introduction to polymer theory - universiteit utrecht

Introduction to Polymer Theory

Gert Jan Vroege

Van ’t Hoff Laboratory for Physical andColloid Chemistry

Utrecht University

The Netherlands

Introductionchemical details vs. universal properties

A polymer is a statistical mechanical system, for which the role of entropy is very important

Programme

1. Ideal polymers: • conformations: Gaussian coil• in an external field• in a Self Consistent Field (SCF)

2. Non-ideal polymers• excluded volume• attractions

3. Concentrated solutions:• Flory-Huggins theory• scaling theory (semi-dilute solutions)

Polymer conformations

1

End-to-end vector: N

ii

R r=

=

0R =

End-to-end vector

bond vectorbond length

segment

Polymer conformations

2

1 1

2

1 1

N N

i ji j

N N

i i ji i j i

R R R r r

r r r

= =

= = ≠

= =

= +

1

End-to-end vector: N

ii

R r=

=

N∝ N∝ smaller if j-i larger2

2 1/ 2

R N

R N

∝

∝

Chain models (1)Freely jointed chain:

2 2

0 ( )

i i i

i j

r r r b

r r i j

= =

= ≠

( )2 22

1 1

0N N

i i ji i j i

R r Nbr r= = ≠

= += +

Chain models (2)

2 2i i ir r r b= =

Freely rotating chain:

( )121 cosi ir r b θ+ =

( )222 cosi ir r b θ+ =

( )2 cos j ii jr r b θ −=

Chain models (3)

( ) ( ) ( ) 2

2

1 2 32 2

1

1 cos1 cos

1 co

1 cos cos cos

with s

1 c s

o

N

eff eff

i

NbR

Nb b b

bθθ

θθ

θ θ θ=

+−

= + + + + =

= +≡−

General result when NO INTERACTION between segments

2 2 with more general eff effR Nb b=

End-to-end distribution (1)( , ) : probability of finding after segments?P R N R N

recursion relation:

( , ) ( , 1)N

N rP R N P R r N= − −

Taylor expansion ( 1 and ):NN R r

( ) ( )

( )( )

,, ,

2

, ,, , , ,

( , 1) ( , ) 1

12

N Nx y z

N Nx y z x y z

P PP R r N P R N r

N

Pr r

αα

α βα α

α

α β

=

= =

∂ ∂− − ≈ + − + −∂ ∂

∂+ − − +∂ ∂

End-to-end distribution (2)

2 2 2 2

apply : 0

0 ( )

13

N N

NN N

N N N

Nr r

N N N Nrr r

Nx N y Nzr r r

r

r r r r

r r r b

α β α β α β

=

= = ≠

= = =

End-to-end distribution (3)( )

2 2 2

2 2 2

1( , ) ( , ) 1

6P

P R N P R N PN x y z

∂ ∂ ∂ ∂≈ + − + + + + ∂ ∂ ∂ ∂

2( , )( , )

6P R N b

P R NN

∂ = ∆∂

2 2 2

2 2 2With the definition of the Laplacian

we thus find that ( , ) is the solution of:

x y z

P R N

∂ ∂ ∂∆ ≡ + +∂ ∂ ∂

( , )( , )

c R tD c R t

t∂ = ∆

∂

cf. the diffusion equation for ( , ) :c R t

End-to-end distribution (4)Cf. one diffusing colloidal particle (Einstein):

2 6R Dt=

2b N

An (ideal) polymer is like the trajectory of a diffusing particle!

End-to-end distribution (5)2( , )

( , ) :6

The solution of is (by analogy)P R N bP R N

N∂ = ∆

∂

3/ 2 2

2 2

3 3( , ) exp

2 2R

P R NNb Nbπ

= −

Central Limit Theorem

2

Consider the sum of independent, stochastic variables ( is large).

This sum has a normal (= Gaussian) distribution with .

N N

Nσ ∝

( ) ( )22 2 2

1

0N

i effRi

R r R R N bσ=

= = − = ∝ ×

2Variation in :R

( )2

22 2 22

3RR R Nbσ = − =

A Gaussian coil is a strongly fluctuating object!

Conclusion: many (global) properties of polymers do notdepend on the (local) details of the model.

Entropy of a Gaussian coil

22

( ) ln

cst ln ( )

3cst

2

B

B

B

S R k W

k P R

kR

Nb

=

= +

= −

( ) 22

3( ) cst ENTROPIC SPRING

2Bk T

A R TS RNb

= − = +

Additional polymer modelsGaussian bond model:

( )2

2

3 Every single bond Gaussian exp -

2 1irb

∝

Bead-spring model:

( ) 2

3spring constant:

2 1Bk Tb

(used in the Rouse/Zimm models for polymer dynamics)

Continuous model:

permits the use of path integrals

A polymer in an external field (1)assume a segment at position has energy ( )R Rϕ

the recursion relation now changes to:

( )( , ) ( , 1) exp

NN r

B

RP R N P R r N

k Tϕ

= − − −

( )

Taylor expansion:

( )( , 1) ( , ) 1 etc. 1N

B

P RP R r N P R N

N k Tϕ ∂ − − ≈ + − + × − + ∂

2 ( )6 B

P b RP P

N k Tϕ∂ = ∆ −

∂

cf. diffusion in an external field

A polymer in an external field (2)

2 ( )6 B

G b RG G

N k Tϕ∂− = − ∆ +

∂

similarly for the conditional probability ( | )NG R R′

is a parameter, but: R R R′ ′↔

( )

2

cf. ( )2

QM: time-dependent Schroedinger equation for ,

i V Rt m

R t

ψ ψ ψ

ψ

∂− = − ∆ +∂

linear, partial differential equations; solution method:

separation of variables

Separation of variables2 ( )

6 B

G b RG G

N k Tϕ∂− = − ∆ +

∂

assume can be written as: ( ) ( )G G f N Rψ=

2( ) ( )( ) ( ) ( ) ( ) ( )

6 B

f N b RR f N R f N R

N k Tϕψ ψ ψ∂− = − ∆ +

∂

divide by ( ) ( ) :f N Rψ

21 ( ) ( ) ( )( ) 6 ( ) B

f N b R Rf N N k TR

ψ ϕψ

λ∂ ∆− = − + =∂

2 ( )( ) ( ) ( )

6 B

b RR R R

k Tϕψ ψ λψ− ∆ + =

( )( ) expf N c Nλ= −

nneigenvalues , complete set (orthogonal)

eig

eig

envalue equat

enfunctions

ion

( )Rλ ψ

( )( )expn nG R Nψ λ= −

or any linear combination for different n

A polymer in an external field (3)( )linear combination: ( | ) ( ) expN n n n

n

G R R c R Nψ λ′ = −

using R R′↔

( )bilinear expansion: ( | ) ( ) ( ) expN n n nn

G R R R R Nψ ψ λ′ ′= −

2 ( )where ( ) ( ) ( )

6 n n n nB

b RR R R

k Tϕψ ψ λ ψ− ∆ + =

2 216example: ( ) 0

here we need all eigen Gaussianfunctio cns o l i

and ik Rk k

R e b kϕ ψ λ= = =

1) continuous spectrum of eigenvalues


2) discrete spectrum of eigenvalues

0 lowest eigenvalue dominates for large

GROUND STATE DOMINANCE

Nλ

( )0 0 0( | ) ( ) ( ) ex

chain ends are decou d

p

ple !NG R R R R Nψ ψ λ′ ′ −

( )bilinear expansion: ( | ) ( ) ( ) expN n n nn

G R R R R Nψ ψ λ′ ′= −


integrate over: beginning end ν

( )0

2( ) cf. QM: bound statec R N Rψ

segment density ( ) ?c R

Lifshitz entropy: derivation(for ground-state dominance)

Partition function : # conformations

(weighed with Boltzmann factors exp( ( ) / ))B

Z

R k Tϕ−

( | )NZ dR dR G R R′ ′=

( )0 0 0( | ) ( ) ( ) expNG R R R R Nψ ψ λ′ ′ −

( )00

2( )

end effects

NZ e dR Rλ ψ

0

free energy:ln end effectsB BA k T Z k T Nλ= − +

0

entropy:1

( ) ( ) B

U AS R c R dR k N

T Tϕ λ−= = −

0

0

2use ( )

and the eigenvalue equationand eliminate

c R Nψ

λ

Lifshitz entropy: result2

0 0

1( ) ( )

6BS Nk b R R dRψ ψ= ∆

20

21( ) partial integration using

6BS Nk b R dRψ = − ∇ ∆ ≡ ∇ ∇

[ ]22 ( )

( )

1

24B

c R

c RS k b dR

∇= −

02using ( )c R Nψ

• independent of• also valid for a collection of polymers• S decreases because of concentration gradients•

( )Rϕ

[ ( )] ( is a functional of ( ))S S c R S c R=

Self-consistent field methodthis method incorporates inter-segment interactions:

free energy:

[ ( )] [ ( )] [ ( )]

[ ( )] represents e.g. a non-ideal gas of segments

A c R U c R TS c R

U c R

= −

THIS APPROACH NEGLECTS FLUCTUATIONS/CORRELATIONS

( ) is then obtained by functional minimization

!eqc R

Non-ideal polymer chains

second virial

[ ( )] ( )

is the ( 0 repulscoeffici ion)entBU c R Nk TB c R

B B

=>

2 6/5Edwards (1965): R N∝

swellingPURE REPULSION

22

2

total number of configurations (depending on R)

3( , ) 4 exp

2R

P R N RNb

π

∝

∝ ∝ −

2

3 3

( 1) / 2

but a certain fraction of these configurations is "forbidden":

( ) 1 exp2

c cN N

Np R

R Rν ν−

≈ − −

≈

Non-ideal (repulsive) polymer chains

[ ]2

2 3

2

free energy:( ) ( )

ln ( ) ( , )

3cst 2 ln

2 2

B B

c

A R S Rp R P R N

k T k T

NRR

Nb Rν

= − = −

≈ − + +

minimize ( ) with respect to A R R* 1/ 2

00 :c R N bν ∝=* *

* *0 0

1/ 23

5 3

Flory0 : cc

R RN

R R bνν

≈

≠ −

* * 1/100

3/51 SWELLING: !N NR R N b≈ ∝

* 0.588RG theory / simulations : NR b∝

Repulsion combined with attraction (1)Assumptions:• only attraction between nearest neighbours• coordination number: z• solvent-solvent

solvent-polymerpolymer-polymer

• random pair contacts:

Repulsion combined with attraction (2)attractive energy within a coil:

( ) ( )

( ( ) is the probability to find a neighbouring segment)attr c

c

U N c R z

c R

ν εν

= −∆

2

3 where ( )chi-parameterattrc

B B B

U N zk T

zk kR T T

ε χεν ∆ ≡∆= − usually > 0χ

3

2( )Compare with repulsive term in , which was:

2c

B

NA Rk T R

ν

( )1 2c cν ν χν −→ ≡

( )* *

* *0 0

1/ 23

5 3

Fl y2 or1cR RN

R R bν χ

≈ −

−

Repulsion combined with attraction (3)

( )* *

* *0 0

1/ 23

5 3

Fl y2 or1cR RN

R R bν χ

≈ −

−

*

* *0

3/5

1/ 2

CONCLUSION:

at 0 swollen chain

at 1/ 2 ( - temperature): ideal chain

1 cst right-hand side only small if = i.e. abrupt change if large

2

R N

R R N

NN

χχ θ

χ

• = ∝

• = = ∝

• −

* 1/31/ 2 : (bound state, cf. QM)

in general

glob

*

ule R N

R b Nν

χ > ∝

•

Repulsion combined with attraction (4)

polystyrene in cyclohexane

Concentrated polymer solutions

( )3 3 3

1/ 2 4/5

* very small !( )

N NN Nc

R bN bν

− −↔≈ ≈ ≈

Lifshitz

FLORY-HUGGINS:

concentrated systems: ( ) homogeneous systems:

random mixing ( places)

S R

S

••• Ω

*c

NO

NOYES

Flory-Huggins theory (1)

( ),

,

,

translational entropy: ln species

polymer: ln

solvent: (1 ) ln(1 )

tr sp B sp sp

m p B

m s B

S k sp

A k TN

A k T

φφ φ

φ φ

≈ − Ω =

≈ Ω

≈ Ω − −

1ln (1 ) ln(1 ) (1 )m BA k T

Nφ φ φ φ χφ φ ≈ Ω + − − + −

Flory-Huggins theory (2)

c

1 (highly) asymmetric

11 1

follows from 2

for large , i.e. near the coil globule transition

note that fluctuations are neglected!

c

c

c

N

TN

T N

φ

χ

θ

• =+

• ≈ +

→ →•

Polystyrene in methylcyclohexane

Scaling theory

Semi-dilute solution (good solvent)3 3

34/

/5

3 5* * still very small !( )

Nb c b N

bNφ −≈ ≈ ≈

OSMOTIC PRESSURE TYPICAL LENGTH SCALE

DILUTE

SEMI-DILUTE(power law)

B

ck T

NΠ =

*

4/51

B

B

m

m m

ck T

Nc

k T NN

φφ

φ

Π

independent of N

m = 5/4

35/4 9/4

des Cloizeaux

BB

k Tk Tc

bφ φΠ

( ) 3/5R bNξ = =

3/5

3/5

*

4/5

m

m m

bN

bN N

φφ

ξ

φ

m = -3/4

3/4

de Ge nnesbξ φ−

What is the meaning of ksi ? (1)

3 39/4 5/4des Cloizeaux : B Bk T k T

b bφ φ φΠ ×

3 32 Flory-Huggins: B Bk T k T

b bφ φ φΠ ×

probability of segment probability of contact w×

probability of contact w: lower for scaling theory (correlations!)

Flory-Huggins:

des Cloizeaux:

What is the meaning of ksi ? (2)on one chain: number of segments between contacts?

( )1 5/ 4

3/53/5 5/ 4 3/ 4 distance between chain con

monomers:

acts

t !

g g w

bg b b

φ

φ φ ξ

− −

− −

3 5/ 4 3

3 3/ 4 3

1) volume fraction within one b

blobs

lob:

tou( )

ch !gb b

bφ φ

ξ φ

−

−

What is the meaning of ksi ? (3)2) a SEMI-DILUTE SOLUTION

is a collection of BLOBS !

3 39/43 blobs are the osmotic unitsdes Cloizeaux: ) B Bk T k T

bφ

ξΠ

screening le4) is also the for the excluded vngth e olumξ

3/ 45) 1: ( ) chains i melts an polymer ! (Floryre id eal )

b bφ ξ φ −→ →

3/4de Gennes: bξ φ−

introduction to polymer theory - universiteit utrecht

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