Subsections

• Problem
• Landau Hamiltonian and Normal Coordinates
• Fluctuations in Fourier Space. Scattering
• Correlation in Real Space. Ornstein-Zernicke Function
• Ginsburg Number

Inhomogeneous Systems

Problem

In inhomogeneous systems G depends on the value of x in all points $\mathbf{r}$. G is a functional $G\bigl\{x(\mathbf{r})\bigr\}$--it depends on a function $x(\mathbf{r})$.

Analogy with multi variable case:

we have variables $x(\mathbf{r}_1)$,$x(\mathbf{r}_2)$,..., and want to minimize G.

Idea:

Introduce normal coordinates that diagonalize G

Trick:

Use Fourier transform--it leads to normal coordinates in linear problems

Landau Hamiltonian and Normal Coordinates

For inhomogeneous system the free energy is

$$G= G_0 + \int\left[aM^2+dM^4-HM+g\left(\nabla M\right)^2\right]\,d\mathbf{r}$$

Assumptions and simplifications:

1.

We are above critical point, so a>0 (for negative a we expand around M₀)

2.

We neglect the term dM⁴, This is the most important approximation!!!

Then we make Fourier transform:

$$M(\mathbf{r}) = \sum_{\mathbf{k}} M_{\mathbf{k}}e^{i\mathbf{r}\mathbf{k}}, \quad M_{-\mathbf{k}} = M_{\mathbf{k}}^{\ast}$$

Quadratic term:

$$M^2(\mathbf{r})=\sum_{\mathbf{k}}\sum_{\mathbf{k}'}M_{\mathb... ...M_{\mathbf{k}'}e^{i\mathbf{k}\mathbf{r}+i\mathbf{k}'\mathbf{r}}$$

Gradient term:

$$\nabla M(\mathbf{r}) = \nabla \sum_{\mathbf{k}} M_{\mathbf{k... ...\mathbf{k}} i\mathbf{k}M_{\mathbf{k}}e^{i\mathbf{r}\mathbf{k}}$$

and

$$\bigl(\nabla M(\mathbf{r})\bigr)^2 = - \sum_{\mathbf{k}}\su... ..._{\mathbf{k}'}e^{i\mathbf{k}\mathbf{r}+i\mathbf{k}'\mathbf{r}}$$

Term with external field:

$$HM(\mathbf{r}) = H\sum_{\mathbf{k}} M_{\mathbf{k}}e^{i\mathbf{r}\mathbf{k}}$$

Mathematical result

$$\int_V e^{i\mathbf{k}\mathbf{r}}\,d\mathbf{r}= \begin{cases} 0,& \mathbf{k}\ne0\ V,& \mathbf{k}=0 \end{cases}$$

Integrals with quadratic & gradient terms contain

$$\int e^{i\mathbf{k}\mathbf{r}+i\mathbf{k}'\mathbf{r}}\,d\mathbf{r}$$

This is non-zero only if $\mathbf{k}'=-\mathbf{k}$. Then

$$M_{\mathbf{k}}M_{\mathbf{k}'} = M_{\mathbf{k}}M_{-\mathbf{k}}= \left\lvert M_{\mathbf{k}}\right\rvert^2$$

and we obtain

$$G=G_0+V\sum_{\mathbf{k}}(a+g\mathbf{k}^2)\left\lvert M_{\mathbf{k}}\right\rvert^2 -VHM_{\mathbf{0}}$$

We represented free energy as a sum over Fourier components--this means $M_{\mathbf{k}}$ are normal coordinates!

Fluctuations in Fourier Space. Scattering

We immediately obtain for fluctuations

$$I(\mathbf{k})=\left\langle \left\lvert M_{\mathbf{k}}\right\rvert^2\right\rangle = \frac{kT}{2V(a+gk^2)}$$

This is exactly the function measured in scattering experiments!

What happens near critical points? Here $a=\alpha(T-T_c)$ is small.

1.

Short wavelength fluctuations do not depend on the temperature:

$$\left\langle \left\lvert M_{\mathbf{k}}\right\rvert^2\right\rangle = \frac{kT}{2Vgk^2},\quad gk^2\gg a$$

2.

Long wavelength fluctuations grow:

$$\left\langle \left\lvert M_{\mathbf{k}}\right\rvert^2\right\rangle = \frac{kT}{2V\alpha(T-T_c)}, \quad gk^2\ll a$$

3.

The characteristic wavelength tends to infinity: $k_0 = \sqrt{a/g}\to0$

This is called critical opalescence

Correlation in Real Space. Ornstein-Zernicke Function

The function

$$I(\mathbf{r}) = \sum_{\mathbf{k}} I(\mathbf{k}) e^{i\mathbf{r}\mathbf{k}}$$

Is called correlation function. It is just

$$I(\mathbf{r})= \left\langle M(\mathbf{0})M(\mathbf{r})\right\rangle$$

Integrating $I(\mathbf{k})$, we obtain:

$$I(r) = \frac{kT_c}{8\pi g r}e^{-r/\xi}$$

with correlation radius

$$\xi = \sqrt{g/\alpha(T-T_c)}$$

This function is called Ornstein-Zernicke function.

What happens at $T\to T_c$? Correlation radius $\xi\to\infty$

1.

At $r\gg\xi$ we have exponential decay with distance

2.

At $r\ll\xi$ correlations decay as 1/r and are independent of the temperature!

Above we obtained the formula

$$\left\langle M^2\right\rangle = \frac{kT}{Va}$$

This works in thermodynamic limit $V\to\infty$. In other words $V\gg\xi^3$. Near T_c this condition becomes very stringent!

Ginsburg Number

Landau theory works if fluctuations are small. Average fluctuation in the volume $\xi^3$ is

$$\left\langle M^2\right\rangle = \frac{kT_c}{\xi^3\left\lvert... ...ight\rvert} = \frac{kT\left\lvert a\right\rvert^{1/2}}{g^{3/2}}$$

Comparing this to the equilibrium value below critical point $M_0^2\sim \left\lvert a\right\rvert/2d$, we obtain that $\left\langle M^2\right\rangle\ll M_0^2$ if

$$\left\lvert a\right\rvert\gg \frac{(kT)^2d^2}{g^3}$$

or, since $a=\alpha(T-T_c)$ 

$$\frac{\left\lvert T-T_c\right\rvert}{T_c}\gg \frac{k^2T_cd^2}{\alpha g^3}=\mathsf{Gi}$$ (4)

$\mathsf{Gi}$ is Ginsburg number.

Landau theory works, if we are both:

1.

Close to critical point, so $\left\lvert T-T_c\right\rvert/T_c\ll 1$ (we used expansion!)

2.

Still not close enough, so $\left\lvert T-T_c\right\rvert/T_c\gg\mathsf{Gi}$.

Why does Landau theory work well for polymers? Coefficient g does not depend on molecular weight N, but $d\sim1/N$ and $\alpha\sim1/N$. We obtain $\mathsf{Gi}\sim1/N$, i.e. Ginsburg number is small.