# 2.5 Thermodynamics of Polymer Solutions (1)

Mechanics

Oct 27, 2013 (4 years and 8 months ago)

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22
2.5 Thermodynamics of Polymer Solutions (1)

Notation: A = solvent; B = solute (polymer)
in case of copolymers or multi-component systems:
1 = solvent; 2,3...polymer

Thermodynamic of low molecular weight solution
(revision):
Gibbs free energy (Free Enthalpy): G = f(p,T,n)
dG =

G
T
dT
G
p
dp
G
n
dn
p n
T n
i
p T n
i
i
i j

+

+

,
,
,,

dG = - S dT + V dp + Σµ
i
dn
i
;

p = const; T = const:  dG = Σµ
i
dn
i

1. + 2. law of thermodynamics (isothermal condition, dT = 0):

dG = dH – T dS + Σµ
i
dn
i

partial molar entropy s
i
: s
i
= -(∂µ
i
/∂T)
p,n

partial molar volume v
i
: v
i
= (∂µ
i
/∂p)
T,n

Pressure dependence of chemical potential µ
i
:

µ
i
id
(p) = µ
i
id
(p
o
) + RT ln (p/p
o
); µ
i
id
(p
o
) = µ
i,o
(standard pot.)

µ
i
re
(p) = µ
i
id
(p
o
) + RT ln (f/f
o
) ; f = fugacity

Concentration dependence of chemical potential µ
i
:

µ
i
id
(p,T,x
i
) = µ
i
*
(p,T,x
i
=0) + RT ln x
i

µ
i
re
(p,T,x
i
) = µ
i
*
(p,T,x
i
=0) + RT ln a
i
; a
i
(activity) = x
i
f
i

f
i

activity coefficient

µ
i
re
(p,T,x
i
) = µ
i
id
(p,T,x
i
) + µ
i
excess
(p,T,x
i
)

Entropy of mixing: )S
id
= -R ∑ n
i
ln x
i
= -R n
A
ln x
A
– R n
B
ln x
B
;

23
Classification of solutions:

∆µ
ex
∆s
ex
∆h
Ideal solutions
Athermic solutions
Regular solutions
Irregular solutions
= 0
≠0
≠0
≠0
= 0
≠0
= 0
≠0
= 0
= 0
≠0
≠0

Entropy of mixing: The Flory-Huggins theory (1)

Deviation of polymer solutions from ideal behavior is mainly due to
low mixing entropy. This is the consequence of the range of difference
in molecular dimensions between polymer and solvent.

 Flory (1942) and Huggins (1942)

Calculation of ∆G
m
= ∆G

(A,B)

- {∆G (A) + ∆G (B)}
∆H = 0  ∆G
m
= -T ∆S
m

Lattice model

volume of solvent molecule: V
A
;
each solvent molecule occupies
1 lattice cell
N
A
= number of solvent molecules

volume of macromolecule: V
B

each macromolecule occupies
V
B
/V
A
= L

lattice cells
N
B
= number of macromolecules

Number of lattice cells: K = N
A
+ L N
B
Coordination number: z (two-dimensional: z = 4)

V
A

V
B
= L V
A
=10 V
A

24
Flory-Huggins theory (2)

• transfer of the polymer chains from a pure, perfectly ordered state
to a state of disorder
• mixing process of the flexible chains with solvent molecules

Calculation of the number of possible ways a polymeric chain can be

1. Macromolecule

1
st
Segment K possibilities of arrangement on lattice
2
nd
Segment z possibilities of arrangement on lattice
3
rd
Segment z – 1 possibilities of arrangement on lattice

L segments of 1. macromolecule:
Λ
1
= K z (z – 1)
L – 2

i. Macromolecule

number of vacant cells: K – (i - 1)L

 probability to find a vacant cell: (K – (i-1)L)/K
( mean-field theory)

L segments of i. macromolecule:
ν
i
= (K – (i-1)L z (K – (i-1)L)/K {(z-1) (K – (i-1)L)/K}
L - 2

thermodynamic probability Ω = (N
B
! 2
NB
)
-1
Π ν
i

entropy (Boltzmann): S(N
A
,N
B
) = k
B
ln Ω
(solvent: only 1 arrangement

∆S
m
= S

(N
A
,N
B
)

- {S(N
A
) + S(N
B
)}

 ∆S
m
= -R (n
A
ln Φ
A
+ n
B
ln Φ
B
)

Φ
A
= volume fraction solvent = N
A
/K = n
A
V
A
/( n
A
V
A
+ n
B
V
B
)
Φ
B
= volume fraction polymer = L N
B
/K = n
B
V
B
/( n
A
V
A
+ n
B
V
B
)

25
Flory-Huggins theory (3); chemical potential

∆µ
A
= RT ln a
A
= RT (ln Φ
A
+ (1 - V
A
/V
B
) Φ
B
)
∆µ
B
= RT ln a
B
= RT (ln Φ
B
+ (1 – V
B
/V
A
) Φ
A
)

 ∆µ
A
= f (M) !

V
A
/V
B
= [ρ
B
M
A

A
M
O
] 1/P  V
A
/V
B
~ 1/P ~1/M
B

Enthalpy change of mixing

quasichemical process: ½ (A-A) + ½ (B-B)  (A-B)

(A-B): solvent-polymer contact

interchange energy per contact: ∆u = ∆ε
AB
= ε
AB
– ½ (ε
AA
+ ε
BB
)

∆U = ∆H if no volume change takes place on mixing

∆H = q ∆ε
AB
; q = number of new contacts

calculation of number of contacts can be estimated from the lattice
model assuming that the probability of having a lattice cell occupied
by a solvent molecule is simply the volume fraction Ν
A
, by a polymer
Ν
B
.
 q = Φ
A
Φ
B
z K

∆H = Φ
A
Φ
B
z K∆ε
AB

with: χ := z ∆ε
AB
/k
B
T (definition of χ!)
and K = N
L
n
A

A
; R = N
L
k
B

∆H = RT n
A
Φ
B
χ

χ = Huggins interaction parameter

26
Gibbs enthalpy based on Flory-Huggins theory:

∆G
m
= RT (n
A
ln Φ
A
+ n
B
ln Φ
B
+ n
A
Φ
B
χ)

(often in literature ΔG
m
/mol (monomer and solvent));

∆G
m
/(n
A
+ Pn
B
0
) = RT (Φ
A
ln Φ
A
+ (Φ
B
/P)

ln Φ
B
+ Φ
A
Φ
B
χ)

 ∆µ
A
= RT (ln Φ
A
+ (1 - V
A
/V
B
) Φ
B
+ Φ
B
2
χ)

Meaning of χ

combinatorial
comb
= entropy according F.-H.
residual
R
= difference to the combinatorial solution, and
excess
ex
= difference to the ideal low-molecular weight solution
term of chemical potential

∆µ
A
= ∆µ
A
comb
+ ∆µ
A
R

 χ = ∆µ
A
R
/RTΦ
B
²

enthalpic and entropic parts of Δµ
A
R
:

∆µ
A
R
= T∆s
A
R
+ ∆h
A
 χ = χ
H
+ χ
S

with χ
H
= ∆h/RTΦ
B
²; χ
S
= Δs
A
R
/RΦ
B
²

Determination of χ
H
and χ
S
: χ
H
= -T(∂χ/∂T)
p

χ
S
= d(Tχ)/dT

χ
S
= 0 (combinatorial solution; F.-H. equ. valid)
 χ

= a/T

experiments:
χ

= a + b/T 
χ
S
 0

in most cases: χ
S
, χ
H
> 0; χ
S
> χ
H
;

χ < 0 means: contacts between A and B are preferred (good solution)

27
Theta-temperature and Phase separation (1)

phase stability conditions:
• temperature (∂²g/∂T²)
p
< 0
• pressure (∂²g/∂p²)
T
< 0
• concentration (∂²g/∂x²)
p,T
> 0

binodal curve (local minima):
spinodal curve (reflection point):
Application on Flory-Huggins:
• binodal curve
∂Δμ
∂Φ
χΦ
A
B
p T
B
A
B
B
RT
V
V

= −

+ −

+

,
1
1
1 2
Φ
(1)

• spinodal curve
(
)

∂Φ
χ
2
2 2
1
1
2
Δμ
Φ
A
B
p T
B
RT

= −

+

,
(2)

critical point: (1); (2) = 0

Φ
Bc
B A
c
A
B
A
B
V V
V
V
V
V
,
/
;=
+
= + +
1
1
1
2 2
χ

polydispers polymer:
(
)
(
)
Φ
Bc
w z
c z w z
P P
P P P
,
/
;//=
+
= + +
1
1
1
2
1 1 1χ

∂Δ

g
x
g
x
m
p T
m
p T

=

>
,
,
;0 0
2
2
Δ

2
2
3
3
0 0
Δ Δg
x
g
x
m
p T
m
p T

=

,,
;

28
Theta-temperature and Phase separation (2)
(
)
Δ Δ
Δ Δ Δ
μ μ
μ
χφ χ χ φ
A
R
A
comb
A A A
RT RT RT
h
RT
s
R
=− + = − = = +
B H s B
2 2

critical point: T = T
c
; χ = χ
c

(*)
χ
φ
χ
χ
φ
χ
c
A
B
A
B
A
c
A
c
V
V
V
2V
h
RT
M ) =
1
h
RT M )
= + + = +
→∞
⇒ =
→∞
+
1
2
2
1
2
2
2
Δ
Δ
B
s
c
B
s
(
(

χ ψ
s
≡ −

1
2

(*)
1
2
1
2
2
+ + = + −
V
V
V
2V
h
RT
A
B
A
B
A
c
Δ
φ
ψ
B

Δh RT M )
A c
= →∞ψφ
B
2
(

RT (M
RT
c B
c B
A
B
A
B
V
V
V
V
→∞
= + +
+
)ψφ
φ
ψ
2
2
2

1
T
1
T M
1
T M
c c c
=
→∞
+
→∞
+

( ) ( )ψ
V
V
V
V
A
B
A
B
2

T M T
c
( )→∞
=
Θ

Theta-temperature

29
Second virial coefficient and χ

Δµ
A
= Δµ
A
id
+ Δµ
A
ex

Osmosis: Δµ
A
= -πV
A
real solution, virial expression:
π/c
B
= RT (1/M + A
2
c
B
+ A
3
c
B
+ …)

Δµ
A
= f(c
B
)

 expanding ln Φ
A
= ln (1 - Φ
B
) as far as the second term in a Taylor
series, Φ
B
= c
B

B

 A
2
= (1/2 - χ)/(ρ
B
2
V
A
)

Δµ
A
ex
= - RTA
2
c
B
²V
A
= - RT(1/2 - χ)Φ
B
²ρ
B
²V
A
/(ρ
B
2
V
A
)

 Δµ
A
ex
= - RT(1/2 - χ)Φ
B
²

Second virial coefficient and Thetatemperatur

A
2
= (1/2 - χ)/(χ
B
2
V
A
);
χ = χ
H

S

χ= ΨT
θ
/T + χ
S
= ΨT
θ
/T + ½ - Ψ

 A
2
= Ψ( 1 - T
θ
/T) /(ρ
B
2
V
A
)

T = T
θ
: A
2
= 0

 pseudo-ideal solution

30
Flory–Krigbaum theory

to overcome the limitations of the lattice theory resulting from the
discontinuous nature of a dilute
polymer solution

 solution is composed of areas
containing polymer which were
separated by the solvent

Polymer areas: Polymer segments
with a Gaussian distribution about the center of mass
 chain segments occupy a finite volume from which all other chain
segments are excluded (long range interaction)

see Excluded Volume Theory

Introduction of two parameters
enthalpy parameter κ
entropy parameter Ψ
to describe long range interaction effects:

Δµ
A
= Δ µ
A
id
+ Δ µ
A
ex
;

Δ µ
A
ex
= - RT(1/2 - χ) Φ²
B
=

Δ h
ex
-T Δ s
ex

Δ h
ex
= RT κ Φ²
B
; Δ s
ex
= R ψ Φ²
B

 (1/2 - χ) = (ψ - κ)

Theta condition Δ µ
A
ex
= 0
 ψ = κ

Δ h
ex
=T Δ s
ex
;

T
θ
:= T (κ/ψ) ;

Δ µ
A
ex
= - RTψ( 1 - T
θ
/T) Φ²
B

Deviations from ideal (pseudoideal) behavior vanish
when T = T
θ
!

31

32

33
2.5 Thermodynamic of Polymer Solution (2)
Solubility Parameter

The strength of the intermolecular forces between the polymer
molecules is equal to the cohesive energy density (CED),
which is the molar energy of vaporization per unit volume.
Since intermolecular interactions of solvent and solute must be
overcome when a solute dissolves, CED values may be used to
predict solubility.

1926, Hildebrand showed a relationship between solubility
and the internal pressure of the solvent;

1931, Scatchard incorporated the CED concept into
Hildebrand´s eq.

δ=
ΔH
V
2

(nonpolar solvent; )H = heat of vaporization)

heat of mixing: ΔH
m
= VΦ
A
Φ
B

A
- δ
B
)² = n
A
V
A

A
- δ
B

Acc. solubility parameter concept any nonpolar polymer will
dissolve in a liquid or a mixture of liquids having a solubility
parameter that does not differ by more than ±1.8 (cal cm
-3
)
0.5
.
Small:

F = Σ F
i
; F
i
= molar attraction constant [in (Jcm³)
1/2
mol
-2
]
-CH
3
438 -CH
2
- 272 ≡CH- 57
=C= -190 -O- 143 -CH(CH
3
)- 495
-HC=CH 454 -COO 634 -CO- 563

“Like dissolves like” is not a quantitative expression!
Problems: polymers with high crystallinity;
polar polymers – hydrogen-bonded solvents or polymers
δ
ρ
= = =
E
V
E F V V
M
coh
Bo
coh
Bo Bo
o
Bamorph,
,,
,
;/;
2

34

The square root of cohesive energy density is called
“solubility parameter”. It is widely used for correlating
polymer solvent interactions. For the solubility of polymer P
in solvent S ( δ
P
- δ
S
)² has to be small!

35
Excluded-Volume-Effect

Dilute gas of random flight chains:
it is physically impossible to occupy the same volume element
in space at the same time
the conformations in which any pair of beads
(segments) overlap were avoided
when a pair of beads come close they exert a repulsion
force F on each other

Dilute solution of random flight chains:

The force that acts between a pair of beads becomes no longer
equal to F.

good solvent: F´ repulsive

The term excluded volume-effect is used to describe any effect
arising from intrachain or interchain segment-segment
interaction.

Excluded-volume of two hard spheres:
(
)
β π π= = =
4
3
2R
4
3
R V
sphere
3
3
8 8

excluded volume

Second virial coefficient A
2
and the
excluded volume:
A
2M
R ~M
2
2
2
3
2
3
2
=
N
hard sphere
L
β
β;:~

A
2
~ M
-1/2

36
Excluded Volume Theory

• volume of segments
• interaction between segments (repulsion forces)

 excluded volume depends on space-filling effects and interaction
forces
 short range, long range interactions

Problems:

Calculation of excluded volume in dependence on molecular
properties;
relation between interaction (A
2
) and excluded volume.

Excluded volume and lattice theory:

number of possibilities, that the molecule mass center is in the volume
V, excluded volume/molecule β, proportionality constant k:
1.molecule: Ω
1
= k V
2. molecule: Ω
2
= k (V - β)
i. molecule: Ω
i
= k [V – (i – 1)β]

 Δµ
A
= - RT V
A
c
B
[1/M
B
+ ((βN
L
)/(2M
B
²))c
B
]

Qualitative Discussion:

excluded
volume

r

hard sphere β < 0; A
2
< 0
β = 0; A
2
= 0
β > 0; A
2
> 0

37
Scaling Law

<h²>
o
= n
s
l
s
²

eq. tell us, how <h²>
o
"scales" with n
s

Global (universal) Properties
properties of polymer chains, which do not depend on
local properties (independent of the monomer structure, nature
of solvent, etc.)
=> very large characteristic lengths
=> small frequencies

It has been found that in the appropriate variables all
macroscopic polymer properties can be plotted on universal
curves (power laws, characteristic exponents).

The Blob-chain, <R²> (<h²>) of a labeled chain

of n/g blobs each of length ξ
(screening length) containing g
segments
 a blob is an effective step
along the contour of the chain
contains g segments

we assume:
- the segments inside the blob
obey the excluded volume
chain statistics, ξ ~ g
3/5
- the n/g blobs obey the random walk statistics such that
<R²> = (n/g) ξ²
ξ is the distance up to which the native self-avoidance due to the
excluded volume interaction is completely correlated and beyond
which it is totally uncorrelated; since g ~ ξ
5/3

 <R²> ~ (n/g) ξ² ~ n ξ
1/3

<R²> ~ n ρ
-1/4
(see ξ = f (c))

38
Scaling Laws for polymer solutions

(good solvent at nonzero concentrations)

We are in search of a dimensionless variable in order to apply
the scaling method.

 fundamental concentration to make the polymer
concentration dimensionless:

We introduce a reduced concentration (ρ/ρ*)
with: ρ = segment concentration (number of
segments/volume); N chains with n
s
segments
ρ* = overlap concentration
~ ( N n
s
)/( N R³
F
) ~ n
s
(1-3ν)
~ n
s
-4/5
; (R
F
~ n
s
3/5
)

 scaling laws:
• concentration dependence of <r²> (Radius of gyration)
ρ = ρ*:

<r²>
1/2
= R
F
solid amorphous polymer, ρ > ρ*:

<r²> ~ n
s
<r²> ~ R
F
² (ρ/ρ*)
x

ρ = ρ*: <r²> ~ R
F
² (ρ/ρ*)
x
~ R
F
²
ρ > ρ*:

<r²> ~ R
F
² (ρ/ρ*)
x
~ n
s

since: ρ*~ n
s
-4/5
; R
F
~

n
s
3/5
;

x = - ¼ *)
<r²>
1/2
~ ρ
-1/4
~ c
B
-1/4
• concentration dependence of the screening length
ρ= ρ*:

ξ
= R
F
ρ > ρ*:
ξ ~
n
s
0
(no molar mass dependence)

ξ
~ R
F
(ρ/ρ*)
y

ρ > ρ*:
ξ
~ R
F
(ρ/ρ*)
y
~ n
s
0

since: ρ*~ n
s
-4/5
; R
F
~

n
s
3/5
y = - 3/4

ξ
~ ρ
-3/4
~ c
B
-3/4

*) <r²> ~ n
s
~ ρ
x
n
s
(6/5+4x/5)
; 6/5 + 4x/5 = 1 ; x = -1/4

39
• osmotic pressure

dilute solution:
π π ρ
R T k T n
B s
= ⇒ =
c
M

ρ > ρ*: π/c
B
show no molar mass dependence
π ρ ρ
ρ
k T n
B s
~
*

z

ρ > ρ*:
π

n
B
s
0
~
㬠 ρ*~ n
s
-4/5

z = 5/4
π ρ~ ~
9
4
n
s
0

experiments, poly-α-methyl-styrene in toluene, different molar mass