[3dem] (mean Inner potential) Re: 3dem Digest, Vol 142, Issue 38

[Philip Köck]

Hi Vladan.


I think I've spotted the problems in your argument.

One is that the formula for the potential from a dipole is only valid at a large distance from the dipole, r much bigger than the distance between + and - in the dipole.
The inside of a slab of matter is not necessarily very far from the surface. Think of a 10 nm carbon film coated with water molecules. A lot of it is quite close to the surface (compared to the charge separation in a water molecule).

The second problem is that you would have to integrate over the surface if you want to apply your reasoning to an extended double layer of dipoles.


I would suggest a very simple model in stead: Just think of two infinitely large charged planes on each face of the slab, both with the negative charge closer to the inside. If you come from the vacuum the potential decreases within the first bilayer and then remains constant within the slab and then increases back to the vacuum value in the opposite bilayer. The value inside the slab will be constant and will not depend on the thickness of the slab, but only on what happens in the surface bilayers.


There's an easy way to calculate potentials numerically with an image processing package (MATLAB maybe). You just create a charge distribution and then apply the Poisson equation by a simple manipulation in Fourier space and you get the potential.

I put the formulas in my recent Ultramicroscopy paper, but you can find them in many other places (papers by Spence, e.g.).

If you want to try this I'd like to here what you come up with.


There's one more thing I might have to point out since I'm not sure about your last statement quoted below:
The phase shift is proportional to the projected potential. This means that it's proportional to the thickness if the potential is constant inside the slab.

If the potential was proportional to the thickness the phase shift would be proportional to the square of the thickness.


All the best,


Philip

________________________________
From: 3dem <3dem-bounces at ncmir.ucsd.edu> on behalf of Vladan Lucic <vladan at biochem.mpg.de>
Sent: Thursday, 27 June 2019 19:08:51
To: 3dem at ncmir.ucsd.edu
Subject: Re: [3dem] (mean Inner potential) Re: 3dem Digest, Vol 142, Issue 38

I don't see how the potential can be proportional to the distance between the dipole layers. If we think of a slab of neutral matter (as a thought experiment), which is covered first by a layer of positive charges and then an equal amount of negative charges on top of that, the following should happen: The negative charge curves the potential upwards and the potential increases, then the positive charge curves the potential down again by an equal amount. The total effect is that the potential is constant inside the slab and independent of the slab's thickness and therefore the phase shift is proportional to the thickness.

Let me try to expand on your model. Adsorbed H2O forms dipoles on the surface, with positive charge towards the vacuum and the negative towards the material. A potential arising from one dipole (having electric dipole moment p, vector) that an electron at a distance (vector) r sees is proportional to:

    scalar_product(r, p) / magnitude(r)^3

So you're right that it is proportional to the distance between the positive and the negative charge (contained in p), but that's not all. H2O dipoles form on both sides of the material and they're oriented in the opposite direction. The contribution to the potential from the "other side dipol" has the same form, but r needs to be replaced by r+d, where (vector) d is the thickness of the material. Adding the two terms (with opposite signs because of the opposite orientation), when (magnitudes) r>>d leaves in the first approximation a term proportional to d. That's the reason for my proportionality statement.

In any case, we agree that it is clear why the phase should be proportional to the thickness, at least for flat surfaces.

Best,
Vladan





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