Liquid-Liquid Phase Equilibria of Polymer Solutions and Blends by Simplified Lattice Fluid Model

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1 HWAHAK KONGHAK Vol. 4, No. 6, Deember, 2003, pp ( , ) Liquid-Liquid Phase Equilibria of Polymer Solutions and Blends by Simplified Lattie Fluid Model Hun Yong Shin Department of Chemial Engineering, Seoul National University of Tehnology, 72, Kongneung-2-dong, Nowon-gu, Seoul , Korea (Reeived 2 July 2003; aepted 2 September 2003) MF-NLF(multi-fluid nonrandom lattie fluid) - (liquid-liquid equilibrium)!. MF-NLF " #$%& '(!' )*+, - './02 34 r-mer+ 5 6,7 +89: A@B4: CDE!. - F GH IJK(binodal) GH LMN OPQ: R S H A@B4: -!. 5PE T" UV W. XY+& Z" [: \, V W.]H ^9> H A@B4: 2, - LM_`7 [^ a b!. Abstrat Simplified lattie model was extended to alulate LLE (liquid-liquid equilibrium) of polymer solutions and blend. The EOS (equation of state) was derived based on the multi-fluid approximation of the nonrandom lattie fluid theory. The EOS requires two moleular parameters for a pure r-mer whih representing moleular sizes and energies. The liquid-liquid phase equilibriums were alulated with the binary parameters whih were obtained by optimization of experimental binodal ompositions. Temperature dependent binary parameters were used for more good results for wide range of temperatures and pressures. The model gives good orrelations for the seleted LLE of polymer solutions and blend systems. Key words: Lattie Theory, Equation of State, Polymer Solution, Liquid-Liquid Equilibira.! " #$% &' ( )*+,-. $/ 0* 34/ , :; <= 2>?@ A* B #C DEF, GH IJK L-L 3MN * 34/5 BD O P%Q RD 2[-3]. ST*, : 7 UV L-L 3MN * B #C WXF YZ [ 34/5 \N ] ^_ 2> K #`a, \b* ] e fg L3 %hijv kl* m= nbop q Flory, Huggins[4-6] UNIQUAC ^_[7] r \N ks t u=(hole)q v ] wx ya+ #/ To whom orrespondene should be addressed. hyshin@snut.a.kr Sanhez Laombe[8, 9] LF ^_z, < {F Or IJK * + } #~ ^_,wop q2[0-6]. You [4-6], NLF (nonrandom lattie fluid) 34/5 Helmholtz * ~Q y BL(athermal XF * ~ ƒ * & 2 ~ ˆ 2 Š Q ddo 2 ~ B #CD9 Œ D Yoo [7-2] NLF(nonrandom lattie fluid) 34/5 y (athermal part)% ŽO (residual part)xf JD, f g N4Q #~ ŽO Wilson[22], 2-\bTn* DO N4Q #~ MF-NLF(multi-fluid nonrandom lattie fluid) 34/5, D / 3MN C #~ + N4 ^_ ~ p *+ \BD MF-NLF 34 /5 š œa% 3MN, 3MN, 3MN* ^_K% žÿdo N4Q #~ + P%Q ROY IJ*+ MF-NLF 34/5 B Q DO -B ƒ Lb-Lb 3MN " Lb-Lb 3MN* B 77

2 78 DO )* B#C D IJ MF-NLF 34/5 NLF(nonrandom lattie fluid) 34 /5[4-6] ŽO (residual part) 2-\bTn* DO N4Q 9Œ D \9%/ ª«[7-2]*+ Xz, 34/5% U 2±% ² kl( P z βv H 2 -- q ln M rm z τ ρ ln( ρ) -- θ 0i = + 2 i i = θ k τ k = 0 ki () µ i = λ RT i ( T) r i ln( ρ) θ --- i q i + ln + r iln rm ρ zq i θ r i θ ln θ 2 q i θ k τ ki βε l ( τ il τ 0l ( r q i i )) + ii k = 0 l = θ k τ k = 0 kl O@+ q M = x q i i, r M = x r i i, ρ = ρ i, ρ i = V *, * i V V i = N a r i V H, τ ji = exp{ βε ( ji ε ii )} 3. MF-NLF 34/5 4 ƒ F³ (z), ]?@(V H ),?@ƒ (r ) <= * ~ƒ (ε ij )Q #µ (z) 0XF P/D, ]?@(V H ) 9.75 m 3 mol X F 3F ¹º+ œ\b* DO,»¼s ƒ r % ε P/ ½ ƒ r % ε 89*+ ¾ À * DO Á( {, 2± 89* & VF³ kl( q M (2) 4. - GÍs L-L 3MN ª, 3=# Gp#Q P/D, 3 Î JD W* IJ*+ ] \b 34/5 B BL L-L 3MN %/ Lee Danner[]#, BD Ypµ 89 :;*+, L3* Gibbs * ~# G3* Gibbs * ~R2 Ï #~ 3=# GpÐ ( 3=# GpÑ { 3 Îs ÒÓÔ(binodal) Î ÎÕ Ö ² ^Ø Ù(i=, 2,..., N)* DO L3 I% II U (hemial potential) ² W* µ i I = Ú, µ i II I II µ i = µ i i=, 2,..., N (6) i=, 2,..., N (7) O@+, µ i œ% i U Q ÛDz, 5 (2)Q d + Ü jåóô Î(spinodal omposition) 2± &5 BDO Ü µ i x TP, = 0 L-L 3MN Z Ypµ 89 :;*+ MN34* L3 Î W*, 3Ý* ¹Þ, (binary system) ÂÃ \9 2 ¹º+ 3MNÎ L-L 3MN &5s 5 (6) BDO J Íf ddo Or ÎÕ 89*+ L3 jåóô% Ò ÓÔ Î J jåóô% ÒÓÔ ß= à Fig. * Èá (8) ε k = E a + E b ( T T 0 ) + E [ Tln( T 0 T) + T T 0 ] r = R a + R b ( T T 0 ) + R [ Tln( T 0 T) + T T 0 ] (3) (4) O@+ T 89F³ K š A ÂÃ 8 9 Ä@:% LbU HÅ aæq BDO Ypµ 89 ƒ Q P/ A ÂÃ œ P-V-T wçaæq BDO Or 89*+ ƒ Q J { 89* & VF³ ÈÉ œ P-V-T ÊÇaÆ Zoller Walsh[23], Rogdgers [24] ª«* Ro Ë, * BD@ + 2±5XF /o ƒ, λ 2 Q ÌF ε 2 = ( ε ε 22 ) 2 ( λ 2 ) (5) O@+ λ 2 œ 3MN ÊÇaÆQ B S ddo P/ Fig.. Algorithm for spinodal and binodal ompositions of polymer solutions. Table. Energy and size parameters of MF-NLF EOS [25] E a E b E R a R b R K - - m 3 /mol * or, m 3 /g ** m 3 /mol K m 3 /mol K Pentane * Hexane * Polyisobutylene ** Polyethylene ** Polybutadiene ** Polystyrene **

3 - 79 Fig. Calulated and experimental temperature-pressure-volume relations of PIB. Fig. 3. LLE of PE (polyethylene)/hexane solution. 5. IJ * nb( š A " A œ * ~ " HÅ ƒ Q Table [25]* Èá PIB(polyisobutylene) PVT P%Q ÊÇaÆ[24] žÿ Fig. 2*+ ß â, -00 MPa :; K 89 * DO œ ƒ Q Ü 4 /Bƒ / L-L 3MN * nb( 89 #~ ƒ ãºûæ Mä å Q Table 2* Èá Fig. 3 PE(polyethylene)/Hexane [26] L-L 3MN P% 3= Îs ÒÓÔæ ÊæXF Èá / çèî s jåóôæ éæxf Èá L-L 3MN *+ 3XF *+ êëjr` P%Q Ü@ DO ƒ * 89 HOD 89 # ~ ƒ Q nbdo LCST(lower ritial solution temperature) ìí R * DO ÊÇP% P%# î G D W ï L-L 3MN * m= n Bo %h ij* ~ ^_K nb ÂÃ* :;* vo~ ðxef, : * :; 7ñ òó v ô 34/5 BD W ÈõD Fig. 4 PIB Fig. 4. LLE of PIB (polyisobutylene)/n-pentane system. Table Binary parameters and error % for liquid-liquid equilibria of polymer solution System Range Fitting parameter for binary parameter (λ 2 =a+bt+t 2 ) Error % T(K) P(MPa) a b P T Ref. PE/hexane [26] PIB/pentane [27] PIB/hexane [27] PE/pentane [27] PBD/PS [28] P = --- ( P exp P al /P exp ) 00 N T = --- ( T exp T al /T exp ) 00 N (N: number of data) HWAHAK KONGHAK Vol. 4, No. 6, Deember, 2003

4 720 Fig. 5. LLE of PIB (polyisobutylene)/n-hexane system. Fig. 7. LLE of PBD (polybutadiene)/ps(polystyrene) blend. P%Q Ü 6. N4 ]\b 34/5s MF-NLF 34/5 B DO /Bƒ % L-L 3MN D L-L 3MN UCST LCST ìí òó kl Xz, öó : UV L-L 3MN * BDO :; 7ñ òó kl ª +ø)ÿ eijž ~ * DO IJo Xz, * ùn ¼!"# Fig. 6. LLE of PE (polyethylene)/n-pentane. (polyisobutylene)/n-pentane [27] L-L 3MN P% K MPa :; 7 L-L 3ì í aæ ÊÇaÆ# î G V s Fig. 5 PIB (polyisobutylene)/n-hexane[27]* ÊÇP%(89 : K, :; :.-5. MPa) P%Q žÿ W, Fig. 6 PE(polyethylene)/n-Pentane L-L 3MN ÊÇ P%[27](89 : K, :; : MPa) P%Q žÿdo 9 wd ÊÇP% 3= Î% ( ÒÓÔ Î î G V ß Fig. 7 PBD(polybutadiene)/PS(polystyrene) [28] L-L 3 MN P%Q ÊÇP% žÿ W UCST(upper ritial solution temperature) ìí R * DO ÒÓÔ Î% jå ÓÔ Î D 0.3 KPa :;% 0.3 MPa :* DO íg ƒ Q BDO êëê ÒÓÔ Î N a : avogadro number P : pressure [Pa] q i : surfae area parameter r i : segment number of moleule [i] R : universal gas onstant [J mol K ] T : temperature [K] V H : volume of unit ell, 9.75 [m 3 mol ] z : lattie oordination number [z=0] β : /kt (J ) ε ij : interation energy for i-j segment ontats [J] λ i : part of hemial potential due to internal degrees of freedom of omponent [i] λ ij : binary interation parameter for i-j ontats µ i : hemial potential for omponent i [J mol ] θ : surfae area fration

5 - 72 ρ : redued density defined by [ ρ = N i r i N r ] i = : nonrandomness fator τ ij i, j, k, l : omponents i, j, k and l ij : interation pairs, [i-j] M : property of mixture 0 : hole $%&. Lee, B. C. and Danner, R. P., Group-Contribution Lattie-Fluid EOS: Predition of LLE in Polymer Solutions, AIChE J., 42(), (996). Wang, W., Tree, D. A. and High, M. S., A Comparison of Lattie- Fluid Models for the Calulation of the Liquid-Liquid Equilibria of Polymer Solutions, Fluid Phase Equilib., 4, 47-62(996). 3. Folie, B. and Radosz, M., Phase Equilibria in High-Pressure Polyethylene Tehnology, Ind. Eng. Chem. Res., 34, 50-56(995). 4. Flory, P. J., Thermodynamis of High Polymer Solutions, J. Chem. Phys., 9, (94). 5. Flory, P. J., Thermodynamis of Polymer Solutions, Dis. Faraday So., 49, 7-29(97). 6. Huggins, M. L., Thermodynami Properties of Solutions of Long- Chain-Compounds, J. Phys. Chem., 46, 5(942). 7. Abrams, D. S. and Prausnitz, J. M. Statistial Thermodynamis of Liquid Mixtures: A New Expression for the Exess Gibbs Energy of Partly or Completely Misible Systems, AIChE J., 2, 6-28(975). 8. Sanhez, I. C. and Laombe, R. H., An Elementary Moleular Thoery of Classial Fluids. Pure Fluids, J. Phys. Chem., 80, (976). 9. Sanhez, I. C. and Laombe, R. H., Statistial Thermodynamis of Fluid Mixtures, J. Phys. Chem., 80, (976). 0. Okada, M. and Nose, T., Quasi-Chemial Treatment of the Hole Theory for r-mers. I. Pure Liquids, Polymer J., 3, (98).. Okada, M. and Nose, T., Quasi-Chemial Treatment of the Hole Theory for r-mers. II. Mixtures, Polymer J., 3, (98). Panayiotou, C. and Vera, J. H., Statistial Thermodynamis of r-mer Fluids and Their Mixtures, Polymer J., 4, (982). 3. Kumar, S. K., Suter, U. W. and Reid, R. C., A Statistial Mehanis Based Lattie Model Equation of State, Ind. Eng. Chem. Res., 26, (987). 4. You, S. S., Yoo, K. -P. and Lee, C. S., Modeling of Superritial Fluid-Phase Equilibria using A new Nonrandom Lattie-Fluid Theory, J. Superrit. Fluids, 6, 69-84(993). 5. You, S. S., Yoo, K. -P. and Lee, C. S., An Approximate Nonrandom Lattie Fluid Theory of Fluids. General Derivation and Appliation to Pure Fluids, Fluid Phase Equil., 93, 93-23(994). 6. You, S. S., Yoo, K. -P. and Lee, C. S., An Approximate Nonrandom Lattie Fluid Theory of Fluids. Mixtures, Fluid Phase Equilib., 93, (994). 7. Yoo, K.-P., Shin, H. Y. and Lee, C. S., Approximate Nonrandom Two-Fluid Lattie Hole Theory. Thermodynami Properties of Real Mixtures, Bull. Korean Chem. So., 8(8), (997a). 8. Yoo, K.-P., Shin, H. Y. and Lee, C. S., Approximate Nonrandom Two-Fluid Lattie Hole Theory. General Derivation and Appliation of Pure Fluids, Bull. Korean Chem. So., 8(9), (997b). 9. Yoo, K.-P., Shin, H. Y. and Lee, C. S., A New Two-Fluid Quasilattie Equation of State for Calulating Physial Properties and Superritial Phase Equilibria, J. Superrit. Fluids, 3, 55-60(998). 20. Shin, H. Y., Yoo, K.-P., Lee, C. S., Tamura, K. and Arai, Y., Rigorous and Simplified Lattie-Hole Equations of State for Calulating Speifi Volumes of Common Pure Polymers, Korean J. Chem. Eng., 5(), 5-9(997). Yoo, K.-P. and Lee, C. S., Redisovering the Lattie-Fluid Theory for Phase Equilibria of Complex Mixtures Redisovering the Lattie- Fluid Theory for Phase Equilibria of Complex Mixtures, Korean J. Chem. Eng., 7(3), (2000). 2 Wilson, G. M., Vapor-Liquid Equilibrium. XI. A New Expression for the Exess Free Energy of Mixing, J. Am. Chem. So., 86, (964). 23. Zoller, P. and Walsh, D. J., Standard Pressure-Volume-Temperature Data for Polymers, Tehnomi Pub. Co. In. Basel(995). 24. Rodgers, P. A., Pressure-Volume-Temperature Relationships for Polymeri Liquids: A Review of Equation of State and Their Charateristi Parameters for 56 Polymers, J. Applied Polymer Siene, 48, (993). 25. Shin, H. Y., Generalized Formulation of New Lattie Theory for Phase Equilibria of Complex System, Ph.D. Dissertation, Sogang University, Seoul(999). 26. Kleintjens, L. A. and Koningsveld, R., Liquid-Liquid Phase Separation in Multiomponent Polymer Systems XIX. Mean-field Lattiegas Treatment of the System (n-alkane/linear-polyethylene), Colloid & Polymer Si., 258, 7-78(980). 27. Lee K. J., Shin, H. Y., Kim, H. S., Joung, S. N., Kim, S. Y. and K.-P., Phase Behavior of Polymer in Superritial Fluid, Theories and Appliations of Chem. Eng., 8(2), (2002). 28. Rostami, S. and Walsh, D. J., Simulation of Upper and Lower Critial Phase Diagrams for Polymer Mixtures at Various Pressures, Maromoleules, 8, (985). HWAHAK KONGHAK Vol. 4, No. 6, Deember, 2003

A multi-fluid nonrandom lattice fluid model: General derivation and application to pure fluids

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