## 1 Introduction
For instance, selectivity of 1500 was reported for CO\({}_{2}\)/O\({}_{2}\)[23; 34]. Despite high selectivity and permeability performances, these membranes lacked severely in terms of stability, mainly because of carrier leakages and deactivation due to evaporation or entrainment with the gas stream [35]. To tackle this instability issue, recent research has been focused on fixed site carrier (FSC) membranes. These membranes have functional carriers that are covalently bonded to the matrix polymer backbone or to a dispersed secondary phase restricting their mobility but drastically improving the membrane stability [21; 36; 31]. Both approaches showed strengths and weaknesses: small mobile carriers tend to leak and evaporate, while fixed carriers can struggle to achieve the same diffusion rates due to their lack of mobility. However, the diffusivity of a swollen FSC membrane should display performances between that of a mobile and a fixed carrier. Among the numerous amine-based FTMs used for CO\({}_{2}\) separation and extensively investigated in last decades [20; 21; 37; 38; 39; 40; 41; 42], polyvinylamine (PVAm) is an attractive aminated common polymer to prepare FSC membranes [29]. Thanks to its hydrophilicity and high amount of amine moieties, PVAm presented good separation performances in various conditions [43; 44; 45]. In presence of water, these amine groups are able to interact with CO\({}_{2}\)[31]. Moreover, as presented in the mechanisms above, the amine groups can also play the role of catalyst in the CO\({}_{2}\) hydration reaction [46]. However, PVAm can also suffer from instability issues, due to the poor mechanical properties achieved at high humidity [47]. This problem is usually overcome by blending PVAm with a'structural' polymer, which can increase the mechanical properties of the matrix under high humidity conditions, still allowing the facilitated transport mechanism. In several works, PVAm has been blended with different materials such as (i) polyvinyl alcohol (PVA) [48; 49; 50], (ii) carbon nanotubes [44; 46; 49; 51], and (iii) native nanofibrillated cellulose (NFC) [52; 53], showing positive results in term of stability and separation performance. Other approaches to obtain more stable materials can rely on the use of a high molecular weight PVAm, as proposed by [PERSON] and [PERSON] [54]. In this work, carboxymethylated nanofibrillated cellulose (cmNFC) was chosen as a'structural' polymer. Actually, for sustainability and green technology development, it appears essential to prepare new membranes for carbon capture using bio-based materials such as cmNFC. Cellulose is the most abundant biopolymer on earth. Extracted most often from wood or annual plants, cellulose fibers are composed of several fibrils which were firstly isolated by [PERSON] et al. [55] in 1983 and are usually used as microfibrilated cellulose (MFC), cellulose nanofibrils (CNF), or nanofibrillated cellulose (NFC) obtained after mechanical disintegration. Since 2008, the development of NFC, which represents one of the most studied bio-based materials [56; 57; 58; 59], attracts exponential interest from researchers, as well as industrial companies, with more than 60 producers worldwide. More recently, in 2016, NFC was identified as the second bio-economy priority in Europe thanks to its properties. Indeed, NFC displays high mechanical resistance and excellent barrier properties, and it is also biodegradable and biocompatible, highlighting this bio-based material as an excellent candidate in several applications such as packaging [58], paper and board [60], composites [61], printed electronics [62], biomedical devices [63], etc. Also, in the field of membranes, nanocellulose is gaining attention; in the last three years, several papers appeared using this material as a base for the production of gas separation membranes [52; 64; 65; 66; 53].