Cloud point pressure in the system polyethylene + ethylene – Impact of branching
Graphical abstract
Introduction
Polyethylene (PE) is one of the most widespread plastic materials, produced commercially on a large scale in a continuous way at relatively high temperatures and pressures by free-radical bulk polymerization in supercritical ethylene [1]. Folie and Radosz [1] explained in detail the significance of the involved phase behavior for the industrial production. These authors [1] emphasize the complexity of the phase behavior of this system, which is caused by a) the materials behaving highly nonideal at high pressure, b) the polymer and the solvent greatly differing in size, and c) commercial polymers being composed of many molecules differing in molar mass and chemical composition. In the case of polyethylene, the last one can be characterized as branching. The phase behavior of this system drastically changes with the polymers molar mass distribution and with branching. Both of these properties are usually not well known, especially in combination, i.e. the distribution of polymers with respect to both molar mass and branching. If anything, number and mass average molar mass are known, and sometimes the average branch content of the whole polymer sample. The latter number can make up a difference of up to 40 MPa in the high pressure cloud point curve in the mixture of ethylene + polyethylene [1]. In addition to the afore mentioned difficulties concerning the molar mass and chemical distribution function, often no pure compound properties are known and therefore the equation of state necessary for the description of high pressure phase equilibria can only be parameterized on mixture data.
Caused by the high industrial importance of this mixture, a large amount of experimental e.g. [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] as well as theoretical work e.g. [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30] has been performed on these systems in the past. For modelling, different approaches, including different equations of state (Perturbed Hard Chain Theory [8]; Sanchez-Lacombe-EOS or their modifications [10], [11], [12], [13], [14], [16], [17], [19], [26]; Statistical Association Fluid Theory [7], [16], [17], [19], [22]; Statistical Association Fluid Theory – Variable Range [19]; Perturbed Chain- Statistical Association Fluid Theory [25], [26]; Group Contribution - Statistical Association Fluid Theory –Variable Range [30]; Sako-Wu-Prausnitz-EOS [26]) or models for the excess Gibbs energy of mixing (GE-model) [6], [15], [26] were applied. In the theoretical framework, the polydispersity was taken into account mostly by the pseudo-component method [9], [11], [15], [16], [17], [22], [26]. Usually, the pure-component parameters of the branched and the linear polymer were fit to experimental data [19]. However, this approach requires a large experimental effort. Only, few papers consider the branching directly in the model using different ideas. Kleintjens et al. [31] suggest the use of topological arguments in their GE-model. Dominik and Chapman [25] developed a thermodynamic model for branched polyolefins within the PC-SAFT approach by hetero-segmentation. Marshall and Chapman [32] suggested a new EOS for branched molecules based on Wertheim’s perturbation theory. Yang et al. [33] developed a theoretical framework, incorporating the degree of branching, by fitting the long-range correlation to simulation data. Using molecular simulation [34], [35], [36] or renormalization group theory [37] also allows studying the impact of branching on the phase behavior.
We focus our attention here to the impact of branching on the cloud-point pressure of the system PE and ethylene. The importance of branching can be recognized by the fact that linear and branched polymers of the same type and approximately the same molecular weight are not even completely miscible [38], [39], [40], [41], [42], [43], [44], [45]. Next to the phase behavior e.g. [46], [47], the branching also has a large impact on other physical properties, for instance crystallization [48], osmotic pressure [49], rheological properties [48] and theta temperature [50]. From experiments it is known that the cloud point pressure decreases with increasing degree of branching [2], [5]. The goal of this contribution is the calculation of this equilibrium using a minimum number of adjustable parameters. To this end, the pure compound parameters are fit only to pure compound properties and derived from arguments where this is not possible. Furthermore, the binary parameters are fit only to the phase equilibrium of one specific polymer sample in solution and transferred to all other polymer samples discussed.
The Lattice Cluster Theory (LCT), developed by Freed and Coworkers [51], [52], [53], can reach this goal in principle. This theory allows the calculation of the Gibbs energy for molecules having an arbitrary architecture. In the last years, the LCT was successfully applied for calculation of phase equilibria of systems containing branched [54], [55], [56] or hyperbranched polymers [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67]. Arya and Panagiotopoulos [34] compare results obtained by grand canonical Monte Carlo simulations with results obtained by a simplified version of LCT. It was observed that the LCT significantly underestimates the impact of polymer branching on the critical behavior of polymers [34]. However, the authors use the coordination number of 26 for a cubic lattice, which should be 6 in the notation of lattice cluster theory for that lattice [34]. Since in LCT, architecture is developed in the inverse coordination number, the underestimation using this too high coordination number is not surprising. By introducing empty lattice sites, an equation of state based on LCT can be derived [68], [69], [70]. However, the theoretical foundation of LCT is based on a lattice theory, which reaches its limitation at very high pressures. Therefore, we suggest in this contribution an alternative approach, which is a combination of a cubic EOS, namely the Sako-Wu-Prausnitz- EOS (SWP-EOS) [71] and the LCT, where the lattice is taken as incompressible. This approach is similar to the idea of Vidal [72], who combined a cubic EOS with a model for the Gibbs energy of mixing. The phase behavior of n-alkane mixtures [73] and polystyrene in different solvents [74], [75] could be modelled using SWP-EOS close to experimental data. Additionally, the equations for the critical point were already derived by Browarzik and Kowalewski [76].
Section snippets
Theory
The SWP-EOS, originally suggested by Sako et al. [71], was used in the theoretical framework of continuous thermodynamics [77], [78], where the polydispersity can be taken into account by a continuous distribution function, by Browarzik et al. [73], [74], [75]. The stability of a semi-continuous mixture, made from a discrete component (solvent, A) and a continuous ensemble (polymer, B), was treated by continuous thermodynamics based on an EOS [73]. Its parameters were assumed to depend on the
Parameter estimation procedure
Since ethylene is rigid, it is considered as the base unit here, hence rA = 1, which also implies cs,A = 1. It can be shown that the segment-molar critical volume Vs,cr,A of ethylene divided by the co-volume bs per segment has to fulfill the equationwhere . The only real root of this equation is approximately:
We choose bs,A = 34.05 cm3/mol such that the calculated critical volume is equal to the experimental critical volume
Model predictions
Since all model parameters are determined, the remaining systems’ cloud point curves can be predicted. This is a quite stringent test of the theory, because molar mass, non-uniformity and branching degree of the respective polymers vary for these systems. For the polymer with the highest non-uniformity (BPE1) the p-T-behavior of a 3 Ma.-% solution in ethylene is depicted in Fig. 3. The agreement is quantitatively correct for almost all temperatures. Only at low temperatures there are some minor
Conclusion
In this contribution we use a combination of a simple cubic equation of state, namely the Sako-Wu-Prausnitz-EOS [71], with the Lattice Cluster Theory of Freed and co-workers [51], [52], [53], where polydispersity is explicitly accounted for by continuous thermodynamics [77], [78], to model the phase behavior of 10 different polyethylene samples in solutions of ethylene. The EOS has three pure compound parameters that are fit to the critical point for ethylene and constructed for polyethylene
Acknowledgment
The authors are grateful for the financial support by the Deutsche Forschungsgemeinschaft (German Science Foundation) in Germany (EN 291/7-2) to accomplish this work.
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