Mesoscopic description of boundary effects in nanoscale heat transport
F.X. Àlvarez; V.A. Cimmelli; D. Jou; A. Sellitto
Nanoscale Systems: Mathematical Modeling, Theory and Applications (2012)
- Volume: 1, page 112-142
- ISSN: 2299-3290
Access Full Article
topAbstract
topHow to cite
topF.X. Àlvarez, et al. "Mesoscopic description of boundary effects in nanoscale heat transport." Nanoscale Systems: Mathematical Modeling, Theory and Applications 1 (2012): 112-142. <http://eudml.org/doc/266781>.
@article{F2012,
abstract = {We review some of the most important phenomena due to the phonon-wall collisions in nonlocal heat transport in nanosystems, and show how they may be described through certain slip boundary conditions in phonon hydrodynamics. Heat conduction in nanowires of different cross sections and in thin layers is analyzed, and the dependence of the thermal conductivity on the geometry, as well as on the roughness is pointed out. We also analyze the effects of the roughness of the surface of the pores on the thermal conductivity of porous silicon. Thermoelectric effects are considered as well. In memory of Professor Carlo Cercignani},
author = {F.X. Àlvarez, V.A. Cimmelli, D. Jou, A. Sellitto},
journal = {Nanoscale Systems: Mathematical Modeling, Theory and Applications},
keywords = {Phonon hydrodynamics; Nanosystems; Phonon-wall collisions; Effective thermal conductivity; Nonlocal heat transport; phonon hydrodynamics; nanosystems; phonon-wall collisions; effective thermal conductivity; nonlocal heat transport},
language = {eng},
pages = {112-142},
title = {Mesoscopic description of boundary effects in nanoscale heat transport},
url = {http://eudml.org/doc/266781},
volume = {1},
year = {2012},
}
TY - JOUR
AU - F.X. Àlvarez
AU - V.A. Cimmelli
AU - D. Jou
AU - A. Sellitto
TI - Mesoscopic description of boundary effects in nanoscale heat transport
JO - Nanoscale Systems: Mathematical Modeling, Theory and Applications
PY - 2012
VL - 1
SP - 112
EP - 142
AB - We review some of the most important phenomena due to the phonon-wall collisions in nonlocal heat transport in nanosystems, and show how they may be described through certain slip boundary conditions in phonon hydrodynamics. Heat conduction in nanowires of different cross sections and in thin layers is analyzed, and the dependence of the thermal conductivity on the geometry, as well as on the roughness is pointed out. We also analyze the effects of the roughness of the surface of the pores on the thermal conductivity of porous silicon. Thermoelectric effects are considered as well. In memory of Professor Carlo Cercignani
LA - eng
KW - Phonon hydrodynamics; Nanosystems; Phonon-wall collisions; Effective thermal conductivity; Nonlocal heat transport; phonon hydrodynamics; nanosystems; phonon-wall collisions; effective thermal conductivity; nonlocal heat transport
UR - http://eudml.org/doc/266781
ER -
References
top- C. C. Ackerman and R. A. Guyer. Temperature pulses in dielectric solids. Annals of Physics 50, 128 (1968).
- F. X. Àlvarez, D. Jou, and A. Sellitto. Phonon hydrodynamics and phonon-boundary scattering in nanosystems. J. Appl. Phys. 105, 014317 (2009).
- F. X. Àlvarez, D. Jou, and A. Sellitto. Pore-size dependence of the thermal conductivity of porous silicon: a phonon hydrodynamic approach. Appl. Phys. Lett. 97, 033103 (2010).
- F. X. Àlvarez, D. Jou, and A. Sellitto. Phonon boundary effects and thermal conductivity of rough concentric nanowires. J. Heat Transfer 133, 022402 (2011).
- V. M. Aroutiounian and M. Zh. Ghulinyan. Electrical conductivity mechanisms in porous silicon. Phys. Status Solidi A 197, 462 (2003).
- M. Asheghi, Y. K. Leung, S. S. Wong, and K. E. Goodson. Phonon-boundary scattering in thin silicon layers. Appl. Phys. Lett. 71, 1798 (1997).
- C. L. Bailey, R. W. Barber, D. R. Emerson, D. A. Lockerby, and J. M. Reese. A critical review of the drag force on a sphere in the transition flow regime. AIP Conf. Proc. 762, 743 (2005).
- J. R. Baird, D. F. Fletcher, and B. S. Haynes. Local condensation heat transfer rates in fine passages. Int. J. Heat Mass Transfer 46, 4453 (2003).
- A. Balandin and K. L. Wang. Effect of phonon confinement on the thermoelectric figure of merit of quantum wells. J. Appl. Phys. 84, 6149 (1998).
- L. E. Bell. Cooling, heating, generating power, and recovering waste heat with thermoelectric systems. Science 5895, 1457 (2008).
- G. Benedetto, L. Boarino, and R. Spagnolo. Evaluation of thermal conductivity of porous silicon layers by a photoacoustic method. Appl. Phys. A: Mater. Sci. Process. 64, 155 (1997).
- G. Bergmann. Conductance of a perfect thin film with diffuse surface scattering. Phys. Rev. Lett. 94, 106801 (2005). [PubMed]
- A. I. Boukai, Y. Bunimovich, J. Tahir-Kheli, J.-K. Yu, W. A. Goddard-III, and J. R. Heath. Silicon nanowires as efficient thermoelectric materials. Nature 451, 168 (2008).
- H. C. Brinkman. A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27 (1947). Zbl0041.54204
- H. Bruus. Theoretical microfluidics. Oxford University Press, Oxford (2007).
- A. Burgdorfer. The influence of the molecular mean path on the performance of hydrodynamic gas lubricated bearings. J. Basic Eng. - T. ASME 8, 94 (1959).
- L. T. Canham. Silicon quantum wire fabrication by electrochemical and chemical dissolution of wafers. Appl. Phys. Lett. 57, 1046 (1990).
- B.-Y. Cao and Z.-Y. Guo. Equation of motion of a phonon gas and non-Fourier heat conduction. J. Appl. Phys. 102, 053503 (2007).
- C. Cattaneo. Sulla conduzione del calore. Atti Sem. Mat. Fis. Univ. Modena 3, 83 (1948). Zbl0035.26203
- C. Cattaneo. A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Comput. Rend. 247, 431 (1958).
- C. Cercignani. Higher order slip according to the linearized Boltzmann equation. California Univ. Berkeley Inst. of Engineering Research, Report AS-64-19, Berkeley (1964).
- C. Cercignani. Slow Rarefied Flows - Theory and Application to Micro-Electro-Mechanical Systems. Birkhäuser Verlag, Basel (2006). Zbl1097.82001
- G. Chen. Nanoscale Energy Transport and Conversion - A Parallel Treatment of Electrons, Molecules, Phonons, and Photons. Oxford University Press, Oxford (2005).
- J. D. Chung, , and M. Kaviany. Effects of phonon pore scattering and pore randomness on effective conductivity of porous silicon. Int. J. Heat Mass Transfer 43, 521 (2000). Zbl1052.74570
- V. A. Cimmelli. Different thermodynamic theories and different heat conduction laws. J. Non-Equilib. Thermodyn. 34, 299 (2009). Zbl1195.80001
- V. A. Cimmelli and K. Frischmuth. Nonlinear effects in thermal wave propagation near zero absolute temperature. Physica B 355, 147 (2005).
- V. A. Cimmelli and K. Frischmuth. Gradient generalization to the extended thermodynamic approach and diffusivehyperbolic heat conduction. Physica B 400, 257 (2007).
- V. A. Cimmelli, A. Sellitto, and D. Jou. Nonlocal effects and second sound in a nonequilibrium steady state. Phys. Rev. B 79, 014303 (2009). Zbl1236.82050
- V. A. Cimmelli, A. Sellitto, and D. Jou. Nonequilibrium temperatures, heat waves, and nonlinear heat transport equations. Phys. Rev. B 81, 054301 (2010). Zbl1213.80011
- V. A. Cimmelli, A. Sellitto, and D. Jou. Nonlinear evolution and stability of the heat flow in nanosystems: Beyond linear phonon hydrodynamics. Phys. Rev. B 82, 184302 (2010). [Crossref] Zbl1213.80011
- J. E. Cornett and O. Rabin. Thermoelectric figure of merit calculations for semiconducting nanowires. Appl. Phys. Lett. 98, 182184 (2011).
- E. Cunningham. On the velocity of steady fall of spherical particles through fluid medium. Proc. R. Soc. Lond. A 83, 357 (1910). Zbl41.0843.05
- V. Dobrosavljevic and G. Kotliar. Mean field theory of the Mott-Anderson transition. Phys. Rev. Lett. 78, 3943 (1997).
- Y. Dong, B.-Y. Cao, and Z.-Y. Guo. Generalized heat conduction laws based on thermomass theory and phonon hydrodynamics. J. Appl. Phys. 110, 063504 (2011).
- S. Volz (ed). Thermal Nanosystems and Nanomaterials. Springer, Berlin (2010).
- M. S. El-Genk and H. H. Saber. High efficiency segmented thermoelectric unicouple for operation between 973 and 300 K. Energy Conversion and Management 44, 1069 (2003).
- J. Fang and L. Pilon. Scaling laws for thermal conductivity of crystalline nanoporous silicon based on molecular dynamics simulations. J. Appl. Phys. 110, 064305 (2011).
- J. Fang and L. Pilon. Tuning thermal conductivity of nanoporous crystalline silicon by surface passivation: A molecular dynamics study. Appl. Phys. Lett. 101, 011909 (2012).
- J. Fang, C. B. Kang, Y. Huang, S. H. Tolbert, and L. Pilon. Thermal conductivity of ordered mesoporous nanocrystalline silicon thin films made from magnesium reduction of polymer-templated silica. J. Phys. Chem. C, 116, 12926 (2012).
- D. K. Ferry and S. M. Goodnick. Transport in Nanostructures. Cambridge University Press, Cambridge, England, second edition (2009).
- Y.-C. Fung. Biomechanics: circulation. Springer-Verlag, New York, second edition (1997).
- A. M. García-García. Classical intermittency and quantum Anderson transition. Phys. Rev. E 69, 066216 (2004).
- G. Gesele, J. Linsmeier, V. Drach, J. Fricke, and R. Arens-Fischer. Temperature-dependent thermal conductivity of porous silicon. J. Phys. D: Appl. Phys. 20, 2911 (1997).
- T. I. Gombosi. Gaskinetic Theory. Cambridge University Press, Cambridge (1994).
- I. Graur and F. Sharipov. Gas flow through an elliptical tube over the whole range of gas rarefaction. Eur. J. Mech. B/Fluids 27, 335 (2008). Zbl1154.76376
- Z.-Y. Guo. Motion and Transfer of Thermal Mass - Thermal Mass and Thermon Gas. J. Eng. Term 27, 631 (2006).
- Z.-Y. Guo and Q.-W. Hou. Thermal wave based on the thermomass model. J. Heat Trans - T. ASME 132, 072403 (2010).
- Z.-Y. Guo, B.-Y. Cao, H.-Y. Zhu, and Q.-G. Zhang. State equation of phonon gas and conservation equations for phonon gas motion. Acta Phys. Sin. 56, 3306 (2007).
- R. A. Guyer and J. A. Krumhansl. Solution of the linearized phonon Boltzmann equation. Phys. Rev. 148, 766 (1966).
- R. A. Guyer and J. A. Krumhansl. Thermal conductivity, second sound and phonon hydrodynamic phenomena in nonmetallic crystals. Phys. Rev. 148, 778 (1966).
- N. G. Hadjiconstantinou. Comment on Cercignani’s second-order slip coefficient. Phys. Fluids 15, 2352 (2003).
- L. D. Hicks and M. S. Dresselhaus. Effect of quantum-well structures on the thermoelectric figure of merit. Phys. Rev. B 47, 12727 (1993). [Crossref]
- L. D. Hicks and M. S. Dresselhaus. Thermoelectric figure of merit of a one-dimensional conductor. Phys. Rev. B 47, 16631 (1993). [Crossref]
- R. J. Hill, D. L. Koch, and A. J. C. Ladd. Moderate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 243 (2001). Zbl0997.76068
- A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. Yang. Enhanced thermoelectric performance of rough silicon nanowires. Nature 451, 163 (2008).
- P. E. Hopkins, P. M. Norris, L. M. Phinney, S. A. Policastro, and R. G. Kelly. Thermal Conductivity in Nanoporous Gold Films during Electron-Phonon Nonequilibrium. Journal of Nanomaterials 2008, 418050 (2008).
- Y. T. Hsia and G. A. Domoto. An experimental investigation of molecular rarefaction effects in gas lubricated bearings at ultra-low clearance. J. Lubr. Technol - T. ASME 105, 120 (1983).
- M.-J. Huanga, R.-H. Yena, and A.-B. Wang. The influence of the Thomson effect on the performance of a thermoelectric cooler. Int. J. Heat Mass Transfer 48, 413 (2005).
- K. Imai. A new dielectric isolation method using porous silicon. Solid-State Electronics 24, 159 (1981).
- G. Joshi, H. Lee, Y. Lan, X. Wang, G. Zhu, D. Wang, R. W. Gould, D. C. Cuff, M. Y. Tang, M. S. Dresselhaus, G. Chen, and Z. Ren. Enhanced Thermoelectric Figure-of-Merit in Nanostructured p-type Silicon Germanium Bulk Alloys. Nano Lett. 8, 4670 (2008). [PubMed]
- D. Jou and L. Restuccia. Mesoscopic transport equations and contemporary thermodynamics: an introduction. Contemp. Phys. 52, 465 (2011).
- D. Jou, J. Casas-Vázquez, and G. Lebon. Extended irreversible thermodynamics revisited (1988-1998). Rep. Prog. Phys. 62, 1035 (1999). Zbl0852.73005
- D. Jou, J. Casas-Vázquez, and G. Lebon. Extended Irreversible Thermodynamics. Springer, Berlin, fourth revised edition (2010). Zbl1185.74002
- D. Jou, M. Criado-Sancho, and J. Casas-Vázquez. Heat fluctuations and phonon hydrodynamics in nanowires. J. Appl. Phys. 107, 084302 (2010).
- D. Jou, G. Lebon, and M. Criado-Sancho. Variational principles for thermal transport in nanosystems with heat slip flow. Phys. Rev. E 82, 031128 (2010). [Crossref]
- D. Jou, A. Sellitto, and F. X. Àlvarez. Heat waves and phonon-wall collisions in nanowires. Proc. R. Soc. A 467, 2520 (2011).
- D. Jou, V. A. Cimmelli, and A. Sellitto. Nonlocal heat transport with phonons and electrons: Application to metallic nanowires. Int. J. Heat Mass Transfer 55, 2338 (2012). Zbl1254.82030
- E. H. Kennard. Kinetic Theory of Gases. McGraw-Hill, New York (1938).
- J. Y. Kim and B. J. Yoon. The effective conductivities of composites with cubic arrays of spheroids and cubes. J. of Composite Materials 33, 1344 (1999).
- V. I. Kushch. Conductivity of a periodic particle composite with transversely isotropic phases. Proc. R. Soc. A 453, 65 (1997). Zbl0888.73037
- G. Lebon, D. Jou, J. Casas-Vázquez, and W. Muschik. Weakly nonlocal and nonlinear heat transport in rigid solids. J. Non-Equilib. Thermodyn. 23, 176 (1998). Zbl0912.73006
- G. Lebon, D. Jou, and J. Casas-Vázquez. Understanding nonequilibrium thermodynamics. Springer, Berlin (2008). [ Zbl1163.80001
- G. Lebon, H. Machrafi, M. Grmela, and Ch. Dubois. An extended thermodynamic model of transient heat conduction at sub-continuum scales. Proc. R. Soc. A 467, 3241 (2011). Zbl1239.80004
- G. Lebon, D. Jou, and P. C. Dauby. Beyond the Fourier heat conduction law and the thermal non-slip condition. Phys. Lett. A 376, 2842 (2012). Zbl1266.82062
- H. Lee, D. Vashaee, D. Z. Wang, M. S. Dresselhaus, Z. F. Ren, and G. Chen. Effects of nanoscale porosity on thermoelectric properties of SiGe. J. Appl. Phys. 107, 094308 (2010).
- J.-H. Lee, J. C. Grossman, J. Reed, and G. Galli. Lattice thermal conductivity of nanoporous Si: Molecular dynamics study. Appl. Phys. Lett. 91, 223110 (2007).
- D. Li, Y. Wu, P. Kim, L. Shi, P. Yang, and A. Majumdar. Thermal conductivity of individual silicon nanowires. Appl. Phys. Lett. 83, 2934 (2003).
- R. L. Liboff. Kinetic theory (classical, quantum and relativistic description). Prentice Hall, Englewood Cliffs, New Jersey (1990).
- W. Liu and M. Asheghi. Phonon-boundary scattering in ultrathin single-crystal silicon layers. Appl. Phys. Lett. 84, 3819 (2004).
- H. Looyenga. Dielectric constants of heterogeneous mixtures. Physica 31, 401 (1965).
- R. Luzzi, Á. R. Vasconcellos, and J. Galv ao Ramos. Predictive Statistical Mechanics: A Nonequilibrium Ensemble Formalism (Fundamental Theories of Physics). Kluwer Academic Publishers, Dordrecht (2002). Zbl0992.70001
- F. Márkus and K. Gambár. Heat propagation dynamics in thin silicon layers. Int. J. Heat Mass Transfer 56, 495 (2013).
- P. Martin, Z. Aksamija, E. Pop, and U. Ravaioli. Impact of phonon-surface roughness scattering on thermal conductivity of thin Si nanowires. Phys. Rev. Lett. 102, 125503 (2009). [PubMed]
- D. Mattia and F. Calabrò. Explaining high flow rate of water in carbon nanotubes via solid-liquid molecular interactions. Microfluid Nanofluid 13, 125 (2012).
- R. A. Millikan. The general law of fall of a small spherical body through a gas, and its bearing upon the nature of molecular reflection from surfaces. Phys. Rev. 22, 1 (1923). [Crossref]
- N. Mingo. Thermoelectric figure of merit and maximum power factor in III-V semiconductor nanowires. Appl. Phys. Lett. 84, 2652 (2004).
- N. Mingo. Thermoelectric figure of merit of II-VI semiconductor nanowires. Appl. Phys. Lett. 85, 5986 (2004).
- G. Mistura. Microfluidica, quando contaminarsi fa bene alla fisica. http://www.pd.infn.it/M5P/abstracts/mistura.htm (accessed 4 December 2012).
- Y. Mitsuya. Modified Reynolds equation for ultra-thin film gas lubrication using 1.5-order slip-flow model and considering surface accommodation coefficient. J. Tribol - T. ASME 115, 289 (1993).
- D. Nakamura, M. Murata, H. Yamamoto, Y. Hasegawa, and T. Komine. Thermoelectric properties for single crystal bismuth nanowires using a mean free path limitation model. J. Appl. Phys. 110, 053702 (2011).
- A. Pattamatta and C. K. Madnia. Modeling heat transfer in Bi2Te3-Sb2Te3 nanostructures. Int. J. Heat Mass Transfer 52, 860 (2009). Zbl1156.82400
- B. Poudel, Q. Hao, Y. Ma, Y. Lan, A. Minnich, B. Yu, X. Yan, D. Wang, A. Muto, D. Vashaee, X. Chen, J. Liu, M. S. Dresselhaus, G. Chen, and Z. Ren. High-Thermoelectric Performance of Nanostructured Bismuth Antimony Telluride Bulk Alloys. Science 320, 634 (2008).
- B. Qui, L. Sun, and X. Ruan. Lattice thermal conductivity reduction in Bi2Te3 quantum wires with smooth and rough surfaces: A molecular dynamics study. Phys. Rev. B 83, 035312 (2011).
- N. A. Roberts, D.G. Walker, and D. Y. Li. Molecular dynamics simulation of thermal conductivity of nanocrystalline composite films. Int. J. Heat Mass Transfer 52, 2002 (2009). Zbl1157.80365
- H. W. Russell. Principles of heat flow in porous insulators. J. Am. Ceram. Soc. 18, 1 (1935). [Crossref]
- A. S. Sangani and A. Acrivos. Slow flow through a periodic array of spheres. Int. J. Multiphase Flow 8, 343 (1982). Zbl0541.76041
- A. S. Sangani and A. Acrivos. The effective conductivity of a periodic array of spheres. Proc. R. Soc. A 386, 263 (1983). Zbl0569.76100
- A. S. Sangani and A. Acrivos. Creeping flow through cubic arrays of spherical bubbles. Int. J. Multiphase Flow 9, 181 (1983). Zbl0569.76100
- C. B. Satterthwaite and R. W. Jr Ure. Electrical and thermal properties of Bi2Te3. Phys. Rev. 108, 1164 (1957). [Crossref]
- A. Sellitto and V. A. Cimmelli. A continuum approach to thermomass theory. J. Heat Trans. T. - ASME 134, 112402 (2012). Zbl1259.80006
- A. Sellitto, F. X. Àlvarez, and D. Jou. Second law of thermodynamics and phonon-boundary conditions in nanowires. J. Appl. Phys. 107, 064302 (2010).
- A. Sellitto, F. X. Àlvarez, and D. Jou. Temperature dependence of boundary conditions in phonon hydrodynamics of smooth and rough nanowires. J. Appl. Phys. 107, 114312 (2010).
- A. Sellitto, F. X. Àlvarez, and D. Jou. Phonon-wall interactions and frequency-dependent thermal conductivity in nanowires. J. Appl. Phys. 109, 064317 (2011).
- A. Sellitto, F. X. Àlvarez, and D. Jou. Geometrical dependence of thermal conductivity in elliptical and rectangular nanowires. Int. J. Heat Mass Transfer 55, 3114 (2012).
- A. Sellitto, V. A. Cimmelli, and D. Jou. Analysis of three nonlinear effects in a continuum approach to heat transport in nanosystems. Physica D 241, 1344 (2012). Zbl1259.80006
- A. Sellitto, D. Jou, and J. Bafaluy. Nonlocal effects in radial heat transport in silicon thin layers and graphene sheets. Proc. R. Soc. A 468, 1217 (2012).
- A. Sellitto, D. Jou, and V. A. Cimmelli. A phenomenological study of pore-size dependent thermal conductivity of porous silicon. Acta Appl. Math. 122, 435 (2012). Zbl1254.74037
- A. Sellitto, V. A. Cimmelli, and D. Jou. Thermoelectric effects and size dependency of the figure-of-merit in cylindrical nanowires. Int. J. Heat Mass Transfer 57, 109 (2013).
- G. J. Snyder and T. S. Ursell. Thermoelectric efficiency and compatibility. Phys. Rev. Lett. 91, 148301 (2003). [PubMed]
- D. Song and G. Chen. Thermal conductivity of periodic microporous silicon films. Appl. Phys. Lett. 84, 687 (2004).
- N. Stojanovic, D. H. S. Maithripala, J. M. Berg, and M. Holtz. Thermal conductivity in metallic nanostructures at high temperature: electrons, phonons, and the Wiedemann-Franz law. Phys. Rev. B 82, 075418 (2010).
- B. Straughan. Heat waves. Springer, Berlin (2011). Zbl1232.80001
- H. Struchtrup. Macroscopic transport equations for rarefied gas flows: approximation methods in kinetic theory - Interaction of Mechanics and Mathematics. Springer, New York (2005). Zbl1119.76002
- J. Sturm, P. Grosse, and W. Theiss. Effective dielectric functions of alkali halide composites and their spectral representation. Z. Phys. B: Condens. Matter 83, 361 (1991).
- P. J. Taylor, M. P. Walsh, and B. E. La Forge. Quantum dot superlattice thermoelectric materials and devices. Science 297, 2229 (2002).
- Z. Tešanovic, M. V. Jaric, and S. Maekawa. Quantum transport and surface scattering. Phys. Rev. Lett. 57, 2760 (1986).
- D. Y. Tzou. Macro to micro-scale heat transfer. The lagging behaviour. Taylor and Francis, New York (1997).
- D. Y. Tzou. Nonlocal behavior in phonon transport. Int. J. Heat Mass Transfer 54, 475 (2011). Zbl1205.80051
- D. Y. Tzou and Z.-Y. Guo. Nonlocal behavior in thermal lagging. Int. J. Thermal Sci. 49, 1133 (2010).
- A. Uhlir. Electrolytic shaping of germanium and silicon. Bell Syst. Techn. J. 35, 333 (1956).
- F. T. Vasko and O. E. Raichev. Quantum Kinetic Theory And Applications: Electrons, Photons, Phonons. Springer, New York (2005). Zbl1125.82002
- Y. Wanatabe, Y. Arita, T. Yokoyama, and Y. Igarashi. Formation and properties of porous silicon and its applications. J. Electrochem. Soc: Solid-State Science and Technology 122, 1351 (1975).
- M. Wang, B.-Y. Cao, and Z.-Y Guo. General heat conduction equations based on the thermomass theory. Frontiers Heat Mass Transfer 1, 013004 (2010).
- M. Wang, N. Yang, and Z.-Y. Guo. Non-Fourier heat conductions in nanomaterials. J. Appl. Phys. 110, 064310 (2011).
- L. Wu. A slip model for rarefied gas flows at arbitrary knudsen number. Appl. Phys. Lett. 93, 253103 (2008).
- Z. M. Zhang. Nano/Microscale heat transfer. McGraw-Hill, New York (2007).
- V. M. Zhdanov and V. I. Roldughin. Non-equilibrium thermodynamics and kinetic theory of rarefied gases. Phys. Usp. 41, 349 (1998).
- J. M. Zide, D. O. Klenov, S. Stemmer, A. C. Gossard, G. Zeng, J. E. Bowers, D. Vashaee, and A. Shakouri. Thermoelectric power factor in semiconductors with buried epitaxial semimetallic nanoparticles. Appl. Phys. Lett. 87, 112102 (2005).
- J. M. Ziman. Electrons and Phonons. Oxford University Press, Oxford (2001). Zbl0983.82002
Citations in EuDML Documents
topNotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.