Examples of different random coil conformations

Conformation = stereostructure of the molecule defined by its sequence of bonds and torsion angles

C

H

CC

CC

CC

CC

H

H

H

H

H

H

H

H

HH H H H

HHHH

Conformation of polymers

22

6

1rRg

r

Rg

1) Average of end-to-end distance (mean square end-to-end distance) 2r

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

Gaussian coil

n

ii

n

iii

g

m

rmR

1

1

2

2

ir 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

Polymer coil size

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 .

r

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

22 Cnlr

C = 1 freely jointed chainC 2 freely rotating chainC 3.4 hindered rotationC 6.7 Advanced models (statistical weight matrices) Experimental value for polyethylene 6.7 C 4 – 10 Experiments for different polymers

Different conformations

Example 1: individual C-C bonds, simplest case ethane CH3-CH3

CH

HH

CH H

H

Both end form tetrahedrons.What is the relative orientation of these two tetrahedrions ?

Example 2: n-butane 4 carbons

H H

HH H

H H

H H

H

H H

HH H

H H

H H

H

60 o

H H

HH H

HH

H

HH

60 o

Trans-conformation

All carbon atoms in the same plane

Gauche-conformation

G

G’

CH3 CH2 CH2 CH3

TT

TG

TG'

GT

GG

GG'

G'T

G'G

G'G'

9 potential combination: but only six are independentTG = GT, TG ’= G’T, and GG’ = G’G

33 nConformations … if the polymer have 10 000 carbons thus

onsconformati103 47709997 In general

(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

Example 3: pentane 5 carbons

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’

T T

G

T

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

Models for polymers

Individual polymer chains

• 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:

Low concentration – no interaction between the different chains

High concentration:Entanglemets etc..

But:In low concentration there is still internal interactions within the single polymer chain and interactions with the solvent molecules...

Dilute solutions

Example

Good solvent (T > 35 oC)”open coil” Rg large

CH2 CH n

Rg

But if solvent is changed to ethanol, which is ”bad” solvent for polystyrene ”compact coil’’ Rg small

Rg

Good solvent

”open coil””extended coil”

”Bad” solvent

”compact coil”

In general: -q solvent

Polymer-polymer and polymer-solvent interactions compensate each others...

1.

2. 1.

2.

Polymer repeat unit

Solvent molecule

polymer-polymer 1) interaction

>> polymer-solvent 2) interactionpolymer-polymer 1) interaction

<< polymer-solvent 2) interaction

CH2 CH2

CH2

CH2 CH2

CH2Low concentration 1% polystyre 99% cyklohexane

T > -q Temperature

”open coil””extended coil”

T < -q Temperature

”compact coil”

One can define for polymer-solvent system -q Temperature

T = -q Temperature

”in between” where Polymer-polymer and polymer-solvent interactions compensate each others

Polymer solubility depents typically also on Temperature .. In general solubility is better in high Temperatures (typical case but not always..)

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

q-Temperature

r

r1

r2 r3 ri

rn

General model:

Assume that the each mainchain bond (or repeat unit) in the polymer is a vectore segment ri

CH2 CH2

ri

Chain consist of these segments:n = total number of segmentsl = lenght of each segment

End-to-end vector:

n

iirr

1

vector: magnitude and direction

Vector magnitude

n

ij

n

ji

n

i

n

jji rrrrrrr

1 11 1

2 )()(

n

jiji

n

iii rrrr

1

cosABBA

Remember: where q is the angle between vectors

n

jiji

n

jiji

n

i

rrnlrrlr 2

1

22

Models for statistical chains

…Take large ensemble of different conformations

mean square end-to-end distance

n

jiji

n

jiji

n

i

rrnlrrlr 2

1

22

and end-to-end distance

n

jiji rrnlr22

ijji lrr cos2

This result is still a general formulation and is valid for any continuos polymer chain. (Unperturbed phantom chains, q-solvents and also polymer melt)

Average value of <r2>

1) The freely jointed chainThe 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

n

jiij

n

jiji lnlrrnlr cos2222

There is no correlation between the segments: angle between the two bond vector qij (= 180-t) can have all values

0cos jiij

22 nlr

mean square end-to-end distance

average end-to-end distance 212 nlr

22 Cnlr now C = 1 C 4 – 10 Experiments for different polymers …

i.e. bond angle t is fixed but torsion angle f [0, 360] can have any value.

Carbon-carbon bond angle 110o

ii-1 i+1

jiforalsoij ,0cos

)180cos(21 lrr ii

222 )180cos( lrr ii

323 )180cos( lrr ii

.

.

. ijji lrr )180cos(2

ri ri+1

180-

ri ri+1

180-

ri+2

180-

n

jiji rrnlr22

ijk and180cos( inserting

summation can be performed over single variable (k) by

1

1

222 )(2n

k

kknlnlr

1

1

1

1

2 221

n

k

kn

k

k kn

nl

2) The freely rotating chain

1

1

1

1

22 221

n

k

kn

k

k kn

nlr m

m

k

km xxxxxS

...321

1

113211

11 ...

m

mmm

m

k

km xSmxxxxxS

On the other hnd)1()...1( 321

1 mm

m SxxxxxxS

mmm

m xSxSxxS )1(1

x

xxxSxxxSS

mm

k

km

mmm

1

1

1

1

Differentiate previous

)1(*)1(1

)1(12

1

1

1

x

xx

x

xmkx

mmm

k

k

2

1

1

1

)1(1

)1(1

x

xx

x

xmkx

mmm

k

k

Multibly by x

2

2

2

221

1 )1(

)1(

1

)1(1

)1(1

)1(

x

xx

x

xmx

x

xx

x

xmxkx

mmmmm

k

k

Geometrical series sum

1

1

1

1

2 221

n

k

kn

k

k kn

nl

))1(1

(2

121

2

122

nnn n

nnl

)11)1(

(2

121

2

122

nnn n

nnl

)1)1(

)(

2

121

2

2122

nnn n

nnl

)1)1(

(2

121

2

12

nnn n

nnl

1

2

)1(

2

121

2

12

nnn

nnl

2

2

)1(

)1(2

1

21

n

nnl

infinitely long chains (n=)

1

1

1

21 222 nlnlr

)180cos(1

)180cos(12

nl

if t=110o 222 2)70180cos(1

)70180cos(1nlnlr

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

x

xxx

mm

k

k

1

1

1

2

221

1 )1(1

)1(

x

xx

x

xmxkx

mmm

k

k

ii-1 i+1

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

n

jiji

n

jiji

n

i

rrnlrrlr 2

1

22

Again we can use same equation:

but this case it means a lot of work … see Example Gedde pages 26-27

cos(1

cos(1

)180cos(1

)180cos(122 nlr.. result

First term is the freely rotation result and second term is the correction due to hindered rotation

)cos( is the average value for cosf, where f is the torsion angle. And it is temperature dependend

3) The hindered rotation model

How to calculate average values?

i = 1 i =2 i = 3

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

i

RTEieZ / At T = 0 this equals 1 and it increases with increasing temperature

Take any parameter f, which takes a value fi for the state i.

i

RTE

i

RTEi

i

i

e

eff

/

/

AverageNow for polyethylene 3 states:

i = 1 (Gauche) f1 = -120, E1 = Eg

i = 2 (Trans) f2 = 0 E2 = E t=0i = 3 (Gauhe’) f3 = 120, E3 = Eg

RTERTERTE

i

RTE gggi eeeeeZ //0/3

1

/ 21

Averages in statistical mechanics

cos(1

cos(1

)180cos(1

)180cos(122 nlr

121

121

)180cos(1

)180cos(1

211

1

211

1

)180cos(1

)180cos(1 22 nlnl

21

1

21

)120cos()0cos()120cos(/

/0/

RTE

RTERTE

g

gg

e

eee

3

2

)180cos(1

)180cos(12nl

RTEge /

3

1

/

3

1

/)(coscos

i

RTE

i

RTEi

i

i

e

e

For hindered rotation square end-to-end average

For example polyethylene at 140 oC, Et =0 and Eg 2.1,

54.0/ RTEge

2222 14.354.0*3

54.022

3

2

)180cos(1

)180cos(1nlnlnlr

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

where

Average for cos(f), polyethylene

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

'GGTGGTTGT EEEEEE

4) The chains with interdependent bonds: statistical weight matrices

ii-1 i+1

The chains with interdependent bonds

1

1

1

'

'

G

G

T

uU

GGT

ii

…average square end-to-end distance …

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

Previous models for chain conformation predicts quite well the polymer chain end-to-end distance scaling

21

nlr

But they does not lead any further analysis – for example what is the distribution of end-to-end distances r ?

Random flight analyzis – The Gaussian Chain

Also similar random process problem – rifle shots on a target

The function which models random processes is the Gaussian function

2)/(

)(xe

xp

1D The probability that polymer end-to-end lenght lying between x and x+dx is the product of p(x) and dx

dx

edxxp

x

2)/(

)(

correspondingly in 3D

dxdydz

edxdydz

e

dxdxdyzpypxpdxdydzzyxPrzyx

3

/

3

/)(

)()(

)()()(),,(222222

The probability that chain end-to-end lenght lies between r and r+dr is the product of P(r) and dV

drrdV 24

(4pr2 is the area of the sphere)

drr

edVrP

r2

3

/

4)(

)(22

Mean square value of r ?

1

4)(

1

)(

)(2/

30

2

0

0

2

2

22

drrer

dVrP

dVrPrr

r

drer r 22 /

0

43

)(

4

aa

ndrex

nnaxn

10

2

2

)12...(5312

definite integral from mathematics tables

now n = 2 and a = 1/r2

2

22

33

/

0

43 /1)

1(2

31

)(

4

)(

4 22

drer r

2

38

34

4

2/1

2/332 r

And from random freely jointed chain 22 nlr

ln

nlr2/1

22

2

3

2

3

Crystallization:High density polyethylene HDPE

•Crystallization

•Liquid crystals or stiff chains

•helices

-Chains with preferred conformations

nC CF

F

F

F

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

poly(tetrafluoroethylene) (PTFE)

13:1 helix

Vinyl polymers: example Isotactic PP (i-PP)

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

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

From Rubinstein talk

From Rubinstein talk

Diffusion of Branched Chains We study now the diffusion of branched chains. Lets start with a chain with 1 side group having Ns monomers.

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 ).

Ns

tubeNstube eNN

D

N

T

L

eNN

211

22

t21

1 and

The effect of branching is exponential. Suppose N2=1.

τis dominated by branching as soon as Ns>1/α*ln(N1)

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

From Mezzenga lecture

From Rubinstein talk