Nucleation of Ordered Phases in Block Copolymers

Cheng, Xiuyuan; Li, Gang; Weinan, E; Zhang, Pingwen; Shi, An‐Chang

doi:10.1103/physrevlett.104.148301

Cited by 118 publications

(131 citation statements)

References 30 publications

(32 reference statements)

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“…Importantly, the model has been shown to provide a representation of copolymer thin films that is in good agreement with available experimental data (19,20). We rely on the string method to identify the MFEP between two metastable states on a free energy landscape (21)(22)(23)(24)(25)(26)(27)). An initial string is constructed by linear interpolation between the two end states.…”

Section: Model and Methodsmentioning

confidence: 62%

Molecular pathways for defect annihilation in directed self-assembly

Hur

Thapar

Ramírez-Hernández

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

140

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Over the last few years, the directed self-assembly of block copolymers by surface patterns has transitioned from academic curiosity to viable contender for commercial fabrication of nextgeneration nanocircuits by lithography. Recently, it has become apparent that kinetics, and not only thermodynamics, plays a key role for the ability of a polymeric material to self-assemble into a perfect, defect-free ordered state. Perfection, in this context, implies not more than one defect, with characteristic dimensions on the order of 5 nm, over a sample area as large as 100 cm 2 . In this work, we identify the key pathways and the corresponding free energy barriers for eliminating defects, and we demonstrate that an extraordinarily large thermodynamic driving force is not necessarily sufficient for their removal. By adopting a concerted computational and experimental approach, we explain the molecular origins of these barriers and how they depend on material characteristics, and we propose strategies designed to overcome them. The validity of our conclusions for industrially relevant patterning processes is established by relying on instruments and assembly lines that are only available at state-of-the-art fabrication facilities, and, through this confluence of fundamental and applied research, we are able to discern the evolution of morphology at the smallest relevant length scales-a handful of nanometers-and present a view of defect annihilation in directed self-assembly at an unprecedented level of detail.directed self-assembly | copolymer | defect | minimum free energy path | string method O ver the last decade, the directed self-assembly (DSA) of block copolymers has rapidly evolved from mere intellectual curiosity (1-4) to a potentially crucial step in the commercial fabrication of next-generation electronic circuits. Indeed, the characteristic length scale of ordered self-assembled copolymer domains is in the range of 5-50 nm. Furthermore, their size and shape can be manipulated through simple processing steps, thereby making them attractive for the production of semiconductor devices, nanofluidic devices, or high-density storage media (5, 6). The general idea behind copolymer DSA is that a surface patternchemical or topographic-can be used to guide the assembly of a polymeric material into an ordered, device-like structure that is free of defects. In so-called "density multiplication" patterning strategies (7,8), the spacing or pitch of the surface features can be much larger than the characteristic dimensions of the copolymer of interest. One can thus prepare coarse surface patterns, which are easier to create, and rely on the copolymer to self-assemble into features whose density is considerably larger. Fig. 1 shows a schematic representation of the process for obtaining a lamellar morphology on a stripe-patterned substrate under a one-to-three (or 3X) density multiplication strategy. Patterned stripes interact preferentially with one of the blocks and guide the assembly of thin copolymer films into ordered lam...

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Section: Model and Methodsmentioning

confidence: 62%

Molecular pathways for defect annihilation in directed self-assembly

Hur

Thapar

Ramírez-Hernández

et al. 2015

Proc. Natl. Acad. Sci. U.S.A.

140

View full text Add to dashboard Cite

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“…The SCFT approach is based on the assumption that coarse-grained polymer chains in dense melts are Gaussian (already mentioned as Flory's theorem [1]), and on the MF approximation which selects the dominant contribution in the appropriate partition function, thus neglecting uctuations. The SCFT can be applied to a variety of diblock copolymer phenomena, for example Cheng et al [20] have studied the nucleation of ordered phases and the minimum energy paths.…”

Section: A-b-b Melts Can Self-assemble Inmentioning

confidence: 99%

Diblock Copolymer Melt in Spherical Unit Cells of Higher Dimensionalities

et al. 2012

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Using self-consistent eld theory in spherical unit cells of various dimensionality, D = 1, 2, 3, and 4, we calculate phase diagram of a diblock, A-b-B, copolymer melt in 4-dimensional space, d = 4. The phase diagram is parameterized by the chain composition, f , and incompatibility between A and B, quantied by the product χN . We predict 4 stable nanophases: layers, cylinders, 3D spherical cells, and 4D spherical cells. We also calculate orderdisorder and orderorder transition lines. In the strong segregation limit, that is for large χN , the orderorder transition compositions are determined by the strong segregation theory in its simplest form, for D = 1, 2, 3, and 4.

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“…Several kinds of stable ordered phases, such as the lamellar phase, cylinder phase, and spherical phase, etc., have been discovered in experiments, and lots of work has been done to determine the phase diagram [33,23]. For the second issue, people try to understand the dynamical behaviors of the diblock copolymers, for example, the interfacial motion [19,24], spinodal decomposition phenomena [28], and nucleation events between different kinds of ordered phases [31,21,5].In this paper, we study the nucleation rate calculation for diblock copolymers between the lamellar and cylinder phases with a Landau-Brazovskii energy functional, and the random fluctuation is assumed to be of stochastic Cahn-Hilliard type. In this model, it is supposed that diblock copolymer melts contain lots of chains, each of which contains monomers of types A and B linked together (see Figure 1.1).…”

mentioning

confidence: 99%

Nucleation Rate Calculation for the Phase Transition of Diblock Copolymers under Stochastic Cahn--Hilliard Dynamics

Li¹,

Zhang²,

Zhang³

2013

Multiscale Model. Simul.

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Abstract. We focus on the nucleation rate calculation for diblock copolymers by studying the two-dimensional stochastic Cahn-Hilliard dynamics with a Landau-Brazovskii energy functional. To do this, we devise the string method to compute the minimal energy path of nucleation events and the gentlest ascent dynamics to locate the saddle point on the path in Fourier space. Both methods are combined with the semi-implicit spectral method and hence are very effective. We derive the nucleation rate formula in the infinite-dimensional case and prove the convergence under numerical discretizations. The computation of the determinant ratio is also discussed for obtaining the rate. The algorithm is successfully applied to investigate the nucleation from the lamellar phase to the cylinder phase in the mean field theory for diblock copolymer melts. The comparison with projected stochastic Allen-Cahn dynamics is also discussed.Key words. Cahn-Hilliard equation, diblock copolymer, Landau-Brazovskii energy, string method, nucleation rate, functional determinant ratio AMS subject classification. 65C20 DOI. 10.1137/120876307 1. Introduction. In the past few decades, diblock copolymers [22,12] have been extensively investigated due to their interesting physical and chemical features as well as widespread applications in material science [13,6]. Roughly speaking, the studies of copolymers mainly focus on two kinds of issues: determining equilibrium phases and studying the dynamical behaviors. For the first issue, it has been known for a long time that the diblock copolymers admit various ordered phase separation in the microscale. Several kinds of stable ordered phases, such as the lamellar phase, cylinder phase, and spherical phase, etc., have been discovered in experiments, and lots of work has been done to determine the phase diagram [33,23]. For the second issue, people try to understand the dynamical behaviors of the diblock copolymers, for example, the interfacial motion [19,24], spinodal decomposition phenomena [28], and nucleation events between different kinds of ordered phases [31,21,5].In this paper, we study the nucleation rate calculation for diblock copolymers between the lamellar and cylinder phases with a Landau-Brazovskii energy functional, and the random fluctuation is assumed to be of stochastic Cahn-Hilliard type. In this model, it is supposed that diblock copolymer melts contain lots of chains, each of which contains monomers of types A and B linked together (see Figure 1.1). On each chain, the composition ratios of these two kinds of monomers are f A and f B , respectively, which satisfy f A + f B = 1. The order parameters φ A (r) and φ B (r) characterize

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Nucleation of Ordered Phases in Block Copolymers

Cited by 118 publications

References 30 publications

Molecular pathways for defect annihilation in directed self-assembly

Molecular pathways for defect annihilation in directed self-assembly

Diblock Copolymer Melt in Spherical Unit Cells of Higher Dimensionalities

Nucleation Rate Calculation for the Phase Transition of Diblock Copolymers under Stochastic Cahn--Hilliard Dynamics

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