COSMO-RS || The wider range of COSMO-RS applicability

Chapter 10

The wider range of COSMO-RS appl icabi l i ty

Since COSMO-RS allows for a rather fundamental calculation of the chemical potential of molecules in various chemical environ- ments, there are many application areas that go beyond the calculation of activity and partition coefficients, which are directly accessible from the differences of chemical potentials in different phases. Often, such applications require the addition of some empiricism to the model because they involve free-energy contributions that are not directly accessible by COSMO-RS. Never- theless, in many cases, the COSMO-RS is a robust starting point for such empirical extensions, and the resulting models are still less empirical and more fundamental than many other approaches. Without going into details we will describe some extended application areas in this chapter.

10.1 COSMO-RS FOR REACTION MODELING IN THE LIQUID PHASE

Although substantial progress has been achieved in the past dec- ade, even advanced ab initio quantum chemical methods, like coupled cluster calculations, still have problems in achieving an accuracy of 1 kcal/mol in the calculation of the total free-energy differences of starting materials and products of a reaction. But such "chemical accuracy" is required for many practical decisions in reaction design and development. Therefore, the application of quantum chemistry in reaction engineering is still a very demanding, but definitely evolving technique. Even more demanding is the accurate location and energetic quantification of transition states, which is required for the prediction of the kinetics of chemical reactions. However, with some computational and chemical experience such transition-state searches are

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feasible to a certain extent, and have sufficient accuracy to dis- tinguish between reasonable and impossible reaction mechan- isms, and also to give a rough estimate of the expected activation energies. Nevertheless, most quantum chemical reaction modeling tools are restricted, so far, to gas-phase reactions. Rigorous tools for the transfer of such techniques to the technically and biologically important fluid phase are still rare.

While, at first glance, the chances for a successful quantum chemical reaction modeling in the fluid phase appear to be small, since it is still extremely complicated in the gas phase, it turns out that by using COSMO-RS, the prediction of the changes of the reaction free energies, reaction enthalpies, and activation barriers of the same reaction between different solvents is much simpler than the prediction of absolute reaction quantities in the gas phase. Therefore, COSMO-RS can be used well for the selection of appropriate solvents meeting the required solubility properties for reagents and products, and lowering of the activation barrier of a reaction. The basic concepts of the solvent free energy changes in reaction kinetics are shown in Fig. 10.1.

Indeed, prediction of the change of the equilibrium constants of a chemical reaction in a variable liquid environment requires nothing other than the prediction of the chemical potentials or activity coefficients of the starting materials and products in the liquids. Thus, this task can be performed simply by using the standard COSMO-RS capabilities. Successful applications of this

Fig. 10.1. Schematic visualization of the free energies relevant for reactions in solution.

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kind have been reported by several industrial users, and quite promising results of the application of COSMO-RS to the TAME synthesis reaction in varying liquid compositions will be published in the near future [117].

Since most industrial reactions are not performed under equilibrium conditions, the prediction of kinetic constants and activation barriers in different solvents is much more important than the prediction of the equilibrium constants. Unfortunately this is also much more demanding and requires more approximations. The search for the transition state is restricted at present to the gas phase, and yet cannot be performed reliably in the COSMO CSM. Nevertheless, the first applications to the prediction of the solvent shifts of the kinetic constants of different reactions have yielded promising results, although single-point DFT/COSMO calculations based on the gas-phase transition states have been used for the generation of the transition-state COSMO files. Thus it appears that meaningful predictions of the change of kinetic constants under varying liquid conditions can be done with COS- MO-RS, and hence appropriate solvents with low activation barriers can be selected for a reaction. As an example, the results for the Menshutkin reaction [C32] are shown in Fig. 10.2. Although

Fig. 10.2. Predictions of the logarithmic relative rate constants of the Menschutkin reaction in various solvents [C32]. The COSMO polarization charge densities a of the transition state are visualized in the inset.

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not yet put into practice, the prediction of the right solvent for lowering the free energy of a transition state leading to a desired product relative to other transition states leading to unwanted side products, may be of great importance in enhancing the yield of a reaction. This should also be possible with COSMO-RS. There is a growing interest in the prediction of liquid-phase reaction thermodynamics with COSMO-RS and the first results have been published in the literature [118].

10.2 COSMO-RS PREDICTIONS OF pKa" A MYSTERIOUS SUCCESS

Proton transfer and the associated dissociation constants of acids and bases in water and in other solvents are of great importance in many areas of organic, inorganic, and biological chemistry. Therefore the first principle prediction of pKa values has been a subject of many research projects and scientific papers. In the gas phase, dissociation constants are experimentally difficult to measure and good quantum chemical calculations, i.e., post- Hartree-Fock ab initio calculations or state-of-the-art density functional methods, can nowadays be as good or even more accurate than experimental measurements. In the aqueous phase-- which is of real importance--a large number of reliable data for dissociation constants is available, but here a theoretical prediction is extremely challenging because of the very strong solvation effects of the neutral and ionic species involved in the reaction. Thus the prediction of pKa in some way is the ultimate challenge for quantum chemically based solvation models. A large number of attempts in this direction have been reported applying all types of solvation approaches, but a consistent and broadly applicable solution is still lacking.

According to the fundamental relationship,

AGdiss pKa - s + const (10.1) RTln(10)

the calculation of pKa primarily means the calculation of the Gibbs free-energy of dissociation, which is typically defined by the reaction

AH + H20 ~ A- + H30 + (10.2)


where HA is an arbitrary acid. The slope s in Eq. (10.1) has to be unity if AGdiss is calculated without any systematic error, and the constant is expected to be - l o g [ H 2 0 ] - - 1 . 7 4 . Thus the computational task is just the accurate calculation of the total free energies of the four species occurring in Eq. (10.2). Here it should be noted that the total free energies of the neutral and protonated water species are the same for all acid dissociations, and hence an error in these values, or even their total omission, would only affect the value of the constant in Eq. (10.1), but not the slope or the quality of the prediction model. This is important, because the true state of the aqueous proton is not really known, and the species H30 + is just a plausible model for it. As an illustration of a typical aqueous dissociation reaction, the four species involved are shown in Fig. 10.3 for the case of formic acid.

Before continuing the issue of pKa prediction, I should point out a peculiarity visible in Fig. 10.3. The hydronium ion H30 + appears to be an amphiphilic molecule with one extremely polar face resulting from the three hydrogens on the one side, and a hydrophobic face at the electron-depleted oxygen lone pair on the other side. I became aware of this unexpected feature during the Conference on Current Trends in Computational Chemistry in Jackson, MS, in November 2002, when Srinivasan Iyengar reported results on some new quantum molecular dynamics simulations on water clusters including one hydronium ion [130,131]. After showing some movies of such simulations, he pointed out that the hydronium had moved from the center of the cluster to the surface, with the oxygen finally sticking outside. He concluded that hydronium must be in some way hydrophobic. Immediately I turned on my laptop computer, visualized the a-surface of hydronium as shown in Fig. 10.3, and realized that it is green, i.e., hydrophobic, in the presence of oxygen. Showing this figure to the

Fig. 10.3. Species involved in the aqueous dissociation of formic acid.

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audience during the subsequent discussion generated much interest, since the same unexpected finding was now immediately supported by two absolutely independent approaches. While it is not clear to me whether this one-sided hydrophobicity, i.e., this amphiphilic behavior of hydronium, has any physicochemical importance, e.g., by some preferential association of hydronium to hydrophobic surfaces, it would be interesting to learn about experimental consequences.

Because of its wide practical importance, I tried the prediction of pKa in the very early phase of my development of the solvation model, which was based on MOPAC/AM1/COSMO calculations in 1993, in collaboration with my colleague Jannis Batoulis, at Bayer [132]. While we found quite a good correlation of the calculated differences in total heat of formation with the experimental pKa values of a large number of alcohols and carboxylic acids, we were not satisfied with the slope of the regression curve, which was close to s = 0.5 in Eq. (10.1). Having no experience on the charge dependence of COSMO radii, we tried to achieve a physical slope closer to unity by changing the COSMO radii of the anions. To our surprise, we needed to change them to values much smaller than the Bondi radii in order to approach the expected slope, while our physical intuition would suggest that anions should be slightly larger than neutral species, owing to their additional electron. Furthermore, making such a change worsened the correlation with experiment. We concluded that these changes were unphysical, and that our approach was too crude to be meaningful.

After this failure, I did not consider pKa calculations with COSMO or COSMO-RS for many years. In 2002, Michael Beck from Bayer CropScience reported very impressive COSMO-RS results on the calculation of pKa values of heterocyclic pesticide compounds [133] (see Fig. 10.4), for which all other methods failed to give good predictions. We started a joint study on pKa prediction [C27]. This time we chose a wide variety of 65 alcohols, phenols, carboxylic acids, inorganic acids, and heterocyclic acidic compounds, spanning an experimental pKa values in the range 0 to 18 as shown in Fig. 10.5. We calculated AGdiss as the total free-energy difference of the four species of Eq. (10.2) according to our COSMO-RS high-level standards. However, obviously, we made no change to the radii for the ions, because we then had sufficient experience to be reasonably sure that element-specific


Fig. 10.4. Correlation of pKa of some pesticide compounds with COSMO- RS dissociation energies [133]. Confidential parts of the molecules are hidden. The three yellow symbols denote three carboxylic acids that were treated for comparison. All other methods failed on this data set.

18 17 16 15 14 13 12

z "' 11

10 w " 9 x w< 8

,,., 7 6 5 4 3

13 Z~ alcohols

o carboxylic acids

�9 inorganic acids

�9 subst, phenols

0 N-acids (uracils, imines)

-2.5 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.0 27.5 30.0 32.5 35.0 37.5 40.0

AGdiu(kcal/moI )

Fig. 10.5. Experimental pKa values vs. calculated free energy of dissociation from COSMO-RS [C27].

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COSMO radii should also apply for charged species. A linear regression yielded the result

pKa - 0.58(• AGdiss RTln(10)

+ 1.66(=t=0.10) (10.3)

with a very good correlation of r 2 = 0.983 and rms = 0.5. The constant is not far from the expected value o f - 1.74, at least if we consider the large free-energy differences, of the order of 300 kJ/ mol, involved in this reaction. However, the slope is again much closer to 0.5 than to unity. Next we confirmed this unexpected and somewhat unwanted, "wrong slope" by trying different quantum chemical methods (HF, various DFT functionals, and MP2) and different basis sets, but the slope was almost uninflu- enced by these changes. A literature study yielded the result that many papers on pKa predictions based on a solvation model had shown similar results, with a slope near 0.6 in the pKa regression. Some workers had tried to enforce "better" slopes by modifying their models, while others just reported the regression equation. However, none of them had such a high statistical significance for the low slope as obtained by us. Our result for the slope is specially significant, because it was achieved without any pKa- specific model adjustments.

As a final attempt to understand the origin of the unexpectedly low slope, we split AGd~ into four different parts, i.e., the quantum chemical gas-phase dissociation, the COSMO-interaction energies of the neutral and the ionic species and the chemical potential difference from COSMO-RS. On the basis of these reasonably independent descriptors, we performed a multilinear regression of pKa yielding an almost identical regression coefficient and rms error as before with slopes ranging from 0.55 to 0.62 for the four contributions. Thus all contributions--although very different in nature--show the same unphysical slope with respect to the pKa.

At this point, we had to conclude that the "wrong" slope must have a mysterious reason outside our calculation methods, either in the experimental pKa scale or in the theoretical interpretation of pKa, although both alternatives seem equally strange and un- likely. Nevertheless, from a practical point of view, Eq. (10.3) is the most widely applicable and most accurate quantum chemically based PKa equation presently available, and it may be of great practical value in predicting the dissociation constants of


Fig. 10.6. Results of pKa prediction in DMSO.

new acidic compounds with demanding chemical functionality, where conventional pKa prediction models fail.

Meanwhile, the approach has been extended to base protona- tion reactions. The respective regression equation based on 43 bases reads as

AGdiss p g a = 0.75(• - 1.8(• (10.4)

RTln(10)

with r 2 = 0.98 and a RMS deviation of 0.56. The constant is in extremely good agreement with the expected value of -1 .74 , but slightly different from the regression for the acids. The slope is a bit higher than the one found for acids, but still significantly lower than the theoretically required slope of unity. It must be noted that special parameterization is required for secondary and tertiary amines.

The rather fundamental and almost unparameterized COS- MO-RS approach also allows for the calculation of pKa in aqueous mixtures and other solvents. As an example we applied it to DMSO (see Fig. 10.6), achieving about the same accuracy as in water, but a slightly higher slope. It is not clear whether the difference in slope is of physical significance. Thus, COSMO-RS may be used for pKa prediction in different solvents, but unfortunately a parameterization is required for each solvent.

10.3 COSMO-RS FOR POLYMER SIMULATIONS

The state of polymers above the glass transition is quite similar to a highly viscous liquid state. Therefore it is a straightforward idea

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to apply COSMO-RS for the calculation of the solubility of molecules in amorphous polymers. Since large polymer molecules cannot be treated by DFT programs, the a-profile of a polymer has to be approximated by a sum of the partial a-profiles of the central repeat units of oligomer molecules. For most polymers it is possible to calculate at least trimer molecules, and the a-profiles of such central trimer units have been proved to be reasonably accurate for the representation of the polarity distribution of long polymer chains. By the definition of atom-weight factors, the COSMOtherm program provides the technical features to extract partial a-profiles of molecules and thus allow for the simple definition of pseudo-polymers based on oligomer COSMO files, as shown schematically in Fig. 10.7. It should be noted that co-polymers can also be "constructed" easily in the same way, if the net composition with respect to the monomer units A and B is known. Even AB transitions can be mimicked as oligomers and these

Fig. 10.7. Flowchart for polymer treatment in COSMO-RS.


units can be considered in the right concentration. It should be noted that the combinatorial contribution has to be switched off in such polymer calculations, since it is not suited for the large molecular sizes of polymers.

The calculation of the solubility of gases and other small molecules in various polymers showed that for each polymer the relative solubility of the compounds was well predicted, but one polymer-specific correction had to be fitted. The results of this study are shown in Fig. 10.8. The correction constant may have to do with the missing combinatorial term, or with free-volume effects in polymers, or with s t ructural differences of polymers, or with a combination of several such effects. The lack of a tempera ture dependence suggests that it is mainly of entropic nature. Unfortunately, no clear relationship between the value of

Fig. 10.8. Calculation of the solubility of gases in 24 different polymers, ranging from polyethylene to nitrocellulose [120]. One polymer-specific constant has to be adjusted in order to achieve good results. Experimental data are taken from Pauli [121] (HDPE = high density polyethylene, POP = poly(oxyethylene glycol), PDMS = poly(dimethylsiloxane),

PTFE = poly(tetrafluoroethylene)).

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the polymer correction and structural parameters of the polymers has been detected so far.

Since most other modeling techniques for polymers are extremely demanding, the limited capabilities of COSMO-RS for efficient prediction of solubilities in polymers can be of great help in practical applications when suitable polymers with certain solubility requirements are desired. One application may be the selection of appropriate membrane polymers for certain separation processes. Predictions of drug solubility in polymers are sometimes of interest for pharmaceutical applications. Further- more, it is most likely that COSMO-RS can also be used to in- vestigate the mutual compatibility of polymers for blends. This aspect, and many other aspects of the potential of COSMO-RS for polymer modeling, still awaits systematic investigation.

10.4 COSMO-RS FOR SURFACTANTS, MICELLES, AND BIOMEMBRANES

Surfactants, i.e., amphiphilic molecules with strongly polar, hy- drophilic head groups and non-polar lipophilic tails, play a major role in many areas of industrial chemistry and biochemistry. The special feature of such molecules is their tendency to self-organize at liquid interfaces in form of spherical, cylindrical, or lamellar micelles. Biomembranes, which separate biological cells from the surrounding environment, are such lamellar micelle structures formed from phospholipid surfactants. Tensides, which are the most substantial parts of soaps, shampoos, and most other clean- ing agents, are other examples of surfactants having practical importance in many areas of liquid-phase chemistry.

There are three quite different classes of surfactants. While the tail mostly consists of one or several hydrocarbon chains with a few unsaturated bonds, the polar head group may either be a strongly polar neutral group, such as the hydroxyl group, or a zwitterionic fragment as in the case of the phospholipids, or even a positively or negatively charged group with an appropriate counterion.

Even though surfactants can have long lipophilic chains, and often have molecular weights of 500-1000 a.u., the DFT/COSMO calculations for these molecules do not normally present major


problems. Some examples of a-colored COSMO surfaces are shown in Fig. 10.9. Hence COSMO-RS calculations can be pre- ferred for dilute surfactant molecules in homogeneous solvents with COSMO-RS. However, since molecules with pronounced surfactant properties tend to assemble at all kinds of surfaces and phase boundaries, there are almost no experimental data available for such strong surfactants dissolved homogeneously at low concentrations, and thus a direct proof of the applicability of COSMO-RS at this end is not possible. On the other hand, many compounds with weak or moderate surfactant tendency, like long- chain alcohols, have been considered in the parameterization and validation studies with no apparent problems. Unfortunately, COSMO-RS by itself is not capable of treating inhomogeneous liquid systems with micelles and surfactants. This is a result of the basic approximation of neglecting the 3D geometry of the a-surface and reducing it to an ensemble of independently pairwise interacting surface segments. Without additional

Fig. 10.9. Collection of ionic surfactants (without counterions). It can clearly be seen that the polarization charge densities of the charges are localized on the head groups, while the tails already look like normal alkane or fluoroalkane chains after two carbon units from the head

group.

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assumptions, there is no way to describe a behavior in which the probability of finding polar surface segments is higher in certain space regions of a liquid system, while less polar, lipophilic surface segments have a higher concentration in other regions.

It would be very desirable to have an efficient tool for the modeling of surfactant micelles or biomembranes, e.g., for the investigation of the partitioning and solubilities of fragrances or other small additives in aqueous surfactant systems, or for the calculation of the affinity of pharmaceutical agents for cell membranes. While MD/MC force-field methods have been applied successfully to surfactants and biomembranes in order to build reasonable 3D models of the structures of such systems [125,126], the computational demands of such calculations are extremely high. Thermo- dynamically equilibrated MD/MC calculations including additional solutes would be almost prohibitively expensive, if not impossible. This essentially prohibits the calculation of membrane partition coefficients of drug candidates with MD/MC. Also, it is questionable whether standard non-polarizable force fields can handle the cross- over from polar to non-polar regions of such systems with the accuracy that is required for a reliable description of the distribution of partially polar molecules in such systems (Fig. 10.10).

Since it is well known that micelles and biomembranes have a large degree of internal dynamics and disorder, it appears plausible

Fig. 10.10. MD simulation results for a lamellar biomembrane and a typical spherical tenside micelle, respectively [122,123].


to consider them as layered liquids with a composition changing as a function of the distance from the center of the micelle or membrane. Therefore, in collaboration with Bayer AG and the Univer- sity of Cologne, we tried to model phospholipid biomembranes with COSMO-RS as a series of liquid layers with ~-profiles pM((~,z) changing as a function of the distance z from the center of the bilayer of phospholipids molecules [126]. After performing the required DFT/COSMO calculations for typical phospholipids, for which we selected the lecithin zwitterion dimyristyrolphos- phatidylcholin (DMPC), we used experimental information about the probability of finding certain atoms of these molecules at different distances from the membrane center. In this way we constructed the z-dependent a-profilePM(a,z ) and converted it into a z- dependent a-potential pM(a,z) by solving the COSMOSPACE equations. Special care had to be taken, because the individual layers were not neutral---owing to the charge separation within the DMPC molecule. The resulting electrostatic ~-potential was taken into account as an additional potential energy during the calculation of the a-potential.

The a-potential pM(6,Z) reflects the chemical potential of a surface segment of kind a in a certain depth z of the membrane. For a given center position and orientation of a solute molecule X in this field of the membrane, we know the exact position and hence the z-coordinate of each segment and can easily integrate the respective values of pM(a,z) for the entire solute surface, to yield the residual part of the free energy of X in this special position and orientation. By a systematic sampling of the different center positions and orientations of X we yield the entire partition sum of X, and this easily gives us the probability of finding the solute X at a certain depth and orientation in the membrane/ water system. In this way, we were able to achieve promising results for a set of membrane-water partition coefficients. A schematic illustration of this approach is given in Fig. 10.11. Re- cently, we extended the same concept to spherical geometries in order to simulate the distribution of fragrance molecules in tenside micelles.

Our preliminary results suggest that the described concept of the simulation of micelles and membranes as layered COSMO-RS liquids is a promising new approach to an efficient simulation of such systems, but more development and validation work is

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Fig. 10.11. Schematic illustration of solute molecules in the field of a surfactant bilayer.

required to establish it as a standard tool in this area. A clear disadvantage is that it depends on external information about the geometrical composition of the inhomogeneous system and cannot simulate the self-organization directly. When this technique is developed it might be possible to achieve the information about the micelle or membrane composition recursively by starting from a reasonable guess at the composition, to yield a a-potential PM(a,z), applying this to the surfactant molecules, and thus gen- erating a partition function and a new composition of the membrane or micelle with respect to the surfactant molecules.

Finally, it should be mentioned that a combination of COSMO- RS with tools such as MESODYN [127] or DPD [128] (dissipative particle dynamics) may lead to further progress in the area of the mesoscale modeling of inhomogeneous systems. Such tools are used in academia and industry in order to explore the complexity of the phase behavior of surfactant systems and amphiphilic block-co-polymers. In their coarse-grained 3D description of the long-chain molecules the tools require a thermodynamic kernel


that evaluates the residual part of the chemical potential of each chain bead in a volume element with a given overall bead composition. While simple Flory-Huggins terms are currently used for this purpose, they require a lot of parameterization and even composition-dependent interaction parameters for a reasonable description; COSMO-RS could easily provide such free-energy contributions based on the partial a-profiles of the beads.

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COSMO-RS || The wider range of COSMO-RS applicability