Diatoms (a member of the algal class Bacillariophyceae) appear as yellow green or brown patches, and/or as stringy brown mass. Diatoms can be found in virtually every freshwater and saltwater body, and are one of the most ubiquitous algae in the world, making up a quarter of the world's biomass. Photosynthesis by diatoms in the oceans produce more than 25% of the world's oxygen. They can be found in small amounts on the surfaces of plant leaves and the substrate - and are found in virtually every aquarium, regardless of how clean a tank may look.
Diatoms multiply rapidly when conditions are right, doubling in just 24 hours, however, the life span of individual cells are short, usually less than a week. Diatoms construct cell walls out of silica, and this has lead some to draw the conclusion that it is due to silicates from sand or soil that causes diatoms to bloom. Studies have shown (Kilham, 1971), that the silicate levels required for diatoms to thrive in freshwater are very low and most freshwater systems are rich in silicates - so the total removal of silicates does not seem like a practical approach in controlling diatoms. Most tap water contain silicates, as does substrates such as sand or aquasoil.
In planted aquariums, diatoms are especially common in new setups in the first couple of weeks. Thankfully, diatoms tend to fade by themselves, even in planted tanks with no changes in Silicate levels, as the plants settles in and the tank becomes biologically matured over 3 to 4 weeks. Tank maturity and plant dominance are stronger contributing factors to a tank being free from visible diatoms rather than specific water parameters or silicate levels.
Why do new tanks suffer from Diatoms more readily? Some have suggested ammonia or metabolites from stressed, freshly planted plants as possible triggers, while others have claimed that elevated phosphorous levels can be a trigger. Ion-exchange resins that absorb the above seem to work in some cases, but not work in other cases; perhaps they can be used in tanks where there are persistent cases as a last ditch effort. As with most biological systems, the trigger factors for algae are complex and often tied to more than a couple of variables.
Diatoms reproduce very quickly, and many algicides are not particularly affective against them. Plant husbandry, and getting new plants adapted to the tank, does have a big impact on algae. To see how the tank at top, transforms into the tank below, with no use of algicides, read this page.
Biologically immature planted tanks
Sustained presence often caused by tank instabilities that resets/disrupts the biological cycle (exotic substrates, especially for DIY dirt substrate users can face this)
Poor water quality from tap
Elevated ammonia or phosphorous levels in a biologically immatured tank
Actions that help speed up tank cycling
Plant husbandry techniques to get plants adapted more quickly as described on this page
Manually siphon off mild cases
Ion-exchange resin that removes ammonia/phosphorous in desperate cases
"But many papers out there show a link between silicates and diatoms"
Just as many papers show links between phosphates and algae. Concepts from surveying natural systems do not necessarily transfer directly onto how our aquariums work. Silicates are in most tap waters and soils; and most aquariums have no lack of them. A newly set up planted aquarium with low silicates from the tap water can still suffer from a diatoms outbreak. On the other end, a well planted, biologically matured aquarium will see no diatoms even with elevated Silicates or phosphorous levels. It is much more practical to focus on overall plant health and plant dominance in a planted tank.