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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
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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
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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
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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
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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( )
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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
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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).
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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
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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.
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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
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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)
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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"
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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.
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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.
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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
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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.
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Phase Separation:
Nucleation and Growth Mechanism
T
v20
v2P
v2P
1 Phase
2 Phases
U MM
S
binodal
spinodal
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Phase Separation:
Spinodal Decomposition Mechanism
T
v20
v2P
v2P
1 Phase
2 Phases
U MM
S
binodal
spinodal
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Polymer Blend Phase Separation:
Morphologies
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Gel Structure:
Polymer/Solvent (3/97)
Spinodal Decomposition
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Block Copolymer Morphologies
Diblock
Copolymer
(AB)
Triblock
Copolymer
(ABC)