EUROPHYSICS LETTERS

15 June 2000

Europhys. Lett., 50 (6), pp. 762–768 (2000)

Deformation and tribology of multi-walled
hollow nanoparticles
U. S. Schwarz, S. Komura and S. A. Safran
Department of Materials and Interfaces, Weizmann Institute of Science
Rehovot 76100, Israel(received 17 January 2000; accepted in ﬁnal form 14 April 2000)
PACS. 61.48.+c – Fullerenes and fullerene-related materials.
PACS. 81.40.Pq – Friction, lubrication, and wear.

Abstract. – Multi-walled hollow nanoparticles made from tungsten disulphide (WS2 ) show
exceptional tribological performance as additives to liquid lubricants due to eﬀective transfer
of low shear strength material onto the sliding surfaces. Using a scaling approach based oncontinuum elasticity theory for shells and pairwise summation of van der Waals interactions,
we show that van der Waals interactions cause strong adhesion to the substrate which favors
release of delaminated layers onto the surfaces. For large and thin nanoparticles, van der Waals
adhesion can cause considerable deformation and subsequent delamination. For the thick WS2
nanoparticles, deformation due to van der Waals interactions remains small and the main
mechanism for delamination is pressure which in fact leads to collapse beyond a critical value.
We also discuss the eﬀect of shear ﬂow on deformation and rolling on the substrate.

Introduction. – Graphite and layered material made from metal disulphides (MoS2 ,WS2 ) and similar composites (MoSe2 , WSe2 , BN, etc.) are good lubricants since under shear
the layers can easily slide over each other due to atomic smoothness and weak van der Waals(vdW) interactions [1]. Therefore they are widely used as solid lubricants or additives to liquid
lubricants. However, ﬁnite-sized crystallites in powders have edges with dangling bonds which
can react chemically with the sliding surfaces. This problem can be avoided by using hollow
nanoparticles made from the same material, where the layers are not planar, but are bent into
closed shells. It has been shown recently that multi-walled WS2 nanoparticles perform very
well as additives [2,3]. Closer investigations using the Surface Force Apparatus (SFA) revealed
that WS2 layers are transfered onto the sliding surfaces by delamination of the nanoparticles[4]. These peeled layers form islands on which the Friction Force Microscope measured muchsmaller friction than on the surrounding mica. Hence multi-walled nanoparticles can act as
reservoirs which release layers of low shear strength exactly where needed while avoiding too
many dangling bonds.
A theoretical understanding of the mechanical properties of multi-walled hollow nanoparticles is important for future tribological applications. In this paper, we use scaling arguments
based on continuum elasticity theory and pairwise summation of vdW interactions in order
to investigate theoretically the eﬀect of adhesion, pressure and shear ﬂow on the deformationc EDP SciencesU. S. Schwarz et al.: Deformation and tribology etc.

Fig. 1

(a)

(b)

(c)

(d)

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Fig. 2

Fig. 1 – Several aspects of friction experiments with multi-walled nanoparticles: (a) weak deformation
due to adhesion onto a substrate; (b) strong deformation due to pressure; (c) deformation in shear
ﬂow and (d) rolling of adhering particles in shear ﬂow.
Fig. 2 – High-resolution transmission electron micrograph of a multi-walled WS2 nanoparticle. The
slightly faceted shape is typical and can be attributed to the assembly of defects into grain boundaries [7].

and mechanical stability of hollow nanoparticles of spherical shape. Figure 1 schematically
depicts some of the aspects discussed in the following. We show that vdW adhesion to thesubstrate (as well as to each other) can be several orders of magnitude larger than thermalenergies and scales linearly with radius, but is almost independent of thickness. Despite thelarge energy of adhesion, we ﬁnd that for typical WS2 nanoparticles, coherent and even incoherent deformations due to the vdW adhesion to the substrate are only in the Angstrom
range and will leave the nanoparticles basically intact. Thus for WS2 nanoparticles the vdW
adhesion favors the release of delaminated layers onto the surfaces, but does not trigger thedelamination itself. The main mechanism for delamination of WS2 nanoparticles is shown
to be pressure which leads to a mechanical instability in linear elasticity theory. This might
explain why damage of WS2 nanoparticles was found to occur only beyond a critical load [3].
We also show that shear ﬂow does not lead to considerable deformation of WS2 nanoparticles
and that very large shear rates are needed in order to make adhering particles roll in shear
ﬂow.
Similar methods have been used before to predict the shape of fullerenes [5, 6] and toaccount for the faceted shape of metal disulphide particles [7, 8]. Compared with ab initio
methods [9], tight-binding schemes [10] and molecular simulations [11] which have been used
before to investigate the mechanical properties of fullerene-like material, our approach has
the advantage that it is asymptotically correct for large systems and universal in the sensethat diﬀerent material systems enter on the level of their elastic and vdW constants. In
particular we discuss C, MoS2 and WS2 , although we focus on the WS2 nanoparticles used in

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the tribological experiments mentioned above. Our scaling approach allows us to predict how
tribological performance depends on radius R, shell thickness h and layer thickness a of thenanoparticles. We show that scaling with h does not result from the vdW contributions, butrather from the scaling of the elastic constants with h. Several critical quantities derived scalestrongly with R/h for coherent deformation and with R/a for incoherent deformation. Hence
tribological performance can be tuned by varying the concentration of defects (which switches
between the coherent and incoherent regime) and the ratios R/h and R/a, respectively.
Preliminaries. – In the following we consider hollow nanoparticles with outer radius R
and thickness h which results from nesting several elastic shells of thickness a each. Typical
WS2 particles used in tribological experiments have R ≈ 60 nm and h ≈ 9.3 nm (15 layerswith an interlayer distance of a = 0.62 nm, R/h ≈ 6, R/a ≈ 100). In ﬁg. 2 a high-resolution
transmission electron micrograph of a WS2 nanoparticle with 11 layers is shown. The elastic
shells are closed since their formation is driven by the energy reduction due to the absenceof dangling bonds. In order to achieve closure, topology dictates that for carbon shells 12
pentagons have to be inserted into the graphite network of hexagons. For hexagonally layered
composites, the sheets have a more complicated molecular structure (typically triple layers)
and diﬀerent kinds of defects have to occur to ensure closure. In our continuum approach, we
assume that defects are distributed in a homogeneous way to give a spherical shape with acertain preferred radius R [7]. Similar approaches to fullerene-like particles have considered
sheets which are planar in equilibrium; in order to account for their curvature, they explicitly
considered defects [5–8]. However, in this approach the elastic response of a spherical shell isvery complicated and no unifying scaling approach is possible.
The new feature of the elasticity of a closed shell with a preferred radius R is that stretchingis a ﬁrst-order eﬀect and the spherical shell cannot be bent without being stretched [12]. This
interplay between bending and stretching has been studied before for thermal ﬂuctuations ofpolymerized vesicles [13].
 The elastic behavior of the shell is determined by two contributions:bending energyEb ∼ κ dA c2 with bending rigidity κ and mean curvature c, and stretching
stretching modulus (or two-dimensional Young modulus)
energy Es ∼ G dA e2 with in-planeG and in-plane strain e. Here dA represents the surface integral; the elastic contributionsin shell theory follow by integrating over the thickness of the shell. Consider a sphericalshell of radius R which is expanded by ∆R. Then c changes by ∆R/R2 and e by ∆R/R,
thus both contributions are ﬁrst-order eﬀects. Since Eb /Es ∼ κ/GR2 , the relative strengthof bending and stretching depends both on material parameters and radius; on large lengthscales R  (κ/G)1/2 and stretching will always dominate.
In the framework of continuum elasticity theory, the elastic moduli κ and G can be calculated from the in-plane stretching elastic constant C11 of the corresponding hexagonal layeredmaterial as κ = C11 h3 /12 and G = C11 h [7]. The values for C11 are 1060, 238 and 150 × 1010
erg/cm3 for C, MoS2 and WS2 , respectively [14]. The same scaling with h is found for thinﬁlms made from isotropic elastic material [12]. It only applies for coherent bending of the different layers; for incoherent bending, slip occurs between adjacent layers and overall bending
becomes easier. The bending rigidity then follows as κ ∼ C11 a3 (h/a) = C11 a2 h, where a isthe eﬀective thickness of a single layer [7]. It should be of the same order of magnitude, butsomewhat smaller than the interlayer distance (which is 3.4 Å for C and 6.2 Å for MoS2 andWS2 ). Note that the in-plane modulus G ∼ C11 a(h/a) = C11 h stays the same since it scales
linearly with thickness. The change from coherent to incoherent bending can be considered to
be defect-mediated; dislocations will proliferate for increasing thickness and ﬁnally will leadto grain boundaries which account for the typical faceted shape seen in ﬁg. 2 [7].

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U. S. Schwarz et al.: Deformation and tribology etc.

h

h
R

R'

r
H

H

(a)

(b)

Fig. 3 – Possible deformations of a spherical elastic shell on contact with a substrate: (a) ﬂattening and(b) contact disc. R is the initial particle radius, R the radius after ﬂattening, h the shell thickness, H
the indentation and r the radius of the contact disc. For small H, one ﬁnds geometrically r2 = 2HR .

Adhesion. – We consider a substrate which interacts by attractive vdW forces with aﬁlm of thickness h which is a distance D away from the substrate. Pairwise integration of thepotential −A/π 2 r6 (where the Hamaker constant A is typically of the order of 10−12 erg) over
the two volumes yields the vdW energy per unit areau=

A h(h + 2D).
12π D2 (h + D)2(1)

Here D is an atomic cutoﬀ for the vdW interaction which in the following is chosen to be
1.65 Å [15]. Although the adhesion energy u scales linearly with h for small h, for more than
one layer we have h > D and u saturates at a constant value u = A/12πD2 ≈ 100 erg/cm2 .
For the case of a single layer, one can replace the volume integral over the ﬁlm by a surface
integral times the eﬀective thickness h; this is equivalent to considering h  D in eq. (1) andleads to u = Ah/6πD3 . Since D and h are of the same order of magnitude, we essentially
recover the result for a multi-layered ﬁlm. Hence we conclude that due to the rapid saturationof the vdW binding energy with increasing h, for multi-layered ﬁlms under adhesion conditions
no considerable scaling with h is expected from the vdW terms.
It can be shown by similar arguments (e.g., by using eq. (1) within the Derjaguin approximation) that the energy of a hollow nanoparticle adhering to a substrate does not depend> D. Then the adhesion energy equals the vdW energy of close approachon h as long as h ∼between a sphere and a substrate, EA = AR/6D [15]. Using values for A, D and R as given
above yields EA = 6 × 10−11 erg = 1400 kT. The energy of adhesion between two particles
is only a factor of 2 smaller than the one between a particle and the substrate, and hence
well above thermal energies as well. We thus conclude that the vdW interaction leads toconsiderable adhesion of the particles to the substrate and to each other. The same holds true
for peeled oﬀ outer layers: with R = 60 nm, the energy of adhesion can be estimated to be
R2 u = 105 kT.
Deformation under adhesion. – The onset of delamination can be estimated by considering large deformations of the nanoparticles. We do not oﬀer a theory for fracture and estimate
the limit of mechanical stability by an internal criterion within our continuum description:
fracture and delamination sets in when so much stress has accumulated that the deformation
becomes of the order of the radius. Determining the deformation of a spherical elastic shell
is a diﬃcult problem which depends on prestress, elastic constants and size as well as on thenature of the deforming force. Here we discuss the deformation on a substrate for two cases
which can be treated in the framework of our scaling approach. In the following H is theindentation. For small deformations H < h (ﬁg. 3a), the shell ﬂattens at the bottom. For
large deformations H > h (ﬁg. 3b), a contact disc with radius r develops and the elastic

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energy is localized in the circular fold surrounding it. The respective elastic energies are [12]Esmall ∼G1/2 κ1/2 2
H ,R

Elarge ∼G1/4 κ3/4 3/2
H
.
R

(2)

Here we neglect a bending term ∼ κH/R. We also assumed that due to low friction, slip canoccur between the contact disc and the substrate; otherwise compression energy ∼ GH 3 /R
would accumulate in the contact disc. The rest of the relaxing shell is assumed to keep aspherical form with radius R > R in order to keep area constant and to avoid compression
energy. However, the diﬀerence is (R − R) ∼ H 2 /R and can be shown to be negligible in thefollowing.
If the deformations are driven by vdW adhesion, the energy gained on adhesion can beshown to scale as EA ∼ uRH for both small and large deformations. Setting this adhesion energy equal to the estimates for the elastic energies from eq. (2) yields estimates forindentation: 2  3
u
u
R
R
R2 u
R 4 u2R . (3)
Hsmall ∼ 1/2 1/2 ∼R, Hlarge ∼ 1/2 3/2 ∼
C11 h
h
C11 h
h
G κ
G κ
Here and in the following we will always give the result both in terms of the elastic moduli
and their scaling with thickness for coherent bending. The dimensionless quantity u/C11 h
is the ratio of van der Waals adhesion energy to two-dimensional Young modulus and will
be of order 10−4 for the WS2 nanoparticles. In this case we are in the regime H < h anddeformations will be in the Angstrom range. Thus delamination cannot be expected to becaused by adhesion. For incoherent bending, R/h has to be replaced by R/a in both regimes.
One system for which vdW adhesion leads to strong deformations are single-walled hollow
nanoparticles. In this case, care has to be taken to use the correct values for G and κ. Since
G is a purely two-dimensional quantity, G = C11 h can be used where h is identiﬁed with theinterlayer distance. However, κ has to be extracted from molecular calculations [5,10,11]. For
carbon, one ﬁnds G = 3.6 × 105 erg/cm2 and κ ≈ 1.6 × 10−12 erg ≈ 40 kT. For both MoS2 andWS2 , G is smaller by a factor 4 and κ is larger by a factor 10. In all three cases, for radii larger
than a few nm considerable deformations arise. The critical radius Rc where indentation H
and radius R become of the same magnitude follows from the second relation of eq. (3) as
Rc ∼ G1/6 κ1/2 u−2/3 . For C, we ﬁnd Rc ≈ 5 nm and for the metal disulphides Rc ≈ 10 nm.
For nanotubes, we ﬁnd similar results which are in good agreement with experiments andmolecular calculations [16, 17].
Deformation under pressure. – We now proceed to show that pressure can lead to an
instability which could account for the observed delamination. We consider the formation ofa contact disc for a shell which is pressed onto the substrate by pressure in the surrounding
liquid. The corresponding energy is Ep ∼ −pRH 2 , thus it scales more strongly with H than
the restoring elastic energy Elarge in eq. (2). This scaling indicates an instability: small
deformations are suppressed, but large deformations grow without limits beyond the criticalindentation Hc where Elarge + Ep attains a maximum [12]. Setting Hc = R gives an estimate
for the critical pressure for delamination:hG1/4 κ3/4
∼ C11
pc ∼
5/2
R
R



h
R

3/2
.

(4)

Using typical values gives pc ≈ 1.7 GPa. Incoherent deformations can be treated by replacing(h/R)3/2 by (a/R)3/2 in eq. (4) and hence would decrease the estimate to 25 MPa. AlthoughU. S. Schwarz et al.: Deformation and tribology etc.

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a full treatment of the instability requires a theory for the non-linear elasticity, we can stillsuggest that pressure will be a very likely mechanism to cause delamination. This prediction
agrees nicely with the experimental observation that damage sets in at a ﬁnite threshold of theload [3]. Note that for single-walled buckyballs C60 with R = 3.55 Å, eq. (4) gives pc = 15 GPa
which is in surprisingly good agreement with the value 20 GPa found experimentally [18].
Although the pressure values estimated here seem to be high, they refer to particles of
nm-size, and the corresponding forces are of the order of 10−3 dyn. In friction experiments,
the pressures and forces needed for delamination are most likely to occur near asperities.
This might explain why delamination has been observed in the SFA only when shearing thetwo surfaces [4]: the nanoparticles then behaves as granular material which gets jammed andpressure and forces become localized.
Shear ﬂow. – Since the WS2 nanoparticles are immersed in liquid lubricant in strong
motion, we ﬁnally consider the eﬀect of shear ﬂow. Balancing the deforming viscous force
with the elastic restoring force and setting the resulting deformation equal to the radius gives
an estimate for the shear rate at which delamination sets in. For h < R, we ﬁnd that only
stretching is relevant and that the critical shear rate is
γ̇c ∼

C11 h
G
∼
.
ηR
η R

(5)

Using viscosity η ≈ 1 cP (which applies both to water and mineral oil), we ﬁnd γ̇c ∼ 2.5 ×
1013 Hz. Such high shear rates are unattainable even near contacting asperities in macroscopic
friction experiments, thus delamination is not likely to occur due to shear ﬂow alone.
Shear ﬂow past nanoparticle adhering to a substrate can cause them to roll. The particle
begins to roll if the viscous drag evaluated at the midpoint, FS = 6πηR2 γ̇, equals the friction
force FF caused by the adhesive load FA = 2πRu. Assuming Amontons’ law for rolling friction,
FF = µFA , we ﬁnd
µu
,
(6)
γ̇c =
3ηR
where the coeﬃcient of rolling friction µ can be taken to have the typical value 10−3 ; thenγ̇c is of the order of 5 × 105 Hz. This value is higher than typical shear rates probed with
the SFA (v = 10 µm/s, γ̇ = 104 Hz on nm-separation), but certainly lower than shear rates
occurring near the asperities of macroscopic friction experiments (v = 1m/s, γ̇ = 109 Hz on
nm-separation).
Conclusion. – We have shown that in friction experiments with multi-walled hollow
nanoparticles, delamination is likely to be caused by pressure. When nanoparticles delaminate, dangling bonds develop as they do for powders from layered material. However, there
are two reasons why they should be less problematic. First delamination takes place with
one or two layers and the density of dangling bonds remains low. Second we showed thatdelamination may occur under adhering conditions. The adhesion (and possibly the shear)
will cause the layers to align parallel to the substrate, in contrast to the powders where theorientation is preferentially perpendicular to the substrate. It has been shown experimentally that delaminated layers from nanoparticles form islands while platelets form a thicker,
more disorganized ﬁlm [4]. Our scaling approach shows that there are two control parameterswhich might be used for tribological optimization: the ratio of radius to thickness and theconcentration of defects (that determine the regimes of coherent and incoherent bending).
Using nanoparticles as additives to liquid lubricants raises interesting questions about therole of wetting. The lubricant used in the friction experiments mentioned is a bad solvent

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for the nanoparticles, which provides an additional force to drive them onto the substrate. A
dewetted region is expected to form between the adhering particle and the substrate; like acapillary bridge in a good wetting situation, it might enhance adhesion.
∗∗∗
We thank C. Drummond, Y. Golan, J. Israelachvili, J. Klein, G. Seifert and especially R. Tenne for helpful discussions. R. Tenne kindly provided the electron micrographshown in ﬁg. 2. SAS thanks the Gerhard M. J. Schmidt Minerva Center for SupramolecularArchitecture for support. SK thanks the Ministry of Education, Science and Culture, Japanfor providing ﬁnancial support during his visit to Israel. USS gratefully acknowledges support
by the Minerva Foundation.
REFERENCES[1] Singer I. L. and Pollock H. M., Fundamentals of Friction: Macroscopic and Microscopic
Processes (Kluwer, Dordrecht) 1992.
[2] Rapoport L., Bilik Y., Feldman Y., Homyonfer M., Cohen S. R. and Tenne R., Nature,387 (1997) 791.
[3] Rapoport L., Feldman Y., Homyonfer M., Cohen H., Sloan J., Hutchison J. L. andTenne T., Wear, 225-229 (1999) 975.
[4] Golan Y., Drummond C., Homyonfer M., Feldman Y., Tenne R. and Israelachvili J.,
Adv. Mat., 11 (1999) 934; Drummond C., Alcantar N., Israelachvili J., Tenne R. andGolan Y., preprint (1999).
[5] Tersoff J., Phys. Rev. B, 92 (1992) 15546.
[6] Witten T. A. and Li H., Europhys. Lett., 23 (1993) 51.
[7] Srolovitz D. J., Safran S. A. and Tenne R., Phys. Rev. E, 49 (1994) 5260.
[8] Srolovitz D. J., Safran S. A., Homyonfer M. and Tenne R., Phys. Rev. Lett., 74 (1995)
1779.
[9] Adams G. B., Sankey O. F., Page J. B., O’Keeffe M. and Drabold D. A., Science, 256(1992) 1792.
[10] Hernandez E., Goze C., Bernier P. and Rubio A., Phys. Rev. Lett., 80 (1998) 4502; Seifert
G., Terrones H., Terrones M., Jungnickel G. and Frauenheim T., preprint (1999).
[11] Yakobson B. I., Brabec C. J. and Bernholc J., Phys. Rev. Lett., 76 (1996) 2511.
[12] Landau L. D. and Lifshitz E. M., Theory of Elasticity (Pergamon Press, Oxford) 1970.
[13] Komura S. and Lipowsky R., J. Phys. II, 2 (1992) 1563; Zhang Z., Davis H. T. and Kroll
D. M., Phys. Rev. E, 48 (1993) R651.
[14] Landolt and Börnstein, Units and Fundamental Constants in Physics and Chemistry(Springer, Berlin) 1991.
[15] Israelachvili J. N., Intermolecular and Surface Forces (Academic Press, London) 1992.
[16] Ruoff R. S., Tersoff J., Lorents D. C., Subramoney S. and Chan B., Nature, 364 (1993)514.
[17] Our calculation for nanotubes is not given here since deformations of cylindrical shells areconceptually diﬀerent from the ones of spherical shells: for spherical shells, the balance ofstretching and bending leads to localization of deformation. For cylindrical shells, stretching
can be avoided and non-local bending has to be considered.
[18] Nunez-Regueiro M., Mod. Phys. Lett. B, 6 (1992) 1153.