Polymers as Long Molecular Chains

− − − − − CH2 CH2 CH2 CH2 Poly(ethylene)

− − − − − CH2 CH CH2 CH Poly(styrene)

− − − − − CH2 CH CH2 CH Poly(vinyl chloride) Cl Cl

Number of monomer units in the chain N >> 1. For synthetic macromolecules usually N ~ 10 2 ÷ 10 4 . For DNA macromolecules N ~ 10 9 ÷ 10 10 . Polymers as Long Molecular Chains

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.) 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

H H H H H H H H H H C C C C C C C C C C HH HH HH HH HH

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 U p(U ) ~ exp(− ) kT . Rotational-Isomeric Flexibility Mechanism

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

C C

however the rotation with fixed ϕ (changing the angle of internal rotation γ ) is possible. Any value γ ≠ 0 gives rise to the deviations from rectilinear conformation, i.e. to chain flexibility. ≈ U U1 3 kcal/ mol ∆ <1 kcal/ mol U 1 kT ≈ 0.6 kcal/ mol (at room temperature) ∆ ϕ 0 1200 2400 3600 Positions corresponding to ϕ = 120º and 240º- gauche rotational isomers, ϕ = 0º - trans rotational isomers. Gauche isomers induce sharp bends of the chain and give dominant contribution to chain flexibility. Persistent Flexibility Mechanism

In the case when rotational isomers are not allowed (e.g. for α -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. Freely-Jointed Flexibility Mechanism

l

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. Portrait of a Polymeric Coil

(freely-jointed chain of N segments; interactions between the segments are not taken into account). A typical con- formation of a polymer coil. The freely jointed chain of 10 4 segments has been simulated on a computer in three- dimensional space.

• 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. Types of Polymer Molecules 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 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 Possible Physical States for Polymer Materials 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)). 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 Θ - conditions (see below). Let us consider ideal N-segment freely-jointed chain ( each segment of length l ).

& u& & u2 3 u u& 4 1 & uN O & R & & R - end-to-end vector, R = 0

The size of the coil is characterized by R ∝ R 2 & u& & u2 3 u u& 4 1 & uN O & R N  N  N N 2 =  ∑ &  ∑ & = ∑∑& & R  ui  u j  uiu j  i=1  j=1  i==11j N N N N N 2 = ∑∑ & & = ∑ &2 + ∑∑ & & R uiu j ui uiu j i==11j i=1 i==11j j ≠ i N N ∑∑ & & = ⇒ uiu j 0 (different segments are uncorrelated) i==11j j ≠ i N 2 = ∑ &2 = 2 = = R ui Nl Ll, L Nl i=1 L - contour length of the chain

1 R ∝ R2 = N 2l, R << L

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. Model with fixed valency angle The conclusion R~N1/2 is valid for ideal chain with any flexibility mechanism. E.g., let us consider the model with fixed valency angle γ between the segments of length b and free internal rotation ( u ( ϕ ) = 0 ). ϕ

b γ

N N N 2 = ∑ 2 + ∑∑ & & 2 = 2 As before R ui ui u j , ui b i=1 i==11j j ≠ i & & ≠ but now ui u j 0 & & = 2 θ θ angle between segments ui u j b cos ij , ij - i and j ⇓ N N 2 = 2 + 2 ∑∑ θ R Nb b cos ij i ==11j i +1 γ θ = γ i cos i,i+1 cos

θ = γ γ = ()γ 2 cos i,i+2 cos cos cos i +1 i + 2 i

By continuing these arguments, we have θ = ()γ k cos i,i+k cos N N −i 2 = 2 + 2 ∑∑ θ = R Nb 2b cos i,i+k i=11k= N N −i k = Nb2 + 2b2 ∑∑()cosγ = i=11k= N cosγ = Nb2 + 2b2 ∑ = i=11− cosγ cosγ = Nb2 + 2Nb2 ⇒ 1− cosγ

1+ cosγ R2 = Nb2 1− cosγ Conclusions

1+ cosγ • R ∝ R2 = N 1 2b . 1− cosγ

Average size of the macromolecule is proportional to N 1 2 ⇒ for this model we have entangled coil as well. This is a general property of ideal polymer chains independently of the model.

• At γ < 90 0 the value of R is larger than for the freely-jointed chain. At γ > 90 0 the relationship is reverse. Persistent Length of a Polymer Chain

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

θ = ()γ k = (γ )= cos i,i+k cos exp k ln cos  kb  = exp()− k ln cosγ = exp−  =  b lncosγ  ~ ~ = exp()− s l , l = b lncosγ

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

u(0)

s u(s)

θ ∝ (− ~) cos u&(o),u&(s) exp s l θ ∝ (− ~) cos u&(o),u&(s) exp s l

This formula was derived for the model with fixed valency angle γ , but it is valid for any model: orientational correlations decay exponentially along the chain. The characteristic ~ length of this decay, l , is called a persistent length of the chain. ~ At s << l the chain is approximately rectilinear, ~ at s >> l the memory of chain orientation is lost. ~ Thus, different chain segments of length l can be considered as independent, and

2 L ~2 ~ R ∝ R ∝ ~ l ∝ Ll l Therefore, R is always proportional to L 1 .2 Kuhn Segment Length of a Polymer Chain We know that for ideal chain R2 ~ L Kuhn segment length l is defined as l = R2 L (at large L) Thus the equality R 2 = Ll is exact by definition. Advantage of l : it can be directly experimentally measured. ~ Advantage of l : it has a direct microscopic meaning. ~ We always have l ∝ l . Let us examine this relationship for the model with fixed valency angle. 1+ cosγ 1+ cosγ Since R2 = Nb2 = Lb ⇒ 1− cosγ 1− cosγ 1+ cosγ l = b 1− cosγ ~ b On the other hand, l = ⇒ lncosγ ~ 1+ cosγ l l = lncosγ 1− cosγ Stiff and Flexible Chains Now we have a quantitative parameter that characterizes the chain stiffness: Kuhn segment ~ length l (or persistent length l ∝ l ) . The value of l is normally larger than the contour length per monomer unit l 0 .

The ratios l l 0 for most common polymers are shown below.

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 d

l

From macroscopic viewpoint a polymer chain can be represented as a filament characterized by two lengths: • Kuhn segment l • characteristic chain diameter d Stiff chains: l >> d (DNA, helical polypeptides, aromatic polyamides etc.)

Flexible chains: l ∝ d (most carbon backbone polymers) Polymer Volume Fraction Inside Ideal Coil

End-to-end vector is R ∝ R 2 ∝ () Ll 1 2 ⇒

4 the volume of the coil V ∝ π R3 ∝ ()Ll 3 2. 3

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

πd 2 L 4 d 2  l 1 2  d 2 Φ ∝ ∝ ∝     <<1 ()Ll 3 2 L1 2l 3 2  L   l  Radius of Gyration of Ideal Coil

N & = 1 ∑ & Center of mass of the coil r 0 r i , N = & i 1 where r i is the coordinate of the i-th monomer unit.

Radius of gyration, by definition, is

N 2 = 1 ∑ ()& − & 2 S ri r0 . N i=1

It can be shown that for ideal coils

1 1 S 2 = R2 = Ll . 6 6

The value of S 2 can be directly measured in the light scattering experiments (see below). Gaussian Distribution for the End-to-End Vector for Ideal Chain

& P N ( R ) -probability distribution for the end-to- end vector of N - segment freely-jointed chain. Since each step gives independent contribution to R & , by analogy with the trajectory of a Brownian particle

3 3 3R2 P (R&) = ( ) 2 exp(− ) N 2πNl 2 2Nl 2

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

& 3 = ∫ PN (R)d R 1.

& = Also, P N ( R ) P N ( R x ) P N ( R y ) P N ( R z ) . 2 3 1 3R P (R ) = ( ) 2 exp(− x ) N x 2πNl 2 2Nl 2

PN (Rx ) The value of R& undergoes strong fluctuations. 1 R N 2l x

For other models, since orientational correlations decay exponentially, Gaussian distribution is still valid: 3 3 3R2 P (R&) = ( ) 2 exp(− ) = N 2πNl 2 2Nl 2 3 3 3R2 = ( ) 2 exp(− ). 2π 〈R2 〉 2〈R2 〉

& This form of P N ( R ) is independent of any specific model. Elasticity of a Single Ideal Chain

& f

∆l 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). & & R f

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. According to Boltzmann, the entopy & = & S(R) k lnWN (R) & Where k is the Boltzmann constant and WN (R) is the number of chain conformations compa- & tible with the end-to-end distance R . & = ⋅ & ⇒ WN (R) const PN (R) & = & + S(R) k ln PN (R) const 3 2 &  3   3R2  But, P (R) =   exp−  N  2πLl   2Ll  & 3kR2 Thus, S(R) = − + const 2Ll The free energy F : 3kTR2 F = E −TS = −TS = + const 2Ll & & & ∂F 3kT & fdR = dF ⇒ f = & = R ∂R Ll