This paper presents a generalized Gronwall inequality with singularity. Using the inequality, we study the dependence of the solution on the order and the initial condition of a fractional differential equation.

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Thick silica antireflective (AR) coatings with controlled thickness and durable AR performance were prepared by a base-catalyzed sol-gel process using tetraethyl orthosilicate (TEOS) as precursor and hydroxyl-terminated polydimethylsiloxane (PDMS) as a modifier. The addition of PDMS greatly increased the controllable viscosity range of the silica sol but did not obviously affect the particle size. This phenomenon is attributed to a "compulsive aggregation" process of the sol, which involves the formation of "PDMS bridges" between silica particles in the sol. The mechanism of "PDMS bridge" formation is proposed based on sol viscosity, sol particle size changes, and FTIR identification. The increased controllable viscosity range provided a convenient way to prepare AR coatings with controlled thickness and therefore with controlled wavelength of maximum transmittance. The introduction of PDMS into the silica sol also increased the hydrophobicity and hence the durability of the AR coatings in wet environments.