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
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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)
Nano Lett., Vol. 9, No. 5, 2009
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
Nano Lett., Vol. 9, No. 5, 2009
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
Nano Lett., Vol. 9, No. 5, 2009
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.
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