brief communications

Euler’s disk and its finite-time singularity

I
t is a fact of common experience that if a ez
ed
circular disk (for example, a penny) is and make the ‘adiabatic’ assumption that
spun upon a table, then ultimately it c z equation (1) continues to hold. Because the
comes to rest quite abruptly, the final stage air moves a distance of order a in a time
of motion being characterized by a shudder Ω 2p /V, the horizontal velocity uH in the layer
and a whirring sound of rapidly increasing has order of magnitude rV sinf ; and as this
e
frequency. As the disk rolls on its rim, the velocity satisfies the no-slip condition on
point P of rolling contact describes a circle z40 and on z4h (4O(a a)), the vertical
with angular velocity V. In the classical a
O shear |!uH /!z| is of the order (rV /a a)|sinf|.
(non-dissipative) theory1, V is constant and α r z=0
The rate of viscous dissipation of energy F
the motion persists forever, in stark conflict is given by integrating m(!uH/!z)2 over the
with observation. Here I show that viscous P volume V of the layer of air: this easily gives
acosα
dissipation in the thin layer of air between Föpmga2/a2, using equation (1). The fact
the disk and the table is sufficient to that F →÷ as a →0 should be noted.
account for the observed abruptness of the The energy E now satisfies dE/dt41F
settling process, during which, paradoxical- Figure 1 A heavy disk rolls on a horizontal table. The point of (neglecting all other dissipation mecha-
ly, V increases without limit. I analyse the rolling contact P moves on a circle with angular velocity V. Owing nisms). Hence, with E given by equation
nature of this ‘finite-time singularity’, and to dissipative effects, the angle a decreases to zero within a finite (2), it follows that
show how it must be resolved. time and V increases in proportion to a11/2.
_3 Mgada /dtö1pmga2/a 2 (3)
2
Let a be the angle between the plane of This integrates to give
the disk and the table. In the classical v4v üe41V sina. a 342p (t01t)/t1 (4)
description, and with the notation defined Euler’s equation for the motion of a where t14M/ma, and t0 is a constant
in Fig. 1, the points P and O are instanta- rigid body is here given by d h/dt4 of integration determined by the initial
neously at rest in the disk, and the motion is V ` h4G, where G4Mgaez ` e is the grav- condition: if a4a 0 when t40, then
therefore instantaneously one of rotation itational torque relative to P (` indicates t04(a30 /2p)t1. What is striking here is that,
about line PO with angular velocity v, the vector product). This immediately gives according to equation (4), a does indeed go
say. The angular momentum of the disk is the result V 2sina44g/a, or, when a is to zero at the finite time t4t0. The corre-
therefore h4Av e(t), where A4_41 Ma2 is the small, sponding behaviour of V is Vö(t01t)11/6,
moment of inertia of the disk of mass M V 2a ö4g/a (1) which is certainly singular as t→t0.
about its diameter; e(t) is a unit vector The energy of the motion E is the sum Of course, such a singularity cannot be
in the direction PO; ez, ed are unit vectors of the kinetic energy _12 Av24_12 Mgasin a, realized in practice: nature abhors a singu-
in the directions Oz, OC, respectively (see and the potential energy Mgasin a, so larity, and some physical effect must inter-
Fig. 1). In a frame of reference rotat- E 4_32 Mgasin a ö _32 Mga a (2) vene to prevent its occurrence. Here it is not
ing with angular velocity V d4V ez, the disk In the classical theory, a, V and E are all difficult to identify this effect: the vertical
rotates about its axis OC with angular constant, and the motion continues indefi- acceleration |h ˙˙ |4|aȧ˙| cannot exceed g in
velocity V d4V d ed ; hence the nitely. As observed above, this is utterly magnitude (as the normal reaction at P
rolling condition is V d4V cos a. The unrealistic. must remain positive). From equation (4),
absolute angular velocity of the disk is thus Let us then consider one of the obvious this implies that the above theory breaks
v 4V (ed cosa1ez), and so mechanisms of energy dissipation, namely down at a time t before t0, where
that associated with the viscosity m t 4t0 1t ö(2a/9g)3/5(2p/t1)1/5 (5)
of the surrounding air. When A toy, appropriately called Euler’s
a is small, the dominant disk2, is commercially available (Fig. 2; Tan-
contribution to the gent Toys, Sausalito, California). For this
viscous dissipation disk, M4400 g, and a43.75 cm. With
comes from the these values and with m 41.7821014 g
layer of air cm11 s, t14M/maö 0.82106 s, and, if we
between the take a040.1(ö6°), we find t0 ö100 s. This
disk and the is indeed the order of magnitude (to within
table, which is 520%) of the observed settling time in
subjected to many repetitions of the spinning of the disk
strong shear (with quite variable and ill-controlled initial
when V is large. conditions), that is, there is no doubt that
We may estimate dissipation associated with air friction is
the rate of dissipation sufficient to account for the observed
of energy in this layer as behaviour. The value of t given by equation
follows. Let (r,u) be polar (5) is 1012 s for the disk values given above;
coordinates with origin at O. For that is, the behaviour described by equation
Figure 2 Euler’s disk is a chrome- small a, the gap h(r,u,t) between the (4) persists until within 1012 s of the singu-
plated steel disk with one edge machined to a smooth radius. If it disk and the table is given by larity time t0. At this stage, aö0.521012,
were not for friction and vibration, the disk would spin for ever. h(r,u,t)öa (a&rcos f), where f4u 1V t. h04a a ö0.2 mm, V ö500 Hz (and the
Photo courtesy of Tangent Toys. See http://www.tangenttoy.com/. We now concede that a is a slowly varying adiabatic approximation is still well
NATURE | VOL 404 | 20 APRIL 2000 | www.nature.com © 2000 Macmillan Magazines Ltd 833
brief communications
satisfied). This is as near to a ‘singularity’ as 20 Clarkson Road, Cambridge CB3 0EH, UK to a potential that is significantly modulat-
this simple toy can approach. The effect is e-mail: hkm2@newton.cam.ac.uk ed, creating a sequence of barriers for the
nevertheless striking in practice. 1. Pars, L. A. Treatise on Analytical Dynamics (Heinemann, hole carriers (Fig. 1b). Similar SGPI mea-
London, 1965).
H. K. Moffatt 2. Euler, L. Theoria Motus Corporum Solidorum Seu Rigidorum
surements on metallic tubes did not show
Isaac Newton Institute for Mathematical Sciences, (Greifswald, 1765). any contrast.
The effect of increased bias voltage is
illustrated in the lower panels of Fig. 1d,e.
The emphasis of the dot pattern appears to
Molecular transistors distance above the device surface, we scan shift towards the electrode with the lower
the tip of an atomic-force microscope potential. Existing dots vanish (particularly
Potential modulations (AFM), on which a positive tip voltage, Vt, clear for very large bias; see Fig. 1e) and new
along carbon nanotubes is applied. The AFM tip acts as a local gate
and induces a local potential barrier (Fig.
dots appear (bottom right of Fig. 1d; bot-
tom left of Fig. 1e) that were not present for
low bias. These trends may be due to the

T
rue molecular-scale transistors have 1b) when it is above the tube. This can
been realized using semiconducting reduce the hole current through the nano- effective gate voltage near the right and left
carbon nanotubes1–5, but no direct tube, which is recorded as a function of the electrodes being different at higher bias
measurements of the underlying electronic tip position (Fig. 1c). The main point of voltages. Contrast also diminishes when the
structure of these have been made. Here we SGPI is that the current reduction depends tube potential approaches the tip voltage.
use a new scanning-probe technique to on the original local potential of the tube. The electronic properties of semicon-
investigate the potential profile of these In this way, SGPI maps out the local poten- ducting nanotubes have been proposed to
devices. Surprisingly, we find that the tial profile of the nanotube. be sensitive to perturbations by local disor-
potential does not vary in a smooth, The top images in Fig. 1d,e show regular der2,6–8. Our results confirm the occurrence
monotonic way, but instead shows marked AFM images of two different samples with of such electronic disorder by direct spatial
modulations with a typical period of about their corresponding SGPI measurements at images. The microscopic origin of this dis-
40 nm. Our results have direct relevance for different bias voltages underneath. The order is still unclear, however. The most
modelling this promising class of molecular most remarkable feature of these images is likely causes are localized charges near the
devices. that they show a sequence of current dips nanotube, or mechanical deformations.
The principle of our scanning-gate along the tubes. The dips appear to be con- Detailed height measurements by AFM did
potential-imaging (SGPI) technique is as fined to the region between the electrodes. not reveal any correlation between height
follows. An individual semiconducting sin- Surprisingly, they are rather evenly spaced, and electronic features.
gle-wall carbon nanotube is connected to with a distance of about 40 nm for the sam- Our results shed a new light on other
two metal electrodes, and this transistor is ple in Fig. 1d, and 36 nm for the sample in reported transport data. Step features have
switched to a conducting state by a negative Fig. 1e. These observations indicate that the been found in current–voltage (I–V) curves
gate voltage, Vg (Fig. 1a). A current flows edge of the valence band does not vary in a of TUBEFET devices (ref. 3, and S. J. T. et
when a bias voltage, Vb, is applied. At a close smooth monotonic way. Instead, they point al., unpublished results). Our findings may
explain these observations because an
increasing bias voltage can bring down the
potential barriers one at a time, leading to
step-like features. Reported asymmetries in
I–V curves1,3,4 can now be corroborated by
the asymmetries in the potential profile
along tubes at low bias. In conduction
experiments at low temperatures (ref. 8,
and Z. Yao et al., unpublished results), phe-
nomena related to multiple metallic islands
have been observed, which can be explained
by the barrier sequence seen in our SGPI
images. Near the tube on top of the elec-
trodes, no contrast could be found in the
SGPI images, even for large tip voltages (up
to ±3 V). This indicates that Schottky barri-
ers do not exist at the metal interface, as was
suggested earlier4.
Figure 1 Scanning-gate potential imaging (SGPI) along a semiconducting carbon nanotube. a, SGPI measurement set-up. Variable bias New scanning techniques that give a
voltages, Vb, and a gate voltage, Vg, of 16 V are applied to the TUBEFET device. An atomic-force microscope (AFM) tip at 500 mV is direct view of the potential landscape, such
scanned at a constant height of about 10 nm above the surface by retracing each line taken in regular tapping mode AFM while setting a as the one presented here, provide a
certain height offset and the cantilever amplitude to zero. b, Potential landscape of the device. In the conducting state, the valence band promising starting point for a better under-
edge is horizontal and pinned to the edge of the Fermi level of the electrode1,9. The tip voltage creates a potential dip (yellow) which pro- standing of the electronic structure of
vides a probe for the local potential. SGPI measurements (d,e) show that the band edge of the nanotube is not smooth but strongly mod- nanotube devices. It should, for example, be
ulated. c, Corresponding SGPI measurement. The device current (colour) is displayed as a function of tip position. Current is reduced feasible to study the effect of deliberate
when the AFM-tip-induced barrier aligns with minima in the original potential profile. The spatial resolution of the SGPI measurements, bending of nanotubes, different substrate
which we estimate to be of the order of 10 nm, is determined by the tip–sample distance and the tip radius. d, AFM image of the first and electrode materials, and the different
sample and the corresponding SGPI images for Vb values of 110, 1100, 1500 and 1750 mV (top to bottom). The sample consists of geometry of devices such as intramolecular
an individual single-wall carbon nanotube (horizontal line) on top of two 25-nm-high platinum electrodes (on the left and right) that are kinked-nanotube diodes5 and nanotube
spaced by 650 nm. e, AFM image of a second sample and the corresponding SGPI images for Vb values of 400, 500, 700 and 1,000 mV crossings.
(top to bottom). The sample consists of an individual single-wall carbon nanotube on top of two 750-nm-spaced gold electrodes. The Sander J. Tans*†, Cees Dekker*
electrodes of this sample are embedded in the SiO2 substrate to create a flat surface5 (see a). *Delft University of Technology, Department of

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