Mechanical properties of graphite oxides:Ab initio simulations and continuum theory

Andrei Incze,1 Alain Pasturel,1 and Philippe Peyla1,2

1Laboratoire de Physique et de Modélisation des Milieux Condensés, UJF, CNRS, Maison des Magistères, Boîte Postale 166,38042 Grenoble Cedex 9, France

2Laboratoire de Spectrométrie Physique, UJF, CNRS, Saint-Martin d’Hères 38402 Cedex, France(Received 10 May 2004; published 30 December 2004)

The mechanical properties of oxidized graphitic layers are calculated by atomicab initio simulations andcontinuum theory of elasticity. The adsorption of atomic oxygen atoms induces a nanocorrugation as well as alocal bonding reinforcement that modify the bending coefficient of a graphite layer. As a matter of fact, asurface oxygen composition of 12.5% leads to an increase of the rigidity coefficient by more than a factor of40. We thus demonstrate that oxidation also has a considerable impact on the mechanical properties of graphitestructures.

DOI: 10.1103/PhysRevB.70.212103 PACS number(s): 73.22.2f, 46.70.2p, 68.60.2p

INTRODUCTION

In recent years, the physical and chemical propertiesof graphite oxide sGOd have been studied in greatdetail, both theoretically and experimentally. One of themost important characteristics of GO is its high faradaiccapacity in lithium cell systems1,2 in which it has beendemonstrated that this material has remarkable swellingproperties as a layered host for Li intercalation. GO isalso used in multilayer electronically conducting films,3 inencapsulation of metallic and ferromagnetic nanoparticles,4,5

and in tethered membranes.6 Graphite oxide fibers havebeen obtained7 and used in reinforcement of plastics. GO,also known as graphitic acid, has been used for over acentury in very different fields. However, relatively littlehas been reported regarding the mechanical properties ofGO structures despite their increasing use in compositematerials.

There are different ways of GO preparation, all involvingtreating graphite with strong oxidizing agents.6 Thus a widevariety of composition can be obtained making the structureanalysis very difficult. However, Phaner-Goutorbeet al.8

have shown with x-ray photoelectron spectroscopy and scan-ning tunneling microscop experiments that a soft oxidationof a highly oriented pyrolitic graphite sample can give rise toa very reproducible nano-organization of adsorbed oxygenatoms with the occurrence of epoxy bonds. The presenceof epoxy groups was also proposed by Lerfet al.9 andby Heyong et al..10 In the following, we use this pictureto modelize a generic GO and to study its mechanicalproperties.

In this Brief Report, we show that the presence of atomicoxygen atoms chemisorbed on a graphite basal plane surfacecan change the mechanical properties of the carbon atomicplanes drastically. The elastic bending coefficientD is en-hanced by more than a factor of 40 for only an oxygen con-centration of 12.5%. In the following, we show that this ri-gidification is due to a nanostructuration of the basal planesurface induced by oxygen atoms. Results are obtained bycouplingab initio calculations with the continuum elasticitytheory.

AB INITIO CALCULATIONS

Calculations were performed using the ab-initio total en-ergy and molecular dynamics program VASP(ViennaAb ini-tio Simulation Package).11,12 Ultrasoft Vanderbiltpseudopotentials11,13 were used to approximate the electron-ion interactions. A cutoff energy of 270 eV was used forcarbon and oxygen. The generalized gradient approximationwas used to allow for exchange and correlation in thePerdew-Wang form.14 We model a graphite surface by intro-ducing a supercell ofs839d graphitic sheet with periodicboundary conditions applied along the planar directions tosimulate an infinite graphitic layer. The surface is simulatedby periodically repeated slabs of a graphitic monolayer par-allel to it with a vacuum region of 9 Å. The equilibriumC-C distance has been found to be equal to 1.42 Å. This slabcontains 288 carbon atoms with oxygen atoms located onone side of the surface or on both sides. The number ofoxygen atoms varies from 12(4.16%) to 36 (12.5%). Eachoxygen is located in the bridge position(epoxy bonds) whichis known to be the most stable site.15 For each composition,the oxygen sites were constrained to be as far as possiblefrom each other to avoid any chemical interactions betweenO adatoms. No symmetry was imposed to build the oxygennetwork. For each concentration, total energy calculationswere performed by using onek point at theG point.

Relaxation of the whole structure leads to a corrugation ofthe hexagonal basal plane(Fig. 1). However, the hexagonalsymmetry is preserved and no topological defects are re-vealed. Let us mention that a folding of the graphite planesby oxygen atoms was suggested by Riley in 1945 and Clausset al.16 in 1957 from geometrical considerations aboutchemical bonds involved in GO. However, this folding hasnever been measured nor calculated. This configuration rep-resents our reference state. This GO structure is deformed inorder to obtained its mechanical properties. A displacementalong thez axis [perpendicular to the basalsx,yd plane] isimposed on each atom

z = z0coss2px/Lxd, s1d

whereLx and Ly are the lateral dimensions of the periodicstructure. This deformation bends the structure by creating

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curvature. It also extends the structure by increasing the in-teratomic distances. Calculations are carried out with differ-ent values of the amplitudez0 of the displacement rangingfrom 0 to 1.8 Å. The biggest value ofz0 has been carefullychosen in order to preserve the harmonicity of each inter-atomic potential, i.e. only elastic effects are considered here.Let us mention that in the deformed states, the C-O bondlengths have been kept equal to those found in the referencestate. For the oxygen composition of 4.16% and forz0=1.8 Å, a local relaxation related to o atom displacementshas been performed. We have checked that such displace-ments are indeed negligible, which confirms the above as-sumption. The variation of the total energyDEab (ab meansab initio) for each value ofz0 has been plotted in Fig. 2 forboth a pure graphite sheet and a GO plane with a 4.15%oxygen concentration;z0=0 is the reference state.

We have calculatedDEab for different oxygen concentra-tions. Our results, displayed in Fig. 3 forz0=1.8 Å, showthat the energetic cost of the deformation Eq.(1) increasesdrastically with oxygen concentration.

In order to get further insights into the strong modificationof mechanical properties of this GO structure we will con-sider a continuum elastic model of a thin elastic plate that isstrained with a displacement given by Eq.(1).

CONTINUUM MODEL

Let us consider a thin elastic flat plate parallel to asx,ydplane with lateral dimensionsLx and Ly. If this plate is de-formed according to the displacement given by Eq.(1), wecall k its curvature and«xx the in-plane deformation. Forsmall values of the deformation amplitudez0, k<d2z /dx2,and exx<sdz /dxd2. The elastic energy density(energy perunit surface) is17

E = 12Dk2 + 1

2L«xx2 , s2d

whereD is the elastic bending coefficient andL is the elasticextension coefficient of the plate. The first and the secondterm are usually denominated as the bending and the mem-brane term, respectively. Using Eq.(1) and integrating ex-pression(2) on the whole surface, we get the total variationof elastic energy of the deformed plate(the reference state isthe one for whichz=0)

DEc = 4p4Ly

Lx3Dz0

2 + 6p4Ly

Lx3Lz0

4. s3d

Within this expression ofDEc ( c meaning “continuum”), thecost in bending energy(curvature) is proportional toz0

2 whilethe extensional energy is proportional toz0

4.For each concentration of oxygen, we fit theab initio

energyDEab as a function of the amplitudez0 with expres-sion (3) of DEc. In Fig. 2, we show the variation ofDEab forthe 4.16% oxygen GO as well as for the pure graphite layer.An excellent fit is obtained from which precise values ofDandL coefficients are obtained for the different oxygen con-centrations. In Fig. 4, we see that the elastic extension coef-ficient L is not affected by the presence of oxygen atomswhile the elastic flexion coefficientD is drastically modified.

FIG. 1. Graphite basal plane in presence of adsorbed oxygenatoms (epoxy groups). A nanocorrugation of the atomic plane iscreated(gray represents carbon, dark gray represents oxygen). Theout-of-plane atomic displacements have been magnified by a factorof 7 in order to vizualize the corrugation.

FIG. 2. Variation ofDEab with respect to the deformation am-plitudez0 given by Eq.(1) for a pure graphite plane(circles) and forthe GO structure(squares) represented in Fig. 1(4.15% of oxygenconcentration). A perfect agreement is found by fitting theab initioresults with Expression(3) of our continuum model(see below).

FIG. 3. Variation of the elastic energyDEab created by displac-ing each atom according to Eq.(1) with z0=1.8 Å for differentoxygen concentrations.

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We note that an oxygen concentration of 12.5% leads to anincrease of the flexion coefficient by more than a factor of40. We also reportD andL values for two different oxygenconfigurations at 8.33%. Our results emphasize thatD ismainly affected by the oxygen concentration. A value ofD=0.9 eV is obtained for pure graphene, in very good agree-ment with a previous estimation.18

ORIGIN OF THE RIGIDIFICATION

We now study the effects of the rigidification in moredetail. As shown below, the surface oxidation gives rise totwo different effects that contribute to the enhancement ofrigidity. First, each oxygen atom waves the whole carbonplane(it is a long range elastic effect.19 ) Thus, the carbonplane is nanostructured(corrugated) at atomic scale by oxi-dation(Fig. 1). This “corrugated cart board effect” increasesthe bending coefficient. Second; each epoxy bond rigidifiesthe C-C bond locally. Thus, the latter effect tends to alsoincrease the bending coefficient. The first mentioned contri-bution to the bending rigidification is a global effect whilethe second one is a local effect. We callDG (global) andDL (local) the two contributions to the bending coefficient:D=DG+DL. In order to evaluate the global contribution, theoxygen atoms are eliminated, but the corrugation created bythem is kept. Thus, for each oxygen concentration, we get acorresponding free-oxygen corrugated graphite plane(FOCG). Each FOCG is characterized by its corrugationZsx,yd. By using the same procedure as before, each FOCGis deformed by imposing the displacement given by Eq.(1)and we calculate the corresponding elastic bending coeffi-cient, i.e.,DG. As the corrugation is a global effect, we de-velop DG as a function of the first moments of the heightdistribution

DG = akZl + bkZ2l + ckZ3l, s4d

with

kZnl =1

LxLyE

Lx

ELy

Zndx dy. s5d

As the symmetryZ→−Z does not affect the elastic proper-ties of the structure, odd moments do not affectDG. Thus, wecan conclude that at the lowest order,DG is a linear functionof kZ2l. The plot ofDG in Fig. 5 confirms such a behavior.

Now it is easy to obtain the local contribution to the bend-ing coefficient, i.e.,DL which is given byDL=D−DG. Asthis effect is local(each oxygen contributes to locally ri-gidify the GO structure) DL should vary linearly with theoxygen concentration. Indeed, Fig. 6 reveals such a lineardependence ofDL.

We note that the oxygen configuration(either oxygen at-oms on one side or on both sides) at 8.33% does not affectDL. As a matter of fact, C-C bonds are rigidified in flexionindependently of the side of the oxygen adsorption(except if

FIG. 4. Variation of the bending coefficientD (squares) and theextension coefficientL (circles) with respect to the oxygen concen-tration. Lines are just guides to the eye. The two points at the same8.3% concentration represent two configurations of adsorbed oxy-gens, namely on one side or on the two sides(marked by a star) ofthe graphite plane.

FIG. 5. The corrugation of the graphite plane by oxygen atomscontributes to the bending coefficient enhancement. Each secondmoment corresponds to a corrugation created by a given oxygenconcentration. The line is a guide to the eye. The star indicates theconfiguration where oxygen atoms are adsorbed on both faces of thegraphite plane.

FIG. 6. Bending coefficientDL due to the local rigidification ofC-C bonds by an oxygen atom in each epoxy group. Due to thatlocal aspect the associated bending coefficient varies linearly withrespect to the oxygen concentration. The line is a guide to the eye.The star indicates the configuration where oxygen atoms are ad-sorbed on both faces of the graphite plane.

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two oxygens were placed in front of each other on each sideof the plane, which is not the case here).

CONCLUSION

In this paper we have shown that the presence of oxygenadatoms can significantly modify the mechanical behavior ofgraphitic systems. Atomic oxygens induce both a nanocorru-

gation and a local bonding reinforcement that increases dras-tically the bending coefficient of the graphite plane. More-over, our study which combinesab initio calculations and thecontinuum theory of elasticity allows us to extract relevantmacroscopic mechanical coefficients(bending and extensioncoefficients) that could be used in experiments on such ma-terials. We speculate that such a rigidification could help incomposite reinforcement.

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