Polymers as Long Molecular Chains

Number of monomer units in the chain N >> 1.

For synthetic macromolecules usually ~ .

For DNA macromolecules ~ .

4

2 10

10 ÷÷÷÷

N

10

9 10

10 ÷÷÷÷

N

Polymers as Long Molecular Chains

Poly(ethylene)

−

−

−

−

−CH2 CH2 CH2 CH2

Poly(styrene)

−

−

−

−

−CH₂ CH CH₂ CH

Poly(vinyl chloride) Cl

Cl

CH CH

CH

CH₂− − ₂− −

−

Electron microscope picture of bacterial DNA partially released from its native shell. (Source: Dictionary of Science and

Technology, Christopher Morris, ed. , San Diego, CA: AcademicPress, 1992.)

Polymers as Long Molecular Chains

Physical properties of polymers are governed by three main factors:

• Number of monomer units in the chain, N, is large: N >> 1.

• Monomer units are connected in the chain.

⇒⇒⇒

⇒ They do not have the freedom of independent motion (unlike systems of disconnected particles, e.g. low molecular gases and liquids). ⇒⇒⇒⇒ Polymer systems are poor in entropy.

• Polymer chains are generally flexible.

History of Polymer Physics

• Discovery of chain structure of polymer molecule

H.Staudinger, 1920-1930

• First papers in polymer physics:

molecular explanation of rubber high elas- ticity

W.Kuhn, E.Guth, H.Mark, 1930-1935

•“Physico-Chemical” Period (1935-1965) P.Flory, V.A.Kargin

• Discovery of DNA double helix Watson and Crick, 1953

• Penetration of physical methods to polymer science (from 1965)

I.M.Lifshitz (Russia), P.de Gennes (France),

S.Edwards (England)

Now polymer physics is an important sub- field of condensed matter physics, basis for

“Soft Condensed Matter Physics”

Flexibility of a Polymer Chain

Rectilinear conformation of a poly(ethylene) chain corresponding to the minimum of the energy. All the monomer units are in trans- position. This would be an equilibrium conformation at T = 0.

At T 0 due to thermal motion the deviation from the minimum-energy conformation are possible. According to the Boltzmann law the probability of realization of the conformation with the excess energy U over the minimum- energy conformation is

) exp(

)

( kT

U U

p ^~ −−−−

.

C

C H H

H H C

C H H

H H

C

C H H

H H C

C H H

H H C

C H H

H H

≠

Positions corresponding to = 120º and 240º- gauche rotational isomers, = 0º - trans rotational isomers. Gauche isomers induce sharp bends of the chain and give dominant contribution to chain flexibility.

For carbon backbone the valency angle is fixed (for different chains 50º< <80º ),

Rotational-Isomeric Flexibility Mechanism

however the rotation with fixed (changing the angle of internal rotation ) is possible.

Any value gives rise to the deviations from rectilinear conformation, i.e. to chain flexibility.

U

ϕ

0 120⁰ 240⁰ 360⁰ U1

∆ ^{(at room temperature)}

/ 6

. 0

/ 1

/

1 3

mol kcal

kT

mol kcal

mol kcal

U

≈

<

∆

≈ C

C

C γ

ϕ

γγγγ γγγγ

γγγγ ϕϕϕϕ

≠≠≠≠ 0 γγγγ

ϕϕϕϕ ϕϕϕϕ

Persistent Flexibility Mechanism

In the case when rotational isomers are not allowed (e.g. for -helical polypeptides or DNA double helix) small thermal vibrations around the equilibrium position of atoms are still possible; accumulation of these vibrations over large distances along the chain gives rise to the deviations from the straight conformation to the chain flexibility. This is a persistent flexibility mechanism, it is analogous to the flexibility of a homogeneous elastic filament.

⇒ α

Freely-Jointed Flexibility Mechanism

The flexibility is located in the freely- rotating junction points. This mechanism is normally not characteristic for real chains, but it is used for model theoretical calculations.

l

Portrait of a Polymeric Coil

(freely-jointed chain of N segments;

interactions between the segments are not taken into account).

• Chain trajectory is analogous to the trajectory of a Brownian particle.

• The volume fraction occupied by the monomer units inside coil is very small.

Inside the coil there are many “holes”.

Polymer coil conformations can be realized in dilute polymer solutions when macromolecules do not overlap.

A typical con- formation of a polymer coil. The freely jointed chain of segments has been simulated on a computer in three- dimensional space.

104

1 Homopolymers: all monomer units are the same.

Copolymers : monomer units of different types.

(for example, proteins - 20 types of units DNA - 4 types of units ).

Sequence of monomer units along the chain is called primary structure.

2 Branched macromolecules

a) Comb-like c) Randomly branched b) Star-like d) Polymer network

Types of Polymer Molecules

3 Ring macromolecules

a) unknotted ring macromolecule b) knotted ring macromolecule

c) tangling of two ring macromolecule d) olympic gel

e) tangling of two complementary strands into a double helix

Topological restrictions

Traditional classification of physical states (gases, liquids, crystals) is not informative for polymer materials.

partially crystallised liquid perfect crystal polymer (polymer melt)

Classification for polymer materials:

1 Partially crystalline state

2 Viscoelastic state (polymer melt) 3 Highly elastic state ( e.g. rubbers)

4 Glassy state ( e.g. organic glasses from poly(styrene), poly(methylmethacrylate), poly(vinyl chloride)).

Possible Physical States for Polymer

Materials

Polymer Solutions

a) Dilute polymer solution;

b) Crossover from dilute to semidilute solution;

c) Semidilute solution;

d) Concentrated solution;

e) Liquid-crystalline solution;

Ideal polymer сhain: interactions of monomer units which are far from each other along the chain are neglected.

Polymer chains behave as ideal ones in the so- called - conditions (see below).

Let us consider ideal N-segment freely-jointed chain ( each segment of length l ).

Θ

- end-to-end vector,

The size of the coil is characterized by

R &

= 0 R&

R 2

R ∝

R &

O

u &

u &

u &

u &

u &

(different segments are uncorrelated) ⇒ Nl

L Ll

Nl u

R ^N

i i = = =

= ^∑

= ,

1

2 2

2 &

L - contour length of the chain

Thus, the conformation of an ideal chain is far from the rectilinear one. Ideal chain forms an

entangled coil. The chain trajectory is equivalent to the trajectory of a Brownian particle.

l N R

R ∝ ² = ¹² , ^R _<< ^L

0

1 1

1 1 1

2 1 1

2

1 1 1

1 2

=

+

=

=

 =



 









= 

∑ ∑

∑ ∑

∑

∑ ∑

∑ ∑

∑

∑

= =

= =

=

= =

= =

=

=

N i

N

j i j

N i

N

j i j

N

i i

N i

N

j i j

N i

N

j i j N

j j

N

i i

u u

u u u

u u R

u u u

u R

&

&

&

&

&

&

&

&

&

&

&

i j ≠

i j ≠

R &

O

u &

u &

u &

u &

u &

The conclusion R~N^1/2 is valid for ideal chain with any flexibility mechanism. E.g., let us consider the model with fixed valency angle between the segments of length b and free internal rotation ( ).

γγγγ

0 )

(ϕϕϕϕ = u

ij ij

j

iu b

u& & = ² cosθθθθ , θθθθ - angle between segments

i and j

⇓

Model with fixed valency angle

ϕϕϕϕ γγγγ

b

As before

but now 0

, ^{2 2}

1 1 1

2 2

≠

= +

= ∑ ∑ ∑

= =

=

j i

i N

i N

j i j

N

i i

u u

b u

u u u

R

&

&

&

&

i j ≠

∑ ∑= =

+

=

i

N

j ij

b Nb

R

2 2

2

cos θθθθ

( )

2 ,

1 ,

cos cos

cos cos

cos cos

γγγγ γγγγ

γγγγ θθθθ

γγγγ θθθθ

=

=

=

+ +

i i

i i

By continuing these arguments, we have

( )

k i

i γγγγ

θθθθ cos

cos _{, +} =

( )

cos 1

2 cos

cos 1

2 cos

cos 2

cos 2

2 2

1 2 2

1 1 2

2

1 1 ,

2 2

2

− ⇒ +

=

− = +

=

= +

=

= +

=

∑

∑ ∑

∑ ∑

=

=

−

=

=

−

= +

γγγγ γγγγ

γγγγ γγγγ

γγγγ θθθθ

Nb Nb

b Nb

b Nb

b Nb

R

N i

N k i

i N

k N i

i N

k i i k

γγγγ γγγγ cos 1

cos

21

2

−

= Nb + R

i i +1 i + 2

i i +1 γγγγ

•

Average size of the macromolecule is

proportional to for this model we have entangled coil as well. This is a

general property of ideal polymer chains independently of the model.

Conclusions

cos . 1

cos

2 1

1 2

γγγγ γγγγ

−

= +

∝ R N b

R

2

N1 ⇒

• At the value of R is larger than for the freely-jointed chain. At the

relationship is reverse.

900

γγγγ <

900

γγγγ >

Persistent Length of a Polymer Chain

( ) ( )

( )

( )

γγγγ

γγγγ γγγγ

θθθθ

cos

~ ln

~ , exp

cos exp ln

cos ln

exp

cos ln

exp cos

cos _,

b l

l s

b k kb

k k

k i i

=

−

=

 =

 

−

=

−

=

=

=

+ =

Let us return to the formula derived for the model with fixed valency angle

Here s=kb is the contour distance between two monomer units along the chain.

(

)

s u o u

exp ~

cosθθθθ ^&_{( ),}^&_{( )} ∝ −

) 0 ( u

) (s s u

This formula was derived for the model with fixed valency angle , but it is valid for any model: orientational correlations decay

exponentially along the chain. The characteristic length of this decay, , is called a persistent

length of the chain.

At s << the chain is approximately rectilinear, at s >> the memory of chain orientation is lost.

Thus, different chain segments of length can be considered as independent, and

γγγγ

~l

l L l l

R L

R ~ ~

~ ²

2 ∝ ∝

∝

~l

~l

~l

Therefore, R is always proportional to ._L^{1 2}

(

)

s u o u

exp ~

cosθθθθ^&_{( ),}^&_{( )} ∝ −

Advantage of l : it can be directly

experimentally measured.

Advantage of : it has a direct

microscopic meaning.

We always have . Let us examine this relationship for the model with fixed valency angle.

Kuhn Segment Length of a Polymer Chain

We know that for ideal chain R² ~ L Kuhn segment length l is defined as

L R

l = ² (at large L)

Thus the equality is exact by definition.R² = Ll

Since ⇒

−

= +

−

= +

cos 1

cos 1

cos 1

cos

21

2

γγγγ γγγγ γγγγ

γγγγ Lb Nb

R

γγγγ γγγγ cos 1

cos 1

−

= b + l

On the other hand, = ⇒ cos

ln

~

γγγγ l b

γγγγ γγγγ γγγγ

cos 1

cos cos 1

~ ln

−

= + l l

~l

l l ∝ ~

Stiff and Flexible Chains

Now we have a quantitative parameter that

characterizes the chain stiffness: Kuhn segment length l (or persistent length ) .

The value of l is normally larger than the contour length per monomer unit .

The ratios for most common polymers are shown below.

l l ∝

~

l0

l0

l

Poly(ethylene oxide) 2.5

Poly(propylene) 3

Poly(ethylene) 3.5

Poly(methyl methacrylate) 4

Poly(vinyl chloride) 4

Poly(styrene) 5

Poly(acrylamide) 6.5

Cellulose diacetate 26

Poly(para-benzamide) 200

DNA (in double helix) 300

Poly(benzyl-l-glutamate) (α-helix) 500

From macroscopic viewpoint a polymer chain can be represented as a filament characterized by two lengths:

• Kuhn segment l

• characteristic chain diameter d

(DNA, helical polypeptides, aromatic polyamides etc.)

(most carbon backbone polymers)

Stiff chains: l >> d

Flexible chains: l ∝ d l

d

Polymer Volume Fraction Inside Ideal Coil

End-to-end vector is ⇒

the volume of the coil .

Polymer volume fraction within the coil is very small for long chains.

( )

2 Ll

R

R ∝ ∝

( )

3

3

4 R Ll

V ∝ π ∝

( )

4 ^{12 2}

2 3 2 1

2 2

3

2  <<



 



 



 



∝ 

∝

∝

Φ l

d L

l l

L d Ll

L πd

Radius of Gyration of Ideal Coil

Center of mass of the coil ,

where is the coordinate of the i-th monomer unit.

Radius of gyration, by definition, is .

It can be shown that for ideal coils .

The value of can be directly measured in the light scattering experiments (see below).

∑=

= ^N

i ri

r⁰ N ₁

1 &

&

r&i

( )

∑= −

= ^N

i ri r

S N

1

2 0

2 1 & &

Ll R

S 6

1 6

1 ₂

2 = =

S2

Gaussian Distribution for the End-to-End Vector for Ideal Chain

-probability distribution for the end-to- end vector of N - segment freely-jointed chain.

Since each step gives independent contribution to , by analogy with the trajectory of a Brownian particle

) (R

P_N &

R&

2 ) exp( 3 2 )

( 3 )

( ₂

2 32

2 Nl

R R Nl

P_N = −

π

&

-Gaussian distribution. Therefore, the ideal coil is sometimes called a Gaussian coil. Since is a probability distribution,P_N (R&)

Also, .P_N (R&) = P_N (R_x)P_N (R_y )P_N (R_z) 1

)

( ³ =

∫ P_N R& d R

.

2 ) exp( 3 2 )

( 3 )

( ₂

2 12

2 Nl

R R Nl

P_{N x} = − ^x

π

For other models, since orientational correlations decay exponentially, Gaussian distribution is still valid:

The value of

undergoes strong fluctuations.

R&

2 ).

exp( 3 2 )

( 3

2 ) exp( 3 2 )

( 3 )

(

2 2 32

2

2 2 32

2

〉

− 〈

〉

= 〈

=

−

=

R R R

Nl R R Nl

P_N

π π

&

This form of is independent of any specific model.

) (R

P_N &

) ( _x

N R P

l

N¹² R^x

Elasticity of a Single Ideal Chain

For crystalline solids the elastic response ap- pears, because external stress changes the equilibrium inter-atomic distances and

increases the internal energy of the crystal (energetic elasticity).

Since the energy of ideal polymer chain is equal to zero, the elastic response appears by purely entropic reasons (entropic elasticity).

Due to the stretching the chain adopts the less probable conformation its entropy

decreases. ⇒

R&

f&

f&

∆l

According to Boltzmann, the entopy )

( ln

)

(R k W R

S & _N &

=

Where k is the Boltzmann constant and

is the number of chain conformations compa- tible with the end-to-end distance .

) (R

W_N &

R&

const R

P k

R S

R P

const R

W

N

N N

+

=

⇒

⋅

=

) ( ln

) (

) ( )

( &

&

&

&

const Ll

TS kTR TS

E

F = − = − = +

2

3 ²

The free energy F :

Ll R kT R

f F &

&

& 3

∂ =

= ∂ dF

R d

f& & =

⇒



 

−



 



= 

Ll R R Ll

P_N

2 exp 3

2 ) 3

(

2 2 3

ππππ

&

But,

const Ll

R kR

S = − +

2 ) 3

(

& 2

Thus,