Why does the titanium go to the tungsten filament in the crystal bar process?
By &quot;gaseous titanium&quot;, do you mean &quot;titanium tetraiodide ($\ce{TiI4}$) vapor&quot;?
Van-Arkel De-Boer process is a purification process of titanium and zirconium. Basically, what happens is that the impure metal, let's say titanium is heated in iodine environment at a temperature of $\ce{250 ^\circ C}$ to form volatile titanium tetraiodide ($\ce{TiI4}$) vapor. The impurities are left behind, as they do not react with iodine. This vapor is then  passed over a hot tungsten filament at $\ce{1400 ^\circ C}$ for which the vapor gets decomposed and pure titanium is deposited on the filament and is removed. The iodine is reused. The overall reaction is:
$$\ce{Ti _{(impure)} + 2I2 -&gt;[250 ^\circ C] TiI4 -&gt;[1400 ^\circ C][W filament] Ti_{(pure)} + 2I2}$$
The thermodynamics of this reaction is discussed in Buttonwood's comment.