The elastic properties of materials have been of interest to mankind since ancient times. Significant progress in the understanding of elasticity was achieved in the works of Robert Hooke, an English researcher who worked in the second half of the 17th century. It is believed that along with Isaac Newton, Hooke is “responsible” for the law of universal gravitation.
In the theory of elasticity, he is famous for the discovery of Hooke’s law, which states that the force required to stretch a material (see figure) is proportional to the amount of tension. The aspect ratio characterizes a specific material and is called Young’s volumetric modulus. For thin plates, it is convenient to work with Young’s modulus Y, which is obtained by multiplying the bulk Young’s modulus by the plate thickness. (Young’s modulus in SI units is measured in newtons per meter (N / m).) In addition to stretching or compression in a plane, thin plates have the possibility of bending. The energy associated with bending is determined by the amount of bending stiffness. (Flexural stiffness is measured in SI units in joules.)
The object that arises when the thickness of the plate decreases to one atomic layer is usually called a crystalline membrane. The most famous example is graphene, which is a single layer of carbon atoms arranged at the nodes of hexahedral (honeycomb) cells. Graphene was obtained in the laboratory by Andrey Geim and Konstantin Novoselov in 2004. For this discovery, they were both awarded the Nobel Prize in Physics in 2010. Currently, graphene is widely used in various fields, including biomedicine (molecular sensors), chemistry (molecular filters), and electronics (graphene capacitors and transistors). It is often said that graphene is a very durable material. This is due to the fact that its Young’s modulus is 340 n / m. A monoatomic layer of aluminum has 50 times less Young’s modulus! Graphene is also characterized by a fairly high flexural stiffness of 1.6 per 10 ^ –19 joules.
A bar to one end of which a tensile force F is applied. This force causes the bar to stretch longitudinally. Young’s modulus Y is related to the volumetric Young’s modulus E as Y = E t
As is known from the school physics course, atoms in crystals perform thermal vibrations around their equilibrium positions. The same thermal vibrations can cause bending deformations of the membrane. It is possible to estimate how the magnitude of bending deformations caused by thermal fluctuations increases with an increase in the longitudinal size of the membrane, and to find at what size it compares with the thickness of the membrane. The latter for a monatomic layer is of the order of fractions of a nanometer (nm). The corresponding length, called the Ginzburg length, turns out to be extremely small for graphene. It is equal to 1 nm at room temperature. Therefore, for all graphene samples studied in laboratories, thermal flexural fluctuations are of fundamental importance. For comparison, you can estimate the length of Ginzburg for a sheet of paper 1 mm thick. It turns out to be equal to 1000 km at room temperature! That is why in ordinary life we, as a rule, do not encounter the manifestation of thermal flexural fluctuations.
Despite the fact that the role of the Ginzburg length in the physics of crystalline membranes was only recognized in the mid-1980s, the importance of thermal flexural fluctuations has been understood since the 1930s. At this time, Lev Landau and Rudolf Peierls, an English theoretical physicist, one of the pioneers of the concept of hole conductivity in semiconductors and the theory of excitons, independently showed that infinite crystalline membranes should collapse due to thermal bending fluctuations, turning into a shapeless lump of atoms … In other words, it turns out that their result, which is still included in standard textbooks on theoretical physics, for example, in the course of Landau and Lifshitz, forbids the existence of graphene!
How to try on the classic Landau-Peierls statement and graphene flakes that are currently available in many laboratories around the world. As is usually the case with many questions, the clue is hidden in the details. Note that, according to the Landau – Peierls result, there are no crystal membranes of infinite size. At any given temperature, there is a limiting longitudinal dimension of the membrane, when bending thermal fluctuations are insufficient to destroy the membrane. We emphasize that this size should not be confused with the Ginzburg length! If we estimate this limiting size for graphene at room temperature, then we get a gigantic length. Since in laboratories graphene flakes are usually only a few micrometers in size, it would seem that the clue has been found!
As is often the case in physics, the simplest explanation of the phenomenon does not always turn out to be correct. The Landau-Peierls assertion is based on the assumption that flexural thermal fluctuations do not interact with each other, as they say, are harmonic. It turns out that for graphene this idealization is valid only for very small sizes, less than the Ginzburg length. For micron-sized graphene, it is fundamentally incorrect to ignore the interaction of flexural fluctuations with each other. Their interaction leads to the fact that the bending stiffness of the membrane begins to grow in a power-law manner with an increase in the longitudinal size, that is, the larger the size of the membrane, the more difficult it becomes to bend it. This effect was predicted theoretically in the mid-1980s. It leads to the fact that an infinite crystalline membrane is stable at temperatures below a certain critical value. For graphene, the critical temperature turns out to be above 10 thousand Kelvin. Such a high critical temperature means that thermal bending fluctuations never destroy graphene, since at such high temperatures the crystal lattice will already melt.
The interaction of flexural fluctuations leads to a number of interesting phenomena in crystalline membranes, which are termed anomalous elasticity. In the region of small longitudinal deformations, Hooke’s law is violated: the deformation of the membrane as a whole becomes a power function of the applied force with an exponent less than one, which, moreover, does not depend on the type of membrane atoms, that is, it is universal. Linear Hooke’s law is restored only for sufficiently large deformation values. Recently, such a nonlinear Hooke’s law for graphene was measured experimentally. Also, graphene has a negative Poisson’s ratio in a wide range of parameters. Recall that the positive Poisson’s ratio characterizes how, when stretched in one direction, the material is compressed in the transverse direction. Graphene is stretched instead of being compressed in the transverse direction! Finally, down to ultra-low temperatures, the thermal expansion coefficient of graphene turns out to be negative, that is, graphene contracts when heated, instead of expanding, as most substances do.
In conclusion, we can say that the physics of crystalline membranes, in particular graphene, once again confirms that there are no absolute truths in real science: all laws are derived within the framework of certain assumptions that should not be forgotten. In the theory of crystalline membranes, there are still a significant number of issues related to anomalous elasticity that require theoretical and experimental study. In particular, this is being done by scientists from the Physico-Technical Institute. AF Ioffe RAS, Institute for Theoretical Physics. LD Landau RAS and Skolkovo Institute of Science and Technology within the framework of the RFBR grant 20-52-12019.
.