Statistical associating fluid theory-dimer of the solid phase of the pearl-necklace model

Transcription

1 Indian Journal of Pure & Applied Physics Vol., epteber 00, pp tatistical associating fluid theory-dier of the solid phase of the pearl-necklace odel Vivek Kuar ingh & K N Khanna Departent of Physics, V D College, Kanpur Received 10 March 00, revised February 00; accepted 7 May 00 tatistical associating fluid theory-dier (AFT-D of chain olecules for liquid phase is extended to odel the solid phase of chains. The odel of freely jointed chains of tangent hard sphe requi the equation of state of hard sphere onoers in solid phase and contact values of the radial distribution function of the dier olecules for solid phase. The theoretical ults for the equation of state and free energies for the chains up to 8 ers are copared with siulation ults. The agreeent between theory and siulation is iproved over the recent ults of Vega and MacDowell (Vega C and MacDowell L G, J Che Phys, 11( eploying therodynaics perturbation theory of first order (TPT-1. [Keywords: Fluid theory, Dier, Pearl- necklace odel, Perturbation theory] IPC Code: G 01 K 1 Introduction A theoretical description of the solid phase of flexible chain olecules is of intet, both fro a fundaental and fro a practical point of view. olid phases of long flexible olecules exist at roo teperature and psure, for instance all linear alkanes with ore than 0 carbon atos are solid at roo teperature and psure and the sae is true for polyethylene. In any industrial processes one has to deal with fluid - solid separation of alkane ixtu. It is iportant to consider the solid structure of freely jointed flexible chains. In flexible chains, there is neither bending nor torsional potential between the onoers of the chains. Therefore, no extra energy is required when the atos of the chain adopt a close packed structure such as f.c.c. In the solid phase the atos are arranged in closed packed structure as f.c.c. solid but, there is no longer range orientation order between the bond vectors of the chains. The fluid phase of fully flexible chains has been studied extensively in recent years 1-7. Werthei 8-11 pented a very successful theory to study the therodynaic properties of fluids. In this odel, the equation of state is derived for a chain of freelyjointed tangent hard segents using only therodynaic inforation of the onoer reference fluid. Thus, Werthei s theory, which is known as therodynaic perturbation theory (TPT-1 requi the equation of state of onoer fluid and its pair correlation function at contact. The equations of state are used to predict the properties of solid, liquid and gas. While there are nuerous equations of state capable of describing the properties of fluids, that is, both liquids and gases, there are only a few equations of state capable of describing the properties of solids. More iportantly, there are no known equations of state capable of describing the properties of both fluids and solids accurately. One such effort was ade by Vega and Macdowell 1. They have extended Werthei s first order perturbation theory to the solid phase by freezing the Pearl-necklace odel obtaining a fair agreeent with the siulation ults of Malanoski and Monson 1. This work is extended by Blas et al. 1 to deal with fully flexible hard-chain olecules with segent-segent attractive interactions treated at the ean-field level of Van der Walls. The properties of tangent sphere dier odels in the solid phase have been the subject of recent investigation as it was found successful in fluid phase 18. Wojciechowski et al have also shown that the stable solid structu in two diensions are fored by a periodic closed packed arrangeent of atos siilar to, as in three diensions 1. Recently, Vega et al. carried out the coputer siulations of dier Lennard-Jones olecules in the solid phase. Their theoretical ults for the equation of state and the internal energy of the Lennard-Jones dier odel in the solid phase are in excellent agreeent with the

2 INGH & KHANNA: AFT-D THEORY OF CHAIN MOLECULE 689 siulation data. In this work we apply the idea of dier in solid phase for the hard chain olecules eploying our equation of state 6 cobined with the pair correlation function of solid dier obtained fro statistical associating fluid theory of Werthei 8-11 (AFT. Theory Let us consider a chain of tangent hard sphe coposed of pairs of diers. We note that for fully flexible chains, we need not consider the actual nature of phase. All that we needed for the Pearl-necklace odel in solid phase is the reference syste of hard sphe in the solid phase. The solid phase of hard sphe is described quite accurately by the equation of state proposed by Hall, which will describe the reference syste of the solid hard sphe chain. Vega and MacDowell 1 extended this reference syste for the equation of state of Pearl-necklace odel of the onoeric segents in solid phase while in the pent work, we consider dierized segents of the Pearl-necklace. Thus, the equation of state of dier in solid phase can be written as in fluid phase, firstly applied by Ghonasgi and Chapan 18. (1 where, s H will describe the EO of the reference syste of hard solid sphe, proposed by Hall that are least square fit to the siulation ults of hard sphe obtained by Alder, H H ( 1 α α H ln g α α α.81861α α α ( where, α has been defined as α 1 cp ( π cp here, 6 is the volue fraction at close packing for the hard sphe. The pairs of di-sphere can for a chain, which has four segents i.e., tetraers. iilarly, the equation of state of 8 ers and 16 ers can be written. The details can be seen elsewhere 6. A general expsion can be written as lng H ( 1 H ln g HT HO ( σ lng ( σ lng ( σ 8 16 ( Although, Chiew 5-7 derived the interolecular site-site correlation function (g f (σ of freely jointed -er hoonuclear hard sphere chains based on a particle-particle description of the chain syste in fluid phase, but for solid phase g s (σ will be different than g f (σ. ince the pair correlation function for higher ers are not known for solid phase, we assue g s (σ g HT s (σ g HO s (σ, and hence we can write a general expsion for the equation of state of - er chain fluid as H H ln g ( 1 ln g 1 (5 The pair correlation function at contact for onoers can be calculated using the relation g H H 1 (6 The pair correlation function of solid dier at contact is deterined by the equation of the disphere ( in AFT theory 1 ln g 1 g (7 (8

3 690 INDIAN J PURE & APPL PHY, VOL, EPTEMBER 00 We consider first three ters of Eq ( for as other ters alost cancel each other. Thus to siplify the expsion of g (σ, we consider (9 olving Eqs (7 and (8, we obtained the expsion of pair correlation function of solid dier as g ( 1 α.0 9 α α ( ( (10 Let us now deterine the free energy of the solid hard sphe chain. The idual free energy of the hard sphere solid can be written as 1 H A H A H H ( A ( 0 ( 1 d 0 (11 (1 where A H ( 0 is the idual free energy of a reference hard sphere solid at a certain reference A H ( 0 volue fraction 0 such that , for a reference packing fraction ( (Ref. 8. The total free energy of the dierized solid chain can be derived as H ( ( d the analytical expsion by integrating second ter of Eq. (1 as A H ( 65.9 ln( ln Results and Discussion (1 The equation of state for freely jointed tangent hard sphere chains in solid phase has been calculated using dierized chain (Eq. 5 foring pearl necklace odel in solid phase by eploying the reference syste of hard sphe in solid phase described by Hall and the pair correlation function of solid dier eploying AFT theory. The Eq. (5 corponds to any value of. Thus, it can be ipleented for chains of odd nuber of segents. The ults are shown in Figs 1 and corponding to the chain length,, 5, 6, 7 and 8. Recently, Malanoski and Monson 1 perfored the siulation ults for the Pearl necklace odel in solid phase with chain length ranging fro to 8. Thus, we have obtained the siulation ults by evaluating equation A ln ( ρσ ( H A 1 ln g where H ( A H coponent of 1 ln g (1 can be derived by integrating the H s given by Eq. (.We have derived Fig. 1 Equation of state of chains for, and 5 (fro botto to top in the solid phase (a AFT-D theory (, (b TPT-1 theory ( and (c siulation ults (

4 INGH & KHANNA: AFT-D THEORY OF CHAIN MOLECULE 691 dier theory predicts good agreeent with siulation ults in coparison to TPT1 theory particularly for longer chain length. However, AFT-D theory needs further iproveent for higher chain length 7 and 8. A ore accurate estiation of site-site correlation function at contact for the dier in the solid phase ay leads to better ults. The deterination of free energy of the chain olecules in solid phase is described. The free energy of hard sphere solid chain can be obtained fro Eq. (1. The analytical expsion of the idual free energy of a reference hard sphere solid described by Eq. (1 has been derived. It is lengthy and tedious job. The free energy obtained by eploying Eq. (1 for various ers ranging fro to 8 are pented in Table 1. The ults are copared with TPT-1 values and siulation data in Table 1. The pent work reveals that AFT-D theory predicts better agreeent with Monte Carlo values in coparison to TPT-1 theory particularly for higher ers 7 and 8. The lower values of free energy eploying AFT-D theory in coparison to TPT-1 theory ay predict the higher degree of stability of the chain olecules. Fig. Equation of state of chain for 6, 7 and 8 (fro botto to top in the solid phase (a AFT-D theory (, (b TPT-1 theory ( and (c siulation ults ( Table 1 Free energy of the pearl- necklace odel in the solid phase as obtained fro AFT-D theory, TPT-1 odel and siulation A/ AFT-D TPT-1 iulation (.10 of the Malanoski and Monson 1 for solid phase at various densities. The equation of state eploying References 1 Werthei M, J Che Phys, 87 ( Chapan W G, Jackson G & Gubbins, Mol Phys, 65 ( Johnson J K, J Che Phys, 10 ( Dickan R & Hall C K, J Che Phys, 85 ( Honnell K G & Hall C K, J Che Phys, 90 ( Kushwaha K B & Khanna K N, Mol Phys, 97 ( rivastava V & Khanna K N, Mol Phys, 100 ( Werthei M, J tat Phys, 5 ( Werthei M, J tat Phys, 5 ( Werthei M, J tat Phys, ( Werthei M, J tat Phys, ( Vega C & Mac Dowell L G, J Che Phys, 11 ( Malanoski A P & Monson P A, J Che Phys, 107 ( Blas F J, Galindo A & Vega C, Mol Phys, 101 ( Vega C, Paras E P A & Monson P A, J Che Phys, 97 ( Bowles R K & peedy R J, Mol Phys, 87 ( ear R P & Jackson G, J Che Phys, 10 ( Ghonasgi D & Chapan W J, J Che Phys, 100 (

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