Do proteins bind to carbon? If so how?
I can't tell you for sure what is happening, but I can tell you what could be happening.
Physical Adsorption
It is possible that the proteins are not chemically adsorbed, but rather physically adsorbed. The article you cited implies that this is the case.
Many proteins in the presence of nano-scale hydrophobic &quot;features&quot; will partially denature, and the hydrophobic core will physically adsorb (attach via van der Waals forces). This effect is amplified if water is present, since the hydrophilic parts of the protein will be stabilized in water, allowing more of the hydrophobic parts to attach to the carbon feature.
Now you are probably wondering, &quot;Why doesn't it attach to the PEG surface as well, since that is hydrophobic?&quot; There are two possible reasons.
PEG is not as hydrophobic as you think

The PEG chains have oxygen in each monomer, which would reduce the non-polar character of the chains. It is possible that the difference is enough to make adsorption of non-polar amino acids on PEG thermodynamically unfavorable, while on pure-carbon structures it would be favorable.
Shape matters
Another possibility is that the actual shape of the features makes the difference. In order for a protein to denature and &quot;attach&quot; to carbon feature via hydrophobic interactions, it has to first expose the hydrophobic groups in the core. For a flat surface, the amount of initial denaturization (maybe not a real word) would need to be higher than for a &quot;sharp&quot; feature. Sort of like how a needle can slip into your skin without doing much damage. So, the smaller the characteristic length of the features, the more likely the protein would adsorb. There are also protein-protein effects on the surface, which are again reduced by smaller-scale features. In this scenario, you could think of the proteins as kinetically trapped in solution when near a regular PEG surface, while the activation energy for adsorption to a small feature is smaller.
I can think of two ways to test this idea:

Use different proteins on the same type of surface features - the ones with less exposed hydrophobic groups should adsorb less easily
Use different sizes of features - there should be a feature size that gives a maximum adsorption for a given protein.

Chemical Bonding
The other possibility is chemical bonding between the carbon features and the protein. It is unlikely that this will happen if the carbon features are really pure carbon (graphene, for example). However, if they are creating them by ionizing PEG with an electron beam, then it is highly likely that they are functionalized with something. Carbonyl groups would be my guess, but depending on the conditions there could be nitro groups as well. In this case, there is a possibility that hydrogen bonds would form, and if the surface is reactive enough, chemical bonds might form as well.
To test this idea, I would do the same thing as before - only this time, if we are right:

Proteins with more exposed hydrophobic groups should be less likely to adsorb
Feature size should not matter at all

If this turned out to be the case, you could then systematically test polypeptides made up of different amino acids to verify it and determine whether hydrogen bonding or covalent bonding was more likely.