Mat. Res. Soc. Symp. Proc. 651: T5.3.1-T5.3.6 (2001)

arXiv:cond-mat/0201109v1 [cond-mat.soft] 8 Jan 2002

Mechanical, adhesive and thermodynamic
properties of hollow nanoparticles
U. S. Schwarz1 , S. A. Safran and S. Komura2
Department of Materials and Interfaces, Weizmann Institute, Rehovot 76100, Israel
1Max-Planck-Institute of Colloids and Interfaces, 14424 Potsdam, Germany
2
Department of Chemistry, Tokyo Metropolitan University, Tokyo 192-0397, Japan

Abstract
When sheets of layered material like C, WS2 or BN are restricted to finite sizes, theygenerally form single- and multi-walled hollow nanoparticles in order to avoid dangling
bonds. Using continuum approaches to model elastic deformation and van der Waals
interactions of spherical nanoparticles, we predict the variation of mechanical stability,
adhesive properties and phase behavior with radius R and thickness h. We find thatmechanical stability is limited by forces in the nN range and pressures in the GPa range.
Adhesion energies scale linearly with R, but depend only weakly on h. Deformation due
to van der Waals adhesion occurs for single-walled particles for radii of few nm, but isquickly suppressed for increasing thickness. As R is increased, the gas-liquid coexistence
disappears from the phase diagram for particle radii in the range of 1-3 nm (dependingon wall thickness) since the interaction range decreases like 1/R.

Introduction
The prototypical hollow nanoparticle is the buckyball C60 , which crystallizes into fullerite.
Many other morphologies for carbon sheets have been found since the early 1990s, including fullerenes Cn of varying size, multi-walled fullerenes (carbon onions), and single- andmulti-walled carbon nanotubes. Closed structures of carbon are formed in order to avoid
dangling bonds at the edges of finite sized carbon sheets. This mechanism is generic foranisotropic layered material of finite size, and up to now more than 30 other materials
have been prepared as hollow nanoparticles (see the reviews [1] and references therein).
This includes the metal dichalcogenides MeX2 (Me = W, Se, X = S, Se), BN, and GaN.
Usually inorganic fullerenes like WS2 and MoS2 are multi-walled. In Fig. 1 we showtransmission electron micrographs of MoS2 -particles which were synthesized by solid-gasreaction. Control of size and shape is rather difficult, as evidenced by the irregularly
faceted shapes. However, the methods used for the synthesis of hollow nanoparticles aredeveloping very rapidly, and it is to be expected that control of size, thickness and shapewill become much better in the future.
Hollow nanoparticles combine covalent in-plane strength with flexible out-of-plane bending of thin films, which results in high mechanical stability. They interact by van derWaals (vdW) forces, which for example determine the material properties of fullerite [2]and the phase behavior of buckyballs [3]. VdW interactions become even more important
for larger fullerenes, carbon onions and inorganic fullerenes; these materials, therefore,

hr
R(a)
V

hard core
diameterr0

interaction
range δ = (r1-r0)/r0
r1
ε/100

potential
depth ε(b)rFigure 1: Left: Transmission electron micrographs of hollow MoS2 -nanoparticles (courtesyof R. Tenne). Van der Waals interactions leads to strong adhesion. The faceted shapeis due to grain boundaries. Right: (a) Model for two spherical nanoparticles of radius Rand thickness h each, interacting over a distance r. (b) The effective interaction potential
V as a function of distance r is characterized by the hard core diameter 2R, potential
depth ǫ ∼ R and interaction range δ ∼ 1/R.
feature a large degree of non-specific adhesion. In particular, vdW adhesion can lead to
deformation of the hollow nanoparticles. For example, it has been found that carbon
nanotubes adhering to each other or to flat surfaces show observable deformations forradii as small as 1 nm [4]. In the presence of local surface features like crystal steps,
adhering nanoparticles will have spacially varying electronic properties due to spaciallyvarying deformation.
Hollow nanoparticles may lead to new applications in nanoelectronics and -optics. In
contrast to filled nanoparticles, they can also be used for storage and delivery systems.
They show superior mechanical properties, like high flexibility, high tensile strength andlight weight, which will lead to applications as ultra-strong fibers. Larger fibers canbe produced using the cohesiveness provided by vdW interactions. Inorganic fullerenes
have also been shown to offer favorable tribological properties [5]. Detailed investigations
with the Surface Force Apparatus demonstrated that low friction and wear is caused bymaterial transfer onto the sliding surfaces [6].
For these applications, a good theoretical understanding of the physical properties of hol-low nanoparticles is important. Here we discuss structural properties of spherical hollow
nanoparticles in a continuum approach that allows us to focus on unusual generic properties which result mainly from geometrical effects. In particular, we address mechanicalproperties, van der Waals interaction, thermodynamic behavior and deformation by adhesion. In this paper, we focus on the main results of our work; more details can be foundelsewhere [7, 8].

Mechanical properties
During recent years, mechanical properties of fullerenes and carbon nanotubes have been
extensively studied at the level of molecular calculations, that is by first principles, tightbinding and force field techniques. The advantage of molecular models is that they canprovide detailed quantitative predictions. For larger systems like multi-walled nanoparticles, their implementation becomes difficult due to computer time requirements. Here weuse classical continuum elasticity theory, which is asymptotically correct for large systems
and allows us to treat all these systems using the same framework. The large advantageof this approach is that it provides insight into the generic properties of these systems.
Molecular calculations for carbon nanotubes have shown that the predictions of continuum elasticity theory persist well into the limit of radii smaller than 1 nm [9]. Recently
this observation could be verified also for BN and MoS2 nanotubes [10].
The layered material discussed here usually has a hexagonal lattice structure, for whichelasticity theory [11] predicts that there are only two in-plane elastic constants and thesheet is elastically isotropic. Its deformation energy consists of stretching and bending
terms. For single-walled carbon sheets, the in-plane stretching modulus G ≈ 3.6 × 105
erg/cm2 and the out-of-plane bending rigidity κ ≈ 3×10−12 erg. For both MoS2 and WS2 ,G is smaller by a factor 4 and κ is larger by a factor 5. For multi-walled nanoparticles,
the elastic constants scale with thickness h as G ∼ C11 h and κ ∼ C11 h3 , where C11 is thelargest in-plane elastic constant of the corresponding layered material [12]. The values for
C11 are 1060, 238 and 150 × 1010 erg/cm3 for C, MoS2 and WS2 , respectively.
Stretching can be avoided completely by bending the sheet into a cylindrical nanotube.
However, this is not possible for spherical nanoparticles, which we discuss here. The sameholds true for saddle-like structures, and in general for all surfaces with non-zero Gaussian curvature. On a microscopic level, the requirement for non-zero Gaussian curvature
also means that a flat sheet cannot be bent into a sphere without introducing defects.
Continuum elasticity approaches to fullerenes therefore explicitly treated the 12 pentagonal defects necessary to close a sheet of carbon hexagons [13]. In order to avoid thiscomplication, here we assume that the curvature generating defects are distributed in a
homogeneous manner and result in a vanishing internal strain (like for a ping-pong ball).
In this case, the hollow nanospheres can be modeled as elastic shells with thickness h andprefered radius R.
Mechanical collapse in tribological applications is likely to be caused by either direct
forces or high pressure. In the latter case, an inward buckling instability occurs, since
the pressure energy scales with a higher power of indentation H than the restoring elastic
energy (H 2 versus H 3/2 ) [7]. Mechanical collapse is expected when indentation H becomesof the order of radius R. The critical pressure for collapse can then be estimated to bepc ∼ G1/4 κ3/4 /R5/2 . For buckyballs and typical inorganic fullerenes, this yields values

of 15 and 2 GPa, respectively. For nanoparticles, the corresponding forces are in thenN-range.

Van der Waals interaction and phase behavior
Since vdW interactions are very difficult to treat on a molecular level, a continuum approach is even more useful in this case. It is well known from colloid science that vdW
interactions strongly depend on the geometry of the system under investigation. Geometrical aspects of vdW interactions are well treated in the Hamaker approach, in which anatomic Lennard-Jones interaction is summed in a pairwise manner over the relevant geometry and the molecular details of the interaction are lumped into the effective LennardJones parameters σ and ǫ. For carbon, σ = 3.5Å and ǫ = 4.6 × 10−15 erg [2]. The
corresponding Hamaker constant A is in the order of 10−12 erg. For two buckyballs, theHamaker approach for thin sheets leads to the Girifalco potential [2], which is considerably shorter ranged than a Lennard-Jones potential. Since a small range of attractionleads to the disappearance of the gas-liquid coexistence from the phase diagram [14], it
has been suggested that buckyballs constitute the first non-colloidal system for which
the gas-liquid critical point does not appear in the phase diagram. However, large scale
simulations have shown that buckyballs indeed do show a gas-liquid coexistence, which
extends roughly from 1900 to 2000◦ C [3].
For hollow nanoparticles in general, the effect of a finite thickness h becomes important.
However, for nanoparticles in a crystal or adhering to a flat substrate or to each other, thedistance of closest approach D between interacting particles is on an atomic scale. Under
such conditions, the vdW interaction saturates on the atomic scale D with increasing
thickness h. For example, the surface energy for the interaction with a halfspace scales asǫh/D3 and ǫ/D 2 for thin films and halfspaces, respectively, with the crossover occuring
for h = D. Since h & D in our case, the surface energy is ǫ/D 2 independently of h andno significant scaling with thickness is expected.
For a more detailed investigation, one has to consider the general case of two nanospheresof finite thickness interacting in a pairwise manner by a Lennard-Jones potential [8]. Theinteraction energy can be written as V = VR,R + VR−h,R−h − 2VR,R−h , where VR1 ,R2 is theinteraction energy between two filled balls of radii R1 and R2 , respectively, which can becalculated analytically for a Lennard-Jones potential. In order to predict phase behavior,
three quantities characterizing the resulting potential are especially important: the hardcore diameter 2R sets the density scale, the potential depth sets the temperature scale,and the interaction range determines if a gas-liquid coexistence occurs (compare Fig. 1).
Our full analysis has shown that for hollow nanoparticles of radius R and thickness h, thepotential depth and interaction range scale linearly and inversely with R. Both quantities
depend only weakly on h. In Fig. 2 we show our numerical results for the interaction
range as a function of R and h. Recent work for the Double-Yukawa potential suggeststhat the gas-liquid coexistence disappears for an interaction range smaller than δ = 0.4[15]. Our analysis predicts that this happens for R = 12Å and R = 35Å for single- andmulti-walled nanoparticles, respectively. For larger R, the critical point will disappear
from the phase diagram.

100

δ = 0.12Rσ

δ

80

0.2

1.0
0.5
60
0.2
0.1

40
0.3

0.05
0.02

202

5

102050

0.4

100

2R

σ00(a)

102030(b)

40h50

σ

Figure 2: Numerical results for the interaction range δ (a) for single-walled nanoparticles
as a function of radius R and (b) for multi-walled nanoparticles as a function of radius Rand thickness h. σ = 3.5 Å is the atomic length of the Lennard-Jones potential. The gasliquid coexistence is expected to disappear from the phase diagram for δ < 0.4 (dashedlines).

Deformation due to adhesion
We consider the case where a spherical hollow nanoparticle is indented a distance H due toadhesion to a flat substrate. The vdW energy gained on adhesion can be estimated to beEvdW ∼ (A/D2 )DH where A is the Hamaker constant and D an atomic cutoff [16]. Theenergy of deformation can be estimated to be Eel ∼ G1/2 κ1/2 H 2 /R. With increasing R,deformation becomes more likely, since both the vdW energy and deformability increase.
However, deformability decreases rapidly with increasing thickness h, since G1/2 κ1/2 ∼ h2 .
Solving for the indentation H and using the values given above, it is found that for hollow
carbon nanoparticles with few walls, H can be in the nm-range [7]. However, for typicalinorganic fullerenes deformations are suppressed by the larger shell thickness h.
Finally we briefly discuss a nanotube adhering to a flat substrate. As explained above,nanotubes can be bent without being stretched. On deformation, one has to pay a bending
energy per length of Eel /L ∼ κH 2 /R3 . The gain in vdW energy scales as EvdW /L ∼AH/D 3/2 R1/2 . It follows that for carbon nanotubes with small thickness h, the indentation
H can be in the nm-range for even smaller values of radius R than for hollow carbon
nanoparticles.

Acknowledgments
It is a pleasure to thank R. Tenne for many stimulating discussions. USS was supportedby the Minerva Foundation and SK by the Ministry of Education, Science and Culture ofJapan.

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[16] This estimate is valid for weak adhesion. In [7], we used an estimate for strong
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