Opaque Layer Solutions!

My next test consisted of 2 separate particle sets, an opaque layer and a transparent layer.  For the opaque layer, I still needed to address the falloff into the transparent areas- they needed to be transparent enough that there wouldn't be any weird black junk like there was before.  

To fix this, I remapped the original alpha value using this equation: new_alpha = (old_alpha)^4.  This seemed to create the look I wanted. 

With the quadric falloff applied.

On a background- note the speckliness

After that, I needed a way to soften the falloff into transparency- I was getting a sprinkling of more opaque particles, creating a speckled look.  By manipulating the radius size based on the alpha amount, I was able to soften up the edges.  I tried both going larger with the radius and getting smaller (all the way to 0).  I like the larger method better.

Radius goes to 0 as alpha gets smaller.

Radius gets larger as alpha gets smaller.