# Transport phenomena in micro- and nanoscales

Transport phenomena at dimensions between 1 and 100 μm are different from those at larger scales. At these scales, phenomena that are negligible at larger scales become dominant, but the macroscopic transport theory is still valid. One example of these phenomena in multiphase systemss is surface tension. In larger scale systems, the hydrostatic pressure or dynamic pressure effects may dominate over pressure drops caused by surface tension, but at smaller scales, the pressure drops caused by surface tension can dominate over hydrostatic and dynamic pressure effects. Transport phenomena in these scales are still regarded as macroscale, because classical theory of transport theory is still valid.

As systems scale down even further to the nanoscale, 1-100 nm, or for ultrafast process (e.g., materials processing using picosecond or femtosecond lasers) the fundamental theory used in larger scale systems breaks down because of fundamental differences in the physics. The purpose of research in micro-/nanoscale heat transfer is to exploit their differences from the macroscopic systems to create and improve materials, devices and systems. It is also the objective of this research to understand when system performance will begin to degrade because of the adverse effects of scaling. One large field of interest in which microscale heat transfer is of great interest is energy and thermal systems. Microscale energy and thermal systems include thin film fuel cells, thin film electrochemical cells, photon-to-electric devices, micro heat pipes, bio-cell derived power, and microscale radioisotopes (Peterson, 2004). Other devices that are of a slightly larger scale but require components on the microscale include miniaturized heat engines as well as certain combustion-driven thermal systems. These energy systems can power anything from MEMS sensors and actuators to cell phones. Thermal management of thermally based energy systems is extremely important. To emphasize its importance, consider a micro heat engine. The most general components of a micro heat engine are a compressor, a combustor, and a turbine. Generally, the combustor runs much cooler than the turbine, because the gases must expand in the combustor to drive the turbine’s shaft. However, the turbine and the compressor are linked by this shaft. The shorter the distance between the turbine and the compressor, the less thermal resistance through the shaft. Therefore, the temperature difference between the turbine and the compressor is much less, which decreases the system’s efficiency. Another form of energy conversion is thermoelectric energy conversion, in which thermal energy is directly converted to electricity. There are three thermoelectric effects: the Seebeck effect, the Peltier effect, and the Thomson effect. The Seebeck effect occurs when electrons flow from the hot side to the cold side of a material under a temperature gradient. The result is an electric potential field (voltage) balancing the electron diffusion. The Peltier effect occurs when heat is carried by electrons in an electrical current in a material held at a constant temperature. The Thomson effect is seen when current flows through a conductor under a temperature gradient. The suitability of a thermoelectric material for energy conversion is based on the figure of merit Z,

$Z = \frac{{{\alpha ^2}}}{{{R_e}k}}\qquad \qquad(1)$

where the Seebeck coefficient, electrical resistivity and thermal conductivity are α, Re and k, respectively. Materials with a high figure of merit are difficult to find in bulk form, therefore, nanostructures provide additional parameter space (Chen et al., 2004). To manipulate the nanostructures of certain materials, the electron and phonon thermoelectric transport must first be understood.

Conduction heat transfer in a solid at a small scale is analogous to the kinetic theory of gases (see Section 1.3.2). However, instead of molecules transporting momentum, there are electrons and phonons transferring heat. The classical heat conduction theory is a macroscopic model based on the equilibrium assumption. From the microscopic perspective, the energy carriers in a substance include phonons, photons, and electrons. Depending on the nature of heating and the structure of the materials, the energy can be deposited into materials in different ways: it can be simultaneously deposited to all carriers by direct contact, or only to a selected carrier by radiation (Qiu and Tien, 1993). For short-pulsed laser heating of metal, the energy deposition involves three steps: (1) deposition of laser energy on electrons, (2) exchange of energy between electrons, and (3) propagation of energy through media. If the pulse width is shorter than the thermalization time, which is the time it takes for the electrons and lattice to reach equilibrium, the electron and lattice are not in thermal equilibrium, and a two-temperature model is often used. If the laser pulse is shorter than the relaxation time, which is the mean time required for electrons to change their states, the hyperbolic conduction model must be used. More insights about the microscale heat transfer can be found in Tzou (1996) and Majumdar (1998). In addition to the two-temperature and hyperbolic models, which are still continuum models, another approach is to understand heat and mass transfer at the molecular level using the molecular dynamics method (Maruyama, 2001).

The conduction of heat is caused by electrons and by phonons. Therefore, the thermal conductivity, k, in solids can be broken into two components, the thermal conduction by electrons (ke) and by phonons (kph).

$k = {k_{{e^ - }}} + {k_{ph}}\qquad \qquad(2)$

From kinetic theory, the thermal conduction of each component is given by (Flik et al. 1992),

${k_{{e^ - }}} = \frac{1}{3}{c_{p,{e^ - }}}{\bar c_{{e^ - }}}{\lambda _{{e^ - }}}\qquad \qquad(3)$
${k_{ph}} = \frac{1}{3}{c_{p,ph}}{\bar c_{ph}}{\lambda _{ph}}\qquad \qquad(4)$

The subscripts e- and ph refer to an electron and a phonon, respectively. The specific heats of the electron and the phonon are ${c_{p,{e^ - }}}$ and cp,ph, respectively. The average velocities of an electron and a phonon are ${\bar c_{{e^ - }}}$ and ${\bar c_{ph}}$. A phonon is a sound particle, and the speed at which it travels is the speed of sound in that material. Finally, the mean free paths of an electron and a phonon are ${\lambda _{{e^ - }}}$ and λph, respectively. The mean free path is the distance one of the conduction energy carriers (electrons or phonons) travels before it collides with an imperfection in a material. Defects, dislocations, impurities and boundaries within a solid structure all have an effect on the phonon transport in a solid. The effect of impurities in a solid material can be described in terms of the acoustic impedance (Z) of the lattice waves (Huxtable et al., 2004). The acoustic impedance is

$Z = \rho {\bar c_{ph}}\qquad \qquad(5)$

The speed of sound (mean velocity of a phonon) is related to the elastic stiffness of a chemical bond (E) by

${\bar c_{ph}} = \sqrt {\frac{E}{\rho }}\qquad \qquad(6)$

The acoustic impedance is a function of both the stiffness of a bond and the density. When a phonon encounters a change in the acoustic impedance, it may scatter. Scattering of this nature can change the direction in which a phonon is traveling and also its mean free path or wavelength. The wavelength or mean free path of a material also has temperature dependence that can be approximated by

${\lambda _{ph}} \approx \frac{{h{{\bar c}_{ph}}}}{{3{k_B}T}}\qquad \qquad(7)$

where h is Planck’s constant. From this equation, it can be seen that the mean free path has an inverse dependence on temperature. Therefore, with a decrease in temperature, the wavelength will increase, which increases the probability that a particle will be affected by either an imperfection or a boundary.

Mean free path of electrons or phonons (a) far away from boundaries and (b) in the presence of boundaries.

Physical boundaries in a material have the effect of scattering the energy carriers. The effect of a boundary on the mean free path is presented in the figure on the right. The scattering characteristics of boundaries either reflect or transmit energy carriers. The probability that a phonon will transmit from material A to material B, ${P_{A \to B}}$, at normal incidence is a function of the impedance of both materials.

${P_{A \to B}} = \frac{{4{Z_A}{Z_B}}}{{{{\left( {{Z_A} + {Z_B}} \right)}^2}}}\qquad \qquad(8)$

The effect of boundaries can reduce the effective mean free path in the vertical and horizontal directions for phonons traveling at all incident angles. Examining eqs. (3) and (4) indicates that reducing the mean free path will reduce the thermal conductivity. Physical boundaries in a system can be grain boundaries or the surfaces of an extremely thin solid film. The effective conductivity in a thin film was related to the isotropic bulk conductivity of a material by Flik and Tien (1990). The effective conductivities normal to the thin film keff,n and along the thin film layer keff,t are:

$\frac{{{k_{eff,n}}}}{k} = 1 - \frac{\lambda }{{3\delta }}\qquad \qquad(9)$
$\frac{{{k_{eff,t}}}}{k} = 1 - \frac{{2\lambda }}{{3\pi \delta }}\qquad \qquad(10)$

where δ is the film thickness.

The electrical component of thermal conductivity in a solid can also be approximated by the Wiedemann-Franz law, which is valid up to the metal-insulator transition (Castellani et al., 1987). This transition occurs when the electrical conductivity of a metal suddenly changes from high to low conductivity due to a decrease in temperature. This equation is valid because, in a highly electrically conductive material, essentially all of the thermal transport is carried through electrons.

$k = {\sigma _{{e^{ - 1}}}}{L_0}T\qquad \qquad(11)$

where L0 is the Lorenz number and ${\sigma _{{e^ - }}}$ is the electrical conductivity. For semi-conductor materials, it is expected that the contribution to heat conduction by the electrons will be small, because the electrical conductivity is small. Therefore, molecular dynamics simulations have been used to predict the effect of system size on the thermal conductivity by considering only the phonon transport. Such analyses were done by Petzsch and Böttger (1994), Schelling et al. (2002) and McGaughey and Kaviany (2004). The methods used in these studies are nonequilibrium molecular dynamics (NEMD) or equilibrium molecular dynamics (EMD). The NEMD approach to computing thermal conductivity is called the “direct method,” which imposes a temperature gradient across a simulation cell and is analogous to the experimental set-up. However, due to computational capacity, the cell is very small, resulting in extremely high temperature gradients in which Fourier’s law of heat conduction may break down. A commonly used approach to EMD is the Green-Kubo method, which uses current fluctuations to compute thermal conductivity by the fluctuation-dissipation theorem. This theorem captures the linear response of a system subjected to an external perturbation and is expressed in terms of fluctuation properties of the thermal equilibrium.

Heat conduction in a liquid is a combination of the random movement of molecules, similar to the kinetic theory of gases, as well as the movement of phonons and/or electrons. Therefore, thermal transport through conduction is much more complex in the liquid state than the gaseous or solid states; the theory is in its infancy, and is therefore not included.

The discussion of microscale and nanoscale systems thus far has been primarily theoretical. Therefore, a discussion of experimental measurements of these systems is needed. The most common tool for sensing and actuating at the nanometer scale is atomic force microscopy (AFM) (King and Goodson, 2004). The basic structural design of an AFM consists of a micromachined tip at the end of a cantilever beam. A motion control stage is attached to the cantilever beam. The motion control stage brings the tip into contact with a surface, and moves the tip laterally over the surface. As the tip follows the surface, small changes in the vertical position of the cantilever beam are detected. The result is a topographic map of a surface with a resolution as good as 1 nm. The AFM can be used to intentionally modify a surface over which it scans in order to study the effects of certain modifications. This research includes local chemical delivery, thermally-assisted indentation of soft materials, direct indentation of soft materials, and guiding electromagnetic radiation into photoreactive polymers.

## References

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Chen, G., 2004, Nano-To-Macro Thermal Transport, Oxford University Press.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Flik, M.I., Choi, B.I. and Goodson, K.E., 1992, “Heat Transfer Regimes in Microstructures,” ASME Journal of Heat Transfer, Vol. 114, pp. 666-674.

Huxtable, S.T., Abramson, A.R. and Majumdar, A., 2004, “Heat Transport in Superlattices and Nanowires,” Heat and Fluid Flow in Microscale and Nanoscale Structures, eds. Faghri, M. and Sundén, B., Chapter 3, Southampton, UK.

King, W.P. and Goodson, K.E., 2004, “Thermomechanical Formation and Thermal Detection of Polymer Nanostructures,” Heat and Fluid Flow in Microscale and Nanoscale Structures, eds. Faghri, M. and Sundén, B., Chapter 4, Southampton, UK.

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Maruyama, S., 2001, “Molecular Dynamics Method for Microscale Heat Transfer,” Advances in Numerical Heat Transfer, edited by Minkowycz, W.J., and Sparrow, E.M., Taylor & Francis, New York, pp. 189-226.

McGaughey, A.J.H. and Kaviany, M., 2004, “Quantitative Validation of the Boltzmann Transport Equation Phonon Thermal Conductivity Model under the Single-Mode Relaxation Time Approximation,” Physical Review B, Vol., 69, 094303.

Peterson, R.B., 2004, “Miniature and Microscale Energy Systems,” Heat and Fluid Flow in Microscale and Nanoscale Structures, eds. Faghri, M. and Sundén, B., Chapter 1, Southampton, UK.

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