## 1 Introduction  
Current research by the scientific-technical community looks for alternative solutions to problems related to energy consumption and CO\({}_{2}\) emissions.  
Global infrastructure demands require massive concrete and cement production, causing a significant negative environmental impact since its production is responsible for 5-7% of global anthropogenic CO\({}_{2}\) emissions [1; 2; 3]. In addition, Portland cement production releases about 0.8 tons of carbon dioxide for every ton of clinker produced [4]. This represents a considerable fraction of total CO\({}_{2}\) emissions. For this reason, it is important to propose alternative solutions with second-generation materials, remains of other productive cycles (waste), which reduce the environmental impact and avoid the use of new reserves. Therefore, the perspective for the future is to allow the adaptation of new materials pertaining to concrete and Portland cement, but from a new perspective of durability, as well as economic and social sustainability.  
Fly ash (FA), metakaolin, and silica fume have been used as additions in the cement industries and as pozzolanic materials that improve the physical, chemical, and mechanical properties of concrete [5; 6; 7; 8]. In the last years, FA with high silica and aluminum content (Class F, ASTM C618 [9]) has made important advances in the search for new applications [10; 11; 12]. Currently, the annual production of fly ash worldwide is estimated to be around 700 million tons [13; 14]. These achievements are aimed at the development of binding materials due to their wide range of possible potential applications [15; 16; 17]. Alkaline cements, also called geopolymers, are obtained through a chemical process between fly ash and an alkaline solution at temperatures below 100 \({}^{\circ}\)C. This reaction results in the dissolution of reactive phases of aluminosilicates. With the progress of the reaction, water is gradually removed, and the tetrahedral groups of SiO\({}_{4}\) and AlO\({}_{4}\) form the polymeric precursors [18; 19; 20].  
According to various studies, the physicochemical properties of geopolymers depend mainly on the Si/Al, Na/Al ratios and water content [21; 22]. Often, these ratios are investigated indirectly, for example, by varying the activation ratio and curing conditions [23; 24; 25]. The chemical composition in mass is largely related to the kinetics of the geopolymer matrix; therefore, FA reactivity and the geopolymers chemical behavior depend specifically on the particle sizes and surface area [26; 27]. The activation of these materials is also promoted by mechanical means, such as grinding, where results that favor geopolymerization are obtained at low curing temperatures. Through FT-IR it is demonstrated that FA with finer particles produce high reactivity indexes in the geopolymers and, therefore, an increase in the reaction product of the alkaline aluminosilicate amorphous gel. This produces an increase in compressive strength [28; 29].  
On the other hand, the use of X-ray diffraction (XRD) allows us to observe the formation of new phases which are characteristics of the FA geopolymerization process, such as semicrystalline and polycrystalline phases of inorganic polymers (especially zeolites) [30; 31].  
In this work, the influence of three different particle sizes of Mexican FA were comparatively studied in the physicochemical behavior of the geopolymers, using XRD, FT-IR, and SEM.