Interpretation of the Angular Distribution

[Alec]

Good afternoon Marton,

I have a question regarding the interpertation of the angular distribution of particles traversing a surface.

As you mentioned in your response to "Diffuse reflections; distribution type" in January 2017, confusion can exist about the definition of appropriate angular distributions for scattering from a surface. Rather than trying to examine the behavior of a single facet, I am confused about the meaning of the angular distribution as sampled by a transparent facet at some distance from a gas inlet. The transparent facet is perpendicular to the inlet pipe; when I use the Profile Plotter to read out the angular distribution of particles traversing that facet, the distribution does respond more or less as I would expect to changes in the inlet geometry.

The relevant property of the flow for my application is the number of particles as a function of their angle theta with respect to the axis of the inlet pipe. The azimuthal distribution is not important.

If I want to integrate the distribution from the Profile Plotter, do I need to account for the 2 pi sin(theta) dtheta dependence of the solid angle on theta? Or has the Profile Plotter code already done so in order to display the distribution as a function of just theta and not also phi? I would like to calculate a value like the number of particles between, say, 30 and 40 degrees from the axis of the inlet, at any azimuthal position.

Many thanks,
Alec

[maarton (Marton Ady)]

Hello Alec,

To answer, I made a simple geometry where particles are effusing from a narrow pipe until they hit the outer wall, similar to your problem:

If you view the profile plotter for this geometry, and select "Angle" as normalization, you will see a plot that consists of 100 bin values corresponding to incident angle 0.45 deg, 1.35 deg, ... 89.55 deg (the range of 0..90 degrees divided into 100 pieces and taking the mid-values):

Each value is a fraction of 1, the proportion of particles in the 0..0.9 deg, 0.9...1.8 deg range, so the sum of all values is 1. (To visualize the values, click on points in the Profile Plotter or right-click and copy the values as a table to Excel.)

If you switch Normalize to None, then instead of fractions you'll see absolute hit number, i.e. number of Monte Carlo hits in each bin. The sum should equal the number of hits on the facet.

Now here comes the hard part: these are the values as sampled by the facet, which means that they are weighed by sin(theta) because of the solid angle dependence. As you see above there are relatively few perpendicular values (at 0 deg) because the solid angle is very small.

If you turn on the "Surface->Volume conversion" checkbox under the plot, then Molflow divides each value with sin(theta) and renormalizes the distribution so that the sum is again 1:

This is the angular distribution in the gas, which as you can see is different from the hit direction distribution on a facet (the difference is exactly the solid angle dependence). In the latter case most hits are perpendicular as one would expect.

So to answer your question, most likely you're looking for this corrected "in gas" distribution. To get the ratio of particles between 30 and 40 degrees relative to the inlet axis, align the sampler facet so that its normal is on the axis, enable angle profile on it, then sum all values between 30 and 40 degrees of the plot.

(As you asked: whether or not you work with the values corrected with the solid angle, all bin values always sum for all azimuth angles between 0 and 2PI.)

Finally, let me recommend a new feature in Molflow: you can create a custom resolution angle map for any facet (introduced in Molflow 2.6.52, the goal of this feature is to allow desorbing from a sampler facet according to the sampled distribution). The screenshot below (from Advanced facet parameters) would create a profile with 1000 elements, if required:

As plotted in Excel, you can see that this function doesn't correct for the solid angle:

Good luck, get back if I wasn't clear.

Marton

[10187173966221302116 (Chao LI)]

Molflow: Surface->Volume conversion

Very helpful explanation! Thanks!

[maarton (Marton Ady)]

A slightly better explanation, with visuals is found in this document, see Figures 6, 7 and section 2.8.0.5.