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.