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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.
• Following the standard treatment for small molecules (x1 = x2 = 1)
using Stirling’s approximation:
No =N1 + N2 = n1x1 + n2x2
V =Nov
Ω1,2 =N0!
N1!N2!
1MforMMlnMM !ln >>−=
ΔSm = k(−N1 lnφ1 −N2 lnφ2) φ1 = ?
Entropy Change on Mixing ΔSm
ΔSmNo
= + k(−φ1 ln φ1 −φ2 ln φ2)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
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.
Remember this isfor small moleculesx1 = x2 = 1
Note the symmetry
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 =υ12 = N1z φ2
υ11 = N1 zφ1 / 2
υ22 = N2 zφ2 / 2
H1,2 = υ12 ε12 + υ11ε11 + υ22 ε22
H1,2 = z N1φ2 ε12 +z2
N1 φ1ε11 +z2
N2 φ2 ε22
H1 =z N1
2ε11 H2 =
z N22
ε22
# of i,j interactions
N1 + N2 = N0
( ) 212211120 21 φφεεε ⎥⎦
⎤⎢⎣⎡ +−= NzHMΔ
ΔHM = z N1φ2ε12 +N1ε11
2φ1 −1( )+
N2ε22
2φ2 −1( )
⎡ ⎣ ⎢
⎤ ⎦ ⎥
Mixed state enthalpicinteractions
Pure state enthalpicinteractions
recall
Some math Note the symmetry
χ Parameter• Define χ: ( )⎥⎦
⎤⎢⎣⎡ +−= 221112 2
1 εεεχkTz
χ represents the chemical interaction between the componentsΔHM
N0= k T χ φ1 φ2
• ΔGM : ΔGM = ΔHM −T ΔSM
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.
[ ]2211210
lnln φφφφφχφ ++= kTNGMΔ
[ ]2211210
lnln φφφφφχφ −−−= kTkTNGMΔ
Ned: add eqn for Computing Chi from Vseg (delta-delta)2/RT
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 xsegments into a single molecule:
• Systems of Interest- solvent – solvent x1 = 1 x2 = 1- solvent – polymer x1 = 1 x2 = large- polymer – polymer x1 = large, x2 = large
⎥⎦
⎤⎢⎣
⎡+−= 2
2
21
1
1
0
lnln φφφφxx
kNSMΔ
⎥⎦
⎤⎢⎣
⎡++= 2
2
21
1
121
0
lnln φφφφφχφxx
kTNGMΔ
( )⎥⎦⎤
⎢⎣⎡ +−= 221112 2
1 εεεχkTz
Need to examine variation of ΔGM with T, χ, φi, and xi to determine phase behavior.
For polymers
Note possible huge asymmetry in x1, x2
Flory Huggins Theory• Many Important Applications
1. Phase diagrams2. Fractionation by molecular weight, fractionation by composition3. Tm depression in a semicrystalline polymer by 2nd component4. Swelling behavior of networks (when combined with the theory
of rubber elasticity)The two parts of free energy per site
• ΔHM can be measured for low molar mass liquids and estimated for nonpolar, noncrystalline polymers by the Hildebrand solubility approach.
ΔSM = −φ1x1
ln(φ1) −φ2x2
ln(φ2)
ΔHM = χ φ1 φ2
ΔGMN0 k T
S
H
Solubility Parameter• Liquids
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 polymersTake 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.
δ =ΔEV
⎛ ⎝ ⎜
⎞ ⎠ ⎟ 1/ 2
=cohesiveenergydensity
=ΔHν − RT
Vm
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
1/ 2
Estimating the Heat of MixingHildebrand equation:
Vm = average molar volume of solvent/monomersδ1, 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 specificinteractions, for which ΔHM can be negative.
ΔHM = Vm φ1 φ2 δ1 − δ2( )2 ≥ 0
Influence of χ on Phase Behavior
ΔSm
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ΔHm
-0.4
-0.2
0
0.2
0.4
0.6
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ΔGm
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0 0.2 0.4 0.6 0.8 1
χ = -1,0,1,2,3
χ = -1,0,1,2,3
Assume kT = 1 and x1 = x2
At what value of chi does thesystem go biphasic?
Expectsymmetry
-T
What happens to entropyfor a pair of polymers?
Polymer-Solvent Solutions • Equilibrium: Equal Chemical potentials: so need partial
molar quantities
jnPTii n
G
,,,⎥⎦
⎤⎢⎣
⎡∂∂
=μ μ1o μ2
o μ1, μ2
state standard in the potential chemical theis and
activity theis a whereln
0i
,,,
0
μ
φφ
μμi
i
i
m
jnPTi
miii n
GnGaRT
∂∂
∂∂
=⎥⎦
⎤⎢⎣
⎡∂
∂≡≡−
ΔΔ
μ1 − μ1
0 = RT ln φ1 + 1− 1x2
⎛
⎝ ⎜
⎞
⎠ ⎟ φ2 + χ φ2
2⎡
⎣ ⎢
⎤
⎦ ⎥ solvent
μ2 − μ2
0 = RT ln φ2 − x2 −1( )φ1 + x2 χ φ1
2[ ] polymerNote: For a polydisperse system of chains, use x2 = <x2> the number average
Recall at equilibrium μiα = μi
β etc
P.S. #1
pure mixed
Construction of Phase Diagrams
• Chemical Potential
• Binodal - curve denoting the region of 2 distinctcoexisting phases μ1' = μ1" μ2' = μ2"or equivalently μ1' – μ1
o = μ1" – μ1o
μ2' – μ2o = μ2" – μ2
o
⎥⎦
⎤⎢⎣
⎡++= 2
2
21
1
121
0
lnln φφφφφχφxx
kTNGmΔ
jnPTii n
G
,,,⎥⎦
⎤⎢⎣
⎡∂∂
=μ
and since2,2
μΔΔ
=⎥⎦
⎤⎢⎣
⎡∂
∂
PT
m
nG
⎥⎦
⎤⎢⎣
⎡∂∂
⎥⎦
⎤⎢⎣
⎡∂
∂=⎥
⎦
⎤⎢⎣
⎡∂
∂
2
2
22 nG
nG mm φ
φΔΔ
Phases called primeand double prime
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
• Spinodal (Inflection Points)
• Critical Point
∂2ΔGm
∂φ22
⎡
⎣ ⎢
⎤
⎦ ⎥ = 0 and ∂3ΔGm
∂φ23
⎡
⎣ ⎢
⎤
⎦ ⎥ = 0
∂2ΔGm
∂φ22
⎡
⎣ ⎢ ⎤
⎦ ⎥
α
= 0∂2ΔGm
∂φ22
⎡
⎣ ⎢ ⎤
⎦ ⎥
β
= 0
χ c
φ1,c
Two phases
Equilibrium
Stability
00.0 0.2 0.4 0.6 0.8 1.0
1
2
3χN
4
5
6
One phase
A
φ'A
A B
B
φ"A
A'
B
B'
Critical point
φ
Tlow
Thigh
hot
Figure by MIT OCW.
Dilute Polymer Solution• # moles of solvent (n1) >> polymer (n2) and n1 >> n2x2
• For a dilute solution:
φ1 =n1v1
n1v1 + n2x2v2
φ2 =n2x2v2
n1v1 + n2x2v2
~ n2x2
n1
Useful Approximations
φ2 ≡n2x2
n1
~ X2x2
ln(1+ x) ≅ x −x 2
2+
x 3
3− .....
ln(1− x) ≅ − x −x 2
2−
x 3
3− .....
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −+−=− 2
22
2011 2
1 φχφμμx
RT
Careful…
Let’s do the math:
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
• Comparing to the 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
– This term has 2 parts due to • Contact interactions (solvent quality) χ φ2
2 RT• Chain connectivity (excluded volume) -(1/2) φ2
2 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 compensatedby the solvent quality effect.
Previously we called this the θ condition, so χ = 1/2 is also the theta point
2
2221
011 )1ln(ln
xRTRTXXRTXRT φμμ −=−≅−==−
μ1 − μ10 = RT −
φ2
x2
+ χ −12
⎛ ⎝ ⎜
⎞ ⎠ ⎟ φ2
2⎡
⎣ ⎢
⎤
⎦ ⎥
μ1 − μ10
F-H Phase Diagram at/near θ condition
• x1 = 1, x2 >> 1 and n1 >> n2x2
χ c =12
1x1
+1x2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
φ1,c =x2
x1 + x2
Note strong asymmetry!
Find:
Symmetric when x1 = 1 = x2
TX2 = 1000
X2 = 100
X2 = 30
X2 = 10
X1= 1
Binodal
Two phases
0.0 0.6
Figure by MIT OCW.
Critical Composition &Critical Interaction Parameter
x1 , x2General
0.5x1 = x2 = NSymmetricPolymer-Polymer
x1 = 1; x2 = N
Solvent-polymer
20.5x1 = x2 = 12 low molar mass liquids
χcφ1,cBinary System
x21+ x2
12
1+1x2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
2N
x2x1 + x2
12
1x1
+1x2
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
2
Polymer BlendsGood 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) BehaviorWell accounted for by F-H theory with χ = a/T + b
• Lower Critical Solution Temperature (LCST) BehaviorFH 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.
x1 = x2 = N
Phase Diagram for UCST Polymer Blend
A polymer x1 segments v1 ≈ v2 & x1 ≈ x2B polymer x2 segments
χ = AT
+ B
∂ΔGm
∂φ= 0 binodal
∂2ΔGm
∂φ 2 = 0 spinodal
∂3ΔGm
∂φ 3 = 0 critical point
Predicted from FH Theory
Figure by MIT OCW.
Two phases
Equilibrium
Stability
00.0 0.2 0.4 0.6 0.8 1.0
1
2
3χN
4
5
6
One phase
A
φ'A
A B
B
φ"A
A'
B
B'
Critical point
φ
Tlow
Thigh
hot
2 Principal Types of Phase Diagrams
Konigsveld, Klentjens, SchoffeleersPure Appl. Chem. 39, 1 (1974)
Nishi, Wang, Kwei Macromolecules, 8, 227 (1995)
Figure by MIT OCW.
Two-phaseregion
Single-phase region
UCST
0 1φ2
Tc
TTwo-phase
region
Single-phase region
LCST
0 1φ2
Tc
T
φ
600 0.2 0.4 0.6 0.8 1.0
80
100
120
140
160
180
T(oC)
P1 PS
Wps
0.2 0.4 0.6 0.8
100
140
2700
2100
T(oC)
180
PS-PVMEPS-PI
PS PVME
Assignment - Reminder
• Problem set #1 is due in class on February 15th.