Mineralogical studies of silicate features emitted by dust grains in protoplanetary disks and Solar System bodies can shed light on the progress of planet formation. The significant fraction of crystalline material in comets, chondritic meteorites and interplanetary dust particles indicates a modification of the almost completely amorphous ISM dust from which they formed. The production of crystalline silicates thus must happen in protoplanetary disks, where dust evolves to build planets and planetesimals. Different scenarios have been proposed, but it is still unclear how and when this happens. This paper presents dust grain mineralogy of a complete sample of protoplanetary disks in the young Serpens cluster. These results are compared to those in the young Taurus region and to sources that have retained their protoplanetary disks in the older Upper Scorpius and Eta Chamaeleontis stellar clusters, using the same analysis technique for all samples. This comparison allows an investigation of the grain mineralogy evolution with time for a total sample of 139 disks. The mean cluster age and disk fraction are used as indicators of the evolutionary stage of the different populations. Our results show that the disks in the different regions have similar distributions of mean grain sizes and crystallinity fractions (~10-20%) despite the spread in mean ages. Furthermore, there is no evidence of preferential grain sizes for any given disk geometry, nor for the mean cluster crystallinity fraction to increase with mean age in the 1-8 Myr range. The main implication is that a modest level of crystallinity is established in the disk surface early on (< 1 Myr), reaching a equilibrium that is independent of what may be happening in the disk midplane. These results are discussed in the context of planet formation, in comparison with mineralogical results from small bodies in our Solar System. [Abridged]
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