General approach for solving the density gradient theory in the interfacial tension calculations
Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite...
Ausführliche Beschreibung
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| Autor*in: |
Liang, Xiaodong [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2017transfer abstract |
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| Schlagwörter: |
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| Umfang: |
12 |
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| Übergeordnetes Werk: |
Enthalten in: Fabrication and compressive behaviour of an aluminium foam composite - Li, Yong-gang ELSEVIER, 2015, an international journal, New York, NY [u.a.] |
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| Übergeordnetes Werk: |
volume:451 ; year:2017 ; day:15 ; month:11 ; pages:79-90 ; extent:12 |
| Links: |
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| DOI / URN: |
10.1016/j.fluid.2017.07.021 |
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ELV030502713 |
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| 245 | 1 | 0 | |a General approach for solving the density gradient theory in the interfacial tension calculations |
| 264 | 1 | |c 2017transfer abstract | |
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| 520 | |a Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. | ||
| 520 | |a Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. | ||
| 650 | 7 | |a Density gradient theory |2 Elsevier | |
| 650 | 7 | |a CPA |2 Elsevier | |
| 650 | 7 | |a Interfacial tension |2 Elsevier | |
| 650 | 7 | |a PC-SAFT |2 Elsevier | |
| 650 | 7 | |a Direct optimization |2 Elsevier | |
| 700 | 1 | |a Michelsen, Michael Locht |4 oth | |
| 773 | 0 | 8 | |i Enthalten in |n Science Direct |a Li, Yong-gang ELSEVIER |t Fabrication and compressive behaviour of an aluminium foam composite |d 2015 |d an international journal |g New York, NY [u.a.] |w (DE-627)ELV013241125 |
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10.1016/j.fluid.2017.07.021 doi GBVA2017012000002.pica (DE-627)ELV030502713 (ELSEVIER)S0378-3812(17)30289-3 DE-627 ger DE-627 rakwb eng 660 540 660 DE-600 540 DE-600 670 VZ 540 VZ 630 VZ Liang, Xiaodong verfasserin aut General approach for solving the density gradient theory in the interfacial tension calculations 2017transfer abstract 12 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. Density gradient theory Elsevier CPA Elsevier Interfacial tension Elsevier PC-SAFT Elsevier Direct optimization Elsevier Michelsen, Michael Locht oth Enthalten in Science Direct Li, Yong-gang ELSEVIER Fabrication and compressive behaviour of an aluminium foam composite 2015 an international journal New York, NY [u.a.] (DE-627)ELV013241125 volume:451 year:2017 day:15 month:11 pages:79-90 extent:12 https://doi.org/10.1016/j.fluid.2017.07.021 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_31 GBV_ILN_40 GBV_ILN_100 GBV_ILN_136 AR 451 2017 15 1115 79-90 12 045F 660 |
| spelling |
10.1016/j.fluid.2017.07.021 doi GBVA2017012000002.pica (DE-627)ELV030502713 (ELSEVIER)S0378-3812(17)30289-3 DE-627 ger DE-627 rakwb eng 660 540 660 DE-600 540 DE-600 670 VZ 540 VZ 630 VZ Liang, Xiaodong verfasserin aut General approach for solving the density gradient theory in the interfacial tension calculations 2017transfer abstract 12 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. Density gradient theory Elsevier CPA Elsevier Interfacial tension Elsevier PC-SAFT Elsevier Direct optimization Elsevier Michelsen, Michael Locht oth Enthalten in Science Direct Li, Yong-gang ELSEVIER Fabrication and compressive behaviour of an aluminium foam composite 2015 an international journal New York, NY [u.a.] (DE-627)ELV013241125 volume:451 year:2017 day:15 month:11 pages:79-90 extent:12 https://doi.org/10.1016/j.fluid.2017.07.021 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_31 GBV_ILN_40 GBV_ILN_100 GBV_ILN_136 AR 451 2017 15 1115 79-90 12 045F 660 |
| allfields_unstemmed |
10.1016/j.fluid.2017.07.021 doi GBVA2017012000002.pica (DE-627)ELV030502713 (ELSEVIER)S0378-3812(17)30289-3 DE-627 ger DE-627 rakwb eng 660 540 660 DE-600 540 DE-600 670 VZ 540 VZ 630 VZ Liang, Xiaodong verfasserin aut General approach for solving the density gradient theory in the interfacial tension calculations 2017transfer abstract 12 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. Density gradient theory Elsevier CPA Elsevier Interfacial tension Elsevier PC-SAFT Elsevier Direct optimization Elsevier Michelsen, Michael Locht oth Enthalten in Science Direct Li, Yong-gang ELSEVIER Fabrication and compressive behaviour of an aluminium foam composite 2015 an international journal New York, NY [u.a.] (DE-627)ELV013241125 volume:451 year:2017 day:15 month:11 pages:79-90 extent:12 https://doi.org/10.1016/j.fluid.2017.07.021 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_31 GBV_ILN_40 GBV_ILN_100 GBV_ILN_136 AR 451 2017 15 1115 79-90 12 045F 660 |
| allfieldsGer |
10.1016/j.fluid.2017.07.021 doi GBVA2017012000002.pica (DE-627)ELV030502713 (ELSEVIER)S0378-3812(17)30289-3 DE-627 ger DE-627 rakwb eng 660 540 660 DE-600 540 DE-600 670 VZ 540 VZ 630 VZ Liang, Xiaodong verfasserin aut General approach for solving the density gradient theory in the interfacial tension calculations 2017transfer abstract 12 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. Density gradient theory Elsevier CPA Elsevier Interfacial tension Elsevier PC-SAFT Elsevier Direct optimization Elsevier Michelsen, Michael Locht oth Enthalten in Science Direct Li, Yong-gang ELSEVIER Fabrication and compressive behaviour of an aluminium foam composite 2015 an international journal New York, NY [u.a.] (DE-627)ELV013241125 volume:451 year:2017 day:15 month:11 pages:79-90 extent:12 https://doi.org/10.1016/j.fluid.2017.07.021 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_31 GBV_ILN_40 GBV_ILN_100 GBV_ILN_136 AR 451 2017 15 1115 79-90 12 045F 660 |
| allfieldsSound |
10.1016/j.fluid.2017.07.021 doi GBVA2017012000002.pica (DE-627)ELV030502713 (ELSEVIER)S0378-3812(17)30289-3 DE-627 ger DE-627 rakwb eng 660 540 660 DE-600 540 DE-600 670 VZ 540 VZ 630 VZ Liang, Xiaodong verfasserin aut General approach for solving the density gradient theory in the interfacial tension calculations 2017transfer abstract 12 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. Density gradient theory Elsevier CPA Elsevier Interfacial tension Elsevier PC-SAFT Elsevier Direct optimization Elsevier Michelsen, Michael Locht oth Enthalten in Science Direct Li, Yong-gang ELSEVIER Fabrication and compressive behaviour of an aluminium foam composite 2015 an international journal New York, NY [u.a.] (DE-627)ELV013241125 volume:451 year:2017 day:15 month:11 pages:79-90 extent:12 https://doi.org/10.1016/j.fluid.2017.07.021 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U SSG-OLC-PHA GBV_ILN_31 GBV_ILN_40 GBV_ILN_100 GBV_ILN_136 AR 451 2017 15 1115 79-90 12 045F 660 |
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Enthalten in Fabrication and compressive behaviour of an aluminium foam composite New York, NY [u.a.] volume:451 year:2017 day:15 month:11 pages:79-90 extent:12 |
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general approach for solving the density gradient theory in the interfacial tension calculations |
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General approach for solving the density gradient theory in the interfacial tension calculations |
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Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. |
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Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. |
| abstract_unstemmed |
Within the framework of the density gradient theory, the interfacial tension can be calculated by finding the density profiles that minimize an integral of two terms over the system of infinite width. It is found that the two integrands exhibit a constant difference along the interface for a finite planar interface, and this difference becomes smaller as the interface width increases. These findings naturally lead to a solution procedure that consists of an inner loop and an outer loop for calculating the interfacial tension of a planar interface. The outer loop deals with the relationship between the interfacial tension and the interface width, and it permits us to obtain accurate results from finite width calculations. The inner loop minimizes the interfacial tension for a given interface width by adjusting the density profiles, in which the integrals are evaluated by a combination of Gauss-Lobatto quadrature and Lagrange interpolation based polynomial approximation. A better approximation of the interfacial tension is derived by a path integration along the density profiles. These strategies enable us to obtain accurate solutions with looser tolerance criteria and a fewer number of thermodynamic property evaluations compared to other methods. The performance of the algorithm with recommended parameters is analyzed for various systems, and the efficiency is further compared with the geometric-mean density gradient theory, which only needs to solve nonlinear algebraic equations. The results show that the algorithm is only 5–10 times less efficient than solving the geometric-mean density gradient theory. |
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General approach for solving the density gradient theory in the interfacial tension calculations |
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