Important Points from Last Lecture• The Peclet number, Pe, describes the competition between particle

disordering because of Brownian diffusion and particle ordering under a shear stress.

• At high Pe (high shear strain rate), the particles are more ordered; shear thinning behaviour occurs and decreases.

• van der Waals’ energy acting between a colloidal particle and a semi- slab (or another particle) can be calculated by summing up the intermolecular energy between the constituent molecules.

• Macroscopic interactions can be related to the molecular level. Forces are found by differentiating the interaction energy with respect to the distance of separation.

• The Hamaker constant, A, contains information about molecular density () and the strength of intermolecular interactions (via the van der Waals’ constant, C): A = 22C

PH3-SM (PHY3032)

Soft MatterLecture 8

Introductions to Polymers and Semi-Crystalline Polymers

29 November, 2011

See Jones’ Soft Condensed Matter, Chapt. 5 & 8

Definition of PolymersPolymers are giant molecules that consist of many repeating units. The molar mass (molecular weight) of a molecule, M, equals moN, where mo is the the molar mass of a repeat unit and N is the number of units.

Synthetic polymers never have the same value of N for all of its constituent molecules, but there is a Gaussian distribution of N.

Polymers can be synthetic (such as poly(styrene) or poly(ethylene)) or natural (such as starch (repeat units of amylose) or proteins (repeat unit of amino acids)).

Synthetic polymers are created through chemical reactions between smaller molecules, called “monomers”.

The average N (or M) has a huge influence on mechanical properties of polymers.

Examples of Repeat Units

Molecular Weight Distributions

In both cases: the number average molecular weight, Mn = 10,000

M M

Fraction of molecules

Molecular Weight of Polymers

The molecular weight can be defined by a number average that depends on the number of molecules, ni, having a mass of Mi:

The polydispersity index describes the width of the distribution. In all cases:

MW/MN > 1

The molecular weight can also be defined by a weight average that depends on the weight fraction, wi, of each type of molecule with a mass of Mi:

ii

iiii Mn

MnMw

2

==MW

MN i

ii

nMn

== Total mass divided by the total number of molecules

Types of Copolymer MoleculesWithin a single molecule, there can be “permanent order/disorder” in copolymers consisting of two or more different repeat units.

Diblock

Alternating

Random orStatistical

Can also be multi (>2) block.

Semi-Crystalline Polymers

• It is nearly impossible for a polymer to be 100% crystalline.

• Typically, the level of crystallinity is in the range from 20 to 60%.

• The chains surrounding polymer crystals can be in the glassy state, e.g. in poly(ethylene terephthalate) or PET

• The chains can be at a temperature above their glass transition temperature and be “rubbery”, e.g. in poly(ethylene) or PE

• The density of a polymer crystal is greater than the density of a polymer glass.

15 m x 15 m

Poly(ethylene) crystal

5 m x 5 m

Polymer crystals can grow up to millimeters in size.

Crystals of poly(ethylene oxide)

Examples of Polymer Crystals

•The unit cell is repeated in three directions in space.

•Polyethylene’s unit cell contains two ethylene repeat groups (C2H4).

•Chains are aligned along the c-axis of the unit cell.

Crystal Lattice Structure

Polyethylene

From G. Strobl, The Physics of Polymers (1997) Springer, p. 155

Structure at Different Length Scales• Chains weave back and forth to create crystalline sheets,

called lamella.• A chain is not usually entirely contained within a lamella:

portions of it can be in the amorphous phase or bridging two (or more) lamella.

• The lamella thickness, L, is typically about 10 nm.

L

From R.A.L. Jones, Soft Condensed Matter, O.U.P. (2004) p. 130

Lamella stacks

Structure at Different Length Scales• Lamella usually form at a nucleation site and grow outwards.• To fill all available space, the lamella branch or increase in

number at greater distances from the centre.• The resulting structures are called spherulites.• Can be up to hundreds of micrometers in size.

From G. Strobl, The Physics of Polymers (1997) Springer, p. 148

Hierarchical Structures of Chains in a Polymer Crystal

• Chains are aligned in the lamella in a direction that is perpendicular to the direction of the spherulite arm growth.

• Optical properties are anisotropic.From I.W. Hamley, Introduction to

Soft Matter, p. 103

Crossed polarisers:

No light can pass!

Crossed Polarisers Block Light Transmission

http://www.kth.se/fakulteter/TFY/kmf/lcd/lcd~1.htm

Parallel polarisers:

All light can pass

• An anisotropic polymer layer between crossed polarisers will “twist” the polarisation and allow some light to pass.

• The pattern is called a “Maltese cross”.

Observing Polymer Crystals Under Crossed Polarisers

Light is only transmitted when anisotropic optical properties “twist” the polarisation of the light.

Temperature, T

Free

ene

rgy,

G

Free Energy of Phase Transitions

Crystalline state

Liquid (melt) state

Tm()

• The state with the lowest free energy is the stable one.

• Below the equilibrium melting temperature, Tm(), the crystalline state is stable.

•The thermodynamic driving force for crystallisation, G,

increases when cooling below the equilibrium Tm ().G

Undercooling, T, is defined as Tm – T.

Thermodynamics of the Phase Transition

•Enthalpy of melting, Hm: heat is absorbed when going from the crystal to the melt.

•Enthalpy of crystallisation: heat is given off when a molten polymer forms a crystal.

•The melting temperature, Tm, is always greater than the crystallisation temperature.

•The phase transitions are broad: they happen over a relatively wide range of temperatures.

Heat flows in

Heat flows out

From G. Strobl, The Physics of Polymers (1997) Springer

Hm

Xu et al., Nature Materials (2009) 8, 348.

Crystals from small molecules grow one molecule at a time.

Entire chains must join the polymer crystal at one time.

Crystal Growth Mechanisms

L

Melt to crystal: Increase in Gibbs’ free energy from the creation of an interface between the crystal and amorphous region. When a single chain joins a crystal:

At equilibrium: energy contributions are balanced and G = 0.

22 aG f

Melt to crystal (below Tm): Decrease in Gibbs’ free energy because of the enthalpy differences between the states

)()(2

mm

mm T

TLaH

T

TVHG

(Enthalpy change per volume, H m) x (volume) x (fractional undercooling)

Thermodynamics of the Crystallisation/Melting Phase Transition

L

a2

f is an interfacial energy

fm

m aLaT

TH 22 2)(

Thermodynamics of the Crystallisation/Melting Phase Transition

From G = 0:

LHT

LTT

m

f

m

mm

2

)(

)()(Re-arranging and writing undercooling in terms of Tm(L):

LH

TLTT

LT

LH m

fmm

m

m

m

f 21)()(

)(

)(21Solving for Tm(L):

Conclusion: We see that a chain-folded crystal (short L) will melt at a lower temperature than an extended chain crystal (very large L).

a

Crystal growth is from the edge of the lamella.

The lamella grows a distance a when each chain is added.

Lamellar growth directionLamella thickness, L

Lamellar Crystal Growth

LFrom U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 145

From U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 161

Free energy

G

TS

The Entropy Barrier for a Polymer Chain to Join a Crystal

Re-drawn from R.A.L. Jones, Soft Condensed Matter, O.U.P. (2004) p. 132

Melted state

Crystalline state

Melt to crystal: the rate of crystal growth is equal to the product of the frequency 1) of “attempts” and the probability of going over the energy barrier (TS):

Crystal to melt: the rate of crystal melting is equal to the product of the frequency 1) of “attempts” and the probability of going over the energy barrier (TS + G):

k

S

kT

STumc expexp 11

kT

G

k

S

kT

GSTucm expexp

)(exp 11

Net growth rate, u: the net rate of crystal growth, u, is equal to the difference between the two rates:

)exp(1expexpexpexp 111

kT

G

k

S

kT

G

k

S

k

Suu cmmc

The Rate of Crystal Growth, u

a

Crystal growth is from the edge of the lamella.

The lamella grows a distance a when each chain is added.

Lamellar growth directionLamella thickness, L

Lamellar Crystal Growth

LFrom U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 145

From U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 161

The velocity of crystal growth can be calculated from the product of the rate of growth (u, a frequency) and the distance added by each chain, a. Also, as G/kT << 1, exp(-G/kT) G/kT:

kT

G

k

Sa

kT

G

k

Sauav exp)exp(1exp 11

fm

m aT

TLaHG 22 2

)(

But from before (slide 19) - G is a function of L:

a

LS ~

The entropy loss in straightening out a chain is proportional to the number of units of size a in a chain of length L:

)2)(

(exp1

~ 22f

mm a

T

TLaH

a

cL

kTv

The Velocity of Crystal Growth,

We see that the crystal growth velocity is a function of lamellar thickness, L.

Finally, we find:

The Fastest Growing Lamellar Thickness, L*

)2)(

(exp~ 22f

mm a

T

TLaH

a

cLv

L dependence

To find the maximum , set the differential = 0, and solve for L = L*.

))((

)(2*

TTH

T

c

aL

mm

mf

a

c

a

cLLa

T

THaa

T

TH

a

cL

dL

d

m

mf

m

m exp)(

2)(

exp0 222

Solve for L = L*:

fm

m

m

m caT

THLc

T

TH 2)()(

L

L*

Tmconst

TTH

T

c

aL

mm

mf

1

.~))((

)(2*

Tm()-T

TTT m )(

Lamellar Thickness is Inversely Related to Undercooling

Original data from Barham et al. J. Mater. Sci. (1985) 20, p.1625

Jones, Soft Condensed Matter, p. 134

Experimental data for polyethylene.

Xu et al., Nature Materials (2009) 8, 348.

Chains Can Re-organise to Reduce the Number of Folds

Temperature Dependence of Crystal Growth Velocity,

The rate at which a chain attempts to join a growing crystal, , is expect to have the same temperature-dependence as the viscosity of the polymer melt ( ~ G):

0

0 expTT

B

kT

G

a

cL

TT

Ba

kT

G

a

cLa

expexpexp0

10

1

This temperature-dependence will contribute to the crystal growth velocity:

Recall that G depends on T and on L as:

fm

m aT

TLaHG 22 2

)(

Finally, recall that the fastest-growing lamellar size, L = L*, also depends on temperature as:

TH

T

c

aL

m

mf

)(2

*

We see that G(L*) becomes:

cT

TaHLG

m

3

*)(

Temperature Dependence of Crystal Growth Velocity,

kT

LG

a

cL

TT

Ba

*)(*expexp

0

10

We can evaluate when L = L* and when G = G(L*):

We finally find that:

THa

Tc

TT

B

T

TH

c

a

ekT

a

m

mf

m

m)(2

expexp)( 0

310

Recall that T0 is approximately 50 K less than the glass transition temperature.

Describes molecular slowing-down as T decreases towards T0

Describes how the driving force for crystal growth is smaller with a lower amount of undercooling, T.

Experimental Data on the Temperature Dependence of Crystal Growth Velocity

T-Tm (K)T-Tm (K) T-Tm (K)

Tm = crystal melting temperature

From Ross and Frolen, Methods of Exptl. Phys., Vol. 16B (1985) p. 363.

(c

m s

-1)

Data in Support of Crystallisation Rate Equation

J.D. Hoffman et al., Journ. Res. Nat. Bur. Stand., vol. 79A, (1975), p. 671.

TTTT

B

m )(

1expexp~

0

V-F contribution: describes molecular slowing down with decreasing T

Undercooling contribution: considers greater driving force for crystal growth with decreasing T

ex

p (B

/(T-

T 0))

[cm

s-1

]1/(T(Tm()-T)) [10-4 K-2]

Why Are Polymer Single Crystals (Extended Chains) Nearly Impossible to Achieve?

• Crystal with extended chains are favourable at very low levels of undercooling, as L* ~ 1/T

• But as temperatures approach Tm(), the crystal growth velocity is exceedingly slow!

T-Tm (K)

T-Tm (K)

T-Tm (K)

(c

m s

-1)

Factors that Inhibit Polymer Crystallisation

1. Slow chain motion (associated with high viscosity) creates a kinetic barrier

2. “Built-in” chain disorder, e.g. tacticity

3. Chain branching

Tacticity Builds in Disorder

Isotactic: identical repeat units

Syndiotactic: alternating repeat units

Atactic: No pattern in repeat units

Easiest to crystallise

Usually do not crystallise

R.A.L. Jones, Soft Condensed Matter (2004) O.U.P., p. 75

Linear

Star-branched

Branched

Side-branched

Polymer Architecture

Linear Poly(ethylene) Branched Poly(ethylene)

Effects on Branching on Crystallinity

Lamella are packed less tightly together when the chains are branched. There is a greater amorphous fraction and a lower overall density.

From U.W. Gedde, Polymer Physics (1995) Chapman & Hall, p. 148

Determining Whether a Polymer Is (Semi)-Crystalline

Raman Spectra

“Fully” crystalline

Amorphous

Partially crystalline

From G. Strobl, The Physics of Polymers (1997) Springer, p. 154

Xu et al., Nature Materials (2009) 8, 348.

Crystal Nucleation from “Seeds”

Original crystal

Re-crystallised from “seed crystals”

Summary

Further Reading1. Gert Strobl (1997) The Physics of Polymers, Springer

2. Richard A.L. Jones (2004) Soft Condensed Matter, Oxford University Press

3. Ulf W. Gedde (1995) Polymer Physics, Chapman & Hall

• Polymer crystals have a hierarchical structure: aligned chains, lamella, spherulites.

• Melting point is inversely related to the crystal’s lamellar thickness.

• Lamellar thickness is inversely related to the amount of undercooling.

• The maximum crystal growth rate usually occurs at temperatures below the melting temperature (Tm) but above the glass transition temperature, Tg.

•Tacticity and chain branching prevents or interrupts polymer crystal growth.

Problem Set 5This table lists experimental values of the initial lamellar thickness for polyethylene crystallised at various temperatures. The equilibrium melting temperature was independently found to be 417.8 K.

(a) Are the data broadly consistent with the predictions of theory?

(b) Predict the melting temperature of crystals grown at a temperature of 400 K.

Temperature, T (K) Lamellar thickness, L (nm)358.95 8.9368.95 9.9385.75 12.0396.15 14.1397.55 16.1399.15 15.9400.85 17.3401.65 18.2403.05 17.9404.15 20.1405.55 22.2