## 1 Introduction  
Global warming and climate change are being caused by the increase in greenhouse gases, primarily carbon dioxide (CO\({}_{2}\)), in the atmosphere. CO\({}_{2}\) emissions were estimated to be more than 33 Gtons in 2018 [1]. In order to achieve the \"2 \({}^{\circ}\)C target\" proposed by the Intergovernmental Panel on Climate Change (IPCC), CO\({}_{2}\) may need to be captured at a rate of 25 Gt/yr in 2050 [2]. About half of these emissions are due to diffuse sources, belonging to the transport sector and small sources (homes and offices) [3]. CO\({}_{2}\) can be captured by using a negative emission technology such as Direct Air Capture (DAC), proposed first as a mitigation option by [PERSON] in 1999 [4, 5, 6]. By using this technology, the captured CO\({}_{2}\) can be verifiably removed from the atmosphere. Potential technologies include Direct Air Carbon Capture and Storage (DACCS) processes, or the CO\({}_{2}\) could potentially be used in a Direct Air Carbon Capture and Utilization (DACCU) system [7, 8, 9]. Direct Air Capture technologies might include absorption, adsorption, mineral carbonation, membrane, photocatalysis, cryogenic separation, electrochemical approaches, and electrodialysis [4, 10, 11, 12, 13, 14, 15]. However, among these, absorption and adsorption are the most mature and most investigated in the literature [4]. For defined advantages of adsorption system, potential applications for CO\({}_{2}\) capture using this technology are growing significantly and this work is focused on this technology addressing the relative merits of different systems (among physisorbents and chemisorbents), from a performance point of view, regarding capacity, energy efficiency, and economic viability [16,17]. For physisorption, only a physical interaction occurs between CO\({}_{2}\) and surface sorbent, weak Van Der Waals forces are established, and low CO\({}_{2}\) adsorption capacities are exhibited, owing to the relatively low adsorption heat [18]. Potential sorbents include zeolites, activated carbons, metal-organic frameworks (MOFs), and other new materials (e.g., boron nitride nanosheets and nanotubes) [10,19]. Particular interest has recently been paid towards MOFs, as underlined by publications growing rapidly in the last decade [20, 21, 22, 23, 24, 25, 26]. In fact, MOFs have a specific surface area much higher than other porous materials such as activated carbons and zeolites with values up to 7140 m\({}^{2}\)/g [27,28]. Chemisorption involves a chemical reaction between CO\({}_{2}\) and surface sorbent, for example, an amine functionalized sorbent, making the CO\({}_{2}\) capture capacity larger [29,30]. Amine-functionalized adsorbents are divided in three different classes [4]. In class I, there is a physical impregnation of amines on a porous support. In class II, amines are covalently bonded on the walls of porous support, while in class III, amine monomers are polymerized in situ, resulting in a polyamine structure bonded to the walls of support. Chemisorbents are investigated in more detail than MOFs for air capture, due to their high uptake capacity and selectivity, resilience to humidity (which has a positive effect on CO\({}_{2}\) capture), and the possibility of regeneration under relatively mild conditions [3,10]. Generally, research in this field has explored and optimized adsorption capacities, kinetics and regenerability for amine-functionalized sorbents individually [31, 32, 33, 34, 35, 36] and this is also the case for MOFs [37, 38, 39, 40, 41]. Experimentally, as shown in Table 1, it has been found that for amine-based sorbents of class I, CO\({}_{2}\) uptakes can be up to 3.36 molCO\({}_{2}\)/kg of sorbent for a polyethyleneimine (PEI) functionalized hierachical bimodal meso/microporous silica support in the presence of 19% of relative humidity [32]. A mixture of PEI and poly(ethylene glycol) (PEG) impregnated on a fumed silica support has been reported in [PERSON] et al. [42] with an adsorption capacity of 6.8 \(\times\) 10\({}^{-1}\) molCO\({}_{2}\)/kg.