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Unformatted text preview: 990 REPORTS Fig. 2. Temperature dependence of the
thermal conductivity of the W/AlZC)3
nanolaminate deposited at 177°C when
8 = 2.9 nm (open circles). Data for a
fully dense amorphous AlZO3 film pre
pared by ion—beam sputtering (solid tri—
angles) (7) are included for comparison.
The dashed line is the calculated mini—
mum conductivity Amin for alumina; the
solid line is A :: SGDMM, where 8 a 2.9
rim and GDMM is the calculated conduc
tance of W/Alzo3 interfaces under the
diffuse mismatch model (19). Thermal conductivity (W m"1 K‘l) expected thermal conductivity, assuming a con
stant G a? 260 MW m"2 K" and ﬁxed values
for the thermal conductivities of the individual
alumina and W layers. For the AU) nanolami
nate with 1/6 = 0.35 nm" and a deposition
temperature of 177°C, the thermal conductivity
was almost wholly dominated by this interface
conductance. i.e.. A as 86. This experimental value for G is close to the
prediction of the diffuse mismatch model
(DMM) (19). in this model of interface thermal
transport, lattice vibrations are assumed to be
scattered strongly at the interface and to have a
transmission coefficient given by the ratio of the
densities of vibrational states on either side of the
interface. Using a Debye model for the densities
of states. we calculated that GDMM = 320 MW
m‘2 K“ for W/Al203. Typically, the DMM
overestimates the conductance near room tem
perature. because this model. when based on a
Debye density of states, does not take into ac
count the dispersion of the vibrational modes. We continued the comparison between our
data and the GDMM by examining the tempera—
ture dependence of the thermal conductivity
(Fig. 2). The solid line in Fig. 2 is A = bGDMM;
that is. the solid line shows the thermal conduc
tivity of a hypothetical nanolaminate in which 8
is 2.9 nm and the themial conductivity is domi
nated by the diﬁ‘bse mismatch value of the ther
mal conductance of the W/AIZO3 interfaces. The
temperature dependence of the data and GDMM
are similar, giving ﬂirthcr support to our asser
tion that thermal transport in the nanolaminates
is mostly controlled by the conductance of
the interfaces. Interfaces between dissimilar materials such
as W and A1203 are effective in reducing the
thermal conductivity of nanostmcmred materi
als. but the relatively high interface energy will
limit the stability of these materials at the high
service temperatures typically required of ther 50 100 500
Temperature (K) mal barrier coatings. Applications of nanolaini
hates as thermal barriers at temperatures higher
than l000°C would require the development of
material interfaces that satisfy the conﬂicting
demands of low thermal conductance and excep»
tional thermal stability. References and Notes 1. D. o. Cahill et 3L. j. Appl. Phys. 93, 793 (2003). 2. N. P. Padture, M. Gell. E. H. Jordan. Science 296, 280
{2002). 3. S. T. Huxtable et a!.,Appl. Phys. Lett. 80, 1737 (2002). 4. K. E. Goodson. 1. Heat Transfer 118, 279 (1996). S. C.~W. Nan. R. Birringer, D. R. Clarke, H. Gleiter,
1. Appl. Phys. 81. 6692 (1997). 6. R. J. Stoner. H. J. Maris, Phys. Rev. B 48. 16373 (1993). 7. S.M. Lee, D. G. Cahill, T. H. Allen, Phys. Rev. B 52, 253
(1995). B. D. P. H. Hasselman et al.. Am. Ceram. Soc. Bull. 66,
799 (1987). 9. K. W. Schlichting. N. P. Padture, P. G. Klemens, j.
Mater. Sci. 36. 3003 (2001). 10. G. Soyez et al.. Appl. Phys. Lett. 77. 1155 (2000). 11. Materials and methods are available as supporting
material on Science Online. 12. A. C. Dillon. A. W. Ott. J.'D. Way. S. M. George, Surf.
Sci. 322. 230 (1995). 13. J. W. Klaus. S. J. Ferro. S. M. George. Thin Solid Films
360, 145 (2000). 14. C A Paddodc G. L Eesley, 1. Appl. Phys. 60, 285 (1986). 15. D. A. Young, C. Thomsen, H. T. Grahn, H. J. Maris. J.
Tauc, in Phonon Scattering in Condensed Matter, A. C.
Anderson. J. P. Wolfe. Eds. (Springer, Berlin, 1986). p.
49—51. 16. D. G. Cahill. K. E. Goodson, A. Majumdar. 1. Heat
Transfer 124, 223 (2002). 17. R. M. Costescu, M. A Wall. D. G. Cahill. Phys. Rev. B
67. 54302 (2003). 18. A Feldman, High Temp. High Pressures 31, 293 (1999). 19. E. T. Swartz, R. O. Pohl, Rev. Mod. Phys. 61, 605
(1989). 20, Supported by NSF grant no. CTS0319235, U.S. De
partment of Energy (DOE) grant no. DEFGOZcOl
ER45938, and the Air Force Ofﬁce of Scientific Re
search. Sample characterization used the Laser Facil.
ity of the Salt: Materials Research Laboratory and the
facilities of the Center for Microanalysis of Materials.
which is partially supported by DOE under grant no.
DEFGOZQlER4S439, Supporting Online Material
sciencemag.org/cgilcontent/full/BOBISGSOIQBS/DC1
Materials and Methods References and Notes 17 November 2003; accepted 29 December 2003 Improving the Density of
Jammed Disordered Packings
Using Ellipsoids Aleksandar Donev,” Ibrahim Cisse.z'5 David Sachs,z
Evan A. Variano,2'6 Frank H. Stillinger.3 Robert Connelly.’
Salvatore Torquato,1’3"* P. M. Chaikinz'4 Packing problems, such as how densely objects can fill a volume. are among the
most ancient and persistent problems in mathematics and science. For equal
spheres. it has only recently been proved that the facecentered cubic lattice has
the highest possible packing fraction q; = ﬁ/V’Té z 0.74. It is also well known that
certain random (amorphous) jammed packings have to z 0.64. Here. we show
experimentally and with a new simulation algorithm that ellipsoids can randomly
pack more denselyup to tp Z 0.68 to 0.71 for spheroids with an aspect ratio close
to that of M&M's Candies—and even approach to a 0.74 for ellipsoids with other
aspect ratios. We suggest that the higher density is directly related to the higher
number of degrees of freedom per particle and thus the larger number of particle
contacts required to mechanically stabilize the packing. We measured the number
of contacts per particle 2 e 10 for our spheroids, as compared to Z k 6 for spheres.
Our results have implications for a broad range of scientiﬁc disciplines, including
the properties of granular media and ceramics, glass formation, and discrete geometry. The structure of liquids, crystals. and glasses
is intimately related to volume fractions of
ordered and disordered (random) hardsphere packings, as are the transitions between these
phases (1). Packing problems (2) are of cur
rent interest in dimensions higher than three 13 FEBRUARY 2004 VOL 303 SClENCE www.5ciencemag.org Downloaded from www.sciencemag.org on January 25, 2007 for insulating stored data from noise (3), and
in two and three dimensions in relation to
flow and jamming of granular materials (4—6)
and glasses (7). Of particular interest isyran—
dom packing, which relates to the ancient
(economically important) problem of how
much grain a barrel can hold. Many experimental and computational algo
rithms produce a relatively robust packing frao
tion (relative density) «4: ~— 0.64 for randomly
packed monodisperse spheres as they proceed to
their limiting density (8). This number, widely
designated as the random close packing (RCP)
density, is not universal but generally depends
on the packing protocol (9). RCP is an illde
ﬁned concept because higher packing fractions
are obtained as the system becomes ordered. and
a deﬁnition for randomness has been lacking. A
more recent concept is that of the maximally
random jammed (MR1) state, corresponding to
the least ordered among all jammed packings
(9). For a variety of order metrics. it appears that
the MR1 state has a density of tp % 0.637 and is
consistent with what has traditionally been
thought of as RCP (10). Henceforth. we refer to
this random form of packing as the MR] state. We report on the density of the MRI state
of ellipsoid packings as asphcricity is intro
duced. For both oblate and prolate spheroids.
q: and Z (the average number of touching
neighbors per particle) increase rapidly. in a
cusplike manner, as the particles deviate
from perfect spheres. Both reach high densi—
ties such as c an 0.71. and general ellipsoids
pack randomly to a remarkable tp e6 0.735,
approaching the density of the crystal with
the highest possible density for spheres
(11) t; = «N173 z 0.7405. The rapid in
creases are unrelated to any observable in
crease in order in these systems that develop
neither crystalline (periodic) not liquid crys
talline (nematic or orientational) order. Our experiments used two varieties of
M&M‘s Milk Chocolate Candies: regular and
baking (“mini”) candies ( 12 ). Both are oblate
spheroids with small deviations from true
ellipsoids, Ar/r < 0.01. Additionally.
M&M"s Candies have a very low degree of
polydispersity (principal axes 20 r: 1.34 i
0.02 cm, 217 = 0.693 t 0.018 em, at?) =
1.93 i 0.05 for regular; 20 = 0.925 t 0.011
cm, 21) = 0.493 1* 0.018 cm. a/b 2 1.88 i
0.06 for minis). Several sets of experiments
were performed to determine the packing
fraction. A square box, 8.8 cm by 8.8 cm, was W
1Program in Applied and Computational Mathematics,
2Department of Physics, 5Department: of Chemistry,
Princeton University, Princeton. NJ 08544, USA.
‘Princeton Materials institute, Princeton, NJ 08544,
USA. 5North Carolina Central University, Durham, NC
27707. USA. 6Department of Civil and Environmental
Engineering, 7Department of Mathematics. Cornell
University, Ithaca. NY 14853, USA. *To whom correspondence should be addressed. E
mail: [email protected] www.5ciencemag.org SClENCE VOL 303 ﬁlled to a height of 2.5 cm while shaking and
tapping the container. The actual measure
ments were performed by adding 9.0 cm to
the height and excluding the contribution
from the possibly layered bottom. After mea
suring the average mass, density. and volume
of the individual candies, the number of can
dies in the container and their volume frac
tion could be simply determined by weigh—
ing. These experiments yielded up = 0.665 t
0.01 for regulars and (p = 0.695 1' 0.01 for
minis. The same technique was used for
3.175 2 mm ball bearings (spheres) and
yielded a; = 0.625 t 0.01. A second set of
experiments was performed by ﬁlling 0.5,
l«. and 5liter round ﬂasks (to minimize or—
dering due to wall effects) with candies by
pouring them into the ﬂasks while tapping (5
liters corresponds to about 23,000 minis or
7500 regulars) (Fig. 1A). The volume frac»
tions found in these more reliable studies
were to = 0.685 r 0.01 for both the minis
and regulars (13). The same procedure for
30,000 ball hearings in the 0.5—liter ﬂask
yielded tp = 0.635 1' 0.01. which is close to
the accepted MR] density. A 5liter sample of regular candies similar
to that shown in Fig. 1A was scanned in a
medical magnetic resonance imaging device
at Princeton Hospital. For several planar slic
es, the direction 0 (with respect to an arbitrary
axis) of the major elliptical axis was manua
ally measured and the twodimensional
nematic order parameter $2 = (2 cos:2 0 — 1)
was computed, yielding 52 ~ 0.05. This is
consistent with the absence of orientational
order in the packing (14). Our simulation technique generalizes the
LubachevskyStillinget (LS) spherepacking
algorithm (15, 16) to the case of ellipsoids.
The method is a hardparticle molecular dy
namics (MD) algorithm for producing dense
disordered packings. Initially. small ellip
soids are randomly distributed and randomly
oriented in a box with periodic boundary
conditions and without any overlap. The cl
lipsoids are given velocities and their motion
followed as they collide elastically and also
expand uniformly. Aﬁer some time a jammed
state with a diverging collision rate is reached
and the density reaches a maximal value. A
novel eventdriven MD algorithm (17) was
used to implement this process efﬁciently,
based on the algorithm used in (15) for
spheres and similar to the algorithm used for
needles in (18). A typical conﬁguration of
1000 oblate ellipsoids (aspect ratio at =
b/a == 19" re 0.526) is shown in Fig. 113,
with density of q: 7: 0.70 and nematic order
parameter S m 0.02 to 0.05. We have veriﬁed that the sphere packings
produced by the LS algorithm are jammed
according to the rigorous hierarchical deﬁni
tions of local. collective. and strict jamming
(19, 20). Roughly speaking, these deﬁnitions REPORTS are based on mechanical stability conditions
that require that there be no feasible local or
collective particle displacements and/or
boundary deformations. On the basis of our
experience with spheres (10). we believe that
our algorithm (with rapid particle expansion)
produces final states that represent the MR5
state well. The algorithm closely reproduces the
packing fraction measured experimentally.
The density of simulated packings of 1000
particles is shown in Fig. 2A. Note the two clear
maxima with tp e: 0.71, already close to the
0.74 for the ordered face~centered cubic (fcc)/
hexagonal closepacked (hep) packing, and the
cusplike minimum near or = 1 (spheres). Pre
vious simulations for random sequential addi—
tion (RSA) (21 ). as well as gravitational depo
sition (22), produce a similarly shaped curve,
with a maximum at nearly the same aspect
ratios (X z 1.5 (prolate) or or m 0.67 (oblate), but with substantially lower volume fractions
(such as to r 0.48 for RSA). Why does the packing fraction initially in
crease as we deviate from spheres? The rapid
increase in packing fraction is attributable to the
expected increase in the number of contacts
resulting from the additional rotational degrees Fig. 1. (A) An experimental packing of the
regular candies. (B) Computergenerated pack—
ing of 1000 oblate ellipsoids with or = 1.9”. ‘13 FEBRUARY 2004 991 Downloaded from www.50iencemagorg on January 25, 2007 992 REPORTS Fig. 2. (A) Density (p 0.74
versus aspect ratio or
from simulations, for
both prolate (circles) and oblate (squares) 0.72 ellipsoids as well as ,5
fully aspherical (dia E 07
monds) ellipsoids. The '1:
most reliable experi— g
mental result for the E 0‘68 regular candies (error
bar) is also shown;
this likely underpre
diets the true density
(38). (8) Mean contact 0.64
number 2 versus 35 g
pect ratio or from sim ulations [same sym» .0
m
on 0.5 bols as in (A)], along with the experimental result for the regular candies (cross). to rotate and escape the cage of neighbors. Fig. 3. Shearing the dens
est packing of ellipses. of freedom of the ellipsoids. More contacts per
particle are needed to eliminate all local and
collective degrees of freedom and ensure jam—
ming. and forming more contacts requires a
denser packing of the particles. in the inset in
Fig. ZB, the central circle is locally jammed. A
uniform vertical compression preserves tp. but
the central ellipsoid can rotate and free itself
and the packing can densify. The decrease in
the density for very asphcrical particles could
be explained by strong exclusionvolume ef
fects in orientationally disordered packings
(23). Results resembling those shown in Fig.
2A are also obtained for isotropic random pack—
ings of spherocylinders (23. 24), but an argu
ment based on “caging” (not jamming) of the
particles was given to explain the increase in
density as asphericity is introduced. Spherocyl
inders have a very different behavior for or
dered packings from ellipsoids (the conjectured
maximal density is n/VE z 0.91. which is
signiﬁcantly higher than for ellipsoids), and
also cannot be oblate and are always axisym
metric. The similar positioning of the maximal
density peak for different packing algorithms
and particle shapes indicates the relevance of a
simple geometrical explanation. By introducing orientational and transla—
tional order, it is expected that the density of
the packings can be further increased, at least
up to 0.74. As shown in Fig. 3 for two
dimensions. an afﬁne deformation (stretch)
of the densest disk packing produces an el
lipse packing with the same volume fraction.
However, this packing. although the densest
possible. is not strictly jammed (i.e., it is not
rigid under shear transformations). The figure 1 (D an Average coordination \j 1.5 2 2.5 3 Aspect ratio .03
tn
0 shows through a sequence of frames how one
can distort this collectively jammed packing
(20), traversing a whole family of densest
conﬁgurations. This mechanical instability of
the ellipse packing as well as the three
dimensional ellipsoid packing arises from the
additional rotational degrees of freedom and
does not exist for the disk or sphere packing. There have been conjectures (25. 26) that
frictionless random packings have just
enough constraints to completely statically
deﬁne the system (27). Z = 2f (i.e.. that the
system is isostatic). where f is the number of
degrees of freedom per particle (f = 3 for
spheres, f = 5 for spheroids. and f 2 6 for
general ellipsoids) (28). If friction is strong,
then fewer contacts are needed. Z = f + l
(29). Experimentally, Z for spheres was de
termined by Bemal and Mason by coating 3
system of ball bearings with paint, draining
the paint. letting it dry, and counting the
number of paint spots per particle when the
system was disassembled (30). Their results
gave Z k 6.4. surprisingly close to isostatic—
ity for frictionless spheres (31). We performed the same experiments with
the M&M’s, counting the number of true con—
tacts between the particles (32). A histogam of
the number of touching neighbors per particle
for the regular candies is shown in Fig. 4. The
average number is Z = 9.82. In simulations 3
contact is typically deﬁned by a cutoff on the
gap between the particles. Fortunately. over a
wide range (10—9 to 10“) of contact toleranc—
es, Z is reasonably constant. Superposed in Fig.
4 is the histogram of contact numbers obtained
for simulated packings of oblate ellipsoids for 0.5 1 1.5 2 Aspect ratio 2.5 3 3.5 inset: introducing asphericity makes a locally jammed particle free P
to Probability
P
M .0
.t 0.0
6 7 8 9 10 1 ‘l 12 13
Number of contacts Fig. 4. Comparison of experimental (black bars,
from 489 regular candles) and simulated (white
bars, from 1000 particles) distribution of parti
cle contact numbers. or = 0.526. from which we found Z m 9.80. In
Fig. 28 we show Z as a function of aspect ratio
0. (33). As with the volume fraction. the contact
number appears singular at the sphere value and
rises sharply for small deviations. Unlike tp.
however, Z does not decrease for large aspect
ratios, but rather appears to remain constant.
We expect that fully aspherical ellipsoids,
which have f = 6. will require even more
contacts for jamming (Z = 12 according to
the isostatic conjecture) and larger to. Results
from simulations of ellipsoids with axes a =
o' ‘, b = l, and c = or (where or measures the
asphericity) are included in Fig. 2A. At or ~ 1.3 we obtain a surprisingly high density of tp k 0.735, with no signiﬁcant orientational
ordering. The maximum contact number ob
served in Fig. 28 is Z w 11.4. It is interesting
that for both spheroids and general ellipsoids.
Z reaches a constant value at approximately
the aspect ratio for which the density has a
maximum. This supports the claim that the
decrease in density for large at is due to
exclusion volume effects. The putative nonanalytic behavior of Z and
q; at or = l is striking and is evidently related to
the randomness of die jammed state. Crystal
close packings of spheres and ellipsoids show
no such singular behavior, and in fact (p and Z
are independent of or for small deviations from
unity. On the other hand, for random packings. 13 FEBRUARY 2004 VOL 303 SClENCE www.5ciencemag.org Downloaded from www.sciencemag.0rg on January 25, 2007 the behavior is not discontinuous, whereas the number of degrees of freedom jumps from three . to ﬁve (or six) as soon as or deviates from 1. In
several industrial processes such as sintering
and ceramic formation, interest exists in in
creasing the density and number of contacts of
powder particles to be fused, if ellipsoidal in
stead of spherical particles are used, we may
increase the density of a randomly...
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 Spring '18
 Randall J. Scalise
 Conductivity, Solid State Physics, ellipsoids, packing, Random close pack, general ellipsoids

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