers and symmetric confinement, the MFEP is symmetric around the highest barrier at α = 1/4. We conclude that the free energy barriers associated with altering the domain topology, i.e., breaking and reconnecting domains, are not the only relevant ones. Instead, defect motion and annihilation may additionally involve small-amplitude but long-range domain rearrangements that do not alter the domain topology but may result in qualitatively different free energy barriers of comparable or even larger heights.

■

SUMMARY AND CONCLUSIONS Using SCMF simulations and SCFT calculations, we have studied defect motion and annihilation in symmetric diblock copolymers confined into a thin film with symmetric boundaries. The motion of dislocations along the stripe patternclimb motiondoes not involve a free energy barrier because of the continuous translational invariance along the lamellae. In contrast, the motion of a dislocation perpendicular to the stripe patternglide motioninvolves a significant free energy barrier. Therefore, the motion of two apposing dislocations with opposite Burgers vectors in response to strain-field-mediated interactions predominately occurs along the stripes, whereas the perpendicular distancethe impact parameter Δis approximately conserved. For the system sizes considered in SCMF simulations and SCFT calculations, we observe that the attractive strain-fieldmediated forces to leading order do not depend on the distance along the lamellae. This finding is in marked contrast to the Peach−Koehler force40 for infinite systems but it can be rationalized by a compression of the lamellae between the two opposing defect cores. Whereas this compression force may become negligible for unconfined systems, we find that it is larger than or comparable to the Peach−Koehler force for systems that contain fewer than 48 lamellaea size range that is relevant to grapho- or chemoepitaxy DSA techniques. The attractive force results in a collision of the dislocations. For the evaporation climb, Δ = 0, the dynamics is protracted because the motion of the defect cores toward each other involves the tunneling of the dislocation-core blocks through a domain of the opposite segment species. This “evaporation” does not involve a barrier in the collective order parameter the composition fieldthat characterizes the morphology and therefore does not feature in the MFEP because the free energy continuously decreases as the dislocation cores approach each other. Instead, it involves a barrier in the single-chain dynamics. For larger impact parameters, Δ ≥ d0, no “evaporation” is required for the climbing of the two dislocations, and in the SCMF simulations, we observe a time difference of about a factor of 25 for defect annihilation with Δ = 0 and Δ = d0. In principle, the intrinsic slowness of the “evaporation climb” can be captured by dynamic SCFT calculations or other continuum descriptions of structure formation like the Ohta−

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (M.M.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the European Union FP7 under grant agreement 619793 CoLiSA.MMP (Computational Lithography for Directed Self-Assembly: Materials, Models, and Processes) and the National Natural Science Foundation of China (NSFC) (Grants 21322407 and 21574026). We thank the GWDG Göttingen, the HLRN Berlin/Hannover, and the Jülich Supercomputing Centre for ample computing time.

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REFERENCES

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