# Why doesn't the diffusivity of a particle in a fluid depend on the particle's density?

From this answer and from the Stokes-Einstein equation the diffusivity of a particle of radius $$R$$ in a fluid of viscosity $$\eta$$ is

$$D=\frac{k_B T}{6 \pi \eta R}$$

where $$\xi=6 \pi \eta R$$ is a coefficient of friction Stokes' law such that for velocity $$v$$ the viscous drag force is

$$F_D=\xi v.$$

To me it looks like the diffusivity $$D$$ is independent of the particle's density. A 1 micron sphere of a dense metal like gold and a 1 micron thin spherical shell would have the same diffusivity as long as other forces were not present if for example the experiment were done in a microgravity environment.

This bothers me because I want to believe that since the same number of fluid atoms are hitting both particles transferring the same random amounts of momentum to them, why wouldn't the heavier particle diffuse more slowly?

I'm looking for an "Aha!" type answer rather than just additional mathematics. Is there some way to understand why particles that differed dramatically in density could diffuse in the same way?

## 2 Answers

In general mass should be considered, but the equations you give assume mass is negligible, so you cannot base your reasoning of the relation between mass and diffusivity on these equations.

As stated in the first sentence in the first Wikipedia article section you give a link to, this equation is derived assuming a low Reynold's number. One can interpret the Reynold's number as the ratio between inertial forces and drag forces. Therefore at low Reynold's number the viscous forces are much larger than the inertial forces, and thus mass is assumed to not play a huge role in describing the diffusion.

If your density is large enough to make the Reynold's number large enough then you cannot use this equation.

• Okay so the answer lies within the derivation of the Stokes-Einstein equation at the point where the limit of low Reynolds number comes into play. Then that would be the "Aha!" that I'm looking for. – uhoh Mar 22 '19 at 14:59
• @uhoh I think there is more than one way to arrive at these equations, but essentially yes. You have to find where it is assumed that mass, inertial forces, etc. are negligible. If something doesn't make sense in an equation you find then you should always go back to the derivation and understand all assumptions being made about the system. – BioPhysicist Mar 22 '19 at 15:01
• @uhoh My answer to your question of "Why doesn't the diffusivity of a particle in a fluid depend on the particle's density?" is that it actually does, but the equation you are basing your question on does not apply to thinking about objects with large mass. I think looking up and learning the derivations yourself is something that would be very useful. If you want someone to walk you through explaining the derivation, then ask a new question. I feel like I have answered this one sufficiently. – BioPhysicist Mar 22 '19 at 15:08
• Thanks, this is a step in the right direction at least. I'll read further... – uhoh Mar 22 '19 at 15:11
• @uhoh Yes, that is what I was going for. Happy learning :) Typically down-votes are for answers that are not useful, but it seems like you are saying it was useful. oh well. – BioPhysicist Mar 22 '19 at 15:13

Let's look for a way for the OP to have the requested "Aha!" moment.

To me it looks like the diffusivity 𝐷 is independent of the particle's density. A 1 micron sphere of a dense metal like gold and a 1 micron thin spherical shell would have the same diffusivity as long as other forces were not present if for example the experiment were done in a microgravity environment.

Yes, under conditions where this is valid (smoke in air, bacteria in water) this is so.

This bothers me because I want to believe that since the same number of fluid atoms are hitting both particles transferring the same random amounts of momentum to them, why wouldn't the heavier particle diffuse more slowly?

So there's no conflict here. If the particle is at rest and a molecular collision knocks it to the side, the recoil velocity will be inversely proportional to the 1 micron particle's density.

The OP's "Aha!" will come when they realize they've tried to reconcile two different worlds. A microscopic model of individual collisions with atoms, and the collective effects of a fluid.

The diffusivity D can only be applied for time scales much longer than the time between individual collisions. The momentum imparted by each collision needs time to damp due to several more collisions.

If the lighter particle moves faster after one collision than the heavier one would, the subsequent collisions on average will slow it down a bit faster since their relative speeds are faster. On a macroscopic scale, that's where one can start thinking about the origin of the drag force.

What is that collision time? For a 1 micron particle in water or air its of order $$\text{10}^{-19}$$ and $$\text{10}^{-16}$$ seconds. So even if you are watching your particles using single pulses from nanosecond lasers, you're averaging over millions or billions of collisions.

It's time for the OP to throw away their microscopic thinking and start to embrace statistical mechanics and fluid dynamics.

A very good way to ease one's way into this picture slowly would be to read Life at Low Reynolds Number. E. M. Purcell Am J. Phys. J45, 3 (1977); doi: 10.1119/1.10903