Mean Field Flory Huggins Lattice Theory

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.

• Following the standard treatment for small molecules (x₁ = x₂ = 1)

using Stirling’s approximation:

N

_o

=

N

₁

+

N

₂

=

n

₁

x

₁

+

n

₂

x

₂

V

=

N

_o

v

Ω_1,2= N0! N₁!N₂! 1 M for M M ln M M ! ln = − >>

ΔS

_m

=

k

(

−

N

₁

ln

φ

₁

−

N

₂

ln

φ

₂

)

φ

1 = ?

Entropy Change on Mixing

Δ

S

m

Δ

S

m

N

o

= +

k

(

−φ

1

ln

φ

1

−φ

2

ln

φ

2

)

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 ΔG_m is thus seen to always

favor mixing if the random mixing approximation is used.

Remember this is for small molecules x₁ = x₂ = 1

Enthalpy of Mixing

Δ

H

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

Δ

H

m

cont’d

υ

ij

=

υ₁₂ = N₁z φ₂

υ

₁₁ = N₁zφ₁ / 2

υ

₂₂ = N₂ zφ₂ / 2 H_1,2 =

υ

₁₂

ε

₁₂ +

υ

₁₁

ε

₁₁ +

υ

₂₂

ε

₂₂ H_1,2 = z N₁φ₂ε₁₂ + z 2 N1φ1ε11 + z 2 N2 φ2 ε22 H₁ = z N1 2 ε11 H2 = z N₂ 2 ε22 # of i,j interactions N₁ + N₂ = N₀

(

_{11 22}

)

_{1 2} 12 0⎢⎣⎡ε − 1₂ ε +ε ⎥⎦⎤φφ = z N H_M Δ ΔH_M = z N₁φ₂ε₁₂ + N1ε11 2

(

φ1−1

)

+ N₂ε₂₂ 2

(

φ2 −1

)

⎡ ⎣ ⎢ ⎤ _⎦ ⎥

Mixed state enthalpic interactions Pure state enthalpic

interactions

recall

χ

Parameter

• Define

χ

:

(

)

⎥⎦

⎤

⎢⎣

⎡

−

+

=

_{12 11 22}

2

1

ε

ε

ε

χ

kT

z

χ

represents the chemical interaction between the components

Δ

H

_M

N

₀

=

k T

χ

φ

1

φ

2

•

Δ

G

_M

:

Δ

G

_M

= Δ

H

_M

−

T

Δ

S

_M

Note: Δ_{GM is symmetric in} φ_{1 and} φ_2.

This is the Bragg-Williams result for the change in free energy for the mixing of binary metal alloys.

[

_{1 2 1 1 2 2}

]

0

ln

ln

φ

φ

φ

φ

φ

χφ

+

+

=

kT

N

G

_M

Δ

[

_{1 1 2 2}

]

2 1 0

ln

ln

φ

φ

φ

φ

φ

χφ

−

−

−

=

kT

kT

N

G

_M

Δ

Ned: add eqn for Computing Chi from V_seg (delta-delta)2_/RT

Δ

G

M

(T,

φ

,

χ

)

• Flory showed how to pack chains onto a lattice and correctly evaluate

Ω_1,2 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 (1/x) due to connectivity of x

segments 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 2 1 1 1 0 ln ln

φ

φ

φ

φ

x x k N S_M Δ

⎥

⎦

⎤

⎢

⎣

⎡

+

+

=

₂ 2 2 1 1 1 2 1 0

ln

ln

φ

φ

φ

φ

φ

χφ

x

x

kT

N

G

_M

Δ

(

)

⎥⎦

⎤

⎢⎣

⎡

−

+

=

_{12 11 22}

2

1

ε

ε

ε

χ

kT

z

Need to examine variation of ΔG_M with T, χ, φ_i, and x_i to determine phase behavior.

For polymers

Note possible huge asymmetry in x1, x2

Flory Huggins Theory

•

Many Important Applications

1. Phase diagrams

2. Fractionation by molecular weight, fractionation by composition

3. T_m depression in a semicrystalline polymer by 2nd _component

4. Swelling behavior of networks (when combined with the theory of rubber elasticity)

The two parts of free energy per site

• ΔH_M can be measured for low molar mass liquids and estimated for

nonpolar, noncrystalline polymers by the Hildebrand solubility approach. ΔS_M = − φ1 x₁ ln(φ1) − φ₂ x₂ ln(φ2) ΔH_M = χ φ₁ φ₂ ΔG_M N₀ k T

S

H

Solubility Parameter

• Liquids

Δ

H

_ν

= molar enthalpy of vaporization

Hildebrand proposed that compatibility between

components 1 and 2 arises as their solubility parameters

approach one another

δ

₁

→ δ

₂

.

•

δ

_p

for polymers

Take

δ

_P

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

δ

_P

by chemical group

contributions for a particular monomeric repeat unit.

δ = ΔE V ⎛ ⎝ ⎜ ⎞ _⎠ ⎟ 1/ 2 = cohesive energy density = ΔHν − R T V_m ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ 1/ 2

Estimating the Heat of Mixing

Hildebrand equation:

V

_m

= average molar volume of solvent/monomers

δ

₁

,

δ

₂

= 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.

Influence of

χ

on Phase Behavior

ΔSm -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ΔH m -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 ΔG m -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

x

₁

= x

₂

At what value of chi does the system go biphasic?

Expect

symmetry

-T

What happens to entropy for a pair of polymers?

Polymer-Solvent Solutions

• Equilibrium: Equal Chemical potentials:

so need partial

molar quantities j n P T i i

n

G

, , ,

⎥

⎦

⎤

⎢

⎣

⎡

∂

∂

=

μ

μ

1o

μ

2o

μ

1

,

μ

2

state

standard

in the

potential

chemical

the

is

and

activity

the

is

a

where

ln

0 i , , , 0

μ

φ

φ

μ

μ

i i i m j n P T i m i i i

n

G

n

G

a

RT

∂

∂

∂

∂

=

⎥

⎦

⎤

⎢

⎣

⎡

∂

∂

≡

≡

−

Δ

Δ

μ

₁

−

μ

₁0

=

RT

ln

φ

₁

+

1

−

1

x

₂

⎛

⎝

⎜

⎞

⎠

⎟

φ

2

+

χ φ

2 2

⎡

⎣

⎢

⎤

⎦

⎥

solvent

μ

₂

−

μ

₂0

=

RT

[

ln

φ

₂

−

(

x

₂

−

1

)

φ

₁

+

x

₂

χ φ

₁2

]

polymer

Note: For a polydisperse system of chains, use x₂ = <x₂> the number average

Recall at equilibrium

μ

_iα

=

μ

_iβ etc

P.S. #1

Construction of Phase Diagrams

• Chemical Potential

• Binodal - curve denoting the region of 2 distinct

coexisting phases

μ

₁

' =

μ

₁

"

μ

₂

' =

μ

₂

"

or equivalently

μ

₁

' –

μ

₁o

=

μ

₁

" –

μ

₁o

μ

₂

' –

μ

₂o

=

μ

₂

" –

μ

₂o

⎥

⎦

⎤

⎢

⎣

⎡

+

+

=

₂ 2 2 1 1 1 2 1 0

ln

ln

φ

φ

φ

φ

φ

χφ

x

x

kT

N

G

_m

Δ

j n P T i i

n

G

, , ,

⎥

⎦

⎤

⎢

⎣

⎡

∂

∂

=

μ

and since 2 , 2

μ

Δ

Δ

=

⎥

⎦

⎤

⎢

⎣

⎡

∂

∂

P T m

n

G

⎥

⎦

⎤

⎢

⎣

⎡

∂

∂

⎥

⎦

⎤

⎢

⎣

⎡

∂

∂

=

⎥

⎦

⎤

⎢

⎣

⎡

∂

∂

2 2 2 2

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 (x₁ = x₂) (can use volume

Construction of Phase Diagrams

cont’d

• Spinodal (Inflection Points)

• Critical Point

∂

2

Δ

G

_m

∂

φ

₂2

⎡

⎣

⎢

⎤

⎦

⎥ =

0

and

∂

3

Δ

G

_m

∂

φ

₂3

⎡

⎣

⎢

⎤

⎦

⎥ =

0

∂

2

Δ

G

m

∂

φ

₂2

⎡

⎣

⎢

⎤

⎦

⎥

α

=

₀

∂

2

Δ

G

m

∂

φ

₂2

⎡

⎣

⎢

⎤

⎦

⎥

β

=

₀

χ

c

φ

1,

c

Two phases Equilibrium Stability 0 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 χ N 4 5 6 One phase A φ_'A A B B φ_"A A' B B' Critical point φ T_low T_high hot

Dilute Polymer Solution

• # moles of solvent (n₁) >> polymer (n₂) and n₁ >> n₂x₂

• For a dilute solution:

φ

₁ = n1v1 n₁v₁ + n₂x₂v₂

φ

2 = n₂x₂v₂ n₁v₁ + n₂x₂v₂ ~ n₂x₂ n₁ Useful Approximations

φ

₂ ≡ n2x2 n₁ ~ X2x2 ln(1+ x) ≅ x − x 2 2 + x3 3 −... ln(1− x) ≅ − x − x 2 2 − x3 3 −...

⎥

⎦

⎤

⎢

⎣

⎡

⎟

⎠

⎞

⎜

⎝

⎛ −

+

−

=

−

₂2 2 2 0 1 1

χ

1

2

φ

φ

μ

μ

x

RT

Careful…

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

• 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 φ₂2 RT

• Chain connectivity (excluded volume) -(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

2 2 2 2 1 0 1 1

ln

ln(

1

)

x

RT

RTX

X

RT

X

RT

φ

μ

μ

−

=

=

−

≅

−

=

−

μ₁−μ₁0 ₌ RT −φ2 x₂ + χ − 1 2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ φ₂2 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ μ₁ −μ₁0

F-H Phase Diagram at/near

θ

condition

•

x

₁

= 1, x

₂

>> 1 and n

₁

>> n

₂

x

₂

χ

_c

=

1

2

1

x

₁

+

1

x

₂

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

2

φ

_1,c

=

x

2

x

₁

+

x

₂

Note strong asymmetry!

Find:

Symmetric when x₁ = 1 = x₂ T X2 = 1000 X₂ = 100 X2 = 30 X2 = 10 X1= 1 Binodal Two phases 0.0 0.6

Critical Composition &

Critical Interaction Parameter

x₁ , x₂ General 0.5 x₁ = x₂ = N Symmetric Polymer-Polymer x₁ = 1; x₂ = N Solvent-polymer 2 0.5 x₁ = x₂ = 1 2 low molar mass liquids

χ

_c

φ

_1,c

Binary System

x

₂

1

+

x

₂

1

2

1

+

1

x

₂

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

2

2

N

x

₂

x

₁

+

x

₂

1

2

1

x

₁

+

1

x

₂

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

2

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

x₁ = x₂= N

Phase Diagram for UCST Polymer Blend

A polymer x₁ segments v₁ ≈ _{v2 & x1} ≈ _x2

B polymer x₂ segments

χ

=

A

T

+

B

∂Δ

G

_m

∂

φ

=

0

binodal

∂

2

Δ

G

_m

∂

φ

2

=

0 spinodal

∂

3

Δ

G

_m

∂

φ

3

=

0

critical point

Predicted from FH Theory

Figure by MIT OCW.

Two phases Equilibrium Stability 0 0.0 0.2 0.4 0.6 0.8 1.0 1 2 3 χ N 4 5 6 One phase A φ_'A A B B φ_"A A' B B' Critical point φ T_low T_high hot

2 Principal Types of Phase Diagrams

Konigsveld, Klentjens, Schoffeleers

Pure Appl. Chem. 39, 1 (1974) Nishi, Wang, Kwei _{Macromolecules, 8, 227 (1995)}

Figure by MIT OCW.

Two-phase region Single-phase region UCST 0 _φ 1 2 T_c T Two-phase region Single-phase region LCST 0 _φ2 1 T_c T φ 60 0 0.2 0.4 0.6 0.8 1.0 80 100 120 140 160 180 T(oC) P1 PS W_ps 0.2 0.4 0.6 0.8 100 140 2700 2100 T(oC) 180 PS-PVME PS-PI PS PVME

Assignment - Reminder

• Problem set #1 is due in class on

February 15th.