Chain Models

Home , Ideal chain

Johannes Gutenberg-University Mainz

Institute for Physics

Chapter 3: CHAIN MODELS

3.1 Definitions

Configuration is the arrangement of a polymer in space.

bond angle θ bond-rotation angle φ

bond length l0

Conformations are all accessible configurations a polymer can attain as a consequence of its flexibility.

We can reduce the polymer to the chain backbone. It is well defined by N bond vectors

Ri (i=1,…, N) or (N+1) position vectors ri (i=0,..., N). N is the degree of polymerization.

PD Dr. Silke Rathgeber Johannes Gutenberg-University Mainz phone:phone: +49 +49 (0) (0) 6131/ 6131/ 392-3323 392-3323 Institute for Physics – KOMET 331 Fax:Fax: +49 +49 (0) (0) 6131/ 6131/ 392-5441 392-5441 Staudingerweg 7 email:email: [email protected] [email protected] 55099 Mainz webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/ Germany

polar coordinates (θi,φi,|Ri|)

Degree of freedom (θ1, φ1, φ2) and the start vector r0 are undefined ⇒ global degree of freedom of rotation and translation ⇒ all others internal degree of freedom

Restrictions in the degree of freedom: 1. fixed bond length: covalent bond energy >> thermal energy 2. fixed bond angle: determined by electronic configuration of atoms building the backbone 3. bond-rotation angle: relatively unrestricted

Comparison of single-bond in ethane and double-bond in ethene:

single-bond double-bond

PD Dr. Silke Rathgeber Johannes Gutenberg-University Mainz phone: +49 (0) 6131/ 392-3323 Institute for Physics – KOMET 331 Fax: +49 (0) 6131/ 392-5441 Staudingerweg 7 email: [email protected] 55099 Mainz webpage: http://www.cond-mat.physik.uni-mainz.de/~rathgebs/ Germany

3.2 Characteristic quantities

N End-to-End distance: REN=−=rr0 ∑ R j ji= Thermal movements: polymers adopt with time a large number of conformations, thus, thermal averages are considered. For an isotrope system: RRE = E = 0 look at the mean-square average (norm):

NN N N N 2 22 R E ==RRR = RRR = +2 RR E EE∑∑ jk ∑ j ∑ ∑ jk jk==11 j = 1 jkj ==+ 1 1

General definition of the radius of gyration Rg :

The radius of gyration of a mass about a given axis is a distance y z Rg from the axis at which an equivalent mass is thought of as a point mass. The moment of inertia of this point mass and original mass about the axis are the same. x

22 2 2 Rg =++RRRgx,,, gy gz

Radius of gyration Rg is defined as the (average) of second moment (relative to the center of mass (rcm)) of the mass distribution:

N 2 Ri-1 ∑ mrrjjcm()− Ri 2 2 j=0 RRg ==g N Ri+1 ∑ m j j=0 ri rij N Rj ∑ mrj j j=0 with rcm = N rj Rj+1 ∑ m j cm j=0

N N 2 1 2 1 In case all masses are identical mj = m: Rrrg = ∑ ()jcm- with rrcm= ∑ j N +1 j=0 N +1 j=0 NN 2 1 2 2 2 With the help of the Lagrange Theorem: R g = 2 ∑∑ rij with rrrij=−() i j (1)N + iji==+11

Characterization of the stiffness of a polymer:

How is the orientientation of the end-to-end vector relative to the first bond vector? e1 Projection HN of RE on the unit vector e1 R (pointing in the direction of R1) 1

HN is the sum of the projections of all bond RE vectors Rj on the direction of the first bond vector R1.

NNR R RR11E R 1 j HeRNE==1 =∑∑ R j = R11RRjj==11 1

The persistence length lP is defined as the thermal average in the limit of infinate long chain:

∞ R1Rj lHPN≡=lim lim NN→∞ →∞ ∑ j=1 R1

3.3 Chain models

2 For the calculation of the correlations between bond vectors Rj Rk and position vectors rij detailed assumptions about the chain statistics are necessary.

3.3.1 Freely rotating chain

φi-1 φi+1 φi+3

φi+4 φi φi+2

bond angles are fixed = θ0 bond lengths are fixed = l0 bond-rotation angles are evenly distributed over 0 ≤ φ≤ 2π

average over all orientations ϕk of vector Rk ⇒ projection on vector Rk−1 : Rkk=×RR-1cosθ 0 j ϕk φk

average over remaining ϕi : RRkj= R k-1 R j cosθ 0 θ0 Rk-1 Rk continue until k-1=j:

2 kj- 22 RRkj== R k-1 R j cosθθ 0 l 0 cos 0 with Rj = l0

( cosθ0 since θ0 fixed average not necessary but useful for freely jointed chain model)

End-to-End distance: N NNN 2 2 Nl ⎡ 2cosθθ [1− cos ]⎤ RR=+2 21cos RR =0 ⎢ +−θ 00⎥ E ∑∑∑jjk1cos−−θθ0 N (1cos) jjkj===+111 00⎣⎢ ⎦⎥

2 2 1cos+ θ0 For N>>1 ()cosθ0 < 1 : R EE=⇒∝Nl0 R N 1cos− θ0

2 22 Fully stretched chain ()cosθ0 = 1 : R EE=⇒=Nl00 R Nl

and the end-to-end distance of a chain with |i-j| elements replace N by ij− : ijl−−2 ⎡⎤2cosθθ [1 cosij− ] r 2 =+−000⎢⎥1cosθ ij 1cos−−−θθ0 ij (1cos) 00⎣⎦⎢⎥

Persistence length: ∞ lim RR1 j l0 lP ==∑ N →∞ j=1 R101cos− θ

Radius of gyration: NN 2 11222⎛⎞N + ⎛⎞1cos+ θ0 RrNlg =≈ 2 ∑∑ ij 0 ⎜⎟⎜⎟ (1)NN++−jij==+11 6⎝⎠ 11cos⎝⎠θ0

2 1122⎛⎞1cos+ θ0 For N>>1: R g ≈=Nl R 0 ⎜⎟E 61cos6⎝⎠− θ0

3.3.2 Freely jointed chain

θi-1 θi+1 θi+3

θ θi-2 θi i+2 θi+4

bond angles are evenly distributed over 0 ≤ θ ≤ 2π

bond lengths are fixed = l0 bond-rotation angles are evenly distributed over 0 ≤ φ ≤ 2π

⇒ Special case of a freely rotating chain with cosθ = 0

End-to-End distance: 2 N 2 Nl ⎡⎤2cosθθ [1− cos ] R =+−0 ⎢⎥1cosθ 00 =Nl 2 E 1cos−−θθ00N (1cos) 00⎣⎦⎢⎥ and for the end-to-end distance of a chain with |i-j| elementsreplace N by ij−

22 rijlij =− 0

Persistence length:

l0 llP = = 0 1cos− θ0

Radius of gyration: N 2 11(1)(1)22NN−+ g R =−=2 ∑ ijl00 l (1)NN+ ij< 6

22112 For N>>1: R g ≈=Nl RE 660

2 2 The end-to-end distance R E = Nl0 is in analogy to a 3-dimensional random walk!

Brownian motion:

Any minute particle suspended in a liquid (or gas) moves chaotically under the action of collisions with surrounding molecules. The intensity of this chaotic motion is increased with an increase in temperature. (R.Brown in 1827)

With a random velocity, a Brownian particle of size R will move in a tangled zigzag path, and will progress with time away from its initial location. The mean-square displacement of a Brownian particle is described by: 2 2 2 2 2 Δ=rrtrtt[]() −= ( 0) ∝ compare to R E =−[]rrN 00 = Nl

Δt l ≡ distance covered in Δ t 0 Nt≡

snap shots of the random flight in time intervals Δt

Probability distribution

To determine the probability distribution of any global property of a polymer we need to know the single probabilities for the occurrence of a certain conformation. The bond vectors in the freely jointed chain are not correlated, thus, the total probability is given by the product of the single probabilities. N Ψ=Ψ(){}Rjj∏ ()R , j=1 1 where the single probabilities are given by: Ψ()Rjj=−2 δ (Rl0 ) 4πl0

Question: What is the probability that a given vector R is equal to the end-to-end vector?

Ansatz: Φ=Ψ()R (RRRdR )(δ − )3N ∫ { j } Ej{ }

Calculation: Φ=Ψ()RRRRdR ( )(δ − )3N ∫ { jEj} { } N =−ΨdR dR... dRδ ( R R ) ( R ) ∫∫12 ∫Njj∑ {} j=1 1 with the definition of the delta-function δ ()redk= ikr (2π )3 ∫

1 N Φ=(R )dk dR dR ... dR exp[ ik ( R − R )] Ψ ( R ) 3 ∫∫∫12 ∫Njj∑ {} (2π ) j=1 1 N =−Ψdkexp( ikR ) dR ... dR exp( ikR ) ( R ) 3 ∫∫∫1 Njj∏ (2π ) j=1 1 N =−Ψdkexp( ikR )⎡⎤ dR exp( ikR ) ( R ) 3 ∫∫⎣⎦ (2π ) 14444244443 sin(kl ) = 0 kl0 up to here: same for the gaussian chain

with polar coordinates (θii φ r ) N 1 ⎡⎤sin(kl ) Φ=(RdkikR ) exp( ) 0 3 ∫ ⎢⎥ (2π ) ⎣⎦kl0 142 43 small apart for small kl0 approximation for large N: N 22N ⎡⎤sin(kl00 )⎛⎞⎛⎞ ( kl ) N ( kl 0 ) replace ⎢⎥≈−⎜⎟⎜⎟1 ≈ exp − ⎣⎦kl0 ⎝⎠⎝⎠66 vanishs also for large N! 1 ⎛⎞Nkl()2 Φ=(RdkikR ) exp( )exp − 0 3 ∫ ⎜⎟ (2π )⎝⎠ 6 2222 with kR=++ kxx R k yy R k zz R and k =++ k x k y k z 3 ⎛⎞⎛⎞332 R2 Φ=()R ⎜⎟⎜⎟22 exp − ⎝⎠⎝⎠22π Nl002 Nl

Result: 3/2 ⎛⎞⎛33R2 ⎞ Φ=()R ⎜⎟⎜22 exp − ⎟ ⎝⎠⎝222π Nl00 Nl ⎠ Gaussian distribution

Effective bond length

In case only local correlations are effective, so that

Nj− c Ψ=Ψ(R ) (RR , , R ,..., R ) with j << N {}jjjjjjc∏ ++12 +c j=1 the real chain can always be described by a freely jointed chain with larger effective bond length, 1 so that the relations R 22= Nl and R 22= R for large N still hold. E g 6 E

Characteristic ratio 22 RE l ClCl∞∞==⇒=lim 0 N →∞ 22 Nl00 l

1cos+ θ0 e.g. freely rotating chain C∞ = 1cos− θ0

Kuhn segment

is defined by a transformation of the real chain onto a freely jointed chain under 2 lk, Nk preservation of the end-to-end distance RE and the maximal possible end-to-end distance of the fully stretched chain RE,max

R 22==Nl and R Nl l0, N Ekk E,max kk

⇒ the number and the length of the (effective) segments are rescaled.

2 2 RE RE,max lNkk== and 2 REE,max R

Segments of length lk are the smallest units being statistically uncorrelated. The Kuhn segment is like the persistence length a measure of the chain stiffness. e.g. freely rotating chain

θ0 =°70.53 (tetrahedron angle)

1cos+ θ RNl22= 0 E 0 1cos− θ 0 l 0 θ0 1cos+ θ RNl==cos()θ 2 Nl 0 E,max 0 0 0 2 θ0/2 2 RE l0 ⇒=llk =2(1 + cosθ00 ) ≈ 2.45 × RE,max1cos− θ 0

2 RE,max1cos− θ 0 NNk ==2 ≈×0.33 N RE 2

3.3.3 Gaussian chain

On length scales much larger than the Kuhn length, the local nature of the chain segments plays no role. Many segments contribute to a Kuhn segment.

Assume that the effective segments are variable in length and follow a Gaussian-distribution with a standard deviation corresponding to the effective segment length:

3/2 2 ⎛⎞3 ⎛⎞3Rj Ψ=()R exp withlR22 = =Ψ () RRdR 23 j ⎜⎟22⎜⎟ jjjj∫ ⎝⎠22πll⎝⎠

Apart from this, the orientations of the bond vectors are completely free as for the freely jointed chain 3/2 ⎛⎞3 ⎛⎞3r 2 ⇒Φ()r = exp − ij Gaussian distribution ij ⎜⎟22⎜⎟ ⎝⎠22π ijl−−⎝⎠ ijl 1 ⇒=R 22Nland R 2 ≈ Nl 2 Eg6 22 in Φ(rRRij )and E, g the global properties are preserved!

⇒ easy calculation of characteristic quantities when using independent variable following Gaussian distributions

General description of chain configuration

Global properties can be calculated once the probability distribution of the chain configurations is known. Statistical mechanics: in thermal equilibrium Ψ(){Rj } can be obtained from the

Hamiltonian HR(){ j } ⎛⎞HR() ⎛⎞ HR () 1 { j } { j } 3N Ψ=({}R j )eZedR xp⎜⎟ − with normalization (state sum) = xp ⎜⎟ − {}j ZkT⎜⎟∫ ⎜⎟ kT ⎝⎠B ⎝⎠B

Enables the introduction of binding potential (see RIS model) and the calculation of the chain conformations in the presence of external fields, such as walls, electric, and stress fields.

Question: Which Hamiltonian describes the Gaussian chain?

3 3 N N − R2 N 2 ∑ j ⎛⎞3 2 2l Ψ=Ψ=()RR () ej=1 {}jj∏ ⎜⎟2 j=1 ⎝⎠2πl N 1 3kTB 2 ⇒=HR({}jj )2 R (harmonic potential) 2 l ∑ { j=1 spring constant

The beads of the chain appear to be connected by harmonic springs with spring constant 2 kkTl= 3/B see dynamics: Rouse model (overdamped coupled harmonic oscillator)

Stretching of the chain results in a reduction of accessible conformations and consequently in a reduction of the entropy of the polymer chain. The elastic forces between the segments are purely of entropic nature and are not due to changes in the internal energy. Thus, the Hamiltonian of a Gaussian chain is considered to describe a pseudo-potential rather than a real interaction potential.

Characteristic ratio for different polymers

polymer solvent freely rotating chain:

polyethylen 1cos+ θ0 C∞ = 1cos− θ0

with:

polystyrene, θ = 70.53o atactic ⇒≈C∞ 2

However, real chains polydimethylsiloxan are much stiffer!

various to to

RIS

Bond rotational potentials

The picture of unrestricted freedom of bond-rotation angles is oversimplified. Finite extension of atoms and side groups lead to "long-ranged" (excluded volume interactions, prohibition of chain intersections) as well as "short-ranged interactions". Due to sterical hindrance not all bond rotation angles have the same probability. Some bond- rotation angles are compared to other energetically favorable.

Bond rotational potentials HH=Φ=ΦΦΦΦΦ({ iiiiN} ) H (11 ,...,−+ , , 1 ,..., )

1 ⎛⎞HH(){ΦΦ} ⎛⎞ (){ } ΨΦ( ) =eZed xp −i with normalization = xp −i N Φ {}i ⎜⎟∫ ⎜⎟{}i ZkT⎝⎠BB ⎝⎠ kT

RIS (rotational isomeric state)-model (l and θ fixed) replaces the continuous variable Φi by a discret set of angles Φi,ν

e.g. ethane, methane, azethaldehyde

staggered conformation H H ()Φ=0 [1cos(3)] − Φ rotation barrier H as a measure of the sterical interaction 2 0

HkT0 >> B localized rotational states equal minima

Barrier height thermal energy at 300K ≈ 0.6 kcal/mole (1 Joule = 2.390 ×10−4 kcal)

ethane propylene azethaldehyde butane methane dimethylether methanethiol methylphosphine

e.g. n-butane

trans gauche

o minima are not equal transΦtt==180 : H 0 ++o + gaucheΦ gg=>60 : H 0 −−o − + gaucheΦ ggg==>300 : H H 0

1 ⎛⎞HHνν ⎛⎞ with the probability ΨΦ()ν = exp⎜⎟ − mit Zexp =∑ ⎜⎟ − Z ⎝⎠kTBBν ⎝⎠ kT

Independent bond rotation potentials

N 1cos++φθ 1cos HH(){}Φ=iii∑ () Φ⇒=C∞ × i=1 1cos−−φθ 1cos 142 43 ≈2 for freely rotating chain

o ±±oo e.g. polyethylene @ T =140 C : H gg=Φ≈±=0.5 kcal/mole; 120 ;θ 70.53 o Htt= 0;Φ≈ 0 1 ⎛⎞H cosθσ==−= ; exp⎜⎟ν 0.54 3 ⎝⎠kTB

1 ⎛⎞HHν σσ ⎛⎞ν cosΦ=∑∑ exp⎜⎟ − cos Φν =1- - =1-σ with Z= exp ⎜⎟ − =1+σσ + =1+2 σ ZkTν ⎝⎠B 22 ν ⎝⎠ kTB 1−σ ⇒Φ=≈⇒≈cos 0.22C 3.14 12+ σ ∞

exp still much smaller than experimental value: C∞ ≈ 6.7 => real chains are much stiffer!

Bond rotation potentials are not independent!

rotation around single bond: rotation around two bonds:

trans

CH3 CH3

n-pentane

gauche

n-butane energetically unfavourable

N Interdependent rotation potentials of adjascent bonds HH(){}Φiii=ΦΦ∑ ()−1, i=2 Ab-initio calculations: microscopic, quantum mechanic problem. Semi-empirical methods: "molecular force field" calculations based on Newton's mechanics and electrostatics. Atoms are replaced by beads and bonds by springs. Goal: find 3-dim structure with minimal energy.

e.g. polyethylene:

3.3.4 Worm-like chain (Kratky-Porod model)

Examples

lL<< lLP >> P lLP ≈ contour length L = maximum end-to-end distance of a polymer

bond lengths are fixed = l0 bond-rotation angles are evenly distributed over 0 ≤ φ≤ 2π bond angles {θi} subsequent segments should posses a certain stiffness against bending

Question: What is the persistence length and end-to-end distance of a worm-like chain? discrete case: tangential unit vector of the chain at position rj is given by unit vector ej parallel to the bond Rj

Curvature at the position rj: change in tangential unit vectors (ej-ej-1)

(ej-ej-1) (ej-ej-1)

θj ej ej ej-1 θj ej-1

rj x cm 2 Pythagoras (eej −=−jj−1 )(2 1 cosθ )

Ansatz: κ NN2 Bending energy is given by harmonic potential: HeeBjj=−=−∑∑()(−1 κ 1cosθ j ) 2 jj==22

If bending energy κ is large κ>>kBT, the bond angles θj will be small:

NN 1 22κ series expansion: cos()θ j ≈− 1θκθθjB + ... ⇒H =∑() 1 − cos jj = ∑ 22jj==22

⇒ for the distributions of the conformations

N −1 2 N ⎛⎞κκ⎛⎞2 ⇒Ψ(){}θ j =⎜⎟exp⎜⎟ − ∑θ j Gaussian distribution ⎝⎠22π kTBB⎝⎠ kT j=2 since the Φi=Φ are free, the results of the freely-rotating chain can be taken:

2 N 2 ⎡⎤ Nl0 2cosθθ [1− cos ] l0 RlE =+−⎢⎥1 cosθ and P = 1cos−−−θ ⎣⎦⎢⎥N (1cos)θθ 1cos the problem reduces to the calculation of cosθ

N −1 ⎛⎞kTB cosθθθθ=Ψ(){}ii cosd {} i = exp⎜⎟ − ∫ ⎝⎠2κ

l0 2κ llkTPB=≈0 for κ >> ⎛⎞kTB kT 1exp−−⎜⎟B ⎝⎠2κ

⎡ ⎛⎞kTBBN ⎛⎞ kT ⎤ 2 ⎢ 2exp⎜⎟−−− [1 exp ⎜⎟ ]⎥ 2 Nl ⎛⎞kT 22κκ 0 ⎢ B ⎝⎠ ⎝⎠⎥ R E =+−−1exp⎜⎟ ⎛⎞kTBB⎢ ⎝⎠2κ ⎛⎞ kT ⎥ 1exp−−⎜⎟N (1exp −− ⎜⎟ ) ⎝⎠22κκ⎣⎢ ⎝⎠ ⎦⎥

2 R E expressed as a function of lP, l0 and LNl= 0 N 2 ⎡ ⎤ ⎛⎞l0 RLllE =−−−−−()22()11PPP00 lll⎢ ⎜⎟⎥ l ⎣⎢ ⎝⎠P ⎦⎥

N ⎛⎞lNl ⎛ ⎞ for large N replace ⎜⎟1exp−≈−00 ⎜ ⎟ ⎝⎠llPP ⎝ ⎠

2 ⎡ ⎛⎞L ⎤ ⇒=RLllE ()22()1expPPP −−00 lll −⎢ −⎜⎟ −⎥ ⎣ ⎝⎠lP ⎦

e.g. DNA length 10-100μm extension to full length persistence length of 50nm Kuhn segments: 100nm dashed line: freely-jointed chain model solid line: worm like chain model

3kT z2 free energy: F = B 2 2 RE,0

∂Fz required force: fkT==3 B 2 ∂z RE,0

3.4 Polymer solutions

Up to now we neglected any "long-ranged" interactions involving chain segments well separated along the chain. Backfolding of the chain due to the high chain flexibility has the consequence that distant chain segments can become very close to each other. However, chain segments have finite extension and interpenetration of volume occupied by other segments is forbidden

This effect is called excluded volume interaction with the consequence that the extension of the coil will be larger than that of an "ideal chain" without excluded volume interactions. Excluded volume interactions lead to swelling! In the melt the interaction between segments in one chain is equal to the interaction with neighboring chains. The segment concentration is high and a segment of one chain is in the average surrounded by many segments belonging to other chains

⇒ excluded volume interaction is screened!

In the melt the chains behave like "ideal chains" and the simple chain model introduced so far hold. In solution the interaction between solvent molecules and monomers is in the general case different to the monomer-monomer interactions. Solvent-monomer interaction is temperature dependent and only at the so called θ temperature, both solvent-monomer as well as monomer- monomer interaction are equal.

Radius of PS in cyclohexane as a function of the solvent quality

Gaussian chain with excluded volume interaction

Also excluded volume interaction is short-ranged (of local nature), it effects distant segments on the chain. The concrete nature might be complicated but unimportant on larger scales.

1 HkTrrij, excl=υδ B( i−⇒ j ) in total H excl = υ kTrr B∑∑ δ ( i − j ) 2 ji==00

Here υ represents the excluded volume and δ the delta function.

For a Gaussian chain with excluded volume interactions we obtain for the probability distribution: ⎛⎞ ⎡⎤⎜⎟ HrG(){} j+ H excl(){} r j ⎜⎟3 2 υ Ψ∝−rrrrrexp⎢⎥ ∝− exp −−δ − (){}j ⎜⎟2 ∑∑∑()jj−1 () ji ⎢⎥kTB 22 l jij ⎣⎦⎜⎟144424443144424443 ⎜⎟interaction between interaction between ⎝⎠neighbouring segments distant segments Rigorous solution of the problem not possible!

Flory theory

Question: 2 What is the end-to-end distance RE of a chain with excluded volume interaction?

We start with a chain with radius R and N segments for a particle with arbitrary dimension d.

Excluded volume interaction is proportional to the number of pair contacts ∝ c2 where c is the concentration. N We obtain for the internal chain concentration: c = int Rd 2 d Energy for one pair contact fexcl≈ kT B υ() Tcl (υ excluded volume has dimension of a d- dimensional volume)

In mean field theory the specific correlation between monomers (local heterogenities) are 222 neglected ⇒ assumption: ccc→∝int

Total free energy by integration over total volume Rd :

2 d F≈= kTTRclυυ()dd2 kTT ()Nl excl Bint B Rd

Entropy contributes as elastic term relative to ideal chain:

RR22 FkTel≈= B2 kT B 2 RNlE () 2 d 2 Fel υ()TNl kTRB Total free energy: ≈+d 2 ()kTB R Nl

Minimum for: Rdd++223∝⇔∝∝lN RlN 3/(2) d + Nlν

=> Flory exponent ν =+3/(d 2) ν= 3/5 for d=3 3/4 d=2 1 d=1

Mean field approach (MFT)

The main idea of MFT is to replace all interactions to any one body with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. A many-body system with interactions is generally very difficult to solve exactly, except for extremely simple cases (Gaussian field theory, 1D Ising model.) The great difficulty (e.g. when computing the partition function of the system) is the treatment of combinatorics generated by the interaction terms in the Hamiltonian when summing over all states.

The goal of mean field theory (MFT, also known as self-consistent field theory) is to resolve these combinatorial problems. The main idea of MFT is to replace all interactions to any one body with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving MFT problems means that some insight into the behavior of the system can be obtained at a relatively low cost.

In field theory, the Hamiltonian may be expanded in terms of the magnitude of fluctuations around the mean of the field. In this context, MFT can be viewed as the "zeroth-order" expansion of the Hamiltonian in fluctuations. Physically, this means a MFT system has no fluctuations, but this coincides with the idea that one is replacing all interactions with a "mean field". Quite often, in the formalism of fluctuations, MFT provides a convenient launch-point to studying first or second order fluctuations.

In general, dimensionality plays a strong role in determining whether a mean-field approach will work for any particular problem. In MFT, many interactions are replaced by one effective interaction. Then it naturally follows that if the field or particle exhibits many interactions in the original system, MFT will be more accurate for such a system. This is true in cases of high dimensionality, or when the Hamiltonian includes long-range forces. The Ginzburg criterion is the formal expression of how fluctuations render MFT a poor approximation, depending upon the number of spatial dimensions in the system of interest.

While MFT arose primarily in the field of Statistical Mechanics, it has more recently been applied elsewhere, for example for doing Inference in Graphical Models theory in artificial intelligence.

3.6 Self-similarity and fractal dimension of polymers

Looking at ensemble averages the relation between n and R2 stays the same for any subsection of the chain with n segments as long as N>>n>>1.

The relation between mass and volume defines the fractal dimension Df

mass ∝∝ volume R Df

Df nR∝ ( 2 )

solid body 3-dim body Df = 3

2-dim Df = 2

1-dim Df =1

22 Polymers: ideal chain RNlD=⇒=f 2

3/5 with excluded volume interaction 3-dim RNl∝⇒= Df 53 3/4 2-dim RNl∝⇒= Df 43

1-dim RNlD∝ ⇒=f 1

3.7 Summary

characteristic quantities: Rg, RE, lP describe dimension & stiffness in an ensemble of polymers. All orientations have same probabilty in isotropic system ⇒ , =0

2 2 ⇒ look at the norms (mean square values) , 2 2 2 ⇒ geometrical considerations lead to , and lP in terms of and <(ri-rj) > ⇒ for the calculation of and <(ri-rj)2> assumptions about the chain statistics are necessary / probability to find a certain l,φ, θ

Determination of the probability distribution of any global property requires the knowledge of the single probabilities for the occurance of a certain conformation.

⇒ chain models

1. freely rotating chain l, θ = fixed & φ totaly free 2. freely jointed chain l = fixed & θ, φ = totaly free

2 2 and 2 2 ⇒ =Nl =Nl and its analogy to random walk (chaotic motion) of Brownian particles.

N total probability Ψ=Ψ(){}Rjj∏ ()R j=1

Nj− c only local correlations Ψ=Ψ(R ) (RR , , R ,..., R ) mitj << N {}jjjjjjc∏ ++12 +c j=1 ⇒.real chain can be transformed onto freely jointed chain with different segment length (and) segment number under preservation of global properties.

⇒ this led us to definition of characteristic ratio & Kuhn length (measure of chain stiffness). one effective segment is made up by several real bonds ⇒

3. Gaussian chain θ, φ = totaly free effective segments are variable in length and follow a Gauss-distribution with a standard deviation corresponding to the effective segment length

3/2 2 ⎛⎞3 ⎛⎞3Rj Ψ=()R exp withlR22 = =Ψ () RRdR 23 jjjjj⎜⎟22⎜⎟ ∫ ⎝⎠22πll⎝⎠ 3 N N ⎛⎞332 ⎛⎞N Ψ=Ψ=()R ()RR exp − 2 {}j ∏ jj⎜⎟22⎜⎟∑ j=1 ⎝⎠22πll⎝⎠j=1

global properties are preserved! statistical mechanics: Ψ(){Rj } can be obtained from the Hamiltonian HR(){ j }

⎛⎞HR() ⎛⎞ HR () 1 { jj} { } 3N Ψ=({}Rj )eZedR xp⎜⎟ − normalization = xp ⎜⎟ − {}j ZkT⎜⎟∫ ⎜⎟ kT ⎝⎠BB ⎝⎠ which Hamiltonian describes the Gaussian chain? N 1 3kTB 2 ⇒ harmonic potential HR(){}j = 2 Rj 2 l ∑ { j=1 spring constant chain appears to be made up by beads connected by harmonic springs ⇒ Rouse model (overdamped coupled harmonic oscillator)

4. RIS (Rotational Isomeric State model) l, θ = fixed replaces the continuous variable Φi by a discret set of angles Φi,ν where adjacent bonds are interdependent.

5. worm-like chain model l = fixed & φ = totally free restrictions to θ: subsequent segments posses a certain stiffness against bending. so far only local interactions involving close-by bonds excluded volume interactions

• interpenetration of volume occupied by other segments is forbidden. • short-ranged but can involve distant chain segments on one chain. • rigorous calculation of global properties is not possible.

⇒ Mean-Field Theory of Flory for the free energy one obtains for the extension of a polymer chain under excluded volume interactions R ∝ Nlν with the Flory-Huggins exponent ν = 3/5 for d=3 3/4 d=2 1 d=1 compare to ideal chain R ∝ Nl1/2

Looking at ensemble averages polymers are self-similar structures i.e. fractal objects as long as subsections of the chains are considered which comprise much more than only one and much less than N segments.

Literature

1. U.W. Gedde, “ Polymer Physics”, Chapman & Hall, London, 1995. 2. G. Strobl, “ The Physics of Polymers”, Springer-Verlag, Berlin, 1996. 3. H.G. Elias, “ Makromoleküle Band 1: Chemische Struktur und Synthesen”, Wiley-VCH, Weinheim, 1999. 4. S. Hoffmann, “Conformations of Polymer Chains”, Lecture Manuscripts of the 33th IFF Winter School on “Soft Matter – Complex Materials on Mesoscopic Scales”, 2002. 5. H. Frielinghaus, “ Flexible Polymers”, Lecture Manuscripts of the 35th IFF Winter School on “ Physics meets Biology – From Soft Matter to Cell Biology”, 2004. 6. K. Sturm, “Konformationen”, Lecture Manuscripts of the 22th IFF Winter School on “Physik der Polymere”, 1991.