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GENERAL RESEARCH Free-Volume Activity Coefficient Models for Dendrimer Solutions Irene A. Kouskoumvekaki,† Ralf Giesen,‡ Michael L. Michelsen,† and Georgios M. Kontogeorgis*,† Engineering Research Center IVC-SEP, Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark; and Lehrstuhl fuer Technische Thermodynamik, Rheinisch-Westfa¨ lische Technische Hochschule Aachen, D-52056 Aachen, Germany
The ability of the Unifac-FV and Entropic-FV models to predict phase equilibria for dendrimer systems is investigated in this work. Different approaches are considered for the estimation of the density of the dendrimers, which is required as input parameter in the free-volume models. The density is not always available for such polymers.The predictions obtained by the two models are compared with recent vapor-liquid equilibrium experimental data for dendrimer systems. It is shown in this study that both methods represent satisfactorily the experimental data in many cases, whereas Unifac-FV performs better for PAMAM dendrimers (where some of the interaction parameters are missing). However, Unifac-FV is the model that is most influenced by the dendrimer’s density. Both models yield better results when experimental densities are employed (for AR dendrimers), but in those cases where such data are not available, predicted densities can be reliably obtained via the van Krevelen method (for C12 and A4 dendrimers). 1. Introduction Dendrimers (also called cascade polymers) are a special type of highly branched macromolecules with a branch point at each monomeric unit. They consist of a central core and an external surface. The branches (dendrons) are built of repeat units or cells, which are connected in a precise architectural arrangement that produces a series of regular, radially concentric layers, called generations, organized around the core, as can be seen in Figure 1. Because of such an organized architecture, dendrimers can be prepared with a high degree of synthetic control and monodispersity and thus provide a unique combination of high molecular weights and molecular shapes similar to ideal spherical particles.1 This unusual combination is responsible for their unique properties, such as perfectly Newtonian flow even at high molecular weights,1 high reactivity, and good solubility in solvents.2 As a consequence of these properties, dendrimers have found applications in a variety of fields such as photovoltaic devices, medicine, and biotechnology. Previous efforts for modeling dendrimer solutions have been orientated toward modifying the mean-field theory of Flory-Huggins.2-4 Other developments focus on the lattice cluster theory (LCT).5 Mio et al.3 used a simplified version of this theory with one adjustable parameter to correlate VLE data of dendrimer/solvent systems. Lieu et al.4 used a more complex version of LCT for similar correlations, again using one adjustable parameter. Jang et al.2 proposed a lattice model based
on LCT, which, using three adjustable parameters, takes into account the specific interactions encountered between solvent and end groups of the dendrimer. Earlier studies have been reviewed recently,3 and thus they are not repeated here. In this study, we investigate the predictive performance of two free-volume activity coefficient models for dendrimer solutions, namely the Unifac-FV and Entropic-FV models. Oishi and Prausnitz6 proposed the Unifac-FV model for polymer solutions. The activity coefficient is given by the following expression: res ln γi ) ln γcomb + ln γfv i i + ln γi
(1)
ln γcomb and ln γres account for the combinatorial and i i energetic effects, respectively, and are obtained from the Unifac7 group contribution method. The free-volume term:
(
ln γfv i ) 3c ln
V ˜ i1/3 - 1
V ˜m
1/3
-1
)
-c
[(
)(
V ˜i 1 - 1 1 - 1/3 V ˜m V ˜ i
)
-1
]
(2)
Here
Vi bVi,w
(2a)
∑ wiVi b∑ wiVi,w
(2b)
V ˜i ) and
* Corresponding author. Telephone: ++45252859. Fax: ++45882258. E-mail:
[email protected]. † Technical University of Denmark. ‡ Rheinisch-Westfa¨lische Technische Hochschule Aachen.
V ˜m )
10.1021/ie020034a CCC: $22.00 © 2002 American Chemical Society Published on Web 08/24/2002
Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4849
Figure 1. Schematic representation of the structure of the dendrimer PAMAM, generation 2.
where c ) 1.1 and b ) 1.28 for all solvents and polymers, is derived from the Flory equation of state.8 In the residual term, the revised by Hansen et al.9 temperature-independent parameters are employed. The Entropic-FV model is given by the expression:
ln γi ) ln γcomb-fv + ln γres i i
(3)
When Entropic-FV is applied to nonathermal systems, such as the dendrimer systems in this study, the same residual term as in the Unifac-FV model is used,6 in order to compare both models with the same number of parameters. The combinatorial-free volume expression used in the Entropic-FV model is somewhat simpler than that of Unifac-FV and has been originally suggested by Hildebrand10 and many years later put into a working form and tested by Elbro et al.:11
) ln ln γcomb-fv i
φfv φfv i i +1xi xi
(4)
Here
φfv i
)
xiVfv i
∑j
(4a)
xjVfv j
and
Vfv i ) Vi - Vi* ) Vi - Vi,w
(4b)
Equation 4 is essentially identical to the well-known Flory-Huggins expression,12-13 except that free-volume fractions are used instead of segment or volume fractions. The van der Waals volume (Vw) is estimated from the group increments of Bondi.14 Elbro et al.11 showed
Table 1. Absolute Percentage Deviation between Experimental and Predicted (by the van Krevelen Method) for Densities the AR and PAMAM Dendrimers at 20 °C dendrimer
dexp, g/cm3
g/cm3
dpred % deviation
d ) 1, % deviation
ARG-3 ARG-4 ARG-5 ARG-6 PAMAMG1 PAMAMG2 PAMAMG4
1.197 1.227 1.237 1.196 1.196 1.214 1.224
1.145 1.154 1.158 1.160 1.048 1.052 1.054
4 6 6 3 12 13 16
16 18 19 16 16 17 18
that the Entropic-FV model performs much better than Flory-Huggins (segment or volume-based) for nearly athermal polymer solutions. The purpose of this work is to evaluate the performance of the above two free-volume activity coefficient models in this new category of systems, as well as to estimate the influence of the dendrimer’s density in the performance of the models. Since experimental densities are usually not known, we are interested in evaluating the performance of the models based on predicted values of densities. Under this scope, the predictive capabilities of the van Krevelen method for estimating dendrimer densities are also evaluated. Furthermore, since many of the dendrimers have newly characterized groups in their molecules, for which not all the interaction parameters have been estimated, the influence of assigning zero values to all the missing interaction parameters is also investigated. 2. Database and Results The models Unifac-FV and Entropic-FV were evaluated against experimental data of certain dendrimer solutions at intermediate concentrations, obtained by Mio et al.3 and Lieu et al.4
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Table 2. Absolute Percentage Deviation (% ADD) between Predicted and Experimental Solvent Activitiesa Entropic-FV dexp ARG3-acetone ARG4-acetone ARG4-acetone ARG5-acetone ARG3-chloroform ARG4-chloroform ARG4-chloroform ARG5-chloroform ARG5-chloroform ARG3-toluene ARG4-toluene ARG4-toluene ARG5-toluene ARG4-tetrahydrofuran ARG5-tetrahydrofuran overall overall (SINC excluded) a
(g/cm3)
1.176 1.204 1.189 1.216 1.176 1.204 1.189 1.216 1.202 1.162 1.189 1.175 1.202 1.189 1.202
dpred
(g/cm3)
1.124 1.132 1.119 1.137 1.124 1.132 1.119 1.137 1.123 1.110 1.119 1.106 1.123 1.119 1.123
Unifac-FV
T (°C)
dexp
dpred
dexp
dpred
50 50 70 50 50 50 70 50 70 70 70 89 70 70 70
58b 34 31 34 82b 38 49 44 38 84b 30 12 36 4 48 41 30
61b 40 39 41 83b 43 54 49 44 86b 37 17 38 6 22 44 33
40b 25 10 23 78b 29 40 34 27 84b 27 12 34 13 33 34 24
52b 29 26 30 82b 40 53 47 42 87b 41 22 42 6 22 41 30
dpred is the density of the dendrimers predicted via the van Krevelen method. b SINC of the dendrimer.
Figure 2. Temperature dependence of experimental and predicted (via the van Krevelen method) densities of AR dendrimers.
Figure 4. Experimental and predicted activities of acetone in A4 with the Entropic-FV and the Unifac-FV models. Table 3. Absolute Percentage Deviation (% ADD) between Predicted and Experimental Solvent Activities
Figure 3. Experimental and predicted activities of methanol in PAMAM-G2 with the Entropic-FV and the Unifac-FV models.
The database used in this work includes the following: • Benzyl Ether dendrimers with aromatic termination ring (AR) of generations 3-5 with polar and nonpolar solvents. • Poly(amidoamine) (PAMAM) dendrimers of generations 1, 2, and 4 with polar solvents • A-series poly(imidoamine) dendrimers (A) of generation 4, with polar and nonpolar solvents. • Benzyl ether dendrimers with dodecyl alkane termination ring (C12) of generation 3 with polar and nonpolar solvents Both Unifac-FV and Entropic-FV require, as can be seen by eqs 2 and 4, the knowledge of the molar volumes of all the components of a system. For the solvents, these values were calculated from the correlations in the DIPPR database.15 For the dendrimers, since the performance of the models may be very sensitive to the value of the molar volume16 and since experimental data are not always available, three different approaches were considered:
A4-acetone A4-chloroform A4-hexane A4-heptane A4-octane A4-nonane overall C12G3-acetone C12G3-chloroform C12G3-cyclohexane C12G3-pentane C12G3-toluene overall
dpred (g/cm3)
T (°C)
0.963 0.963 0.954 0.954 0.954 0.954
50 50 65 65 65 65
0.952 0.952 0.946 0.958 0.941
50 50 60 40 70
Entropic-FV
Unifac-FV
13 11 19 18 27 40 21 25 20 8 19 15 17
12 14 22 21 31 43 24 23 24 14 19 8 18
a d pred is the density of the dendrimers predicted via the van Krevelen method.
1. In the papers of Lieu et al.4 and Tande et al.,17 all dendrimers were assumed to have a density of 1 g/cm3. The influence of this rather simplified assumption in the performance of the models is evaluated. 2. The second is use of a predictive method for the calculation of the molar volume such as the van Krevelen equation18
V ) Vw(1.3 + 10-3T)
(5)
where the van der Waals volume (Vw) is estimated from the group increments of Bondi14 or the GCVOL method19
V)
∑i ni(Ai + BiT + CiT2)
(6)
Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4851 Table 4. Absolute Average Deviation (% AAD) between Predicted and Experimental Solvent Activities for the PAMAM Dendrimers, with Missing Interaction Parameters Set to Zeroa Entropic-FV dexp
dpred
d ) 1 (g/cm3)
dexp
dpred
d ) 1 (g/cm3)
35 35 35
33 34 38 35 73 66 73 71 61 59 67 62 71 73 77 74 65 71 72 69
44 46 50 47 80 76 81 79 70 70 77 72 80 81 84 82 71 77 78 75
47 49 54 50 82 78 83 81 73 73 79 75 82 83 86 84 73 78 80 77
10 14 18 14 40 33 48 40 27 18 41 29 26 28 31 28 51 52 49 51
41 43 48 44 80 76 81 79 70 70 76 72 80 81 84 79 72 78 79 76
45 48 52 48 82 79 84 82 73 74 80 76 83 84 87 85 74 80 81 78
PAMAMG1-methanol PAMAMG2-methanol PAMAMG4-methanol overall PAMAMG1-propylamine PAMAMG2-propylamine PAMAMG4-propylamine overall PAMAMG1-acetone PAMAMG2-acetone PAMAMG4-acetone overall PAMAMG1-acetonitrile PAMAMG2-acetonitrile PAMAMG4-acetonitrile overall PAMAMG1-chloroform PAMAMG2-chloroform PAMAMG4-chloroform overall a
Unifac-FV
T (°C)
35 35 35 35 35 35 40 40 40 35 35 35
dpred is the density of the dendrimers predicted via the van Krevelen method.
where the Ai, Bi, Ci parameters are taken from the GCVOL parameter table.19 For the dendrimers studied here, however, not all necessary group parameters are included in the GCVOL table. The evaluation is thus limited to the van Krevelen method. 3. The third is use of the experimental molar volume of the dendrimer, but such data are scarce and limited to few types of dendrimers. The experimental density data for PAMAM dendrimers provided by Uppuluri et al.1 and for AR dendrimers provided by Hay et al.20 are used in this work in order to evaluate both the effect of density on the predictions of both activity coefficient models, as well as the accuracy of the van Krevelen method in predicting the density of these systems. The performance of the van Krevelen method is shown in Table 1 and Figure 2. The results of the evaluation, for both predicted and experimental density of the dendrimers (when the latter are available), are presented in Figures 3 and 4 and Tables 2-4, in the form of absolute average percent deviations (% AAD) between the calculated and the experimental activities:
% AAD ) 100
(
)
|R exp - Rcal| R exp
(7)
The values of the experimental solvent activities were obtained by the relation (assuming ideal vapor phase)
Ri )
P Ps
(8)
where Ps is the saturated pressure and P the pressure of the system. 3. Discussion On the basis of the results shown in Tables 1-4 and Figures 2-4, the following points summarize our conclusions: 1. Prediction of the Dendrimer Density. The predicted densities with the van Krevelen method are
in the area between unity and the experimental density of the PAMAM and AR dendrimers (Table 1). The deviation from the experimental values increases for increasing generation number. This could be attributed to the characteristic structure of dendrimers, which, unlike other polymers, are flexible and allow a more compact packing. However, although the density of the AR dendrimer increases from G-3 to G-5, it suddenly decreases for G-6. This is attributed by the authors20 to the fact that a structural rearrangement occurs on an increase in the dendrimer molecular weight and hence the generation number. This packing is much closer to the packing of linear polymers and this may be the reason the van Krevelen method can predict the density in this case with much accuracy. From the plot of the dendrimer’s volume against temperature (Figure 2), we see that the van Krevelen method overestimates the volume, but has the same T dependence as the experimental data. The assumption that the density is equal to unity is, thus, a rather crude simplification. The use of a predictive tool, such as the van Krevelen equation, seems more appropriate. 2. Sensitivity of the Models to the Value of Density. The Unifac-FV model exhibits higher sensitivity to the value of the density than Entropic-FV, as can be seen in Tables 2 and 4 and Figures 3 and 4. In the case of Entropic-FV, the results are quite similar when the predicted density or a value equal to unity is employed. They are improved when the value of the experimental density is employed. This behavior permits the application of the model for predicting the solvent activities in systems where the density of the dendrimer is not known. 3. Influence of the Dendrimer’s Generation Number. There is no influence of the dendrimer’s generation number on the performance of the models. The differences in the deviation among systems with the same solvent and dendrimer of different generation number are mainly due to the different concentration range of the experimental data. As expected, when the concentration of the solvent in the system approaches the
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infinite dilution region, maximum nonideality is encountered and higher deviations are expected. 4. Solvent-Induced Crystallization. Because of solvent-induced crystallization (SINC) phenomena present in some of the studied systems (ARG3/acetone, ARG3/chloroform, ARG3/toluene), the dendrimer rejects the solvent, when the solvent mass fraction exceeds a certain value, leading to a two-phase liquid system.3 The models cannot predict this behavior and this leads to higher deviations than the models’ average performance, as can be seen in Table 2. 5. Polar Solvents. The predictions of the solvent activity in systems with a nonpolar solvent are more accurate than in systems with polar solvents. This could be due to interactions (e.g., hydrogen bonding) present in polar systems, which are not explicitly taken into account in the Entropic-FV and Unifac-FV models. 6. Influence of Missing Interaction Parameters. Table 4 presents systems where experimental densities are known but some of the interaction parameters between the groups of the solute and the solvent are missing (e.g., CHCl3-CON, CHOH-CH2NH2) and were assigned here the value of zero. The results show that there is an increase in the deviations from the experimental data, possibly due to the missing interaction parameters. 4. Conclusions Dendrimers constitute a new type of polymeric molecules, where the predictive performance of free-volume models that use the group contribution approach has not been extensively evaluated so far. In this work, we show that the Entropic-FV and the Unifac-FV models, with no extra fitting parameters, can predict the solvent activity in dendrimer systems with acceptable accuracy in many cases. However, due to the special structure of dendrimers, dependable tools such as the van Krevelen method are necessary for the prediction of their molar volume, which is a parameter that influences significantly the overall performance of free-volume models. Furthermore, the Unifac-FV model is more sensitive to the value of density than the Entropic-FV model, which indicates deficiencies in these cases of the free-volume expression derived from the Flory equation of state. Adjusting the b and c parameters appearing in the freevolume term of the Unifac-FV model may improve the performance upon sacrificing the predictive character of the model. 5. List of Symbols Ai, Bi, C ) GCVOL parameters b, c ) parameters in Unifac-FV d ) density P ) pressure T ) temperature V ) molar volume V ˜ ) reduced volume V* ) hard-core volume w ) weight fraction x ) mole fraction Greek Letters R ) activity γ ) mole based activity coefficient φ ) (volume/segment) fraction
Abbreviations A ) A-series poly(imidoamine) dendrimer AAD) absolute average deviation AR ) benzyl ether dendrimer with aromatic termination ring C12 ) benzyl ether dendrimer with dodecyl alkane termination ring FV ) free volume G ) generation LCT ) lattice cluster theory PAMAM ) poly(amidoamine) dendrimer SINC ) solvent-induced crystallization Unifac ) universal functional activity coefficient VLE ) vapor-liquid equilibria Subscripts cal ) calculated value exp ) experimental value i ) component index m ) mixture pred ) predicted s ) saturated w ) van der Waals Superscripts comb ) combinatorial comb-fv ) combinatorial-free volume fv ) free volume res) residual
Acknowledgment The authors wish to thank the J. C. Hempel foundation and the Danish Research Council for financial support of this work in the framework of a grant entitled: “Thermodynamic properties of polymer solutions related to paints and coatings”. Literature Cited (1) Uppuluri, S.; Keinath, S. E.; Tomalia, D. A. S.; Dvornic, P. R. Rheology of Dendrimers. I. Newtonian flow Behavior of Medium and Highly Concentrated Solutions of Polyamidoamine (PAMAM) Dendrimers in Ethylenediamine (EDA) Solvent. Macromolecules 1998, 31, 4498. (2) Jang, J. G.; Bae, Y. C. Vapor-liquid Equilibria of Dendrimer Solutions: the Effect of Endgroups at the Periphery of Dendrimeric Molecules. Chem. Phys. 2001, 285. (3) Mio, C.; Kiritsov, S.; Thio, Y.; Brafman, R.; Prausnitz, J.; Hawker, C.; Malmstrøm, E. E. Vapor-Liquid Equilibria for Solutions of Dendritic Polymers. J. Chem. Eng. Data 1998, 43, 541. (4) Lieu, J. G.; Liu, M.; Frechet, J. M. J.; Prausnitz, J. M. VaporLiquid Equilibria for Dendritic-Polymer Solutions. J. Chem. Eng. Data 1999, 44, 613. (5) Nemirovsky, A. M.; Bawendi, M. G.; Freed, K. F. Lattice Models of Polymer Solutions: Monomers Occupying Several Lattice Sites. J. Chem. Phys. 1987, 87, 12, 7272. (6) Oishi, T.; Prausnitz, M. Estimation of Solvent Activities in Polymer Solutions Using a Group-Contribution Methodol. Ind. Eng. Chem., Process Des. Dev. 1978, 17, 3, 333. (7) Fredenslund, Aa.; Jones, R. L.; Prausnitz, J. M. GroupContribution Estimation of Activity Coefficients in Nonideal Liquid Mixtures. AIChE J. 1975, 21, 1086. (8) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid Phase Equilibria, 3rd ed.; Prentice Hall International: Englewood Cliffs, NJ, 1999. (9) Hansen, H. K.; RasmuEssen, P.; Fredeslund, Aa.; Schiller, M.; Gmehling, J. Vapor-Liquid Equilibria by UNIFAC Group Contribution. 5. Revision and Extension. Ind. Eng. Chem. Res. 1991, 30, 2352.
Ind. Eng. Chem. Res., Vol. 41, No. 19, 2002 4853 (10) Hildebrand, J. H.; Scott, R. L. The Solubility of Nonelectrolytes; Dover: New York, 1964. (11) Elbro, H. S.; Fredenslund, Aa; Rasmussen, P. New Simple Equation for the Prediction of Solvent Activities in Polymer Solutions. Macromolecules 1990, 23, 4707. (12) Flory, P. J. Thermodynamics of High Polymer Solutions. J. Chem. Phys. 1941, 9, 660. (13) Huggins, M. L. Solutions of Long Chain Compounds. J. Chem. Phys. 1941, 9, 440. (14) Bondi, A. Physical Properties of Molecular Crystals, Liquids and Glasses; Wiley and Sons: New York, 1968. (15) Danner, R. P.; High, M. S. Handbook of Polymer Solution Thermodynamics; Design Institute for Physical Property Data, AIChE: New York, 1993. (16) Pappa, G. D.; Voutsas, E. C.; Tassios, D. P. Prediction Of Activity Coefficients in Polymer and Copolymer Solutions Using Simple Activity Coefficient Models. Ind. Eng. Chem. Res.. 1999, 38, 4975. (17) Tande, B. M.; Deitcher, R. W., Jr.; Sandler, S. I.; Wagner, N. J. Unifac-FV Applied to Dendritic Macro-
molecules in Solution: Comment on ‘Vapor-Liquid Equilibria for Dedritic-Polymer Solutions’. J. Chem. Eng. Data 2002, 47, 2, 376. (18) Van Krevelen, D. W. Properties of polymers. Their correlation with chemical structure; their numerical estimation and prediction from additive group contributions; Elsevier: Amsterdam, 1990. (19) Elbro, H. S.; Fredenslund, Aa.; Rasmussen, P. Group Contribution Method for the Prediction of Liquid Densities as a Function of Temperature for Solvents, Oligomers and Polymers. Ind. Eng. Chem. Res. 1991, 30, 2576. (20) Hay, G.; Mackay, E.; Hawker, C. J. Thermodynamic Properties of Dendrimers Compared with Linear Polymers: General Observations. J. Polym. Sci. 2001, 39, 1766.
Received for review January 10, 2002 Revised manuscript received July 9, 2002 Accepted July 17, 2002 IE020034A
