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.
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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?
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.
• 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