User login

You are here

Journal Club Theme of March 2015: Mechanics of 2D Solid and Fluid Crystals

Sulin Zhang's picture

 

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 [1]. 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.  

(2) Thickness

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) [1] 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 [4]. 

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. 

(1) Edges

Edges underlie the electronic properties of graphene. Large stress can be generated at unreconstructed edges, inducing graphene warping [5]. At high temperatures, double-layer edges [6] are more stable than monolayer edges [7]. 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. 

(2) Folds

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 [8], 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) [9].

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 [10], 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) [12], and local mechanical environment regulative [13], 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 [14]. 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 [15] (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 [10]. 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 [16]. 

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) [3]. 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.  

References:

1. Yakobson, B. I., Brabec, C. J. and Bernholc, J. Nanomechanics of carbon tubes: Instabilities beyond linear response. Physical Review Letters 76, 2511-2514, (1996).

2. Fasolino, A., Los, J. H. and Katsnelson, M. I. Intrinsic ripples in graphene. Nature Materials 6, 858-861, (2007).

3. Yuan, H. Y., Huang, C. J., Li, J., Lykotrafitis, G. and Zhang, S. L. One-particle-thick, solvent-free, coarse-grained model for biological and biomimetic fluid membranes. Physical Review E 82, (2010).

4. Wei, Y. J., Wang, B. L., Wu, J. T., Yang, R. G. and Dunn, M. L. Bending Rigidity and Gaussian Bending Stiffness of Single-Layered Graphene. Nano Letters 13, 26-30, (2013).

5. Shenoy, V. B., Reddy, C. D., Ramasubramaniam, A. and Zhang, Y. W. Edge-Stress-Induced Warping of Graphene Sheets and Nanoribbons. Physical Review Letters 101, (2008).

6. Huang, J. Y., Ding, F., Yakobson, B. I., Lu, P., Qi, L. and Li, J. In situ observation of graphene sublimation and multi-layer edge reconstructions. Proceedings of the National Academy of Sciences of the United States of America 106, 10103-10108, (2009).

7. Liu, Y. Y., Dobrinsky, A. and Yakobson, B. I. Graphene Edge from Armchair to Zigzag: The Origins of Nanotube Chirality? Physical Review Letters 105, (2010).

8. Reynwar, B. J., Illya, G., Harmandaris, V. A., Muller, M. M., Kremer, K. and Deserno, M. Aggregation and vesiculation of membrane proteins by curvature-mediated interactions. Nature 447, 461-464, (2007).

9. Kushima, A., Qian, X. F., Zhao, P., Zhang, S. L. and Li, J. Ripplocations in van der Waals layers. Nano Letters 15, 1302-1308, (2015).

10. Huang, X., Yang, H., van Duin, A. C. T., Hsia, K. J. and Zhang, S. L. Chemomechanics control of tearing paths in graphene. Physical Review B 85, (2012).

11. Zhang, S. L., Li, J., Lykotrafitis, G., Bao, G. and Suresh, S. Size-Dependent Endocytosis of Nanoparticles. Advanced Materials 21, 419-425, (2009).

12. Huang, C. J., Zhang, Y., Yuan, H. Y., Gao, H. J. and Zhang, S. L. Role of Nanoparticle Geometry in Endocytosis: Laying Down to Stand Up. Nano Letters 13, 4546-4550, (2013).

13. Huang, C. J., Butler, P. J., Tong, S., Muddana, H. S., Bao, G. and Zhang, S. L. Substrate Stiffness Regulates Cellular Uptake of Nanoparticles. Nano Letters 13, 1611-1615, (2013).

14. Yuan, H. Y., Li, J., Bao, G. and Zhang, S. L. Variable Nanoparticle-Cell Adhesion Strength Regulates Cellular Uptake. Physical Review Letters 105, (2010).

15. Geim, A. K. and Grigorieva, I. V. Van der Waals heterostructures. Nature 499, 419-425, (2013).

16. Arroyo, M. and Belytschko, T. An atomistic-based finite deformation membrane for single layer crystalline films. Journal of the Mechanics and Physics of Solids 50, 1941-1977, (2002). 

 

Comments

Rui Huang's picture

Hi Sulin,

Apparently you have done a lot of works on a variety of 2D materials. I learnd a lot from this summary. However, I am surprised that you still use B ~ E*h^3 in the discussion of thickness of graphene. The bending modulus of graphene has nothing to do with its thickness! Thinking about how B ~ E*h^3 is derived for a beam, you would understand why this relation does not apply for an atomic monolayer. The physical origin of bending modulus of an atomic monolayer (graphene or not) is the multibody interactions between the atoms. For a more detailed discussion, please take a look at the following short paper.

Q. Lu, M. Arroyo, R. Huang, Elastic bending modulus of monolayer graphene. J. Phys. D: Appl. Phys. 42, 102002 (2009).

Regards,

Rui

Sulin Zhang's picture

Rui,

Your point is well received, and I have read your paper before. As I stated, thin shell theory has been frequently "borrowed" to simulate these 2D crystals, including graphene. Then we would ask wether this connection between the bending modulus and thickness still holds phenomelogically. It turns out that graphene is not far from this scaling. 

Sulin

Rui Huang's picture

Sulin, what do you mean by "graphene is not far from this scaling"? You can always fit the bending modulus with a thickness, but that does not give you a physical thickness. It is not physical at all! The point is, there is no reason for this scaling to hold for a monolayer of atoms. Can stress be distributed across the thickness of an atom? On the other hand, the nominal thickness is meaningful only for multilayers and depends on the van der Waals interactions between the layers (thus not a fundamental property of the monolayer). However, using the nominal thickness for graphene (~0.335 nm), the B ~ E*h^3 scaling works pretty well down to bilayer graphene, but is two orders of magnitude off for monolayer. A simple formula for the bending modulus of monolayer to multilayer graphene was given as Eq. (6) of another short paper (W. Gao and R. Huang, Effect of surface roughness on adhesion of graphene membranes. J. Phys. D: Appl. Phys. 44, 452001, 2011).

Thanks.

Rui

Sulin Zhang's picture

Dear Rui,

I used the physical thickness for the scaling. 

Bending rigidity = 1.1eV = 1.76x10^(-19) J

E = 1.0 TPa = 10^12 N/m^2

h = 0.06nm; h^3 =2.16x10^(-31) m^3

Eh^3 = 2.16x10^(-19) 

 

Sulin

 

Sulin

Rui Huang's picture

Where do you get 0.06 nm as the "physical" thickness? Sorry, but it is irritating to see people do this. You know where Eh^3 comes from if you teach undergraduate mechanics.

Rui

Sulin Zhang's picture

Rui,

I wrote in my summary. It appears that you did not acknowledge the difference between the nominal and physical thickness. It makes sense to me. 

Thanks,

Sulin

Rui Huang's picture

I guess the physical thickness by Sulin has something to do with the atomic radius of carbon. So I checked and found the radius of carbon to be about 0.07 nm. With this radius, you may argue that the "physical" thickness of a monolayer graphene is twice of the radius, about 0.14 nm, which is close to the C-C bond length (unsurprisingly). But again, this thickness should never be used in Eh^3 to calculate the bending modulus of graphene.

Rui

Sulin Zhang's picture

Rui: 

This is not the definition of the physical thickness I refered to. Please read the paper by

Wang LF, Zheng QS et al, PRL 2005. 

Thanks.
Sulin

Rui Huang's picture

Sulin: please give the full citation information of this paper. I will read and discuss further. Another paper of interest may be:

Y. Huang, J. Wu and K.C. Hwang, Thickness of graphene and single-wall carbon nanotubes. Physical Review B 74: 245413, 2006.

Rui

Sulin Zhang's picture

Lifeng Wang, Quanshui Zheng, Jefferson Z. Liu, and Qing Jiang, Size Dependence of the Thin-Shell Model for Carbon Nanotubes, Physical Review Letters, 95, 105501, 2005. 

 

Thanks for the paper by YG Huang. I have read it before. 

Rui Huang's picture

After reading the above paper (Phys. Rev. Lett. 95, 105501, 2005), I realize now the so-called physical thickness of graphene is nothing but a force-fitting of the energy function into a continuum thin-shell model (YBB) with a classical defintion of bending modulus, Young's modulus, etc.  Apparently, Y. Huang et al in their 2006 paper addressed this issue quite sufficiently. In their words, "The present analysis also provides an explanation of Yakobson’s paradox that the very high Young’s modulus reported from the atomistic simulations together with the shell model may be due to the not-well-defined CNT thickness." It is unfortunate that after almost 10 years this "not-well-defined" thickness is still being used (not only by many physists but by some mechanicians too).

Rui

Sulin Zhang's picture

0.34nm is not physical as well. I tried to distinguish 0.066nm from the nominal thickness.

Sulin Zhang's picture

I would like to thank for Rui to raise up the heated discussions. Sorry for irratating Rui for not being clear or specific in our summary and discussions. Hopefully my following comments would make it clearer.

1. My decription "physical thickness" may not be appropriate. I tended to use "physical" to distinguish "nominal".

2. We do not need to define a thickness of monolayer crystals to proceed mechanics analysis. A definition of Y* (= Yh, Y: Young's modulus, h: thickness) would be sufficient. 

3. The nominal thickness 0.34nm was inconsistent with the in-plane and bending moduli. The thickness "0.066nm" is chosen such that the thin shell formula for Young's modulus and the flexural stiffness becomes consistent (if you phenomenologically regard a monolayer as a thin shell). This thickness can be regarded as the "bending effective thickness" . 

Thanks.

 

Rui Huang's picture

Dear Sulin:

Thank you for the summary and I agree. I may have misunderstood what you mean by "physical thickness". I hope this discussion made it clearer what it means and how we may use it (or not use it) to model graphene. We can now move on to more interesting topics.:)

Rui

Sulin Zhang's picture

Dear Rui,

I should have made this clearer and worded better. I am pretty sure most of mechanicians who walked through the carbon nanotube era understand the sublty well. 

Indeed this is a small point. It appears that mechanics research of 2D crystals is fading. We shall discuss more what our mechanicians can contribute to the 2D research. 

Sulin

Igor Berinskii's picture

I also believe that it is not possible to use a formula from classic mechanics to determine the graphene bending rigidity. We also have a paper on this topic. See I.E. Berinskii, A. M. Krivtsov, A. M. Kudarova. Bending stiffness of graphene sheet. Physical Mesomechanics 2014Vol. 17pp 356-364 I can send it if you need.

Igor Berinskii's picture

Sorry for the hyperlinks. Igor 

Sulin Zhang's picture

You cannot bending a monolayer as the classic way to generate compression on one side and tension on the other, because there is only one atomic plane. I thought this has been well accepted and did not make a big deal of this. Bending stiffness of graphene can be mapped out from the thermal flunctuation spectrum or from the rolling energy of a graphene into a carbon nanotube. 

Teng zhang's picture

Dear Prof. Zhang and Hongyan,

 

Thank you very much for this excellent summary on the mechanics of 2D crystal. As you discussed, the 2D crystalline structures are generally very flexible and often exhibit complicated shapes. Therefore, the conventional numerical solver such as Newton-Raphson (NR) method may have convergence problems. In fact, we found that the damped dynamics methods (i.e. fire http://dx.doi.org/10.1103/PhysRevLett.97.170201) usually work better than the NR solver based on our studies on defect controlled wrinkles in graphene (doi:10.1016/j.jmps.2014.02.005). However, it still takes a quite long time and is sensitive to the initial perturbation. For example, the symmetry of the final configuration may be broken when using random perturbations.

 

What's your experience and suggestion on the numerical solvers for the wrinkle/fold deformation of the 2D crystalline structures?

 

Thanks.

 

Teng

Sulin Zhang's picture

Dear Teng,

Thank you for your excellent question. 

Convergency is always a problem in numerical simulations. For dynamics simulations, the problem is less servere - at least you get some results (sometimes are not fully converged though). For static simulations, optimizer may fail in the middle of the simulation. 

The difficulty for simulating shape evolution of 2D crystals arises from the fact the energy landscape is highly irregular, and minimizer goes through many valleys, owing to the ease of bending into the 3rd dimension. If your initial perturbation is far from the targeted valley, it may take a lot of time because of the kinetic barriers along the long path. 

I have limited experience in dealing with the convergency problem. Instead of using NR method, we use conjugate gradient methods as static minimizer, which generally works quite well. Preconditioned CG works even better.

Sulin

 

 

 

Fan Xu's picture

Dear Teng,

You may find a very brief introduction on path-following perturbation techniques in my post for last journal club by Mazen (http://imechanica.org/node/17889). For static simulation, this asymptotic numerical method has been successfully applied to solve very strong nonlinear response problems such as post-buckling of film-substrate and graphene from continuum to atomic scale, without encountering any convergent difficulty. It would be interesting to see someone using this approach for dynamic simulation of crystal. I hope you may find some inspirations from the following references.

[1] Xu, F., Potier-Ferry, M., Belouettar, S., Cong, Y., 2014. 3D finite element modeling for instabilities in thin films on soft substrates. Int. J. Solids Struct. 51, 3619–3632. http://dx.doi.org/10.1016/j.ijsolstr.2014.06.023

[2] Xu, F., Potier-Ferry, M., Belouettar, S., Hu, H., 2015. Multiple bifurcations in wrinkling analysis of thin films on compliant substrates. Int. J. Nonlin. Mech.http://dx.doi.org/10.1016/j.ijnonlinmec.2014.12.006

[3] Cong, Y., Yvonnet, J., Zahrouni, H., 2014. Simulation of instabilities in thin nanostructures by a perturbation approach. Comput. Mech. 53, 739-750. http://link.springer.com/10.1007/s00466-013-0927-7

 

Best regards,

Fan

Sulin Zhang's picture

Thanks, Fan. Great comments. I will look into the references. 

Sulin

Teng zhang's picture

Dear Fan,

 

Thanks a lot for sharing this promising method. I was wondering whether you could post some codes for a simple example related to  searching the configuration of instability? That will make people easier to understand and follow your method. 

I know that it is difficult to obtain the really global minimum configuration, and sometimes the local minimum configurations are meaningful too.  For the current method, will it be trapped in some local minimum configuration? Have you ever encountered such kind of problems?

Best,

Teng

Fan Xu's picture

Dear Teng,

Thanks for your interest. Actually, this method was initiated in the early 90s in the group of Prof. Michel Potier-Ferry, which seems to have received limited attentions outside France. You may find many application examples in various fields in the book Cochelin, B., Damil, N., Potier-Ferry, M., 2007. Méthode asymptotique numérique. Hermès Science Publications, although it is in French. Here is a simple coding example to solve Bratu problem by using this method from the course of Prof. Hamid Zahrouni: https://www.dropbox.com/s/n3j17iry1xb7im4/ANM.pdf?dl=0

From my experience, I have never been bothered by such convergent problems probably as you mentioned even coping with multiple local minimum solutions in continuum, see paper [2] in my last post. I think it works well also in atomistic scale for example in paper [3], at least much better than NR algorithm.

Best,

Fan

Teng zhang's picture

Dear Fan,

Thanks very much for the further explaination. I will look into the literature you suggested.

 

Best,

Teng

Hongyan Yuan's picture

Dear Fan, Teng,

Thank you for bringing up the challenging topic of developing effective numerical methods for nonlinear equations or opitmization problems, and for letting us know the path-following perturbation technique that you are working on.  

For the numerical methods in solving nonlinear equations, please allow me to deviate a little bit and raise another question about the protein folding problem in biology. As far as I know, so far, when predicting the configuration of a folded protein, the homology modeling method (a method that is NOT based on mechanics or physics) is still better than the first-principle methods that predict the protein configuration by minizing the total free energy. What we, mechanicians, can do to improve the accuracy of the mechanics-based first-principle methods? 

Best,

Hongyan

Rui Huang's picture

The generally accepted bending modulus of monolayer graphene is about 1.5 eV, which is around 60 k_b*T for T = 300 K. This is about four times the 15 k_b*T condition for biocompaibility as noted above. It is same order of magnitude, but the difference may be significant from biocompatibility point of view.

Sulin Zhang's picture

Researchers have seeded cells on graphene and the cells can survive. But whether graphene is biologically toxic was inclusive, based on my reading several years back. 

Sulin

Rui Huang's picture

Sulin raised this question above. I could only share my personal opinion here. First of all, 2D crystals have attracted extensive interests over a wide range of areas including mechanics, and it is likely to continue for a few more years at least. While most simple mechanics has been worked out through the works of carbon nanotubes and graphene, there remains many interesting questions as more and more 2D crystals are being made or discovered. Sulin already summarized a few good examples. One more example I could add is about the interfaces. All 2D crystals interact with other materials through interfaces (adhesion, friction, penetration, etc). The mechanics of interface is rich and dirty, with the effects of water, contamination, surface roughness, defects, and so on. For 2D crystals, the interfaces are inevitable for most applications (if not all), and the mechanics of these interfaces are important for many applications (e.g., transfer, integration, reliability). Below are a couple of recent works in case you are interested:

S.R. Na, J.W. Suk, R.S. Ruoff, R. Huang, K. M. Liechti, Ultra Long-Range Interactions between Large Area Graphene and Silicon. ACS Nano 8, 11234-11242 (2014).

S.R. Na, J.W. Suk, L. Tao, D. Akinwande, R.S. Ruoff, R. Huang, K. M. Liechti, Selective Mechanical Transfer of Graphene from Seed Copper Foil Using Rate Effects. ACS Nano 9, 1325-1335 (2015).

Thanks.

Sulin Zhang's picture

Rui,

I do hope that mechanicians can share a good slice of the cake of the 2D research. I have been talking to some 2D peoples (in Penn State alone, there are three EFRI awards for 2D), they care more about electrical and optical properties than mechanical properties, very different from carbon nanotube research era. 

I agree there are indeed some interesting mechanics problems existent in 2D crystals, as you have listed. But how we could make a case to the main caker sharers (materials scientists & physicists) and those funding agencies ? 

 

 

Sulin Zhang's picture

Rui,

I have read your recent interesting papers. Graphene remains to one of the important 2D crystals, and a lot of research remains to be done for it.

 

People in Penn State have started to talk about "beyond graphene". There has been a workshop alternatively held in Penn State and Rice, titled "Beyond Graphene". Also please see the excellent review article by Geim I cited in the summary. I am wondering if you have stepped into other 2D crystals, and you could comment on what our mechanicians can contribute, to popularize the research among mechanics community.

Thanks, 

Sulin

 

Teng Li's picture

Sulin and Hongyan,

I find the concept of ripplocation intriguing. I wonder if you can comment on the similarity and dissililarity between ripplocations and dislocations in terms of other properties, e.g., annihilation, intersections, jogs and kinks. Any further literatures on experimental observations of such surface phenomena are welcome too. Thanks.

Sulin Zhang's picture

Teng,

Indeed the finding of ripplocation is quite interesting to us as well. The similarity between ripplocation in 2D crystals and conventional dislocation (edge dislocation) in bulk crystals is that they are both orginated by inserting line of atoms. The first difference is morphology. Owing to the ease of out-of-plane bending, ripples form when ripplocations are introduced. 

Perhaps the most important difference is their behaviors. When same-sign dislocations are in close vicinity, they repel to each other; whereas same-sign ripplocations attract to each other to merge. This conforms to the "curve-to-attract" law, a phenemonon that has been widely observed in cell membrane, wherein intergral proteins create curved domains.

As a first contribution to this area (see our Nano Letter paper), we did not get a chance to discuss other properties of ripplocations, as you mentioned. Annihilation of a ripplocation can occur; but I think it can only be eliminated when it reaches the edges of the 2D crystals, different from annihilation of two dislocations of different signs. Intersections between ripplocations are even more interesting, if you think about two straight ropes traveling in different directions and intersect. What the intersection point of two ripplocations is like remains to be explored. 

Another variation of ripplocation is surface ripplocation versus buried ripplocation, i.e., ripplocations buried in multi-layered crystals. The interlayer van der Waals interactions would reduce the mobility of the buried ripplocations, and possibly the wavelength.

As an additional note, because ripplocation is a very narrow, straight line defect and highly mobile, we can regard it as a directional thermal load that can generate motion in dedicated directions, in contrast to the random motion generated by thermal noise. In this regard, I was thinking that we can further regard a ripplocation as a thermal mop for cleanning up surface and interlayer dirts. 

As this is the first paper, I am not aware of any other relevant papers. I do hope there are follow-up papers by our groups and others.

 

Sulin

 

 

 

 

Dear Prof. Zhang and Hongyan,

Thank you for this excellent review on the mechanics of 2D crystals. As you pointed out, cell membranes are an important member of the family of 2D crystals and their mechanical properties (e.g. bending stiffness and membrane tension) play critical roles in the endocytosis and budding of nanoparticles. Here I would like to share our theoretical work on the cellular uptake of nanomaterials with emphasis on the effects of particle elasticity and shape.

1. Cellular uptake of an elastic fluid liposome or a solid thin-shelled nanocapsule

Fluid vesicular nanoparticles (e.g. liposomes) and solid nanocapsules have found broad biomedical applications as drug delivery vehicles with tunable geometrical and mechanical properties. We investigated the adhesive wrapping of these deformable particles by a lipid membrane. In our theoretical studies, the vesicular nanoparticle is modeled as an elastic fluid vesicle [1] and the solid nanocapsule is modeled as an elastic spherical thin shell [2]. We found that there exist a maximum of five distinct wrapping phases based on the stability of wrapping states including full wrapping, partial wrapping, and no wrapping states, and these wrapping phases strongly depend on the nanoparticle size, adhesion energy, surface tension of membrane, and bending rigidity ratio between the nanoparticle and lipid membrane. It has also been found that stiffer nanoparticles require less adhesion energy to achieve full wrapping than softer ones for both fluid vesicles [1] and solid capsules [2]. Further calculations suggest that solid nanocapsules can achieve full wrapping more easily than fluid vesicles with the same bending stiffness [2].

[1] X. Yi, X. Shi and H. Gao, Cellular uptake of elastic nanoparticles. Phys. Rev. Lett. 107, 098101 (2011). (http://dx.doi.org/10.1103/PhysRevLett.107.098101)
[2] X. Yi and H. Gao, Cell membrane wrapping of a spherical thin elastic shell. Soft Matter 11, 1107-1115 (2015). (http://dx.doi.org/10.1039/c4sm02427c)

2. Cell membrane wrapping of (elastic) nanorods

Besides the effect of nanoparticle elasticity, cellular uptake exhibits strong shape effect. We perform a 2D theoretical study on the cell membrane wrapping of an elastic rod-shaped nanoparticle [3]. Similar wrapping phases as reported in [1,2] are observed. While symmetric morphologies are observed in the early and late stages of wrapping, in between the soft rod-shaped nanoparticle undergoes a dramatic symmetry breaking morphological change while stiff and rigid nanoparticles experience a sharp reorientation from a parallel configuration of their long axes in the early wrapping stage to a perpendicular configuration during the late stage of wrapping. This sharp reorientation of a rigid nanorod is consistent with the MD simulations reported in Ref.[12] in Prof. Zhang and Hongyan's review. Such reorientation of short nanoparticles seems ubiquitous in the budding and endocytosis of virus such as fowlpox and pigeonpox viruses.

[3] X. Yi and H. Gao, Phase diagrams and morphological evolution in wrapping of rod-shaped elastic nanoparticles by cell membrane: A two-dimensional study. Phys. Rev. E 89, 062712 (2014). (http://dx.doi.org/10.1103/PhysRevE.89.062712)

3. Cell membrane interaction with 1D or 2D nanomaterials

There has been increasing interest in the cell interaction with 1D nanomaterials (e.g. nanotubes) and 2D nanomaterials (e.g. graphene nano-/micro-sheets). In our recent study [4], we found that perpendicular tip entry and parallel surface adhering are two basic modes of cell interaction with 1D nanomaterials, controlled by the dimensionless normalized membrane tension that scales with the membrane tension and radius of the nanomaterial and inversely with the membrane bending stiffness. At small membrane tension, the uptake of 1D nanomaterials follows a near-perpendicular entry mode but it switches to a near-parallel interaction mode at large membrane tension. This tension-dependent behavior can also be used to explain the size limit of filopodia [4].

In the case of cell interaction with 2D (graphene) microsheets, we studied two modes of interaction between the cell membrane and micro-sized rigid 2D nanomaterials: near-perpendicular transmembrane penetration and parallel attachment onto a membrane [5]. Our analysis indicates that, driven by the membrane splay and tension energies, a hydrophobic 2D microsheet would adopt a near-perpendicular configuration with respect to the membrane in the transmembrane penetration mode, whereas the membrane bending and tension energies would lead to parallel attachment of a hydrophilic 2D microsheet.

[4] X. Yi, X. Shi and H. Gao, A universal law for cell uptake of one-dimensional nanomaterials. Nano Lett. 14, 1049-1055 (2014). (http://dx.doi.org/10.1021/nl404727m)
[5] X. Yi and H. Gao, Cell interaction with graphene microsheets: near-orthogonal cutting versus parallel attachment. Nanoscale. (http://dx.doi.org/10.1039/c4nr06170e)

Best wishes,

Xin

Sulin Zhang's picture

Dear Xin, 

Thanks for the thorough review of your work on NP-cell membrane interactions. Indeed many biologists think that ligand-receptor based chemotargeting is the only way to bias the cellular uptake of NPs by diseased cells. Your work and Hongyan's made the case clear that  mechanics also plays a critical role in dictating the uptake. 

While I have read your papers before, I still have some questions about the uptake of elastic spherical NPs. Intuitively, a soft NP would be more difficult to be internalized than a solid NP because both the NP and cell membrane deform. But is this the reason ? I have done some simulations on this, though never published. We found that for soft spherical NPs, wrapping breaks the symmetry. In other words, cell membrane wets the soft NP, and the NP becomes a disk alike (like a normal red blood cell). Further wrapping of course is more difficult because of the increased curvature. I wonder if you have any comment on this - it seems to me that you did not mention this in your papers. 

Thanks,

Sulin

Xinghua Shi's picture

<p>Dear Sulin,</p>
<p>Thanks for this timely post.</p>
<p>I think your intuition is right that soft NP would becomes a disk alike on the cell membrane. Recently we published one experimental work on investigating the role of NP's rigidity on the cellular uptake.</p>
<p><a href="http://onlinelibrary.wiley.com/doi/10.1002/adma.201404788/abstract">http...
<p>It shows soft liposomes stay on the cell membrane while rigid NPs (with lipid shell) can be internalized. This work actually corroborates the prediction by Yi et al (Physical Review Letters, 107, 098101 (2011).</p>
<p>Interestingly, almost at the same time, Parak Walfgang's group published one similar paper</p>
<p><a href="http://onlinelibrary.wiley.com/doi/10.1002/anie.201409693/abstract">http...
<p>Yet the conclusion is almost opposite. They find the soft NPs transport faster into lysosomes than rigid ones. Since in their experiments, they used NPs with large size (~micrometer), so this contradiction is actually well understood that the internalization process is mainly hindered by the cytoskeleton of cells. On the surface of membrane, I think it has minor differnce for soft and rigid NPs (for such a large size) in the wrapping process.</p>
<p>Xinghua</p>

Sulin Zhang's picture

Dear Shi,

I fully agree with your comments on this. Indeed if the NP size is over 200nm, the role of the cytoskeleton becomes non-negligible. 

I will enjoy reading your papers. 

Sulin

Dear Professor Zhang,

Thank you very much for your comments. As you mentioned, the cell membrane indeed wets the soft nanoparticle in the early wrapping stage. In our work [1,2], we observed similar phenomena that "a very soft particle would initially spread along the membrane without significant membrane deformation" and the membrane undergoes large deformation to wrap around the particle at a later wrapping stage [1]. Configuration of the membrane and nanoparticle at selective wrapping degrees also demonstrate this feature.

To undertand why soft particles are less prone to wrapping than stiff ones, we made analysis from the view of elastic energy variation [1]. In the wrapping around a rigid or stiff nanoparticle undergoing small deformation, a relatively gentle rise in elastic energy (mainly in the membrane) is involved. When wrapping around a very soft particle, the membrane does not deform initially but then needs to catch up to almost the same configuration at full wrapping. This means a more abrupt rise in elastic energy at the late stage of wrapping. Consequently, larger adhesion energy is required to balance that rapid rise in elastic energy. Detailed discussion can be found on the last page of [1]. This stiffness-dependent wrapping process is like climbing a mountain (see figure below). Climbers, who take a more gentle pathway at the mountain base, can save energy initially but will spend and need more energy than others when they are approaching the top.

energy_elastic-variation

 

 

In the studies [1,2], we assume that both the membrane and nanoparticle undergo axisymmetric deformation. This can be regarded as a good approximation for spherical nanoparticles with water permeable surfaces. We haven't performed analysis for soft particles with a given reduced volume or of large pressure difference. It is highly possible that these particles (with strong constraints) would exhibit symmetry breaking morphologies in the wrapping, as observed in your simulations. In our recent two-dimensional study [3], we found that soft particles with a given reduced volume undergoes a dramatic symmetry breaking morphological change and particles without such a volume constraint exhibit symmetric configurations through the wrapping process. It would be extremely interesting to develop a fully three-dimensional continnum model capable of capturing the general deformation of both the nanoparticle and membrane.

Best wishes,

Xin

Sulin Zhang's picture

Xin:

I agree with your points. But to me, the explanation is simple. When the soft NP (of radius R) is wet on the cell membrane, the radius of the curvature of the unwet edges of the NP becomes much smaller than R, just thinking of a spherical vesicle versus a red blood cell like configuration (with the same volume). Further wrapping the wetted NP involves a very large energy barrier (The bending energy penalty scale with 1/R^2). I wonder if this is the case in your analysis.

Thanks,

Sulin 

 

Dear Professor Zhang,

In our axisymmetric studies, the soft NP transits from an 'oblate' shape in the early wrapping stage to a 'droplet' shape in the later stage and finally to a sphere at full wrapping. The 'oblate' shape with high curvature around the contact edge only remain in the early stage. If the NP can maintain a certain shape with high curvature parts (e.g. a rigid nanorod in your Nano Letters paper [Nano Letters 13, 4546 (2013)] or a rigid oblate NP [A.H. Bahrami, Soft Matter 9, 8642 (2013)]), the NP would rotate (after a certain wrapping degree) to reduce membrane deformation energy and to achieve further internalization. In the 3D case where the soft NP can display a general asymmetric configuration and undergo morphological evolution, things become extremely complicated. The wrapping configurations might strongly depend on the size and shape of NP, adhesion energy, membrane tension, and bending rigidity ratio between the NP and membrane. Detailed analysis of such a general case remain to be undertaken. It would be nice and timely if you would publish your simulation results. We will enjoy reading them.

Best wishes,

Xin

Sulin Zhang's picture

Dear Xin,

I think we are on the same page. The high curvature at the contact edge makes further wrapping difficult. Thermodynamically, wrapping is still possible because the end state has a much lower energy. But kinetically it invovles a very high barrier. This is what we observed in our 3D simulations. We also observed that it is very difficult for a wetted vesicle to rotate, though the imposed adhesion energy density is not very high.

I think this is a small point but might worth a paper to clarify. I will look into that. 

Thanks,

Sulin

 

 

 

 

Hongyan Yuan's picture

Dear Xin,

 

Thank you for letting us know your series of important theoretical/computational studies on the mechanics of nanomaterial-cell membrane interactions. 

It is also exciting to see that your theoretical predictions concerning the effect of rigidity of nanoparticles on cellular uptake (Physical Review Letters, 107, 098101) has been verified in Xinghua and his collaborators’ experimental and computational data (http://onlinelibrary.wiley.com/doi/10.1002/adma.201404788/abstract).  Impressive!

A friend of mine just graduated from Harvard last year, her PhD work was to study the effect of the diffusivity of the lipid molecules in liposomes on the uptake rate of these liposomes into living cells, for developing more efficient/effective techniques in the targeted drug delivery. I think the diffusivity of the lipid molecules not only affects how fast the ligands move, but also changes the rigidity of the liposomes, for which your model can be well used to understand and explain her experimental data.    

Best,

 

Hongyan

Dear Hongyan,

Thank you for your kind comments and for letting me know the work of your friend. I agree with you that the mechanical behavior of a liposome in the cell uptake can be strongly affected by the diffusivity of its forming lipid molecules. It is interesting that Schubertová et al observed a similar phenomenon (Fig.3) in their molecular dynamics simulations published in Soft Matter [Schubertová et al., Influence of ligand distribution on uptake efficiency. 2015 (http://dx.doi.org/10.1039/C4SM02815E)].

Best wishes,

Xin

Marino Arroyo's picture

Dear Sulin, Hongyan,

Thanks for putting together this interesting discussion. I wanted to raise a point relating this and the previous Journal Club discussion on “ruga mechanics”: what happens when a thin film weakly interacting with a substrate is compressed, but now this thin film is a fluid membrane? 

A physical realization may be a lipid bilayer supported on a PDMS substrate, and the motivation the observation that membranes in cells are highly confined, e.g. by other membranes (in mitochondria) or by the extra-cellular matrix. 

This question was examined experimentally and theoretically in PNAS, 108:9084 (2011) and PRL, 110:028101 (2013). Because fluid membranes can accommodate shear without storing elastic energy, rather than linear wrinkles they develop isolated spherical or tubular protrusions, whose shape can be precisely controlled by strain and osmolarity.

But is this relevant to cells? Microscopically, the membrane is highly inextensible. However, cells can significantly change their surface area to adapt to external conditions. This paradox is resolved by the existence of membrane reservoirs in the form of folds, wrinkles, protrusions of various kinds, in addition to endo/exocytosis. Control of membrane area and tension is thought be highly regulated by interactions between bilayer mechanics, membrane proteins, and the cytoskeleton. In unpublished work, however, we have found that living cells accommodate sudden stretch and osmotic shocks by the same passive mechanisms found in supported bilayers, followed by slower biological responses. 

Regards,

Marino

 

Hongyan Yuan's picture

Dear Marino,

Thank you for letting us know your recent work on the mechanics of supported bilayer memrbanes.  I look forward to the publication of your work on the passive mechanisms of living cells to avoid being ruptured under sudden stretch. I can imagine without these passive mechanisms the damages to the neuron membranes in our brains in the situations of even mild traumatic brain injuries would be devastating.  

Best regards,

Hongyan

Sulin Zhang's picture

Dear Marino,

Thanks for posting this inspiring note. I have enjoyed reading your work, as always. Thanks to Hongyan for pointing out the critical biological relevance and significance of the work. 

As you mentioned, cell membrane is highly inextensible, and a cell changes its area by releasing membrane reserviors. I was wondering how sudden the stretch is applied ? Does kinetics play a role here ? That is, the time scale of the loading comaring to the time scale of the membrane reservior release.

In slow stretching mode, after all the membrane reserviors are used out, I wonder if cells would fabricate and insert lipids (there are more lipids in cell interior than in the cell membrane) to the mother membrane, and how we can detect the event.

Thanks,
Sulin 

 

 

Marino Arroyo's picture

Thanks for your questions.

In our experiments, cells adhered to a flexible substrate, which was then stretched. Cells were also seeded on a pre-stretched substrate, allowing us to compress them. Kinetics plays a major role. If you change stretch gently (in 15 sec), then tension redistributes on the cell surfaces and area requirements on the plasma membrane are accomodated by a smooth overall deformation. However, at fast strain rate (stretch release in ~1 sec), friction of the bilayer and the substrate/cortex hinders such global relaxation, and the membrane locally releases tension by forming protrusions. The same happens with water at the substrate-cell interface. Active mechanisms of area/tension regulation seem to take minutes.

If you very slowly stretch the cell (which we did not do), I guess that it may remodel adhesions to relax stretch. 

regards

 

Harold S. Park's picture

Hi Sulin - thanks for initiating this timely, and very interesting discussion.  I think there are a few areas where mechanicians have, and will continue to make contributions in this field, where admittedly most people are far more interested in the non-mechanical properties.  One area is that of elastic strain engineering, where mechanical strain is used to tailor, or enhance or physical properties, primarily the electronic ones (http://www.sciencemag.org/content/336/6088/1557.full, http://pubs.acs.org/doi/abs/10.1021/nl400872q, http://www.nature.com/nphoton/journal/v6/n12/abs/nphoton.2012.285.html).  Another is in adhesion and friction, since these 2D materials will almost certainly be used in situations where they interact with a substrate (http://scitation.aip.org/content/aip/journal/jap/107/12/10.1063/1.3437642, http://iopscience.iop.org/0022-3727/43/7/075303, http://pubs.acs.org/doi/abs/10.1021/nl4007112).  Finally, mechanicians can contribute new ideas to overcome known mechanical limitations, such as the brittle nature of graphene (http://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.245437, http://scitation.aip.org/content/aip/journal/apl/104/17/10.1063/1.4874337). 

Sulin Zhang's picture

Hi Harold, 

Thanks for sharing your vision with the board, and I cannot agree more. Strain engineering has been quite popular in the last several years and will continue to be an effort-worth topic among our mechanicians. To enable these 2D crystals as next generation electronics, adhesion, friction, fracture, and other mechanical properties of 2D crystals need to be settled by our mechanicians. I think as of now, most chemists, physicists, and materials scientists are being focused on synthesis and device fabrication, leaving mechanical properties aside. We shall explore interesting mechanics of 2D crystals and prove that mechanics is an indispensable component in 2D research, just like CNT/graphene era.

Sulin 

 

Teng Li's picture

Thanks, Harold, for referring to our recent publications on strain engineering of graphene. The recent studies in your group in this area are of great interest as well!

I second Harold's comment on the significant potential of elastic strain engineering and its application to 2D materials. I'd like to point out some existing efforts in this nascent area. In the 2013 MRS Fall Meeting, I gave a talk in Symposium YY: Elastic Strain Engineering for Unprecedented Materials Properties, organized by Ju Li (MIT), Evan Ma (Johns Hopkins University), Zhiwei Shan (Xi'an Jiaotong University) and Oden L. Warren (Hysitron, Inc.). This symposium covered a wide range of topics in elastic strain engineering and was a success. The organizers intend to continue this symposium series in near future. So stay tuned if interested.

 

Subscribe to Comments for "Journal Club Theme of March 2015: Mechanics of 2D Solid and Fluid Crystals"

Recent comments

More comments

Syndicate

Subscribe to Syndicate