- Email: [email protected]
Equation of state for Lennard–Jones flexible ring fluids
Fluid Phase Equilibria 219 (2004) 61–65
Equation of state for Lennard–Jones flexible ring fluids Jichul An, Hwayong Kim∗ School of Chemical Engineering & Institute of Chemical Process, Seoul National University, SAN 56-1, Shilim-Dong, Gwanak-Gu, Seoul 151-742, Republic of Korea Received 8 September 2003
Abstract We present a new equation of state for Lennard–Jones (LJ) flexible ring fluids. We perform Monte-Carlo simulations for freely-jointed Lennard–Jones chain fluids (3-, 6- and 8-mer) in the canonical ensemble and obtain the intramolecular end-to-end pair correlation function data under extensive density and temperature conditions. We correlate these as a function of the density, the temperature and the number of segments in a chain. We apply this function to thermodynamic perturbation theory (TPT) and obtain a new equation of state for Lennard–Jones flexible ring fluids. We also compare existing simulation data [J. Chem. Phys. 104 (1996) 1729] with the results obtained using the newly derived equation of state. © 2004 Elsevier B.V. All rights reserved. Keywords: Lennard–Jones flexible ring; Molecular simulation; Equation of state; Monte-Carlo simulation; Thermodynamic perturbation theory; Intramolecular end-to-end correlation function
1. Introduction Little interest has been shown recently in cyclic molecules because of their complexity to describe the thermodynamic behavior, although a number of cyclic molecules are widely used in the chemical and petrochemical industries. Thus, we undertook to develop an equation of state for cyclic molecules. Sear and Jackson [2,3] and Ghonasgi and co-workers [4,5] independently developed extensions to the Zhou–Stell and Wertheim theories that describe ring formation in hard chain fluids. Sear and Jackson considered only the formation of athermal ring molecules starting from Zhou–Stell zeroth-order theory, in which no distinction is made between the bonds that make up the linear chain and the bond that forms a ring from a linear chain [2]. Ghonasgi et al. considered an athermal chain with bonding sites on the terminal segments, and Johnson extended this theory to Lennard–Jones (LJ) monomer and LJ dimer reference fluids [1]. However, Johnson assumed that the intramolecular end-to-end correlation function is the same as the reference fluid radial distribution function.
∗
Corresponding author. Tel.: +82-2-880-7406; fax: +82-2-888-6695. E-mail address: [email protected] (H. Kim).
0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2004.01.016
In the present study, we performed NVT Monte-Carlo simulations of the freely-jointed LJ flexible chains composed of three, six and eight segments and obtained intramolecular end-to-end correlation function data under extensive conditions. We correlated these as a function of density, temperature and the number of segments in a chain. We applied this function to thermodynamic perturbation theory (TPT) and obtained a new equation of state for Lennard–Jones flexible ring fluids.
2. Thermodynamic perturbation theory (TPT) for ring The equation of state for LJ chains is derived in the same manner as that for hard sphere chains [6,7]. The first-order thermodynamic perturbation theory solution only requires knowledge of the pair distribution function of the reference fluid. For chains, the resulting expression for the residual Helmholtz free energy of a linear chain fluid is Achain Aref = + (1 − m)ln[yref (r)] NC kT NC kT
(1)
where NC is the number of chain molecules, k is Boltzmann’s constant, T is the temperature in Kelvin, Aref is the reference fluid residual free energy, m is the number of reference fluid
62
J. An, H. Kim / Fluid Phase Equilibria 219 (2004) 61–65
monomers in a chain, and yref (r) is the reference fluid cavity correlation function evaluated at a separation distance of r. The Eq. (1) may also be obtained from Stell and Zhou [8–10] study of an ionic fluid. The resulting equation of the Stell and Zhou theory is Achain Aref = − ln[yref(m) (1 · · · m)] NC kT NC kT
(2)
where yref(m) (1 · · · m) is the m-body cavity correlation function. The use of the linear approximation [11] of the yref(m) (1 · · · m) and the constant bond length (r) lead to the expression that is equivalent to TPT1. The cavity correlation function in Eq. (1) is as follows: y(r) = exp[−βφ(r)]g(r)
(3)
where β=
1 kT
If we assume that the reference fluid potential is that of an atom with a diameter σ equal to the distance at which the pair potential is exactly zero, and hold the bond length (σ) constant, Eq. (1) becomes Achain Amono = + (1 − m)ln[gmono (σ)] NC kT NC kT
(4)
where gmono (σ) is the radial distribution function of the monomer fluid, evaluated at a separation distance σ. Sear and Jackson developed extensions that account for ring formation in hard chain fluids from the Zhou–Stell zeroth-order theory [2] and the Kirkwood superposition approximation. Ghonasgi et al. considered the more general problem of intramolecular association in a fluid of freely-jointed hard sphere chains [4]. Johnson extended this theory to LJ monomer reference fluid and LJ dimer reference fluid [1]. The expression for the free energy of a fluid of rings in terms of a monomer reference fluid is Aring Amono = + (1 − m)ln[gmono (σ)] − ln[ωee (σ)] NC kT NC kT
(5)
where ωee (σ) is the intramolecular end-to-end correlation function evaluated at contact. The expression for the free energy of a fluid of rings in terms of a dimer reference fluid is Aring Amono m = − ln[gmono (σ)] NC kT NC kT 2 m dimer + 1− ln[gee (σ)] − ln[ωee (σ)] 2
(6)
Johnson assumed that the intramolecular end-to-end correlation function is the same as the reference fluid radial distribution function, and thus used the form of Jackson [1]. In Eqs. (5) and (6), Amono is calculated from the LJ equation of state derived by Nicolas et al. [12], gmono (σ) is provided dimer (σ) in Johnson [1]. in Johnson et al. [13] and gee
In this work, we performed NVT Monte-Carlo simulations to obtain the intramolecular end-to-end correlation function data for the freely-jointed LJ chains under various conditions. We obtained the compressibility factor from Eqs. (5) and (6) with the thermodynamic relation equation. The expression for the compressibility factor of a fluid of rings in terms of a monomer reference fluid is ∂ ln gmono (σ) mρ ∂Amono − (m − 1) 1 + ρ Z =m+ T ∂ρ ∂ρ ∂ ln wee (σ) −ρ (7) ∂ρ and in terms of a dimer reference fluid is mρ ∂Amono m ∂ ln gmono (σ) − (m − 1) − ρ T ∂ρ 2 ∂ρ dimer m ∂ ln gee (σ) ∂ ln wee (σ) − 1− ρ −ρ 2 ∂ρ ∂ρ
Z =m+
(8)
3. Computer simulations In this study, we consider three systems of tangent, freely-jointed Lennard–Jones flexible chains composed of three, six and eight segments. Each segment in a chain is represented by a LJ sphere; nonbonded segment–segment interactions are given by the 6–12 Lennard–Jones potential σ 12 σ 6 φLJ (r) = 4ε − (9) r r where r is the intermolecular distance, ε is the potential well depth, and σ is the distance at which the LJ potential is zero. The bonded interactions are modeled using simple harmonic potential φbond (r) = 21 κ(r − l)2
(10)
where r is the center–center distance between two bonded segments, κ is the spring constant, and l is the bond length; in our simulations we chose l = σ and κσ 2 /ε = 3000. Chain simulations were performed in the canonical (NVT) ensemble, and the simulations for the chain molecules were performed by using the configurational-bias Monte-Carlo technique, which reduces the time-consuming problem and speeds up the sampling of the complex conformations, using the standard Metropolis Monte-Carlo technique. The simulation cell is a cube with edges of length L. Periodic boundary conditions are employed in all three directions. The system consists of N = 128 chains in the case of 3-, 6- and 8-mers. The periodic length is adjusted to half the box length. Using the above conditions, we performed NVT Monte-Carlo simulations to obtain intramolecular end-to-end correlation function(wee (σ)) data for three, six and eight segment chains.
J. An, H. Kim / Fluid Phase Equilibria 219 (2004) 61–65
63
Table 1 Simulation results (wee (σ)×102 ) for flexible chains containing three beads
Table 4 The correlation parameters(aijk ) in the Eq. (11)
ρ∗
T∗
j
2
3
4
5
6
7
8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.138 1.188 1.209 1.254 1.309 1.438 1.656 1.889 2.361
1.501 1.536 1.617 1.698 1.797 1.981 2.205 2.462 3.120
1.754 1.828 1.886 1.998 2.148 2.321 2.638 2.954 3.564
1.938 2.009 2.096 2.231 2.365 2.596 2.867 3.287 3.948
2.069 2.169 2.256 2.377 2.544 2.752 3.083 3.498 3.996
2.184 2.274 2.380 2.534 2.685 2.887 3.201 3.600 4.121
2.280 2.357 2.473 2.606 2.807 3.018 3.313 3.713 4.213
Table 2 Simulation results (wee (σ) × 103 ) for flexible chains containing six beads ρ∗
i 0
1
2
k =0 0 1 2
0.02956 −0.22653 0.52199
−0.21574 2.23415 −5.44341
0.61900 −6.34200 15.4801
−0.48334 4.97624 −12.2936
k =1 0 1 2
−0.34187 2.39480 −5.36216
2.15486 −22.3815 54.7138
−6.16867 63.3703 −155.451
4.84641 −49.8115 123.699
k =2 0 1 2
1.02111 −5.74899 11.9253
−4.52352 48.0432 −116.720
13.2151 −136.700 332.039
4
5
6
7
8
2.027 2.143 2.333 2.589 3.073 3.568 4.405 6.015
2.191 2.237 2.532 2.846 3.294 3.882 4.845 6.186
2.244 2.424 2.673 2.958 3.508 4.205 5.027 6.386
2.238 2.514 2.751 3.128 3.513 4.335 5.101 6.701
2.344 2.501 2.888 3.207 3.650 4.484 5.365 6.777
The intramolecular end-to-end correlation function is proportional to the probability density of finding two end sites i and j on the same chain a distance σ apart, where σ is the bonding length. The simulations for each chain composed of three, six and eight segments were performed at various densities ranging from 0.1 to 0.9 and temperatures ranging from two to eight. Our simulation results are shown in Tables 1–3. We correlated our simulation results as a function of density, temperature and the number of segments in a chain, as follows:
where T∗ is the reduced temperature, ρ∗ is the reduced density, m is the segment number of a chain and aijk is the correlation parameter, given by Table 4.
4. Results and discussion We performed NVT Monte-Carlo simulations to obtain the intramolecular end-to-end correlation function data for Lennard–Jones flexible chains under extensive conditions. Eq. (11) correlated the simulation data with an AAD of 2.0% over the temperature and density range 2 ≤ T ∗ ≤ 8, 0.1 ≤ ρ∗ ≤ 0.9 for the 3-mer, 4 ≤ T ∗ ≤ 8, 0.1 ≤ ρ∗ ≤ 0.8 for the 6-mer, 3 ≤ T ∗ ≤ 8, 0.1 ≤ ρ∗ ≤ 0.9 for the 8-mer. 16 14 at T* = 2 at T* = 5
12
from eq. (7) with
wee (σ) =
2 2 3
ωee(σ) = gmono(σ)
from eq. (7)
10
aijk ρ∗i T ∗−j m−k
(11)
i=0 j=0 k=0
P*
Table 3 Simulation results (wee (σ)×103 ) for flexible chains containing eight beads
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
−10.2445 107.559 −264.621
T∗
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
ρ∗
3
8 6 4
T∗ 3
4
5
6
7
8
1.153 1.164 1.236 1.306 1.645 1.791 2.491 2.808 3.249
1.194 1.270 1.327 1.558 1.852 2.056 2.904 3.082 5.212
1.212 1.284 1.485 1.578 1.917 2.399 2.989 3.570 4.066
1.185 1.309 1.493 1.700 2.047 2.395 3.221 3.786 5.855
1.213 1.362 1.556 1.775 2.150 2.606 2.924 4.423 4.861
1.231 1.433 1.607 1.901 2.207 2.792 3.524 4.255 5.018
2 0 -2 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ρ* Fig. 1. Pressures on two isotherms for the flexible 3-mer ring fluid. The symbols are simulation data [1]. The lines are the monomer reference theory.
64
J. An, H. Kim / Fluid Phase Equilibria 219 (2004) 61–65 16
16
14
14
at T* = 3 at T* = 4 at T* = 6
12
from eq. (8) with ω ee(σ) = gdimer(σ)
at T* = 2 at T* = 5
12
from eq. (8) with ω ee(σ) = gdimer(σ
10
P*
from eq. (8) 10
from eq. (8)
8
P*
8
6
6
4
4
2
2
0
0
-2
-2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
Fig. 2. Pressures on two isotherms for the flexible 3-mer ring fluid. The symbols are simulation data [1]. The lines are the dimer reference theory.
We applied the correlated function to the monomer reference fluid theory for rings, Eq. (7), and the dimer reference fluid theory for rings, Eq. (8), to develop the equation of state for Lennard–Jones flexible rings. The performances of these equations were compared with the MD data of Johnson [1] for each ring fluid composed of three and eight segments. Figs. 1 and 2 show the pressures on two isotherms for a flexible 3-mer ring fluid. In Fig. 1, the lines were ob-
16 14
at T* = 3 at T* = 4 at T* = 6
12
from eq. (7) with ω ee(σ) = gmono(σ from eq. (7)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 4. Pressures on three isotherms for the flexible 8-mer ring fluid. The symbols are simulation data [1]. The lines are the dimer reference theory.
tained from the monomer reference theory, and show that the solid line (calculated from the Eq. (7)) is in much better agreement with the simulation data than the dashed double dotted line(calculated from the Eq. (7), which is assumed ωee (σ) = gmono (σ)). In Fig. 2, the lines were obtained from dimer reference theory. For rings composed of three beads, the monomer theory is more accurate than the dimer theory, except at highest densities. Figs. 3 and 4 show the pressures on three isotherms for a flexible 8-mer ring fluid. In Fig. 3, the solid line is in much better agreement with the simulation data than the dashed double dotted line. In Fig. 4, the solid line also show a good result. For rings composed of eight segments, the dimer theory is more accurate than the monomer theory.
5. Conclusions
10
P*
0.2
ρ∗
ρ*
We performed NVT simulations for freely-jointed Lennard–Jones chains and obtained intramolecular endto-end correlation function data, which is correlated as a function of temperature, density and the number of segments in a chain. We applied it to TPT and obtained an equation of state which shows good agreements with the MD data of Johnson [1] for the Lennard–Jones flexible ring fluid.
8 6 4 2 0 -2 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ρ∗ Fig. 3. Pressures on three isotherms for the flexible 8-mer ring fluid. The symbols are simulation data [1]. The lines are the monomer reference theory.
List of symbols aijk correlation parameter in Eq. (11) Achain residual Helmholz free energy of Amono residual Helmholz free energy of Aref residual Helmholz free energy of Aring residual Helmholz free energy of
chain fluid monomer fluid reference fluid ring fluid
J. An, H. Kim / Fluid Phase Equilibria 219 (2004) 61–65
the percent absolute average deviation for a property X is given by 100/n ni=1 |(Xi − Xicalc )/Xi |, where Xi is the “exact” value, Xicalc is the correlated value, and n is the number of data points g(r) radial distribution function gmono (σ) radial distribution function of the monomer fluid, evaluated at a separation distance σ dimer (σ) dimer site–site pair correlation function gee evaluated at contact k Boltzmann’s constant L edge length of the simulation box m number of segments a chain NC number of chain molecules P∗ reduced pressure (Pσ 3 /ε) T temperature (K) T∗ reduced temperature (kT/ε) yref (r) reference fluid cavity correlation function evaluated at a separation distance of r yref (m) (1 · · · m) m-body cavity correlation function
ωee (σ)
Greek letters β 1/kT ε potential well depth φ(r) potential energy κ spring constant ρ density ρ∗ reduced atomic number density (mρσ 3 ) σ bonding length
[8] [9] [10] [11]
AAD
65
intramolecular end-to-end correlation function evaluated at contact
Acknowledgements The authors are grateful to the support of BK21 project of Ministry of Education and the National Research Laboratory Program of Korea Institute of Science & Technology Evaluation and Planning.
References [1] [2] [3] [4] [5] [6] [7]
[12] [13]
J.K. Johnson, J. Chem. Phys. 104 (1996) 1729–1742. R.P. Sear, G. Jackson, Mol. Phys. 81 (1994) 801–811. R.P. Sear, G. Jackson, Phys. Rev. E50 (1994) 386–394. D. Ghonasgi, V. Perez, W.G. Chapman, J. Chem. Phys. 101 (1994) 6880–6887. D. Ghonasgi, W.G. Chapman, J. Chem. Phys. 102 (1995) 2585–2592. (a) M.S. Wertheim, J. Chem. Phys. 85 (1986) 2929–2936; (b) M.S. Wertheim, J. Chem. Phys. 87 (1987) 7323–7331. W.G. Chapman, G. Jackson, K.E. Gubbins, Mol. Phys. 65 (1988) 1057–1079. G. Stell, Y. Zhou, J. Chem. Phys. 91 (1989) 3618–3623. Y. Zhou, G. Stell, J. Chem. Phys. 96 (1992) 1504–1506. Y. Zhou, G. Stell, J. Chem. Phys. 96 (1992) 1507–1515. G. Stell, in: H.L. Frisch, J.L. Lebowitz (Eds.), The Equilibrium Theory of Classical Fluids, Benjamin, New York, 1964. J.J. Nicolas, K.E. Gubbins, W.B. Street, D.J. Tildesley, Mol. Phys. 37 (1979) 1429–1454. J.K. Johnson, E.A. Muller, K.E. Gubbins, J. Phys. Chem. 18 (1994) 6413–6419.
Equation of state for Lennard–Jones flexible ring fluids Jichul An, Hwayong Kim∗ School of Chemical Engineering & Institute of Chemical Process, Seoul National University, SAN 56-1, Shilim-Dong, Gwanak-Gu, Seoul 151-742, Republic of Korea Received 8 September 2003
Abstract We present a new equation of state for Lennard–Jones (LJ) flexible ring fluids. We perform Monte-Carlo simulations for freely-jointed Lennard–Jones chain fluids (3-, 6- and 8-mer) in the canonical ensemble and obtain the intramolecular end-to-end pair correlation function data under extensive density and temperature conditions. We correlate these as a function of the density, the temperature and the number of segments in a chain. We apply this function to thermodynamic perturbation theory (TPT) and obtain a new equation of state for Lennard–Jones flexible ring fluids. We also compare existing simulation data [J. Chem. Phys. 104 (1996) 1729] with the results obtained using the newly derived equation of state. © 2004 Elsevier B.V. All rights reserved. Keywords: Lennard–Jones flexible ring; Molecular simulation; Equation of state; Monte-Carlo simulation; Thermodynamic perturbation theory; Intramolecular end-to-end correlation function
1. Introduction Little interest has been shown recently in cyclic molecules because of their complexity to describe the thermodynamic behavior, although a number of cyclic molecules are widely used in the chemical and petrochemical industries. Thus, we undertook to develop an equation of state for cyclic molecules. Sear and Jackson [2,3] and Ghonasgi and co-workers [4,5] independently developed extensions to the Zhou–Stell and Wertheim theories that describe ring formation in hard chain fluids. Sear and Jackson considered only the formation of athermal ring molecules starting from Zhou–Stell zeroth-order theory, in which no distinction is made between the bonds that make up the linear chain and the bond that forms a ring from a linear chain [2]. Ghonasgi et al. considered an athermal chain with bonding sites on the terminal segments, and Johnson extended this theory to Lennard–Jones (LJ) monomer and LJ dimer reference fluids [1]. However, Johnson assumed that the intramolecular end-to-end correlation function is the same as the reference fluid radial distribution function.
∗
Corresponding author. Tel.: +82-2-880-7406; fax: +82-2-888-6695. E-mail address: [email protected] (H. Kim).
0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2004.01.016
In the present study, we performed NVT Monte-Carlo simulations of the freely-jointed LJ flexible chains composed of three, six and eight segments and obtained intramolecular end-to-end correlation function data under extensive conditions. We correlated these as a function of density, temperature and the number of segments in a chain. We applied this function to thermodynamic perturbation theory (TPT) and obtained a new equation of state for Lennard–Jones flexible ring fluids.
2. Thermodynamic perturbation theory (TPT) for ring The equation of state for LJ chains is derived in the same manner as that for hard sphere chains [6,7]. The first-order thermodynamic perturbation theory solution only requires knowledge of the pair distribution function of the reference fluid. For chains, the resulting expression for the residual Helmholtz free energy of a linear chain fluid is Achain Aref = + (1 − m)ln[yref (r)] NC kT NC kT
(1)
where NC is the number of chain molecules, k is Boltzmann’s constant, T is the temperature in Kelvin, Aref is the reference fluid residual free energy, m is the number of reference fluid
62
J. An, H. Kim / Fluid Phase Equilibria 219 (2004) 61–65
monomers in a chain, and yref (r) is the reference fluid cavity correlation function evaluated at a separation distance of r. The Eq. (1) may also be obtained from Stell and Zhou [8–10] study of an ionic fluid. The resulting equation of the Stell and Zhou theory is Achain Aref = − ln[yref(m) (1 · · · m)] NC kT NC kT
(2)
where yref(m) (1 · · · m) is the m-body cavity correlation function. The use of the linear approximation [11] of the yref(m) (1 · · · m) and the constant bond length (r) lead to the expression that is equivalent to TPT1. The cavity correlation function in Eq. (1) is as follows: y(r) = exp[−βφ(r)]g(r)
(3)
where β=
1 kT
If we assume that the reference fluid potential is that of an atom with a diameter σ equal to the distance at which the pair potential is exactly zero, and hold the bond length (σ) constant, Eq. (1) becomes Achain Amono = + (1 − m)ln[gmono (σ)] NC kT NC kT
(4)
where gmono (σ) is the radial distribution function of the monomer fluid, evaluated at a separation distance σ. Sear and Jackson developed extensions that account for ring formation in hard chain fluids from the Zhou–Stell zeroth-order theory [2] and the Kirkwood superposition approximation. Ghonasgi et al. considered the more general problem of intramolecular association in a fluid of freely-jointed hard sphere chains [4]. Johnson extended this theory to LJ monomer reference fluid and LJ dimer reference fluid [1]. The expression for the free energy of a fluid of rings in terms of a monomer reference fluid is Aring Amono = + (1 − m)ln[gmono (σ)] − ln[ωee (σ)] NC kT NC kT
(5)
where ωee (σ) is the intramolecular end-to-end correlation function evaluated at contact. The expression for the free energy of a fluid of rings in terms of a dimer reference fluid is Aring Amono m = − ln[gmono (σ)] NC kT NC kT 2 m dimer + 1− ln[gee (σ)] − ln[ωee (σ)] 2
(6)
Johnson assumed that the intramolecular end-to-end correlation function is the same as the reference fluid radial distribution function, and thus used the form of Jackson [1]. In Eqs. (5) and (6), Amono is calculated from the LJ equation of state derived by Nicolas et al. [12], gmono (σ) is provided dimer (σ) in Johnson [1]. in Johnson et al. [13] and gee
In this work, we performed NVT Monte-Carlo simulations to obtain the intramolecular end-to-end correlation function data for the freely-jointed LJ chains under various conditions. We obtained the compressibility factor from Eqs. (5) and (6) with the thermodynamic relation equation. The expression for the compressibility factor of a fluid of rings in terms of a monomer reference fluid is ∂ ln gmono (σ) mρ ∂Amono − (m − 1) 1 + ρ Z =m+ T ∂ρ ∂ρ ∂ ln wee (σ) −ρ (7) ∂ρ and in terms of a dimer reference fluid is mρ ∂Amono m ∂ ln gmono (σ) − (m − 1) − ρ T ∂ρ 2 ∂ρ dimer m ∂ ln gee (σ) ∂ ln wee (σ) − 1− ρ −ρ 2 ∂ρ ∂ρ
Z =m+
(8)
3. Computer simulations In this study, we consider three systems of tangent, freely-jointed Lennard–Jones flexible chains composed of three, six and eight segments. Each segment in a chain is represented by a LJ sphere; nonbonded segment–segment interactions are given by the 6–12 Lennard–Jones potential σ 12 σ 6 φLJ (r) = 4ε − (9) r r where r is the intermolecular distance, ε is the potential well depth, and σ is the distance at which the LJ potential is zero. The bonded interactions are modeled using simple harmonic potential φbond (r) = 21 κ(r − l)2
(10)
where r is the center–center distance between two bonded segments, κ is the spring constant, and l is the bond length; in our simulations we chose l = σ and κσ 2 /ε = 3000. Chain simulations were performed in the canonical (NVT) ensemble, and the simulations for the chain molecules were performed by using the configurational-bias Monte-Carlo technique, which reduces the time-consuming problem and speeds up the sampling of the complex conformations, using the standard Metropolis Monte-Carlo technique. The simulation cell is a cube with edges of length L. Periodic boundary conditions are employed in all three directions. The system consists of N = 128 chains in the case of 3-, 6- and 8-mers. The periodic length is adjusted to half the box length. Using the above conditions, we performed NVT Monte-Carlo simulations to obtain intramolecular end-to-end correlation function(wee (σ)) data for three, six and eight segment chains.
J. An, H. Kim / Fluid Phase Equilibria 219 (2004) 61–65
63
Table 1 Simulation results (wee (σ)×102 ) for flexible chains containing three beads
Table 4 The correlation parameters(aijk ) in the Eq. (11)
ρ∗
T∗
j
2
3
4
5
6
7
8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1.138 1.188 1.209 1.254 1.309 1.438 1.656 1.889 2.361
1.501 1.536 1.617 1.698 1.797 1.981 2.205 2.462 3.120
1.754 1.828 1.886 1.998 2.148 2.321 2.638 2.954 3.564
1.938 2.009 2.096 2.231 2.365 2.596 2.867 3.287 3.948
2.069 2.169 2.256 2.377 2.544 2.752 3.083 3.498 3.996
2.184 2.274 2.380 2.534 2.685 2.887 3.201 3.600 4.121
2.280 2.357 2.473 2.606 2.807 3.018 3.313 3.713 4.213
Table 2 Simulation results (wee (σ) × 103 ) for flexible chains containing six beads ρ∗
i 0
1
2
k =0 0 1 2
0.02956 −0.22653 0.52199
−0.21574 2.23415 −5.44341
0.61900 −6.34200 15.4801
−0.48334 4.97624 −12.2936
k =1 0 1 2
−0.34187 2.39480 −5.36216
2.15486 −22.3815 54.7138
−6.16867 63.3703 −155.451
4.84641 −49.8115 123.699
k =2 0 1 2
1.02111 −5.74899 11.9253
−4.52352 48.0432 −116.720
13.2151 −136.700 332.039
4
5
6
7
8
2.027 2.143 2.333 2.589 3.073 3.568 4.405 6.015
2.191 2.237 2.532 2.846 3.294 3.882 4.845 6.186
2.244 2.424 2.673 2.958 3.508 4.205 5.027 6.386
2.238 2.514 2.751 3.128 3.513 4.335 5.101 6.701
2.344 2.501 2.888 3.207 3.650 4.484 5.365 6.777
The intramolecular end-to-end correlation function is proportional to the probability density of finding two end sites i and j on the same chain a distance σ apart, where σ is the bonding length. The simulations for each chain composed of three, six and eight segments were performed at various densities ranging from 0.1 to 0.9 and temperatures ranging from two to eight. Our simulation results are shown in Tables 1–3. We correlated our simulation results as a function of density, temperature and the number of segments in a chain, as follows:
where T∗ is the reduced temperature, ρ∗ is the reduced density, m is the segment number of a chain and aijk is the correlation parameter, given by Table 4.
4. Results and discussion We performed NVT Monte-Carlo simulations to obtain the intramolecular end-to-end correlation function data for Lennard–Jones flexible chains under extensive conditions. Eq. (11) correlated the simulation data with an AAD of 2.0% over the temperature and density range 2 ≤ T ∗ ≤ 8, 0.1 ≤ ρ∗ ≤ 0.9 for the 3-mer, 4 ≤ T ∗ ≤ 8, 0.1 ≤ ρ∗ ≤ 0.8 for the 6-mer, 3 ≤ T ∗ ≤ 8, 0.1 ≤ ρ∗ ≤ 0.9 for the 8-mer. 16 14 at T* = 2 at T* = 5
12
from eq. (7) with
wee (σ) =
2 2 3
ωee(σ) = gmono(σ)
from eq. (7)
10
aijk ρ∗i T ∗−j m−k
(11)
i=0 j=0 k=0
P*
Table 3 Simulation results (wee (σ)×103 ) for flexible chains containing eight beads
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
−10.2445 107.559 −264.621
T∗
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
ρ∗
3
8 6 4
T∗ 3
4
5
6
7
8
1.153 1.164 1.236 1.306 1.645 1.791 2.491 2.808 3.249
1.194 1.270 1.327 1.558 1.852 2.056 2.904 3.082 5.212
1.212 1.284 1.485 1.578 1.917 2.399 2.989 3.570 4.066
1.185 1.309 1.493 1.700 2.047 2.395 3.221 3.786 5.855
1.213 1.362 1.556 1.775 2.150 2.606 2.924 4.423 4.861
1.231 1.433 1.607 1.901 2.207 2.792 3.524 4.255 5.018
2 0 -2 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ρ* Fig. 1. Pressures on two isotherms for the flexible 3-mer ring fluid. The symbols are simulation data [1]. The lines are the monomer reference theory.
64
J. An, H. Kim / Fluid Phase Equilibria 219 (2004) 61–65 16
16
14
14
at T* = 3 at T* = 4 at T* = 6
12
from eq. (8) with ω ee(σ) = gdimer(σ)
at T* = 2 at T* = 5
12
from eq. (8) with ω ee(σ) = gdimer(σ
10
P*
from eq. (8) 10
from eq. (8)
8
P*
8
6
6
4
4
2
2
0
0
-2
-2
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.0
0.1
Fig. 2. Pressures on two isotherms for the flexible 3-mer ring fluid. The symbols are simulation data [1]. The lines are the dimer reference theory.
We applied the correlated function to the monomer reference fluid theory for rings, Eq. (7), and the dimer reference fluid theory for rings, Eq. (8), to develop the equation of state for Lennard–Jones flexible rings. The performances of these equations were compared with the MD data of Johnson [1] for each ring fluid composed of three and eight segments. Figs. 1 and 2 show the pressures on two isotherms for a flexible 3-mer ring fluid. In Fig. 1, the lines were ob-
16 14
at T* = 3 at T* = 4 at T* = 6
12
from eq. (7) with ω ee(σ) = gmono(σ from eq. (7)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 4. Pressures on three isotherms for the flexible 8-mer ring fluid. The symbols are simulation data [1]. The lines are the dimer reference theory.
tained from the monomer reference theory, and show that the solid line (calculated from the Eq. (7)) is in much better agreement with the simulation data than the dashed double dotted line(calculated from the Eq. (7), which is assumed ωee (σ) = gmono (σ)). In Fig. 2, the lines were obtained from dimer reference theory. For rings composed of three beads, the monomer theory is more accurate than the dimer theory, except at highest densities. Figs. 3 and 4 show the pressures on three isotherms for a flexible 8-mer ring fluid. In Fig. 3, the solid line is in much better agreement with the simulation data than the dashed double dotted line. In Fig. 4, the solid line also show a good result. For rings composed of eight segments, the dimer theory is more accurate than the monomer theory.
5. Conclusions
10
P*
0.2
ρ∗
ρ*
We performed NVT simulations for freely-jointed Lennard–Jones chains and obtained intramolecular endto-end correlation function data, which is correlated as a function of temperature, density and the number of segments in a chain. We applied it to TPT and obtained an equation of state which shows good agreements with the MD data of Johnson [1] for the Lennard–Jones flexible ring fluid.
8 6 4 2 0 -2 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
ρ∗ Fig. 3. Pressures on three isotherms for the flexible 8-mer ring fluid. The symbols are simulation data [1]. The lines are the monomer reference theory.
List of symbols aijk correlation parameter in Eq. (11) Achain residual Helmholz free energy of Amono residual Helmholz free energy of Aref residual Helmholz free energy of Aring residual Helmholz free energy of
chain fluid monomer fluid reference fluid ring fluid
J. An, H. Kim / Fluid Phase Equilibria 219 (2004) 61–65
the percent absolute average deviation for a property X is given by 100/n ni=1 |(Xi − Xicalc )/Xi |, where Xi is the “exact” value, Xicalc is the correlated value, and n is the number of data points g(r) radial distribution function gmono (σ) radial distribution function of the monomer fluid, evaluated at a separation distance σ dimer (σ) dimer site–site pair correlation function gee evaluated at contact k Boltzmann’s constant L edge length of the simulation box m number of segments a chain NC number of chain molecules P∗ reduced pressure (Pσ 3 /ε) T temperature (K) T∗ reduced temperature (kT/ε) yref (r) reference fluid cavity correlation function evaluated at a separation distance of r yref (m) (1 · · · m) m-body cavity correlation function
ωee (σ)
Greek letters β 1/kT ε potential well depth φ(r) potential energy κ spring constant ρ density ρ∗ reduced atomic number density (mρσ 3 ) σ bonding length
[8] [9] [10] [11]
AAD
65
intramolecular end-to-end correlation function evaluated at contact
Acknowledgements The authors are grateful to the support of BK21 project of Ministry of Education and the National Research Laboratory Program of Korea Institute of Science & Technology Evaluation and Planning.
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