Why do we have two standard filament sizes, 1.75 mm and 3 mm? Does it really make a difference when printing? Or is the 1.75 mm just for smaller printers?
In what situations should I be using 1.75 mm?
When should I be using 3 mm?
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There's no appreciable difference. Just use the filament that fits your particular printer.
If you don't yet have a printer, then I'd get one that uses 1.75 mm filament:
1.75 mm is increasingly becoming the "standard", thus being easier to get. Some filaments are not available as 3 mm.
1.75 mm filament allows for finer control, because feeding in 1 mm of filament corresponds to less plastic extruded.
1.75 mm filament requires less force to extrude. Compressing 1.75 mm down to 0.3 mm takes less force than doing the same to 3 mm filament.
However, the advantages are fairly minor. I don't see any reason to replace a functioning 3 mm extruder with a 1.75 mm one (yet).
There are a few factors to consider those two:
I generally agree with the points in masteusz's and Tom van der Zanden's answers, but I would add a bit more detail. Generally, the differences are minimal, however:
Ultimately, you can tune your printer and slicer settings to create or negate most any of the differences in print quality caused by filament diameter, but you may have an easier time with certain prints and printers with one diameter over the other.
I think (as do a lot of others) that the differences are rather minor. So just my 2 things I know from experience. We've been using 3 mm for some years and now we are moving towards 1.75.
1.75 mm filament is very easy to entangle, especially, if it is not on spool. You can even easily create knots on the filament and it is very hard to untangle. As long as you keep it on spool and always buy on spool, you should be fine.
3 mm filament on the other hand, creates very high tension when the spool is about to end. Sometimes the last couple of meters are unusable because of this and you have to throw the rest of the spool away.
As I read the history, 3 mm filament was an accident of the supply chain when 3D filament printers were first being developed by hobbyists. There was a product called a "plastic welder" which consisted of a melting device and a source of filler material. This filler was 3 mm plastic.
As the techniques and equipment developed, the market for filament grew to a size where it could support companies producing filament specifically for 3D printing. The benefits of 1.75 mm filament over 3 mm were, IMO, huge -- especially the easier melting and lower force needed by the extruder.
Except for special purposes like pushing soft plastics through Bowden tubes, it seems from the marketplace that 1.75 mm filament as completely overtaken 3 mm filament.
A possible second-order disadvantage of 1.75 mm filament can be water absorption. The surface-to-volume ratio is higher -- there is more surface per unit of the filament through which water vapor can be absorbed. It is important to keep filament dry, and sometimes necessary with both 3 mm and 1.75 mm to dry the filament in an oven before use.
One thing I haven't seen anyone mention yet is the issue of the size vis-a-vis time of the final print. 3 mm filament allows the printer to spit out a lot of plastic at once, letting you build must taller prints much faster. Larger filament can also provide a much wider base for the next layer to rest on; smaller filament needs to print multiple times side-by-side to get the same width.
Of course the fineness of the layers will suffer, but you can go about twice as high in one pass compared to when you use the smaller filament. On the flip side, the smaller filament will more easily print extremely fine layers than the bigger filament. So it's a trade off.
Consider the printers building items feet or tens of feet in height. They use "filaments" that are measured in centimeters or decimeter a instead of millimeters.