Does a shrinking disk grow in the z-direction?

[wonfish]

This is not a homework question! I am observing that a thin, flat disk made from a material whose density is gradually increasing, shrinks in the radial direction (of course) but appears to grow very slightly in the z-direction. The change in density is supposed to be isotropic.

Is growth in the z-direction predicted or an artifact of my experiment? I don't know enough continuum mechanics to solve this problem or even how to search for a solution. Is the solution to this problem somewhere in the literature?

 

[sophiecentaur]

I'm not clear about your model. How is the density increasing? Is it actually accreting material or is it changing dimensions? Are you thinking of a massive astronomical body?

 

[wonfish]

Actually its a chemical reaction where the product are denser than the reactants. And the size of my disk is about a cm in diameter and a few mm in height. We can detect size differences down to a couple nm.

 

[sophiecentaur]

If the material remains isotropic / amorphous then I can't see what could change the ratio of its dimensions.

 

[rollcast]

Is the reaction possibly exothermic and the disc is heating up and expanding in the z direction from the heat?

 

[sophiecentaur]

So it's anisotropic then?

 

[wonfish]

The reaction is exothermal, but the disc is kept isothermal. And the material is isotropic. The reaction takes days. It's the geometry that puzzles me. The problem is then a disc that is subjected to constant forces in the radial direction that also has a time changing density. (The internal forces caused by the change is density can be equated to external forces pushing on the disc -- I think.)

If it were a parallelpiped I would agree that the shrinkage would be proportional to the dimensions since the forces would be proportional. But it's the radial shrinkage that puzzles me. Can all the material fit into the shrinking circle, or does some have to pop out into the z-direction, even with the shrinkage forces in the z-direction.

 

[sophiecentaur]

If the material were truly isotropic then it would expand and contract by the same amount in all directions. But, because it is exothermic, I imagine that the periphery could be at a different temperature from the centre temperature (i.e. cooling down faster). This could lead to the rim contracting and squeezing the centre - prior to solidifying or increasing its modulus. This would / could induce stresses

Could you do the experiment in some sort of thermal jacket which would cool very slowly? This would / could relieve the differential stresses. Would it be worthwhile heating the disc up again and seeing if it changed shape as these stresses, if they exist, could even out - a sort of annealing/

 

[krysith]

If you are using a chemical reaction which has an end-product which has a higher density than the initial reactants, then some transfer of material is occurring as a result of the chemical action. Even if it is internal to the structure, there is some movement of atoms, even if only from one crystal structure to another.

Is the disk made of one of the reactants, and it is immersed in the other reactant, or is the disk made of a mixture of reactants? This is very relevant, because the main asymmetry in the disk from the standpoint of a chemical reaction is the much larger surface area of the top of the disk relative to the surface area of its sides.

I suspect that the reason why you are seeing an increase in the z-direction is either because the larger surface area on top of the disk allows more reactants to meet there or because it allows more heat transfer there (since you said you are cooling an exothermic process).

 

[wonfish]

Its a very uniform mixture of reactants. It is kept isothermal. The question comes down to continuum mechanics. Even though we can assume the material to be isotropic, due to the requirement of the disc being continuus, the geometry of the system is not. If the restraint of continuity weren't there, the material would shrink into a powder. But molecular forces keep the material together. What I am looking for is the solution to the continuum mechanics problem of a disk with nonconstant density.