An extrusion system for providing a foamed material in which a material such as a polymer material is supplied to an extruder for movement through a rotating screw member. The material is placed in a molten state and a foaming agent, such as a supercritical fluid, is introduced into the extruder at a selected pressure so that a two-phase mixture of the molten material and the foaming agent is formed. The foaming agent is then diffused into and dissolved in the molten material to form a single-phase solution which is forwarded from a solution formation to a nucleation device. A thermodynamic instability is induced through a rapid pressure drop, e.g., higher than 0.9 GPa/s in the nucleation device to nucleate microcells in the solution. A further shaping device, e.g., a die, can be used to produce a foamed material of a desired shape.