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Statistical mechanical calculations on a periodic cubic system containing two molecules in the unit cell were reported [4], where the model potential was the step function. The minimum image convention [5] was assumed, resulting in the canonical partition function. When the interaction energy between the molecule 1 and the other molecules is calculated, only the closest periodic images of the other molecules interact with the molecule 1 in this approximation. The fluid-fluid phase transition-like behavior and negative thermal expansion was observed in the system. The theoretical

This work assumes a spherical cell without the periodic boundary condition. This shape is convenient because the canonical partition function can be obtained from statistical mechanics. There are two identical spherical molecules in the cell. The repulsive step function is assumed outside the hard sphere wall between the molecules. The present work is compared with the previous one in Table 1. The intent of this study is to provide basic information in the research on nanoporous materials [6]. The theoretical

Table 1. The present work is compared with the previous one [4].

The present work | The previous paper [4] | |
---|---|---|

the potential function | the repulsive step function outside the hard sphere wall | the repulsive step function outside the hard sphere wall |

the boundary condition | non-periodic | periodic |

the shape of the cell | spherical | cubic |

the number of molecules in the unit cell | 2 | 2 |

the approximation | no | the minimum image convention [5] |

The assumed potential function contains three constant parts and depends only on the molecular distance. The proposed method can be applied easily to other potential systems, like the hard sphere and the square well model, to obtain the thermodynamic properties of the two-molecule system in the spherical cell.

Here, quantities e and s are the potential constants, which have the dimensions of energy and length, respectively. Figure 1 shows the potential function

Figure 1. Step-function potential *u*(*r*) plotted as a function of the intermolecular distance *r*.

Figure 2. Cell **V _{c}** and the sphere centered at

Figure 3. The space is classified into three regions where the Boltzmann factor is constant.

**Partition Function of the Two-Molecule System with the Step-Function.** Two identical spherical molecules were assumed, and the canonical ensemble of the two-molecule system in the spherical cell at temperature *T* and volume *V* with the above potential function was considered, without the periodic boundary condition. The canonical partition function *Q* is the product of the ideal gas part *Q _{id}* and the interaction part

In Equation (3), b = 1/

The cell is referred to as

The common volume

Because the interaction energy depends only on the molecular distance and the function

The final stage is the integration with respect to

The Helmholtz free energy

The following form for

This means that the internal energy and the entropy have the following forms, respectively:

Because the system is a two-molecule system, the pressure

The interaction part

The remaining thermodynamic properties can be calculated the standard way [10].

Figure 4. The isotherm of pressure *p* in the step-function system is plotted as a function of the volume *V*.

Figure 5. The spinodal line is depicted by the solid line in the volume-temperature domain. The positive and negative thermal expansion coefficient regions are divided by the solid line and the solid line with solid circles. The sign of the thermal expansion coefficient is also displayed.

Figure 5 shows the spinodal line (k

The line with the solid circles will be described in the next section.

It can be seen that the thermal expansion coefficient is negative in the range 0.21 e/

The physical reason for this negative expansion coefficient is evident from the pressure

The zero points of the thermal expansion coefficient a are depicted by the lines with and without the solid circles in Figure 5. This means that the spinodal line coincides with one of the zero lines of the thermal expansion coefficient. In these two regions, the thermal expansion coefficient is negative, as shown in Figure 5 and Figure 8. One of them is located in the unstable states, inside the spinodal line, while the other is outside. The reason why the thermal expansion coefficient is negative in the two regions can be determined from Equation (15).

This equation shows that the thermal expansion coefficient may be zero for one of two reasons, which are represented by the two types of solid lines in Figure 5. The spinodal line corresponds to the zero points indicated by arrows in Figure 8. The other line corresponds to the minimum and the maximum of the pressure as a function of temperature under the constant volume condition, as shown in Figure 7.

The

Figure 6. Thermal expansion coefficient vs. the temperature plot at constant volume *V* = 2.3 s^{3}. The zero points indicated by arrows correspond to those shown by the solid line with the solid circles in Figure 5.

Figure 7. The interaction part of the internal energy *U _{e}* and pressure

Figure 8. Thermal expansion coefficient vs. the volume plot at constant temperature *T* = 0.3 e/*k*. The zero points indicated by arrows correspond to the spinodal line, and the others correspond to the other line with the solid circles in Figure 5.

Figure 9 shows the

Next, the finite system is compared with the periodic, effectively infinite system simulated by the MC method. For this purpose, the cubic periodic boundary condition is assumed and the minimum image convention is applied in the MC. The corrections on the long tail terms are not adopted for the present purpose of comparison. For this case, the initial configuration is FCC. The other conditions in the MC simulation are essentially the same as for the

The MC results for the periodic

Figure 9. The theoretical isotherm of pressure *p* of *N* = 2 with the step-function system is compared with the MC results as a function of the volume at *T* = 0.2 e/*k*. The pressure of the ideal gas *p _{id}* is also shown for comparison.

Figure 10. The MC results on the periodic cubic *N* = 108 system: the interaction part of the internal energy *U _{e}* and pressure

The authors thank the Research Center for Computational Science for access to their super computer. Computations were also performed at the Computational Science Research Center, Hosei University.

[ 2] Y. Yoshimura,

[ 3] S. V. Buldyrev, G. Franzese, N. Giovambattista, G. Malescio, M. R. Sadr-Lahijany, A. Scala, A. Skibinsky, and H. E. Stanley,

[ 4] Y. Kataoka and Y. Yamada,

[ 5] M. P. Allen and D. J. Tildesley,

[ 6] G. Q. Lu and X. S. Zhao (Ed.),

[ 7] J. P. Hansen and I. R. McDonald,

[ 8] D. A. McQuarrie,

[ 9] Mathematica, Wolfram Research, Inc. 100 Trade Center Drive Champaign, IL 61820-7237 USA

URL: http://www.wolfram.com/

[10] P. W. Atkins,

[11] A. Rotenberg,

[12] A. Awazu,

[13] T. Munakata and G. Hu,

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