In my fantasy setting, slimes are a species of giant (about eight cubic feet), single-celled amoeba. They have the ability to freely change their state and viscocity, being able to "melt" into a slurpy fluid close to water, harden into a dense putty, or anything inbetween.
What organic mechanism might allow the slime to change state like this?
If your slimes are single cells, you're going to run into some huge problems. Cells are filled with a fluid called cytoplasm, which is contained within the cellular membrane. In essence, cells are like little water-filled bags. If your slime is only one cell, it can't be very slimy, because you don't want that bag to burst and leak the slime's guts everywhere. Your slime's ability to move, and its ability to "flow" or "leak" will be constrained by its outer "skin" as long as it's a true cell. I'm going to answer your question as best I can while maintaining the "one cell" constraint.
If you really want to have just one cell, I suggest evolving a very strong cytoskeleton. This is the organelle that gives cells structure, and helps them expand, contract, divide, and move. A microscopically-thin cytoskeleton might not work on the scale of a massive slime, but you can always make it thicker. This is a really simplified explanation of a complex structure, but it's a start.
By adjusting its cytoskeleton, a slime can achieve locomotion. It can conceivably change its shape - and by relaxing or tensing this organelle, it may be able to act more fluid or more rigid. Various cytoskeletal components are shown below in red and green.
Sticking with the "giant cell needs need giant organelles" theme, another direction to move in is vacuoles - the large, storage organelles that plant cells use to hold water. In plants, full vacuoles provide structural support, while empty vacuoles leave plants sagging.
You could give your slimes thousands of small vacuoles that similarly allow for different degrees of cellular rigidity. Slimes could move water in and out of their vacuoles to go from rigid to soft. Keep in mind that moving water into or out of the cytoplasm wouldn't cause a net change in viscosity. Your slimes would need to soak up ambient moisture or leak some water to become more or less fluid.
If you want your slimes to be "cells" in that they aren't composed of smaller cells, you might be able to get away with stripping away the outer membrane that keeps the cytoplasm contained. You would need to thicken the cytoplasm (inner fluid) a LOT so that the cell doesn't just turn into a puddle. No cell ever seen on Earth has done this, but if your slimes are large enough and thick enough, they might be able to pull it off. Getting your cell to move with no membrane to push or pull on would be really difficult, unless you developed "muscle" organelles. But it would be runnier - and thus a lot more "slimey" - than a single-celled organism.
Starting with this definition of viscosity — a quantity expressing the magnitude of internal friction, as measured by the force per unit area resisting a flow in which parallel layers unit distance apart have unit speed relative to one another.
I think that physics of friction — causation not results — is not well understood. But one theory of a cause of friction is coulombic attraction. This also is used to explain ideas like wet-ability and why teflon works.
If your slimes molecules could vary their charge by generating a dangling hydrogen bond, then your slimes molecules would repel each other and reduce the Van der Waals forces (attractive force between molecules). This could explain how they become less viscous.
To become more viscous, they either selectively absorb or generate dangling hydrogen bonds causing the molecules to attract each other, strengthening the attractive forces.
To reach maximum viscosity, they might make every other molecule postive and the others negative, maximizing the coulombic force which would pull them together and maximize the van der waal force.
They might also twist their molecules to make them ‘smoother’ or ‘kinkier’ so that the molecules interlock tighter when closer or slide easier when far away.
I doubt any of this is actually true, but it has the virtue of sounding plausible. Hopefully, that is enough for your need.
