Why does a laboratory centrifuge cause heavier particles to go to the bottom of the tube? From what I have read online, I can understand that there is a "centrifugal force" that pushes the heavier objects away from the center, but I cannot understand why.

I asked my teacher and she told me that I have to think about what changes between a particle with larger mass vs. smaller mass. I had thought about this, and decided that it must be the radius, since the particles with lower mass are pulled towards the center, and must therefore have a smaller radius of rotation than the higher mass particles. I then tried to plug in numbers into equations in order to see if I was correct. I used the net force equation I was taught in class
$$
\sum \mathbf F = 4\pi^2 r f^2 m
$$
which I rearranged for $r$
$$
r = \frac{\sum \mathbf F}{4\pi^2 f^2 m}\,.
$$
I solved for $r$ with one mass (say 10 kg) and then a larger mass (say 15 kg), keeping all other variables constant. To my surprise, the radius of the larger mass was smaller than the one with the smaller mass. This would mean that the smaller particles go to the bottom of the tube, not the heavier ones. I assume that I did something wrong since this is not what happens in real life. I think it might have something to do with the variables that I kept constant actually changing when the mass does. In any case after much thought I was not able to come up with a reasonable explanation for how the apparatus works. What did I do wrong? What is the correct reasoning and explanation?
 A: It is not mass but density which is the important parameter.
What you have is a "local" value of $g$, the "gravitational field strength" which is $R \omega^2$ where $R$ is the radius of the orbit and $\omega$ is the angular speed - this provides your centrifugal force which is the weight of a mass in this local gravitational field.
The all you need to do is to use Archimedes principle to find the upthrust on a particle which is equal to the weight of fluid displaced.  
The denser material will "sink" ie move towards the outside of the rotation.
A: In a centrifuge, a vial or tube is spun very quickly. Think about when you spin a a bucket full of water or a full grocery bag in the air, the contents of the bag or bucket don't fall out. Lets look at a snapshot in this scenario:

At this moment, the water in the bucket has velocity moving in tangent to the circle and has an inertia that is pushing it away from the center of the circle. If you have ever been on one of those fair rides where you are strapped to the wall of a large drum and pushed into the wall as the drum spins, you know the kind of force that the water is experiencing.
So, we confirm that there is a constant and large downward force on the water in the bucket. Everything in the bucket is being pushed downward. It's as if the bucket has it's own gravity, pulling everything inside towards the bottom.
Now, think about a pond or a puddle, in earths gravity. Lets say you kick up some mud at the bottom. For a moment, the water will go all cloudy and it will take a while for the mud to settle back down to the bottom. Now, if you multiplied earths gravitational field by 1000, the mud would settle faster, wouldn't it?
Now, the faster we spin the bucket, the more force the things inside the bucket will experience, the greater the 'gravity' in the bucket. If we spin it really really fast, the 'mud' will settle to the bottom really quickly. This is essentially how it works.
A: Nonsense. Nothing “pushes” down. In fact every thing is ‘pushed’ to the centre-hence CENTRI petal force. The particles want to travel in a strait tangential line as indicated by the sketch but are prevented by the bucket bottom pushing (forcing) them in a circular path. Exactly the same as when sitting in a car going round a corner. It feels as though you are being ‘pushed’ against the door by a force (that was called centrifugal force - a term no longer used in physics) whereas the car’s door is actually forcing you to go round the corner. Thus only an inward force is being exerted against you. Also note that Gravity is ALWAYS a downwards force acting towards the centre of the Earth and never a horizontal force. 
