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Journal Club Theme of March 2015: Mechanics of 2D Solid and Fluid Crystals
Sulin Zhang, Pennsylvania State University
Hongyan Yuan, University of Rhode Island
2D crystal is a widely existing material form with intriguing mechanical properties. Examples include polymersomes, liposomes, cell walls, endoplasmic reticulum, graphene, carbon nanotubes (CNTs), hexagonal boron nitride, transition metal dichalcogenides, etc. By virtue of their huge lateral dimension to thickness ratio, they are 2D surfaces deforming in 3D space. They are highly flexible, able to undergo large conformational and topological changes, yet extremely robust, able to maintain their structural integrity without fracture even under severe deformation (e.g., red blood cells pass through small capillaries). They exist in different phases (fluid, gel, and solid) and frequently undergo phase transitions. In performing their functions of various kinds, they act as interfaces interacting with matters of different length scales, and/or provide a medium for small inclusions of sizes on the order of their thickness. Understanding the mechanics of these 2D crystals provides rational guidance for tailoring their functional performance.
I. Basic properties
Mechanics research of 2D crystals triggered by the discovery of CNTs, dates back to late 1990’s, motivated by the high strength and high flexibility of the single-crystal cylinders . Results and concepts of thin shell theories were frequently borrowed to characterize the mechanical properties of CNTs; yet some unique properties go beyond the classical shell theories. A few of them are listed below.
(1) Isotropic or anisotropic?
For liquid crystals, isotropy is expected as the basic building blocks diffuse within the plane. Graphene, as a hexagonal lattice, is isotropic in the infinitesimal strain regime, but not isotropic at finite strain. Instead, it displays 6-fold symmetry and chirality-dependent properties.
A 2D crystal has geometrical (nominal) and physical thicknesses. If the 2D crystal is sufficiently thick (i.e., more than one atom in the thickness direction), these two definitions coincide. For instance, lipid bilayer is ~5 nm thick, measured by the distance between the head groups of the inner and outer lipid molecules. Yet defining the thickness of a monolayer, such as graphene, is not straightforward, and its nominal (0.34 nm) and physical (0.067nm)  thicknesses should not be confused.
If a 2D crystal is biologically compatible, it should be excitable by physiologically relevant energy, about 15 k_b*T. Let its bending modulus B ~ E*h^3 ~ 15 k_b*T, one could estimate its thickness h with known Young’s modulus E of the crystal, or vise versa. Given the thickness of the lipid bilayer is ~ 5 nm, one arrives at a Young’s modulus of ~10 MPa. Given the Young’s modulus of graphene is ~ 1 TPa, the thickness of the graphene is ~ 0.05 nm, on the same order of the theoretical value. The consistency suggests that graphene is biologically compatible.
(3) Bending modulus and Gaussian bending modulus
Bending modulus of 2D crystals can be accurately calculated by measuring the undulation spectra under thermal excitation [2,3]. For a single-layer crystalline sheet, the bending modulus is related to the Young’s modulus by , ignoring the effect of Poisson’s ratio. While for a multi-layer crystalline sheet, the relationship between the bending modulus and Young’s modulus depends on the nature of inter-layer interaction.
Gaussian bending energy arises when a surface undergoes topological change. During virus budding, each budding from the mother lipid membrane gives rise to a Gaussian bending energy of 4*pi*B_G. For 2D solid crystals, topological changes generally involve not only Gaussian bending energy, but also bending and stretching energies. A careful separation of these energy terms is essential to determine the Gaussian bending modulus B_G .
II. Defect-mediated phenomena
2D layered crystals support new classes of defects that are absent or unimportant in bulk crystals, such as 1D edges, pervasive surface functionalization, 2D grain boundaries, doping functionalization sites, folds, curvature-induced topological defects, and rationally controlled stacking faults, in-plane heterojunctions between two different components and stacking with different layers. While these defects are typically seen as imperfections that could significantly degrade performance of materials, they can be exploited to produce novel physical phenomena, hereby holding promise beneficial for creating new classes of devices.
Edges underlie the electronic properties of graphene. Large stress can be generated at unreconstructed edges, inducing graphene warping . At high temperatures, double-layer edges  are more stable than monolayer edges . Owing to the strong hydrophobicity of the lipid bilayer, open edges in lipid bilayer rarely exist, which underlies the self-healing properties of the cells, the basic building block of living bodies.
In lipid bilayer, transmembrane proteins of different shapes impose local curvatures to the membrane, forming curved domains. These curved domains weakly attract to each other, a unique phenomenon in biology: curve to attract , rendering protein aggregation and sorting and subsequent functions possible. Owing to the ease of out-of-plane bending, surface dislocations in 2D layered crystals display a ripple morphology, termed as ripplocations (Figure 1) .
Interestingly, same-sign ripplocations are short-range attractive, in distinct contrast to the repulsion of same-sign dislocation in bulk crystals. MWCNTs ripple under compression and bending. Doping can change the energy landscape of the folds, resulting in controlled origami. Due to the in-plane fluidity of the 2D liquid crystal, vesicles (i.e., closed biomembranes) can undergo large out-of-plane deformation and morphological changes (Figure 2), which is essential for living cells.
(3) Fracture mode
Because of its thinness, one rarely sees the mode-I fracture for a 2D crystal. Tearing a 2D crystal in vacuum, just as tearing a piece of paper in air, naturally results in a mixed fracture mode , even though mode III is intended. This is because the material is always bent at the crack tip, owing to the large ratio of in-plane to out-of-plane rigidities. The fracture path might be chirality dependent, and chemical additives may alter the fracture path. It would be interesting to tear a piece of lipid bilayer and see how it self heals.
(4) Endocytosis and virus budding
It is understood that the uptake of nanoparticles by living cells is size dependent [5, 11], shape sensitive (Figure 3) , and local mechanical environment regulative , all of which depend on the mechanical properties (bending and Gaussian bending moduli, membrane tension) of the lipid bilayer and the lipid-nanoparticle interactions. In the case that the adhesion is protein-mediated, the adhesion strength is thermodynamically controlled and biochemically regulated . This unique property renders viral infection a highly effective and robust process, and sheds light on the design of nanoparticle-based therapeutic agents.
(5) Building Legos with 2D crystals
With a large library of 2D crystals available by scotch-tape exfoliation, each with its own beauty and unique character, it is now possible to assemble these isolated atomic planes into 3D van der Waals (vdW) homostructures and heterostructures through layer-by-layer stacking, just like building Legos  (Figure 4). With rationally chosen building blocks (2D crystals) and stacking sequence, these 3D heterostructures may compensate for the weakness of its constituent 2D crystals and possess exceptional multifunctionalities that enable multi-tasking (mechanical, optical and electronic) applications. Such new materials with great diversity are subverting the classical materials synthesis, leading to a new paradigm of “complex materials on demand”.One expects to see Moiré patterns, lattice mismatch induced dislocations, surface and buried ripplocations, trapped interfacial atomic dirt, etc., in these homo and heterostructures.
III. Computational/theoretical modeling
The intrinsic multiscale organization of the 2D crystals has naturally been studied via an equally broad range of methods, spanning from all-atom molecular dynamics (MD) simulations, coarse-grained simulations, to continuum shell theory.
(1) Atomistic simulations
Molecular dynamics with empirical force fields have become a routine tool to simulate properties and phenomena. The accuracy of these simulations relies on the quality of the force fields. Care must be taken for phenomena involving complicated bond orders, bond forming and bond breaking, as most empirical force fields are incapable of treating these bonding conditions. Reactive force fields (reaxFF), though one order of magnitude more computationally expensive than typical ones, are designed to treat these bonding conditions in non-equilibrium states. For instance, The Tersoff-Brenner potential predicts that a graphene would fracture along the zigzag direction upon tearing, whereas reaxFF predicts along the armchair direction . The latter is consistent with tight-binding predictions.
(2) Coarse-grained simulations
Coarse-graining crystals relies on the Cauchy-Born rule that couples lattice and continuum deformation kinematics. Yet the standard Cauchy-Born hypothesis is deficient to establish the constitutive relations for a 2D crystal deforming in 3D space, which motivated the development of exponential Cauchy-Born kinematics .
Coarse-graining liquid crystals involves lumping a group of atoms into a site (a coarse grain). A lipid can be lumped into 10 sites or 3 sites. A recent approach coarse-grains a small patch of lipids into a grain, thereby enabling the highest possible coarse-graining (Figure 5) . However, stabilizing the grains in the 2D fluid phase while capturing the essential materials properties is proven to be challenging.
(3) Continuum modeling
Given the established constitutive relations from the exponential Cauchy-Born kinematics, simulation of the deformation morphologies of 2D solid crystals seems to be straightforward. One notes, however, the computational affordability is limited by the interlayer interactions (or self interactions). A large number of Gaussian points are required in order to prevent inter-penetration to occur between the crystal sheets. For 2D liquid crystals such as lipid membranes, the Canham-Helfrich free energy functional can be used to simulate conformational and topological changes. Additional numerical burden is however involved to addressing membrane fluidity.
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