Lecture - Institut Prozess- & Verfahrenstechnik: Thermodynamik … · 2009-04-17 · 1 Polymer Thermodynamics Prof. Dr. rer. nat. habil. S. Enders Faculty III for Process Science

1

Polymer Thermodynamics

Prof. Dr. rer. nat. habil. S. Enders

Faculty III for Process ScienceInstitute of Chemical EngineeringDepartment of Thermodynamics

Lecture

0331 L 337

6. Thermodynamics of Polymer Solutions

2Polymer Thermodynamics

6. Thermodynamics of Polymer Solutions6.1. Introduction

P = constant

( )

F F I I II II F I IIA A A

F I IF I II I I II II A AA A A II I II

A A

x n x n x n n n n

x x nx n n x n x nx x n

= + = +

−+ = + → =

−

0.0 0.2 0.4 0.6 0.8 1.0250

260

270

280

290

300

310

320

tie line

critical point

meta-stable

meta-stable

stable

stableinstable

spinodalebinodale

T / K

XA



phase equilibrium conditions

1 2

1 2

1 2

L LA AL LB BL LP P P

μ μ

μ μ

=

=

= = S2

T,P=constant

S1II

I

free

mix

ing

enth

alpy

composition

instability limit

2

2

( ) 0MIX g xx

⎛ ⎞∂ Δ=⎜ ⎟∂⎝ ⎠

3

3

( ) 0MIX g xx

⎛ ⎞∂ Δ=⎜ ⎟∂⎝ ⎠

critical conditions



0( , , ) ( , ) ln lni i i i iT P x T P RT x RTμ μ γ= + +

( ) ( )1 1 1 2 2 2

0 0

1 1 2 2 1 1

1 1 2

2 2

2

( , ) ln ln ( , ) ln ln

ln ln ln ln ln ln

L L L L L LA A A A A A

L L L L L L L LA A A

L L L L

A A A

A A A A

A A

T P RT x RT T P RT x RT

x x

x x

x x

μ γ μ γ

γ γ γ γ

γ γ

+ + = + +

+ = + →

=

=

gE-model for the determination of the activity coefficients is necessary

( )( ) ( ) 1E

A B A Ag A T x x A T x xRT

= = −2

2

ln ( )

ln ( )B A

A B

A T x

A T x

γ

γ

=

=i.e.



( ) ( ) ( )

( ) ( ) ( )

1 2 21 2 2 12

1 2 21 2 2 12

ln ( ) 2

ln ( ) 2

LL L L LBB B B BL

B

LL L L LAA A A AL

A

x A T x x x xx

x A T x x x xx

⎛ ⎞ ⎡ ⎤= − +⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠⎛ ⎞ ⎡ ⎤= − +⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠

system of equations made from 2 equations → 2 quantities can be calculated

1) T and xAL1→ fixed xA

L2

2) T and xAL2→ fixed xA

L1

3) xAL1 and xA

L2→ fixed T

3 possibilities:



spinodale2

2

( ) 0MIX g xx

⎛ ⎞∂ Δ=⎜ ⎟∂⎝ ⎠

( )

( )2

2

1 1 2 ( ) 0

ln ln ln ln

1 1 2 ( ) 21 1

1 ( )

MIX A A B B A A B B

MIXA A

AA A A A

g RT x x x x x x

g A T x A T xx

Ax x

Tx x

γ γΔ = + + +

⎛ ⎞ ⎛ ⎞∂ Δ= + − − − =⎜ ⎟ ⎜ ⎟∂ −

−⎠ −⎠

+⎝

=⎝

2 2ln ( ) ln ( )B A A BA T x A T xγ γ= =

critical point

( )

3

23 2

( ) 1 1 01

12

CMI

A AA

X g x xx x x

⎛ ⎞∂ Δ= − + = →⎜ ⎟∂ −⎝ ⎠

=

( ) 2 ( ) ( ) spontaneous demixing( ) ( ) stable or metastable

C C

C

A T A T A TA T A T

= > ⇒< ⇒



1. Why are polymers usually do not dissolve in normal solvent ?2. Do we have phase separation (LLE) for polymer solutions ?3. How depends phase separation on the molecular weight?

Problems:

Differences between low-molecular weight mixtures and polymer solution

structure of polymer depends on solvent concentration

the structure of both components do not depend on concentration

both components show a big difference in size→ additional parameter molecular weight distribution

both components have approximately the same size

i.e. PS in cyclohexanei.e. benzene + toluene

polymer solutionlow-molecular weight mixture

6.1. Introduction



Taking into account the differences in size:

division of polymers into segments

standard

ii

LrL

=

Properties1. possibility L= molar volume

disadvantage: volumes depend on temperature2. possibility L = molar mass3. possibility L = degree of polymerization4. possibility L = van der Waals-volume (quantity b in the van der

Waals – equation of state)advantage: available by experiments using X-ray diffraction or by prediction

from molecular data (group-contribution method by Bondi)do not depend on temperature

Lstandard = property of an arbitrarily chosen standard segment (i.e. solvent or monomer unit)

6.1. Introduction



division of polymers into segments

standard

ii

LrL

=segment-related thermodynamic quantities(segment-molare quantities)

polymer solution made from solvent (A) + polymer (B)

AA A

A A B B A A B B

r x zx zr x r x r z r z

= ⇒ =+ +

i.e. cyclohexane + polystyrene MPS=100000g/mol xB=0.00001 Lstandard= molar mass of solvent =84.16g/mol-1

100000 / 84.16 /1188.2 184.16 / 84.16 /PS CH

g mol g molr rg mol g mol

= = = =

1188.2*0.00001 0.011741*(1 0.00001) 1188.2*0.00001PSx = =

− +

Taking into account the differences in size:6.1. Introduction



the structure of polymer depends on concentration

diluted polymer solutionno entanglements of the polymer chain

semi-diluted polymer solution,entanglements of the polymerchain

overlapping concentration c‘‘

6.1. Introduction



vapor pressure of polymer solution

exp. data: B. A. Wolf, et al. University of Mainz, 1995

ideal mixturevery strong

derivation from the behavior of ideal

mixture

?

6.1. Introduction

CH/PS 233

0 0.2 0.4 0.6 0.8 1φPS

4

8

12

16

P [a.u.]

35°C45°C

55°C

65°C



phase diagramcyclohexane + PSat different molecular weightsin [kg/mol] of PS

example: MPS=37000 g/molwPS=0,1 T=280 K

Experimental result: demixing

2

2

( )/ 0mix

PS

meta stableg RT spinodale

xinstable

>⎛ ⎞∂ Δ

=⎜ ⎟∂⎝ ⎠ <

6.1. Introduction



real low-molecular weight mixture

WWAB≠WWAA≠WWBB

derivations from the behavior of an ideal mixture are caused by intermolecular interaction

model of regular mixtureHildebrand (1933)

The real behavior can be explained completelyby enthalpic effects.

0E E Es g h= → =

AB AA BBWW WW WW=

6.1. Introduction



Hildebrand model of regular mixtureThe real behavior can be explained completely by enthalpic effects.

0E E Es g h= → =E

A B ABg v DRT

φ φ=

0

01

i

i ik

i ii

x v

x vφ

=

=

∑0

1 10

k kE

i i i ii i

v v x v x v= =

= → = =∑ ∑

( )2A B

ABDRT

δ δ−=

δi solubility parameter

δCH=16.7 (J/cm3)1/2 δPS=17.4 (J/cm3)1/2

( )2

3 3

16.7 17.4 mol0.000218.314 280 cmCHPS

JmolKD

cm J K−

= =

6.1. Introduction



E

A B ABg v DRT

φ φ= 3

mol0.00021cmCHPSD =

2 2 20 0

2 3

/ 21 011

Mix AB A B

B B B

g RT D v vx x x v

⎛ ⎞∂ Δ= + −⎜ ⎟∂ −⎠

<⎝

3

3

3 3

111 1.0556

37000 350511.0556

CH PS

PSPS

PS

cm gvmol cm

M g cm cmvmol g mol

ρ

ρ

= =

= = =

Table book:

CH CH PS PSv x v x v= +

example: MPS=37000 g/molwPS=0,1 T=280 K

Hildebrand model of regular mixtureThe real behavior can be explained completely by enthalpic effects.

6.1. Introduction



3 3

3

mol0.00021 111 35051cmCHPS CH PS

cm cmD v vmol mol

= = =

2 2 2

2 3

/ 21 1 01

Mix CHPS CH PS

PS PS PS

g RT D v vx x x v

⎛ ⎞∂ Δ= + −⎜ ⎟∂ −⎠

<⎝

CH CH PS PSv x v x v= +

example: MPS=37000 g/mol wPS=0.1 T=280 K

3 2

2

/ 2cm0.00025267 v=119.828m

6o

65l

3.

PS PS PS

PS PS PS PSPS

PS CH PS CH PS CHPS CH

PS CH PS CH PS CH

PMix

PS

S

m w m wn M M Mx m m w m w m w wn n

M M M M M

T

M

gx

x R⎛ ⎞∂ Δ=⎜ ⎟∂⎝ ⎠

= = = =+ + + +

=

conflict to experimental

findings

interaction are not the reasons for the strong real effects

6.1. Introduction



real polymer solution

WWAB=WWAA=WWBB

derivation from ideal behaviorcaused by differences in size

model of ideal-athermic mixture

J.P. Flory (1930)M.L. Huggins

The real behavior can be explained completely by entropic effects.

0E E Eh g Ts= → = −

→ Flory-Huggins mixture

6.2. Flory-Huggins Theory



WWAB=WWAA=WWBB


0E E Eh g Ts= → = −

Polymer chains will be arranged on latticesites of equal sizes (i.e. v0A). The number of occupied lattice site dependson segment number r.

Boltzmann’s definition of entropy

lnBS k= Ω

kB Boltzmann’s constantΩ thermodynamic probability

monomer solution

polymer solution

polymer coil




Statistical Interpretation of Entropy

ε1

ε2

ε3

Number of possible configuration for 3 distinguishable particles in 3 rooms

P=N! = 3*2*1 = 6

Number of possible configurations corresponds to the permutations of the number of particles:

probability:

1 2 3

!! ! ! !i

NWN N N N

=…

W is called „thermodynamic probability“ and it is the number of possible configuration for a given macroscopic state.




Example: distribution of 4 particles with identical energy in 4 volumes.

WC= 4!/(2!1!1!0!)= 4!/2! =12

WA= 4!/(4!0!0!0!)= 4!/4! =1

C

D

A

B WB= 4!/(3!0!1!0!)= 4!/3! =4

WD= 4!/(1!1!1!1!)= 4!/1! =24

The macroscopic state D can be realized by the most configurations.

The macroscopic state D has the highest thermodynamic probability.




Model of ideal-athermic mixture

0E E Eh g Ts= → = − lnBS k= Ω

Ω represents the number of distinguishable possibilities of the system to fill the lattice places with particles.

size of the lattice place: v0A

Every lattice place is occupied either with polymer segmentor with a solvent molecule.The polymer segments of a polymer chain are connected bychemical bond and hence they can not move freely.

NA-solvent molecules + NB- polymer chains (every chain consist of r-segments)

number of total lattice places: N0=NA+r*NBevery lattice place is surrounded by q next neighbors (coordination number)

cubic lattice q=6





0E E Eh g Ts= → = − lnBS k= Ω

( ) 21 0 1 rN q qν −= −the first polymer chain has ν1 possibility to arrange

0 ( 1)fN N i r= − −number of empty lattice places Nf, if the lattice is filled with i-1 polymer chains

the probability to find an empty place is equal to Nf/N0

number of possibilities to arrangefor the chain i: ( )

12

0

1r

rfi f

NN q q

Nν

−−⎛ ⎞

= −⎜ ⎟⎝ ⎠

1

!2

Bi N

ii

rBN

υ=

=

⎡ ⎤⎢ ⎥⎣ ⎦Ω =∏

(1 ) ln(1 ) lnBMix B B B

xs R x x xr

⎡ ⎤Δ = − − − +⎢ ⎥

⎢ ⎥⎣ ⎦




(1 ) ln(1 ) lnBMIX B B B

xg RT x x xr

⎡ ⎤Δ = − − +⎢ ⎥

⎢ ⎥⎣ ⎦

(1 ) ln(1 ) lnBMix B B B

xs R x x xr

⎡ ⎤Δ = − − − +⎢ ⎥

⎢ ⎥⎣ ⎦

( ) ( )( )

( )2

2

/ 1 1ln 1 1 ln

1/ 1 1 10 011 1

MIXB B

B

B BMIX

B

B B B BB

g RT x xr rx

rx xg RT xrx rx x rxx

⎛ ⎞∂Δ⎜ ⎟ = − − − + +⎜ ⎟∂⎝ ⎠

⎛ ⎞ + −∂ Δ⎜ ⎟ = + = → = → = −⎜ ⎟ −− −∂⎝ ⎠

r ³1 negativesegment mole fractionis nonsense

ideal-athermicmixture can not demixed

model of ideal-athermic mixture6.2. Flory-Huggins Theory



model of real polymer solutions

(1 ) ln(1 ) ln RBMIX B B B

xg RT x x x gr

⎡ ⎤Δ = − − + +⎢ ⎥

⎢ ⎥⎣ ⎦

Rg description of the derivation of a real polymer solution from anideal-athermic mixture

[ ]12AB A B

B

A B

qk T

ε ε ε ε

εχ

Δ = − +

Δ=

interaction energy between lattice places

χ Flory-Huggin‘s interaction parameter

bad solventχ>0

Ideal - athermic mixtureχ=0

good solventχ<0

0 (1 )R RB Bg h N q x x ε= = − Δ




model of real polymer solutions

(1 ) ln(1 ) ln RBMIX B B B

xg RT x x x gr

⎡ ⎤Δ = − − + +⎢ ⎥

⎢ ⎥⎣ ⎦

0 0(1 ) (1 ) (1 )

B

B

R R BB B B B B B

q k Tk T q

k Tg h N q x x N q x x RT x xq

ε χχ ε

χε χ

Δ= Δ =

= = − Δ = − = −

χ Flory-Huggins interaction parameter

(1 ) ln(1 ) ln (1 )MIX BB B B B B

g xx x x x xRT r

χΔ= − − + + −

Flory-Huggins-equation




(1 ) ln(1 ) ln (1 )MIX BB B B B B

g xx x x x xRT r

χΔ= − − + + −

( ) ( ) ( )

( ) ( )

2

2

3

2 23

/ 1 1ln 1 1 ln 1 2

/ 1 1 2 01

/ 1 1 01

MIXB B B

B

MIX

B BB

MIX

B B B

g RT x x xr rx

g RT

x rxx

g RT

x x r x

χ

χ

⎛ ⎞∂Δ⎜ ⎟ = − − − + + + −⎜ ⎟∂⎝ ⎠⎛ ⎞∂ Δ⎜ ⎟ = + − =⎜ ⎟ −∂⎝ ⎠⎛ ⎞∂ Δ⎜ ⎟ = − =⎜ ⎟∂ −⎝ ⎠

spinodal condition

critical condition




(1 ) ln(1 ) ln (1 )MIX BB B B B B

g xx x x x xRT r

χΔ= − − + + −

( ) ( ) ( ) ( ) ( )

( )( ) ( ) ( ) ( )

( ) ( )( )

3 2 2 2

2 23

2 2

1,2 2

1,2

/ 1 1 0 1 1 21

1 20 1 2 1 01 1

2 4 1 1 1 11 12 1 1 1 1 14 1

0 1 11

11B

MIXB B B B

B B B

BB B

B

c

B

B

g RT r x x x xx x r x

xx r x xr r

rx rr r r r rr

r rxr r

x

⎛ ⎞∂ Δ⎜ ⎟ = − = → = − = − +⎜ ⎟∂ −⎝ ⎠

→ = − + − → = − +− −

±= ± − = ±

− −= =

−

− − =− − − − −−

< <−

→




0 1000 2000 3000 4000

0,00

0,05

0,10

0,15

0,20

0,25

Flory-Huggins theory

criti

cal c

once

ntra

tion

segment number r

11B c

rxr−

=−




(1 ) ln(1 ) ln (1 )MIX BB B B B B

g xx x x x xRT r

χΔ= − − + + −

( )

11 12lim 01 1 2B Bc cr

r rx für r xr r→∞

−= →∞⇒ = = =

−

2

2

/ 1 1 12 0 21 1

0 1/ 2

MIX

B B

C

BB

B

g RT für rx rx xx

x

χ χ

χ

⎛ ⎞∂ Δ⎜ ⎟ = + − = →∞⇒ =⎜ ⎟ − −∂⎝ ⎠

→ ⇒ =0.5

instablespinodalstable

χ>=<for polymers with r →∞




1.3725rubber + acetone

0.4425rubber + benzene

0.2820rubber + CCl4

0.1526PVC + THF

0.4425polystyrene + toluene

0.525polyisobutylene + benzene

0.2725cellulose nitrate + acetonecT [°C]system

Flory-Huggins parameter for polymer + solvent - systems




B. A. Wolf, University of Mainz, 1995

χ can also depends on polymer concentrationand molecular weight

conflict to FH-theory


χ

0.0

0.2

0.4

0.6

0.8

1

φPVME

0.0 0.2 0.4 0.6 0.8 1

CH/PVME

MW [kg/mol]

81 51

28

T = 65°C



B. A. Wolf, University of Mainz, 1995


χ

0.4

0.5

0.6

0.7

0.8

0.9

0.2 0.4 0.6 0.8

TL / PDMS 170

T=35°C

T=45°C

T=55°C

wPDMS



formal division of χ in a enthalpic- and a entropic part

( )

22

2

H S

S

H

h Ts h sRT RT R

s hh RT

R T RT T

RTh T TRT RT T

χ χ χ χ

χ χχ

χ

χ χ χ

χ χχ

χχχ

−= + = = −

∂ ∂⎛ ⎞ ⎛ ⎞= − = − → = −⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠∂⎛ ⎞− ⎜ ⎟ ∂∂ ⎛ ⎞⎝ ⎠= = = − ⎜ ⎟∂⎝ ⎠

temperature dependence of χ available

special case: 0H Sχ χ χ− = ⇒ = pseudo-ideal state = Theta-state

1H S

h s h hs s

RT R R T Tχ χ χ χ

χ χχ χ χ⎛ ⎞

= + = − = − ⇒ =⎜ ⎟⎝ ⎠

Theta-temperature




0.2

0.03

0.03

0.03

0.04

-0.02

-0.08

cH

0.310.51m-xylole

0.450.48Acetone

0.420.45toluene

0.420.45THF

0.390.43dioxane

0.450.43benzene

0.440.36chloroform

cScsolvent

Flory-Huggins parameter for PMMA + solvent systems

entropic effectsare more importantthan enthalpic effects




B. A. Wolf, Universität Mainz, 1995


c

fPDMS

0.2 0.4 0.6 0.8-3

-2

-1

0

1

2

3

4

cS

cH

TL/PDMS 170T = 35°C


6. Thermodynamics of Polymer Solutions6.2. Flory-Huggins Theory

Prediction of χ

Cohesive Energy eCoh: internal energy of the material if all of its intermolecularforces are eliminated

solubility parameter δ1/ 2

3/ 2cohe Jv mol

δ δ⎡ ⎤

= ⎢ ⎥⎣ ⎦

solubility of polymer P in solvent S ( )2P Sδ δ− has to be small,

as small as possible

( ) ( )22

(1 ) (1 ) BR

BB B B

AAB

h x x xvRT

xRT vRT

δ δχχ

δ δ−=

⎡ ⎤−= − ≈ − →⎢ ⎥

⎢ ⎥⎣ ⎦

2

(298 )CohFe

v K= F molar attraction constant

v molar volume



Prediction of χ example: solubility of polystyrene at 25°CFirst step: Calculation of atomic indices δ and valence atomic δV

Second step: Calculation of connectivity indicesV V V

ij i j ij i jβ δ δ β δ δ= =

0 0 1 1

0 0 1 1

1 1 1 1

5.3973 4.6712 3.9663 3.0159

V V

V Vvertices vertices edges edges

V V

χ χ χ χδ βδ β

χ χ χ χ

≡ ≡ ≡ ≡

= = = =

∑ ∑ ∑ ∑



Prediction of χ example: solubility of polystyrene at 25°CFourth step: Calculation of molar volume

1.42 0.15Wg

Tv vT

⎛ ⎞⎛ ⎞= +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

( )

( )

30 1

3 3

3.861803* 13.748435*

3.861803*5.3973 13.748435*3.0159 62.31

VW

cmvmol

cm cmmol mol

χ χ= +

= + =

Tg=373.15K

3 329862.31 1.42 0.15 95.95373

cm K cmvmol K mol

⎛ ⎞⎛ ⎞= + =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠



Prediction of χ example: solubility of polystyrene at 25°CThird step: Calculation of molar attraction constant F

( )( )( )

( )( )( )

0 0 1 1 3

2

3

2

3

2

97.95 2

134.61

97.95 5.3973 2 4.6712 3.9663 3.0159

134.61 0 0 0

1754.2

V V

Si Br Cyc

Jcmwhere FmolN N N

JcmFmol

JcmFmol

χ χ χ χ⎡ ⎤− + + +⎢ ⎥=⎢ ⎥+ − −⎣ ⎦

⎡ ⎤− + + += ⎢ ⎥

+ − −⎢ ⎥⎣ ⎦

=



Prediction of χ example: solubility of polystyreneFifth step: Calculation cohesive energy eCoh

( )

2

3 3

2

2 3

2 3

(298 )

1754.2 95.95

1754.232.07

95.95

Coh

Coh

Fev K

Jcm cmF vmol mol

Jcm mol kJemol cm mol

=

= =

= =

3 3

32.07 18.2895.95

cohe kJmol Jv mol cm cm

δ = = =

Sixth step: Calculation of solubility parameter δ



Prediction of χ example: solubility of polystyrene

Seventh step: Calculation of interaction parameter χ( )2

B A

vRTδ δ

χ−

=

4,96690,17690,00300,00050,4520

χ solvent qualityδA [J/cm3]solvent

poor-LLE29,60methanolgood20,42dioxane

excellent18,56benzeneexcellent18,17toluenemiddle14,86n-hexane



Application of Flory-Huggins Theory to LLEphase equilibrium conditions

low molecular weight mixtures polymer solutions1 2

1 2

1 2

L LA AL LB BL LP P P

μ μ

μ μ

=

=

= =

1 2

1 2

1 2

L LA A

L LB BL LP P P

μ μ

μ μ

=

=

= =

(1 ) ln(1 ) ln (1 )MIX BB B B B B

g xx x x x xRT r

χΔ= − − + + −

0 0 0 0

, , A

A BB MIX A B MIXA B

B T P n

G G G G G n g n g n gn

μ⎛ ⎞∂⎜ ⎟= → = + + Δ = + + Δ⎜ ⎟∂⎝ ⎠

43Polymer Thermodynamics6. Thermodynamics of Polymer Solutions


( ) ( )

( ) ( )

( ) ( )

0 0

2

0

, ,

0

2

,

0

,

(1 ) ln(1 ) ln (1 )

1ln 1

1 1l

1/ / ln 1

n 1

B

A

BA B B B B B BA B

A A B BA

A T P n

B B A AB

B T P n

AA A B B

xG n g n g RT n x x x x xr

G g RT x x xrn

G g RT x x xr

RT RT x x x

n

r

r

μ μ

χ

μ χ

μ

χ

⎡ ⎤= + + − − + + −⎢ ⎥

⎢ ⎥⎣ ⎦⎛ ⎞∂ ⎛ ⎞⎛ ⎞⎜ ⎟= = + + − +⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠∂⎝ ⎠

⎛ ⎞∂ ⎛ ⎞⎜ ⎟= = + − − +⎜ ⎟⎜ ⎟ ⎝ ⎠∂

⎛ ⎞= + + +⎜⎝ ⎠

⎝ ⎠

− ⎟

( ) ( )2

0

2

1 1/ / ln 1BB B A ART RT x x xr r

μ μ χ

χ

⎛ ⎞

⎡ ⎤⎢ ⎥⎣ ⎦

= + − − +⎜ ⎟⎝ ⎠

Application of Flory-Huggins Theory to LLE



10LA

RTμ ( ) ( )2 2

1 1 1 01ln 1L

L L L AA B Bx x x

r RTμχ⎛ ⎞+ + − + =⎜ ⎟

⎝ ⎠ ( ) ( )( ) ( ) ( )

22 2 2

1

2 212 2 1

2

0

1

1ln 1

1ln 1L

L L L LAB B B B

L

L L LB

A

A B

LB

x x x x xrx

x x xr

RT

χ

χ

μ

⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ = − − + −⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠

⎛ ⎞+ + − +⎜ ⎟⎝ ⎠

⎝→

⎠

( ) ( )2 21 1 1 01 1ln 1

LL L L BB A Ax x x

r r RTμχ⎛ ⎞+ − − + =⎜ ⎟

⎝ ⎠ ( ) ( )( ) ( ) ( )

22 2

2 211 2

2

2 1

2

1 1ln 1

1 11

L L LB A A

LL L L LBA A A A

LB

x x x x xr rx

x x xr r

χ

χ⎛ ⎞ ⎛ ⎞⎛ ⎞⎜

⎛ ⎞+ − − +⎜ ⎟⎝ ⎠

→ ⎟ = − − + −⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠


2 equations and 3 unknowns → select one concentration in one phase→ calculation of the concentration in the other phase and the parameter χ using

both equations, where a numerical procedure must be appliedor reduce the system of equations to one equation

equation 1

equation 2



( )( ) ( )

( )( ) ( )

1 12 1 1 2

2 2

2 2 2 22 1 2 1

1 1 1ln 1 1L L

L L L LA BB B A A

L LA B

L L L LB B A A

x xx x x xr r rx x

x x x x

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟− − − − − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠=⎛ ⎞ ⎛ ⎞

− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


reduce the system of equations to one equation → rearrangement of both equationsaccording χ → to equate both results

from equation 1 from equation 2

equation 3

Now: solve equation 3 in order to calculate the unknown concentration usingroot-searching method, like Newton-Procedurecalculate χ using equation 1 or equation 2


6.2. Flory-Huggins TheoryApplication of Flory-Huggins Theory to LLE

321 2( )T

T Tββχ β= + +

special cases:β3=0 only one critical point can be calculated

β2>0 UCSTβ2<0 LCST

β3≠0 two critical points can be calculatedclosed miscibility gap or UCST+LCST

T / K

XA

UCST

XA

T/K

LCST

T / K

XA

XA

T [K

]



0 5 10 15 20 25

30

40

50

60

70

80

χ(T)=1.186-207.87/T

MW=500 kg/mol

PAA + Dioxan

T / °

C

wPAA [%]




vapor pressure of polymer solutions

liquid mixture (A+B) « pure vapor phase (A)

solvent A + solute B PBVL ≈ 0

equilibrium condition: ( , , ) ( , )L VA A Ad T P x d T Pμ μ= T = constant

0 0

0

00

1V L L VVL

V VA A A A AA AVL V

A A A

VLL L LA A AVL

A

ideal gas

a xP with

P a

P

xP

ϕ γ ϕ ϕ ϕϕ ϕ

γ

= = =

=

=

→ =




0

VLLAVL

A

P aP

=application to polymer solution

( ) ( )2

01ln 1AA A B BRT x x xr

μ μ χ⎡ ⎤⎛ ⎞= + + − +⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

low-molecular weight mixtures0( , , ) ( , ) lni i i iT P x T P RT aμ μ= +

solvent in polymer solution

( ) ( )( ) ( )

2

2

0

1ln ln 1

1ln ln 1

LA A B B

VL

A B BVLA

a x x xr

P x x xP r

χ

χ

⎛ ⎞⎛ ⎞ = + − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞→ = + − +⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠possibility to determine χ from experimental data, like vapor pressure of polymer solutions




liquid mixture (A+B) « pure vapor phase (A)

solvent A + solute B PBVL ≈ 0

equilibria condition: ( , , ) ( , )L VA A Ad T P x d T Pμ μ= T = constant

0 0

0

00

1V L L VVL

V VA A A A AA AVL V

A A A

VLL L LA A AVL

A

ideal gas

a xP with

P a

P

xP

ϕ γ ϕ ϕ ϕϕ ϕ

γ

= = =

=

=

→ =

( ) ( )2

0

1ln ln 1VL

A B BVLA

P x x xP r

χ⎛ ⎞ ⎛ ⎞= + − +⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

Flory-Huggins theory





0,0 0,2 0,4 0,6 0,8 1,00

5

10

15

20

T = 298K

r=1 χ=0 ideal mixture r=100 χ=0 ideal-athermic mixture r=100 χ=0.452 real mixture

PS + n-hexanePVL

[kPa

]

segment mole fraction of PS




measurement of χ: inverse gas chromatography (IGC)IGC refers to the characterization of the chromatographic stationary phase (polymer)using a known amount of mobile phase (solvent).The stationary phase is prepared by coating an inert support with polymer and packingthe coated particles into a conventional gas chromatography column.The activity of the given solvent can be related to its retention time on the column.

→ infinity dilution regarding to the solvent

experimental information: N R GV V V= −VR retention volume [m3]VG gas holdup [m3]VN net retention volume [m3]

0NV net retention volume extrapolated to zero column pressure

00

NL GAL G L

A

Vn VkV n V

= = at infinity dilution: A AA A

A A

Pax xγ ϕ

∞ ∞

∞ ⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

fugacity coefficient =1

0

LA A A

A NA A

P x P RTnax x V

∞ ∞

∞ ⎛ ⎞ ⎛ ⎞= = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠




experimental estimation of χ shows −χ depends on temperature−χ depends on polymer concentration−χ depends on molecular weight

more complex models are needed

321 2( )T

T Tββχ β= + +




Flory-Krigbaum Model (1950): a diluted polymer solution is considered as adispersion of clouds consisting of polymer segments surrounded byregions of pure solvent

( ) ( ) ( )( )2

1 11ln ln 1A A B Ba x x xr

κ ψ⎛ ⎞= + − + −⎜ ⎟⎝ ⎠

enthalpic parameter entropic parameter

1

1

T Theta Temperatureκθ θψ

= = −

Difficulty: the interaction parameter depends not on concentration

6.3. Extensions of Flory-Huggins Theory



( )(1 ) ,B BR

Bg RT x x Txχ= −

1. Possibility: ( ) ( )2

0 1 2, ( ) ( ) ( ) ...B B Bx T T T x T xχ χ χ χ= + + + +

advantage: very flexibledisadvantage: empirical, a lot of parameters

2. Possibility: ( ) ( ),1

B

B

Tx Tx

βχγ

=−

advantage: very flexible, semi-empiricaleasy parameter fitting procedure using the critical point

Koningsveld-Kleintjens-Eq.




0,00 0,02 0,04 0,06 0,08 0,10287,5

288,0

288,5

289,0

289,5

290,0

290,5

TD/PMMA 1770

γ = 0.32281 β1 = 0.24844 β2 = 143.307 / K

T/K

wPMMA

( ) ( ),1

B

B

Tx Tx

βχγ

=−




M. Schnell, S. Stryuk, B.A. Wolf, Ind. Eng. Chem. Res. 43 (2004) 2852.

high-pressure production of PE

equation of state is necessary




Flory-Equation of State

Starting point: partition function

( ) ( )31/3* 1 exprncrnc

combrncZ Z gv vvT

⎛ ⎞= − ⎜ ⎟⎝ ⎠

combinatory factor Zcomb geometrical factor g characteristic volume v*

3rnc = total number of degrees of freedom ( r = segment-number, n = number of molecules, c = mean number of external degrees per segment)

* * *v Tv T P

PP

v T= = =

v*, T* and P* are characteristic quantities = parameters of the polymer




Flory-Equation of StateStarting point: partition function

( ) ( )31/3* 1 exprncrnc

combrncZ Z gv vvT

⎛ ⎞= − ⎜ ⎟⎝ ⎠


ln

T

ZP kTV

∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠

from statistical thermodynamics:

1/3

1/3

11

Pv vT v Tv

= −−





chemical potential for solvent:

1/3* ** *1 1 1

1 1 2 1 1 1* 1/31

22

2

12 1 1 1ln 1 3 ln1

Xv v vRT v P Tv v v v v

μ φ θφ⎡ ⎤⎡ ⎤ ⎛ ⎞⎛ ⎞ −

Δ = + − + + + −⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟ −⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

Flory-Huggins term

binary parameters

free volume terminteraction term

1/3* ** *2 2 2

2 2 1 2 2 2* 1/32

21

1

12 1 1 1ln 1 3 ln1

Xv v vRT v P Tv v v v v

μ φ θφ⎡ ⎤⎡ ⎤ ⎛ ⎞⎛ ⎞ −

Δ = + − + + + −⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟ −⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦

chemical potential for polymer:





calculation of the mixing volume1 1 2 21E v vv v

vφ φ+⎡ ⎤= −⎢ ⎥⎣ ⎦

other equations of state useful in polymer thermodynamics

Sanchez-Lacombe Equation of state (SL-EOS)lattice theory, which allows additionally empty sites

Statistical associated fluid theory (SAFT-EOS)takes into account additionally specific interactions like hydrogen bonding

and associationPerturbed-Chain Statistical associated fluid theory (PC-SAFT-EOS)

further development of SAFT-EOS, takes the chain connectivity into account

( )2

1 1/ 1ln1

rP VT V V V T

−= − −

−




Polydispersity effects

three liquid phases for the system polyethylene + diphenyletherat constant temperature and pressure ???

R. Koningsveld, W.H. Stockmayer, E. Nies,Polymer Phase Diagrams, Oxford University Press, 2001

Gibbs ‘s phase rule: F = K – Ph + 2binary system: K=2 F=4-Phfixed P,T: F=2 ® PhMax=2

polymer solutions arenot binary systems



0,00 0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20295

300

305

310

315

320

T / K

X Ensemble

critical point

spinodal

cloud-point curve

shadow-curve

coexisting curve

6.3. Extensions of Flory-Huggins TheoryPolydispersity effects



0 200 400 600 800 1000 1200 1400 1600 1800 20000

5

10

15

20

25

30

Feed Phase I Phase II

103 W

(r)

r

fractionation effectPolydispersity effects


phase Ipolymers withhigher chain lengths

phase IIpolymers withlower chain lengths

In both phasesare different polymers.


6.4. Swelling Equilibria

solvent

polymer network

V/V00.2 0.5 1 2

T [°C]

20

30

40

50




Application to drug-delivery systems



T = constant

polymer gelP1

P2liquid phase

+

+

+

+

--

- -+

+-

-

''' ''ln ( , ', ') ln ( , '', '') i eli i

T

v Fa T P n a T P nRT V

∂Δ⎛ ⎞= + ⎜ ⎟∂⎝ ⎠

MIX

MIX

el

deformed relaxed

FF F

F FF

Δ = Δ += Δ +

Δ−

ΔelF elastic free energy (deformation of rubber, rubber elasticity)

i.e. Flory-Huggins-theory

2/3

10

1el F VCRT V

⎡ ⎤⎛ ⎞Δ ⎢ ⎥= −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦


first approximation

more information available: G. Maurer, J.M. Prausnitz, Fluid Phase Equilibria 115 (1996) 13-133.

C1 – fit parameterV volume of swollen networkV0 volume of non-swollen network



starting point:Li

Si ff = phase equilibrium condition

iso-fugacity criteria

Li0

Li

Li

Li fxf γ=liquid phase:

Si0

Si

Si

Si fxf γ=solid phase:

00 0

0

L S SL L L S S S i i ii i i i i i S L L

i i i

f xx f x ff x

γγ γγ

= → =

Calculation of pure-component properties

0 00 0 0 0

00 0 0

0

( , ) ln ( , ) ln

ln

S LS id L idi ii i i i

LL S i

LS i i i Si

f fg g T P RT g g T P RTP P

fg g g RTf

= + = +

→ Δ = − =

6.5. Solid-Liquid Equilibria



0

0

L S Si i iS L Li i i

f xf x

γγ

=0

0 0 0 00

00 0 0

00 0

0

ln

1

1ln

Li

LS i LS i LS i LS iSi

LS iLS i LS i LS iLS LS

LS iL LSi LS iSi

fg RT g h T sf

h Tg h T hT T

Thf g Tf RT RT

Δ = Δ = Δ − Δ

Δ ⎛ ⎞Δ = Δ − = Δ −⎜ ⎟⎝ ⎠

⎛ ⎞Δ −⎜ ⎟Δ ⎝ ⎠= =

0 1ln

LS i S SLSi iL Li i

ThxT

RT xγγ

⎛ ⎞Δ −⎜ ⎟ ⎛ ⎞⎝ ⎠ = ⎜ ⎟⎝ ⎠

SLE for binary systems




solids are pure-components 1S Si ix γ =

0

0

ln 1L L LS ii i LS

i

h TxRT T

γ⎛ ⎞Δ

= − −⎜ ⎟⎝ ⎠

SLE

Application to polymer solution using Flory-Huggins theory

( )

0

0

20

0

ln 1 ln

ln 1

L LLS iB BLS

i

L LLS iB ALS

i

h TxRT T

h Tx xRT T

γ

χ

⎛ ⎞Δ= − − −⎜ ⎟

⎝ ⎠

⎛ ⎞Δ= − − −⎜ ⎟

⎝ ⎠




solvent1) xylene only SLE2) amyl acetate or nitrobenzene

A-B SLEC-B LLEor eutectic mixture

6.5. Solid-Liquid EquilibriaT[°C]

80

100

120

140

160

180

200

0.2 0.4 0.6 0.8 wPE

xylene

amylacetate

nitrobenzene

AB

B

C

C

Documents

Lecture - Institut Prozess- & Verfahrenstechnik: Thermodynamik … · 2009-04-17 · 1 Polymer Thermodynamics Prof. Dr. rer. nat. habil. S. Enders Faculty III for Process Science