고분자용액에서의 기액평형의 예측: 모델의 비교와 평가

이 병 철, 이 재 민

한남대학교 공과대학 화학공학과

Prediction of Vapor-Liquid Equilibria in Polymer Solutions:

15

* Corresponding author.

Evaluation of Current Models

Byung-Chul Lee and Jae Min Lee

Dept. of Chem. Eng., Hannam University, Taejon, Korea

Introduction

Calculations involving phase equilibrium behavior in polymer solutions are required in the design and operation of polymer processes. The most important of the equilibrium behavior is how the solvent will distribute itself between the polymer and solvent phases. When experi- mental data are not available at a specific condition, one has to use a thermodynamic model which can predict the equilibrium behavior. Polymer-solvent vapor-liquid equilibrium (VLE) calculations are to determine the equilibrium weight fraction of solvent in a polymer at a given pressure and temperature. Since the polymer is present only in the liquid phase, the VLE problem reduces to calculating the activity coefficient of the solvent in the liquid phase.

Lee and Danner [1-3] developed a group-contribution lattice-fluid equation of state (GCLF-EOS), which was capable of predicting the VLE in polymer solutions. In this paper the performance of the GCLF-EOS is compared with the UNIFAC-free volume (FV) model [4], the entropic-FV model [5], and the revised group-contribution (GC) Flory EOS [6]. Danner and High [7] collected a large amount of experimental VLE data for many polymer- solvent systems from the literature. In this work, using the entire available experimental data, the above prediction models were evaluated and extensive statistical analyses were performed.

Description of Current Prediction Models

The GCLF-EOS is a group-contribution version of the lattice-fluid EOS of Panayiotou and Vera [8]. Using the lattice statistics and quasichemical theory, Panayiotou and Vera derived a lattice-fluid EOS which corrects for the nonrandom mixing arising from the interaction energies between molecules. The pure component EOS contains two adjustable parameters: molecular interaction energy and molecular reference volume parameters. For a binary mixture, it has an additional binary interaction parameter. To make the model predictive, Lee and Danner [1] developed the group-contribution mixing rules for these three parameters. The group parameters were estimated from the properties of low molecular weight compounds only: saturated vapor pressure and liquid density data of pure substances [9], and VLE data of binary pairs [10]. Tables of the group parameters for a variety of functional groups are available from the references [1,2]. Once the molecular structures of the components in a mixture are known with respect to their functional groups, one can calculate the molecular parameters from the group parameters and then predict the equilibrium behavior of polymer solutions as well as of low molecular weight compounds.

The UNIFAC-FV method [4] is a group-contribution activity coefficient model based on the well-known UNIFAC model [11]. The free volume contribution, derived from the Flory EOS [12], was added to the combinatorial and residual terms of the UNIFAC model to account for the free volume differences between the polymer and solvent molecules. The com- binatorial term considers size and shape effects, while the residual contribution accounts for interactions among groups. These two terms are identical to those in the original UNIFAC method. This method is less reliable in the dilute regions especially for highly nonideal systems which are characterized by large infinite dilution activity coefficients. Since this model is an activity coefficient model, it cannot take into account the pressure dependence on the phase equilibria. This model uses the extensive UNIFAC tables which contain many more groups than the other prediction models. A big disadvantage of this model is that it requires the liquid densities of both the solvent and polymer at the temperature of the solution.

The entropic-FV method [5] is an activity coefficient model consisting of two terms, the "combined" combinatorial and free-volume term by Elbro et al. [13] and the residual term of the UNIFAC method. The free volume fraction was used instead of volume fraction as a simple modification of the combinatorial term in the Flory-Huggins equation. The residual term is identical to that of the UNIFAC method. The entropic-FV model differs in several ways from the UNIFAC-FV method. The combinatorial-free volume term is much simpler than the rather complicated term in the UNIFAC-FV model. No adjustable parameters for the free volume part are used in this model. The linear temperature-dependent interaction parameters are used in this method instead of the temperature-independent parameters of the UNIFAC-FV model. As in the UNIFAC-FV method, however, this method also requires the molar volumes of both pure solvent and polymer.

The revised GC-Flory EOS [6] is a group-contribution extension of a modified form of the Flory EOS [12]. This model consists of the combinatorial, the interaction or enthalpic, and the free volume contributions. The group parameters of the model were estimated from PVT and enthalpy of vaporization data of pure components and also from VLE data of low molar mass binary mixtures. In the GCLF-EOS and revised GC-Flory EOS, no pure component or mixture properties of the polymer and solvent are required other than the molecular structures of the solvent and polymer repeat unit.

Results and Discussion

For various polymer-solvent systems, the solvent weight fraction activity coefficients (WFAC) at a given temperature and polymer molecular weight were predicted both at infinite dilution and finite concentrations of the solvent by the above four models. The WFAC of a solvent in a polymer-solvent system, [eqn: OMEGA_1], is defined as [eqn: OMEGA_1 w_1 `=` gamma_1 x_1], where [eqn: w_1] is the weight fraction of the solvent, [eqn: gamma_1] is the mole fraction based activity coefficient. The complete expressions of the WFAC for each model can be found in [3]. The number average molecular weight of polymer was used, if available. When the polymer molecular weight was not known, the number of the polymer repeat units was set to 200.

Graphical illustrations of the predictions of the WFACs, which are shown in Lee and Danner [1,3], are not given in this paper due to space limitation. This paper focuses only on statistical analyses on prediction of VLE in polymer solution. The polymer-solvent systems were classified into three categories with respect to the three solvent groups; nonpolar, weakly polar and strongly polar, as shown in Table 1.

Table 2 summarizes the statistical analyses on the prediction of the WFAC at infinite

Table 1. Polymers and solvent groups studied in this work.

Polymers	BR, HDPE, IR, LDPE, PBA, PBMA, PDMS, PEA, PEMA, PEO, PHA, PHMA, PIB, PIBMA, PMA, PMMA, PoMS, PpMS, PP, i-PP, PPA, PpeMA, PPMA, PS, PVAC, PVAL, PVC
Solvent Groups Nonpolar Weakly polar Strongly polar	Normal and branched alkanes, Cyclic hydrocarbons, Aromatic hydrocarbons, Alkenes, Chlorinated hydrocarbons Ethers, Ketones, Esters Acids, Alcohols, Water

dilution of solvents by all the models for the three groups of solvents. It contains the number of polymer-solvent systems and the total number of data points tested in each model and solvent group. Much more data were available for the nonpolar solvents than for the weakly and strongly polar solvents. (AD%)total indicates the average percentage deviation for the total number of systems included in each solvent group. (AAD%)total is the average absolute percentage deviation, and (STD)total is the standard deviation of (AAD%)total.

[eqn: (AD%)_total ``=`` 1 over N_total SUM from i to N_total { \{ OMEGA_1,i (pred) ``-`` OMEGA_1 (exp) \} } over { OMEGA_1 (exp) } TIMES 100``]

[eqn: (AAD%)_total ``=`` 1 over N_total SUM from i to N_total { LEFT | OMEGA_1,i (pred) ``-`` OMEGA_1 (exp) RIGHT | } over { OMEGA_1 (exp) } TIMES 100``]

Here [eqn: OMEGA_1 (pred)] and [eqn: OMEGA_1 (exp)] are the predicted and experimental values of the WFAC of the solvent, respectively, and [eqn: N_total`] is the total number of the data points for a solvent group. The GCLF-EOS gave the best predictions for nonpolar and weakly polar solvents. The entropic-FV model resulted in a big deviation for weakly polar solvents, due to huge overpredictions for some systems such as polyacrylates-acetates and polymethacrylates-acetates. The free-volume contribution is considered to be excessively overestimated for those systems. However, it gave the best result for strongly polar solvents. The GCLF-EOS was slightly worse than the entropic-FV model for strongly polar solvents. Overall the GCLF-EOS provided more accurate results than other models in predicting the VLE at infinite dilution of solvents.

Table 3 summarizes the statistical analyses on the prediction of the WFAC at finite concentrations of solvents by all the models for the three groups of solvents. For nonpolar solvents, all the models showed similar performance. The UNIFAC-FV and the revised GC-Flory EOS models gave unrealistic overflow errors or huge amounts of WFACs when applied to most of strongly polar systems such as water-containing systems. This means those models predicts phase separation for those systems. These systems are not included in Table 3. The revised GC-Flory EOS gave poor predictions even for weakly polar solvents. It is not

Table 2. Summary of prediction results of WFAC at infinite dilution by prediction models for polymer-solvent systems.

Models	Nonpolar Solvents	Weakly Polar Solvents	Strongly Polar Solvents
GCLF-EOS No. of Systems No. of Data Points (AD%)total (AAD%)total (STD)total	248 1525 -1.8 14.7 12.9	61 250 -7.4 18.0 12.6	33 237 -24.4 27.5 15.6
UNIFAC-FV No. of Systems No. of Data Points (AD%)total (AAD%)total (STD)total	244 1512 -9.6 20.3 20.7	60 244 5.5 27.1 56.5	32 226 63.1 64.4 41.8
Entropic-FV No. of Systems No. of Data Points (AD%)total (AAD%)total (STD)total	248 1524 -8.2 18.9 22.1	61 250 106.2 134.5 321.1	33 237 11.8 22.9 25.8
Revised GC-Flory No. of Systems No. of Data Points (AD%)total (AAD%)total (STD)total	215 1160 8.0 24.4 41.3	60 243 1.6 19.7 15.9	33 237 14.7 31.4 51.3

Table 3. Summary of prediction results of WFAC at finite concentrations of solvents by prediction models for polymer-solvent systems.

Models	Nonpolar Solvents	Weakly Polar Solvents	Strongly Polar Solvents
GCLF-EOS No. of Systems No. of Data Points (AD%)total (AAD%)total (STD)total	131 2040 1.4 13.0 15.0	29 332 -8.9 14.5 20.2	9 192 11.5 38.1 24.8
UNIFAC-FV No. of Systems No. of Data Points (AD%)total (AAD%)total (STD)total	116 1911 -2.1 14.8 15.9	16 238 -0.3 8.6 14.8	-* - - - -
Entropic-FV No. of Systems No. of Data Points (AD%)total (AAD%)total (STD)total	128 1985 -0.7 14.1 16.6	25 296 -4.7 12.6 14.1	10 190 -22.5 22.5 16.4
Revised GC-Flory No. of Systems No. of Data Points (AD%)total (AAD%)total (STD)total	97** 1487** -0.3 11.9 17.0	23 340 20.9 48.7 120.5	5*** 46*** -42.2 49.7 24.4

* The model predicts overflow errors or phase separations for all water-containing systems.

** The model is not applicable for PDMS, because its group parameters is not available.

*** The model predicts overflow errors or phase separation for systems such as PEO/water, which are not included in analysis.

applicable for PDMS system due to the lack of SiO group in it. Conclusively, the GCLF-EOS and entropic-FV models showed the best performance for all kinds of solvent groups in the prediction of VLE at finite concentrations of solvents.

References

1.	Lee, B.-C., Prediction of Phase Equilibria in Polymer Solutions, Ph.D. Thesis, The Pennsylvania State Univ., University Park, PA (1995).

2.	Lee, B.-C.; Danner, R.P., AIChE J., 42, 837 (1996).

3. Lee, B.-C.; Danner, R.P., Fluid Phase Equilibria, 128, 97 (1997).

4.	Oishi, T.; Prausnitz, J.M., Ind. Eng. Chem. Res. Dev., 17, 333 (1978).

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7.	Danner, R.P.; High, M.S., Handbook of Polymer-Solution Thermodynamics, DIPPR Data, AIChE, New York, NY (1993).

8.	Panayiotou, C.; Vera, J.H., Polym. J., 14, 681 (1982).

9.	Daubert, T.E.; Danner, R.P., Physical and Thermodynamic Properties of Pure Compounds: Data Compilation, Taylor and Francis, New York (extent 1994).

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13.	Elbro, H.S.; Fredenslund, Aa.; Rasmussen, P., Macromolecules, 23, 4707 (1990).