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You don't need to PWM a peltier, but they do last longer if you control them that way. A peltier I got from a mini fridge, when hooked up to a power supply, produced a bit of ice on the surface, so it really depends how quickly you want to cool the samples and if they're producing heat or not.
A solid-state relay is just a transistor and a transistor driver, any MOSFET with a gate driver should be fine, the gate driver is required to boost the instantaneous current to the gate of the MOSFET , which sucks up charge like a capacitor. The reason you need the driver is so the charge up time is minimal (happen quickly) so you avoid the in-between time of not being fully on or off. The MOSFET is less efficient in that region of being turned on, so to consume less power you need to switch fast. 20khz or more should be fine for a PWM frequency. The idea is that you change the temperature in a little burst, then give it the off-duty time to diffuse the heat a bit, then repeat.
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