18. Flory theory.Edwards model.
IN MY LAST LECTURE:

• Began talking about correlated random walks.
• Showed that if correlations are short range the Gaussian nature of walk prevails, with a different persistence length.
• Started to talk about the self-avoiding random walk. If R_N is the end to end distance of a N-step self-avoiding walk one finds empirically that the mean square end-to-end distance of self-avoiding walks scales as
  
  $$\langle{\bf R}_N^2\rangle\propto N^{2\nu}$$
  
  
  where $\nu$ depends on the dimensionality d of the medium in which the medium takes place, but is otherwise universal.
  
  
• Conceptually easiest way to determine $\nu$ for a lattice model is by counting walks.
  
  • Generate all self avoiding walks of length N by considering all possible ways one can add one step to walks of length N-1.
  • Compute the average value of R²_N.
  • Make a log-log plot of $\langle{\bf R}_N^2\rangle$ vs N.
  • Estimate $\nu$ from fit of slope.
• Today I want to discuss some approximate analytical methods to estimate $\nu$.
  • Flory theory.
  • Edwards model.

FLORY THEORY
Consider length R of chain as thermodynamic variable which at equilibrium fluctuates about its most probable value. This is somewhat analogous to how one can treat magnetization M in the Heisenberg or x-y models where in the ordered phase there are many equivalent directions of M.
Recalling that the polymer is made up of N monomers we find find for the density of monomers in a polymer environment:

$$\rho\propto\frac{N}{\langle R_N^2\rangle^{d/2}}$$

Represent self-avoidance constraint by repulsive monomer-monomer interaction:

$$U\propto\frac{N}{\langle R_N^2\rangle^{d/2}}$$

For a Gaussian walk the probability that chain has length $\vec{R}$ given by

$$P_N(\vec{R})=\left(\frac{3}{2\pi Na^2}\right)\exp\left(-\frac{3R^2}{2Na^2}\right)$$

Rewrite as

$$=\frac{\char93 of\; walks\; of \; ending\; at\; \vec{R}}{\char93 walks}\equiv\frac{n_{\vec{R}}}{n_{tot}}$$

Boltzmann entropy given by:

$$S_{\vec{R}}=k\ln n_{\vec{R}}$$

Hence

$$S_{\vec{R}}\propto -\frac{R^2}{N}+const.$$

Free energy

$$F=U-TS=c_1 \frac{N^2}{R^d}+c_2\frac{R^2}{N}+const.$$

Most likely value of R near minimum of F

$$\frac{\partial F}{\partial R}=0=- c_1d \frac{N^2}{R^{d+1}}+2c_2\frac{R}{N}$$

or



Obtain Flory formula $\nu=\frac{3}{d+2}$. For

• d=1 we find $\nu=1$ which is the exact value.
• d=2 we find $\nu=3/4$ which is believed to be exact.
• d=3 Flory theory gives $\nu=3/5$ close to best estimate $\nu\approx 0.588\cdots$.
• d=4 find $\nu=1/2$ again exact.
• d>4 find $\nu<1/2$ which is absurd. Interprete this by saying self-avoidance "irrelevant" for d>4.

Flory theory example of "mean field theory" in the spirit of van der Waals theory for gas liquid transition. Success of Flory theory due to cancellation of errors. For further discussion see Bouchaud and Georges [1990].

EDWARDS MODEL
Go back to result for Gaussian chain

$$P_N(\vec{R})=\left(\frac{3}{2\pi Na^2}\right)\exp\left(-\frac{3R^2}{2Na^2}\right)$$

This result can be obtained by Gaussian model where

$$P(\vec{r}_i)=\left(\frac{3}{2\pi a^2}\right)\exp\left(-\frac{3r_i^2}{2a^2}\right)$$

for each link in chain.
Express this in terms of Boltzmann factor e^-H₀/kT where

$$H_0=\frac{3kT}{2Na^2}\sum_{i=1}^{N}(R_i-R_{i-1})^2$$

This is the free energy of a set of coupled "springs" with a spring constant proportional to the temperature.
This phenomenon called entropic elasticity.
In continuum limit:

$$\frac{H_0}{kT}=K\int_0^Nds(\frac{\partial\vec{R}(s)}{\partial s})^2$$

Next take into account self-avoidance constraint by including repulsion term

$$\frac{H_1}{kT}=w\int_0^Nds_1\int_0^Nds_2\delta_d(\vec{R}(s_1)-\vec{R}(s_2))$$

Provision should be made to exclude $s_1\approx s_2$ from region of integration (Oono [1985]). Ignore this problem here (but we will come back to it later).
We write

H=H₀+H₁

for the free energy of the system.
We analyze the relative importance of the two terms by a scaling argument:

The d-dimensional Dirac $\delta-$function has the scaling behavior

$$\delta_d(a\vec{x})=a^{-d}\delta_d(\vec{x})$$

Assume that lengths along the chain are rescaled according to

$$s=ls';\;ds=lds'$$

N=l^-1N'

We asume vectors R rescale as

$$\vec{R}(s)=l^\nu\vec{R}(s')$$

so that

$$\frac{H}{kT}=Kl^{2\nu-1}\int_0^{N'}ds(\frac{\partial\vec{R}(s')}{\partial s'})^2$$

$$+wl^{2-d\nu}\int_0^{N'}ds'_1\int_0^{N'}ds'_2\delta_d(\vec{R}(s'_1)-\vec{R}(s'_2))$$

or

$$\frac{H}{kT}=l^{2\nu-1}H_0'+l^{2-d\nu}H_1'$$

We note that H₀ contains a scaling term $l^{2\nu-1}$. H₁ scales as $l^{2-d\nu}$.

• The first term retains it form under rescaling if $\nu=1/2$ as expected for a Gaussian process.
• l>1 if s represents the coarser scale.
• The second term grows under rescaling to a coarser scale if $2-d\nu>0$. In that case the perturbation caused by self-avoidance term is relevant.
• Since the Gaussian term is invariant when $\nu=1/2$, self-avoidance is relevant if d<4!
• Self-avoidance is irrelevant if $d\ge 4$. (logarithmic corrections to scaling if d=4).
• Recover Flory theory
  
  $$\nu=\frac{3}{d+2}$$
  
  
  if require that H₀ and H₁ scale the same way!

Then why is the Flory formula for the exponents not exact in the Edwards model? The problem is that the region |s₁-s₂|<a should be excluded from the integral. The resulting correction is not negligible, but it will in general scale differently than the rest of the integral for H₁. The problem is not 3-point and higher point interactions which could be added to the expression for H₁. It can be shhown that these shrink under rescaling with l>1 when compared to the H₁ term.

SUMMARY

• We have discussed self-avoiding random walk as example of random process with long range correlations.
• Mean field model of Flory gave good agreement with observed exponents
  
  $$\langle R_N^2\rangle \propto N^{2\nu}$$
  
  

• Some insight into why Flory model works so well found by studying Edwards model.
• Edwards model suitable starting point for renormalization group analysis.

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