1 Prof. Dr. rer. nat. habil. S. Enders Faculty III for Process Science Institute of Chemical Engineering Department of Thermodynamics Lecture Polymer Thermodynamics 0331 L 337
6.1. Introduction x n = x n + x n n = n + n F F I I II II F I II A A A F I II I I II II xa xa n x ( n + n ) = x n + x n = x x n F I I A A A II I II A A 30 critical point T / K 310 300 90 80 stable instable spinodale stable binodale P = constant 70 60 tie line metastable metastable 50 0.0 0. 0.4 0.6 0.8 1.0 X A
phase equilibrium conditions Polymer Thermodynamics 6.1. Introduction 3 μ μ = μ L1 L A A = μ L1 L L1 L P P P T,P=constant = = S instability limit ΔMIX gx x ( ) critical conditions = 0 free mixing enthalpy I S 1 II ( ) 3 ΔMIX gx 3 x = 0 composition
6.1. Introduction 4 μ ( T, P, x ) = μ ( T, P) + RTlnx + RTlnγ i i 0i i i μ ( T, P) + RT ln x + RT ln γ = μ ( T, P) + RT ln x + RT ln γ L1 L1 L1 L L L 0A A A 0A A A ( ) ( x = x ) A A L1 L1 L L L1 L1 L L ln xa + ln γ A = ln xa + ln γ A ln Aγ A ln γ x γ = x γ L1 L1 L L A A A A g E -model for the determination of the activity coefficients is necessary i.e. g ln γ = A( T) x E A( T) xax A( T) xa( 1 xa) RT = = A ln γ = A( T) x A
6.1. Introduction L1 x ( ) ( ) ( ) L1 L L L1 AT x L x x x = + x ln ( ) L1 x ( ) ( ) ( ) A L1 L L L1 AT x L A xa xa x = + A x A ln ( ) 5 system of equations made from equations quantities can be calculated 3 possibilities: 1) T and x A L1 fixed x A L ) T and x A L fixed x A L1 3) x A L1 and x A L fixed T
spinodale Polymer Thermodynamics 6.1. Introduction ΔMIX gx ( ) = 0 x ( ln ln ln γ ln γ ) Δ g = RT x x + x x + x + x MIX A A A A ΔMIX g 1 1 1 1 ( ) = + AT ( 1 xa) AT ( ) xa = + xa xa 1 xa xa 1 xa critical point 3 ΔMI ln γ = A( T) x ln γ = A( T) x 6 A A X gx ( ) 1 1 3 = + = 0 ( ) x xa 1 x A A( T ) = 0 AC( T) = A( T) > AC( T) spontaneous demixing AT ( ) < A ( T) stable or metastable C x C A = 1
7 Problems: 6.1. Introduction 1. Why are polymers usually do not dissolve in normal solvent?. Do we have phase separation (LLE) for polymer solutions? 3. How depends phase separation on the molecular weight? Differences between low-molecular weight mixtures and polymer solution low-molecular weight mixture i.e. benzene + toluene both components have approximately the same size the structure of both components do not depend on concentration polymer solution i.e. PS in cyclohexane both components show a big difference in size additional parameter molecular weight distribution structure of polymer depends on solvent concentration
6.1. Introduction 8 Taking into account the differences in size: division of polymers into segments r i = L L i standard L standard = property of an arbitrarily chosen standard segment (i.e. solvent or monomer unit) Properties 1. possibility L= molar volume disadvantage: volumes depend on temperature. possibility L = molar mass 3. possibility L = degree of polymerization 4. possibility L = van der Waals-volume (quantity b in the van der Waals equation of state) advantage: available by experiments using X-ray diffraction or by prediction from molecular data (group-contribution method by ondi) do not depend on temperature
6.1. Introduction Taking into account the differences in size: division of polymers into segments segment-related thermodynamic quantities (segment-molare quantities) polymer solution made from solvent (A) + polymer () x A rx A A = z = rx A A + rx rz A A + rz z r i = L L i standard 9 i.e. cyclohexane + polystyrene M PS =100000g/mol x =0.00001 L standard = molar mass of solvent =84.16g/mol -1 r PS = 100000 g / mol 84.16 g / mol 1188. rch 1 84.16 g / mol = = 84.16 g / mol = 1188.*0.00001 x PS = = 0.01174 1*(1 0.00001) + 1188.*0.00001
6.1. Introduction 10 the structure of polymer depends on concentration diluted polymer solution no entanglements of the polymer chain semi-diluted polymer solution, entanglements of the polymer chain overlapping concentration c
P [a.u.] Polymer Thermodynamics 6.1. Introduction vapor pressure of polymer solution 11 16 1 8 4 CH/PS 33 ideal mixture 65 C 55 C 45 C 35 C very strong derivation from the behavior of ideal mixture? 0 0. 0.4 0.6 0.8 1 φ PS exp. data:. A. Wolf, et al. University of Mainz, 1995
6.1. Introduction 1 phase diagram cyclohexane + PS at different molecular weights in [kg/mol] of PS example: M PS =37000 g/mol w PS =0,1 T=80 K Experimental result: demixing Δmix xps g/ RT > = < 0 ( meta) stable spinodale instable
6.1. Introduction real low-molecular weight mixture 13 WW A WW AA WW model of regular mixture Hildebrand (1933) The real behavior can be explained completely by enthalpic effects. s = 0 g = h E E E WW derivations from the behavior of an ideal mixture are caused by intermolecular interaction = WW WW A AA
Hildebrand model of regular mixture The real behavior can be explained completely by enthalpic effects. φ = i D A = s = 0 g = h E E E k i= 1 x v i 0i x v i 0i ( δ δ ) A RT 6.1. Introduction D CHPS E g RT = vφφd A A k E v = 0 v= xv = xv δ i solubility parameter i i0 i i i= 1 i= 1 δ CH =16.7 (J/cm 3 ) 1/ δ PS =17.4 (J/cm 3 ) 1/ ( ) 16.7 17.4 JmolK mol = = 0.0001 3 3 cm 8.314J 80K cm k 14
E g RT Polymer Thermodynamics 6.1. Introduction Hildebrand model of regular mixture The real behavior can be explained completely by enthalpic effects. mol D = 0.0001 cm = vφφ A DA CHPS 3 15 Δ g/ RT 1 1 D v v Mix A 0A 0 = + 3 x 1 x x v Table book: v v CH PS 3 cm g = 111 ρps = 1.0556 3 mol cm 3 3 M PS 37000gcm cm = = = 35051 ρ mol 1.0556g mol PS v= xch vch + xpsvps < 0 example: M PS =37000 g/mol w PS =0,1 T=80 K
6.1. Introduction 16 x x example: M PS =37000 g/mol w PS =0.1 T=80 K 3 3 mol cm cm DCHPS = 0.0001 v 111 35051 3 CH = vps = cm mol mol v= x v + x v ΔMixg/ RT 1 1 DCHPSvCH vps 0 = + < 3 xps 1 xps xps v PS PS mps wpsm wps nps MPS MPS MPS = = = = n m m w m w m w w PS + nch + + + M M M M M M PS CH PS CH PS CH PS CH PS CH PS CH 3 cm ΔMixg / RT = 0.000567 v=119.88 63. 65 m = ol xp S CH CH PS PS interaction are not the reasons for the strong real effects conflict to experimental findings
real polymer solution Polymer Thermodynamics 6.. Flory-Huggins Theory model of ideal-athermic mixture Flory-Huggins mixture J.P. Flory (1930) M.L. Huggins 17 The real behavior can be explained completely by entropic effects. h = 0 g = Ts E E E WW A =WW AA =WW derivation from ideal behavior caused by differences in size
model of ideal-athermic mixture h = 0 g = Ts E E E 6.. Flory-Huggins Theory WW A =WW AA =WW 18 monomer solution polymer coil polymer solution Polymer chains will be arranged on lattice sites of equal sizes (i.e. v 0A ). The number of occupied lattice site depends on segment number r. oltzmann s definition of entropy S = k ln Ω k oltzmann s constant Ω thermodynamic probability
6.. Flory-Huggins Theory 19 Statistical Interpretation of Entropy Number of possible configuration for 3 distinguishable particles in 3 rooms ε 3 ε ε 1 probability: Number of possible configurations corresponds to the permutations of the number of particles: P=N! = 3**1 = 6 W N! = N! N! N! N! 1 3 W is called thermodynamic probability and it is the number of possible configuration for a given macroscopic state. i
6.. Flory-Huggins Theory 0 Example: distribution of 4 particles with identical energy in 4 volumes. A W A = 4!/(4!0!0!0!)= 4!/4! =1 W = 4!/(3!0!1!0!)= 4!/3! =4 C D W C = 4!/(!1!1!0!)= 4!/! =1 W D = 4!/(1!1!1!1!)= 4!/1! =4 The macroscopic state D can be realized by the most configurations. The macroscopic state D has the highest thermodynamic probability.
Model of ideal-athermic mixture 6.. Flory-Huggins Theory E E E h = 0 g = Ts S = k ln Ω 1 Ω represents the number of distinguishable possibilities of the system to fill the lattice places with particles. size of the lattice place: v 0A Every lattice place is occupied either with polymer segment or with a solvent molecule. The polymer segments of a polymer chain are connected by chemical bond and hence they can not move freely. N A -solvent molecules + N - polymer chains (every chain consist of r-segments) number of total lattice places: N 0 =N A +r*n every lattice place is surrounded by q next neighbors (coordination number) cubic lattice q=6
model of ideal-athermic mixture 6.. Flory-Huggins Theory E E E h = 0 g = Ts S = k ln Ω the first polymer chain has ν 1 possibility to arrange ν ( ) 1 N0 q q 1 r = number of empty lattice places N f, if the lattice is filled with i-1 polymer chains N f = N0 ( i 1) r the probability to find an empty place is equal to N f /N 0 number of possibilities to arrange f for the chain i: ν i = N f q( q 1) Ω= i= N i= 1 N υi r! N N 0 r 1 r x Δ s = R (1 x ) ln(1 x ) + ln x r Mix
model of ideal-athermic mixture 6.. Flory-Huggins Theory x Δ s = R (1 x ) ln(1 x ) + ln x r Mix x Δ g = RT (1 x ) ln(1 x ) + ln x r MIX Δ / ( ) 1 ( ) MIX g RT 1 = ln 1 x 1+ ln x + x r r ( ) 1 x ( ) Δ rx MIX g RT + / 1 1 1 = + = 0 = 0 x = x 1 x 1 r 1 rx x rx 3 ideal-athermic mixture can not demixed r ³1 negative segment mole fraction is nonsense
model of real polymer solutions 6.. Flory-Huggins Theory x Δ MIX g = RT (1 x ) ln(1 x ) + ln x + g r R 4 g R description of the derivation of a real polymer solution from an ideal-athermic mixture 1 Δ ε = ε ε A + ε qδε χ = kt [ ] A A interaction energy between lattice places χ Flory-Huggin s interaction parameter χ<0 good solvent R R g = h = N0 qx(1 x) Δε χ=0 χ>0 Ideal - athermic mixture bad solvent
model of real polymer solutions 6.. Flory-Huggins Theory x R Δ MIX g = RT (1 x ) ln(1 x ) + ln x + g r qδε ktχ χ = Δ ε = χ Flory-Huggins interaction parameter kt q R R kt χ g = h = N0qx(1 x) Δ ε = N0qx(1 x) = RT x(1 x) χ q 5 ΔMIX g RT x = (1 x) ln(1 x) + ln x + x(1 x) χ r Flory-Huggins-equation
6.. Flory-Huggins Theory 6 ΔMIX g RT x = (1 x) ln(1 x) + ln x + x(1 x) χ r ΔMIX g/ RT 1 1 = ln 1 1+ ln + + 1 x r r ( x ) ( ) ( ) x x MIX g/ RT Δ 1 1 = + χ = 0 x 1 x rx 3 MIX g/ RT Δ 1 1 = = 0 3 x x r x ( 1 ) ( ) χ spinodal condition critical condition
6.. Flory-Huggins Theory 7 ΔMIX g RT x = (1 x) ln(1 x) + ln x + x(1 x) χ r g/ RT Δ 1 1 = = = = + ( 1 ) ( ) ( )( ) ( ) ( ) ( ) ( ) 41 ( r) ( ) ( ) ( ) 3 MIX 0 r x 1 x 1 x x 3 x x r x 1 x 0= 1 x + 1 r x 0= + x 1 r 1 r x 1, 4 1 1 1 1± r = ± = ± 1 ( 1 r) = 1 r 1 r 1 r 1 r 1 r 0 < x < 1 1, x c 1 r r 1 = = 1 r r 1
6.. Flory-Huggins Theory 8 0,5 critical concentration 0,0 0,15 0,10 0,05 0,00 Flory-Huggins theory x c = r 1 r 1 0 1000 000 3000 4000 segment number r
6.. Flory-Huggins Theory 9 ΔMIX g RT x = (1 x) ln(1 x) + ln x + x(1 x) χ r 1 1 r 1 x = für r lim x r = = = 0 c c 1 r 1 r ( r) MIX g/ RT Δ 1 1 1 = + χ = 0 für r = χ x 1 x rx 1 x x 0 χ = 1/ C for polymers with r > instable χ = 0.5 spinodal < stable
6.. Flory-Huggins Theory 30 Flory-Huggins parameter for polymer + solvent - systems system cellulose nitrate + acetone polyisobutylene + benzene polystyrene + toluene PVC + THF rubber + CCl 4 rubber + benzene rubber + acetone T [ C] 5 5 5 6 0 5 5 c 0.7 0.5 0.44 0.15 0.8 0.44 1.37
6.. Flory-Huggins Theory 31 χ 1 0.8 0.6 T = 65 C CH/PVME M W [kg/mol] χ can also depends on polymer concentration and molecular weight 0.4 0. 81 51 0.0 8 conflict to FH-theory 0.0 0. 0.4 0.6 0.8 1 φ PVME. A. Wolf, University of Mainz, 1995
6.. Flory-Huggins Theory 3 χ 0.9 TL / PDMS 170 0.8 0.7 0.6 0.5 T=35 C T=45 C T=55 C 0.4 0. 0.4 0.6 0.8. A. Wolf, University of Mainz, 1995 w PDMS
6.. Flory-Huggins Theory 33 formal division of χ in a enthalpic- and a entropic part ( ) hχ Tsχ hχ sχ χ = χh + χs = = RT RT R sχ χ hχ χ χs = = h RT χ = R T RT T χ RT h χ T χ χh = = = T RT RT T special case: H S 0 temperature dependence of χ available χ = χ χ = pseudo-ideal state = Theta-state hχ sχ 1 hχ hχ χ = χh + χs = = sχ = s RT R R T T χ Theta-temperature
6.. Flory-Huggins Theory 34 Flory-Huggins parameter for PMMA + solvent systems solvent c c H c S chloroform 0.36-0.08 0.44 benzene dioxane THF 0.43 0.43 0.45-0.0 0.04 0.03 0.45 0.39 0.4 entropic effects are more important than enthalpic effects toluene 0.45 0.03 0.4 Acetone 0.48 0.03 0.45 m-xylole 0.51 0. 0.31
6.. Flory-Huggins Theory 35 4 c 3 TL/PDMS 170 T = 35 C c H 1 0-1 c S - -3 0. 0.4 0.6 0.8. A. Wolf, Universität Mainz, 1995 f PDMS
6.. Flory-Huggins Theory 36 Prediction of χ Cohesive Energy e Coh : internal energy of the material if all of its intermolecular forces are eliminated e Coh solubility parameter δ 1/ δ e coh δ J 3/ solubility of polymer P in solvent S ( δ δ ) R h RT F v(98 K) = F molar attraction constant v molar volume = v mol P S has to be small, as small as possible ( δ δ ) ( δ δ ) A = x(1 x) χ x (1 x) χ = vrt vrt A
6.. Flory-Huggins Theory Prediction of χ example: solubility of polystyrene at 5 C First step: Calculation of atomic indices δ and valence atomic δ V 37 Second step: Calculation of connectivity indices β = δδ β = δ δ V V V ij i j ij i j 1 1 1 V 1 χ χ χ χ V V vertices δ vertices δ edges β edges β 0 0 V 1 1 V χ = 5.3973 χ = 4.671 χ = 3.9663 χ = 3.0159 0 0 V 1 1
6.. Flory-Huggins Theory 38 Prediction of χ example: solubility of polystyrene at 5 C Fourth step: Calculation of molar volume v W T T v= vw 1.4 + 0.15 g =373.15K T g 3 ( 0 1 V 3.861803* 13.748435* ) cm = χ + χ mol 3 3 cm cm = ( 3.861803*5.3973+ 13.748435*3.0159) = 6.31 mol mol 3 3 cm 98K cm v = 6.31 1.4 0.15 95.95 mol + = 373K mol
6.. Flory-Huggins Theory 39 Prediction of χ example: solubility of polystyrene at 5 C Third step: Calculation of molar attraction constant F where F F ( 0 ( 0 V 1 1 V χ + χ + χ + χ )) 97.95 = 134.61( N ) Si Nr N + Cyc ( ) ( ) ( + + + ) 97.95 5.3973 4.671 3.9663 3.0159 = + 134.61 0 0 0 F = 1754. Jcm mol 3 Jcm mol 3 Jcm mol 3
6.. Flory-Huggins Theory 40 Prediction of χ example: solubility of polystyrene Fifth step: Calculation cohesive energy e Coh e F e Coh Coh F = v(98 K) 3 3 Jcm cm = 1754. v= 95.95 mol mol ( 1754.) 3 Jcm mol = = 95.95 3.07 kj 3 mol cm mol Sixth step: Calculation of solubility parameter δ ecoh 3.07kJmol J δ = = = 18.8 v mol95.95cm cm 3 3
6.. Flory-Huggins Theory 41 Prediction of χ example: solubility of polystyrene Seventh step: Calculation of interaction parameter χ χ = ( δ δ ) vrt A solvent δ A [J/cm 3 ] χ solvent quality n-hexane 14,86 0,450 middle toluene 18,17 0,0005 excellent benzene 18,56 0,0030 excellent dioxane 0,4 0,1769 good methanol 9,60 4,9669 poor-lle
μ Polymer Thermodynamics 6.. Flory-Huggins Theory Application of Flory-Huggins Theory to LLE phase equilibrium conditions low molecular weight mixtures polymer solutions μ μ = μ L1 L A A L1 L = μ L1 L P = P = P μ μ = μ L1 L A A = μ L1 L L1 L P = P = P G = G = G + G +Δ G = n g + n g + nδ g n 0A 0 MIX A 0A 0 MIX T, P, na 4 ΔMIX g RT x = (1 x) ln(1 x) + ln x + x(1 x) χ r
T, P, n Polymer Thermodynamics 6.. Flory-Huggins Theory x G = nag0a + ng0 + RTn (1 x) ln(1 x) + ln x + x(1 x) χ r ( ) ( ) G 1 μa = = g0 A + RT ln xa + x 1 + x χ n r A μ Application of Flory-Huggins Theory to LLE ( ) ( ) A 0 A A G 1 1 = = g + RT ln x x 1 + x n r r ( ) ( ) 0 A A T, P, n 1 μ / RT = μ / RT + ln x + x 1 + x r A μ 1 ( ) 1 / ( ) RT μ = 0 / RT + ln x xa 1 + xa r r χ χ χ 43
L 1 0 A Polymer Thermodynamics 6.. Flory-Huggins Theory Application of Flory-Huggins Theory to LLE μ ( ) 1 ( μ ( ) 1 ( ) + ln x + x 1 + x χ = + ln x + x 1 + x RT r RT r ) L L1 L1 L1 0 A A L L L A χ 44 L1 ( ) ( ) ( ) x L L1 1 A L L1 ln = x x 1 + χ x x L x r A equation 1 μ RT L1 0 ( ) ( μ 1 ( ) 1 ( ) x x x χ + ln x x 1 + x r r 1 1 + ln 1 + = r r RT ) L L1 L1 L1 0 A A 1 ( ) ( ) ( ) 1 L x L1 L 1 L L1 = xa xa 1 + χ xa x A r L x r L L L A A equation equations and 3 unknowns select one concentration in one phase calculation of the concentration in the other phase and the parameter χ using both equations, where a numerical procedure must be applied or reduce the system of equations to one equation χ
6.. Flory-Huggins Theory Application of Flory-Huggins Theory to LLE 45 reduce the system of equations to one equation rearrangement of both equations according χ to equate both results L1 ( ) L1 x ( ) 1 1 1 A L L x L1 L 1 ln x x 1 xa xa 1 L L x r r A x r = ( ) ( ) ( ) ( ) L L1 L L1 x x xa xa equation 3 from equation 1 from equation Now: solve equation 3 in order to calculate the unknown concentration using root-searching method, like Newton-Procedure calculate χ using equation 1 or equation
6.. Flory-Huggins Theory Application of Flory-Huggins Theory to LLE χ( T ) β β3 = β1 + + T T special cases: β 3 =0 only one critical point can be calculated β >0 UCST β <0 LCST β 3 0 two critical points can be calculated closed miscibility gap or UCST+LCST T/K T / K UCST X A LCST 46 X A T / K T [K] X A X A
6.. Flory-Huggins Theory 47 80 70 PAA + Dioxan M W =500 kg/mol 60 T / C 50 40 χ(t)=1.186-07.87/t 30 0 5 10 15 0 5 w PAA [%]
6.. Flory-Huggins Theory 48 vapor pressure of polymer solutions liquid mixture (A+) «pure vapor phase (A) solvent A + solute P VL 0 equilibrium condition: L V dμ ( T, P, x ) dμ ( T, P) P P VL V L L V A A0 A A A0 VL V 0 A ϕa ϕa VL 0 A A A A = T = constant a ϕ x γ ϕ V V = = with ϕa0 = ϕa = 1 VL P = P x γ = a L L L A A A ideal gas
6.. Flory-Huggins Theory vapor pressure of polymer solutions application to polymer solution low-molecular weight mixtures solvent in polymer solution VL P L VL A P = a 0 A μi( T, P, xi) = μ0i( T, P) + RTlnai ( ) 1 μ ( ) A = μ0 A + RT ln xa + x 1 + x χ r 1 ln L a A = ln xa + x 1 + x r VL P 1 ln ln 1 VL = + + P0 A r ( ) ( ) ( x ) ( ) A x x possibility to determine χ from experimental data, like vapor pressure of polymer solutions χ χ 49
6.. Flory-Huggins Theory 50 vapor pressure of polymer solutions liquid mixture (A+) «pure vapor phase (A) solvent A + solute P VL 0 equilibria condition: L V dμ ( T, P, x ) dμ ( T, P) A A A = T = constant P P a ϕ x γ ϕ V V = = with ϕa0 = ϕa = 1 VL V L L V A A0 A A A0 VL V 0 A ϕa ϕa VL P = P VL 0 A x γ = a L L L A A A ideal gas Flory-Huggins theory VL P 1 ln ln 1 VL = + + P0 A r ( x ) ( ) A x x χ
6.. Flory-Huggins Theory 51 vapor pressure of polymer solutions 0 PS + n-hexane 15 P VL [kpa] 10 T = 98K 5 r=1 χ=0 ideal mixture r=100 χ=0 ideal-athermic mixture r=100 χ=0.45 real mixture 0 0,0 0, 0,4 0,6 0,8 1,0 segment mole fraction of PS
6.. Flory-Huggins Theory 5 measurement of χ: inverse gas chromatography (IGC) IGC refers to the characterization of the chromatographic stationary phase (polymer) using a known amount of mobile phase (solvent). The stationary phase is prepared by coating an inert support with polymer and packing the coated particles into a conventional gas chromatography column. The activity of the given solvent can be related to its retention time on the column. infinity dilution regarding to the solvent V R retention volume [m 3 ] N R G experimental information: V = V V V G gas holdup [m 3 ] V N net retention volume [m 3 ] N V0 k 0 net retention volume extrapolated to zero column pressure L G N = nv A V0 L G L V n = at infinity dilution: γ P a A A A = = A V xa x A L P A xp A RTnA aa = = = N xa xa V0 ϕ A fugacity coefficient =1
6.. Flory-Huggins Theory 53 experimental estimation of χ shows χ depends on temperature χ depends on polymer concentration χ depends on molecular weight χ( T ) β β3 = β1 + + T T more complex models are needed
6.3. Extensions of Flory-Huggins Theory 54 Flory-Krigbaum Model (1950): a diluted polymer solution is considered as a dispersion of clouds consisting of polymer segments surrounded by regions of pure solvent ln ln 1 1 r ( ) ( ) ( )( a = x + x + κ ψ x ) A A 1 1 enthalpic parameter entropic parameter θ κ T ψ 1 = θ = 1 Theta Temperature Difficulty: the interaction parameter depends not on concentration
6.3. Extensions of Flory-Huggins Theory 55 ( x ) R g = RT x (1 x ) χ, T χ x, T = χ ( T) + χ ( T) x + χ ( T) x +... + 1. Possibility: ( ) ( ) 0 1 advantage: very flexible disadvantage: empirical, a lot of parameters. Possibility: χ ( x, ) T = β ( T ) 1 γ x Koningsveld-Kleintjens-Eq. advantage: very flexible, semi-empirical easy parameter fitting procedure using the critical point
6.3. Extensions of Flory-Huggins Theory 56 90,5 90,0 89,5 γ = 0.381 β 1 = 0.4844 β = 143.307 / K T/K 89,0 88,5 88,0 TD/PMMA 1770 χ ( x, ) T = β ( T ) 1 γ x 87,5 0,00 0,0 0,04 0,06 0,08 0,10 w PMMA
6.3. Extensions of Flory-Huggins Theory 57 high-pressure production of PE equation of state is necessary M. Schnell, S. Stryuk,.A. Wolf, Ind. Eng. Chem. Res. 43 (004) 85.
6.3. Extensions of Flory-Huggins Theory 58 Flory-Equation of State Starting point: partition function rnc ( ) ( 1/3 * 1) 3 rnc rnc Z = Zcomb gv v exp vt combinatory factor Z comb geometrical factor g characteristic volume v* 3rnc = total number of degrees of freedom ( r = segment-number, n = number of molecules, c = mean number of external degrees per segment) v T P v = T = P = v* T* P * v*, T* and P* are characteristic quantities = parameters of the polymer
6.3. Extensions of Flory-Huggins Theory 59 Flory-Equation of State Starting point: partition function rnc ( ) ( 1/3 * 1) 3 rnc rnc Z = Zcomb gv v exp vt from statistical thermodynamics: P ln Z = kt V T Pv 1/3 v = 1 1/3 T v 1 Tv
6.3. Extensions of Flory-Huggins Theory 60 Flory-Equation of State chemical potential for solvent: binary parameters Δ μ = + + + + * * 1/3 v1 θ v1x 1 * * v1 1 1 1 1 RT lnφ1 1 * φ v1p1 3T 1ln 1/3 v v v 1 v 1 v Flory-Huggins term interaction term free volume term chemical potential for polymer: Δ μ = + + + + * * 1/3 v θ1 vx1 * * v 1 1 1 RT lnφ 1 * φ1 vp 3T ln 1/3 v1 v v 1 v v
Flory-Equation of State Polymer Thermodynamics 6.3. Extensions of Flory-Huggins Theory calculation of the mixing volume E φ v v 1 v + φ = v v 1 1 other equations of state useful in polymer thermodynamics 61 Sanchez-Lacombe Equation of state (SL-EOS) lattice theory, which allows additionally empty sites ( r) P V 1 1/ 1 = ln T V 1 V V T Statistical associated fluid theory (SAFT-EOS) takes into account additionally specific interactions like hydrogen bonding and association Perturbed-Chain Statistical associated fluid theory (PC-SAFT-EOS) further development of SAFT-EOS, takes the chain connectivity into account
6.3. Extensions of Flory-Huggins Theory 6 Polydispersity effects three liquid phases for the system polyethylene + diphenylether at constant temperature and pressure??? Gibbs s phase rule: F = K Ph + binary system: K= F=4-Ph fixed P,T: F= Ph Max = polymer solutions are not binary systems R. Koningsveld, W.H. Stockmayer, E. Nies, Polymer Phase Diagrams, Oxford University Press, 001
63 30 315 6.3. Extensions of Flory-Huggins Theory cloud-point curve Polydispersity effects critical point shadow-curve 310 T / K 305 spinodal coexisting curve 300 95 0,00 0,0 0,04 0,06 0,08 0,10 0,1 0,14 0,16 0,18 0,0 X Ensemble
6.3. Extensions of Flory-Huggins Theory 64 30 5 0 Polydispersity effects fractionation effect phase II polymers with lower chain lengths 10 3 W(r) 15 10 Feed Phase I Phase II phase I polymers with higher chain lengths 5 In both phases are different polymers. 0 0 00 400 600 800 1000 100 1400 1600 1800 000 r
6.4. Swelling Equilibria 65 T [ C] 50 polymer network 40 30 solvent 0 0. 0.5 1 V/V 0
66 6.4. Swelling Equilibria Application to drug-delivery systems
67 P + - liquid phase - + - polymer gel - + P 1 + + - + '' ' '' i el ln ai( T, P', n') ln ai( T, P'', n'') i.e. Flory-Huggins-theory 6.4. Swelling Equilibria - v Δ F = + RT V T = constant Δ F =Δ MIX F + Δ =Δ F + F MIX T el F deformed F relaxed Δ el F elastic free energy (deformation of rubber, rubber elasticity) Δ RT elf /3 V = C1 1 V0 first approximation C 1 fit parameter V volume of swollen network V 0 volume of non-swollen network more information available: G. Maurer, J.M. Prausnitz, Fluid Phase Equilibria 115 (1996) 13-133.
68 6.5. Solid-Liquid Equilibria starting point: f = S i f L i phase equilibrium condition iso-fugacity criteria liquid phase: f L i = x L i γ L i f L 0i solid phase: f S i = x S i γ S i f S 0i L S S L L L S S S 0i i i i γi 0i = i γi 0i = S L L f0i xi γ i x f x f f x γ Calculation of pure-component properties f g g T P RT g g T P RT P f Δ = = S L S id 0i L id 0i = 0i(, ) + ln 0i = 0i(, ) + ln 0i L L S 0i LS g0i g0i g0i RTln S f0i f P
f f x γ L S S 0i i i = S L L 0i xi γ i Polymer Thermodynamics 6.5. Solid-Liquid Equilibria f Δ = Δ =Δ Δ L 0i LSg0i RTln LSg0i LSh S 0i T LSs0i f0i ΔLSh0 i T Δ LS g0i =ΔLSh0i T =ΔLSh0i 1 LS LS T T ln T Δ h 1 Δ L LS 0i LS f0i LSg0i T = = S f0i RT RT 69 Δ h T = LS 0i 1 LS S S T xi γ i ln L L RT xi γ i SLE for binary systems
70 6.5. Solid-Liquid Equilibria S S solids are pure-components x γ = 1 SLE Δ L L LS 0i ln xi γ i = 1 LS RT T0 i h i i T Application to polymer solution using Flory-Huggins theory x Δ = h T L LS 0i ln 1 ln LS RT T0 i Δ h T = χ LS RT T0 i L LS 0i ln x 1 γ L ( ) x L A
T[ C] 00 180 160 140 10 100 80 C C amyl acetate Polymer Thermodynamics 6.5. Solid-Liquid Equilibria nitrobenzene xylene A solvent 1) xylene only SLE ) amyl acetate or nitrobenzene A- SLE C- LLE or eutectic mixture 71 0. 0.4 0.6 0.8 w PE