Geoffrey Mitchell Use of nano enhancers to control ...

Use of nano enhancers to control morphology and hence properties

Geoffrey Mitchell March 2012

www. cdr-sp.ipleiria.pt

Acknowledgements •

Dr Donatella Duraccio, Marilena Pezutto Prof Clara Silvestre ICTP Pozzouli and EU Napolynet

•

Dr Supatra Wangsoub, Mr Robert Olley, Dr Fred Davis, Dr Peter Harris, Dr Laura Felisari, Dr Chris Stain – Centre for Advanced Microscopy, University of Reading Artur Mateus and Paulo Bartolo CDRSP Leiria Portugal Dr Janette Dexperte CNRS Tolouse, France Dr Evgeni Ivanov, Prof Rumiana Kotsilkova Bulgarian Academy of Sciences Dr Aurora Nogales Dr Daniel Lopez Garcia ICTP Madrid Spain Juris Bitenieks Riga Technical University Latvia

• • • • •

Beam-line Scientists Dr Sigrid Bernstoff (Elettra) , Dr Francois Fauth (ESRF), Dr Sergio Funari (Hasylab), Dr Jen Hiller and Dr Nick Terrill (Diamond), Dr Steve King, Dr Sarah Rogers, Dr Ann Terry, Dr ) Richard Heenan (ISIS), Dr Florian Meneau (Soleil) Funding

COST FA0904, EU – Napolynet , EU – ELISA programme, EU Marie Curie, EPSRC, STFC

2


Leiria


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Nanoparticles

• Plate-like • Rod-like – Carbon Nanotubes – Halloysite (clay) – Self-assembling fibrils

• Sphere-like

– Montmorillonite (clay) – Graphene platelets

– ZnO

Food Packaging External Atmosphere

Food item

Internal Atmosphere

• Barrier material – Block the interior/exterior atmosphere from the exterior/interior atmosphere – transparent

Food Packaging

External Atmosphere

Internal Atmosphere

Nanoparticles as barriers

Bayer – Polyamide plus silicate nanparticles from Nanocor

• Need Platelets • Need preferred orientation 9

Nanoparticles

Polymer Matrices • Polyethylenes, Polypropylenes, Poly(εcaprolactone) • All semi-crystalline polymers – properties entirely dependent on the presence of the crystals and the arrangements • Yes – there are real crystals as evidenced by sharp peaks in the wide-angle x–ray scattering (WAXS)

• It is the development of lamellar chain folded crystals which is the dominant feature, length scale 10-30nm (SAXS)

Polymer Matrices • Polyethylenes, Polypropylenes, Poly(εcaprolactone) • All semi-crystalline polymers – properties entirely dependent on the presence of the crystals and the arrangements • Yes – there are real crystals as evidenced by sharp peaks in the wide-angle x–ray scattering (WAXS) • It is the development of lamellar chain folded crystals which is the dominant feature, length scale 10-30nm (SAXS) • Polymer crystals also provide platelets • Need preferred orientation

Nanoparticles + Polymers? • Nanoparticles + Polymers – what could happen? – Lets consider we have overcome the challenges of dispersion and inhibiting aggregation

• A: Nothing , the nanoparticles and polymer matrix are unnconnected • B: The polymer matrix influences the nanoparticle as a consequence of forces exerted during flow • C: The nanoparticles influences the polymer matrix by altering or directing the organisation of the polymer chains

How to observe? • Nanoparticles – use transmission electron microscopy and FEG scanning electron microscopy – Need to consider both starting nanoparticulates and the nanoparticles within the matrix when we are assembling or preparing the particles in-situ

• Nanocomposites but will need to reveal the nanoparticles and the polymer matrix by differential etching – use FEG Scanning electron microscopy • The reorganisation processes – very help to follow in real-time – largely excludes microscopy – use scattering techniques

Nanoparticles Halloysite nanotubes a)

Modified Montmorillonite in isotactic polypropylene

Graphene flakes

Polyethylene + Clay platelets

Scanning electron microscope of etched surfaces of a polyethylene blend with 10% of MMT There appears to be little interaction between the MMT platelets and the polymer matrix Mitchell, Green, Vaughan and Liu

16


Scanning electron microscope of etched surfaces of a polyethylene blend with 10% of MMT There appears to be little interaction between the MMT platelets and the polymer matrix

Mitchell, Green, Vaughan and Liu

• A: Nothing , the nanoparticles and polymer matrix are unnconnected 17

Scattering Structure and Scattering

| Q | = 4 π sin(

|Q | 2 θ*

Å

< 0.1 < 2 °

-1

Incident beam

θ )/ λ

~1 Å -1

>2 Å

~ 15 °

~

Morphology

Intersegment

Lamellar thickness

Correlations,

inanoparticles

Crystal planes

Chain

2 θ

-1

>8 Å

30 °

>

-1

160 °

conformations, Bonds

Crystal planes

C C

Small Angle Scattering * calculated using CuK radiation

Wide-Angle Scattering

C

Clay +iPP

Clay layers show exfoliation and anisotropy

iPP Scattering Measured using Couette geometry

SAXS/WAXS

WAXS

SAXS

BEAM Simultaneous SAXS/WAXS BM16 at ESRF in Grenoble Time-resolving 1 to 10s time slices

SAXS/WAXS Simultaneous SAXS/WAXS SAXS Detector

BM16 at ESRF in Grenoble Time-resolving 1 to 10s time slices

In-situ SAXS/WAXS with shear flow

Experimental System y

Velocity gradient ∇v

d

v x Flow direction

z

Vorticity vector

Parallel plate arrangement means we sample a range of shear rates

Shear rate =v/d (s-1) Shear strain = shear rate x time

Experimental System


Experimental System


Shear Flow

y

Remember flow is 3d

• Shear Flow Velocity gradient ∇v d

v x Flow direction

z

Vorticity vector



SAXS - Basic Geometry

• •

The structure probed depends on the geometry of scattering In the parallel plate shear cell the scattering vector Q probes the structure in the plane containing the flow direction and the vorticity vector

Q

Flow direction

θ

α

2θ Incident beam

Sample Scattered beams


SAXS

Shear flow has a 3-d nature

How to ‘look’ in this direction

This is how!

This system probes the plane containing the flow direction and the velocity gradient

Temperature

Shear Flow

Shear /Temperature Profile

Time

29

Shear rate =v/d (s-1) Shear strain = shear rate x time Shear rate 10s-1 Shear strain 1000su

Tcryst

Directed Crystallisation

j05

π / 2 Qmax

Ω=

∫ ∫

2

Q I ( Q , α ) sin α dQdα

0 Q =0

Derivative gives rate of crystallisation cf DSC

30

Influence of shearing temperature Ts

Polymers will crystallise Chains extended in the flow field Extended chains act as row nuclei for matrix melt The variation simply relates to whether this row nuclei remain when crystallisation is initiated The formation of the row nuclei is directly related to the molecular architecture The relaxation rate of the row nuclei is directly related to the molecular architecture

Polymer Crystallisation

• Shear flow in the melt above the equilibrium melting point yields a barely anisotropic melt with ~ 0.01 but the subsequent crystals have a very high level of preferred orientation

Directing the Crystallisation of Polymers


Anisotropic crystal texture

• What is the origin of anistropy?

Flow Directing the Crystallisation of Polymers

Flow generates a number of nucleation sites with a common axis Level of anisotropy depends on competition of growth between anisotropic and isotropic sites www. cdr-sp.ipleiria.pt

Polypropylene and modified MMT

Memory of shear retained

Temperature too high memory lost

Chains extended in the flow field, Extended chains act as row nuclei for matrix melt. The variation simply relates to whether this row nuclei remain when crystallisation is initiated Here the addition of the clay platelets has stabilised the memory of the shear flow by ~ 5°C

Dibenzylidene Sorbitol O O

O

•Use Dibenzylidene Sorbitol as a self-assembing fibril to template crystallisation •1% DBS to copolymer (CPP) of PP/PE Mw 2.2 x105 •1% DBS to linear Polyethylene (PE) Mw 1.1. X105 •1% DBS to Polycaprolactone (PCL) Mw 8 X104

O

OH OH

Dibenzylidene Sorbitol (DBS)

Results PCL/DBS


Results PCL/DBS


Phase behaviour for different nanoparticle forming systems •Phase behaviour dependent on chemical configuration of additive • Para, meta and ortho substitution gives different behaviour

Shear Flow

Templating •The high level of preferred orientation of the fibrils means we can consider this using a 2-d model rf Templated zone

rt

Fibril

Templating •The high level of preferred orientation of the fibrils means we can consider this using a 2-d model

N=

rf rt

f= fraction of additive fc=solubility limit of additive

( f − fc )

πr

2 f

2

rt ft = ( f − fc ) 2 rf < P2 >= 1.0 f t +(1 − f t )0.0 = f t

Templating •The high level of preferred orientation of the fibrils means we can consider this using a 2-d model rf rt

rt 2 ft = ( f − fc ) 2 rf Full templating reached with ~ 3%DBS Solubility of DBS in PCL at 80ºC ~ 1% Radius of DBS fibril 140Å


Gives Template zone radius ~ 990Å

Dispersion Dispersion of the nanoparticles aided by in-situ formation

Sample disc

43

SEM micrograph of an etched PE sample with 1% w/w DBS Showing distribution of sites of DBS


44

iPP

α form

iPP (HCPP1) Ts=210°C (J14)

iPP + β nucleator Ts = 210C

Shear Flow Decreasing temperature iPPN+ (0.3%) N'-Dicyclohexyl-2,6-Naphthalene dicarboxamide Ts=210C (j05)


Shear Flow iPPN+ N'-Dicyclohexyl-2,6-Naphthalene dicarboxamide Ts=210C (J05)


Shear Flow B: the nanoparticles aligned by polymer matrix iPPN+ N'-Dicyclohexyl-2,6-Naphthalene dicarboxamide Ts=210C (J05) • C: Polymer crystals templated by nanoparticles •


Shear Flow Decreasing temperature iPPN+ N'-Dicyclohexyl-2,6-Naphthalene dicarboxamide Ts=170C m04


Shear Flow iPPN+ N'-Dicyclohexyl-2,6-Naphthalene dicarboxamide Ts=170C m04


Shear Flow •

B: the

nanoparticles aligned by polymer matrix

•iPPN+ C: but polymer crystals templated by polymer based row nuclei N'-Dicyclohexyl-2,6-Naphthalene dicarboxamide Ts=170C m04

Directed Crystallisation •Self – assembling nanoparticles • Dibenzylidene sorbitol (DBS) and many derivatives • Polycaprolactone (PCL) Mw 8 X104 •Carbon Nanotubes •1% SW CNT •Polycaprolactone (PCL) Mw 8 X104

52

Small-angle X-ray Scattering

No shear

Shear (shear rate 10/s 1000su


No shear •

B: the




Shear Flow 3% Cl-DBS in PCL.

1% CNT in PCL.


Shear Flow •

C: Polymer crystals templated by nanoparticles 3% Cl-DBS in PCL.

1% CNT in PCL.

Looking in the right direction

Shear flow has a 3-d nature This is how! How to ‘look’ in this direction

Graphene Flakes +PCL

Flow

A

Flow

Vorticity Vector

‘Standard view parallel plates

B

∇V

Couette cell 0.5% Graphene Flakes in PCL

Flow

A

Vorticity Vector

Flow

B

∇V

•

B: the


•

C: Polymer crystals templated by nanoparticles

Flow

∇V

0.1% 12nm Graphene Flakes

Flow

∇V


Flow

∇V


Flow

∇V


Flow

∇V


Flow

∇V


Flow

∇V


Graphene Flakes/PCL

67

Graphene Flakes/PCL

Ideally crystals parallel to graphene flake

68

Rods and Plates and Shear Flow

• Plate-like • Rod-like – – – –

Align parallel to the flow field Carbon Nanotubes Halloysite (clay) Self-assembling fibrils

– Parallel to flow field and normal to the velocity gradient – Clay and graphene

Rods and Plates and Shear Flow •

Rod-like – Align parallel to the flow field – Carbon Nanotubes – Halloysite (clay) – Self-assembling fibrils

But what drives the directing of crystallisation? Is just a surface? Is there atomistic epitaxy?

•

Plate-like – Parallel to flow field and normal to the velocity gradient – Clay and graphene

Summary We have observed all three possibilities: A: Nothing , the nanoparticles and polymer matrix are unnconnected B: The polymer matrix influences the nanoparticle as a consequence of forces exerted during flow C: The nanoparticles influences the polymer matrix by altering or directing the organisation of the polymer chains •The alignment of the nanoparticles is related to the shape •The aligned nanoparticles may template the crystallisation in competition to other nuclei , ie polymer row nucli •The balance of these competing effects determines the outcome.

Summary

• Nanoparticles can be aligned in modest flow fields • Nanoparticles may template the crystallisation • The type of nanoparticle dictates the alignment of both nanoparticle and polymer crystals

72


Clay +iPP










•


Summary

We have observed all three possibilities: A: Nothing , the nanoparticles and polymer matrix are unnconnected B: The polymer matrix influences the nanoparticle as a consequence of forces exerted during flow C: The nanoparticles influences the polymer matrix by altering or directing the organisation of the polymer chains

•The alignment of the nanoparticles is related to the shape •The aligned nanoparticles may template the crystallisation in competition to other nuclei , ie polymer row nucli •The balance of these competing effects determines the outcome.


Clay +iPP




rt

Fibril


N =

rf rt


( f − fc )

πr

2 f

2


Wide angle X-ray Scattering

•WAXS shows aligned objects are crystalline • Temperature variation shows that these crystal peaks disappears at the liquidus temperature identified by SANS and SAXS • Comparison of averaged intensities shows that no crystals are generated or destroyed

82


83

Phase diagram generated by SANS, SAXS and DSC `Data

PCL/DBS

Shear induced inhibition of nucleation

π / 2 Qmax

Ω=

∫∫

2

Q I ( Q,α)sinαdQdα

0 Q=0



Shear supressed nucleation Modest shear flow initiated in the single phase region inhibits nucleation and growth of the DBS crystals Cessation of shear flow leads to growth of crystals






Acknowledgements • •

89

The neutron scattering based measurements were performed at – ISIS (LOQ , SANS2D and NIMROD) – Funding EPSRC and CCLRC The synchrotron-based x-ray scattering measurements were made at the – Daresbury SRS (16.1, 2.1) – Funding EPSRC and CCLRC – Diamond (I22) – Funding EPSRC and CCLRC – Hasylab (A2) - European Community's Seventh Framework Programme (FP7/20072013) under grant agreement n° 226716." – Elettra (SAXS) – Funding European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement nº 226716”. – ESRF (BM16) – Funding ESRF – SOLEIL Funding European Community’s Seventh Framework Programme (FP7/20072013) under grant agreement n° 226716


Structure from SAXS Preferred Orientation Structural Scale Invariant

Shear Flow

π / 2 Qmax

Ω=

∫ ∫

2


0 Q =0

From Diamond I22

Q scattering vector |Q| = 4πsinθ/λ α - angle between scattering vector and the flow axis


β form

iPPN+ N'-Dicyclohexyl-2,6-Naphthalene dicarboxamide Ts=210C (J05)

Controlling the 3-d Structure in Polymer Nanocomposites Geoffrey Mitchell

www.cdr-sp.ipleiria.pt

Physical Aspects of Polymer Science September 2011

www.reading.ac.uk/cfam




•

Dr Supatra Wangsoub, Mr Robert Olley, Dr Fred Davis, Dr Peter Harris, Dr Laura Felisari, Dr Chris Stain – Reading • Artur Mateus and Paulo Bartolo CDRSP Leiria Portugal • Dr Janette Dexperte CNRS Tolouse, France • Dr Evgeni Ivanov, Prof Rumiana Kotsilkova Bulgarian Academy of Sciences • Dr Aurora Nogales Dr Daniel Lopez Garcia ICTP Madrid Spain • Juris Bitenieks Riga Technical University Latvia Beam-line Scientists • Dr Sigrid Bernstoff (Elettra) , Dr Francois Fauth (ESRF), Dr Sergio Funari (Hasylab), Dr Jen Hiller and Dr Nick Terrill (Diamond), Dr Steve King, Dr Sarah Rogers, Dr Ann Terry, Dr Richard Heenan (ISIS), Dr Florian Meneau (Soleil) Funding EU – Napolynet , EU Marie Curie, EPSRC, CCLRC,

93


Nanoparticles


• Sphere-like

– Montmorillonite (clay) – Graphene platelets There are many others

– ZnO

Nanoparticles •

Safe – –

Eg Montmorillonite already used in pharmaceutical products Self-assembling nano fibrils approved for food use

•

Anti-bacterial

•

Property Enhancement and Property Change

– – –

•

Eg Montmorillonite, ZnO Clay platelets decrease gas permeation due to tortuous gas path Carbon Nanotubes and Graphene Platelets increase conductivity

Modify Polymer Matrix – –

Act as nucleant for formation of crystals Direct crystallisation

Polymer Matrices •

Polyethylenes, Polypropylenes, Poly(ε-caprolactone)

•

All semi-crystalline polymers properties entirely dependent on the presence of the crystals and the arrangements Yes – there are real crystals as evidenced by sharp peaks in the wide-angle x–ray scattering (WAXS)

•

•

–

It is the development of lamellar chain folded crystals which is the dominant feature, length scale 10-30nm

Nanoparticles + Polymers? •

Nanoparticles + Polymers – what could happen? –

• • •

Lets consider we have overcome the challenges of dispersion and inhibiting aggregation

A: Nothing , the nanoparticles and polymer matrix are unnconnected B: The polymer matrix influences the nanoparticle as a consequence of forces exerted during flow C: The nanoparticles influences the polymer matrix by altering or directing the organisation of the polymer chains

How to observe? •

Nanoparticles – use transmission electron microscopy and FEG scanning electron microscopy –

• •

Need to consider both starting nanoparticulates and the nanoparticles within the matrix when we are assembling or preparing the particles in-situ

Nanocomposites but will need to reveal the nanoparticles and the polymer matrix by differential etching – use FEG Scanning electron microscopy The reorganisation processes – very help to follow in real-time – largely excludes microscopy – use scattering techniques



Graphene flakes



100



• A: Nothing , the nanoparticles and polymer matrix are unnconnected


101

Scattering Structure and Scattering |Q | = 4 π sin( θ )/ λ

|Q |

Å -1

< 0.1

Incident beam

~1 Å -1

>2 Å -1

< 2 °

~ 15 °

~ 30 °

Morphology Lamellar thickness inanoparticles

Intersegment Correlations, Crystal planes

2 θ*

2 θ

>8 Å -1 >

Chain conformations, Crystal planes

160 ° Bonds

C C



C

SAXS/WAXS

WAXS

SAXS



Shear Flow

y

Remember flow is 3d


v x Flow direction

z

Vorticity vector




• •


Q

Flow direction

θ

α

2θ Incident beam



SAXS



This is how!


ZnO +iPP

iPP Scattering

Scattering dominated by ZnO particles (high atomic number) Nanoparticles can act as a nucleant

ZnO Scattering

Temperature

Shear Flow


Time

109


Tcryst


j05

π / 2 Qmax

Ω=

∫ ∫

2


0 Q =0


110







iPP

α form













Shear Flow •

B: the




120


No shear



No shear •

B: the





1% CNT in PCL.


Shear Flow •


1% CNT in PCL.




Flow

A

Flow

Vorticity Vector


B

∇V


Flow

A

Vorticity Vector

Flow

B

∇V

•

B: the


•


Flow

A

Vorticity Vector

Flow

B

∇V

•

B: the


•



rt

Fibril


N =

rf rt


( f − fc )

πr

2 f

2




131


132


PCL/DBS


π / 2 Qmax

Ω=

∫∫

2


0 Q=0










138



Structure from SAXS Preferred Orientation Structural Scale Invariant

Shear Flow

π / 2 Qmax

Ω=

∫ ∫

2


0 Q =0

From Diamond I22

Q scattering vector |Q| = 4πsinθ/λ α - angle between scattering vector and the flow axis


β form

iPPN+ N'-Dicyclohexyl-2,6-Naphthalene dicarboxamide Ts=210C (J05)

Controlling the 3-d Structure in Polymer Nanocomposites Geoffrey Mitchell

www.cdr-sp.ipleiria.pt

Physical Aspects of Polymer Science September 2011

www.reading.ac.uk/cfam




•

Dr Supatra Wangsoub, Mr Robert Olley, Dr Fred Davis, Dr Peter Harris, Dr Laura Felisari, Dr Chris Stain – Reading • Artur Mateus and Paulo Bartolo CDRSP Leiria Portugal • Dr Janette Dexperte CNRS Tolouse, France • Dr Evgeni Ivanov, Prof Rumiana Kotsilkova Bulgarian Academy of Sciences • Dr Aurora Nogales Dr Daniel Lopez Garcia ICTP Madrid Spain • Juris Bitenieks Riga Technical University Latvia Beam-line Scientists • Dr Sigrid Bernstoff (Elettra) , Dr Francois Fauth (ESRF), Dr Sergio Funari (Hasylab), Dr Jen Hiller and Dr Nick Terrill (Diamond), Dr Steve King, Dr Sarah Rogers, Dr Ann Terry, Dr Richard Heenan (ISIS), Dr Florian Meneau (Soleil) Funding EU – Napolynet , EU Marie Curie, EPSRC, CCLRC,

142


Nanoparticles


• Sphere-like

– Montmorillonite (clay) – Graphene platelets There are many others

– ZnO

Nanoparticles •

Safe – –

Eg Montmorillonite already used in pharmaceutical products Self-assembling nano fibrils approved for food use

•

Anti-bacterial

•

Property Enhancement and Property Change

– – –

•

Eg Montmorillonite, ZnO Clay platelets decrease gas permeation due to tortuous gas path Carbon Nanotubes and Graphene Platelets increase conductivity

Modify Polymer Matrix – –

Act as nucleant for formation of crystals Direct crystallisation

Polymer Matrices •

Polyethylenes, Polypropylenes, Poly(ε-caprolactone)

•

All semi-crystalline polymers properties entirely dependent on the presence of the crystals and the arrangements Yes – there are real crystals as evidenced by sharp peaks in the wide-angle x–ray scattering (WAXS)

•

•

–

It is the development of lamellar chain folded crystals which is the dominant feature, length scale 10-30nm

Nanoparticles + Polymers? •

Nanoparticles + Polymers – what could happen? –

• • •

Lets consider we have overcome the challenges of dispersion and inhibiting aggregation

A: Nothing , the nanoparticles and polymer matrix are unnconnected B: The polymer matrix influences the nanoparticle as a consequence of forces exerted during flow C: The nanoparticles influences the polymer matrix by altering or directing the organisation of the polymer chains

How to observe? •

Nanoparticles – use transmission electron microscopy and FEG scanning electron microscopy –

• •

Need to consider both starting nanoparticulates and the nanoparticles within the matrix when we are assembling or preparing the particles in-situ

Nanocomposites but will need to reveal the nanoparticles and the polymer matrix by differential etching – use FEG Scanning electron microscopy The reorganisation processes – very help to follow in real-time – largely excludes microscopy – use scattering techniques



Graphene flakes



149



• A: Nothing , the nanoparticles and polymer matrix are unnconnected


150

Scattering Structure and Scattering |Q | = 4 π sin( θ )/ λ

|Q |

Å -1

< 0.1

Incident beam

~1 Å -1

>2 Å -1

< 2 °

~ 15 °

~ 30 °

Morphology Lamellar thickness inanoparticles

Intersegment Correlations, Crystal planes

2 θ*

2 θ

>8 Å -1 >

Chain conformations, Crystal planes

160 ° Bonds

C C



C

SAXS/WAXS

WAXS

SAXS



Shear Flow

y

Remember flow is 3d


v x Flow direction

z

Vorticity vector




• •


Q

Flow direction

θ

α

2θ Incident beam



SAXS



This is how!


ZnO +iPP

iPP Scattering

Scattering dominated by ZnO particles (high atomic number) Nanoparticles can act as a nucleant

ZnO Scattering

Temperature

Shear Flow


Time

158


Tcryst


j05

π / 2 Qmax

Ω=

∫ ∫

2


0 Q =0


159







iPP

α form













Shear Flow •

B: the




169


No shear



No shear •

B: the





1% CNT in PCL.


Shear Flow •


1% CNT in PCL.




Flow

A

Flow

Vorticity Vector


B

∇V


Flow

A

Vorticity Vector

Flow

B

∇V

•

B: the


•









•


Summary

We have observed all three possibilities: A: Nothing , the nanoparticles and polymer matrix are unnconnected B: The polymer matrix influences the nanoparticle as a consequence of forces exerted during flow C: The nanoparticles influences the polymer matrix by altering or directing the organisation of the polymer chains

•The alignment of the nanoparticles is related to the shape •The aligned nanoparticles may template the crystallisation in competition to other nuclei , ie polymer row nucli •The balance of these competing effects determines the outcome.


Clay +iPP




rt

Fibril


N =

rf rt


( f − fc )

πr

2 f

2




184


185


PCL/DBS


π / 2 Qmax

Ω=

∫∫

2


0 Q=0










191

