Chemical Engineering 160/260 Polymer Science and Engineering. Lecture 10 - Phase Equilibria and Polymer Blends February 7, 2001

Transcription

1 Checal Engneerng 60/60 Polyer Scence and Engneerng Lecture 0 - Phase Equlbra and Polyer Blends February 7, 00

2 Therodynacs of Polyer Blends: Part Objectves! To develop the classcal Flory-Huggns theory for the free energy of xng of polyer solutons based on a statstcal approach on a regular lattce.! To descrbe the crtera for phase stablty and llustrate typcal phase dagras for polyer blends and solutons.

3 ! Phase Equlbra Outlne " Free energy of xng for a phase-separated syste " Spnodal and bnodal curves n the Flory-Huggns odel " Qualtatve results fro equaton-of-state theory

4 Geoetrc Mean Mxng Rule For the systes n whch H > 0, t s coon to express H n ters of the cohesve energy densty, or solublty paraeter. Analyze by frst consderng the (,) and (,) nteractons. Change n nternal energy of vaporzaton of one ole of () U ZN w v A, = To evaluate the (,) cross nteractons, nvoke the geoetrc ean xng rule: w w w Ths approxaton wll be ost accurate when both the (,) and (,) forces are ether London dsperson or dpole-dpole. It breaks down for hydrogen bondng or strong specfc nteractons. w = w + w w w ( ) w = w w

5 Recall that w = Uv ZN Cohesve Energy Densty H = ZNϕϕ w Uv ZN,, Rewrte on a per unt volue bass H = Vϕϕ U V Uv o V v,, o U = H P V v v ( ) U = H P V V v v g l U H PV v v g U H RT v U V v, o v Cohesve energy densty (CED)

6 Solublty Paraeter U V v, / CED o ( ) δ Solvent CED (cal c -3 ) δ (cal c -3 ) / Cyclohexane Carbon tetrachlorde Toluene Benzene Methyl acetate Acetone Cyclohexanone Acetc acd Cyclohexanol Methanol Water

7 Polyer Solublty Paraeters Solublty paraeters for polyers ust be deterned ndrectly, typcally by ntrnsc vscosty easureents of solutons or by swellng of a crosslnked polyer. Polyer δ (cal c -3 ) / Polysobutylene Polyethylene Natural rubber Polystyrene Poly(ethylene terephthalate) Polyacrylontrle Nylon-6,

8 Crteron for Phase Equlbru Consder the ncreent n Gbbs free energy G assocated wth transfer of dn oles of () fro phase α to phase β at constant teperature T and pressure P. dg = VdP SdT + µ dn At equlbru α dg = 0 = dg + dg 0 = µ + dn µ β dn α α β β ( ) 0 = µ µ dn α β α α µ = µ β

9 Free Energy of Mxng for a Sngle Phase Recall G = x µ + x µ G x µ x µ µ x µ µ = where µ = µ µ o ( ) + = + ( ) Plot G vs x : ( ) Slope = µ µ 0 G x P µ µ Intercept at x = 0 s µ Plot G vs x : Slope = ( µ µ ) Intercept at x = 0 s µ Sngle Phase at P G = H T S <0 >0 G < 0

10 G for a Phase-Separated Syste Suppose that H > 0 and S > 0. Ths ay lead to the followng: C x 0 D G µ µ P A B Q The coon tangent to the G curve at P and Q ples that phases of copostons (x P, x P ) and (x Q, x Q ) have the sae values of µ and µ,.e. µ P = µ Q and µ P = µ Q. Thus, P and Q are n equlbru, and any xture havng coposton between P and Q wll phase separate nto phases wth copostons x P and x Q.

11 Stable Mxtures Regons of Phase Stablty For C-P and Q-D regons, all xtures are stable aganst separaton nto dfferent phases. Metastable Mxtures For P-A and B-Q regons, where A and B denote nflecton ponts, G > 0 and the xture wll be stable aganst separaton nto neghborng phases dfferng only slghtly n coposton but not aganst separaton nto phases of coposton P and Q. Unstable Mxtures G For the A-B regon, < 0 and all xtures are unstable to nfntesal perturbatons.

12 Spnodal and Bnodal Curves Spnodal Curve The boundary between the absolutely unstable and the etastable regons s defned by G = 0 The locus of all such A, B nflecton ponts as a functon of teperature generates the spnodal curve. Bnodal Curve The locus of all ponts of coon tangency P and Q as a functon of teperature generates the bnodal curve. Crtcal Pont The crtcal pont s characterzed by convergence of the ponts of coon tangency and the nflecton ponts such that 3 G = 0 3

13 Phase Equlbra n Polyer Solutons G = 0 T 3 G = 0 3 Spnodal Curve Bnodal Curve T o Crtcal Pont x

14 Phase Behavor for the Flory-Huggns Model [ ( )] G = RT Nϕϕ χ+ x lnϕ + x lnϕ G N = µ G N Note that N = Vϕ and ϕ Bnodal Curves = = µ N nn + nn µ µ ϕ ϕ o = + χϕ + RT ln( ) n o µ µ = RT ln ϕ + ( ϕ )( n)+ χn( ϕ ) [ ] Note that dfferentaton wth respect to x s equvalent to ϕ. Also = ϕ ϕ

15 Phase Behavor for the Flory-Huggns Model Spnodal Curve G µ = 0 = = ϕ ϕ + + χϕ n Crtcal Pont χ 3 G 3 c = ϕ, c µ = 0 = = ϕ ( ϕ ), c + χ + + ϕ, c = 0 ϕ n ϕ, c, c c χ c = ( + n ) /

16 Post Flory-Huggns Therodynacs Flory equaton-of-state and Sanchez lattce flud theores LCST behavor s characterstc of exotherc xng (whch could arse fro specfc checal nteractons) and negatve excess entropy (whch arses due to densfcaton of the polyers on xng). Nether one of these phenoena s ncluded n the Flory-Huggns theory.

17 Typcal Phase Dagras Typcal for Polyer Blend Typcal for Polyer Soluton T Upper Crtcal Soluton Tep. Bnodal T Lower Crtcal Soluton Tep. Coposton Spnodal Coposton

18 Mechanss of Phase Separaton Nucleaton and Growth Intal fragent of ore stable phase fors Free energy deterned by work requred to for the surface and the work ganed n forng the nteror Concentraton n the edate vcnty of the nucleus s reduced and dffuson s downhll (dffuson coeffcent s postve) Droplet sze ncreases by growth ntally Requres actvaton energy T Nucleaton and growth between the bnodal and spnodal curves Coposton

19 Mechanss of Phase Separaton Spnodal Decoposton Intal sall-apltude coposton fluctuatons Apltude of wavelke coposton fluctuatons ncreases wth te Dffuson s uphll fro the low concentraton regon nto the doan (negatve dffuson coeffcent) Unstable process; no actvaton energy requred Phases are nterconnected at early te Spnodal decoposton nsde the spnodal curve T Coposton