Abstract

GROUP CONTRIBUTION LATTICE FLUID EQUATION OF STATE: APPLICATION TO POLYMER+SOLVENT SYSTEMS

S. Mattedi^1** Mailing Address: DEQ- Escola Politécnica - UFBA Aristides Novis 2, Federação 40210-630 - Salvador, BA - e-mail: Mailing Address: DEQ- Escola Politécnica - UFBA Aristides Novis 2, Federação 40210-630 - Salvador, BA - e-mail: silvana@ufba.br , F.W. Tavares² and M. Castier²

¹Programa de Engenharia Química-COPPE, Universidade Federal do Rio de Janeiro, C.P.68502, Rio de Janeiro-RJ, CEP-21949-970, Brazil

²Escola de Química, Universidade Federal do Rio de Janeiro, C.P. 68542, Rio de Janeiro-RJ, CEP-21949-900, Brazil

(Received: January 14, 1998; Accepted: August 7, 1998)

Abstract - A group contribution equation of state recently proposed by the authors is used to calculate the activity of solvents in polymer+solvent solutions. The model is based on the generalized van der Waals theory and combines the Staverman-Guggenheim combinatorial term with an attractive lattice gas expression. The results show that the equation correlates the properties of pure solvents and polymers accurately. Furthermore, the equation satisfactorily represents the vapor-liquid equilibrium of solutions containing polymer and solvents of different polarities.

Keywords:

INTRODUCTION

Group-contribution equations of state (GC-EOS) for polymer systems offer the promise of using a single set of parameters to model the behavior of polymers with different molecular weights and to describe copolymer solutions. Lattice fluid theories have been used to derive GC-EOS, and examples of previous papers in this area are Saraiva et al. (1995), Lee and Danner (1996), and Yoo et al. (1996) including the references therein. In a previous work, Mattedi et al. (1994) studied the EOS obtained by using the combinatorial term of Staverman (1950) and the attractive lattice gas term of Tavares and Rajagopal (1992). A GC version of this EOS (Mattedi et al., 1996) is tested here using calculations of pure polymer densities and vapor-liquid equilibrium (VLE) of polymer solutions.

MODEL

In the development of the model, we assumed that a fluid of total volume V is represented by a lattice of coordination number Z containing M cells of fixed volume V*. In group contribution form, the EOS is:

(1)

where z is the compressibility factor, is the number of groups of type a in a molecule of type i, is the area parameter of group a, and Y is a constant of the lattice structure (Mattedi et al., 1994). The average number of segments occupied by a molecule in the lattice (r), the average number of first neighbors (Zq) and the reduced volume () are given by:

(2)

(3)

(4)

(5)

(6)

Here, and are the group contributions to the number of segments and hard core volume, respectively. is the molar hard core volume parameter of a group of type . We also define:

(7)

(8)

(9)

where is the interaction energy between groups m and a. The fugacity coefficient derived from the EOS is:

(10)

As suggested by Chen and Kreglewski (1977), and are calculated using:

(11)

(12)

In summary, our GC-EOS has five parameters for each group (, , , and ) and two parameters for interactions between unlike groups (and ).

EQUATION OF STATE PARAMETERS AND GROUP PARTITION

In this work, the lattice coordination number Z was set equal to 10, as usually done in the derivation of lattice-based thermodynamic models (e.g., Abrams and Prausnitz, 1975). The empirical characteristic constant Y was set equal to 1, as suggested in previous work by Tavares and Rajagopal (1992) and Mattedi et al. (1994); a value of 5 cm³/mol was used for the cell volume on a molar basis (v*). Although this value is smaller than the values used by Panayiotou and Vera (1980) (9.75 cm³/mol) and Abrams and Prausnitz (1975) (15.17 cm³/mol), preliminary tests made by Paredes et al. (1994) showed that our model provides better fits of the vapor pressure of polar compounds when this lower value is used.

Solvents were considered as single groups; for polymers, the monomers were considered as groups. Solvent parameters were fitted using pure vapor pressure data from Boublík et al. (1980) and Danner (1992). Polymer properties were computed by using group contribution from their monomers. The polymer parameters were fitted using liquid volume pseudo-data at 1 atm computed using the Tait correlation (Rodgers, 1993). We used 100 data points, evenly distributed in the temperature range of the correlation. A specific molecular weight was used for fitting, but the same group parameters can be used for any chain length formed by a monomer. The interaction parameters were fitted using weight fraction activity coefficient data (Danner and High, 1993). We studied fifteen binary mixtures, that were distributed into four classes combining polymers and solvents of apolar or polar nature. A least square fit of relative deviations was used.

RESULTS

Average deviations were lower than 2% and 0.002 % in solvent vapor pressures and in pure polymer liquid volumes from the Tait equation, respectively (Table 1). Table 2 presents the interaction parameters for polymer+solvent systems. Table 3 summarizes the results of the VLE calculations. For PIB and PS solutions, the deviations are usually lower than in Pretel and Danner (1996), but higher for PVAC solutions. However, their deviations refer to a single temperature and fewer data points. The results are similar to those of Yoo et al. (1996). Deviations in solvent activities from Harismiadis et al. (1994) for PIB and PS systems using four G^E models are larger than ours. In Table 3, the largest deviations occur for the system PEO+water, which contains a polar polymer and is the only aqueous system studied. A possible way to improve the results for this system might be splitting the water molecule into groups in order to include the hydrogen bonding effect, as previously done in the molecular version of this equation of state (Mattedi et al., 1994).

Table 1: Parameters for pure solvents and monomer groups

Table 2: Energy interaction parameters for polymer+solvent systems

Figures 1-4 illustrate the results for systems with different combinations of apolar and polar polymers and solvents. For PIB+benzene (Figure 1), similar curves were calculated using the three experimental conditions, and only one curve is shown. Figs. 2-4 present an apolar polymer in a polar solvent (PS+acetone), a polar polymer in an apolar solvent (PEO+benzene) and a polar polymer in a polar solvent (PVAC+1-propanol). Although for some systems such as PEO+benzene (Figure 3), the effect of temperature seems to be underpredicted, for other systems such as PVAC+1-propanol (Figure 4) this effect is satisfactorily predicted. When compared with other models available in the literature, the model provides satisfactory predictions of the effects of both temperature and polymer molecular weight on the values of activity coefficients.

Table 3: Deviations in the weight fraction activity coefficient of the solvent for polymer solutions

¹

²Results published by Yoo et al. (1996); only mean deviations were reported.

³T is the temperature range, N is the number of data points and W is the weight fraction activity coefficient.

CONCLUSIONS

A GC-EOS proposed by Mattedi et al. (1996) was used to model polymer+solvent systems. The equation satisfactorily correlates polymer densities and vapor pressures of pure solvents and is able to predict the activity coefficients of solvents in binary solutions containing solvents and polymers of different polarities. The results compare favorably with those from G^E models and are at least similar to those of other EOS.

Figure 1: Prediction of the weight fraction activity coefficient of benzene in PIB (see text for details).

Figure 2: Prediction of the weight fraction activity coefficient of acetone in PS (MW 15700).

Figure 3: Prediction of the weight fraction activity coefficient of benzene in PEO (MW 4000000).

Figure 4: Prediction of the weight fraction activity coefficient of n-propanol in PVAC (MW 170000).

ACKNOWLEDGEMENTS

The authors acknowledge Prof. S.I. Sandler from the University of Delaware where part of this work was developed and the support of CNPq, CAPES and COPENE. This research received the support of the "Programa Nacional de Excelência" (PRONEX/Brazil, Project no. 41.96.0878.00).

NOMENCLATURE

Temperature dependence parameter of the hard core volume of type a groups

Temperature dependence parameter of the interaction energy between groups of types a and b

Bulkiness factor

Number of components

Number of group types

Number of molecules

Superficial area of type a groups

Universal gas constant

Number of segments occupied by type a groups

Number of segments

Area fraction of type a groups on a void free basis

Temperature

Molar interaction energy between groups of types b and a

Temperature independent molar interaction energy between groups of types b and a

Molar volume

Reduced molar volume

Molar volume of lattice cells

Molar hard core volume characteristic of type a groups

Temperature independent molar hard core volume of type a groups

Volume of a lattice cell

Total volume

Hard core volume of each type a group

Molar fraction

Lattice coordination number

Compressibility factor

Mean number of nearest neighbors

Greek letters:

Fugacity coefficient of component i in the mixture

D P/P Mean relative deviation between experimental and calculated pressure

Mean relative deviation between experimental and calculated weight fraction activity coefficient

Mean relative deviation between experimental and calculated liquid volume

Empirical universal constant of the lattice structure

Number of type a groups present in a molecule of type

Danner, R.P., Private Communication (1992).

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