Is Silicone rubber a family of materials with different types like FKM, Viton, or Fluorocarbon, or are all silicone rubbers essentially the same with some minor tweaks? Can someone point me to a list of available silicone rubbers on the market?
Dow Corning, Wacker, and Momemtive (formerly GE) are the worlds dominant silicone manufacturers. There are others who play niche roles. If not truly a definition, the common feature of silicone rubber is an Si-O-Si backbone. Different silicons will have varying lengths, degree of cross linking and substituents on the Si atoms.
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All generalizations are false, including this one.
As indicated by others (and like all other polymers), the only thing common to all silicone rubbers is the base polymer. The final characteristics are determined by compounding to obtain the desired properties (but I wouldn't describe compounding as minor tweaks).
There are MQ, methyl siloxane, VMQ - Vinyl methy siloxane based, PVMQ, a Phenyl based siloxane. They each have their unique properties. It will take a long time to explain these. So google it under "Types of Silicone Rubber"