Polymer models

Talk given for:

Hauptseminar in statistical physics

8/5 2006

Peter Bjødstrup Jensen

� Polymer basics and definitions

� The ideal chain

� Freely jointed chain (FJC)

� Freely rotating chain (FRC)

� Kuhn length and persistence length

� End-to-end vector distribution function

� Gaussian chain

� Force extension relation

� Worm-like-chain

� Real chains

� Conclusion

Polymers

Overview

Polymers

Definition:

Polymers are usually long molecules (high molecular weight) consisting of repeated units (monomers) of relatively small and simple molecules covalently bonded.

The process of covalent joining of the monomers is known as polymerization

Ethylene monomer Polyethylene

Deoxyadenosinemonophosphate

Synthetic polymers: Biopolymers:Plastics, fibers, glues… Proteins, polysaccharides,

Actin, DNA…

Polymers

Other structural factors

� Polymers can have different kinds of branched architectures

� Structural isomerisms (double bonds) � hindered rotations

� Stereoisomerism: Orientation of –R and –H groups to carbon plane

� Different types of monomer repeats: homopolymers, alternating, random….

The ideal chain

No correlation between polymer monomers seperated by longdistances along the polymer.

� Short range correlations between neighboring monomers are not excluded

� Ideal chain models do not take interactions caused by conformations in space into account

� Ideal chains allow the polymer to cross itself

Polymers

Modelling a polymer

Imagining a blown up picture of a section of the polymer polyethylene in a certain conformation, could look like this:

Conformations:

� Torsion angle φ

� Bond angle θ

Bond vectors:

Starting from one end we use vectors ri to represent the bonds

End-to-end vector:

The sum of all bond vectors

The ensamble average of <Rn>=0 due to isotropy

Mean square end-to-end distance:

Simplest non-zero average

1rr

2rr

3rr

4rr

5rr

5Rr

∑=

=n

iin rR

1

rr

∑∑= =

⋅=⋅=n

i

n

jjinn rrRRR

1 1

2 rrrr

Polymers

Freely jointed chain

∑∑

∑∑

= =

= =

=⇒=⋅

⋅=⋅=

n

1i

n

1jij

22ijji

n

1i

n

1jjinn

2

cosθlRllcosθrr

rrRRR

rrr

rrrrr

nR

R2∝

= 2nl

No correlation between different bond vectors, i≠j

Polymers

No correlation between the directions of different bond vectors. Θ and φ are free to rotate. All bond vectors have length l

0=⋅=⋅ jiji rrrrrrrr

Freely rotating chain

Bond angle θ is fixed. Torsion angle φ still free to rotate.

?1 1

2 =⋅→⋅=∑∑= =

ji

n

i

n

jji rrrrR

rrrrr

1rr

2rr

3rr

θcos⋅l

( )2cosθ⋅l

Ex: what is the correlation between vector r3 and r0?

Due to the free rotation around the torque angle, only the perpendicular component of r3 is passeddown.

0rr

( ) ( )32203 coscoscos θθθ lllrr =⋅=⋅ rr

The generel expression becomes:

( ) jiji lrr −=⋅ θcos2rr

Polymers

( ) jiji lrr −=⋅ θcos2rr

∑∑∑∑= =

−

= =

=⋅=n

i

n

j

jin

i

n

jji lrrR

1 1

2

1 1

2 )(cosθrr

Inserting this expression in our equation for <R2>

Polymers

This is solved by manipulating sums, and by writing the rapidlydecaying cosine terms as an infinite series.

For calculation see Rubinstein p.56

The end result is:

θθ

cos1

cos122

−+= nlR

Polymers

For range limited interactions this will always be the case

θθ

cos1

cos1

−+=∞C

C∞ is called Flory’s characteristic ratio, and can be seen as a measureof the stiffness of the polymer in a given ideal chain model. For therotating chain we have:

We see, that the introduction of correlation has not changed the n½

proportionality. We have just added a constant >1

∞= = =

=∞→−

===

=⇒=

∑∑ ∑

∑

CnlCllR

C

n

i

n

j

n

iiij

i

n

jijij

ij

2

1 1 1

'222

'

1

cos

cos0coslim

θ

θθ

Kuhn length:

N

Nb

∝

=

R

R2 2

NbR =max

222 NbCnlR == ∞

ideal chains can be rescaled into a freely jointed chain, as long as thechain is long compared to the scale of short range interactions

New segment length b is choosen so long, that neighbooring segments arenon-correlated � New chain is a freelyjointed chain

b is called the Kuhn length, and obviously holds information onshort scale interactions and stiffness.

max

2

max

2

R

Cnl

R

Rb ∞==

Polymers

( )

( ) [ ]

)ln(cos

1

exp)ln(cosexpcos

cos2

θ

θθ

θ

=

−−=−=

=⋅

−

−

p

p

ji

jiji

s

s

ijij

lrrrr

Polymers

Persistence length

The vector correlation term from the freely rotating chain, decays quicklyand can be written in terms of an exponential function

This is a consequence of the range limited interactions, and will alwaysbe the case

Sp is the number of bonds in a persistence segment

The persistence length lp is the length of the persistence segment

The persistence length lp is the length scale with which the decayoccurs

pp sll ⋅=

End-to-end vector distribution:

We use the Central Limit Theorem which states:

CLT: Given a series of random variables; X1,X2,…Xn sampled from thesame pool of probability with a defined mean µ and variance σσσσ2, thedistribution of the sum S=X1+X2+…Xn will converge to a gaussiandistibution.

Mean and variance of the end-to-end vector is already known:

Mean µ=<R>=0

Variance σσσσ2=<R2>-<R>2=<R2>=Nb2

we get the probability distribution function in 3D:

−

=2

2/3

23 2

3exp

2

3),(

Nb

R

NbRNP D

rr

π

Polymers

The Gaussian chain:

The gaussian chain is a chain made up of kuhn bonds that are assumedgaussianly distributed

−

=2

22

3

2 2

3exp

2

3)(

b

r

brP

rr

π

{ }( )

−

=

−

∏=Ψ

∑=

=

N

n

n

N

nN

nn

b

r

b

b

r

br

12

22

3

2

2

22

3

21

2

3exp

2

3

2

3exp

2

3

r

r

π

π

We can now create the conformational distribution function of the entirechain, by multiplying each bond distribution

Polymers

The bead-spring model is a mechanical representation of the Gaussianchain

Each spring represents a Gaussianly distributed Kuhn segment

If the spring potential between two beads is defined as:

2

20 2

3)( nBn rTk

brU

rr =

Polymers

The bond distribution function for a single segment can be found.

=

−∝

2

20

2

3expexp)(

b

r

Tk

UrP n

Bn

rr

Normalizing we get the known Gaussian distribution.

One will also find the same mean quare end-to-end distance.

Force extension relations:

What happens when we apply a force F to stretch thepolymer

Looking at first at one segment

FrE isegment

rr ⋅−=

Orientations are boltzmann weigted

∫∑

⋅=

⋅==

ϕθθ ddTk

Fr

Tk

FrZ

Bnsorientatio

states Bsegment sinexpexp

rrrr

Polymers

Partition function for the entire chain

N

N

iiB

i

kT

fbkT

fb

FNZ

drFTk

FNZ

=

⇓

⇓

⋅= ∫

sinh4),(

1expsin2),(

π

θθπ rr

i

N

iii

N

ii

B

ddrFTk

FNZ ϕθθ∏∫ ∑==

⋅=

11

sin1

exp),(rr

This can be factorized and solved to give:

Polymers

∑=

⋅=⋅=n

iichain FRFrE

1

rrrr

Energy for the entire chain

The free energy is found in the standard way from the partitionfunction, and the average end-to-end distance for a given force can finally be found by differentiating the free energy

=

−

=

∂∂−=

−

−=−=

Tk

FbbNL

Tk

FbTk

FbbN

F

GR

Tk

Fb

Tk

FbTNkFNZTkNFG

B

B

B

BBBB

1coth

lnsinh4ln(),(ln),( π

53

)(453

1)coth()( x

xx

xxxL Ο+−=−= Langevin function

=

Tk

FbL

R

R

Bmax

Polymers

One important limit to the force extension expression is thatof a small force. Taking only the first order of the Langevinfunctions gives us:

RR

TkR

Nl

TkF

Tk

FlNlR

BB

B

22

33

3

==⇒

=

We obtain spring like behavior with the spring constant

2

3

R

Tkk B

spring =

Spring constant is proportional to temperature. Highertemperatures � greater forces necessary to stretch

Entropic effect � entropic spring

Polymers

The worm like chain:Polymers

Continous development of the freely rotating chain for small bondangles θ, used for polymers with high stiffness

The meaningfull limits to take in this development are:

l � 0, θ � 0, but contour length nl and persistence length lp remainthe same

We calculate the mean square end-to-end distance:

We change the sum over segments into an integral over contour

( )

∑ ∫∑ ∫

∑∑∑∑∑∑

==

= == =

−

= =

→→

−−==⋅=

n

j

Rn

i

R

n

i

n

j p

n

i

n

j

jin

i

n

jji

dsldsl

ll

ijllrrlR

10

'

10

1 1

2

1 1

2

1 1

22

maxmax

and

expcosθrr

''

expmax max

0 0

2 dsdsl

ssR

R R

∫ ∫

−−=

Polymers

The integral can be solved to give the result:

−−−=

ppp l

RlRlR max2

max2 exp122

recovered ischain jointedfreely the2For

for 2 maxmax2

p

pp

lb

lRRlR

=

>>≅

The two interesting limits are for the maximum end-to-end distance Rmax >>lp and Rmax <<lp

The ideel chain limit:

The rod like limit:

2max

2

2

p

max

p

maxmaxmax

R R

l

R

2

1

l

R-1exp For

≅⇒

+≅

−→<<

pp l

RlR

That is, it is fully extended

Polymers The worm like chain as a space curve:

r(s) is the radius vector of an arbitrary point on thespace curve s. The tangentvector u(s) and thecurvature are then:

( )

∂∂=

∂∂=

∂∂=

2

2

)( and s

r

s

usc

s

rsu

rrr

rr

The energy per unit length of a bended beam is proportional to theinverse of the radius of curvature squared which is the curvaturesquared

∫

∂∂=⇒

∂∂==

s

contour

unitlengthbending

dss

uU

s

uU

0

2

22-

/

2

1

2

1curvature) of radius(

2

1

r

r

ε

εε

Real chains:Polymers

� Long range interactions between monomers are taken into account

� Interactions between solute and polymer and between differentpolymers

� Excluded volume and self avoiding random walks

� Dynamics

�

�

�

Polymers Conclusion:

� For ideal chains correlations are finite. � for long chains we generallyhave the expression: <R2>=C∞nl2

� For chains much longer than range of correlation � rescaling into freelyjointed chain with chain length b (Kuhn length): <R2>=Nb2

� Probability distribution function for end-to-end vector is Gaussian

� The force-extension relation for the freely jointed chain is well discribedby the Langevin function

� For low forces, a Hook’s relationship with an entropic spring constantdescibes the extension

Polymers References:

[1] Michael Rubinstein and Ralph H. Colby.

Polymer physics

[2] Masai Doi.

Introduction to polymer physics

[3] Hiromi Yamakawa.

Modern Theory of Polymer Solutions.

http://www.molsci.polym.kyotou.ac.jp/archives/redbook.pdf

[4] Justin Bois

Rudiments of polymer physics

www.its.caltech.edu/~bois/pdfs/poly.pdf

[5] Thomas Franosch.

Polymers

[6] Fredrik Wagner. Diploma thesis

Polymers in confined geometry