The current concentration of carbon dioxide in the atmosphere demands for development of negative emission solutions such as direct carbon dioxide removal from the atmosphere (air capture). Many well established processes can remove carbon dioxide from the atmosphere but the real technological challenge consists of concentrating and compressing carbon dioxide at the conditions for long term geological storage, with efficient use of non-fossil energy sources. A thermally-driven, negative-carbon adsorption process for capture, purification and compression of carbon dioxide from air is proposed. The process is based on a series of batch adsorption compressors of decreasing size to deliver a compressed carbon dioxide stream to a final storage. Thermodynamic analysis of the process shows that, by exploiting the equilibrium properties of commercial and non-commercial materials, carbon dioxide can be produced at specifications appropriate for geological storage. By operating the process with zeolite 13X at regeneration temperature of 95 degrees C a final storage vessel can be pressurized with carbon dioxide at purities >0.95 mol fraction and specific energy consumption <2.2 MJ(th), mol(CO2)(-1). Tailored materials provide a step-change in performance. When the process operates with zeolite NaETS-4, carbon dioxide can be purified at values >0.97 mol fraction. (C) 2018 Elsevier Ltd. All rights reserved.