Development new correlations for NRTL Parameters in Polymer Solutions

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1 Rhodes, Greece, August 0-, 008 Development new correlations for NRTL Parameters in Polymer Solutions A. Saatchi, M. Edalat* Oil and Gas Center of Excellence, Department of Chemical Engineering University of Tehran Enghelab Ave, Tehran, P.O.Box: IRAN Abstract: In this study correlations for NRTL parameters, as a function of temperature and degree of polymerization (Mn) are proposed in order to use NRTL for polymer system. Analysis of variance (ANOVA) is applied to investigate significant effect of temperature and degree of polymerization on NRTL Parameters. Recognizing the effects of temperature and degree of polymerization on these parameters is first step in improving NRTL for polymers. A genetic algorithm is used to obtain the best possible correlations for NRTL parameters as a function of temperature and Mn. Obtained results demonstrate that α 1 as a function of temperature and degree of polymerization can be use in NRTL and predicts activity coefficients of polymer solution within the range of acceptable accuracy. Solutions of polydimethylsiloxane in various solvents and temperatures are considered and τ 1, τ 1 and α 1 are calculated for these systems.. Existence of meaningful relation between them and temperature and Mn are investigated with a statistical analysis method. Obtained results from statistical analyses demonstrate that α 1 can be a function of temperature and degree of polymerization, τ 1 as a function of temperatures and τ 1 as a function of degree of polymerization. The proposed correlation for α 1 is applied to calculating activity coefficient; results are demonstrated that proposed correlation for α 1 activity coefficient accurately. Key-Words: activity coefficient, polymer, NRTL, temperature, degree of polymerization, Analysis of variance 1 Introduction An understanding of the thermodynamics of the polymer solutions is important in practical applications such as polymerization and the incorporation of plasticizers and other additives. Diffusion phenomena in polymer melts and solutions are strongly affected by nonideal solution behavior, since the chemical potential rather than the concentration provides the driving force for diffusion. Proper design and engineering of many polymer processes depend greatly upon accurate modeling of thermodynamic parameters such as solvent activities. Analysis of variance (ANOVA) used in this article to investigating that variation of temperature and degree of polymerization have significant effect on NRTL parameters or variation of NRTL parameters are independent of variation of temperature and degree of polymerization. NRTL is one of the most successful equations proposed to prediction of nonpolymer systems activity coefficient but about polymer systems, practically became useless and some equations like UNIQUAC and combination of group contribution with existence equations are preferred [1].The NRTL equation is applicable for to partially miscible as well as completely miscible systems.nrtl for excess Gibbs energy is: E g = xx 1 RT where τ1g1 τ1g1 + x + x G x + xg g 1 g , g τ = τ g 1 = () RT RT G = exp(- α τ ), G = exp(- α τ ) (3) (1) In these relations, subscripts 1 and stand for solvent and polymer, respectively also x and γ stand for mole or mass fraction and activity coefficient, respectively. The significance of g 1 is an energy parameters characteristic of 1- interaction. Parameter α 1 is related to the nonrandomness in the mixture. The parameters (g 1 -g ) and (g 1 -g 11 ) are liner functions of temperature for nonpolymer systems. NRTL equation is readily generalized to multi-component mixture. From equation (1) the activity coefficient get as a equation (4). Equations are proposed for prediction of activity coefficient can be dividing to two models. In first model, equations are suggested based on parameters that have significant impact on the behavior of activity coefficient. According to this model, meaningful parameters are recognized then equations are found as a function of these meaningful parameters. These models have been ISSN: ISBN:

2 Rhodes, Greece, August 0-, 008 extended and applied to Vapor-Liquid Equilibrium in aqueous solutions of various glycols and poly(ethylene glycols), 3poly (ethy1ene glycols). Mordechay Herskowltz, Moshe Gottlleb and also Ali Eliassi and Hamid Modarress suggested polynomial function for Activity of water in aqueous poly (ethylene glycol) solutions [,3]. On the other hand Johann Gaube assumed activity coefficient is a function of specific volume of water and average molecular weight of polymers for binary polymer solutions such as aqueous PEG or dextran solutions [4]. These works generally led to unacceptable results or results which are acceptable in limited systems. G τ G ln γ = x τ +, 1 1 x + x G 1 1 ( x + x G 1 1) G x + x G 1 1 τ G ln γ = x τ ( x + x G ) 1 1 (4) In second model some governing equations like NRTL and Flory-Huggins are combined with group contribution concept and established new method [1-5].Non-Random Two Liquid (NRTL) proposed by Henri Renon and J. M. Prausnitz[1], these equation is very flexible and predict acceptable activity coefficient for nonpolymer systems. Efforts have done for expansion of NRTL for polymer solutions. These efforts faced problems during expansion of NRTL parameters because NRTL parameters have theoretical bases and expansion of them should be done with attention to their bases. Chau-Chyun Chen improved NRTL and suggested Polynrtl with combination of the Flory-Huggins description for the configurational entropy of mixing molecules of different sizes and the NRTL theory for the local composition contribution from mixing solvents and polymer segments [6-5]. All of the prior works are performed in this field assumed that α 1 has a constant value between. and.3 also in segment based equations are assumed. The excess Gibbs energy of a polymer solution expressed as the sum of the local composition contribution, g ex,lc, and the configurational entropy of mixing, g ex,config : g ex = g ex,config + g ex,lc (5) In this relation, superscripts ex, LC and config stand for excess, local composition and configuration, respectively Following Wu et al. [7], they used the truncated Freed correction to Flory Huggins expression as first correction for the configurational entropy of mixing [8]. ex, config g φ φ p 1 1 = x ln w s + xpln + xwφpα RT xw x p rw r p (6) Where xii r φ i = (7) x i ir In these relations, subscripts w and p stand for solvent and polymer, respectively; r i and x i are the number of segments and the mole fraction of the species i, respectively. The number of polymer segments, r p, approximates the ratio of the molar volume of the polymer and that of the solvent molecules and for the solvent r w = 1. Parameter α is the nonrandomness factor. The first two terms on the right hand side of equation () are originally from the Flory Huggins expression. The third term accounts for the correction to the Flory Huggins mean-field approximation, which can be understood as the contribution of solution structure and size dissimilarity between two components or as the local composition effect from the inner connections of the polymer chain. In this article NRTL parameters are evaluated to find out that these parameter could be presented as function of temperature and degree of polymerization. General forms of Development correlations for NRTL Parameters Analysis of variance (ANOVA) is applied to investigate significant effect of temperature and degree of polymerization on NRTL Parameters. A genetic programming is used to obtain the best possible correlation for α 1 as function of temperature and degree of polymerization. The following correlation is proposed for α 1 : 3 3 α (8) 1 = A+ Blog( Mn) + Clog( Mn) + Dlog( Mn) + E/ T+ F/ T + G/ T In these relations, T and Mn stand for temperature and degree of polymerization, respectively. In the above correlations A, B, C, D, F, J, and I are universal constants. Also a genetic programming is used to obtain the best possible correlations for τ 1 and τ 1 as function of temperature and degree of polymerization. The following correlations are proposed for τ 1 and τ 1 : ISSN: ISBN:

3 Rhodes, Greece, August 0-, 008 B C D E F τ = A GT 1 M n Mn Mn Mn Mn HT + IT + JT (9) B C D E F G H I 1 A M n Mn M T n Mn Mn T T τ = (10) In the above correlations A, B, C, D, F, J, and I are universal constants. 3 Results and discussion Experimental data of activity coefficient is prepared, and then NRTL parameters are calculated from these experimental data. The experimental data are presented in Table.1. The experimental data are from DECHEMA Chemistry Series [9].One-way ANOVA are used to investigate that which parameters have meaningful effect on NRTL parameters. Sufficient experiment data for using in ANOVA are used and NRTL parameters are calculated for these data. 3.1 Interpreting the results of ANOVA The default one-way output contains an analysis of variance table, a table of level means, individual 95% confidence intervals, probability and the pooled standard deviation but we just present probability. When probability is than.01 indicates that there is a highly statistically significant and for probability between.01 and.1 there is a statistically significant. Also for probability value more than.1 there isn t statistically significant. The results of ANOVA are presented in Table. Table. One-way ANOVA: NRTL parameters versus Temperature and degree of polymerization NRTL parameters Consider parameters Probability α 1 Temperature.08 α 1 M n.01 τ 1 Temperature τ 1 M n.075 τ 1 Temperature.054 τ 1 M n On a basis of results are presented in Table, α 1 is a function of temperature and degree of polymerization and τ 1 is a function of temperatures also τ 1 is a function of degree of polymerization. In this study, existence meaningful effects of temperature and degree of polymerization on NRTL parameters are demonstrated. This difference between Table1. Calculated NRTL parameters for solvent activity in Polydimethylsiloxane solutions Number solvent Mn Temperature α 1 Toluene Toluene Toluene Toluene Benzene Benzene Benzene Benzene Benzene Benzene Hexane Hexane Hexane Hexane Heptane Heptane Heptane Octane Octane Octane Octane Octane Octane Pentane Pentane Pentane Pentane Pentane obtained results and preceding opinions can be issued from calculation of NRTL parameters. These parameters have physical meanings, α 1 is equal to Z-1 where Z is the coordination number of the lattice and about g 1 and g 1 for NRTL are used of two-liquid theory of Scott. Two-liquid theory of Scott assumes that there are two kinds of cells in a binary mixture: one for molecules 1 and one for molecules, as shown in Fig.1 [10]. In Fig.1 subscripts 1 and stand for solvent and polymer respectively. The NRTL parameters are calculated in this work without assuming any boundary limitation or initial value, it means that in this study physical meaning of parameters are considered but in other works with assuming constant value for α 1, results are converted to mathematical values are missed physical meaning. In ISSN: ISBN:

4 Rhodes, Greece, August 0-, 008 Fig.1 Two types of cells according to Scott s twoliquid theory of binary mixtures. fact in polymer systems with increasing degree of polymerization the coordination number of the lattice changes then Z changes and it led to changing α 1. According to Flory Huggins expression it's obviously that effects of degree of polymerization are considered in this part then NRTL parameters are not shown meaningful significant with variation of degree of polymerization. It seems that existence of several expressions for prediction of activity coefficient leading to condition that statistically significant is not shown between temperature, degree of polymerization and NRTL parameters in segment based models but in our study NRTL parameters are calculated without any cramped condition. 3. coefficient and parameters Developed correlations A genetic algorithm is used to obtain Universal constants for equations (8), (9) and (10), these constants are presented in Table 3, Table 4 and Table 5. Obtained results from statistical analyses are demonstrated that α 1 dependent on temperature and degree of polymerization and τ 1 dependent on temperatures also τ 1 dependent on degree of polymerization but we used temperature and Mn for all parameters because our analysis specified a value of significant each parameters and on a basis of performed statistical analysis we can't conclude that we must eliminate some parameters. We determine a value effect of each parameter. By applying the proposed correlation for α 1, average predicted error for considered systems is about 9%. Also for τ 1 and τ 1 average predicted error for considered systems is about 7% and 11% respectively. Fig., Fig.3 and Fig.4 shows the plot of the proposed correlations for α 1, τ 1 and τ1respectively. For calculating NRTL parameters we need to have three information's about system, boundary condition of activity coefficient is applied as a first information (in Table 3. Value of universal constants for proposed correlation for α 1 A B C D 1.97E-0 E F G Table 4. Value of universal constants for proposed correlation for τ 1 A B C D E+11 E E+14 F -1.61E+17 G H I J 1.17E-04 Table 5. Value of universal constants for proposed correlation for τ 1 A B C D E+1 E -6.99E+15 F.69E+18 G H I E+11 Fig. Proposed function for α 1 and dimensionless experimental α 1 ISSN: ISBN:

Rhodes, Greece, August 0-, 008 correlation for α 1 to calculating activity coefficient. The results are showing good agreements between experimental data and calculated activity coefficient.

4 Proposed function for τ 1 and dimensionless experimental τ 1 pure substance is equal to one ), for second information proposed correlation for α are applied and for last information we applied

5 Rhodes, Greece, August 0-, 008 correlation for α 1 to calculating activity coefficient. The results are showing good agreements between experimental data and calculated activity coefficient. The mean result deviations are about 1.3% for considered systems. Fig.3 Proposed function for τ 1 and dimensionless experimental τ 1 Fig.4 Proposed function for τ 1 and dimensionless experimental τ 1 pure substance is equal to one ), for second information proposed correlation for α are applied and for last information we applied experimental data for infinity dilution. The deviations of calculated activity with experimental data are reported in Tables 6. The results are showing good agreements between experimental data and calculated activity coefficient. The mean result deviations are about 1.3% for considered systems. 4 Conclusion The significant effect of temperature and degree of polymerization on NRTL Parameters with using Analysis of variance are demonstrated. In this article we indicated that NRTL parameters are as function of Temperature and degree of polymerization. Obtained results from statistical analyses demonstrate that α 1 as a function of temperature and degree of polymerization and τ 1 as a function of temperatures also τ 1 as a function of degree of polymerization. On a basis of performed analysis, a genetic programming is used to obtain the best possible correlation for NRTL parameters as function of temperature and degree of polymerization. These correlations predict NRTL parameters of polymer solution within the range of acceptable accuracy (about 10%). We applied proposed Table 6.Deviation of new model with experimental data system number Deviation NP: number of experimental point, Deviation: (1/NP) Σ ( a exp - a cal ) References: [1]H.Renon,J.M.Prausnitz, Local Compositions in Thermodynamic Excess Functions for Liquid Mixtures,AlChE Journal, Vol.14,No.1,1968, []M.Herskowltz,M.Gottlleb, Vapor-Liquid Equilibrium in Aqueous Solutions of Various Glycols and Poly (ethylene glycols), 3. Poly (ethy1eneglycols), Chem. Eng. Data, Vol. 30, 1985, ISSN: ISBN:

6 Rhodes, Greece, August 0-, 008 [3] A.Eliassi,H.Modarress,Measurement of Activity of Water in Aqueous Poly(ethylene glycol) Solutions (Effect of Excess Volume on the Flory-Huggins1.χ- Parameter), J. Chem. Eng. Data,Vol.44,1999, [4]J.Gaube, A.Pfennig,M.Stumpf,Vapor-Liquid Equilibrium in Binary and Ternary Aqueous Solutions of Poly(ethy1ene glycol) and Dextran, j. Chem. Eng. Data,Vol.38,1993, [5]Flory, P. J. Principles of Polymer Chemistry; Cornell University Press.New York, [6]J.C.Hsu,Multiple Comparisons, Theory and methods. Chapman & Hall., [7] Y.-T. WU, Z.-Q. ZHU, D.-Q. LIN, L. H. MEI. FLUID PHASE EQUILIB. 11, 15 (1996). [8]J.Dudowicz, K.F.Freed, W.G.Madden, Role of Molecular Structure on the Thermodynamic Properties of Melts, Blends, and Concentrated Polymer Solutions. Comparison of Monte Carlo Simulations with the Cluster Theory for the Lattice Model, Macromolecules,Vol.3, 1990, [9]W.Hao, H.S.Elbro, P.Alessi, DECHEMA Chemistry Data Series.Volume XIV.Polymer Solution Data Collection, DECHEMA,199 [10] R.L.Scott, Corresponding States Treatment of Nonelectrolyte Solutions, J. Chem. Phys.,Vol.5, 1956,193. ISSN: ISBN:

A new segmental local composition model for calculation of thermodynamic properties of binary polymer solutions

Scientia Iranica C (204) 2(6), 2087{2097 Sharif University of Technology Scientia Iranica Transactions C: Chemistry and Chemical ngineering www.scientiairanica.com A new segmental local composition model