Physics Letters A 375 (2011) 2437–2440

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Physics Letters A

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Gaussian curvature energy of graphene sheets

J.C. Martinez, M.B.A. Jalil ∗

Information Storage Materials Laboratory, Electrical and Computer Engineering Department, National University of Singapore, 4 Engineering Drive 3, Singapore 117576, Singapore

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 February 2011Received in revised form 25 April 2011Accepted 1 May 2011Available online 7 May 2011Communicated by R. Wu

The energy density of an elastic sheet is the sum of a mean curvature H term, 2kc H2, and a Gaussiancurvature K term, kK , where kc , k are constants. The sign of kc is commonly understood to be positive,but the sign of k and even the role of K are subject to debate. We give reasons for the Gaussiancontribution to be k|K | instead, with k positive. This is used to obtain the mean amplitude of thermalfluctuations of a graphene sheet.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The discovery of ripples in graphene [1], initially only of theo-retical interest, is now poised to usher into modern technology astrain-based graphene engineering thanks to the creation of peri-odic ripples in suspended graphene sheets [2]. Exploiting thermalproperties of graphene is an additional tool in manipulating itselectronic properties [3]. As a manifestation of elastic phenom-ena in nature’s ultimate electronic membrane, these ripples areassociated with the curvature energy, a subject still enveloped incontroversy and lack of consensus. In the absence of spontaneouscurvature, the curvature energy per unit area of a membrane iswritten in terms of the local principal radii of curvature r1, r2 as

E = 2kc H2 + kK , (1)

where H = 12 (1/r1 + 1/r2) and K = 1/r1r2 are the mean and Gaus-

sian curvatures, respectively; kc is the bending rigidity and k theGaussian rigidity. Originally derived for thin elastic plates [4], thisformula has been successfully applied to biomembranes [5], nano-structures [6], liquid crystals [7], and other low-dimensional con-tinua [8]. That kc > 0 follows since energy is required to roll anelastic sheet into a cylinder (for which H > 0, and K = 0) [9]. Thesign of k is less accessible to common experience. Experimentswith membranes separating similar (bilayer membranes) or dis-similar (monolayer membranes) material have been interpreted sothat for positive-K surfaces k takes a negative sign while thosewith negative K it takes the positive sign [10]. But for lipid bilayersit had been argued that the sign depends on the molecular asym-metry [11]. Recent molecular-dynamics calculations of the energyof various toroidal forms of graphitic carbon reveal that the signof Gaussian bending energy is always positive for either sign of K

* Corresponding author.E-mail addresses: elemjc@nus.edu.sg (J.C. Martinez), elembaj@nus.edu.sg

(M.B.A. Jalil).

0375-9601/$ – see front matter © 2011 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2011.05.002

[12]. In this Letter we give four arguments, the last on graphene,whose conclusion is that a consistent form for the Gaussian bend-ing interaction should be k|K | instead. Thus we replace the secondterm in Eq. (1) by the absolute Gaussian curvature and assign apositive sign to k. While, perhaps, no one argument below is con-clusive on its own, we believe that the four offer strong supportfor our assertion.

Two initial remarks are pertinent. First, Helfrich distinguishedthree elastic strains: stretching, tilt and curvature [5]. While he ar-gued that the only elasticity controlling nonspherical vesicles wascurvature, it is also true that curvature (mean and Gaussian) hasbeen derived in classical elasticity starting from elastic stretchingand compression [4]. We regard curvature and strain energies ascomplementary and overlapping. That the energy depends only onthe mean and Gaussian curvatures is clear from the fact that theseare the only basis independent quadratic forms [13, p. 146]. Inthis work, we focus on the curvature energy, i.e. the energy associ-ated with out-of-plane deformations, for which changes in densityconstitute higher-order terms which can thus be neglected. Thus,stretching, which gives rise to density change, is not directly con-sidered here, nor is tilting, which is not applicable to a one-atom-thick planar sheet like graphene (see Section 5), [14]. Curvatureenergy is a non-extensive quantity (i.e., one which is independenton the amount of matter). Second, in classical elasticity the rigid-ity constants are not independent of each other [4]. Our assertionabove says nothing directly for or against these relations. In fact,these relations are derived by appealing to further assumptions(e.g., the strain is a symmetric tensor, Hooke’s law, homogeneityof deformations), which we do not make use of.

The surface integral of K for a compact surface M is propor-tional to the Euler characteristic χ(M) of M (Gauss–Bonnet theo-rem), a topological invariant (but this is not the case for surfaceswith a boundary) [13]. By Gauss’ theorema egregium, K is intrin-sic, i.e. it depends only on measurements made on the surfacewithout relying on information about the surface’s embedding inspace. (Twisting a flat sheet or rolling it into a cylinder will not

2438 J.C. Martinez, M.B.A. Jalil / Physics Letters A 375 (2011) 2437–2440

change its Gaussian curvature.) For surfaces of fixed topology itis natural to disregard K altogether where energy alone is con-cerned1 and recent treatments of the elasticity in membranes andgraphene have all but ignored it [8,15,16]. But the integral of |K |,or the total absolute curvature, satisfies the Chern–Lashof inequal-ity,

∫ |K |dσ � 2π(4−χ(M)) [17] and when it attains its minimumvalue, M is said to be tight [18]. The total absolute curvature is notan invariant. Our study of the Gaussian curvature elastic energy ismotivated by the fact that graphene sheets (unlike the closed sur-faces, i.e. surfaces without a boundary, encountered in membranes)can rarely be regarded as closed two-dimensional surfaces; theyare open surfaces which have a boundary. The Gaussian curvatureplays an important role in open surfaces, a role that sometimescan be put aside for closed surfaces [5].

2. Torus

We begin by considering a torus-shaped membrane T 2: x =(a+b cosσ 1) cosσ 2, y = (a+b cosσ 1) sinσ 2, z = b sinσ 1, (σ 1, σ 2)

being in the range [0,2π ] and c ≡ b/a < 1. The curvatures are

[13]: H = a+2b cosσ 1

2b(a+b cosσ 1)and K = cosσ 1

b(a+b cosσ 1). As shown in Fig. 1,

the circles at the top and bottom of the torus divide it into twohalves, one having positive K and the other negative K . In fact∫

T 2 K d2σ = 0 by virtue of the cancellation of the contributionscoming from these halves (this is really the Euler characteristic).Thus the bending energy due to the Gaussian curvature term iszero and the rigidity term gives

2kc

∫

T 2

H2 d2σ = 2kc

∫H2b

(a + b cosσ 1)dσ 1 dσ 2

= 2kcπ2

c

1√1 − c2

. (2)

The two halves of the torus contribute equally to the energy (2)as is the case with the Gaussian curvature term (but with oppo-site signs). If we take Eq. (1) as the expression for elastic energy,we infer that one of the halves should have higher energy thanthe other on account of the sign of the Gaussian interaction. How-ever we can think of the two halves as being the compressed andstretched configurations of a ‘half-cylindrical’ shell. We expect inthe elastic limit (harmonic approximation) that both of these con-figurations have the same energy (cf. remark above that regardlessof the physical mechanism responsible for energy, it would haveto depend of H2 and K ). Thus Eq. (1), without the modificationsuggested, appears inconsistent with this interpretation.

At present, the above claim is purely theoretical as there isno known experimental measurement which can conclusively sep-arate the energy contributions of the compressed and stretchedportions of a torus. Current studies have instead focused on con-formational and entropic aspects of curvature energy of tori [12].

3. Soap film

Consider next minimal surfaces, i.e. surfaces with vanishingmean curvature, H = 0, and negative Gaussian curvature, K < 0[13]. One of these, the helicoid, can be formed from a soap so-lution and a helix wire (see Fig. 2) [19,20]. The helicoid can beparametrized by x = u cos v , y = u sin v, z = av , u ∈ R , v ∈ R ,where a = h/2π and h is the constant pitch. We have H = 0 andK = −a2/(a2 + u2)2. It is known that if we start with a helicoidof large pitch and slowly contract the spiral wire, i.e., decrease the

1 Since it depends only on boundary conditions, it gives rise to stress but pro-duces no force or torque [5], and hence no dynamically observable change.

Fig. 1. Radially outward from the dark circle, the torus has K > 0, while inward ithas K < 0.

Fig. 2. (a) The helicoid, a minimal surface formed by dipping a helix wire into soapsolution. (b) If the helicoid’s pitch is continuously decreased, it eventually ceases assuch, becoming a film instead. (c) Plots of the sum (dashed) and difference (solid)of Eqs. (4) and (5) as functions of pitch a. The dashed curve, decreasing with de-creasing a, implies that the helicoid continues to exist even for small pitch.

pitch, then there is a point beyond which the helicoid ceases tobe such and turns into a different sort of film altogether as shownschematically in Fig. 2b [19]. We explain this phenomenon by ap-pealing to Eq. (1).

Without loss of generality we take the interval 1 � u � 0 andone spiral. We calculate the energy due to surface tension, γ S (S =surface area, γ = surface tension),

γ S = γπ

[√1 + a2 + a2 ln

1 + √1 + a2

a2

], (3)

and similarly the energy due to the (unsigned) Gaussian curvatureis

k

∫helicoid

K d2σ = − kπ√1 + a2

. (4)

The mean curvature energy is zero (since H = 0). Taking the un-signed form of K (i.e. Eq. (1) without modification), the sum ofEqs. (3) and (4) is a monotonically decreasing function of a as thepitch approaches zero (dashed curve in Fig. 2c). On the other hand,if we replace K by its absolute value, the total energy increasesmonotonically as pitch approaches zero. The former case is notconsistent with observations.

4. Surface reconstruction theory

We can explain why |K | and not K should be used in Eq. (1)by appealing to the theory of surfaces. Suppose initially that agiven closed surface is represented by a discrete set of points (inEuclidean space E3) and that we would like to reconstruct thesurface with the sites as points on the surface. In general thesites are randomly scattered (i.e. not forming a regular grid). Ge-ometrically we may represent each data point in the Monge form(xi

1, xi2, f i = f (xi

1, xi2)), but one cannot usually expect a single func-

tion f to cover the entire surface. Alternatively one can represent

J.C. Martinez, M.B.A. Jalil / Physics Letters A 375 (2011) 2437–2440 2439

the surface by a triangulation of the data set and approximate itby a polyhedron. But difficulties prevent this from being the sim-ple answer: (a) the triangulation may not be unique; (b) the areaof the polyhedron may not converge to the area of the surface;(c) the triangulation may not preserve convexity. As a topologi-cal invariant,

∫K d2σ cannot give a criterion for choosing a ‘best’

triangulation (since for any fixed-genus closed surface undergoingdeformation, it would give the same constant value). However, itis known that demanding that a triangulation be tight (i.e., mini-mal total absolute curvature) yields the ‘smoothest’ possible choice[21]. Thus the total absolute curvature encodes information aboutthe embedding of the surface in E3 which the total curvature doesnot [22]. For instance the fluctuations of a non-flat sheet must bemeasured with reference to the ambient space; fluctuations parallelto the sheet surface would go unnoticed on the sheet itself. Sincean elastic sheet can undergo similar fluctuations this explains why|K | – not K – is the proper quantity that should enter into thesecond term of Eq. (1).

5. Graphene sheet

Consider finally a freely suspended graphene sheet, almost flatwith area L2. We compute its mean thermal fluctuation ampli-tude. Employing the Monge form (u, v, w(u, v)), we compute H ∼=12 ∇2 w to lowest order in w . Expanding w(u, v) in a Fourier series

w(u, v) = 1

L

∑q

φqeiq·x (5)

the partition function corresponding to mean curvature fluctua-tions is (β = 1/kB T is the Boltzmann factor)

Z H ∝∫

(dφ)exp(−βE[φq]

)

=∫ ∏

q

dφq exp

{−1

2kcβq4|φq|2

}=

∏q

√2π/kcβq4 (6)

where the exponent, E[φq] = 12 kc

∑q q4|φq|2, is the contribution

from the H2-bending energy. To evaluate the contribution from|K |, we appeal to a theorem of Grossman’s which states that foran orientable (not necessarily closed) surface S with a connectedboundary curve C (arc element ds) embedded in E3

∫S

|K |d2σ +∫C

κ ds � 2πr, (7)

κ is the Frenet curvature of C viewed as a space curve in E3

and r an integer or half integer at least equal to unity [23]. Ac-cording to him, consideration of the Gaussian curvature of thesurface must include the curvature of the boundary. Assumingan almost flat surface (K ∼= 0), we write the |K |-partition func-tion Z |K | ∝ ∫

(dφ)exp{−βk∫

C κ ds}. This fails to converge if k isnegative so we assign it a positive sign, as noted above. The re-maining calculation can be carried out as above. If we consider forsimplicity a circular almost flat boundary of radius ρ and circum-ference C , with a small component z(s) in the normal direction,then to first order, κ ∼= κ0 + z2/2κ0 v4, where κ0 = 1/R is theaverage local curvature and v the ‘velocity’ z at that point. Em-ploying the Fourier expansion z(s) = 1√

L

∑q(ζqx eiqxx + ζqy eiqy y),

we get analogous to Eq. (6), Z |K | ∝ ∫(√

R dζ )exp(−βE[ζr]) ∼=∫ ∏qi=qx,qy

√R dζqi exp{−kβ( 1

2 Rq4i )|ζqi |2} = ∏

i=x,y

√2π/|k|βq4

i .Thus the contribution from the Gaussian curvature can be givenas a boundary contribution E |K | = 1 |k|R ∑

q ,q q4|ζqi |2. For closed

2 x y i

Fig. 3. Plot of some points of the sum in Eq. (9) versus x ≡ |k|/kc .

surfaces this is absent since there is no boundary. Note the relation|ζqi |2 R = |φqi |2. Then by the equipartition theorem we have⟨

1

2

{kcq4 + |k|(q4

x + q4y

)}|φq|2⟩= kB T , (8)

〈.〉 denoting an ensemble average. The sheet’s mean displacementis derived (x ≡ |k|/kc) using qx = 2πn/L, n = 1,2,3, . . . , etc.,

⟨w2⟩ = 1

L2

∑q

⟨|φq|2⟩

= kB T L2

(2π)2kc

∑n,m

2

(n2 + m2)2 + x(n4 + m4). (9)

The sum is plotted in Fig. 3. Taking a typical value of x ≈ 0.7, wefind 0.6 for the sum, and using kc ≈ 0.5 eV [6], kB T ≈ 0.025 eV (atT = 300 K), L ≈ 25 nm, we estimate

√〈w2〉 ≈ 0.7 nm, comparedwith an observed maximum amplitude of 1 nm [1]. This estimateis better (by 25%) and more consistent than an earlier one, whichcorresponds to x = 0 in Eq. (9) [8]. That is, the earlier estimate wasbased on the H2 part of Eq. (1).

6. Conclusion

To sum up, we have presented several reasons for advocatingthe form k|K |, k > 0, for the Gaussian curvature elastic energy den-sity of a membrane and verified its consistency with the observedthermal fluctuation amplitude of a graphene sheet.

Acknowledgements

The support of the ASTAR SERC Grant No. 092 101 0060 (R-398-000-061-331) is gratefully acknowledged.

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[23] N. Grossman, J. Diffl. Geom. 7 (1972) 611, we note the significance of r: whenr = 1, S is a planar disk and C a convex curve (this is not of interest as itimplies that K = 0 everywhere in S). If r � 5/2, S is diffeomorphic to a closeddisk and C is unknotted.