# Convergence of Coupling-Parameter Expansion-Based Solutions to Ornstein–Zernike Equation in Liquid State Theory

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Coupling-Parameter Expansion—General

#### 2.1. Method of Solution

^{th}order. Furthermore, the linear operator in Equation (9) (for all orders) is identical to that in Equation (8). Finally, the derivatives ${B}_{n}^{*}$ are expressed as follows.

#### 2.2. Thermodynamic Functions

## 3. Coupling Parameter Expansion—AHS Model

## 4. Coupling Parameter Expansion—MSA Model

## 5. Coupling Parameter Expansion—LJ Potential

#### 5.1. Phase Diagram of LJ Potential

#### 5.2. Phase Diagrams of Generalized LJ(n,m) Potentials

#### 5.3. Solution-Spaces and Bifurcations of Solutions

## 6. Summary

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**A**) Upper part shows ${\Gamma}^{*}\left(1\right)$ versus $\eta $ from CPE of 20th order for ${T}^{*}=0.5$ (curve-1), ${T}^{*}=0.4$ (curve-2), and exact results (symbols). The range ${\eta}_{1}<\eta <{\eta}_{2}$ is the ‘no-real-solution-region’ for ${T}^{*}=0.4$. Lower part of the figure for ${T}^{*}=0.45$ shows compressibility pressure P versus $\eta $, from CPE of 20th order (curve-1), and exact results (symbols). The blank portion is the ‘no-real-solution-region’. (

**B**) The exact results for phase diagram of AHS model (blue line and symbols) are compared with those obtained using a CPE of 50th order (curve-2) and 20th order (curve-3). The ‘no-real-solution-region’, lying within the dashed lines (curve-1), is also shown.

**Figure 2.**(

**A**) Lower part (for $K=0.8,\zeta =2)$ shows $\Phi \left(1\right)$ versus $\eta $ obtained from CPE of 20th order (curve-1) and the exact results (curve-2). In the upper part (for $K=1.2,\zeta =2)$, the range ${\eta}_{1}<\eta <{\eta}_{2}$ is ‘no-real-solution-region’, as ${K}^{*}=1.1354$. Graphs show $\Phi \left(1\right)$ versus $\eta $ obtained from CPE of 20th order (curve-1) and real (curve-2) and imaginary (curve-3) parts of the exact results. (

**B**) Graphs show (reduced) inverse compressibility ${\chi}^{-1}$ versus $\eta $ obtained from CPE (curve-1) and exact results [11] (curve-2) for $K=1.2$. The inset figure is similar, but it is for $K=1.4$.

**Figure 3.**(

**A**) Pair distribution function ${g}_{0}(r/\sigma )$ versus scaled distance $(r/\sigma )$ (curve-1) for the reference system of LJ potential, as per the WCA prescription (see text), at reduced temperature ${T}^{*}=0.7290$ and density ${\rho}^{*}=0.8446$. Symbols are molecular dynamics simulation results [32]. (

**B**) Similar results for the parameters ${T}^{*}=1.095$ and density ${\rho}^{*}=0.6257$. The inset figure shows compressibility factor $Z=\beta P/\rho $ versus volume fraction $\eta =\pi {\rho}^{*}/6$ for reduced temperature ${T}^{*}=0.75$ (top curve) and ${T}^{*}=2.0$ (bottom curve). Symbols are simulation results [33].

**Figure 4.**Convergence of direct correlation function $c(r/\sigma )$ and pair distribution function $g(r/\sigma )$ for the LJ system at the phase point ${\rho}^{*}=0.3$ and ${T}^{*}=1.45$, which is above the critical temperature. (

**A**) Graphs show $c(r/\sigma )$ (curve-c), reference function ${c}_{0}(r/\sigma )$ (curve-0), and successive derivative terms ${c}_{n}(r/\sigma )/n!$ (curve-1 to curve-6). (

**B**) Similarly, the graphs show $g\left(r\right)$ (curve-g), reference function ${g}_{0}(r/\sigma )$ (curve-0), and derivative terms ${g}_{n}(r/\sigma )/n!$ (curve-1 to curve-6) for the same phase point.

**Figure 5.**Convergence of direct correlation function $c(r/\sigma )$ and pair distribution function $g(r/\sigma )$ for the LJ system at the phase point ${\rho}^{*}=0.3$ and ${T}^{*}=1.15$, which is in the spinodal region. (

**A**) Graphs show $c(r/\sigma )$ (curve-c), reference function ${c}_{0}(r/\sigma )$ (curve-0), and successive derivative terms ${c}_{n}(r/\sigma )/n!$ (curve-1 to curve-6). (

**B**) Similarly, the graphs show $g(r/\sigma )$ (curve-g), reference function ${g}_{0}(r/\sigma )$ (curve-0), and derivative terms ${g}_{n}(r/\sigma )/n!$ (curve-1 to curve-6) for the same phase point.

**Figure 6.**Graphs show inverse compressibility ${\chi}^{-1}$ versus reduced density ${\rho}^{*}$ obtained at successive orders of series expansion. Thus, ${\chi}^{-1}$ calculated with just ${c}_{0}(r/\sigma )$ (curve-0), ${c}_{0}(r/\sigma )+{c}_{1}(r/\sigma )$ (curve-1), ${c}_{0}(r/\sigma )+{c}_{1}(r/\sigma )+(1/2){c}_{2}(r/\sigma )$ (curve-2), etc., and finally $c(r/\sigma )$ (curve-6) is shown. (

**A**) Graphs for reduced temperature ${T}^{*}=1.45$ in one-phase region. (

**B**) Similarly, for ${T}^{*}=1.15$ when crossing the spinodal region.

**Figure 7.**(

**A**) Comparison of ${\chi}^{-1}$ versus ${\rho}^{*}$ obtained by using CPE (curve-1) and independent computations (symbols) [12] on the isotherm for ${T}^{*}=1.33$, which is just above the critical temperature. The inset graphs show pressure $\beta P$ (curve-1) and chemical potential $\beta \mu $ (curve-2) (both obtained via compressibility route) on the same isotherm. (

**B**) Comparison of co-existence lines using thermodynamic conditions (curve-1) and Maxwell’s construction (curve-2) with the simulation results (symbols) [34], obtained with CPE. The spinodal lines (curve-3) on gaseous and liquid sides are also shown.

**Figure 8.**(

**A**) Comparison of co-existence lines, obtained with CPE, for potentials: LJ(7,6) (curve-1), LJ(9,6) (curve-2), LJ(12,6) (curve-3), LJ(15,6) (curve-4), and LJ(20,6) (curve-5), with simulation data (symbols) [35]. (

**B**) Reduced isotherms for LJ(7,6) (curve-1) using virial (solid line), compressibility (dashed line), and free energy (dash-dot line) routes corresponding to ${T}^{*}=1$. Other sets of graphs are for LJ(12,6) (curve-3) at ${T}^{*}=1$ and LJ(20,6) (curve-5) at ${T}^{*}=0.75$.

AHS Model | ${\mathsf{\Gamma}}_{\mathit{n}}^{*}/{\mathsf{\Gamma}}_{\mathit{n}-1}^{*}$ | ${\mathit{T}}^{*}$ = 0.45 | |
---|---|---|---|

n | $\mathbf{\eta}$= 0.05 | $\mathbf{\eta}$= 0.15 | $\mathbf{\eta}$= 0.25 |

1 | 2.11588 | 1.90467 | 1.69443 |

6 | 0.769139 | 1.93929 | 3.16261 |

11 | 0.823916 | 1.00009 | 0.764407 |

16 | 0.881579 | 0.948853 | 0.716068 |

21 | 0.90604 | 0.959392 | 0.842613 |

26 | 0.918982 | 0.971881 | 0.878875 |

31 | 0.927508 | 0.981006 | 0.880897 |

36 | 0.933696 | 0.987649 | 0.884413 |

41 | 0.938408 | 0.992688 | 0.889093 |

46 | 0.942117 | 0.996646 | 0.892829 |

51 | 0.945112 | 0.999839 | 0.895698 |

MSA-HCY Model | ${\mathit{\varphi}}_{\mathit{n}}/{\mathit{\varphi}}_{\mathit{n}-1}$ | K = 1.2 | |
---|---|---|---|

n | $\mathbf{\eta}=\mathbf{0.1}$ | $\mathbf{\eta}=\mathbf{0.2}$ | $\mathbf{\eta}=\mathbf{0.3}$ |

2 | 0.203451 | 0.233279 | 0.187291 |

6 | 0.568111 | 0.612217 | 0.454637 |

11 | 0.648161 | 0.689816 | 0.499086 |

16 | 0.678529 | 0.719187 | 0.51365 |

21 | 0.694581 | 0.734819 | 0.520534 |

26 | 0.704522 | 0.744575 | 0.524387 |

31 | 0.711287 | 0.751261 | 0.526771 |

36 | 0.71619 | 0.756136 | 0.528354 |

41 | 0.719906 | 0.75985 | 0.529465 |

46 | 0.722821 | 0.762777 | 0.53029 |

51 | 0.725169 | 0.765143 | 0.530938 |

Order | Inverse Compressibility | ${\mathit{T}}^{*}=1.15$ | Inverse Compressibility | ${\mathit{T}}^{*}=1.45$ | ||
---|---|---|---|---|---|---|

${\mathbf{\rho}}^{*}=\mathbf{0.091}$ | ${\mathbf{\rho}}^{*}=\mathbf{0.361}$ | ${\mathbf{\rho}}^{*}=\mathbf{0.631}$ | ${\mathbf{\rho}}^{*}=\mathbf{0.091}$ | ${\mathbf{\rho}}^{*}=\mathbf{0.361}$ | ${\mathbf{\rho}}^{*}=\mathbf{0.631}$ | |

1 | 1.47757 | 4.53543 | 13.6363 | 1.46285 | 4.35925 | 12.6832 |

2 | 0.434523 | −1.13238 | 2.19019 | 0.632217 | −0.091533 | 3.77993 |

3 | 0.396093 | −0.532527 | 2.79679 | 0.606776 | 0.283586 | 4.16096 |

4 | 0.369497 | −0.491079 | 2.85451 | 0.592575 | 0.304848 | 4.19282 |

5 | 0.352317 | −0.459612 | 2.88001 | 0.585209 | 0.318156 | 4.20459 |

6 | 0.342099 | −0.43545 | 2.89254 | 0.581667 | 0.326561 | 4.20943 |

7 | 0.335684 | −0.416063 | 2.89917 | 0.579865 | 0.332105 | 4.21159 |

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**MDPI and ACS Style**

Menon, S.V.G. Convergence of Coupling-Parameter Expansion-Based Solutions to Ornstein–Zernike Equation in Liquid State Theory. *Condens. Matter* **2021**, *6*, 29.
https://doi.org/10.3390/condmat6030029

**AMA Style**

Menon SVG. Convergence of Coupling-Parameter Expansion-Based Solutions to Ornstein–Zernike Equation in Liquid State Theory. *Condensed Matter*. 2021; 6(3):29.
https://doi.org/10.3390/condmat6030029

**Chicago/Turabian Style**

Menon, S. V. G. 2021. "Convergence of Coupling-Parameter Expansion-Based Solutions to Ornstein–Zernike Equation in Liquid State Theory" *Condensed Matter* 6, no. 3: 29.
https://doi.org/10.3390/condmat6030029