Polymer Solution Thermodynamics. 5. Polymer Blends. Flory-Huggins Model

Transcription

1 Polymer Soluton Thermodynamcs 5. Polymer Blends Flory-Huggns Model Up to ths pont we have consdered polymer chans n solvent at varous concentratons. In dlute soluton, we observed that the segmental densty s localzed around the center of mass of the cols; the local concentraton gradent experenced by each chan drves the swellng of the cols. By contrast, n polymer melts, ths swellng force s exactly cancelled by analogous forces exerted by neghborng chans -- the consequence s that polymer chans n the melt (even blends of polymers) are well descrbed by Gaussan col models! For a blend of A and B polymers, havng N A and N B segments per chan, respectvely, the Flory-Huggns free energy of mxng per ste s calculated to be: F mx,ste /kt = φ A /N A ln φ A + φ B /N B ln φ B - χφ A φ B (5.1) where χ s the A-B nteracton parameter. The phase dagram and lmts of stablty of the mxture can be readly calculated from ths expresson: d 2 F ste /dφ A 2 = 0 1/(N A φ A,s ) + 1/(N B φ B,s ) = 2χ s Spnodal lne d 3 F ste /dφ 3 = 0 Crtcal pont φ A,c = N B 1/2 /[N B 1/2 + N A 1/2 ] χ c = [N B 1/2 + N A 1/2 ] 2 / (2N B N A ) 80

2 From these expressons we see that for fxed N A, ncreasng N B reduces χ c (or rases T c,) and shfts φ A,c, the crtcal composton, to hgher fractons of A, as B becomes relatvely less mscble n A than A n B. N A N B χ N A < N B 0 1 φ Α Model Lmtatons Fluctuaton Effects on Blend Stablty In bnary mxtures of smple fluds, t s emprcally known that mean feld descrptons break down as one approaches the crtcal pont (eg., T T c ), due to large local fluctuatons n concentraton (These are analogous to the local fluctuatons n magnetc moment experenced near the Cure pont of ferromagnets.) For polymer blends, however, mean-feld analyss often yelds an adequate descrpton of the system, even near the crtcal pont. Ths s because, for long chans, the average feld experenced by a sngle polymer col s generated by a large number of both A and B chans that occupy the col volume. To see ths more quanttatvely, we can wrte n A,col as the average number of A chans that occupy the col volume for a blend concentraton φ A : n A,col = φ A R A 3 /N A b 3 = φ A N A 1/2 81

3 Thus, at the crtcal composton φ A,c : n A,col = N B 1/2 N A 1/2 /[N B 1/2 + N A 1/2 ] We can look at the magntude of ths expresson for dfferent ratos of N B to N A : N B N A n A,col ~ N A 1/2 /2 mean-feld vald N B << N A n A,col ~ N B 1/2 mean-feld vald for large N B N B 1 n A,col ~ 1 mean-feld breaks down Excluded Volume Effects Note that for N B 1, the expresson for F ste for the blend reduces to the regular soluton model for polymer/solvent mxtures. In ths lmt, we have already dscussed how excluded volume effects lmt the applcablty of the expresson to concentrated solutons. An nterestng queston to consder, then, s: at what value of N B do we expect a crossover from soluton to melt chan statstcs? To answer ths queston we reexamne the excluded volume contrbuton to the free energy of a swollen polymer col as derved from Flory theory: F nt (R eq )/kt 1/2 v c col 2 R eq 3 = 1/2 v N A 1/2 /b 3 82

4 For a solvent wth N B > 1, an approprate expresson must be developed for v. Ths can be estmated from eq. (5.1) by calculatng the osmotc pressure n the lmt of φ A 0, followng the analyss on pg. 69: Πb 3 = φ A 2 F ste / φ A Π b 3 /kt = φ A /N A + ½ (1/N B - 2χ ) φ A or Π /kt = c/n A + ½ b 3 (1//N B - 2χ)c From ths expresson the 2nd vral coeffcent s dentfed as: ½ v = ½ b 3 [1/N B -2χ] Let us look for smplcty at the case of an athermal blend, e.g., a blend wth equal A-A, A-B, and B-B segmental nteractons (χ = 0). Then: v = b 3 /N B F nt (R eq )/kt ½ N A 1/2 /N B Let s compare the magntude of ths expresson for varous values of N B : N B N A F nt /kt 1/N A 1/2 excluded volume neglgble N B N A 1/2 F nt /kt 1 excluded volume mportant An mportant concluson from ths analyss s that, even n polymer blends, excluded volume effects can be mportant for mxtures wth large molecular weght dspartes -- e.g., a blend of 1M g/mol and 10,000 g/mol polystyrene should exhbt sgnfcant swellng of the longer PS chans at low blend concentratons. 83

5 Equaton of State Models In the Flory-Huggns regular soluton model for polymer mxtures, the parameter χ represents the net enthalpc nteracton between solvent and monomer segments: χ = z[ε 12 - (ε 11 + ε 22 )/2] /kt In many polymer blends and solutons, the nteracton energes ε j < 0 are the result of weak van der Waals attractve forces, and one typcally fnds that χ > 0. Hence the phase separaton of mxture components s predcted to occur wth decreasng temperature, when χ exceeds χ c, or equvalently, T decreases below T c. A sgnfcant lmtaton of ths model s ts nablty to account for the frequent observaton of phase separaton n polymer-contanng mxtures (both solutons and blends) wth ncreasng temperature. Such systems are sad to undergo a lower crtcal soluton transton (LCST), whereas mxtures that exhbt a conventonal crtcal pont are sad to have an upper crtcal soluton temperature (UCST). Polymer-contanng systems can exhbt the LCST, UCST, or both, or more exotc phase dagrams. T LCST T UCST φ φ 84

6 The LCST and ts relatves result from entropc contrbutons omtted n the regular soluton model and typcally consdered neglgble for atomc or small molecule mxtures. Such contrbutons can arse from strong polar nteractons or more smply from molecular packng consderatons that nduce dfferences n the thermal expanson coeffcent between the two components and ther mxture. In contrast to the UCST, whch s an enthalpcally drven phase separaton, LCST behavor s always the result of an entropc drvng force for demxng. Ths ncrease n entropy s related to a larger number of avalable confguratons at elevated temperatures, and emprcally s always assocated wth a negatve volume change on mxng. To see ths more fundamentally, let s reconsder the stablty condton for a bnary mxture of fxed composton usng the Gbbs free energy: The dervatve s: G = F + PV = H - TS dg = dh - TdS - SdT For constant pressure: dh = de + PdV = (TdS - PdV) + PdV = TdS ( G/ T) P = - S takng the Gbbs free energy per mole, g = G/n tot, we can wrte: ( g xx / T) P = - s xx where g xx 2 g/ x 2 and x s the mole fracton of one mxture component. The condton for phase stablty of the mxture s: ( 2 g/ x 2 ) T,P > 0 LCST behavor mples that ( g xx / T) P s negatve along the spnodal lne, so that s xx > 0, as shown below. 85

7 T 1 [g xx (T 3 ) - g xx (T 2 )] / [T 3 -T 2 ] g xx =0 = (-)/(+) < 0 g T 2 for T 3 > T 2 T 3 s xx > 0 0 x 1 Snce g xx = h xx - Ts xx = 0 at the spnodal, we must also have h xx > 0. Ths ndcates that entropy destablzes the mxture! We can further wrte: dg xx = ( g xx / x)dx + ( g xx / T)dT + ( g xx / P)dP = 0 = (0) + s xx dt - v xx dp (at fxed composton) (dt s /dp) x = v xx /s xx Expermentally, the spnodal temperature T s for systems exhbtng an LCST s always seen to ncrease wth appled pressure, one must conclude that: v xx > 0 for demxng, or V mx < 0 Thus, the entropy ganed on mxng s assocated wth an ncrease n the total volume of the system. 86

8 To account for the effects of compressblty on the phase behavor of polymer systems, equaton of state models have been developed by a number of groups, the most well known beng the Flory, Orwoll and Vrj model (J.A.C.S., 86, 3507 (1964) and the Sanchez-Lacombe lattce flud (LF) model (J. Phys. Chem., 80, 2352 (1976); Macromol., 11, 1145 (1978)). Such models have been successful n helpng to elucdate the molecular orgn of thermodynamc propertes pecular to polymer systems, such as neglgble vapor pressure and LCST transtons. Here we wll restrct our bref dscusson to the LF model, though a source lst for other equaton of state models can be found n Chapter 3 of Models for thermodynamc and phase equlbra calculatons (S.I. Sandler, ed., Marcel Dekker, Inc. 1994). As orgnally proposed, the LF model accounts for system compressblty through the ntroducton of an addtonal component, a hole component, to a lattce contanng the polymers or polymers and solvent. One defnes the system n terms of reduced PVT parameters: ρ! = ρ/ρ = 1/ v! reduced densty (= 1 at 0 K) T! = T/T* = kt/ε* P! = P/P* = Pv*/ε* reduced temperature reduced pressure where ε*, v* and ρ* are obtaned from the nteracton energes (ε j *), hard core volumes (v *), and zero Kelvn denstes (ρ *) of the system s components: ε* = v* = ε j * φ φ j j v * φ LF mxng rules for mult-component systems ρ* = w /ρ * where φ represents the ste fracton of the th component n the absence of vacances, and w represents a weght fracton. 87

9 Values of v *, ρ * and ε * are determned by fttng expermental PVT data for the pure components. The number of lattce stes occuped per chan (or per solvent molecule) s gven by r calculated as: r = M /ρ *v * where M s the number average molecular weght of the th speces. Note that n the lattce-based LF model, r=1 for holes. The reduced densty ρ! represents the total volume of the system that s occuped by the hard core volume of the system components. The fractonal free volume of the system s consequently 1 - ρ!. The general effect of the holes s to 1) add translatonal entropy; 2) dlute parwse nteractons. The Gbb s free energy per ste n the LF model s gven by: G/rn = (E + PV -TS)/rn = ρ! ε* + P v! v* + kt v! [(1- ρ! ) ln(1- ρ! ) + ρ! /r ln( ρ! ) ] + kt (φ /r ) ln(φ /ω ) where ω = rδ r 1 / e, wth δ = ( ) r z as a geometry-related constant, and r = r n /n wth n = n Here, n s the total number of the th speces present on the lattce, occupyng n r stes. The value of r s thus equvalently calculated by: 1/r = φ /r 88

10 The LF equaton of state s obtaned by mnmzng the free energy wth respect to the reduced volume: G/ v!! TPφ,, = 0 ρ! 2 + P! + T! [ ln(1- ρ! ) + (1-1/r) ρ! ] = 0 (5.2) At equlbrum the average densty s gven by solutons to eq. (5.2). For fxed values of T and P, there are 3 solutons to ths expresson, two of whch are mnma. The hgh-densty mnmum gves the densty of the lqud phase whle the low-densty mnmum corresponds to the vapor phase. The P-T values for whch the mnma are equal yeld the coexstence lne between lqud and vapor. At the crtcal pont, the lqud and vapor phases are no longer dstngushable, and there exsts a unque soluton to the equaton of state: ρ! c = 1/(1 + r 1/2 ) T! c = 2r ρ! c 2 P! c = T! c [ ln (1+r -1/2 ) + (1/2 - r 1/2 )/r ] These equatons demonstrate the dependence of the phase behavor on molecular weght. Note that n the lmt r, the value of P! c 0;.e., nfnte chans have no lqud-vapor transton and ther vapor pressure s zero! 89

11 1.0 - ρ! lq r= r=10 crtcal ponts r= T! The LF EOS accurately reproduces PVT for polymer melts and solvents. Fts to PVT data for such sngle components to the LF equaton of state allow one to determne the parameters T *, P * and ρ *, and thus extract the molecular parameters v *, ε *, and r. Alternatvely, one can calculate these hard core parameters from thermal expanson (α) or compressblty (β) data, usng the followng approxmate relatonshps for polymers: (P 0 at 1atm; r ) αt = [T! /(1- ρ! ) - 2] -1 P* = αt/ ρ! 2 β 90

12 Knowng α, β and ρ for a sngle T = T ref, the above equatons can be used to predct the PVT phase dagrams for a sngle component. If such data are not avalable, t s stll possble to estmate the hard core parameter values for a partcular component based on group contrbuton methods (see D. Boudours, et al., Ind. Eng. Chem. Res. 36, 1997, p. 3968), or from molecular dynamcs smulaton. Phase Dagrams for Mult-component Systems To determne the equlbrum phases and stablty lmts for compressble mxtures from the LF model, one must calculate the bnodal and spnodal lnes from the Gbbs free energy expresson. The calculatons are rather tedous, though not dffcult. The reader s referred to Chapter 3 of Sandler s text, or to the orgnal artcles by Sanchez and coworkers, for more detals. The LF model along wth other EOS models not dscussed here have provded needed nsght nto the molecular factors at play n the orgn of LCST behavor. The LF model has even been successful n estmatng the LCST temperatures for some nonpolar polymer-solvent mxtures from pure component propertes only, employng the geometrc mean approxmaton for ε 12. More typcally, however, EOS models provde lttle predctve capacty n practce, as the calculated phase dagrams are hghly senstve to the value of the cross-nteracton term, and thus mxture data s requred to acheve a reasonable representaton of the expermental phase dagram. To address the need for a more predctve tool for phase dagram analyss, we very recently developed a somewhat smpler free energy model (not yet a true EOS) that appears to have sgnfcant capablty for capturng the man features of the phase behavor of weakly nteractng polymer pars, usng only sngle component parameters. Ths model s ntroduced n the next secton of these notes. 91

13 Lattce Flud EOS Parameters for Polymer Lquds [from I.C. Sanchez & C.G. Panaylotou, n Models for thermodynamc and phase equlbra calculatons, S.I. Sandler, ed., Marcel Dekker, Inc. NY, 1994, pp ) Polymer lqud T* (K) P* (MPa) ρ* (kg/m 3 ) T range P max (MPa) lnear polyethylene branched polyethylene polypropylene polysobutylene poly(1-butene) poly(1,4-cs butadene) polystyrene poly(o-methyl styrene) polyvnyl acetate polymethyl methacrylate polyethyl methacrylate poly (n-butyl methacrylate) poly(cyclohexyl methacrylate) polyethylene oxde poly(2,6-dmethyl phenylene oxde) poly(vnyl methyl ether) poly(tetrahydrofuran) bsphenol-a polycarbonate) poly(ethylene terephthalate) poly(methyl acrylate)

14 Polymer lqud T* (K) P* (MPa) ρ* (kg/m 3 ) T range P max (MPa) poly(ethyl acrylate) poly(ε-caprolactone) polyacrylontrle poly(vnyl chlorde) poly(chloroprene) poly(epchlorohydrn) polydmethyl sloxane polysulfone polytetrafluoroethylene