Supplementary material to Simultaneous description of bulk and interfacial properties of fluids by the Mie potential

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1 To appear in Molecular Physics Vol. 00, No. 00, Month 2016, 1 11 Supplementary material to Simultaneous description of bul and interfacial properties of fluids by the Mie potential Stephan Werth 1, Katrin Stöbener 2, Martin Horsch 1 & Hans Hasse 1 1 Laboratory of Engineering Thermodynamics, Department of Mechanical and Process Engineering, University of Kaiserslautern, Erwin-Schrödinger-Str. 44, Kaiserslautern, Germany 2 Fraunhofer Institute for Industrial Mathematics, Department for Optimization, Fraunhofer-Platz 1, Kaiserslautern, Germany (submitted May 2016) 1. Long range correction for multi-site Mie models and planar interfaces The Mie potential is given by [1, 2] u n [( σ ) n ( σ ] ɛ, (1) n m m s s where σ and ɛ are the size and energy parameter, n and m are the repulsive and dispersive exponents and s represents the distance between two interaction sites, which may deviate from the distance r between the centers of mass. Fig. 1 shows the three possible cases for the interaction type. The center-site (CS) and site-site (SS) cases are discussed in the Appendix. The CC case is discussed in the main text. [Figure 1 about here.] According to previous wor for the multi-site Lennard-Jones models, the angleaveraging scheme by Lustig [3] is used for multi-site Mie models Center-site case The angle-averaged value of s 2a has been derived by Lustig [3]. s 2a (r τ)2a2 (r τ) 2a2, (2) 4rτ(a 1) where τ is the elongation from the center of mass. Corresponding author. martin.horsch@mv.uni-l.de ISSN: print/issn online c 2016 Taylor & Francis DOI: / YYxxxxxxxx

2 U LRC i 2πρ(y ) y πρ(y ) y n dr τ r n m m σ m (r τ) m2 (r τ) m2 ] ( m 2) πρ(y ) yɛσ 3 τ n dr ɛ [ σ n s n σ m s m] r n m m n n m m σ m 3 (r τ) m3 ( τ) m3 ɛ [σ n (r τ) n2 (r τ) n2 ( n 2) [σ n 3 (r τ) n3 ( τ) n3 ] (3) F LRC i 2πρ(y ) y πρ(y ) yɛσ 2 τ dr u ξ r r r σ m 2 (r τ) m2 ( τ) m2 ( m 2) ξ n [σ n 2 (r τ) n2 ( τ) n2 n m m ( n 2) ] (4) Π LRC N,i πρ(y ) y πρ(y ) yɛσ 2 2τ dr u ξ 2 r r r σ m 2 (r τ) m2 ( τ) m2 ( m 2) ξ 2 n [σ n 2 (r τ) n2 ( τ) n2 n m m ( n 2) ] (5) 2

3 Π LRC T,i 1 2 πρ(y ) y πρ(y ) yɛσ 2 4τ dr u r r 2 ξ 2 r r n [σ n 2 (r τ) n2 ( τ) n2 n m m ( n 2) ] (r 2 ξ 2) σ m 2 (r τ) m2 ( τ) m2 ( m 2) πρ(y ) yɛσ 3 n [σ n 3 (r τ) n3 ( τ) n3 2τ n m m σ m 3 (r τ) m3 ( τ) m3 ] (6) 1.2. site-site case The sites are separated from their respective center of mass by the distances τ 1 and τ 2, respectively. The angle-averaged value of s 2a has been derived by Lustig [3]. s 2a (r τ ) 2a3 (r τ ) 2a3 (r τ ) 2a3 (r τ ) 2a3, (7) 8rτ 1 τ 2 (a 1)(2a 3) where τ τ 1 τ 2 and τ τ 1 τ 2. Ui LRC 2πρ(y ) y n dr ɛ [ σ n s n σ m s m] r n m m πρ(y ) y n dr ɛ 2τ 1 τ 2 r n m m [σ n (r τ ) n3 (r τ ) n3 (r τ ) n3 (r τ ) n3 σ m (r τ ) m3 (r τ ) m3 (r τ ) m3 (r τ ) m3 πρ(y ) yɛσ 4 n 2τ 1 τ 2 n m m [σ n 4 (r τ ) n4 ( τ ) n4 ( τ ) n4 ( τ ) n4 ( n 4) σ m 4 (r τ ) m4 ( τ ) m4 ( τ ) m4 ( τ ) m4 ] ( m 4) (8) ] 3

4 F LRC i 2πρ(y ) y dr u ξ r r r πρ(y ) yɛσ 3 ξ n 2τ 1 τ 2 n m m [σ n 3 (r τ ) n3 ( τ ) n3 ( τ ) n3 ( τ ) n3 σ m 3 (r τ ) m3 ( τ ) m3 ( τ ) m3 ( τ ) m3 ] (9) Π LRC N,i πρ(y ) y dr u ξ 2 r r r πρ(y ) yɛσ 3 ξ 2 n 4τ 1 τ 2 n m m [σ n 3 (r τ ) n3 ( τ ) n3 ( τ ) n3 ( τ ) n3 σ m 3 (r τ ) m3 ( τ ) m3 ( τ ) m3 ( τ ) m3 ] (10) Π LRC T,i 1 2 πρ(y ) y dr u r r 2 ξ 2 r r πρ(y ) yɛσ 3 n 8τ 1 τ 2 n m m [σ n 3 (r τ ) n3 ( τ ) n3 ( τ ) n3 ( τ ) n3 σ m 3 (r τ ) m3 ( τ ) m3 ( τ ) m3 ( τ ) m3 ] (r 2 ξ 2) πρ(y ) yɛσ 4 n 4τ 1 τ 2 n m m [σ n 4 (r τ ) n4 ( τ ) n4 ( τ ) n4 ( τ ) n4 ( n 4) σ m 4 (r τ ) m4 ( τ ) m4 ( τ ) m4 ( τ ) m4 ] ( m 4) (11) 4

5 2. Pareto Optimization of CO 2 A multidimensional Pareto set can be visualized in two dimensions by a selforganizing patch plot [4]. Fig. 2 shows the three dimensional Pareto set of the single-site Mie model for CO 2 in the parameter and objective space. Each patch in Fig. 2 represents a molecular model. The color indicates the magnitude of the given parameter or objective function. The results are the same as in Fig. 15 of the main article, but in the self-organizing patch plot similar solutions are grouped close together and therefore simplify the choice of a molecular model from the Pareto set. [Figure 2 about here.] Selected Pareto optimal solutions for the three different molecular model classes are given in Tables 1-3. [Table 1 about here.] [Table 2 about here.] [Table 3 about here.] References [1] G. Mie, Ann. Phys. 316 (8), 657 (1903). [2] E. Grüneisen, Ann. Phys. 344 (12), 257 (1912). [3] R. Lustig, Mol. Phys. 65 (1), 175 (1988). [4] K. Stöbener, P. Klein, M. Horsch, K.H. Küfer and H. Hasse, Fluid Phase Equilib. 411, 33 (2015). [5] S. Werth, G. Rutai, J. Vrabec, M. Horsch and H. Hasse, Mol. Phys. 112 (17), 2227 (2014). 5

6 CC CS τ τ SS 1 τ 2 Figure 1. Illustration of the three different cases discussed here. Sites in the center of mass interact with each other as a center-center interaction (top), as opposed to the center-site interaction (middle) and the site-site interaction (bottom), cf. Werth et al. [5]. The distance of the sites from the center of mass of their molecules is denoted by τ. The bullets indicate the center of mass, while the crosses denote the site positions. 6

7 (ε/) / K σ / Å n δρ / % 12 δp s / % 25 δγ / % Figure 2. Self-organizing patch plot of the Pareto set of the Mie potential for CO 2 : Parameter space (top) and objective space (bottom). 7

8 Table 1. Selected parameter combinations of the Mie fluid for the description of CO 2 for the three criteria optimization. σ / Å ɛ/ B K n δρ / % δp S / % δγ / %

9 Table 2. Selected parameter combinations of the LJ fluid for the description of CO 2 for the two criteria optimization. σ / Å ɛ/ B K δρ / % δp S / % δγ / %

10 Table 3. Selected parameter combinations of the 2CLJQ fluid for the description of CO 2 for the two criteria optimization. σ / Å ɛ/ B K L / Å Q / DÅ δρ / % δp S / % δγ / %