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Phase equilibria of carbohydrates: the study of a series of glucose oligomers from glucose to maltopentaose in aqueous solution
Fluid Phase Equilibria 194–197 (2002) 947–956
Phase equilibria of carbohydrates: the study of a series of glucose oligomers from glucose to maltopentaose in aqueous solution Experimental versus predicted data using various UNIQUAC/UNIFAC models S.A. Cooke a , S.Ó. Jónsdóttir a,∗ , P. Westh b a
b
The Department of Chemistry, Technical University of Denmark, DK-2800 Lyngby, Denmark The Department of Life Sciences and Chemistry, Roskilde University, DK-4000 Roskilde, Denmark Received 9 March 2001; accepted 29 August 2001
Abstract Vapour pressures above solutions of glucose (Glc) in water and maltose in water at both 298.06 and 317.99 K have been measured. Vapour pressures above aqueous solutions of maltotriose, maltotetraose and maltopentaose at 317.99 K have also been measured. The experimental results are compared with predictions from two existing UNIFAC type models available in the literature. Further to this a theoretical model is examined in which existing interaction parameters, calculated for the 1,2-ethanediol/water system using a molecular mechanical approach, are incorporated into the UNIQUAC equation to describe the vapour pressures of the aforementioned series of ␣-(1–4) linked Glc oligosaccharides in aqueous solution. This so-called transference principal is found to be adequate for these systems when used in conjunction with an adjustment to the coordination number used for the lattice based model. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Vapour pressure; Model; Data; Glucose oligomers
1. Introduction The importance and prevalence of starch in the natural world cannot easily be overstated. Starch is a high molecular weight homo-polysaccharide, which is built from a series of ␣-(1–4) linked glucose (Glc) units. A statistical thermodynamic model describing the aqueous behaviour of starch would be of considerable interest to researchers involved in a broad range of disciplines and also to a variety of industries. At present the complexity of the problem and lack of experimental data precludes the successful development of such a model. A reasonable route forward, however, involves undertaking a study of the ∗
Corresponding author. Tel.: +45-4525-2383; fax: +45-4588-3136. ´ J´onsd´ottir). E-mail address: [email protected] (S.O. 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 6 8 9 - 6
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thermodynamic properties of aqueous Glc solutions and then systematically increasing the chain length of the saccharide to include aqueous solutions of short chain ␣-(1–4) linked Glc oligomers and hence, step by step, gain an insight into the aqueous behaviour of starch itself. With respect to the vapour liquid equilibrium of Glc and maltose, several models are at present available in the literature. The majority of these are UNIQUAC [1,2] or UNIFAC [3] type models, which are based on expressions for the excess Gibbs free energy. These models are advantageous due to the requirement for only a small number of adjustable parameters to be known, together with structural parameters that may be readily calculated or read from tables. Four different models based on the UNIFAC approach can be found in the literature. Two of the most recent UNIFAC type models, together with a third, a model having a theoretical origin, are discussed in this paper. The predictions arising from these models are compared to new experimental data. The first of these models has been proposed by Peres and Macedo [4] and is referred to herein as P&M UNIFAC. Here the modified group contribution UNIFAC model of Larsen et al. [5] is employed. In this model four new groups are defined these are a pyranose group, a furanose group, an osidic bond group and a ring hydroxyl group. Secondly, the S-UNIFAC model (S for saccharide) has been proposed by Spiliotis and Tassios [6]. This model is based on the LLE version of UNIFAC [7] and introduces two new main groups with several sub-groups. The first main group introduced is CHOHsugar , this main group has sub-groups describing axial hydroxyl groups, CHOHax , and equatorial hydroxyl groups, CHOHeq , along with the primary alcohol group, CH2 OH. The second new main group is the CH–O–CH group describing the osidic bond. A third model of interest, but not considered here, has been proposed by Catté et al. [8]. Here an approach that endeavours to take into account the equilibrium between different sugar conformers in solution is developed. The physical part of the model is based on the modified group contribution UNIFAC model of Larsen et al. [5]. Whilst the benefits of such an approach are clearly evident the absence of appropriate information concerning the conformational equilibrium of the oligosaccharide solutions of interest, means that this approach is not feasible in this study. A fourth UNIFAC type model for calculating the activity coefficients of biological compounds dissolved in water alone, or in aqueous solution containing salt, was proposed by Kuramochi et al. [9]. We refer to this model as bio-UNIFAC. In this model, which covers both carbohydrates and amino acids, a long range Pitzer–Debye–Hückel term was added to the original UNIFAC equation. New groups were defined within this model. With respect to sugars these new groups included whole monosaccharides such as Glc and fructose (Fru). The authors state explicitly that the UNIFAC model may not be satisfactory for the higher molecular mass solutes considered in this study. All of the above models have been demonstrated to adequately predict a variety of thermodynamic properties over a moderate temperature range for aqueous solutions of a number of mono and disaccharides. Despite these successes the numerous and necessary modifications to the UNIFAC and UNIQUAC models together with the apparent requirement for more and more new groups to be defined for each problem highlights the phenomenological origin and empirical tendencies of these methods. An alternative approach has been developed by Jónsdóttir et al. [10] and is explored further in this paper [11,12]. Here, a molecular mechanical approach is used to access the two interaction parameters of the original UNIQUAC equation; this equation is then employed to predict phase equilibria properties of a variety of mixtures. A further development in this model involves the applicability of a so-called ‘transference principle’ where interaction parameters, calculated for a more simple system, are used to describe a more complicated, related system. This approach has been used to good effect for a variety of glycols and small saccharides using interaction parameters calculated for the related
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water/1,2-ethanediol system [11,12]. The method is discussed further below. An interesting question concerns whether this simple approach can be used to predict vapour liquid equilibria properties of aqueous oligosaccharides? As part of this study, vapour pressures of water above solutions of varying concentrations of either Glc or maltose at 298.06 and 317.99 K have been recorded. Further to this, vapour pressures of water above solutions of either maltotriose, maltotetraose and maltopentaose have been measured at 317.99 K.
2. Methods Vapour pressures were measured using an electronic barometer system (Barocel 570 capacitance pressure sensor, Edwards high vacuum, Crawley, UK) connected to an electronic bench top manometer (Edwards high vacuum, Crawley, UK). The technique involves the measurement of a differential pressure between pure water in a reference cell and the solution of interest in the sample cell both housed in a thermostatically controlled water bath. The accuracy of the measurements is estimated to be within 0.5 kPa. d(+)-Glc monohydrate, d(+)-maltose monohydrate, maltotriose, maltotetraose and maltopentaose were purchased from Fluka and despite catalogue lists indicating purities as low as 95% consultation of the actual lot certificate of analysis indicated purities of >99.5, >99, >99, >99.4 and >99.4%, respectively. In-house FORTRAN programs were written to process the experimental data and make use of the P&M UNIFAC and S-UNIFAC models, respectively. 2.1. The theoretical model In the original work by Jónsdóttir et al. UNIQUAC interaction parameters for a binary mixture were determined by adhering to the following process. Initially, equilibrium structures for all conformers for both molecules are determined by potential energy minimization. Secondly, interaction energies are determined for molecular complexes formed between molecules of (a) type 1 and type 1, (b) type 2 and type 2 and (c) between molecules of type 1 and type 2. Inherent in this second step is a criteria for the sampling of conformational space for the molecular complex this is provided via the Boltzman Jump Procedure which is based on a Monte Carlo technique, followed by an energy minimization. From these energies UNIQUAC interaction parameters are then calculated. The development and details of this method can be found elsewhere [10–12]. In this work, the so-called ‘transference principal’ is of particular interest. Maltose, the prototype ␣-(1–4) linked Glc oligomer, has 59049 possible conformers for every chosen orientation of the glycosidic linkage, this vast number of possible conformers indicates the complexity of a ‘simple’ disaccharide to the molecular modeller [13]. The benefits of taking interaction energies calculated for a smaller system with less degrees of freedom and ‘transferring’ it to a more complicated system are clearly evident. The validity of this approximation has been evidenced by the successful prediction of vapour liquid equilibria data via the UNIQUAC equation for aqueous solutions of 1,2-propandiol, glycerol, Glc and sucrose using interaction parameters, calculated using the molecular mechanical method outlined above, for the water/1,2-ethanediol system [11,12]. The model also shows good predictive ability for several other aqueous polyol systems [14]. It may be postulated that the success of the transference principle in these cases is a consequence of the difference in size of the solute/solvent molecules. Hence, the portion
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Table 1 Showing the structural parameters used in the UNIQUAC equation Molecule
ri
qi
Water Glucose Maltose Maltotriose Maltotetraose Maltopentaose
0.9200 5.8028 10.7985 15.7762 20.7629 25.7496
1.400 4.840 8.752 12.664 16.576 20.488
of the larger molecule with which the water interacts closely resembles the 1,2-ethanediol molecule, a similar approach to considering 1,2-ethanediol to be a repeat unit in a polymer. As mentioned above, vapour pressures are calculated in this theoretical model using the original UNIQUAC equation, to give the activity coefficient of water, which is then used in turn in the modified Raoult’s law expression P = P1sat x1 γ1 where x1 , γ 1 and P1sat are the mole fraction, the activity coefficient and vapour pressure of pure water respectively. This relationship gives the vapour pressure above the solution with the reasonable assumption that Glc and the higher oligomers have negligible vapour pressures and the water vapour behaves as an ideal gas. The structural parameters, ri and qi , used in this work are given in Table 1 and calculated using Bondi’s method [15]. For all systems the interaction parameters A12 = 396 and A21 = −334 K are used in this theoretical model.
Fig. 1. Water/glucose at 298.06 K.
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Fig. 2. Water/glucose at 317.99 K.
3. Results and model calculations Figs. 1–7 show the experimental data recorded for aqueous solutions of Glc, maltose, maltotriose, maltotetraose and maltopentaose at the stated temperatures. Error bars, although small, are indicated on the data points, a more rigorous treatment of this data together with thermodynamic data for several other saccharides in aqueous solution can be found elsewhere [16]. It was found that measurements of vapour pressures could be taken well into the supersaturated region where solubility data was known
Fig. 3. Water/maltose at 298.06 K.
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Fig. 4. Water/maltose at 317.99 K.
[17,18]. Measurements within the supersaturated region are believed to occur for all of the systems studied here even where solubility data is not known. The maltotriose, maltotetraose and maltopentaose systems were measured only at the higher temperature, increasing the solubility of the sugars and hence providing more data points with which to test the various models. On each plot the predictions obtained from P&M UNIFAC and S-UNIFAC are also given. Also on each plot the results of the theoretical model are given both with the coordination number traditionally employed in the UNIQUAC model, i.e. z = 10, and also with the coordination number altered in a method discussed below. Other workers have
Fig. 5. Water/maltotriose at 317.99 K.
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Fig. 6. Water/maltotetraose at 317.99 K.
previously measured vapour pressures above aqueous solutions of Glc and maltose [17,18] and whilst their results are not shown in these figures, these new data are in good accord with their findings. All of the systems exhibit a negative deviation from ideal behaviour, in good accordance with other experimental observations and indicative of the strong interactions that occur between the sugars and the surrounding water. Comparison of activity coefficients indicates that the deviation from ideal behaviour increases with increasing molecular size of the solute, as one would expect.
Fig. 7. Water/maltopentaose at 317.99 K.
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4. Discussion Vapour pressures above aqueous solutions of Glc and maltose at 298.06 and 317.99 K and vapour pressures above aqueous solutions of maltotriose, maltotetraose and maltopentaose at 317.99 K have successfully been recorded. The experimental data is presented, as are the predicted values from the P&M UNIFAC, S-UNIFAC and theoretical models. The S-UNIFAC model predicts the vapour pressures of aqueous solutions of Glc at 298.06 and 317.99 K very well. For the maltose and higher oligomer systems this model does not perform well. These results are probably a consequence of the osidic bond group interaction parameters in S-UNIFAC having been developed around just one disaccharide molecule, namely sucrose. Expectation that this method, in its current form, should perform well for oligosaccharides containing as many as four osidic bond groups in different environments to that of sucrose is therefore optimistic. The requirement for a model which distinguishes between the many carbohydrate isomers is, however, very necessary. The P&M UNIFAC model performs very well for the aqueous Glc and maltose systems at 298.06 K. At the higher temperature of 317.99 K and for the oligosaccharide systems predictions using this model are satisfactory only within the very dilute regions with the exception of the water/maltopentaose system. Again it should be pointed out that P&M UNIFAC has not been developed for the prediction of properties of aqueous solutions of oligosaccharides (in fact no literature method has) and advantages of P&M UNIFAC, S-UNIFAC and, although not shown, bio-UNIFAC include their successful representation of binary, ternary and even quaternary mixtures of mono and disaccharides. Figs. 1–7 show that an important factor in the success of the theoretically based model is the value chosen for the coordination number, z. In the UNIQUAC equation the coordination number of the lattice represents the number of nearest neighbours to each segment. Further to this, the product of the pure component area parameter, q, and z gives the number of external contacts per molecule. It is perhaps surprising that in the UNIQUAC model z is often fixed at a value of 10, with reasonable alternatives seldom being used (6 ≤ z ≤ 12) [1]. Figs. 1–7 show the out comes of using the traditional value of z = 10 in the theoretical model. For the water/Glc system at 298.06 K and the water/maltose system at 317.99 K the model performs very well. For the other systems predictions are only satisfactory in the dilute regions. The performance of this model is in most cases similar to the performance of the P&M UNIFAC model. Altering the z number to values indicated in the respective figures does give a significant improvement. It is true that the coordination number, z, may well be taking on the rôle of a fitting constant in the theoretical model, however, the following trends should be considered. The coordination number is required to decrease with an increase in temperature possibly consistent with the lattice becoming more diffuse. Not immediately obvious is the requirement for the coordination number to increase from 8 in the water/Glc system through to 11 in the water/maltotriose system and decreasing to 10 in the water/maltopentaose system. It is possible that the conformational behaviour of the oligosaccahrides and the effect of this on the structure of the water are involved in the requirement for this trend. Given the complexity of the systems that are studied in this work, the performance of all the models in this work indicates the advances that have been made to both the UNIFAC and UNIQUAC methods since its original proposal. However, it is clear that further model development is required specifically addressing many of the assumptions made in these models such as the coordination number.
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5. Conclusion In this paper the predictive properties of two UNIFAC models and a theoretically based UNIQUAC model have been examined for a series of ␣-(1–4) linked Glc oligosaccharides in aqueous solution. The S-UNIFAC model in its present form cannot usefully be applied to theses systems whereas the P&M UNIFAC model has predictive ability only within the very dilute regions. The simple theoretical model used in this work can be shown to adequately reproduce vapour liquid equilibrium data for the systems studied here. However, it is clear that a satisfactory statistical thermodynamic model describing aqueous solutions of long chain saccharides is unlikely to be provided by, or via, the over-simplified UNIQUAC or UNIFAC methods in their current form. This work suggests that a lattice based model may be able to be utilised in such a project with the addendum that the coordination number of the lattice is likely to be a function of several variables including the temperature of the system and the molecular size and shape of both the solute and solvent. List of symbols Aij interaction parameter P vapour pressure qi surface area parameter ri volume parameter xi mole fraction γi activity coefficient z coordination number Acknowledgements The financial support for this project has been provided by the Danish Technical Research Council and is greatly appreciated. We also gratefully acknowledge the support of the Novo Nordisk Foundation and the Danish Natural Science Research Council, and we would like to thank Peter Rasmussen and Kjeld Rasmussen for many useful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
D.S. Abrams, J.M. Prausnitz, AlChE J. 21 (1) (1975) 116–128. G. Maurer, J.M. Prausnitz, Fluid Phase Equilib. 2 (1978) 91–99. Aa, Fredenslund, J. Gmehling, P. Rasmussen, Vapour–Liquid Equilibra Using UNIFAC, Elsevier, New York, 1977. A.M. Peres, E.A. Macedo, Fluid Phase Equilib. 139 (1997) 47–74. B.L. Larsen, P. Rasmussen, Aa. Fredenslund, Ind. Eng. Chem. Res. 26 (1987) 2274–2286. N. Spiliotis, D. Tassios, Fluid Phase Equilib. 173 (2000) 39–55. T.F. Anderson, J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev. 17 (4) (1978) 561–567. M. Catté, C.G. Dussap, J.B. Gros, Fluid Phase Equilib. 105 (1995) 1–25. H. Kuramochi, H. Noritomi, D. Hoshino, K. Nagahama, Fluid Phase Equilib. 130 (1997) 117–132. S.Ó. Jónsdóttir, K. Rasmussen, A. Fredenslund, Fluid Phase Equilib. 100 (1994) 121–138. S.Ó. Jónsdóttir, R.A. Klein, Fluid Phase Equilib. 132 (1997) 117–137. S.Ó. Jónsdóttir, P. Rasmussen, Fluid Phase Equilib. 158/160 (1999) 411–418. K.J. Naidoo, J.W. Brady, Chem. Phys. 224 (1997) 263–273.
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S.A. Cooke et al. / Fluid Phase Equilibria 194–197 (2002) 947–956 S.Ó. Jónsdóttir, S.A. Cooke, in preparation. A. Bondi, Physical Properties of Crystal, Liquids and Glasses, Wiley, New York, 1968. S.A. Cooke, S.Ó. Jónsdóttir, P. Westh, J. Phys. Chem. B, submitted. J.B. Taylor, J.S. Rowlison, Trans. Faraday Soc. 51 (1955) 1183–1192. M. Ruegg, B. Blanc, Lebebensm. Wiss.u.-Technol. 14 (1981) 1–6.
Phase equilibria of carbohydrates: the study of a series of glucose oligomers from glucose to maltopentaose in aqueous solution Experimental versus predicted data using various UNIQUAC/UNIFAC models S.A. Cooke a , S.Ó. Jónsdóttir a,∗ , P. Westh b a
b
The Department of Chemistry, Technical University of Denmark, DK-2800 Lyngby, Denmark The Department of Life Sciences and Chemistry, Roskilde University, DK-4000 Roskilde, Denmark Received 9 March 2001; accepted 29 August 2001
Abstract Vapour pressures above solutions of glucose (Glc) in water and maltose in water at both 298.06 and 317.99 K have been measured. Vapour pressures above aqueous solutions of maltotriose, maltotetraose and maltopentaose at 317.99 K have also been measured. The experimental results are compared with predictions from two existing UNIFAC type models available in the literature. Further to this a theoretical model is examined in which existing interaction parameters, calculated for the 1,2-ethanediol/water system using a molecular mechanical approach, are incorporated into the UNIQUAC equation to describe the vapour pressures of the aforementioned series of ␣-(1–4) linked Glc oligosaccharides in aqueous solution. This so-called transference principal is found to be adequate for these systems when used in conjunction with an adjustment to the coordination number used for the lattice based model. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Vapour pressure; Model; Data; Glucose oligomers
1. Introduction The importance and prevalence of starch in the natural world cannot easily be overstated. Starch is a high molecular weight homo-polysaccharide, which is built from a series of ␣-(1–4) linked glucose (Glc) units. A statistical thermodynamic model describing the aqueous behaviour of starch would be of considerable interest to researchers involved in a broad range of disciplines and also to a variety of industries. At present the complexity of the problem and lack of experimental data precludes the successful development of such a model. A reasonable route forward, however, involves undertaking a study of the ∗
Corresponding author. Tel.: +45-4525-2383; fax: +45-4588-3136. ´ J´onsd´ottir). E-mail address: [email protected] (S.O. 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 6 8 9 - 6
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thermodynamic properties of aqueous Glc solutions and then systematically increasing the chain length of the saccharide to include aqueous solutions of short chain ␣-(1–4) linked Glc oligomers and hence, step by step, gain an insight into the aqueous behaviour of starch itself. With respect to the vapour liquid equilibrium of Glc and maltose, several models are at present available in the literature. The majority of these are UNIQUAC [1,2] or UNIFAC [3] type models, which are based on expressions for the excess Gibbs free energy. These models are advantageous due to the requirement for only a small number of adjustable parameters to be known, together with structural parameters that may be readily calculated or read from tables. Four different models based on the UNIFAC approach can be found in the literature. Two of the most recent UNIFAC type models, together with a third, a model having a theoretical origin, are discussed in this paper. The predictions arising from these models are compared to new experimental data. The first of these models has been proposed by Peres and Macedo [4] and is referred to herein as P&M UNIFAC. Here the modified group contribution UNIFAC model of Larsen et al. [5] is employed. In this model four new groups are defined these are a pyranose group, a furanose group, an osidic bond group and a ring hydroxyl group. Secondly, the S-UNIFAC model (S for saccharide) has been proposed by Spiliotis and Tassios [6]. This model is based on the LLE version of UNIFAC [7] and introduces two new main groups with several sub-groups. The first main group introduced is CHOHsugar , this main group has sub-groups describing axial hydroxyl groups, CHOHax , and equatorial hydroxyl groups, CHOHeq , along with the primary alcohol group, CH2 OH. The second new main group is the CH–O–CH group describing the osidic bond. A third model of interest, but not considered here, has been proposed by Catté et al. [8]. Here an approach that endeavours to take into account the equilibrium between different sugar conformers in solution is developed. The physical part of the model is based on the modified group contribution UNIFAC model of Larsen et al. [5]. Whilst the benefits of such an approach are clearly evident the absence of appropriate information concerning the conformational equilibrium of the oligosaccharide solutions of interest, means that this approach is not feasible in this study. A fourth UNIFAC type model for calculating the activity coefficients of biological compounds dissolved in water alone, or in aqueous solution containing salt, was proposed by Kuramochi et al. [9]. We refer to this model as bio-UNIFAC. In this model, which covers both carbohydrates and amino acids, a long range Pitzer–Debye–Hückel term was added to the original UNIFAC equation. New groups were defined within this model. With respect to sugars these new groups included whole monosaccharides such as Glc and fructose (Fru). The authors state explicitly that the UNIFAC model may not be satisfactory for the higher molecular mass solutes considered in this study. All of the above models have been demonstrated to adequately predict a variety of thermodynamic properties over a moderate temperature range for aqueous solutions of a number of mono and disaccharides. Despite these successes the numerous and necessary modifications to the UNIFAC and UNIQUAC models together with the apparent requirement for more and more new groups to be defined for each problem highlights the phenomenological origin and empirical tendencies of these methods. An alternative approach has been developed by Jónsdóttir et al. [10] and is explored further in this paper [11,12]. Here, a molecular mechanical approach is used to access the two interaction parameters of the original UNIQUAC equation; this equation is then employed to predict phase equilibria properties of a variety of mixtures. A further development in this model involves the applicability of a so-called ‘transference principle’ where interaction parameters, calculated for a more simple system, are used to describe a more complicated, related system. This approach has been used to good effect for a variety of glycols and small saccharides using interaction parameters calculated for the related
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water/1,2-ethanediol system [11,12]. The method is discussed further below. An interesting question concerns whether this simple approach can be used to predict vapour liquid equilibria properties of aqueous oligosaccharides? As part of this study, vapour pressures of water above solutions of varying concentrations of either Glc or maltose at 298.06 and 317.99 K have been recorded. Further to this, vapour pressures of water above solutions of either maltotriose, maltotetraose and maltopentaose have been measured at 317.99 K.
2. Methods Vapour pressures were measured using an electronic barometer system (Barocel 570 capacitance pressure sensor, Edwards high vacuum, Crawley, UK) connected to an electronic bench top manometer (Edwards high vacuum, Crawley, UK). The technique involves the measurement of a differential pressure between pure water in a reference cell and the solution of interest in the sample cell both housed in a thermostatically controlled water bath. The accuracy of the measurements is estimated to be within 0.5 kPa. d(+)-Glc monohydrate, d(+)-maltose monohydrate, maltotriose, maltotetraose and maltopentaose were purchased from Fluka and despite catalogue lists indicating purities as low as 95% consultation of the actual lot certificate of analysis indicated purities of >99.5, >99, >99, >99.4 and >99.4%, respectively. In-house FORTRAN programs were written to process the experimental data and make use of the P&M UNIFAC and S-UNIFAC models, respectively. 2.1. The theoretical model In the original work by Jónsdóttir et al. UNIQUAC interaction parameters for a binary mixture were determined by adhering to the following process. Initially, equilibrium structures for all conformers for both molecules are determined by potential energy minimization. Secondly, interaction energies are determined for molecular complexes formed between molecules of (a) type 1 and type 1, (b) type 2 and type 2 and (c) between molecules of type 1 and type 2. Inherent in this second step is a criteria for the sampling of conformational space for the molecular complex this is provided via the Boltzman Jump Procedure which is based on a Monte Carlo technique, followed by an energy minimization. From these energies UNIQUAC interaction parameters are then calculated. The development and details of this method can be found elsewhere [10–12]. In this work, the so-called ‘transference principal’ is of particular interest. Maltose, the prototype ␣-(1–4) linked Glc oligomer, has 59049 possible conformers for every chosen orientation of the glycosidic linkage, this vast number of possible conformers indicates the complexity of a ‘simple’ disaccharide to the molecular modeller [13]. The benefits of taking interaction energies calculated for a smaller system with less degrees of freedom and ‘transferring’ it to a more complicated system are clearly evident. The validity of this approximation has been evidenced by the successful prediction of vapour liquid equilibria data via the UNIQUAC equation for aqueous solutions of 1,2-propandiol, glycerol, Glc and sucrose using interaction parameters, calculated using the molecular mechanical method outlined above, for the water/1,2-ethanediol system [11,12]. The model also shows good predictive ability for several other aqueous polyol systems [14]. It may be postulated that the success of the transference principle in these cases is a consequence of the difference in size of the solute/solvent molecules. Hence, the portion
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Table 1 Showing the structural parameters used in the UNIQUAC equation Molecule
ri
qi
Water Glucose Maltose Maltotriose Maltotetraose Maltopentaose
0.9200 5.8028 10.7985 15.7762 20.7629 25.7496
1.400 4.840 8.752 12.664 16.576 20.488
of the larger molecule with which the water interacts closely resembles the 1,2-ethanediol molecule, a similar approach to considering 1,2-ethanediol to be a repeat unit in a polymer. As mentioned above, vapour pressures are calculated in this theoretical model using the original UNIQUAC equation, to give the activity coefficient of water, which is then used in turn in the modified Raoult’s law expression P = P1sat x1 γ1 where x1 , γ 1 and P1sat are the mole fraction, the activity coefficient and vapour pressure of pure water respectively. This relationship gives the vapour pressure above the solution with the reasonable assumption that Glc and the higher oligomers have negligible vapour pressures and the water vapour behaves as an ideal gas. The structural parameters, ri and qi , used in this work are given in Table 1 and calculated using Bondi’s method [15]. For all systems the interaction parameters A12 = 396 and A21 = −334 K are used in this theoretical model.
Fig. 1. Water/glucose at 298.06 K.
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Fig. 2. Water/glucose at 317.99 K.
3. Results and model calculations Figs. 1–7 show the experimental data recorded for aqueous solutions of Glc, maltose, maltotriose, maltotetraose and maltopentaose at the stated temperatures. Error bars, although small, are indicated on the data points, a more rigorous treatment of this data together with thermodynamic data for several other saccharides in aqueous solution can be found elsewhere [16]. It was found that measurements of vapour pressures could be taken well into the supersaturated region where solubility data was known
Fig. 3. Water/maltose at 298.06 K.
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Fig. 4. Water/maltose at 317.99 K.
[17,18]. Measurements within the supersaturated region are believed to occur for all of the systems studied here even where solubility data is not known. The maltotriose, maltotetraose and maltopentaose systems were measured only at the higher temperature, increasing the solubility of the sugars and hence providing more data points with which to test the various models. On each plot the predictions obtained from P&M UNIFAC and S-UNIFAC are also given. Also on each plot the results of the theoretical model are given both with the coordination number traditionally employed in the UNIQUAC model, i.e. z = 10, and also with the coordination number altered in a method discussed below. Other workers have
Fig. 5. Water/maltotriose at 317.99 K.
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Fig. 6. Water/maltotetraose at 317.99 K.
previously measured vapour pressures above aqueous solutions of Glc and maltose [17,18] and whilst their results are not shown in these figures, these new data are in good accord with their findings. All of the systems exhibit a negative deviation from ideal behaviour, in good accordance with other experimental observations and indicative of the strong interactions that occur between the sugars and the surrounding water. Comparison of activity coefficients indicates that the deviation from ideal behaviour increases with increasing molecular size of the solute, as one would expect.
Fig. 7. Water/maltopentaose at 317.99 K.
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4. Discussion Vapour pressures above aqueous solutions of Glc and maltose at 298.06 and 317.99 K and vapour pressures above aqueous solutions of maltotriose, maltotetraose and maltopentaose at 317.99 K have successfully been recorded. The experimental data is presented, as are the predicted values from the P&M UNIFAC, S-UNIFAC and theoretical models. The S-UNIFAC model predicts the vapour pressures of aqueous solutions of Glc at 298.06 and 317.99 K very well. For the maltose and higher oligomer systems this model does not perform well. These results are probably a consequence of the osidic bond group interaction parameters in S-UNIFAC having been developed around just one disaccharide molecule, namely sucrose. Expectation that this method, in its current form, should perform well for oligosaccharides containing as many as four osidic bond groups in different environments to that of sucrose is therefore optimistic. The requirement for a model which distinguishes between the many carbohydrate isomers is, however, very necessary. The P&M UNIFAC model performs very well for the aqueous Glc and maltose systems at 298.06 K. At the higher temperature of 317.99 K and for the oligosaccharide systems predictions using this model are satisfactory only within the very dilute regions with the exception of the water/maltopentaose system. Again it should be pointed out that P&M UNIFAC has not been developed for the prediction of properties of aqueous solutions of oligosaccharides (in fact no literature method has) and advantages of P&M UNIFAC, S-UNIFAC and, although not shown, bio-UNIFAC include their successful representation of binary, ternary and even quaternary mixtures of mono and disaccharides. Figs. 1–7 show that an important factor in the success of the theoretically based model is the value chosen for the coordination number, z. In the UNIQUAC equation the coordination number of the lattice represents the number of nearest neighbours to each segment. Further to this, the product of the pure component area parameter, q, and z gives the number of external contacts per molecule. It is perhaps surprising that in the UNIQUAC model z is often fixed at a value of 10, with reasonable alternatives seldom being used (6 ≤ z ≤ 12) [1]. Figs. 1–7 show the out comes of using the traditional value of z = 10 in the theoretical model. For the water/Glc system at 298.06 K and the water/maltose system at 317.99 K the model performs very well. For the other systems predictions are only satisfactory in the dilute regions. The performance of this model is in most cases similar to the performance of the P&M UNIFAC model. Altering the z number to values indicated in the respective figures does give a significant improvement. It is true that the coordination number, z, may well be taking on the rôle of a fitting constant in the theoretical model, however, the following trends should be considered. The coordination number is required to decrease with an increase in temperature possibly consistent with the lattice becoming more diffuse. Not immediately obvious is the requirement for the coordination number to increase from 8 in the water/Glc system through to 11 in the water/maltotriose system and decreasing to 10 in the water/maltopentaose system. It is possible that the conformational behaviour of the oligosaccahrides and the effect of this on the structure of the water are involved in the requirement for this trend. Given the complexity of the systems that are studied in this work, the performance of all the models in this work indicates the advances that have been made to both the UNIFAC and UNIQUAC methods since its original proposal. However, it is clear that further model development is required specifically addressing many of the assumptions made in these models such as the coordination number.
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5. Conclusion In this paper the predictive properties of two UNIFAC models and a theoretically based UNIQUAC model have been examined for a series of ␣-(1–4) linked Glc oligosaccharides in aqueous solution. The S-UNIFAC model in its present form cannot usefully be applied to theses systems whereas the P&M UNIFAC model has predictive ability only within the very dilute regions. The simple theoretical model used in this work can be shown to adequately reproduce vapour liquid equilibrium data for the systems studied here. However, it is clear that a satisfactory statistical thermodynamic model describing aqueous solutions of long chain saccharides is unlikely to be provided by, or via, the over-simplified UNIQUAC or UNIFAC methods in their current form. This work suggests that a lattice based model may be able to be utilised in such a project with the addendum that the coordination number of the lattice is likely to be a function of several variables including the temperature of the system and the molecular size and shape of both the solute and solvent. List of symbols Aij interaction parameter P vapour pressure qi surface area parameter ri volume parameter xi mole fraction γi activity coefficient z coordination number Acknowledgements The financial support for this project has been provided by the Danish Technical Research Council and is greatly appreciated. We also gratefully acknowledge the support of the Novo Nordisk Foundation and the Danish Natural Science Research Council, and we would like to thank Peter Rasmussen and Kjeld Rasmussen for many useful discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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