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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. CTS-0319235, 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. DEFGOZ-Ql-ER4S439, 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 face-centered 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 densely-up 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 Downloaded from 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 ill-de- 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 cusp-like 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 0854-4, 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] 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 5-liter 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 5-liter 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 two-dimensional 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 Lubachevsky-Stillinget (LS) sphere-packing algorithm (15, 16) to the case of ellipsoids. The method is a hard-particle 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 event-driven 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 close-packed (hep) packing, and the cusp-like 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) Computer-generated pack— ing of 1000 oblate ellipsoids with or = 1.9”. ‘13 FEBRUARY 2004 991 Downloaded from 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 0-7 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 exclusion-volume 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 Downloaded from 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 poured and compacted powder to a value approaching that of the densest (fcc) lattice packing. Refere...
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