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1 Polymer Thermodynamics Prof. Dr. rer. nat. habil. S. Enders Faculty III for Process Science Institute of Chemical Engineering Department of Thermodynamics Lecture 0331 L 337 5. Thermodynamics of Pure Polymers
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• 1

Polymer Thermodynamics

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

Faculty III for Process ScienceInstitute of Chemical EngineeringDepartment of Thermodynamics

Lecture

0331 L 337

5. Thermodynamics of Pure Polymers

• 2Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.1. Thermodynamics of Polymerization

Stoichiometric coefficients νi

1 H2S + 2 NaOH 1 Na2S + 2 H2O

Example.: ν = -1 ν = -2 ν = +1 ν = +2negative = educts positive = products

reaction variable λ

Goal: Reduction of ni variables to only one variable λ

() (0)i int nt= =+

i

i

dn dλν

= i idn dν λ=or

( ) ( 0)i i in t n t ν λ= = +The reaction variable gives information about the progress of the chemical reaction.l [mol]

• 3Polymer Thermodynamics5. Thermodynamics of Pure Polymers5.1. Thermodynamics of Polymerization

Example:

Calculation of ni and xi as function of reaction variable λ:

2

2

21 12 2

1 0

1 1

H O

H

O

n

n mol

n mol

λ

λ

λ

= +

= − +

= − +

( )

( )( )

( )

2

2

2

12

12

1 12 2

12

1 23 3

2 113 3

13 3

H O

H

O

xmol mol

molmolxmol mol

mol molxmol mol

λ λλ λ

λλλ λλ λλ λ

= =− −

−−= =

− −

− −= =

− −

3 mole fraction can be reduced to one λ( )

312 2

12 3

jn mol

mol

λ

λ

= − +

= −∑

2 2 212

H O H O+

• 4Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.1. Thermodynamics of Polymerization

NaOH + H2O AlCl3 + H2O

heating cooling

exothermic reaction endothermic reaction

0RHΔ < 0RHΔ >

• 5Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

equilibrium

dG=0

free

enth

alpy

G

state variable

λpure educts pure products

equilibrium

spontaneous

spontaneous

free

enth

alpy

5.1. Thermodynamics of Polymerization

i

i

dn dλν

=

reaction variable λ

• 6Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.1. Thermodynamics of Polymerization

Spontaneous run of chemical reactions R R Rg h T sΔ = Δ − Δ

at higher temperature preferred→entropy drivenT>ΔRh/ΔRs spontaneousT=ΔRh/ΔRs equilibriumT0>0d

at low temperature preferred→ enthalpy-drivenTΔRh/ΔRs not spontaneous

• 7Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.1. Thermodynamics of Polymerization

Calculation of enthalpy of reaction

Kirchhoff Law

with

( ) ( ) 00 0 ( )iT

B i B i pT

h T h T c T dT+

+ + +Δ = Δ + ∫

0( ) ( )R i B ii

h T h Tν+ +Δ = Δ∑

( )0B ih T+Δ( )0B ih T+ +Δ

0( ) ( )R i B ii

h T h Tν+ + + +Δ = Δ∑

( )Rh T+ +Δ( ) ( ) ( )

T

R R R pT

h T h T c T dT+

+ + +Δ = Δ + Δ∫( )Rh T+Δ

0 iR p i pi

c cνΔ = ∑

• 8Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.1. Thermodynamics of Polymerization

Calculation of entropy of reaction

( )Rs T+ΔCalculation of free enthalpy of reaction

0iR R R ii

g h T s gν+ + + +Δ = Δ − Δ = ∑

( )0is T+ + ( ) ( )0

0 0

( )i

Tp

i iT

c Ts T s T dT

T++ + += + ∫ ( )0is T+

0( ) ( )R i ii

s T s Tν+ +Δ = ∑0( ) ( )R i i

i

s T s Tν+ + + +Δ = ∑

( )Rs T+ +Δ( ) ( ) ( )

TR p

R RT

c Ts T s T dT

T++ + + ΔΔ = Δ + ∫

( )Rs T+Δ

• 9Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.1. Thermodynamics of Polymerization

00

iR R R ii

g h T s gν+ + + +Δ = Δ − Δ =

• 10Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.1. Thermodynamics of Polymerization

00

iR R R ii

g h T s gν+ + + +Δ = Δ − Δ = Δ >

lower limit in temperature = Floor – temperature TF

entropy-driven reaction

reason: during polymerization the number of degree of freedom regarding rotationincrease

example: ring-opening polymerization of oxepines

• 11Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.2. Physical Properties of Pure Polymersfundamental characteristics of polymer

chemical structurenature of repeating unitsnature of end groupscomposition of possible branches and cross-linksnature of defects in the structure sequence

molecular mass distributionaverage molecular sizepolydispersity

both controlcohesive forcespacking densitymolecular mobilitymorphologyrelaxation phenomena

} properties of polymers

• 12Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Application of topological formalism in developing of structure-property correlationPrediction of thermodynamic properties using only chemical composition

Introduction of connectivity indices via graph theoretical concepts

Starting Point: construction of hydrogen-suppressed graph of the moleculeexample: vinyl fluoride

vertex

C CH

H H

F edgeδ number of non-hydrogen atoms

to which a given non-hydrogen atom is bonded

δV valence connectivity indexelectronic configuration of eachnon-hydrogen atom= lowest oxidation state of the

elements

5.2. Physical Properties of Pure Polymers

• 13Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Application of topological formalism in developing of structure-property correlation

1

VV H

V

Z NZ Z

δ −=− −

ZV number of valence electrons of an atomNH number of hydrogen atoms bonded to itZ atomic numberZ=ZV + number of inner shell electrons

Examples-CH3

4 3VVZ

δ −=2 VZ+ −

1 11

Vδ= =−

-CH2-4 2V

VZδ −=

2 VZ+ −2 2

1Vδ= =

=O6 0V

VZδ −=

2 VZ+ −6 1

1Vδ= =

5.2. Physical Properties of Pure Polymers

• 14Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Application of topological formalism in developing of structure-property correlation

Bond indices βij and βijV for each bond (edge) do not involved a hydrogen atom

V V Vij i j ij i jβ δ δ β δ δ= =

vertex

C CH

H H

F edge

0 0 1 11 1 1 1V VV V

vertices vertices edges edgesij ij

χ χ χ χδ βδ β

≡ ≡ ≡ ≡∑ ∑ ∑ ∑

5.2. Physical Properties of Pure Polymers

• 15Polymer Thermodynamics5. Thermodynamics of Pure Polymers

Application of topological formalism in developing of structure-property correlation

V V Vij i j ij i jβ δ δ β δ δ= =

vertex

C C

H

H H

Fedge

0

0

1

1

1 1 1 1 2.70711 2 1

1 1 1 1 1.66252 3 7

1 1 1 1.41411*2 1*2

1 1 1 0.62642*3 3*7

vertices

V

Vvertices

edges ij

V

Vedges ij

χδ

χδ

χβ

χβ

≡ = + + =

≡ = + + =

≡ = + =

≡ = + =

Example

5.2. Physical Properties of Pure Polymers

δ number of non-hydrogen atomsto which a given non-hydrogen atom is bonded

δV valence connectivity indexelectronic configuration of eachnon-hydrogen atom= lowest oxidation state of the

elements

• 16Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Application of topological formalism in developing of structure-property correlationApplication to polymers:

literature: J. Bicerano, Prediction of Polymer Properties, Marcel Dekker, 1993

5.2. Physical Properties of Pure Polymers

0

1

1 1 1 1.66252 3 71 1 1 1.03472*3 2*3 3*7

V

V

χ

χ

= + + =

= + + =

• 17Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

phase transition

critical point

triple point

solid liquid

vapor

P

T

low-molecular weight componentpolymers

polymers cannot be evaporated sincethey decompose before boiling

no vapor pressureliquid state: very high viscosity,

viscoelasticity

solid state: very complexpartially or totally amorphous

The typical state of polymers are rubbery, glassy or semicrystalline, which are thermodynamically metastable.

5.2. Physical Properties of Pure Polymers

• 18Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

phase transition

SRO – short range orderLRO – long range orderSTS – short time stiffnessLTS – long time stiffness

low-molecular weight molecules

polymers

5.2. Physical Properties of Pure Polymers

• 19Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

phase transitions

SRO LRO

STS LTS

SRO LRO

STS LTS

SRO LRO

STS LTS

SRO LRO

STS LTS

SRO LRO

STS LTS

SRO LRO

STS LTS

SRO – short range order LRO – long range order STS – short time stiffness LTS – long time stiffness

blue – not presentred – presentmagenta - partly

gas liquid(polymer melt)

rubbery state

glass

semicrystalline crystalline

5.2. Physical Properties of Pure Polymers

• 20Polymer Thermodynamics5. Thermodynamics of Pure Polymers

amorphous polymer

semicrystalline polymer

5.2. Physical Properties of Pure Polymers

T

T

M

M

viscousliquid

viscousliquid

thermal decomposition

thermal decomposition

rigid (semi) crystalline

glassy

glass transition

glass transitionmelting point

rubbery

rubbery

leathery

diffuse transition state

• 21Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Formability at higher temperature: glass transition temperature Tgmelting ⇔ glass transition

crystal

glass

super cooledmelt

melt

First-order phase transitions- exhibit a discontinuity in the firstderivative of the G with respect to a thermodynamic variable

- latent heat is involved

Second-order phase transitionshave a discontinuity in a second derivative of the free energy.

Ehrenfest classification

ii

G ρμ

⎛ ⎞∂=⎜ ⎟∂⎝ ⎠

5.2. Physical Properties of Pure Polymers

2

2, i

P

P n

G CT T

⎛ ⎞∂= −⎜ ⎟∂⎝ ⎠

i.e.

i.e.

• 22Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

volume

thermal expansion

specific heat

heat conductivity

modulus ln(G)

5.2. Physical Properties of Pure Polymerspr

oper

ty

T

amorphous crystalline semi-crystalline

• 23Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Tg is dependent on the viscoelastic materials properties, and so varies with rate of applied load.

rubber plateau

5.2. Physical Properties of Pure Polymers

Tg TM T

E [Pa]

E1

E2

• 24Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

glass transition temperature

durability at higher temperature

i.e. coffee cup from PS

glass transition temperature TgThe glass transition temperature is characterizedby the transition from amorphous or semi-crystallinestate to a rubbery state.

Reason: Below the glass transition temperature themolecules have very little relative mobility.Above Tg, the secondary, non-covalent bondsbetween the polymer chains become weak in comparison to thermal motion, and the polymer becomes rubbery and capable of elastic or plastic deformation without fracture.

Consequences: large change in viscosity, hardness, modulus, volume, enthalpy, entropy

Tg>100°C

5.2. Physical Properties of Pure Polymers

• 25Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

The glass transition temperature depends on the structure of the chain molecules. (i.e. flexibility, side-groups [softener])

Flexibility of the backbone: high flexibility → low Tg

H3C Si O

CH3

CH3

Si O Si

CH3

CH3

CH3

CH3

CH3

nPDMS Tg=-127°C

i.e.

liquid at room temperatureApplication: hair shampoo

CH2 CH

COOH n

Poly acryl acid Tg= 106 °C

5.2. Physical Properties of Pure Polymers

• 26Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Measurement of Tg

thermal methodsi.e. expansion coefficient

heat capacity

Attentionresult depends on heating rate

mechanic methodsi.e. rheology

AttentionTg depends on load respectively frequency

5.2. Physical Properties of Pure Polymers

• 27Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Glass transition temperature Tg

Measurement – Differential-Scanning-Calorimetry (DSC)

Differential scanning calorimetry or DSC is a thermoanalytical technique in which the difference in the amount of heat required to increase the temperature of a sample and reference are measured as a function of temperature. Both the sample and reference are maintained at very nearly the same temperature throughout the experiment.

Crystallization → exothermic procedure → positive signalMelting → endothermic procedure → negative signal

5.2. Physical Properties of Pure Polymers

• 28Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

sample reference

sample holder

oven

thermocouple

oven temperaturecontrol

Measurement of Glass Transition Temperature via DSC

5.2. Physical Properties of Pure Polymers

computer recorder

• 29Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Measurement of Glass Transition Temperature via DSC

5.2. Physical Properties of Pure Polymers

exo-thermic effect

endothermic effect

endo-thermic

effect

Hea

t flo

w [m

W]

T [°C]

glass transition

crystallization

melting

• 30Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Glass transition temperature Tg

The glass transition temperaturedepends on molecular weight.

( ) ( )g gn

AT M T MM

= → ∞ −

5.2. Physical Properties of Pure Polymers

1 2 3 4 5 105 M [g/mol]

450

420

380

340

T [K]

• 31Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Prediction of Glass Transition Temperature5.2. Physical Properties of Pure Polymers

13

1 2 3 4 5 6 7 8

9 10 11 12

351 5.63 31.68 23.94

15 4 23 12 8 4 8 5

11 8 11 4

g

g

Tg

T

i

NT x

NN x x x x x x x x

x x x xx structural parameters given in textbooks

δ= + + −

= − + + − − − +

+ + − −

231213poly(ε-caprolactone)202203polyisoprene

187195polyethylene

Tg [K] (pred.)Tg [K] (exp.) polymerexamples

rule of thumb

0.52 / 3

g

M

T for symmetrical polymersfor unsymmetrical polymersT

⎧= ⎨

• 32Polymer Thermodynamics5. Thermodynamics of Pure Polymers

The most polymers consist of an amorphous and a crystalline part.Experimental Investigations: X-ray-measurements (X-ray diffraction)

The degree of crystallization depends onthe following factors:

cooling ratemelting temperaturechemical compositiontacticitymolar massdegree of branchingtype of additives

ideal crystal PIB semi-crystalline amorphous

0.4artificial silk0.7cotton

0.1PVC

0.6PE (branched)

0.8 – 0.95PE (linear)

Degree of crystallinitypolymer

5.2. Physical Properties of Pure Polymers

• 33Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

fringe-crystallite folding-crystallite

high regularity→ application as fibersi.e. polyamide, polyester

Mixture (crystalline + amorphous)i.e. cellulose, protein

The totality of orientation of the crystallites are called Texture.

exp. estimation: electron microscopy, tunneling microscopy

Crystallinity

5.2. Physical Properties of Pure Polymers

• 34Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

liquid crystal's

Gan

ghöh

e p

( )

nematic phase smectic phase cholesteric phase

5.2. Physical Properties of Pure Polymers

• 35Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Crystallinity

192

185 ± 15

165 ± 18

193 ± 28

TM [°C]

0.946triclinic

0.922rhombic

6.10.93trigonal

4.20.931hexagonal

8.40.94monoclinic

ΔMH [kJ/mol]density [g/cm3]Crystal structure

i.e. polypropylene

5.2. Physical Properties of Pure Polymers

• 36Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Estimation of degree of crystallinity

( )1

crys amorph crys amorph

crys amorph crys crys amorph amorph

crys amorph cry

cryscrys crys crys

crys amorph

amorphcrys

c

s am

rys amorp

orph

crys amo

h

rph

V V V m m m

m m V VmV V V

V V

V V

Vφ φ φ

ρ ρφ

ρ ρ

ρ ρρ

ρ ρ ρ≡+

−=

= + = +

+ += = =

+ +

= + −

→Measurement of density ρ (flotation experiments)

available from the crystal structure

transformation in amorphous state via cooling down very rapidly; measurement of the density

ordensity measurements of polymer and extrapolation to crystallization temperature

5.2. Physical Properties of Pure Polymers

• 37Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

semicrystalline polymers consist of a portion of amorphous properties and a portionof crystalline properties

5.2. Physical Properties of Pure Polymers

at glass transition temperature: the amorphous portion will “melt” or “soften”the crystalline portion remain “solid” up to the melting temperature

semicrystalline polymers can be treated as a solid below Tgcan be treated as composite consisting of solid and rubbery phase

above Tg but below Tmcan be treated as fluid above Tm

• 38Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Approximations: ( )( )

( ) (298 ) 0.106 0.003 / /( )

( ) (298 ) 0.64 0.0012 / /( )

S SP P

L LP P

c T c K T K J molK

c T c K T K J molK

≈ +

≈ +

5.2. Physical Properties of Pure Polymers

polypropylene

Isobaric heat capacitiescP [J/(molK)]

50

100

T [K]

Tg Tm

100 200 300 400 500

amorphous

crystalline

• 39Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

polypropylene

Enthalpy and Entropy

( 0) ( 0)AACh T h TΔ → = →from T→0 to Tg both lines run parallel ( 0) ( ) 0AC AC gh T h T T T TΔ → = Δ → ≤ ≤

( )( )( )

SL M

AC M

M M

h Th Th T

Δ= Δ= Δ

heat offusion

5.2. Physical Properties of Pure Polymers

amorphouscrystalline

h [kJ/mol]

0

10

20

30

40

T [K]100 200 300 400 500

Tg Tm

• 40

100 200 300 400 500

Polymer Thermodynamics5. Thermodynamics of Pure Polymers

Enthalpy and Entropy

0

0

0

0

( ) ( )

( ) ( )

T

P p iiP T

TpP

iiP T

h c h T h T c dT hT

cs c s T s T dT sT T T

∂⎛ ⎞ = → = + + Δ⎜ ⎟∂⎝ ⎠

∂⎛ ⎞ = → = + + Δ⎜ ⎟∂⎝ ⎠

∑∫

∑∫first-order phase transitions

5.2. Physical Properties of Pure Polymers

polypropylene

Enthalpy and Entropy

amorphouscrystalline

h [kJ/mol]

0

10

20

30

40

T [K]

• 41Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Example: poly (ethylene terephthalate)1) molar heat capacity of the solid and liquid

polymer at 25°Cgroup – contribution method

M=n*192.2g/mol

table book: i.e. D.W. van Krevelen, Properties of Polymers, Elsevier 1990.

(298 ) 221.5 /( )

(298 ) 304.0 /( )

SPLP

c K J molK

c K J molK

=

=exp. value for solid polymer: 223.9 J/(molK)exp. value for liquid polymer: 321.2 J/(molK)

5.2. Physical Properties of Pure Polymers

• 42Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Example: poly (ethylene terephthalate)1) molar heat capacity of the solid and liquid

polymer at 25°C

exp. value for solid polymer: 223.9J/(molK)

5.2. Physical Properties of Pure Polymers

0 1(298 ) 8.985304* 20.920972* 7.304602( 5 )S VP ROT SiJc K N N

molKχ χ⎡ ⎤= + + +⎣ ⎦

NROT= rotation degree

for PET0 19.9663 4.2152 7 0

(298 ) 228.9 /( )

VROT Si

SP

N N

c K J molK

χ χ= = = =

→ =

topological method

• 43Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Example: poly (ethylene terephthalate)1) molar heat capacity of the solid and liquid

polymer at 25°C

exp. value for liquid polymer: 321.2J/(molK)

5.2. Physical Properties of Pure Polymers

0 08.162061* 23.215188*(298 )

8.47737 5.350331

VLP

BBrot SGrot

Jc KmolKN N

χ χ⎡ ⎤+= ⎢ ⎥+ +⎣ ⎦

NBBrot= rotation degree into backboneNSGrot= rotation degree into side groups

for PET0 09.9663 7.3566 7 0

(298 ) 311.5 /( )

VBBrot SGrot

LP

N N

c K J molK

χ χ= = = =

→ =

topological method

• 44Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Example: poly (ethylene terephthalate)2) molar heat capacity of the solid polymer at 277°C

(spinning temperature)

(298 ) 304 /( )LPc K J molK=

( )( )

( ) (298 ) 0.64 0.0012 / /( )

304 0.64 0.0012*550 /(550 ) 395.2 /( )

L LP P

LP

c T c K T K J molK

K K Jc K J molK

molK

≈ +

+= =

exp. value for liquid polymer at 550K: 386.3J/(molK)

5.2. Physical Properties of Pure Polymers

• 45Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Example: poly (ethylene terephthalate)3) heat of fusion at melting temperature

(TM=543K)

exp. value: 26.9 kJ/(molK)25 /M h kJ molΔ =

5.2. Physical Properties of Pure Polymers

• 46Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Example: poly (ethylene terephthalate)4) enthalpy difference between the solid and the

rubbery form at the glass transition temperature(TM=543K, Tg=343K)

0

0( ) ( )T

pT

h T h T c dT= + ∫general:application:

( )

0

( ) ( ) ( )

( ) 25 /

g g

M M

M g

T TL S

M g M M M p M M P PT T

M M

T T T T

h T h T c dT h T c c dT

h T kJ mol

= =

Δ = Δ + Δ = Δ + −

Δ =

∫ ∫

( )( )

( ) (298 ) 0.106 0.003 / /( )

( ) (298 ) 0.64 0.0012 / /( )

S SP P

L LP P

c T c K T K J molK

c T c K T K J molK

≈ +

≈ +

with

5.2. Physical Properties of Pure Polymers

• 47Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Example: poly (ethylene terephthalate)4) enthalpy difference between the solid and the

rubbery form at the glass transition temperature(TM=543K, Tg=343K)

( )( )

( )( )

( ) ( )2 2

(298 ) 0.64 0.0012 / /( )( ) ( )

(298 ) 0.106 0.003 / /( )

( ) ( )

(298 )*0.64 (298 )*0,106

(298 )*0.0012 / (298 )*0.003 /2

g

M

LTP

M g M M ST P

M g M M

L SP P g M

g ML SP P

c K T K J molKh T h T dT

c K T K J molK

h T h T

c K c K T T

T Tc K T K c K T K

⎛ ⎞+Δ = Δ + ⎜ ⎟

⎜ ⎟− +⎝ ⎠Δ = Δ

⎡ ⎤− −⎢

+ ⎢ −⎢ + −⎢⎣ ⎦

/( )

( ) 17.34 /M g

J molK

h T kJ mol

⎥⎥⎥⎥

Δ =

5.2. Physical Properties of Pure Polymers

• 48Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Example: poly (ethylene terephthalate)5) entropy difference between the solid and the

rubbery form at the glass transition temperature(TM=543K, Tg=343K)

0

0( ) ( )T

p

T

cs T s T dT

T= + ∫general:

application:

( )0

( ) ( ) ( )g g

M M

M g

L ST TP PM p

M g M M M MT T

T T T T

c ccs T s T dT s T dT

T T

= =

−ΔΔ = Δ + = Δ +∫ ∫

( )( )

( ) (298 ) 0.106 0.003 / /( )

( ) (298 ) 0.64 0.0012 / /( )

S SP P

L LP P

c T c K T K J molK

c T c K T K J molK

≈ +

≈ +

( ) ( ) / 46 /( )M M M M Ms T h T T J molKΔ = Δ =with

5.2. Physical Properties of Pure Polymers

• 49Polymer Thermodynamics5. Thermodynamics of Pure Polymers

Example: poly (ethylene terephthalate)5) entropy difference between the solid and the rubbery form at the glass transition

temperature (TM=543K, Tg=343K)

( )( )

( )

(298 ) 0.64 0.0012 /

(298 ) 0.106 0.003 /( ) ( )

(298 )*0.64 (298 )*0.106( ) ( )

(298 )*0.0012 (298 )*0.003

(

g

M

g

M

LP

T SP

M g M MT

L SP PT

M g M MT L S

P P

M g

c K T K

c K T K Js T s T dTT molK

c K c KJTs T s T dT

molKc K c K

s T

⎛ ⎞+⎜ ⎟⎜ ⎟− +⎝ ⎠Δ = Δ +

⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟

Δ = Δ + ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟+ −⎝ ⎠

Δ

( )

( )( )

) ( ) (298 )*0.64 (298 )*0.106 ln

(298 )*0.0012 (298 )*0.003 27.39 /( )

gL SM M P P

M

L SP P g M

T Js T c K c KT molKJc K c K T T J molK

molK

⎛ ⎞= Δ + − ⎜ ⎟

⎝ ⎠

+ − − =

5.2. Physical Properties of Pure Polymers

• 50Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Example: poly (ethylene terephthalate)

07.9Free enthalpy [kJ/mol]

yesnoEquilibrium

4627.39Entropy [J/(molK)]

2517.34Enthalpy [kJ/mol]

TM=543 KTg=343 KTemperature [K]MeltingGlass transitionQuantity

summary

The knowledge of the chemical structure, glass transition temperature and melting temperatureallows the estimation of thermodynamic quantities,like enthalpy and entropy.

5.2. Physical Properties of Pure Polymers

• 51Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Coefficient of volumetric thermal expansion, a, at 25°C

2.9 10-4PE8 10-5PVC7 10-5PS

3.7 10-4CS2

3.8 10-7Quarzglasa [K-1]Material

Reason: strong covalent bonds within the polymer chainweak van-der Waals forces between polymer chains

a depends strongly on the chemical bond strength

1( )P

VTV T

α ∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠

5.2. Physical Properties of Pure Polymers

Impact on: injection molding and extrusion process

• 52Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Coefficient of volumetric thermal expansion

1( )P

VTV T

α ∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠ 3αβ ≡

molar volume v W P Tv v v v= + +vW is the space truly occupied by its molecules, often called van der Waals volume,

it is impenetrable to other moleculesvP packing volume = amount of additional “empty space” due to packing constraints

imposed by the sizes and shapes of moleculesvT expansion volume resulting from the thermal motions of molecules;

is the difference between the molar volume at the temperature of interest and themolar volume at absolute zero temperature

empirical correlation

0.15 11.42 0.159.47

WW

Pg g g

vT vv vT T T T T

α⎛ ⎞⎛ ⎞ ∂⎛ ⎞= + → = → =⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ∂ +⎝ ⎠⎝ ⎠⎝ ⎠

Coefficient of linear thermal expansion

Tg = glass transition temperature

5.2. Physical Properties of Pure Polymers

• 53Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Estimation of vWliterature: J. Bicerano, Prediction of Polymer Properties, Marcel Dekker, 1993.

5.2. Physical Properties of Pure Polymers

( )3

0 13.861803* 13.748435* VWcmvmol

χ χ= +or

( )0 1 3

( )

2.28694* 17.14057* 1.369231 /

0.5 2 3 42.5 2 7 8 4

VW vdW

vdW menomar mear alamid OH cyanide carbonate cyc

fused C C Si S Br

v N cm mol

N N N N N N N NN N N N N

χ χ

= − −

= + +

= + + + + − −

− + + − −

number of methyl groups attached to non-aromatic ring

number of methyl groups attached to aromatic ring

number of non-aromatic rings with no double-bond

number of rings in fused ring structures

• 54Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Concept of Simha and Boyer 19625.2. Physical Properties of Pure Polymers

( ) ( 0) 0.001298

( ) ( 0) ( ) ( 0)298 2980.00045

( ) ( ) 0.00055

( 0) 1.3 (298 ) 1.435(298 ) 1.6

L LW

g g c c

W

g g c W g

c w c w

g w

v T v T v

v T v T v T v T

vv v T v T v T

v T v v K vv K v

− →≈

− → − →=

≈Δ = − ≈

→ ≈ ≈≈

V

TTg TM

VW

crystalline solid

glass

crystallization range

undercooledliquid

liquid

excess volume

VCVC

Vg

VL

ΔVg

ΔVm

• 55Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Tait-relation (1888)( 0, ) ( , ) ln 1

( 0, ) ( )v P T v P T PC

v P T B T⎛ ⎞= −

= +⎜ ⎟= ⎝ ⎠

C dimensionless constant B(T) dimension of pressure v [cm3/g]

Simha-relation (1973)( )( )

( ) ( )1 2

20 1 2

0.0894 ( ) exp 273.15

(0, ) 273.15 273.15

C B T b b T K

V T A A T A T

= = − −

= + − + −

4.083.16polycarbonate4.142.44polystyrene

4.151.91polyisobutylene

5.11.99polyethylene

10-3 b2 [°C]103 b1 [bar]polymer

J. Appl. Phys. 42 (1971) 4592.

5.2. Physical Properties of Pure Polymers

• 56

100 120 140 160 180 200

0,98

1,00

1,02

1,04

Polystyrene

V [cm3/g]=0.92351+0.00053158 T/[°C]

V [c

m3 /g

]

T [°C]

Polymer Thermodynamics5. Thermodynamics of Pure Polymers

Example: polystyrene

( ) ( )3 7 3 7 3 20 1 28 3 1

1 2

0.000938 / 3.31*10 / 6.69*10 /

2.5*10 4.18*10 0.0894

A cm g A cm Kg A cm K g

b Pa b K C

− −

− −

= = =

= = =

P=0.1 MPa

5.2. Physical Properties of Pure Polymers

• 57Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Cohesive Energy ecohCohesive Energy eCoh: internal energy of the material if all of its intermolecular

forces are eliminatedCohesive Energy Density εCoh: is the energy required to break all intermolecular

physical links in a unit volume of material

CohCoh

ev

ε =

Cohesive Energy plays a role in the prediction of many other physical propertiessolubility parameterglass transition temperaturesurface tensiondielectric constantmechanical propertiespermeability

low molecular weight substances

0VL

Coh VL i VLe h P v= Δ − ΔProblem: polymers do not evaporate

5.2. Physical Properties of Pure Polymers

• 58Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Cohesive Energy ecoh

( )( )( )

2

0 0 1 1 3

2

(298 )

97.95 2

134.61

Coh

V V

Si Br Cyc

Fev K

Jcmwhere FmolN N N

χ χ χ χ

=

⎡ ⎤− + + +⎢ ⎥=⎢ ⎥+ − −⎣ ⎦

F molar attraction constantv molar volume

1. Possibility

5.2. Physical Properties of Pure Polymers

• 59Polymer Thermodynamics

5. Thermodynamics of Pure Polymers

Cohesive Energy ecoh

2. Possibility Coh D P He e e e= + +

dispersioninteraction

polar interaction

hydrogen bonding

5.2. Physical Properties of Pure Polymers

3. Possibility

( ) ( )( )0 0 1 110570.9 9072.8 2 1018.2V VCoh VKH Je N molχ χ χ χ= − + − +NVKH group contribution method

Hanson method

• 60Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Prop

erty

molar mass

Some thermic (i.e. expansion coefficient) and some mechanical properties (i.e. loss modulus) depend for M>MC only very slightly on molecular weight.The critical molar weight, MC, depends on the type of polymers.

i.e. PE MC≈20000g/mol PET MC ≈ 5000 g/mol

MC

All quantities of the second law of thermodynamics (entropy, free enthalpy, free energy) depends on the molecular weight and on the molecular weight distribution.

• 61Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Estimation of molecular weight – thermodynamic method→ use colligative properties → membrane osmometry

,

,

,

,1

B

spB B BB

B B

sp B

sp Bsp B

B

c RTcn mc

V M V Mc

RT ideal diluted s

RT B c polymer solutionc M

olution

M

M

Π=

=

Π ⎛ ⎞= + +⎜ ⎟

= =

Π

=

Van't Hoff`sche equation

B second osmotic virial coefficient

• 62Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Estimation of molecular weight – thermodynamic method→ use colligative properties → membrane osmometry

solution

semipermeable membrane

solvent

The difference in height Δh corresponds to the osmotic pressure π.

• 63Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Estimation of molecular weight – thermodynamic method→ use colligative properties → membrane osmometry

,,

1 sp Bsp B

RT B cc M MΠ ⎛ ⎞= + +⎜ ⎟

⎝ ⎠…

• 64Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Estimation of molecular weight – thermodynamic method→ use colligative properties → vapor pressure osmometry

solution pure solvent

T2, PLV T1,

Two thermistors are located in a chamber saturated with pure solvent vapor. The difference between PLV and caused a new equilibrium between the two drops. Solvent from the vapor phase will condense on the thermistor connected to the solution. The formed condensation heat leads to a measurable temperature difference DT=T2-T1.

0LVAP

0LVAP

• 65Polymer Thermodynamics5. Thermodynamics of Pure Polymers

5.3. PolydispersityEstimation of molecular weight – thermodynamic method→ use colligative properties → vapor pressure osmometry

syringe

Al-block for thermostatingthe syringes

thermistors

measurement cell (Al)

windowglass container for solvent

sealing

vapor pressure osmometry

1. saturation of sample chamberwith solvent vapor

2. filling with solution and puresolvent using thermostatic syringes

3. measurement of ΔT

• 66Polymer Thermodynamics5. Thermodynamics of Pure Polymers

data analysis of vapor pressure osmometry experiments:vapor pressure depletion:

0

0

0

VL VLA

VLA BB B B

BVLA A B A B A

P P P

P mP n n mx PP n n M n M n

Δ = −

Δ= = ≈ = → Δ =

temperature dependence of vapor pressure – Clausius-Clapeyron equation

( )0 0 0 0

0 0 0 0

0

00 0

0 0 0 0 0

0 0 0 0

VL A VL A VL A VL AVL VL V VL VVL V LA A A A AA A

VL VL VL VVL A A B B A A A

VL V VLA A B A A

B AB

A

h h h h TdP P PdT T v T v T T vT v v

m Kh T P m m P T vT v M n

MTn h T

Δ Δ Δ Δ ΔΔ= = ≈ ≈ → Δ =

Δ Δ−

Δ Δ= ==

Δ Δ Δ→

→ for the estimation of molecular weight is only one experimental value isnecessary

5.3. Polydispersity

• 67Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Wi(M

) 1

1

( )

( )

i ik

iik

ii

n W M M

n n

n W M M

=

=

= Δ

=

= Δ

ΔM

molecular weight

• 68Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

2

1 2

1

,

0

( )

( )

M

M MM

n W M dM

n W M dM∞

=

=

W(M

)

molecular weight

M1

M2

extensive

intensive

2

1 2

1

,

0

( )

1 ( )

M

M MM

x w M dM

w M dM∞

=

=

• 69Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

differential distribution function cumulative distribution function

0

( ) ( )x

F x f x dx= ∫

• 70Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Characterization of the distribution function using moments

( ) ( )

1 0

( )k

n n n ni i

iM w M M M w M M dM

=

= Δ ⇔ =∑ ∫

n=0 normalization condition

( (0)

1

0)

0

1 ( ) 1k

ii

M w M M w M dM∞

=

= Δ = ⇔ = =∑ ∫

n=1 average value

( ) ( )

1 0

1 1 ( )k

i ii

M w M M M w M MdM∞

=

= Δ ⇔ =∑ ∫

• 71Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

( ) ( )

1 0

( )k

n n n ni i

iM w M M M w M M dM

=

= Δ ⇔ =∑ ∫

n=3 asymmetry

( ) ( )

1 0

2 2 2 2( )k

i ii

M w M M M w M M dM∞

=

= Δ ⇔ =∑ ∫

( ) ( )

1 0

3 3 3 3( )k

i ii

M w M M M w M M dM∞

=

= Δ ⇔ =∑ ∫

Characterization of the distribution function using moments

• 72Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Relation between the moments and experimental quantities

( ) ( )

1 0

( )k

n n n ni i

i

M w M M M w M M dM∞

=

= Δ ⇔ =∑ ∫

number-average molar mass

(1)1

(0)1

1

(1)0

(0)0

0

( )( )

( )

k

i i ki

n i iki

ii

n

w M MMM w M MM w M

w M MdMMM w M MdMM

w M dM

=

=

=

Δ= = = Δ

Δ

= = =

∑∑

∫∫

Experiment:methods, which are proportionalto the number of molecules

colligativeproperties

vapor pressure osmosesmembrane osmoses

• 73Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

( ) ( )

1 0

( )k

n n n ni i

i

M w M M M w M M dM∞

=

= Δ ⇔ =∑ ∫

mass-average molar mass

2 2(2)

1 1(1)

1

1

2 2(2)

0 0(1)

1

0

( ) ( )

( )

k k

i i i ii i

w kn

i ii

wn

w M M w M MMMM Mw M M

w M M dM w M M dMMMM M

w M M dM

= =

=

∞ ∞

Δ Δ= = =

Δ

= = =

∑ ∑

∫ ∫

Experiment:methods, which are proportionalto the mass of molecules

light scattering

Relation between the moments and experimental quantities

• 74Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

( ) ( )

1 0

( )k

n n n ni i

i

M w M M M w M M dM∞

=

= Δ ⇔ =∑ ∫

z-average molar mass 3 3(3)1 1

(2)2

1

3 3(3)

0 0(2)

2

0

( ) ( )

( )

k k

i i i ii i

z kw n

i ii

zw n

w M M w M MMMM M Mw M M

w M M dM w M M dMMMM M M

w M M dM

= =

=

∞ ∞

Δ Δ= = =

Δ

= = =

∑ ∑

∫ ∫

Experiment:ultracentrifugeestimation via the distribution

GPC

Relation between the moments and experimental quantities

• 75Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

( ) ( )

1 0

( )k

n n n ni i

i

M w M M M w M M dM∞

=

= Δ ⇔ =∑ ∫

viscosity-average molar mass 1/ 1/( )

(0)1

1/1/( )

(0)0

( )

0,5 0,9

a aa ka

i ii

aaaa

MM w M MM

MM w M M dMM

η

η

α

=

⎛ ⎞ ⎛ ⎞= = Δ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞= = ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠≤ ≤

∫Experiment:rheology

Staudingerindex

Relation between the moments and experimental quantities

• 76Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersitydefinition of uniformity

polydispersity D

( )

2

21

1

0

1

0

k

i i ki

i in i

nn n

n w

w

n

z

n z

w

n

w

MDM

M

w M Mw M M

MM M

M M M M

Ufor U monodispers polymer

M M

UM

M M

η

η

=

=

ΔΔ

= =

=

= − ≤ ≤

⇒ ≥=

⇒ = = =

∑∑

uniformity U

• 77Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

0 200 400 600 800 1000 1200 1400 1600 18000,0

0,5

1,0

1,5

2,0

2,5

103 W

(M)

M [g/mol]

0,097942878

800blue curve

0,035U854Mz

828Mw

800Mn

red curvequantity

• 78Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

molar weight distributionGPC field-flow-fractionationMALDI -TOF

average values of distributionnumber-average: exp. methods which are proportional to the number of

moleculesend-group analysis, colligative properties

mass-average: exp. methods which are proportional to the mass of moleculesLight-scattering, SANS, SAXS

z-average: ultracentrifugeviscosity-average: rheology

• 79Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

GPC – gel permeation chromatographySize-Exclusion Chromatography (SEC)

• 80Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

( )22

1( ) exp22

M Mw M

σσ π

⎛ ⎞−⎜ ⎟= −⎜ ⎟⎝ ⎠

normal distribution function= bell curve= probability density functionfor random distributed variables(i.e. to play dice)

C.F. Gauß (1777-1855)

0 500 1000 1500 20000,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

103 W

(M)

M [g/mol]

symmetrical distribution

M average values standard derivation

• 81Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Schulz-Flory-Distribution( ) exp

( )

kkk r rw r kk r r r

⎛ ⎞ ⎛ ⎞= −⎜ ⎟ ⎜ ⎟Γ ⎝ ⎠ ⎝ ⎠

r – segment number polymer molecules will be divided into segments of equal size

J.P. Flory (1910-1985) Nobel Prize 1974

description of kinetics of chain-growth polymerization (statistical, anionic and cationic polymerization)

most probable distribution

w(r

)

segment numberhttp://nobelprize.org

• 82Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Calculation of moments n=0 normalization condition

( )

( )

(0)

0 0 0

(0)

0

10

(0)

( ) exp exp( ) ( )

exp( )

( 1)exp

( 1( )

k kk k

kk

mm

k

k r r k r rM w r dr k dr k drk r r r k r r r

r dr k rsubstitution x dx dr rdx M x kx dxr r k r

mtextbook of mathematics x axa

k r kMk r

∞ ∞ ∞

+

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

= = = → = −Γ

Γ +− =

Γ +=

Γ

∫ ∫ ∫

1

) ( 1) ( ) 1( ) ( )k

k k kk k k k k+

Γ + Γ= = =

Γ Γ

( )( ) 1 !( 1) ( )x x x Nx x x

Γ = − ∈

Γ + = Γ

Distribution function fulfills the normalization condition.

• 83Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Schulz-Flory-distribution

Calculation of moments n=1 average value

( )

( ) ( )

(1)

0 0 0

(1)

0

2(1) 1

0 0

( ) exp exp( ) ( )

exp( )

( 1)exp exp( )

k kk k

kk

kk m

k r r r k r rM rw r dr k dr r k drk r r r k r r r

r dr k rsubstitution x dx dr rdx M xrx kx dxr r k r

k r mM x kx dx textbook x axk r a

∞ ∞ ∞

∞ ∞+

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

= = = → = −Γ

Γ += − − =

Γ

∫ ∫ ∫

∫ ∫ 1

(1)1 1 2 2 2

(1)

( 1 1) ( 2) ( 1) ( 1) ( 1) ( )( ) ( ) ( ) ( )( 1)

m

k

k

k r k r k r k k r k k kMk k k k k k k k

r kMk

+

+ +

Γ + + Γ + + Γ + + Γ= = = =

Γ Γ Γ Γ+

=

( )( ) 1 !( 1) ( )x x x Nx x x

Γ = − ∈

Γ + = Γ

• 84Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Schulz-Flory-distribution

( )

( ) ( )

2(2) 2 2

0 0 0

(2) 2 2

0

2(2) 2

0 0

( ) exp exp( ) ( )

exp( )

(exp exp( )

k kk k

kk

kk m

k r r r k r rM r w r dr k dr r k drk r r r k r r r

r dr k rsubstitution x dx dr rdx M x r x kx dxr r k r

k r mM x kx dx textbook x axk

∞ ∞ ∞

∞ ∞+

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

= = = → = −Γ

Γ= − − =

Γ

∫ ∫ ∫

∫ ∫ 12 2 2

(2)2 1 2 1 3

2 2 2(2)

3 3 2

1)

( 2 1) ( 2 1) ( 2) ( 2)( ) ( ) ( )

( 2)( 1) ( 1) ( 2)( 1) ( ) ( 2)( 1)( ) ( )

m

k

k

a

k r k r k r k kMk k k k k k

r k k k r k k k k r k kMk k k k k

+

+ + +

+

Γ + + Γ + + + Γ += = =

Γ Γ Γ

+ + Γ + + + Γ + += = =

Γ Γ

( )( ) 1 !( 1) ( )x x x Nx x x

Γ = − ∈

Γ + = Γ

• 85Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Schulz-Flory-distribution

Calculation of moments n=2 asymmetry

( )

( ) ( )

3(3) 3 3

0 0 0

(3) 3 3

0

3(3) 3

0 0

( ) exp exp( ) ( )

exp( )

(exp exp( )

k kk k

kk

kk m

k r r r k r rM r w r dr k dr r k drk r r r k r r r

r dr k rsubstitution x dx dr rdx M x r x kx dxr r k r

k r mM x kx dx textbook x axk

∞ ∞ ∞

∞ ∞+

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

= = = → = −Γ

Γ= − − =

Γ

∫ ∫ ∫

∫ ∫ 13 3 3

(3)3 1 3 1 4

3 3(3)

4 3

1)

( 3 1) ( 3 1) ( 3) ( 3)( ) ( ) ( )

( 3)( 2)( 1) ( ) ( 3)( 2)( 1)( )

m

k

k

a

k r k r k r k kMk k k k k k

r k k k k k r k k kMk k k

+

+ + +

+

Γ + + Γ + + + Γ += = =

Γ Γ Γ

+ + + Γ + + += =

Γ

( )( ) 1 !( 1) ( )x x x Nx x x

Γ = − ∈

Γ + = Γ

• 86Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Schulz-Flory-distributionCalculation of moments n=-1

( )

( )

1( 1) 1 1

0 0 0

( 1) 1 1

0

( 1) 1

0 0

( ) exp exp( ) ( )

exp( )

exp exp( )

k kk k

kk

kk m

k r r r k r rM r w r dr k dr r k drk r r r k r r r

r dr k rsubstitution x dx dr rdx M x r x kx dxr r k r

kM x kx dx textbook xk r

∞ ∞ ∞−− − −

∞− − −

∞ ∞− −

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

= = = → = −Γ

= −Γ

∫ ∫ ∫

∫ ∫ ( ) 1

( 1)1 1

( 1)

( 1 1) ( ) 1( ) ( )

m

k k

k k

maxa

k k k kMk r k k r k r

+

−− +

Γ +− =

Γ − + Γ= =

Γ Γ

( )( ) 1 !( 1) ( )x x x Nx x x

Γ = − ∈

Γ + = Γ

The characterization variable segment number requires the (-1) moment.

• 87Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

calculation of experimental available average values number-average

(1)0

(0)0

0

( )( )

( )n

w M MdMMM w M MdMM

w M dM

∞= = =∫

∫∫

characterization variable: molar mass

characterization variable: segment number

(0)0

( 1)

0

( )1

1/( )n

w r drMr rM rw r dr

r

∞−= = = =∫

• 88Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

calculation of experimental available average values weight-average

characterization variable: molar mass

characterization variable: segment number

2(2)

0(1)

0

( )

( )w

w M M dMMMM

w M MdM

∞= =∫

(1)0

(0)

0

( )( 1)( 1)

( )

nw

w r rdrr kM r kr

M k kw r dr

++= = = =

• 89Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

calculation of experimental available average values z-average

characterization variable: molar mass

characterization variable: segment number

3(3)

0(2)

2

0

( )

( )z

w M M dMMMM

w M M dM

∞= =∫

22(2) 2

0(1)

0

( 2)( 1)( )( 2)

( 1)( )

z

r k kw r r drM r kkr r kM k

w r rdr k

+ ++

= = = =+

• 90Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Uniformity U

1Uk

= nr r=

( )(1/ 1) 11/

nw n

r Ur r UU

+= = +

( )(1/ 2) 1 21/z

r Ur r UU

+= = +

After the estimation of the average values of the segment-molar distributionfunction (rn, rw or rz) the parameters U respectively k can be calculated.

The parameters U (respectively k) can also be estimated using kinetic data.

• 91Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

0 200 400 600 800 10000,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

4,0

rn=100 k=1rw=200 rz=300

103 w

(r)

r

Schulz-Flory-distribution

• 92Polymer Thermodynamics5. Thermodynamics of Pure Polymers

5.3. Polydispersitygeneral log-normal distribution

( )22

ln( ) ln( )1( ) expr r

w rAAr π

⎛ ⎞−= −⎜ ⎟

⎜ ⎟⎝ ⎠

( )2 ln Aσ =

( )21 2

ln( ) ln( )1( ) expz

z

r rrw rr y AA π +

⎛ ⎞−= −⎜ ⎟

⎜ ⎟⎝ ⎠

special cases: z=-1 y=1 Wesslau-distribution

20 exp( / 2)z y σ= = Lansing-distribution

( )222

ln( ) ln( )1 exp( / 2)( ) expr r

w rr AAσ

π

⎛ ⎞−= −⎜ ⎟

⎜ ⎟⎝ ⎠

( )( )

( )

2

21

ln( ) ln( )1( ) expln 2 2 ln

z

z

r rrw rr yσ π σ+

⎛ ⎞−= −⎜ ⎟

⎜ ⎟⎝ ⎠

• 93Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

Wesslau-distribution

( )22

ln( ) ln( )1( ) expr r

w rAAr π

⎛ ⎞−= −⎜ ⎟

⎜ ⎟⎝ ⎠

( )( )( )

2

2

2

exp / 4

exp / 4

exp 3 / 4

n

w

z

r r A

r r A

r r A

= −

=

=

suitable for high polydispersity (i.e. PE)

• 94Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. PolydispersityPoisson-distribution

( ) 11 exp(1 )( )( )

Pn nP Pw P

P

−− −=

Γ

2

1 11wn n n

PP P P

= + −

suitable for polymers with small polydispersity

P = degree of polymerization

In the limiting case of high molar mass the uniformity goes to zero.

polymers produced with anionic or living polymerization

• 95Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

degree of polymerization

Poisson distribution

Schulz-Flory-distribution

102

W(P

)

• 96Polymer Thermodynamics

5. Thermodynamics of Pure Polymers5.3. Polydispersity

0 200 400 600 800 1000 1200 14000

2

4

6

8

W(M

)

M [kg/mol]

variation of molar masses and of molar mass distribution functionlow molecular weight (oligomere) → polymer → ultra-high molecular weight

small distribution → broad distribution → mono-modal → bimodal → multimodal