Chemical Engineering 160/260 Polymer Science and Engineering. Model for Polymer Solutions February 5, 2001

Transcription

1 Chemical Egieerig 60/60 Polymer Sciece ad Egieerig Lecture 9 - Flory-Huggis Model for Polymer Solutios February 5, 00 Read Sperlig, Chapter 4

2 Objectives! To develop the classical Flory-Huggis theory for the free eergy of mixig of polymer solutios based o a statistical approach o a regular lattice.! To describe the criteria for phase stability ad illustrate typical phase diagrams for polymer bleds ad solutios.

3 Outlie! Lattice Theory for Solutios of Small Molecules " Thermodyamic probability ad the Boltzma Equatio " Ideal solutio! Flory-Huggis Theory of Polymer Solutios " Placemet of a ew polymer molecule o a partially filled lattice " Etropy of mixig " Ethalpy of mixig (for dispersive or dipole-dipole iteractios) " Cohesive eergy desity ad solubility parameter " Free eergy of mixig

4 Lattice Theory for Solutios of Small Molecules Assume that a solutio may be formed by distributig the pure compoets o the sites of a regular lattice. Further assume that there are molecules of Type, molecules of Type, ad that Type ad Type molecules are idistiguishable but idetical i size ad iteractio eergy. Small molecule of Type (e.g. solvet) Small molecule of Type (e.g. solute)

5 Thermodyamic Probability Place the molecules o the = + sites of a threedimesioal lattice. Total o. differet ways of arragig molecules of Types ad o the lattice Total umber of arragemets of molecules Ω =!!! Iterchagig the s or s makes o differece. Ω is the thermodyamic probability, which couts the umber of ways that a particular state ca come about.

6 Boltzma Equatio The thermodyamic probability (or the umber of ways that the system may come about )may be related to the etropy of the system through a fudametal equatio from statistical thermodyamics that is kow as the Boltzma Equatio. S = klω

7 S = S S = k Cofiguratioal Etropy Apply the Boltzma Equatio to the mixig process: Ω l Ω Cosider the etropy of the mixture: Ω Ω > Ω < Ω ( ) Smix = klω = k l! l! l! Stirlig s approximatio: Smix = k l + l l y! yl y y S = mix R x l x + x l x Sm = Smix S S = R xi xi ( ) Multiply r.h.s.by l S = S S > 0 S < 0 S mix is that part of the total etropy of the mixture arisig from the mixig process itself. This is the cofiguratioal etropy. mix m A A A

8 Ideal Solutio of Small Molecules What etropy effects ca you evisio other tha the cofiguratioal etropy? Sm = R xil xi If the -, -, ad - iteractios are equal, the Gm = RT xil xi H Η m = 0 If the solute ad the solvet molecules are the same size, V m = 0 (athermal mixig) The thermodyamics of mixig will be govered by the Gibbs free eergy of mixig. Do you expect a polymer solutio to be ideal?

9 Lattice Approach to Polymer Solutios To place a macromolecule o a lattice, it is ecessary that the polymer segmets, which do ot ecessarily correspod to a sigle repeat uit, are situated i a cotiguous strig. Small molecule of Type (solvet) Macromolecule of Type (solute)

10 Flory-Huggis Theory of Polymer Solutios Assume (for ow) that the polymer-solvet system shows athermal mixig. Let the system cosist of solvet molecules, each occupyig a sigle site ad polymer molecules, each occupyig lattice sites. + = What assumptio about molecular weight distributio is implicit i the system chose?

11 Placemet of a ew Polymer Molecule o a Partially Filled Lattice Let (i) polymer molecules be iitially placed o a empty lattice ad determie the umber of ways that the (i + )st polymer molecule ca be placed o the lattice. How ca we get the (i+)st molecule to fit o the lattice?

12 Placemet of Polymer Segmets o a Lattice Placemet of first segmet of polymer (i + ): i = umber of remaiig sites umber of ways to add segmet Placemet of secod segmet of polymer (i + ): Let Z = coordiatio umber of the lattice Z i = umber of ways to add segmet umber of lattice sites adjacet to the first segmet umber of sites occupied by iitial i polymer molecules Average fractio of vacat sites o the lattice as a whole (Whe is this most valid?)

13 Probability of Placemet of the (i)th Molecule Placemet of third segmet (ad all others) of polymer (i + ): ( Z ) i = umber of ways to add segmet 3 Igore cotributios to the Oe site o the coordiatio sphere average site vacacy due to is occupied by the secod segmet segmets of molecule (i + ) Thus, the (i + )st polymer molecule may be placed o a lattice already cotaiig (i) molecules i ω i+ ways. ω i + i Z i = ( ) ( Z ) ω i + ZZ i ( ) = i ( ) For the (i)th molecule: ω i = ZZ ( ) i

14 Total umber of Ways of Placig Polymer Molecules o a Lattice ωω Lω i Lω Ω = =!! Apply Boltzma s Equatio:S Substitute for ω i to obtai: Ω = Z Examie the product: = ( Z )! i= mix ( ) ( ) = kl i=! ω i= i ω ( ) [ i ] i ( ) i= i= i [ ] = + i

15 Etropy of the Mixture Write out several terms i the product expressio: = L ote that: = ()( )() + ()( )()!! 3 3 L L L Thus =!! Ω = ( ) ( ) ( ) Z Z!!! Apply Stirlig s approximatio to obtai: S k Z Z mix = + + ( ) ( )+ ( )+ [ ] l l l l l

16 Flory-Huggis Etropy of Mixig Calculate etropy of pure solvet ad pure polymer: Pure solvet: = 0 S = 0 Pure polymer: = 0 Etropy of the disordered polymer whe it fills the lattice S = k l Z + ( ) l( Z )+ ( )+ l [ ] Sm = Smix S S S = m k l + l Multiply ad divide r.h.s. by + ad assume + = A Calculate etropy of mixig: S = m R x l + x l = =

17 Etropy of mixig: Flory-Huggis Theory for a Athermal Solutio [ l ] S = m R x l + x Ethalpy of mixig: H m = 0 Gibbs free eergy of mixig: [ l ] G = m RT x l + x

18 Cocetratio Coversios = = x = + = + = + x x + = + = x = + = + x x

19 Flory-Huggis Ethalpy of Mixig Use the same lattice model as for the etropy of mixig, ad cosider a quasi-chemical reactio: (,) + (,) (,) represets a solvet ad represets a polymer repeat uit The iteractio eergy is the give by: w = w w w w = Chage i iteractio eergy per (,) pair Defie the system to be a filled lattice with Z earest eighbors. Each polymer segmet is the surrouded by Z polymer segmets ad Z solvet molecules.

20 Cotributios to the Iteractio Eergy Cotributios of polymer segmets Iteractio of a polymer segmet with its eighbors yields Z w + Z w The total cotributio is Remove double coutig Cotributios of solvet molecules Z ( ) w + w [ ] Each solvet molecule is surrouded by Z polymer segmets ad Z solvet molecules. Iteractio of the solvet with its eighbors the yields Z w + Z w The total cotributio is Remove double coutig Z[ w + ( ) ] w

21 Flory-Huggis Ethalpy of Mixig H = m Z w w w ( ) H = m Z w Let Z w = χrt Flory-Huggis iteractio parameter (the chi parameter) χ = χ > χ < For athermal mixtures For edothermic mixig For exothermic mixig Hm = χrt = χrt

22 Flory-Huggis Free Eergy of Mixig: Geeral Case G = H T S m m m H χrt m = [ lφ ] S = R x lφ + x m [ ( )] Gm = RT χ+ x l + x l >0 0 <0 <0