## 3 Results
### Particle Populations in Pea Powders  
Regardless of the type of mill and the grinding conditions, the grinding of pea cotyledons produces powders with a wide range of particle diameters between 1 and 2000 \(\upmu\)m. The particle size distribution curve for the pea powders shows a multimodal shape, characterized by several peaks of different intensity depending on the grinding conditions. This trend was observed for the powders generated by the two experimental designs with ball and roller mills, as well as for the grinding kinetic with the ball mill (Figure 2). Multimodal particle size distributions of legume powders, especially pea powders, have already been reported in the literature [8; 28; 29]. Figure 1: Scanning electron micrographs of pea cotyledon structure. (**a**) Cotyledon matrix structure; (**b**) cotyledon cell structure. Figure 2: Particle size distributions of pea powders at different grinding times in the ball mill. The particle size distribution of the powders (Figure 2) was studied as a function of the grinding time using the ball mill. The effect of grinding time can be seen in the evolution of peak intensities. The intensity of peaks associated with coarser particle size ranges decreases with increasing grinding time, while the intensity of peaks associated with finer particle size ranges increases (Figure 2). We propose an original approach to the analysis of multimodal distribution curves, considering that pea powders are formed by the coexistence of several particle populations, differentiated by their diameters and their characteristics. [PERSON] et al. (2006) [PERSON] et al. (2006) proposed a probabilistic deconvolution method for the analysis of multimodal particle size distribution curves of powders obtained after the milling of durum wheat grains. Their probabilistic solutions considered 4-5 particle populations to deconvolute the particle size curves, with overlapping diameters between the populations. However, they did not identify relationships between the different populations and the comminution mechanisms. In the present study, the multimodal analysis is based on a simplified approach by considering the assembly of particle populations without diameter overlap. We assume that each of the log-beta laws identifiable on the particle size distribution curve can be associated with a specific population of particles within the powder. The decomposition of the particle size distribution curves by deconvolution of several log-beta laws allows us to identify four different populations whose presence is either systematic (i.e., in all powders) or dependent on the grinding conditions. This method made it possible to identify the limits for each population (as described in the next section). However, in the case of population 2, we chose to define the upper limit as a technological value with the maximum particle size of standard quality wheat semolina (\(\sim\)470 \(\upmu\)m). This criterion was chosen as the boundary between populations 1 and 2. The multimodal analysis makes it possible to identify the different populations of particles in the pea powders in order to describe the specificities of their microstructure (Figure 2). 1. Population 1 is associated with very large particles with a diameter greater than 470 \(\upmu\)m (Figure 2). This population is made up of preground particles that are usually considered too large for classical technological applications. Particles of population 1 consist of large tissue fragments of the cotyledon matrix with a high proportion of intact cells and only a few open cells in the periphery (Figure 3a). The diameter range of the very large particles (>470 \(\upmu\)m) is largely higher than the diameter of the cotyledon cells (84 \(\upmu\)m). 2. Population 2 is associated with coarse particles with diameters ranging between 55 and 470 \(\upmu\)m (Figure 2). This particle diameter range is classically suitable for technological food applications of pea powders as flour and/or semolina.