In this Paper we analyze the mid-infrared (mid-IR) emission of very small dust particles in a sample of 12 protoplanetary disks to see how they are connected to interstellar dust particles and to investigate the possibility that their emission can be used as a probe of the physical conditions and evolution of the disk. We define a basis made of three mid-IR template spectra PAH$^0$, PAH$^+$ and VSGs that were derived from the analysis of reflection nebulae, and an additional PAH$^x$ spectrum

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... PAH$^x$ spectrum that was introduced by Joblin et al. (2008) for the analysis of the spectra of planetary nebulae. From the optimization of the fit of 12 star+disk spectra, using a linear combination of the 4 template spectra, we found that an additional small grain component with a broad feature at 8.3 $\mu$m is needed. We find that the fraction of VSG emission in disks decreases with increasing stellar temperature. VSGs appear to be destroyed by UV photons at the surface of disks, thus releasing free PAH molecules, which are eventually ionized as it is observed in photodissociation regions. On the opposite, we observe that the fraction of PAH$^x$ increases with increasing star temperature except in the case of B stars where they are absent. We argue that this is compatible with the identification of PAH$^x$ as large ionized PAHs, most likely emitting in regions of the disk that are close to the star. Finally, we provide a UV-dependant scheme to explain the evolution of PAHs and VSGs in protoplanetary disks. We show that A stars modify the size spectrum of PAHs and VSGs in favor of large PAHs while B stars destroy even the largest PAHs up to large radii in the disk. These results allow us to put new constrains on the properties of two sources: IRS 48 and "Gomez's Hamburger" which are poorly characterized.