I'm not sure I understand your question, but here's what I think you're asking: If I place hot (or cold) water into a container, which type of container will keep the water warm (or cold) the longest?

The variable in this experiment is the material holding the water. One important property of materials is how well they conduct heat. Generally, metals conduct heat very well while ceramics, foams, and even wood are usually good insulators. In a metal, heat travels quite easily: if you put one end of a metal spoon in a flame, the other end will get warm fairly quickly. Insulators do not conduct heat as quickly, and are used for pot handles: they stay cool even while the metal pot gets hot. It's important to note that there are several different ways that hot water can lose its heat to the environment. One of the strongest is the steam rising off the surface of the water, so it would be important to either compare closed objects (like sealed thermoses) or open cups with the same exposed area.

Newton's Law of Cooling would be a good place to start investigating the science behind cooling. One important part of this law is that, no matter how good your insulating material is, as soon as you place the hot water filled container in a cooler room, the temperature of the water will start to drop. A very good insulator keeps the rate of the temperature change very low, but cannot keep the water at its original temperature. The change may be so small it doesn't register on your thermometer for a while, but it's still happening.

All subjects dealing with changing temperature fall under the general category of thermodynamics, but in this case the emphasis is on the thermal properties of the material holding the liquid. A material scientist, especially one who deals with thermal insulation, would be the best mentor.

Foam is better than metal, it might be interesting to try different shapes with the same material.

I got some information for you from the Web sites below:

"Our material is by far the world's worst conductor of thermal energy and the world's best thermal insulation medium, and it provides excellent protection against rust and corrosion, including Corrosion Under Insulation (CUI).

nanotech

"I have just done an experiment on what makes the best insulator. I used 4 materials (cups) polystyrene, paper, porcelain and metal. Polystyrene was the best insulator followed by paper, porcelain and then metal. I understand it has something to do with conduction convection and radiation but am not sure exactly what or why."physics