Lecture - thermodynamik.tu-berlin.de · 5. Thermodynamics of Pure Polymers Application of...

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= =+

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:

21 12 2

= − +

( )( )

1 12 2

1 23 3

2 113 3

xmol mol

molmolxmol mol

mol molxmol mol

λ λλ λ

λλλ λλ λλ λ

= =− −

−−= =

− −

− −= =

− −

3 mole fraction can be reduced to one λ( )

jn mol

= − +

= −∑

2 2 212

H O H O+

NaOH + H2O AlCl3 + H2O

heating cooling

exothermic reaction endothermic reaction

0RHΔ < 0RHΔ >

equilibrium

state variable

λpure educts pure products

equilibrium

spontaneous

5.1. Thermodynamics of Polymerization

dn dλν

reaction variable λ

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

at higher temperature preferred→entropy drivenT>ΔRh/ΔRs spontaneousT=ΔRh/ΔRs equilibriumT<ΔRh/ΔRs not spontaneous

at low temperature preferred→ enthalpy-drivenT<ΔRh/ΔRs spontaneousT=ΔRh/ΔRs equilibriumT>ΔRh/ΔRs not spontaneous

at no temperature possible ΔRg>0<0>0b

at any temperature possible ΔRg<0>0<0a

Spontaneous reactionΔRsΔRh

Calculation of enthalpy of reaction

Kirchhoff Law

( ) ( ) 00 0 ( )i

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 = ∑

Calculation of entropy of reaction

( )Rs T+Δ

Calculation of free enthalpy of reaction

0iR R R ii

g h T s gν+ + + +Δ = Δ − Δ = ∑

( )0is T+ + ( ) ( ) 00 0

c Ts T s T dT

+ + += + ∫ ( )0is T+

0( ) ( )R i ii

s T s Tν+ +Δ = ∑0( ) ( )R i i

s T s Tν+ + + +Δ = ∑

( )Rs T+ +Δ

( ) ( ) ( )TR p

c Ts T s T dT

+ + + ΔΔ = Δ + ∫

( )Rs T+Δ

iR R R ii

g h T s gν+ + + +Δ = Δ − Δ = <∑

for the most polymerization reactions is valid:

0 0R Rh and s+ +Δ < Δ <

upper limit in temperature = Ceiling – temperature TC

a-methyl styrene: TC=60°Ctrichloroacetaldehyde : TC<0°C

enthalpy-driven reaction

iR R R ii

g h T s gν+ + + +Δ = Δ − Δ = <∑for rare polymerization reactions is valid:

0 0R Rh and s+ +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

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

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

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

Application of topological formalism in developing of structure-property correlation

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 1

1Vδ= =

-CH2-4 2V

VZδ −

=2 VZ+ −

Vδ= =−

=O6 0V

VZδ −

=2 VZ+ −

Vδ= =−

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

F edge

0 0 1 11 1 1 1V V

V Vvertices vertices edges edgesij ij

χ χ χ χδ βδ β

≡ ≡ ≡ ≡∑ ∑ ∑ ∑

15Polymer Thermodynamics5. Thermodynamics of Pure Polymers

V V Vij i j ij i jβ δ δ β δ δ= =

vertex

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

Vvertices

edges ij

Vedges ij

≡ = + + =

≡ = + =

Example

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

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

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

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

= + + =

phase transition

critical point

triple point

solid liquid

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.

phase transition

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

low-molecular weight molecules

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

semicrystalline crystalline

amorphous polymer

semicrystalline polymer

viscousliquid

thermal decomposition

rigid (semi) crystalline

glassy

glass transition

glass transitionmelting point

rubbery

leathery

diffuse transition state

Formability at higher temperature: glass transition temperature Tg

melting ⇔ glass transition

crystal

super cooledmelt

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

G ρμ

⎛ ⎞∂=⎜ ⎟∂⎝ ⎠

G CT T

⎛ ⎞∂= −⎜ ⎟∂⎝ ⎠

volume

thermal expansion

specific heat

heat conductivity

modulus ln(G)

5.2. Physical Properties of Pure Polymerspr

amorphous crystalline semi-crystalline

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

rubber plateau

Tg TM T

E [Pa]

glass transition temperature

durability at higher temperature

i.e. coffee cup from PS

glass transition temperature Tg

The 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

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

Si O Si

nPDMS Tg=-127°C

liquid at room temperatureApplication: hair shampoo

CH2 CH

COOH n

Poly acryl acid Tg= 106 °C

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

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

sample reference

sample holder

thermocouple

oven temperaturecontrol

Measurement of Glass Transition Temperature via DSC

computer recorder

Measurement of Glass Transition Temperature via DSC

exo-thermic effect

endothermic effect

endo-thermic

effect

T [°C]

glass transition

crystallization

melting

Glass transition temperature Tg

The glass transition temperaturedepends on molecular weight.

( ) ( )g gn

AT M T MM

= → ∞ −

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

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

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

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

T for symmetrical polymersfor unsymmetrical polymersT

⎧= ⎨

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

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

liquid crystal's

nematic phase smectic phase cholesteric phase

Crystallinity

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

Estimation of degree of crystallinity

crys amorph crys amorph

crys amorph crys crys amorph amorph

crys amorph cry

cryscrys crys crys

crys amorph

amorphcrys

rys amorp

crys amo

V V V m m m

m m V VmV V V

Vφ φ φ

ρ ρφ

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

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

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

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

polypropylene

Isobaric heat capacitiescP [J/(molK)]

100 200 300 400 500

amorphous

crystalline

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 → ≤ ≤

( )( )( )

h Th Th T

Δ= Δ= Δ

heat offusion

amorphouscrystalline

h [kJ/mol]

T [K]100 200 300 400 500

100 200 300 400 500

Polymer Thermodynamics5. Thermodynamics of Pure Polymers

( ) ( )

P p iiP T

h c h T h T c dT hT

cs c s T s T dT sT T T

∂⎛ ⎞ = → = + + Δ⎜ ⎟∂⎝ ⎠

∑∫

∑∫first-order phase transitions

polypropylene

amorphouscrystalline

h [kJ/mol]

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 /( )

c K J molK

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

polymer at 25°C

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

0 1(298 ) 8.985304* 20.920972* 7.304602( 5 )S VP ROT Si

Jc K N NmolK

χ χ⎡ ⎤= + + +⎣ ⎦

NROT= rotation degree

for PET0 19.9663 4.2152 7 0

(298 ) 228.9 /( )

VROT Si

c K J molK

χ χ= = = =

topological method

polymer at 25°C

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

0 08.162061* 23.215188*(298 )

8.47737 5.350331

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

c K J molK

χ χ= = = =

topological method

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

c T c K T K J molK

K K Jc K J molK

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

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

(TM=543K)

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

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

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

0( ) ( )T

h T h T c dT= + ∫general:

application:

( ) ( ) ( )

( ) 25 /

T TL S

M g M M M p M M P PT T

T T T T

h T h T c dT h T c c dT

h T kJ mol

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

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

( )( )

( ) ( )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

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

⎥⎥⎥⎥

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

0( ) ( )T

cs T s T dT

T= + ∫general:

application:

( ) ( ) ( )g 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

−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

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

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

M g M MT

L SP PT

M g M MT L S

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

⎛ ⎞+⎜ ⎟⎜ ⎟− +⎝ ⎠Δ = Δ +

⎛ ⎞⎛ ⎞−⎜ ⎟⎜ ⎟

Δ = Δ + ⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟+ −⎝ ⎠

( )( )

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

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

gL SM M P P

L SP P g M

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

⎛ ⎞= Δ + − ⎜ ⎟

⎝ ⎠

+ − − =

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.

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

α ∂⎛ ⎞= ⎜ ⎟∂⎝ ⎠

Impact on: injection molding and extrusion process

Coefficient of volumetric thermal expansion

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 molecules

vP packing volume = amount of additional “empty space” due to packing constraintsimposed by the sizes and shapes of molecules

vT 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

Pg g g

vT vv vT T T T T

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

Coefficient of linear thermal expansion

Tg = glass transition temperature

Estimation of vW

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

0 13.861803* 13.748435* VW

cmvmol

χ χ= +

( )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

= − −

= + + + + − −

− + + − −

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

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

g g c c

g g c W g

c w c 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 = − ≈

→ ≈ ≈≈

TTg TM

crystalline solid

crystallization range

undercooledliquid

liquid

excess volume

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.

100 120 140 160 180 200

Polystyrene

V [cm3/g]=0.92351+0.00053158 T/[°C]

T [°C]

Polymer Thermodynamics5. Thermodynamics of Pure Polymers

Example: polystyrene

( ) ( )3 7 3 7 3 20 1 2

8 3 11 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

Cohesive Energy ecoh

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

Cohesive Energy Density εCoh: is the energy required to break all intermolecularphysical links in a unit volume of material

CohCoh

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

low molecular weight substances

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

( )( )( )

0 0 1 1 3

(298 )

97.95 2

134.61

Si Br Cyc

Jcmwhere FmolN N N

χ χ χ χ

⎡ ⎤− + + +⎢ ⎥=⎢ ⎥+ − −⎣ ⎦

F molar attraction constantv molar volume

1. Possibility

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

dispersioninteraction

polar interaction

hydrogen bonding

3. Possibility

( ) ( )( )0 0 1 110570.9 9072.8 2 1018.2V VCoh VKH

Je Nmol

χ χ χ χ= − + − +

NVKH group contribution method

Hanson method

5. Thermodynamics of Pure Polymers5.3. Polydispersity

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

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

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

spB B BB

sp Bsp B

c RTcn mc

V M V Mc

RT ideal diluted s

RT B c polymer solutionc M

olution

Π ⎛ ⎞= + +⎜ ⎟

Van't Hoff`sche equation

B second osmotic virial coefficient

solution

semipermeable membrane

solvent

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

1 sp Bsp B

RT B cc M MΠ ⎛ ⎞= + +⎜ ⎟

⎝ ⎠…

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.

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

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

VL VLA

VLA BB B B

BVLA A B A B A

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 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

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

n W M M

molecular weight

n W M dM

n W M dM∞

molecular weight

extensive

intensive

x w M dM

w M dM∞

differential distribution function cumulative distribution function

( ) ( )x

F x f x dx= ∫

Characterization of the distribution function using moments

( ) ( )

n n n ni i

iM w M M M w M M dM

= Δ ⇔ =∑ ∫

n=0 normalization condition

1 ( ) 1k

M w M M w M dM∞

= Δ = ⇔ = =∑ ∫

n=1 average value

( ) ( )

1 1 ( )k

M w M M M w M MdM∞

= Δ ⇔ =∑ ∫

( ) ( )

n n n ni i

iM w M M M w M M dM

= Δ ⇔ =∑ ∫

n=2 broadness

n=3 asymmetry

( ) ( )

2 2 2 2( )k

M w M M M w M M dM∞

= Δ ⇔ =∑ ∫

( ) ( )

3 3 3 3( )k

= Δ ⇔ =∑ ∫

Characterization of the distribution function using moments

Relation between the moments and experimental quantities

( ) ( )

n n n ni i

= Δ ⇔ =∑ ∫

number-average molar mass

( )( )

i i ki

n i iki

w M MMM w M MM w M

w M MdMMM w M MdMM

w M dM

Δ= = = Δ

∑∑

∫∫

Experiment:methods, which are proportionalto the number of molecules

colligativeproperties

vapor pressure osmosesmembrane osmoses

( ) ( )

n n n ni i

= Δ ⇔ =∑ ∫

mass-average molar mass

2 2(2)

1 1(1)

2 2(2)

0 0(1)

( ) ( )

i i i ii i

w M M w M MMMM Mw M M

w M M dM w M M dMMMM M

w M M dM

∞ ∞

Δ Δ= = =

∑ ∑

∫ ∫

Experiment:methods, which are proportionalto the mass of molecules

light scattering

( ) ( )

n n n ni i

= Δ ⇔ =∑ ∫

z-average molar mass 3 3(3)

1 1(2)

3 3(3)

0 0(2)

( ) ( )

i i i ii i

z kw 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= = =

∑ ∑

∫ ∫

Experiment:ultracentrifugeestimation via the distribution

( ) ( )

n n n ni i

= Δ ⇔ =∑ ∫

viscosity-average molar mass 1/ 1/( )

1/1/( )

0,5 0,9

a aa ka

MM w M MM

MM w M M dMM

⎛ ⎞ ⎛ ⎞= = Δ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞= = ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠≤ ≤

∫Experiment:rheology

Staudingerindex

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

polydispersity D

i i ki

i in i

w M Mw M M

M M M M

Ufor U monodispers polymer

= − ≤ ≤

⇒ ≥=

⇒ = = =

∑∑

uniformity U

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

M [g/mol]

0,097942878

800blue curve

0,035U854Mz

red curvequantity

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

GPC – gel permeation chromatographySize-Exclusion Chromatography (SEC)

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

M [g/mol]

symmetrical distribution

M average value

s standard derivation

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

segment numberhttp://nobelprize.org

Calculation of moments n=0 normalization condition

( ) exp exp( ) ( )

exp( )

( 1)exp

( 1( )

k kk 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 +− =

∫ ∫ ∫

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

k k kk k k k k+

Γ + Γ= = =

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

Γ = − ∈

Γ + = Γ

Distribution function fulfills the normalization condition.

Schulz-Flory-distribution

Calculation of moments n=1 average value

( ) ( )

2(1) 1

( ) exp exp( ) ( )

exp( )

( 1)exp exp( )

k kk k

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 += − − =

∫ ∫ ∫

∫ ∫ 1

(1)1 1 2 2 2

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

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

Γ + + Γ + + Γ + + Γ= = = =

Γ Γ Γ Γ+

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

Γ = − ∈

Γ + = Γ

calculation of moments n=2 broadness

( ) ( )

2(2) 2 2

(2) 2 2

2(2) 2

( ) exp exp( ) ( )

exp( )

(exp exp( )

k kk k

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= − − =

∫ ∫ ∫

∫ ∫ 1

2 2 2(2)

2 1 2 1 3

2 2 2(2)

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

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

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 + += = =

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

Γ = − ∈

Γ + = Γ

Calculation of moments n=2 asymmetry

( ) ( )

3(3) 3 3

(3) 3 3

3(3) 3

( ) exp exp( ) ( )

exp( )

(exp exp( )

k kk k

k r mM x kx dx textbook x axk

∞ ∞ ∞

∞ ∞+

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

= = = → = −Γ

Γ= − − =

∫ ∫ ∫

∫ ∫ 1

3 3 3(3)

3 1 3 1 4

3 3(3)

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

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

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 + + += =

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

Γ = − ∈

Γ + = Γ

Schulz-Flory-distributionCalculation of moments n=-1

1( 1) 1 1

( 1) 1 1

( 1) 1

( ) exp exp( ) ( )

exp( )

exp exp( )

k kk k

kM x kx dx textbook xk r

∞ ∞ ∞−− − −

∞− − −

∞ ∞− −

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

= = = → = −Γ

= −Γ

∫ ∫ ∫

∫ ∫ ( ) 1

( 1)1 1

( 1 1) ( ) 1( ) ( )

k k k kMk r k k r k r

−− +

Γ +− =

Γ − + Γ= =

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

Γ = − ∈

Γ + = Γ

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

calculation of experimental available average values number-average

( )( )

w M MdMMM w M MdMM

w M dM

∞= = =∫

∫∫

characterization variable: molar mass

characterization variable: segment number

1/( )n

w r drMr rM rw r dr

∞−= = = =∫

calculation of experimental available average values weight-average

w M M dMMMM

w M MdM

∞= =∫

( )( 1)( 1)

w r rdrr kM r kr

M k kw r dr

++= = = =

calculation of experimental available average values z-average

w M M dMMMM

w M M dM

∞= =∫

22(2) 2

( 2)( 1)( )( 2)

( 1)( )

r k kw r r drM r kkr r kM k

w r rdr k

= = = =+

Uniformity U

= nr r=

( )(1/ 1) 11/

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.

0 200 400 600 800 10000,0

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

5.3. Polydispersitygeneral log-normal distribution

ln( ) ln( )1( ) expr r

w rAAr π

⎛ ⎞−= −⎜ ⎟

⎜ ⎟⎝ ⎠

( )2 ln Aσ =

ln( ) ln( )1( ) expz

r rrw rr y AA π +

⎛ ⎞−= −⎜ ⎟

⎜ ⎟⎝ ⎠

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

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

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

w rr AAσ

⎛ ⎞−= −⎜ ⎟

⎜ ⎟⎝ ⎠

( )( )

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

r rrw rr yσ π σ+

⎛ ⎞−= −⎜ ⎟

⎜ ⎟⎝ ⎠

Wesslau-distribution

ln( ) ln( )1( ) expr r

w rAAr π

⎛ ⎞−= −⎜ ⎟

⎜ ⎟⎝ ⎠

( )( )( )

exp / 4

exp 3 / 4

suitable for high polydispersity (i.e. PE)

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

( ) 11 exp(1 )( )

Pn nP P

−− −=

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

degree of polymerization

Poisson distribution

0 200 400 600 800 1000 1200 14000

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

Lecture - thermodynamik.tu-berlin.de · 5. Thermodynamics of Pure Polymers Application of...

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