There are many different ways to achieve this effect. Here is how I would go about it.
You could use metaballs, but just make the shape of the "welded part" I just used a box.
Create your "cooled off" material and mix it with your heated material. Then you will add a gradient texture and mapping node. Animate the location of the mapping node to "wipe off the hot material with the cool material".
I didnt do it here but I would also add on to the shader by animating in a transparent node so it looks like the material is being added and not already there. This is no different to what we did above except in we are just switching out a cooling down material to being transparent.
That covers the cooling off of the material, Then I would just add a separate object to shoot out the particles like below, and animate it to follow the change in material.