Application of Wertheim's thermodynamic perturbation ...

MOLECULAR PHYSICS, 1999, VOL. 97, NO. 3, 307± 319

Application of Wertheim’s thermodynamic perturbation theory to dipolar hard sphere chains PRASANNA K. JOG and W. G. CHAPMAN* Rice University, Chemical Engineering Department, PO Box 1892, Houston, Tx 77005, USA (Received 17 December 1998; revised version accepted 5 February 1999) We present results from molecular simulation and statistical mechanics based theory for dipolar hard sphere chains. We consider tangent hard sphere chains with dipoles on alternate segments. The equation of state is obtained by applying Wertheim’s associating ¯ uid theory in the total bonding limit to a mixture of non-polar and dipolar hard spheres. The equation of state of the chain requires the compressibility factor and pair correlation functions of this mixture which is the reference ¯ uid. Computer simulation shows that the distribution function between the non-polar and polar hard spheres perpendicular to the dipole can be closely approximated by the hard sphere distribution function. This result is consistent with results from the mean spherical approximation and from reference linear hypernetted chain theory. With this approximation we propose an equation of state for dipolar hard sphere chains. 1.

Introduction

Multipolar interactions are important in a wide range of systems including those containing ketones, aldehydes, ethers and esters. Multiple multipolar sites are present in polyethers, polyesters and various polar copolymers. Although theories have been developed for chain molecules and for nearly spherical polar molecules, accounting simultaneously for multipolar interactions and shape e ects has remained an unsolved problem. In this paper, we develop a theory for a ¯ uid of dipolar chains with multiple dipolar sites. Our approach is based on Wertheim’s thermodynamic perturbation theory [1± 4]. Since Wertheim’s theory is the basis for the statistical associating ¯ uid theory (SAFT), our approach is similar to that used in developing the SAFT equation of state. SAFT[5, 6]has a similar form to group contribution theories in that the ¯ uid of interest is initially considered to be a mixture of unconnected groups or segments. SAFT then includes a chain connectivity term to account for the bonding of various groups to form polymers and an explicit intermolecular hydrogen bonding term. Both of these terms come from extensions of Wertheim’s ® rst order perturbation theory. A mean ® eld dispersion term is added to this perturbation expansion. SAFT has been successfully applied to the study of phase behaviour of ¯ uids of chain molecules [7± 10]. The advantage of SAFT is that it systematically accounts for di erent interactions like hard core repulsion between chain segments, chain connectivity and *Author for correspondence. e-mail:[email protected]

dispersion forces. However, the inability to account for multipolar interactions limits the predictive ability of the equation of state. In previous work, the most studied is quadrupolar hard dumbbells with charges along the molecular axis [11]. Wojcik and Gubbins [11] have investigated the mixtures of such non-polar and quadrupolar hard molecules and calculated the full angular distribution functions by molecular simulation. However, an accurate theory was not found. The present chain equations of state either do not include the e ect of the dipolar termor include it in an approximate way. Vimalchand and Donohue [12] modi® ed the perturbed hard chain theory (PHCT) of Donohue and Prausnitz [13] to include anisotropic multipolar interactions by application of the u-expansion. The PACT approach derived by Walsh et al. [14] uses an interaction site perturbation theory with polar interactions which accounts for the shape of the molecule through an interaction volume parameter that depends on the shape of the molecule. However, it does not account for the e ect of orientation of the dipole relative to the molecular axis. Sear [15] applied a diagrammatic expansion technique similar to that of Wertheimto study the e ect of chain formation by dipolar hard spheres at low density, which is energetically favourable. However, this approach is valid only at low temperature and low density because it neglects the long range interactions. SAFT is known to give poor results for chain molecules having dipolar groups. One way of including dipolar interaction is to model it as dispersion with short range, directional bonding interactions. The later

Molecular Physics ISSN 0026± 8976 print/ISSN 1362± 3028 online Ñ 1999 Taylor & Francis Ltd http://www.tandf.co.uk/JNLS/mph.htm http://www.taylorandfrancis.com/JNLS/mph.htm

308

P. K. Jog and W. G. Chapman

can be treated by Wertheim’s theory of associating ¯ uids and the dispersion interactions are dealt with at mean ® eld level. However, in some cases this approach leads to arti® cially large dispersion energy and a large binary interaction parameter is required to model mixtures which makes the approach less predictive. For example, Hasch et al. [16]have attempted to adjust the mean ® eld parameter in the SAFT equation of state whichaccounts for dispersion, to ® t the homopolymer density data. They found that as a consequence of the necessarily large values of the mean ® eld parameter u0/ k, the calculated mixture cloud-point curves are not even in qualitative agreement with the experimental data prior to ® tting a binary interaction parameter. The binary interaction parameters (kij ) in the mixing rule for dispersion energies can be adjusted, but that is an ad hoc solution. Hence, it is necessary to account for dipolar interactions explicitly. A similar situation is seen with the SAFT-VR approach [17] that accounts for the dispersion interactions through a variable range square well potential. But in the case of multipolar interactions, the variable range potential does not include the e ect of dipolar orientation. Xu et al. [18] applied the u-expansion with PadeÂ approximation to account for the dipolar interaction. However, they did not consider the e ect of location and orientation of the dipole nor the range of applicability of the u-expansion. As has been shown in the literature, the u-expansion does not accurately predict the properties of polar ¯ uids with shape and with high reduced dipole moments (i.e. high dipole moment or very low temperature) [19]. In this work we propose a method to account for dipolar interactions in chain equations of state. We demonstrate the range of validity of the theory through comparisons with new molecular simulation results. 2.

Approach

Wertheim’s theory [1± 4] has been successfully used to bond hard spheres [20], Lennard Jones [21, 22], and square well segments [17] together to form chains. More complex segments have not previously been studied. In the present work, we apply Wertheim’s TPT1 to hard sphere chains with alternate segments having an ideal dipole at the centre. The dipole on a particular segment is perpendicular to the line joining the centre of that segment with that of the immediately preceding segment. This can be considered to be a generic model of a copolymer chain with alternating dipolar pendant groups. All the segments are of the same diameter. Our choice of the TPT1 approach is based on the observation that Wertheim’s theory works well for association and chemical strength bonding because the monomer± monomer distribution function in a mixture of chains and monomers remains nearly the same as in

the reference ¯ uid of all monomers. Also, we have found in a variety of ¯ uids that, if two components interact with a hard sphere potential, then, regardless of their interaction potential with other components in the ¯ uid, the pair correlation function between these two components is hard sphere like [23]. This has not been tested for multipolar ¯ uids. However, for mixtures of non-polar hard spheres and dipolar hard spheres, this is correct in the mean spherical approximation and reference linear hypernetted chain theory [24, 25]. The hypothesis is that a dipolar chain ¯ uid can be formed by bonding non-polar hard spheres to dipolar hard spheres and that the bonding contribution will be the same as for a hard sphere ¯ uid. This should be approximately true if the dipole vector is perpendicular to the bond, but may also be true for other orientations of the dipole moment. Our approach is to determine the orientation dependence of the hard sphere/dipolar hard sphere distribution function, test our hypothesis and test the bonding term in Wertheim’s theory. We study these chains for a reduced dipole moment (¹ = ¹/ (s 3kT )1/ 2) of up to 1.25 and a chain length of up to eight (octamer). We follow the same methodology as in the derivation of the equation of state for hard chain ¯ uids[20]. The ® rst step is to calculate the thermodynamic properties (free energy, compressibility factor) and structure (distribution functions) of the reference ¯ uid which is an equimolar mixture of non-polar and polar hard spheres. Then we bond these spheres to form a chain. The thermodynamic properties of the reference ¯ uid are calculated by computer simulation and uexpansion which is a perturbation theory. The distribution function between the non-polar and polar hard sphere at a particular density is a function of the separation of the spheres and also the orientation of the dipole with respect to the non-polar sphere. This distribution function, calculated from simulation, is used to test our hypothesis. The equation of state is given by combining reference ¯ uid properties with the bonding term based on Wertheim’s theory. The chain equation of state is also validated by comparing with computer simulation data for dipolar chains. It is found that the bonding term given by the modi® ed form of Wertheim’s theory, the SAFT-D theory, is in agreement with the computer simulation results even for high dipole moments (up to ¹ = 1. 25 studied in this work). 3.

Theory

Wertheim’s ® rst order perturbation theory is applied to dipolar tangent hardsphere chains. Consider a hard sphere chainwith dipoles on alternate segments as shown in® gure 1. We de® ne a dimensionless dipole moment by ¹ = ¹/ (s 3kT )1/ 2, (1)

309

Application of Wertheim’s thermodynamic perturbation theory

Figure 1. The model of the dipolar chain. Note the orientation of the dipoles.

where s is the hard sphere diameter and ¹ is the dipole moment. The contribution of the dipole moment to the thermodynamic properties is determined by this single combined parameter as can be seen fromthe formof the interaction potential. As a model, we consider the case where the dipole moment on a segment is perpendicular to the vector joining the centre of the segment bearing the dipole to the previous segment. This can be thought of as a model for chain molecules with polar pendant groups. To form a ¯ uid of chain molecules, we consider an equimolar mixture of non-polar and dipolar hard spheres as our reference ¯ uid. Let uHS be the hard sphere potential with hard sphere diameter s . Then, by labelling the hard sphere as component 1 and the dipolar hard sphere as component 2, the pair potential uij between components of type i and j can be written, u11(r) = uHS(r), u12(r) = uHS(r), u22(r, X

1, X

) = uHS(r)

2

Figure 2. Orientation dependence of pair correlation function g = g( r, µ, u , q ).

® gure 2. For our model, g(r, µ, u ) is independent of the azimuthal angle u . As explained by Chapman et al. [20], the bonding term is obtained by considering association sites on the hard spheres with a suitably chosen association potential. Consider a chain with N segments in which every other segment has a dipole. There are two types of bonds in the chain. Firstly, the dipolar segment bonds to the preceding segment such that the dipole vector is perpendicular to the bond. Secondly, the dipolar segment bonds to the following hard sphere segment with no angular constraints so that the dipole is freely rotating around the bond. These two types of bonds are alternating (® gure 1). To get the bonding termfor the ® rst kind of bonds, consider the association potential whose Mayer f function is given by, fHB(r, µ) = d (r s )d (µ p / 2) .

(3)

For the second type of bond, ¹2 r3

[3(u^ 1 r^)(u^ 2 r^)

u^ 1 u^ 2],

fHB(r, µ) = d (r s ).

( 2)

(4)

where r^ is a unit vector in the direction of r, u^ i is a unit vector parallel to the dipole moment of molecule i. The development of the equation of state of dipolar hard sphere chains consists of two parts. First, calculation of the thermodynamic properties of the reference ¯ uid and second, calculation of the bonding term using Wertheim’s theory which requires the structure of the reference ¯ uid (i.e. pair correlation function g). We ® rst derive the bonding termin terms of the pair correlation function.

The sites are assumed to be independent of each other. The contribution to the bonding termdue toa particular type of bond is given by [20]

3.1. Derivation of the bonding term The reference ¯ uid is a mixture of non-polar and polar hard spheres. The distribution function between a non-polar hard sphere (1) and a polar hard sphere (2) depends on the magnitude of the relative position vector r and the polar and azimuthal angles between the position vector and the dipole moment vector as shown in

is the bonding integral in Wertheim’s theory, gR(12) is the reference ¯ uid pair correlation function between non-polar and dipolar hard spheres and nbondi is the number of bonds of type i. fHB is the Mayer f function for the association potential. Substituting the association potential from equations (3) and (4) in equation (5), we get

Zchaini

= nbondi 1 + q

¶ ln ¶ q

,

(5)

where

=

gR(12)fHB(12) d(12)

(6)

310

P. K. Jog and W. G. Chapman Zchain = n1 1 + q

¶ ln gR(s , p / 2, q ¶ q

¶ ln hgR(s , µ, q ¶ q

n2 1 + q

)

)iµ

,

( 7)

where n1 and n2 are the corresponding number of bonds. The angular brackets h iµ in equation (7) represents an unweighted angle average. The total compressibility factor is given by Z = mZref + Zchain, ( 8) m being the chain length. For the residual Helmholtz free energy we can write, Ares = mAres ( 9) ref + Abond, with Abond = n1 ln (yR(s , p / 2)) n2 ln (hyR(s , µ)iµ), (10) where yR is the cavity correlation function between nonpolar and dipolar hard spheres for the reference ¯ uid. The internal energy change due to bonding can be calculated by thermodynamic di erentiation. Thus, we have to calculate the compressibility factor and the above mentioned distribution functions for the reference ¯ uid. This is done by computer simulation and statistical mechanics. 4.

Computer simulation

Metropolis Monte Carlo simulations [26] were performed to study the structure and thermodynamic properties of the reference ¯ uid mixture of non-polar and polar hard spheres and also of chain molecules. The simulations were performed in the NPT ensemble, except the calculation of the pair correlation function in the reference ¯ uid is done in the NV T ensemble. The advantage of NPT ensemble simulations is that the equation of state can be easily obtained for discontinuous potentials. In the simulation of the reference ¯ uid, 108 segments were used. For dimers and tetramers, 108 molecules were used. For octamers, 54 molecules were used in order to save some computation time, since the CPU time varies roughly as the square of the chain length at constant total number of molecules. In the simulation, a molecule is chosen and displaced randomly within a cube. The molecule is reoriented randomly following the random displacement. The change in orientation of the bonds which are constrained to be perpendicular to the dipole is achieved using the method described by Allen and Tildesley [26]. In this method, a change in orientation is achieved by making small random displacements in each of the three Euler angles of the molecule chosen. The translational-jiggling algorithmof Dickman and Hall [27]was used to reorient the free bonds. The same method is applicable for simu-

lation of molecules of any length. In all the cases the reorientation and displacement parameters were adjusted so that the overall acceptance probability was around 50% . After each molecule was displaced and reoriented, the cell length was varied randomly. The cell length change parameter was adjusted to give about 50% rate of acceptance for volume changes. Initially, the system was equilibrated for 10 106 con® gurations, and a further 10 106 con® gurations were run to accumulate the averages. For higher densities or higher dipole moments more con® gurations were used. The canonical ensemble simulations were performed to calculate the full distribution function between nonpolar and polar molecules in the reference ¯ uid mixture, as a function of radial distance and angle between the centre to centre vector and the dipole. The angular bin width was suitably adjusted to balance detail and reliable statistics. If the angular bin width is too small, there is more noise in the data, but if it is too large, there is more discretization error. An angular bin width of around 2± 5ë was selected. To get good statistics, the NV T ensemble simulations were run for 100 106 con® gurations with an initial 25 106 steps used for equilibration. In all of the above simulations, periodic boundary conditions were used. The potential of interaction was truncated at the minimumof 3s and half the box length. Our goal in this work is to test the validity of Wertheim’s bonding term for dipolar chains. Since the dipolar long range contribution is expected to be similar in the reference ¯ uid and for the chains, no long range interactions were calculated. As pointed out by MuÈ ller and Gubbins [28], in the case of ideal dipoles, the long range correction to the energy is not large and corresponds to values on the order of 1% of the total con® gurational energy. Simulations were performed for the reference ¯ uid mixture and for dipolar chains of dimers, tetramers and octamers at reduced dipole moments, ¹ , of 0.75 and 1.25. The simulations provide a direct test of the bonding term for dipolar chains. A useful result from computer simulation is that up to a reduced dipole moment of 1.25, gR(q , s , p / 2) gHS(q , s ). ( 11) At moderate dipole moments (up to around 1.0 or so) ( 12) hgR(q , s , µ)iµ gHS(q , s ).

Figures 3 and 4 show that the approximation works well for moderate dipole moments. These results con® rmour hypothesis that at moderate dipole moments we can replace gR in the bonding term by gHS. This implies that the bonding term is same as in the hard chain.

311


Figure 3. Pair correlation function between nonpolar and dipolar hard spheres at contact and perpendicular to the dipole moment as a function of reduced density for an equimolar mixture of non-polar and dipolar hard spheres for reduced dipole moments of 0.75 and 1.25 from simulation. 5.

ua b

A = A0 + A2 + A3 +

Development of the equation of state

In the last section we described the computer simulation of polar hard sphere chains. In order to predict the PVT behaviour of these chains, we need a statistical mechanical equation of state for these molecules. We will use equation (7) for the bonding term with the approximations presented above for the reference ¯ uid distribution function. So, it remains to calculate the compressibility factor of the reference ¯ uid which is an equimolar mixture of non-polar and dipolar hard spheres. We use the u-expansion tocalculate the thermodynamic properties because it has an analytical form and has been previously applied [29, 30]. The u-expansion is a perturbation expansion for ¯ uids with an anisotropic potential about a spherically symmetric reference ¯ uid. For mixtures,

( r, X

1, X

) = u0a b (r) + uaa b (r, X

2

),

1, X

2

(13)

The ® rst order term A1 vanishes due to the particular choice of reference potential [29]. The second and third order terms for a mixture with a multipolar potential are given by A2 =

b

( r, X

1, X

)i

2 X

1,X

2

.

(14)

For all the three pair interactions (1± 1, 1± 2 and 2± 2), the isotropic potential is the hard sphere potential because the orientational average of the dipole± dipole potential is zero. The Helmholtz free energy is expanded in powers of u0a b / kT to give

1 4kT

q a q

a b

a hua b (R12, X

dR1 dR2 b

)2iX

1, X

2

1X

2

g0a b (R12).

( 16)

The third order term is given by

( 17)

A3 = A3A + A3B,

where A3A

=

where a and b denote components in a mixture. The isotropic potential is given by u0a b (r) = hua

( 15)

.

A3B =

1

12(kT )2

q a q

a b

a hua b (R12, X

1

6(kT )

2

a b g

uaa g (R13X

)3iX

1, X

q a q bq 1X

dR1 dR2 b

g

2

1X

( R12, R13, R23).

g0a b ( R12),

dR1 dR2 dR3huaa b ( R12X

)uab g (R23X

3

2

2X

)i

3 X

1X

2X

3

)

1X 2

g0a b g

( 18)

312


(a)

(b) Figure 4. (a) Angle averaged pair correlation function between non-polar and dipolar hard spheres at contact as a function of reduced density for an equimolar mixture of non-polar and dipolar hard spheres for various reduced dipole moments from simulation. (b) Semilog plot of the angle averaged pair correlation function between non-polar and dipolar hard spheres at contact as a function of reduced density for an equimolar mixture of non-polar and dipolar hard spheres for various reduced dipole moments from simulation.

Application of Wertheim’s thermodynamic perturbation theory The PadeÂ approximant to the perturbation expansion of equation (15), proposed by Rushbrooke et al. [30] is 1

A = A0 + A2

(19)

.

1 A3/ A2

For the reference ¯ uid mixture under consideration, de® ne y=

4p q ¹2 , 9 2 kT

(21)

q

= q s 3,

T

s = ¹2 . kT

(20)

3

(22)

To calculate the Helmholtz free energy, we de® ne the following integrals. 3

I2(q

) = 34s p

I3(q

) = 35ps 2

3

) r16 dr,

gHS(r, q ghs (123, q

(23)

)u(123) dr2 dr3,

(24)

where u(123) =

1 + 3 cos a

1

cos a

(r12r23r13)

2 3

cos a

(25)

3

is the Axilrod± Teller three-body interaction. Rushbrooke et al. [30] have shown that 0.3618q 0.3205q (1 0.5236q

2

I2(q

) = 1

I3(q

0.62378q 0. 11658q ) = 1+ 1 0.59056q + 0. 20059q

3

+ 0.1078q

)2

2

,

(26) (27)

2.

The Helmholtz free energy is given by A A0 = NkT

p 2

1

2

q x2p I2(q 9 T I (q 5p 1 q xp 3 1+ I2(q 36 T

) ) )

.

(28)

xp is the mole fraction of polar hard spheres in the mixture. Note that A0 in equation (28) is the hard

sphere free energy. Thus, the perturbation term is actually the contribution due to the dipole. The other thermodynamic quantities can be obtained by taking proper derivatives of the free energy. For example, the compressibility factor is given by P q ¶ (A/ NkT ) . (29) q kT = ¶ q The internal energy is calculated from

313

¶ (A/ T ) . ( 30) ¶ T Thus, equation (28) is the equation of state of the reference ¯ uid. Combining this equation of state for the reference ¯ uid with the chain term of equation (7) produces the equation of state for dipolar chains given by equation (8). U = T2

6.

Modi® ed chain term

In the last couple of sections we concluded from the computer simulation and statistical mechanical theory that the distribution functions in the bonding term can both be approximated by the hard chain bonding term up to a dipole moment of around 1.0 (as tested by the simulations). Ghonasgi and Chapman [31]and independently Chang and Sandler [32] showed that for chains longer than dimers, the bonding term obtained by using a dimer reference gives better agreement with simulation data. We apply this SAFT-D bonding term to the dipolar chains. The SAFT-D equation is [31] ´(5 2´) m m Zm = mZref 1+ 2 (1 ´)(2 ´) 2 1 1+ ´

2 + 163.1984´5.17 2 + . 6 17 ´ ´ + + 1 ´ 1 2 26. 45031

.

( 31) The compressibility factor of the reference ¯ uid is calculated analytically using the u-expansion described earlier. This completes the equation of state for the dipolar chains. 7.

Results and discussions

Figures 3± 6 show the results for the pair correlation function between non-polar and dipolar hard spheres in the reference ¯ uid mixture. From ® gure 5 it is seen that at a dipole moment of 1.25, the pair correlation function between non-polar and polar hard spheres at contact shows signi® cant dependence on the angle µ de® ned as the angle between the centre to centre vector and the dipole vector. The angular dependence is a result of the preference of dipolar hard spheres to align in a head to tail con® guration, reducing the density of nonpolar hard spheres in those con® gurations. Figure 6 shows a plot of the angle averaged distribution function at a density of 0.45. It is seen that the contact values are reasonably close to the corresponding hard sphere values even at ¹ = 1. 25 (® gure 4(a)). Figure 4(b) shows a plot of the logarithm of the angle averaged correlation function versus density. It is seen that the plots for ¹ = 0.75, ¹ = 1. 25 and hard spheres are nearly parallel. Hence the bonding term ¶ log hgi/¶ q in equation (7) is nearly equal to the hard sphere

314


Figure 5. Angular dependence of pair correlation function between non-polar and dipolar hard spheres at contact for an equimolar mixture of non-polar and dipolar hard spheres at ¹ = 1.25 at q = 0.45 and q = 0. 9. At a density of 0.9, the angular dependence is very pronounced.

Figure 6. Radial dependence of angle averaged pair correlation function between non-polar and dipolar hard spheres for an equimolar mixture of non-polar and dipolar hard spheres at q = 0.45 at the reduced dipole moments of 0.75 and 1.25 calculated by simulation. The hard sphere radial distribution function is shown for reference.


315

(a)

(b) Figure 7. (a) Compressibility factor of reference ¯ uid mixture at ¹ = 0.75. Results fromsimulation and theory. Hard sphere result is shown for comparison. (b) Compressibility factor for reference ¯ uid mixture at ¹ = 1.25. Results from simulation and theory. Hard sphere result is shown for comparison.

316


Figure 8. Compressibility factor for dipolar dimer. Results from simulation and theory. Hard dimer result is shown for comparison.

Figure 9. Compressibility factor for dipolar tetramer. Results from simulation and theory. Hard tetramer result is shown for comparison.


317

Figure 10. Compressibility factor for dipolar octamer. Results from simulation and theory. Hard octamer result is shown for comparison.

bonding term. Figure 3 shows that the pair correlation function between non-polar and dipolar hard spheres at contact at µ = 90 (perpendicular to the dipole moment) is reasonably close to the contact value of the hard sphere correlation function. To apply the present equation of state to the dipolar chains, we need to know the thermodynamic properties of the reference ¯ uid. The u-expansion gives an analytical expression for the Helmholtz free energy of the reference ¯ uid (equation (28)). The compressibility factor can be calculated from equation (29). Figure 7 shows that the u-expansion results agree very well with the simulation data at reduced dipole moments of 0. 75 and 1.25. The bonding term is assumed to be the same irrespective of the dipole moment. The compressibility factor for dimers, tetramers and octamers calculated using the SAFT-D bonding term is compared with direct simulation results in ® gures 8± 10. It is seen that the agreement is very good at ¹ = 0. 75 and also at ¹ = 1.25. Since we showed that the bonding term for polar chains is the same as the SAFT-D bonding term, it follows that the bonding term is independent of temperature. Application of equation (30) shows that there should be no change in internal energy due to bonding. To test this, we compare the internal energies of the reference mixture and the dimers, tetramers and

octamers at the same dipole moment on the same plot. We plot all results versus the segment density in ® gure 11. The data for all systems (dimers, tetramers etc.) is not available at the same segment densities, but we can see that they show a trend towards an upward shift as chain length increases. The e ect, however, is small in magnitude indicating that the approximation made in assuming the bonding term equal to the hard chain bonding term is reasonable. 8.

Conclusion

An equation of state is obtained for hard chain molecules having ideal dipoles on alternate segments. The location and orientation of dipoles are considered explicitly. It is shown that the bonding term can be well approximated by the hard chain bonding term. Speci® cally, the SAFT-D bonding term gives very good agreement with the simulation results. The properties of the reference ¯ uid can be calculated from the u-expansion up to a reduced dipole moment of 1.25, rendering the equation of state in analytical form. In this paper, we have considered chains up to a length of eight (octamers) and it is found that the theory works well even for octamers. It is shown from the simulation data that there is no change in internal energy due to bonding. This is consistent with the conclusion that the bonding term is identical to that in a non-polar hard chain.

318


(a)

(b) Figure 11. Comparison of internal energies of dipolar hard chains with di erent chain lengths (a) ¹ = 0.75 and (b) ¹ = 1.25. Results from simulation and u-expansion.

Application of Wertheim’s thermodynamic perturbation theory Though we have considered the case of perpendicular dipole moments and dipoles alternating in the chain, the method can be generalized to di erent densities of dipoles and di erent orientations. We thank Dr Keshawa P. Shukla for his help, especially for providing important references. We are grateful to Dr John Walsh for suggesting this problem of dipolar hard chains. We also thank the Petroleum Research Fund and Robert A. Welch Foundation for their ® nancial support. References

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