Polymer Percus-Yevick Ideal Chain Approximation For The Lennard-Jones Chain Fluid

The structure and thermodynamic properties of the freely jointed linear chain uid with monomers interecting by the Lennard-Jones potential are studied using the polymer Percus-Yevick (PPY) ideal chain approximation. The theory is based on the Wertheim's multi-density integral-equation theory for associating uids, appropriately modiied to describe the polymer uids. In the case of the chain system composed of 4,8 and 16 monomeric units the PPY ideal chain approximation is found to yield good agreement with corresponding computer simulation results for the internal energy at the entire range of the density and temperature studied. For the longer chains composed of 50 and 100 segments the agreement becomes slightly worse, especially at low values of the density. Predictions of the theory for the structure properties are less satisfactory, i.e. theoretical results for the intermolecular everage site-site distribution function of 16-mer uid is only in semiquantitative agreement with Monte Carlo simulation results. It is expected, that the thery will be more accurate in predicting overall everage distribution functions and for the uid of shorter chain length.


Introduction
During the last decade several o -lattice models for the structure and thermodynamic properties of the polymeric uids have been proposed.Most of theoretical descriptions developed recently are based on the extension of the integral equation techniques developed for the uids of small interactionsite molecules.These include the polymer reference interaction site model (PRISM) theory (see Refs. 1,2] and references therein), polymer Born-Green-Yvon (PBGY) theory 3,4], a version of PY theory for a mixture of associating species that associate into polymer uids 5], theory based on the Chandler-Silbey-Ladanyi PY approximation for the site-site uid 6] and multidensity polymer PY (PPY) theory for associating uids 7,8] appropriatly modi ed to describe the polymer uids 9,10,2,11,12].
However the majority of the applications of these theories are restricted to the case of the hard-sphere chain uids.Although this model re ects the most characteristic features of the polymer uids, i.e. excluded volume e ects and e ects of chain connectivity, it is still highly simpli ed due to the absence of attractive forces.The model, which in addition to the chain connectivity provides a realistic description for both repulsive and attractive interactions between the chain segments is the freely jointed Lennard-Jones chain (LJC) model.
The goal of the present paper is to study the structure and thermodynamic properties of the LJC uid using the numerical solution of the PPY ideal chain approximation.The paper is orgenized as follows.In Section 2 we discuss the model and present the corresponding version of the PPY approximation together with the details of its numerical solution.Numerical results and their comparison with the corresponding computer simulation results rae discussed in Section 3 and in Section 4 our conclusions are collected.

The model and PPY ideal chain approximation
Let us consider the system which consists of the freely jointed LJ monomer chains with xed bond length L, chain length of m monomer units and number density of monomers .All the nonbonded monomers, regardless of wether they belong to the same momecule or to the di erent chain momecules interact via the following pair potential where is the potential well depth and is the distance at which the potential is equal zero.The present model is studied using the PPY ideal chain approximation 7{9].The approach is based on the multidensity integral equation theory for the associating uids 7,8] appropriately modi ed to describe the chain polymer systems 9].The derivation of the PPY ideal chain approximation for the hard-sphere linear chain uid has been discussed earlier 9].Its extension to the case of the LJ chains is straightforward and we shall therefore omit any detailes and present here only the nal results, speci ed for the model at hand.
The theory consists of the Ornstein-Zernike (OZ) like integral equation where Ĉ(k), Ĥ(k) and are the matrices de ned by and PPY ideal chain approximation, relating the direct C (r) and total H (r) partial correlation functions Y (r) = H + 0 0 ?C (r) (2.3) where the partial cavity correlation functions are de ned by 4 L 2 (r?L) (2.4)Here e R (r) = exp ?u(r)], = 1=kT , Ĥ (k) and Ĉ (k) are the Fourier transforms of the functions H (r) and C (r), respectively, and the lower indeces and take the values 0; A; B and denote the bonding states of the corresponding particles 7{9].Elimination of the cavity correlation functions Y (r) between (2.3) and (2.4) yields In the present version of the theory the equilibrium properties of the system is described via everage overall g(r) and intermolecular g (inter) (r) pair distribution functions, which are related to the partial correlation functions where i and j denote the site species and take the values 1; :::; m, g (inter) (r) is the site-site distribution function between the sites i and j belonging to the di erent chains and g ij (r) is the overall particle-particle distribution function between the pair of monomers, regardless of wether they belong to the di erent chains or to the same chain.In eq.(2.7) the symmetry property of the model is used, due to which H 0A (r) = H 0B (r) and H AA (r) = H BB (r).One can also de ned the intramolecular distribution function !(r) via the following relation 9] !(r) = 4 r 2 h g(r) ?g (inter) (r) i (2.8) The set of the OZ equation (2.2) together with its closure conditions (2.5) form a closed set of equations to be solved.However direct application of the numerical methods of solution to this set of equations is not convenient.A form more appropriate for the numerical calculations follows from (2.2) and (2.5) written in terms of the correlation functions which remains nite when m approaching in nity.In addition it is necessary to eliminate the delta-function term in the closure (2.5) and perform the correspondent renormalization of the OZ equation (2.2).With this goal in mind To eliminate the delta-function term let us present these relations in the following form where the matrices Q(k) and ^ (k) contain the elements which are the Fourier transforms of the elements Q (r) and (r), respectively.
Solution of this set of equations gives where ŝ(k) = sin kL kL Substituting Eqs.(2.13) and (2.14) into the OZ equation (2.10) and making use of equation (2.15) we obtain the OZ equation in the renormelized form ĥ(k) = Ŝ(k)ĉ(k) ŜT (k) + Ŝ(k)ĉ(k) ĥ(k) (2.17) and corresponding closure relation where ŜT (k) is the matrix transposed to the matrix Ŝ(k), ŝ(k) = 1+ Q(k) , t (r) = h (r) ?c (r) and W(r) is represented by its Fourier transform Now the initial set of equations is written in the form suitable for the numerical calculations.Solution of this set of equations is obtained using the method of direct iterations.

Results and discussion
The equilibrium properties of the model at hand are de ned by the reduced density = 3 , bonding length L, which is choosed to be L = , chain length m and reduced temperature T = kT= .We will present rst the results for the residual internal energy E, de ned by the standart relation E NkT = 2 Z 1 0 r 2 u(r)g(r) dr: (3.20)where N is the number of monomers, i.e. = N=V .
In gures 1-5 the results of the theory for the polymer systems with the chain length m = 4; 8; 16; 50 and 100 at di erent values of the temperature T and density are compared with the corresponding Monte Carlo (MC) simulation results 13].
For relatively short chains (m = 4; 8; 16) the agreement is very good at all values of the temperature and density, while for the longer chains (m = 50; 100) the agreement becomes slightly worse, although it is still reasonably good.For the values of the density approaching zero the agreement between theory and simulation for longer chains (m = 50; 100) is not that good, although the theory properly predict the nonzero value of E due to the intramolecular interactions.
Comparison of theoretical results and corresponding MC simulation results 14] for the intermolecular distribution function g (inter) (r) for 16-mers at three values of the density = 0:898; 0:5; 0:102 and di erent values of the temperature T = 8; 4; 2 is demonstrated in gures 6 and 7.
In general the agreement here is only semiquantitative, although in the case of = 0:102 and T = 4 the agreement becomes quantitative.For the sake of completeness on the same gures we show the overall pair distribution function g(r).Unfortunately computer simulation results are not available for g(r).As one would expect, with the decrease of the density the di erence between g(r) and g (inter) (r) increases.Similar, as in the case of the hard-sphere chain uid 9], at r = 2L g(r) has jump discontinuity, which re ects the rigidity of bonding between the adjacent monomers.di erent values of the temperature.Again, computer simulation results are not available here.Similar, as in the earlier studies 9] the intramolecular distribution function !(r) obtained within PPY ideal chain approximation is not sensitive to the changes in the density, at r = 2L it has a jump discontinuity and for r > mL !(r) is nonzero, although its value is relatively small there.The increase in temperature causes the overall decrease of !(r).

Conclusions
In this paper the internal energy, everage site-site intermolecular and particle-particle overall pair distribution functions of the uid of freely jointed Lennard-Jones monomers are studied within the PPY ideal chain approximation.Theoretical predictions for the internal energy of the systems consisting of the chains of the length m = 4; 8; 16; 50; 100 are in good agreement with the corresponding computer simulation predictions.The structure properties are predicted less accurate, i.e. the results of the present approach for the intermolecular distribution functions of 16-mers are only in semiquantitative agreement with MC simulation results.The accuracy of the present theory for the structure properties has been assesed only in the case of 16-mer system, since computer simulation results are not available for the shorter chains.We would expect, on the basis of previous studies 9,10] and studies carried out here, that the PPY ideal chain approximation will be of higher accuracy for the overall pair distribution function and for the chains of shorter length, but this remains to be tested.

Figure 1 .
Figure 1.Internal energy mE=NkT of LJ 4-mers.Symbols are the MC results 13] for T = 4; 3; 2 from the top to the bottom, respectively.Solid lines are the present theory.

Figure 4 .Figure 5 .
Figure 4. Internal energy mE=NkT of LJ 50-mers.Symbols are the MC results 13] for T = 5; 4; 3 from the top to the bottom, respectively.Solid lines are the present theory.

Finally in gure 8 Figure 6 .Figure 7 .Figure 8 .
Figure 6.Average intermolecular distribution function g(inter) (r) from the present theory (solid lines), MC simulation (symbols) and everage overall distribution function g(r) from the present theory (dashed lines) for 16-mer LJ chains at T = 2; = 0:898, T = 2; = 0:5 and T = 4; = 0:102 from the top to the bottom 14) and Q (r) is de ned such that, together with (r) it satis es the equation