육각형 형태의 분자에 대한 열역학적 섭동 이론

하 경 진, 김 순 호, 김 화 용

서울대학교 응용화학부

Thermodynamic Pertur

bation Theory for Cyclic Hexamers

K. Ha, S. Kim, and H. Kim

School of Chemical Engineering, Seoul National University

INTRODUCTION

The perturbation theories of fluids allow to describe the behaviour of real gases and liquids, i.e., of systems with relatively complicated intermolecular interactions, on the basis of thermodynamic and distribution functions of a simpler system chosen as a reference. Hence the perturbation methods are analogous to classical thermodynamic treatments which employ, for the description of real systems, appropriately selected standard states together with expressing the excess functions through expansions; for a method to be successful, the proper choice of the reference and a knowledge of its behaviour as accurately as possible are extremely important.

Recently there has been shown no interest in the cyclic molecules because of their complexity to describe the thermodynamic behaviour. In the present theory we made attempts to apply perturbation theory to cyclic hexamers which could be expected to be useful in the petroleum chemical industry where the aromatic chemicals are commonly treated.

THEORY

Thermodynamic Perturbation Theory(TPT) for L-J ring polymers

Sear and Jackson and Ghonasgi et al. have independently developed extensions to the Zhou-Stell and Wertheim theories that account for ring formation in hard chain fluids. The work of Sear and Jackson initially considerd only the formation of athermal ring molecules starting from the Zhou-Stell zeroth-order theory. This is because the number of bonds in a ring molecules is one more than the number of bonds in a linear chain. The superposition approximation for a ring of m beads is

[eqn: y^(m) (1...m) = [y(l)]^m]

In the work of Sear and Jackson no distinction is made between the bonds that make up the linear chain and the bond that closes the linear chain into a ring. Ghonasgi et al. considerd the more general problem of intramolecular association in a fluid of freely-jointed hard sphere chains. Their result for ring molecules is the special case of association in the limit of total bonding. The starting point is and athermal chain with bonding sites on the therminal segments. They note that the correct term for the intramolecular bonding contains the intramolecular site-site correlation function for the ends of the chain. Here we extend their theory to the LJ monomer reference fluid. So doing, we directly obtain

[eqn: {A^ring } over {N_c kT } = {A^mono } over {N_c kT } +(1-m)ln[g^mono ( sigma )]-ln[w_ee ( sigma )]]

where [eqn: w_ee ( sigma )] is the intramolecular end-end correlation function evaluated at contact. If one assumes that the intramolecular end-end correlation function is the same as the reference fluid radial distribution function then the result of Sear nad Jackson is recovered. Extending their results to rings of freely-jointed, tangently bonded LJ spheres we have

[eqn: {A^ring } over {N_c kT } = {A^mono } over {N_c kT } -m ln[g^mono ( sigma )]]

where [eqn: A^mono] is calculated from the LJ equation of state of Johnson et al. (1993) and [eqn: g^mono ( sigma )] is given by the correlation of Johnson et al.(1994). While the assumption that [eqn: w_ee (sigma) = g^mono (sigma)] is undoubtedly a poor one, Ghonasgi et al. point out is required th calculate the equation of state is that the derivatives of the intramolecular and reference correlation functions be the same.

Both Sear and Jackson and Ghonasgi et al. present original Monte Carlo calculations of athermal rigid ring molecules and use these data to test their equation for ring molecules. Their combined simulations provide the compressibility for ring containing from 3 to 6 beads. They find good agreement for the compressibility factor from simulation and theory over a wide density range. The treatment of intramolecular association was later extended to include competition intra and intermolecular association.

We now develop an ezpression for the free energy of a fluid of rings in terms of a dimer reference fluid. We use the superposition approximation. Setting s=2 and [eqn: l=sigma] we have

[eqn: {A^ring } over {N_c kT } = {A^mono } over {N_c kT } - m over 2 ln[g^mono ( sigma )] - m over 2 ln[g_ee ^dimer ( sigma )]]

This equation implicity assumes that [eqn: w_ee (sigma) = g_ee ^dimer (sigma)]. If we explicitly include the intramolecular correlation function we obtain

[eqn: {A^ring } over {N_c kT } = {A^mono } over {N_c kT } - m over 2 ln[g^mono ( sigma )] + (1- m over 2 )ln[g_ee ^dimer ( sigma )] - ln [w_ee (sigma) ]]

Above equation and this equation are the first dimer reference equation for rings.

RESULT AND DISCUSSION

The equation of state for Lennard-Jones cyclic hexamers is developed based on the perturbation theory. In fig. 1 and fig. 2, the internal energy and pressure calculated from the present theory are compared with MC-NVT simulation data for Lennard-Jones cyclic hexamers at the reduced temperature 3, 5. The simulated system is constant-NVT and we use periodic boundary conditions in a cubic box which length of each side is adjusted as each density and size parameter [eqn: sigma] is 1. Also the number of molecules is adjusted like length of box and all molecules are rigid planar cyclic hexamers of which pair potential are Lennard-Jones potential. The simulation has performed 600,000 times each density and temperature.

The calculation from the present theory does not show good agreement with the simulation data as seen in fig.1 and fig. 2. When the reduced temperature is 5, the present theory predicts better than lower temperature 3. The calculation of TPT1M show better agreement with the simulation data than that of TPT1D. The present EOS cannot predict accurately in this reduced temperatures.

From the comparison with the simulation data, we can consider whether or not the thermodynamic perturbation theory is appropriate for describing the thermodynamic behaviour of the molecules that have many segments such as cyclic hexamer we studied. Though the perturbation theories for monomer or dimer have predicted accurately, the theory fails to predict well for cyclic hexamer molecules.

fig. 1 The reduced pressure

fig. 2 The reduced internal energy

REFERENCE

·J. K. Johnson, E. A. Muller, and K. E. Gubbins, J. Phys. Chem. 98,     6413 (1994)

·R. P. Sear and G. Jackson, Mol. Phys. 81, 801 (1994)

·J. K. Johnson, E. A. Muller, and K. E. Gubbins, Mol. Phys. 78, 591     (1993)

·R. P. Sear and G. Jackson, Phys. Rev. E50, 386 (1994)

·D. Ghonasgi, V. Perez, and W. G. Chapman, J. Chem. Phys. 101, 6880    (1994)

·D. Ghonasgi and W. G. Chapman, J. Chem. Phys. 102, 2585 (1995)