The liquid-gas phase diagram for polydisperse dipolar hard-sphere fluid with polydispersity in the hard-sphere size and dipolar moment is calculated using extension of the recently proposed thermodynamic perturbation theory for central force (TPT-CF) associating potential. To establish the connection with the phase behavior of ferrocolloidal dispersions it is assumed that the dipole moment is proportional to the cube of the hard-sphere diameter. We present and discuss the full phase diagram, which includes cloud and shadow curves, binodals and distribution functions of the coexisting daughter phases at different degrees of the system polydispersity. In all cases studied polydispersity increases the region of the phase instability and shifts the critical point to the higher values of the temperature and density. The larger size particles always fractionate to the liquid phase and the smaller size particles tend to move to the gas phase. At relatively high values of the system polydispersity three-phase coexistence is observed. \keywordsTPT, associating fluid, polydispersity, ferrocolloids, phase diagram \pacs64.10.+h, 64.70.Fx, 82.70.Dd
Проведений розрахунок фазової дiаграми рiдина-газ полiдисперсної сумiшi дипольних твердих сфер з полiдисперснiстю як по розмiрах твердих сфер, так i по величинi дипольних моментiв, використовуючи узагальнення термодинамiчної теорiї збурення для систем з центральним характером асоцiативної взаємодiї. Для того, щоб встановити зв’язок з фазовою поведiнкою фероколоїдних дисперсiй, було зроблено припущення про те, що дипольний момент частинки є пропорцiйний до кубу її дiаметра. Ми представили та обговорили повну фазову дiаграму, яка включає кривi ‘хмари’ та ‘тiнi’, бiнодалi та функцiї розподiлу спiвiснуючих дочiрнiх фаз при рiзних значеннях полiдисперсностi системи. У всiх випадках, якi дослiджувалися, полiдисперснiсть збiльшує область фазової нестабiльностi та змiщує критичну точку в область вищих температур та густин. Частинки бiльшого розмiру завжди фракцiонують у рiдинну фазу, а частинки меншого розмiру вiддають перевагу газовiй фазi. У випадку вiдносно високого значення полiдисперсностi системи було вiдмiчено наявнiсть спiвiснування трьох фаз. \keywordsТТЗ, асоцiйована рiдина, полiдисперснiсть, фероколоїди, фазова дiаграми
Polydisperse dipolar hard-sphere fluid]Liquid-gas phase behavior of polydisperse dipolar
hard-sphere fluid: Extended thermodynamic perturbation theory for central
force associating potential
Yu.V. Kalyuzhnyi, S. Hlushak, P.T. Cummings]Yu.V. Kalyuzhnyi\refaddrlabel1,
S. Hlushak\refaddrlabel1,label2, P.T. Cummings\refaddrlabel2,label3
\addrlabel1 Institute for Condensed Matter Physics of the National
Academy of Sciences of Ukraine,
1 Svientsitskii Str., 79011 Lviv, Ukraine \addrlabel2 Department of Chemical Engineering, Vanderbilt University, Nashville, Tennessee 37235 \addrlabel3 Nanomaterials Theory Institute, Center for Nanophase Material Sciences, Oak Ridge National Laboratory,
Oak Ridge, Tennessee 37830 \authorcopyrightYu.V. Kalyuzhnyi, S. Hlushak, P.T. Cummings, 2012
In this paper we consider the liquid-gas phase behavior of polydisperse dipolar hard-sphere mixture. Recently, liquid-gas phase equilibria in monodisperse dipolar hard-sphere fluid, Yukawa dipolar hard-sphere fluid and Shtockmayer fluid were studied using thermodynamic perturbation theory for central force (TPT-CF) associating potential [1, 2, 3, 4]. In this study we propose an extension of the TPT-CF, which enables us to investigate the phase behavior of polydisperse mixture of the dipolar hard spheres with polydispersity in both hard-sphere size and dipole moment. We call this extension as extended TPT-CF (ETPT-CF). Similar to our previous study , ETPT-CF combines Wertheim’s TPT [7, 8] for associating fluid with association due to off-center attractive sites, and TPT-CF [2, 6], which permits a multiple bonding of one site. In our theory we have several Wertheim’s types of associating sites with the possibility for each site to be multiply bonded (in Wertheim’s TPT each site is only singly bondable). Final expressions for thermodynamical properties of polydisperse dipolar hard-sphere fluid is written in terms of the finite number of distribution function moments, i.e., in the framework of ETPT-CF this system belongs to the family of the so-called truncatable free energy models (see  and references therein). This property enables us to calculate the full liquid-gas phase diagram (including cloud and shadow curves and binodals) and to study the effects of fractionation on the level of the distribution functions of coexisting daughter phases.
2 Extended thermodynamic perturbation theory for central force associative potential
2.1 Analysis and classification of diagrams
We consider a multicomponent fluid mixture consisting of species with a number density at a temperature (), where is the density of the particles of species. The particles of the species and interact via the pair potential , which can be written as a sum of the reference and associative parts
where and denote positions and orientations of the particles and . We assume that the associative part of the potential can be represented as a sum of terms, i.e.
where the lower indices and take the values and , respectively. These values specify the splitting of the total associating potential into several particular pieces. For example in the case of the models utilized by Wertheim  these indices denote off-center attractive sites and in the case of the Mercedes-Benz (MB) type of models  or cone models  they stand for the type of hydrogen bonding arms. Hereafter we will refer to these indices as the site indices, keeping in mind that they may have a more general meaning. Here and are the number of such sites on the particles of and species, respectively. According to (1) and (2) the Mayer function for the total potential (1) takes the following form:
where we use the usual notation:
For the sake of diagrammatic analysis we will follow Wertheim  and instead of
circles we introduce hypercircles to represent particles in diagrammatic expansions.
Each hypercircle is depicted as a large open circle with small circles inside denoting
the sites. Corresponding cluster integrals are represented by the diagrams built on a
hypercircles connected by and bonds
and site circles connected by
the associating bonds .
Due to the decomposition of the Mayer function
(3) we will have the following diagrammatic expressions for the logarithm of a
grand partition function and for the one-point density in terms of the
sum of all topologically distinct connected diagrams consisting of field hypercircles, , and bonds. Each bonded pair of hypercircles has either , or and one or more bonds.
sum of all topologically distinct connected diagrams obtained from by replacing in all possible ways one field hypercircle by a circle labeled .
Here , denotes position and orientation of the particle , and is an external field. For a uniform system . Following [7, 8, 1, 2] we introduce the definition of the -mer diagrams. These are the diagrams consisting of hypercircles, which all are connected by the network of bonds. The site circles, which are incident with more than bonds are called oversaturated site circles. We consider now the set of oversaturated site circles with each pair connected by at least one path formed by the circles from the same set. The subdiagram involving this set of circles, together with the site circles adjacent to them and bonds connecting all these circles , we call the oversaturated subdiagram. The set of all possible -mer diagrams can be constructed in three steps: (i) generating the subset of all possible connected diagrams with only bonds, (ii) inserting combined bond between all pairs of hypercircles with the site circles, which belong to the same maximal oversaturated subdiagram and (iii) taking all ways of inserting an bond between the pairs of hypercircles, which were not connected during the previous two steps. As a result the diagrams, which appear in and , can be expressed in terms of the -mer diagrams:
= sum of all topologically distinct connected diagrams consisting of -mer diagrams with and bonds between pairs of hypercircles in distinct -mer diagrams.
The procedure for obtaining the expression for from remains unchanged.
The diagrams appearing in the expansion of the singlet density can be classified with respect to the number of bonds associated with the labeled hypercircle. We denote the sum of the diagrams with associating bonds connected to the site (), which belongs to the particle of species as . Any site K, which is connected to associating bons, will be denoted as In what follows we will use also a condensed version of the notation, i.e.
where . The set with all indices, except one index , equal 0, will be denoted as , i.e. , so that for any quantity we have
Thus can be written as follows
2.2 Topological reduction
The process of switching from the activity to a density expansion goes in the same fashion as in Refs. [8, 1, 2]. However, to proceed it is convenient to use an operator form of notation. The operators are introduced in a manner similar to that presented in references [8, 1] to which we refer the reader for more details. We associate with each labeled hypercircle an operator with the following properties:
where . The one-point quantities, which, for convenience, are denoted by , can be presented as illustrated below:
The operators are straightforward generalization of the operators introduced earlier [8, 1, 12]. Thus, the rules of manipulation with the new quantities are similar to that discussed before. In particular, the usual algebraic rules apply to these quantities and analytical functions of are defined by the corresponding power series. Similar, as in references [8, 1, 12], it is convenient to use the angular brackets to specify the operation
In the case of several labeled circles the subscripts on the brackets denote the circle to which procedure (10) is to be applied.
Analyzing the connectivity of the diagrams in , at a labeled hypercircle we have
where with denotes the sum of diagrams in for which the labeled 1 hypercircle is not an articulation circle. Similarly denotes the sum of diagrams in for which hypercircle 1 is not an articulation circle. Elimination of the diagrams containing field articulation circles can be achieved by switching from an activity to a density expansion. To do so we adopt the following rule: each field hypercircle , with bonding state of its sites represented by the set , in all irreducible diagrams is replaced by a hypercircle, where () for and for . The new quantities are connected to the densities via the following relation:
This relation can be inverted expanding into a power series, i.e.
Now the diagrammatic expansions for can be expressed in terms
of the irreducible diagrams. To present this result in compact and convenient form we introduce
a sum of the diagrams defined as follows:
= sum of all topologically distinct irreducible diagrams consisting of -mer diagrams with and bonds between pairs of hypercircles in distinct -mer diagrams. All hypercircles are field circles carrying the -factor according to the rule formulated above
Functional differentiation of with respect to gives an expression for :
2.3 Extended thermodynamic perturbation theory for central force associating
Now we are in a position to rewrite the regular one-density virial expansion for the pressure in terms of the density parameters . Following the scheme, proposed earlier [7, 8, 2, 4] we have expression for the pressure in operator form
where is the volume of the system. Similarly, as in [7, 8] one can verify that these expressions satisfy the regular thermodynamic relation where and is the chemical potential. This can be achieved by taking a variation of (15) (or (16)) and combining (11), (13) and (14). The corresponding expression for Helmholtz free energy is
where is the thermal de Broglie wavelength. This expression is derived using the regular thermodynamic expression for Helmholtz free energy together with relation
which follows from (11), written for . Here is the number of particles of species in the system.
Helmholtz free energy in excess to its reference system value is obtained by subtracting corresponding expression for from (17), i.e.
where is the corresponding sum of the diagrams for the reference system. Ordering the virial expansion (19) with respect to the number of associating bonds and neglecting the terms with more than one associating bond we have
where is the reference system distribution function and
Due to the single bond approximation for all values of the set , except for and with . This property together with (11) yield the following relations:
2.4 Extended TPT-CF for two sites with double-bonding condition
The theory presented in the previous section is quite general and can be applied to a number of different situations. However, in the present study we are interested in the version of the theory for the model with two sites both of which can be bonded twice. More specifically, we are interested in the extension and application of the theory to the study of the phase behavior of polydisperse dipolar hard-sphere mixture.
We assume that each of the particles in the system has two doubly bondable attractive sites, and , i.e. we have: and . We also assume that attractive interaction is acting only between the sites of the same sort. Using these suggestions, relations (11) and (12), and taking into account that the system is uniform, the density parameters and can be expressed in terms of
where takes the values and and . These two equations give
In turn, using (21), for we have
which satisfy the following set of equations:
Chemical potential and pressure in excess of their reference system values can be obtained using standard thermodynamical relations:
Finally, the average size of the clusters, which appear in the system, can be characterized by the average length of the chain formed by either -bonded () or -bonded () particles. Following [13, 14] we defined this quantity by the following expression
where is the fraction of singly -bonded particles (fraction of the chain ends), is the fraction of doubly -bonded particles (fraction of the chain middles) and is the fraction of nonbonded particles. For these fractions we have
we get the final expression for in terms of :
where if then and if then .
3 Liquid-gas phase behavior of polydisperse dipolar hard-sphere fluid
3.1 The model
We consider a polydisperse dipolar hard-sphere fluid mixture with a number density and a polydispersity in both the hard-sphere diameter and the dipolar moment . We assume, that the dipole moment is proportional to the particle volume, i.e., . Thus, the type of the particle is completely defined by its hard-sphere size and hereafter we will be using instead of the indices to denote the particle species. We also assume that hard-sphere size of the particles is distributed according to a normalized distribution function ,
Interaction between particles of species and in our system is described by the following pair potential:
where is the hard-sphere potential and is the dipole-dipole potential, given by
Here and denote the angles between the dipole vectors and the vector that joins the centers of the two particles, and and are the azimuthal angles about this vector. To proceed we have to split the total potential (40) into the reference and the associative pieces. We assume that the reference part of the potential is represented by the hard-sphere part and the associative part by the dipole-dipole potential . At the contact distance , the latter potential has two equal potential minima of the depth at ‘‘nose-to-tail’’ configuration (). These minima are responsible for the formation of chains of particles in the system. In addition, there are twice less deep minima at antiparallel configuration with . The latter minima cause the formation of the network connecting the chains. According to the earlier theoretical and computer simulation studies [15, 16] competition between the chain formation and network formation defines the existence of the liquid-gas phase transition in the dipolar hard-sphere fluid. To account for this effect we propose the following splitting of the total associative potential :
where and is the Heaviside step function. Here plays a role of the potential splitting parameter. For we have that and . On the other hand gives: and . In both limiting cases the theory developed will treat the system as a polydisperse mixture of the hard-sphere chains. For the intermediate values of , the energy minima at ‘‘nose-to-tail’’ configuration are included into and network forming minima appear in . We have chosen here . With this value of , our results for monodisperse version of the model are in good agreement with the results of the previous studies [16, 2].
3.2 Thermodynamic properties
For a general multicomponent dipolar hard-sphere mixture, thermodynamic properties can be obtained using the solution of a set of nonlinear equations (33) and an expression for the Helmholtz free energy (32). However, even for the multicomponent case, solution of this equation rapidly becomes involved as the number of components increases. As we proceed to the polydisperse case, solution of the polydisperse version of equation (33) becomes intractable, since now we have to deal with the following integral equation:
where we have dropped the lower index 1, i.e. . In order to solve this equation we propose here to interpolate the key quantity of the theory, the volume integral , using a sum of Yukawa terms. Since the reference system pair distribution function is independent of mutual orientation of the particles for the integral (31) we have
where is an orientation averaged Mayer function for associative potential . We assume, that can be represented in the following form:
where is the Laplace transform of the radial distribution function . We will use here Percus-Yevick approximation for the hard-sphere radial distribution function, since the analytical expression for its Laplace transform is known 
Here are the moments and are the generalized moments of the distribution function . Expression for these moments can be symbolically presented as follows:
Hereafter all the quantities denoted as with certain set of indices will represent the generalized moments defined by (51). Corresponding expressions for are collected in the appendix B. Inserting (47) into equation (44) and using (48), we find
where satisfies the following set of equations:
Thus, solution of the integral equation (44) for the unknown function now is reduced to the solution of a set of equations for unknown constants . This solution can be used to calculate , which in turn can be utilized to calculate thermodynamical properties of the system via Helmholtz free energy (32). Generalizing the expression for Helmholtz free energy (32) for a polydisperse system, we have
Now we can use the standard relation between Helmholtz free energy and chemical potential (34), generalized to polydisperse case