2. Conformation of polymers
Conformation = stereostructure of the molecule defined by its sequence of bonds and torsion angles
Examples of different random coil conformations

3. Polymer coil size Gaussian coil
1) Average of end-to-end distance (mean square end-to-end distance)
Note: if the Contour lenght of the chain is L maximum of rmax = L.
denotes the average calculated from a large number of conformations
2) Radius of gyration Rg Root mean square distance of the collection of atoms from their common centre of gravity
is the vector from the centre of the gravity to atom i
Mean square end-to-end distance and Radius of gyration Rg are related

4. The essential message …
a) Polymer chains tend to retract to coils (unless there exist opposing factors..) and the chain conformation can be characterized as a random coil (solution, polymer melt and also glassy amorphous state)
The polymer chain size can be evaluated based on average end-to-end distance r and radius of gyration Rg .
b) Individual polymer chains can be studied as dissolved in solvents. Unperturbed state is characterized by the absence of long-range interactions (q-solvent)
c) Models for polymer chain dimensions are developed. Mean square of end-to-end distance scales like
C = 1 freely jointed chain
C  2 freely rotating chain
C  3.4 hindered rotation
C  6.7 Advanced models (statistical weight matrices)
Experimental value for polyethylene  6.7
C  4 – 10 Experiments for different polymers

5. Different conformations
Example 1: individual C-C bonds, simplest case ethane CH3-CH3
Both end form tetrahedrons.
What is the relative orientation of these two tetrahedrions ?

6. Example 2: n-butane 4 carbons
Trans-conformation
All carbon atoms in the same plane
Gauche-conformation G G’

7. Example 3: pentane 5 carbons
9 potential combination: but only six are independent
TG = GT, TG ’= G’T, and GG’ = G’G
In general
Conformations … if the polymer have carbons thus
(number of distingguishable conformations is less due to symmetry and also some comformations have low probability e.g. GG’)
Statistical treatment:
Flory: Rotation isomeric state approximation. Each repeat unit can only be in T, G, or G’. Fluctuations around minima are ignores

8. Models for polymers
We have demostrated that the allowed conformations of the consequtive C-C bonds are T, G, G’.  We can start constructing chains just by taking random orders of T, G, G’
{TTGGTG’T….}
= polymer chain conformation
Random coil model
ideal ”phantom” chain
But long range interactions..
Polymer chain tends to revisit a spatial point that is already occupied.  We have to take also account the chain segments that are ”far apart” - long range interactions.
Self avoiding walks – more open coils

9. Dilute solutions Individual polymer chains Gas: Polymer solutions:
We want to eliminate interactions between the polymer chains
Analogy with the idal gas model: ideal gas = no interactions between the gas molecules - real gas (interactions: repulsion and attractions)
Gas:
Low concentration:
No interactions – like ideal gas
High concentration:
Interaction between the molecules ”real gas”
Polymer solutions:
High concentration:
Entanglemets etc..
Low concentration – no interaction between the different chains
But:
In low concentration there is still internal interactions within the single polymer chain and interactions with the solvent molecules...

10. Example Good solvent (T > 35 oC) ”open coil” Rg large
Low concentration 1% polystyre
99% cyklohexane
Good solvent (T > 35 oC)
”open coil” Rg large
But if solvent is changed to ethanol, which is ”bad” solvent for polystyrene  ”compact coil’’  Rg small
In general:
”Bad” solvent
”compact coil”
Good solvent
”open coil”
”extended coil”
q-solvent
Polymer-polymer and polymer-solvent interactions compensate each others...
polymer-polymer 1) interaction
>> polymer-solvent 2) interaction
polymer-polymer 1) interaction
<< polymer-solvent 2) interaction

11. q-Temperature
Polymer solubility depents typically also on Temperature .. In general solubility is better in high Temperatures (typical case but not always..)
One can define for polymer-solvent system q-Temperature
T < q-Temperature
”compact coil”
T > q-Temperature
”open coil”
”extended coil”
T = q-Temperature
”in between” where Polymer-polymer and polymer-solvent interactions compensate each others
Compare ideal gas Boyle-Temperature where attractive interaction between the gas molecules are equal to repulsive ”hard core” interactions. In this temperature gas molecules behave like ideal gas

12. Models for statistical chains
General model:
Assume that the each mainchain bond (or repeat unit) in the polymer is a vectore segment ri ri
Chain consist of these segments:
n = total number of segments
l = lenght of each segment
End-to-end vector:
vector: magnitude and direction
Vector magnitude
Remember:
where q is the angle between vectors

13. Average value of <r2>
Take large ensemble of different conformations …
mean square end-to-end distance
and end-to-end distance
This result is still a general formulation and is valid for any continuos polymer chain. (Unperturbed phantom chains, q-solvents and also polymer melt)

14. 1) The freely jointed chain
The freely jointed chain consist of a chain of bonds: the orientation of the different bonds is completely uncorrelated and no direction is preferred.
i.e. bond angle t [0, 180] and torsion angle f [0, 360] can have any value
There is no correlation between the segments: angle between the two bond vector qij (= 180-t) can have all values  
mean square end-to-end distance
average end-to-end distance
now C = 1
C  4 – 10 Experiments for different polymers …

15. 2) The freely rotating chain
i.e. bond angle t is fixed but torsion angle f [0, 360] can have any value.
Carbon-carbon bond angle  110o .
summation can be performed over single variable (k) by

16. Geometrical series sum
On the other hnd
Differentiate previous
Multibly by x

17. infinitely long chains (n=)
if t=110o
freely rotating model gives C  2, for polyethylene chain good approximation at high T when T,G, and G’states are almost ”equally” populated.
At low T trans is more populated and chain conformation is more extended

18. 3) The hindered rotation model
i.e. bond angle t is fixed and also torsion angle f can only have three values T, G, G’.
Relative population of T, G and G’ states depends on the energy levels (figure) and they are temperature dependent
Again we can use same equation:
but this case it means a lot of work … see Example Gedde pages 26-27
.. result
First term is the freely rotation result and second term is the correction due to hindered rotation
is the average value for cosf, where f is the torsion angle. And it is temperature dependend

19. Averages in statistical mechanics
Now we have a system which have a fixed number of states (3: G, T, and G’).
System partition function Z measures the number of different states the system can adopt at the tremperature of interest
At T = 0 this equals 1 and it increases with increasing temperature
How to calculate average values?
Take any parameter f, which takes a value fi for the state i.
Now for polyethylene 3 states:
i = 1 (Gauche) f1 = -120, E1 = Eg
i = 2 (Trans) f2 = 0 E2 = E t=0
i = 3 (Gauhe’) f3 = 120, E3 = Eg
Average

20. Average for cos(f), polyethylene
where
For hindered rotation square end-to-end average
For example polyethylene at 140 oC, Et =0 and Eg  2.1,
Note that in this case the constant C is higher than freely rotating model- hindered rotation includes some rigidity to the chain.. but still less than experimentally found C = 6.7 for polyethylene

21. 4) The chains with interdependent bonds: statistical weight matrices
i.e. bond angle t is fixed and also torsion angle f can only have three values T, G, G’.
And potential energy of a given bond i depends on the states of the adjacent bonds i-1 and i+1
The conformation of n bond each having three possible torsion angles may specified by n-2 componenets. For example heptane 7 bonds: one possible conformation TGTGG’ Total conformation energy

22. The chains with interdependent bonds
…average square end-to-end distance …

23. Random flight analyzis – The Gaussian Chain
Previous models for chain conformation predicts quite well the polymer chain end-to-end distance scaling
But they does not lead any further analysis – for example what is the distribution of end-to-end distances r ?
Statistical problem - What is the propability that the chain displacement vector reaches from origin to the point r and lies within the volume element dV=dxdydz ?
Similar random process problems - Brownian particles

24. Also similar random process problem – rifle shots on a target

25. The function which models random processes is the Gaussian function
The probability that polymer end-to-end lenght lying between x and x+dx is the product of p(x) and dx
correspondingly in 3D
The probability that chain end-to-end lenght lies between r and r+dr is the product of P(r) and dV
(4pr2 is the area of the sphere)

26. Mean square value of r ? definite integral from mathematics tables
now n = 2 and a = 1/r2
And from random freely jointed chain

27. -Chains with preferred conformations
Crystallization
Liquid crystals or stiff chains
helices
Crystallization:
High density polyethylene HDPE

28. poly(tetrafluoroethylene) (PTFE)
If hydrogen atoms (from PE) are replaced by fluorine atoms with van der Waals diameter 270 pm – to accommodate the fluorine atoms a rotation around each C-C bond of about 20o is induced. This is accompanied by a slight opening of the C-C chain bond angle to about 116o
13:1 helix

29. Vinyl polymers: example Isotactic PP (i-PP)

31. Dense Polymer Systems: Entanglements
How can we sketch a melt of polymers ?
Given the red polymer with N monomers, we call N’ the number of monomers in between two entanglements
In a polymer melt where N>>N’, a chain will see N/N’ obstacles or entanglements . It is the number of entanglements (or N/N’) which rules motion of macromolecules in melt.
We now aim at describing a theory which captures sufficient details of a polymer motion, but with a picture as simple as possible:
Problems should be made as simple as possible….but not any simpler !!! (A. Einstein)
From Mezzenga lecture

32. Reptation Model (in simple linear chains)
Suppose to Sketch a chain in a melt of chains constrained within obstacles O1, O2. These obstacles represent the entanglements, and the chain is not allowed to cross them.
How can the chain move within this network ? The obstacles are fixed. The chain can only move tangentially to its countour, since transversal motion is impeeded.
The chain moves as a snake, a reptile and the movement is called reptation (after DeGennes)
From Mezzenga lecture

33. From Rubinstein talk

34. From Rubinstein talk

35. Diffusion of Branched Chains
We study now the diffusion of branched chains. Lets start with a chain with 1 side group having Ns monomers.
The motion of the main red chains is identical as before. However, the mobility of J, the monomer to which the side chain is attached is not longer μ1 but μ2
If N2 monomers now carry a side chain with having Ns monomers (and N1 are free) we can work out the total force f to apply to the chain to have a contour velocity v: f=v(N1μ1-1 + N2μ2-1)
In order to be mobile a branch unit with Ns monomers needs to refold back. If z is the coordination of the lattice, this may happen with zNs possibilities.
The probability for the branch chain to get back to the branch starting point is then P≈e-αNs with α≈1. Thus we can write μ2=μ1 e-αNs and f=vμ1-1(N1+ N2 eαNs ).
The effect of branching is exponential. Suppose N2=1.
τis dominated by branching as soon as Ns>1/α*ln(N1)
From Mezzenga lecture

36. From Rubinstein talk