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A numerical bifurcation analysis of the ornstein-zernike equation with hypernetted chain closure

Beardmore, R. E. ; Peplow, A. T. LU orcid and Bresme, F. (2007) In SIAM Journal on Scientific Computing 29(6). p.2442-2463
Abstract

We study the codimension-one and -two bifurcations of the Ornstein-Zernike equation with hypernetted chain (HNC) closure with Lennard-Jones intermolecular interaction potential. The main purpose of the paper is to present the results of a numerical study undertaken using a suite of algorithms implemented in MATLAB and based on pseudo arc-length continuation for the codimension-one case and a Newton-GMRES method for the codimension-two case. Through careful consideration of the results of our computations, an argument is formulated which shows that spinodal isothermal solution branches arising in this model cannot be reproduced numerically. Furthermore, we show that the existence of an upper bound on the density that can be realized on a... (More)

We study the codimension-one and -two bifurcations of the Ornstein-Zernike equation with hypernetted chain (HNC) closure with Lennard-Jones intermolecular interaction potential. The main purpose of the paper is to present the results of a numerical study undertaken using a suite of algorithms implemented in MATLAB and based on pseudo arc-length continuation for the codimension-one case and a Newton-GMRES method for the codimension-two case. Through careful consideration of the results of our computations, an argument is formulated which shows that spinodal isothermal solution branches arising in this model cannot be reproduced numerically. Furthermore, we show that the existence of an upper bound on the density that can be realized on a vapor isothermal solution branch, which must be present at a spinodal, causes the existence of at least one fold bifurcation along that vapor branch when density is used as the bifurcation parameter. This provides an explanation for previous inconclusive attempts to compute solutions using Newton-Picard methods that are popular in the physical chemistry literature.

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author
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publishing date
type
Contribution to journal
publication status
published
subject
keywords
Liquid phase, Ornstein-Zernike equation, Spinodal, Vapor phase
in
SIAM Journal on Scientific Computing
volume
29
issue
6
pages
22 pages
publisher
Society for Industrial and Applied Mathematics
external identifiers
  • scopus:55549103717
ISSN
1064-8275
DOI
10.1137/060650659
language
English
LU publication?
no
id
1162a66b-5f35-49bb-8d0a-595527d2739e
date added to LUP
2021-02-15 19:53:17
date last changed
2022-02-01 20:15:56
@article{1162a66b-5f35-49bb-8d0a-595527d2739e,
  abstract     = {{<p>We study the codimension-one and -two bifurcations of the Ornstein-Zernike equation with hypernetted chain (HNC) closure with Lennard-Jones intermolecular interaction potential. The main purpose of the paper is to present the results of a numerical study undertaken using a suite of algorithms implemented in MATLAB and based on pseudo arc-length continuation for the codimension-one case and a Newton-GMRES method for the codimension-two case. Through careful consideration of the results of our computations, an argument is formulated which shows that spinodal isothermal solution branches arising in this model cannot be reproduced numerically. Furthermore, we show that the existence of an upper bound on the density that can be realized on a vapor isothermal solution branch, which must be present at a spinodal, causes the existence of at least one fold bifurcation along that vapor branch when density is used as the bifurcation parameter. This provides an explanation for previous inconclusive attempts to compute solutions using Newton-Picard methods that are popular in the physical chemistry literature.</p>}},
  author       = {{Beardmore, R. E. and Peplow, A. T. and Bresme, F.}},
  issn         = {{1064-8275}},
  keywords     = {{Liquid phase; Ornstein-Zernike equation; Spinodal; Vapor phase}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{2442--2463}},
  publisher    = {{Society for Industrial and Applied Mathematics}},
  series       = {{SIAM Journal on Scientific Computing}},
  title        = {{A numerical bifurcation analysis of the ornstein-zernike equation with hypernetted chain closure}},
  url          = {{http://dx.doi.org/10.1137/060650659}},
  doi          = {{10.1137/060650659}},
  volume       = {{29}},
  year         = {{2007}},
}