1

Overview of the lectures in polymer physics

Topics: The amorphous state The cr stalline state The crystalline state Thermal transitions and properties Mechanical properties Rubber elasticity Polymer blends and IPNs Polymer composites and nanocompositesy p p Polymer processing and rheology

Lecurer: Patric JannaschInstitute of Chemistry, Division of Polymer & Materials Chemistrypatric.jannasch@polymat.lth.se

Solid-State Properties

Different chain conformations in different phases

AmorphousDissordered, entangled state

Semi-crystallinePartly ordered, partly folded state

p. 153

PS PVC PEPMMA PP

Increasing crystallinity

2

The Amorphous State

• Randomly coiled interpenetrating chains

• Entanglements at sufficiently high molecular weight

• Not all volume is occupied: free volume concept

• Diffusion of small molecules

• Segmental and chain mobility strongly dependent on temperature and free volume

A h hi h l l b i th

p. 154

• Amorphous high molar mass polymers can be in the glassy state, rubbery state or melt state going from low to high temperature

spaghetti analogy

Chain Entanglements

• Sufficiently long molecules

• Critical molecular weight

• Depend on chain flexibility• Depend on chain flexibility

• Restrict flow

Mc critical molecular weight for formation of stable entanglements

Me molecular weight between entanglements

M ≈ 2M

p. 154

Mc ≈ 2Me

Flexible chains give high Mc, aromatic main chains give low Mc

The molecular weight of commercial polymers significantly above Mc

3

Reptation model in the melt state

How do polymers move in the entangled state?

• long range movements of chains

• snakelike motion within a virtual tube

f l b l

Theory of De Gennes

p. 156

• frictional resistance by entanglements

• successful in predicting viscous properties of entangled polymer melts

The Glass Transition

Different theories – isoviscous state (1012 Pa s)

- isofree volume state

- isoentropic state: conformational entropy goes to zerop py g

Free volume Vf is the difference between actual volume V and occupied volume V0.

Vf = V - V0

p. 156

The glass transition depends on- chain flexibility- interaction

4

Secondary-Relaxation Processes

Small scale molecular motions that occur in the glassy state:

- Limited rotations in the main chains

- Movements of side groups

Important for glassy state properties (impact strength, gas permeation)

5 bonds rotate aroundthe main chain

Example: crankshaft motion

p. 158

Summary

Temperaturelow highTemperature

Mobility and free volume

Glassy state ---- Glass rubber transition ---- Rubbery state ----- Melt state

low high

Small scale molecular motions - large scale segmental motions – polymer chain reptation

5

The Crystalline state

Many important polymers are partially crystalline

• Polyethylene

• Polypropylene

• Polyamides; Nylon 6, Nylon 6.6, Nylon 4.6

• Linear polyesters; PET, PBT

Crystallinity influences

• stiffness and brittleness

p. 158

stiffness and brittleness

• fracture strength and elongation at break

• solubility

• permeability of gases and water sorption

• many other properties

High thermal energy favours a large number of conformations

Lower-energy conformations are favoured during cooling

Ordering of polymer chains

Eventually the polymers are able to attain their lowest-energy conformation,often the extended chain or the planar zigzag (e.g., polyethylene, nylons)

folded polymer chain

polymer helix

The lowest-energy conformation of syndiotactic polymers,and polymers with large substituents, is usually a helix (e.g., polypropylene, polyisobutylene)

The packing of polymers in ordered structures is favouredby stereo regular symmetrical chain structures and specificinteractions (tacticity, trans configuration, hydrogen bonding)

p y

6

Crystalline structures

Polymer crystallisation by chain folding

crystalline lamellae

folded polymer chain

p. 158

Chain folding in lamellae

Three idealized models for chain folding in lamellae

A. Nonadjacent reentry

B. Regular adjacent reentry

p. 158

C. Irregular adjacent reentry

Lamellae thickness: 10-20 nm (in PE 40-80 repeating units)

7

Thermal Transitions

Hydrogen bonding between the

amide groups in nylon 6.6

p. 162

For many polymers Tg is one-half to two thirds of the melting temperature Tm ( in Kelvin )

Crystalline-Melting Temperature

Free energy of fusion per repeating unit:

Gu = Hu TSu

Tm0 = Hu/Su

Equilibrium melting temperature at Gu=0

(crystalline lamellae are destroyed as fast as they are formed)

Tm0 > Tm

p. 163

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Crystallization Kinetics

Linear growth rate of spherulites in PET

• Thermodynamic driving force below Tm, and Tg = 69 oC, Tm = 265 oC

y g m,necessary mobility above Tg

• Crystallisation possible between Tg and Tm

• Increasing viscosity at low temperatures

• Possible to quench polymers with slow crystallisation rates

Avrami equation: the

p. 164

Avrami equation: the fractional crystallinity is

= 1 – exp(-ktn)

k is a temperature dependent parameter

n varies between 1 and 4 depending on the nature of growth process

Techniques to Determine Crystallinity

Non-destructive

• Density measurements: fractional crystallinity

= ( – a)/(c - a) c from single crystals, a from quenched samples

• X-Ray diffraction (WAXS): weight fraction of crystalline phase

Wc = 1 – Iam/ Iam0

HDPE

p. 167

From thermal transitions

• Differential Scanning Calorimetry

• Dilatometry

Bragg peaks

Amorphous halo

9

Measurement Techniques - Dilatometry

bulb

capillaryPrinciple:

Specific volume as a function of temperature

increasing T

bulb

polymer sample

mercury

p. 173

function of temperature of a semicrystalline polymer

Change in thermal expansion coefficient at Tg:= l - g

Measurement Techniques - Differential Scanning Calorimetry (DSC)

S is sample and R is reference pan

Individual heaters to keep T=0 during a temperature scan

Principle

during a temperature scan. Difference in the required heat flow is measured.

DSC thermogram of PET

• glass transition near 75 oC

• recrystallization above 143 oC

p. 175

recrystallization above 143 C

• melting endotherm around 250 oC

• crystallinity = Q/Hf with Q the heat of fusion measured and Hf the heat of fusion at 100% crystallinity

10

Other Measurement Techniques

Many properties change drastically at the glass transition temperature and can be used to determine Tg such as:

Mechanical properties

Dielectric properties

Optical properties

load pressure

p. 177

Temperature at 0.25 mmdeflection = HDT

Structure-Property Relationships

Influence of flexibility of the polymer chain on the melting temperature for an analogous series of polyesters

p. 178

Tm0 = Hu/Su

Tm0 governed by Su. Flexible polymers have higher Su

11

Structure-Property Relationships

Influence of hindered chain rotation of the polymer chain on the glass-transition temperatures of selected vinyl polymers

inceasedpolarity

OO

OO

rod polymer

flexible polymer

p. 179

R

R'

R

R'

rod polymer

ladder polymer

Structure-Property Relationships

Effect of increasing size of the substituent groups on the glass-transition temperature of polymethacrylates

Increasing flexibilityof the side chain

p. 179

i-PMMA, Tg = 45 oCa-PMMA, Tg = 105 oCs-PMMA, Tg = 115 oC

Influence by tacticity

12

Effect of molecular weight on Tg

The glass transition temperature is dependent on the number average molecular weightaverage molecular weight

The effect levels off at high Mn

Fox-Flory equation:

Tg = Tg∞ – K/Mn in Kelvin

p. 180

The constant K is polymer-specific

The effect can be related to the free volume contribution of chain ends.

Effect of composition on Tg

The glass transition temperature of a homogeneous mixture is dependent on the amount of each component present, and their respective Trespective Tg.

Rule of mixtures: Tg = W1 Tg,1 + W2 Tg,2

Fox equation: 1/Tg = W1 / Tg 1 + W2 / Tg 2

p. 181

Fox equation 1/Tg W1 / Tg,1 W2 / Tg,2

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Mechanical Properties

How are polymers deformed? - mechanisms of deformation

• elastic, viscous, viscoelastic

• time dependent, frequency dependentAt small deformations:

At ’large’ • crazing

p. 183

At large deformations: • shear banding

• fracture

Crazing

Crazes consist of polymer microfibrils (0.6-30 nm in diameter) bridging two surfaces of a crack.

d l l Crazes develop at a certain critical strain

True cracks appear after degradation of crazes

direction ofdeformation

crazepropagation

p. 184

propagation

14

Crazing

p. 184

Crazes in a polycarbonate dogbone

Crazes in poly(phenylene oxide)

Shear banding

Shear banding: g

- occurs in some glassy amorphous polymers instead of, or together with, crazing.

- is the dominant mode of deformation of ductile polymers during tensile testing

- provides larger energy

p. 185

provides larger energy dissipation in, e.g., polycarbonate and SAN

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Stiffness and Strength

Methods of testingMethods of testing

In tension, shear or hydrostatic

• static: deformation rate is constant

p. 186

• transient: creep and stress relaxation

• impact: Izod and Charpy

• cyclic: fatigue

Static tensile testing

Dogbone sample

engineering (nominal) stress = F/A0

engineering (nominal) strain = L / L0

p. 187

Alternatively: true stress and true strain

true stress T = F/A actual cross section A

true strain T = ln (L/L0)

16

Static tensile testing

During uniaxial tensile deformation, glassy amorphous polymers increase in volume V.

V = V – V = ( 1-2 VV = V – V0 = ( 1-2 V0

• V0 is the initial (unstrained) volume

• is the true strain

• is Poisson’s ratio, defined as the ratio of true strain in transvers direction and true strain in longitudinal direction.

p. 187

= - trans / axial = - x / y

Poisson’s Ratio

Molecular origin:contacted

strain

= - trans / axial

relaxedextended

p. 188

For the majority of polymers ≈ 0.4

Incompressible polymers have = 0.5

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Static testing

Determination of materials properties in tension

Hooke’s law = Elinear elastic behaviour

= stress E = tensile modulus = strain= stress, E = tensile modulus, = strain

Alternatively: = D D = 1/E = tensile compliance

In reality:

effects of

p. 189

time, rate and

temperature

Modulus as a function of temperature

• Glassy modulus typically 1 GPa

• Rubbery plateau modulus typically 1 MPa1 MPa

• Entanglements responsible for rubbery plateau (physical crosslinks)

• Chemical crosslinks have the same effect

• The rubbery plateau modulus Ep is inversely proportional to the

l l b t

p. 190

molecular mass between entanglements Me

Ep proportional to RT/Me

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Materials properties in shear

Engineering (nominal) shear stress,

= F/A0

Shear strain,

= tan = X/C

p. 191

Hooke’s law: = GG = shear modulus

= JJ = 1/G = shear compliance

Stress-strain as a function of temperature

1. Brittle low temperature behaviour

2. Ductile behaviour with yield stress

”x” marks the stress and strain at failure

stress

3. Ductile behaviour with yield stress, necking, cold drawing and orientation hardening

4. Rubbery behaviour with strain-induced crystallisation

y

p. 193

crystallisation

19

Simulation of the development of a neck via shear bandingEffective plastic strain

Necking of ductile polymers

A polyethylenesample with astable neck

Mechanism for the deformation of a semicrystalline polymer

(a) Two adjacent chain folded lamellae and interlamellar

h

(e) Orientation of the block segments and tie chains with the tensile axis in the final deformation stage

amorphous material before deformation (d) Separation of the

crystalline block segments during the third stage

(b) Elongation of amorphous tie chains during the first stage of deformation

(c) Tilting of lamellar chain folds during the second stage

20

Mechanical properties of representative polymers

p. 194

Tensile response depends on: - polymer structure and architecture- MW and MW distribution- sample preparation- crystallinity- temperature- rate of deformation

Time dependent behaviour: creep

Creep : constant stress 0 and measuring the time dependent strain (t)

Result: compliance D(t) = (t)/0

I t t f l th t t t i l d f l i dImportant for polymers that must sustain loads for long periods

F0 F0

t’(0) (t’)

p. 196

= D0

21

Time dependent behaviour: stress relaxation

Stress relaxation : constant strain 0 and measuring the time dependent relaxing stress (t)

Result: modulus E(t) = (t) / 0 t´0

(t’)

0

(0)

(0)

p. 196

0t0

Stress and strain

• Static tensile testing: constant strain rateOb h h i i h h i

ddt = const.

F(t)

Observe how the stress varies with the strain

• Creep testing: constant stressObserve how the strain varies with time

F0

(t)

• Stress relaxation: constant strainObserve how the stress varies with time (t)0

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Relations between moduli and compliances

Modulus in tension E

Compliance in tension D

Modulus in shear G

Compliance in shear J

Bulk modulus in compression K

For isotropic materials two independent material properties

E = 2(1+)G J = 2(1+)D K = E/3(1-2)

p. 192

With Poisson’s ratio = 0.5

E = 3G J =3D K =infinite (incompressible)

Impact and fatigue testing

Measures energy expended up tofailure under conditions ofrapid loading

Fatigue testing

Impact strength critical inmany applications

Oscillative stress

p. 200

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Rubber elasticity

p. 249

Chain conformations and entropy

Which dynamic chain has the highest (conformational) entropy, S?

a)

b)

c) S = k lnW

d)

24

Rubber elasticity – the Gough-Joule effect

elongation

Unloaded coiled chains in anentropically favoureddissordered state

Stretched chains in a lessentropically favoured state

heating

A rubber band acts

Stretched chains with a moreentropically favoured state

like an entropy spring

Rubber elasticity - models

Elastic force = f = G0(-

= L/L0 = 1 +

G = shear modulus proportional to T and G0 = shear modulus, proportional to T and the crosslink density

- Good fit at low

- Overestimation at moderate because of deviation from Gaussian distribution

- Underestimation at high because of strain-induced crystallisation

p. 254

Fillers in rubbers:

Guth-Smallwood equation

Ef/E0 =1 + 2.5f + 14.1f2