Better explanation found on the web
Well, actually it's both. First, not all sunflowers are phototropic. It's usually the sunflowers with a single-head (the ones with many heads facing in all directions their heads don't move with the sun).
The phototropic ones only follow the sun for a period. When the sunflower is growing the stem & material is soft and pliable, the unbloomed bud is light and will follow the sun during this phase. However, when the sunflower blooms the sunflower slowly stops growing, instead putting the energy into the seed head. At this point the rest of the sunflower (stem, etc.) starts to solidify, and will eventually solidify facing in an eastwardly direction and the bloom will no longer follow the sun. So, the phototropic ones do follow the sun but only for a time and once bloomed will eventually "freeze" facing in an eastwardly direction.
My guess is, they solidify as a mechanism to help support the head which gets bigger and heavier once bloomed. If they didn't solidify they would eventually collapse under the weight of the seed head as it matures.