(203c) Extension of Co-Oriented Fluid Functional Equation for Electrostatic Interactions to Include Multi-Body Repulsive Effects

Phase equilibria and thermophysical properties are essential input data for industrial separation processes. Especially their prediction for mixtures is important. For this purpose e.g. equations of state (EOS) can be used. EOS enable a fast calculation of e.g. thermodynamic equilibria, which are consequently of great importance in industry. Developments over the last few decades have led to major improvements in the EOS, allowing the free energy to be described much more precisely. Most are similar to the Statistical Associating Fluid Theory (SAFT) [e.g. Cha90, Emb11, Gro01, Gro02, McC10, Tan08]. Nevertheless, problems arise in the description of polar + non-polar mixtures. There, it is often not possible to draw conclusions about the phase equilibrium of mixtures based on knowledge of the pure substance data. One reason for this is that orientation behavior of molecules is important for their thermodynamic behavior and not included in SAFT-type EOS.

In recent years, Langenbach has developed an EOS for this purpose: Co-Oriented Fluid Functional Equation for Electrostatic interactions (COFFEE), which is to date the only EOS, that describes molecular orientation explicitly [Lan17]. COFFEE has already been developed for pure components and binary mixtures [Lan 17, Mar23a] and reproduces molecular dynamics simulations of polar + unpolar mixtures accurately. It achieves results comparable to SAFT-type theories for real fluids. Due to competition of two dipole molecules for the more favorable place, multi-body effects arise at higher density. This cooperativity causes a reduction of the orientational order. Currently, multi-body effects, e.g. van der Waals attractions and repulsion in the first coordination shell due to orientation and favorable positions are not included in COFFEE. In the first coordination the repulsion represents the strongest effect between the molecules. [Mar23a]. The purpose of this contribution is to extend COFFEE by multi-body repulsive effects by adding a repulsive van der Waals term. Since COFFEE does not currently consider multi-body effects, there are notable deviations to the MD simulation results in the orientation distribution function (ODF), as well as in the angular distribution function (ADF).

The ODF is used as a measure for the probability of mutual molecular orientation in a 3D orientation space. In figure 1 the angles and other variable of two dipolar molecules are shown, where r is the distance vector, µ the dipole vector, the angle between the distance vector and the dipole vector of the dipolar molecules 1 and 2 respectably and the torsion angle of the two dipole vectors around the distance vector.

In the first shell of molecules with large dipole moments, a significant cooperativity is found, which influences the thermodynamic behavior. Due to high densities, not all molecules can occupy the most favorable configuration, i.e. the orientation behavior becomes frustrated. In binary mixtures this leads to different local concentrations of the different species in the first coordination shell in dependence on orientation state, which can be best seen in ADF. The ADF depends on the ODF, as it arises through a double integral form that function [Mar23a, Mar23b]. In figure 2 the ADF of binary mixture calculated with COFFEE without a repulsive free energy contribution (blue line) is compared with the MD simulation results (black circles), which show opposite behavior due to orientation frustration.

Methods

To include local repulsive multi-body effects into COFFEE, a van der Waals repulsion term is added to the free energy functional [Waa73], including a prefactor J that compensates for simplifications made during the derivation for keeping numerical tractability. This term depends on the ODF or the ADF respectively and can also be calculated using different (non-local) measures. COFFEE in its original version [Lan17] has a fitted parameter I depending on dipole strength. This parameter and the pre-factor J are fitted to molecular dynamics simulation data of the ODF and ADF of pure Stockmayer fluids in this contribution. Since both goals are conflicting, different weights are chosen to approximate the Pareto-front in the goal-function-space.

Results

Using the approach described above, the pre-factors I and J are fitted to ODF and ADF data. The results in the goal-function-space is shown for two different states in figure 4 and figure 5. For the low density results in figure 4, the extreme compromises for ODF and ADF least square sum differ by a factor of roughly 2 for the ODF, but by two orders of magnitude for the ADF. The pareto front is (at least approximately) convex. Though there is no way to define an optimal pareto point, the range of 30%-70% weight factor for the ODF seems most promising, as it retains most of the accuracy for the ODF, while significantly improving the ADF. For the high density results of figure 5, there is a distinct “pareto-knee” between at least 1% and 90% weight factor for the ODF. At this knee, the value of the least square function for the ADF is down by a factor of three compared to its worst value, while the ODF representation barely changes compared to the extreme compromise favoring the ODF.

In figure 3 the calculated ADF including the fitted the van der Waals term for the weight factor of 1% for the ODF of a high-density pure component is shown in comparison to molecular dynamics data. For high densities the inclusion of the van der Waals term can lead to a significant improvement of the local density field, while essentially retaining the quality of ODF fit even for pure compounds. Since the range of weight factor is very large, a single state-independent value may be retained. In the low density limit, the van der Waals term is practically inactive, while in the high density range, it has a significant contribution.

Conclusion & Outlook

The perturbation theory Co-Oriented Fluid Functional Equation for Electrostatic interactions (COFFEE), which describes molecular orientation explicitly and is suited to describe polar non-polar mixtures, is extended to include repulsive multi-body effects. Thereby, more accurate results especially for higher densities are achieved for the angular distribution function, because in a crowded environment the significance of multi-body effects rises. A pareto analysis shows that the ADF can be strongly improved for pure components, especially in the high density region, while retaining the quality of the ODF description. While these effects are small for pure compounds, the inclusion of such effects is likely to allow the description of cooperativity also for mixtures.

This is achieved by adding the repulsive free energy from a local van der Waals term to the free energy from COFFEE and after a few mathematical conversions the ODF and ADF are fitted to the results of molecular dynamic (MD) simulations. This has led to huge improvements in the ADF. The results of COFFEE including the repulsive free energy fit one order of magnitude better than COFFEE while retaining almost the same quality in the ODF. Even a local van der Waals approximation of repulsive effects leads to this improvement of COFFEE.

In the next steps, it is planned to include a non-local functional dependence of the repulsive functional on the ADF in order to further improve the description of high-density states based on Rosenfeld’s fundamental measure theory [Ros89]. Furthermore, an extension to mixtures is planned, where the expected effects of the repulsive interaction on the different ADFs is much larger, leading to co-operative effects especially in polar + non-polar mixtures.

Literature

[Cha90] W. G. Chapman, K. E. Gubbins, G. Jackson, M. Radosz, Industrial & Engineering Chemistry Research 29: 1709-1721 (1990).

[Emb11] C. O. Emborsky, Z. Feng, K. R. Cox, W. G. Chapman, Fluid Phase Equilibria 306: 15-30 (2011).

[Gro01] J. Gross, G. Sadowski, Industrial & Engineering Chemistry Research 40: 1244-1260 (2001).

[Gro02] J. Gross, G. Sadowski, Industrial & Engineering Chemistry Research 41: 5510-5515 (2002).

[Lan17] K. Langenbach, Chemical Engineering Science 174: 40-55 (2017).

[Mar23a] J. Marx, M. Kohns, K. Langenbach, Journal of Chemical & Engineering Data 69: 400-413 (2024).

[Mar23b] J. Marx, M. Kohns, K. Langenbach, Journal of Chemical & Engineering Data 69: 400-413 (2024).

[McC10] C. McCabe, A. Galindo, SAFT Associating Fluids and Fluid Mixtures, Royal Society of Chemistry, Cambridge, 2010.

[Ros89] Y. Rosenfeld, Physical Review Letters 63: 980-983 (1989).

[Tan08] S. P. Tan, H. Adidharma, M. Radosz, Industrial & Engineering Chemistry Research 47: 8063-8082 (2008).

[Waa73] J.D. van der Waals, Docterat Dissertation: Over de continuiteit van den gasen vloeistofiocstand, 1873.