This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Hyperbranched polymers show an outstanding potential for applications ranging from chemistry over nanotechnology to pharmacy. In order to take advantage of this potential, the underlying phase behaviour must be known. From the thermodynamic point of view, the modelling of these phase diagrams is quite challenging, because the thermodynamic properties depend on the architecture of the hyperbranched polymer as well as on the number and kind of present functional end groups. The influence of architecture can be taken into account via the lattice cluster theory (LCT) as an extension of the well-known Flory–Huggins theory. Whereas the Flory–Huggins theory is limited to linear polymer chains, the LCT can be applied to an arbitrary chain architecture. The number and the kind of functional groups can be handled via the Wertheim perturbation theory, applicable for directed forces between the functional groups and the surrounding solvent molecules. The combination of the LCT and the Wertheim theory can be established for the modelling or even prediction of the liquid-liquid equilibria (LLE) of polymer solutions in a single solvent or in a solvent mixture or polymer blends, where the polymer can have an arbitrary structure. The applied theory predicts large demixing regions for mixtures of linear polymers and hyperbranched polymers, as well as for mixtures made from two hyperbranched polymers. The introduction of empty lattice sites permits the theoretical investigation of pressure effects on phase behaviour. The calculated phase diagrams were compared with own experimental data or to experimental data taken from literature.

Number of branching points

Contributions to the Helmholtz energy within the lattice cluster theory

Corrections to the Flory–Huggins theory, connectivity factor (Equation (20))

Internal energy

Helmholtz energy

Mayer functions

Gibbs energy

Generation number

Enthalpy or summands in Equation (37)

Summands in Equation (38)

Grand thermodynamic potential

Factors describing the architecture of the polymer, defined in Equations (78,79)

Ratio of nearest-neighbour positions with a proper orientation to all possible orientations

Interaction parameter (Equation (91))

Molecular weight or number of segments

Number of chains in the system

Topological coefficient (

Amount of mole

Pressure

Counting variable

Summands in Equation (41)

Position of the segments

Entropy

Temperature

Interaction potential

Volume

Specific volume

Microcanonical partition function

Mass fraction

Mole fraction

Partition function

Coordination number

Phase a or b

Athermic mixture

Liquid-vapour equilibrium

Mean field approach

Regular mean field energetic contribution

_{i}

Non-bonded segment to the association site A

Association

Attractive part of the interaction potential

Boltzmann constant

Solvent cyclohexane

Pure compound

Flory–Huggins theory

Component i or counting variable

Lattice

Lattice cluster theory

Polymer

Repulsive part of the interaction potential

Void lattice site

Flory–Huggins interaction parameter

Factor in the polynomial series in Equation (29)

Vector pointing to the next neighbour

Difference or association strength

Kronecker Delta function

Interaction energy

Volume fraction

Segment molar fraction

Corrections to the Flory–Huggins theory, combinatorial factor (Equation (20))

Association volume in the original Wertheim theory

Chemical potential

Density

Length of a cubic cell

Changing the polymer architecture from that of conventional linear to partially or highly branched is one of the methods available to tailor a material’s properties for a specific application where high performance or a specific functionality is required. Highly branched polymers are dendritic polymers, including dendrimers with a perfectly branched, monodisperse structure, imperfectly branched polymers, or hyperbranched polymers (HBP). These advanced materials are gaining more and more interest in recent years because of their tailor-made properties. Due to their architecture, HBP show a lower viscosity in melt and solution compared to their linear analogue and the rich amount of various functional groups offers a tuneable solubility in different solvents. Based on these advantages of HBP different applications have been suggested, for instance possible implementations of HBP are discussed in the field of medicine [

However, one should recognize that the break-through in terms of industrial application is still a promising vision with few exceptions. At present, an exact determination of the structure and molar mass characteristics of hyperbranched polymers using available characterization techniques is not completely feasible (e.g., [

Polymer theory has started to deal with the effects of different molecular architectures of polymer a long time ago [

The classical way of describing the phase behaviour of polymer containing systems [

In addition to the architecture, the present functional groups also have an influence on polymer phase behaviour. Moorefield and Newkome [

The well-established way of dealing with phase equilibria including polymers is the FH theory, which was developed by Flory and Huggins [

While the FH model describes successfully the fact of immiscibility of long chain polymers in solution, other aspects can only be described qualitatively. In order to improve the FH theory, Freed and co-workers [

Phase equilibria such as VLE or LLE do play an important role in separation processes as well as in drug delivery or other pharmaceutical applications. As this section is concerned with the thermodynamic phase behaviour of HBP the main focus lies on the thermodynamic description of LLE.

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 [

The thermodynamic potentials are entropy:

Helmholtz free energy:

and the grand thermodynamic potential

The observables are internal energy

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

Here

Here entropy and energy change their roles as potential and observable. The potentials are: internal energy

In calculations of equilibria between two separated phases

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:

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.

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

is also fulfilled. Hence, the problem to solve is a (typically nonlinear) system of equations with

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

Vapour Pressure of polymer solutions

Random walk of a polymer chain in solute on a two dimensional lattice with coordination number four after Flory’s original drawing [

Flory [

This is done by Flory [

(a) a polymer chain is composed of

(b) the polymer segments size equals that of the solvent.

(c) the polymer is inserted randomly, but can fill the lattice completely (

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

Here

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

Usually the interaction energy is expressed in terms of

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

In 1985 Freed [

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

where

By introducing these cluster expansions and fundamental statistical thermodynamics, the Helmholtz free energy of a

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

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 [

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

where

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

where

Here the segment fraction

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

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 [

The segment molar Helmholtz energy can be calculated as follows:

The first two terms on the right hand side of Equation (29) represent the mean field entropic contribution (see Equation (15)) and

where the corrections made by Dudowicz

LCT for a Ternary Polymer Solution

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

where the contributions of entropy (

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

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 (

where the contributions

For compressible systems the LCT can be extended to account for free volume. The proposal of Freed and co-workers [

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

Here

The coefficients of this polynomial depend on the molecule’s structure and on the lattice coordination number

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

Here

The same procedure for the second order energy contribution leads to [

where the polynomial coefficients can be calculated as [

Helmholtz free energy can now be expressed as [

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

Here,

The chemical potential of the species is given by [

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 [

Differences to

The multi-component expressions are somewhat more extended. Recently [

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

All other topological coefficients of

Topological coefficients of the LCT for an arbitrary polymer chain.

Number of repeating units in a polymer chain | |

Number of bonds in a polymer chain | |

Number of two consecutive bonds in a polymer chain | |

Number of three consecutive bonds in a polymer chain | |

Number of four consecutive bonds in a polymer chain | |

Number of distinct ways of selecting two non-sequential bonds on the same chain | |

Number of distinct ways of selecting two sequential bonds and one non-sequential bond on the same chain | |

Number of distinct ways of selecting two non-sequential double consecutive bonds on the same chain | |

Number of ways in which three bonds meet at a lattice site for a polymer chain | |

Number of ways in which four bonds meet at a lattice site for a polymer chain | |

Number of ways in which three bonds meet at a lattice site for a polymer chain and one bond is at this lattice site |

The total number of branching points is [

where

As an example the determination of the coefficient

where

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 (

Number of two consecutive bonds up to a branching degree of four.

Number of additional possibilities of choosing two consecutive bonds on a polymer chain with branching degree up to seven.

Branching degree | Additional possibilities of choosing two consecutive bonds |
---|---|

3 | 1 |

4 | 3 |

5 | 6 |

6 | 10 |

7 | 15 |

The factor

where

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

Number of ways of three bonds meeting at one lattice site up to a branching degree of four.

Number of ways of choosing three bonds meeting at one lattice site up to branching degree of seven.

Branching degree | Ways of choosing three bonds at one lattice site |
---|---|

3 | 1 |

4 | 4 |

5 | 10 |

6 | 20 |

7 | 35 |

With help of

Here, the factor

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

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

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

The architecture of the polymer can be described with help of the number of segments and the number of branching points with degree,

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

Schematic sketch of a hyperbranched polymer of generation number

As an example for determining the architectural parameters, three different hyperbranched polyesters with the generation numbers _{2}_{2}_{5})(_{2}_{2}]_{2}. Depending on the generation number _{3})(_{2}_{2}. and _{3})(_{2}_{2}. The general formulae of the polymers are for

To describe the architecture of a hyperbranched polymer two specifications in addition to the generation number

Architectural parameters describing hyperbranched polyester [

Separator length |
4 |

Number of core segments |
7 |

Generation number |

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 [

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:

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 [_{2}_{4}, 12 separator groups _{3})(_{2}_{2}. and 16 groups _{3})(_{2}_{2}_{3}–(_{2})_{14}_{B}_{3} = 28) and one branching point of degree 4 (_{4} = 1).

The hyperbranched polymers of the Boltorn family carry hydroxyl groups able to form hydrogen bonds, not only associates between two polymer molecules and between other polar groups of the same polymer molecule but also with solvents present in the solution. Žagar and Grdadolnik [

A model for a fluid with directional attractive forces is a fluid of hard particles with an off-center spot that is the origin of an attractive potential. The formation of an attractive interaction needs the orientation of two particles towards each other in such a way that the attractive potentials are within each other’s reach. In the case of a short ranged attractive potential originated near the edge of a spherical particle, so that there can only one bond per particle, the directional attraction has the character of a bond between two particles.

The formalism used here was developed by Wertheim [

In Wertheim’s approach [

In a fluid of particles with only one attractive spot, bonded and non-bonded particles are present. The total 2-particle distribution function

To get a multiple density approach, the pair interaction potential

with

and

It has been proven [_{R}_{R}

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 _{00}(1,2), _{10}(1,2) and _{11}(1,2) with help of Orstein–Zernicke equations and on the other hand the _{10}(1,2) and _{11}(1,2) determine the values of

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 [

where

where the summation over

with

Association model of a water molecule (A–D).

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 [

and

The parameter

The presented theoretical framework can be applied to investigate the miscibility of polymer solutions and polymer mixtures.

Within a certain polymer concentration range, a polymer-poor solvent solution phase separates into a polymer-lean and a polymer-rich phase to minimize its overall free energy depending on the enthalpic interactions and the mixing entropy. Like mentioned in _{1} = 500 dissolved in solvent occupying only one lattice site (_{2} = 1) were performed. Using the FH theory for this purpose mostly leads to a prediction of a too narrow miscibility gap. This problem can be solved, if the

The mixing entropy and enthalpy are complex functions of the polymer structure. Intuitively, one expects that a branched polymer will display fewer unfavorable polymer-solvent interactions than a linear polymer with an identical molecular weight [

Calculations of LLE for the system linear polymer _{B}_{B}_{B}

Yokoyama

We will focus our attention to commercial available hyperbranched polymers from the Boltorn-family.

Pure-component parameters for the Lattice Wertheim Theory.

Component | Ref. | ||
---|---|---|---|

Boltorn H20a ^{1} |
0.023 | 1,200 | [ |

Boltorn H20b ^{1} |
0.023 | 1,200 | [ |

Boltorn U3000 | 0.023 | 1,200 | [ |

Water | 0.01 | 1,800 | [ |

Propan-1-ol | 0.011 | 1,745 | [ |

Butan-1-ol | 0.01 | 1,710 | [ |

^{1} These polymers have different lot numbers [

Parameters of the mixture.

Component i | Component j | ^{2} |
Ref. | ||
---|---|---|---|---|---|

Boltorn H20a | Water | 46.842 | 11.65 | 0.06 | [ |

Boltorn H20b | Water | 45.27 | 18.05 | 0.02 | [ |

Boltorn H20a | Propan-1-ol | 18.96 | 10.55 | 0.04 | |

Boltorn H20b | Butan-1ol | 14.983 | 9.01 | 0.035 | [ |

Boltorn U3000 | Propan-1-ol | 12.59 | 3.9 | 0.03 | |

Boltorn U3000 | Butan-1-ol | 10.54 | 2.03 | 0.02 | |

Propan-1-ol | Water | 64 (fitted to binary VLE) |
[ |
||

Butan-1-ol | Water | 184.622 | 57.5 | 0.03 | [ |

^{1} These parameters are valid, if only the LCT is used; ^{2} These parameters are valid, if the LCT in combination with the Wertheim theory is used.

The phase behaviour of Boltorn H20 in different polar solvents is shown in

Phase behaviour of Boltorn H20 in different solvents (water: black triangles [

Similar to the results obtained for the system Boltorn H20 in polar solvents (

The performance of the theory in describing the mixing behaviour in non-polar solvent, for instance

Phase behaviour of Boltorn U3000 in different solvents (propan-1-ol: black squares [

The calculation of LLE of a ternary system using the LCT requires the knowledge of all three interaction parameters, _{23}/_{B}_{23}/_{B}_{23}/_{B}

Experimental (squares, [

For the other ternary system, built up from Boltorn H20 + water + butan-1-ol (

Experimental (solid squares, [

Recently [_{B}_{2} = 10) the miscibility gap occurs at very low temperatures, which are not relevant in practical applications. A slight increase of the segment number from _{2} = 10 to _{2} = 30 shifts the demixing curve over 300 K to higher temperatures, where the polymers are hardly stable. This situation can be relevant in technical applications, because the demixing is undesirable. In all phase diagrams an upper critical solution temperature (UCST) occurs. If a homogeneous phase must be prepared, then the systems must be heated above the UCST. However, at this high temperature most polymers start to degrade. If the molecular mass of the linear counterpart increases further, then the critical point is above all realistic temperatures. Maybe, the cloud-point curve can be shifted to lower temperature by a small change of the enthalpic effects via incorporation of functional groups in the linear polymer. However, the dominant effect is the entropic penalty for these kinds of mixtures.

LCT model calculations of the mixing (symbols: critical point, lines: cloud point curves) behaviour of hyperbranched polymer Boltorn U3000 + linear polymers having different segment numbers (solid line: _{2} = 10, dashed line: _{2} = 30, dotted line: _{2} = 50), where _{B}

LCT model calculations of the mixing (symbols: critical point, lines: cloud point curves) behaviour of hyperbranched polymer Boltorn U3000 + hyperbranched polymer H20 (solid line), of Boltorn U3000 + H30 (dashed line) and Boltorn U3000 + H40 (dotted line), where _{B}

_{B}_{1}) further leads to an UCST which is above the degradation temperature of the hyperbranched polymers. In other words, the LCT predicts that it is not possible to prepare a homogenous mixture made from Boltorn H40 and Boltorn U3000.

In this context it appears worth mentioning that blends of branched and linear polyisoprene exhibit a relatively broad miscibility gap at room temperature [

Experimental (black open squares: ^{−10}_{B}^{−10}_{B}^{−10 }_{B}

In

In summing up the theoretical investigations we can state that the LCT in combination with the Lattice Wertheim approach is a promising tool to calculate the thermodynamic properties of polymer containing systems, especially if hyperbranched polymers are involved. In this way it could be shown, that besides the architecture of the hyperbranched polymer, the influence of functional groups also has to be considered in the theoretical framework.

For financial support the authors thank the German Science Foundation (DFG, En 291/7-1) and the Max-Buchner Research Foundation (Nr. 2864). The authors thank the company Perstorp Speciality Chemicals AB for donating the hyperbranched polymer as gift and Johannes Sailer for performing some of the model calculations.

^{®}series hyperbranched aliphatic polyester and folic acid

_{2}O solutions: Small-Angle neutron-scattering experiments and wertheim lattice thermodynamic perturbation theory predictions

_{2}O investigated by small-angle neutron scattering

_{9}to C

_{15}