Hierarchical Nanostructures Are Crucial To Mitigate Ultrasmall

Mar 26, 2009 - elements show a reasonable thermal performance, they are not qualified to solve the point-load thermal management problem, because ther...
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NANO LETTERS

Hierarchical Nanostructures Are Crucial To Mitigate Ultrasmall Thermal Point Loads

2009 Vol. 9, No. 5 2065-2072

Zhiping Xu† and Markus J. Buehler*,†,‡,§ Laboratory for Atomistic and Molecular Mechanics, Department of CiVil and EnVironmental Engineering, Center for Computational Engineering, Center for Materials Science and Engineering, Massachusetts Institute of Technology, 77 Massachusetts AVenue, Cambridge, Massachusetts 02139 Received February 6, 2009; Revised Manuscript Received March 14, 2009

ABSTRACT Here we show that hierarchical structures based on one-dimensional filaments such as carbon nanotubes lead to superior thermal management networks, capable of effectively mitigating high-density ultrasmall nanoscale heat sources through volumetric heat sinks at micrometer and larger scales. The figure of merit of heat transfer is quantified through the effective thermal conductance as well as the steady-state temperature distribution in the network. In addition to providing an overall increased thermal conductance, we find that hierarchical structures drastically change the temperature distribution in the immediate vicinity of a heat source, significantly lowering the temperature at shorter distances. Our work brings about a synergistic viewpoint that combines advances in materials synthesis and insight gained from hierarchical biological structures, utilized to create novel functional materials with exceptional thermal properties.

Miniaturization and higher integration of components in microelectromechanical systems (MEMS), nanoelectromechanical systems (NEMS), and optical microdevices such as laser diodes lead to high density, point-load singular heat sources that often induce catastrophic device failure, thereby significantly reducing their operational reliability.1,2 Conventional thermal management strategies to mitigate heat sources by using convection driven heat fins, fluids, heat pastes, or metal wiring fail in micro- and nanodevices because of the limited area of heat dissipation, the high energy densities, and the dynamically changing or a priori unknown locations of heat sources.3 To solve these problems, a high-performance heat transfer material must be designed that satisfies the following two conditions: (1) is comprised of high thermal conductivity components, and (2) being capable of migrating heat from a small confined space (on the order of several nanometers) to larger-scale heat sinks (micrometers and larger, which is more efficient to maintain at a lower temperature). Recent advances in nanotechnology have resulted in great achievements in the synthesis of nanostructures with ultrahigh thermal conductivities such as nanowires,4 carbon * To whom correspondence should be addressed. E-mail: mbuehler@ MIT.EDU. Phone: +1-617-452-2750. Fax: +1-617-324-4014. † Laboratory for Atomistic and Molecular Mechanics, Department of Civil and Environmental Engineering. ‡ Center for Computational Engineering. § Center for Materials Science and Engineering. 10.1021/nl900399b CCC: $40.75 Published on Web 03/26/2009

 2009 American Chemical Society

nanotubes (CNTs) (3000 to 6600 Wm-1K-1)5 and monolayer graphene sheets (5000 Wm-1K-1).6 The summary provided in Table 1 shows that these nanomaterials possess ultrahigh thermal conductivity in comparison with bulk materials, which satisfies requirement 1. However, although the asproduced bulk materials7,8 or thin films9,10 based on these elements show a reasonable thermal performance, they are not qualified to solve the point-load thermal management problem, because there is a lack of an efficient structural link between the bulk material and a nanoscale point-load. For the point-load heat management problem, the number of heat-conducting fibers connected to the point-load itself is limited by the physical size of the heat source, while the heat sink is usually an open space with much larger dimensions. To effectively mitigate the generated heat, a network bridging these two scales, from nano to macro, is essential. However, it remains an open question which structural arrangement is most suitable to provide this function. In biological structures such as actin or intermediate filaments in the cell’s cytoskeleton, bone, or collagen fiber networks, the use of hierarchical structures has been identified as an effective way to utilize nanostructures to form functional elements that bridge nano to macro.11-14 Here we show that for the case of thermal management, both requirements 1 and 2 can simultaneously be satisfied through introducing hierarchical structures, similar as found in

Figure 1. Geometry of the hierarchical heat management network. (a) Heat management network for a point-load heat source problem. The network should be able to bridge the scale between heat source of dimensions below 100 nm to the heat sink at much larger scale (on the order of micrometers). To this end, a network of thermal conductors is introduced. Hierarchical structures are created through forming branches between self-assembled nanoscale heat-conducting fibers (for example, covalent carbon nanotubes Y-junctions, as shown (b). Detailed geometry of the heat management network without (c) and with (d) hierarchical structures (i.e., branches), to migrate heat from point-load heat source with temperature Th to volumetric heat sink at temperature Tc. The black bars resemble heat-conducting fibers assembled into the network. The network can be characterized by the number of heat-conducting fibers connected to the heat source (N0), the number of branches (Nb) and the number of heat-conducting fibers (m) along the heat transfer path from heat source to heat sink. The heat resistor network comprising identical elements with thermal resistance R0 (e) is equivalent to the hierarchical structure shown in panel d.

Table 1. Intrinsic Thermal Conductivity Κ of Typical Materials (Data Taken from Reference 29) material

thermal conductivity (W K-1 m-1)

single wall carbon nanotubes multiwall carbon nanotubes silicon nanowires diamond (bulk) HOPG (bulk) SiC (bulk) copper (bulk)

6600 3000 40 3000 2000 (in-plane) 325 400

biological protein materials, where networks are utilized that comprise of highly heat-conducting nanofibers such as CNTs or graphene materials. One of the most promising hierarchical structures for this purpose is the branched tree structure (see Figure 1a,b). In this structure, the heat source is located at the root, and heat is dissipated into the area surrounding the tips of the branches. The structure is comprised of identical elements, that is, each segment of the structure has the same geometry (such as length l, diameter d), and the same physical properties (thermal conductivity, specific heat, and density). The structure of the network hierarchy is specified by three characteristic numbers: (1) N0, the number of heat-conducting fibers connected to the point-load heat source, (2) Nb, the number of branches at each joint, and (3) m, the number of heat-conducting fibers along one heat path from source to 2066

sink, equivalent to the number of hierarchical level (which is a measure for the radial distance of heat transport). For direct comparison, in Figure 1c,d we present structures without hierarchy (Figure 1c) and with hierarchy (Figure 1d). Both structures have the same N0 ) 3 and m ) 2, but the hierarchical structure has Nb ) 2 and the nonhierarchical structure has Nb ) 1. Please see Supporting Information, Table 1 for a list of all variables used with a brief explanation. In the thermal management network introduced above, the overall heat transfer performance is determined by both the intrinsic thermal conductivity of heat-conducting fibers R0 and the interfacial thermal conductance Rc, which depends on the chemical structure at the interface.15-17 As shown in Figure 2, the effects of interfacial thermal conductance Rc can be included by renormalization of the thermal conductance of heat-conducting fibers R0 to R0e. For hierarchical structure with Nb branches at the junction, the interfacial resistance Rc can be subdivided into Nb + 1 individual resistors (Rc′) that are grouped with associated heat-conducting fibers. On the basis of the equivalent resistor network shown in Figure 2b, we obtain Rc ) Rc′ + Rc′ /Nb

(1)

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Figure 2. Effects of interfacial thermal resistance at the junctions between heat-conducting fibers. The effects of interfacial thermal conductance Rc can be included by renormalization on the thermal conductance of heat-conducting fibers (R0) to R0e. (a) Part of the hierarchical network for thermal management, near a junction between heat-conducting fibers. A thermal resistor Rc is added into the network for the interfacial thermal resistance. (b) For hierarchical structure with Nb (Nb ) 2 in the figure) branches at the junction, the interfacial resistance Rc can be subdivided into Nb + 1 individual resistors (Rc′) that are grouped together with associated heat-conducting fibers, where Rc ) Rc′ + Rc′/Nb. (c) The equivalent thermal resistance including the interfacial effects is given by R0e ) R0 + 2Rc′ ) R0 + 2RcNb/(Nb + 1).

Consequently, the equivalent thermal resistance including the interfacial effects can be written as R0e ) R0 + 2Rc′ ) R0 + 2RcNb /(Nb + 1)

(2)

Therefore, the interfacial thermal resistance and can be taken into account by simply substituting R0 (associated with properties of an individual element) with an effective R0e (associated with properties of individual elements including interfacial effects). Therefore, in the following discussion we will develop all results dependent on an effective R0 and not explicitly consider the thermal interfaces between heatconducting fibers. The interfacial thermal conductivity can lower than overall thermal conductivity as illustrated in the discussion above. However, recent breakthroughs in experimental synthesis of hierarchical nanostructures suggest that materials with extremely low interfacial resistance can be synthesized, for example, based on metal junctions,16 polymer wrapping,15,18 the formation of chemically cross-linked CNT structures,19,20 patterned heterojunctions of graphene,21,22 or even the asproduced hierarchical branched covalently bonded structures of nanotubes and nanowires.23 Covalently bonded structures are particularly suitable for thermal applications as proposed here, as they provide a seamless coupling of phonon modes relevant for thermal properties and thus represent a negligible thermal interface resistance. Useful assemblies might also be created through interlinking or patterning of CNTs/ graphene materials into hierarchical heat paths.15,17,21,22 To calculate the overall thermal conductance, the hierarchical structure is considered as a resistor network as shown in Figure 1e (where the structure shown in Figure 1e resembles the one shown in Figure 1d). A similar approach Nano Lett., Vol. 9, No. 5, 2009

was applied earlier for a fractal network in ref 24 and is adapted here to describe hierarchical structures. The equivalent resistor network contains m series of N0Nbm-1 parallel heat resistors. Thus, for the nonhierarchical network, the total number of heat-conducting fibers is M ) mN0, and the overall thermal resistance is R ) mR0/N0, where R0 is the intrinsic resistance of each heat-conducting fiber (for example, values for carbon nanotubes see Table 1). For a hierarchical network with Nb (>1) branches, the total number of heat-conducting fibers is M ) N0(1 - Nbm)/(1 - Nb), and the overall thermal resistance is given by R ) R0(1 - cm)/(1 - c)/N0,

where c ) 1/Nb

(3)

To characterize the influence of hierarchy on the overall thermal transfer performance, we first compare the effective thermal conductance of the entire structure, λ ) 1/R. Consider the two structures shown in Figure 3a,b. The nonhierarchical structure (linear chain of heat resistors plotted in black) in Figure 3a) has a lower thermal conductance 0.33λ0 (where λ0 ) 1/R0) than 0.57λ0 for the hierarchical structure (branched tree in Figure 3b). To achieve a thermal conductance as high as for the corresponding hierarchical structure, several parallel chains must be used, which effectively increases the width of each filament. In the example shown in Figure 3a, we need at least two parallel nonhierarchical chains, which can be achieved by including the gray shaded structure, where the resulting structure has a thermal conductance of 0.67λ0. In general, with the number of parallel chains defined as w (corresponding to the number of channels connected to a heat source, N0), the minimum width to reach a thermal conductance identical to that of the hierarchical structure is 2067

wmin ) m(1 - c)/(1 - cm)

(4)

where c ) 1/Nb. As shown in Figure 3c, hierarchical structures have a considerably higher thermal conductance than corresponding nonhierarchical structures with the same number of N0 elements that connect to the heat source. Through introducing thicker filaments by using w (>wmin) elements to form multiple parallel structures, nonhierarchical networks can reach a better performance than their hierarchical counterparts; however, at the cost of requiring thicker filaments. The associated minimum number of parallel paths (wmin) increases as the hierarchy level (Nb, m) is enhanced, as depicted in Figures 3d,e. These results show that hierarchies provide an effective approach to mitigate heat from very small sources while using very few connections, in particular for large distances (that is, large m). We provide a specific example. In order to feature the same thermal performance as a hierarchical structure with m ) 100 and Nb ) 2, a nonhierarchical structure (with Nb ) 1) should

have a width of w ) 50, where this thicker filament must be directly connected to the heat source. For a carbon nanotube with a diameter of 2 nm, the characteristic length scale associated with each connecting element in the nonhierarchical structure is 100 nm (versus 2 nm in nonhierarchical structures). This characteristic length limits the applicability of nonhierarchical structures in high performance thermal management networks applied to ultrasmall heat sources. This shows that nonhierarchical networks are not capable of mitigating heat from very small sources in particular for large distances and thus cannot satisfy requirement 2 described above. In the structures shown in Figure 3a,b, the total number of elements M is different (it is 6 (for two parallel chains) in panel a and 7 in panel b). In order to characterize the performance per unit material mass, we define the thermal conductance density as D ) λ/M. The expression of the thermal resistance given in eq 3 shows that with the same number of N0 connecting elements, the hierarchical network

Figure 3. Quantification of the effect of hierarchical structures on overall thermal properties, as function of radial distance and number of hierarchical branches. Subplots (a) and (b) display geometries of two example structures. (a) Nonhierarchical structures (Nb ) 1) (black part) or parallel nonhierarchical structure with width w ) 2 (including the gray part). (b) Hierarchical structure with number of branches Nb ) 2. (c) Panel shows the thermal conductance of nonhierarchical (Nb ) 1) and hierarchical (Nb ) 2, 4) structures with the same number of connections with the heat source N0 ) 1. The result shows that the use of hierarchical structures provides a means to maintain a high thermal conductance at large distances. (d,e) Panels display a comparison of the overall thermal conductance between nonhierarchical and hierarchical structures (although the results are plotted continuously, number m, Nb, and w can only be integers). The plots show the dependence of the minimum width wmin of an equivalent nonhierarchical structure as a function of the number of fibers along heat transfer path (that is, the minimum width of elements to reach an equivalent thermal performance as with a hierarchical structure), m, and the number of branches, Nb. For nonhierarchical networks, a higher minimum number of parallel chains wmin is needed to achieve the identical overall thermal conductance of the hierarchical network. The result shows that hierarchical networks have considerably higher thermal conductance than nonhierarchical networks with the same number of connections N0 with the heat source. This analysis shows that through introducing more parallel elements into thicker filaments, nonhierarchical structures can reach a similar performance as hierarchical structures. However, this comes at a cost that individual filaments are much thicker that can no longer reach ultrasmall heat sources. 2068

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Figure 4. Comparison of thermal conductance density () thermal conductance divided by the total number of fibers) between nonhierarchical and hierarchical structures. (a,b) Panels show the dependence on the number of fibers along heat transfer path, as a function of m (panel a) and Nb (panel b). For nonhierarchical networks, a minimum number of parallel chains wmin is needed to achieve comparable overall thermal conductance of the hierarchical network with number of branches Nb. In comparison with results shown in Figure 3, these results show that when taking into account the number of fibers used in the network, hierarchical structures have much higher thermal conductance density than nonhierarchical networks with the same number of connections N0 to heat source. For large Nb and m less than 10, nonhierarchical networks need unrealistic parallel numbers (up to the order of 1000) to reach comparable performance. This analysis shows, similar to that shown in Figure 3, that through introducing more parallel elements into thicker filaments, nonhierarchical structures can reach a similar performance as hierarchical structures. However, this comes at a cost that individual filaments are much thicker that can no longer reach ultrasmall heat sources.

always has a resistance reduced by Nb1-m, in comparison with nonhierarchical chains, which is reduced by 1/w2 (where w parallel chains are put together in parallel). In analogy to the previous discussion, we calculate the minimum width wmin so that a corresponding nonhierarchical network has the same thermal conductance density D as hierarchical networks with Nb branches. The calculation results are depicted in Figure 4. As the number of branches Nb increases, the number of heat-conducting fibers that must be directly connected to the heat source in nonhierarchical networks increases very rapidly (quickly reaching thousand and more for m > 10 and Nb > 4). This trend is more pronounced as the length of the heat transfer path m increases. For example, to compare with a hierarchical structure with m ) 20 and Nb ) 2, the nonhierarchical structure should have a width of w ) 725. If we connect multiple nonhierarchical structures to obtain the same thermal conductance density of a hierarchical structures (with m ) 20 and Nb ) 2), for a carbon nanotube with a diameter of 2 nm the characteristic length scale associated with the nonhierarchical structure would be 1450 nm and the applicable heat source is thus limited by this dimension. This example provides further evidence that nonhierarchical networks are not capable of effectively mitigating heat from ultrasmall sources. Most importantly, in comparison with the results shown in Figure 3, the results depicted in Figure 4 illustrate that taking into account the amount of material used in the network, hierarchical structures have a much greater thermal conductance density than nonhierarchical networks with the same number of connections N0 to the heat source. In summary, the number of connections N0 to a heat source is limited by the actual physical size of the heat source and heat-conducting fibers. Thus, for a specific point-load problem, introducing branches into the network is an effective method to improve its performance. Nano Lett., Vol. 9, No. 5, 2009

There exists a limitation on the number of hierarchy levels (and thus reachable radial length-scales) that can be introduced, due to spatial confinement. The number of heatconducting fibers at a specific distance r from the heat source increases drastically as function of m (i.e., Nbm), while the available space at r increases linearly as 2πr (in twodimensional (2D) systems) and 4πr2 (in three-dimensional systems). Therefore, at large distances, the fibers will quickly occupy all available space when Nb > 1. As can be seen in Figure 5, for a two-dimensional problem, the radius of the hierarchical network (distance from heat source to sink) is r ) ml(1 + cos θ)/2

(5a)

when m is an even number, or r ) (m - 1)l(1 + cos θ)/2 + l

(5b)

when m is an odd number, l is the length of each heatconducting fiber, and θ is half of the branch angle. Because of this geometric confinement of the 2D space, for a heatconducting fiber with diameter d the circumference of the network 2πr must be larger than the space occupied by heatconducting fibers, which is given by N0Nbm-1d. Thus the number of heat-conducting fibers along heat transfer path m is limited by m < 1 + log(2πr/N0d)/logNb. For example, for a carbon nanotube (with diameter 2 nm) network with extent 1 cm from the heat source to heat sink, if the point-load has a diameter of 10 nm and only five connections to carbon nanotubes are permitted, a maximum hierarchical level m is calculated to be limited by 24 for number of branches Nb ) 2. This requirement can be satisfied if the carbon nanotubes used to build this structure have a length on the order of micrometers. Hierarchical structures are highly effective in quickly reducing the temperature in the immediate vicinity of heat 2069

Figure 5. Geometrical constraints of hierarchical networks. The black and gray bars are heat-conducting fibers and the black bars show one of the heat transfer path from heat source (red) to heat sink (blue), which is composed by 4 heat-conducting fibers. The parameter θ denotes half of the branch angle, and r is the radius of the network. For the two-dimensional problem, the radius of the hierarchical network is r ) ml(1 + cosθ)/2 when m is an even number, or r ) (m - 1)l(1 + cosθ)/2 + l when m is odd number, where l is the length of each heat-conducting fiber and θ is half of the branch angle. Because of the confinement in the 2D space, for heat-conducting fiber with diameter d the perimeter of the network 2πr should be larger than the space occupied by heat-conducting fibers, N0Nbm-1d. The number of heat-conducting fibers along heat transfer path m is limited by m < 1 + log(2πr/N0d)/log Nb.

sources. In order to investigate the steady-state thermal transfer process, we consider a hierarchical thermal manage-

ment network between a point-load heat source with temperature Th and a heat sink with temperature Tc. A reduced temperature Tr ) (T - Tc)/(Th - Tc) can be defined to describe the temperature distribution within the network. Using the conservation of heat flux J in the network and a linear temperature distribution solution in uniform structures, the Fourier law J ) λi(Ti - Ti+1)/l can be written for each fiber i, where i ) 1, 2,..., m, λi ) λ0/Nbi-1 is the thermal conductance and Ti, Ti+1 is the temperature at the hot and cold end of fiber i. With the boundary condition of T1 ) Th and Tm+1 ) Tc, an analytic expression can be derived as Tr ) (cx*m - cm)/(1 - cm), where c ) 1/Nb and x* is the reduced coordinate variable from 0 (source) to 1 (sink). The result is shown in Figure 6a. From this result we observe that the temperature profile is linear in the nonhierarchical network (Nb ) 1), as predicted by Fourier’s law. However the introduction of hierarchical branches results in a concave profile and thus significantly lowers the temperature in the region near the point-load heat source. This effect can be further improved by increasing Nb, as shown in Figure 6a. We also investigate this problem by solving the onedimensional heat transfer equation FCpdT/dt ) κd2T/dx2

(6)

by using a finite difference method, where F, Cp, and κ are the density, specific heat, and heat conductivity of the materials,respectively, and T ) T(x, t) is the temperature as a function of spatial position x and time t. In the calculation, the heat source and sink are maintained at temperature Th ) 373.15 K and Tc ) 273.15 K, respectively, then the heat transfer equations are solved using the Crank-Nicolson method.25 The steady state is reached when the temperature distribution profile between the heat source and sink converges. The reduced temperature Tr is then plotted in Figure 6b as a function of spatial coordinates x (corresponding to

Figure 6. Temperature distribution profile between the heat source and sink (where x denotes the spatial coordinate along the radial heat transfer path). (a) Analytic solution, where x* denotes the reduced position (x* ) 0 for the heat source and x* ) 1 for the heat sink). (b) Numerical solution using the finite-difference method for hierarchical structure (m ) 10) made of carbon nanotubes with length of 100 nm, and x measures the distance from the point-load heat source. The temperature at the heat source and the heat sink are kept constant at 373.15 and 273.15 K, respectively. For a nonhierarchical network (Nb ) 1, top inset in panel a) the temperature distribution is linear as predicted by Fourier equation. In contrast, in hierarchical networks (bottom inset in panel a), the introduction of branches effectively reduces the temperature in the region close to the point-load heat source. The effect can be further improved by increases of Nb, the number of branches. 2070

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the radial distance from the heat source, where m ) 10). The numerical solution confirms the analytical solution of the temperature distribution and reveals a linear temperature profile within each hierarchical level. These results clearly illustrate the advantage of hierarchical structures by protecting devices from overheating near point-loads, providing the basis for a rapid decay of the temperature near ultrasmall heat sources. Figure 6b shows results for a thermal management network with a radial size of 1 µm, which is made of heat-conducting fibers with a length of 100 nm. The hierarchical structure (with Nb ) 2, m ) 10) can reach a 50% reduction of the temperature at 100 nm, in comparison to only 10% reduction in the nonhierarchical structure. This represents a performance increase by a factor of 5, providing a significant advantage that could be crucial to decrease the probability of device failure. Carbon nanotubes and graphene, low-dimensional and single atomic layer materials, are outstanding electronic and thermal conductors5,6 with many potential applications in thermal management and energy science and technology. These unique properties result not only from their graphitic lattice but also their low-dimensional overall structure. Our analysis provides a novel approach in utilizing these nanostructures to bridge multiple length-scales (from nano- to micrometer length-scales) through the formation of hierarchical networks. The effects of hierarchical structures on the performance of thermal management network were investigated here through theoretical analysis and numerical calculation. We found that in thermal management networks composed of identical high conductivity fibers, the overall thermal conductance of the network can be dramatically improved through introducing hierarchical structures (see, e.g., Figures 3 and 4). A detailed comparison with nonhierarchical networks and geometrical constraints on the networks showed that for a point-load heat management problem, hierarchical structures are able to lower the temperature in the region close to heat source much more effectively, as shown in the analysis presented in Figure 6. Despite recent progress in creating hierarchical carbon nanotube based networks,15-19,23 further studies are necessary to identify strategies to ensure seamless links between individual carbon nanotube elements to reduce the interfacial thermal resistance. Challenges also arise in terms of the manufacturability of hierarchical networks. Hierarchical assemblies of carbon nanotubes based on peptide coatings might be a promising strategy to provide a directed design approach at multiple levels.18 Hierarchical structures are also very widely observed in biological materials, where they often help to mitigate mechanical force loads (e.g., at crack tips, which represent stress singularities). In materials such as bone, the existence of a hierarchy of structural levels contributes to these materials’ extraordinary capacity to robustly mitigate these extreme mechanical loads.26-28 The key physical effect is their ability to dissipate mechanical energy effectively, through multiple levels in the hierarchical structure, leading to a rapid mitigation of mechanical stresses. This results in Nano Lett., Vol. 9, No. 5, 2009

a very high fracture toughness, as well as a generally great robustness against catastrophic failure (as seen, e.g., in bone or nacre). Our approach applied to the design of hierarchical structures to mitigate thermal point loads follows similar concepts, and provides the structural basis for effective mitigation of thermal energy near point sources (see, e.g., Figure 6). The analysis of biological materials further demonstrates that these structures are formed by a highly controllable process of self-assembly that results in linear, kinked, branched, regular, and random network structures,11 depending on the specific area of application. This provides a possible means to identify assembly strategies for the structures discussed here. Acknowledgment. This work was supported by DARPA (award number HR0011-08-1-0067) and the MIT Energy Initiative (MITEI). This work was supported in part by the MRSEC Program of the National Science Foundation under award number DMR-0819762. Supporting Information Available: Supplementary Table 1 contains a list of all variables used with a brief explanation. This material is available free of charge via the Internet at http://pubs.acs.org. References (1) Chen, G. Nanoscale Energy Transport and ConVersion; Oxford University Press: New York, 2005. (2) Tien, C. L. Microscale Energy Transfer. In Chemical and Mechanical Engineering; CRC: Boca Raton, FL, 1997. (3) Simons, R. E.; Antonetti, V. W.; Nakayama, W.; Oktay, S. Heat Transfer in Electronic Packages. In Microelectronics Packaging Handbook; Chapman and Hall: New York, 1997. (4) Ponomareva, I.; Srivastava, D.; Menon, M. Thermal Conductivity in Thin Silicon Nanowires: Phonon Confinement Effect. Nano Lett. 2007, 7 (5), 1155–1158. (5) Dresselhaus, M. S.; Dresselhaus, G.; Avouris, P. Carbon nanotubes: synthesis, structure, properties and applications; Springer: New York, 2000. (6) Balandin, A. A.; Ghosh, S.; Bao, W.; Calizo, I.; Teweldebrhan, D.; Miao, F.; Lau, C. N. Superior Thermal Conductivity of Single-Layer Graphene. Nano Lett. 2008, 8 (3), 902–907. (7) Chatterjee, T.; Jackson, A.; Krishnamoorti, R. Hierarchical structure of carbon nanotube networks. J. Am. Chem. Soc. 2008, 130, 6934– 6935. (8) Chen, Q.; Chen, T.; Pan, G. B.; Yan, H. J.; Song, W. G.; Wan, L. J.; Li, Z. T.; Wang, Z. H.; Shang, B.; Yuan, L. F.; Yang, J. L. Structural selection of graphene supramolecular assembly oriented by molecular conformation and alkyl chain. Proc. Natl. Acad. Sci. U.S.A. 2008, 44, 16849-16854. (9) Biercuk, M. J.; Llaguno, M. C.; Radosavljevic, M.; Hyun, J. K.; Johnson, A. T.; Fischer, J. E. Carbon nanotube composites for thermal management. Appl. Phys. Lett. 2002, 80 (15), 2767–3. (10) Kordas, K.; Toth, G.; Moilanen, P.; Kumpumaki, M.; Vahakangas, J.; Uusimaki, A.; Vajtai, R.; Ajayan, P. M. Chip cooling with integrated carbon nanotube microfin architectures. Appl. Phys. Lett. 2007, 90 (12), 123105–3. (11) Dong, H.; Paramonov, S. E.; Hartgerink, J. D. Self-assembly of R-helical coiled coil nanofibers. J. Am. Chem. Soc. 2008, 130, 13691– 13696. (12) Fratzl, P.; Weinkamer, R. Nature’s hierarchical materials. Prog. Mater. Sci. 2007, 52 (8), 1263–1334. (13) Alberts, B.; Johnson, A.; Lewis, J.; Raff, M.; Roberts, K.; Walter, P. Molecular biology of the cell; Taylor & Francis: 2002. (14) Keten, S.; Buehler, M. J. Geometric Confinement Governs the Rupture Strength of H-bond Assemblies at a Critical Length Scale. Nano Lett. 2008, 8 (2), 743–748. (15) Zhang, Y. C.; Broekhuis, A. A.; Stuart, M. C. A.; Landaluce, T. F.; Fausti, D.; Rudolf, P.; Picchioni, F. Cross-linking of multiwalled carbon nanotubes with polymeric amines. Macromolecules 2008, 41, 6141–6146. 2071

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