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 No 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 xi adjacent cells. No =N1 + N2 = n1x1 + n2x2
V =Nov
• Following the standard treatment for small molecules (x1 = x2 = 1) N0! Ω1,2 = N1!N2! using Stirling’s approximation: !ln = − >>1MforMMlnMM
φ ΔSm = k(−N1 lnφ1 −N2 lnφ2) 1 = ? Entropy Change on Mixing ΔSm
ΔSm =+k(−φ1 lnφ1 −φ2 lnφ2) Remember this is No for small molecules x1 = x2 = 1 0.8
0.7
0.6
0.5
0.4
0.3
0.2 Note the symmetry 0.1
0 0 0.2 0.4 0.6 0.8 1
φ2
The entropic contribution to ΔGm is thus seen to always favor mixing if the random mixing approximation is used. Enthalpy of Mixing ΔHm
• 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: φi. ΔHm cont’d
υij = # of i,j interactions υ12 = N1z φ2 υ11 = N1 zφ1 /2 υ22 = N2 zφ2 /2
Mixed state enthalpic H1, 2 = υ12 ε12 + υ11ε11 + υ22 ε22 interactions z z H = zN φ ε + N φ ε + N φ ε 1, 2 1 2 12 2 1 1 11 2 2 2 22 z N z N Pure state enthalpic 1 2 H1 = ε11 H2 = ε22 interactions 2 2 ⎡ ⎤ N1ε11 N2ε22 ΔHM = zN⎢ 1φ2ε12 + ()φ1 −1 + ()φ2 −1 ⎥ ⎣ 2 2 ⎦
recall N1 + N2 = N0 ⎡ 1 ⎤ Δ M NzH ()+−= φφεεεNote the symmetry Some math ⎣⎢ 120 2 ⎦⎥ 212211 χ Parameter
z ⎡ 1 ⎤ • Define χ: ()+−= εεεχ 12 2211 ⎥ kT ⎣⎢ 2 ⎦ χ represents the chemical interaction between the components ΔH Ned: add eqn for M = kTχ φ φ Computing Chi from 1 2 V (delta-delta)2/RT N0 seg ΔG = ΔH −T ΔS • ΔGM : M M M
ΔGM kT 21 kT[]−−−= lnln φφφφφχφ2211 N0
ΔGM = kT[]++ lnln φφφφφχφ221121 N0
Note: ΔGM is symmetric in φ1 and φ2. This is the Bragg-Williams result for the change in free energy for the mixing of binary metal alloys. ΔGM(T, φ, χ)
• Flory showed how to pack chains onto a lattice and correctly evaluate Ω1,2 for polymer-solvent and polymer-polymer systems. Flory made a complex derivation but got a very simple and intuitive result, namely that ΔSM is decreased by factor (1/x) due to connectivity of x segments into a single molecule:
ΔSM ⎡ φ1 φ2 ⎤ For polymers −= k⎢ φ1 + lnln φ2 ⎥ N0 ⎣ 1 xx 2 ⎦ • Systems of Interest
- solvent – solvent x1 = 1 x2 = 1
- solvent – polymer x1 = 1 x2 = large - polymer – polymer x1 = large, x2 = large
ΔG ⎡ φ φ ⎤ M 1 2 Note possible huge = kT φχφ21 φ1 ++ lnln φ2 ⎢ ⎥ asymmetry in x1, x2 N 0 ⎣ 1 xx 2 ⎦ z ⎡ 1 ⎤ ()+−= εεεχ ⎢ 12 2211 ⎥ kT ⎣ 2 ⎦
Need to examine variation of ΔGM with T, χ, φi, and xi to determine phase behavior. Flory Huggins Theory
• Many Important Applications 1. Phase diagrams 2. Fractionation by molecular weight, fractionation by composition nd 3. Tm depression in a semicrystalline polymer by 2 component 4. Swelling behavior of networks (when combined with the theory of rubber elasticity) ΔG The two parts of free energy per site M N0 kT
φ1 φ2 S ΔSM =− ln(φ1) − ln(φ2) x1 x2
H ΔHM = χ φ1 φ2
• ΔHM can be measured for low molar mass liquids and estimated for nonpolar, noncrystalline polymers by the Hildebrand solubility approach. Solubility Parameter
cohesive 1/2 • Liquids ⎛ ΔE⎞ 1/2 ⎡ ΔH − RT⎤ δ=⎜ ⎟ = energy = ⎢ ν ⎥ ⎝ V ⎠ ⎣⎢ V ⎦⎥ density m
ΔHν = molar enthalpy of vaporization Hildebrand proposed that compatibility between components 1 and 2 arises as their solubility parameters approach one another δ1→δ2.
• δp for polymers Take δP as equal to δ solvent for which there is: (1) maximum in intrinsic viscosity for soluble polymers (2) maximum in swelling of the polymer network or (3) calculate an approximate value of δP by chemical group contributions for a particular monomeric repeat unit. Estimating the Heat of Mixing
Hildebrand equation: 2 ΔHM = Vm φ1 φ2 (δ1 − δ2 ) ≥ 0
Vm = average molar volume of solvent/monomers δ1, δ2 = 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 ΔHM can be negative. Influence of χ on Phase Behavior
Assume kT = 1 and x1 =x2 Expect ΔH symmetry
m 0.8
0.6 ΔG 0.2 0.4
0.2 m
0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.2 0 0.2 0.4 0.6 0.8 1
-0.4 χ = -1,0,1,2,3 -0.2
-0.4
-T ΔSm -0.6 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 χ = -1,0,1,2,3 -0.2 -0.3 -0.8 -0.4 -0.5 -0.6 -1 -0.7 -0.8 At what value of chi does the system go biphasic? What happens to entropy for a pair of polymers? Polymer-Solvent Solutions
• Equilibrium: Equal Chemical potentials: so need partial molar quantities pure mixed ⎡∂G ⎤ μi = ⎢ ⎥ o o ∂n μ1 μ2 μ1, μ2 ⎣ i ⎦ ,,, jnPT
0 ⎡∂ΔGm ⎤ ∂ΔGm ∂φi μμii ≡− ln aRT i where a theis activity ≡ ⎢ ⎥ = ∂n ∂φ ∂n ⎣ i ⎦ ,,, jnPT i i 0 state standard in the potential chemical theis and μi theis chemical potential in the standard state ⎡ ⎛ ⎞ ⎤ 0 1 2 μ1 − μ1 = RT ⎢ lnφ1 + ⎜1 − ⎟ φ2 + χφ2 ⎥ solvent P.S. #1 ⎣ ⎝ x2 ⎠ ⎦ μ − μ 0 = RT φ − x − φ + x χφ 2 2 2 []ln 2 ()2 1 1 2 1 polymer
Note: For a polydisperse system of chains, use x2 =
ΔGm ⎡ φ1 φ2 ⎤ = kT ⎢ φχφ21 φ1 ++ lnln φ2 ⎥ N0 ⎣ 1 xx 2 ⎦ • Chemical Potential ⎡∂G ⎤ μi = ⎢ ⎥ ∂n ⎣ i ⎦ ,,, jnPT • Binodal - curve denoting the region of 2 distinct
coexisting phases μ1' = μ1" μ2' = μ2" o o Phases called prime or equivalently μ1' – μ1 = μ1" – μ1 o o and double prime μ2' – μ2 = μ2" – μ2
⎡∂ΔGm ⎤ ⎡∂ΔGm ⎤ ⎡∂ΔGm ⎤ ⎡∂φ2 ⎤ ⎢ ⎥ = Δμ2 and since ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ∂n ∂n ∂φ ∂n ⎣ 2 ⎦ ,PT ⎣ 2 ⎦ ⎣ 2 ⎦ ⎣ 2 ⎦
Binodal curve is given by finding common tangent to ΔGm(φ) curve for each φ, T combination. Note with lattice model (x1 = x2) (can use volume fraction of component in place of moles of component Construction of Phase Diagrams cont’d
6
T low Two phases • Spinodal (Inflection Points) 5 α β Equilibrium 2 2 ⎡ ∂ ΔGm ⎤ ⎡ ∂ ΔGm ⎤ Stability = 0 = 0 4 ⎢ 2 ⎥ ⎢ 2 ⎥ ⎣ ∂φ2 ⎦ ⎣ ∂φ2 ⎦ N
χ 3 A' B' •Critical Point 2 2 3 Critical point ⎡ ∂ ΔG ⎤ ⎡ ∂ ΔG ⎤ hot m m A B χ c ⎢ ⎥ = 0 and ⎢ ⎥ = 0 1 2 3 φ φ" ⎣ ∂φ ⎦ ⎣ ∂φ ⎦ 'A One phase A 2 2 T high 0 φ 0.0 0.2 0.4 0.6 0.8 1.0 1,c A B φ B
Figure by MIT OCW. Dilute Polymer Solution
• # moles of solvent (n1) >> polymer (n2) and n1 >> n2x2
n1v1 n2 x2v2 n2 x2 φ1 = φ2 = ~ n1v1 + n2 x2v2 n1v1 + n2 x2v2 n1
Useful Approximations n x 2 2 Careful… φ2 ≡ ~ X 2 x2 n1 x 2 x 3 ln(1+ x) ≅ x − + − ..... 2 3 x 2 x 3 ln(1− x) ≅−x − − − ..... 2 3 • For a dilute solution: 0 ⎡ φ ⎛ 1 ⎞ 2 ⎤ μμ RT 2 ⎜ −+−=− ⎟φχ Let’s do the math: 11 ⎢ 2 ⎥ ⎣ x2 ⎝ 2 ⎠ ⎦ 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, Xi 0 φ2 μμ11 =− ln 1 XRTXRT 2 )1ln( RTX 2 −=−≅−= RT x2 0 • Comparing to theμ1 − μ1 expression we have for the dilute solution, we see the first term corresponds to that of an ideal solution. The 2nd term is called the excess chemical potential ⎡ ⎤ 0 φ ⎛ 1⎞ 2 μ − μ = RT⎢− 2 + ⎜ χ − ⎟φ ⎥ 1 1 ⎣ x ⎝ 2⎠ 2 ⎦ – This term has 2 parts due to 2 2 • Contact interactions (solvent quality) χ φ2 RT 2 • Chain connectivity (excluded volume) -(1/2) φ2 RT • Notice that for the special case of χ = 1/2, the entire 2nd 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 = 1/2 is also the theta point F-H Phase Diagram at/near θ condition
T
X2 = 1000 X1= 1
•x1 = 1, x2 >> 1 and n1 >> n2x2 X2 = 100 Find: Binodal ⎛ ⎞ 2 1 1 1 X2 = 30 χ c = ⎜ + ⎟ 2⎝ x1 x2 ⎠ Two phases x 2 X2 = 10 φ1,c = x1 + x2 Note strong asymmetry! 0.0 0.6
Symmetric when x1 = 1 = x2 Figure by MIT OCW. Critical Composition & Critical Interaction Parameter
Binary System φ1,c χc
2 low molar x = x = 1 0.5 2 mass liquids 1 2
⎛ ⎞ 2 Solvent- x = 1; x 1 1 1 2 ⎜ + ⎟ polymer x = N ⎜ 1 ⎟ 2 1+ x2 2⎝ x2 ⎠ Symmetric 2 x = x = N 0.5 Polymer- 1 2 N Polymer x ⎛ ⎞ 2 2 1⎜ 1 1 ⎟ General x1 , x2 ⎜ + ⎟ x1 + x2 2⎝ x1 x2 ⎠ Polymer Blends
Good References on Polymer Blends: • O. Olasbisi, L.M. Robeson and M. Shaw, Polymer-Polymer Miscibility, Academic Press (1979).
• D.R. Paul, S. Newman, Polymer Blends, Vol I, II, Academic Press (1978).
• 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. Phase Diagram for UCST Polymer Blend
Predicted from FH Theory 6
T A low B Two phases χ = + 5 T Equilibrium
Stability x1 = x2 = N 4 ∂ΔG
m N
χ 3 = 0 binodal A' B' ∂φ 2 2 ∂ ΔGm Critical point hot 2 = 0 spinodal ∂φ 1 A B φ φ 'A One phase "A 3 T high ∂ ΔGm 0 = 0 critical point 0.0 0.2 0.4 0.6 0.8 1.0 3 A B ∂φ φ B
Figure by MIT OCW.
A polymer x1 segments v1 ≈ v2 & x1 ≈ x2 B polymer x2 segments 2 Principal Types of Phase Diagrams
T T Single-phase region Two-phase region
Tc UCST
LCST Tc Two-phase region Single-phase region
0 1 0 1 φ2 φ2
PS-PI PS-PVME 2700 180
180 160
140 T(oC) T(oC) 120 140
100
2100 80 100 W ps 60 P1 0.2 0.4 0.6 0.8 PS 0 0.2 0.4 0.6 0.8 1.0 PS φ PVME
Figure by MIT OCW.
Konigsveld, Klentjens, Schoffeleers Nishi, Wang, Kwei Pure Appl. Chem. 39, 1 (1974) Macromolecules, 8, 227 (1995) Assignment - Reminder
• Problem set #1 is due in class on February 15th.