Hello Antonin,

it seems that this dependence between area and volume is quite a geometric fact. Although the geometry of particle itself also has a great influence, it seems that at the beginning this relationship is slightly logarithmic, but as the volume increases the relationship becomes quite linear. As an example, I attach here the A-V dependence for the ideal cube and sphere. As you can see, the dependencies are very similar to yours.

Sincerely

Pavlo

Hi Antonin,

A PhD project has two objectives. First, to achieve a useful and interesting piece of research that advances human knowledge in a non-trivial way – so it is not merely a collection of facts, but an advance in the interpretation and understanding of those facts. Second, it is to teach the student how to do research: how to plan and execute a programme of work, how to interpret the findings, how to apply scientific method in proposing, testing and developing hypotheses, and how to place all this in the context of previous work and current understanding.

So your question is one you should first be asking yourself, not us!

You will not learn anything from the graph alone; you need to examine these pores – and CT scanning provides an excellent way of doing this. Are they all the same? Can you group them by size, shape, or location? What types of pores would you expect in such a casting? Do your observations match these expectations? What have previous workers found, and how have they interpreted their data?

I assume that your graph displays the results of a single CT scan of a casting, with each point corresponding to a pore. In fitting a straight line to your data, you are introducing some important assumptions – that the pores form a single, homogeneous set, and that a linear relationship is reasonable.

If there are different types of pores – gas bubbles, shrinkage voids, for instance – then they should be separated and analysed as sets, because they will have different characteristics. Look at the images, identify the different types and relate typical examples to the area/volume measurements. Can you distinguish the types by size, by sphericity, by aspect ratio, or by some other parameter? Can this then be automated to ease analysis?

It is tempting to fit a straight line to your data, but what does it mean? The equation you give states that a pore of zero volume nonetheless has a surface area of 1554 square microns. That is physically impossible. Any proposed relationship must pass through the origin. Secondly, neither area nor volume can be considered an independent variable. Thus, the usual regression equation, of y upon x, is inappropriate. A symmetric regression model should be employed.

Pavlo Pavliuchenko has reminded us that geometric relationships are fixed. A series of objects, identical in shape but differing in size, will show an area-volume relationship that follows a power law, with exponent 2/3, as he demonstrates. Any other relationship implies that the pore shape varies with pore size. A linear relationship would result if all the pores were discs of constant thickness, but varying radius, or tubes of constant radius, but varying lengths. However, casting pores do not have these geometries. It is much more likely that several pore types are represented in your data. You need to separate these types and analyse each set individually.

It would also be useful to plot the size distributions of each set. You clearly have a lot of small pores, with sphericity >0.5 – these could well be gas bubbles with a lognormal size distribution. You also have a few large pores, with sphericity 0.4 – these could well be shrinkage voids with a near-gaussian size distribution.

I hope these thoughts help! Please message me if you wish to discuss.

John

Hello Pavlo Pavliuchenko

Hello John Brear

Thank you for your help !

Don't worry I already spent many time to understand this finding with my supervisors and in the litterature but I've never find something that looks like my population of porosities ... That's why now I try to find help somewhere else !

For sure my question was very basic and I didn't explain my all PhD project.

We use CT-scan to characterize porosities in fatigue test specimen, in order to characterize the population (size, shape etc ...) and to find which porosity initiate crack (by comparison with fracture area observed with SEM)

I already tried to compare my porosities to platonic solids and sphere. (see picture attached)

My porosities seem to be first close to a sphere/cube, then with increasing the volume they become more angulous like a thetrahedron and after ... ? That's where I don't find solution !

I think you're right John Brear when you say "A linear relationship would result if all the pores were discs of constant thickness, but varying radius, or tubes of constant radius, but varying lengths". I also think about my porosites that when we increase the volume, they grow in only one direction (thickness, length or another direction)

I also have a look at shape factors like flatness & elongation but there is no apparent link with the volume (see picture attached)

But there is something interesting with two variables (see pictures attached) :

- sphericity

- ratio between volume of the porosity & volume of the bounding box

Concerning sphericity, biggest porosities (in volume) have the smallest sphericity so it confirms that with increasing the volume, porosities grow following a preferential direction. But it's weird that something with the same meaning when we look at flatness or elongation ratios.

Concerning the ratio between volume of the porosity & volume of the bounding box, we see that the bigger the porosity is, the less it fills its bounding box.

So with increasing volume, porosities increase their footprint (and not their thickness). It's logical because shrinkage porosities (which are the biggest porosities in my data set) have a "branch form". Other porosities like bubble gas are smaller and are likely a sphere as you said John Brear

To sum up, we have a good idea about what our porosities look like but we wan't to go further and we thought this "constant ratio" between volume and surface area was the key.

I think I will try to split my data set between biggest porosities (shrinkage porosities) and the others. I just have to find the good criterion to distinguish them (as you said John Brear , by size or shape or other parameter).

Thank you again for your help.

Antonin