Polymer models Talk given for: Hauptseminar in statistical physics 8/5 2006 Peter Bjødstrup Jensen
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