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I'd like to understand the physics of dynamic light scattering experiment. In particular I want to understand the basic relation between relaxation time $\tau_q$ and the diffusion coefficient $D$:

$\tau_q=(q^2 D)^{-1}.$

I suppose that more or less the following steps are necessary:

1) Calculate the scattered wave from Maxwell equations. 2) Find a relation between the scattered intensity autocorrelation function and the brownian motion of the particles.

I found this book by Berne and Pecora, but at least for point 1 I got lost because there seem to a strange version (or an approximation) of maxwell equations that I don't understand (it is citing Landau but looking at this book was not such an help).

Is there any book were I can find what I need or at least point 1 written with some detail?

thank you

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2 Answers 2

Yes, Berne and Pecora is typically used as the canonical reference for Dynamic Light Scattering.

I want to make sure I understand your question correctly. You're asking about the derivation in Appendix 3.A, and specifically equations 3.A.3-6, correct? The only thing that looks weird to me is that they look like they are written in Heaviside-Lorentz units, but I don't see that fact specified anywhere in the book. I've never tried to read Landau and Lifshitz, so I don't know what units they use; if they are using SI or Gaussian, that might be the source of your confusion. One other source to check is Jackson. Most of 3rd Edition Jackson uses SI, but an appendix has a chart to convert the Maxwell equations between the different unit systems. Jackson also has a derivation of the scattering equations that you might find helpful. I can't find it in the copy in my office, but my home copy is more marked up and it might be easier to find that when I get home.

One other possible source of confusion is that Berne and Pecora are assuming no charges and currents, so they are setting $\mathbf{J} = 0$ and $\rho = 0$. They at least make that explicit.

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Yes, correct. Also if I don't care about units, there is a point I don't understand. Let us take as a reference for the maxwell equation this page (en.wikipedia.org/wiki/Maxwell_equations), than eq. 3.A.3 reads $\nabla \times E = - \partial B/\partial t$, and in fact in note 14 of B.P. H is the magnetic field. But then $\nabla \times B = \partial E / \partial t$, not $D$ as in eq. 3.A.4. A hint comes from Landau and Lifshitz which puts the magmetization $M=0$. in this case $B=H$. Now the question is: which is the physical reason for this? –  user7669 Jun 8 '12 at 7:41
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(I'm posting a new Answer because editing my previous Answer would be an enormous re-write. I hope that this is not too bad a violation of SE etiquette. If it is, please correct me.)

Physical assumptions

  • $\rho = 0$. There are no charges in the system. Yes, there are charges on the protons and electrons in the molecules, but those aren't relevant on the scale of the scattering event.
  • $\mathbf{J} = 0$. There are no currents in the system.
  • $\mathbf{P} = \chi_e\mathbf{E}$. The medium is isotropic, so the induced polarization is parallel to the $\mathbf{E}$ field from the light wave. This allows you to write $\mathbf{D} = \epsilon\mathbf{E}$. [EDIT: this is now correct.]
  • $\mathbf{M} = 0$. The light wave's $\mathbf{B}$ field doesn't magnetize anything in the system. This allows you to write $\mathbf{H} = \frac{1}{\mu}\mathbf{B}$.
  • $\mu = \mu_0$, (or $\mu = 1$ if you prefer). This is approximately true for most materials. In the DLS experiment at least, all of the scattering comes from variations in $\epsilon$ rather than $\mu$.

Those physical assumptions give you the form of Maxwell's Equations that B&P uses.

the Scattered Wave

As I said, one way of understanding the scattering is that the light wave scatters as a result of a patch of volume with a different dielectric constant $\epsilon$. That in turn comes from the solute you have in the solution. For a given solute/solvent pair, you can relate the concentration of the solute to the change in the local dielectric constant. That gives you the total scattering.

That total scattering is useful, and it is used in Static Light Scattering (SLS). But in DLS you're not directly interested in the total scattering; you're interested in how the total scattering fluctuates with time.

Autocorrelation and Einstein Relations

Your scattering detector sees the concentration of solute in a particular volume in your sample. The signal changes with changes in the local concentration at that volume. Those changes in local concentration are due to diffusion. The other post that @Ron linked to has a good description of how the diffusion constant $D$ relates to the relaxation time $\tau$ (he also provides a lot of background on how that idea relates to other systems). This is a very general process, and (as Ron says) is completely independent of how you probe your system. All you need is a probe that correlates with local concentration.

The local concentration that you measure is random (about some mean value) over "long" times. Over short times, the local concentration at one time is highly correlated with the local concentration at another time. The autocorrelation function is the formal definition of this idea. The relaxation time constant $\tau$ is a parameter in the autocorrelation function. In the DLS experiment, there will be a fall off in the value of the autocorellation at a particular value of $\tau$. That tells you the diffusion coefficient of your solute.

References

  • Jackson, Classical Electrodynamics (3 ed). Section 10.2 in particular does the EM derivation, starting from the same point that B&P starts from.
  • Sun, Physical Chemistry of Macromolecules (2 ed). Chapter 10 discusses diffusion. Section 16.3 discusses DLS, calling it "Laser Light Scattering," and discusses the autocorrlation function.
  • Teraoka, Polymer Solutions. Ch 3 discusses the dynamics of polymer solutions, autocorelation, and the dynamic light scattering experiment.

Any of those should be a useful supplement to B&P.

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