Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation
Field-theoretic simulations (FTS) provide an efficient technique for investigating fluctuation effects in block copolymer melts with numerous advantages over traditional particle-based simulations. For systems involving two components (i.e., A and B), the field-based Hamiltonian, <inline-formula&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-07-01
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| Series: | Polymers |
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| Online Access: | https://www.mdpi.com/2073-4360/13/15/2437 |
| _version_ | 1797525237567324160 |
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| author | Mark W. Matsen Thomas M. Beardsley |
| author_facet | Mark W. Matsen Thomas M. Beardsley |
| author_sort | Mark W. Matsen |
| collection | DOAJ |
| description | Field-theoretic simulations (FTS) provide an efficient technique for investigating fluctuation effects in block copolymer melts with numerous advantages over traditional particle-based simulations. For systems involving two components (i.e., A and B), the field-based Hamiltonian, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo stretchy="false">[</mo><msub><mi>W</mi><mo>−</mo></msub><mo>,</mo><msub><mi>W</mi><mo>+</mo></msub><mo stretchy="false">]</mo></mrow></mrow></semantics></math></inline-formula>, depends on a composition field, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mo>−</mo></msub><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">r</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, that controls the segregation of the unlike components and a pressure field, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mo>+</mo></msub><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">r</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, that enforces incompressibility. This review introduces researchers to a promising variant of FTS, in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mo>−</mo></msub><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">r</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> fluctuates while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mo>+</mo></msub><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">r</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> tracks its mean-field value. The method is described in detail for melts of AB diblock copolymer, covering its theoretical foundation through to its numerical implementation. We then illustrate its application for neat AB diblock copolymer melts, as well as ternary blends of AB diblock copolymer with its A- and B-type parent homopolymers. The review concludes by discussing the future outlook. To help researchers adopt the method, open-source code is provided that can be run on either central processing units (CPUs) or graphics processing units (GPUs). |
| first_indexed | 2024-03-10T09:09:55Z |
| format | Article |
| id | doaj.art-3440901b272246b0867fdc94d113e98a |
| institution | Directory Open Access Journal |
| issn | 2073-4360 |
| language | English |
| last_indexed | 2024-03-10T09:09:55Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Polymers |
| spelling | doaj.art-3440901b272246b0867fdc94d113e98a2023-11-22T06:02:54ZengMDPI AGPolymers2073-43602021-07-011315243710.3390/polym13152437Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point ApproximationMark W. Matsen0Thomas M. Beardsley1Department of Chemical Engineering, University of Waterloo, Waterloo, ON N2L 3G1, CanadaWaterloo Institute for Nanotechnology, University of Waterloo, Waterloo, ON N2L 3G1, CanadaField-theoretic simulations (FTS) provide an efficient technique for investigating fluctuation effects in block copolymer melts with numerous advantages over traditional particle-based simulations. For systems involving two components (i.e., A and B), the field-based Hamiltonian, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mi>f</mi></msub><mrow><mo stretchy="false">[</mo><msub><mi>W</mi><mo>−</mo></msub><mo>,</mo><msub><mi>W</mi><mo>+</mo></msub><mo stretchy="false">]</mo></mrow></mrow></semantics></math></inline-formula>, depends on a composition field, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mo>−</mo></msub><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">r</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, that controls the segregation of the unlike components and a pressure field, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mo>+</mo></msub><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">r</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, that enforces incompressibility. This review introduces researchers to a promising variant of FTS, in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mo>−</mo></msub><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">r</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> fluctuates while <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mo>+</mo></msub><mrow><mo stretchy="false">(</mo><mi mathvariant="bold">r</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> tracks its mean-field value. The method is described in detail for melts of AB diblock copolymer, covering its theoretical foundation through to its numerical implementation. We then illustrate its application for neat AB diblock copolymer melts, as well as ternary blends of AB diblock copolymer with its A- and B-type parent homopolymers. The review concludes by discussing the future outlook. To help researchers adopt the method, open-source code is provided that can be run on either central processing units (CPUs) or graphics processing units (GPUs).https://www.mdpi.com/2073-4360/13/15/2437block copolymer meltsfield-theoretic simulationsmolecular self-assemblyphase diagramsorder-disorder transitionsgyroid phase |
| spellingShingle | Mark W. Matsen Thomas M. Beardsley Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation Polymers block copolymer melts field-theoretic simulations molecular self-assembly phase diagrams order-disorder transitions gyroid phase |
| title | Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation |
| title_full | Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation |
| title_fullStr | Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation |
| title_full_unstemmed | Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation |
| title_short | Field-Theoretic Simulations for Block Copolymer Melts Using the Partial Saddle-Point Approximation |
| title_sort | field theoretic simulations for block copolymer melts using the partial saddle point approximation |
| topic | block copolymer melts field-theoretic simulations molecular self-assembly phase diagrams order-disorder transitions gyroid phase |
| url | https://www.mdpi.com/2073-4360/13/15/2437 |
| work_keys_str_mv | AT markwmatsen fieldtheoreticsimulationsforblockcopolymermeltsusingthepartialsaddlepointapproximation AT thomasmbeardsley fieldtheoreticsimulationsforblockcopolymermeltsusingthepartialsaddlepointapproximation |