Diamonds are among the hardest things in nature. In fact they define 10, the top of the Moh’s Hardness Scale. You can scratch all things with a number lower then your number.
Glass is pretty fragile but very uniform. If you introduce an imperfection, like a scratch, it will fail along that line much earlier. So diamond, being super hard, is useful for doing the cut and break sequence on glass without dulling very fast.
Cutting brittle materials is done by stabbing into it and then dragging your ‘blade’ to make a scratch. Ten thousand scratches is a cut.
Stabbing into a material requires something harder than it. Glass is pretty hard, so that limits us to 2 common tool materials; diamond and carbide. Diamond is the hardest thing out there, and because glass is so easily broken you want to choose the absolute best tool for the job. So, diamond dust applied to a steel tool is used, and the cut is made very slowly and carefully.
Diamond is a state of carbon formed under intense pressure and heat. The carbon atoms are all interlocked in a matrix where each bond angle is 109.5 degrees. This provides a level of stability and hardness that is second to almost nothing.
Graphite is a counter example. Instead of this matrix I describe, graphite carbon is bonded in planes or sheets that shave easily. This is why pencils leave behind substance on paper.
The 109.5 degree bond angle is a motif present in a lot of chemistry, and as molecular structures have bonds that approach this number they become more stable. This angle occurs a lot in organic compounds as well.
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