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.
N o = N1 + N 2 = n1 x1 + n 2 x 2
V = N ov
•
Following the standard treatment for small molecules (x1 = x2 = 1)
using Stirling’s approximation:
lnM ! = M ln M − M
Ω1,2 =
N 0!
N1!N 2!
for M >> 1
ΔSm = k(−N1 ln φ1 − N 2 ln φ 2 )
φ1 = ?
Entropy Change on Mixing ΔSm
ΔSm
= + k(−φ1 ln φ1 − φ 2 ln φ 2 )
No
Remember this is
for small molecules
x1 = x2 = 1
0.8
0.7
0.6
0.5
0.4
0.3
Note the symmetry
0.2
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
Mixed state enthalpic
interactions
Pure state enthalpic
interactions
recall
Some math
υij = # of i,j interactions
υ12 = N1 z φ2
υ11 = N1 z φ1 / 2
υ 22 = N2 z φ 2 / 2
H1,2 = υ12 ε12 + υ11 ε11 + υ22 ε 22
z
z
H1,2 = z N1 φ 2 ε12 + N1 φ1 ε11 + N2 φ2 ε 22
2
2
z N1
z N2
H1 =
ε11
H2 =
ε 22
2
2
⎡
⎤
Nε
Nε
ΔH M = z ⎢N1φ 2ε12 + 1 11 (φ1 −1) + 2 22 (φ 2 −1)⎥
⎣
⎦
2
2
N1 + N2 = N 0
1
⎡
⎤
ΔH M = z N 0 ⎢ε12 − (ε11 + ε 22 ) ⎥φ1φ2
2
⎣
⎦
Note the symmetry
χ Parameter
• Define χ:
χ=
z
kT
1
⎡
⎤
(
)
−
+
ε
ε
ε
⎢⎣ 12 2 11 22 ⎥⎦
χ represents the chemical interaction between the components
ΔH M
= k T χ φ1 φ 2
N0
• ΔGM :
Ned: add eqn for
Computing Chi from
Vseg (delta-delta)2/RT
ΔGM = ΔH M −T ΔSM
ΔGM
= kT χφ1φ2 − kT [− φ1 ln φ1 − φ2 ln φ2 ]
N0
ΔGM
= kT [χφ1φ2 + φ1 ln φ1 + φ 2 ln φ 2 ]
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:
⎡ φ1
⎤
ΔS M
φ2
= − k ⎢ ln φ1 + ln φ2 ⎥
N0
x2
⎣ x1
⎦
•
Systems of Interest
- solvent – solvent
- solvent – polymer
- polymer – polymer
For polymers
x1 = 1
x2 = 1
x2 = large
x1 = 1
x1 = large, x2 = large
⎡
⎤
ΔG M
φ1
φ2
ln φ 2 ⎥
= kT ⎢ χφ1φ 2 + ln φ1 +
N0
x1
x2
⎣
⎦
χ=
z
kT
1
⎡
⎤
(
)
ε
ε
ε
−
+
⎢ 12 2 11 22 ⎥
⎣
⎦
Need to examine variation of ΔGM with T, χ, φi,
and xi to determine phase behavior.
Note possible huge
asymmetry in x1, x2
Flory Huggins Theory
• Many Important Applications
1.
2.
3.
4.
Phase diagrams
Fractionation by molecular weight, fractionation by composition
Tm depression in a semicrystalline polymer by 2nd component
Swelling behavior of networks (when combined with the theory
of rubber elasticity)
ΔG M
The two parts of free energy per site
N0 k T
S
H
•
φ
ΔSM = − 1 ln(φ1 ) −
x1
φ2
ln(φ 2 )
x2
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
• Liquids
⎞1/ 2
⎛ ΔE
δ=⎜ ⎟
⎝ V ⎠
cohesive
= energy =
density
⎡ ΔH − R T ⎤1/ 2
⎥
⎢ ν
⎥⎦
⎢⎣ Vm
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:
ΔH M = Vm φ1 φ 2 (δ1 − δ2 ) ≥ 0
2
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
ΔH
m
ΔG
0.8
0.6
Expect
symmetry
0.2
0.4
m
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
-0.2
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
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
χ = -1,0,1,2,3
1
-0.2
-0.3
-0.8
-0.4
-0.5
-0.6
-1
-0.7
-0.8
What happens to entropy
for a pair of polymers?
At what value of chi does the
system go biphasic?
Polymer-Solvent Solutions
• Equilibrium: Equal Chemical potentials: so need partial
molar quantities
pure
mixed
⎡ ∂G ⎤
μi = ⎢ ⎥
⎣ ∂ni ⎦ T , P ,n , j
0
μ i − μ i ≡ RT ln ai
μ1, μ2
μ1o μ2o
⎡ ∂ΔGm ⎤
∂ΔGm ∂φi
where a is the activity ≡ ⎢
=
⎥
∂φi ∂ni
⎣ ∂ni ⎦ T , P ,n , j
0
and μ i is the chemical potential in the standard state
⎡
⎤
⎛ 1⎞
2
μ1 − μ1 = RT ⎢ ln φ1 + ⎜1− ⎟ φ 2 + χ φ 2 ⎥
⎝ x2 ⎠
⎣
⎦
solvent
0
μ 2 − μ 2 = RT [ln φ 2 − (x 2 −1)φ1 + x 2 χ φ1
0
2
]
P.S. #1
polymer
Note: For a polydisperse system of chains, use x2 = <x2> the number average
Recall at equilibrium
μiα = μiβ etc
Construction of Phase Diagrams
⎡
⎤
ΔGm
φ1
φ2
= kT ⎢ χφ1φ2 + ln φ1 + ln φ2 ⎥
N0
x1
x2
⎣
⎦
• Chemical Potential
⎡ ∂G ⎤
μi = ⎢ ⎥
⎣ ∂ni ⎦ T , P ,n , j
• Binodal - curve denoting the region of 2 distinct
coexisting phases
or equivalently
μ1' = μ1"
μ2' = μ2"
Phases called prime
μ1' – μ1o = μ1" – μ1o
and double prime
μ2' – μ2o = μ2" – μ2o
⎡ ∂ΔGm ⎤
⎡ ∂ΔGm ⎤ ⎡ ∂ΔGm ⎤ ⎡ ∂φ2 ⎤
=
Δ
μ
2 and since ⎢
⎢
⎥
⎥=⎢
⎥⎢
⎥
n
∂
∂
∂
∂
φ
n
n
2 ⎦ T ,P
⎣
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
• Spinodal (Inflection Points)
α
5
β
⎡ ∂ ΔGm ⎤
⎡ ∂ ΔGm ⎤
=0
=0
2
2
⎢⎣ ∂φ 2 ⎥⎦
⎢⎣ ∂φ 2 ⎥⎦
2
Equilibrium
Stability
4
χN
2
• Critical Point
χ
⎡∂2ΔG ⎤
⎡ ∂3ΔG ⎤
m
m
=
0
and
c ⎢
⎥
⎢
2
3 ⎥=0
⎣ ∂φ 2 ⎦
⎣ ∂φ 2 ⎦
φ1,c
low
Two phases
3
A'
B'
2
Critical point
A
1
φ'
B
φ"
One phase
A
0
hot
A
Thigh
0.0
A
0.2
0.4
0.6
φ
0.8
1.0
B
B
Figure by MIT OCW.
Dilute Polymer Solution
•
# moles of solvent (n1) >> polymer (n2) and n1 >> n2x2
φ1 =
n1v1
n1v1 + n 2 x 2v 2
φ2 =
Useful Approximations
n x
φ2 ≡ 2 2 ~ X 2 x2
n1
n 2 x 2v 2
n x
~ 2 2
n1v1 + n 2 x 2v 2
n1
Careful…
x2 x3
ln(1+ x) ≅ x − +
− .....
2
3
x2 x3
ln(1− x) ≅ − x − − − .....
2
3
•
For a dilute solution:
Let’s do the math:
⎡ φ2 ⎛
1⎞ 2⎤
μ1 − μ1 = RT ⎢ − + ⎜ χ − ⎟φ2 ⎥
2⎠ ⎦
⎣ x2 ⎝
0
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
μ1 − μ10 = RT ln X 1 = RT ln(1 − X 2 ) ≅ − RTX 2 = − RT
•
x2
0
Comparing to the μ1 − 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
⎡ φ2 ⎛
1 ⎞ 2⎤
0
μ1 − μ1 = RT⎢−
– This term has 2 parts due to
• Contact interactions (solvent quality)
• Chain connectivity (excluded volume)
•
φ2
⎣ x2
+ ⎜ χ − ⎟φ 2 ⎥
⎝
2⎠ ⎦
χ φ22 RT
-(1/2) φ22 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, x2 >> 1 and n1 >> n2x2
Find:
χc
X1= 1
X2 = 100
Binodal
1⎛ 1
1 ⎞
= ⎜⎜
+
⎟⎟
2 ⎝ x1
x2 ⎠
φ1,c =
2
X2 = 30
Two phases
x2
X2 = 10
x1 + x 2
Note strong asymmetry!
Symmetric when x1 = 1 = x2
0.0
0.6
Figure by MIT OCW.
Critical Composition &
Critical Interaction Parameter
φ1,c
χc
x1 = x2 = 1
0.5
2
x1 = 1;
x2 = N
x2
1 + x2
2
⎛
⎞
1 ⎟
1⎜
1+
⎜
x2 ⎟⎠
2⎝
0.5
2
N
Binary System
2 low molar
mass liquids
Solventpolymer
Symmetric
PolymerPolymer
General
x1 = x2 = N
x1 , x2
x2
x1 + x2
2
⎛
⎞
1 ⎟
1⎜ 1
+
⎜
x2 ⎟⎠
2 ⎝ x1
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
A
χ= +B
T
5
Equilibrium
x1 = x2 = N
binodal
2
∂ ΔGm
=0
2
∂φ
3
∂ ΔGm
=0
3
∂φ
Stability
4
χN
∂ΔGm
=0
∂φ
Tlow
Two phases
3
A'
B'
2
spinodal
Critical point
A
1
φ'
critical point
B
φ"
A
One phase
A
0
hot
Thigh
0.0
A
0.2
0.4
0.6
φ
0.8
1.0
B
B
Figure by MIT OCW.
A polymer
B polymer
x1 segments
x2 segments
v1 ≈ v2 & x1 ≈ x2
2 Principal Types of Phase Diagrams
T
T
Single-phase region
Two-phase
region
Tc
UCST
LCST
Tc
Two-phase
region
0
Single-phase region
1
φ2
PS-PI
0
2700
1
φ2
PS-PVME
180
160
180
T(oC)
T(oC)
140
140
120
100
P1
80
2100
100
Wps
0.2
0.4
0.6
60
0.8
PS
0
PS
0.2
0.4
0.6
φ
0.8
1.0
PVME
Figure by MIT OCW.
Konigsveld, Klentjens, Schoffeleers
Pure Appl. Chem. 39, 1 (1974)
Nishi, Wang, Kwei
Macromolecules, 8, 227 (1995)
Assignment - Reminder
• Problem set #1 is due in class on
February 15th.