Polymer Solutions and Blends

Why do some things mix and others don’t?.

A Phase Separated System

Δ S total = + Δ S system Δ S surroundings Remove Barrier

The second law

- T Δ S total = - T Δ S Δ H

The Gibbs Free Energy

Would Δ Stotal be > 0 ? Heat

Δ S total > 0

Δ G = - T Δ S total = - T Δ S Δ H

= - Δ Q system T = Δ H system

T Δ S surroundings -

The Free Energy of Mixing

Mix Δ S total > 0 Δ Gmix < 0

Polymer + Solvent For Mixing

The Mean Field Assumption

Each molecule is assumed to be acted upon by a potential that is an average taken over all the interactions in the system, rather than one determined by local composition . This allows us to treat the entropy and enthalpy changes upon mixing as separate and additive quantities. We will start by treating the entropy.

The Entropy of Mixing

First examine the simplest possible problem, mixing spherical molecules of equal size. If we can figure out the total number of ways all the molecules can be arranged, Ω, then we can use Boltzmann’s equation;

S = k ln Ω to calculate the entropy. But where on earth do you even start on a problem like this, when you have the jumbled up, randomly arranged mess that is the liquid state ?

The Entropy of Mixing

Lattice Model

The Entropy of Mixing Small Molecules of Equal Size

If we start with nA molecules of type A and nB molecules of type B, then also assume that there are no “holes” on the lattice*, then there are just n0 = nA + nB lattice sites. Now we take all the molecules off the lattice and put them back on randomly one at a time, not caring, at this point, if it’s an “A” or a “B”. How many ways are there to put the first molecule on the lattice? A. nA B. nB C. n0

*This is equivalent to assuming that there is no free volume, or the fluid is incompressible.

The Entropy of Mixing Small Molecules of Equal Size

How many ways are there of putting two molecules on the lattice?

The Entropy of Mixing Small Molecules of Equal Size

It must be (n0)(n0 - 1). OK, how many ways can you put 3 molecules back on the lattice? - you’ve got it, (n0)(n0 - 1) )(n0 - 2). So, the number of ways of putting all n0 molecules back on the lattice is, (n0)(n0 - 1) )(n0 - 2) (n0 - 3) - - - - (3)(2)(1). or n0 factorial, n0!

There are n0 - 1 arrangements of

the second molecule with first molecule placed

here

Another n0 - 1 arrangements with first molecule now

placed here

The Entropy of Mixing Small Molecules of Equal Size

Do we get a new distinguishable

arrangement if we let these two molecules

switch positions?

The Entropy of Mixing Small Molecules of Equal Size

Ω = ( n A + n B )!

n A ! n B !

- Δ S m = k ( n A ln x A + n B ln x B )

x A = n A

n A + n B x B =

n B n A + n B

ln( n A ! ) = n A ln n A - n A , etc

S = k lnΩ

- Δ S m = R ( n A ln x A + n B ln x B )

- Δ S m = k ( n A ln x A + n B ln x B )

The Entropy of Mixing Small Molecules of Equal Size

Molecules

Moles

The Entropy of Mixing Molecules of Different Size

The Entropy of Mixing a Polymer and a Solvent

Model -the polymer is a chain of connected

segments, each equal in size to a solvent

molecule.

The Entropy of Mixing a Polymer and a Solvent

A much more complex problem. Flory’s result;

- Δ S m = R ( n A ln φ A + n B ln φ B ) where φA and φB are the volume fractions of the A and B components (say solvent and polymer), respectively. This is often called the combinatorial entropy of mixing.

- Δ S m = R

( n A ln x A + n B ln x B )

- Δ S m = R

( n A ln φ A + n B ln φ B )

The Entropy of Mixing

Regular Solution Theory

Flory - Huggins Theory

Mole Fraction and Volume Fraction

The mole fraction of polymer is;

x p = n p

n p + n s = ������ 1

76

φp = = ������ 25 100 np mVs + nsVs

np mVs

The volume fraction is;

The Entropy of Mixing

- ΔSm = k(25 ln0.25 +75 ln0.75)

- ΔSm = k(1 ln0.25 +75 ln0.75)

The Entropy of Mixing

- Δ S m R

V r V =

V r V

n A ln φ A + n B ln φ B [ ]

n A V r V

= n A m A V r Vm A

= φ A m A

n B V r V =

n B m B V r Vm B

= φ B m B

- Δ S m R

V r V

= - Δ S ‘ m R

= φ A m A

ln φ A + φ B m B

ln φ B ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

The small entropy of mixing polymers is more seen if we express it on a per mole of lattice sites basis (i.e. divide by the number of moles of lattice sites = V/Vr)

Using

The Enthalpy of Mixing

•  Dispersion Forces •  Dipole/dipole Interactions •  Hydrogen Bonding •  Coulombic Interactions

Increasing Interaction Strength Non Polar

Highly Polar

Focus on dispersion and weak polar

forces }

I can feel these guys But not

these

Nearest Neighbor Assumption

B B

B B

B B B

B

Let the attractive interaction energy between a pair of small A molecules be εAA. If we assume that we can simply add the interactions between all pairs, then we can say that the interaction of a chosen A molecule with all its nearest neighbors is zεAA, where z is the number of nearest neighbors. A

A

A A

A A

A A

Interaction energy

Do the same for the B molecules

B B What we really want, however, is the change in energy on going from the pure to the mixed state. In order to calculate this we first consider the change in energy when we replace interactions between a pair of A molecules, AA, and pairs of B molecules, BB, with AB pairs. The energy change per AB pair is given by;

A A

B A B A Δ ε AB = ε AB -

1 2 ε AA + ε BB [ ]

Change in Interaction Energy

Δ H m = n A φ B z Δ ε AB [ ] A

A A

A

B B

B A

Probability that a B is next to a chosen A is φB

The Heat of Mixing

Δ H m = n AB Δ ε AB

Now all we have to do is multiply this by the number of AB pairs

Then

The Flory Parameter χ

χ = z Δ ε AB kT

Δ H m = kT n A φ B χ [ ]

Flory defined the following interaction parameter, χ, which he made dimensionless by dividing by kT;

The heat of mixing is then simply

The Flory-Huggins Equation

Δ G m = Δ H m - T Δ S m = kT n A φ B χ + n A ln φ A + n B ln φ B [ ]

Δ G m RT

= n A ln φ A + n B ln φ B + n A φ B χ

Cohesive Energy Density

A

A A

A A

A A

A

B B

B B

B B B

B

C AA = Δ E vap

V = n A z ε AA

2 V A

C BB = Δ E vap

V = n B z ε BB 2 V B

Cohesive Energy Density

A A

A A

A A

A B

C AB = ?

C AB = C AA 0 . 5 C BB

0 . 5

Δ E ~ - ( 2 C AB - C AA - C BB ) = ( C AA 0 . 5 - C BB 0 . 5 ) 2

δ A = C AA 0 . 5

δ B = C BB 0 . 5

Δ E ~ ( δ A - δ B ) 2

Δ H m = ( n A + n B ) V m φ A φ B ( δ A - δ B ) 2

χ ≈ 0 . 34 + V r RT

δ A - δ B [ ] 2

Solubility Parameters

χ ≈ V r RT

δ A - δ B [ ] 2

The Phase Behaviour of Polymer Solutions and Blends

Example of a phase separated or immiscible

system

Δ S total > 0 Δ Gmix < 0

One Condition for Mixing

The Phase Behaviour of Polymer Solutions and Blends

Δ G ' m RT

= Δ G m RT

V r V

= φ A m A

ln φ A + φ A m A

ln φ B + φ A φ B χ

χ ≈ V r RT

δ A - δ B [ ] 2 Always Positive

Positive

Δ G ' m RT

= Δ G m RT

V r V

= φ A m A

ln φ A + φ A m A

ln φ B + φ A φ B χ

Small

Polymer Blends

Phase Behaviour

Single Phase

(Miscible Mixture)

Tem

pera

ture

Composition

Phase Separated

(Immiscible)

χ ≈ V r RT

δ A - δ B [ ] 2

χ = a T

+ b

xB1 xB2

B1

B2 Q

Q*

xB

ΔG

0 0

Phase Behaviour

ΔG1

ΔGtotal = ΔGQ = ΔG1 + ΔG2

ΔG2

xB1 xB2

B1

B2 Q

Q*

xB

ΔG

0 0

Phase Behaviour

ΔGQ*

ΔGQ* < ΔGQ

xB1 xB2

B1

B2 Q

Q*

xB

ΔG

0 0

The Chemical Potential and the Conditions for Phase Separation

ΔG1 ΔG2

xB1 xB2

B1

B2

xB

ΔG

0 0

The Chemical Potential and the Conditions for Phase Separation

∂ Δ G ∂ n B

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ n B 1

= ∂ Δ G ∂ n B

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ n B 2

∂ Δ G ∂ x B

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ x B 1

= ∂ Δ G ∂ x B

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥ x B 2

Δ µ B 1 = Δ µ B

2

or

The Chemical Potential

Δ µ B 1 = Δ µ B

2 Composition B1

Composition B2

Δ µ A 1 = Δ µ A

2

The Flory-Huggins Chemical Potential

Composition B1

Composition B2

Δ G m RT

= n s ln φ s + n p ln φ p + n s φ p χ

Δ µ s RT = ln( 1 - φ p ) + 1 -

1 M

⎡ ⎣

⎤ ⎦ ⎥ φ p + φ p

2 χ ⎥

Phase Diagrams

Δ µ s 1 = Δ µ s

2 B1

B2

xB

ΔG Te

mpe

ratu

re

UCST

Composition

xB1 xB2

B1

B2

xB

ΔG

S1 S2 C

0 0

More Phase Behaviour

B1

B2

xB

ΔG

S1 S2

Tem

pera

ture

Composition

0 0

UCST Binodal

Spinodal

The Binodal and Spinodal

∂ 2 Δ G

∂ x B 2 = 0

∂ 3 Δ G

∂ x B 3 = 0

At the UCST we also have

Tem

pera

ture

Composition

The Conditions for Forming a Miscible Mixture

Single Phase

Phase Separated

We can now summarize our conditions for forming a single phase or miscible mixture; a) The free energy change on mixing should be negative. b) The second derivative of the free energy of mixing should be positive (which means a point of inflection on the free energy curve has not been reached and it is concave upwards across the composition range).

Tem

pera

ture

Tc

[φp]c Composition

The Critical Value of χ

∂ 2 ( Δ G m ' / RT )

∂ φ A 2 =

1 φ A m A

+ 1

φ B m B - 2 χ = 0

∂ 3 ( Δ G m

' / RT ) ∂ φ A 3

= - 1

φ A 2 m A +

1 φ B 2 m B

= 0

χ c = 1 2

1 + 1

m p 1 / 2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

2

The Critical Value of - Blends χ

∂ 2 ( Δ G m ' / RT )

∂ φ A 2 =

1 φ A m A

+ 1

φ B m B - 2 χ = 0

∂ 3 ( Δ G m

' / RT ) ∂ φ A 3

= - 1

φ A 2 m A +

1 φ B 2 m B

= 0

φ A [ ] c = m B

1 / 2

m A 1 / 2 + m B

1 / 2

χ c = 1 2

1 m A

1 / 2 + 1

m B 1 / 2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

2

UCST

Upper Critical Solution Temperature

Tem

pera

ture

Composition

The Critical Value of the Solubility Parameter Difference for Polymer Solutions

χ c = 0 . 34 + V r RT

δ p - δ s [ ] c 2

= 0 . 5

δ p - δ s [ ] ≈ ± 1

χ c = 1 2

1 + 1

m p 1 / 2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

2

χ c ~ 1 / 2

0.15 0.10 0.05 0.00 -10 -5 0 5

10 15 20

Volume Fraction of Polymer

T (0

C)

Flory - Huggins Theory

Experimental Data points

A Comparison of Theory and Experiment

Values of for Solutions of ���Polystyrene in Cyclohexane

χ

1.0 0.8 0.6 0.4 0.2 0.0 0.4

0.6

0.8

1.0

1.2

1.4

VOLUME FRACTION OF POLYMER

χ

FLORY - HUGGINS

χ = χ 1 + χ 2 φ p + χ 3 φ p 2 + - - - - -

The Lower Critical Solution Temperature

Tem

pera

ture

Composition

LCST - Lower Critical Solution Temperature

Phase Separated

Single Phase

10,300 4800

19,800 10,300

4800 0.1 0.2 0.3 Polymer volume fraction

Tem

pera

ture

(°C) 190

140 100

50

-50 0

The Lower Critical Solution Temperature

The Lower Critical Solution Temperature

UCST - Upper Critical Solution Temperature

Tem

pera

ture

Composition

LCST - Lower Critical Solution Temperature

An aside is in order here. The fact that the lower critical solution temperature is at a higher temperature than the upper critical solution temperature can be a bit confusing. The word “lower” in LCST is chosen to mean at the bottom (lowest temperature) of a two phase region, while the word “upper” in UCST designates the top of a two phase region.

Phase Separated

Phase Separated

Some Limitations of the Flory - Huggins Theory

1. Based on a lattice model that uses various approximations in the “counting” process 2. Ignores "free volume” and only accounts for combinatorial entropy 3. Assumes random mixing of chains in calculating the entropy and segments in calculating the enthalpy 4. Only applies to non - polar molecules 5. Does not Apply to Dilute Solutions

Dilute Solutions

• There is an intermolecular excluded volume effect in a good solvent (meaning χ << 0.5, the critical value). • There is also an intramolecular excluded volume effect, again in a good solvent, such that the chain expands its dimensions relative to the usual end-to-end distance found in concentrated solutions or the melt. • The chain expansion varies with χ, hence temperature, and there is a temperature, the theta temperature, where the chain again becomes ideal, or can be described by Gaussian statistics.

Tem

pera

ture

The Theta Temperature

α 5 ~ m 0 . 5 1 2 - χ ⎡ ⎣ ⎢

⎤ ⎦ ⎥ +

φ p 3 + - - - - - -

R 2 0 . 5

~ α m 0 . 5 ~ m 0 . 6

At high temperatures

At the θ temperature

1 2

χ ∼

R 2 0 . 5

~ m 0 . 5