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A Highly Convergent Algorithm for Phase Equilibria Calculations of Ternary Polymer/Solvent Systems Using an Equal-Area Rule Method Ming Yu,* Robert Sauve´ , Mohammad K. Khoshkbarchi, and Ensheng Zhao AEA Technology Engineering Software-Hyprotech, Suite 800, 707 8th Avenue SW, Calgary, Alberta T2P 1H5, Canada
Solubility diagrams of ternary polymer systems are usually analyzed using calculated critical points and spinodals rather than tie lines and binodals. This is partly because achieving convergence in rigorous calculations of tie lines and binodals is difficult. To overcome this problem, we have extended the equal-area rule (EAR) method for calculating phase equilibria of systems containing small molecules to calculate the tie lines and binodals of polymer 1/polymer 2/solvent and polymer/solvent 1/solvent 2 systems. An advantage of the EAR method over those available in the literature for calculating the tie lines and binodals is that this method does not require an initial guess for the equilibrium concentrations to begin the calculation process. This capability is particularly useful for prediction purposes in the absence of experimental data. The method has been applied to several polymer/solvent ternary systems using the Flory-Huggins model to represent the chemical potentials of the components. The results show that the EAR method is highly efficient and reliable in calculating the tie lines and binodals in ternary polymer/solvent systems. Introduction
3
OBJ ) Knowledge of tie lines and binodals provides important information about the solubility diagrams of polymer systems. However, numerical calculation of tie lines and binodals of ternary polymer systems is “notoriously difficult”1 and requires “long and tedious approximation methods”.2 For this reason calculation of tie lines and binodals is usually avoided during the solubility diagram calculations, which has a major impact on their accuracy. Despite the availability of a rigorous method for tie line and binodal calculations, proposed by Hsu and Prausnitz,3 solubility diagrams have been routinely estimated from the position of critical points and spinodals,4-9 which may result in inaccurate conclusions. Although the method proposed by Hsu and Prausnitz has a theoretically sound basis and has been adopted by several researchers,10-15 its successful convergence depends to a large extent on the initial guess for the equilibrium concentration of the phases. As mentioned by Sˇ olc,16 the method of Hsu and Prausnitz still “leaves something to be desired”, and since the work of Scott,2 our understanding of “ternary systems solvent/ polymer 1/polymer 2 is scant, and it has not seen much progress over the past 35 years”. Sˇ olc16 developed an efficient method to calculate the cloud-point curve, which is a quasi-binary section of the ternary multidimensional temperature diagram. However, in Sˇ olc’s method the phase separation, induced by changing the composition of components, cannot be conveniently calculated. The binodal and tie-line calculation method of Hsu and Prausnitz requires the minimization of the following objective function:
* To whom correspondence should be addressed.
wti ∑ i)1
(∆µRi - ∆µβi )2 Fi
(1)
which is equivalent to solving the following nonlinear algebraic set of equations simultaneously:
∆µRi ) ∆µβi
(2)
where ∆µRi and ∆µβi represent the difference of the chemical potential of component i at the temperature, pressure, and composition of the system and its chemical potential at a standard state in phases R and β, respectively. In eq 1, wti and Fi are the weighting factor and the penalty function for species i, respectively. The weighting factor in eq 1 is introduced to evenly distribute the contribution of both large and small molecules on the objective function, and the penalty function is introduced to avoid convergence to trivial solutions. As mentioned before, the method of Hsu and Prausnitz is highly sensitive to the value of the initial guess for the equilibrium compositions. To successfully achieve convergence in the algorithm proposed by Hsu and Prausnitz, it is essential to provide an initial guess reasonably close to the true equilibrium compositions.3,12,13 This is because the response surface of the chemical potential is rather flat and only shows a narrow and steep well near the equilibrium point. As a result, an initial guess that is not close enough to the equilibrium value will lead the algorithm to converge to an unstable solution. Durrani et al.11 have also reported frequent failures in the convergence of the algorithm proposed by Hsu and Prausnitz. It is also important to mention that this problem arises from the calculation method and not the underlying model. The equal-area rule (EAR) method, developed by Eubank and co-workers17-19 for small-molecule systems, is a Gibbs energy minimization technique for solving phase equilibrium problems. A unique feature of this method is that it ensures that the calculations converge
10.1021/ie000088p CCC: $19.00 © 2000 American Chemical Society Published on Web 08/25/2000
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Figure 2. Flowchart of the algorithm to calculate an initial value for D12. Figure 1. Variation of (dg/dx1)D12 with x1 in a polymer 1/polymer 2/solvent system using the Flory-Huggins model with m1 ) m2 ) 1000, m3 ) 1, χ12 ) 0.002, χ13 ) 0.4, χ23 ) 0.3, φ1 ) 0.5, φ2 ) 0.25, and φ3 ) 0.25.
to a stable solution. The EAR method, unlike the tangent line method, finds the equilibrium compositions by integrating the Gibbs energy curve. Compared to the tangent plane method, the EAR method because of its higher sensitivity is a better method for calculations near critical points, near critical end points, and in the retrograde region. Since its proposal, the EAR method has been tested for a variety of binary and ternary systems with promising results.17-19 Ease of implementation in calculation algorithms and fast and reliable convergence are among other advantages of the EAR method. The purpose of this work is to extend the EAR method to calculate the phase equilibrium in polymer 1/polymer 2/solvent and polymer/solvent 1/solvent 2 systems. Throughout this work, subscripts 1-3 for polymer 1/polymer 2/solvent systems respectively refer to polymer 1, polymer 2, and solvent, and those for polymer/solvent 1/solvent 2 systems respectively refer to polymer, solvent 1, and solvent 2. Calculation Method The Flory-Huggins expression for the Gibbs energy of mixing of a ternary system, ∆gM, as a function of volume fractions is given as
function of x1 and x2 as
∆gM ) G+(x1,x2)
To apply the EAR method, which was initially developed for binary systems, to ternary systems, Shyu et al.18,19 introduced a new parameter D12. Parameter D12 geometriclly corresponds to the slope of the tie line in phase diagrams and is defined as
D12 )
φi )
ximi x1m1 + x2m2 + x3m3
(6)
g ) G++(x1,D12)
(7)
As shown by Shyu et al.18,19 when the slope of the tie eq , the line, D12, is close to its equilibrium value, D12 function (dG++/dx1)D12 exhibits a van der Waals loop behavior. This behavior is also shown in Figure 1. It has been proven that18,19 if a value for (dG++/dx1)D12 can be found such that the two areas L and U shown in Figure 1 are equal and the two conditions
[ ] [ ] [ ] [ ] ∂G+ ∂x1 ∂G+ ∂x2
R
)
x2 R
)
x1
∂G+ ∂x1 ∂G+ ∂x2
β
(8)
x2 β
(9)
x1
hold, the corresponding equilibrium compositions will be
{
xR1 R xR2 ) Deq 12(x1 - z1) + z2
(4)
where x is the mole fraction of component i, mi represents the ratio of the molar volume of component i to that of a reference component, and χij is the Flory interaction parameter. For a ternary system at constant temperature and pressure and subject to the mass balance constraint ∑xi ) 1, the Gibbs energy of mixing can be expressed as a
x2 - z 2 x 1 - z1
where zi is the mole fraction of component i in the feed solution. When eqs 5 and 6 are combined, the Gibbs energy of mixing can be written in terms of x1 and D12 as
φ2 φ3 ∆gM φ1 ) ln φ1 + ln φ2 + ln φ3 + (χ12φ1φ2 + RT m1 m2 m3 χ13φ1φ3 + χ23φ2φ3) (3) where T is the absolute temperature and R is the universal gas constant. The term φi in eq 3 is the volume fraction of component i defined as
(5)
and
{
xβ1 β xβ2 ) Deq 12(x1 - z1) + z2
(10)
Parameter D12 is the key parameter for the convergence of the calculation in the EAR method. The EAR method can only be successful when a van der Waals
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Figure 3. Flowchart of the algorithm of the EAR method to calculate tie lines and binodals for ternary polymer/solvent systems.
loop can be found. As mentioned above, a van der Waals loop only appears when the value of the parameter D12 eq 18,19 is close to its equilibrium value, D12 . For systems containing small molecules, it was suggested18,19 that eq the value of the parameter D12 is close to -1 and this value can serve as an initial value to begin the calculations. However, for systems containing polymers and depending on the molecular weights of the polymers eq present in the system, the value of D12 may be significantly different from -1. In this study a method is proposed to evaluate an initial value for D12. This method requires the minimization of the second derivative of the Gibbs energy with respect to mole fraction x1. The results obtained from several calculations for
polymer 1/polymer 2/solvent systems showed that the negative ratio of the molar volumes of the immiscible pair in the system, -m1/m2, is a suitable choice for searching an initial guess for D12. The flowchart of this procedure is presented in Figure 2. The initial guess for D12 obtained using this algorithm can be used to begin the flash calculation. The flowchart for the flash calculation of a feed consisting of polymer 1/polymer 2/solvent or polymer/solvent 1/solvent 2 with composition zi is presented in Figure 3. Results and Discussion The extension of the EAR method to polymer systems, proposed in this study, was applied to calculate the
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Figure 4. Calculated tie lines/binodals for a polymer 1/polymer 2/solvent system with m1 ) m2 ) 1000, m3 ) 1, χ12 ) 0.004, and the polymer-solvent interaction pairs: (a) χ13 ) χ23; (b) χ13 ) 0.4, χ23 ) 0.44; (c) χ13 ) 0.4, χ23 ) 0.46; (d) χ13 ) 0.4, χ23 ) 0.48.
Figure 5. Calculated tie lines/binodals for a polymer 1/polymer 2/solvent system with m1 ) m2 ) 1000 and m3 ) 1: (a) χ12 ) -0.004, χ13 ) 0.3, χ23 ) 0.45; (b) χ12 ) 0.002, χ13 ) 0.4, χ23 ) 0.48; (c) χ12 ) 0.0015, χ13 ) 0.4, χ23 ) 0.48.
solubility phase diagrams of several polymer 1/polymer 2/solvent and polymer/solvent 1/solvent 2 systems. The calculations were performed over a wide range of polymer molecular weight ratios and Flory interaction parameters. Some typical results are presented in Figures 4-9. Figure 4 shows the calculated tie lines and binodals for polymer 1/polymer 2/solvent systems containing polymers with equal molar volumes and the effect of the Flory interaction parameter, χ12, on the miscibility of the two polymers. In all cases the value of the Flory parameter is greater than the critical value, C : χ12
χC12 )
(x
1 2
1 m1
+
1
xm2
)
(11)
In the figure curve a corresponds to a symmetrical case with equal Flory parameters χ23 ) χ13 and shows that for a symmetrical case, as expected, the binodal is
Figure 6. Calculated tie lines/binodals for a polymer 1/polymer 2/solvent system with m1 ) 4000, m2 ) 250, m3 ) 1, χ12 ) 0.02, and the polymer-solvent interaction pairs: (a) χ13 ) 0.1, χ23 ) 0.15; (b) χ13 ) 0.15, χ23 ) 0.1; (c) χ13 ) 0.2, χ23 ) 0.3; (d) χ13 ) 0.3, χ23 ) 0.2; (e) χ13 ) 0.3, χ23 ) 0.45; (f) χ13 ) 0.45, χ23 ) 0.3.
Figure 7. Calculated tie lines/binodals for a polymer 1/polymer 2/solvent system with χ12 ) 0.02, χ13 ) 0.3, χ23 ) 0.2, m1 ) 4000, and m3 ) 1 and (a) m2 ) 150, (b) m2 ) 200, (c) m2 ) 300, and (d) m2 ) 500.
independent of χ13. As can be seen from the figure, the poor solvents, which correspond to larger χ values between the polymer and the solvent, have a more drastic effect on the miscibility of the two polymers than the good solvents. It can also be seen that an increase in the χ12 parameter reduces the miscibility of the two polymers. Figure 5 shows the solvent effect on the miscibility of two polymers in polymer 1/polymer 2/solvent systems containing polymers with equal molar volumes and χ12 parameters smaller than the critical value and χ13 * χ23. It can be seen that unsymmetrical solvent effect on the two polymers can cause immiscibility even with χ12 smaller than the critical value. Figures 6 and 7 show the calculated tie lines and binodals for polymer 1/polymer 2/solvent systems containing polymers with different molar volumes. The solvent effect is shown in Figure 6 with constant χ12 parameters and polymer molar volumes. The effect of the polymer molar volume on the miscibility of the two
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when the feed composition lies in the single-phase region but in the extrapolation of a tie line. Using the Hsu and Prausnitz method or other flash calculation methods, the feed is concluded to be a single-phase system when no solution for the set of eq 2 can be found after many rounds of iteration. However, that no solution can be found for the set of eq 2, although a necessary condition, is not sufficient to conclude that the feed is a single-phase system. Using the EAR method proposed here if the feed composition lies in the single-phase region but on the extension of a tie line, convergence can also be achieved. In this case because
z2 )
Figure 8. Calculated tie lines/binodals for a polymer/solvent 1/solvent 2 system with χ12 ) 1.0, χ13 ) 0.05, χ23 ) 0.05, m1 ) 4000, and m3 ) 1 and (a) m2 ) 1.5, (b) m2 ) 2.0, (c) m2 ) 3.0, and (d) m2 ) 6.0.
Figure 9. Calculated tie lines/binodals for a polymer/solvent 1/solvent 2 system with m1 ) 4000, m2 ) m3 ) 1, χ12 ) 0.05, and χ13 ) 0.5 and (a) χ23 ) 2.5, (b) χ23 ) 3.0, (c) χ23 ) 4.0, and (d) χ23 ) 5.5.
polymers is shown in Figure 7, where all of the Flory interaction parameters are kept constant. It can be seen from this figure that the larger the polymer molar volumes are, the more immiscible the two polymers become. The new algorithm developed in this study for calculating the initial value for D12 using the negative molar volume ratio of the immiscible pair can also be applied to the polymer/solvent 1/solvent 2 systems. Figures 8 and 9 show the results obtained for the typical phase behavior in polymer/mixed solvent systems using this new algorithm. These figures show the effect of the molar volume of the poor solvent on the polymersolvent mutual solubility and the effect of the solventsolvent interaction on the polymer partitioning in the two solvent phases. In all calculations performed using our method with feed compositions in the two-phase region, convergence was achieved without any difficulty. The advantage of our new method over other flash calculation methods proposed in the literature becomes even more clear
xβ2 - xR2 (z - xR1 ) + xR2 β R 1 x1 - x1
(12)
and z1 is either smaller than or larger than both xR1 and xβ1, it can be concluded that phase R with composition {xR1 , xR2 } is in equilibrium with phase β with composition {xβ1, xβ2} and the feed is a single-phase system. Therefore, provided that the feed composition is in the extrapolation of a tie line in the single-phase region, the EAR method provides a sufficient condition to conclude that a feed solution is a single-phase system. This is another important feature of this method. It should be noted that there is a fundamental difference between the present approach and that of Sˇ olc.16 In the present work the constant-temperature binodal curves and tie lines are calculated and the three-dimensional temperature-composition phase diagram of ternary systems can be built by bringing together all of the binodals and tie lines, whereas in Sˇ olc’s16 method the cloud-point curves, which are quasi-binary sections of a ternary multidimensional temperature diagram, are calculated. Although the cloud-point curve is a better representation of temperature-induced phase separation, it is more convenient and less computationally intensive to use the binodals and tie lines to study the phase separation induced by changing component compositions. The binodals and tie lines show not only the phase diagram boundaries but also the composition and fraction of the resulting phases and provide more information about the phase behavior of ternary systems. Conclusions The EAR method proposed by Eubank et al. for small molecules has been extended to calculate the tie lines and binodals of polymer 1/polymer 2/solvent as well as polymer/solvent 1/solvent 2 systems. The method was applied to several ternary polymer/solvent systems, and the results indicated its efficiency and reliability to calculate the phase behavior of polymer systems under different conditions. It was found that using this method convergence is achieved fast and reliably. An important feature of this method is that unlike other flash calculation methods it does not require an initial guess for the equilibrium compositions to begin the calculation process. This makes the method useful for prediction purposes in the absence of experimental data and for correlation purposes when accurate Flory interaction parameters are not available. Calculation using the EAR method also provides a sufficient condition to conclude that a feed solution is a single-phase system when its composition lies in the extrapolation of a tie line. This is particularly important to distinguish con-
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vergence to a real single-phase system from a failure of the algorithm to find a solution in a two-phase region. Literature Cited (1) Zeman, L.; Patterson, D. Effect of the Solvent on Polymer Incompatibility in Solution. Macromolecules 1972, 5, 513. (2) Scott, R. L. The Thermodynamics of High Polymer Solutions. V. Phase Equilibria in the Ternary System: Polymer 1-Polymer 2-Solvent J. Chem. Phys. 1949, 17, 279. (3) Hsu, C. C.; Prausnitz, J. M. Thermodynamics of Polymer Compatibility in Ternary Systems. Macromolecules 1974, 7, 320. (4) Okazawa, T. On the Narrow Miscibility Gap in Polymer 1-Polymer 2-Solvent Ternary Systems. Macromolecules 1975, 8, 371. (5) Robard, A.; Patterson, D. Temperature Dependence of Polystylene-Poly(vinyl methyl ether) Compatibility in Trichloroethene. Macromolecules 1977, 10, 1021. (6) Rostami, S.; Walsh, D. J. Miscibility of Ethylene-Vinyl Acetate Copolymers with Chlorinated Polyelectrolytes. 3. Simulation of the Spinodal Using the Equation of State. Macromolecules 1984, 17, 315. (7) Frufier, D.; Audebert, R. Interaction Between Oppositely Charged Low Ionic Density Polyelectrolytes: Complex Formation or Simple Mixture? In Macromolecular Complexes in Chemistry and Biology; Dubin, et al., Eds.; Springer-Verlag: Berlin, 1994. (8) Schaink, H. M.; Smit. J. A. M. Toward an Integrated Analytic Description of Demixing in Ternary Solutions of Nonideal Uncharged Lattice Polymers, Hard Spheres, and Solvents. Macromolecules 1996, 29, 1711. (9) Schaink, H. M.; Smit. J. A. M. Mean Field Calculation of Polymer Segment Depletion and Depletion Induced Demixing in Ternary Systems of Globular Proteins and Flexible Polymers in a Common Solvent. J. Chem. Phys. 1997, 107, 1004. (10) Kang, C. H.; Sandler, S. I. Phase Behaviour of Aqueous Two-Polymer Systems. Fluid Phase Equilib. 1987, 38, 245.
(11) Durrani, C. M.; Prystupa, D. A.; Donald, A. M.; Clark, A. H. Phase Diagram of Mixtures of Polymers in Aqueous Solution Using Fouier Transform Infrared Spectroscopy. Macromolecules 1993, 26, 981. (12) Yu, M.; de Swaan Arons, J. Phase Behaviour of Aqueous Solutions of Neutral and Charged Polymers. Polymers 1994, 35, 3499. (13) Figari, G.; Costa, C. On the Reduction of the Number of Unknowns in Flory-Huggins Calculations of Polymeric Equilibria. Macromol. Theory Simul. 1994, 3, 649. (14) Boom, R. M.; van dan Boomgaard, Th.; Smolders, C. A. Equilibrium Thermodynamics of a Quaternary Membrane-Forming System with Two Polymers. 1. Calculations. Macromolecules 1994, 27, 2034. (15) Sargantanis, I. G.; Karim, M. N. Prediction of Aqueous Two-Phase Equilibrium Using the Flory-Huggins Model. Ind. Eng. Chem. Res. 1997, 36, 204. (16) Sˇ olc. K. Phase Separation in Ternary Systems SolventPolymer 1-Polymer 2. 1. Cloud Point and Critical Concentration. Macromolecules 1986, 19, 1166. (17) Eubank, P. T.; Hall, K. R. Equal Area Rule and Algorithm for Determining Phase Compositions. AIChE J. 1995, 41, 924. (18) Shyu, G. S.; Hanif, N. S. M.; Alvarado, J. F. J.; Hall, K. R.; Eubank, P. T. Equal Area Rule Methods for Ternary Systems. Ind. Eng. Chem. Res. 1995, 34, 4562. (19) Shyu, G. S.; Hanif, N. S. M.; Alvarado, J. F. J.; Hall, K. R.; Eubank, P. T. Maximum Partial Area Rule for Phase Equilibrium Calculations. Ind. Eng. Chem. Res. 1996, 35, 4348.
Received for review January 24, 2000 Revised manuscript received June 7, 2000 Accepted June 13, 2000 IE000088P
