I've read a bunch of pages for what seems like a simple question but haven't been able to find a definitive answer. My main confusion comes from the (in my newbie mind) conflicting statement, where wool presumably is warmer, but acrylic doesn't breathe. The intended use is indoor slipper socks (not for use with shoes).
My understanding is that acrylic not breathing should make it warmer, since it's better at storing body heat.
Am I missing something?
Warmth in clothing comes mostly from trapped air, which is a good insulator. Water is a good conductor of heat, and water vapor can transfer heat.
With acrylic, depending on the characteristics of how the fiber was made and how the fiber is used to create a fabric, it can be a better insulator than wool. However, clothing presents a challenge because the body perspires and gives off moisture. Depending on what happens to that moisture, it can defeat the insulation properties of the trapped air.
"Breathability" is kind of a misnomer. Fabrics aren't an air-tight seal around your body. The term really describes more what happens to the moisture. In the summer, you will be coolest if the perspiration is evaporated, removing body heat. Synthetics, like acrylic, aren't as good as many natural fibers at getting the moisture away from your body and evaporating it, so you feel hot and sticky. It isn't so much the insulation value of the fabric preventing the hot air from warming you, as not getting much wicking to keep you drier, and the benefit of evaporation to make you cooler.
It would seem logical that holding that moisture in better would help keep you warm in the winter, but there's more going on. Heat moves from a hotter area to a cooler area. The bigger the temperature difference, the more heat moves. Its ability to do that depends on the material its doing it through. An insulator slows it way down. During the summer, there typically isn't a huge difference in temperature between your body and the surrounding air, so the heat isn't under a lot of "pressure" to move, and the insulation value of the fabric doesn't play a big role. It's ability to wick and evaporate moisture is the most important characteristic in keeping you comfortable.
A different process is at work when you're outside in the winter or inside using slipper socks at floor level and want to keep warm. There's a much bigger temperature difference between your body and the surrounding air, and your body heat wants to move to the surrounding air. The moisture is very efficient at transferring heat away from your body, whether or not it evaporates. Think of something like water-cooling that uses a closed loop of water to transfer heat without evaporating it.
A synthetic material like acrylic can become saturated with moisture, so it's basically just holding a heat conductor against your body. Wool has a unique property. The fibers have a crimped structure that traps a lot of tiny air pockets. At the same time, it has the ability to absorb and wick a lot of moisture. Eventually, there will be enough moisture to reach the outer surface and evaporate, but there is a very long, convoluted, capillary path between you and the outside air, so the moisture can't do a good job of siphoning heat.
This makes wool warmer for a purpose like slipper socks. There are also blends of wool (particularly Merino wool), and synthetic fibers, like acrylic, that do a good job. The wool manages the moisture, allowing the synthetic to provide good insulation. The synthetics are much less expensive, so you get the warmth at a lower cost.
I want to add to the answer of fixer1234 that wool is better at regulating the temperature than purely insulating the wearer, though I don't know the mechanics of how this works.
Imagine you could make socks from a perfect insulator. Your feet would never ever be cold again, but because our body doesn't stop producing heat, your feet would soon overheat.
By more regularly wearing woolen clothes I noticed that wool seems to insulate just to the right degree (to a comfortable body temperature). If the surrounding air temperature rises, wool seems to insulate a little less than in colder temperatures, allowing you to still feel comfortable. Of course there are limits to that property, but in changing temperatures I feel more comfortable in wool than in other fibres like cotton or synthetics.
One disadvantage of wool is that it can be scratchy. Either make the slippers from special sock yarn, use a wool cotton blend or put on a thin layer of cotton underneath (like a pair of thin cotton socks inside a woolen sock slipper).
And acrylics has another disadvantage when it comes to making slipper socks - it tends to cling to smell. If the recipient of the socks has smelly feet, you can expect the socks to intensify that smell in no time at all.
