Modelling of liquid crystalline polymer...

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Course MP10 – Lecture 3 25/2/2005 1 Dr James Elliott Modelling of liquid crystalline polymer processing Directors, disclinations and fingerprints Course MP10 – Lecture 3 25/2/2005 3.1.1 Liquid crystals These are materials containing molecules with rigid, bulky groups which behave more or less like molecular ‘rods’. These rods can form crystals with long range orientational or positional order. [from Chaikin & Lubensky (CUP, 1995)] The orientation of each molecule is represented by a molecular director which lies parallel to the molecular long axis. Areas of coherently aligned directors give rise to phase contrast due to the anisotropic optical properties of the molecule.

Transcript of Modelling of liquid crystalline polymer...

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Course MP10 – Lecture 325/2/2005

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Dr James Elliott

Modelling of liquid crystalline polymer processing

Directors, disclinations and fingerprints

Course MP10 – Lecture 3

25/2/2005

3.1.1 Liquid crystals

These are materials containing molecules with rigid, bulky groups which behave more or less like molecular ‘rods’. These rods can form crystals with long range orientational or positional order.

[from Chaikin & Lubensky (CUP, 1995)]

The orientation of each molecule is represented by a molecular director which lies parallel to the molecular long axis.

Areas of coherently aligned directors give rise to phase contrast due to the anisotropic optical properties of the molecule.

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3.1.2 Liquid crystals

Isotropic-nematic transition

In particular, the rods canorient to form a nematicphase in which there is netorientational order about abulk director n, but no netpositional order.

Other ordered phases arecholesteric and smectic.

[from Chaikin & Lubensky (CUP, 1995)]

3.2.1 Liquid crystalline polymers

Polymers containing rigid units also display liquid crystalline order, and are referred to as liquid crystalline polymers (LCPs)LCPs show a wide range of ordered mesophases which are very similar to those of small molecule liquid crystals.

[from Donald & Windle (CUP, 1992)]

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3.2.2 Liquid crystalline polymers

For LCPs in the nematic phase, thread-like textures (νηµα)are observed using phase contrast light microscopy.

[from Donald & Windle (CUP, 1992)]

3.2.3 Liquid crystalline polymers

The threads are due to discontinuities in the director field of the polymer. The nature of these discontinuities can be seen more clearly under polarised light.

Between crossed polarisers,a so-called Schlieren textureis observed, in which the darkareas occur when one of theprincipal optical axes isparallel to the polariser oranalyser. Note the pointsingularities (A, B).[from Donald &

Windle (CUP, 1992)]

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3.2.4 Liquid crystalline polymers

There are three main classes of distortion in the director field: splay, twist and bend.

Non-deformed splay twist bendspecimen distortion distortion distortion

3.2.5 Liquid crystalline polymers

( )2splay divn∝E

( )2twist curlnn ⋅∝E

( )2bend curlnn×∝E

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3.2.6 Liquid crystalline polymers

The distortion energies are combined to form the Frank free energy density of the director field:

For LCPs, the magnitudes of the elastic constants are usually in the ratio k11 > k33 > k22, i.e. twist distortions are most favoured and splay distortions are least favoured.This is due to the chain connectivity of the liquid crystal units, which strongly disfavours any divergence of the director field.

( ) ( ) ( )[ ]233

222

211 curlcurldiv

21 nnnnn ×+⋅+= kkkF

3.3.1 Disclinations

These correspond to the point singularities in the director fields observed in the Schlieren patterns earlier, and are analogous to dislocations in crystals with positional order.They can be classified according to their strength and character.The disclination strength is the number of rotations undergone by the director around a closed loop centered on (but avoiding) the defect core.The disclination character is given by the relative amounts of splay, bend and twist distortions which it contains.For elastically isotropic systems (i.e. k11 = k22 = k33), the disclination energy is proportional to the square of the strength, and independent of the character.

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3.3.2 Disclinations

Strength s = ±1 topological defects.

Pure splay

Pure bend

Mixed character

Mixed character

3.3.3 Disclinations

Strength s = ±½ topological defects.

You often see these on fingerprints (whorls). Try adding up the total strength of the disclinations on one finger! What do you find?

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3.3.4 Disclinations

The character of the disclinations changes in systems with strong elastic anisotropy, such as LCPs.

This is due to the spontaneous minimisation of strain energy in the structure. The defect microstructure evolves in time to a state of where its free energy is a minimum.

Archway Sunrise

k11 > k33 k11 = k33 k11 < k33

s = +½ defect

3.4.1 The lattice director model

Can we model the LCP microstructure, in order to control the degree of orientation and defect structure?Employ a lattice director model, in which the director field is spatially coarse-grained onto a multi-dimensional grid.

The aims are to reveal the microstructure in a bulk sample, understand how this evolves during processing and how the microstructure affects the properties of the final product.

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3.4.2 The lattice director model

We faced with a choice of using either a deterministic or Monte Carlo relaxation algorithm. Either can be used, but the deterministic method has a number of advantages:

– Each point in the trajectory is related to adjacent points in time, which gives rigorous dynamical information as well as the equilibrium microstructure.

– The treatment of external fields, in particular the flow field, is more straightforward.

A tensorial form for the director field is used, which has nematic symmetry, i.e.

( )( )jiij nnnn −−= nn −→is invariant under

3.4.3 The lattice director model

The algebraic details of deterministic model are beyond the scope of this lecture, but are contained (for the two dimensional case) in e.g. Gruhn and Hess, ZeitschriftNaturforsch, 51a, 1 (1996).The principle is to minimise the curvature elastic torque obtained from the texture field h:

where F is the Frank free energy density.The governing equations of motion, including external fields, are called the Ericksen-Leslie equations.

nh

dd F

−=

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3.5.1 Texture coarsening in the lattice model

0 300 600

1200 1800 2400

3.5.2 Texture coarsening in the lattice model

3000 3600 4200

4800 5400 6000

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3.6 Effects of splay-bend anisotropy

Free of Elastic Anisotropy

SplayDominate

BendDominate

Initial Pattern

3.7 Modelling of simple shear forces

Recall that simple shear has both rotational and extensional components.Therefore, for an LCP melt in a shear field we expect the molecules to both rotate and extend. The relative magnitude of the rotational and extensional components of the field is represented by the parameter λ.

– High λFlow is predominantly extensional. There is alignment of the molecules at an angle to the domain walls (the Leslie angle).

– Low λFlow is predominantly rotational. The molecules tumble freely inthe shear field.

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3.8 Effects of simple shear on LCPs

Velocity gradient

Vorticity

Velocity

Flow Aligning Log-rolling Tumbling (λ≥1.0) (λ<1.0)

3.8.1 Disclination interaction in simple shear

04

7 12

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3.8.2 Disclination interaction in simple shear

15 16

1718

3.8.3 Disclination interaction in simple shear

27 32

3433

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3.8.4 Disclination interaction in simple shear

35 36

37 50

3.9 Impact of modelling results on processing

The most significant effects of liquid crystalline polymer flow behaviour are:

– Lower viscosity as a result of liquid crystallinity, especially at high strain rates due to log-rolling and tumbling.

– Very efficient molecular orientation in extensional strain fields.

These affect injection and extrusion processes:– Strong shear thinning leads to plug flow characteristics (see

lecture 1).– LCPs give very low die swell (see lecture 2) as the molecular

alignment is much more uniform than for normal polymers.– LCPs also give very good mould filling profiles for the same

reason, although there are problems when two mould fronts meet to form a weld line.

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Lecture 3 summary

In this lecture, we started by discussing liquid crystals and liquid crystalline polymers, and the fact that they form highly oriented microstructures. We saw that there were 3 distinct types of distortion in these structures: splay, bend and twist.For LCPs, the splay energy is considerably higher than the other two, and the defective nematic microstructures which evolve from the high temperature isotropic phase show this clearly.Finally, we saw how a lattice director model can be used to predict the observed microstructures, and understand how processing conditions affect the melt flow properties.