Optimizing the design of nanostructures for improved thermal conduction within confined spaces
© Kou et al; licensee Springer. 2011
Received: 23 March 2011
Accepted: 14 June 2011
Published: 14 June 2011
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 micro-spaces 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 single-walled 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 single-walled 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
With the miniaturization of electronic devices and the increased integration density, the effective dissipation of heat becomes an important requirement for ensuring trouble-free 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 nano-devices , make it difficult to apply conventional thermal management strategies and techniques of heat transmission, such as convection-driven heat fins, fluids, heat pastes, and metal wiring . 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, single-walled CNTs exhibit extraordinary strength , high electrical conductivity (4 × 109 Acm-2)  and ultra-high thermal conductivity (3,000 to 6,600 Wm-2 K-1) [6, 7], which make them potentially useful in many applications in nano-technology, 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 ; 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 micro-structure or 'point' to bulk, which occurs from the interior to the exterior of electronic devices. Thus, for example, it is not possible to use single-walled 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  presented a novel concept involving the use of hierarchical structures as an effective means to create a bridge from the nano- to the macro-scale. Either from the confined interior to the exterior of electronic devices or from nano- to micro-spaces, 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 micro-spaces.
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 micro-spaces of electronic devices. Lastly, some concluding remarks are given in the Conclusions section.
Description of structure
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 micro-spaces
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 nano-scale). It is therefore desirable to channel the heat from the nano-scale exterior surface of the electronic device the micro- or larger space. Bifurcate single-walled 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 . 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 convection-driven 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 single-walled CNTs for efficiently conducting heat from nano- to micro-spaces.
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 single-walled 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 single-walled 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 single-wall 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 single-wall 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.
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 nano-scaled surface of electronic devices to the micro-space, 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 single-walled 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 micro-structures. It must be noted that the present work is an improvement from the Ref. , 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 single-walled 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 single-walled CNT elements of the branched structures, thereby reducing the interfacial thermal resistance [35–37], as well as to precisely control the scale of nanostructures.
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.
- Chen G: Nanoscale energy transport and conversion. New York: Oxford University Press; 2005.
- Tien CL: Microscale energy transfer, chemical and mechanical engineering. Boca RAton: CRC; 1997.
- Simons RE, Antonetti VW, Nakayama W, Oktay S: Heat transfer in electronic packages. In Microelectronics Packaging Handbook. New York: Chapman and Hall; 1997.
- Hong SH, Myung S: Nanotube electronics: a flexible approach to mobility. Nat Nanotechnol 2007, 2: 207–208. 10.1038/nnano.2007.89View Article
- Berber S, Kwon YK, Tománek D: Unusually high thermal conductivity of carbon nanotubes. Phys Rev Lett 2000, 84: 4613. 10.1103/PhysRevLett.84.4613View Article
- Pop E, Mann D, Wang Q, Goodson K, Dai HJ: Thermal conductance of an individual single-wall carbon nanotube above room temperature. Nano Lett 2006, 6: 96–100. 10.1021/nl052145fView Article
- Helgesen G, Knudsen KD, Pinheiro JP, Skjeltorp AT, Svåsand E, Heiberg-Andersen H, Elgsaeter A, Garberg T, Naess SN, Raaen S, Tverdal MF, Yu X, Mel TB: Carbon nanocones: a variety of non-crystalline graphite. Mater Res Soc Symp Proc 2007, 1057: 24–29.View Article
- Brinkmann G, Fowler PW, Manolopoulos DE, Palser AHR: A census of nanotube caps. Chem Phys Lett 1999, 315: 335347.View Article
- Shenderova OA, Lawson BL, Areshkin D, Brenner DW: Predicted structure and electronic properties of individual carbon nanocones and nanostructures assembled from nanocones and nanostructures assembled from nanocones. Nanotechnolo 2001, 12: 191–197. 10.1088/0957-4484/12/3/302View Article
- Reich S, Li L, Robertson J: Structure and formation energy of carbon nanotube caps. Phys Rev B 2005, 72: 165423.View Article
- Gong XJ, Li JY, Lu HJ, Wan RZ, Li JC, Hu J, Fang HP: A charge-driven molecular water pump. Nat Nanotechnol 2007, 2: 709–712. 10.1038/nnano.2007.320View Article
- Yang N, Zhang G, Li BW: Carbon nanocone: a promising thermal rectifier. Appl Phys Lett 2008, 93: 243111. 10.1063/1.3049603View Article
- Tu YS, Xiu P, Wan RZ, Hu J, Zhou RH, Fang HP: Water-mediated signal multiplication with Y-shaped carbon nanotubes. Proc Natl Acad Sci USA 2009, 106: 18120–18124. 10.1073/pnas.0902676106View Article
- Song J, Whang D, McAlpine MC, Friedman RS, Wu Y, Lieber CM: Scalable interconnection and integration of nanowire devices without registration. Nano Lett 2004, 4: 915–919. 10.1021/nl049659jView Article
- Kordas K, Tóth G, Moilanen P, Kumpumäki M, Vähäkangas J, Uusimäki A, Vajtai R, Ajayan PM: Chip cooling with integrated carbon nanotube microfin architectures. Appl Phys Lett 2007, 90: 123105. 10.1063/1.2714281View Article
- Gannon CJ, Cherukuri P, Yakobson BI, Cognet L, Kanzius JS, Kittrell C, Weisman RB, Pasquali M, Schmidt HK, Smalley RE, Curley SA: Carbon nanotube-enhanced thermal destruction of cancer cells in a noninvasive radiofrequency field. Cancer 2007, 110: 2654–2665. 10.1002/cncr.23155View Article
- Prasher R: Predicting the thermal resistance of nanosized constrictions. Nano Lett 2005, 5: 2155–2159. 10.1021/nl051710bView Article
- Dong H, Paramonov SE, Hartgerink JD: Self-assembly of α-helical coiled coil nanofibers. J Am Chem Soc 2008, 130: 13691–13695. 10.1021/ja8037323View Article
- Fratzl P, Weinkamer R: Nature's hierarchical materials. Prog Mater Sci 2007, 52: 1263–1334. 10.1016/j.pmatsci.2007.06.001View Article
- Alberts B, Johnson A, Lewis J, Raff M, Roberts K, Walter P: Molecular biology of the cell. New York: Garland Sciences, Taylor and Francis; 2002.
- Keten S, Buehler MJ: Geometric confinement governs the rupture strength of H-bond assemblies at a critical length scale. Nano Lett 2008, 8: 743–748. 10.1021/nl0731670View Article
- Xu Z, Buehler MJ: Hierarchical nanostructures are crucial to mitigate ultrasmall thermal point loads. Nano Lett 2009, 9: 2065–2072. 10.1021/nl900399bView Article
- Bejan A: Heat transfer. New York: Wiley; 1993.
- Ledezma GA, Bejan A, Errera MR: Constructal tree networks for heat transfer. J Appl Phys 1997, 82: 89100.View Article
- Li J, Papadopoulos C, Xu J: Growing Y-junction carbon nanotubes. Nature 2000, 402: 253–254.
- Wei DC, Liu YQ: Review-The intramolecular junctions of carbon nanotubes. Adv Mater 2008, 20: 2815–2841. 10.1002/adma.200800589View Article
- Li N, Chen XX, Stoica L, Xia W, Qian J, Aßmann J, Schuhmann W: The catalytic synthesis of three-dimensional hierarchical carbon nanotube composites with high electrical conductivity based on electrochemical iron deposition. Adv Mater 2007, 19: 2957–2960. 10.1002/adma.200602625View Article
- Xu P, Yu BM, Yun MJ, Zou MQ: Heat conduction in fractal tree-like branched networks. Int J Heat Mass Transfer 2006, 49: 3746–3751. 10.1016/j.ijheatmasstransfer.2006.01.033View Article
- Maruyama S: A molecular dynamics simulation of heat conduction in finite length SWNTs. Phys B 2002, 323: 193–195. 10.1016/S0921-4526(02)00898-0View Article
- Zhang G, Li BW: Thermal conductivity of nanotubes revisited: effects of chirality, isotope impurity, tube length, and temperature. J Chem Phys 2005, 123: 014705. 10.1063/1.1949166View Article
- Chang CW, Okawa D, Garcia H, Majumdar A, Zettl A: Breakdown of Fourier's Law in Nanotube Thermal Conductors. Phys Rev Lett 2008, 101: 075903.View Article
- Krishnan A, Dujardin E, Treacy MMJ, Hugdahl J, Lynum S, Ebbesen TW: Graphitic cones and the nucleation of curved carbon surfaces. Nature 1997, 388: 451–454. 10.1038/41284View Article
- Naess SN, Elgsaeter A, Helgesen G, Knudsen KD: Carbon nanocones: wall structure and morphology. Sci Technol Adv Mater 2009, 10: 065002. 10.1088/1468-6996/10/6/065002View Article
- Hu JQ, Bando Y, Zhan JH, Zhi CY, Xu FF, Golberg D: Tapered carbon nanotubes from activated carbon powders. Adv Mater 2006, 18: 197–200. 10.1002/adma.200501571View Article
- Meng GW, Han FM, Zhao XL, Chen BS, Yang DC, Liu JX, Xu QL, Kong MG, Zhu XG, Jung YJ, Yang YJ, Chu ZQ, Ye M, Kar S, Vajtai R, Ajayan PM: A general synthetic approach to interconnected nanowire/nanotube and nanotube/nanowire/nanotube heterojunctions with branched topology. Angew Chem Int Ed 2009, 48: 7166–7306. 10.1002/anie.200901999View Article
- Hobbie EK, Fagan JA, Becker ML, Hudson SD, Fakhri N, Pasquali M: Self-assembly of ordered nanowires in biological suspensions of single-wall carbon nanotubes. ACS Nano 2009, 3: 189–196. 10.1021/nn800609yView Article
- Xu ZP, Buehler MJ: Nanoengineering heat transfer performance at carbon nanotube interfaces. ACS Nano 2009, 3: 2767–2775. 10.1021/nn9006237View Article
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.