What affects heat transfer within a material?

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so I'm told that temperature difference is ultimately what results in heat transfer. But when considering what happens inside a uniform material, if temperature difference was solely responsible for heat transfer, then different heat transfer rates wouldn't exist among different materials at the same temperature difference. For example, Styrofoam is not a good conductor of heat, and metals are a good conductor of heat. What properties of a material give rise to heat conductivity?
 
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That's a very good question, to which I don't have a complete answer, but what I can say is this.

In almost every case, heat takes the form of the randomized translational, vibrational, and rotational kinetic energy of the atoms making up the material. In materials where the atoms are tightly bound to one another, like the atoms in chunks of metal or in diamonds, the vibrations on one atom more affect its neighbors because they are so tightly bound to one another.
On the other hand, good thermal conductivity also requires the opportunity for adjacent atoms of the material to interact in the first place. Gases are poor thermal conductors because they can only interact by collision. Atoms in gases are only in close proximity with one another for a relatively short period of time, while atoms in metals are always right next to one another.

Styrofoam is a poor conductor of heat because:
- It is made up of a foam of trapped gases and solid bubbles.