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Phase-Separating Binary Polymer Mixtures: The Degeneracy of the Virial Coefficients and Their Extraction from Phase Diagrams
[Image: see text] The Edmond–Ogston model for phase separation in binary polymer mixtures is based on a truncated virial expansion of the Helmholtz free energy up to the second-order terms in the concentration of the polymers. The second virial coefficients (B(11), B(12), B(22)) are the three parame...
Autores principales: | , , |
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Formato: | Online Artículo Texto |
Lenguaje: | English |
Publicado: |
American Chemical Society
2021
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7992149/ https://www.ncbi.nlm.nih.gov/pubmed/33778298 http://dx.doi.org/10.1021/acsomega.1c00450 |
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author | Bot, Arjen Dewi, Belinda P. C. Venema, Paul |
author_facet | Bot, Arjen Dewi, Belinda P. C. Venema, Paul |
author_sort | Bot, Arjen |
collection | PubMed |
description | [Image: see text] The Edmond–Ogston model for phase separation in binary polymer mixtures is based on a truncated virial expansion of the Helmholtz free energy up to the second-order terms in the concentration of the polymers. The second virial coefficients (B(11), B(12), B(22)) are the three parameters of the model. Analytical solutions are presented for the critical point and the spinodal in terms of molar concentrations. The calculation of the binodal is simplified by splitting the problem into a part that can be solved analytically and a (two-dimensional) problem that generally needs to be solved numerically, except in some specific cases. The slope of the tie-lines is identified as a suitable parameter that can be varied between two well-defined limits (close to and far away from the critical point) to perform the numerical part of the calculation systematically. Surprisingly, the analysis reveals a degenerate behavior within the model in the sense that a critical point or tie-line corresponds to an infinite set of triplets of second virial coefficients (B(11), B(12), B(22)). Since the Edmond–Ogston model is equivalent to the Flory–Huggins model up to the second order of the expansion in the concentrations, this degeneracy is also present in the Flory–Huggins model. However, as long as the virial coefficients predict the correct critical point, the shape of the binodal is relatively insensitive to the specific choice of the virial coefficients, except in a narrow range of values for the cross-virial coefficient B(12). |
format | Online Article Text |
id | pubmed-7992149 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2021 |
publisher | American Chemical Society |
record_format | MEDLINE/PubMed |
spelling | pubmed-79921492021-03-26 Phase-Separating Binary Polymer Mixtures: The Degeneracy of the Virial Coefficients and Their Extraction from Phase Diagrams Bot, Arjen Dewi, Belinda P. C. Venema, Paul ACS Omega [Image: see text] The Edmond–Ogston model for phase separation in binary polymer mixtures is based on a truncated virial expansion of the Helmholtz free energy up to the second-order terms in the concentration of the polymers. The second virial coefficients (B(11), B(12), B(22)) are the three parameters of the model. Analytical solutions are presented for the critical point and the spinodal in terms of molar concentrations. The calculation of the binodal is simplified by splitting the problem into a part that can be solved analytically and a (two-dimensional) problem that generally needs to be solved numerically, except in some specific cases. The slope of the tie-lines is identified as a suitable parameter that can be varied between two well-defined limits (close to and far away from the critical point) to perform the numerical part of the calculation systematically. Surprisingly, the analysis reveals a degenerate behavior within the model in the sense that a critical point or tie-line corresponds to an infinite set of triplets of second virial coefficients (B(11), B(12), B(22)). Since the Edmond–Ogston model is equivalent to the Flory–Huggins model up to the second order of the expansion in the concentrations, this degeneracy is also present in the Flory–Huggins model. However, as long as the virial coefficients predict the correct critical point, the shape of the binodal is relatively insensitive to the specific choice of the virial coefficients, except in a narrow range of values for the cross-virial coefficient B(12). American Chemical Society 2021-03-11 /pmc/articles/PMC7992149/ /pubmed/33778298 http://dx.doi.org/10.1021/acsomega.1c00450 Text en © 2021 The Authors. Published by American Chemical Society Permits non-commercial access and re-use, provided that author attribution and integrity are maintained; but does not permit creation of adaptations or other derivative works (https://creativecommons.org/licenses/by-nc-nd/4.0/). |
spellingShingle | Bot, Arjen Dewi, Belinda P. C. Venema, Paul Phase-Separating Binary Polymer Mixtures: The Degeneracy of the Virial Coefficients and Their Extraction from Phase Diagrams |
title | Phase-Separating Binary Polymer Mixtures: The Degeneracy
of the Virial Coefficients and Their Extraction from Phase Diagrams |
title_full | Phase-Separating Binary Polymer Mixtures: The Degeneracy
of the Virial Coefficients and Their Extraction from Phase Diagrams |
title_fullStr | Phase-Separating Binary Polymer Mixtures: The Degeneracy
of the Virial Coefficients and Their Extraction from Phase Diagrams |
title_full_unstemmed | Phase-Separating Binary Polymer Mixtures: The Degeneracy
of the Virial Coefficients and Their Extraction from Phase Diagrams |
title_short | Phase-Separating Binary Polymer Mixtures: The Degeneracy
of the Virial Coefficients and Their Extraction from Phase Diagrams |
title_sort | phase-separating binary polymer mixtures: the degeneracy
of the virial coefficients and their extraction from phase diagrams |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7992149/ https://www.ncbi.nlm.nih.gov/pubmed/33778298 http://dx.doi.org/10.1021/acsomega.1c00450 |
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