COSMO-RS || The basic COSMO-RS

Chapter 6

The basic COSMO-RS

We have now collected almost all the pieces required for a first version of COSMO-RS, which starts from the QM/COSMO calculations for the components and ends with thermodynamic properties in the fluid phase. Although some refinements and generalizations to the theory will be added later, it is worthwhile to consider such a basic version of COSMO-RS because it is sim- pler to describe and to understand than the more elaborate com- plete version covered in chapter 7. In this model we make an assumption that all relevant interactions of the perfectly screened "COSMO" molecules can be expressed as local contact energies, and quantified by the local COSMO polarization charge densities

and a' of the contacting surfaces. These have electrostatic misfit and hydrogen bond contributions as described in Eqs. (4.31) and (4.32) by a function for the surface-interaction energy density

~' (T,)2 e(a, a') - -~(a + + Chbfhb(T) min(0, rain(a, a') + (Thb)

max(0, max(a, a') - ahb) (6.1)

The parameters appearing in Eq. (6.1) are known. The exact values used in this section are a - 1385 kJ nm2/mol/e 2, (~hb- 0.79 e/nm 2, and 19424kJ nm2/mol/e 2, and a temperature-dependence:

Tln(1 + exp{-20 kJ/mol/(kT)}/200) fhb(T) - 298.15 Kln(1 + exp{-20 kJ/mol / (k298.15 K)}/200) (6.2)

which is derived from plausible physical assumptions about the energy gain and the entropy loss during the formation of a hydrogen bond. The size of a thermodynamically independent contact, aeff, is 0.0767 nm 2. These values correspond to a basic COSMO-RS parameterization which is based on DFT/COSMO calculations with Becke-Perdew (BP) [150,151] functional and a triple-zeta valence plus polarization (TZVP) basis set. A simple expression for

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the combinatorial part of the chemical potential is used. For details see Appendix C.

6.1 6-AVERAGING

The polarization charge-density, a, on the molecular COSMO surface is the only property required for the evaluation of the surface interaction energies according to Eq. (6.1). In principle, the charge density qi/si from COSMO could be used for the value of a on a segment i. However, the COSMO segments have areas that range from 5 �9 10 -3 nm 2 down to about 5 �9 10 -5 nm 2. Thus, the COSMO charge densities are much more local than the average values on the effective contact surfaces, a e f f - 0.0767 nm 2. Hence, it appears to be reasonable to use values of a that are averaged over larger areas. For this, we introduce an averaging radius rav and define the COSMO-RS polarization charge density by a local average

qJ exp{ - d 2 / d v } j sj-t-Sav

ai = (6.3) E sj expl- / / vl j Sj-'l-Sav

where Sav is the area of a circle of radius rav. Compared to the most trivial averaging, this slightly more complicated formula bet ter takes into account the very different finite areas sj of the segments j contributing to the average.

Since thermodynamically independent contact segments have an area of about 0.07 nm 2 corresponding to a radius of about 0.15nm, it would be most satisfying to use this value for the averaging radius, rav, i.e., if the averaging would be over an area corresponding to a thermodynamic contact area. Unfortunately, in repeated a t tempts to average over so wide an area, a signif- icantly lower accuracy in the COSMO-RS parameter izat ion has resulted. It appears that by use of such large radii, many of the relevant details of the a-distribution get averaged out. We found that the opt imum value of rav is about a factor of 3 smaller, i.e., at a b o u t rav ~0.05 nm, but we do not have any good explanation for this discrepancy between the radius of a thermodynamic contact and the opt imum value for the averaging radius. In a re-imple- mentat ion of COSMO-RS, Lin and Sandler [91] claimed to have

The basic COSMO-RS 85

removed this problem, but it turned out that they had missed the fact that in the COSMO files the segments coordinates were re- ported in Bohr units. Thus, the value they used was 1.5/~ for the thermodynamic radius, but a value of 1.5 Bohr ~ 0 . 8 , ~ - 0.08 nm for the averaging radius. Hence, they in fact used a radius much closer to our optimum than to the effective contact radius. Con- sidering their results further, it appears that they may have lost some detail by using a larger averaging radius than 0.05 nm. This confirms our finding that an averaging radius of about 0.05 nm is quite optimal and thus we are using this value in all COSMO-RS parameterizations since 1998.

6.2 q-PROFILES

Since the interaction energies of the surfaces depend only on the local polarization charge-densities, a, only the net composition of the surface of a molecule X with respect to a is of importance for the statistical thermodynamics of local pair-wise surface interactions. Thus, we have to reduce the full 3D information about on the molecular surface to a histogram pZ(a), which tells us how much of a surface we find in a polarity interval [ ~ - d~/2,

+ da/2]. We will refer to such a histogram as the a-profile of the molecule X. As an example, let us consider water, the most important solvent of all. Its surface polarization charge density and the resulting a-profile are shown in Fig. 6.1. The entire a-profile of water spans the range of • 2 e/nm 2, and we will see that this is about the range of a-values generally found for stable organic and inorganic molecules, including most ions. It is dominated by two major peaks arising from the strongly negative polar regions of the electron lone-pairs of the oxygen atom and from the strongly positively polar hydrogen atoms, respectively. In the color coding of the surfaces these regions can be recognized clearly as deep red and deep blue. Note that owing to the sign inversion of the polarization charge density, a, compared to the molecular polarity, the peak from the negative lone-pairs is located on the right side of the a-profile at about 1.5 e/nm 2, while the peak arising from the positively polar hydrogens is located on the left side, at about -1 .5e /nm 2. Both peaks are beyond the hb threshold of _ O ' h b - • 2, i.e., large parts of the surface of water

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Fig. 6.1. a-Profile of water.

molecules are able to form more or less strong hb's. Since hydrogen bonding is weak up to • 2, we will generally consider the a-regions beyond • 1 e/nm 2 as strongly polar and potentially hydrogen bonding, and the rest as weakly polar or non-polar. Very few surface areas on the water molecule are located in less polar ~-regions, i.e., between - 1 and 1 e/nm 2.

One important feature of the a-profile is its remarkable sym- metry with respect to a. There is about the same amount of strongly positive and equally strong negative surface area. This enables energetically very favorable pairings of positive and negative surfaces and formation of strong hb's without any lack of adequate partners. As we will see later, this is an almost unique feature of the liquid water, which causes its relatively high boiling point for such a small molecule. The polarization charge densities and a-profiles of additional characteristic solvents are shown in Figs. 6.2-6.4, and are always together with the a-profile of water, which acts as a type of reference. For the reader interested in an even more vivid 3D visualization of a on the surface of the molecules, all virtual reality mark-up language (VRML) files of all


Fig. 6.3. a-Profiles of common compounds.

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Fig. 6.4. a-PROFILES of common compounds.

these molecules are given as supplementary material [92]. It is worthwhile, and hopefully also interesting to consider these in more detail, in order to become familiar with the typical features of molecular surface polarity.

Let us start with hexane, which is a typical representative of alkanes. As can be seen in the inset, the top left of Fig. 6.2, the surface of hexane is mainly non-polar, i.e., green, with a tendency to blue-green on the hydrogens and to yellow-green on the carbon regions. The corresponding a-profile of hexane ranges from -0 .5 to +0.6 e/nm 2, with a maximum at -0 .1 e/nm 2 arising from the hydrogens, and a shoulder at about +0.2 arising from the exposed surfaces of carbon atoms. The a-profiles of other alkanes look very similar, and mainly differ in height according to the differences in the total surface area.

The surface of benzene (inset below hexane in Fig. 6.2) shows more pronounced polarity. The negatively charged u-face appears yellow, corresponding to a moderately positive ~, and the


hydrogens are light blue, corresponding to moderately negative a. In the a-profile, this results in two clearly separated peaks centered at _+0.6e/nm 2. The a-profile of benzene is almost exactly symmetric with respect to a.

Next we consider methanol. On the surface (top right inset in Fig. 6.2) we can dearly identify the polar hydrogen as a deep blue area, the area of the oxygen lone-pairs as a deep red region, and the methyl group, which appears slightly more blue than the methyl groups of hexane. In the a-profile we see that methanol has about the same shape as water in the positive a-range, arising from the sp3-oxygen in both molecules. In the strongly negative a-region, methanol has about half the intensity of water because it has only one polar hydrogen compared with two in water. The methyl group appears alkane-like, with one higher peak from the hydrogens and a shoulder from the carbon, but the position is dearly shifted to the negative a-range, expressing the polarization of this methyl group by the neighboring oxygen.

Acetone (propanone, see inset below methanol in Fig. 6.2) has an sp2-oxygen, which clearly shows deeply red polarity maxima in the direction of the two lone-pairs. The shape of the a-profile of this oxygen is very different from that of the sp3-oxygens in methanol and water. The six hydrogens are well polarized and give a peak at -0 .6e /nm 2, almost identical with the hydrogen peak of benzene, while the three carbon atoms form a peak at 0.0 e/nm 2. The a-profile of acetone is very asymmetric.

The a-surface of acetic acid is most colorful. The sp2-oxygen is slightly less polar than that of acetone, and the hydroxyl oxygen is less polar than in water, but the acidic hydrogen is much more polar than those of water and alcohols, causing a change into violet on the a-surface. The methyl group is quite similar to those of acetone.

In Fig. 6.3, we see a-profiles of some nitrogen compounds and a sulfur-containing compound. Methylamine shows a very strong lone-pair on the nitrogen. In the a-profile we see that its polarization charge-density a ranges up to 2.7 e/nm 2, while water ends at about 2.1 e/nm 2. This causes extremely strong hb's of this lone-pair with polar hydrogens, and we will see that these extremely strong polarities cause some problems in the correct quantification of the hydrogen-bond energy of amines. We can also see that the two hydrogens of the amine group are moderately polar with a values

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at about -1 .0e/nm 2. Compared with the hb donors of water or alcohol, they are only weakly hydrogen bonding. The methyl group of methylamine is less polarized than that of methanol.

Nitromethane is shown below methylamine in Fig. 6.3. The two sp2-oxygens appear mainly yellow, each showing two more polar, red regions in the lone-pair directions. The shape of the oxygen peak is similar to that of the sp2-oxygen of acetone, but it is clearly shifted to lower a-values, i.e., lower polarity. On the other hand, the methyl group is strongly polarized and hence centered at almost -0 .8 e/nm 2 in the a-profile. The ~-orbitals of the nitrogen are electron-depleted by the two oxygens and appear slightly blue. Although the shapes on the positive and negative sides are a little different, the a-profile of nitromethane is overall quite symmetric.

In acetamide the carbonyl oxygen has the typical shape of an sp2-oxygen in the a-profile, but is located at higher a-values than in acetone. It is about as polar as the oxygen of water. The two N- hydrogens appear as deeply blue as the hydrogens in water, and indeed their peak is located at about the same position in the a-profile. The methyl group is moderately polarized, the ~-orbital of the carbonyl carbon is quite neutral (green), and that of the nitrogen is slightly negatively polar, resulting in a somewhat positive a and a yellow color.

Methanethiol, despite being iso-electronic with methanol, has a a-surface and a-profile, which look quite different. The sulfur lone-pairs are much weaker than those of the oxygen atom in methanol, but they are still weak hb acceptors. The hydrogen bound to sulfur barely overcomes the hb threshold.

Some halogenated compounds are shown in Fig. 6.4. First we consider the non-polar compound tetrachloromethane. As can be seen in the upper left inset, the a-surface mainly appears green, but in the opposite bond directions the chlorine atoms show some blue regions, owing to the fact that the electrons of the px-orbitals of the chlorine atoms are involved in the covalent bond and hence are shifted toward the central carbon atom. As a result, the a-profile of tetrachloromethane is dominated by a strong peak at 0.2 e/nm 2, balanced by a peak at -0 .2 e/nm 2 and another peak at -0 .6 e/nm 2.

Chloroform (trichloromethane, see inset below tetrachloromethane) has three chlorine atoms showing basically the same


features as those in tetrachloromethane, but being slightly more polar because the electron density is withdrawn from the hydrogen. As a result, the hydrogen is strongly polarized and appears as deep blue on the a-surface. This is centered at about 1.4e/nm 2, and is thus almost as polar as the hydrogens of water. The a-profile of chloroform is therefore very asymmetric.

Bromodifluoromethane is very similar to chloroform, but it can be seen that the smaller fluorine atoms do not show an electron lack (blue area) in the opposite bond direction, while the larger bromine clearly shows a yellow ring and a blue spot at the cap.

Finally we consider formyl fluoride. Here, we see a relatively weakly polar carbonyl oxygen with two lone-pairs in orange. The fluorine is competing for electrons with the oxygen atom and becomes slightly more polar than in chloroform. The two electron- withdrawing neighbors cause the ~-orbital of the carbon to be electron-deficient and thus it appears as a clearly blue region on the a-surface. The hydrogen atom is strongly polarized and gives a peak similar to that in chloroform, but is slightly less polar.

We could continue this discussion of a-surfaces and a-profiles with many other interesting and colorful examples, but this would exceed the limits of this book. From the representative examples discussed so far, the basic principles of the surface polarities of organic compounds expressed by the polarization charge densities, a, should have become clear. We leave it to the reader to study additional examples in the supplementary material.

6.3 WHY DO SOME MOLECULES LIKE EACH OTHER AND OTHERS NOT?

Before turning to more quantitative applications, it is worthwhile to consider a few examples of binary mixtures on a qualitative basis, just by viewing their a-profiles. From such considerations we can learn quite well why some molecules like each other so much, while others do not.

We start with the most striking example, of a mixture of acetone and chloroform. Although they have been discussed sepa- rately before, their a-profiles are shown together in Fig. 6.5. Both have very asymmetric a-profiles, so they do not feel very comfortable in their pure liquids, because the oxygen in acetone does

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Fig. 6.5. a-Profiles and mixture properties for the mixture of propanone and chloroform.

not find appropriate polar counterparts, and the polar hydrogen in chloroform does not find a partner with a reasonably positive a. Hence, in each pure liquid there is a considerable amount of electrostatic misfit. But if we mix both liquids, the a-profiles complement each other in an almost ideal way. The polar hydrogen of chloroform matches very well with the polar oxygen surface polarity of acetone, and even the other peaks in both a-profiles find almost perfectly corresponding peaks. In this way, the misfit energy is strongly reduced in the mixture, resulting in a strongly negative heat of mixing (see inset in Fig. 6.5), i.e., the mixture becomes warm, when both liquids are poured together. In some way this acetone-chloroform example appears to contradict the ancient chemical recipe of similis similibus solvuntur, i.e., "like dissolves like," since acetone and chloroform dissolve each other so exceptionally well as a result of their complementarity, and not their similarity.

A typical example of similis similibus solvuntur is the mixture of acetone and dimethyl ether (see Fig. 6.6). The a-profiles of both solvents are very similar. Both are asymmetric in the same sense, and hence they do not feel very well in themselves, nor in the mixture. However, due to the entropy gain of the mixing process


Fig. 6.6. a-Profiles and mixture properties for mixtures of propanone and dimethyl ether.

both dissolve each other quite well. Since the energetic situation regarding the misfit does not change much during mixing, there is no significant enthalpy gain or loss in this mixture. Such mixtures, in which the situation for the molecules does not change, either in a positive or negative direction, are generally called "ideal" mixtures, although the situation for the molecules need not be ideal at all, as seen in this case. Another example of similis similibus solvuntur is the mixture of benzene and 1,3-pentadiene, as shown in Fig. 6.7. Here, both solvents have reasonably similar, symmetric a-profiles, and hence both feel well in themselves, as well as in the mixture.

In a mixture of acetone and benzene, as shown in Fig. 6.8, the ~-profiles of both solvents are quite dissimilar. Acetone has a rather polar "unmatched" oxygen and does not feel well in itself, while benzene is only weakly polar, being quite comfortable in its pure liquid. Since in the negative region the a-profiles of both are quite similar, the pairing situation for the surface segments with positive a stays quite unchanged if both acetone and benzene are mixed. Therefore, these solvents form an ideal mixture for en- tropic reasons although they have quite different polarities.

As a last example we consider the mixture of water and hexane, as shown in Fig. 6.9. As already discussed, water has a broad and

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Fig. 6.8. a-Profiles and mixture properties for the mixture of propanone and benzene.

symmetric a-profile, and hexane has a narrow a-profile around a = 0. Both solvents are very satisfied in their pure liquids. If we try to put a hexane molecule into water at ambient temperature, the non-polar surface pieces of hexane cannot break the strong


Fig. 6.9. a-Profiles and mixture properties for the mixture of water and hexane.

pairs between water segments with strongly positive and negative a, which in addition to their good electrostatic match also form strong hydrogen bonds. There is no sufficient thermal energy available to break these pairs. Hence, the only choices for the hexane segments are the few less-polar surface segments of water. With these, hexane can pair without significant energy costs, and hence the excess heat of solution of hexane in water indeed is almost zero. However, the restriction to a small portion of the segments means a strong loss of entropy for hexane in water, and hence there is a strongly negative mixing entropy, resulting in the low solubility of hexane in water. On the other side, a water molecule in hexane does not find any good partners for its polar surface segments. All hexane segments are almost equally inap- propriate for the polar water segments. This results in a large positive excess heat of solution and a low solubility of water in the alkane, but a positive excess entropy of solution, because the partner choice is much less selective than it was in pure water. As shown in more detail in [22], this COSMO-RS picture of the un- usual mixing behavior of alkane and water is in good qualitative and quantitative agreement with experimental data.

To summarize, we have seen that the a-profiles provide a de- tailed and realistic understanding of the mutual solubilities of

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solvents and overcomes old heuristic recipes, such as simil is s imil ibus solvuntur.

6.4 6-POTENTIALS

We shall now consider the solvent properties in a more quantitative way by applying the COSMOSPACE thermodynamics to an ensemble of solvent surface segments of composition

n

xip~(6)A6 Os(a) - ps(a)Aa - i=1

n

~-~.xiqi i = 1

(6.4)

with respect to the polarization charge density, a. Here, X i is the mole fraction of a component i of the solvent, pi(6) the corresponding a-profile, and qi the surface area. Apparently, for a pure solvent, the solvent a-profile, ps(a), is nothing but the a-profile of the solvent normalized to unity, i.e., a-profile divided by the total surface area qs. Applying Eq. (5.12) in combination with the COSMOSPACE Equation (5.32), we thus obtain

where the final identity follows in the limit of infinitely small a intervals, Aa. Indeed, this was the original form of the COSMO- RS thermodynamics. Here, ps(a) is the chemical potential of a unit piece of surface of kind a in the solvent S. We call this the a-potential of the solvent. It tells us how much a solvent S likes polarity of kind a. As noted for the COSMOSPACE equation, Eq. (6.5) also, in general, requires an iterative numerical solution, but it converges well starting with the initial assumption of ps(a') on the right-hand side. Only in the case of negligible hydrogen bonding and for a Gaussian-shaped solvent a-profile ps(a) an analytical solution is available. In that case, as shown in the appendix of [C9], the a-potential corresponds to a simple parabola.

Obviously, the a-potential is a function of composition and temperature. In Fig. 6.10 we compare the a-potentials of some


Fig. 6.10. a-Potentials of representative solvents at 25~ (and 100~C).

representative solvents at room temperature. The plot of the a-potential of hexane is close to a parabola. Since hexane is not hydrogen bonding, and since the a-profile of hexane is a narrow, almost Gaussian peak, this parabola corresponds reasonably to the analytical solution of Eq. (6.5) derived for Gaussian a-profiles. A parabolic a-potential also corresponds to "dielectric behavior" in the sense that any solvent behaving like a dielectric on a molecular scale would show a parabolic a-potential, and the curvature would depend on the dielectric constant, e. Since we have already stated in section 4 that alkanes should behave like real dielectrics on a molecular scale, because of the absence of relevant permanent molecular electrostatic moments and the quite linear behavior of the electronic polarizability up to molecular electrostatic field-strengths, we may identify the curvature of the a-potential of hexane of 26.6 kJ nm2/e 2 with a dielectric constant of e ~ 2.1, i.e., the macroscopic dielectric constant of alkane. Apparently tetrachloromethane has almost the same a- potential, because it is also rather non-polar. The slightly larger extension on the negative side of the a-profile cause a slightly smaller curvature on the positive side of the a-potential. The asymmetry of the a-potential is much more pronounced for chloroform, because this has a much more asymmetric a-profile, as discussed before.

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The a-potential of benzene also appears to be an almost perfect parabola. Apparently, the parabolic solution holds even for non- Gaussian a-profiles, if these are reasonably symmetric and not hydrogen bonding. The curvature of the a-potential parabola of benzene is smaller than that of an alkane. Because of its broader a-profile, benzene is more tolerant to polar surfaces than hexane, resulting in a curvature of the a-potential of 20.6 kJ/mol nm2/e 2. By using the rough approximation

curvature = const(1 - f(D) ~ const (1 - ~ + 0_5) ~ - 1 (6.6)

and determining the proportionality constant from the values for hexane, we find an effective dielectric constant of ~ e f f - - 2.85 for benzene, while (because it has no dipole moment) its macroscopic dielectric constant is equal to its optical dielectric constant, and has the value 2.1. Thus, a dielectric continuum solvation model itself would have led to the wrong result that the solvents benzene and alkane behave almost identically, while COSMO-RS correctly reflects the fact that benzene is more tolerant with respect to electrostatic polarity than alkane.

The a-potential of water shows an almost linear behavior in the non-hydrogen-bonding region. The very small curvature corresponds to a high effective dielectric constant, in agreement with its high macroscopic dielectric constant. For non-hydrogen-bonding compounds this almost linear behavior would extend further out, as indicated by the dashed line in Fig. 6.10. The small effective curvature caused by the almost symmetric a-profile, means that electrostatically water behaves roughly like a strong dielectric. This is the reason for the macroscopic dielectric con- t inuum solvation models working roughly for the solvent water. As we will see later, the constant value of PH20 (~ --0) corresponds to a surface tension, which is usually fitted into the non- electrostatic part of CSMs, and the slight slope of the curve does not affect the chemical potential of neutral solutes. Nevertheless, it is apparent from Fig. 6.10 that water has an exceptionally high ~-potential in the non-polar a-region. All other solvents have much lower values in this range and, as a result, a non-polar surface hates to be in water and likes to escape into any other solvent. This is exactly the behavior that is usually called "hydrophobicity." In most other models, hydrophobicity has to be


introduced as an additional type of interaction, but in COSMO-RS it follows directly from the thermodynamics of the surface pairs as a result of the exceptionally strong and balanced interactions of the water segments, without any additional assumptions. Fur- thermore, we see that in both hydrogen-bonding regions the a- potential of water turns into an almost linear descent, expressing water 's ability to form some additional hydrogen bonds with the hb-donors and hb-acceptors of solute molecules.

The a-potential of methanol clearly expresses strong differences from water. Owing to the presence of the weakly polar methyl group, it has good partners available for non-polar surfaces and hence has quite a low a-potential in that region. Owing to the lack of one polar hydrogen compared with water, it is more attractive for hb-donors than is water, which is expressed by the lower a-potential, in the range a < - 1 e/nm 2, but it does not like additional acceptors as much as water does. As can be seen in Fig. 6.10, only acceptors with a > 1.5e/nm 2 have a chance of making hydrogen bonds in methanol, because this solvent has an excess of strongly hydrogen-bonding acceptor surface area itself.

It can also be seen in Fig. 6.10 that the solvent acetone has an almost parabolic a-potential over a wide ~-range, but this turns into a strong descent in the donor region, because acetone has a strong lack of hb-donors. As can be seen in the a-profiles, its hb-acceptors are less polar than those of methanol, and therefore acetone is less attractive for very strong hb-donors than is methanol.

In the a-potential plot for methylamine it can be clearly seen that the moderately polar hydrogens of the amine group are not sufficiently polar to cause any significant attraction for hb-acceptors. However, the extreme ~-hotspot on the lone-pair of the amine nitrogen causes very attractive interactions with hb- donors. While, without any doubt, this is qualitatively correct it must be emphasized again that this very strong hydrogen bonding of amines brings the hb-interaction parameters of COSMO-RS to its quantitative limits and hence causes larger errors.

As a result of its broad and symmetric, but still mainly non- hydrogen-bonding, a-profile, the solvent nitromethane has an almost zero a-potential in the non-hydrogen-bonding a-range. Here it behaves closest to the perfect conductor limit. Because it has no

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hb-donors, but very weak hb-acceptors, it shows a slight descent in the hb-donor ~-range, and a slight increase in the opposite region.

In Fig. 6.10 we also see the a-potentials of water and hexane at 100~ (dashed lines). Hexane shows almost no temperature dependence, in good agreement with its almost temperature-independent macroscopic dielectric constant. For water, we see that the non-hydrogen-bonding part is also quite insensitive to temperature. In contrast, in the hydrogen-bonding regions the ~r-potential increases strongly with rising temperature, because hydrogen-bonding becomes less important at high temperatures, as expressed by the temperature-dependent scaling factor fhb(T) in Eq. (6.2).

To summarize, in this chapter, we have seen that the a-potentials ps(a) are significant fingerprints of the solvent behavior of pure and mixed solvents. They express the affinity of a solvent to electrostatic polarities, hb-donors, hb-acceptors, and to non- polar surfaces, and include the concept of hydrophobicity in a natural way.

6.5 CHEMICAL POTENTIAL OF SOLUTES AND PHASE EQUILIBRIA

Translating the thermodynamic concept of interacting surfaces to our basic COSMO-RS, the residual part of the chemical potential of a compound i in a solvent S is found by a summation of the chemical potentials of the surface segments of i. Starting from Eq. (5.10) we have

pR(s; T) - ~ ni'tzv(S; T) - jpi(a)t,s( )d l '

(6.7)

where pi((r) is the a-profile of the solute i and ps((r) is the cr- potential of the solvent S at temperature T. Since pi(~r) gives the amount of surface with polarity a, Eq. (6.7) expresses the residual part of the chemical potential of compound i in solvent S as a surface integral of the a-potential over the surface of the solute i. Combining this with a simple, and usually small, combinatorial contribution (see Appendix C) and with the trivial concentration


dependence kT In X i we obtain the expressions

~i(S; T) - - Ft~(S; T) -~- kT lnx i - pR(s; T) + pC(s; T) + kT lnx i

- fpi(cr)ps(~; T)da + pCi(s; T) + kT lnx i (6.8)

for the chemical potential p i ( S ; T ) and the pseudo-chemical potential p~(S; T) of an arbitrary solute i in a pure or mixed solvent S. Since the pseudo-chemical potential p~(S; T), as introduced by Ben Naim [93], does not include the contribution proportional to lnxi, it can be calculated irrespective of whether the solute i is a minor or major part of the solvent S, or whether it is just a solute at infinite dilution, a fact that makes it more usable than the chemical potential itself in many situations.

Having the chemical potential of arbitrary compounds in arbitrary solvents and mixtures, we are able to express any kind of liquid-liquid equilibria (LLE) between different liquid phases, S and S', using the necessary condition that the chemical potential of each compound i must be equal in both phases. Partition coefficients of a compound i between two phases S and S' can be calculated directly from Eq. (6.8). If we assume that the mole fractions of i in both phases is very low, i.e., close to infinite dilution, we may disregard it in the composition of the phases S and S'. Then the partition coefficient, i.e., the ratio of the mole fractions, is

, , fx (S) } , , k T l n K i ( S , S ; T) - ln[xi(S, ) - pi(S'; T) - pi(S'; T)

_ / pi(o.)(pS,(a; T) - ps(O-; T)) da

+ T) - C(S; T) (6.9)

It should be noted that Eq. (6.9) gives the partition coefficient in terms of mole fractions, as is common in the chemical engineering literature. We therefore denoted it as K*. In chemistry it is usual to report partition coefficients in mol/1 units, and to denote them by K. Although the partition coefficient is dimensionless in both conventions, it differs by the ratio of the molar volumes of the two solvents. As we will see in more detail in the next chapter, logarithmic partition coefficient data have been used heavily for the parameterization of COSMO-RS. The octanol-water partition

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coefficient, Kow, is the most widely considered and best-measured partition coefficient [94]. It should be noted that the octanol phase in equilibrium with water (wet octanol) contains about 25 mol% of water. The logarithmic octanol-water partition coefficient, log- Kow is often used as a measure of molecular lipophilicity. From Eq. (6.9) we see that logarithmic partition coefficients in general, and also log Kow are mainly given as a surface integral, because the differences of the combinatorial contributions are usually small. Hence, the difference in the a-potentials between the two phases can be considered as the local contribution to the partition coefficient, and hence the partition coefficient can be visualized as a surface property, as is done for the molecules o-cresol and methylimidazole in Fig. 6.11. The total logKow of o-cresol is calculated as 1.73 (exp. 1.95), and for methylimidazole, basic COSMO-RS gives 0.26 (exp. 0.24). From the difference of the two a-potentials shown in Fig. 6.11, we see that negative, hydrophilic contributions to the logarithmic octanol-water partition coefficient arise only from the positive polarization charge-densities a, i.e., mainly from strong hb-acceptors, while neutral and positively polar surface regions give a constant positive contribution favo- ring the octanol phase. Hence, only the acceptor regions of the oxygen in o-cresol and, in a more extreme form, the nitrogen

Fig. 6.11. a-Potential difference of water and wet (water saturated) octanol.


Fig. 6.12. Surfaces of o-cresol and methylimidazole colored by log Kow, i.e., by the o-potential difference of octanol and water.

lone-pair in methylimidazole are visible as green and blue areas, i.e., as being hydrophilic on the molecular surfaces in Fig. 6.12.

For normal mixtures the activity coefficient of a compound i in a mixture S can be calculated as

~' ~S'~ , T ) - e x p { p*(S; T) - T)} (6.10)

where p~(i, T) refers to the pseudo-chemical potential of the compound i in the pure liquid i. Here, it is assumed that this pure liquid exists in the same conditions as the mixtures. This, at least, is required for the experimental determination of the activity coefficient, while we would be able to treat the virtual pure liquid in any case with COSMO-RS. Owing to this experimental difficulty, activity coefficients are defined with other reference states for gases and for electrolyte solutions. Some caution is therefore warranted when the expression "activity coefficient" is used in chemical or chemical engineering literature.

If we also know the chemical potential of the compounds in the gas phase or in a solid phase, then we can also calculate the equilibria between the vapor and the liquid phase (VLE), or between solids and the liquid phase (SLE). If the vapor pressure pvap(T) of the pure compound i is known, the partial pressure of i in the gas phase is given by

_ v a p , , Pi - P i (T) ",i(S; T)*xi (6.11)

104 A. Klamt

as long as the gas is sufficiently ideal, so that fugacity corrections are small. For SLE calculations, one can usually use an extrap- olation of the free energy of fusion, AGfus, s tar t ing from the melting tempera ture Tfus, the enthalpy of fusion AHfus, and change in the heat capacity Acp,fu~ between the liquid and the solid, that gives

/ /tsolirr~ �9 �9 (

i (.~ ~ - Pi (t, T) - AGfus ~ (AHfus + TfusAcp,fus)

+ T fu~Acp , fu~ In ~

1

(6.12)

For tempera tures reasonably close to the melt ing point, Acp,fus often can be neglected.

6.6 SOME EXAMPLES OF BINARY MIXTURES

As a demonstrat ion of quanti tat ive LLE calculations, we now consider in more detail some of the binary mixtures that we have discussed qualitatively in section 6.3. In Fig. 6.13 we see the excess Gibbs free energy of mixing G ex, the heat of mixing H ~x, and the excess entropy of mixing S ex for mixtures of acetone and

Fig. 6.13. Binary mixture data for acetone and chloroform at 298 K.


chloroform. The figure shows the large negative heat of mixing of about -1.35 kJ/mol and the exceptionally strong negative values of the logarithmic activity coefficients. The excess entropy of the mixture is negative (i.e., - T S E is positive), which results from the higher degree of order introduced by the very favorable, and hence more specific, interactions in the acetone-chloroform mixture.

Fig. 6.14 shows the same excess mixture properties and logarithmic activity coefficients for a mixture of acetone and dimethyl ether. Since this mixture behaves ideally (see discussion in section 6.3) the excess properties and logarithms of activity coefficients are very close to zero (note the scale of the axes) in this c a s e .

The situation is quite different for a mixture of 1-propanol and n-hexane (see Fig. 6.15). Here, we find a significant positive excess heat of mixing. The excess entropy of mixing shows a change of sign at about 10% n-hexane. This feature has also been found experimentally in many alcohol-alkane mixtures.

As a last example, we consider the binary phase diagram of water and 1-butanol (Figs. 6.16 and 6.17). There is a negative heat of mixing, H E, but a positive excess Gibbs energy of mixing, G E. The infinite dilution activity coefficient of 1-butanol in water is very

Fig. 6.14. Binary mixture data for acetone and dimethyl ether at 298 K.

106 A. Klamt

Fig. 6.16. Binary mixture data for water and 1-butanol at 298 K.

high. Experimentally, the 1-butanol-water is the first in the alkane-water series to show a miscibility gap at room temperature. The existence of a miscibility gap is reproduced well by the COSMO-RS calculations, as shown in Fig. 6.17. Since a miscibility


Fig. 6.17. Plot of mutual activities for the binary mixture of water and 1-butanol at 298 K.

gap, i.e., a LLE, means that there are two compositions, X l and x~' for which both compounds have the same activities, the existence of such an LLE can easily be checked by plotting the activities a2 vs. a~, where ai is x:,'i. If the result ing curve shows a closed loop, the binary mixture has a miscibility gap, and the two concentrations corresponding to the self-intersection point of the curve are the LLE compositions x~ and x~'. From Fig. 6.16 we can see that, in this case, there is an LLE between x ~ t ~ ~ 0.38 and Xwate r ' ~_ 0.992

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