Structure and Morphology

?

•  Into what types of overall shapes or conformations can polymer chains arrange themselves? •  How do polymer chains interact with one another. •  Into what types of forms or morphologies do the chains organize •  What is the relationship of conformation and morphology to polymer microstructure. •  What is the relationship of conformation and morphology to macroscopic properties.

Gas

Liquid

Solid (Crystalline)

Solid (Glass)

Evaporation Condensation

Crystallization Melting

Glass Transition

Temperature States of Matter

•  Solids •  Liquids •  Gases

Usually consider;

“1st-Order” Transitions

Gas

Liquid

Solid (Crystalline)

Small Molecules

States of Matter Vo

lume

Temperature!T c

Cool!Gas!

Liquid!

Solid!

Polymers No Gaseous State

Viscoelastic liquid

Semi-crystalline Solid Glassy Solid

Crystallization

Melting

Glass Transition

Temperature

More complex behaviour

Crystallizable materials can form metastable glasses. What about polymers like atactic polystyrene that cannot crystallize?

Observed Behavior depends on: • Structure • Cooling Rate • Crystallization Kinetics

The Glassy State

Glass Transition Liquid

Gas

Glass

Crystal Vo

lume

Temperature!T g T c

Cool!

Liquid or Melt!Glassy

Solid!

Crystalline Solid!

The Issues

•  Bonding & the Forces between Chains •  Conformations

•  Ordered •  Disordered

•  Stacking or Arrangement of Chains in Crystalline Domains

•  Morphology of Polymer Crystals (and things like Block Copolymers)

Polymer Structure

Bonding and Intermolecular

Interactions

What are the forces between chains that provide cohesion in the solid state?

What determines how close these chains pack?

Get out of my space

Pot

enti

al E

nerg

y

0 Distance

I can’t feel you

This is awesome

Repulsive and Attractive Forces

Potential due to Repulsive Forces

Potential due to Attractive Forces

Need to consider balance between; •  Attractive and repulsive forces •  Attractive forces and thermal energy (motion). This will be discussed later.

Get out of my space

Pot

enti

al E

nerg

y

0 Distance

I can’t feel you

Potential due to Attractive Forces

Potential due to Repulsive Forces

This is awesome

•  Not as well understood as attractive forces. •  Often assumed that the repulsive part of the potential varies as 1/r12.

Repulsive Forces

Repulsive and Attractive Forces

•  Better understood •  In many non-polar or weakly polar systems the potential goes as -1/r6.

Attractive Forces

•  Dispersion Forces •  Dipole/dipole Interactions •  Hydrogen Bonding •  Coulombic Interactions

Increasing Interaction Strength

Intermolecular and Intersegmental Interactions

Non Polar

Highly Polar

Polar Forces

- CH - CH2 -

- C

N

- - -

δ-

δ+

- CH2 - CH - - C N

- - - δ+

δ-

Segments of poly(acrylonitrile) chains

Dispersion Forces

δ+ δ- δ+ δ-

δ+ δ-

δ+ δ-

No change in shape of a symmetric electron distribution

Instantaneous fluctuation inducing a dipole in a neighbor

Hydrogen Bonds

H - N C = O H - N

C = O H - N C = O

There is no simple, universally accepted definition of a hydrogen bond, but the description given by Pauling** comes close to capturing its essence; . . . under certain circumstances an atom of hydrogen is attracted by rather strong forces to two atoms instead of only one, so that it may be considered to be acting as a bond between them. This is called a hydrogen bond The N-H and C=O groups of nylon (and the polypeptides and proteins) interact in this manner;

** L. Pauling, The Nature of the Chemical Bond. Third Edition. Cornell University Press, Ithaca, New York, 1960.

O - H O - H O - H

O - H - C

H - O C -

Hydrogen bonds form between functional groups A–H and B such that the proton usually lies on a straight line joining A–H -- B. The atoms A and B are usually only the most electronegative, i.e., fluorine, oxygen and nitrogen

Hydrogen Bonds

H - N C = O

H - N C = O

The hydrogen bond is largely electrostatic in nature

- CH2 - C - CH3 -

-

C

- CH2 - C - CH3 -

-

C - Zn++

- CH2 - C - CH3

--

C

-

Zn++

Coulombic Interactions - Ionomers

+ - - - +

+

+ - - - +

+

+ - - - +

+

+ - - - +

+

The structure of ionomers is actually far more complicated than this and the ionic domains phase separate from the non - polar parts of the chains into some form of cluster.

Ionomers

Type of Type of Interaction Interaction

Characteristics Characteristics Approximate Approximate Strength Strength

Examples Examples

Dispersion Forces

Dipole/dipole Interactions (Freely Rotating)

Strong Polar Interactions and Hydrogen Bonds

Coulombic Interactions (Ionomers)

Short Range Varies as -1/r 6

Short Range Varies as -1/r 6

Complex Form but also

Short Range

Long Range Varies as 1/r

About 0.2 - 0.5 kcal/mole

About 0.5 - 2 kcal/mole

About 1 - 10 kcal/mole

About 10 - 20 kcal/mole

Poly(ethylene) Polystyrene (simple hydrocarbon polymers)

Poly(acrylonitrile) PVC

Nylons Poly(urethanes)

Surlyn

Increasing Interaction Strength Increasing Interaction Strength

SUMMARY

Conformations

Ordered

Disordered

Staggered

Eclipsed

Staggered Staggered Staggered

Eclipsed Eclipsed

Pot

enti

al E

nerg

y (A

rbit

rary

Uni

ts)

-180 -120 -60 0 60 120 180°

Rotation Angle

Conformations

Conformations; Or how do Chains

Fold gauche

trans

trans

trans

Polyethylene

trans

Pot

enti

al E

nerg

y (

Arb

itra

ry U

nits

)

-180 -120 -60 0 60 120 180° Rotation Angle

gauche gauche

•  Why Doesn't the Chain Just Sit in its Minimum Energy Conformation? –  e.g. polyethylene

•  What is the Effect of Thermal Motion ?

•  How Many Shapes or Conformations are Available to a Chain ?

Interesting Questions

gauche

trans

trans

trans

How Many Shapes or Conformations are Available to

a Chain ?

•  Assume each bond in the chain is only allowed to be in one of three conformations, trans, gauche and the other gauche •  Assume each of these conformations has the same energy

A Simple Estimate

Just one of many conformations or configurations

How Many Conformations are Available to a Chain?

The first bond can therefore be found in any one of three conformations, as can the second, the third, and so on. How many configurations are available to the first two bonds taken together (ignoring redundancies). A. 3+3=6 B. 3x3=9

This has 3 possible conformations

So has this

And this

Another 3

And another 3

3 3 3

3

This has 3 possible conformations

So has this

And this

Another 3

And another 3

3 3 3

3

How Many Conformations are Available to a Chain?

Pascal’s triangle

Bond 1

G

T

G’

G

G’

T

G

G’

T

G

G’

T

Bond 2

How many arrangements are there for a chain with 10,000 bonds ?

For this chain of 10 bonds there are 3.3.3.3.3.3.3.3.3.3 = 310 possible arrangements. Or have we over-counted?

How Many Conformations are Available to a Chain Consisting of 10,000 Bonds ?

The answer would seem to be simple; 310,000 =104,771

But you have to be careful in doing these types of calculations. You have to account for redundancies.

Crucial Point; even after accounting for redundancies there are one hell of a lot of (distinguishable) configurations available to a chain. So, how on earth can we relate structure, or in this case the absence of structure, to properties?

The Chain End-to-End Distance

R

•  It is the enormous number that saves us as it permits a statistical approach. •  But we will need a parameter that tells us something about the shape of the chain.

The Chain End-to-End Distance

R

R

The distance between the ends will be equal to the chain length if the chain is fully stretched out;

But may approach zero if the chain is squished in on itself forming a compact ball.

Intuitively, one would expect most chains to lie somewhere between these extremes.

Redrawn from J. Perrin, Atoms, English translation by D. L. Hammick, Constable and Company, London, 1916. .

Random Walks and Random Flights

What is the distance between the starting point (first observation) and the finishing point (last observation) after a walk of N steps ?

Random Walks and Random Flights

R!

Random Walks and Random Flights

•  Consider steps of equal length, defined by the chemical bonds.

For a polymer chain;

Complications; •  A polymer chain is sterically excluded from an element of volume occupied by other bits of itself. •  The “steps” taken by a chain are constrained by the nature of the covalent bond and the steric limitations placed on bond rotations.

The Freely Jointed Chain

“Freely Hinge”

Rotate

What we will assume and still get the right answer . . . more or less

The way it really is . . . more or less

Fix

Rotate

•  To begin with we are only going to consider a one- dimensional walk. •  Actually, that is all we really need.

•  Imagine a three dimensional walk projected onto (say) the x-axis of a Cartesian system •  There will be some average value of the bond length <l>, that we can use*. •  We can then sum the contributions of projections in all three spatial directions (remember Pythagoras ?) to get the end -to - end distance R

* This is actually calculated using the same arguments as we are going to use for the random walk!

Random Walks and Random Flights

WEST EAST +R -R Ye Olde

English Pub

A One-Dimensional Drunken Walk

What is the average distance traveled from the Pub for drunken walks of N steps? (Assume each step is 1 unit in length).

Q: WALK R 1 +30 2 +50 3 -40 4 -50 5 +40 6 -30 . . . .

!!

.!.!

500 300 100 -100 -300 -500 0!

100!

200!

300!

400!

500!

600!

R

Nu

mb

er

of

Wa

lks o

f d

ista

nc

e R

0

<R> = 0

Intuitive Answer

500 300 100 -100 -300 -500 0.0000!

0.0002!

0.0004!

0.0006!

0.0008!

0.0010!

0.0012!

R

P(R

) =

A e

xp

(-B

R2)

x 1

04

0

<R> = 0

Probability Distributions

•  Another way to graph this is to plot the fraction of walks that end up a distance R from the pub. •  Each of these values then also represents the probability that a walk of N steps will have an end - to-end distance R. •  This is a probability distribution, P(R), and if you know some statistics you may guess that the shape of the curve will be Gaussian (for large N). (See equation on y-axis). We’ll come back to this.

What We Need To Do

+R

-R START

•  Determine Distance Traveled Regardless of Direction

•  Method: –  Determine R for a Whole

Bunch of Walks –  Determine <R2> –  Calculate <R2>0.5

The Root-Mean-Square End-to-End Distance*

*See Feynman, Lectures on Physics, Vol. 1, Chapt. 6

ONE DIMENSIONAL WALK

For a Walk of One Step:

For the Nth Step:

<R21> = 1

or

RN = RN-1 + 1 RN = RN-1 - 1

RN-1

or

Feynman’s Method

<R2N> = <R2

N-1> +1

Square:

Average of Squares:

Recalling:

Then:

<R21> = 1

<R22> = <R2

1> + 1 = 2 <R2

3> = 3 - - - etc. <R2

N> = N

R2N = R2

N-1 + 2RN-1 + 1 R2

N = R2N-1 - 2RN-1 + 1

One dimensional walk

The Root-Mean-Square End-to-End Distance

<R2> = N l 2

<R2>0.5 = N0.5 l

If N = 10,000, l = 1;

<R2>0.5 = 100 ! ! !

R!

Random Coils and Rubber Elasticity

•  What is the most probable state ? •  How do we calculate a probability distribution for the end - to - end distance regardless of direction ?

The Radial Distribution Function

A distribution function that describes the end-to-end distance regardless of direction can be obtained by fixing one end of the chain at the origin of a coordinate system and then finding the probability that the other end lies in an element of volume 4πR2dR. This is the radial distribution function and is simply given by

R!

dR!

W(R) = P(R). 4πR2

The Radial Distribution Function

•  If a chain in its most probable state is stretched, then it enters a less probable state

•  An entropic driving force to return the chain to its most probable state is created

•  More on this in mechanical properties !

500 400 300 200 100 0 0.000

0.001

0.002

0.003

0.004

R

<R > 2 1/2

P ( R ) x 1 0 4

o r W ( R )

P ( R )

W ( R ) = 4 π R 2 P ( R )

• We have figured out a way to Describe a collection of random Chains • A qualitative understanding of Rubber elasticity immediatly follows • A pathway to more rigorous and quantitative work is opened up

Crucial Points

Real Chains

< R 2 > = Nl

2 1 + cos θ 1 - cos θ ⎛ ⎝ ⎜

⎞ ⎠ ⎟

1 + η

1 - η

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

< R 2 > = Nl

2 1 + cos θ 1 - cos θ ⎛ ⎝ ⎜

⎞ ⎠ ⎟

Fix (θ)

Freely Rotate

Fix

Hindered Rotation

•  Even sterically allowed bond rotation angles will not be independent of one another. Local overlap can occur

Real Chains

Overlap - ouch!!

•  Incorporate corrections (using the Rotational Isomeric States Model) into a general factor C∞ , and we can now write;

< R 2

>= C ∞ Nl 2

The Kuhn Segment Length

C ∞ = < R 2 >

Nl 2

The Flory characteristic ratio

The Kuhn segment length KUHN SEGMENT

N K l K 2 = < R

2 > = C ∞ Nl

2

But what about overlap between parts of the chain that are topologically distant?

R

Self-Avoiding Walks and Intramolecular Excluded Volume

A one dimensional self - avoiding walk

R = Nl

R

A two dimensional self - avoiding walk

R = Nl

<R2>0.5 = N0.75l

Self-Avoiding Walks and Intramolecular Excluded Volume

While in three dimensions;

In general;

< R 2 > SAW

0 . 5 = cons tant. N

v

Where;

v = 3

d + 2

<R2>0.5 = N0.6l