Prepare a thermoplastic polymer foam having a porosity of 70% or more and at least one of: (i) an average cell size of 200 nanometers or less; and (ii) a nucleation density of at least 1×1015 effective nucleation sites per cubic centimeter of foamable polymer composition not including blowing agent using a foamable polymer composition containing a thermoplastic polymer selected from styrenic polymer and (meth)acrylic polymers, a blowing agent comprising at least 20 mole-percent carbon dioxide based on moles of blowing agent and an additive having a Total Hansen Solubility Parameter that differs from that of carbon dioxide by less than 2 and that is present at a concentration of 0.01 to 1.5 weight parts per hundred weight parts thermoplastic polymer.