Many scientific papers suggest that the mere existence of free-standing graphene sheets violates theoretical expectations, that it is an anomaly that demands an explanation because flat graphene sheets would somehow destroy themselves. A paper in the current Nature describes “Ultraflat graphene”, but this graphene resides safely on mica surfaces. The paper mentions the numerous observations of microscopic corrugations in graphene, and notes that “This rippling has been invoked to explain the thermodynamic stability of free-standing graphene sheets”, referencing a much-cited 2007 paper in Nature Materials, “Intrinsic ripples in graphene”.
The basis for this expectation of instability is the Mermin-Wagner theorem, which says that crystalline order is impossible in two-dimensional structures at non-zero temperatures, and has been interpreted as saying that a free-standing, two-dimensional crystal would be disrupted by thermodynamic forces (hence “2D materials were presumed not to exist”).
In graphene, structural disruption would require that something (unspecified) somehow impart enough energy to break covalent bonds. This made no physical sense to me.
The stubborn existence of free-standing graphene has been explained by arguing that low-amplitude ripples protect graphene from disruption by perturbing it away from exact flatness, thereby making the Mermin-Wagner theorem technically inapplicable. This struck me as making even less physical sense, so I dug deeper.
The Mermin-Wagner theorem says nothing of the kind
It turns out that there’s been a basic misreading of the Mermin-Wagner theorem. The theorem establishes a scaling property of in-plane, long-wavelength thermal fluctuations — elastic distortions — in 2D structures. This has nothing to do with disrupting the structure, which can be what a materials scientist (or anyone else, for that matter) would call a 2D crystal. There is therefore nothing anomalous about an AFM observation that reveals a sheet of graphene stretched over a gap.
In an infinite 3D crystal, there is a regular lattice such that thermal fluctuations displace atoms by finite distances from mean positions centered on the lattice points. The Mermin-Wagner theorem tells us that, in an infinite 2D crystal, thermal fluctuations will obliterate long-range alignment with a lattice, and hence there can be — in this very special sense — no crystalline order.
In short, 2D structures are compatible with crystalline order except in the abstract sense that they cannot maintain strict periodicity in the limit of infinite size. That’s all. There is no need to worry about a covalent structure from being ripped apart by energy conjured from nowhere.
Nanoscale mechanics is fascinating, but not quite that exotic.
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Okay, so if I am understanding you correctly, at any sheet size below infinity, Graphene will remain stable?
Which raises the possibility of extremely long lasting materials made of graphene.
Slightly off topic, but still on the subject of graphene, I recently came across a couple of articles which to me seemed related:
http://nextbigfuture.com/2009/11/self-asse…tubes-into.html
An article on using DNA lattices to position CNs into precisely defined positions
and this one
http://www.nanowerk.com/news/newsid=13456.php
An article on the ability to use current photolithography techniques to create graphene components for electronics.
Reading the two together it seemed feasible that between the two techniques, a completely carbon based computer could be possible in the near future, and seeing your article here on graphene made me curious as to your opinion.
Re. graphene stability, I simply pointed out that the theorem isn’t about the stability of structures in the first place: It’s about a subtle property of long-range elastic deformation relative to a fixed, perfectly-regular grid.
Re. carbon nanotube and graphene technologies, I’ve been impressed by the rate of progress. I say more in a recent post, “Carbon Nanotube Transistors through DNA Origami”.
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