A short and oversimplified answer is that the keratin in horn is not as densely linked with bonds as diamond is, and consequently the atoms are less confined (in a way diamond is sort of like a maximally crosslinked material, though it’s not usually described that way).
Generally speaking, crosslinking polymers (including proteins) tends to increase their rigidity. To use a non-living example, when latex is treated with sulfur, the polymer chains also get crosslinked with those same disulfide bonds, producing “vulcanized” rubber which is harder and tougher.
The crosslinks are why you’ll sometimes see people say that vulcanized rubber is “one big molecule” (though in practice it’s hard to tell if the crosslinking was actually so complete and to link every polymer chain). This is also why vulcanized rubber doesn’t really melt, increasing the temperature will cause chemical changes instead (and while I’m not sure, my educated guess it that something similar would happen if you try to melt animal horns).
P.S. I didn’t bring it up earlier, but I don’t think your earlier statement about the way the masses of the atoms affect the bond strength is accurate. As a counterexample I’d point out that the deuterium-oxygen bond in heavy water is actually a little stronger than that of the protium-oxygen bond in regular water, and in that case the only change is using a more massive form of hydrogen.
What I’m pointing at there is that for strength/weight purposes, using big calcium nuclei to create stronger individual bonds in bone, is like making a steel beam stronger by putting more steel into it; the strength costs weight.
As a physicist, I think your understanding of bonds is a little off here.
using big calcium nuclei to create stronger individual bonds in bone
Using bigger nuclei usually makes an atom bond weaker, rather than stronger, for reasons to do with the quantum mechanical bond natures. See this explanation for why si-si bonds are weaker than C-C bonds. The simplest explanation for why is simply that the bonding electrons are much further out in heavier elements, because more atomic shells have been filled, so there is correspondingly less force of attraction between them.
I was a little confused learning this initially because i thought that the extra protons in the nuclei would have a bigger attractive effect, but then I remembered that the extra protons come along with extra electrons, so overall the effect is much more complicated and averages out to bigger atom= weaker bond.
But as @Thomas Pilgrim and the linked post pointed out, there are exceptions to this rule due to the intricacies of particular types of bonds, and you really have to dig into the quantum mechanical nature of things to be sure.
They have the same type of bonds, and the exact same structure. Diamond is harder because not every type covalent bond is equally strong (as you already noted when discussing bone).
Diamond is (close to) the hardest material in the world, because C-C bonds are quite strong, and each carbon atom has four of them. Diamond has C-C bonds densely packed in every direction.
I don’t know as much about keratin specifically. This source says some keratin-associated proteins have as much as 41% of their structure consisting of cystine (the amino acid with sulfur attached), so presumably it is also densely packed with Di-Sulfide covalent bonds.
It also says:
Disulfides are also a dominant structural and stabilizing element within larger proteins made up of tandem repeats of smaller, internally cross-linked units. Low density lipoprotein receptor-like repeats and epidermal growth factor-like (EGF-like) repeats are two examples of disulfide-rich modules that are linked in tandem to generate long, rod-shaped structures acting as cell-surface receptors or key components of the extracellular matrix (ECM)
So there you go, a clearcut case of protein being held together with covalent bonds. I mean, I still think “the primary structure is 100% covalently bonded” is sufficient to say this, but whatever.
Why is this not stronger than diamond? Well, i would guess that while the dominant bonds are covalent, they are weaker covalent bonds than C-C bonds, and there are not as many of them per atom as in diamond.
Also, we’ve been talking a lot about hardness here, but it’s not the only measure of “strength” you can use. If I cherry-picked fracture toughness, I could say that diamond is weaker than wood, because the fracture toughness of wood is higher than diamond. Check out this video of a diamond being shattered with a regular hammer! Being able to deform and then rebound back into place offers advantages in many situations, and it’s why wood and metal doesn’t similarly shatter.
To be clear, I obviously still think diamond is stronger than wood along most other measures, such as melting temp, hardness, etc. But there is not zero cost to the rigidity and stiffness of diamond.
This is a relatively minor (but interesting!) point though, please do not only respond to the last two paragraphs.
Why is horn weaker than diamond?
A short and oversimplified answer is that the keratin in horn is not as densely linked with bonds as diamond is, and consequently the atoms are less confined (in a way diamond is sort of like a maximally crosslinked material, though it’s not usually described that way).
Generally speaking, crosslinking polymers (including proteins) tends to increase their rigidity. To use a non-living example, when latex is treated with sulfur, the polymer chains also get crosslinked with those same disulfide bonds, producing “vulcanized” rubber which is harder and tougher.
The crosslinks are why you’ll sometimes see people say that vulcanized rubber is “one big molecule” (though in practice it’s hard to tell if the crosslinking was actually so complete and to link every polymer chain). This is also why vulcanized rubber doesn’t really melt, increasing the temperature will cause chemical changes instead (and while I’m not sure, my educated guess it that something similar would happen if you try to melt animal horns).
P.S. I didn’t bring it up earlier, but I don’t think your earlier statement about the way the masses of the atoms affect the bond strength is accurate. As a counterexample I’d point out that the deuterium-oxygen bond in heavy water is actually a little stronger than that of the protium-oxygen bond in regular water, and in that case the only change is using a more massive form of hydrogen.
What I’m pointing at there is that for strength/weight purposes, using big calcium nuclei to create stronger individual bonds in bone, is like making a steel beam stronger by putting more steel into it; the strength costs weight.
As a physicist, I think your understanding of bonds is a little off here.
Using bigger nuclei usually makes an atom bond weaker, rather than stronger, for reasons to do with the quantum mechanical bond natures. See this explanation for why si-si bonds are weaker than C-C bonds. The simplest explanation for why is simply that the bonding electrons are much further out in heavier elements, because more atomic shells have been filled, so there is correspondingly less force of attraction between them.
I was a little confused learning this initially because i thought that the extra protons in the nuclei would have a bigger attractive effect, but then I remembered that the extra protons come along with extra electrons, so overall the effect is much more complicated and averages out to bigger atom= weaker bond.
But as @Thomas Pilgrim and the linked post pointed out, there are exceptions to this rule due to the intricacies of particular types of bonds, and you really have to dig into the quantum mechanical nature of things to be sure.
Why is crystalline silicon weaker than diamond?
They have the same type of bonds, and the exact same structure. Diamond is harder because not every type covalent bond is equally strong (as you already noted when discussing bone).
Diamond is (close to) the hardest material in the world, because C-C bonds are quite strong, and each carbon atom has four of them. Diamond has C-C bonds densely packed in every direction.
I don’t know as much about keratin specifically. This source says some keratin-associated proteins have as much as 41% of their structure consisting of cystine (the amino acid with sulfur attached), so presumably it is also densely packed with Di-Sulfide covalent bonds.
It also says:
So there you go, a clearcut case of protein being held together with covalent bonds. I mean, I still think “the primary structure is 100% covalently bonded” is sufficient to say this, but whatever.
Why is this not stronger than diamond? Well, i would guess that while the dominant bonds are covalent, they are weaker covalent bonds than C-C bonds, and there are not as many of them per atom as in diamond.
Also, we’ve been talking a lot about hardness here, but it’s not the only measure of “strength” you can use. If I cherry-picked fracture toughness, I could say that diamond is weaker than wood, because the fracture toughness of wood is higher than diamond. Check out this video of a diamond being shattered with a regular hammer! Being able to deform and then rebound back into place offers advantages in many situations, and it’s why wood and metal doesn’t similarly shatter.
To be clear, I obviously still think diamond is stronger than wood along most other measures, such as melting temp, hardness, etc. But there is not zero cost to the rigidity and stiffness of diamond.
This is a relatively minor (but interesting!) point though, please do not only respond to the last two paragraphs.