Mean Field Flory Huggins Lattice Theory

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1 Mean Field Flory Huggins Lattice Theory Mean field: the interactions between molecules are assumed to be due to the interaction of a given molecule and an average field due to all the other molecules in the system. To aid in modeling, the solution is imagined to be divided into a set of cells within which molecules or parts of molecules can be placed (lattice theory). The total volume, V, is divided into a lattice of N o cells, each cell of volume v. The molecules occupy the sites randomly according to a probability based on their respective volume fractions. To model a polymer chain, one occupies x i adjacent cells. N o =N + N = n x + n x V =N o v Following the standard treatment for small molecules (x = x = ) Ω, = N! N!N! using Stirling s approximation: ln M! = Mln M M for M >> ΔS m = k( N lnφ N lnφ ) φ =?

2 Entropy Change on Mixing ΔS m ΔS m N o =+k( φ ln φ φ ln φ ) Remember this is for small molecules x = x = Note the symmetry φ The entropic contribution to ΔG m is thus seen to always favor mixing if the random mixing approximation is used.

3 Enthalpy of Mixing ΔH m Assume lattice has z nearest-neighbor cells. To calculate the enthalpic interactions we consider the number of pairwise interactions. The probability of finding adjacent cells filled by component i, and j is given by assuming the probability that a given cell is occupied by species i is equal to the volume fraction of that species: φ i.

4 ΔH m cont d υ ij = # of i,j interactions υ = N z φ υ = N zφ / υ = N zφ / Mixed state enthalpic interactions Pure state enthalpic interactions H, = υ ε + υ ε + υ ε H, = zn φ ε + z N φ ε + z N φ ε H = z N ε H = z N ε ΔH M = zn φ ε + N ε ( φ )+ N ε ( φ ) recall Some math ΔH M N + N = N ( ) = z N ε ε + ε φφ Note the symmetry

5 χ Parameter z χ = ε ε + ε kt Define χ: ( ) χ represents the chemical interaction between the components ΔG M : ΔH M N = ktχ φ φ ΔG M = ΔH M T ΔS M ΔG M N ΔG M N = kt χφ φ kt = kt [ χφ φ + φ lnφ + φ lnφ ] Note: ΔG M is symmetric in φ and φ. This is the Bragg-Williams result for the change in free energy for the mixing of binary metal alloys. [ φ lnφ φ lnφ ] Ned: add eqn for Computing Chi from V seg (delta-delta) /RT

6 ΔG M (T, φ, χ) Flory showed how to pack chains onto a lattice and correctly evaluate Ω, for polymer-solvent and polymer-polymer systems. Flory made a complex derivation but got a very simple and intuitive result, namely that ΔS M is decreased by factor (/x) due to connectivity of x segments into a single molecule: ΔS M N φ φ = k lnφ + lnφ x x For polymers Systems of Interest - solvent solvent x = x = - solvent polymer x = x = large - polymer polymer x = large, x = large ΔG M N φ φ = kt χφφ + lnφ + lnφ x x z χ = ε kt ( ε + ε ) Need to examine variation of ΔG M with T, χ, φ i, and x i to determine phase behavior. Note possible huge asymmetry in x, x

7 Flory Huggins Theory Many Important Applications. Phase diagrams. Fractionation by molecular weight, fractionation by composition 3. T m depression in a semicrystalline polymer by nd component 4. Swelling behavior of networks (when combined with the theory of rubber elasticity) ΔG The two parts of free energy per site M N kt S H ΔS M = φ x ln(φ ) φ x ln(φ ) ΔH M = χ φ φ ΔH M can be measured for low molar mass liquids and estimated for nonpolar, noncrystalline polymers by the Hildebrand solubility approach.

8 Solubility Parameter Liquids δ= ΔE V / = cohesive energy = ΔH ν RT V m density / ΔH ν = molar enthalpy of vaporization Hildebrand proposed that compatibility between components and arises as their solubility parameters approach one another δ δ. δ p for polymers Take δ P as equal to δ solvent for which there is: () maximum in intrinsic viscosity for soluble polymers () maximum in swelling of the polymer network or (3) calculate an approximate value of δ P by chemical group contributions for a particular monomeric repeat unit.

9 Estimating the Heat of Mixing Hildebrand equation: ΔH M = V φ φ ( δ m δ ) V m = average molar volume of solvent/monomers δ, δ = solubility parameters of components Inspection of solubility parameters can be used to estimate possible compatibility (miscibility) of solvent-polymer or polymer-polymer pairs. This approach works well for non-polar solvents with non-polar amorphous polymers. Think: usual phase behavior for a pair of polymers? Note: this approach is not appropriate for systems with specific interactions, for which ΔH M can be negative.

10 Influence of χ on Phase Behavior ΔH Assume kt = and x =x Expect symmetry m. ΔG m χ = -,,,, T ΔS m What happens to entropy for a pair of polymers? At what value of chi does the system go biphasic? χ = -,,,,3

11 Polymer-Solvent Solutions Equilibrium: Equal Chemical potentials: so need partial molar quantities pure mixed G μi = μ o μ o μ, μ μ μ i i and μ μ μ μ μ i n RT ln a i i T, P, n, j ΔG where a is the activity ni is the chemical potential in the standard state = RT ln φ + φ + χφ x = RT ln φ ( x )φ + x χφ m T, P, n, j = solvent ΔG φ [ ] polymer Note: For a polydisperse system of chains, use x = <x > the number average Recall at equilibrium μ iα = μ i β etc i m φi n P.S. # i

12 Construction of Phase Diagrams Chemical Potential Binodal - curve denoting the region of distinct coexisting phases μ ' = μ " μ ' = μ " or equivalently μ ' μ o = μ " μ o μ ' μ o = μ " μ o + + = ln ln φ φ φ φ χφ φ x x kt N ΔG m j n P T i i n G,,, = μ and since, Δμ Δ = P T m n G = n G n G m m φ φ Δ Δ Phases called prime and double prime Binodal curve is given by finding common tangent to ΔG m (φ) curve for each φ, T combination. Note with lattice model (x = x ) (can use volume fraction of component in place of moles of component

13 Construction of Phase Diagrams cont d 6 Spinodal (Inflection Points) ΔG m φ α = ΔG m φ β = 5 4 Two phases Equilibrium Stability T low χ N 3 A' B' Critical Point χ c φ,c ΔG m φ = and 3 ΔG m φ 3 = Critical point A B φ' A One phase φ" A A B φ B T high hot Figure by MIT OCW.

14 Dilute Polymer Solution # moles of solvent (n ) >> polymer (n ) and n >> n x φ = n v n v + n x v φ = n x v n v + n x v ~ n x n Useful Approximations For a dilute solution: Let s do the math: φ n x n ~ X x ln(+ x) x x ln( x) x x μ μ + x x φ = RT x Careful + χ φ

15 Dilute Polymer Solution cont d Recall for an Ideal Solution the chemical potential is proportional to the activity which is equal to the mole fraction of the species, X i φ μ μ = RT ln X = RT ln( X ) RTX = RT x Comparing to the μ μ expression we have for the dilute solution, we see the first term corresponds to that of an ideal solution. The nd term is called the excess chemical potential This term has parts due to Contact interactions (solvent quality) χ φ RT Chain connectivity (excluded volume) -(/) φ RT μ μ = RT φ + χ φ x Notice that for the special case of χ = /, the entire nd term disappears! This implies that in this special situation, the dilute solution acts as an Ideal solution. The excluded volume effect is precisely compensated by the solvent quality effect. Previously we called this the θ condition, so χ = / is also the theta point

16 F-H Phase Diagram at/near θ condition T X = X = Find: x =, x >> and n >> n x χ c = + x x X = X = 3 Two phases Binodal φ,c = x x + x Note strong asymmetry! X =..6 Symmetric when x = = x Figure by MIT OCW.

17 Critical Composition & Critical Interaction Parameter Binary System φ,c χ c low molar mass liquids x = x =.5 Solventpolymer Symmetric Polymer- Polymer General x = ; x = N x = x = N x, x x + x + x.5 x x + x N x + x

18 Polymer Blends Good References on Polymer Blends: O. Olasbisi, L.M. Robeson and M. Shaw, Polymer-Polymer Miscibility, Academic Press (979). D.R. Paul, S. Newman, Polymer Blends, Vol I, II, Academic Press (978). Upper Critical Solution Temperature (UCST) Behavior Well accounted for by F-H theory with χ = a/t + b Lower Critical Solution Temperature (LCST) Behavior FH theory cannot predict LCST behavior. Experimentally find that blend systems displaying hydrogen bonding and/or large thermal expansion factor difference between the respective homopolymers often results in LCST formation.

19 Phase Diagram for UCST Polymer Blend Predicted from FH Theory 6 χ = A T + B 5 Two phases T low x = x = N 4 Equilibrium Stability ΔG m φ = binodal χ N 3 A' B' ΔG m φ = spinodal Critical point A B hot 3 ΔG m φ 3 = critical point A B φ B T high φ' A One phase φ" A Figure by MIT OCW. A polymer x segments v v & x x B polymer x segments

20 Principal Types of Phase Diagrams T Single-phase region T Two-phase region UCST T c Two-phase region LCST T c Single-phase region φ φ PS-PI 7 8 PS-PVME 8 6 T( o C) 4 T( o C) 4 P W ps PS PS φ PVME Figure by MIT OCW. Konigsveld, Klentjens, Schoffeleers Pure Appl. Chem. 39, (974) Nishi, Wang, Kwei Macromolecules, 8, 7 (995)

21 Assignment - Reminder Problem set # is due in class on February 5th.