Modelling of activity coefficents by comp. chem. 1
Modelling of activity coefficients using computational chemistry
Eirik Falck da Silva
Report in DIK 2099 Faselikevekter
June 2002 Modelling of activity coefficents by comp. chem. 2
SUMMARY ...... 3
INTRODUCTION ...... 3 background ...... 3
The use of computational chemistry ...... 4
REVIEW...... 4
Introduction ...... 4
Free energy of solvation ...... 5 Cosmo-RS ...... 7 SMx models...... 7 Application of infinite dilution solvation energy...... 8
Equation of state (EOS) approaches...... 8 UNIQUAC...... 8 UNIFAC ...... 10 General issues...... 10 Direct approaches ...... 11 Indirect methods ...... 11
Explicit solvent modelling...... 12
RESULTS AND PROPOSED PROCEDURES...... 13
Activity coefficients predicted with free energy of solvation ...... 13
UNIQUAC parameters from Sandlers method...... 14
Ab Initio Fitting to UNIFAC parameters...... 15
Monte Carlo modelling of UNIFAC parameters ...... 17
DISCUSSION ...... 21
CONCLUSION...... 22
LITERATURE ...... 24
APPENDIX 1 ...... 26
APPENDIX 2 ...... 29
APPENDIX 3 ...... 30
APPENDIX 4 ...... 32 Modelling of activity coefficents by comp. chem. 3
Summary
In this work I have looked at various approaches to the estimation of activity coefficients by computational methods. I have worked on binary systems consisting of neutral molecules. I have looked at several approaches described in the literature. Some I only discuss briefly, while others I have studied and applied. I have focused mostly on using computational methods to obtain parameters for the UNIQUAC and UNIFAC equations. Use of Monte Carlo modelling to obtain energy parameters for UNIQUAC has been fairly successful, supporting an interpretation of UNIQUAC energy parameters as a residual form of the free-energy of solvation.
Introduction
background
My group is working on the processes of CO2 removal from exhaust gases. We are looking at processes using chemical absorption of CO2 from a gas phase into a liquid phase. In order to understand, control and improve on the absorption process a detailed understanding of the chemistry is necessary. This understanding should include a model for the activities in the liquid phase. Use of models such as UNIFAC and UNIQUAC on these absorption processes has been reported in literature1,2. Published data does not seem to be very reliable and there is no consistency in reported data. The uncertainty is not surprising since almost all of the work is based on limited experimental data. Often only temperature, pressure, the partial pressure of CO2 and initial concentration of different species in the system is known. Concentration of various species are often unknown and even the understanding of the reaction chemistry can be limited. Various assumptions must be made and the activity coefficients are only one of a larger set of properties that must be fitted to the data. Group contribution models such as UNIFAC can be used, but their reliability for the systems in question is unknown and the group contribution factors available to us do not cover all molecules formed. Any form of modelling that can generate independent estimates of the activity coefficients would obviously be useful to improve the understanding and modelling of the absorption processes. The main goal of my own work is to predict the performance of new absorbers for these processes. A predictive model for the activity of these new absorbers is then desirable. Modelling of activity coefficents by comp. chem. 4
The use of computational chemistry
Computational chemistry is a term used to refer to molecular Monte Carlo methods, molecular dynamics and the so called "Ab Initio" methods, based on solving some (simplified) form of the Schrødinger equation. All these methods are well established. Ab Initio methods have been very successful in modelling of molecular geometry, particularly in the gas-phase. Most molecular properties can be calculated with reasonable precision using these methods.
The behaviour of a liquid mixture is a function of molecular properties and it is therefore natural to ask if computational chemistry can not be used to model activity in solvents. While I have not found much work on the modelling of activity in the literature, modelling of solvation effects is a major topic in computational chemistry. The methods in use can be used to calculate activities and use theory and concepts that would seem applicable to overall activity modelling.
Review
Introduction
Activity coefficients are defined as the ratio between activity and some measure of the concentration, often the mole fraction. Relating this to the chemical potential we get:
0 RT ln(x ) 1
It is usual to relate activity to the excess Gibbs energy, defined as:
E V G G(actual solution at T,P and x) - G (ideal solution at T,P and x) 2
Using this we get that:
RG E . / RT ln( ) 3 R / i N i 0 T ,P,n j
Borrowing from Tomasi et al3 we can divide the approaches to modelling activity into three categories:
S Methods based on the continuum model.
S Elaboration of physical functionals, here we find models such as UNIQUAC and NRTL.
S Methods based on computational simulations of liquids. Modelling of activity coefficents by comp. chem. 5
While I will not go further into this approach I will finally mention the possibility of using quantum chemical descriptors and using statistical methods to find their correlation with activity4.
I will discuss these various approaches and their application. Tomasi did note that the division of these methods was slightly dated since there was a convergence of these methods taking place and in this work I will also show possible combinations of these methods.
Free energy of solvation
One quantity often used in computational chemistry is the free energy of solvation (Gsol), giving the energy in going from (ideal) gas phase to a given solvent phase. If we can calculate Gsol for two solvent states, we in fact have data to calculate equation 1, i.e. differences in solvation free energy between two systems with the same reference state (gas-phase) is equivalent to differences in chemical potential. One main form of modelling Gsol is the Continuum models, so called because they treat the solvent as a continuum. The solvated molecule is placed in a cavity and the solvent effects are represented by the interaction with the cavity wall and the imposition of an electrostatic field. Ben Naim5 has developed the following equation to express the solvation energy based on statistical mechanics:
q q . n 3 . G W (M / S) RT ln rot,g vib,g / RT ln M ,g M ,g / 4 sol / 3 / qrot,s qvib,s 0 nM ,s M ,s 0
The equation is based on separating the process in two parts. First you have the work to move the solute to a fixed point in the solvent W(M/S). The final terms give energy contribution of molecular motion. The q's are microscopic partition functions of rotation and vibration in the gas phase and solution. nM,g and nM,s are the numerical densities of M molecules and finally the are momentum partition functions.
The work to add a molecule M to a fixed point in a solvent ( W(M/S) )can be decomposed into the following elements.
W (M / S) Gel Gcav Gdis Grep 5
Gel is the change in the electronic energy of the solute in the presence of the electrostatic field. This can be computed straight-forwardly in Ab Initio calculations. A general illustration is shown in figure 1. Modelling of activity coefficents by comp. chem. 6
Figure 1
Gcav is the cavitation energy. The energy needed to generate a cavity in the solvent. This quantity is not experimentally observable nor can it be calculated from Ab Inito calculations (on a single molecule). This value for this quantity is therefore based on (indirect) experimental data and some guesswork. A further difficulty is determining the size and shape of the cavity. Obviously it relates to the size of the solute, but that still leaves a lot unresolved. Older models used spherical cavities, while newer models use the molecular shape using definitions of atomic radii with some scaling factor. At some level this involves fitting to the experimental data. This limits these models to known solvents and temperature.
Gdis is the dispersion energy resulting from electronic interactions between the solute and solvent molecules. This is again not available from Ab Initio calculations. It can be estimated from more elaborate calculations and can then be assumed to scale with molecular size. Modelling of activity coefficents by comp. chem. 7
Grep is the repulsion potential. It relates to the quantum mechanical effects of mutual penetration of electron charge distribution. This presents problems similar to those for the dispersion energy.
Returning to equation 4 there is still the last two terms. The last term can be viewed as entropy generated by releasing the molecule. While this is a readily calculable property it is assumed to be zero in continuum models.
The second term in equation 4 relates to the changes in the internal motions of the solute as it enters the solvent. The gas-phase motions can be calculated fairly well by Ab Initio frequency calculations. There is however at present no models that calculate this reliably in a solvent. This term will therefore be omitted. It should be clear from my discussion above that the continuum models are to some extent based on fitting to experimental data. This suggests that they are less reliable for untried solvents and their performance outside room temperature is also unpredictable. It should also be noted that while these models are suited for solvation in pure solvents they can not reliably be extended to other concentrations. While the continuum models are in wide use there is doubts as to their ability to model all aspects of solvation. One specific issue is their failure to take account of hydrogen bonding. Finally we can note that these models has some similarities with the equation of state approaches to prediction of activity.
Cosmo-RS Cosmo-RS6 is a model which is based on the continuum models. The model is based on calculating a discrete surface around a molecule embedded in virtual conductor. Each surface segment has an area and a so-called screening charge density. A liquid is then considered as an ensemble of closely packed ideally screened molecules. Unlike the continuum models this model is able to take account of a solvent consisting of different molecules. The model is a proprietary development, only available in the commercial Cosmo packages, despite publications on it is difficult to compare it's performance to that of other models. The problems of determining cavity sizes found in the continuum models would also be present here.
SMx models The SM models are parameterised models for calculating the aqueous free energies of solvation7. One of the central equations in these models is:
Gsol k Ak rk 6 k
Here k denotes atom, Ak is the van der Waals surface area and k is a proportionality constant which is referred to as an atomic surface tension. k is a functional dependent on atomic species. These have been parameterised to experimental data. Modelling of activity coefficents by comp. chem. 8
This model is simpler and based on a more direct parameterisation then is the case for continuum models. While less general (it only works for infinite dilution in water), I have also found it to be precise.
Application of infinite dilution solvation energy
The models above give us solvation energies in infinite dilution, this in itself is however not enough to calculate activity coefficients as we do not have a reference state. Lazaridis8 used the following equation:
A 0 A A f kT ln 1 V , V , kT ln 1 7 A 1 2 0 2 f 2
V f1 and f2 being the fugacity of the pure components and is the residual chemical potential relative to an ideal gas mixture. The equation shows the possibility of combining relative free energies of solvation (computational) with fugacities of the pure components (experimental data). While this only gives relative activity coefficients it can also be used to find any single activity coefficient by using the same species as one of the solutes and solvent. Lazardis applied this equation on Monte Carlo free energy simulations in combination with experimental data (to which I will return later). I have tried to do something similar using the SM model to obtain the free energies of solvation, this work is shown in the results section. While this method works well it is limited to activity at infinite dilution and room temperature.
Equation of state (EOS) approaches
Before discussing some attempts at using computational methods to give parameters for equations of state it is appropriate to give a very brief summary of the equations of state modelling of activity coefficients.
UNIQUAC
Uniquac9 is one of the classical models for EOS modelling of activity coefficients. A derivation of the model is given in appendix 1, here I will only summarise the main features of the equations. Initially we can note that activity coefficients relate to the excess Gibbs free energy (gE) in the following way:
E nT g RT ni ln i 8 i Where nT is total number of moles. In the UNIQUAC model the excess Gibbs free energy is composed of two parts: Modelling of activity coefficents by comp. chem. 9
E E E g g combinatorial g residual 9
The combinatorial part is the entropy contribution from mixing a set of molecules, while the residual part expresses the energy interactions between different molecular species weighted over their surface contact areas. The following expression were derived for these quantities:
g E . z . / x ln 1 x ln 2 q x ln 1 q x ln 2 / 10 / 1 2 1 1 2 2 / RT 0combinatorial x1 x2 2 1 2 0
Where the 's are local area fractions, i.e. the fraction of area around a species filled by molecules of the same species. is the total volume fraction of a species. q is the surface area of a species. z is the coordination number for a molecule, which is simply assumed to be 10.
E . g / / q1 x1 ln 1 2 21 q2 x2 ln 2 1 12 11 RT 0 residual
Where the is a expression for molecular interaction energy
'$ !u u 1'4 ij jj ij exp% " 25 12 &' # RT 36'
Where u is the averaged binding energy for neighbouring molecules.
Finally there is a parameter that I have not shown but that is used in calculating the volume fraction(), this is a parameter for molecular size usually written as r. In the UNIQUAC model the q and r values are general parameters which have been calculated for a large number of species. The energy parameters uij and uji are then fitted to experimental data.
Some attention should be given to the assumptions underpinning UNIQUAC. This concerns the concentration of molecules of different species around one molecules of one species ( ji - concentration of particles j around species i):
!u u 1 ji i exp" ji ii 2 13
ii j # RT 3
According to this equation the probability that atoms of type i are interacting with molecules of type j is independent of the interaction between molecules of type j (ujj). If molecules of type j interacted strongly with each other they would be less likely to interact with molecules of type j and the assumption is therefore wrong. A brief discussion on this is given in appendix 4. Modelling of activity coefficents by comp. chem. 10
The assumption that the interaction parameters are independent of concentration is also an important assumption and a probably a weakness in the model.
The assumptions regarding mixing can be improved upon quite readily, while a better model of interaction parameters requires a more detailed understanding of molecular interactions.
A further issue is that in the derivation of UNIQUAC there is assumed to be only binary interactions in a liquid. This is a very crude approximation. When fitted to experimental data it is likely that the energy parameters will also capture these long- range energy contributions, but those contributions do not have the same dependency on concentrations as assumed in UNIQUAC.
UNIFAC
UNIFAC is a group contribution method. While being based on essentially the same model as UNIQUAC it views molecules as a set of groups. By obtaining data for the groups one would be able to predict the activity of any molecules composed of groups of which parameters are known. This gives UNIFAC predictive properties that UNIQUAC does not have. The main underlying assumptions is that the residual part of of excess gibbs free energy is a summation over the individual contributions of the groups. This leads to group energy interaction parameters. The combinatorial energy is calculated using the same method as in UNIQUAC, molecular volumes and surfaces being sums of group volumes and surfaces. The energy of molecular conformers in the solvent phase can vary by several kcal/mol, especially for large flexible molecules the orientation of the molecule is important. This kind of energy contribution can not easily be included in a group contribution model, this is one serious short-coming of group contribution models. In my work I will use UNIFAC with groups corresponding to whole molecules in which case it is the same as UNIQUAC. I will therefore use these terms interchangeably.
General issues
All aspects of the UNIQUAC model can be related to concepts and quantities in computational chemistry. The volumes and areas of species in UNIQUAC can be related to the cavities used in continuum models and binding energies is something readily available from computational models. While the UNIQUAC and other EOSs calculation of activity are based on models have a physical interpretation but they are in the end fitted to experimental data. So it is not clear to what extent the parameters do in fact capture what they were intended as. This is something that must be kept in mind when combining computational results with these models. Modelling of activity coefficents by comp. chem. 11
Direct approaches
One of the earlier efforts to use computational chemistry methods to obtain parameters for an EOS (UNIQUAC) was done by Jonsdottir et al10. They took a straightforward approach. They took the pair of molecules in question and used Molecular Mechanics to optimise their interaction. To include temperature effects the Gibbs free energy was calculated. The energy found here was then interpreted as the binding energy in UNIQUAC. While they reported good results there seems to be a fundamental flaw in this approach. Say that you had a mixture of an alcohol and water. The optimised structure would involve the alcohol facing the water molecule. Changing the size of the alkane group would not effect the binding energy to any significant effect. Yet the average binding energy to water must be lower for an alcohol with a large alkane group. A similar approach was described by Sandler11. He used a cluster with 8 molecules, 4 of each species. First this cluster was optimised and then molecular pairs were drawn out from the molecular cluster. Binding energies were calculated for these pairs and the averages of these energies were then used as the parameters in UNIQUAC. Only overall phase-diagrams were shown in Sandlers results, while these data seemed good it is difficult to tell from this how good the estimates for the activity coefficients were. I have attempted to reproduce Sandlers method, details and comments are given in the result section. Here it will suffice to say that the method would seem to suffer from several problems; one thing is that in such a small cluster one would have the same problem as with Jonsdottirs method, poorly binding parts of a molecule would face out of cluster and not be included in the binding energies.
Indirect methods
Sandler12 proposed a more developed method. First of all not to directly model the parameters in an EOS, but rather to calculate the infinite dilution coefficients and then to use these activity coefficients to fit the UNIQUAC parameters. His method is fairly elaborate and I will only try to give a summary of the essentials here. Change in molecular motions upon solvation is ignored in the model, the basic equation being used is:
* * * Gsol Gcav Gchg 14
G* here refers to energy to move to and from fixed points in gasp-phase and solvent. Using the same theory as in equation 2. Gcav refers to the energy of creating a empty cavity in the solvent, while Gchg is the energy arising from the insertion of a molecule with electrons and charges in that cavity. This is in itself equivalent to equation number 3 that we encountered in the continuum models. Sandler then combines the free energies of charging from the continuum models with the UNIQUAC theory, essentially using UNIQUAC to derive the cavitation energy. The energy terms in UNIQUAC he is also able to derive from the charging energy from the continuum models, most of the volume and area parameters were derived from Ab Initio calculations suing van der Waals radius (the exception is water where UNIQUAC parameters were used). Modelling of activity coefficents by comp. chem. 12
While Sandler essentially derives the cavitation energy from the UNIQUAC model, the cavity is still needed in the calculations. As I mentioned previously the size of the cavities in continuum models are not a priori given. Here Sandler uses van der Waals radius of atoms but adds a scaling factor . Here an analogy to the UNIFAC method is invoked. The scaling factors are determined as a group contribution. All atoms in a group will be given the same scaling factor. These scaling factors are then set by fitting to experimental data for activity coeffiecents. While the results presented are very good, most results are reported in terms of infinite dilution partition coefficients and average errors for these are reported to be 22%, compared with 237% for UNIFAC. This is however less impressive when one keeps in mind that the model is to some extent fitted to the same set of experimental data. Furthermore it is noted that the results are very sensitive to the setting of the scaling factor. In the result section I show a less ambitious effort at fitting UNIFAC parameters to data from Ab Initio calculations.
Explicit solvent modelling
All methods so far are based on the solvent being treated as a continuum or as geometrical shapes with some form of electrostatic properties. The other alternative is to explicitly model solute and solvent molecules. The main approaches to doing this is using molecular dynamics (MD) and Monte Carlo simulations (MC). Both usually use some potential function to represent molecular interactions. While these models are simpler then full Ab Initio representations they would present relatively advanced models for a solvent system. When being applied to the modelling of real solvent-solute systems the potential functions used would rely on some form of parameterisation. But as opposed to what we have seen for other solvent models this would be general parameterisations. One general problem with these methods is if one is in fact sampling the entire phase-space, i.e. if one is in fact computing real averages of the properties one is looking at. It is widely thought that properties such as overall energy for a system can not be calculated accurately. One can however calculate the effects of substituting one molecule in a system with another. This is usually done by using the Free Energy Perturbation method, which is based on gradually replacing one molecule by another. Lazardis et al9 used equation 7 to use free energy perturbations to calculate ratio of infinite dilution activity coefficients. They used a function of the potential energy based on Lennard-Jones and Coulomb potential functions, all with empirically fitted parameters. In their work they used the Monte Carlo program BOSS13. Their results show quantitative agreement with experimental data, performing somewhat worse then UNIFAC. I have used the same kind of calculation to generate parameters for the UNIFAC model. Details and results are shown in result section. The UNIQUAC model is based on assumptions about the interaction of molecules. All these assumptions can to some extent be tested using molecular modelling. This would however require a more extensive study then I am undertaking here. Modelling of activity coefficents by comp. chem. 13
Results and proposed procedures
Activity coefficients predicted with free energy of solvation
Using equation 7 I have attempted to model activity coefficients at infinite dilution using differences in free energy of solvation using the SM model. To give activity coefficients in water, water was set as component 2 with an activity coefficient of 1.
Molecules were optimised at HF/3-21G* level. This is not a particularly high level of calculation for molecular geometries, but since the SM model is empirically fitted at this level it's performance does not scale with level of calculation. All calculations were done using the Spartan program package14. All data were used as outputted from the program, except the free energy of solvation of water. This I shifted by 2.3 kcal/mol to give the best overall fit with experimental data. This only shifts the results linearly and does not effect the correlation of the results Experimental activity coefficient data are from Sandler12 and references therein, Bergmann et al15 and Kojima16. Data for vapour pressure over pure components were taken from CambridgeSofts database on the internet17. Results are given in table 1.
Table 1 A A Species Solvation energy Vapour pressure Calc. Exp. [kcal/mol] [kPa] Water -6.56 3.17 Hexane 1.765 130 1500165 400000 Isopropanol -3.543 33 756 6.6 n-octanol -3.505 0.1 266155 11600 Methanol -5.266 16.9 1.65 1.3 Ethanol -4.712 7.87 9.1 3.8 n-Propylamine -4.119 248 0.78 2.6 n-butanol -3.305 4 192 50 3-methyl-1-butanol -3.574 2 244 208 Acetone -3.091 181 6.1 7.56 Pyridine -5.658 16 0.9 19.14 Butyl-ether 0.763 4.8 153886 47180
A A plot of ln experimental versus calculated is shown in figure 1. Modelling of activity coefficents by comp. chem. 14
Figure 2
The overall fit is quite good, especially keeping in mind the very different kind of molecules I have looked at. ln-ln plot was partly chosen because of the large spread in data, but also because in the computational work we are calculating energies and uncertainty in results would be in energy (and proportional to ln).
UNIQUAC parameters from Sandlers method
Sandler11 proposed a method for calculating the parameters in UNIQUAC. I have described this work in the review section. I have attempted to reproduce his work on methanol. The described method was followed in detail, the only difference being that I did not correct for the so-called basis set superposition error. I have done some calculations on the correction Sandler used for basis-set superposition error and it does not seem to effect trends in the results.
In figure 3 I show the cluster I obtained from the optimised calculation.
Figure 3 Modelling of activity coefficents by comp. chem. 15
In table 2 I show Sandlers reported parameters, my own parameters and reported parameters based on experimental data found in Gmehling et al18.
Table 2 Interaction parameters for UNIQUAC. Component 1 is methanol, 2 is water UNIQUAC Sandler This work Experimental parameter u12 [kJ/mol] -0.395 0.90 1.66 u21 [kJ/mol] 0.053 0.31 -1.19
I should note that my own numbers varied a lot from calculation to calculation. In fact it would seem that a very large set of calculations would be needed for this method to produce statistically stable results. Agreements between Sandlers data and experimental data would seem to be fortuitous.
Ab Initio Fitting to UNIFAC parameters
One possible approach in using computational chemistry together with EOS models is to find correlation between properties calculated computationally and values for parameters in the EOS models. This could then be used to make estimates of the parameters for new molecules and groups. In table 3 I give UNIFAC data from Fredenslund et al19 together with calculated properties. To study the energy interaction parameter I have used interactions with water, this because of I have an available model for calculating solvation energies in water.
Table 3 Group UNIFAC data Computational data Rk Qk a-H2O* Volume Surface Solvation [Å3] area [Å2] energy [kcal/mole] CH3NH2 1.596 1.544 357.5 52.99 74.58 -5.83 CH2Cl2 2.256 1.988 370.4 75.34 96.6 -1.84 CH3NO2 2.009 1.868 -19.44 69.51 92.07 -4.47 HCOOH 1.528 1.532 225.4 53.11 73.79 -4.81 CH3CN 1.87 1.724 112.6 61.85 82.83 -3.90 CH3OH 1.431 1.432 289.6 48.83 69.8 -5.27 * Interaction parameter between species and water.
While I could derive group contributions from my molecular calculations I have chosen to only use whole molecules found in the UNIFAC data set. In this case UNIFAC and UNIQUAC are in fact the same.
In the figures below I show the plot of UNIFAC volume and surface parameters plotted against computes volumes and sizes. Modelling of activity coefficents by comp. chem. 16
80
75
70
65
60
55
Computed volume 50
45
40 1.4 1.6 1.8 2 2.2 2.4 UNIFAC volume parameter
Figure 4
80
75
70
65
60
55
Computed volume 50
45
40 1.4 1.6 1.8 2 2.2 2.4 UNIFAC volume parameter
Figure 5
The fit is extremely good, in itself no more then a reminder that the volume and surface parameters are parameters calculated directly from molecular structure. It is nevertheless good to note that they can be easily calculated. I did here omit the water molecule, its parameters in UNIFAC and UNIQUAC were obtained by fitting to experimental data and I would therefore expect it to deviate from the others Modelling of activity coefficents by comp. chem. 17
Monte Carlo modelling of UNIFAC parameters
There are several ways in which Monte Carlo Free Energy perturbation calculations can be applied to finding UNIFAC energy interaction parameters. I will give a brief outline of some possibilities and make some observations on the behaviour of UNIFAC.
At infinite dilution UNIFAC gives:
. A !u u 1 !u u 1. 21 11 12 22 // ln 1,residual q1 " 2 q1 1 exp " 2// 15 # RT 3 # RT 300
If on the other hand we had modified UNIFAC assuming random mixing of molecules, for example:
E u x1q1 u21 u11 x2 q2 u12 u22 16
we would have obtained the equation below:
A !u u 1 21 11 ln 1,residual q1 " 2 17 # RT 3
We see that the nature of the mixing rules effect the activity coefficient at infinite dilution. At infinite dilution the local concentrations are however given and the activity should here be a direct function of energy parameters. That this is not the case reveals a flaw in UNIFAC equations.
In terms of free energies of solvation, the activity coefficient at infinite dilution can be given as:
A 0 A 1,2 1,1 RT ln 1 18
Comparing equation 15 and 18 we find that the UNIQUAC interaction energies are not identical with (residual) differences in the free energies of solvation.
The formulation of the energy parameters in both the original UNIQUAC paper and in alternative derivation is nevertheless consistent with the interaction parameters corresponding to the free energy of solvation, albeit not including the entropy. In the original UNIFAC paper the following equation is also proposed:
u vap i uii 19 qi
Finally I should note that UNIFAC is reported to over-correct for deviations from random mixing. Modelling of activity coefficents by comp. chem. 18
The free energy perturbation calculations gives differences in free-energies between molecules in the same solvent. Which means that it can not directly be used to solve equation 18.
It is not clear to what extent the free energy perturbations include entropy effects and I have chosen to interpret them as residual free energies of solvation, i.e. not containing entropy.
I have come up with two main ways of using the free energy perturbations to obtain UNIFAC parameters:
We can first use free energy perturbation method to calculate the activity coefficients at infinite dilution and then set the interaction energy parameters in UNIFAC to give the same activity coefficients at infinite dilution. This we can do by using equation 7, or borrowing an assumption from UNIFAC:
A A 1,2 2,1 20
Using this assumption and equation 18 we get
A 0 A 2,1 1,1 RT ln 1 21
Alternatively we can use the free energies of perturbation directly as interaction parameters. This is the method that seems the most promising and which I will use from here on.
Calculations were done using BOSS version 4.3 developed by Professor William Jorgensen13. The program uses Lennard-Jones and Coulomb potential functions to express the potential energy. These parameters are to some extent empirical in nature. Boxes with 267 solvent molecules were used. All solvent boxes were equilibrated for at least 10 million configurations before use. The perturbation was performed over 5 steps, each with 500000 configurations of equilibration and 500000 configurations of averaging. Further details are given in Appendix 2.
Results for 6 molecules are shown below
Table 4
Component 2* u12-u22 [Kcal/mol] u21-u11 [Kcal/mol] CH3OH -0.328 1.662 CH3NH2 -1.832 2.918 CH2Cl2 1.005 7.015 CH3CN 0.818 2.622 CH3CH2OH 0.067 3.753 CHOOH 0.532 -0.766 * Component 1 is water
Using data in table 4 directly gave fairly good quantitative fits but consistently to high activity coefficients at infinite dilution. Modelling of activity coefficents by comp. chem. 19
To improve the fits I changed the UNIQUAC equation in the following way:
. ln q ln k q ln(1 k) q ji ij / 22 i residual i i j ji i j i / i j ji j i ij 0
The UNIFAC form is what we would have if k was equal to 0. Setting k as 0.4 I obtained the data shown below.
H2O-CH3OH H2O-CH3NH2 4 1.2 3.5 1 3 0.8 2.5 0.6 2 1.5 0.4
1 coefficients Activity 0.2 Activity Coefficients 0.5 0 0 00.51 0 0.2 0.4 0.6 0.8 1 x1 x1
Figure 6 ( )exp. data, ( )MC model Figure 7 ( )exp. data, ( )MC model
H2O-CH3CN 60
50 40
30 20
activty coefficents activty 10 0 0 0.2 0.4 0.6 0.8 1 x1
Figure 8 ( )experimental data, ( )Monte Carlo based model Modelling of activity coefficents by comp. chem. 20
H2O CH2Cl2
300
250
200
15 0
10 0 Activity coefficients Activity 50
0 0 0.2 0.4 0.6 0.8 1
x1
Figure 9 ( )experimental data, ( )Monte Carlo based model
The fits can in general be said to be good, but it must be noted that the modification in UNIQUAC that I used is arbitrary in nature.
Below I show calculated isothermal x-y plots for four of the systems together with experimental data from Gmehling et al18. Gas-phase ideality has been assumed.
CH3OH-H2O CH3CN-H2O 1 1
0.8 0.8
0.6 0.6 y1 y1 0.4 0.4
0.2 0.2
0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x1 x1
Figure 10 (T=298.15) Figure 11 (T=303.15) ( )experimental data, ( )Monte Carlo based model Modelling of activity coefficents by comp. chem. 21
H2O-CHOOH CH3CH2OH-H2O 1 1
0.8 0.8
0.6 0.6 y1 0.4 0.4
0.2 0.2
0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 x1 x1
Figure 12 (T=298.14) Figure 13 (T=303.15) ( )experimental data, ( )Monte Carlo based model
The overall quality of the results must be said to be fairly good. The computational parameters showing overall quantitative agreement with UNIQUAC parameters and experimental data. The only result that is rather poor is the data for ethanol, but even here we find some quantitative agreement with experimental data. It should here be noted that water, methanol (CH3OH) and acetonitrile (CH3CN) have been parameterised as solvents in BOSS, while for formic acid and ethanol I had to create the solvent input myself. The relatively poor performance for these molecules might reflect that the solvent molecule parameters are not as carefully set and also the fact that these molecules are slightly more complex and therefore more difficult to model accurately. What can be seen from the plots is that the fits are quite good at low/high concentrations and poorer fit at molar ratios of 1:1. This is not surprising when we consider that the energy parameters were estimated at infinite dilution. Considering the case of ethanol: We measure the interaction parameter with water at infinite dilution. Under this conditions the alkane group of ethanol will be exposed to water, contributing to a low overall interaction energy. At other compositions we expect the alkane groups of the ethanol molecules to interact with each other leaving to the water molecules to interact with the OH groups, thereby giving much stronger interaction. This explanation does account for the nature of the deviations we see for ethanol. This problem is not easily handled in the present model where the energy parameters are assumed to be constant. Considering the relatively simple molecular representation used in the modelling, this is maybe as good results as one can hope for with the present models.
Discussion
In this work I have looked at using computational chemistry methods to generate parameters for equation of state models. There are two main challenges in succeeding with this. One issue is the equation of state models themselves. When the equation of state models are used together with experimental data, parameters fitted to experimental data can capture behaviour that is not covered in the Modelling of activity coefficents by comp. chem. 22 model. Using computational chemistry we are calculating specific quantities, properties not included in the model will not be corrected for. We are therefore setting greater demands on the equation of state model itself. In my work I chose the UNIQUAC model because work reported in the literature had focused on this model. While my work with UNIQUAC has shown fairly good results, there would seem to several weaknesses in the model. The energy parameters are ambiguous; it is not clear how that they should be calculated and they are based on assumptions of a liquid with only binary interactions. The problem is also connected to the assumption that the energy interactions are independent of concentration, which as a general assumption is not true. Finally the equations regarding local composition around a given molecule are based on approximations that are maybe not very good. Secondly we have the issue of the quality of computational methods. As should be clear from my initial review, modelling of solvent effects is an ongoing project in computational chemistry. The handling of interactions between many molecules is a difficult issue. Explicit simulation of a high number of molecules is expensive while simplified approaches tend to rely on fitting to experimental data and are often limited to well-studied solvents. The Monte Carlo simulations I performed were done using a model that has been in wide use. The model used is semi-empirical in that atomic parameters have been fitted to experimental data, it is also uses a relatively simple atom representation in that electron effects are not included. There is therefore limits to how good results one can obtain with such a model. More advanced models are today available, but most have to my knowledge not been tested for this kind of activity coefficient modelling. The present work shows that it is possible to use computational chemistry to make estimations on activity and liquid-vapour phase behaviour. Results presented in this report show the possibilities of such methods but also show the limitations. The assumption in UNIQUAC (and other EOS models) that energy interaction parameters are independent of concentration does not hold, and must be improved upon if better results are to be achieved. Better estimation of activity coefficients will require an activity coefficient model with a more sound physical model then offered by UNIQUAC at the same time good computational models are required.
Conclusion
Using Monte Carlo methods I obtained reasonably good estimates for the parameters in the UNIQUAC equations for modelling activity coefficients. The results support an interpretation of UNIQUAC interaction parameters being a form of the free energy of solvation. Best results were obtained with an arbitrary adjustment of the UNIQUAC equations. Further improvements of the model can probably be achieved by improving the level of calculation for the Monte Carlo calculations. One must however also look more closely at the underlying assumptions of UNIQUAC and modifications of UNIQUAC might be required to obtain better results. Modelling of activity coefficents by comp. chem. 23
List of symbols Equations only appearing in the review section are not covered here x = liquid-phase mol fraction y = gas-phase mol fraction A ijV ij = chemical potential of i at infinite dilution in j 0 iiV ii = chemical potential of pure component i = activity coefficient T = temperature f = fugacity G = Gibbs free energy V = - Ideal gas mixture(T,x,) Vap = Free energy of vaporisation in UNIQUAC/UNIFAC nT = total number of moles ni = moles of component i i = area fraction ij = local area fraction of sites belonging to molecule i around sties belonging to molecule j = Given by equation 12 i = Segment fraction uij = binary interaction parameter r = pure-component volume parameter q = pure component area parameter z = coordination number for molecule uE = molar excess energy Modelling of activity coefficents by comp. chem. 24
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2. Kaewischan, L, Al-Bofersen, O., Yesavage, V. F., Selim, M. S. "Predictions of the solubility of acid gases in monoethanolamine (MEA) and methyldiethanolamine (MDEA) solutions using the electrolyte-UNIQUAC model. Fluid Phase Equil. , 183, (2001), p 159-171
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15. Bergmann, D. L. and Eckart, C. A., "Measurement of Limiting Activity Coefficients for Aqueous Systems by differential Ebulliometry, " Fluid Phase Equi., 63, 141 (1991) Modelling of activity coefficents by comp. chem. 25
16. Kojima, K., Zhang, S. and Hiaki, T. , "Measuring Methods of infinite Dilution Activity Coefficients and a Database for systems including water", Fluid Phase Equi.,131 145 (1997)
17. http://chemfinder.cambridgesoft.com/
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19. Fredenslund, A. Ghmeling,J and Rasmussen, P. "Vapor-liquid equilibrium using UNIFAC" Elsevier, Amsterdam, 1977 Modelling of activity coefficents by comp. chem. 26
Appendix 1 The UNIQUAC equation
Rather then using the original presentation of the UNIQUAC equation, I will here use the one chosen by Prauznitz et al*.
Here I will look at the case of a binary mixture. Surface area of a molecule 1 is given by a paramter q1. The number of interactions is assumed to be zq1, where z is the coordination number. We consider one molecule of component 1 being isothermally vaporized from its pure liquid denoted by superscript (0) and then condensed into a fluid mixture (1). We assume that that a molecule has z(0) nearest neighbours. We assume that intermolecular forces are short range and therefore only consider nearest neighbour interactions; the energy of (0) (0) (0) vaporization per molecule is 1/2z U11 where U11 characterizes the potential energy of two nearest neighbours in pure liquid 1. (1) The central molecule in the fluid mixture is surrounded by z 11 molecules of species (1) 1 and z 21 of species 2, where 11 is the local surface fraction of component 1, about central molecule 1, and 21 is the local surface fraction of component 2, about central molecule 1 ( (0) (1) Note that 11+ 21=1). We now assume z is the same as z . The energy released by the (1) (1) condensation process is 1/2z( 11U11 + 21U21 ), where the superscript on z has been dropped. We make a similar transfer for a molecule 2 from the pure liquid, denoted by supersicript (0), to a fluid mixture, denoted by superscript (2). We now consider a mixture of x1 moles of fluid (1) and x2 moles of fluid (2). From the two-fluid theory we have that for extensive configurational properties (M):
(1) (2) M mixture x1M x2 M A1
The total change in energy in transferring x1 moles of species 1 from pure liquid 1 and x2 moles of species 2 from pure liquid 2 into the "two-liquid" mixture, i.e., the molar excess energy uE, is given by:
1 1 u E zx N q U (1) U (1) U (0) zx N q U (2) U (2) U (0) A2 2 1 A 1 11 11 21 21 11 2 2 A 2 22 22 12 21 22 where NA is Avogadro's number. Since the local surface fractions must obey the conservation equations
1 21 11 A3 12 22 1 (1) (0) (2) (0) and assuming that U11 =U11 and U22 =U22 equation 2 simplifies to 1 u E zN x q U U x q U U A4 2 A 1 21 1 21 11 2 12 2 12 22 where we have now dropped the superscripts. We now assume
! 1 1 " z U 21 U11 2 21 2 exp" 2 2 A5
11 1 " kT 2 #" 32
*Prausnitz, J., Lichtentaler, R. and Gomes de Azevedo, E. "Molecular thermodynamics of Fluid- Phase Equilibria" third edition, Prentice Hall (1999) Modelling of activity coefficents by comp. chem. 27
and ! 1 1 " z U12 U 22 2 12 1 exp" 2 2
22 2 " kT 2 #" 32 where is the surface fraction: x q x q 1 1 and 2 2 A6 1 2 x1q1 x2 q2 x1q1 x2 q2 When these assumptions are coupled we obtain: E u x1q1 21 u21 x2 q2 12 u12 A7 and exp(u / RT) 2 21 21 exp(u / RT) 1 2 21 A8 exp(u / RT ) 1 12 12 2 1 exp( u12 / RT ) where 1 1 u zU U N and u zU U N 21 2 21 11 A 12 2 12 22 A A9
E E We use the approximation (a )T,VW(g )T,P and use d(a E /T ) u E A10 d(1/T)
Solving for gE from equation A7 we get the residual excess free energy:
g E . ! u .1 ! u .1 / 21 / 12 / / x1q1 ln" 1 2 exp 2 x2 q2 ln" 2 1 exp 2 A11 0 0 RT 0 residual # RT 3 # RT 3
The entropy contribution is taken from equations by Guggenheim** and is referred to as combinatoral energy:
g E . z . / x ln 1 x ln 2 q x ln 1 q x ln 2 / A12 / 1 2 1 1 2 2 / RT 0combinatorial x1 x2 2 1 2 0
Where is the volume fraction
x r x r 1 1 and 2 2 A13 1 2 x1r1 x2 r2 x1r1 x2 r2
r being the molecular volumes.
** Guggenheim, E. A:, Mixtures. Oxford: Oxford University Press, 1952. Modelling of activity coefficents by comp. chem. 28
Using:
Rn g E . T / RT ln A14 R / i ni 0 T ,P,n j
We finally obtain the expressions for activity coefficents:
. ln q ln q ji ij / A15 i residual i i j ji j i / i j ji j i ij 0
'$ !u u 1'4 ij jj Where ij exp% " 25 A16 &' # RT 36'
And z ln ln i ln i i combinatorial xi 2 i A17 z r z .. i // j ri qi ri 1 rj q j rj 1 / 2 r j 2 00 Modelling of activity coefficents by comp. chem. 29
Appendix 2
The potential energy in BOSS is expressed by the following equation:
A A C C q q i j i j i j U 12 6 6 i j rij i j rij i j rij
12 1/2 6 1/2 where sums are over all pairs of interaction sites, Ai=(4ii ) , Ci=(4ii ) , qi is the partial electric charge of the interaction site i and rij is the separation between interaction sites.
In the modelling I have use the united atom approach in which CH groups are treated as one atom. For the water molecule two representations were used, one as solvent (TIP4P) and one as solute (TIP3P). The TIP4P uses a four sites representation of the water molecule, one of these sites just carrying the negative charge. The atomic parameters were set according to recommended values provided with BOSS, values are given in table below.
Molecule (United) Atom q [Å] [kcal/mol] [Electron units] H2O /Water O 3.15365 0.155 0.000 (TIP4P) H 0.000 0.000 0.520 M 0.000 0.000 -1.040 H2O /Water O 3.15061 0.1521 -0.834 (TIP3P) H 0.0 0.0 0.417 CH3OH/ O 3.070 0.170 -0.700 Methanol CH3 3.775 0.207 0.265 H 0.0 0.0 0.435 CH2Cl2/ CH2 3.800 0.118 0.500 Dichloromethane Cl 3.400 0.300 -0.250 CH3CN/ C 3.650 0.150 0.280 Acetonitrile CH3 3.775 0.207 0.150 N 3.200 0.170 -0.430 CH3NH2/ N 3.300 0.170 -0.900 Methylamine H(N) 0.0 0.0 0.360 CH3 3.500 0.066 0.000 CHOOH/ C 3.750 0.105 0.520 Formic Acid O 2.960 0.210 -0.440 H(C) 2.420 0.015 0.000 O(H) 3.000 0.170 -0.530 H(O) 0.000 0.000 0.450 CH3CH2OH/ CH3 3.905 0.175 0.000 Ethanol CH2 3.905 0.118 0.265 O 3.070 0.170 -0.700 H 0.000 0.000 0.435 Modelling of activity coefficents by comp. chem. 30
Appendix 3 Pressure mol fraction diagrams. ( ) is experimental data. (-----) is computational model Same model and experimental results as used in result section.
CH3CH2OH-H2O
10 9 8 7 6 5 4
3 [kPa] Pressure 2 1 0 0 0.2 0.4 0.6 0.8 1 x1,y1 (T=298.14)
CH3CN-H2O
20 18 16 14 12 10 8
6 Pressure [kPa] 4 2 0 0 0.2 0.4 0.6 0.8 1 x1,y1 (T=303.15) Modelling of activity coefficents by comp. chem. 31
CH3OH-H2O
18 16 14 12 10 8 6
Pressure [kPa] 4 2 0 0 0.2 0.4 0.6 0.8 1 x1,y1 (T=298.15)
H2O-CHOOH
8 7 6 5 4 3
Pressure [kPa] 2 1 0 0 0.2 0.4 0.6 0.8 1 x1,y1 (T=303.15) Modelling of activity coefficents by comp. chem. 32
Appendix 4 Rules of mixing
In UNIQUAC the following rules are assumed for mixing: !u u 1 ji i exp" ji ii 2 1
ii j # RT 3
where ji is surface fraction of molecule j around central molecule i (as in Appendix 1).
Here I will propose an alternate form for this quantity borrowing from a model developed by Pelton and Blander*. In their work they use bondfractions (Xij): n X ij 2 ij n11 n22 n33
where nij is number of bonds between molecules of type i and j. I assume that the surface fractions of molecule j around i is proportional to the ratio of bonds of type j-j to bonds of type i-i. The exact form would be: X ji 3 ji X ji 2X ii In Pelton and Blanders work one thinks of bond forming and breaking as a reaction:
11 2 2 G 2 1 2 4 based on this they develop reaction-energy and entropy, and finally also a equilibrium- constant. Using their model and derivations we can write equation 3 as: 2X j 5 ji 1 1 4X i X j exp 2 T z / RT 1 where z is the coordination number of a molecule (as in UNIQUAC) , is the reaction enthalpy, is reaction entropy and X are molefractions. The enthalpy is given as: N z A 2u u u 6 2 ij ii jj where NA is Avogadros number. Following UNIQUAC I assume the entropy term to be 0. We can then write equation 5 as: 2X j 7 ji 1 1 4X i X j exp 2uij uii u jj N A / RT 1 The corresponding form in UNIQUAC (given in Appendix 1) is: exp(0.5u u N z / RT ) j ij ii A 8 ji i j exp( 0.5 uij uii N A z / RT ) When comparing these equations we can assume molar fractions and volume fractions to be
the same. Setting Xi and j as 1 or 0 we see that both equations do go to 1 and 0, as we would expect them to. If we consider the case of component 1 and 2 being the same ( uij=uii=ujj) the exponential terms disappear and both functions are reduced to ji=xj. There are two other examples we can look at uij>>(uii and ujj) and uij<< (uii and ujj).
*Pelton, A. D., Blander, M. "Thermodynamic Analysis of Ordered Liquid Solutions by a Modified Quasichemical Approach-Application to Silicate Slags" Metall. Trans. B, Vol 17B 1986, p 805-815 Modelling of activity coefficents by comp. chem. 33
Assuming uij>>(uii and ujj): Equation 7 goes towards: 2X j 9 ji 1 1 4X i X j While equation 8 (UNIQUAC) will go towards 1, which would in some cases seem odd, if the overall fraction of a component 2 is very low, there would not be enough of that species to obtain a high fraction of that component around molecules of type 1. Equation 8 would therefore seem to have a slightly deficient mass-conservation.
Assuming uij<< (uii and ujj): Both go towards 0.
If for example only ujj is large we see that in equation 8 this does not effect the surface fractions around molecule i, something that is captured in equation 7.
We see that equation 7 based on the model by Pelton and Blander gives more reasonable behaviour then the local composition rule currently in use in UNIQUAC (equation 1 and 8). This does however not guarantee that introducing equation 7 in the UNIQUAC model will give a better model. Further work is needed to draw