Syringe pumps, valve pumps and peristaltic pumps are classified as positive-displacement pumps.
When a solenoid valve opens or closes, there is usually a small amount of liquid moved. This effect can be deliberately used as a pump. There are solenoid valves that are also specified for use as precision pumps, but they have to be at least 2-way (3-port) valves since you need a source to aspirate from and a destination to dispense into. They are very precise but you can't control the volume of the dispense except by making multiple dispenses.
Syringe pumps as you know use the motion of a piston in a cylinder to aspirate or dispense a fluid. They can be extremely precise and accurate if built properly. I used to work with some that were accurate below 10 microliters and precise, IIRC to within about a microliter. However, they were also very expensive. Syringe pumps are generally slow moving as high speed can cause the pump to cavitate, which reduces precision and introduces air into the fluid path. Cavitation is a release of dissolved gas (air) into the pumped fluid when the fluid pressure drops below the vapor pressure of the gas. This causes the gas to be released as small bubbles.
One common technique that I saw when I was involved in the medical device industry is to build a motorized fixture that a syringe can fit into and and use that to control the piston. It gives a high precision and also solves the problem of contamination: just throw the syringe away when you're done.
Peristaltic pumps are also an option for a positive-displacement pump, but they are typically an order of magnitude less accurate than syringe pumps. They have the advantage that material compatibility is mainly an issue of the what the tubing is made from since the pump impeller never actually touches the fluid being pumped. They do also require periodic maintenance and must be adjusted properly to work well.
Hmmm, that ended up being a lot longer than I expected
Solenoids are common for fluids but they can be a bit rough in terms of timing. What you could do is use it anyway but with a small sized hole capping the exit. Then control the valve with a MOSFET or transistor circuit and calibrate this to probably the second (or millisecond) range to get the right size droplets. You would likely need some pressure on the ink depending on the hole size.
This is true. When we used the syringe pumps I mentioned in my post above, we adjusted the height of the dispenser so at the end of the dispense a small drop would be touching the surface of the dispensed fluid. Then when the dispenser moved up, the surface tension would pull that drop off the tip. This was a major factor in getting the precision we needed.
MY reference was chemistry lab where we calibrated the size of drops from a dropper with a rubber bulb on the end by dropping some number of drops, perhaps 20, into a graduated cylinder and measuring the volume of 20 drops. Then we knew how many drops of reagent to use in an experiment.