I don't think Putin has any other expression. Pretty much just the one face and that's it.
The face that means "if this doesn't meet my expectations, no one will ever knew you existed at all". I mean, let's face it, for being a wee little guy, Putin is scary as a tiger...wearing a suicide vest...in late stage rabies...advancing on you ahead of a lava flow...and you are holding her cubs....and shaking them....and you are nude...,and covered in BBQ sauce.