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

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 – (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

/ρ

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

/Φ

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

/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

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

(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 – hydrogen-bonded 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

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.

interaction bead-solvent > interaction bead-bead => good solvent

interaction bead-solvent < interaction bead-bead => poor solvent

=> solvent-bead (segment) interactions: F´

good solvent: F´ repulsive

bad solvent: F´ attractive

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

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 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

歔

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

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