A new analytical perturbed equation of state for hard chain fluids with attractive potentials of variable range

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Chemical Physics 348 (2008) 1–10 www.elsevier.com/locate/chemphys

A new analytical perturbed equation of state for hard chain fluids with attractive potentials of variable range Hossein Farrokhpour *, Elham Satarinezhad Department of Chemistry, Faculty of Science, Yazd University, Yazd 89195-741, Iran Received 3 October 2007; accepted 6 February 2008 Available online 9 February 2008

Abstract A completely analytical perturbation theory equation of state for hard chain fluids is derived. The derived equation of state can represent the thermodynamic properties of many kinds of hard chain fluids such as the square-well chain and Yukawa chain fluid, by varying of its parameters. The predicted results are in good agreement with both the molecular simulation data and the well-known equations of state. In this paper the hard chain molecules are modelled as a pearl necklace of freely jointed spheres that interact via site–site intermolecular potential. Our first- and second-order perturbation terms are based on the Barker–Henderson local compressibility approximation and Gulati–Hall’s perturbation theory, respectively. To obtain the perturbation terms, we do not require knowledge of site–site radial distribution function of the reference hard chain fluid, which is obtained from molecular dynamic simulation, or any explicit mathematical form of it. This perturbation method yields a simple and general analytical expression for each thermodynamic property of hard chain fluids. The most important feature of this equation of state is that it has no adjustable parameters and in some regions in which there is no simulation data for such fluids, such equation may be used to predict the needed data. Ó 2008 Elsevier B.V. All rights reserved. Keywords: Equation of state; Square-well chain fluid; Yukawa chain fluid; Barker–Henderson perturbation theory; Radial distribution function

1. Introduction The development of an analytic equation of state (EOS) for pure fluids and their mixtures is of theoretical and practical relevance for modelling the thermodynamic properties of simple and complex molecules. In recent years there has been increasing interest in molecular-based theories in which idealized chain fluids, such as hard-sphere chains [1–7], square-well chains [8–10], and Lennard–Jones chains[11–18], are used to describe the thermophysical properties of chainlike molecules. The advantage of using these simplified molecular models is the possibility of obtaining better understanding of the size, shape and attractive potential contributions to thermodynamic properties, and to test the statistical mechanical approximations used to obtain the equation of state by comparing the *

Corresponding author. Tel.: +98 351 8211670; fax: +98 351 8210644. E-mail address: [email protected] (H. Farrokhpour).

0301-0104/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2008.02.002

resulting thermophysical properties to molecular simulation data. Hard-core potential models have been widely used to theoretically describe a broad range of fluids in liquid state physics [19]. It has found success in modelling the thermodynamics of simple liquids, colloidal suspensions, electrolytes and molten salts. Hard-core potential functions, such as the hard sphere (HS), square-well (SW), hard-core Lennard–Jones (HCLJ) [20] and hard-core Yukawa (HCY) are empirical in nature, and approximate to the intermolecular forces in real fluids. Recently a new extended SW potential function (ESW) has been introduced which is a compromise between simplicity, reality and flexibility [21]. The potential function is defined by 8 1 r > < e r 6 r 6 kr  uðrÞ ¼ ð1Þ r3r r 6 ae kr < r 6 3r > > : ð3kÞr r 0 r > 3r

2

H. Farrokhpour, E. Satarinezhad / Chemical Physics 348 (2008) 1–10

Here, r is the distance between two atoms, e represents the well-depth (minimum potential energy), r is the hard-core diameter, k is the reduced well width and a is an additional parameter introduced to make the potential more flexible by changing the steepness of the potential tail. In our previous study, using the ESW potential along with the Barker–Henderson perturbation theory [22], a simple perturbed EOS for hard-core monomer fluids was derived, showing a good agreement with available simulation data for the compressibility factor (Z) and the internal energy of the hard-core monomer fluid models. The derived EOS is as follows [21]:     e oH 24aGy Z ¼ Z0  y þ 8k3 y  kT oy 3k     2   6yH oI 1 oI 2 oH  I þy I þy  6y 2 oy oy kT oy ðkT Þ  2   1 oI o2 I  6y 2 H 2 þy 2 ð2Þ kT oy oy in which 1 þ y þ y2  y3

ð3Þ ð1  y Þ3 1 1 1 G¼ 2 3 ð4Þ 54 2k k   e2 a2 e2 1 1 3 1  ðH þ 8k3 y  1Þ  I¼ þ  7 24y ð3  kÞ2 13; 024 k9 4k8 7k Z0 ¼

ð5Þ 4

H¼

ð1  yÞ 1 þ 4y þ 4y 2  4y 3 þ y 4

ð6Þ

where y ¼ p=6qr3 (q is the number density) stands for the so-called packing fraction. This EOS has some advantages; the first advantage is the unique feature of the EOS. It contains the same number of parameters and uses the same expression for different hard-core fluids that is due to the flexibility of the proposed potential [21]. For example, by varying parameter a in addition to parameters r, e and k, the attraction can be easily tuned to mimic different interactions encountered in many physical systems. To obtain good values for the ESW potential parameters, the attraction part of the ESW potential is mapped to the potential model encountered in physical system [21]. The secondly, this EOS has no adjustable parameters; therefore, in some regions that there is no simulation data for such hard-core fluids, it may be used to predict the needed data. So it is worthwhile to extend Eq. (2) to molecular fluids such as the hard chain fluids. In this work we are interested in applying the ESW potential function to hard chain fluid models and the extension of the Eq. (2) to them. In this paper the hard chain fluid is considered in which each chain molecule consists of n tangentially connected segments that interact via site–site ESW intermolecular potential [23]. Throughout this work we assume that the non bonded segments on the same chain interact via hard sphere potential.

In general, the microscopic structure of an isotropic fluid is described in terms of the radial distribution function (RDF) [24,25]. For simple monoatomic fluids, the RDF function gives the probability of finding two particle separated by a given distance. In the case of molecular fluids, the two particle distribution function contains information about both positional and orientational correlation between molecules and is thus, no longer a simple radially symmetric function. In either case, the full two particle correlation function is sufficient to determine the thermodynamic properties of the molecular fluid. However, the task may be simplified by the use of the site–site radial distribution function (SSRDF), which describes positional correlations between specified atoms comprising the molecules. This SSRDF accounts for both intra- as well as interchain correlations. This radially symmetric SSRDF is easier to calculate than the full molecule–molecule distribution function and can be directly probed in X-ray and neutron scattering experiments. Although the SSRDF contains less information than the full molecular pair function, but it is sufficient to calculate many thermodynamic properties of molecular fluids [24,25]. Therefore, in this work, the Helmholtz free energy of the hard chain fluid is written using the SSRDF in the framework of perturbation theory. The Helmholtz free energy of the hard chain fluid is given based on the Barker–Henderson local compressibility approximation and the Gulati–Hall’s perturbation theory [26], by using the ESW potential function. The Gulati–Hall’s perturbation theory is the extension of the Chang and sandler perturbation theory approach to the square-well diatomic fluids [26]. In this perturbation approach the hard sphere diatomic fluid is taken as the reference system. The first and the second-order perturbation terms for the Helmholtz free energy of square-well diatomic fluid are written based on the number of attractive site–site interactions for each pair of molecule and the average SSRDF of the reference fluid. In this paper we have used ESW potential along with the Barker–Henderson perturbation theory similar to the Gulati–Hall’s perturbation theory applied for square-well diatomic fluids. This perturbation theoretical approach differs from that used in the statistical associating fluid theory (SAFT) [18] in that the latter considers nonassociated and nonbonded segments as reference system and introduces the perturbation term to account for the influence of segment–segment connectivity. In contrast in the perturbative scheme adopted in this work, the segments are already connected. The ability of this EOS to predict thermodynamic properties is evaluated by comparing the calculated compressibility factors from the EOS to Monte Carlo (MC) simulation results and well known equations of state. All of the simulation data, used for comparison, are outside the liquid–vapour phase envelope and above the critical point. 2. Perturbation theory and equation of state In this work, the hard sphere chain fluid is considered as reference fluid. The reference fluid consists of chain

H. Farrokhpour, E. Satarinezhad / Chemical Physics 348 (2008) 1–10

where AHSC represents contribution to the Helmholtz free energy from the reference hard sphere chain potential, A1 and A2 are the first- and second-order perturbation terms and represents Helmholtz free energy from the attractive perturbation potential. N c is the total number of hard chain molecules in the system; k denotes Boltzmann constant; and T is absolute temperature. In the absence of any intramolecular interactions (other than the excluded volume interactions present in the reference system), the first- and the second-order Barker–Henderson perturbation contributions to the Helmholtz free energy of the hard chain fluid can be written in a similar way to the work of Gulati and Hall [26,29,30] as following forms: Z 1 A1 2 ¼ 2n2 pqc r3 u1 ðx; e; k; aÞgHSC ð11Þ inter ðx; n; gÞx dx N c kT 1   A2 oqc o ¼ n2 pqc r3 kT HSC N c kT op T og  Z 1  2 2  g ðu1 ðx; e; k; aÞÞ gHSC ðx; n; gÞx dx ð12Þ inter

molecules formed from spherical segments with intra and intermolecular interaction that interact through the hard sphere potential [27,28]. The ESW potential model may be written as uij ðrÞ ¼ uij0 ðrÞ þ uij1 ðr; e; k; aÞ

3

ð7Þ

where uij ðrÞ is the potential between sites i and j on the molecules 1 and 2 at separation r. Here, uij0 ðrÞ denotes the hard sphere interaction and is defined by  1 r < e  ij r3r r 6 kr < r 6 3r ð9Þ u1 ðr; e; k; aÞ ¼ ae ð3kÞr r > : 0 r > 3r In Eqs. (8) and (9), r represents the distance between two nonbonded segments in two different molecule; e, k and a are parameters of the introduced potential model as Eq. (1). The concept that underlies the perturbation approach is that the Helmholtz free energy of a system can be expressed as an expansion in the inverse temperature around the free energy of a reference system whose structure and thermodynamic properties are known. In the framework of the second order perturbation theory, the Helmholtz free energy A of the hard chain fluid may be written as    2 A AHSC 1 A1 1 A2 ¼ þ þ ð10Þ N c kT N c kT kT N c kT kT N c kT

1

where x is defined as r=r and gHSC inter ðx; n; gÞ is the average hard sphere interchain SSRDF and depends on chain length n and segment packing fraction g ¼ npqc r3 =6. The function gHSC inter ðx; n; gÞ accounts for interchain correlations in a hard sphere chain system. The average interchain SSRDF is related to the site–site radial distribution function gab inter ðx; gÞ by n X n 1 X ðx; gÞ ¼ gab ðx; gÞ ð13Þ gHSC inter n2 a¼1 b¼1 inter

60 30

10

n= 2

n=4

n=8

50

25 8

*

*

*

T =3

T =3

T =3

T =2

20

*

40

*

*

T =2

T =2

6

Z

30 15 4 20 10 2 10

5 0

0

0 0.0

0.1

0.2

0.3

0.4

0.5

0.0

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

η Fig. 1. Comparison of the compressibility factor of the square-well dimer, 4-mer and 8-mer chains with k ¼ 1:5 calculated by this work (—) to MC simulation data (symbols) at different temperatures. The MC simulations data of square-well dimer, is taken from Ref. [36] and the MC simulation results of 4-mer and 8-mer chains are from Ref. [41].

4

H. Farrokhpour, E. Satarinezhad / Chemical Physics 348 (2008) 1–10

Eqs. (11) and (12) are similar to the perturbation terms developed for spherical molecules except that the average hard sphere interchain SSRDF, gHSC inter ðx; n; gÞ is used in lieu of the hard sphere RDF. The function gHSC inter ðx; n; gÞ for hard sphere chain has been solved for in the context of the Percus–Yevick (PY) integral equation theory [31,32]. For simplicity and in order to avoid the numerical evaluation of the integrals in Eqs. (11) and (12), using PY numerical solutions for gHSC inter ðx; n; gÞ obtained by Chiew [31–33], and to have an analytical EOS for the hard chain fluids, gHSC inter ðx; n; gÞ is simply given by 8 x<1 <0 gHSC ðx; n; gÞ ¼ g ðx; n; gÞ 16x6k ð14Þ inter : 1 1 x>k

Helmholtz free energy can be written in the following forms: ! A1 k3  27k þ 54 ¼  12nge I 1 þ a ð15Þ N c kT 54k3 ð3  kÞ   A2 oqc oðgI 2 Þ 2 ð16Þ ¼  6nge kT N c kT opHSC T og

In Eq. (14) we have considered a general form for gHSC inter ðx; n; gÞ and there is no need to know the mathematical form of it. By using the general form of gHSC inter ðx; n; gÞ, we can calculate the integrals in Eqs. (11) and (12) analytically. By substituting of Eqs. (9) and (14) into Eqs. (11) and (12), the first- and second-order perturbation contribution to the

ð18Þ

where Z I1 ¼

k

g1 ðx; n; gÞx2 dx 1   a2 1 I2 ¼ I1 þ 2 ð3  kÞ 551; 124   78; 732k2  413; 343k þ 551; 124  k9  k9 and   oqc 1

kT ¼ oZ opHSC T g HSC þ Z HSC og

ð17Þ

ð19Þ

T

Table 1 Calculated compressibility factor of SW dimer fluid with k ¼ 1:5, compared to that given by the Generalized Flory of Yethiraj and Hall EOS (GFYH) [37], Gulati and Hall perturbation theory EOS (GHPT) [26], thermodynamic perturbation theory EOS by Tavares (TPTT) [38], the local composition model using the generalized flory dimer theory EOS (LCGFD) [39], the statistical associating fluid theory with variable range EOS (SAFT-VR) [40] and the MC simulation data of Kim et al. [36] Simulation datab

This work

1.105 1.269 1.521 1.937 2.654 3.905 6.036 9.543

1.11(1) 1.28(2) 1.55(3) 2.01(5) 2.78(7) 4.07(10) 6.25(12) 9.75(22)

1.126 1.266 1.553 2.077 2.953 4.341 6.490 9.797

1.019 1.108 1.279 1.600 2.209 3.336 5.343 8.800

1.015 1.081 1.216 1.489 2.048 3.145 5.162 8.631

1.02(2) 1.11(3) 1.27(3) 1.59(5) 2.19(8) 3.32(9) 5.31(14) 8.95(22)

1.049 1.082 1.253 1.659 2.412 3.677 5.701 8.884

0.588 0.575 0.474 0.542 0.968 2.049 4.231

0.824 0.732 0.689 0.741 1.044 1.863 3.621

0.821 0.691 0.602 0.608 0.860 1.653 3.442

0.81(2) 0.72(4) 0.69(5) 0.77(6) 1.07(10) 1.83(12) 3.60(17)

0.886 0.707 0.657 0.831 1.344 2.363 4.139

0.816 0.141 0.197 0.399 0.245 0.608 2.677 6.729

0.604 0.339 0.096 0.104 0.097 0.421 1.934 5.191

0.606 0.280 0.016 0.256 0.294 0.200 1.760 5.071

0.56(4) 0.32(5) 0.18(9) 0.12(15) 0.08(11) 0.41(13) 1.97(15) 5.22(21)

0.711 0.325 0.065 0.015 0.293 1.069 2.597 5.280

T*

q*a

GH

4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1.127 1.308 1.586 2.045 2.823 4.131 6.302 9.832

1.118 1.293 1.564 2.010 2.770 4.067 6.248 9.846

0.278 1.203 1.470 1.950 2.789 4.211 6.562 10.375

1.108 1.289 1.574 2.034 2.800 4.084 6.218 9.726

3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1.052 1.139 1.306 1.636 2.28 3.458 5.526 9.012

1.037 1.115 1.265 1.569 2.173 3.320 5.392 8.963

0.377 0.996 1.139 1.481 2.182 3.490 5.785 9.645

2

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.892 0.795 0.754 0.855 1.249 2.184 4.057

0.864 0.748 0.668 0.698 1.001 1.856 3.711

1.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.721 0.445 0.214 0.112 0.288 1.008 2.699 5.998

0.675 0.370 0.069 0.158 0.143 0.431 2.075 5.536

a b

TPTT

LCGFD

GFYH

q ¼ nqr3 . The values in parenthesis represent the standard deviation in the last two significant digits.

SAFT-VR

H. Farrokhpour, E. Satarinezhad / Chemical Physics 348 (2008) 1–10 Table 2 Calculated compressibility factor of SW dimer fluids with k ¼ 1:3, compared to that given by TPTT and SAFT-VR equations of state and the MC simulation data of Kim et al. [36] T*

q*a

TPTT

SAFT-VR

Simulation datab

This work

4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1.232 1.543 1.964 2.552 3.400 4.668 6.625 9.732

1.221 1.519 1.921 2.480 3.291 4.519 6.445 9.533

1.22(1) 1.52(2) 1.94(3) 2.54(5) 3.39(6) 4.69(10) 6.70(12) 9.84(23)

1.253 1.513 1.918 2.563 3.561 5.072 7.344 10.775

3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1.192 1.450 1.798 2.286 3.004 4.111 5.883 8.803

1.175 1.416 1.744 2.203 2.886 3.956 5.703 8.615

1.18(1) 1.43(3) 1.77(3) 2.28(4) 3.03(8) 4.15(8) 5.98(13) 8.92(21)

1.221 1.410 1.739 2.303 3.219 4.647 6.837 10.186

2

0.1 0.2 0.3 0.4 0.5 0.6 0.7

1.105 1.256 1.461 1.755 2.219 3.010 4.418

1.073 1.198 1.378 1.646 2.082 2.847 4.245

1.08(2) 1.21(3) 1.42(5) 1.73(7) 2.23(10) 3.07(13) 4.55(14)

1.149 1.200 1.380 1.789 2.543 3.807 5.832

0.8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

6.967 1.010 1.054 1.119 1.225 1.443 1.926 2.977 5.159

6.803 0.959 0.963 0.998 1.084 1.287 1.762 2.819 5.024

7.12(19) 0.96(3) 0.97(4) 1.03(6) 1.17(7) 1.51(11) 2.01(13) 3.17(25) 5.38(25)

9.015 1.069 0.983 1.021 1.280 1.877 2.979 4.839 7.857

1.5

a

q ¼ nqr3 . The values in parenthesis represent the standard deviation in the last two significant digits.

5

where qc , pHSC and Z HSC are the number density, the pressure and the compressibility factor of the hard sphere chain fluid, respectively. Several models for the compressibility factor of the hard sphere chain fluid have been proposed in the literature. A common equation of state for hard sphere chain fluid is the SAFT equation of state, which is given below [34] Z HSC ¼ n

1 þ g þ g2  g3 ð1  gÞ

3

 ðn  1Þ

1 þ g  g2 =2 ð1  gÞð1  g=2Þ

ð20Þ

To obtain the analytic expression for the Helmholtz free energy A of the hard chain fluid, we need to solve the integral in Eq. (17). A number of thermodynamic properties of hard sphere chain fluid, such as isothermal compressibility, can be computed directly from the total average SSRDF HSC gHSC total ðx; n; gÞ [35]. The function gtotal ðx; n; gÞ accounts for both intra as well as interchain correlations in the hard sphere chain system. The isothermal compressibility is given directly as integral over the gHSC total ðx; n; gÞ via the isothermal compressibility equation [24,25]   Z 1  HSC  oqc kT ¼ 1 þ 4pqc r3 gtotal  1 x2 dx opHSC T 0 Z 1  HSC   1 þ 4pqc r3 ginter  1 x2 dx ð21Þ 0 HSC where we have approximate gHSC total ðx; n; gÞ with ginter ðx; n; gÞ in the isothermal compressibility equation. Using Eq. (14) along with the isothermal compressibility equation, we may obtain the following result:     Z k n 1 8

g1 ðx; n; gÞx2 dx ¼ þ gk3  1 24g g oZ HSC þ Z HSC n 1 og T

b

ð22Þ

Table 3 Calculated compressibility factor of SW 4-mer and 8-mer fluids with k ¼ 1:5, compared to that given by TPTT, LCGFD, GFYH and SAFT-VR equations of state and the MC simulation data of Paredes et al. [41] n

T*

ga

TPTT

LCGFD

GFYH

SAFT-VR

Simulation datab

This work

4

3

0.05 0.15 0.25 0.35 0.45 0.15 0.25 0.35 0.45 0.15 0.25 0.35 0.45 0.15 0.25 0.35 0.45

1.041 1.251 2.321 6.785 20.889 0.233 0.194 3.501 17.426 1.276 2.953 11.106 38.294 0.648 1.215 4.506 31.209

0.685 0.858 2.244 7.558 22.738 0.344 0.047 4.637 20.364 0.349 2.760 12.787 42.020 1.995 1.520 7.096 37.396

0.996 1.292 2.424 6.655 19.747 0.320 0.391 3.390 16.000 1.376 3.191 10.759 35.348 0.421 0.715 4.278 27.637

0.979 1.128 1.998 6.182 19.804 0.065 0.156 2.807 16.177 0.9995 2.215 9.718 35.942 1.024 2.011 2.916 28.563

1.00(11) 1.26(10) 2.34(16) 6.59(13) 20.62(39) 0.37(11) 0.62(19) 3.10(28) 16.60(32) 1.47(17) 3.19(29) 10.48(51) 36.90(93) 0.12(25) 0.19(22) 4.37(62) 29.05(122)

1.403 1.184 2.608 7.542 20.078 0.143 0.582 4.542 16.121 1.253 3.375 12.633 36.943 0.673 0.727 6.549 28.962

2

8

3

2

a b

g ¼ pnqr3 =6 is the segment packing fraction. The values in parenthesis represent the standard deviation in the last two significant digits.

6

H. Farrokhpour, E. Satarinezhad / Chemical Physics 348 (2008) 1–10

The compressibility factor of hard chain fluid can be obtained from the Helmholtz free energy through the following thermodynamic relation 0 1 0 1   o A1  2 o A2 N kT 1 c A þ 1 g@ N c kT A g@ Z ¼ Z HSC þ kT kT og og T

T

ð23Þ The most important advantage of the potential function and average interchain SSRDF introduced in this work is that the integrals in the perturbation terms A1 and A2 is easily calculated analytically. The integrals in Eqs. (11) and (12) are simplified to expressions containing I 1 (Eq. (17)). In fact I 1 is proportional to the number of hard sphere segments in r < r < kr attraction region about a central segment. As mentioned before, I 1 is easily calculated using isothermal compressibility equation. In the following sections of this paper, the thermodynamic properties of SWC and YC fluid is calculated by mapping Eq. (1) to the SW and Yukawa potential. In this case the Helmholtz free energy of the SWC and YC fluid is not derived directly, and we will not require numerical calculating Eqs. (11) and (12) by using SW and Yukawa potentials and their SSRDF numerically.

actions, and MC simulation data of Paredes et al. [41] for 4- and 8-mer SW chain fluid are included in Fig. 1. Tables 1 and 2 compare the compressibility factor of SW dimer fluid with k ¼ 1:5 and 1.3 with well known equations of state. The theoretical predictions based on the Generalized Flory equation of state of Yethiraj and Hall (GFYH) with intermolecular interaction [37], Gulati and Hall perturbation theory equation of state (GHPT) with intermolecular interaction [26], analytical thermodynamic perturbation theory equation of state by Tavares et al. (TPTT) with both intra and intermolecular interaction [38], the local composition model using the generalized Flory dimer theory (LCGFD) of Bokis et al. [39] and the statistical associating fluid theory with variable range (SAFT-VR) equation of state of Villages et al. [40] are included in Table 1 for k ¼ 1:5 and the predictions of TPTT and SAFT-VR equa-

Table 4 Calculated compressibility factor for Yukawa dimer, 4-mer and 8-mer fluid with k1 ¼ 1:8 at different reduced temperatures, compared to that given by first-order perturbation theory (FPT) [30], SAFT-VR equation of state and the MC simulation data of Wang et al. [30] n

T*

q*a

FPT

SAFT-VR

Simulation datab

This work

2

3

0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6

1.023 1.089 1.257 1.61 2.263 3.386 0.865 0.729 0.658 0.738 1.093 1.897 1.147 1.356 1.751 2.502 3.845 6.12 0.753 0.388 0.062 0.024 0.396 1.697 1.596 2.446 3.775 5.895 9.249 14.49 1.249 1.54 2.142 3.408 5.822 10.08

0.989 1.043 1.205 1.545 2.173 3.258 0.799 0.647 0.577 0.658 1.000 1.775 1.062 1.242 1.625 2.349 3.626 5.786 0.530 0.120 0.172 0.218 0.196 1.413 1.494 2.285 3.551 5.564 8.736 13.700 1.025 1.252 1.845 3.073 5.359 9.353

0.989(0.003) 1.089(0.012) 1.277(0.007) 1.656(0.008) 2.310(0.035) 3.520(0.001) 0.819(0.008) 0.720(0.010) 0.720(0.005) 0.852(0.009) 1.230(0.008) 2.105(0.009) 1.083(0.015) 1.282(0.009) 1.692(0.020) 2.488(0.016) 3.866(0.043) 6.228(0.041) 0.624(0.087) 0.382(0.058) 0.213(0.070) 0.337(0.131) 0.879(0.085) 2.203(0.152) 1.429(0.019) 2.255(0.034) 3.486(0.034) 5.484(0.035) 8.819(0.058) 14.039(0.042) 1.082(0.014) 1.446(0.054) 1.989(0.025) 3.273(0.056) 5.787(0.058) 10.006(0.056)

0.97 1.003 1.194 1.631 2.422 3.727 0.784 0.598 0.563 0.771 1.334 2.411 1.116 1.205 1.602 2.478 4.04 6.595 0.689 0.083 0.237 0.072 0.789 2.65 1.717 2.328 3.615 5.853 9.433 14.957 1.489 1.305 1.822 3.325 6.191 11.015

3. Calculation of the thermodynamic properties of hard chain fluids using the derived equation of state In this section, we present comparisons between the thermodynamic properties obtained by the proposed EOS and the results of MC simulations and well known equations of state. The resultant EOS can be easily applied to many kinds of hard chain fluids, for the prediction of their thermodynamic properties, due to the flexibility of the potential function. The application of the proposed EOS to the SWC and YC is presented as follows.

2

4

4

3.1. Square-well chain fluid 2

The resultant EOS may be used to calculate the compressibility factor Z of the SWC fluid. The value of k in Eq. (23) was taken to be 1.5, which has been used by many researchers to simulate the properties of this fluid. But we also consider the effect of the well width on the thermodynamic properties. To calculate the properties of this fluid, the value of a in Eq. (9) should be taken equal to zero since there is no tail for the SW potential. At any segment packing fraction g ¼ npqc r3 =6 and reduced temperature T  ¼ kT =e, the value of the fluid compressibility factor can be easily calculated from Eq. (23) without requiring any additional parameters. In Fig. 1 we compare the calculated compressibility factor of the SW dimer, 4-mer and 8-mer fluid with k ¼ 1:5 by the proposed EOS with MC simulation results at T  ¼ 2 and T  ¼ 3. The MC simulation results for the SW dimer fluid of Kim et al. [36] with intermolecular inter-

8

8

4

a

q ¼ nqr3 . The values in parenthesis represent the standard deviation in the last three significant digits. b

H. Farrokhpour, E. Satarinezhad / Chemical Physics 348 (2008) 1–10

tions of state are included in Table 2 for k ¼ 1:3. Table 3 shows the comparison of the compressibility factor of the proposed EOS with the TPTT, SAFT-VR, LCGFD and GFYH equations of state for 4- and 8-mer SWC fluid with k ¼ 1:5, respectively. In general, there is a good agreement between the compressibility factor obtained from our EOS for SWC fluid and MC simulations is good. Furthermore the prediction of our EOS is in good agreement with the predictions of the other equations of state in Tables 1–3. However, on careful examination it is observed that the deviations of our proposed EOS from MC simulation data will increase at lower temperatures, owing to the decrease in the accuracy of the perturbation theory at lower temperatures. In fact perturbation theory is not accurate near the critical point where the convergence is slow. The agreement between the calculated compressibility factor for SW dimer fluid and the MC simulations results for k ¼ 1:5 is better than k ¼ 1:3 because the perturbation theory becomes more accurate as the well width of the SW potential gets larger [36]. Using TPTT and SAFT-VR equations of state to obtain the compressibility factor of SWC fluids requires the contact value of the RDF of SW monomers. The GHPT equation of state is special only for SW dimers with k ¼ 1:5 and to obtain the compressibility factor by GHPT equation of state, we need to calculate SSRDF by MC simulations. The GFYH and LCGFD equations are only for SWC fluid with k ¼ 1:5. The SWC fluid studied here differs from real alkanes in two important ways: attractive interactions between segments on the same chain are neglected, and the chain backbone is assumed to consist of tangent rather than fused spheres. The presence of attraction between nonbonded

8

segments on the same chain can considerably alter the properties of the fluid especially in long length chains and at low density. The contribution of attractive intramolecular interaction between nonbonded segments in the thermodynamic properties of real n-alkanes will increase at low density because the separation between molecules increases at low density and the contribution of attractive intermolecular interaction decreases. To extend this approach to real fluids such as n-alkanes, we can consider the attractive intramolecular interactions by optimizing the parameters r, e, k and n of the SWC model and substitution the reference fluid, tangent hard sphere chain fluid, with fused hard sphere chain fluid. The optimization of the parameters can be done by fitting the calculated thermodynamic properties to the experimental data of n-alkanes. 3.2. Yukawa chain fluid The Yukawa potential is represented by ( 1 r
Here eY represents the energy at contact and k1 represents the Yukawa tail screening length. It reduces to the hardspheres model in the limit k1 ! 1. If the depth of the attraction well is allowed to become infinite as k1 1 ! 0, the adhesive hard sphere model is covered. The success and widespread use of the Yukawa potential model can be attributed to the following reasons. First, the model possesses a hard-core repulsion as well as a long range attraction and describes many physical phenomena involving

20

n=4

n=2

*

T =4

*

T =4

*

T =2

6

n=8

*

10

T =4

ð24Þ

r

12

*

7

*

T =2

T =8

15

Z

8

4

6

10

4 2

5 2

0

0 0.2

0.4

0.6

0.8

0 0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

ρ∗ Fig. 2. Comparison of the compressibility factor of the Yukawa dimer, 4-mer and 8-mer chains with k1 ¼ 3 calculated by this work (—) to MC simulation data (symbols) at different temperatures. The MC simulations results of Yukawa chain fluids are from Ref. [30].

8

H. Farrokhpour, E. Satarinezhad / Chemical Physics 348 (2008) 1–10

screened interactions, for example, strong electrolytes and polymer solutions [42]. Second, by varying a single parameter k1 , the Yukawa attraction can be easily tuned to mimic different interactions encountered in many physical systems. With k1 ¼ 1:8, the model has been found appropriate to approximate the tail of the Lennard–Jones potential. For large value of k1 , the potential has been suitably used to model short-range attraction in the colloidal systems [43–45]. In order to apply Eqs. (15) and (16) to the YC fluid, the potential parameters of Eq. (9) have to obtained suitably because they appear in our EOS. In general the parameters of an EOS which are used to calculate the fluid properties are considered as adjustable parameters, and their values are chosen in such a way that it gives the best fit with the simulation data. In this work we use an analytical method to obtain the values of the potential parameters in Eq. (9) (k, a and e) [21]. Strictly speaking, the value of k in our EOS should vary with density [27]. However, the exact variation of k with density is not known, and therefore, its density dependency is discarded. In our calculation k is taken to be 1.895 which nearly satisfies the following approximation (Eq. (25)) for all densities because gHSC inter ðx; n; gÞ ¼ 1 for x > k [30]. Z 1  HSC  24g ginter  1 x2 dx ¼ 0 ð25Þ

for the parameters were used to calculate the compressibility factor of the YC fluid. The compressibility factor of YC fluid was calculated for dimer, 4- and 8-mer that interact through a Yukawa tail parameter k1 ¼ 1:8. The compressibility factors obtained in this work are compared to the predictions from the first-order perturbation theory EOS of Wang et al. (FPT) with interchain interaction [30] and the SAFT-VR model with inter and intramolecular interactions [40]. In the SAFT-VR model, the compressibility factor of YC fluid is written as chain Z ¼ 1 þ Z mono SAFT-VR þ Z SAFT-VR

where represents the compressibility factor of Yukawa spheres, while Z chain SAFT-VR correspond to the compressibility factor change for the formation of Yukawa chains from Yukawa spheres. The calculation of Z chain SAFT-VR

Table 5 Calculated compressibility factor for Yukawa dimer, 4-mer and 8-mer fluid with k1 ¼ 3 at different reduced temperatures, compared to that given by FPT equation of state [30], SAFT-VR equation of state and the MC simulation data of Wang et al. [30] n

T*

q*a

SAFT-VR

Simulation datab

This work

2

4

0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6

1.208 1.500 1.917 2.524 3.422 4.768 1.061 1.178 1.380 1.725 2.310 3.291 1.308 1.757 2.421 3.424 4.961 7.340 1.038 1.159 1.414 1.911 2.833 4.484 1.731 2.792 4.345 6.646 10.086 15.276 1.508 2.270 3.429 5.225 8.040 12.483

1.173(0.006) 1.446(0.022) 1.823(0.014) 2.379(0.018) 3.228(0.054) 4.559(0.027) 1.070(0.027) 1.242(0.017) 1.486(0.015) 1.922(0.024) 2.590(0.013) 3.715(0.025) 1.318(0.011) 1.752(0.017) 2.514(0.056) 3.551(0.027) 5.181(0.018) 7.678(0.095) 1.120(0.016) 1.313(0.025) 1.696(0.043) 2.311(0.046) 3.496(0.069) 5.218(0.05) 1.624(0.020) 2.686(0.028) 4.156(0.026) 6.522(0.021) 10.206(0.066) 15.745(0.044) 1.482(0.015) 2.355(0.030) 3.467(0.081) 5.360(0.049) 8.429(0.072) 13.242(0.047)

1.166 1.427 1.856 2.532 3.560 5.101 0.988 1.044 1.259 1.716 2.524 3.845 1.288 1.626 2.279 3.408 5.219 8.021 1.028 0.927 1.122 1.794 3.153 5.507 1.836 2.724 4.279 6.775 10.607 16.380 1.724 2.095 3.151 5.171 8.541 13.861

k

In addition we have assumed that r in Eqs. (9) and (8) is equal to that of Eq. (23). The values of a and e=eY are determined in a straightforward analytical procedure [21]. For a given fluid, whose potential function, ud ðxÞ is known, we require that the following two equal-area constraints to be approximated as [21] Z k Z k uðxÞ dx ¼ ud ðxÞ dx ð26Þ 1 1 Z 3 Z 3 uðxÞ dx ¼ ud ðxÞ dx ð27Þ k

T ¼ R T Y ek1 1 1  a¼

2

4

4

k

In fact the Helmholtz free energy of YC fluid is not derived directly form Yukawa potential and the compressibility factor of YC fluid is calculated by mapping chain fluid interacting via Eq. (1) to Yukawa potential. For the YC fluid, we substitute Eqs. (9) and (24) into Eqs. (26) and (27). Eqs. (26) and (27) is solved simultaneously and we would get following equations ðk  1Þ ek1 kt t

  1620k

T TY

5

dt þ ek1

R1 1

ek1 t t

R R1 1 k kt ð3  kÞek1 1 e 1t dt  1  5  k  972 þ 405k

2

8

8

ð28Þ

dt

ek1 t t

dt

4

ð29Þ where T  ¼ kT =e and T Y ¼ kT =eY are the reduced temperatures based on the Eq. (9) and the Yukawa potential, respectively. If we use k ¼ 1:895 in Eqs. (28) and (29) we would get a ¼ 21:217 and T  =T Y ¼ 2:598. These values

ð30Þ

Z mono SAFT-VR

a

q ¼ nqr3 . The values in parenthesis represent the standard deviation in the last three significant digits. b

H. Farrokhpour, E. Satarinezhad / Chemical Physics 348 (2008) 1–10

requires the contact value of the RDF of Yukawa monomers. Table 4 compares the compressibility factor of Yukawa dimer, 4- and 8-mer of our EOS with the predictions of FPT equation of state, SAFT-VR model and MC simulation data of Wang et al. [30] with k1 ¼ 1:8 at different temperatures. We now consider chains that interact through a shorter Yukawa range potential with k1 ¼ 3. If we use k1 ¼ 3 and k ¼ 1:895 in Eqs. (28) and (29) we will get a ¼ 6:890 and T  =T Y ¼ 3:555. Fig. 2 shows the compressibility factor of Yukawa dimer, 4- and 8-mer calculated from our EOS and simulation results of Wang et al. [30] with k1 ¼ 3 at different temperatures. Table 5 compares the compressibility factor of Yukawa dimer, 4- and 8-mer of our EOS with the predictions of SAFT-VR model and MC simulation results of Wang et al. [30]. The influence of the potential parameter k1 on thermodynamic properties is depicted in Fig. 3, where the compressibility factor of 4-mer chains along with MC simulation data are plotted against reduced density q for k1 ¼ 1:8 and 3 at T  ¼ 4. At a fixed density the pressure or compressibility factor of the YC fluid increases as the range parameter k1 increases indicating that molecules interacting through short range potential results in higher pressure. The prediction of our EOS is in good agreement with MC simulation results. However, it will be seen that for lower temperatures the deviations of our EOS from MC simulation result will increase. In order to obtain the compressibility factor of YC fluid from SAFT-VR requires the contact value of the RDF of Yukawa monomers and from FPT equation of state, we need to have gHSC inter from MC simulation. It is to be noted that our EOS is simpler

9

than the FPT and the SAFT-VR equation of state and it is easily applicable to different hard chain fluids. Overall, we find that the FPT, SAFT-VR and our EOS are able to present the compressibility factor of YC fluid very well. 4. Conclusions We have proposed a completely new perturbed EOS for hard chain fluids such as SWC fluid of variable well width and YC fluids with variable Yukawa tail screening length. The Yukawa potential is important because it can be used to model electrolyte systems. The proposed EOS has some advantages, the first advantage is the unique feature of the EOS which is the fact that it contains the same number of parameters and uses the same expressions for different hard chain fluids that are due to the flexibility of the Eq. (9). Secondly, this EOS has no adjustable parameters; therefore in some regions that there is no simulation data for such hard chain fluid, it may be used to predict the needed data. Thirdly, for deriving the EOS based on the perturbation theory there is no need to know any explicit mathematical form of gHSC inter in the 1 < x < k region. As shown in tables and figures, the results of our EOS are in good agreement with the MC simulation results. The reason for this may be due to the fact that the value of a is chosen in such a way that the surface underneath of the ESW potential is equal to that of the model potential (like SW and Yukawa potential). As shown in tables, the results of our EOS is compatible with mentioned equations of state specially SAFT-VR equation of state. Acknowledgement The authors are grateful to the Yazd University for its support.

12

References 10

[1] [2] [3] [4]

Z

8

6

[5] [6] [7] [8] [9] [10] [11]

4

2

[12]

0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ρ∗

Fig. 3. Comparison of the compressibility factor of the 4-mer Yukawa fluid for different values of k calculated by this work (—) with MC simulation results of Wang et al. [30] at T  ¼ 4 for k ¼ 1:8 (j) and k ¼ 3 (d).

[13] [14] [15] [16] [17]

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