22
2.5 Thermodynamics of Polymer Solutions (1)
Notation: A = solvent; B = solute (polymer)
in case of copolymers or multicomponent 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 FloryHuggins 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 (twodimensional: z = 4)
V
A
V
B
= L V
A
=10 V
A
24
FloryHuggins 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
added to a lattice:
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 – (i1)L)/K
( meanfield theory)
L segments of i. macromolecule:
ν
i
= (K – (i1)L z (K – (i1)L)/K {(z1) (K – (i1)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
FloryHuggins 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
/ρ
A
M
O
] 1/P V
A
/V
B
~ 1/P ~1/M
B
Enthalpy change of mixing
quasichemical process: ½ (AA) + ½ (BB) (AB)
(AB): solventpolymer 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
/Φ
A
; R = N
L
k
B
∆H = RT n
A
Φ
B
χ
χ = Huggins interaction parameter
26
Gibbs enthalpy based on FloryHuggins 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 lowmolecular 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
Thetatemperature 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 FloryHuggins:
• 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
Thetatemperature 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
Thetatemperature
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
/ρ
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
+χ
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
pseudoideal 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
(δ
A
 δ
B
)² = n
A
V
A
(δ
A
 δ
B
)²
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 – hydrogenbonded solvents or polymers
additional terms
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
ExcludedVolumeEffect
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.
interaction beadsolvent > interaction beadbead => good solvent
interaction beadsolvent < interaction beadbead => poor solvent
=> solventbead (segment) interactions: F´
good solvent: F´ repulsive
bad solvent: F´ attractive
The term excluded volumeeffect is used to describe any effect
arising from intrachain or interchain segmentsegment
interaction.
Excludedvolume 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 spacefilling 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 Blobchain, <R²> (<h²>) of a labeled chain
the labeled chain is made
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 selfavoidance 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
(13ν)
~ 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
歔
n
B
s
0
~
㬠 ρ*~ n
s
4/5
z = 5/4
π ρ~ ~
9
4
n
s
0
experiments, polyαmethylstyrene in toluene, different molar mass
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