TECHNOLOGY FOCUS - CHEMICAL ENGINEERING - Preferred Method: In separating carbon from at least one lithium:transition metal material, the lithium:transition metal material comprises lithium cobalt oxide (LiCoO2). The method further comprises adding water to the mixture prior to bubbling. The frothing agent comprises formally hydrated terpenes selected from terpineols, citronellol, menthol, linalool, borneol, isoborneol, fenchyl alcohol, dihydromyrcenol, nerol, and combinations, where the frothing agent comprises at least one terpineol. The formulation is substantially devoid of methyl isobutyl carbinol. The collecting agent comprises unsaturated hydrocarbon terpenes selected from limonene, phellandrenes, terpinenes, pinene, camphene, car-3-ene, sabinene, thujenes, allo-ocimene, ocimenes, myrcene, dihydromyrcene, and combinations. The collecting agent comprises limonene. The formulation is substantially devoid of kerosene. Separating comprises scraping, skimming of overflowing of a rim of a container. The method also comprises separating the lithium:transition metal material from the liquid layer; separating a fraction from other components of a lithium-ion battery, where the fraction comprises at least one lithium:transition metal material, carbon and polymer binder, and separating the polymer binder from the fraction to yield a solid mass comprising at least one lithium:transition metal material and carbon; and relithiating the lithium:transition metal material to obtain a Li:M atomic ratio of 1:1, where M is a transition metal. The froth layer is generated by bubbling gas through the mixture. The liquid layer is aqueous. The formulation is substantially devoid of strong acid. The method does not require the use of a strong acid, where the strong acid is selected from sulfuric acid, nitric acid, perchloric acid, hydrochloric acid, hydrobromic acid, and hydroiodic acid. Alternatively, separating carbon from at least one lithium:transition metal material comprises: (a) contacting a solid mass comprising the carbon and at least one lithium:transition metal material with a formulation, the formulation comprising at least one frothing agent and at least one collecting agent to form a mixture; (b) generating a froth layer and a liquid layer; and (c) separating the froth layer from a liquid layer, where the froth layer comprises the carbon and the liquid layer comprises the lithium:transition metal material. The froth layer is generated by bubbling gas through the mixture.