Thermodynamics of Polymer Mixtures

7/30/2019 Thermodynamics of Polymer Mixtures

1/23

Thermodynamics of Polymer Mixtures

For mixing of two components 1 (solvent or polymer) and2 (polymer) to be spontaneous one must have Gmix < 0

Gmix = Hmix - T Smix

However, phase separation can occur when Gmix < 0

Gmix < 0 implies that a homogeneous (1-phase) mixture ismore stable than the two pure components alone.Even when Gmix < 0 a system with two mixed phases may bemore stable than a homogeneous system.


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Ideal Mixing for a Binary Mixture of Small Molecules

Raoults law applicable for each components: ai = Pi/ Pi* = xi

Intermolecular interactions between like and unlike components are identical.

Hmix = 0

Entropy of mixing is easily calculated using a cubic lattice of cells, assumingrandom mixing and using the Boltzmann equation S = k ln .

=N1 + N2( )!N1!N2!

=N0!

N1!N2!

N0 = 25N1 = NW = 16

x1 = 16/25 = 0.64

N2 = NG = 9

x2 = 9/25 = 0.36

= 1,081,575

S1+2 = k N1 lnN1

N0

+ N2 ln

N2

N0

= k N1 ln x1( )+ N2 ln x2( )[ ]

Smix = S1+2 = kN0 x1 ln x1( ) + x2 ln x2( )[ ]

S1 = klnN1!

N1!0!= ln(1) = 0

S2 = 0


3/23

Ideal Solution versus Non-Ideal Solution

Ideal Solution:

Random mixing (Smix = Sideal = -N0 R [x1 ln x1 + x2 ln x2 ]) Identical interactions between polymer segments and polymer segments,

solvent and solvent and solvent and polymer segments. (H = 0)

Gmix = Hmix TSmix = N0 kT x1 ln x1( )+ x 2 ln x2( )[ ]< 0

Non-ideal Mixing:

Athermal Solution: Hmix = 0, Smix differs from Smixideal

Regular Solution: Hmix differs from 0, Smix = Smixideal

Irregular Solution: Hmix differs from 0, Smix differs from smixid

Mixing of Polymer and Solvent: Irregular Solutions

Entropy of mixing differs greatly from ideal entropy change

Gmix = Hmix TSmix = N0kTv1

r1

ln v1( ) +v2

r2

ln v2( )

+ v1v2 FH


4/23

Flory-Huggins Theory of Polymer Solutions

Calculation of the Entropy of Mixing N1 solvent molecules with N2 polymer

molecules on a 3-D lattice containing N0 cells:

Consider a chain having r repeat units (r = 13).

r = V2/V1, is the ratio of polymer to solvent molar volumes

(assuming the repeat unit and the solvent molecules occupy the

same space (1 cell). Assume we have already added i

polymer molecules to the lattice and wish to add the i+1 thpolymer chain.

The i+1 th chain is added one repeat unit at a time. Number of vacant cells is

N0-ri. The probability pi to find a vacant cell is (N0-ri)/N0. The probability that a cell

adjacent to the first is vacant is z pi . For the third and each subsequent repeat units,the probability is (z-1) pi .The total number of ways to place the i+1 th chain isi+1

i+1 = N0 ri( )z z 1( )r2 N0 ri

N0

r1

N0 ri( )r z 1

N0

r1


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The total number of ways to place the N2 polymer molecules in the lattice is then :

= 1234........i ...N2 = ii=1

i=N2

Since we cannot distinguish between the N2 polymer molecules then should be

given by:

=i

i=1

i=N2

N2!

The entropy of the polymer solution is then given by the Boltzmann law :

SM = kln ( ) = N1 lnN1

N1+rN2

N2 ln

rN2

N1+rN2

+ N2 r1[ ] ln

z1e

+ln r

S1 = 0 S2 = N2 r 1[ ]lnz 1

e+ ln r


6/23

SM = SM S1 S2 = k N1 lnv1 + N2 lnv2[ ]= kN0 v1 lnv1 + v2r

lnv2

The entropy change upon mixing N1 molecules of solvent with N2 polymer

molecules having r repeat units is:

The entropy change in mixing N1 molecules of polymer 1 with N2 polymer

molecules to make a mixture of volume v is given below where Vo is the

lattice cell volume (polymer repeat unit or solvent molecular volume).

For a small molecule solution, r1 = r2 = 1 SM is large and positive.

For a polymer solution, r1 = 1 r2 = r (large) SM is lower and positive.

For a mixture of two polymers, r1 and r2 are large, SM is negligibly small positive.

SM = kv

V0

v1

r1

ln v1 +v2

r2

lnv2


7/23

Enthalpy of Mixing

The calculation of the enthalpy of mixing HM is carried out on the basis of theregular solution theory using the lattice approach. Upon mixing, some 1-1 and

2-2 contacts are replaced by 1-2 contacts. The enthalpy of mixing reflects the

change in the number of interactions (1-1, 1-2, 2-2).

1/2 (1-1) + 1/2 (2-2) (1-2)

The change in internal energy upon breaking a 1-1 contact and a

2-2 contact to make a 1-2 contact is given by the interchange energy, 12.

Since the lattice model does not account for changes in volume, we can consider

HM = UM = N1212 where N12 is the number of 1-2 contacts formed during

mixing.

12 = 12 1

211 + 22( )


8/23

The probability that a lattice cell is occupied by a solvent molecule is given

by v1. Therefore, each polymer molecule is surrounded on the average by

v

1rz solvent molecules. H

M= N

2v

1r z

12

Using the fact that v1 = N1 /N0 and v1 = r N2 /N0 then HM = N1v1 (z 12)

To eliminate z and include some information we know about the temperaturedependence of12 , a dimensionless parameter, F-H, known as the Flory-

Huggins Interaction parameter is introduced (per solvent molecule).F-H is a function of T. kT F-H = z 12

Therefore HM = kT N1v1F-H and the Free Energy of Mixing for the

polymer solution, GM is finally given by:

GM = kTN0 v1 lnv1 +v2

rln v2 + v1v2 FH


9/23

Limitations of the Flory-Huggins Equation

The following approximations were made:

Monodiperse polymer chains (i.e. same molar mass).

Concentrated polymer solution (pi is uniform).

Very flexible polymer chains.

Effect of specific interactions between polymer repeat units and solventmolecules on the entropy of mixing is neglected (random mixing).

Concentration dependence of the Flory-Huggins interaction parameter isneglected.

No volume change upon mixing.

The following expression for F-H is often used to account for the effect ofinteractions between polymer and solvent on the non-combinatorial entropy of

mixing.F-H = H + S where the first term accounts for HM and the second termaccounts for the effect of interactions on the entropy of mixing (non-combinatorial).


10/23

Application of Flory-Huggins Theory to Phase Behavior

We will now discuss the conditions under which the mixing of a polymer

with a solvent leads either to a one-phase or to a two-phase mixture.

Four different morphologies can result from the mixing of a polymer

with a solvent (D: dispersed (minor phase); M: matrix, (major phase)).

D: Polymer Rich

M: Solvent Rich

D: Solvent Rich

M: Polymer Rich

Single Phase

Mixture

Bicontinuous

Morphology


11/23

Polymer-Solvent Phase Diagram:

How many phases can coexist ?

The number of phases existing at any concentration, temperature and pressure is

given by the Gibbs Phase Rule. This rule states that there is at equilibrium a

relationship between the variance F, the number of components C, and the number

of phases P:

F = C - P + 2

For single component systems: (C = 1), F = 3 - P (F can take values 0, 1, 2).F = 0 triple point, F = 1 L/S, L/V, S/V coexistence, F = 2 single phase (S,L,V).

For binary mixtures: (C = 2), F = 4 - P (F can take values 0, 1, 2, 3). The

maximum number of coexistent liquid phases at a fixed pressure and temperature

is therefore two.

The variance, F, is the number of intensive variables(vi , T, P) that can be changed independently without

disturbing the number of phases in equilibrium.


12/23

Phase Diagram of Polymer Solutions

Whether a polymer/solvent mixture exhibits one or two phases under given P,

T, vi conditions, is decided by whether G is lowest for the one-phase or for the

two-phase system.

To answer this question, we will plot the free energy of mixing as a function of

composition. We note that when a mixture of two components (1 and 2) is

biphasic (phases P and P), the two phases must coexist under equilibrium

conditions:

Therefore: 1P = 1

P and 2P = 2

P

Since iP - 1

0 is proportional to the derivative ofGM with respect to Ni , then

two phases can exist under equilibrium if the slope of GM vs i is the

same for the 2 phases.

GM = kTN0 v1 lnv1 + v2r

ln v2 + v1v2 FH


13/23

Small Molecule

Solution

F-H = -1 + 1200 / T

r = 1

1.5

1.75

2

2.25

2.5

350 375 400 425 450

T (K)

F-H

- 9 0 0

-7 0 0

-5 0 0

-3 0 0

-1 0 0

100

300

0 0.5 1

2

GMp

ermole(J)

300

345

390

405

420

T (K)


14/23

T

2

20

2P

2P

1 Phase

2 Phases

405

critical point

( 2c , Tc)

binodal curve

UCST Phase Diagram

(Upper Critical Solution Temp.)

-450

-400

-350

-300

-250

-200

-150

-100

-5 0

0

0 0.5 1

2

GM

permole(J)

405

2P'

2P"


15/23

Determination of the Critical Point of a Polymer Solution

(UCST case)

The critical point is the point (v2c

, Tc) above which a one-phase solution isalways the stable state for any composition. It is defined mathematically as the

point where the second and third derivatives of the Gibbs free energy with

respect to the polymer volume fraction is zero.

The following relationships can be easily derived:

FHc =

1

21+

1

r

2

v2c =

1

1+ r

If r = 1 (small molecule solution), then F-Hc = 2 and v2

c = 1/2.

If r = 2, then F-Hc = 1.457 and v2

c = 0.414

If r = 3, then F-Hc = 1.244 and v2c = 0.366If r = infinity, then F-H

c = 0.5 and v2c = 0

The critical point for an infinite molecular weight solution is called the point

or -state. The temperature is the value of Tc for infinite molar mass.


16/23

UCST versus LCST

Phase diagrams can exhibit either an Upper Critical Solution Temperature

(UCST) or a Lower Critical Solution Temperature (LCST).

The UCST behavior (see previous slides) is obtained when phase separationoccurs upon cooling (the F-H interaction parameter decreases with increasingtemperature). In this case, F-H = A + B/T where A is negative and B is

positive.

Phase diagrams exhibiting the LCST behavior are for solutions which phaseseparate upon heating (the F-H interaction parameter increases with increasingtemperature). In this case, F-H = A + B/T where A is positive and B is

negative.

In all cases, phase separation occurs when the interaction parameter exceeds itscritical value, F-H

c (which depends only on the degree of polymerization ofthe polymer chain).

It is often seen that LCST behavior originates in the existence of specificinteractions between the polymer and the solvent.


17/23

Phase diagram for a solution

exhibiting LCST Behavior

0

0.5

1

1.5

2

2.5

3

250 300 350 400 450

T (K)

F-H

r=1

r=2

r=3

v 2

T

v 20

v 2P

v 2P

1 Phase

2 Phases

405

critical point

(v2c , Tc)

binodal curve

Evolution of F-Hc and

Tc with chain length

F-Hc = A + B/T

with A = 5 and B = -1200


18/23

Polymer Blend Thermodynamics

FHc =

1

2

1

r1

+1

r2

2

v2c =

r1

r1 + r2

Both UCST and LCST behaviors have been observed in

polymer blends.

UCST: Low molar mass polymers with only

dispersive interactions.

LCST: Polymers with either specific interactions

or differences in free volume.


19/23

Phase Separation:

Nucleation and Growth Mechanism

T

v20

v2P

v2P

1 Phase

2 Phases

U MM

S

binodal

spinodal


20/23

Phase Separation:

Spinodal Decomposition Mechanism

T

v20

v2P

v2P

1 Phase

2 Phases

U MM

S

binodal

spinodal


21/23

Polymer Blend Phase Separation:

Morphologies


22/23

Gel Structure:

Polymer/Solvent (3/97)

Spinodal Decomposition


23/23

Block Copolymer Morphologies

Diblock

Copolymer

(AB)

Triblock

Copolymer

(ABC)

Thermodynamics of Polymer Mixtures

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