Step-Shear Deformation of Block Copolymers - ACS Publications

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Step-Shear Deformation of Block Copolymers De-Wen Sun and Marcus Müller* Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany

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ABSTRACT: A step-shear deformation imparts an instantaneous, anisotropic stimulus onto a microphase-separated block copolymer mesophase. In response to the concomitant molecular stretching and collective deformation of the morphology, the system relaxes toward a long-lived, pseudometastable state with a minuscule residual shear stress. Using molecular simulations and self-consistent field theory (SCFT) calculations, we systematically explore the variety of ordered, anisotropic mesostructures that have potential applications in engineering functional materials and can be fabricated by this versatile processing strategy. We study three classes of anisotropic ordered mesostructures, including cylindroids, gyroid-like networks, and sloping lamellae, and show that their domain shape and domain orientation can be independently controlled by the magnitude of the step-shear strain, γ, and the composition of the block copolymer as well as by the way in which we implement the step-shear deformation. We also demonstrate that pseudometastability requires a positive curvature of the free energy F(γ) as a function of the step-shear strain. Otherwise, for d2F/dγ2 < 0, the mesostructure spontaneously relaxes back at fixed strain, and we study this relaxation via slippage by dynamic self-consistent field theory (DSCFT).

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INTRODUCTION Block copolymers have the ability to spontaneously selfassemble into various kinds of periodically ordered mesostructures at the length scale of 10−100 nm because they are formed by coupling two or more chemically distinct homopolymers together via covalent bonds.1−3 Because of this fascinating, characteristic ability and the corresponding potential applications,3−8 block copolymers have attracted abiding attention. The stability of these self-assembled mesostructures in the bulk is dictated by the interplay between the interfacial tension between different domains and the conformational stretching entropy.9 Block copolymers can be synthesized in a variety of architectures, the simplest of which is a flexible linear AB diblock copolymer. These deceptively simple block copolymers can form six thermodynamically stable, equilibrium mesophases in a melt10−12close-packed spheres (CPS), body-centered-cubic spheres (BCC), hexagonally packed cylinders (HEX), cubic gyroid networks (GYR), orthorhombic Fddd networks (O70), and lamellae (LAM), where CPS refers to face-centered-cubic spheres (FCC) or hexagonally close-packed spheres (HCP)9as a function of the volume fraction, fA, of A-segments and the incompatibility, χABN, i.e., the product of the Flory−Huggins interaction parameter, χAB, and the total number of segments, N, per molecule. Process-directed self-assembly of block copolymers refers to thermodynamic processes that can reproducibly direct the kinetics of structure formation from an unstable, starting state into a desired, metastable mesostructure.13−15 This strategy allows not only to fabricate equilibrium mesophases, i.e., global free-energy minima, but also provides access to alternate, © XXXX American Chemical Society

metastable, or long-lived pseudometastable mesostructures, i.e., local minima on the rugged free-energy landscape.16 Kinetically trapping structure transformations is a general engineering strategy in materials science, e.g., to tailor the grain microstructure. Here we (i) apply this concept to the molecular scale of a unit cell in a soft, copolymer mesostructure and (ii) exploit the time-scale separation between (1) the relaxation toward a metastable mesostructure and (2) the thermally activated process of jumping between metastable states and reaching equilibrium. Because the process-directed self-assembly only involves structure transformation on the scale of a unit cell, the time scale of the former process is set by the time a macromolecule needs to diffuse its own extension, Re. This rapid time scale, on the order of seconds or less, is in stark contrast to the latter time of achieving equilibrium that may involve protracted thermal annealing at elevated temperatures for minutes or hours because the barriers that separate freeenergy minima typically scale with the invariant degree of polymerization, 5̅ , like 5̅ kBT .8,17 This time-scale separation affords us multiple opportunities to stabilize the metastable mesostructure during its lifetime by, e.g., cooling below the glass-transition temperature, cross-linking, or crystallization. Compared with conventional strategies that aim at making the desired metastable mesostructure thermodynamically stable by either fine-tuning the chain architecture of the block Received: June 10, 2018 Revised: September 16, 2018

A

DOI: 10.1021/acs.macromol.8b01242 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules copolymer18−31 or blending block copolymers with homopolymers,32−35 the process-directed self-assembly of block copolymers avoids the necessity to synthesize complex molecular architectures or adjust the blend composition to prevent macroscopic separation between block copolymers and homopolymers. In previous work, we considered the rapid change of the chemophysical properties of segments by a pressure quench13 or an alchemical transformation,14 and we explored the variety of metastable mesostructures that are accessible by processdirected self-assembly. Because the change of chemophysical properties does not alter the symmetry, we observed that the sofabricated metastable statessuch as Schoen’s F-RD14 and Schoen’s I-WP13,14 networks or the cubic single-diamond (DIA) network14typically inherit their symmetry from the initial, equilibrium mesophase and are stress-free, i.e., the diagonal components of the stress tensor are identical while all offdiagonal components vanish. A step-shear deformation, in turn, imparts an anisotropic stimulus onto the initial, equilibrium mesophase, generating an unstable, starting state with highly stretched macromolecular conformations and collectively deformed morphology. In our recent work,15 applying a step-shear deformation to an initial, equilibrium BCC sphere mesophase, we illustrated that the deformed, unstable, starting state relaxed to a metastable state that consists of ellipsoids at fixed strain. The shape and orientation of the minority domains are dictated by the packing of the ellipsoids and can be tailored by the block copolymer composition, fA, and the step-shear strain, γ. Whereas this state is a local minimum of the free energy at f ixed strain, and the shear stress is reduced by about 2 orders of magnitude compared to the deformed, unstable, starting state, the shear stress is not exactly zero. Thus, the unconstrained system will eventually relax back if not additionally stabilized, and we denote this state as pseudometastable.15 In the present work, we systematically explore the processdirected self-assembly of anisotropic ordered mesostructures in block copolymers via a step-shear deformation. Different classes of unstable, starting states are generated by applying the stepshear deformation onto three different, initial mesophases in equilibrium: HEX, GYR, and LAM. We represent the step-shear deformation as an affine transformation of the macromolecular conformations, giving rise to stretched chain conformations and a sheared domain morphology in the unstable, starting state. Because the stretched chain conformations are not in equilibrium with the instantaneous domain morphology in the sheared, starting state, we investigate the relaxation from this unstable, starting state to the metastable state at fixed strain by molecular simulations using the Single-Chain-in-Mean-Field (SCMF) algorithm.36−39 This technique captures the interplay between the relaxation of the stretched chain conformations and the collective kinetics of the local densities. After the system has relaxed to a pseudometastable state in the molecular simulation, we use the spatial densities of the anisotropic ordered mesostructures as input to self-consistent field theory (SCFT) calculations based on the standard Gaussian chain model.40−42 SCFT calculations at fixed strain identify the free-energy basin43 and are complemented by variable-cell-shape SCFT calculations.42,44 These calculations provide the information about the free energy, F, of the pseudometastable mesostructures as a function of the step-shear strain, γ, and the residual shear stress. We also observe that step-shear-fabricated mesostructures are unstable at f ixed strain if the free energy, F(γ), exhibits a negative

curvature. In this case, the mesostructure relaxes by a slippage mechanism where planes of deformed domains slide past another. Thus, the condition d2F/dγ2 > 0 provides a limit for the maximal, applied strain. A similar relaxation mechanism has been previously observed in continuum models,45 and we study it here by dynamic self-consistent field theory (DSCFT)46−52 for mesostructures of ellipsoids. Our paper is arranged as follows: In the next section, we introduce the particle-based simulations using the SCMF algorithm, the cubic-box SCFT and DSCFT calculations, and the variable-cell-shape SCFT for the melt of linear AB diblock copolymers along with the corresponding numerical implementations. Subsequently, we present our results and discuss which kind of pseudometastable, anisotropic ordered mesostructures is process-accessible via the step-shear deformation. This section also quantitatively compares the results from SCMF simulations and SCFT calculations. Finally, our article concludes with a brief summary.

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MODEL AND COMPUTATIONAL TECHNIQUES Within mean-field theory, flexible, linear AB diblock copolymers with structurally symmetric blocks are characterized by three coarse-grained parameters: the volume fraction, fA, of the Ablock, the incompatibility, χABN, between the blocks, and the unperturbed polymer’s end-to-end distance, Re0, in the absence of nonbonded interactions.9 In this work, we consider a system of n linear AB diblock copolymers in a volume V = LxLyLz with periodic boundary conditions in all three Cartesian directions of extent Lx, Ly, and Lz. The statistical segment length of both structurally symmetric blocks is b0 = R e0/ N − 1 , and all length scales are measured in units of Re0. In the following, we systematically vary fA but restrict ourselves to χABN = 23. This value corresponds to the intermediate segregation regime of a linear AB diblock copolymer melt.40 Table 1 presents the protocol using SCMF simulations as well as static, cubic-box and variable-cell-shape SCFT calculations. Table 1. Protocol Using SCMF Simulations as well as Static, Cubic-Box and Variable-Cell-Shape SCFT Calculations step 1: equilibrate the initial, equilibrium mesophase before the implementation of the step-shear deformation by cubic-box SCMF simulations step 2: implement the step-shear deformation by affine transformation of the spatial positions of coarse-grained segments in SCMF simulations step 3: observe the evolution of morphology and macromolecular conformations at fixed strain by SCMF simulations step 4: check if static, cubic-box SCFT calculations converge to essentially the same morphology (at fixed strain) when using the SCMF-simulation morphology at t ≈ 50τR as initial input and calculate the free energy step 5: verify if the free energy, F, is convex, i.e., d2F/dγ2 > 0, by static, variable-cell-shape SCFT calculations

Single-Chain-in-Mean-Field (SCMF) Simulations. In SCMF simulations, each diblock copolymer is discretized into N effective interaction centerscoarse-grained segments along its linear, molecular backbone. Neighboring segments are connected by harmonic springs. The corresponding Hamiltonian, /b, of bonded interactions represents a bead−spring model36 /b({R i , j}) kBT B

=

3 2b0 2

n

N

∑ ∑ (R i ,j − R i ,j − 1)2 i=1 j=2

(1) DOI: 10.1021/acs.macromol.8b01242 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules with Ri,j being the spatial position of the coarse-grained segment, 1 ≤ j ≤ N, of diblock copolymer, 1 ≤ i ≤ n. kB denotes the Boltzmann constant, while T denotes the temperature. In the present work, we set N = 32. This choice of the chain-contour discretization can provide a compromise between the accuracy of representing the Gaussian molecular architecture and the computational cost. The nonbonded Hamiltonian, /nb, complements the discretized Edwards Hamiltonian in eq 1 and can be expressed via a functional of the two, local microscopic densities, ϕ̂ A(r| {Ri,j}) and ϕ̂ B(r|{Ri,j}), i.e.,

collocation lattice. In the present article, we simply assign a coarse-grained segment to the nearest grid point. Using eqs 4 and 5 in eq 2, we can rewrite the nonbonded Hamiltonian, /nb, into the form /nb({R i , j}) kB T

=

ρ0 (ΔL)3 N

−

∑c

χAB N 4

{

κN [ϕÂ (c|{R i , j}) 2

+ ϕB̂ (c|{R i , j}) − 1]2

[ϕÂ (c|{R i , j}) − ϕB̂ (c|{R i , j})]2

} (6)

/nb({R i , j}) 36−39

The SCMF algorithm exploits the separation between the strong bonded forces and the weak, slowly varying, but computationally costly, nonbonded forces. The latter are temporarily replaced with external fields that can mimic the interaction of a coarse-grained segment with its instantaneous surrounding, while updating the chain conformations via Smart Monte Carlo (SMC) moves.38 These external fields are recalculated from the instantaneous, local microscopic densities after each SMC move, and this quasi-instantaneous field approximation provides an accurate description of fluctuations.36 Because the nonbonded interactions are soft and do not prevent the chain contours from crossing with each other, the single-chain dynamics is accurately described by the Rouse model after less than 10 SMC steps, which is appropriate for the polymer systems without entanglements.38 The Rouse time is

kBT 5̅ dr V R e03

=∫

−

χAB N 4

{

κN [ϕÂ (r|{R i , j}) 2

+ ϕB̂ (r|{R i , j}) − 1]

2

[ϕÂ (r|{R i , j}) − ϕB̂ (r|{R i , j})]2

}

(2)

The first term in eq 2 restrains the fluctuations of the local densities. The strength κN is related to the inverse, isothermal compressibility. In accord with previous studies, we use κN = 50. The second term in eq 2 characterizes the repulsion between unlike blocks. This incompatibility is quantified by the product, χABN, of the Flory−Huggins parameter and chain length. A large value of the invariant degree of polymerization is employed in our simulation. 5̅ = ρ0 R e0 3/N = 12800, where ρ0 = nN/V denotes the segment number density. This large value facilitates the comparison to SCFT calculations and stabilizes (pseudo)metastable states against thermal fluctuations. The two, local microscopic densities, ϕ̂ A(r|{Ri,j}) and ϕ̂ B(r| {Ri,j}), explicitly depend on the spatial positions, {Ri,j}, of the coarse-grained segments and take the form ϕÂ (r|{R i , j}) = =

1 ρ0

n

1 ρ0

n

R

fA N

∑ ∑ δ(r − R i ,j)andϕB̂ (r|{R i ,j}) i=1 j=1

N

∑ ∑

ζjbefore → ζjafter = ζjbefore + ηj tan α

δ(r − R i , j)

i = 1 j = fA N + 1

(3)

ϕÂ (c|{R i , j}) =

1 ρ0 (ΔL)3

choose the step-shear strain, γ = tan α =

-[ϕA , ϕB , ωA , ωB] nkBT

∑ ∑ Π(c|{R i ,j})

1 V 1 − V 1 − V 1 + V

(4)

+

and 1 ϕB̂ (c|{R i , j}) = ρ0 (ΔL)3

n

N

∑ ∑ i = 1 j = fA N + 1

Lζ Lη

, such that the

investigated system before and after the step-shear deformation is compatible with the periodic boundary conditions. The usage of periodic boundary conditions also implies that the step-shear strain, γ, is fixed.15 Self-Consistent Field Theory (SCFT) and Dynamic SelfConsistent Field Theory (DSCFT). Within the SCFT, the free-energy functional, - , for the melt of linear AB diblock copolymers in the canonical ensemble takes the form40−42,53

fA N

i=1 j=1

(7)

where ζ, η = x, y, z and ζ ≠ η. α denotes the shear angle. We

respectively. Such a soft, coarse-grained, particle-based model allows us to capture the interplay between the single-chain dynamics and the kinetics of collective densities, which is difficult in DSCFT46−52 and other continuum models, particularly if the chain conformations are not in equilibrium with the instantaneous, local densities. Both the nonbonded Hamiltonian and the local microscopic densities are evaluated on a collocation lattice with a spatial resolution, ΔL, i.e., at each grid point, c, the two, local microscopic densities can be evaluated from the spatial positions of all coarse-grained segments via n

2

given by τR = e02 ≈ 300 SMC steps in the disordered state, 3π D where D denotes the self-diffusion coefficient. In our SCMF simulations, when the step-shear deformation is applied along the ζ-direction and the shear gradient points along the η-direction, the implementation of the step-shear deformation can be realized by affinely transforming the Cartesian ζ-coordinate of each coarse-grained segment, j, according to

Π(c|{R i , j}) (5)

respectively. Π(c|{Ri,j}) describes how to assign the spatial position of a coarse-grained segment to the grid points on the C

= −ln Q [ωA , ωB]

∫ dr{χAB N[ϕA(r) − fA ][ϕB(r) − fB ]} ∫ dr{ωA(r)[ϕA(r) − fA ]} ∫ dr{ωB(r)[ϕB(r) − fB ]} ∫ dr{ξ(r)[ϕA(r) + ϕB(r) − 1]}

(8)


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Macromolecules with f B = 1 − fA being the volume fraction of B-segments. ϕA(r) and ϕB(r) are the local densities of A-segments and B-segments, respectively, while ωA(r) and ωB(r) denote the potential fields that are conjugated to ϕA(r) and ϕB(r), respectively. Q is the normalized partition function of a single AB diblock copolymer in the spatially varying potential fields, ωA(r) and ωB(r). The local incompressibility constraint is ensured by the Lagrange multiplier, ξ(r). Using the mean-field approximation, we obtain the selfconsistent, saddle-point conditions by minimizing - with respect to all its argument, ϕA(r), ϕB(r), ωA(r), ωB(r), and ξ(r). ωA (r) = χAB N [ϕB(r) − fB ] + ξ(r)

(9)

ωB(r) = χAB N [ϕA (r) − fA ] + ξ(r)

(10)

1 Q [ωA , ωB]

ϕA (r) =

1 Q [ωA , ωB]

ϕB(r) =

∫0 ∫f

fA

1

ds q(r, s)q†(r, s)

dependent Onsager coefficient that accounts for the incompressibility, ϕA(r,t) + ϕB(r,t) = 1. The kinetic equation for this dynamics is ∂ϕA (r, t ) ∂t

(18)

where D is diffusion coefficient, while μ(r) = μA(r) − μB(r) denotes the exchange chemical potential with μA(r) and μB(r) being the chemical potentials of A-segments and B-segments, respectively. We employ a semi-implicit scheme14 to integrate eq 18 and obtain the potential fields, ωA(r,t) and ωB(r,t), that produce the density, ϕA(r,t), and fulfill the incompressibility constraint. Given these potentials fields, we calculate the exchange chemical potential via

(11)

Δμ(r, t ) = χAB N [ϕB(r, t ) − ϕA (r, t ) − 2fA + 1]

ds q(r, s)q†(r, s)

+ [ωB(r, t ) − ωA (r, t )] (12)

A

ϕA (r) + ϕB(r) = 1 1 V

(13)

∫ dr q(r, s)q†(r, s)

(14)

†

q(r,s) and q (r,s) are the two conjugated chain propagators that describe the chain conformations of a linear AB diblock copolymer, giving the probability of finding the segment s at spatial position r when starting from one of the two different free ends. Within the standard Gaussian chain model,40 they satisfy the following two modified diffusion equations:

-*[ϕA , ϕB , ωA , ωB] nkBT

∫ dx{χAB N[ϕA(x) − fA ] × [ϕB(x) − fB ]} − ∫ dx{ωA (x)[ϕA (x) − fA ]} − ∫ dx{ωB(x)[ϕB(x) − fB ]} σ̲ : ϵ̲ + ∫ dx{ξ(x)[ϕA (x) + ϕB(x) − 1]} + (n / V )k T

= −ln Q [ωA , ωB] +

2

R ∂ q(r, s) = e0 ∇2 q(r, s) − ω(r)q(r, s) ∂s 6

−

R 2 ∂ † q (r, s) = e0 ∇2 q†(r, s) − ω(r)q†(r, s) 6 ∂s

(15)

(16)

with initial conditions q(r,0) = 1 and q†(r,1) = 1, respectively. ω(r) = ωA(r) if s < fA and ω(r) = ωB(r) otherwise. We employ the pseudospectral method54,55 to solve the modified diffusion equations in eqs 15 and 16, and a finer 1 discretization of the chain contour is used, i.e., Δs = 128 , than in the particle-based model. The saddle-point equations are solved iteratively to obtain the converged solutions of ϕA(r), ϕB(r), ωA(r), ωB(r), and ξ(r).56 Substituting these saddle-point values into - , we obtain the free energy, F, of a (pseudo)metastable mesostructure. Once the two conjugated chain propagators are obtained, the stress tensor of the fabricated mesostructure can be calculated according to15,44,57 σζη

1 = (n/V )kBT V

∫

l o R e0 2 dro m o 3Q o n

∫0

1

(19)

The value of the time step, Δt, is set to about 4.9 × 10−7τR at the early stage of the dynamics to stabilize the numerical calculation. With the progress of the dynamics, we gradually increase the value of Δt, and the final used value is about 2.5 × 10−4τR. Variable-Cell-Shape Self-Consistent Field Theory. Within the variable-cell-shape SCFT,42,44 the free-energy functional, -*, of a melt of linear AB diblock copolymers in the constant-stress ensemble can be expressed as

with Q=

ij D yz = jjj 2 zzzR e0 2∇[ϕA (r, t )ϕB(r, t )∇μ(r, t )] jR z k e0 {

B

(20)

where σ represents the stress tensor, while ϵ denotes the strain tensor. x = (x1, x2, x3) is a dimensionless, rescaled vector whose components lie in the interval [0, 1], and the relationship between the Cartesian coordinates and the rescaled coordinates is r = hx. Here h is the 3 × 3 shape matrix that describes the shape of the simulation box. The scaling factor, det h = V, represents the volume of the simulation box, via which the integrals over r can be converted into the integrals over x. If we denote the reference configuration of the simulation box by h0, the strain tensor takes the form42,44

| o ∂2 † ds q(r, s) q (r , s )o } o o ∂ζ ∂η ~

ϵ̲ =

1 1 [( h̲ 0T)−1 h̲ T h̲ ( h̲ 0)−1 − 1̲ ] = [( h̲ 0T)−1 G̲ ( h̲ 0)−1 − 1̲ ] 2 2 (21)

(17)

where hT represents the transpose of h and 1 denotes the unit tensor. The metric tensor, G, is constructed via the formula G = hTh. Usually, the reference configuration, h0, is defined as the state under zero applied external stress,44 and in this work, we take the stress-free, equilibrium HEX, GYR, and LAM mesophases as the reference states. Thus,

where ζ, η = x, y, z. When investigating how a shear-deformed structure relaxes at fixed strain via slippage, we utilize dynamic self-consistent field theory (DSCFT).46−52 To this end, we consider the conserved Cahn−Hilliard dynamics (model B)58 with a compositionD


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Figure 1. (a) Free energy, F(γ), of ellipsoidal mesostructures obtained via a step-shear deformation as a function of γ for χABN = 23, fA = 25/128, and L* = 1.8207Re0. Variable-cell-shape SCFT calculations are employed for a single unit cell to compute F(γ). The F−γ curve has negative curvature, d2F/dγ2 < 0, in the interval 0.48 ≤ γ ≤ 1. The two insets in panel a display the initial BCC mesophase at γ = 0 and the ellipsoidal mesostructure at γ = 1 from cubic-box, unit-cell SCFT calculations. (b) Simulation snapshots immediately after the step-shear deformation, γ = 1, of the BCC mesophase at t = 0 as well as at later times, t ≈ 9τR, t ≈ 30τR, and t ≈ 50τR. The SCMF simulation uses the same values of χABN, fA, and cell shape as the cubic-box, unit-cell SCFT calculation. The red arrows in panel b mark the direction, along which the step-shear deformation acts.

ÄÅ É ÅÅ L* 0 0 ÑÑÑ ÅÅ x ÑÑ ÅÅ ÑÑ ÅÅ Ñ h̲ 0 = ÅÅÅ 0 L*y 0 ÑÑÑÑ ÅÅ ÑÑ ÅÅ Ñ ÅÅ 0 0 L*ÑÑÑ zÑ ÅÇ Ö

σij

(22)

3

3

i=1 j=1

R 2 ∂ † q (x, s) = e0 6 ∂s

3

3

i=1 j=1

ds

F F* σ̲ : ϵ̲ = − nkBT nkBT (n/V )kBT

(27)

When calculating the strain dependence of the free energy, F(γ), we refer to F rather than F*.

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RESULTS AND DISCUSSIONS For a melt of linear AB diblock copolymers with χABN = 23 and A-segments being the minority, SCFT calculations10,12 predict four equilibrium mesophasesBCC, HEX, GYR, and LAMin the composition intervals 0.189535 < fBCC < 0.222959 < fHEX < A A LAM 0.330115 < fGYR < 0.362644 < f < 0.5. To systematically A A explore process-accessible structures by a step-shear deformation, we choose five compositions for the initial, equilibrium mesophases before implementing the step-shear deformation, 25 9 11 i.e., fA = 128 for BCC, fA = 32 for HEX, fA = 32 for GYR,

∂2 q(x , s) − ω(x)q(x , s) ∂xi ∂xj

∑ ∑ (G̲ −1)ij

1

(26)

12

16

fA = 32 (asymmetric composition), and fA = 32 (symmetric composition) for LAM. The compositions for HEX, GYR, and LAM are selected using 32 as the denominator to match the chain-contour discretization in our SCMF simulations. Slippage Instability at Fixed Strain. Extending our previous work,15 we study the application of a step-shear deformation to the equilibrium BCC mesophase at χABN = 23 and fA = 25/128. As shown in the left inset of Figure 1a, we consider a cubic system with Lx = Ly = Lz = L* and periodic boundary conditions, where L* = 1.8207Re0 denotes the equilibrium periodicity. This equilibrium BCC mesophase, obtained by a cubic-box, unit-cell SCFT calculation, is used to prepare a particle-based BCC configuration. After relaxing this so-prepared, initial particle configuration, we affinely transform the particle coordinates according to eq 7, thereby generating an

(24)

−

∫ dx ∫0

where h−1 denotes the inverse of h. In addition, the relationship between the canonical free energy, F, and the free energy, F*, in the constant-stress ensemble is

if we apply the step-shear deformation along y-direction and make the shear gradient point along the z-direction. Minimization of -* with respect to ϕA(x), ϕB(x), ωA(x), ωB(x), and ξ(x) yields the saddle-point equations (eqs 9−14) with r being replaced with x. The modified diffusion equations for the two chain propagators, q(x,s) and q†(x,s), are changed to

∑ ∑ (G̲ −1)ij

R e0 2 3Q

ÉÑ| 3 3 Ä ÅÅ l ÑÑo o ÅÅ −1 T ∂ 2 o † o 1 − ÅÅ( h̲ )ik ( h̲ )lj ÑÑÑÑ} q (x , s ) ∑ ∑ × q(x , s)m o Å o k = 1 l = 1 ÅÅÇ o ÑÑÖo ∂xk ∂xl n ~

(n/V )kBT

where Lx*, Ly*, and Lz* are the corresponding equilibrium periodicities, respectively. After the volume-conserving stepshear deformation, eq 7, the simulation box is no longer cubic, and h takes the form ÄÅ ÉÑ ÅÅ L* 0 0 ÑÑ ÅÅ x ÑÑ ÅÅ ÑÑ ÅÅ Ñ h̲ = ÅÅÅ 0 L*y L*y tan α ÑÑÑÑ ÅÅ ÑÑ ÅÅ ÑÑ ÅÅ 0 0 L* ÑÑ z ÅÇ ÑÖ (23)

R 2 ∂ q(x , s) = e0 ∂s 6

=

∂2 † q (x, s) − ω(x)q†(x, s) ∂x i∂xj

(25)

with initial conditions q(x,0) = 1 and q†(x,1) = 1, respectively. ω(x) = ωA(x) if s < fA, whereas ω(x) = ωB(x) otherwise. eqs 24 and 25 are also solved by the pseudospectral method.42,54,55 Substituting the saddle-point values into -*, we obtain the free energy, F*, in the constant-stress ensemble. Once q(x,s) and q†(x,s) are obtained, the stress tensor, σ, can be computed via E


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Figure 2. (a) The main panel and inset illustrate how the asphericity (blue circles), S, and the orientation angle (green diamonds), β, of the ellipsoidal domains vary as a function of time in the course of our DSCFT calculations. The unstable, starting state that is characterized by an uniform shear strain γ(y) = γ̅ = 1 is displayed on the bottom left, whereas the ellipsoidal mesostructure with γ = 1 obtained from the SCFT calculation using a highly symmetric input is shown on the top right. In the latter morphology, all ellipsoidal domains have S ≈ 0.02123 (blue dashed line) and β = 0 (green dots). (b) The four morphologies show the unstable, starting state generated by a nonuniform step-shear strain, γ(y) = γ̅ + (2πa0/Ly) cos(2πy/Ly) with a0 ≈ −0.45Re0, two states in the course of the DSCFT calculation, and the finally obtained BCC mesophase. The main panel depicts how the Fourier amplitudes, |a1(t)| = −a1(t) and |a2(t)| = −a2(t), of the ellipsoidal displacements, Δx(y,t) = yγ̅ + a1(t) sin(2πy/Ly) for the ellipsoids at yc = 0, L*, 2L*, 7 3 5 1 3L*, and Δx(y,t) = yγ̅ + a2(t) sin(2πy/Ly) for the ellipsoids at yc = 2 L*, 2 L*, 2 L*, and 2 L*, vary as a function of time in the course of our DSCFT calculations. yc denotes the center of mass of the ellipsoidal A-domain along the y-direction, and a1(0) = a2(0) = a0. Open symbols correspond to the three morphologies at t = 0, t ≈ 3.07τR, and t ≈ 15.41τR. The maroon dashed line describes the function exp(λt/τR) with λ = 0.0579.

and the ellipsoidal mesostructure at γ = 1. In the latter state, γ = 1, neighboring layers of ellipsoids in horizontal xz-planes are 1 separated by 2 L* in the vertical y-direction and register exactly on top of each other as shown in the right inset of Figure 1a. This packing results in a pronounced squeezing of the shape of the minority domains in the vertical y-direction. By symmetry, the morphology is shear-stress-free, i.e., all off-diagonal components of the stress tensor vanish, but the three diagonal components are not identical. The system may reduce its free energy by either slipping back to the left or forward to the right. Note that F(γ) is convex for γ ≤ 0.47, whereas it has a negative

unstable, starting state, as displayed by the configuration snapshot at t = 0 in Figure 1b. The other three configuration snapshots in Figure 1b illustrate the relaxation of this unstable, starting state at fixed strain, γ, in the course of SCMF simulations. First, the morphology of horizontally extended ellipsoids is formed at t ≈ 9τR. This configuration is maintained up to about t ≈ 30τR. Subsequently, the BCC mesophase is recovered at t ≈ 50τR via the slippage of ellipsoids. We use the local densities of the ellipsoidal morphology from the SCMF simulation as input to initialize the cubic-box, unitcell SCFT calculation. For small strains, the SCFT calculation converges to essentially the same configuration as the SCMFsimulation inputboth morphologies only differ by fluctuations that are included in the SCMF simulation. This combination of SCMF simulations and SCFT calculations directly yields the free energy, F(γ), and allows us to account for molecular dynamics of the relaxation, in particular the effects due to the stretched chain conformations in the unstable, starting state. However, in this way, F(γ) can only be calculated for discrete values of γ that are compatible with the periodic boundary conditions.15 Using the ellipsoidal morphology of the SCMF simulation with γ = 1 at t ≈ 30τR as initial input, the SCFT calculation converges to the BCC mesophase, instead of the ellipsoidal morphology that is used as initial input. The SCFT solution for the ellipsoidal mesostructure, as displayed in the right inset of Figure 1a, can only be obtained by using a carefully designed, highly symmetric, initial input in the SCFT calculation. A detailed analysis of this situation will be presented further below. Alternatively, we use variable-cell-shape SCFT calculations42,44 to compute F as a continuous function of γ. The main panel of Figure 1a presents the canonical free energy, F(γ), as a function of the step-shear strain, γ, in the xy-plane. F(γ) is symmetric with respect to the transformation γ → γ′ = 2 − γ and exhibits a maximum at γ = 1. As the system is sheared, γ < 1, the free energy, F(γ), monotonously increases and the residual shear stress, Vσ = dF/ dγ, is positive for all but the equilibrium BCC mesophase at γ = 0

curvature,

d2F dγ 2

< 0, for larger strains. The latter feature signals

the instability of the mesostructure at fixed strain, i.e., small fluctuations of the local shear strain, γ, around its average, e.g., γ̅ = 1, are amplified and a part of the system slips back toward γ = 0, whereas another part of the system increases γ toward 2 by slipping xz-planes of ellipsoidal domains forward in x-direction. Because

d2F dγ 2

< 0 at γ = 1, the ellipsoidal mesostructure

displayed in the right inset of Figure 1a actually is unstable, and it corresponds to a local maximum of the free-energy landscape rather than a local minimum. Therefore, although the ellipsoidal morphology obtained from our SCMF simulations exhibits vanishingly small residual shear stress, both fluctuations of the local chain conformations and thermal fluctuations of the local densities make the morphology in the SCMF simulation deviate from the highly symmetric SCFT solution. These fluctuations lower the free energy and thus are amplified, finally resulting in the formation of the equilibrium BCC mesophase in the SCMF simulation. This is also the reason why the SCFT calculation using the ellipsoidal morphology from the SCMF simulation at t ≈ 30τR as initial input does not converge to the highly symmetric structure displayed in right inset of Figure 1a. A local, spatial variation of the step strain, γ(y), along the ydirection around its average γ̅ increases the free energy with respect to the uniformly deformed state, γ(y) = γ̅. For slowly F


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on the top right of Figure 2a. The latter is obtained from the SCFT calculation using a highly symmetric initial input and corresponds to the local maximum of the free-energy landscape. The slippage instability of the ellipsoidal mesostructure can not be observed in such a case because (i) the step-shear deformation is uniformly implemented and (ii) our DSCFT calculation does not include thermal fluctuations of the local densities and averages over fluctuations of local chain conformations. In Figure 2b, the morphology at t = 0 displays the unstable, starting state with an average γ̅ = 1 that features a smallamplitude, periodic variation of the local shear strain. In the 1 lower half of the system, y < 2 Ly , the step-shear deformation is slightly smaller than the average, whereas the system is slightly more sheared in the top half. We utilize i 2π y y Δx = x after − x before = yγ ̅ + a0 sinjjj L zzz with a0 ≈ −0.45Re0 k y{ to realize the periodic variation of the local shear strain. Thus, i 2π y y 2π a dΔx γ(y) = dy = γ ̅ + L 0 cosjjj L zzz, and the average strain y k y{ Ly 1 γ ̅ = L ∫ dy γ(y) remains unaltered. Starting from this

varying variations, the free energy can be approximated by a square gradient form - [γ (y )] ≈

∫0

Ly

2| l o o o c(L*)2 dγ o o dy o f ( γ ) + m } o o o o 2 d y o o n ~

(28)

where f(γ) ≈ F(γ)/Ly and c is an elastic constant. Expanding around the average strain, γ̅, and using that ∫ 0Lydy γ(y) = Lyγ̅ at fixed overall strain, we obtain -[δγ(y)] − F(γ ̅ ) ≈

∫0

Ly

l o o 1 d2F d ym 2 o o 2L o n y dγ

δγ 2 + γ̅

c(L*)2 dδγ 2 dy

2|

o o o } o o o ~ (29)

with δγ(y) ≡ γ(y) − γ̅. If the free energy, F(γ), is concave, this functional exhibits a long-wavelength instability. Taking the longest wavelength compatible with the periodic boundary i 2π y y conditions along the y-direction, γ(y) = γ ̅ + δγ0 cosjjj L zzz, we k y{ obtain ΔF(δγ0) ≡ -[δγ(y)] − F(γ ̅ ) ≈

l 2 o 1o od F m 2 o 4o o dγ n

+ γ̅

| o 8π 2c(L*)2 o o 2 δγ0 } o o Ly o ~ (30)

y

i.e., the uniformly deformed state becomes linearly unstable with respect to long-wavelength variations of the local shear strain at fixed overall strain if the free energy, F(γ), has a negative curvature and Ly > 8π2c(L*)2/|d2F/dγ2|. Thus, the condition d2F dγ 2

0

unstable state, the other three morphologies in Figure 2b illustrate how the ellipsoidal domains relax their shapes as well as orientations and slip in the course of our DSCFT calculations, eventually forming the BCC mesophase. The four morphologies displayed in Figure 2b also illustrate that the ellipsoidal A-domains not only change their x-positions but also relax their shapes and orientations in different ways depending on their y-positions. Using yc to denote the center of mass of the ellipsoidal A-domain along the y-direction, we observe that the ellipsoids at yc = 0 and 2L* do not slip but only continuously change their shapes and orientations. In contrast, the ellipsoids at yc = L* and 3L* initially orient their long axis horizontally but thereafter do hardly change their orientations. Their x-positions slip, accompanied by a slight change of their shapes. Shape, orientation, and x-position of the ellipsoids at 7 1 3 5 yc = 2 L*, 2 L*, 2 L*, and 2 L* continuously vary during the slippage of these ellipsoidal domains. These differences indicate that the linear approximation in eq 30 is not quite applicable to the rather large amplitude of strain inhomogeneity. To quantify the slippage of the ellipsoidal domains as a function of time, we first calculate the displacement, Δx, of the center of mass of each ellipsoidal domain as a function of time, t, and coordinate, y, and then fit the data to i 2π y y Δx(y , t ) = yγ ̅ + a1(t ) sinjjj L zzz for the ellipsoids at yc = 0, L*, k y{ i 2π y y 2L*, and 3L* and Δx(y , t ) = yγ ̅ + a 2(t ) sinjjj L zzz for the k y{ 7 1 3 5 ellipsoids at yc = 2 L*, 2 L*, 2 L*, and 2 L*. Two separate fits are employed because (i) these two kinds of ellipsoidal domains have different z-positions and (ii) different relaxation behaviors−predominant displacement without change in orientation vs. coupled variation of shape, orientation, and x-position−are observed for these two kinds of ellipsoidal domains. At the early stage of our DSCFT calculations, the relaxation of the shape and orientation of the ellipsoidal domains is pronounced, whereas the center of mass of each ellipsoidal domain remains virtually unaltered, i.e., a1(t) ≈ a2(t) ≈ a0. After about 3.07τR, since the orientation of the ellipsoidal domains at

> 0 must be fulfilled for a mesostructure to be truly

metastable at fixed strain. In Figure 2, we study the linear instability of the deformed state with γ = 1. Unlike all other studies in this paper, we do not use SCMF simulations to relax from the unstable, affinely deformed, starting state but, instead, employ cubic-box, largecell DSCFT calculations46−52 because the absence of thermal fluctuations of densities and chain conformations allows us (i) to fabricate a highly symmetric initial state and (ii) to clearly observe the very slow but spontaneous slippage of ellipsoids that is driven by very weak thermodynamic driving forces. The unstable, starting state is displayed on the bottom left of Figure 2a and characterized by an uniform shear strain γ(y) = γ̅ = 1. The system geometry is Lx = Ly = 4Lz = 4L* = 7.2828Re0. To this end, first, we divide the system into 128 xz-slabs of thickness 1 ΔL = 128 Ly . Then, the step-shear deformation, γ = 1, is realized by slipping the pth slab (p − 1)ΔL along the x-direction for p = 1, 2, ..., 128. All generated ellipsoids in this unstable, starting state have the same shape and orientation, both of which are only dictated by the affine transformation and are not related to the packing of the ellipsoidal domains. In this unstable, starting state, each ellipsoidal domain has asphericity S ≈ 0.06211 (in units of Re02, see the Supporting Information) and orientation angle β ≈ 31.72° that is defined as the angle between the long axis of the ellipsoid and the x-axis. The main panel and inset of Figure 2a present how the asphericity, S, and the orientation angle, β, of the ellipsoidal domains vary as a function of time in the course of our DSCFT calculations, quantifying the relaxation of the unstable, starting state. The time evolution of both the shape and the orientation reveals that the relaxation of the unstable, starting state approaches values of the ellipsoidal mesostructure displayed G


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Figure 3. (a) Free energy, F(γ), and (b) residual shear stress, i.e., σyx for type I, σxy for type II, σyz for type III, and σxz for type IV, of the obtained mesostructures by variable-cell-shape SCFT calculations. Open symbols mark the results obtained by eq 17 from cubic-box, large-cell SCFT 9 calculations shown in Figures 5c, 7c, 9b, and 10b. The inset of panel a displays the initial HEX mesophase at χABN = 23 and fA = 32 , indicating the Cartesian coordinates, whereas the inset in panel b depicts how the packing lattice of A-domains varies as a function of γ for type-II deformation, illustrating why the corresponding stress−strain relation exhibits a periodic variation.

yc = L* and 3L* relaxes to be nearly flat and hardly changes in the following slippage process, thereafter, the time evolution of the displacement amplitude, |a1(t)| = −a1(t), is compatible with an exponential increase as suggested by the linear dependence of the free energy on a1 in eq 30. At the late stage of the DSCFT calculation, t ≳ 10τR, the morphology approaches the BCC mesophase, and the assumption of Δx(y,t) − γy̅ being small in eq 30 no longer holds. In this time regime, the increase of |a1(t)| is slowed down as it approaches the displacement in the = L* = 1.8207Re0. In equilibrium BCC mesophase, −aBCC 1 contrast, |a2(t)| = −a2(t) does not exhibit an exponential increase because, in contrast to the ellipsoids at yc = L* and 3L*, 7 1 3 5 the ellipsoidal domains at yc = 2 L*, 2 L*, 2 L*, and 2 L* exhibit a pronounced and continuous relaxation of their shapes and orientations during slippage. Finally,

• Additionally, we require that the free energy is a convex function of the strain to rule out the linear instability discussed above. Step-Shear Deformation of the HEX Mesophase. In the following, we apply the step-shear deformation to an initial, 9 equilibrium HEX mesophase at χABN = 23 and fA = 32 , using a 7

cubic cell of dimensions, Lx = 2L* = 3.419Re0 and Ly = 8 Lx , with periodic boundary conditions. The ratio Ly/Lx is compatible with a discretization in a cubic grid with 1 ΔL = 32 L* and approximates the ratio 3 of the hexagonal 2 structure. We do not expect that this slight distortion of the hexagonal structure qualitatively affects our results (see validation in the Supporting Information). Because HEX is not a cubic mesophase, we consider four types of step-shear deformations, i.e., the step-shear deformation is applied along the y-direction with a shear gradient along the xdirection (type I), the step-shear deformation is implemented along the x-direction while the shear gradient points along the ydirection (type II), the step-shear deformation is applied along the y-direction with a shear gradient along the z-direction (type III), and the step shear deformation is implemented along the xdirection while the shear gradient points along the z-direction (type IV). The variable-cell-shape SCFT calculation at γ = 0 employs Lz = 2L* for types I, II, and IV, whereas we utilize 7 Lz = 4 L* ≈ 3 L* for type III. Stress−Strain Relation. The main panel of Figure 3a presents the dependence of the free energy, F(γ), on the strain, γ, for the four types of step-shear deformation of the hexagonal mesophase. The concomitant stress−strain relation, dF Vσ(γ ) = dγ , calculated by eq 26 is shown in the main panel

2

−a 2BCC = 2 L* ≈ 1.2874R e0 . This example illustrates that morphologies obtained by a stepshear transformation will be unstable even when the average strain, γ̅, is fixed by the boundary conditions if the curvature of F(γ) is negative. Thus, we define pseudometastability of a structure after a step-shear deformation by three conditions:

• The morphology is obtained after the affine deformation in a SCMF simulation at f ixed strain. The SCMF simulation duly accounts for the short-time dynamics and the coupling between the relaxation of chain conformations and morphology. • Using the morphology obtained by SCMF simulations as input for a SCFT calculation, the SCFT converges to essentially the same morphology. This condition eliminates the arbitrariness associated with stopping the SCMF simulation at a finite time, and the convergence to a stable morphology at f ixed strain can be well established in a SCFT calculation. Without explicitly stating otherwise, we use the morphology of SCMF simulations at t ≈ 50τR after the step-shear deformation to initialize the SCFT calculations.

of Figure 3b. The residual shear stress, σ(γ), quantifies the external stress that is required to keep the strain, γ, in the pseudometastable state. At small strains, F(γ) always exhibits a positive curvature, and the mesostructures are stable at fixed strain. Upon increasing γ, however, deformations of types I, II, and IV produce structures that are linearly unstable against the slippage of cylindroids. A negative slope of σ(γ) signals this

H


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Figure 4. Morphologies immediately after type-I deformation is applied, t = 0, and at subsequent times, t ≈ 13.33τR and t ≈ 50τR, as well as results from cubic-box, large-cell SCFT calculations. γ =

Ly Lx

=

7 , and the three dimensions of the simulation box are Lx = 8L*, Ly 16

7

= 2 L*, and Lz = 4L* with ΔL ≈

0.107Re0 for SCMF simulations and ΔL ≈ 0.0267Re0 for SCFT calculations. The three red arrows mark the direction, along which the step-shear deformation acts. The rightmost panel displays the packing lattice of A-cylindroids in the yx-plane, which is characterized by two lattice vectors, a1 and a2. β denotes the orientation angle, i.e., the angle that the long axis of the cylindroid in the yx-plane makes with the y-axis, whereas θ1 is defined as the angle between a1 and a2.

SCMF simulation as input in the SCFT calculation, we find that the SCFT calculation essentially converges to the same morphology (as shown in the SCFT panel of Figure 4); however, there are no thermal fluctuations of the domain shape. Immediately after the implementation of the step-shear deformation, the unstable, starting state is also characterized by stretched chain conformations and a large, initial yx-shear stress, i.e., the mean-squared y-components of the bond vector, ⟨by2⟩, and the end-to-end distance, ⟨Rey2⟩, are elongated, and the off-diagonal components, ⟨bybz⟩ and ⟨ReyRez⟩, are nonzero. At t

linear instability, which occurs first for type-I deformation at γ ≥ 0.81, then for type-II deformation at 0.32 ≤ γ ≤ 0.83, and finally at γ ≥ 1.15 for type IV. The residual shear stress monotonously rises for type-III deformation in the considered strain interval, whereas we observe a periodic variation for typ- II deformation. The latter effect, illustrated in the inset of Figure 3b, stems from slippage of cylindroids, gradually changing the packing from hexagonal to rectangular and then back to hexagonal. The qualitative differences of F(γ) and σ(γ) for the different deformation types signal that the different types of step-shear deformation result in different kinds of packing lattice of the Adomains. Type-I Deformation. With three representative configuration snapshots, Figure 4 illustrates how the morphology relaxes in the course of our SCMF simulations after the implementation of 7 type-I deformation with γ = 16 . The packing lattice of the cylindroids is solely dictated by the affine step-shear transformation (provided that the strain remains homogeneous), and it remains unaltered in the course of the SCMF simulation after the step-shear deformation at t = 0. As depicted in the rightmost panel of Figure 4, the packing lattice in the yx-plane is characterized by two lattice vectors, a1 and a2, with lengths a1 ≈ 0.825L* and a2 ≈ 1.0915L*. The value of the angle between a1 and a2 is θ1 ≈76.33°. This packing lattice is also corroborated by our SCFT calculations (see the Supporting Information). In contrast to the packing lattice, the domain shape and orientation of the cylindroids are governed by the interactions of the soft, deformable domains of the minority A-block, and both properties relax after the step-shear deformation. From the unstable, starting state at t = 0 to the final state at t ≈ 50τR, the shape of the A-domains varies from affinely deformed cylinders with acylindricity C ≈ 0.04109 (in units of Re02, see the Supporting Information) to cylindroids with C ≈ 0.05079 in the final state. Likewise, the orientation angle, β (see definition in Figure 4), increases from 36.37° in the unstable, starting state to 48.37° in the final state. This relaxation takes place on the time scale τR, and the morphology of ordered cylindroids is obtained at t ≈ 13.33τR; after this time both the shape and the orientation of the A-domains hardly change in the following course of our SCMF simulations that have been extended up to 50τR (see the Supporting Information). Using the obtained morphology of positionally and orientationally ordered cylindroids from the

= 0, we find

⟨by 2⟩ b0 2 / 3

− 1 ≈ 0.1947 ≈ γ 2 and

⟨bybx⟩ b0 2 / 3

≈ 0.4386 ≈ γ .

According to our SCMF simulations, the relaxation of the chain conformations and the shear stress from the unstable, affinely deformed state is similar to the behavior of Rouse chains after a step strain in a spatially homogeneous system (see the Supporting Information). After 6.67τR, both ⟨bybx⟩ b0 2 / 3

⟨by 2⟩ b0 2 / 3

− 1 and

decrease by about 2 orders of magnitude and then adopt

plateau values that hardly change in the course of the subsequent SCMF simulation. At t ≈ 50τR, we find

⟨bybx⟩ b0 2 / 3

≈ 7.3723 × 10−3.

These simulation results rather nicely agree with our SCFT calculations,15,59 yielding

⟨bybx⟩ b0 2 / 3

=

σyx 1 N (n / V )kBT

≈ 7.863 × 10−3.

Thus, the shear stress does not completely vanish, and the pseudometastable cylindroids are only metastable at a minuscule, external stress that constraints the strain. Figure 5a presents the final morphologies of positionally and orientationally ordered cylindroids obtained by SCMF simulations of various type-I deformations. To illustrate how the shape and orientation of the cylindroids depend on the strain, γ, we fold back the densities of the large-cell morphologies shown in Figure 5a into 4 times smaller boxes. The so-obtained, averaged morphologies, displayed in Figure 5b, are utilized as input in SCFT calculations, which converge to essentially the same morphologies, depicted in Figure 5c. Because the free energy vs. strain is convex for all γ, these six morphologies are pseudometastable. The packing lattice of the cylindroids is geometrically determined by the strain, γ, of the affine transformation, i.e., I


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panels of Figure 6 display the representative configuration snapshots in the course of our SCMF simulations. In this unstable, starting state, a1 ≈ 0.6644L*, a2 ≈ 1.3288L*, i.e., a2 ≈ 2a1, and θ1 ≈ 82.37° (whereas in the HEX mesophase, a1 ≈ 1.0078L*, a2 = L*, and θ1 ≈60.26°, due to the approximate aspect ratio 7 ). 8 From t = 0 to t ≈ 13.33τR, the shape of the A-domains changes from cylindroids with a very large acylindricity to fused cylinders, i.e., A-domains with two, rather well-defined centers. Simultaneously, the value of the bonded shear stress, monotonously decreases from 0.877 to 0.0122. Although

⟨bybx⟩ b0 2 / 3 ⟨bybx⟩

,

b0 2 / 3

has decreased by nearly 2 orders of magnitude, it is still much larger than

⟨bxbz⟩ b0 2 / 3

≈ 1.6396 × 10−5 or

⟨bybz⟩ b0 2 / 3

≈ 2.4265 × 10−4 ,

indicating that the stretched chain conformations have not yet relaxed. Subsequently, 16.67τR< t < 76.67τR, we observe the splitting of the fused cylinders into separated cylinders (with defects) and their recombination. In this time interval,

Figure 5. (a) Final morphologies of cylindroids obtained by SCMF simulations after type-I deformations with various γ. The dimensions of 7 the simulation boxes are Lx = 4mL*, Ly = 2 L*, and Lz = 4L*. m = 2, 3, ...,7, resulting in γ =

Ly Lx

=

7 . 8m

−2

16 + (7 − 4γ )2 8

L*,

(b) Averaged morphologies obtained by

a 2 (γ ) =

1 + γ 2 L*,

observed for both

and

4 7 − 4γ

to 2, a1 decreases from 0.954L* to 0.825L*, a2 increases from 1.0078L* to 1.0915L*, and θ1 increases from 65.52° to 76.33°. The shape and orientation of the cylindroids are also controlled by the strain, γ, but the relation is more complex because it is dictated by the interaction of the cylindroids in the packing lattice, and it additionally depends on the composition. Qualitatively, the larger the strain is, the more the shape of the cylindroids deviates from that of the original cylinders. The quantitative dependence will be discussed in Figure 12. 7

For large step-shear strain, γ = 8 , we find

d2F dγ 2

⟨bybx⟩ b0 2 / 3

and the morphology, and at t ≈ 100τR, a

cylindrical structure with defects is obtained. Because of the existence of these defects, we speculate that it will take a protracted time to finally equilibrate the morphological evolution and completely relax the chain conformations and stress. Type-II Deformation. Figure 7a presents the final morphologies from our SCMF simulations after type-II deformations with various strains, γ. The final morphologies are folded back into the smallest possible, cuboidal boxes with periodic boundary conditions (see panel b), and these averaged morphologies are used as input into SCFT calculations. The converged SCFT morphologies are in Figure 7c. The SCFT converges to essentially the same morphology as the SCMF 8 8 simulation except for γ = 21 and γ = 14 .

( ) + α. Thus, when m varies from 7

θ1(γ ) = 90° − atan

monotonously

decreases from 1.1801 × 10 to 1.604 × 10 , i.e., nearly 1 order of magnitude. After t ≈ 80τR, significant changes are no longer

folding the densities of the large-cell morphologies in panel a back into 7 smaller boxes with Lx = 2mL*, Ly = 4 L*, and Lz = 4L*. (c) Morphologies obtained from SCFT calculations using the averaged morphologies in panel b as initial input.

a1(γ ) =

⟨bybx⟩ b0 2 / 3 −3

8

For type-II deformations with γ ≤ 28 , the four morphologies of cylindroids are pseudometastable. Whereas the shape and orientation of the cylindroids are controlled by the strain, γ, this dependence is different from that of type-I deformation. The packing lattice of the cylindroids for type-II deformation, displayed in the inset of panel a, is also geometrically dictated by the affine transformation and controlled by the strain, γ, i.e.,

< 0, i.e., no

pseudometastable morphology of cylindroids can be obtained. In Figure 6, the configuration snapshot at t = 0 shows the affinely deformed state. Starting from this unstable state, the other

7

Figure 6. Time evolution observed in our SCMF simulations after a type-I deformation with γ = 8 . Different panels display the morphologies at 7

different times as indicated in units of τR. The three dimensions of the simulation box are Lx = Lz = 4L* and Ly = 2 L*. J


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stable, in accord with the variable-cell-shape SCFT calculation d2F

that asserts dγ 2 < 0 for 0.32 ≤ γ ≤ 0.83 (cf. Figure 3b). We have

extended the SCMF simulation up to t ≈ 100τR and still observed cylindroids on an affinely deformed packing lattice, indicating that the relaxation to the HEX equilibrium via slippage of cylindroids45 is protracted. 8 At γ = 14 , the SCMF simulation results in rectangularly ⟨bxby⟩

packed cylindroids (RPC) with b 2 / 3 ≈ −9.8469 × 10−5 at t ≈ 0

50τR, i.e., the driving force out of this morphology nearly vanishes on average, whereas the SCFT calculation that uses this SCMF morphology as initial input yields the equilibrium HEX mesophase. Similar to Figure 1, the converged SCFT solution 7 for this RPC morphology with b1 = L*, b2 = 8 L*, and θ2 = 90° can only be obtained when we employ a highly symmetric input in the SCFT calculation, as shown in the inset of panel c, where all off-diagonal components of the stress tensor vanish, e.g., σxy = 0, but the three diagonal components are not (n / V )k T

Figure 7. (a) Morphologies obtained from SCMF simulations at t ≈ 50τR after type-II deformations with various values of γ. Lx = Lz = 4L* 7 and Ly = 2 mL* are employed with m = 1, 2, ..., 7 such that γ=

Lx Ly

=

8 . 7m

B

The inset in panel a shows the packing lattice of A-

identical. This RPC morphology also corresponds to a local

cylindroids that is characterized by two lattice vectors, b1 and b2, and one angle, θ2, between b1 and b2. (b) Averaged morphologies that are obtained by folding the densities of the large-cell morphologies in panel 7 a back into smaller boxes with dimensions, Lx = 2L*, Ly = 4 mL*, and Lz = 4L*. (c) Morphologies obtained from SCFT calculations using the averaged morphologies in panel b as initial input. The inset in panel c displays the morphology of rectangularly packed cylindroids (RPC) at 8 γ = 14 from converged SCFT calculation using a highly symmetric, initial input. Pseudometastable morphologies of cylindroids are 8 fabricated for γ ≤ 28 with strain-controlled packing lattice, shape, and orientation.

maximum of the free-energy landscape, dγ 2 < 0. Therefore, this

b1(γ)

=

L*,

b 2 (γ ) =

(4 − 7γ )2 + 49 8

L*,

d2F

morphology is not pseudometastable. 8 For even larger strain, γ = 7 , since the affine transformation makes the packing lattice of A-domains hexagonal, the system relaxes back to the equilibrium HEX mesophase in the SCMF simulation after the implementation of type-II deformation. We have extended the SCMF simulation of type-II 8 deformation with γ = 14 to eventually observe the instability of this RPC morphology. Because

d2F dγ 2

< 0, we expect an

instability with a spatially varying strain, similar to that observed for ellipsoids by DSCFT in Figure 2. In Figure 8, the unstable, starting state at t = 0 is generated by uniformly applying the stepshear deformation with thermal fluctuations. Between t = 0 and t

and

( 4 −7 7γ ). Thus, when m varies from 7 to 4,

θ2(γ ) = 90° − atan

≈ 16.67τR, the value of

b2 decreases from 0.9451L* to 0.91L*, and θ2 increases from 67.8° to 74.05°. 8 SCMF simulations and SCFT calculations disagree at γ = 21 : Whereas the SCMF simulation at t ≈ 50τR (still) yields cylindroids on an approximately affinely deformed packing lattice, the SCFT calculation converges to the equilibrium HEX mesophase. Thus, the SCMF morphology is not pseudometa-

−4

⟨bxby⟩ b0 2 / 3

monotonously decreases from

0.573 to 2.0682 × 10 , i.e., by more than 3 orders of magnitude. From t ≈ 16.67τR onward,

⟨bxby⟩ b0 2 / 3

adopts a plateau value that is

already vanishingly small and comparable to both ⟨bybz⟩ b0 2 / 3

Figure 8. SCMF simulation of time evolution after a type-II deformation with γ = The dimensions of the simulation box are Lx = Lz = 4L* and Ly = 7L*. K

⟨bxbz⟩ b0 2 / 3

and

. This observation indicates that the relaxation of the

8 . Times, measured in units of τR, are indicated in different panels. 14


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Macromolecules stretched chain conformations is nearly complete at t ≈ 16.67τR. Simultaneously, the shape of A-domains changes from affinely deformed cylinders at t = 0 to cylindroids at t ≈ 16.67τR, whose acylindricity, C, decreases from 0.05538 to 0.02334 and whose orientation angle, β, decreases from 37.82° to 0. Because the acylindricity and orientation angle both hardly change after t ≈ 16.67τR in the SCMF simulation run that extends up to 50τR (see the Supporting Information), we conclude that the RPC morphology is formed at t ≈ 16.67τR. Compared with the SCFT morphology in Figure 7, where each cylindroid has acylindricity C ≈ 0.021 and zero value of β, rather good agreement between SCMF simulations and SCFT calculations is obtained. Extending the SCMF simulation further, however, we do observe an instability of the packing lattice, i.e., eventually at t ≈ 100τR, a visible zigzag structure of packing lattice in the shear plane, xy, emerges, whose amplitude increases with time. Whereas the type of instability, i.e., a spatially varying γ, is similar to what we have observed for ellipsoids in Figure 2, the onset of the spatial growth of the zigzag structure of the packing lattice in the xy-plane is delayed. Partially, this behavior can be explained by the starting morphology. In the equilibrium mesophase or in the affinely deformed morphology at t = 0, thermal fluctuations of the local center of a cylindroid in the xy-plane as a function of z (undulations) are very small because of the extraordinarily large value of 5̅ in the SCMF simulation. These small undulations at t ≈ 50τR are sufficient to prevent the SCFT calculation to converge toward the RPC morphology. The behavior, however, is in marked contrast to the initial morphology in Figure 2b, where all ellipsoids in the direction perpendicular to the shear plane exhibit the same and rather large deviation from the affine deformation. The instability associated with the negative curvature of F(γ) not only refers to translation of a cylindroid in the xy-plane but also promotes local shifts of the cylindroid position along the z-direction, i.e., the instability also enhances undulations of the cylindroids along the z-direction. These are clearly visible in the configuration snapshot at time t ≈ 100τR. These local shifts reduce the free

Figure 9. (a) Final morphologies of cylindroids after type-III deformations with various values of γ, as obtained by SCMF 7 7 simulations. Lx = 4L*, Ly = 2 L*, and Lz = 2 mL* are utilized with m = 1, 2, ..., 7 for the seven morphologies on the top, resulting in γ=

Ly Lz

1 , m

=

whereas for the other seven morphologies at the bottom, 7

the three dimensions of the simulation boxes are Lx = 4L*, Ly = 2 L*, m̃ L* 4

and Lz =

with m̃ = 7, 8, ..., 13, giving rise to γ =

Ly Lz

=

14 . m̃

(b)

Morphologies obtained from converged SCFT calculations that directly use the large-cell morphologies in panel (a) as initial input without folding-back. The length, L, of each cylindroid is larger than that of the corresponding original cylinder before the step-shear deformation,

d2F

energy for dγ 2 < 0 but hardly change the average position of the

L = Lz 1 + γ 2 . The inset in panel b displays the packing lattice of Acylindroids, which is characterized by two lattice vectors, c1 and c2. θ3 denotes the angle between c1 and c2.

center of a cylindroid in the xy-plane because they occur with equal probability in the positive and negative x-direction. Type-III Deformation. Whereas type-I and type-II deformations of the HEX mesophase (Figures 5 and 7) do not alter the length of the cylindrical domains because the strain and gradient act in the plane normal to the cylinder axis, this is no longer true for the other types of deformations. Figure 9a presents the final morphologies of positionally and orientationally ordered cylindroids that are obtained from our SCMF simulations after a type-III deformation at various strains, γ. Then, without folding-back, we directly use these 14 large-cell morphologies as initial input for SCFT calculations, whose converged solutions are displayed in Figure 9b. Because the morphologies of SCMF simulations and SCFT calculations essentially agree and the type-III deformation does not result in a linear instability (cf. Figure 3b), all obtained mesostructures of cylindroids are pseudometastable. Type-III deformation elongates the cylindroids compared to the original cylinders of the HEX mesophase before the stepshear deformation. After the deformation, the angle of each cylindroid with the z-axis is the shear angle, α, and thus, the

therefore the thinned cylindroids can break up even if F(γ) exhibits positive curvature. For type-III deformation, the packing lattice of the cylindroids in the plane perpendicular to the cylindroid axis is shown in the inset of Figure 9b. Basically the deformation results in a twodimensional packing of cylindroids with c1(γ ) =

1 8

16(1 + γ 2) + 49 1 + γ2

L*,

c2(γ)

=

L*,

and

i 7 yz z. Thus, when γ varies from 1 to 2, c1 θ3(γ ) = atanjjj 2z 7 k4 1+γ { decreases from about 1.0002L* to 0.6349L*, and θ3 decreases from about 60° to 38.05°. Type-IV Deformation. The morphologies obtained by SCMF simulations and SCFT calculations for type-IV deformations with various strains are presented in Figure 10. For small strains, 16 γ ≤ 15 , we obtain oblique cylindroids whose packing lattice is dictated by the geometry of the deformation and that are also

cylindroids are stretched by a factor 1 + γ 2 . Additionally, the elongation of the cylindroid also makes it become thinner, and

stretched by a factor L

1 + γ 2 . For larger strains, γ ≥

16 , 11


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i 7 1 + γ2 y angle between d1 and d2 varies as θ4(γ ) = atanjjj 4 zzz. k { Therefore, when γ varies from 1 to 16 , d1 decreases from 7

15

1.0053L* to 0.9395L*, d2 decreases from 0.9899L* to 0.6839L*, and θ4 increases from 60.5° to 68.65°. Quantitative agreement between SCMF simulations and SCFT calculations can be observed for γ ≤ 16/15 16 (pseudometastable cylindroids) and γ ≥ 11 (lamellae). For 16

Figure 10. (a) SCMF simulations of type-IV deformation with various 7 γ. We employ Lx = 4L*, Ly = 2 L*, and Lz = 4mL* with m = 1, 2, ..., 7 and γ =

Ly =

Lx Lz

7 L*, 2

=

1 m

for the seven morphologies on the top and Lx = 4L*,

and Lz =

m̃ L* 4

with m̃ = 8,9,···,15 and γ =

Lx Lz

=

16 m̃

for the

eight morphologies at the bottom, respectively. (b) Morphologies obtained from SCFT calculations that employ the large-cell morphologies in panel a directly as input. Cylindroids are obtained 16 for γ ≤ 15 , whereas the step-shear deformation results in metastable lamellar morphologies for larger strains. Two lattice vectors, d1 and d2, are employed to characterize the packing lattice of A-cylindroids and are shown in the inset of panel b. The angle between d1 and d2 is denoted as θ4.

7

1 8

16 + 49(1 + γ 2) 1 + γ2

L* and d 2(γ ) =

1 1 + γ2

3

L LAM = 8 L* ≈ 2 L* ≈ 1.4958R e0 . Whereas the off-diagonal components of the stress tensor vanish in SCFT calculations, the three diagonal components (pressure) are not identical. The component σyy is smaller than σxx = σzz, indicating that the lamellar spacing in the obtained morphology is too smallthe equilibrium spacing for this asymmetric lamellar mesostructure is 1.6387Re0. 16 For type-IV deformation with γ = 11 , we illustrate the stepshear-induced transformation from cylinders to lamellae in Figure 11. Immediately after the step-shear deformation, t = 0, we observe that the cylindrical domains of the HEX mesophase instantaneously deform into cylindroids lying in the xz-plane. Subsequently, these highly deformed domains start to connect with each other along the x-direction around t ≈ 3τR and finally form lamellar domains in the xz-plane around t ≈ 6τR. Thereafter, the system only evolves slowly in the following

however, we observe that the deformed system relaxes to a new morphology, i.e., instead of forming cylindroids of the minority component, the system changes its symmetry and forms a lamellar structure. The inset in Figure 10b displays the strain-controlled packing lattice of the cylindroids for type-IV deformation. Two lattice vectors, d1 and d2, are utilized to characterize this packing lattice, i.e., d1(γ ) =

16

the three intermediate values, 12 ≤ γ ≤ 14 , i.e., in the ultimate vicinity of the strain, where the free energy, F(γ), changes curvature, γ* ≈ 1.15 (cf. Figure 3b), SCMF simulations and SCFT calculations disagree. The SCMF simulations at t ≈ 50τR yield morphologies that are neither a perfect cylindroid mesostructure nor a defect-free lamellar mesostructure; it rather resembles a perforated-lamellar morphology. Using this defectriddled morphology as input into the SCFT calculation, some (but not all) of the defects are eliminated in the course of the convergence of the SCFT calculation, and a different morphology with fewer metastable defects is obtained. For 16 16 γ = 12 and γ = 14 , the SCFT converges to defective cylindroid morphologies where the orientation of the packing lattice of the cylindroids differs from the affinely deformed packing lattice of 16 the original HEX mesophase. For γ = 13 , both SCMF simulations and SCFT calculations, give rise to perforated lamellae with different defects. 16 For the lamellar morphologies, γ ≥ 11 , the lamellar domains are stacked along the y-direction, and the lamellar spacing is dictated by the epitaxial relation,

L*. The

Figure 11. SCMF simulation snapshots, illustrating the step-shear-induced transformation from cylinders to lamellae in response to a type-IV 7 16 11 deformation with γ = 11 . The dimensions of the simulation box are Lx = 4L*, Ly = 2 L*, and Lz = 4 L*. Different panels show the morphologies at different times measured in units of τR. M


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Figure 12. (a) Acylindricity, C, and (b) orientation angle, β, of the cylindroids as a function of γ, obtained by SCFT calculations. Open symbols mark the results of cubic-box SCFT calculations (cf. Figures 5c, 7c, 9b, and 10b), whereas solid lines (cylindroids with

d2F dγ 2

> 0) and dots (cylindroids with

2

dF dγ 2

< 0) are obtained from variable-cell-shape SCFT calculations. The two insets in panel a show the morphologies of cylindroids obtained by type-I 29

7

9

deformation with γ = 16 for two compositions, fA = 128 and fA = 32 , respectively. These two cylindroids have acylindricity C ≈ 0.01423 and C ≈ 0.0472, respectively. For comparison, the dashed magenta line in panel b presents the orientation angle in the unstable, starting state immediately the affine step-shear deformation.

For each type of deformation, at fixed value of γ, the obtained cylindroids with larger fA are more acylindrical. For both values of fA, the larger the value of γ is, the more acylindrical the domain shape becomes for type-I deformation, whereas the dependence of the acylindricity on strain γ is nonmonotonous and significantly smaller for type-II deformation. Like the strain in Figure 3, the C−γ curve for type-II deformation is symmetric 8 around γ = 14 , i.e., the strain that results in rectangularly packed cylindroids in Figure 8. For each value of fA, the type-I deformation makes the shape of the cylindroids more sensitive to the variation of γ than for any other type of deformation, and this susceptibility is significantly enhanced for the larger value of fA. The dependence of C on γ for type-III and type-IV deformations is also nonmonotonous, and compared with that for type-I and type-II deformations, it is much weaker. Because the orientation of the cylindroids fabricated by typeIII and type-IV deformations have already been discussed in Figures 9 and 10, in Figure 12b, we only present the results for deformations of types I and II. For type-I deformation, the orientation angle, β, of the cylindroids nonmonotonously depends on the strain, γ, however, it is rather insensitive to the composition, fA. Furthermore, at large strain and independent from fA, the orientation angle of the cylindroids is larger than the corresponding angle in the unstable, starting state, which is geometrically determined by the affine transformation. For typeII deformation, however, the value of β decreases and passes through zero for rectangularly packed cylindroids and continues to decrease. Also for this type of deformation, the dependence of the orientation angle on composition is rather weak. These observations highlight that the shape and orientation of the cylindroids can be independently controlled by the stepshear strain, γ, the composition, fA, and the type of step-shear deformation. Step-Shear Deformation of the GYR Mesophase. In the following, the step-shear deformation is implemented on an 11 initial, equilibrium GYR mesophase at χABN = 23 and fA = 32 ,

SCMF simulation, which has been extended up to 50τR. ⟨b b ⟩ Simultaneously, the value of 2x z monotonously decreases from b0 / 3

1.4567 at t = 0 to 2.4348 × 10−4 at t ≈ 5τR, i.e., by nearly 4 orders of magnitude, and then adopts a plateau value that is already vanishingly small and becomes comparable to both ⟨bybz⟩ b0 2 / 3

⟨bxby⟩ b0 2 / 3

and

in the following course of our SCMF simulations,

indicating that the stretched macromolecular conformations have already relaxed completely. At t ≈ 50τ R , since ⟨bxbz⟩ b0 2 / 3

≈

σxz 1 N (n / V )kBT

= 0 and all off-diagonal components of

the stress tensor obtained by the SCFT calculation vanish though the three diagonal components are not identical, the metastability of the obtained lamellar morphology is corroborated by both the SCMF simulation and the SCFT calculation. Thus, type-IV deformation with a large strain generates a highly asymmetric, compressed, metastable lamellar morphology with fB fA

≈ 2.56.

Shape and Orientation. To quantify the dependence of the shape and orientation of the cylindroids on the step-shear strain, γ, and the composition, fA, of the diblock copolymer, as well as the type of step-shear deformation, we compute their acylindricity, C, and orientation angle, β. To this end, we compute and diagonalize the gyration tensor of each cylindroid, obtained by SCFT calculations. Details about the corresponding computations are deferred to the Supporting Information. Figure 12 displays how the shape and orientation of the cylindroids vary as a function of γ for 2 values of fA and four types d2F

of deformations. The solid lines (cylindroids with dγ 2 > 0) and dots (cylindroids with

d2F dγ 2

< 0) are obtained by variable-cell-

shape SCFT calculations, whereas the open symbols are calculated from the morphologies in Figures 5c, 7c, 9b, and 10b, obtained by cubic-box SCFT calculations. N


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Figure 13. (a) Free energy, F(γ), and (b) residual shear stress, σyz(γ), of the obtained mesostructures as a function of γ from our variable-cell-shape SCFT calculations. The results marked by open symbols are obtained by eq 17 from our cubic-box, large-cell SCFT calculations shown in Figure 14c. 11 The inset in panel a displays the initial GYR mesophase at χABN = 23 and fA = 32 , indicating the Cartesian coordinates.

using a cubic cell with periodic boundary conditions, as displayed in the inset of Figure 13a. All dimensions of the cubic cell are Lx = Ly = Lz = L* = 3.993Re0. Because GYR is a cubic mesophase, like the BCC mesophase, we only consider one step-shear deformation, i.e., specifically, we implement the step-shear deformation along the y-direction with the shear gradient pointing along the z-direction. The main panel of Figure 13a shows how the free energy, F(γ), depends on the strain, γ, for the step-shear deformation of the GYR mesophase. Panel b of Figure 13 presents the dF concomitant stress−strain relation, Vσ(γ ) = dγ , calculated by

eq 26. Because GYR is an equilibrium mesophase at γ = 0, the free energy, F(γ), exhibits a positive curvature at small strains. Upon increasing the value of γ, the residual shear stress of the fabricated mesostructure continuously rises. However, for γ ≥ 0.41, the negative slope of σ(γ) signals the linear instability of the obtained structures, i.e., the obtained morphology is no longer pseudometastable. Applying the step-shear deformation to the GYR mesophase with various strains, γ, we obtain the morphologies depicted in Figure 14a by our SCMF simulations. We fold the densities back into smaller boxes (see panel b) and subsequently employ these averaged densities as input in SCFT calculations. The SCFT calculations converge to the morphologies presented in Figure 14c. Both SCMF simulations and SCFT calculations indicate that the obtained network-morphologies are still gyroid-like, though they deviate from the original network of the GYR mesophase. The degree to which the gyroid-like networks deviate from the original gyroid network is controlled by the strain, γ. Qualitatively, the larger the strain becomes, the more the obtained gyroid-like networks deviate from the original gyroid network. According to the three conditions for pseudometastability, all gyroid-like networks presented in Figure 14 are pseudometastable. Upon increasing the strain to a large value, γ = 1, i.e., more than twice larger than the strain value where the curvature of F(γ) turns negative, we observe the formation of a rather complex morphology. Figure 15 displays the representative configuration snapshots in the course of our SCMF simulations. First, the instantaneous, affine step-shear deformation at t = 0 transforms the original gyroid network into gyroid-like network.

Figure 14. (a) Final morphologies of gyroid-like networks for various values of γ as obtained by SCMF simulations. Lx = Ly = 2L* and Lz = 2mL* are used with m = 3, 4, ..., 7 such that γ =

Ly Lz

=

1 . m

(b) Averaged

morphologies that are obtained by folding the large-cell morphologies shown in panel a back into smaller boxes with dimensions, Lx = Ly = L* and Lz = mL*. (c) Morphologies obtained by SCFT calculations, using the averaged morphologies depicted in panel b as initial input.

Subsequently, additional connections along the y-direction are formed inside the deformed network, and finally, after about t ≈ 33.33τR, a new network morphology is obtained. Meanwhile, the value of

⟨bybz⟩ b0 2 / 3

monotonously decreases from 1.002 in the

unstable, starting state at t = 0 to 1.214 × 10−3 at t ≈ 33.33τR, i.e., by nearly 3 orders of magnitude. After t ≈ 33.33τR,

⟨bybz⟩ b0 2 / 3

continues to slightly decrease further to 4.1434 × 10−4 at t ≈ 200τR, but the morphology remains almost unaltered. Using this morphology at t ≈ 200τR as initial input for the SCFT calculation, however, no converged solution can be obtained for this new network morphology, and the SCFT calculation finally results in the formation of equilibrium GYR mesophase, as shown in the SCFT panel of this figure, where the gyroid O


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Figure 15. SCMF simulations of a large step-shear deformation, i.e., γ = 1, applied to the GYR mesophase with Lx = Ly = Lz = 2L*. Different panels show the morphologies at different times measured in units of τR. The panel obtained from the SCFT calculation utilizes the configuration snapshot at t ≈ 200τR as initial input.

Figure 16. LAM mesophase under step-shear deformation: (a) Free energy, F, and (b) residual shear stress, σyz, as a function of γ obtained by variable16 12 cell-shape SCFT calculations. The two insets in panel a show the two initial LAM mesophases at fA = 32 and fA = 32 and also indicate the orientation of the Cartesian coordinates. The dimensions of the simulation boxes are Lx = Ly = Lz = 2L* = 3.3935Re0 for fA = 3.4255Re0 for

16 fA = 32 . 2 dF

Information) with

dγ 2

12 32

and Lx = Ly = Lz = 2L* =

Open triangle symbols mark the results that are obtained by cubic-box, large-cell SCFT calculations (see the Supporting

> 0 and variable-cell-shape SCFT calculations with

dimensional SCFT calculations using

1 1 + γ2

d2F dγ 2

< 0, whereas open circles and squares mark the results from one-

L* as the dimension of the system for fA =

how the segregation, ϕmax A , varies as a function of the lamellar spacing, L, for fA =

16 , 32

12 32

and fA =

16 , 32

respectively. The inset in panel b illustrates

where the open diamond indicates L = L*.

respectively, are obtained from one-dimensional SCFT 1 L* as the dimension of the system, calculations using 2

networks adopt a different arrangement from that before the step-shear deformation. This observation indicates that this new network morphology is neither metastable nor pseudometastable but rather a long-lived state. Thus, we speculate that our SCMF simulations will also finally lead to the formation of the GYR mesophase, if we continue our simulation runs significantly beyond t ≈ 200τR. Step-Shear Deformation of the LAM Mesophase. Finally, we apply the step-shear deformation to equilibrium 12 LAM mesophases with an asymmetric composition, fA = 32 ,

1+γ

which exactly match the results from variable-cell-shape SCFT calculations, implying that the lamellar domains simply become compressed by a factor 1 + γ 2 . The thicknesses of both Adomains and B-domains are also controlled by the value of γ; f

their ratio remains A . fB

If the curvature of the free energy, F(γ), becomes negative, the system becomes linearly unstable with respect to “bunching” of lamellae, i.e., in one part of the system the lamellar spacing, L, increases whereas in another part the lamellae become even more compressed so that the average lamellar spacing remains unaltered. There is, however, no coexisting between two lamellar structures with different spacings, i.e., the free energy, F(L) ,

16

and a symmetric one, fA = 32 . The lamellar normals point along the y-direction, and consequently, we only consider the situation where the step-shear deformation is applied along the ydirection with the shear gradient pointing along the z-direction. The angle that the lamellar domains make with the xz-plane matches the shear angle, α, in analogy to the cylindroids in Figures 9 and 10. Figure 16a presents the free energy, F(γ), and the residual shear stress, σyz, as a function of the strain, γ. The results marked 16 12 by open circles and squares for fA = 32 and fA = 32 ,

nkBT

exhibits a negative curvature but does not allow for a doubletangent construction because as we decrease the lamellar (see the inset of Figure 16b), spacing, the segregation, ϕmax A 16 decreases as well, and at L ≤ 0.58Re0 for fA = 32 , the disordered P


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Figure 17. SCMF simulations of the relaxation of a LAM morphology after a step-shear deformation. The top row refers to an asymmetric lamellar 16 12 11 morphology, fA = 32 , using γ = 11 , and Lx = Ly = 4L*, Lz = 4 L*. The bottom row depicts the time evolution of a symmetric lamellar system,

fA =

16 , 32

using γ =

16 , 9

9

and Lx = Ly = 4L*, Lz = 4 L*. The times indicated in the different panels are measured in units of τR. 12

phase takes over. In the asymmetric system, fA = 32 , this instability occurs at γ ≥ 1.48, whereas the instability sets in later, at γ ≈ 1.6, for the symmetric system. The time evolution after a large step strain is depicted in Figure 17. In the top panel, we investigate the asymmetric 16 lamellar system with γ = 11 . For this step strain, the curvature of the free energy, F(γ), is still positive (cf. Figure 16). The affinely deformed starting morphology of our SCMF simulations at t = 0 is displayed in the leftmost panel of this figure. Although the value of the bonded stress,

⟨bybz⟩ b0 2 / 3

both

⟨bybz⟩ b0 2 / 3

the value of ⟨bxbz⟩ b0 2 / 3

⟨bybz⟩ b0 2 / 3

⟨bxbz⟩ b0 2 / 3

by about 2 orders of magnitude. In the

⟨bybz⟩ b0 2 / 3

≈ 1.0405 × 10−3 approaches that of

⟨bxby⟩ b0 2 / 3

and

within 1 order of magnitude.

In the bottom panel of Figure 17, we investigate the structure 16 formation after a large step strain, γ = 9 , of symmetric lamellae. In this case, the curvature of the free energy, F(γ), is already negative (cf. Figure 16), and the bunching of the lamellae is an unstable mode of the system. In the three-dimensional SCMF simulation, however, this is not what we observe. Instead, the time evolution is quite similar to the previous case. First, 0 < t

Step-Shear Deformation of Block Copolymers - ACS Publications

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