Fabrication of Ellipsoidal Mesostructures in Block Copolymers via a

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

Fabrication of Ellipsoidal Mesostructures in Block Copolymers via a Step-Shear Deformation De-Wen Sun and Marcus Müller* Institut für Theoretische Physik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D 37077 Göttingen, Germany S Supporting Information *

ABSTRACT: Ellipsoids have attracted abiding attention because of their shape-dependent, anisotropic properties. In some applications, e.g., photonic crystals, both positional and orientational order of the ellipsoidal packing are required. We propose a versatile, facile, and efficient strategy to fabricate positionally and orientationally ordered crystals of soft ellipsoids in block copolymers via a step-shear deformation. Starting from the thermodynamically stable, equilibrium spherical mesophase of the copolymer material, the step-shear deformation provides an instantaneous, anisotropic stimulus to deform the spherical domains into ellipsoids and simultaneously stretches the macromolecular conformations. Subsequently, at fixed strain, the molecular stress relaxes to an equilibrium where the shape and orientation of the obtained ellipsoids are dictated by the packing frustration. Since the residual molecular stress is minuscule, the lattice relaxation via slippage in the absence of external stress is protracted, i.e., the crystal of soft ellipsoids with positional and orientational order is pseudometastable. Our strategy also allows for low volume fractions of ellipsoids (compared to colloidal systems). Both single-chain-in-mean-field (SCMF) simulations and self-consistent field theory (SCFT) calculations are employed to demonstrate the pseudometastability of the obtained ellipsoids. Varying the magnitude of the step-shear strain and the composition of the block copolymer, we can control the asphericity and orientation of the ellipsoidal domains independently. Our study provides a new concept for fabricating soft, positionally and orientationally ordered crystals of ellipsoids with potential applications in engineering functional materials.

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INTRODUCTION Ellipsoids have attracted abiding interest due to their shapedependent, anisotropic properties,1−15 which can be used to improve electronic or optical materials characteristics2,6 and to explore a great number of novel physical phenomena induced by the shape and orientation of the particles.7−15 There are four phases in the phase diagram of hard ellipsoids as a function of packing density and aspect ratio,16−19 including an isotropic fluid, a nematic fluid (liquid crystal), plastic solids, and orientationally ordered solids (crystals). In some applications, e.g., photonic crystals, both positional order and orientational order of the ellipsoidal packing are required.2,6 For instance, positional and orientational order enable three-dimensional (3D) photonic crystals composed of ellipsoidal γ-Fe2O3−SiO2 core−shell nanoparticles to show strong structural colors,2 while magnetically responsive photonic crystals of silica-coated iron nanoellipsoids potentially can provide a platform to fabricate novel active optical components for various color presentation and display applications.6 In hard-ellipsoid © XXXX American Chemical Society

systems, three main strategies have been pursued to simultaneously generate positional and orientational order: the formation of crystals, smectic liquid crystals, and the employment of external fields, such as electric fields and magnetic fields. Although hard-ellipsoid crystals are equilibrium structures at high packing density, high density may lead to jammed states. Smectic liquid crystals, in turn, are not stable phases in hard-ellipsoid systems, and these metastable structures may not be sufficiently robust to exhibit the desired response to external fields. Soft-matter systems, such as block copolymers, may provide alternative routes for fabricating positionally and orientationally ordered structures of ellipsoids. There are two exceptional advantages associated with the spontaneous self-assembly of the block copolymer for fabricating ellipsoids. On the one hand, the Received: September 24, 2017 Revised: December 7, 2017

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DOI: 10.1021/acs.macromol.7b02060 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Illustration of the step-shear deformation, γ = 1: The left panel (a) depicts a snapshot of the initial BCC crystal of spherical A-domains. The bottom face shows a contour plot of the A-density, and green surfaces present the internal AB interfaces. The shape of each A-domain exhibits small shape variations due to thermal fluctuations. The instantaneous, affine step-shear transformation of the particle coordinates fabricates a starting structure of positionally and orientationally ordered ellipsoids, which stretches the chain conformations and imparts a yz shear stress. The right panel (b) sketches the transformation from a sphere (blue) to an ellipsoid (red) in the yz-plane, illustrating that the angle α + θ that the long axis of the starting ellipsoid makes with the z-axis differs from the shear angle α by θ ≈ 13.28°, which agrees with the value observed in our SCMF simulations. β = 90° − α − θ.

(on the order of hours) onto these pseudometastable, softellipsoid crystals, thus, providing ample opportunities for additional stabilization (e.g., by cooling below the glasstransition temperature, crystallization, or cross-linking).

feature size of the self-assembled ellipsoidal domains can be readily tuned by the molecular weight of the block copolymer, and on the other hand, positional and orientational order of the ellipsoids are controlled by the subtle balance between the interfacial tension between different domains and the conformational stretching entropy in block copolymers,20 allowing us to fabricate orientationally ordered crystals even at low volume fraction of ellipsoids. Whereas ellipsoidal domain shapes have been reported in the presence of electric fields21,22 or as nonequilibrium steady states in shear flow,21 these strategies require the presence of static or dynamic external fields, and the structure will rapidly convert back into spherical domain shapes once the external field is switched off. In the following, we will propose a versatile, facile, and efficient strategy to fabricate the soft ellipsoids with both positional and orientational order in diblock copolymers via a step-shear deformation. The initial state is a thermodynamically stable, equilibrium spherical mesophase, e.g., the body-centered-cubic spheres (BCC) of a linear AB diblock copolymer melt.23 Then, a stepshear deformation with a shear angle α is applied. In response to this instantaneous and anisotropic stimulus, the macromolecular conformations are stretched and the spherical microphase-separated domains deform into ellipsoids. Subsequently, by keeping the step-shear strain, we observe that the molecular stress relaxes and the shape and orientation of the obtained ellipsoids are dictated by the packing of the domains. Since the residual molecular stress is minuscule, the lattice relaxation via slippage24 is protracted if the external stress is eliminated; i.e., the obtained ellipsoids are pseudometastable (long-lived but not stress-free). There are two additional, practical aspects: (i) The asphericity and orientation of the soft ellipsoidal domains can be independently tuned by the step-strain amplitude, the molecular architecture of the block copolymer material.25 Moreover, the susceptibility of the domain shape to the anisotropic stimulus can be enhanced by the nonuniform distribution of two types of linear AB diblock copolymers with different chain lengths on the domain interfaces in a binary copolymer blend.26 (ii) The protracted lattice relaxation via slippage in the absence of external stress imparts a long lifetime

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METHODS AND PARAMETERS The initial state is the equilibrium BCC crystal of spherical domains of a linear AB diblock copolymer melt with A volume 7 fraction fA = 32 at intermediate segregation strength. The step-shear deformation is represented by an affine transformation, altering the macromolecular conformations. This results in stretched chain conformations in the starting state that are not in equilibrium with the density distribution. In order to capture the interplay between the relaxation of the stretched molecular conformations and the collective kinetics of the local densities, we use a particle-based simulation algorithm, single-chain-in-mean-field (SCMF) simulations.27−30 The molecular contour is discretized into N = 32 effective interaction centersbeadsthat are connected by harmonic springs. Both blocks are characterized by the same statistical segment length, and all length scales are measured in units of the end-to-end distance, Re0, in the absence of nonbonded interactions. Nonbonded interactions are composed of a repulsion between unlike segments with strength, χABN = 23, and a Helfand term that restrains the fluctuations of the local densities with an inverse, isothermal compressibility, κN = 50. The nonbonded interactions are computed on a collocation grid with resolution, ΔL ≈ 0.1181Re0. We employ a large invariant degree of polymerization, 5̅ = ρ0 R e0 3/N = 12800, where ρ0 = nN/V denotes the segment number density, n the number of diblock copolymers, and V the volume. We consider a cubic system geometry with periodic boundary conditions in all directions. Chain conformations are updated by local, diffusive Monte Carlo (MC) moves, giving rise to Rouse-like single-chain dynamics. Additionally, we use self-consistent field theory (SCFT) based on the standard Gaussian chain model31−34 to compute the equilibrium periodicity L* = 1.8893Re0 of the initial BCC mesophase and to validate the pseudometastability of the fabricated positionally and orientationally ordered crystal of soft B


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Figure 2. Morphologies immediately after the implementation of the step-shear deformation, at t ≈ 16.67τR and t ≈ 50τR after the step-shear deformation is applied, and results from SCFT calculations. γ = 1/2, and the BCC mesophase before the step-shear deformation is prepared for χABN = 23 and fA = 7/32. The three dimensions of the simulation box are Lx = Ly = 3L* and Lz = 6L*. The red arrows mark the direction along which we implement the step-shear deformation.

and orientationally ordered ellipsoids is the equilibrium at fixed strain. The domain shape, position, and orientation of the obtained ellipsoids are all dictated by the packing and interaction of the soft, deformable domains of the minority block. The relaxation of the macromolecular conformations is studied in Figure 3. In the starting state, the mean-squared y-

ellipsoids. Further details about the SCMF simulations and SCFT calculations are compiled in the Supporting Information.

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RESULTS AND DISCUSSION We equilibrate the BCC mesophase in a cubic system with Lx = Ly = 3L* and Lz = 3mL* with m = 1, 2, ..., 7. In our simulations, as illustrated in Figure 1, the step-shear deformation is realized by affinely transforming the Cartesian y-coordinate of every bead, i, according to ybefore → yafter = ybefore + zi tan α. The shear i i i Ly 1 strain γ = tan α = L = m is chosen such that the system z

before and after the step-shear deformation is compatible with the periodic boundary conditions, i.e., the seven different m 1 values correspond to step-shear strains, 7 ≤ γ ≤ 1. The employment of the periodic boundary conditions implies that the step-shear strain is fixed. The configuration snapshots before and after the step-shear deformation in Figure 1a illustrate how the step-shear deformation with γ = 1 alters the domain shape from spherical to ellipsoidal, and Figure 1b sketches the orientation of the ellipsoids in the starting state. The difference, θ, between the shear angle and the long axis of the ellipsoids in this starting state is a property of the geometric transformation and is not related to the packing of the soft ellipsoidal domains of the block copolymer. Immediately after the step-shear deformation, the starting state of positionally and orientationally ordered ellipsoids is characterized by stretched chain conformations and yz shear stress. Figure 2 illustrates with configuration snapshots the relaxation of the morphology in the course of the SCMF 1 simulation for γ = 2 . One can clearly observe that both the shape and the orientation of the A-domains relax from the starting state to the final state at t ≈ 50τR: the shape of the Adomains changes from deformed spheres to end-capped rods with asphericity (in units of Re02, see Supporting Information) varying from 0.036 14 in the starting state to 0.069 74 in the final state, and the angle, θ, increases from 26.01° in the starting state to 30.66° in the final state. The relaxation occurs on the scale of the Rouse time, τR ≈ 300 MC steps, of the disordered state. After t ≈ 16.67τR, a morphology of ordered ellipsoids is obtained that hardly changes in the course of the simulation, which has been extended up to 50τR. This observation demonstrates that the obtained morphology of positionally

Figure 3. Relaxation of macromolecular conformations after a stepstrain deformation when γ = 1/2. The main panel depicts how ⟨bybz⟩ and ⟨by2⟩ decrease as a function of the time, t, measured in units of the Rouse time, τR, of the disordered system. The results from our SCMF simulations are compared with that from the Rouse model using continuous Gaussian (black dashed line) and discrete chain models 2 (green solid line).31 G*(t) = ∑N−1 p=1 exp(−2p t/τR) corresponds to the discrete chain models. The inset presents the time evolution of ⟨ReyRez⟩ and ⟨Rey2⟩ − ⟨Rey2⟩∞, where ⟨Rey2⟩∞ denotes the value in the pseudometastable state. At early times an exponential decay is observed.

components of the bond vector, ⟨by2⟩, and the end-to-end distance, ⟨Rey2⟩, are elongated, whereas the other Cartesian components remain unaltered. In addition, the off-diagonal components, ⟨bybz⟩and ⟨ReyRez⟩, deviate from their values in the initial BCC mesophase. The former quantity is particular important because, for Gaussian chains, it is proportional to the C


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3⟨ReyRez⟩/Re02 significantly exceeds the Rouse time, τR, and stems from two effects: (i) the slow relaxation of the shape of the ellipsoidal domains and (ii) the slowing down of the relaxation of large-scale chain conformations due to the inhomogeneity (i.e., domain interfaces). Figures 2 and 3 demonstrate that the system relaxes into a pseudometastable state after the step-shear deformation, where the long-lived ellipsoidal morphology is observed. Figure 4 presents the final morphologies of both positionally and orientationally ordered crystals of ellipsoids from our

virial stress of the strong bonded forces and thus provides an accurate approximation of the shear stress, σyz. The main panel of Figure 3 presents the relaxation of 3⟨by2⟩/ 2 b0 − 1 and 3⟨bybz⟩/b02 after the step-shear deformation, where b02 = Re02/(N − 1). At t = 0, we find 3⟨by2⟩/b02 − 1 ≈ tan2 α = γ2 = 1/4 and 3⟨bybz⟩/b02 ≈ γ. The latter one quantifies the stress response of the viscoelastic material to the step-shear deformation at t = 0, i.e., the external force per unit area required to deform the sample. After the step-shear deformation, however, the deviation of the bonds from the Gaussian distribution decays approximately like 1/√t in time, similar to the stress relaxation after a step strain in a homogeneous system of Rouse chains, i.e., ∞

2p 2 t τR 31

( )≈

⟨bybz⟩ ∼ G(t ) = ∑ p = 1 exp −

πτR 8t

, where p is the

index of pth Rouse mode. The discrete and continuum predictions for the stress relaxation are plotted together with the SCMF simulation data in Figure 3. Rather good agreement between the simulation data and the Rouse prediction for the discrete chains is observed for t > 10 MC steps ≈0.03τR, demonstrating that the chain conformations and the stress relax similar to the behavior in a homogeneous system. Most notably, after 6.67τR, both quantities are reduced by about 2 orders of magnitude and adopt the values that are already very small. These plateau values in the pseudometastable state indicate a difference between the bond statistics in the ellipsoidal system and the one in the disordered melt. The plateau value of 3⟨by2⟩/b02 − 1 is comparable to the value in the initial BCC mesophase, which is about 1.2193 × 10−3. The value of 3⟨bybz⟩/b02 ≈ 1.5334 × 10−3 at t ≈ 50τR, however, remains about 2 orders of magnitude larger than that of 3⟨bxby⟩/b02 and 3⟨bxbz⟩/b02. This indicates that σyz does not completely relax but attains a minuscule but nonvanishing pseudoequilibrium value. This behavior is corroborated by our SCFT calculations. Using the obtained morphology of ordered ellipsoids from the SCMF simulations as input in the SCFT calculations, we find that the SCFT converges to essentially the same morphology (i.e., without thermal fluctuations of the domain shape) in a periodic, cuboidal box, as shown in the rightmost panel of Figure 2. The calculation of the stress tensor (see Supporting Information), however, shows that σyz ≈ 5.4198 × 10−2 ; i.e., the pseudometastable state is (n / V )k T

Figure 4. Final morphologies of positionally and orientationally ordered crystals of ellipsoids obtained after the step-shear deformation with various values of γ ≤ 1/2 and relaxation for 50τR. These averaged unit-cell morphologies are obtained by folding back the density from our SCMF simulations using the large cell (composed of 27 unit cells) into one unit cell. The aspect ratio and the orientation of the ellipsoidal domains can be controlled by the shear strain, γ. Upon increasing the shear strain to γ = 1, the process finally results in the formation of cylindrical domains along the y-axis, arranged on a square lattice in the xz-plane (SPC mesostructure). The red arrows mark the y-direction, along which the step-shear deformation is applied. χABN = 23 and fA = 7/32.

SCMF simulations with various values of γ ≤ 1/2, which are obtained at t = 50τR after the step-shear deformation is applied. These averaged unit-cell morphologies are obtained by folding back the density of the large simulation box, composed of 27 unit cells, into one unit cell. The original large-cell morphologies are shown in the Supporting Information. Both the aspect ratio and the orientation of the obtained ellipsoidal domains can be controlled by the shear strain, γ. Qualitatively, the larger the value of the shear strain is, the larger the aspect ratio of the ellipsoidal domains becomes. Upon increasing the shear strain to γ = 1, rather than forming the ellipsoids, the process finally leads to the formation of cylindrical domains along the y-axis, arranged on a square lattice in the xz-plane, i.e., square-packed cylinders (SPC), similar to what has been observed under continuous shear.21,36−38 Details about this process are shown in the Supporting Information. Figure 5a presents the morphologies from SCFT calculations, using the morphologies of SCMF simulations (see Figure 4) as initial, input morphologies. We find that all fabricated, positionally and orientationally ordered crystals of ellipsoids with different aspect ratios and orientations are pseudometastable. The morphologies from the SCFT calculations with various 25 values of γ for fA = 128 , i.e., a smaller composition, are presented in Figure 5b. The comparison between Figures 5a

B

not stress-free. Since

3⟨bybz⟩ b0 2

=

σyz 1 , 35 N (n / V )kBT

our SCMF

simulations and SCFT calculations quantitatively agree with each other. The inset of Figure 3 presents the relaxation of the overall chain conformations as quantified by 3(⟨Rey2⟩ − ⟨Rey2⟩∞)/Re02 and 3⟨ReyRez⟩/Re02, where the plateau value, ⟨Rey2⟩∞, is extracted from the simulation data at t = 50τR after the stepshear deformation. Unlike the bond-vector orientation, the endto-end vector distribution relaxes exponentially in time during the early stage, in accord with the prediction of the Rouse model for the disordered system. Importantly, the off-diagonal value, 3⟨ReyRez⟩/Re02, decreases by 2 orders of magnitude; the residual value indicates the slight deformation of the macromolecular conformations due to their anisotropic packing in the morphology of ellipsoidal domains. The diagonal value, 3(⟨Rey2⟩ − ⟨Rey2⟩∞)/Re02, only decreases by 1 order of magnitude after 3.33τR, and after 13.33τR, it decreases by only 2 orders of magnitude (data not shown in Figure 3). This protracted relaxation of both 3(⟨Rey2⟩ − ⟨Rey2⟩∞)/Re02 and D


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Figure 5. Morphologies from SCFT calculations with various values of γ for two different A volume fractions, i.e., (a) fA = 7/32 and (b) fA = 25/128, when χABN = 23. In (b), Lx = Ly = L* and Lz = mL* with L* = 1.8207Re0, where m = 1, 2, ..., 7.

Figure 6. (a) Asphericity, S, and (b) the orientation angle, β, of the obtained ellipsoids from SCFT calculations with various values of γ. The squares and circles are obtained using the morphologies in Figures 5a and 5b, respectively, whereas the solid lines are obtained from variable-cell-shape SCFT calculations39 using an increment of Δγ = 0.01. The two insets in (a) present the morphologies of ordered ellipsoids with γ = 0.5 for fA = 25/ 128 and fA = 7/32, respectively, and the asphericities of these two kinds of ordered ellipsoids are S = 0.022 24 and S = 0.068 87, respectively. In (b), the orientation angle, β, immediately after the step-shear affine deformation is also presented, i.e., the dashed line.

calculations39 using 0.01 as the increment of γ. At fixed value of γ, the larger ellipsoids with bigger fA are more aspherical. 7 For fA = 32 , the larger the value of γ is, the more ellipsoidal

and 5b qualitatively reveals that smaller values of fA not only reduce the size of the ellipsoidal domains but also alter their 1 shape (asphericity) at fixed value of γ ≤ 2 . The biggest

25

25

difference arises for shear strain γ = 1, where for fA = 128 a crystal of ellipsoids with β = 0° is formed because the reduced A volume fraction is insufficient to form cylindrical domains of the SPC mesostructure as the system is farther away from the sphere-to-cylinder boundary of the equilibrium phase diagram. An analogous tendency has been observed in shear flow.36 In order to quantitatively characterize how the shape and orientation of the obtained ellipsoids depend on the composition, fA, of the block copolymer and the step-shear deformation, γ, we compute the gyration tensor of the soft ellipsoids by SCFT calculations and obtain the asphericity, S, and orientation angle, β, via diagonalization. Details of the calculations are deferred to the Supporting Information. We observe that all domains in the unit cell have the same shape and orientation. Figures 6a and 6b display how the shape and orientation of the ellipsoids vary as a function of γ for two values of fA. The squares and circles are obtained using the morphologies in Figures 5a and 5b, respectively, whereas the solid lines are obtained from variable-cell-shape SCFT

the domain shape becomes while for fA = 128 ; however, the asphericity exhibits a nonmonotonous dependence on γ, and the maximum value, S ≈ 0.025 62, arises for γ = 0.69. This nonmonotonous dependence of the asphericity on γ highlights that the shape of the ellipsoids is dictated by the packing and interaction of the soft, deformable domains of the minority 7 block. Additionally, for fA = 32 , although the ellipsoidal morphologies can be obtained from SCFT calculations even for 0.7 < γ ≤ 0.8; in SCMF simulations, the formation of the ellipsoids can only be observed when γ ≤ 0.7 whereas the SPC morphology is formed when γ ≥ 0.8. This indicates that the starting state generated by the step-shear deformation is already located in the free-energy basin of the SPC morphology when γ ≥ 0.8. The orientation angle, β, of the ellipsoids in the starting state immediately after the step-shear deformation is independent from the volume fraction, fA. Similar to the asphericity, the orientation is also dictated by the packing and interaction of the soft ellipsoids. Figure 6b shows how the orientation aligns E


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and asphericity of the ellipsoidal domains formed by the minority component using densities from the SCMF simulations and the SCFT calculations, respectively, (iii) results about the stress tensors and free energies using the variable-cell-shape SCFT, (iv) results from the SCMF simulations and the SCFT calculations using large cells along with the results about the formation of square-packed cylinders (SPC) from SCMF simulations (PDF)

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CONCLUSIONS In summary, we propose and explore a versatile, facile, and efficient strategy to fabricate positionally and orientationally ordered crystals of soft ellipsoids in block copolymers via a step-shear deformation, which allows for low volume fraction of ellipsoids. Using both SCMF simulations and SCFT calculations, we have demonstrated that the fabricated crystals of soft ellipsoids with different shapes and orientations are pseudometastable, i.e., long-lived but not stress-free, and the shape and orientation can be tuned by the magnitude of the step-shear strain and the molecular architecture. The long lifetime is imparted either by keeping the step-shear strain or the protracted lattice relaxation via slippage in the absence of the external stress field after the molecular stress relaxes to a pseudoequilibrium value and thus provides ample opportunities for additional stabilization, e.g., by cooling below the glasstransition temperature, crystallization, or cross-linking. It should be noted here that when the step-shear strain is kept, the pseudometastable state is a local minimum of the freeenergy functional. The origin of the residual molecular stress stems from the fixed shape of the investigated system, which is not the optimal one for the ellipsoidal packing, such that the molecular stress can not completely relax but attains a minuscule, nonvanishing pseudoequilibrium value. In addition, the pseudometastable state should also be characterized by a ∂ 2F positive curvature, ∂γ 2 > 0, of the free energy, F, as a function

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

De-Wen Sun: 0000-0003-0693-2602 Marcus Müller: 0000-0002-7472-973X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge stimulating discussions with L. Schneider, Y. Z. Ren, M. Langenberg, G. J. Zhang, and H. J. Risselada. This work was supported by the Deutsche Forschungsgemeinschaft under Grant DFG Mu1674/14-1 and SFB937 TP A04. Simulations were performed at JSC Jülich, GWDG Göttingen, and HLRN Hannover/Berlin, Germany.

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of the step-shear strain γ. Whereas our study only focuses on a specific spherical packing, i.e., the BCC mesophase in a linear AB diblock copolymer melt, we expect that our strategy is robust and can be straightforwardly extended to other equilibrium sphere packings (e.g., the A15 spherical mesophase in conformationally asymmetric AB-type block copolymers, 40,41 binary mesocrystals in multiblock terpolymers,25,42 and the Frank− Kasper σ-phase in binary blend of two kinds of linear AB diblock copolymers with different chain lengths,26 in which the two different diblock copolymers may exhibit different responses to the step-shear deformation and their nonuniform distribution on the domain interfaces may increase the asphericity of the domains further) to fabricate a variety of positionally and orientationally ordered ellipsoidal packings. We hope that our observations will stimulate corresponding experimental efforts to confirm our predictions and test our strategy to reproducibly fabricate highly ordered ellipsoidal packings.

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AUTHOR INFORMATION

Corresponding Author

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b02060. Details about (i) single-chain-in-mean-field (SCMF) simulations, self-consistent field theory (SCFT) based on the standard Gaussian chain model, and the variablecell-shape SCFT, (ii) how to compute the orientation F


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Macromolecules

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