Binodal Curves in the Flory-Huggins Model " Qualitative Results from Equation-Of-State Theory Geometric Mean Mixing Rule

Chemical Engineering 160/260 Polymer Science and Engineering LectureLecture 1010 -- PhasePhase EquilibriaEquilibria andand PolymerPolymer BlendsBlends FebruaryFebruary 7,7, 20012001 Thermodynamics of Polymer Blends: Part 2 Objectives ! To develop the classical Flory-Huggins theory for the free energy of mixing of polymer solutions based on a statistical approach on a regular lattice. ! To describe the criteria for phase stability and illustrate typical phase diagrams for polymer blends and solutions. Outline ! Phase Equilibria " Free energyenergy ofof mixingmixing forfor aa phase-separatedphase-separated systemsystem " Spinodal and binodal curves in the Flory-Huggins model " Qualitative results from equation-of-state theory Geometric Mean Mixing Rule For the systems in which ∆Hm > 0, it is common to express ∆Hm in terms of the cohesive energy density, or solubility parameter. Analyze by first considering the (1,1) and (2,2) interactions. 1 Change in internal energy of ∆UZNw= vaporization of one mole of (i) vi, 2 A ii To evaluate the (1,2) cross interactions, invoke the geometric mean mixing rule: www12≅ 11 22 This approximation will be most accurate when both the (1,1) and (2,2) forces are either London dispersion or dipole-dipole. It breaks down for hydrogen bonding or strong specific interactions. −∆ww=+11 w 22− 2 ww 11 22 2 −∆ww= ()11− w 22 Cohesive Energy Density 1 Recall that ∆∆HZNw= ϕϕ m 2 12 ∆∆UHPVvv= − ∆ 1 1 2 2 2 UHPVV ∆∆U U ∆∆vvg= −−()l −∆w = vv,,1 − 2 1 1 ZN ZN ∆∆UHPV≅− 2 2 vvg ∆∆UHRTvv≅− Rewrite on a “per unit volume” basis 1 1 2 Cohesive 2 2 ∆Uvi, ∆∆Uv,,1 Uv 2 o energy ∆HVm = ϕϕ12 o − o V V Vi density 1 2 (CED) Solubility Parameter 1 2 ∆Uvi, 12/ o ≡ ()CED ≡ δi Vi Solvent CED (cal cm-3) δ (cal cm-3)1/2 Cyclohexane 67.2 8.2 Carbon tetrachloride 74.0 8.6 Toluene 79.2 8.9 Benzene 84.6 9.2 Methyl acetate 92.2 9.6 Acetone 98.0 9.9 Cyclohexanone 98.0 9.9 Acetic acid 102.0 10.1 Cyclohexanol 130.0 11.4 Methanol 210.3 14.5 Water 547.6 23.4 Polymer Solubility Parameters Solubility parameters for polymers must be determined indirectly, typically by intrinsic viscosity measurements of solutions or by swelling of a crosslinked polymer. Polymer δ (cal cm-3)1/2 Polyisobutylene 7.5-8.0 Polyethylene 7.7-8.2 Natural rubber 8.1-8.5 Polystyrene 9.1-9.4 Poly(ethylene terephthalate) 9.3-9.9 Polyacrylonitrile 12.0-14.0 Nylon-6,6 13.5-15.0 Criterion for Phase Equilibrium Consider the increment in Gibbs free energy G associated with transfer of dni moles of (i) from phase α to phase β at constant temperature T and pressure P. dG= VdP− SdT+ µ dn ∑i i i At equilibrium dG==0 dGα + dGβ 0 =+µµααdnββ dn ∑∑i iii i i 0 α β dnα = ∑()µµii− i i α β µµii= Free Energy of Mixing for a Single Phase o Recall ∆∆∆Gxm =+11µµ x 2 2where ∆µµµiii= − ∆∆∆∆∆∆Gxxm = ()1 − 2122µµµµµ+=+ 122 x()− 1 0 x2 1 ∆Gm Plot ∆Gm vs x2: P ∆µ1 Slope = ()∆∆µµ21− Intercept at x = 0 is 2 ∆µ1 ∆µ2 Single Phase at P Plot ∆Gm vs x1: Slope = −−()∆∆µµ21 ∆∆GHTSmm= − ∆ m ∆Gm < 0 Intercept at x1 = 0 is ∆µ2 <0 >0 ∆Gm for a Phase-Separated System Suppose that ∆Hm > 0 and ∆Sm > 0. This may lead to the following: 0 x2 CD1 ∆Gm ∆µ2 B A Q ∆µ 1 P The common tangent to the ∆Gm curve at P and Q implies that P P Q Q phases of compositions (x2 , x1 ) and (x2 , x1 ) have the same P Q P Q values of ∆µ1 and ∆µ2, i.e. µ1 = µ1 and µ2 = µ2 . Thus, P and Q are in equilibrium, and any mixture having composition between P and Q will phase separate into phases with P Q compositions x2 and x2 . Regions of Phase Stability Stable Mixtures For C-P and Q-D regions, all mixtures are stable against separation into different phases. Metastable Mixtures For P-A and B-Q regions, where A and B denote inflection points, 2 ∂ ∆Gm 2 > 0 ∂x2 and the mixture will be stable against separation into neighboring phases differing only slightly in composition but not against separation into phases of composition P and Q. Unstable Mixtures 2 ∂ ∆Gm For the A-B region, 2 < 0 ∂x2 and all mixtures are unstable to infinitesimal perturbations. Spinodal and Binodal Curves Spinodal Curve The boundary between the absolutely unstable and the metastable regions is defined by 2 ∂ ∆Gm 2 = 0 ∂x2 The locus of all such A, B inflection points as a function of temperature generates the spinodal curve. Binodal Curve The locus of all points of common tangency P and Q as a function of temperature generates the binodal curve. Critical Point The critical point is characterized by convergence of the points of common tangency and the inflection points such that 3 ∂ ∆Gm 3 = 0 ∂x2 Phase Equilibria in Polymer Solutions 2 ∂ ∆Gm 2 = 0 ∂x2 Spinodal Curve T 3 Binodal ∂ ∆Gm 3 = 0 Curve ∂x2 To Critical Point x2 Phase Behavior for the Flory-Huggins Model GRTN xln x ln ∆ m =++[]ϕϕ12 χ() 1 ϕ 1 2 ϕ 2 ∂∆Gm ∂∆Gm = µ1 = µ2 ∂N1 ∂N2 nN Note that NV= ϕ 2 11and ϕ2 = Binodal Curves NnN12+ o 1 2 µµ11− = RT ln(11− ϕϕ 2 ) + 2− + χϕ 2 n o 2 µµ22− =+RT[]ln()ϕϕ 211−− 2() n+ χϕ n () 1− 2 Note that differentiation with respect to x2 is equivalent to ϕ2. ∂ ∂ϕ ∂ Also = ∂xx∂ ∂ϕ Phase Behavior for the Flory-Huggins Model Spinodal Curve 2 ∂ ∆∆Gm ∂ µ1 −1 1 2 ==0 = +1 − + 2χϕ2 ∂xn2 ∂ϕ221 − ϕ Critical Point ∂ 3∆∆G ∂ 2 µ −1 m ==0 1 = + 2χ ∂x3 ∂ϕ 2 2 c 2 2 ()1 − ϕ2,c 1 1 1 1 − ϕ1,c χc = 2 − +1 − + 2 = 0 2ϕ1,c ϕ1,c n ϕ1,c 1 2 χ =+1 n−12/ c 2 () Post Flory-Huggins Thermodynamics • Flory equation-of-state and Sanchez lattice fluid theories • LCST behavior is characteristic of exothermic mixing (which could arise from specific chemical interactions) and negative excess entropy (which arises due to densification of the polymers on mixing). Neither one of these phenomena is included in the Flory-Huggins theory. Typical Phase Diagrams Typical for Typical for Polymer Blend Polymer Solution Upper Critical Solution Temp. T T Binodal Lower Spinodal Critical Solution Temp. Composition Composition Mechanisms of Phase Separation Nucleation and Growth • Initial fragment of more stable Nucleation and growth phase forms between the binodal and • Free energy determined by work spinodal curves required to form the surface and the work gained in forming the interior • Concentration in the immediate T vicinity of the nucleus is reduced and diffusion is downhill (diffusion coefficient is positive) • Droplet size increases by growth initially Composition • Requires activation energy Mechanisms of Phase Separation Spinodal Decomposition Spinodal decomposition inside the spinodal curve • Initial small-amplitude composition fluctuations • Amplitude of wavelike composition fluctuations increases with time T • Diffusion is uphill from the low concentration region into the domain (negative diffusion coefficient) • Unstable process; no activation energy required Composition • Phases are interconnected at early time.

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