Optimizing the design of nanostructures for improved thermal conduction within confined spaces
 Jianlong Kou^{1, 2},
 Huiguo Qian^{1},
 Hangjun Lu^{1},
 Yang Liu^{3},
 Yousheng Xu^{1},
 Fengmin Wu^{1}Email author and
 Jintu Fan^{2}Email author
DOI: 10.1186/1556276X6422
© Kou et al; licensee Springer. 2011
Received: 23 March 2011
Accepted: 14 June 2011
Published: 14 June 2011
Abstract
Maintaining constant temperature is of particular importance to the normal operation of electronic devices. Aiming at the question, this paper proposes an optimum design of nanostructures made of high thermal conductive nanomaterials to provide outstanding heat dissipation from the confined interior (possibly nanosized) to the microspaces of electronic devices. The design incorporates a carbon nanocone for conducting heat from the interior to the exterior of a miniature electronic device, with the optimum diameter, D _{0}, of the nanocone satisfying the relationship: D _{ 0 } ^{ 2 } (x) ∝ x ^{1/2} where x is the position along the length direction of the carbon nanocone. Branched structure made of singlewalled carbon nanotubes (CNTs) are shown to be particularly suitable for the purpose. It was found that the total thermal resistance of a branched structure reaches a minimum when the diameter ratio, β* satisfies the relationship: β* = γ ^{0.25b } N ^{1/k* }, where γ is ratio of length, b = 0.3 to approximately 0.4 on the singlewalled CNTs, b = 0.6 to approximately 0.8 on the multiwalled CNTs, k* = 2 and N is the bifurcation number (N = 2, 3, 4 ...). The findings of this research provide a blueprint in designing miniaturized electronic devices with outstanding heat dissipation.
PACS numbers: 44.10.+i, 44.05.+e, 66.70.f, 61.48.De
Introduction
With the miniaturization of electronic devices and the increased integration density, the effective dissipation of heat becomes an important requirement for ensuring troublefree operation [1, 2]. The limited space available for heat dissipation, the high energy densities and the dynamically changing, and often unknown, locations of heat sources in micro and nanodevices [3], make it difficult to apply conventional thermal management strategies and techniques of heat transmission, such as convectiondriven heat fins, fluids, heat pastes, and metal wiring [3]. It is a challenge to find the best material and structure for providing excellent heat transfer within the severe space constraints.
Nanomaterials have been widely researched and found to possess novel properties [4–10], for example, singlewalled CNTs exhibit extraordinary strength [4], high electrical conductivity (4 × 10^{9} Acm^{2}) [5] and ultrahigh thermal conductivity (3,000 to 6,600 Wm^{2} K^{1}) [6, 7], which make them potentially useful in many applications in nanotechnology, electronics and other fields of material science [11–16]. It therefore follows that nanomaterial should be uniquely suitable for applications requiring exceptional heat transfer properties. Nevertheless, nanomaterials cannot be used directly due to area and volume constraints [17]; particularly in the case of the very small interior of electronic devices which is much smaller than their outside. It is also important to consider the transition from nano to microstructure or 'point' to bulk, which occurs from the interior to the exterior of electronic devices. Thus, for example, it is not possible to use singlewalled CNTs because of severe space constraints at the interior 'point' level. Therefore, it is necessary to design structures to satisfy space constraints, and, furthermore, to optimize the design to also satisfy the heat conduction requirements.
The use of branched nanostructures has been identified as an effective way to form functional elements that bridge nano to macro scale [18–22], for example, actin, cytoskeleton, bone, and collagen fiber networks in biological structures. Recently, Xu and Buehler [22] presented a novel concept involving the use of hierarchical structures as an effective means to create a bridge from the nano to the macroscale. Either from the confined interior to the exterior of electronic devices or from nano to microspaces, the space are limited. So, to find the proper structure is necessary. Nevertheless, no work appears to have been done on the optimum design of the heat conduction structures from the confined interior to the exterior of electronic devices and from nano to microspaces.
The objective of the present work is to propose such an optimum design based on the use of carbon nanocones and carbon nanotubes in the form of a conical and branched structure. In the Description of structure section, we give the detailed description of the heat conduction structure, from the interior of an electronic device to micro space, and in the Optimum design section, we present optimum design for heat conduction from the interior to the exterior and nano to microspaces of electronic devices. Lastly, some concluding remarks are given in the Conclusions section.
Description of structure
Optimum design
Interior to the exterior of electronic devices
By comparing Eqs. 9 and 10, it can be seen that tapering as represented by Eq. 9, produces a 5.6% lower value for T _{0} (0)  T _{0} (L _{0}) than the uniform path design represented by Eq. 10. The optimal designs are illustrated in Figure 2(b). Three curves represent the three shapes of the nanocone corresponding to three different volumes of the nanocone (viz. Vp).
Nano to microspaces
Method
As discussed above, optimum heat conduction pathways made of carbon nanocones can be optimally designed to transfer heat efficiently from the interior to the exterior of a miniaturized electronic device; however, heat may still not be rapidly dissipated into the surrounding space as exterior surface of the miniaturized electronic device is small (possibly in nanoscale). It is therefore desirable to channel the heat from the nanoscale exterior surface of the electronic device the micro or larger space. Bifurcate singlewalled CNTs have been produced and exhibited outstanding performance compared to conventional material [25–27]. The idea is inspired by recent work on concept of using a biologically inspired approach of hierarchical structures [22]. The hierarchical structure is an effective way to provide a bridge between the nano to the macro level in space. Such structures are considered to be highly advantageous over conventional structures, such as convectiondriven heat fins, fluids, heat pastes, and metal wiring, in heat dissipation. However, the optimization of such a branched network of CNTs for heat dissipation has not been analyzed so far. This section thus deals in detail with the optimum design of bifurcate singlewalled CNTs for efficiently conducting heat from nano to microspaces.
where l _{0} and d _{0} are the length and diameter of the 0th branching level.
For given an electronic device, the space may be limited by the design. So the length (L) of the branched structure may be a limiting factor. With (L) being fixed, Eq. (14) implies that, the branched level number m, the length (l _{ 0 } ) of the 0th branched singlewalled carbon nanotube and the length ratio (γ) can be optimized to maximize heat conduction.
R ^{+} represents the ratio of the thermal resistance of the branched structure of singlewalled carbon nanotubes, R _{ t } , to that of the equivalent R _{ s } , under the constraint of total volume, and which is a function of γ, β, N, m, and b. As can be seen, equation (17) involves higher order variables, which makes it difficult to attain the optimum scaling relations analytically.
Results and discussions
By coupling Eqs. 13 and 14 and applying the optimum diameter ratio, the optimum structural parameters of branched singlewall carbon nanotubes can be derived under the constraint of the total volume (V) and length (L). The backgrounds of Figure 4a, b show two optimum designs of the branched singlewall carbon nanotubes with b = 0.3, m = 2, N = 2 and different length ratio γ. The design in the background of Figure 4a has a smaller value of γ, while that of Figure 4b has a greater value of γ. To achieve optimum heat conduction and dissipation under the constraints of the total volume (V) and length (L) of the branched carbon nanotubes structure, the bigger γ, the smaller the length (l _{ 0 } ) of the 0th branch.
Conclusions
In this paper, the optimum design of carbon nanostructure for most efficiently dissipating heat from the confined interior of electronic devices to the micro space is analyzed. It is found that the optimum diameter, D _{0}, of carbon nanocones satisfies the relationship, . For transmitting heat from the nanoscaled surface of electronic devices to the microspace, the total thermal resistance of a branched structure reaches a minimum when the diameter ratio, β*, satisfies β* = γ ^{0.25b } N ^{1/k* }, where, γ is ratio of length, b = 0.3 to approximately 0.4 on the singlewalled CNTS, b = 0.6 to approximately 0.8 on the multiwalled CNTS, k* = 2 and N = the bifurcation number (N = 2, 3, 4,......) under the volume constraints. If space is the only limitation, the optimum diameter remains applicable. These findings help optimize the design of heat conducting media from nano to microstructures. It must be noted that the present work is an improvement from the Ref. [22], which showed hierarchical structure is effective in providing a bridge between the nano to the macro level for heat transfer. The present work provides a theoretical prediction of how such heat dissipater can be optimally designed.
Despite recent progress in synthesizing and manipulating nanocones and branched singlewalled CNTs [25–27, 32–34], further work is necessary to perfect techniques and systems for the fabrication of nanostructures and creation of seamless links between the individual singlewalled CNT elements of the branched structures, thereby reducing the interfacial thermal resistance [35–37], as well as to precisely control the scale of nanostructures.
Abbreviations
 CNTs:

carbon nanotubes.
Declarations
Acknowledgments
This work was partially supported by the Research Grant Council of HKSAR (Project No. PolyU 5158/10E), the National Natural Science Foundation of China under Grant No's 10932010, 10972199, 11005093, 11072220 and 11079029, and the Zhejiang Provincial Natural Science under Grant Nos. Z6090556 and Y6100384.
Authors’ Affiliations
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