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There's never any source on the internet that can simply give me a list of materials that have the kinds of properties I'm looking for. I'm wondering what materials, like for a solid block of something, in general expand either a lot or at least very quickly when exposed to heat. I also can't remember what property this is exactly, but I think it's the coefficient of thermal expansion.
Am I looking for something with a high coefficient or low coefficient to achieve this?
You are looking for thermal expansion and its coefficient. Of course such tables, and diagrams exist
(source)
as it is of relevance in daily live. Thermometers based on liquid mercury, or alcohol, depend on this. Bridges take this into account in expansion joints, too:
(source)
Joining two materials of different expansion coefficient equally is of relevance for the design of bimetallic stripes, found not only in thermometers and mechanical clocks:
(source)
So in reply to your "there is no source available" ...
For solids you are looking for the linear thermal expansion coefficient $α_L$:
$$α_L = \frac{\mathrm{d}L}{L\cdot \mathrm{d}T}$$
where $L$ is length and can be uniform for isotropic materials (amorphous compounds, some polymers) and varied for anisotropic materials (crystals). For crystalline materials $L$ is often given for each axis $a, b, c$ of a crystal's unit cell resulting in direction-dependent thermal expansion. Obviously, the higher the $α_L$, the higher the value of expansion $(\mathrm{d}L/L)$ per temperature change $\mathrm{d}T$, so the expansion of the solid is directly proportional to $α_L$.
The Engineering ToolBox provides a decent list of thermal expansion coefficients for common solids at NTP. Looking for the materials with high thermal expansion coefficients, you probably have to look at polymers such as ethylene ethyl acrylate $(α_L = \pu{205e-6 K-1})$, ethylene vinyl acetate $(α_L = \pu{180e-6 K-1})$ or polyethylene $(α_L = \pu{108e-6 K-1}~\text{to}~\pu{200e-6 K-1})$.
Paraffin wax also has high $α_L = \pu{106e-6 K-1}~\text{to}~\pu{480e-6 K-1}$ (depends on exact composition) and expands drastically when reaches its melting point.
Among crystalline matter, the current record is held by silver(I) hexacyanocobaltate(III), $\ce{Ag3[Co(CN)6]}$ [1]. Due to mobile layer of silver atoms in between $\ce{Co(CN)6}$ sheets the crystal lattice can fold in one direction and stretch in another. Such high anisotropy results in $α_a = \pu{130e-6 K-1}~\text{to}~\pu{150e-6 K-1}$ and $α_c = \pu{-130e-6 K-1}~\text{to}~\pu{-120e-6 K-1}$. This is the record values for both negative and positive thermal expansion in crystals.
