Z. C. Tu and Z. C. Ou-Yang- Elastic Theory of Low-Dimensional Continua and Its Applications in Bio-...

8/3/2019 Z. C. Tu and Z. C. Ou-Yang- Elastic Theory of Low-Dimensional Continua and Its Applications in Bio- and Nano-Struct

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Copyright 2008 American Scientific PublishersAll rights reservedPrinted in the United States of America

Journal ofComputational and Theoretical Nanoscience

Vol. 5, 422448, 2008

Elastic Theory of Low-Dimensional Continua and

Its Applications in Bio- and Nano-Structures

Z. C. Tu12 and Z. C. Ou-Yang3

1Department of Physics, Beijing Normal University, Beijing 100875, China2II. Institut fur Theoretische Physik, Universitat Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany

3Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China

This review presents the elastic theory of low-dimensional (one- and two-dimensional) continua and

its applications in bio- and nano-structures. First, the curve and surface theory, as the geometric

representation of the low-dimensional continua, is briefly described through Cartan moving frame

method. The elastic theory of Kirchhoff rod, Helfrich rod, bending-soften rod, fluid membrane, andsolid shell is revisited. The free energy density of the continua, is constructed on the basis of

the symmetry argument. The fundamental equations can be derived from two kinds of viewpoints:

the bottom-up and the top-down standpoints. In the former case, the force and moment balance

equations are obtained from Newtons laws and then some constitute relations are complemented

in terms of the free energy density. In the latter case, the fundamental equations are derived directly

from the variation of the free energy. Although the fundamental equations have different forms

obtained from these two viewpoints, several examples reveal that they are, in fact, equivalent to each

other. Secondly, the application and availability of the elastic theory of low-dimensional continua in

bio-structures, including short DNA rings, lipid membranes, and cell membranes, are discussed.

The kink stability of short DNA rings is addressed by using the theory of Kirchhoff rod, Helfrich

rod, and bending-soften rod. The lipid membranes obey the theory of fluid membrane. The shape

equation and the stability of closed lipid vesicles, the shape equation and boundary conditionsof open lipid vesicles with free edges as well as vesicles with lipid domains, and the adhesions

between a vesicle and a substrate or another vesicle are fully investigated. A cell membrane is

simplified as a composite shell of lipid bilayer and membrane skeleton, which is a little similar to

the solid shell. The equations to describe the in-plane strains and shapes of cell membranes are

obtained. It is found that the membrane skeleton enhances highly the mechanical stability of cell

membranes. Thirdly, the application and availability of the elastic theory of low-dimensional continua

in nano-structures, including graphene and carbon nanotubes, are discussed. A revised Lenosky

lattice model is proposed based on the local density approximation. Its continuum form up to the

second order terms of curvatures and strains is the same as the free energy of 2D solid shells. The

intrinsic roughening of graphene and several typical mechanical properties of carbon nanotubes

are revisited and investigated based on this continuum form. It is possible to avoid introducing the

controversial concepts, the Youngs modulus and thickness of graphene and single-walled carbonnanotubes, with this continuum form.

Keywords: Elastic Theory, DNA Ring, Biomembrane, Graphene, Carbon Nanotube, MovingFrame Method.

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4232. Fundamentals of Geometric and Elastic Theory on

Low-Dimensional Continua . . . . . . . . . . . . . . . . . . . . . . . 4242.1. Geometric Representation of

Low-Dimensional Continua . . . . . . . . . . . . . . . . . . . . 4242.2. Elastic Theory of 1D Continua . . . . . . . . . . . . . . . . . . 425

Author to whom correspondence should be addressed.

2.3. Elastic Theory of 2D Continua . . . . . . . . . . . . . . . . . . 4283. Application of Elastic Theory in Bio-Structures . . . . . . . . . . 429

3.1. Short DNA Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . 4303.2. Lipid Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . 4303.3. Cell Membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

4. Application of Elastic Theory in Nano-Structures . . . . . . . . . 437

4.1. Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4374.2. Carbon Nanotube . . . . . . . . . . . . . . . . . . . . . . . . . . 4395. Conclusion and Prospect . . . . . . . . . . . . . . . . . . . . . . . . . 443

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445

422 J. Comput. Theor. Nanosci. 2008, Vol. 5, No. 4 1546-1955/2008/5/422/027 doi:10.1166/jctn.2008.002


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Tu and Ou-Yang Elastic Theory of Low-Dimensional Continua and Its Applications in Bio- and Nano-Structures

1. INTRODUCTION

We human beings live in a three-dimensional (3D) spacewhich contains many geometric entities composed of atomsor molecules. The length scale of objects observed withour naked eyes is much larger than the distance betweennearest neighbor atoms or molecules in the objects. Asa result, the objects can be regarded as continua. If onedimension of an object is much larger than the other twodimensions, such as a rod, we call it a one-dimensional(1D) entity. If one dimension of an object is much smallerthan the other two dimensions, such as a thin film, we callit a two-dimensional (2D) entity. In this review, the termlow-dimensional continua represents 1D and 2D entities.

Elasticity is a property of materials. It means that mater-ials deform under external forces, but return to theiroriginal shapes when the forces are removed. Elastictheory, the study on the elasticity of continuum mater-ials, has a long history1 2 which records many geniuses

such as Hooke (16351703), Bernoulli (17001782),Euler (17071783), Lagrange (17361813), Young (17731829), Poisson (17811840), Navier (17851836), Cauchy(17891857), Green (17931841), Lam (17951870),Saint-Venant (17971886), Stokes (18191903), Kirchhoff(18241887), and so on. Now elastic theory has been amature branch of physics and summarized in several excel-lent textbooks.24 Although the classical elastic theory isapplied to macroscopic continuum materials, more andmore facts reveal that it can be also available for bio-or nano-structures such as short DNA rings,513 -helical

coiled coils,14 chiral filaments,1521 climbing plants,22 23bacterial flagella,24 viral shells,2527 bio-membranes,2836

zinc oxide nanoribbons,3739 and carbon nanotubes,4046 tosome extent.

Z. C. Tu was born on February 18, 1977 in Hubei Province of China. He received hisPh.D. in Theoretical Physics in June 2004 at the Institute of Theoretical Physics, ChineseAcademy of Sciences under the supervision of Professor Z. C. Ou-Yang. From 2006 to2007, he worked with Professor U. Seifert at Universitat Stuttgart as an Alexander vonHumboldt fellow. Now he is an Associate Professor of Physics in Beijing Normal University.His research focuses on the physical properties of bio- and nano-materials, such as lipid

membranes, cell membranes, carbon nanotubes, zinc oxide nanobelts, and so on.

Professor Zhong-can Ou-Yang, a theoretical physicist, was born on January 25, 1946 inFujian Province of China. He received his Ph.D. in 1984 at the department of physics ofTsinghua University, Beijing. From 1987 to 1988 he worked with Professor W. Helfrich atFreie Universitat Berlin as an Alexander von Humboldt fellow. He has been a full professorat the Institute of Theoretical Physics, Chinese Academy of Sciences since 1992, and asthe Director of the institute from 1998 to 2007. His research focuses on soft condensed

matter physics and theoretical biophysics. He received the Achievement in Asia Award ofthe Overseas Chinese Physics Association in 1993, and elected as a member of ChineseAcademy of Sciences in 1997, and a member of Third Word Academy of Sciences in 2003.

This review presents the elastic theory of low-dimensional continua and its applications in bio- and nano-structures, which is organized as follows: In Section 2, webriefly introduce the geometric representation and the elas-tic theory of low-dimensional continua including 1D rodand 2D fluid membrane or solid shell. The free energydensity of the continua is constructed on the basis of the

symmetry argument. The fundamental equations can bederived from the bottom-up and the top-down viewpoints.Although they have different forms obtained from thesetwo standpoints, several examples reveal that they are, infact, equivalent to each other. In Section 3, the applicationand availability of the elastic theory of low-dimensionalcontinua in bio-structures, including short DNA rings, lipidmembranes, and cell membranes, are discussed. We inves-tigate the kink stability of short DNA rings, the elasticityof lipid membranes, and the adhesions between a vesi-cle and a substrate or another vesicle. A cell membrane

is simplified as a composite shell of lipid bilayer andmembrane skeleton. The membrane skeleton is shown toenhance highly the mechanical stability of cell membranes.In Section 4, the application and availability of the elas-tic theory of low-dimensional continua in nano-structures,including graphene and carbon nanotubes, are discussed.We propose a revised Lenosky lattice model and fit fourparameters in this model through the local density approx-imation. We derive its continuum form up to the secondorder terms of curvatures and strains, which is the sameas the free energy of 2D solid shells. The intrinsic rough-

ening of graphene and several typical mechanical proper-ties of carbon nanotubes are revisited and investigated byusing this continuum form. Section 5 is a brief summaryand prospect.

J. Comput. Theor. Nanosci. 5, 422448, 2008 423


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Elastic Theory of Low-Dimensional Continua and Its Applications in Bio- and Nano-Structures Tu and Ou-Yang

2. FUNDAMENTALS OF GEOMETRICAND ELASTIC THEORY ONLOW-DIMENSIONAL CONTINUA

In this section, we describe the mathematical basis and theelastic theory of 1D and 2D continua.

2.1. Geometric Representation ofLow-Dimensional Continua

The 1D continuum (rod) and 2D continuum (membrane orshell) can be expressed as a smooth curve and a smoothsurface, respectively.

2.1.1. Curve Theory

Figure 1 depicts a curve C embedded in the 3D Euclidspace. Each point in the curve can be expressed as a vectorr and let s be the arc length parameter. At point rs, one

can take T, N, and B as the tangent, normal and binor-mal vectors, respectively. r T N B is called the Frenetframe which satisfies the Frenet formula:47

r = TT = NN = T + BB = N

(1)

where the prime represents the derivative with respectto s. and are the curvature and torsion of the curve,

respectively.The fundamental theory of curve47 tells us that the bend-ing and twist properties of a smooth curve are uniquelydetermined by the Frenet formula (1).

2.1.2. Surface Theory

Figure 2 depicts a surface M embedded in the 3D Euclidspace. Imagine that a mass point moves on the surface

N

B

C

T

x

y

z

o

Fig. 1. Frenet frame r T N B.

e3

e2

e1

e3

e2

e1

O

r

r

M

Fig. 2. Moving frame r e1 e2 e3 of a surface M.

in the speed of unit and that a right-handed frame, whichconsists of three unit orthonormal vectors with two vec-

tors always in the tangent plane of the surface, adheres tothe mass point. Assume that the mass point is at positionexpressed as vector r and the frame superposes three unitorthonormal vectors e1 e2 e3 with e3 being the normalvector of surface M at some time s. When the mass pointmoves to another position r at time s + s, the framewill superpose three unit orthonormal vectors e1 e

2 e

3.

Thus we call the frame a moving frame and denote it asr e1 e2 e3.

If s 0, we definedr

=lim

s

0r

r

=1e1

+2e2 (2)

and

dei = lims0

ei ei = ijej i = 1 2 3 (3)where 1, 2, and ijij= 1 2 3 are 1-forms, and dis the exterior differential operator.48 49 Here 12 can beunderstood as the infinite rotation angle of vectors e1 ande2 around e3. Similarly, we can understand the physicalmeaning of the other ij. It is easy to obtain ij = jifrom ei ej = ij. Additionally, the structure equations ofthe surface can be expressed as:48 49

d1 = 12 2d2 = 21 1dij = ik kj ij= 1 2 3

(4)

and 13

23

=

a b

b c

1

2

(5)

where represents the wedge production between twodifferential forms. The matrix

a bb c

is the representa-

tion matrix of the curvature tensor

. Its trace and deter-minant are two invariants under the coordinate rotationaround e3 which are denoted by

2H= a + c and K = ac b2 (6)424 J. Comput. Theor. Nanosci. 5, 422448, 2008


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They can be expressed as 2H= 1/R1 +1/R2 and K =1/R1R2 by the two principal curvature radii R1 and R2 ateach point.

Consider a tangent vector m stemming from r. Let bethe angle between m and e1. Then the geodesic curvature,the geodesic torsion, and the normal curvature along thedirection ofm can be expressed:49

kg = d+ 12/dsg = b cos2 + c a cos sin kn = a cos2 +2b cos sin + c sin2

(7)

where ds is the arc length element along m. If m alignswith e1, then = 0, kg = 12/ds, g = b, and kn = a.

2.2. Elastic Theory of 1D Continua

We will elucidate the elastic theory of rod with inextensi-ble centerline. As shown in Figure 3, let us simplify a rodas a curve rs with s being the arc-length parameter, andcut an infinitesimal element (shown in the magnified box)from the rod. There are forces and moments at the twoends of the element which originating from the interactionof other parts of the rod. F and M represent the force andmoment vectors at point rs, while F + dF and M + dMare the force and moment vectors at point rs +ds. FromNewtons laws, we can derive the force and moment bal-ance equations:

F = 0 F = 0 (8)

ds

F

F+dF

T

M

M+dM

r(s)

O

Fig. 3. Force and moment in 1D rod.

N

C

x1

x2

x3

Fig. 4. Rod with rectangle cross section.

and M = 0 M +T F = 0 (9)

where the prime represents the derivative with respect to s.One should add the constitutive relation and boundary con-

ditions to make the above two equations closed.

2.2.1. Kirchhoff Rod Theory

A rod with rectangle cross section and centerline C isshown in Figure 4. Take local coordinates x1 x2 x3 withx1 and x2 paralleling respectively to the two edges of therectangle, and x3 along the tangent of the centerline. N isthe normal of curve C. Let x1 x2 x3 denote the basisof the local coordinates and define 1 = x2 dx3/ds,2

=x1

dx3/ds, and 3

=x2

dx1/ds. Viewed from

geometrical point, 1 and 2 describe the bending of therod around axes x1 and x3, respectively, and 3 representsthe twist of the rod around axis x3. The free energy den-sity G due to the bending and twist can be expressed as afunction of 1, 2, and 3. Expanding G up to the secondorder terms of 1, 2, and 3, we have

G = + k12

1 12 +k22

2 22 +k22

3 32 (10)

where the constant can be interpreted as the line ten-sion.

1 and

2 are interpreted as the spontaneous curva-

tures while 3 the spontaneous torsion. Denote k = 1x1 +2x2 +3x3 and let be the angle between x1 and N. Thenwe have

N = cos x1 sin x2B = sin x1 +cos x2

(11)

where B is the binormal of curve C. From Eqs. (1) and(11), we can derive50 51

k = sin x1 + cos x2 + +x3 (12)

Thus G can be also regarded as the function of.

The moment vector is defined as2

M = Gk

G1

x1 +G

2x2 +

G

3x3 (13)



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which is called the constitutive relation. Eqs. (8), (9) and(13) with some boundary conditions form a group ofclosed equations. They are also available for the rod withcross section different from rectangle if only we take x1and x2 as the two principal axes of inertia. It should benoted that the equivalent form of these equations can bealso obtained from the variational method. This method

is called the top-down method while the former one viaNewtons laws called the bottom-up method.The free energy of a rod with length L can be written

as

=L

0G ds +bd (14)

where bd comes from the contributions of two ends of therod. The general Euler-Lagrange equations correspondingto Eq. (14) are derived as

G G = 0 (15)

G

+2G

/

+ G

/+

2

2

G+2G + G G = 0 (16)G +2G G + 2/G G/ = 0 (17)

where G, G , G and G are the partial derivatives ofGwith respect to , , and , respectively. Additionally,G G, G G, G G. The berief deriva-tion of Eqs. (15)(17) is attached in Appendix A. Theseequations have been employed to investigate helical andtwisted filaments.12 There might be a misprint in Eq. (7)of Ref. [12], corresponding to our above Eq. (17), because

the dimension of its last term is different from that of otherterms.

Now we would give a typical example to reveal theequivalence relation between Eqs. (8), (9), (13) and(15)(17) rather than prove it directly. Let us consider arod with k1 = k2 = k0, k3 = 0, and 1 = 2 = 3 = 0. Thefree energy density (10) is simplified as

G = k0/221 + 21+ = k0/22 + (18)On the one hand, we have M1 = k01 = k0 sin M2 =k02

=k0 cos M3

=0 from Eq. (13). The moment bal-

ance Eq. (9) implies F1 = k013 k02 and F2 = k01 k023. Substituting them into the force balance Eq. (8),we have F3 = F30 k02/2 and

2 + 3/2 F30/k0 = 0 (19)2 + = 0 (20)

where F30 is an integral constant which represents the linetension of the straight ( = 0) rod. On the other hand, wehave G = k0, G = G = G = 0. Equation (15) is triv-ial while Eqs. (16) and (17) are, respectively, transformed

into 2 + 3/2 /k0 = 0 (21)

2 + = 0 (22)

The above equations are the same as Eqs. (19) and (20)obtained from the force and moment balance conditions ifonly we take F30 = . Thus the equations obtained from thetop-down and bottom-up methods are equivalent to eachother.

Substituting the free energy density (10) intoEqs. (15)(17), we obtain the so called shape equations of

Kirchhoff rod as

k1 k22 sin2 2k3+ +2I21 = 0 (23)I12

+ 3 22 2 +2I122 + 2 +2I21

+2k1 k2 sin2 + sin2 I +4k3 +/ +2k3 + /+ k3+ 33+ + 3 = 0 (24)

I1+2 k3+ 3

+k32 + / k3 + /+2k1 k2 sin2+ I21

I12 = 0 (25)

where I1 = k1 sin2 + k2 cos2 , I = k121 + k222, I12 =k11 sin + k22 cos , and I21 = k22 sin k11 cos .

We also suggest that gentle readers consult the work byZhou et al.18 where the above Eqs. (23)(25) and differ-ent kinds of boundary conditions are expressed in another

representation with the aid of Euler angles.

2.2.2. Helfrich Rod Theory

Helfrich rod theory can be regarded as the fourth orderKirchhoff rod theory with circular cross section to someextent. The free energy density is expressed as52

G = 12

k22 +k32+

1

4k22

4 + 12

k42 +22+ (26)

where k2, k3, k22 and k4 are elastic constants while isthe line tension. It is noted that this free energy densityis the simplest stable form including the chirality termbut without spontaneous curvature and torsion. It has beenemployed to investigate the circular DNA in Ref. [6] andthe Euler-Lagrange equations corresponding to

G ds are

given as:

k23/2 2 +

+ k33323 +6 +2 +6

+ k453

2

/2 4

+ 2

/2 2

+62 +12 +4 +32

+ k2235/4 32 +62 +32 = 0 (27)

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k22++ k223 +62

+k362 +6 32 2+k443 +62 32 3 4

6 4 = 0 (28)Here we will not go on the more higher order Hel-

frich rod theory, on which gentle readers can consultRefs. [20, 21].

2.2.3. Theory of Bending-Soften Rod

There are two kinds of rod theory with bending-inducedsoftening. First, let us assume that the bending momentdepends linearly on the curvature for small curvaturebut not on the curvature for large curvature, which isexpressed as

M= k1 < ck1c > c

(29)

where k1 and c are the elastic bending rigidity and thecritical curvature, respectively. Equation (29) describes thebending-induced softening relation of the first kind whichis depicted in Figure 5(a). The corresponding free energydensity can be expressed as

G = + k1/22 c2 0 (30)where is the Heaviside step function. The above form

has been employed by Yan et al. to investigate the loop for-mation mechanism and probability of short DNA rings.53

We conjecture that this model could solve the paradox inthe experiment on the ring closure of single-walled carbonnanotubes with 1,3-dicyclohexylcarbodiimide.54 Fitting theexperiment data with the worm-like chain model,55 the per-sistence length is 800 nm for single-walled carbon nano-tubes in the diameter of 1 nm,54 which is much smallerthan the theoretical value 33 m estimated in terms of theYoungs modulus and thickness of single-walled carbonnanotubes in Ref. [44].

Consider a rod divided into two parts at s = Lc: one part(s < Lc) has curvatures less than c another one larger thanc . In terms of the variational method in Appendix A, wecan derive the equations describing the rod as

M

c c

M

(a) (b)

Fig. 5. Bending-induced softening relation: (a) the first kind in expres-sion of Eq. (29); (b) the second kind in expression of Eq. (38).

k12 22 +3 2 = 0 s < Lc (31)

+2= 0 s < Lc (32)k1cc 22 2 = 0 s > Lc (33)

= 0 s > Lc (34)

At the divided point s=

Lc, we have the joint conditions

as

= + = c (35) = 0 (36)

= + (37)

where and + represent the values of at the leftand right sides of s = Lc.

Secondly, let us assume that the bending momentdepends linearly on the curvature for small curvature but

weaker linearly on the curvature for large curvature, whichis expressed as

M=

k1 < c

k2 c +k1c > c(38)

where k1 > k2 are the elastic bending rigidities while c isthe critical curvature. Equation (38) describes the bending-induced softening relation of the second kind which isdepicted in Figure 5(b). The corresponding free energydensity can be expressed as

G = +k1/22 +k2 k1/2c20 (39)

Consider a rod divided into two parts at s = Lc: one part(s < Lc) has curvatures less than c another one larger thanc. In terms of the variational method in Appendix A, wecan derive the equations describing the rod as

k12 22 + 3 2 = 0 s < Lc (40)

+2= 0 s < Lc (41)2k2

+k2

c

+k1c

2

22

2

+ k2 k1 cc = 0 s > Lc (42)k2

+2 + k1 k2c = 0 s > Lc (43)

At the divided point s = Lc, we have the joint conditionsas

k1 c = k2+ c (44)k1

= k2+ (45)

= + (46)k12 2+ = k2 k1+ c2 (47)

Obviously, the above Eqs. (40)(47) degenerate intoEqs. (31)(37) ifk2 = 0 and into Eqs. (21)(22) ifk2 = k1.



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e1

e2

e3

f

m

t

b

p

C

Fig. 6. Force and moment in a 2D continuum.

2.3. Elastic Theory of 2D Continua

A 2D continuum can be simplified as a surface as shown inFigure 6. At each point, we can select a frame e1 e2 e3.A pressure p is loaded on the surface in the inverse direc-tion of the normal vector e3. Let us cut a region enclosedin any curve C from the surface. t is the tangent vector atpoint of curve C. b is normal to t and in the tangent plane.

The force and moment per length performed by the otherregion on curve C are denoted as f and m, respectively.Through Newtons laws, the force and moment balanceconditions are obtained as

Cfds

pe3 dA = 0 (48)

Cm ds +

C

r fds

r pe3 dA = 0 (49)

where ds and dA are the arc length element of curveC and area element of the region enclosed in curve C,

respectively.Define two second order tensors and such that

b = f b = m (50)

These two tensors can be called as stress tensor and bend-ing moment tensor, respectively. Using the Stokes theo-rem, we can derive

div pe3 dA = 0 (51)

div+e1 1 + e2 2 dA = 0 (52)

where 1 = e1 and 2 = e2. Since the integral isperformed on the region enclosed in an arbitrary curve C,

from the above two equations we obtain the force andmoment balance conditions of 2D continua as:

div= pe3 (53)div=1 e1 +2 e2 (54)

The above two equations are equivalent to Eq. (25) in

Ref. [56], and Eqs. (28) and (57) in Ref. [57]. Equa-tions (53) and (54) with some complement constitutiverelations form the fundamental equations of 2D continua.

2.3.1. Fluid Membranes

A fluid membrane is a 2D isotropic continuum whichcannot withstand in-plane shear strain. Generally, weassume that the fluid is incompressible. The free energydensity, G, of fluid membranes should be invariant underthe in-plane coordinate transformation. In terms of the

surface theory, there are only two fundamental geometricinvariants: the mean curvature 2H and Gaussian curva-ture K. Thus the free energy density should be a functionof 2H and K, that is,

G = G2H K (55)

The free energy of a closed fluid membrane can beexpressed as

=

G dA+ p

dV (56)

where dA is the area element of the membrane and dVis the volume element enclosed in the membrane. p isthe osmotic pressure, the pressure difference between theouter and inner side of the membrane. The general Euler-Lagrange equation of free energy (56) can be derivedthrough the variational method shown in Appendix B as

p 2HG + 2/2 +2H2 KG/H+ +2KHG/K = 0 (57)

As we known, the above equation has been derived byseveral authors such as Ou-Yang et al.58 49 and Giaquintaet al.59 coming from different research fields. It is recentlyemployed to investigate the modified Korteweg-de Vriessurfaces.60 Here can be called as the Laplace operatorof the second class which is also fully discussed by Zhangand Xu.61

We emphasize that (57) can be also derived from thebottom-up method, Eqs. (53) and (54) combining a com-plement constitutive relation

= Gb/2e1e1 e2e2 Gae2e1 + Gce1e2 (58)

where Ga, Gb, and Gc represent the partial derivatives ofG with respect to a, b, and c, respectively. Here a, b, andc are the components of the curvature tensor in Eq. (5).To illuminate this point, we consider an example in which



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the free energy density is taken as G = kc2H 2 +, wherekc and are the bending modulus and surface tension ofthe fluid membrane. It follows that = 2kcH e1e2 e2e1from Eq. (58). Substituting it into Eqs. (53) and (54), wecan derive

p 2H+4kcH H2 K +2kc2H= 0 (59)

which is the same as the result obtained directly from (57).Simultaneously, we have the stress components

1 = 2H2 2aH+ e1 2bHe2 2H1e3 (60)2 = 2bHe1 + 2H2 2cH+e2 2H2e3 (61)

where H1 and H2 are the directional derivatives of Hrespect to e1 and e2. These equations have been alsoderived by Capovilla and Guven,57 from which we seemto arrive at a paradox for fluid membranes: we have men-tioned that fluid membranes cannot withstand in-planeshear strain, however Eqs. (60) and (61) reveals shearstress still exhibits in non-spherical vesicles.

2.3.2. Solid Shells

A solid shell is a 2D isotropic continuum which can endureboth bending and in-plane shear strain. The free energydensity, G, of solid shells should be invariant under thein-plane coordinate transformation. There are only twofundamental geometric invariants, 2H and K, and twofundamental strain invariants: the trace, 2J, and the deter-

minate, Q, of the in-plane strain tensor. Thus free energydensity should be a function of 2H, K, 2J, and Q. Thatis, G = G2H K 2J Q.

If the solid shell has no initial strains and consistsof materials distributing symmetrically with regard to themiddle surface of the shell, we can expand G up to thesecond order terms of curvatures and strains as

G = kc/22H2 kK + kd/22J 2 kQ (62)

where kc and k are the bending moduli while kd and kare the in-plane rigidity moduli. The theory based on theabove free energy density is called Kirchhoffs linear shelltheory.2 Especially, if the shell consists of 3D isotropicmaterials, we have

kc = Yh3/121 2 (63)kd = Yh/1 2 (64)

k/kc = k/kd = 1 (65)

where Y and are the Youngs modulus and Poisson ratiowhile h is the thickness of the shell.3

For a closed shell, its free energy is expressed asEq. (56) with G in Eq. (62). Of course, we can obtain theequations of in-plane strains and shapes through the vari-ational method in Appendix B. The final results are the

same as those obtained from Eqs. (53) and (54) with acomplement constitutive relations (58) and

=i +f (66)with

i

G11 e1e1

+G12 /2e1e2

+e2e1

+G22 e2e2 (67)

where G11 , G12 , and G22 represent the partial derivativesof G with respect to 11, 12, and 22, the componentsof the in-plane strain tensor . Substituting Eq. (62) intoEqs. (58) and (67), and then employing Eqs. (53) and (54),we obtain

di112 i121 i212 i221 21 = 0 (68)di212 i221 i112 i121 12 = 0 (69)

and

p +2kc2H H2 K +2H 4kd kJH k = 0(70)

where i11 = 2kdJ k22, i12 = i21 = k12, and

i22 = 2kdJ k11 are the components of tensor i.

is the curvature tensor related to Eq. (5). The above Eqs.(68)(70) describe the in-plane strains and shapes of solidshells at equilibrium state. The similar equations and thecorresponding dynamics forms have been derived throughthe variational method in Refs. [49] and [62], respectively,with the aid of moving frame method.

The above Eqs. (68) and (69) can be written as onevector equation by introducing a displacement vector u =u1e1 + u2e2 + u3e3, which is related to two invariants 2Jand Q of the in-plane strain tensor as

2J = div u 2Hu3 (71)2Q = div u 2Hu32 + 1/2curl u2 u2 (72)

where u = ue3 u is the in-plane part ofu. Usingthe new variable u, Eqs. (68) and (69) can be be writtenas

k 2kd div u 2Hu3 k2u + Ku + u3 = 0 (73)where u and 2u are the in-plane components of u anddiv u, respectively. is called the gradient operator ofthe second class, which is shown in our previous work.49

In particular, H, K, u3 vanish and 2 degenerates into 2for a flat manifold. Then the above equation degeneratesinto the Cauchy Eq. (2) in 2D plane. Thus Eq. (73) can beregarded as the Cauchy equation in a curved surface.

3. APPLICATION OF ELASTIC THEORYIN BIO-STRUCTURES

In the above section, we have described fundamentals ofgeometric and elastic theory on low-dimensional continua.



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Can this theory be applied to the bio-structures, such asDNA and cell membranes, and so on? DNA is a longchain macromolecule which may be described as an elas-tic rod. A cell membrane is a thin structure whose thick-ness and the size of the microscopic components are somuch smaller than its lateral dimension that it can beregarded as a 2D continuum phenomenologically. We will

discuss the application of the above elastic theory in shortDNA rings, lipid membranes and cell membranes in thissection.

3.1. Short DNA Ring

DNA is a double helical structure whose diameter is about2.5 nm. Its bending rigidity, described as the persistencelength lp, is about 50 nm (150 bp) at the room tempera-ture. The normal DNA is usually flexible enough becauseits length is so much larger than lp that the fluctuations

are quite evident. Thus the rod theory cannot directly beapplied to the normal DNA. The statistical theory com-bining the rod theory is required,63 64 which is out of ourtopic in this review. However, there is a special kind ofshort DNA rings6567 which are in the length scale oflp sothat the fluctuation effect can be neglected. The diameteris still much smaller than the total length. Thus the rodtheory mentioned in Section 2.2 is expected to be availablefor this kind of DNA rings.

Han et al. have used AFM to observe DNA ringsconsisting of several segments connected by kinks in thepresence of Zn2+ ions.65 66 Zhao et al. have analyzed themechanism of this kink instability based on Helfrich rodtheory.6 Their main ideas are sketched as follows. First, acircle is a solution to Eqs. (27) and (28). Next, throughanalyzing the stability of the cycle, it is found that, forthe given elastic constants, there exists a critical radiusabove which DNA circles will be instable. This predic-tion is in good agreement with the experiments,65 66 wherekink deformations were observed in DNA rings of 168 bpbut not 126 bp. Above some thresholds of the chiral mod-ulus, k3 in Eq. (26), the DNA circles turn into elliptical,triangular, square, or other polygonal shapes, respectively.This fact agrees with the experiments if k3 is positivelycorrelated to the condensation of Zn2+ ions.

Interestingly, Zhou and Ou-Yang proposed another inter-pretation based on the dynamic instability of Kirchhoffrod theory8 with 2 = 3 = 0 in Eq. (10). Their result isthe same as that obtained directly from the first and sec-ond order variations of the free energy. We deal with thelatter scenario. First, = 0, = 0, and = 1/R satisfyEqs. (23)(25) derived from the first order variation of thefree energy. That is, a planar circle with radius R is anequilibrium configuration. Next, through the second order

variation of the free energy, we can obtain the character-istic function describing the stability of the circle

gcR = 21 1 1/R n2/R2 0 (74)

where n > 1 is an arbitrary integer and = k3/k1. Fromthe above inequality, we obtain the critical radius

Rc = 8 /1

1+

12 +16 (75)above which the circle is instable. If only the presence ofZn2+ ions tunes the values of and k3/k2 such that Rc isin the range between 63/ (bp) and 84/ (bp), the aboveresult is also in agreement with the experiments,65 66 wherekink deformations were observed in DNA rings of 168 bpbut not 126 bp.

In Section 2.2, we also mention the theory of bending-soften rod. Can this theory also provide an interpretationto the experiments? Let us consider the bending-softenrod theory of the first kind whose free energy density isexpressed as Eq. (30). When the radius R of the ring issmaller than 1/c, any small perturbation will increase thefree energy. IfR > 1/c, the ring might transform into thefictitious configuration shown in Figure 7 which consistsof four arcs AB, BC, CD, DA with the radius R1 and R2.To see conveniently, the joint points are marked as smallcycles in the figure. Obviously, R2 < R < R1. Throughsimple calculations, we find that the fictitious configura-tion is energetically less favorable than the perfect ringwith radius R. Therefore, this coarse analysis revealsthat the theory of bending-soften rod cannot explain theexperiments.

3.2. Lipid Membrane

Lipids are dominant composition of cell membranes. Mostof lipid molecules have a polar hydrophilic head groupand two hydrophobic hydrocarbon tails. When a quantityof lipid molecules disperse in water, they will assemblethemselves into a bilayer vesicle as depicted in Figure 8, inwhich the hydrophilic heads shield the hydrophobic tailsfrom the water surroundings because of the hydrophobicforces. This self-assembly process has been numerically

AB

C D

R1

R2

Fig. 7. A possible configuration of a short DNA ring.



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Water

Water Water

WaterWater

Fig. 8. A lipid bilayer vesicle.

investigated by Lipowsky et al.6870 and Noguchi et al.71

through molecular dynamics simulation based on coarse-grained model or meshless membrane model.The thickness of the lipid bilayer and the size of sin-

gle lipid molecules are much smaller than the scale ofthe whole lipid bilayer. Additionally, at the physiologi-cal temperature, the lipid bilayer is usually at the nematicstate where the hydrocarbon chains of the lipid moleculesare roughly perpendicular to the bilayer surface. Thus thebilayer can be regarded as a 2D fluid membrane whosefree energy density is expressed as Eq. (55). Expanding itup to the second order terms of curvatures, we obtain the

Helfrichs form:29

GH = kc/22H+ c02 kK + (76)

where kc and k are the bending moduli of the lipid bilayer.We emphasize that the minus sign before k in Eq. (76)is opposite to Helfrichs convention. is the surface ten-sion of the bilayer. c0 is called the spontaneous curva-ture that reflects asymmetric factors between two sidesof the bilayer, including the lipid distribution, the chem-ical environment, and so on. kc is about 20 T for lipidbilayers, where the Boltzmann factor is set to 1 and T

the room temperature, from which the persistence lengthof lipid bilayers is estimated about 10 m.32 33 In thissection we only consider the size of lipid bilayers smallerthan 10 m so that the fluctuation effect on the shapeof lipid bilayers can be neglected. The model based onEq. (76) is called spontaneous curvature model. We stillremind gentle readers to note the two similar nonlocalmodelsthe bilayer-coupling model72 73 and the area dif-ference model,74 although we will not touch them in thepresent review.

3.2.1. Closed Vesicles

The free energy of a lipid vesicle under the osmotic pres-sure p (the outer pressure minus the inner one) can be

written as Eq. (56) with G = GH being Helfrichs form(76). Substituting (76) into Eq. (57), we can obtain theshape equation of lipid vesicles:75 76

p 2H+ kc2H+ c02H2 c0H2K +2kc2H= 0(77)

This equation is the fourth order nonlinear equation. It

is not easy to find its special solutions. We have knownthree typical analytical solutions: sphere,75 torus,77 78 andbiconcave discoid shape.79

For a sphere with radius R, we have H = 1/R andK = 1/R2. Substituting them into (77), we arrive at

pR2 +2R kcc02 c0R = 0 (78)This equation gives the sphere radius under the osmoticpressure p.

A torus is a revolution surface generated by a circlewith radius rotating around an axis in the same plane

of the circle. The revolution radius r should be largerthan . A point in the torus can be expressed as a vectorr + cos cos r+ cos sin sin . Throughsimple calculations, we have 2H= r+2 cos /r+ cos , K = cos /r+ cos . Substituting them intoEq. (77), we derive

2kcc20

2 4kcc0 +42 +2P3/3 cos3 + 5kcc202 8kcc0 +102 +6P3/2 cos2 + 4kcc202 4kcc0 +82 +6P3/ cos +2kc/2 +kcc202 1 +2P + 2 = 0 (79)

where = r/. If is finite, then Eq. (79) holds if andonly if the coefficients of 1 cos cos2 cos3 vanish.It follows 22 = kcc04c0, P3 = 2kcc0 and =

2.77 That is, there exists a lipid torus with the ratio ofits two generated radii being

2, which was confirmed in

the experiment.80

To describe the solution of biconcave discoid shape, wewrite the shape Eq. (77) under the axisymmetric condi-tion. If a planar curve z = z revolves around the z-axis,an axisymmetric surface is formed. Each point on the sur-face is expressed as r = cos sin z. Denote = arctandz/d and = sin . Then Eq. (77) is trans-formed into81

12

+ c0

2 c0

kc

+

1 2

+ p

kc= 0 (80)

where the prime represents the derivative with respect to .This equation is called the shape equation of axisymmetric

lipid vesicles. Its first integral, group structure and corre-sponding Hamiltons equations are investigated by Zhengand Liu,82 Xu and Ou-Yang,83 and Capovilla et al.84 85

respectively.



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z

B

z0

Fig. 9. A quarter outline of the biconcave surface.

It is easy to verify that = sin = c0 ln/B witha constant B is a solution to Eq. (80) if p and arevanishing. For 0 < c0B < e, the parameter equation

sin = c0 ln/B

z=

z0 +

0tan d

(81)

corresponds to a curve shown in Figure 9. A biconcavediscoid surface will be achieved when this curve revolvesaround z-axis and then reflects concerning the horizontalplane. The above Eq. (81) can give a good explanation tothe shape of human red blood cell under normal physio-logical conditions.79 If c0B is out of the range between 0and e, Eq. (81) corresponds to a prolate ellipsoid or otherself-intersecting surfaces.86

In the purely mathematical viewpoint, there are also theother solutions to Eq. (77) such as cylinder, constant meancurvature surface, periodic undulation surface,87 pearlingtubule,88 and so on.34 89 However, It is a pity that theyare open surfaces and do not correspond to truly closedvesicles.

As mentioned above, it is fairly difficult to find the ana-lytical solution to Eq. (77). Thus we appreciate the applica-tions of numerical methods to find the equilibrium shapesof closed vesicles. Two kinds of typical numerical frame-works are usually employed. The first one is to use Sur-face Evolver, a software package developed by Brakke,90

to find the configurations minimizing the free energy under

some constraints.9194 The second one is based on thephase field formulation of Helfrichs free energy density(76) and diffusive interface approximation.9598 The abovenumerical methods can obtain lipid vesicles with differentshapes either axisymmetric or asymmetric. Additionally,the finite element method might be a potential methodalthough very sparse literature99 treats lipid bilayers byusing it.

3.2.2. Stability of Closed Vesicles

When the osmotic pressure is beyond some threshold, aclosed vesicle will lose its stability and change its shapeabruptly. The threshold is called the critical pressure. Toobtain it, one should calculate the second order variation

of the free energy (56) with G being Helfrichs form (76),which has been dealt with in the general case as:49 100

2 =

kc23

2 + 2H+ c0 2H 3 3 dA

+

4kc2H2 K2 + kcKc20 4H2

+2K 2Hp23 dA

+

kc14H2 +2c0H4K c20/2

323 dA2kc

2H+c03 3 +23 3 dA

(82)

where 3 is an arbitrary small out-of-plane displacementand the operator is the gradient operator of the secondclass.49

Here we will mention two results for specialconfigurations.First, let us consider a lipid sphere that satisfies

Eq. (78). On the sphere, the function 3 can be expandedby the spherical harmonic functions Ylm as 3 =

l=0m=l

m=l almYlm. Substituting it into Eq. (82), wederive 2 = R/2l m alm2ll + 1 22kcll +1 c0R/R3 p, from which we can obtain the criticalpressure75

pc = 2kc6c0R/R3 (83)Ifp < pc ,

2

0 for any

alm

; on the contrary, 2 can

be negative for the special selection of alm. The aboveequation depends also on c0. Ifc0 > 6/R, then pc is nega-tive, which reveals that a sphere vesicle is always instablefor large enough c0.

Next, let us still regard a long enough lipid tubule as aclosed vesicle. Denoted its radius as . From Eq. (77) wehave

kc/21/2 c20 p = (84)

On the cylindrical surface, 3 can be expanded as Fourierseries 3 =

l= al expil. Substituting it into Eq. (82)

and combining Eq. (84), we derive 2= l= al2l2 1kcl2 1/3 p, from which we can obtain the

critical pressurepc = 3kc/3 (85)

If p < pc, 2 0 for any al; on the contrary, 2 can

be negative for the special selection of al.

3.2.3. Open Vesicles with Free Edges

The opening-up process of lipid vesicles by talin, a pro-tein, has recently been observed101 102 which pushes us

to study the equilibrium equation and boundary condi-tions of lipid vesicles with free exposed edges. Capovillaet al. have addressed this problem and given the equi-librium equation and boundary conditions.103 Inspired by



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the talk moving frame method of Chern,104 we intro-duce exterior differential form to deal with the variationalproblem on open surface and obtain concisely the shapeequation and boundary conditions of open lipid vesicles.105

Numerical solution to the shape equation and boundaryconditions with relaxed method can explain the experi-mental results very well.106 A quantity of open vesicles

with free edges have also been obtained numerically byWang and Du107 with the phase field method. Here we willnot further discussed the dynamical opening process ofthe vesicles, which has been recently investigated by Kagaand Ohta.108

We regard an open lipid vesicle with a free edge asa smooth surface with a boundary curve C, as shown inFigure 10. t is the tangent vector of the curve C. b, inthe tangent plane of the surface, is perpendicular to t andpoints to the opposite side that the surface located in. Thefree energy of the open lipid vesicle is written as

=

GH dA+

Cds (86)

where represents the line tension of the edge and GHhas the Helfrichs form (76).

The first order variation of gives the shape equation

kc2H+c02H2 c0H2K 2H+2kc2H= 0(87)

and the boundary conditions as:105

kc2H+ c0 kknC = 0 (88)2kcH/b+ kn kgC = 0 (89)

GH + kgC = 0 (90)where kn and kg are normal curvature and geodesic cur-vature of the boundary curve C. g is the derivative ofgeodesic torsion g with respect to the arc length ofcurve C. The mechanical meanings of the above four equa-tions are as follows: Eq. (87) is the normal force balanceequation of the membrane; Eq. (88) is the moment bal-ance equation of points in curve C around the directionof t; Eq. (89) is the force balance equation of points incurve C along the normal direction of surface; and Eq. (90)is the force balance equation of points in curve C along

b

t

C

Fig. 10. An open surface with boundary curve C.

the direction of b. It is necessary to emphasize that theboundary conditions are available for open vesicles withmore than one free edge because the edge in our derivationis a general one.

In Ref. [105], we have shown two analytical solutionsto above Eqs. (87)(90): One is a cup-like membrane andanother is the central part of a torus. Several numeri-

cal solutions to these equations are obtained by Umedaet al.106 Their results reveal that the line tension inducedby talin correlates negatively with the concentration oftalin, which is in agreement with the experimental resultthat the hole of vesicle is enlarged with the concentrationof talin.101

3.2.4. Vesicles with Lipid Domains

The above discussion on open lipid vesicles with freeedges can be extended to study a vesicle of several lipidcomponents. The domains usually formed so that eachdomain contains one or two kinds of lipid molecules.The morphology of axisymmetric vesicles with multi-domains has been theoretically investigated by Julicherand Lipowsky.109 It is found that lipid domains facili-tate the budding of vesicles.110 The giant vesicles withlipid domains have been observed in recent experiment.111

There are two kinds of lipid domains which are at theliquid-ordered state and liquid-disordered state, respec-tively. It is natural to assume that different kinds ofdomains have different bending moduli and spontaneouscurvatures. The axisymmetric vesicles in the experimentcan be explained with Julicher-Lipowsky theory throughnumerically method. Baumgart et al. have demonstratedthat the line tension, the osmotic pressure, the relativebending moduli, and the spontaneous curvature have sig-nificant effects on the morphology of a vesicle with twodomains being at the liquid-ordered and disordered states,respectively.112

The asymmetric vesicles are also experimentallyobserved in Ref. [111], which enlightens us to investi-gate the shape equation of each domains and the bound-ary conditions between domains without any axisymmetricassumptions. Let us consider a vesicle with two domainsseparated by curve C sketched in Figure 11. The freeenergy can be expressed as109

=

GIH dA+

GIIH dA+

ds + p

dV (91)

where GIH and GIIH have the Helfrich from (76) with the

bending moduli kIc, kI, kIIc , k

II, the spontaneous curvaturescI0, c

II0 , and the surface tensions

I, II, respectively. The

integrals in the first and second terms of Eq. (91) are per-formed on the domain I and II shown in Figure 11, respec-tively. is the line tension of boundary curve C. p is theosmotic pressure of the vesicle.



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I

II

C

Fig. 11. A vesicle with two domains separated by curve C.

In terms of the physical meanings of Eqs. (87)(90),we can easily write down the shape equation of domainsas:49 113

p 2iH+ kic2H+ c02H2 ci0H2K +2kic2H= 0(92)

where the superscript i = I and II represents the physi-cal quantity of lipid domains I and II, respectively. Addi-

tionally, the boundary conditions between domains are asfollows:49 113

kIc2H+ cI0 kIIc 2H+cII0 kI kIIknC = 0 (93)2kIc + kIIc H/b kI + kIIg +knC = 0 (94)

GI GII + kgC = 0 (95)

where b is perpendicular to the boundary curve C andpoints to the side of domain II.

As we know, there is still no any numerical result on

asymmetric vesicles with domains directly from the aboveequations in the previous literature. Only in Ref. [107],Wang and Du discussed the morphology of asymmetricvesicles with domains through the phase field model.

In the above theory, the detailed architecture of liquid-ordered and disordered phases is neglected. There are spe-cial lipid domains at liquid-ordered phase, so called rafts,which are enriched in cholesterol and sphingolipids.114

Cholesterol is a kind of chiral molecules, which has notbeen included in the above theory. Recently, a concisetheory of chiral lipid membranes developed by Tu and

Seifert115 might be extended to discuss the raft domains.

3.2.5. Adhesions of Vesicles

Cell adhesion is a complex biological process which con-trols many functions of life. It can be understood as afirst-order wetting transition116 and might be simplified asthe adhesion of lipid vesicles. As a model, Seifert andLipowsky have theoretically investigated a lipid vesicleadhering to a flat rigid substrate and found that the vesicleundergoes a nontrivial adhesion transition from the free

state to the bound state, which is governed by the com-petition between the bending and adhesion energies.117 Niet al. have discussed the adhering lipid vesicles with freeedges and the adhesion between a lipid tubule with a rigid

Vesicle

Substrate

(a)

t

b c

Vesicle I

t

b

c

Vesicle II

(b)

Fig. 12. Adhesions. (a) Adhesion between a lipid vesicle and rigid sub-strate with a contact line C. (b) Adhesion between two lipid vesicles witha contact line C.

substrate.118 119 A big progress on this topic is recentlymade by Guven and his coworkers120 121 who obtain thegeneral equations to describe the contact line between thevesicle and the rigid substrate or another vesicle.

The adhesion between a lipid vesicle and a rigid sub-strate is depicted in Figure 12(a) where the contact area isdenoted by A. The free energy of this system is expressedas117

=

GH dA+ p

dV WA (96)where p is the osmotic pressure of the vesicle and W isthe strength of the adhesion potential between the vesi-cle and the substrate. GH is the free energy density ofHelfrichs form (76). For the flat rigid substrate, a char-acteristic radius and the length scale of the vesicle are

defined as Ra =

2kc/W and R = A/4, respectively.If R < Ra, the vesicle is a little stiffer or the attractionis relative weak such that A approaches to zero. Thus thevesicle is unbound to the substrate and this state is calledthe free state. On the contrary, the vesicle is at the boundstate. At this state, let us take t as the tangent vector of thecontact line C, and b perpendicular to t and in the com-mon tangent plane of the lipid vesicle and the substrate.The absolute value of the normal curvature along b forthe point on the contact line is proven to be

2W /kc for

an axisymmetric vesicle adhering to the flat substrate.117

If the rigid substrate is curved, the above conclusion isrevised as121

Vb Sb =

2W /kc (97)

where Vb and Sb are the normal curvatures along b for the

points outside but near the contact line, calculated by usingthe surfaces of the vesicle and the substrate, respectively.

The adhesion between two lipid vesicles is depicted inFigure 12(b). The free energy of this system is expressedas117

= GIH dA

+ pI dV

+ GIIH dA

+pII dV

WA

(98)

where pI and pII are the osmotic pressures of the vesi-cles I and II, respectively. A and W are the contact area



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and adhesion strength, respectively. GIH and GIIH are the

Helfrichs free energy density of vesicle I and II. Thefirst order variation of (98) gives the same shape equa-tion of two vesicles as (92) and the adhesion boundaryconditions:121

1+ kIc/kIIc Ib Ab 2 = 2W /kIc (99)

1+ kIIc /kIcIIb Ab 2 = 2W /kIIc (100)Ib +IIb Ab /b = 0 (101)

where Ib and IIb are the normal curvatures along b for the

points outside the adhesion domain but near the contactline calculated by using the surfaces of vesicles I and II,respectively. Ab is the normal curvature for the pointsinside the adhesion domain but near the contact line cal-culated by using the common surface of vesicles I and II.As we know, there is still lack of numerical solutions tothe above Eqs. (99)(101) in the previous literature. Only

in the recent work, Ziherl and Svetina122 have investigatedthe adhesion between two vesicles by numerically mini-mizing the free energy (98) with kIIc = kIc and various W.

Is the behavior of vesicle adhesion close to that ofcell adhesion? The cell membrane can bear shear strainwhose adhesion behavior might be much closer to theadhesion between a polyelectrolyte microcapsule and thesubstrate.123 Interestingly, beyond the threshold adhesionstrength Wc, the contact length scale increases in pro-portion to (W Wc)1/2, which is the same as the behav-ior of vesicle adhesions except the coefficient before

(W Wc)1/2

.

3.2.6. A Different Viewpoint of Surface Tension

Although the lipid bilayer cannot withstand the in-planeshear strain, it can still endure the in-plane compressionstrain. The in-plane compression modulus, kb, of lipidbilayers is about 0.24 N/m.124 Considering this point, wemay write the free energy of a closed lipid vesicle as

= p

dV+

GB dA+kb/22Jb

2 dA (102)

whereGB = kc/22H+ c02 kK (103)

and Jb is the in-plane compression or stretch strain. Weemphasize that the contribution of chemical potential areomitted when we write the above free energy.

The first order variation of the free energy (102) revealsthat 2Jb is a constant and then

p 22kbJbH+2kc2H+kc2H+ c02H2 c0H2K = 0 (104)

Comparing the above equation with the shape Eq. (77) oflipid vesicles, we deduce that

= 2kbJb (105)

In the discussion on the stability of closed lipid vesicles,we have seen that the surface tensor has no effect onthe critical pressure. The second order variation of the freeenergy (102) can give the same conclusion. 2p

dV+

GB dA has been shown in Eq. (82) with vanishing .The additional term is

2 kb/22Jb

2

dA = kbdivv 2H 32 dA (106)where v = 1e1 + 2e2 + 3e3 represents the infinitesi-mal displacement vector of the vesicle surface. We canalways select the proper deformation modes such thatdiv v 2H 3 = 0 and then 2

kb/22Jb

2 dA vanishs,but 2p

dV + GB dA is not affected. That is, the

critical pressure is determined merely by 2p

dV +GB dA, which is independent on the compression mod-

ulus of lipid bilayer kb.

3.3. Cell Membrane

Cell membrane consists of lipids, proteins, and a smallquantity of carbohydrates and so on. A simple but widelyaccepted model for cell membranes is the fluid mosaicmodel125 proposed by Singer and Nicolson in 1972. In thismodel, the cell membrane is considered as a lipid bilayerwhere the lipid molecules can move freely in the mem-brane surface like fluid, while the proteins are embeddedin the lipid bilayer. Some proteins, so called integral mem-brane proteins, traverse entirely in the lipid bilayer andplay the role of information and matter communicationsbetween the interior of the cell and its outer environ-ment. The others, so called peripheral membrane proteins,are partially embedded in the bilayer and accomplish theother biological functions. Beneath the lipid membrane,the membrane skeleton, a network of proteins, links withthe proteins embedded in the lipid membrane. Maturemammalian and human erythrocytes (i.e., red blood cells)are lack of a cell nucleus. Thus they provide a good exper-imental model for studying the mechanical properties ofcell membranes.126129 On the theoretical side, spontaneouscurvature model,29 rubber membrane model,30 130 131 and

dual network model132 have been employed to investigatethe mechanical and thermal fluctuation properties of ery-throcyte membranes. We will address the elasticity andstability of composite shell model for cell membranes inthis section.

3.3.1. Composite Shell Model of Cell Membranes

A cell membrane can be simplified as a composite shell133

of lipid bilayer and membrane skeleton. The membraneskeleton, inside of the cell membrane, is a network of

protein filaments as shown in Figure 13. The joint pointsof the network are bulk proteins embedded in the lipidbilayer. The whole membrane skeleton seems to float thesea of the lipid bilayer. It can have a global movement



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Inside of cell membraneLipid

Bulk protein

Proteinfilament

Outide of cell membrane

Fig. 13. Local schematic picture of the composite shell model for a cellmembrane.

along the surface of the bilayer but the movement of thejoints along the normal direction is totally coupling withthe bilayer. In the mechanical point of view, the lipidbilayer can endure the bending deformation but hardly bearthe in-plane shear strain. On the contrary, the membraneskeleton can endure the in-plane shear strain but hardlybear the bending deformation. The composite shell over-comes the shortage of the lipid bilayer and the membraneskeleton. It can sustain both bending deformation and in-plane shear strain.

The contour length of protein chain between joints inthe membrane skeleton is about 100 nm which is muchsmaller than the size (10 m) of cell membranes. Thelipid bilayer is 2D homogenous. The membrane skele-ton is roughly a 2D locally hexagonal lattice. As is wellknown, the mechanical property of a 2D hexagonal latticeis 2D isotropic.134 Thus the composite shell of the lipidbilayer plus the membrane skeleton can still be regardedas a 2D isotropic continuum. Its free energy density shouldbe invariant under the in-plane coordinate transformationand can be written as Gcm = Gcm2H K 2J Q. We canexpand Gcm up to the second order terms of curvaturesand strains as

Gcm = GB + kb/22Jb2 + Gsk (107)

where GB results mainly from the bending energy ofthe lipid bilayer, which has the form as Eq. (103).kb/22Jb

2 is the contribution of in-plane compressionof the lipid bilayer where kb and 2Jb are the compres-sion modulus and relative area compression of the lipidbilayer. Gsk = kd/22J 2 kQ is the in-plane compres-sion and shear energy density which comes from theentropic elasticity of the membrane skeleton. kd and k arethe compression and shear moduli of the membrane skele-ton, respectively. Their values are experimentally deter-mined as kd

=k

=48 N/m.129 135 2J and Q are the

trace and determinant of the stain tensor of the membraneskeleton. Because there is no in-plane coupling betweenthe lipid bilayer and the membrane skeleton in the com-posite shell model, thus Jb for the lipid bilayer and J for

the membrane skeleton have no local correlation. In theabove subsection, we have mentioned that the effect ofkb/22Jb

2 can be replaced with the surface tension =2kbJb. Considering a closed cell membrane under osmoticpressure p, the free energy can be written as

=

Gcm dA+p dV (108)

Similarly to Section 2.3, if we define a displacementvector u satisfying Eqs. (71) and (72), we can derive theEuler-Lagrange equations corresponding to the free energy(108) as

k 2kd 2J k2u + Ku + u3 = 0 (109)p +2kc2H+ c02H2 c0H2K +22H

2H+2H k kd2J k u = 0 (110)where u and 2u are the in-plane components of u anddivu, respectively. is the curvature tensor related toEq. (5). is called the gradient operator of the secondclass, which is shown in our previous work.49

Generally speaking, it is difficult to find the analyticalsolutions to Eqs. (109) and (110). But we can verify thata spherical membrane with homogenous in-plane strainssatisfy these equations. The radius R and the homogenousin-plane strain should obey the following relation:

pR2 +2 +2kd kR + kcc0c0R 2 = 0 (111)

3.3.2. Stability of Cell Membranes and

the Function of Membrane Skeleton

When the osmotic pressure is beyond some threshold, aclosed cell membrane will lose its stability and change itsshape abruptly. The threshold is called the critical pres-sure. To obtain it, one should calculate the second ordervariation of the free energy (108) in terms of Appendix B.The variational result is

2 =

kc23

2 + 2H+ c0 2H 3 3 dA

+

4kc2H2 K2 + kcKc20 4H2

+2K 2Hp23 dA

+

kc14H2 +2c0H4K c20/2

323 dA2kc

2H+ c03 3 +23 3 dA

kd

v +2H 3divv 2H 3dA

+ k/2

curl v2 dA k

Kv2 dA

+ k

3 v dA+ k

2H 3div v 2H 3 dA

k

3v dA (112)



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where v = 1e1 + 2e2 + 3e3 is the infinitesimal dis-placement vector of the cell membrane whose in-planecomponent is denoted as v = 1e1 + 2e2.

In terms of the Hodge decomposed theorem,136 v can beexpressed by two scalar functions and as

v dr = d+d (113)

where is the Hodge star.49 136 Then we have div v =2 and curl v = 2. For the spherical cell membranesatisfying Eq. (111), Eq. (112) can be divided into twoparts: one is

21 = k/2

22 + 2/R22dA (114)

another is

22 =

232c0kc/R3 + p/R + 4kd 2k/R2 dA

+ 323kcc0/R +2kc/R2 + pR/2 dA+

kc23

2 dA+ 4kd 2k/R

32 dA

+ kd

22 dA+ k/R2

2 dA (115)

It is easy to verify that 21 is always positive on aspherical surface. Then the stability of the spherical cellmembrane is merely determined by 22. By analogy withour previous work,137 we can prove that 22 is also pos-itive if

p < pl 2k2kd k

kdll +1 kR+ 2kc

R3ll +1 c0R (116)

for any integer l 2. Thus the critical pressure ispc minpll = 2 3 4 (117)

Obviously, if k = 0, i.e., the effect of membrane skeletonvanishes in the cell membrane, pc degenerates into thecritical pressure (83) of a spherical lipid vesicle.

When kkd2kd kR2/kc6kd k2 > 1, the critical

pressure is derived from Eqs. (116) and (117) as

pc = 4/R2

k/kd2kd kkc (118)As an example, let us consider a cell membrane with typ-ical values of k = kd = 48 N/m,129 kc = 1019 J, andR 10 m. Through a simply manipulation, we find thatkkd2kd kR2/kc6kd k2 1, and so Eq. (118) holds,from which we obtain the critical pressure pc = 003 Pa.However, if the membrane skeleton vanishes, k = 0, wecalculate pc = 0001 Pa from Eqs. (116) and (117). Thisexample reveals a mechanical function of membrane skele-

ton: it highly enhances the stability of cell membranes.As a byproduct, Eq. (118) also gives the critical pressure

pc =

4/31 2Yh/R2 (119)

for a spherical thin solid shell of 3D isotropic materials ifwe take kc, kd , and k as Eqs. (63)(65). This formula isthe same as the classic strict result obtained by Pogorelovfrom the other method.138

4. APPLICATION OF ELASTIC THEORYIN NANO-STRUCTURES

In the last section, we have expatiated on the application ofElastic theory in bio-structures. In this section, we will dis-cuss whether and to what extent this theory can be appliedto nano-structures, especially the graphitic structures, suchas graphene and carbon nanotubes.

4.1. Graphene

Graphene is a single layer of carbon atoms with a 2D hon-eycomb lattice as shown in Figure 14(a). It has been a

rapidly rising star in the material science and condensed-matter physics139 since it was successfully cleaved frombuck graphite.140 It is found that the free-standing graphenemight be a strictly 2D atomic crystal which is stable underambient conditions.141 However, Mermin has theoreticallyproved that the 2D crystalline order could not exist atfinite temperature.142 There are two possible ways to solvethis paradox: (i) The graphene might not be a perfect 2D

(a)

(b)

Fig. 14. (a) Graphene. (b) Single-walled carbon nanotube.



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crystal. Recently, Meyer et al. have investigated the elab-orate structure of suspended graphene sheets and foundthat the graphene sheets are not genuine flat.143 They alsoargue that the graphene sheets could be stabilized by theout-of-plane deformation in the third dimension resultingfrom the the thermal fluctuations.143 Fasolino et al. havealso addressed the height fluctuations by means of Monte

Carlo simulations.144

Their result at room temperature isin good agreement with the experiment mentioned above.(ii) Mermin theorem is valid for power-law potentials ofthe Lennard-Jones type while the interaction between near-est neighbor atoms (covalent bond) in the graphene mightnot be of this type.145

To fully understand the experimental result and possiblestable mechanism in theory, we will address the Lenoskylattice model146 and its revised form as follows.

4.1.1. Revised Lenosky Lattice Model and

Its Continuum Limit

We start from the concise formula proposed by Lenoskyet al. in 1992 to describe the deformation energy of a sin-gle layer of curved graphite146

Eg =02

ij

rij r02 + 1

i

j

uij2

+ 2ij

1ni nj + 3ij

ni uijnj uji (120)

The first two terms are the contributions of bond length

and bond angle changes to the energy. The last two termsare the contributions from the -electron resonance. In thefirst term, r0 is the initial bond length of planar graphite,and rij is the bond length between atoms i and j after thedeformations. In the remaining terms, uij is a unit vec-tor pointing from atom i to its neighbor j, and ni is theunit vector normal to the plane determined by the threeneighbors of atom i. The summation

j is taken over

the three nearest neighbor atoms j to atom i, and

ij

taken over all the nearest neighbor atoms. The parame-ters 1 2 3 = 096 129 005 eV were determinedby Lenosky et al.146 through local density approximation.The value of0 was given by Zhou et al. as 0 = 57 eV/2through the force-constant method.147

In the above energy form, the second term requires thatthe energy cost due to in-plane bond angle changes isthe same as that due to out-of-plane bond angle changes.However, the experiment by inelastic neutron scatteringtechniques reveals that the energy costs due to in-planeand out-of plane bond angle changes are quite differentfrom each other.148 To describe this effect, we revise theLenosky lattice model as

Eg =02ij

rij r02

+ 1t i

j


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surface. The vectors Tj and Bj can be expressed by Tj =cos je1 + sin je2 and Bj = sin je1 +cos je2, where jis the rotating angle from e1 to Tj. We have the expres-sions of uij = rij/rij and ni = Nj with rij = rij for thedeformed graphene. Then Eq. (121) is transformed into thecontinuum limit up to the second-order magnitudes of11,22, 12 and r0 as

Eg =

kc2

2H 2 kK + kd2

2J 2 kQ

dA (123)

with four parameters

kc = 91n +62r20 /80 (124)k = 32r20 /40 (125)

kd = 90r20 +31t /160 (126)k = 30r20 +91t /80 (127)

where 0=

3

3r20 /4 is the occupied area per atom. The

continuum form (123) has first derived in our previouswork44 which is, in fact, the natural conclusion of the sym-metry of graphene:154 The curved graphene comprises alot of hexagons which has approximately local hexagonalsymmetry. In fact, 2D structures with hexagonal symmetryare 2D isotropic.134 Thus the elasticity of the graphene canbe reasonably described by the shell theory of 2D isotropicmaterials mentioned in Section 2.3 and so its energy hasthe form of Eq. (123). We also notice that a flaw in thecoefficient before 1 in the expression ofkc in our previouswork.44

Using the values of r0, 1t , 1n, and 2 obtained fromthe first-principles calculations, we have kc = 162 eV, k =072 eV, kd = 2297 eV/2, and k = 1919 eV/2. Becausethe results of first-principles calculation are applicable forzero temperature, only the results derived from the exper-iments at low temperature can be used as reference valuesto compared with them. The value of kc is close to thevalue 1.77 eV estimated by Komatsu155157 at low tempera-ture (less than 60 K). The value k/kd = 083 is quite closeto the experimental value 0.8 derived from the in-planeelastic constants of graphite.158 The elastic properties of

graphene can be described by Eq. (123) with four parame-ters kc, k, kd , and k, where the energy density is the sameas Eq. (62), the free energy density of solid shell with2D isotropic materials. Since k/kc = 044 is much smallerthan k/kd = 083, the graphene cannot be regarded as asolid shell with 3D isotropic materials as Ref. [44].

4.1.2. Intrinsic Roughening in Graphene at

Temperature T

Let us consider the freely suspended graphene which isalmost a flat layer with the area L2. The small out-of-

plane displacement is denoted by w. The energy (123) istransformed into

Eg = kc/2

2w2 d2x (128)

where x x1 x2 represents the point on the grapheneplane before deformations.

Adopting the Fourier series

wx = 1/Lq

wq expiq x (129)

with q

2l/L 2n/L, we transform Eq. (128) into

Eg = kc/2

q

q4wq2 (130)

and then the corresponding partition function is derived as

=

q

dwq expEg/T =

q

2T/kcq4 (131)

where the Boltzmann constant has been set to 1. It followsthat the equipartition theorem:

kc/2q4

wq2

= T ln

/ ln q4

= T /2 (132)where represents the ensemble average. The aboveequation is equivalent to

wq2 = T /kcq4 (133)

Similarly, w2 is derived as

w2 =q

wq2L2

= T L2

164kc

ln

1l2 +n22 (134)

Through simply numerical manipulations, we have159

w2 T L2

150kc(135)

for the graphene contains more than 100 atoms.In terms of Ref. [157], we estimate kc 046 eV at T =

300 K. Substituting it into Eq. (135) and taking L = 25 nmas the experiment,143 we have

w2 05 nm. This value

is a little smaller than the largest out-of-plane deformation1 nm in the experiment. However, they are consistent witheach other because 0.5 nm is the mean square value which

should be smaller than the largest out-of-plane deformationin the experiment.

4.2. Carbon Nanotube

There are two kinds of carbon nanotubes: single- andmulti-walled carbon nanotubes, which are synthesized inthe last decade of 20 century.160 161 Simply speaking, asingle-walled carbon nanotube (SWNT) can be regardedas a seamless cylinder wrapped up from a graphitic sheet,as shown in Figure 14(b), whose diameter is in nanome-

ter scale and length from tens of nanometers to severalmicrometers if we ignore its two end caps. A multi-walledcarbon nanotube (MWNT) consists of a series of coaxialSWNTs with layer distance about 3.4 .



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SWNTs can be expressed as a pair of integers (n m),so called index, in terms of the wrapping rule. They aredivided into two classes: achiral tubes if n = m or nm = 0and chiral tubes for others.162 The electronic propertiesof SWNTs depend sensitively on the index:163 they aremetallic if n m is multiple of 3, else semiconductor.SWNTs also possess many novel mechanical properties,164

in particular high stiffness and axial strength, which arenot sensitive to the tube diameters and chirality. MWNTshave the similar mechanical properties to SWNTs.165 166 Inthis section, we will review the theoretical and numericalresults on the elastic properties of carbon nanotubes, andthen discuss how the low-dimensional elastic theory men-tioned in Section 2 can be applied in carbon nanotubes.

4.2.1. General Review on the Elasticity of

Carbon Nanotubes

The early researches on the elasticity of carbon nano-tubes are focused on their Youngs modulus Y and Poissonratio . A SWNT is a single layer of carbon atoms. Whatis the thickness h of the atomic layer? It is a widelycontroversial question. Three typical values of the thick-ness listed in Table I are adopted or obtained in theprevious literature Refs. [40, 41, 44 and 167180]. Thefirst one is about 0.7 obtained from fitting the atomicscale model with the elastic shell theory of 3D isotropicmaterials.167173 The second one is about 1.4 derivedfrom molecular dynamics or finite element method.174 175

The third one is about 3.4 adopting the layer distanceof bulk graphite.176180 Recently, Huang et al. have inves-tigated the effective thickness of SWNTs and found itdepends on the type of loadings.181

The size- or chirality-dependent elastic properties ofSWNTs have also been discussed by molecular mechanics

Table I. Youngs modulus Y (unit in TPa), Poisson ratio and effectivethickness h (unit in ). (MD = molecular dynamics; TB = tight-binding;SM = structure mechanics; FEM = finite element method; LDA = localdensity approach.)

Authors Y h Method Refs.Yakobson et al. 55 019 066 MD [40]Tu and Ou-Yang 47 034 075 LDA [44]Kudin et al. 39 015 089 ab initio [167]Zhou et al. 51 024 074 TB [168]Vodenitcharova et al. 49 062 Ring theory [169]Pantano et al. 48 019 075 SM & FEM [170,171]Chen and Cao 68 080 SM [172]Wang et al. 51 016 067 ab initio [173]Sears and Batra 25 021 134 MD [174]Tserpes et al. 24 147 FEM [175]Lu 10 028 34 MD [41]Hernandez et al. 12 018 34 TB [176]

Shen and Li 11 016 34 Force-field [177]Li and Chou 10 34 SM [178]Bao et al. 09 34 MD [179]Zhou et al. 08 032 34 LDA [180]

model182184 and ab initio calculations.173 185 The com-mon conclusion is that the Youngs modulus and Poissonratio depend weakly on the diameter and chirality ofSWNTs if the diameter is larger than 1 nm. Only forvery small SWNTs, the size and chirality effect is evident.The SWNTs synthesized in the laboratory have usuallythe diameters larger than 1 nm; thus the size and chirality

effect can be neglected safely.The axial tension properties of MWNTs depend on the

layer number of MWNTs for the small layer number andapproach quickly to the properties similar to the bulkgraphite.44 186 187

The buckling and stability of carbon nanotubes underpressure or bending is a hot topic in the recent researches,where the critical pressure, moment or the equivalent quan-tity, critical strain, are highly concerned. A long enoughcarbon nanotube under an axial loading might be regardedas a Euler rod and the axially critical strain is3

rodzc = 2/AL2 /L2 (136)

where L, and A are the length, radius and cross-sectionalarea of the carbon nanotube, respectively. is the momentof inertia of the nanotube. The value of depends onthe boundary conditions of the carbon nanotube. Thisrelation has been investigated by atomic-scale finite ele-ment method188190 and molecular dynamics method orab initio calculations.191195 The basic numerical result isthat the tube exhibits rod-like buckling behavior as the

right-handed side of Eq. (136) if L . The Timoshenkobeam theory, a more complicated theory than Euler rodtheory, is also employed to discuss the buckling ofMWNTs.196 The difference between the results of boththeories vanishes for large value ofL/.

For a short carbon nanotube under axial loading,the continuous shell model of 3D isotropic materialsare widely used.40 188190 192194 197199 The axially criticalstrain of a short SWNT is138

shellzc

=/31

2h/

1 (137)

where and h are the radius and effective thickness of theSWNT, respectively. is the Poisson ratio of the SWNT.The value of depends on the boundary condition of thecarbon nanotube. For a short MWNT, the above relationis applicable for the outmost layer of the tube becausethe inter-layer interaction of MWNTs is very small.200 Ithas also been investigated by atomic-scale finite elementmethod,188190 198 molecular dynamics method,40 192194 199

and nanoindent experiment.201 202 It is found that the tubedisplays indeed the shell-like buckling behavior as the

right-handed side of Eq. (137) for the tube aspect ratioL/ < 10.

The stability of a long SWNT under radial hydrostaticpressure might also be described by the continuous shell



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model of 3D isotropic materials, and the critical pressureis138

pshellcr 3 (138)where is the radius of the SWNT. This relation hasrecently been confirmed by Hasegawa and Nishidate203

through ab initio calculations. The stability of a MWNTunder radial hydrostatic pressure might also has the simi-

lar relation as Eq. (138) if only we take as the outmostradius of the MWNT, because the transverse elasticity ofMWNTs204 205 is much weaker than the in-plane elasticityof the outmost single layer of tube.

Bending can also result in the buckling of SWNTs. Thekink phenomenon in a SWNT under pure bending has beeninvestigated through molecular dynamics simulations andfinite element method.40 206 The critical curvature can bedescribed as

cr = shellzc / 2 (139)

where is the radius of the SWNT. The kink phenomenonin a MWNT under pure bending satisfies the similar rela-tion to Eq. (139) with small correction due to inter-layervan de Waals interactions207209 if only we take as theoutmost radius of the MWNT.

Here we would not further discuss the problemson the buckling of MWNTs embedded in an elasticmedium,210215 the postbuckling behavior and the plas-tic properties of carbon nanotubes,216222 as well as themechanical properties of nanotube composites,223228 ratherthan recommend gentle readers to consult the correspond-

ing literature.

4.2.2. What are the Fundamental Quantities

for SWNTs?

As mentioned above, different thickness leads to differ-ent Youngs modulus (see Table I), which implies thatthe Youngs modulus and thickness of SWNTs are notwell-defined physical quantities.229 However, the in-planeYoungs modulus Ys = Yh has the similar value 22 eV/2.Thus it is a more well-defined quantity than the Youngsmodulus and the thickness. Here we may ask: what are thefundamental quantities for SWNTs?

A SWNT is also a single layer of graphite, whosedeformation energy can be also described as the revisedLenosky model (121). The corresponding continuum limitis Eq. (123) which contains four elastic constants kc , k,kd , and k. These four quantities avoid the controversialthickness of SWNTs. We suggest to use them as the fun-damental quantities for SWNTs from which we can obtainsome reduced quantities as follows.

Let us consider a cylinder under an axial loading withline density f along the circumference. The corresponding

axial and circumferential strains are denoted as 11 and 22.With Eq. (123), the free energy of this system is written as

2Lkd/211 + 222 k1122 f11 (140)

where L and are the length and radius of the SWNT. Thein-plane Youngs modulus and Poisson ratio can be definedas Ys = f /11 and s = 22/11. From /11 = 0 and/22 = 0, we derive

Ys = k2 k/kd = 2235 eV/2

(141)

s=

1

k/kd=

0165 (142)

where the value ofYs is close to the in-plane Youngs mod-ulus derived from Table I. It is in between 2023 eV/2

obtained by Snchez-Portal et al.230 It is much larger thanthe value 15 eV/2 obtained by Arroyo et al.231 and Zhanget al.,232 and 17 eV/2 by Caillerie et al.,233 but smallerthan 34.6 eV/2 for armchair tube by Wang.234 The valueofs is close to the value 0.160.19 obtained by Yakobsonet al.,40 Kudin et al.,167 Pantano et al.,170 171 Wang et al.,173

Hernandez et al.,176 and Shen et al.177

The other quantity, the bending rigidity D, is also widely

discussed in literature. In terms of Eq. (123), the energyper area of a SWNT without the in-plane strains can beexpressed as

Gg = kc/22 D/22 (143)Thus the bending rigidity

D = kc = 162 eV (144)

which is quite close to the value 1.491.72 eV obtainedby Kudin et al.167 and Snchez-Portal et al.230 throughab initio calculations. It is a little larger than the values

0.851.22 eV obtained Yakobson et al.,40 Pantanoet al.,170 171 and Wang.234

In terms of Eqs. (141)(144), we can infer the valuesof kd, k, kc from the previous literature, which are listedin Table II. There is still lack of literature on k except ourprevious work42 44 154 and the present review. More workon k would be highly appreciated in the future.

We should emphasize that our formula (123) holdsapproximate up to the order of r0/

2 for SWNTs, wherer0 is the C C length and the radius of the SWNT.The omitted terms is in the order of r0/

4. This is the

main reason for the size effect on the elastic constantsin the very small SWNTs found in Refs. [173, 182185].Additionally, we have not considered the effect of Stone-Wales defects on the local properties of carbon nano-tubes. In terms of Refs. [235 and 236], we can deducethat the defects reduce the the elastic constants of carbonnanotubes.

4.2.3. Revisit the Stability of SWNTs

Now we will revisit the stability of SWNTs with the four

fundamental quantities kc, k, kd , and k or the correspond-ing reduced quantities.

First, let us consider a bent SWNT as shown inFigure 15 where and 1/ are the radii of the SWNT



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Table II. The values of Ys , s , kd , k, kc and k. (MD = molecular dynamics; TB = tight-binding; SM = structure mechanics; FEM = finite elementmethod; LDA = local density approach; CTIP = continuum theory of interatomic potential.)

Authors Ys (eV/2) s kd (eV/

2) k (eV/2) kc (eV) k (eV) Method Refs.

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