Self-Consistent Field Theory for Melts of Low-Molecular-Weight Diblock Copolymer
TL;DR: In this paper, the authors apply self-consistent field theory (SCFT) to discrete polymer chains consisting of a finite number of beads, N, joined together by freely jointed bonds of arbitrary potential, b(R).
Abstract: This paper applies self-consistent field theory (SCFT) to discrete polymer chains consisting of a finite number of beads, N, joined together by freely jointed bonds of arbitrary potential, b(R). The numerics of this SCFT can be performed efficiently using spectral or pseudospectral algorithms, permitting its application to complex morphologies. To demonstrate its effectiveness, we examine diblock copolymer melts where the polymer bonds have a fixed length, a, and the nonbonded interactions have a finite range, σ, with a strength controlled by the standard Flory−Huggins χ parameter. Although the results reduce to those of the usual SCFT for Gaussian chains in the limit of large N and small χ, there are some notable differences for short chains with strong interactions. The most significant involves the internal interfaces, which in turn affects the size of the domains. Furthermore, the finite range of the nonbonded interactions, necessary to properly treat the internal interfaces, causes a noticeable shift of the ODT toward larger χN .A sχ becomes very large, particularly at small N, the finite extensibility of the freely jointed chains restricts the size of the domains, which leads to a preference for the lamellar phase.
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01 Mar 1996
TL;DR: In this paper, a mean-field phase diagram for conformationally symmetric diblock melts using the standard Gaussian polymer model is presented, which traverses the weak- to strong-segregation regimes, is free of traditional approximations.
Abstract: A mean-field phase diagram for conformationally symmetric diblock melts using the standard Gaussian polymer model is presented. Our calculation, which traverses the weak- to strong-segregation regimes, is free of traditional approximations. Regions of stability are determined for disordered (DIS) melts and for ordered structures including lamellae (L), hexagonally packed cylinders (H), body-centered cubic spheres (QIm3m), close-packed spheres (CPS), and the bicontinuous cubic network with Ia3d symmetry (QIa3d). The CPS phase exists in narrow regions along the order−disorder transition for χN ≥ 17.67. Results suggest that the QIa3d phase is not stable above χN ∼ 60. Along the L/QIa3d phase boundaries, a hexagonally perforated lamellar (HPL) phase is found to be nearly stable. Our results for the bicontinuous Pn3m cubic (QPn3m) phase, known as the OBDD, indicate that it is an unstable structure in diblock melts. Earlier approximation schemes used to examine mean-field behavior are reviewed, and compa...
1,256 citations
TL;DR: In this article, the authors describe the design and synthesis of a series of lamella-forming, silicon-containing block copolymers (Si-BCPs) and evaluation of these materials as potential candidates for lithographic applications.
Abstract: This report describes the design and synthesis of a series of lamella-forming, silicon-containing block copolymers (Si-BCPs) and evaluation of these materials as potential candidates for lithographic applications. The interaction parameter χ of each Si-BCP is measured by both the mean-field theory predicted order-disorder transition and by analysis of X-ray scattering profiles. The introduction of more-polar methoxy and less-polar methylsilyl moieties increases χ to about 2–3 times that of the reference material, poly(styrene-block−4-trimethylsilylstyrene). The incremental increases appear to be essentially additive in this family of block copolymers, suggesting that improvements in χ can be predicted from appropriate monomer choice. Perpendicularly oriented thin-films of the ordered Si-BCPs generated by thermally annealing between two “neutral” polymeric surfaces and developed by etching on commercial RIE equipment show excellent image fidelity. These images demonstrate the excellent etch contrast of the Si-BCPs and document improvements in pattern fidelity that are realized with more strongly segregated BCPs. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2015, 53, 344–352
128 citations
TL;DR: The importance of end groups in high χ-low N block molecules is shown, where block-copolymer-like self-assembly in the authors' UPy-functionalized oligodimethylsiloxanes relies upon the dominance of phase separation effects over directional end group aggregation.
Abstract: Monodisperse oligodimethylsiloxanes end-functionalized with the hydrogen-bonding ureidopyrimidinone (UPy) motif undergo phase separation between their aromatic end groups and dimethylsiloxane midblocks to form ordered nanostructures with domain spacings of 1.13 results in disorder, showing importance of molecular monodispersity for ultrasmall ordered phase separation. In contrast, oligodimethylsiloxanes end-functionalized with an O-benzylated UPy derivative self-assemble into lamellar nanostructures regardless of volume fraction because of the strong preference of the end groups to aggreg...
89 citations
TL;DR: A coarse-graining methodology for mapping specific block copolymer systems to bead-spring particle-based models is developed, and a semi-empirical correlation between the experimentally determined Flory-Huggins parameter χ and the interaction of the model potential is established.
Abstract: In this contribution, we develop a coarse-graining methodology for mapping specific block copolymer systems to bead-spring particle-based models. We map the constituent Kuhn segments to Lennard-Jones particles, and establish a semi-empirical correlation between the experimentally determined Flory-Huggins parameter χ and the interaction of the model potential. For these purposes, we have performed an extensive set of isobaric–isothermal Monte Carlo simulations of binary mixtures of Lennard-Jones particles with the same size but with asymmetric energetic parameters. The phase behavior of these monomeric mixtures is then extended to chains with finite sizes through theoretical considerations. Such a top-down coarse-graining approach is important from a computational point of view, since many characteristic features of block copolymer systems are on time and length scales which are still inaccessible through fully atomistic simulations. We demonstrate the applicability of our method for generating parameters by reproducing the morphology diagram of a specific diblock copolymer, namely, poly(styrene-b-methyl methacrylate), which has been extensively studied in experiments.
76 citations
TL;DR: In this paper, a series of low-molecular weight (MW) rod-coil block copolymers (BCPs) with the rod blocks of different lengths (LRods) were studied.
Abstract: A remarkably rich variety of nanophase-separated structures and various order–order transitions are observed in a series of low-molecular weight (MW) rod–coil block copolymers (BCPs) with the rod blocks of different lengths (LRod’s). The rod–coil diblock copolymer studied herein is poly(dimethylsiloxane)-b-poly{2,5-bis[(4-methoxyphenyl)oxycarbonyl]styrene} (PDMS-b-PMPCS), in which PMPCS is a rod-like polymer and exhibits an MW-dependent liquid crystalline (LC) phase behavior. When the polymerization degree of the PMPCS rod block (NRod) is less than 32 (LRod < 8 nm), the PMPCS block is always amorphous in the entire temperature range. And the corresponding PDMS-b-PMPCS BCPs with NRod from 11 to 29 and the volume fraction of the PMPCS rod (fRod) from 43% to 67% self-assemble into various equilibrium nanostructures after annealed at temperatures above the glass transition temperatures of the PMPCS blocks. When NRod = 11 and fRod = 43%, the BCP forms a lamellar structure (LAM); when NRod = 15 and fRod = 51%, ...
69 citations
References
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3,434 citations
TL;DR: In this paper, it was shown that the abscissa of the intersecting point of the 1/v2 and 1 /v branches, when extrapolated to the transition region, is connected in a characteristic way with the statistical chain element or the length of persistance, which represent two different measures for the stiffness of the chain molecule.
Abstract: In the X-ray scattering curve of chain-molecules in solution, essentially four regions may be distinguished as is shown by a more detailed mathematical discussion:
(1)The scattering at the smallest angles, showing an approximately Gaussean course, whose ordinate, when extrapolated to zero angle, is proportional to the square of the mol. weight. It is due to the diffraction effect of the whole coiled chain-molecule.
(2)This is followed by a region of v-values where the scattering falls off as 1/v2. It is due to parts of the chain molecule which themselves already occur in all possible orientations.
(3)After a transition region (which is also accounted for by the theory) there follows a scattering which is proportional to 1/v. It is to be considered as the scattering of mostly straight parts of the molecule.
(4)The scattering at large angles, caused by smaller parts of the molecule (monomer group and single atoms), which at the same time indicates the periodicity (= length of structural unit in the direction of the chain axis) by a step, at an angle corresponding to Bragg's law. Its sharpness, too, is a measure for the coiling of the molecule.
It can be shown that the abscissa of the intersecting point of the 1/v2 and 1/v branches, when extrapolated to the transition region, is connected in a characteristic way with the statistical chain element or the “length of persistance”, which represent two different measures for the stiffness of the chain molecule.
Measurements with polyvinyl bromide lead to a length of persistance of 11 ± 1 A, which agrees satisfactorily with the value of 13 A for polyvinyl chloride as derived from the birefringence of flow. At large angles two weak steps were found, one of which corresponds to the monomer length of 2.56 A.
Measurements on solutions of cellulose nitrate gave a curve ∼ 1/v without any steeper ascent up to the smallest angles, corresponding to 300 A. We have, therefore, to conclude that the chain-element of cellulose nitrate is in any case larger than 150 A, which again is in agreement with the results of other physical measurements.
Though only a small number of experimental results is available so far, it seems to be established that the procedure yields a direct evaluation of the shape of chain molecules in solution and, in this respect, appears superior to other methods. However, the experimental difficulties are considerable.
1,668 citations
TL;DR: In this article, a mean-field phase diagram for conformationally symmetric diblock melts using the standard Gaussian polymer model is presented, which traverses the weak- to strong-segregation regimes, is free of traditional approximations.
Abstract: A mean-field phase diagram for conformationally symmetric diblock melts using the standard Gaussian polymer model is presented. Our calculation, which traverses the weak- to strong-segregation regimes, is free of traditional approximations. Regions of stability are determined for disordered (DIS) melts and for ordered structures including lamellae (L), hexagonally packed cylinders (H), body-centered cubic spheres (QIm3m), close-packed spheres (CPS), and the bicontinuous cubic network with Ia3d symmetry (QIa3d). The CPS phase exists in narrow regions along the order−disorder transition for χN ≥ 17.67. Results suggest that the QIa3d phase is not stable above χN ∼ 60. Along the L/QIa3d phase boundaries, a hexagonally perforated lamellar (HPL) phase is found to be nearly stable. Our results for the bicontinuous Pn3m cubic (QPn3m) phase, known as the OBDD, indicate that it is an unstable structure in diblock melts. Earlier approximation schemes used to examine mean-field behavior are reviewed, and compa...
1,506 citations
01 Mar 1996
TL;DR: In this paper, a mean-field phase diagram for conformationally symmetric diblock melts using the standard Gaussian polymer model is presented, which traverses the weak- to strong-segregation regimes, is free of traditional approximations.
Abstract: A mean-field phase diagram for conformationally symmetric diblock melts using the standard Gaussian polymer model is presented. Our calculation, which traverses the weak- to strong-segregation regimes, is free of traditional approximations. Regions of stability are determined for disordered (DIS) melts and for ordered structures including lamellae (L), hexagonally packed cylinders (H), body-centered cubic spheres (QIm3m), close-packed spheres (CPS), and the bicontinuous cubic network with Ia3d symmetry (QIa3d). The CPS phase exists in narrow regions along the order−disorder transition for χN ≥ 17.67. Results suggest that the QIa3d phase is not stable above χN ∼ 60. Along the L/QIa3d phase boundaries, a hexagonally perforated lamellar (HPL) phase is found to be nearly stable. Our results for the bicontinuous Pn3m cubic (QPn3m) phase, known as the OBDD, indicate that it is an unstable structure in diblock melts. Earlier approximation schemes used to examine mean-field behavior are reviewed, and compa...
1,256 citations
TL;DR: Using self-consistent field theory, microphases of diblock copolymers are examined and a stable gyroid phase is found which occurs between the lamellar and hexagonal ones and terminates at a triple point.
Abstract: Using self-consistent field theory, we examine microphases of diblock copolymers and find, in addition to lamellar, hexagonal, and cubic phases, a stable gyroid phase which occurs between the lamellar and hexagonal ones. It terminates at a triple point, with a lamellar to hexagonal transition occurring in the weak-segregation limit. Other phases of experimental interest are studied, and we describe the regions in which they are most nearly stable.
1,235 citations