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Qiao L, Giannakou M, Schmid F. An Efficient and Accurate SCF Algorithm for Block Copolymer Films and Brushes Using Adaptive Discretizations. Polymers (Basel) 2024; 16:1228. [PMID: 38732697 PMCID: PMC11085556 DOI: 10.3390/polym16091228] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/15/2024] [Revised: 04/24/2024] [Accepted: 04/25/2024] [Indexed: 05/13/2024] Open
Abstract
Self-consistent field (SCF) theory serves as a robust tool for unraveling the intricate behavior exhibited by soft polymeric materials. However, the accuracy and efficiency of SCF calculations are crucially dependent on the numerical methods employed for system discretization and equation-solving. Here, we introduce a simple three dimensional SCF algorithm that uses real-space methods and adaptive discretization, offering improved accuracy and efficiency for simulating polymeric systems at surfaces. Our algorithm's efficacy is demonstrated through simulations of two distinct polymeric systems, namely, block copolymer (BCP) films and polymer brushes. By enhancing spatial resolution in regions influenced by external forces and employing finer contour discretization at grafting chain ends, we achieve significantly more accurate results at very little additional cost, enabling the study of 3D polymeric systems that were previously computationally challenging. To facilitate the widespread use of the algorithm, we have made our 1D-3D SCF code publicly available.
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Affiliation(s)
- Le Qiao
- Institut für Physik, Johannes Gutenberg-Universität Mainz, D55099 Mainz, Germany;
| | | | - Friederike Schmid
- Institut für Physik, Johannes Gutenberg-Universität Mainz, D55099 Mainz, Germany;
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2
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Zhu Y, Huang C, Zhang L, Andelman D, Man X. The Process-Directed Self-Assembly of Block Copolymer Particles. Macromol Rapid Commun 2023; 44:e2300176. [PMID: 37071857 DOI: 10.1002/marc.202300176] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/28/2023] [Revised: 04/13/2023] [Indexed: 04/20/2023]
Abstract
The kinetic paths of structural evolution and formation of block copolymer (BCP) particles are explored using dynamic self-consistent field theory (DSCFT). It is shown that the process-directed self-assembly of BCP immersed in a poor solvent leads to the formation of striped ellipsoids, onion-like particles and double-spiral lamellar particles. The theory predicts a reversible path of shape transition between onion-like particles and striped ellipsoidal ones by regulating the temperature (related to the Flory-Huggins parameter between the two components of BCP, χAB ) and the selectivity of solvent toward one of the two BCP components. Furthermore, a kinetic path of shape transition from onion-like particles to double-spiral lamellar particles, and then back to onion-like particles is demonstrated. By investigating the inner-structural evolution of a BCP particle, it is identified that changing the intermediate bi-continuous structure into a layered one is crucial for the formation of striped ellipsoidal particles. Another interesting finding is that the formation of onion-like particles is characterized by a two-stage microphase separation. The first is induced by the solvent preference, and the second is controlled by the thermodynamics. The findings lead to an effective way of tailoring nanostructure of BCP particles for various industrial applications.
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Affiliation(s)
- Yanyan Zhu
- Center of Soft Matter Physics and its Applications, School of Physics, Beihang University, Beijing, 100191, China
| | - Changhang Huang
- Center of Soft Matter Physics and its Applications, School of Physics, Beihang University, Beijing, 100191, China
| | - Liangshun Zhang
- Shanghai Key Laboratory of Advanced Polymeric Materials, Key Laboratory for Ultrafine Materials of Ministry of Education, School of Materials Science and Engineering, East China University of Science and Technology, Shanghai, 200237, China
| | - David Andelman
- School of Physics and Astronomy, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel
| | - Xingkun Man
- Center of Soft Matter Physics and its Applications, School of Physics, Beihang University, Beijing, 100191, China
- Peng Huanwu Collaborative Center for Research and Education, Beihang University, Beijing, 100191, China
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Te Vrugt M, Wittkowski R. Perspective: New directions in dynamical density functional theory. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 35:041501. [PMID: 35917827 DOI: 10.1088/1361-648x/ac8633] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/23/2022] [Accepted: 08/01/2022] [Indexed: 06/15/2023]
Abstract
Classical dynamical density functional theory (DDFT) has become one of the central modeling approaches in nonequilibrium soft matter physics. Recent years have seen the emergence of novel and interesting fields of application for DDFT. In particular, there has been a remarkable growth in the amount of work related to chemistry. Moreover, DDFT has stimulated research on other theories such as phase field crystal models and power functional theory. In this perspective, we summarize the latest developments in the field of DDFT and discuss a variety of possible directions for future research.
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Affiliation(s)
- Michael Te Vrugt
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany
| | - Raphael Wittkowski
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany
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4
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Schmid F. Understanding and Modeling Polymers: The Challenge of Multiple Scales. ACS POLYMERS AU 2022. [DOI: 10.1021/acspolymersau.2c00049] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Affiliation(s)
- Friederike Schmid
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128Mainz, Germany
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5
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Müller M. Memory in the relaxation of a polymer density modulation. J Chem Phys 2022; 156:124902. [DOI: 10.1063/5.0084602] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Marcus Müller
- Institute for Theoretical Physics, Georg August University Gottingen Faculty of Physics, Germany
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Li B, Daoulas K, Schmid F. Dynamic coarse-graining of polymer systems using mobility functions. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 33:194004. [PMID: 33690176 DOI: 10.1088/1361-648x/abed1b] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/23/2021] [Accepted: 03/09/2021] [Indexed: 06/12/2023]
Abstract
We propose a dynamic coarse-graining (CG) scheme for mapping heterogeneous polymer fluids onto extremely CG models in a dynamically consistent manner. The idea is to use as target function for the mapping a wave-vector dependent mobility function derived from the single-chain dynamic structure factor, which is calculated in the microscopic reference system. In previous work, we have shown that dynamic density functional calculations based on this mobility function can accurately reproduce the order/disorder kinetics in polymer melts, thus it is a suitable starting point for dynamic mapping. To enable the mapping over a range of relevant wave vectors, we propose to modify the CG dynamics by introducing internal friction parameters that slow down the CG monomer dynamics on local scales, without affecting the static equilibrium structure of the system. We illustrate and discuss the method using the example of infinitely long linear Rouse polymers mapped onto ultrashort CG chains. We show that our method can be used to construct dynamically consistent CG models for homopolymers with CG chain lengthN= 4, whereas for copolymers, longer CG chain lengths are necessary.
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Affiliation(s)
- Bing Li
- Institut für Physik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany
| | - Kostas Daoulas
- Max-Planck Institut für Polymerforschung, Ackermannweg 10, 55128 Mainz, Germany
| | - Friederike Schmid
- Institut für Physik, Johannes Gutenberg-Universität Mainz, 55099 Mainz, Germany
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Liu W, Wu J, Zhu H, He C, Ngai T. A facile evanescent-field imaging approach for monitoring colloidal gel evolution near a surface. SOFT MATTER 2021; 17:4006-4010. [PMID: 33881131 DOI: 10.1039/d1sm00331c] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
A facile evanescent-field imaging approach is developed to probe the aggregation behavior of near-wall colloids/clusters during colloidal gel evolution. Total internal reflection microscope (TIRM) images are directly utilized to access the structural relaxation time via density-fluctuation theory. The behaviors of cluster-cluster aggregation and physical aging of the colloidal gel networks are resolved in both time and space under fractal scaling criteria.
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Affiliation(s)
- Wei Liu
- College of Chemistry and Environmental Engineering, Shenzhen University, Shenzhen 518060, China. and Key Laboratory of Optoelectronic Devices and Systems of Ministry of Education and Guangdong Province, College of Optoelectronic Engineering, Shenzhen University, Shenzhen 518060, China and College of Food Engineering and Biotechnology, Hanshan Normal University, Chaozhou 521041, China and Department of Chemistry, The Chinese University of Hong Kong, N.T., Shatin, Hong Kong, China.
| | - Jiahao Wu
- Department of Chemistry, The Chinese University of Hong Kong, N.T., Shatin, Hong Kong, China.
| | - Hui Zhu
- College of Food Engineering and Biotechnology, Hanshan Normal University, Chaozhou 521041, China
| | - Chuanxin He
- College of Chemistry and Environmental Engineering, Shenzhen University, Shenzhen 518060, China.
| | - To Ngai
- Department of Chemistry, The Chinese University of Hong Kong, N.T., Shatin, Hong Kong, China.
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Rottler J, Müller M. Kinetic Pathways of Block Copolymer Directed Self-Assembly: Insights from Efficient Continuum Modeling. ACS NANO 2020; 14:13986-13994. [PMID: 32909745 DOI: 10.1021/acsnano.0c06433] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
We introduce a computationally efficient continuum technique to simulate the complex kinetic pathways of block copolymer self-assembly. Subdiffusive chain dynamics is taken into account via nonlocal Onsager coefficients. An application to directed self-assembly of thin films of diblock copolymers on patterned substrates reveals the conditions under which experimentally observed metastable structures intervene in the desired thin-film morphology. The approach generalizes easily to multiblock copolymers and more complex guiding patterns on the substrate, and its efficiency allows for the systematic optimization of guiding patterns and process conditions.
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Affiliation(s)
- Jörg Rottler
- Department of Physics and Astronomy and Quantum Matter Institute, University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
| | - Marcus Müller
- Institute for Theoretical Physics, Georg-August-Universität Göttingen, 37077 Göttingen, Germany
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Schmid F, Li B. Dynamic Self-Consistent Field Approach for Studying Kinetic Processes in Multiblock Copolymer Melts. Polymers (Basel) 2020; 12:polym12102205. [PMID: 32992992 PMCID: PMC7601758 DOI: 10.3390/polym12102205] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/02/2020] [Revised: 09/21/2020] [Accepted: 09/22/2020] [Indexed: 11/24/2022] Open
Abstract
The self-consistent field theory is a popular and highly successful theoretical framework for studying equilibrium (co)polymer systems at the mesoscopic level. Dynamic density functionals allow one to use this framework for studying dynamical processes in the diffusive, non-inertial regime. The central quantity in these approaches is the mobility function, which describes the effect of chain connectivity on the nonlocal response of monomers to thermodynamic driving fields. In a recent study, one of us and coworkers have developed a method to systematically construct mobility functions from reference fine-grained simulations. Here we focus on melts of linear chains in the Rouse regime and show how the mobility functions can be calculated semi-analytically for multiblock copolymers with arbitrary sequences without resorting to simulations. In this context, an accurate approximate expression for the single-chain dynamic structure factor is derived. Several limiting regimes are discussed. Then we apply the resulting density functional theory to study ordering processes in a two-length scale block copolymer system after instantaneous quenches into the ordered phase. Different dynamical regimes in the ordering process are identified: at early times, the ordering on short scales dominates; at late times, the ordering on larger scales takes over. For large quench depths, the system does not necessarily relax into the true equilibrium state. Our density functional approach could be used for the computer-assisted design of quenching protocols in order to create novel nonequilibrium materials.
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