S. CUENOT et al.
PHYSICAL REVIEW B 69, 165410 ͑2004͒
pended nanomaterials are deformed, the surface tension is
solicited. It should be noted that there are very few experi-
mental measurements of surface tension and most values re-
ported in the literature arise from theoretical calculations.
These calculations predict that surface tension values are of
the same order of magnitude and slightly higher than the
surface energy values.14
Wassermann and Vermaak have experimentally deter-
mined the surface tension value for silver from the measure-
ment of the lattice contraction in small Ag spheres as a func-
tion of their radius by electron diffraction.35 The obtained
value for the surface tension was equal to 1.41Ϯ0.30 J mϪ2
at 55 °C. This value is similar to the present result (3.09
Ϯ0.33 J mϪ2). To our knowledge, no data are available for
the surface tension of Pb and PPy. It can, however, be no-
ticed that the obtained surface tension values (0.98Ϯ0.21
and 0.33Ϯ0.01 J mϪ2, respectively, for Pb and PPy͒ are
comparable to published values of the surface energy ͓be-
tween 0.5 and 0.6 J mϪ2 for Pb ͑Ref. 36͒ and 0.145 J mϪ2 for
PPyCl ͑Ref. 37͒ films͔.
ever, expect that only small numerical factors will modify
this ratio, leading to comparable geometrical effects.
The ratio between the apparent elastic modulus of the
smallest nanomaterial and the corresponding macroscopic
modulus of the material is much larger for the PPy nanotubes
than for the metallic nanowires. In both cases, surface effects
explain the increase of the apparent elastic modulus. The
larger effect in the case of PPy nanotubes is essentially due
to the intrinsic lower elastic modulus of the material rather
than to the additional inner surface existing in nanotubes.
Though defect concentration effects cannot be ruled out to
explain the apparent modulus increase observed in our ex-
perimental data, we suggest that surface tension is mainly
responsible for the observed increase.
In conclusion, the elastic modulus of metallic nanowires
and polymer nanotubes with diameters ranging between 30
and 250 nm was measured using resonant-contact AFM. For
the smaller diameters, the measured elastic modulus signifi-
cantly differs from that of the bulk materials. Calculation of
an apparent elastic modulus taking into account the surface
effect shows that the observed increase of the elastic modu-
lus with decreasing diameter is essentially due to surface
tension effects. This model allows the calculation of the in-
trinsic elastic modulus and the surface tension of the probed
material from the measured apparent modulus. For Ag
nanowires a fairly good agreement is obtained with the val-
ues published in the literature. In summary, we showed that,
in the case of metallic nanowires and polymer nanotubes, the
increase of the surface to volume ratio with decreasing size
strongly influences the measured modulus and that AFM
measurements of it enables the evaluation of solid surface
tension.
From relations ͑2͒ and ͑5͒, it is possible to derive a ratio
that could be used to predict the onset of the surface tension
effects. For nanowires, this ratio between the surface stiff-
ness ks and tensile stiffness kt can be expressed as
ks 8 ␥ L2
kt 5 E D3
ϭ
1Ϫ͒.
͑12͒
͑
When this ratio is larger than 1, surface tension effects
prevail. It is important to notice the dependence of this ratio
on the geometrical dimensions of the probed nanostructure.
Depending on the suspended length, surface effects may
arise for different diameters. Moreover, this ratio is estab-
lished for a specific geometry of solicitation ͑clamped ends,
central solicitation͒ and therefore it is not an intrinsic mate-
rial quantity. In others tests, surface tension effects could
show up for different geometrical conditions. We can, how-
´
The authors gratefully acknowledge Dr. J.-P. Aime and
Professor D. Johnson for invaluable discussions. They ac-
knowledge the FRFC and IUAP-V-P03/11 program for fi-
nancial support. S.C. was financially supported by the FSR
of the UCL. B.N. and S.D.C. are supported by the FNRS.
*
Corresponding author. Electronic address: nysten@poly.ucl.ac.be
10 R. E. Miller and V. B. Shenoy, Nanotechnology 11, 139 ͑2000͒.
11 P. Sharma, S. Ganti, and N. Bhate, Appl. Phys. Lett. 82, 535
͑2003͒.
1
´
C. Basire and C. Fretigny, Eur. Phys. J.: Appl. Phys. 6, 323
͑1999͒.
2 F. Dinelli, S. K. Biswas, G. A. D. Briggs, and O. V. Kolosov,
Phys. Rev. B 61, 13 995 ͑2000͒.
12 B. S. Altan, H. A. Evensen, and E. C. Aifantis, Mech. Res. Com-
mun. 23, 35 ͑1996͒.
3 E. Tomasetti, R. Legras, and B. Nysten, Nanotechnology 9, 305
͑1998͒.
13 S. Papargyri-Beskou, K. G. Tsepoura, D. Polyzos, and D. E.
Beskos, Int. J. Solids Struct. 40, 385 ͑2003͒.
4 F. Oulevey, G. Gremaud, A. Semoroz, A. J. Kulik, N. A. Burn-
ham, E. Dupas, and D. Gourdon, Rev. Sci. Instrum. 69, 2085
͑1998͒.
14 R. C. Cammarata, Prog. Surf. Sci. 46, 1 ͑1994͒.
15
´
C. Fretigny ͑unpublished͒.
16
´
S. Cuenot, C. Fretigny, S. Demoustier-Champagne, and B. Nys-
5 J.-P. Salvetat, G. A. D. Briggs, J.-M. Bonard, R. R. Bacsa, A. J.
ten, J. Appl. Phys. 93, 5650 ͑2003͒.
¨
Kulik, T. Stockli, N. A. Burnham, and L. Forro, Phys. Rev. Lett.
17 K. Yamanaka, H. Ogiso, and O. V. Kolosov, Appl. Phys. Lett. 64,
178 ͑1994͒.
82, 944 ͑1999͒.
6 S. Cuenot, S. Demoustier-Champagne, and B. Nysten, Phys. Rev.
Lett. 85, 1690 ͑2000͒.
18 K. Yamanaka and S. Nakano, Appl. Phys. A: Mater. Sci. Process.
66, S313 ͑1998͒.
7 M.-F. Yu, O. Lourie, M. J. Dyer, K. Moloni, T. F. Kelly, and R. S.
Ruoff, Science 287, 637 ͑2000͒.
19 P. Vairac and B. Cretin, Surf. Interface Anal. 27, 588 ͑1999͒.
20 U. Rabe, K. Janser, and W. Arnold, Rev. Sci. Instrum. 67, 3281
͑1996͒.
8 P. Poncharal, Z. L. Wang, D. Ugarte, and W. A. de Heer, Science
283, 1513 ͑1999͒.
21 S. Demoustier-Champagne and P.-Y. Stavaux, Chem. Mater. 11,
829 ͑1999͒.
9 M. E. Gurtin and A. Murdoch, Arch. Ration. Mech. Anal. 57, 291
͑1975͒.
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