LLE often occur, when species are mixed differing strongly in either polarity or molar mass or both. Typical examples are mixtures of a non-polar alkane (e.g., octane) and the highly polar water [100] or mixtures of a long chain polymer (e.g., polyethylene) and a small chain alkane (e.g., hexane) [101]. In phase equilibrium thermodynamics the usual approach to describe phase behaviour is the mean field approach. Using this method the many body problem of describing the kinetic and potential energy of each molecule in a mixture is approximated, mostly using statistical mechanics [102]. The result after the choice of a statistical ensemble is one of three thermodynamic potentials, depending on their respective natural observables [103].

The thermodynamic potentials are entropy:

      (1)

Helmholtz free energy:

      (2)

and the grand thermodynamic potential

      (3)

The observables are internal energy , volume , temperature , amount of substance , , and the chemical potentials of the components , . The derivatives of the respective potentials, with respect to their natural variables, yield all other thermodynamic information. For example the derivative of Helmholtz energy with respect to volume results in the negative system pressure, :

      (4)

It is possible to transform between these potentials with the Legendre transformation. Thereby, one observable is replaced by another in the following way:

      (5)

Here and are the respective potentials and is one of the natural variables of potential, . The list of common transformations is shown below.

      (6)

      (7)

      (8)

      (9)

Here entropy and energy change their roles as potential and observable. The potentials are: internal energy , Helmholtz free energy , Gibbs energy and enthalpy .

In calculations of equilibria between two separated phases and the Helmholtz free energy and the Gibbs energy are of substantial importance. Both these potentials do have a minimum in equilibrium. This means, an optimization approach to the calculation of equilibria is feasible.

      (10)

      (11)

The Gibbs energy minimization, on the one hand, is typically used for phase separations, where pressure does not play a crucial role or is accorded for otherwise. The Helmholtz free energy minimization, on the other hand, is used when compressibility of the system is of concern. In both cases the condition of chemical equilibrium ensues:

      (12)

In case of Helmholtz free energy the optimization yields the additional condition of mechanical equilibrium. This condition is trivial for the Gibbs energy, because it depends directly on the pressure and not on the volume.

      (13)

The equilibrium can be calculated using Equations (12) and (13). This solution ensures that the condition of thermal equilibrium

      (14)

is also fulfilled. Hence, the problem to solve is a (typically nonlinear) system of equations with equations or equations for a compressible system, where is the number of components.

In polymer thermodynamics one is challenged with the immanent huge difference in molar mass between the solute and the solvent. This generally leads to the fact that molar based approaches to calculate the phase behaviour of polymers in solution are predetermined to fail. This can be understood, if the limiting case of infinite molar mass is examined. Here, the mole fraction of the polymer approaches zero implying that Raoult’s law [104] is applicable for the solvent, yet the mixture is highly non-ideal. Figure 1 demonstrates an impressive example, namely the vapour pressure of a polymer solution. According to Raoult’s law, the vapour pressure of the considered solution should change linearly (straight line in Figure 1) from the vapour pressure of the solvent to zero pressure, because the vapour pressure of the pure polymer is zero. However, the experimental data, taken from the literature [61], are far away from Raoult’s law. Additionally, the experimental data in Figure 1 clearly shows the impact of polymer architecture on the thermodynamic properties. Although the same type of polymer from the chemical point of view (linear and branched polyisoprene) in the same solvent (cyclohexane) was used, differences in the vapour pressure are found experimentally. These differences can only be explained by the influence of the chain architecture on the thermodynamic properties. The experimental results demonstrate that cyclohexane is a considerably worse solvent for branched polyisoprene than for the linear analog at all temperatures and at all compositions. This finding is in seeming contrast to the widespread notion that branched polymers are better soluble than their linear counterparts. It may, however, well be that special interactions between the components of the mixture and larger differences between the end groups and the middle groups of the polymer are capable to change the picture.

Flory [84] suggested a way to describe polymer chain solutions avoiding this complication. For this purpose he introduced a lattice on which the solvent occupies a single lattice site and the polymer may occupy several neighbouring lattice sites, as depicted in Figure 2. In order to calculate the micro-canonical partition function of a system containing a solvent and a linear polymer, the number of ways to consecutively insert a polymer chain into a lattice of coordination number , at first fully occupied by solvent beads, must be calculated.

This is done by Flory [84], by assuming that

From this he derived the athermal entropy of mixing per lattice site [84]:

      (15)

Here is the number of lattice sites, is the Boltzmann constant, is the number of segments the polymer is composed of, is the number of solvent segments, and the are the segment mole fractions of components , defined as:

      (16)

In addition a regular mean field energetic contribution of the molecules on the fully occupied lattice is introduced [84]:

      (17)

Usually the interaction energy is expressed in terms of . By using Equation (7) the Helmholtz free energy is derived [84].

      (18)

Though the FH theory is useful for calculating LLE of simple chain polymers, it does neglect the structure of the molecules completely. However, in order to achieve quantitative agreement with experimental data, a concentration-dependent interaction parameter was introduced [81,82,83,84]. The approach, employed to calculate the phase behaviour of hyperbranched polymers, is the extension of FH theory based on physical argumentation.

As the LCT is an extension of the FH theory, both theories have the same fundamental idea of a lattice, which is occupied by a polymer and so the polymer can be divided in different segments. Using this idea, the exact partition function of a polymer blend where two segments and interacts with the energy can be read as [92]:

      (19)

where is the Kronecker delta function and the vector is pointed from a given lattice site to the nearest neighbour lattice sites. A factor has to be introduced for the indistinguishability of polymer chains of the same species and the factor ½ accounts for the symmetry of each chain. The outside summation in Equation (19) prohibits any lattice site from being occupied by two polymer segments. In the outside summation of Equation (19) there are two factors, whereas the first factor accounts for the bonding constraints in the polymer, the second factor describes the Van der Waals interaction between two lattice sites. The expression in Equation (19) represents an exact solution of a component polymer blend on a cubic Bravais lattice, but for using this approach a simplification is desirable. In the FH theory this simplification is the assumption that just the next neighbours of one polymer segment have to be considered. Freed and Dudowicz [89] extend the assumption of the FH theory by the introduction of a cluster expansion. This expansion goes back to the cluster expansion introduced by Mayer [99] for non-ideal gases. In the framework of LCT, the corrections to the Helmholtz free energy are derived in form of a cluster series expansion in the inverse coordination number and in the reduced interaction energy , taking into account the growing correlations between near segments on the same molecule.

By introducing these cluster expansions and fundamental statistical thermodynamics, the Helmholtz free energy of a component polymer blend reads as [89]:

      (20)

where the first two terms in Equation (20) are the mean field contributions to the entropy and the interaction energy of the Helmholtz free energy. The third term of Equation (20) represents the corrections to the mean field approach of the FH theory. It appears in form of a cumulant cluster diagram, which has to be evaluated for different orders of the interaction energy. Usually, the evaluation is truncated at the second order of the reduced interaction energy as suggested by Freed and Dudowicz [89]. One example of evaluating the second order contribution such a diagram will be shown in the following.

      (21)

In term (21) there is a cumulant cluster diagram of three bonded monomers (solid circles with solid line) and two pairs of interacting monomers (stars with dotted lines). The solid lines represent covalent bonds and the dotted lines physical interaction energies. The monomers in this cumulant cluster diagram do not belong to the trimer chain. To evaluate this diagram, it has to be expanded in a series as shown by Dudowicz and Freed [89].

      (22)

In Equation (22) the cumulant cluster diagram is expanded in four diagrams. Two of these diagrams in Equation (22) are vanishing in the thermodynamic limit. The other diagrams have to be evaluated. The analysis of these diagrams needs the knowledge of two factors. One factor is the combinatorial factor , which depends on the number of components and is independent of the polymer architecture. As an example the diagram k1 in Equation (22) is evaluated. For a one component system it is obtained:

      (23)

where and are the number of two successive bonds in a single polymer chain and in that order the number of monomers of one chain, while is the number of chains in the system. The factor 1/5! arises because of the indistinguishability of selecting the chains.

This formalism can be extended to a multi-component system by labelling all monomers in the diagram. In the case with only distinct labels the factor of indistinguishability reduces to unity. The other factor which has to be evaluated is the connectivity factor for each diagram with bonds [89]:

      (24)

where depends only on the number of lattice sites and the number of vertices in the diagram [ 89]. The factor depends only on the lattice, but not on the polymer architecture. The evaluation process is shown by Dudowicz and Freed [89]. By knowledge of the combinatorial factor and the connectivity factor, this diagram can be analysed. The evaluation leads to the contribution [89]:

      (25)

Here the segment fraction is that of the compressible mixture:

      (26)

This segment fraction reduces to the one of the incompressible mixture, if the number of void lattice sites goes to zero, resulting in

      (27)

The evaluation of all cumulant cluster diagrams leads to an expression of the Helmholtz free energy, which will be presented in the following sections.

Lattice Cluster Theory for a Binary Polymer Blend

At first the Helmholtz free energy of an incompressible polymer blend is presented [53,56]. As the polymer blend is regarded as incompressible the three interaction energies can be combined to the single interaction energy (see Equation (17)):

      (28)

The segment molar Helmholtz energy can be calculated as follows:

      (29)

The first two terms on the right hand side of Equation (29) represent the mean field entropic contribution (see Equation (15)) and is the chain length of component . In Equation (29) the corrections to the FH theory appear in form of a power series. Its coefficients depending only on the polymer architecture, which is described with , and the interaction energy can be computed using the following relations [57]:

      (30)

      (31)

      (32)

      (33)

      (34)

      (35)

where the corrections made by Dudowicz et al. [110] were incorporated. In the limit only the parameters and are left, which can be summarized to the well known Flory–Huggins parameter. To characterize the architecture of a molecule the geometric parameters () are important. These parameters will be explained in the section dealing with the application to hyperbranched polymers, but at first the Helmholtz free energy of a ternary solution will be introduced.

LCT for a Ternary Polymer Solution

The starting point for the calculation of the Helmholtz free energy of a ternary polymer solution is [56]:

      (36)

where the contributions of entropy (), as well as the first () and second order () of energy can be calculated using the tables I, II and III published by Dudowicz and Freed [89] and taking into account the corrections introduced by Dudowicz et al. [110]. The entropic part of the Helmholtz free energy reads [56]:

      (37)

where the first terms on the right hand side of Equation (37) represent the contribution to the mean field limit and the following terms are the extensions of the mean field approach. These contributions depend only on the structure of the polymer in terms of and they can found in the literature [56].

In addition to the entropic corrections of the LCT, also the energetic corrections to the FH theory have to be determined. The first order mixing energy () as well as the second order mixing energy () can be expressed as a sum [56]:

      (38)

where the contributions and are given in literature [56]. These contribution depend on the architecture via and additionally from the difference in interaction energy of component and , expressed by the three interaction parameters , , and . In the z→∞ and ε→0 limit this theoretical framework reduces to the Flory–Huggins expression of a ternary polymer solution and it reduces also correctly to the equation describing a binary mixture (Equation (29)). For applying the LCT, the determination of the architectural parameters is necessary. This will be shown in Section 2.2.3.

For compressible systems the LCT can be extended to account for free volume. The proposal of Freed and co-workers [89,90,91] is to introduce void lattice sites in order to do that. Void lattice sites are modelled as single sites on the lattice that interact neither with each other nor with other molecules. This leads to a change in the mean field contribution to the partition function:

      (39)

For a pure compound the contributions to the Helmholtz energy per lattice site are as follows:

      (40)

Here is the segment fraction of void lattice cells. The entropic contribution for pure components can be written as a polynomial in the void segment fractions [111,112]:

      (41)

The coefficients of this polynomial depend on the molecule’s structure and on the lattice coordination number [111,112]:

      (42)

      (43)

      (44)

      (45)

      (46)

The first and second order energy contribution can also be developed in a polynomial of the void segment fractions [111,112]:

      (47)

Here is the interaction energy between two segments of component . Again, the coefficients of the polynomial can be expressed in terms of the molecule’s structure and the lattice coordination number [111,112]:

      (48)

      (49)

      (50)

      (51)

      (52)

      (53)

The same procedure for the second order energy contribution leads to [111,112]:

      (54)

where the polynomial coefficients can be calculated as [111,112]:

      (55)

      (56)

      (57)

      (58)

      (59)

Helmholtz free energy can now be expressed as [111,112]:

      (60)

From the Helmholtz free energy the thermal equation of state (LCT-EOS) can be calculated via standard thermodynamic relationships (Equation (4)) [111,112]:

      (61)

Here, is the length of a cubic cell, a segment of molecule occupies. The specific volume of the substance can be calculated with the following equation:

      (62)

The chemical potential of the species is given by [111,112]:

      (63)

Constant parts do not play a role in equilibrium calculations, as they are equal for both phases. This results in a simpler form for the chemical potential [111,112]:

      (64)

Differences to Section 2.2.1 occur, because of the reversed indexing of the solvent (or in this case the void lattice sites) and the not-specified other compound.

The multi-component expressions are somewhat more extended. Recently [111,112], the first formulation of the multi-component LCT only in terms of was developed. The interaction energy difference between a segment and a void lattice site becomes , because of the vanishing energies and . Using the same strategy discussed above leads to power series allowing the calculation of the Helmholtz energy and all other thermodynamic properties [111,112]. The formulation in terms of and the reformulation of the Helmholtz free energy reduces the number of terms necessary to calculate from 103 [89,90,91] to 21 [111,112]. All coefficients depend only on the species’ chemical architecture and the lattice coordination number. The component indices range from zero to the number of components present in the mixture , where zero is the index of voids.

Modelling phase behaviour of polymers by the LCT requires the estimation of the architectural parameters. These parameters can be determined only by the knowledge of the chemical structure of the polymer.

The number of monomers can be evaluated by counting the repeating unit of a polymer chain. Also the number of bonds is independent of the polymers’ branching architecture and can be calculated as follows [113]:

      (65)

All other topological coefficients of Table 1 depend on the architecture of the polymer. This architecture is described by the number of branching points , where describes the branching degree, that means the number of bonds which meet at one repeating unit.

The total number of branching points is [113]:

      (66)

where is the number of branching points of degree in which bonds meet.

As an example the determination of the coefficient will be shown. For a linear polymer in Figure 3 (first row) with three bonds, there are two possibilities of choosing two consecutive bonds in a linear polymer with three bonds. The chosen bonds are marked by broken lines. This can be generalized for a linear polymer with monomers as follows [113]:

      (67)

where denotes for a linear chain.

If there is one point with branching degree of three, the number of ways of choosing two consecutive bonds is raised by one per branching point in contrast to linear chains (Figure 3, second row). By appearing of a branching point of branching degree of 4, there are three more possibilities in comparison to a linear chain (Figure 3, third row). This can be generalized to higher branching degrees as can be seen in Table 2; whereas it has to be mentioned that in polymer chains only branching points with degree up to four exists. With help of Table 2 an equation can be derived, which can be used to calculate the number of possibilities [113]:

      (68)

The factor is reduced by each chosen bond and the factor 1/2 appears because of the indistinguishability. In an analogous manner the number of ways of choosing three or four bonds can be described [113]:

      (69)

      (70)

where describes the contribution of a linear chain [113]:

      (71)

Equation (69) requires that there are at least two bonds between two branching points and Equation (70) implies that there are at least three bonds between two branching points. Another class of coefficients are the number of bonds meeting at one lattice site. As an example the number of three bonds meeting at one lattice site is derived.

Figure 4 shows different ways of choosing three bonds meeting at one lattice site (monomer). The different ways are shown by broken lines. If there is a point with branching degree of three, there is only one way of choosing three bonds at one lattice site, but for a point of branching degree of four there are four ways. This can be generalized to higher branching degrees as shown in Table 3.

With help of Table 3 or combinatorial considerations the following equation can be derived [113]:

      (72)

Here, the factor is the number of points meeting at one lattice site which is reduced by one for each chosen bond and the factor 6 appears because of the indistinguishability. In a similar way the coefficient can be determined [113]:

      (73)

The number of ways three bonds meeting at one lattice site with one additional bond can then calculated by:

      (74)

The last group of architectural coefficients to be determined are the bonds lying on the same chain but are not sequential. As an example the determination of the coefficient will be shown [113]:

      (75)

On the right hand side the first term describes the number of selecting two bonds on a chain without restriction, whereas the second term excludes the counting of the same bond twice and the third excludes the sequential bonds. The factor 1/2 arises because of the indistinguishabilty of the chosen ways.

In the same way, the factors and the factor can be determined [113]:

      (76)

      (77)

The architecture of the polymer can be described with help of the number of segments and the number of branching points with degree, . With help of the presented parameters, the and the that occur in the incompressible version and for the compressible version can be calculated as follows:

      (78)

      (79)

; ;

; ;

; ;       (80)

where the number as additionally subscript of the architectural parameters indicates the component number.

As an example for determining the architectural parameters, three different hyperbranched polyesters with the generation numbers (Boltorn H20), (Boltorn H30) and (Boltorn H40) will be regarded [53]. All these molecules possess the same core: O[CH₂C(C₂H₅)(CH₂O–)₂]₂. Depending on the generation number g, the molecules additionally include a different number of groups A: COC(CH₃)(CH₂OH–)₂. and B: COC(CH₃)(CH₂OH)₂. The general formulae of the polymers are for g = 2: , for g = 3: and for g = 4: . According to these formulae (neglecting polydispersity) the molar masses take the values 1,642 g/mol (g = 2), 3,498 g/mol () and 7,210 g/mol (g = 4). The number of OH– end groups equals the number of B-units. Figure 5 shows schematically a hyperbranched polyester with the generation number g = 3.

To describe the architecture of a hyperbranched polymer two specifications in addition to the generation number will be used [53]. The separator length is the number of segments between two branching points. It denotes the number of segments of an A unit or a B unit. The number of core segments is given by the quantity, . It is assumed that one water molecule occupies one lattice site, so the number of core segments and the separator length can be determined by dividing the core, A and B units in groups that have a molar mass comparable to a water molecule. The values of the separator length, number of core segments and generation number are collected in Table 4.

Using these architectural parameters, the topological coefficients of the LCT can be calculated. The number of segments of a hyperbranched polyester molecule can be calculated as follows [53]:

      (81)

Each A-unit and each B-unit possesses one branching point of degree 3. The core contains two such branching points. Branching points of degree 4 and higher do not exist in these polymer molecules and hence the number of branching points is calculated as follows:

      (82)

Using these parameters the LCT can be applied to polymer solutions containing one of the presented polymers.

Another polymer, which is considered, is Boltorn U 3000 [57]. This HBP has the molar mass 7,192 g/mol. The molecule consists of the core C(CH₂O–)₄, 12 separator groups A: COC(CH₃)(CH₂OH–)₂. and 16 groups B: COC(CH₃)(CH₂OH)(CH₂OR) with the end groups OH and OR. Here, R origins from the 16-carbon-long alkyl acid (R: CH₃–(CH₂)₁₄CO–). The core is divided into 5 segments and the separator group A into 4 segments. Considering the 16-carbon-long group R the group B has 20 segments. All together a Boltorn U3000 molecule may be described by M_B = 373 segments. Furthermore, there are 28 branching points of degree 3 (b₃ = 28) and one branching point of degree 4 (b₄ = 1).

In Wertheim’s approach [73,74,75,76], the overall density of particles in the fluid is divided in two parts. For a fluid consisting of particles with one attractive spot the formalism is in terms of two densities: the overall particle density is divided into a density of non-bonded particles and a density of particles forming an attractive bond with another particle [73]:

      (83)

In a fluid of particles with only one attractive spot, bonded and non-bonded particles are present. The total 2-particle distribution function is constituted from contributions arising from the correlations between two particles which have not formed an attraction bond , two particles of which one has formed an attraction bond and , and two particles that both have formed an attraction bond . The coordinates of particles 1 and 2 are given by and , where denotes the position and the orientation of particle .

To get a multiple density approach, the pair interaction potential depending on the orientation of particle 1 and 2 can be split in two parts, an isotropic repulsive part and a directional attractive part . The Mayer function can be divided in an attractive and a purely repulsive part [73]:

      (84)

with

      (85)

and

      (86)

It has been proven [73,74] that this division of the Mayer function allows a diagrammatic expansion of in terms of the activity , f_R- and F-bonds similar to LCT. According to the suggested expansion, the overall density is devided into densities of bonded and non-bonded particles. The and -values are then both classified by a different part of the set of diagrams that constitutes . Starting from the grand canonical partition function and applying these expansions of and , Wertheim [73,74] derived an exact diagrammatic expansion of the structural correlations , , and in terms of and , f_R- and F-bonds. He then constituted, along the same lines as the direct correlation function is defined, for fluids consisting of hard spheres, partial direct correlation functions , and and received their diagrammatical expansions in terms of , and and . The partial correlation is related to the Orstein–Zernicke equations. This procedure bears strongly resemblance to the derivation of the Orstein–Zernicke equation for hard spheres [117,118,119]. To obtain the fluid structure, the Orstein–Zernicke matrix equation has to be combined with an appropriate radial distribution function as a closure equation and a self consistency relation based on Equation (83). The self consistency relation is a mass balance equation determining the division in bonded and non-bonded particles. For fluids without directional forces the Orstein–Zernicke equations and a closure equation determine the fluid structure in terms of particle density . For a fluid with directional forces, the Orstein–Zernicke equations and a closure equation generate the correlations , and in terms of and.

These correlations define the distribution of the particles over bonded and non-bonded particles. For fluids consisting of hard spheres this distribution is not possible, because there is only one type of particles left. For fluids with directional attractive forces, the and functions assign g₀₀(1,2), g₁₀(1,2) and g₁₁(1,2) with help of Orstein–Zernicke equations and on the other hand the , g₁₀(1,2) and g₁₁(1,2) determine the values of and . This last step is necessary for internal consistency and is provided by the self-consistency relation. The Wertheim theory has three constituent parts, the Orstein–Zernicke equations, a closure equation and a self-consistence mass balance equation. Wertheim extended this formalism to a multi-component mixture with different interaction sites [75,76]. In the following section the use of Wertheim association theory for a polymer blend will be shown.

As the LCT has no information about the density of the polymer blend, another version of the Wertheim theory than used in the SAFT equations of state [72,77] has to be used. The presented version of the Wertheim theory is a Lattice Wertheim theory suggested by Nies et al. [120,121]. The Wertheim theory was transferred to a fully occupied lattice, so an incompressible fluid is regarded. The following expression for the Helmholtz energy of association was derived [73,74,75,76]:

      (87)

where is the segment molar fraction of the non-bonded polymer segments and is the number of association sites at one molecule. “Non-bonded polymer segments” means in this case that the segments do not contribute to association. The value is given by:

      (88)

where the summation over runs over all molecules and association sites. The association strength, , is given by:

      (89)

with being the ratio of nearest-neighbour positions with the proper orientation to all possible orientations of the component and is the association energy. The difference between the Lattice Wertheim association model and the association model used in the SAFT equation of state lies in the considerations of the quantity occurring in Equation (89) and the similar quantity, , called association volume. The association volume depends on the density and the radial distribution function. In contrast, is a constant. For example a water molecule has four association sites, one at each proton of the hydrogen and one at each lone electron pair of the oxygen (Figure 6).

After the localization of the association sites, the possible interactions of the association sites have to be defined; for instance that only a proton and a lone electron pair can interact with each other and not two proton sites or two electron pair sites. This leads, for multi-component cross associating systems, to a nonlinear equation system of equations of type Equation (88), which has to be solved numerically to calculate the Helmholtz free energy. Recently [122], a notation was developed in order to solve this non linear equation system of equations in an elegant way. In opposite to this, in the case of only one associating molecule, the association can be determined analytically. To consider the cross-association between different molecules mixing rules for the association volume and the association energy are applied to calculate the parameters for the cross-association:

      (90)

and

      (91)

The parameter was introduced to consider the deviation of the interaction energy from the geometrical mixing rule [58,123].