Solution Of The Polymer Percus-Yevick Approximation For The Multiarm Star Polymerization

An analytical solution of the polymer Percus-Yevick approximation for a multicomponent mixture of associating hard spheres, forming star polymer molecules upon association, is derived. Simpliied version of the solution, which combines the polymer Percus-Yevick approximation with the so-called ideal chain approximation is used to describe the equilibrium properties of the two-component associating mixture of hard spheres and linear chain molecules. The structure properties of the model are studied at all degrees of association, including the limit of complete association in which the system is represented by the uid of multiarm polymer star molecules.


Introduction
In recent years a number of integral equation studies of the structure and thermodynamic properties of polymer uids have been published.These include the application of the polymer reference interaction site model theory (PRISM) (see Refs. 1,2] and references therein), the polymer Born-Green-Yvon (PBGY) theory 3,4], a version of the Percus-Yevick (PY) theory extended to polymer uids 5], a theory based on the Chandler-Silbey-Ladanyi PY approximation for the site-site uid 6] and the multidensity polymer PY (PPY) theory for associating uids 7,8] appropriately modi ed to describe polymer uids 9{13].
However, all of these studies are focused on the investigation of the polymer uids represented by the system of monomers that have completely associated into polymers of a xed size, usually into linear freely-jointed chain polymers of a xed length.More recently an analytical solution of the multidensity PPY theory was used to describe an associating uid which polymerizes into freely-jointed tangent hard-sphere linear chain molecules 2,11].The structure properties of such a system was studied at all the degrees of polymerization, including the limit of completely dissociated and completely associated systems.
In this paper an analytical solution of the PPY approximation, formulated for the model of the associating uid which forms polymer star molecules upon association, is derived.We also propose a simpli ed version of the solution, in which the PPY approximation is supplemented by the so-called ideal chain approximation 8,9,12,13].Solution of the PPY ideal chain approximation is utilized to study the structure properties of the associating mixture of hard spheres and linear chain molecules of a xed length 72 Yu.V.Kalyuzhnyi at all the degrees of association.The model of this type can be used to describe the properties of star polymer systems, and polymer coated colloidal systems.

The model and theory
We consider a model represented by the (n p + n c )-component mixture of hard-sphere particles with hard spheres of each of n p species having two sticky points, A and B, randomly placed on the surface, and hard spheres of each of n c species having only one sticky site 0 placed in the center of the hard sphere.The pair potential U ab (12) for this model consists of the hard-sphere term and terms describing the sticky interaction U ab (12) = ab (r) + X ij f ap bp (1 ?ij )(1 ?i0 )(1 ?j0 )+ ap bc (1 ?i0 ) j0 + ac bp (1 ?j0 ) i0 g ab i j (12) (2.1) where the upper indices a and b, each taking the values p or c, and together with the lower indices and , each taking the values 1; 2; :::; n a and 1; 2; ::; n b , stand for the species of the particles.Here ab (r) is the hard-sphere potential ), R a is the hard-sphere diameter, ab is the Kroneker delta, ab i j (12) is the site-site potential of the sticky interaction between the sites i and j belonging to the particles of (a; ) and (b; ) species, respectively, and arguments 1 and 2 denote the positions and orientations of the two particles.
The summation in (2.1) is carried out over i and j taking the values 0; A; B, and we assume that the sticky interaction is valid only between the sites of a di erent type.Thus, in our notation the species of each of the particles is de ned by a pair of indices, (a; ), and the type of each of the attractive sites is de ned by a set of three indices, (a; ; i).
Due to the short-range character of the sticky site-site interaction and due to the random location of the attractive sites, the p-type of the particles polymerizes into freely-jointed tangent hard-sphere linear chain molecules.
The structure of the clusters which involve the c-type of the particles is similar to that formed by the Smith-Nezbeda primitive model of associating uid 14,17,16], i.e. each of the p-type of the particles can simultaneously be bonded to a limited number of the c-type of the particles (in the present case not more than to two c-type particles), while the c-type of the particles can bond an arbitrary number of the p-type of the particles.
The model in question is described using a version of the multidensity integral equation theory for associating uids which combines a four-density theory for linear chain polymerization 2,7,8,11] and a two-density theory developed to treat the systems with strong assymetry in associative interaction 14,16{18].Similarly, as in the earlier studies, we are using an orientationally averaged multidensity Ornstein-Zernike (OZ) equation supplemented by the PPY approximation.The corresponding OZ equation, written in terms of the orientationally averaged partial total h ab i j (r) and direct c ab i j (r) correlation functions reads where ĥab (k), ĉab (k) and (a) are the matrices of the following form: Here ĥab i j (k) and ĉab i j (k) are the Fourier transforms of the partial correlation functions h ab i j (r) and c ab i j (r), respectively, the lower indices i and j in (a)  i , (a)  i and in partial correlation functions denote the bonding states of the corresponding particle.In the case of the p-type of the particle i = 0 corresponds to an unbonded particle, i = A(or B) { to a particle bonded at A(or B) site and i = ?to a particle bonded at both A and B sites.
Since the c-type of the particles does not have o -center attractive sites, we follow the earlier developments 14, 16,18] and do not distinguish between their bonded states.In this case the index denoting the bonding states of the corresponding c-type particle is taking only one value of i = 0.
The PPY closure relations for the present model take the form 4) The sticky potential ab i j ( 12) is entering the closure relation (2.4) via the angle averaged Mayer function fab i j (r) which is proportional to the Dirac delta-function fab i j (r) exp h ?ab (r) i = f ap bp (1 ?ij )(1 ?i0 )(1 ?j0 )+ ap bc (1 ?i0 ) j0 + ac bp (1 ?j0 ) i0 g K ab i j (r ?R ab ) (2.5) where K ab i j is a stickiness parameter related to B i j by ??L K cp 0 L : (2.7) Subindices K and L each take the values A or B, ?? A = B and y ab i j is the contact value of the partial cavity distribution function de ned by g ab i j (r) = e ?ab (r) n y ab i j (r) + ap bp (1 ?ij )(1 ?i0 )(1 ?j0 )+ + ap bc (1 ?i0 ) j0 + ac bp (1 ?j0 ) i0 ] B ab i j (r ?R) o ; (2.8) where g ab i j (r) = h ab i j (r)? i0 j0 .Finally, the relation between the densities of bonded and unbonded particles are found to be ??K ; (2.9) (2.10)

General solution of the PPY approximation
The solution of the present version of the PPY approximation is obtained by using Baxter-Wertheim factorization technique.The general scheme of the solution is quite similar to that derived in 2,11], and we follow these earlier studies closely.The factorized version of the OZ equation (2.3) can be presented in the following form: ?rh ab i j (r) = h q ab i j (r) i 0 ?
where (a) ij is the correspondent elements of the matrix (a) , ?n (a) = (1 ?ac )?, S ab = 1 2 (R a ?R b ) and integration in (3.12) is carried out over the range de ned by S da < t < min R da ; R db ?r].
R ap y ap Thus, solution of the OZ equation (2.3) closed by the PPY closure conditions (2.4) reduces to the solution of a set of algebraic equations formed by relations (2.9), (2.10) and (2.6).

Solution of the PPY ideal chain approximation for a two-component associative polymer-colloidal mixture
The model discussed in two previous sections is quite general and can be used to describe a number of di erent associating macromolecular systems.One of such systems is represented by a two-component associating mixture of chain molecules and hard spheres, which form star polymer molecules From the top to the bottom at r = 1:5 K pc = 0; 1; 5; 15; 30; 1.
upon association.This particular version of the model discussed earlier can be obtained by imposing certain restrictions on the bonding possibilities of the system.The model of this type can be used to describe such systems as polymer coated colloids and star polymer systems.In this section we will illustrate the solution of the PPY approximation, derived in the previous section, by its application to such associating polymer-colloidal systems.In addition, we will simplify the solution by utilizing the version of the so-called ideal chain approximation 8,9] proposed recently 13].
Let us consider the (n p + 1)-component version of the model with (p) = (p) and with the following conditions imposed on the values of the stickeness parameter K ab i j K pp K L = K pp ( KB LA ; ?1 + KA LB ?1; ) K pc K10 = K pc KA 1 (4.25) and restrict our study to the case of an in nitely strong sticky attraction between the p-type of the particles, i.e.K pp ! 1.In this limit the system is represented by a two-component mixture of hard spheres and linear chain molecules which, due to the association between the p-type and the c-type of the particles form star polymer molecules of the type shown in gure 1.
In the complete association limit (K pc ! 1) the average number of arms n arm per each molecule is de ned by the ratio n arm = (p) = (c) .
and the expression for the Baxter q-function derived in the previous section.

Results and discussion
The equilibrium properties of the model at hand are de ned by the total packing fraction , the strength of associative interaction K pp , the number of p-type particles species n p (which is the length of the chain molecules), the ratio between the densities of the p-type and c-type of the particles n arm = (p) = (c) (which is the average number of arms per each star molecule formed in the complete association limit, K pc ! 1).Results presented in this section apply to the version of the model with n arm = 4, n p = 4, R p = R p = 1 and R c 1 = R c = 2.
First we discuss the dependence of the fraction of free (not bonded to the c-type of the particles) chain molecules x = (p) 1B = (p) on the strength of the sticky interaction K pc and packing fraction .Fig. 2 shows x as a function of the system packing fraction for several values of the stickiness.for each of the monomer pair.In addition to these RDFs, for the description of the structure of the polymer system, the so-called averaged RDFs, are often used.For the model in question they are de ned as g pp (r) = 1 (n p ) 2 np X =1 g pp (r); g pc (r) = 1 n p np X g pc 1 (r): (5.48) In gures 3-10 we present various RDFs for a partially associated system at two values of the total packing fraction = 0:1; 0:3 and di erent values of the strength of association.From these gures one can see the changes in the structure with the increase of the degree of association from a nonassociating mixture of hard spheres and chain molecules (K pc =0) to complete the association limit, K pc = 1, represented by the system of polymer star molecules with the average number of arms n arm = 4.
The structure peculiarities of the nonassociating mixture of hard spheres and chain molecules were discussed earlier 10].In brief, the RDF g pp (r) has a jump discontinuity at r = 2R p which is the result of intramolecular correlations, while the shape of the RDF g cc (r) is quite similar to the RDF of the regular hard-sphere mixture.Similarly, as in the case of the sitesite molecular uids 22], the RDF g pc (r) has a cusp at r = 1 2 R c + 3  ( gures 4, 7).With the increase of K pc the cusp of the same origin appeares on the RDF g cc (r) at r = R p + R c ( gures 3, 6).In addition, the RDFs g pp (r) and g pc (r) show a jump discontinuity at r = R p + R c and r = 3 2 R p + 1 2 R c , respectively, re ecting the formation of star molecules ( gures 4, 5, 7, 8).The contact values of all the three average RDFs g cc (r), g pc (r) and g pp (r) change their values from a value larger than 1 to a value smaller than 1 at lower values of the packing fraction ( = 0:1) ( gures 3-5).The corresponding changes at higher values of the packing fraction ( = 0:3) are not so pronounced and are relatively smaller, although the association causes a decrease of the contact values of the RDF.These changes in the contact values result from the screening e ects due to the adjacent bonded spheres.In gure 9 we compare the behaviour of the individual RDFs g pp 11 (r) and g pp 14 (r).At K pc = 0 both RDFs coincide, while with the increase of K pc they show a substantial di erence.At large values of K pc the RDF g pp 11 (r) has a relatively large contact value, and for r = R c + R p shows a jump discontinuity.The corresponding RDF g pp 14 (r) remains almost unchanged.
Similar comparison for the RDFs g pc 11 (r) and g pc 41 (r) is demonstrated in gure 10.Here both functions change their shapes due to the association.With the increase of K pc the contact values of the RDFs g pc 11 (r) and g pc 41 (r) decrease as well.

Concluding remarks
In the present paper an analytical solution of the polymer Percus-Yevick (PPY) approximation for the multicomponent mixture of associating hard spheres, forming polymer star molecules upon association, is derived.A simpli ed version of the solution, which involves in addition the so-called ideal chain approximation, has been used to study the structure properties of the two-component associating mixture of hard spheres and linear chain molecules.The accuracy of the present theory has been assessed only in the complete dissociation limit (K pc = 0) 10], since simulation results for a partially and completely associated system are not available.We expect it to be of the same order of accuracy for all the degrees of association, but this remains to be tested.The corresponding work is in progress and its results will be reported in due cource.

Figure 1 . 4 -
Figure 1.4-arm molecular model, studied in Section 4, in the complete association limit.The c-type of the particles is represented by the central bead, while the rest of the beads belong to the p-type of the particles

Figure 3 .
Figure 3. Radial distribution function g cc (r) at = 0:1 and different values of the strength of association.From the top to the bottom at r = 2 K pc = 0; 1; 5; 15; 30; 1.

Figure 4 .
Figure 4. Average radial distribution function g pc (r) at = 0:1 and di erent values of the strength of association.

Figure 7 .
Figure 7. Average radial distribution function g pc (r) at = 0:3 and di erent values of the strength of association.

pc 11 ac
R b + bc R a ; (4.40) R ap y ap 0 L (R ap +) = ac AL 2

Figure 8 .
Figure 8.Average radial distribution function g pp (r) at = 0:3 and di erent values of the strength of association.

Figure 9 .
Figure 9. Radial distribution function g pp 14 (r) (1) and g pp 11 (r) (2) at = 0:1 and K pc = 0 (solid line), K pc = 5 (long dashed line) and K pc = 1 (short dashed line).As one would expect, with the increase of and/or K pc the fraction of the free chains decreases.The version of the ideal chain approximation utilized in the present study allows one to calculate the individual radial distribution functions (RDF) g ab (r) = nan b X ij g ab i j (r) (5.47)