Distribution of Chain Ends Free Energy of Single Chains

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Lecture 09

Distribution of Chain Ends

Free Energy of Single Chains

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Chain Dimensions

• Problem:

→

So far, we know about average values for the distance between chain ends But, what about the segments between the chain end-points ???

• Motivation:

the distribution of segments between the end-points is related to the number of conformations of a chains while realising a given end-to-end distance

related to chain entropy

→

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Distribution of Chain Ends (1-D)

• Motivation: the detailed distribution gives more information about the “nature of conformational states” than the average value

• Idea: represent chain conformation as a random walk

• Consider: 1D random walk

0 x

-1 +1

N = N+ + N-

x = N+ - N-

• Approach: number of possible trajectories to reach point x after N steps

(N

+

+ N

-

)!

ω (N,x) =

N

+

! N

-

!

(total steps) (displacement)

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Distribution of Chain Ends (3-D)

• Consider: 3D random walk

• In words: P3D (N,R) is probability to find chain end (2) at distance R if chain end (1) is held fixed at origin.

P

1D

(N,x) × P

1D

(N,y) × P

1D

(N,z) P

3D

(N,x,y,z) =

< R

E2

> = < x

²

> + < y

²

> + < z

²

>

< x

²

> = < y

²

> = < z

²

> = < R

E2

>/3

P

3D

(N,R) = 3

2πNb

²

e

( )

^3/2 ⁻ ³² ^Nb^R²²

< R

E2

> = Nb

²

recall,

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Distribution of Chain Ends (3-D)

• Approach: Represent chain as a 3D random walker. Then for a chain of N segments

• Note: definition of probability P3D (N,R)

number of ways to realize a conformation with N segments and end-to-end distance R The probability to find chain ends at distance R is given by Gaussian distribution

total number of conformations of chain with N segments

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Free Energy of Single Chain

• Approach: F = U − TS = −TS

where Ω(N, R) is the number of realisations (conformations) of a chain of N segments and end-to-end distance R

(Helmholtz free energy) for ideal chains (i.e. neglecting interactions)

• Definition: S = kB ln{Ω(N, R)}

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Free Energy of Single Chain

• Starting point:

with Ω(N, R) ≃ number of conformations of a chain with N segments and end- S = kB ln{ Ω(N, R) }

to-end distance R

we have: P3D (N,R) = # conformation of chain with N segments and RE

total # conformation of chain with N segments P3D (N,R) = Ω(N, R)

∫ Ω(N, R) dR Thus, for the chain entropy

S = kB ln{Ω(N, R)} = kB ln{P3D (N,R)} + kB ln{∫ Ω(N, R)dR}

S =

P

3D

(N,R) = 3

2πNb

²

e

( )

^3/2 ⁻ ³² ^Nb^R²²

3

− 2

Nb² R²

kB 3

+ 2kB 3 2πNb²

( )

ln + kB ln{∫ Ω(N, R)dR}

do not depend on R ; S (N, 0)

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Free Energy of Single Chain

Since interest is in relative values of thermodynamic variables, disregard S(N,0) in the following.

Consider: force necessary to extend a chain

Thus, S = 3

− 2

Nb² R² kB

For the Helmholtz Free Energy, F= -TS = 3

2 Nb² R² kBT

force =

∂F =∂R Nb²

3kBT R = constant × distance ≃ Hooke Law Polymers (flexible) are entropy springs

constant ~ T constant ~

Nb² 1

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Free Energy of Single Chain

Linear increase indicates

coil state (Hooke’s law holds) Molecules stretched Breaking of bonds

Hooke’s law no longer holds

Fisher et al. J. Physiol. (1999)

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Lecture 09-1

Scattering Technique

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Scattering Technique

* XRD - atomic crystal structure (d-spacing)

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Scattering Technique

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Scattering Technique

Experiment

the scheme sketched below is particularly frequent in X-ray, neutron scattering

(Wide Angle

X-ray Scattering) (Small Angle X-ray Scattering) scattering angle: 2θ : 5~40° (WAXS)

2θ : < 0.6° (SAXS)

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Scattering Technique

Interference Calculation (general) consider:

Note: the only difference between waves scattered from (1) and (2) is a phase shift

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Scattering Technique

Interference Calculation

For N scattering centers:

Thus:

In continuum representation

* ρ(r) – scattering length density

Scattered field is proportional to the Fourier Transform of the scattering length distribution (holds for light, X-ray, neutron, electron scattering)

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Scattering Technique

For an arbitrary distribution of scattering centeres we have shown that

a) The scattered field is proportional to the Fourier transform of the scattering length density

b) The scattered intensity is proportional to the autocorrelation function of the scattering length density

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Scattering Technique

Conclusions

1. Scattering measures the correlation between scattering centers NOT absolute positions (phase problem)

for X-ray scattering 2 different experiments are distinguished 2. Length scale probed in scattering experiments d ~ (2π/q)

a) small-angle scattering (2θ < 0.6°) : d ~ 1-100 nm

b) wide-angle scattering (2θ > 5°): d ~ 0.1-0.5 nm

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What does “q-range” mean?

A typical small angle scattering intensity is plotted versus q, in the range qmin-qmax defined by the experimental set-up and usually fixed by geometric limitations

High q domain :

The window is very small : there is a contrast only at the interface between the two media. This domain, called the Porod's region, gives information about the surfaces.

Intermediary zone :

The window is of the order of the elementary bricks in the systems. The form factor P(q) can be measured (size, shape and internal structure of one particle).

Low q domain :

When the observation window is very large, the structural order can be obtained : it is the so-called structure factor S(q), which allows to calculate the interactions in the system.

q = 4πsinθ/λ

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Scattering Technique

* Length scale probed in scattering experiments d ~ (2π/q)

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Scattering Technique