Chemical Egieerig 60/60 Polymer Sciece ad Egieerig Lecture 9 - Flory-Huggis Model for Polymer Solutios February 5, 00 Read Sperlig, Chapter 4
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.
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
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)
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.
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
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
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?
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)
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?
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?
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?)
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
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
Etropy of the Mixture Write out several terms i the product expressio: = + + + + 3 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
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 = =
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
Cocetratio Coversios = = x = + = + = + x x + = + = x = + = + x x
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.
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
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 < 0 0 0 For athermal mixtures For edothermic mixig For exothermic mixig Hm = χrt = χrt
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