www.sciencemag.org SCIENCE VOL 326 16 OCTOBER 2009 375
PERSPECTIVES
during chain packing act as if they are large
hollow monomers that can pack together to
form a jammed (shear-resisting) confi gura-
tion. A two-dimensional simulation of granu-
lar chains at low chain density, where the sys-
tem is unjammed (has no resistance to shear)
(see the fi gure, panel A), contains randomly
configured chains. At high chain density
when the system is jammed (see the fi gure,
panel B), semiloop structures appear. Zou et
al. revealed that semiloop formation is cru-
cial to the behavior of the granular chain sys-
tem. The presence of semiloops lowers the
jamming density of the system, and this effect
is more pronounced in longer chains where
there are fewer chains per unit volume.
Real polymer systems undergo a transi-
tion to a glassy state below some temperature.
It is known that this glass transition tempera-
ture increases with polymer length and satu-
rates for long polymers ( 4). In the granular
chain polymer model system, there is no tem-
perature; instead, the inverse density serves
as a quantity analogous to temperature. When
Zou et al. compare the behavior of the inverse
density of the granular chains as a function of
chain length to the glass transition tempera-
ture of a real polymer as a function of poly-
mer length, they fi nd that the behaviors are
nearly identical. This remarkable result could
imply that the glass transition in real polymers
might be a jamming transition governed by
the formation of rigid polymer semiloops. It
would be extremely interesting to determine
whether this type of semiloop actually occurs
in real polymer systems. Semiloops should
form much more easily than complete loops
because the two ends of the polymer do not
need to be near each other. If semiloops play
a role in real polymer systems, this would be
one of the most beautiful examples of where
a macroscopic toy system explains the phys-
ics on a molecular scale. Such a result would
also confi rm that the ideas of jamming origi-
nally proposed for granular assemblies ( 5– 7)
are applicable to a wide variety of molecular
systems.
Granular polymers also represent a new
form of granular matter and most likely will
exhibit new kinds of phenomena that do not
occur in real polymers or in granular systems
composed of individual grains. As the jam-
ming density of the granular polymer sys-
tem can be controlled readily by adjusting
the length of the granular chains, introducing
chains into other systems that exhibit jam-
ming may allow for a much more precise con-
trol of the jamming of these materials.
The experimental demonstration by Zou et
al. that granular polymers can give insight into
molecular polymer systems potentially opens
Observing Monopoles in
a Magnetic Analog of Ice
PHYSICS
Michel J. P. Gingras
Experimental evidence has been found that magnetic poles within metal oxide magnets can be
separated.
A bar magnet has a north and south
pole, and cutting it in half just cre-
ates two new poles, not two sepa-
rated monopoles. However, a recent theo-
retical proposal suggested that defects in the
spin alignment of certain oxide magnets can
create separated effective magnetic mono-
poles ( 1). These materials are called spin ices
because the lowest-energy orientation of the
magnetic spins closely mimics the most sta-
ble arrangement of protons in water ice ( 2).
On pages 415 and 411 of this issue, Fennell
et al. ( 3) and Morris et al. ( 4) report measure-
ments from neutron-scattering experiments
showing that the low-energy excitations in
spin ices are reminiscent of Dirac’s elemen-
tary magnetic monopoles ( 5) that have so far
eluded the searches of high-energy physicists.
These dissociated north and south poles dif-
fuse away from each other ( 6) in these oxides
and leave behind a “Dirac string” of reversed
spins that can be seen as patterns in the inten-
sity of scattered neutrons.
The materials studied, holmium titan-
ate (Ho2Ti2O7) ( 3) and dysprosium titanate
(Dy2Ti2O7) ( 4), are geometrically frustrated
ferromagnets ( 7, 8); they are unusual para-
magnets with strong spin-spin correlations
( 8) that become magnetized in a magnetic
fi eld. These oxides have a pyrochlore struc-
ture; the Ho3+ and Dy3+ ions within these crys-
tals form a lattice of corner-sharing tetrahe-
dra (see the fi gure, panel A). The magnetic
moments of Ho3+ and Dy3+ act like two states,
or Ising model spins, and are constrained by
anisotropic forces to point “in” or “out” of the
tetrahedra. The minimum energy condition,
or rule, is that there must be two spins point-
ing “in” and two spins pointing “out” on each
tetrahedron ( 7, 8).
The “two-in, two-out” rule is analogous to
the Bernal-Fowler ice rule, which states that
for energetic reasons, two protons must be
“close” and two protons must be “far” from
any given oxygen in common water ice ( 2).
As explained by Pauling ( 2), the very large
number of two-in, two-out confi gurations in
a macroscopic ice sample leads to a measur-
able residual entropy in ice at low tempera-
tures ( 9, 10). Correspondingly, Ho2Ti2O7 and
Dy2Ti2O7 also exhibit a residual low-temper-
ature “Pauling entropy” ( 11).
A tetrahedron fulfi lling the two-in, two-
out rule amounts to an effective “magnetic
charge neutrality” where four (two positive,
two negative) magnetic charges cancel out
at the center of the tetrahedron ( 1). A spin
fl ipped by thermal fl uctuations creates two
defective adjoining tetrahedra—a monopole-
antimonopole pair (see the fi gure, panel B).
Once formed, these particles can diffuse away
from each other by reversing spins along the
path they trace as they separate, which recon-
Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario N2L 3G1, Canada, and Cana-
dian Institute for Advanced Research/Quantum Materi-
als Program, Toronto, Ontario MSG 1Z8, Canada. E-mail:
a new avenue of research for polymer sci-
ence. For example, polymer systems under
shear often exhibit very complex dynamics.
Perhaps by shearing granular polymers, the
microscopic motions of the shearing pro-
cess can be studied directly, along with other
analogous phenomena such as entanglement
and polymer mechanical properties ( 1, 2).
One can imagine going even further and cre-
ating increasingly intricate granular poly-
mers to model biological polymers, such as
proteins or DNA.
References
1. M. Rubenstein, R. H. Colby, Polymer Physics (Oxford
Univ. Press, New York, 2003).
2. P. De Gennes, Scaling Concepts in Polymer Physics (Cor-
nell Univ. Press, Ithaca, NY, 1979).
3. L.-N. Zou, X. Cheng, M. L. Rivers, H. M. Jaeger, S. R.
Nagel, Science 326, 408 (2009).
4. T. J. Fox, P. J. Flory, J. Appl. Phys. 21, 581 (1950).
5. A. J. Liu, S. R. Nagel, Nature 396, 21 (1998).
6. C. S. O’Hern, L. E. Silbert, A. J. Liu, S. R. Nagel, Phys. Rev.
E Stat. Nonlin. Soft Matter Phys. 68, 011306 (2003).
7. J. A. Drocco, M. B. Hastings, C. J. Olson Reichhardt, C.
Reichhardt, Phys. Rev. Lett. 95, 088001 (2005).
10.1126/science.1180973
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