A method and electrode assembly for treating a substrate with a non-equilibrium plasma in which the electrode assembly has two or more spaced barrier electrodes and a ground electrode spaced apart from the two spaced barrier electrodes for passage of a substrate to be treated. Plasma fluid medium is introduced between the barrier electrodes and is biased to provide a greater flow to an inlet region of the electrode assembly to help inhibit the ingress of air. Each of the barrier electrodes can be provided with central and leg sections having passages for introducing a cooling fluid into one of the leg sections and discharging said cooling fluid from the other of the leg sections. The central section can be provided with a transverse cross-sectional area less than that of the leg sections to increase velocity in the central section.