Experimental Micro/Nanoscale Thermal Transport E-Book

Xinwei Wang

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 488

Veröffentlichungsjahr: 2012

Beschreibung

This book covers the new technologies on micro/nanoscale thermal characterization developed in the Micro/Nanoscale Thermal Science Laboratory led by Dr. Xinwei Wang. Five new non-contact and non-destructive technologies are introduced: optical heating and electrical sensing technique, transient electro-thermal technique, transient photo-electro-thermal technique, pulsed laser-assisted thermal relaxation technique, and steady-state electro-Raman-thermal technique. These techniques feature significantly improved ease of implementation, super signal-to-noise ratio, and have the capacity of measuring the thermal conductivity/diffusivity of various one-dimensional structures from dielectric, semiconductive, to metallic materials.

Leseprobe

Table of Contents

Title Page

Copyright

Preface

Chapter 1: Introduction

1.1 Unique Feature of Thermal Transport in Nanoscale and Nanostructured Materials

1.2 Molecular Dynamics Simulation of Thermal Transport at Micro/Nanoscales

1.3 Boltzmann Transportation Equation for Thermal Transport Study

1.4 Direct Energy Carrier Relaxation Tracking (DECRT)

1.5 Challenges in Characterizing Thermal Transport at Micro/Nanoscales

References

Chapter 2: Thermal Characterization in Frequency Domain

2.1 Frequency Domain Photoacoustic (PA) Technique

2.2 Frequency Domain Photothermal Radiation (PTR) Technique

2.3 Three-Omega Technique

2.4 Optical Heating Electrical Thermal Sensing (OHETS) Technique

2.5 Comparison Among the Techniques

References

Chapter 3: Transient Technologies in the Time Domain

3.1 Transient Photo-Electro-Thermal (TPET) Technique

3.2 Transient Electrothermal (TET) Technique

3.3 Pulsed Laser-Assisted Thermal Relaxation Technique

3.4 Super Channeling Effect for Thermal Transport in Micro/Nanoscale Wires

3.5 Multidimensional Thermal Characterization

3.6 Remarks on the Transient Technologies

References

Chapter 4: Steady-State Thermal Characterization

4.1 Generalized Electrothermal Characterization

4.2 Get Measurement of Porous Freestanding Thin Films Composed of Anatase TiO2 Nanofibers

4.3 Measurement of Micrometer-Thick Polymer Films

4.4 Steady-State Electro-Raman Thermal (SERT) Technique

4.5 SERT Measurement of MWCNT Bundles

4.6 Extension of the Steady-State Techniques

References

Chapter 5: Steady-State Optical-Based Thermal Probing and Characterization

5.1 Sub-10-nm Temperature Measurement

5.2 Thermal Probing at /SUB- Resolution for Studying Interface Thermal Transport

5.3 Optical Heating and Thermal Sensing using Raman Spectrometer

5.4 Bilayer Sensor-Based Technique

5.5 Further Consideration for Micro/Nanoscale Thermal Sensing and Characterization

References

Index

For further information visit: the book web page http://www.openmodelica.org, the Modelica Association web page http://www.modelica.org, the authors research page http://www.ida.liu.se/labs/pelab/modelica, or home page http://www.ida.liu.se/~petfr/, or email the author at [email protected]. Certain material from the Modelica Tutorial and the Modelica Language Specification available at http://www.modelica.org has been reproduced in this book with permission from the Modelica Association under the Modelica License 2 Copyright © 1998–2011, Modelica Association, see the license conditions (including the disclaimer of warranty) at http://www.modelica.org/modelica-legal-documents/ModelicaLicense2.html. Licensed by Modelica Association under the Modelica License 2.

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Copyright © 2011 by the Institute of Electrical and Electronics Engineers, Inc.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved.

Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Wang, Xinwei, 1948-

Experimental micro/nanoscale thermal transport / Xinwei Wang.

pages cm

Includes bibliographical references.

ISBN 978-1-118-00744-0 (hardback)

1. Nanostructured materials—Thermal properties. 2. Heat—Transmission. I. Title.

TA418.9.N35W365 2012

620.1'1596—dc23

2011047244

Preface

In the past, various books were published to introduce to micro/ nanoscale thermal transport. These books, together with some excellent journal reviews, cover comprehensive knowledge about micro/nanoscale thermal transport, from its unique feature, physics background, and material structure to theoretical analysis, numerical modeling, and experimental characterization. On the other hand, it is realized that this area is still under fast development, partly owing to the emergence of novel materials. Instead of an extended review to cover various technologies developed by researchers to characterize thermophysical properties and thermal phenomena, this book focuses on the novel technology development, material thermal characterization, and thermal transport study conducted by the author and his laboratory. From the perspective of materials, the thermal characterization study covers materials of films (micro- to nanometers thick); single one-dimensional materials, wire/tube bundles, and highly packed and highly aligned one-dimensional materials; and material interface thermal transport phenomena. In terms of technology development for thermal excitation, pulsed, step, and periodic photon and electric excitations have been employed. To measure the thermal response of the material, its electrical resistance, thermal radiation, acoustic vibration, and photon scattering have been used.

This book is designed to cover the details of the novel technology development, from experimental principle, physical model, and experiment conduction to data analysis, result uncertainty assessment, and result physical interpretation. It will help readers adopt the covered technologies, or design specific technologies to characterize their unique materials, and to realize high accuracy thermophysical properties measurement and thermal transport study. Chapter .1 provides a general introduction to thermal transport at micro/nanoscales, including the micro/nanoscale thermal transport constrained by the material dimension or internal structure feature size, thermal transport constrained by time, and thermal transport constrained by the size of physical process. Numerical techniques are discussed on how to predict the thermal conductivity or thermal transport phenomenon at micro/nanoscales, including the molecular dynamics simulation, lattice Boltzmann method (LBM), and direct energy carrier relaxation tracking. Chapter 2 discusses how to characterize thermal transport using thermal excitation and sensing in the frequency domain. The frequency domain photoacoustic technique, photothermal radiation technique, three-omega technique, and optical heating and electrical thermal sensing technique are discussed in detail. These techniques can be used to measure the thermophysical properties of films/coatings and conductive/nonconductive wires. Chapter 3 covers transient technologies in the time domain, involving photon and electric heating. The thermal response of the sample is tracked by observing its electrical resistance change. For nonconductive samples, a metallic coating (e.g., Au) is deposited on the surface of the sample to function as a heater and thermal sensor. In Chapter 4, the focus is on techniques in which the material is subjected to static heating (electric or photon heating), and its temperature is measured by evaluating its electrical resistance or Raman signal. Various materials are discussed for thermal characterization, including microwires, solid films, and films/bundles composed of nanoscale wires. Chapter 5 deals with steady-state thermal characterization but is more focused on temperature measurement, which has very broad applications in evaluating the thermal characteristics of micro/nanoscale structures. Also, in this chapter, several techniques used/developed by other researchers are introduced, in anticipation to broaden the knowledge in this area. A transient photo-heating and thermal sensing technique is proposed in order to eliminate the effect of electrical contact resistance and wire–base connection in technologies involving thermal excitation and sensing based on electrical resistance.

Chapter 1

Introduction

In the past decades, tremendous advancement in nanoscience and nanotechnology has prompted a wide spectrum of unique applications of nanoscale and nanostructured materials. Examples include nanowires and nanostructured materials as novel thermoelectric modules in waste heat recovery, nanowire-based sensors, and composites embedded with carbon nanotubes (CNTs) to achieve superior mechanical strength and significant thermal conductivity enhancement. The unique structure of nanomaterials (nanoscale and nanostructured) makes their physical properties (e.g., thermal conductivity and mechanical strength) differ significantly from the values of the bulk counterparts.

1.1 Unique Feature of Thermal Transport in Nanoscale and Nanostructured Materials

The thermal transport in solid materials is sustained by the transport (movement and collision) of phonons (dielectric and semiconductive materials) and free electrons (metals). A phonon is a quasi-particle representing the quantization of the modes of lattice vibrations of periodic, elastic crystal structures of solid. This is more like a photon represents the quantization of light, which is an electromagnetic wave. To picture the thermal transport by phonons, the movement of phonons in a solid can be thought like gas molecules/atoms filling a space. Phonons move around in the solid, and the hot phonons collide with the cold ones, and energy exchange between them takes place, leading to thermal transport. The thermal conductivity of a solid can be described in a simplified form:

1.1

1.1.1 Thermal Transport Constrained by Material Size

Heat transfer at micro/nanoscales takes place and deserves special attention and treatment due to the rise of several scenarios: the size of the material is comparable to the MFP of phonons in bulk materials, the existence of extensive nanograins and grain boundaries within the material significantly alters the movement and transport of energy carriers, the way heat conduction happens and its characteristic size is constrained by the physical process of interest, and the heat transfer characteristic length is limited to nanometers by the physical process even if the material itself is at macroscale.

Figure 1.1a shows the phonon scattering by the top and bottom boundary of thin films. The MFP l in Equation .1 for bulk materials usually is induced by phonon–phonon scattering, phonon–electron scattering, or scattering by defects in the material. When the material size is reduced to a very small scale, like the thin film shown in Figure 1.1, boundary scattering of phonons becomes more important in comparison with the phonon–phonon scattering inside the material. This is because boundary scattering roughly is proportional to the size of the surface area, while the inside phonon–phonon scattering is proportional to the material volume. When the film gets very thin, its surface-to-volume ratio: As/V − L−1 becomes very large (L, film thickness). As can be seen, when the film is becoming thinner, the surface boundary scattering becomes more dominant, which will significantly increase the scattering events of phonons, thereby reducing their MFP and thermal conductivity. The discussion provided here is intended to illustrate the general physical picture of energy carriers scattering by surfaces. For detailed derivation of surface scattering behavior and its effect on thermal conductivity, readers are encouraged to read the numerous papers published in this area in the past 20 years. When phonons reach the surface of the film, the scattering can be diffusive or reflective, sort of like the situation when a light beam reaches a surface, mirror reflection and diffusive reflection can take place. As a general rule, when the surface roughness is smaller than the phonon wavelength, the scattering tends to be specular, while diffusive scattering will dominate if the surface roughness is larger than the phonon wavelength.

Figure 1.1 Schematic demonstration of (a) phonon scattering by the top and bottom boundary in thin films and (b) phonon scattering at the grain boundary in nanocrystalline materials.

To illustrate how the thermal conductivity becomes anisotropic in nanoscale materials and how it changes with the material size, Figure 1.2 shows the thermal conductivity of a freestanding argon crystal film (2) and nanocrystalline argon (3) (at 30 K) changing with the film thickness and grain size. Figure 1.2a shows that the thermal conductivities in the x, y, and z directions decrease with the decreasing thickness of the film. It reflects the fact that boundary scattering at the top and bottom surfaces introduces diffuse phonon scattering in the three directions. The thermal conductivity in the z direction is more affected by the thickness, which is caused by the smaller size and the free boundary condition applied in this direction. When the thickness is comparable to the MFP of phonons ( ∼ 1.5 nm as calculated later) in argon at 30 K, the thermal conductivity in the z direction varies with thickness significantly due to the strong boundary scattering. After the thickness becomes large, the ratio of boundary scattering events to the internal phonon scattering becomes much smaller. As a result, the thermal conductivity tends to be constant and becomes close to the values in the x and y directions. The thermal conductivity at large film thicknesses is around 0.55 W/m K, which is close to the measured thermal conductivity of argon crystal at 30 K, 0.78 W/m K.

Figure 1.2 (a) Variation of the thermal conductivity of an argon film (at 30 K) with its thickness in the thickness (z) and in-plane (x and y) directions (2) and (b) variation of the thermal conductivity of nanocrystalline argon against the grain size inside. Also shown in (b) is the thermal conductivity of single argon nanoparticles (3).

Figure 1.2b clearly indicates that with the increasing nanograin size, the thermal conductivity of the nanocrystalline material goes up. Compared with our previous result for freestanding nanoparticles consisting of single crystals (also shown in Figure 1.2b) (2), it is found that the thermal conductivity of nanocrystalline materials is a little larger than that of nanoparticles with the same characteristic size. The reason is that the nanograins in the nanocrystalline materials under study are not exactly spheres, but close to cubes, which could have less constraint on the movement of energy carriers than spheres. This will lead to a little longer MFP of phonons in nanocrystalline materials than that in single nanoparticles. Another reason is that for freestanding nanoparticles, the scattering at the boundary is total reflection, whereas for nanograins in nanocrystalline materials, some phonons can penetrate the boundary. As a result, there will be less reduction in thermal transport by boundary scattering in nanocrystalline materials. The thermal conductivity reduction in nanocrystalline argon observed in Figure 1.2b is not induced only by phonon scattering at grain boundaries. In comparison with the bulk counterpart, the nanocrystalline material has a density reduction due to the local disorder at grain boundaries. The density reduction becomes larger for nanocrystalline materials composed of smaller grains. Part of the thermal conductivity reduction observed in Figure 1.2b is caused by the low density of the nanocrystalline material. To rule out the effect of the density on the thermal conductivity reduction, the Maxwell method can be applied to calculate the effective thermal conductivity of nanocrystalline argon assumed full density of the single-crystal counterpart. The equation in the Maxwell method is written as follows (3):

1.2

The result (effective thermal conductivity) is shown in Figure 1.2b as well. It is clear that after taking out the density effect, the effective thermal conductivity of nanocrystalline argon becomes slightly greater but still much less than that of the bulk counterpart. This indicates that the phonon scattering at grain boundaries is the most important factor in the thermal conductivity reduction observed in nanocrystalline material. The presence of phonon scattering at grain boundaries will give rise to a boundary thermal resistance, namely, Kapitza resistance. Assuming the nanograin itself has the same thermal conductivity as the bulk counterpart, the effective thermal conductivity of nanocrystalline materials is related to the Kapitza resistance as follows (3):

1.3

where R is the Kapitza resistance, d is the characteristic nanograin size, and k0 is the thermal conductivity of bulk argon. In this work, k0 takes the value of 0.55 W/m K based on our molecular dynamics (MD) work on thermal transport in nanoscale argon at 30 K (2). It needs to be pointed out that Equation .13 is derived from the Fourier law. When the nanograin size is extremely small, the uncertainty induced by the heat transfer deviation from the Fourier law could be significant. The Kapitza resistance discussed here includes the effect of the non-Fourier thermal transport in nanograins. The variation of the Kapitza resistance versus the grain size is calculated based on the effective thermal conductivity (shown in Figure 1.2b) and is plotted out in Figure 1.3. The result shows that the Kapitza resistance is not constant over the grain sizes studied in this work. It is in the order of 10−9 m2 K/W. For smaller grain sizes, the calculated Kapitza resistance is smaller. On the other hand, the significantly increased boundary interface area in the material overshadows this reduction in the Kapitza resistance, making the overall thermal conductivity smaller.

Figure 1.3 (a) Estimated Kapitza resistance at the nanograin interface for the nanocrystalline material studied in Figure 1.2b (3) and (b) the structure configuration for the nanocrystalline material studied in Reference (3). The nanograin size is about 6.25 nm, and the picture shown here is only for a layer of 0.38 nm thickness in order to clearly show the atom configuration at the grain boundary.

1.1.2 Thermal Transport Constrained by Time

Another kind of micro/nanoscale thermal transport that differs significantly from the classical one is induced by ultrafast thermal excitation, like that in picosecond (10−12 s) and femtosecond (10−15 s) laser–material interaction. In the past, ultrafast laser–material processing has been studied extensively and intensively due to the great advantage of ultrafast lasers in material processing, such as cutting, drilling, welding, sintering, forming, and cleaning. In ultrafast laser–material interaction, the laser heating time is very short, even comparable or shorter than the relaxation time of energy carriers in the material. Under such situations, the Fourier law of heat conduction becomes insufficient and questionable to describe the related heat transfer, especially for the very early stage (including heating and sometime after laser heating) thermal transport. This is because the ultrafast laser heating can quickly bring the material temperature to an elevated level and establish a temperature gradient. On the other hand, heat transfer does not immediately arise in response to the temperature gradient because in order for heat to be conducted, the energy carriers need time to collide with their neighbors, and this time is the energy carrier's relaxation time mentioned above. To account for this special heat transfer, the non-Fourier law of heat conduction has been applied extensively, which usually takes the following form

or in a more straightforward way

A direct consequence of this non-Fourier law of heat conduction is that it eliminates the ambiguity of infinite heat transfer speed encountered within the limit of Fourier's law of heat conduction. In the past, extensive research (mostly theoretical and modeling) has been reported on thermal transport in ultrafast laser heating considering the effect of non-Fourier heat conduction. A thermal wave is usually found inside the material and it decays fast during its propagation. Figure 1.4 shows the temperature distribution predicted by solving the Boltzmann transportation equation (BTE) for phonons in silicon on ultrafast laser heating (4). Also shown in Figure 1.4 is the temperature prediction by the non-Fourier model (hyperbolic heat conduction equation, HHCE) and classical heat transfer model (parabolic heat conduction equation, PHCE). It is obvious that a temperature wave is observed in the prediction using both BTE and HHCE.

1.1.3 Thermal Transport Constrained by the Size of Physical Process

As for the third scenario of micro/nanoscale thermal transport, the way heat conduction happens and its characteristic size is constrained by the physical process of interest. One typical example is the heat conduction in the substrate during surface nanostructuring using laser-assisted scanning probe microprobe (SPM) as shown in Figure 1.5. The SPM tip scans over the substrate surface with a distance of several angstroms (10−10 m) to a few nanometers. A pulsed laser beam is manipulated to be nearly parallel to the sample surface to irradiate the SPM tip. Taking a metallic SPM tip as an example, when the laser irradiates the tip, the tip will act like a receiving antenna to collect the laser (an electromagnetic wave). Such laser beam collection will induce an eddy current in the tip (oscillation of electrons). Then the tip will act like an emitting antenna (just like the emitting tower of a radio station) to emit an extremely focused light as illustrated in Figure 1.5. This near-field focused light exists in a very small region ( ≤ 10 nm), while it is extremely enhanced. Figure 1.6 shows the simulation result about the enhanced optical field when a laser (532 nm wave length) shines on a tungsten tip scanning over a silicon substrate. In the model, the distance between the tip apex and the substrate is 5 nm, the tip apex radius is 30 nm, the laser polarization direction follows the axis direction of the tip, and the incident angle of the laser is 10° with respect to the horizontal direction. It is clear that in a small region less than 10 nm below the SPM tip, the electrical field is enhanced significantly ( > 15 times). The local optical field intensity will be more than 200 times stronger than the original incident laser beam. This extremely focused optical field can heat up the substrate, leading to phase change and surface nanostructuring. Since the size of the heating region by the near-field focused optical field is usually less than the MFP of energy carriers in the substrate, the continuum approach becomes questionable for predicting the local thermal transport, phase change, stress, and structure evolution. Special treatment considering the noncontinuous effect at nanoscales must be taken into account when studying the underlying physical processes.

Figure 1.5 Illustration to demonstrate how the near-field focused optical field is formed when a laser is irradiating an SPM tip scanning over a substrate.

Figure 1.6 The electrical field distribution (a) outside the tip and (b) inside the tip when a 532-nm wavelength laser irradiates a tungsten tip. In (a), it is clear that an extremely enhanced optical field arises below the SPM tip in a very small region ( < 10 nm).

The scenarios discussed above for micro/nanoscale thermal transport that need special treatment are not complete to cover all situations but rather to provide typical senses on why micro/nanoscale thermal transport has come to the researchers' attention and what are their potential applications.

1.2 Molecular Dynamics Simulation of Thermal Transport at Micro/Nanoscales

When the characteristic size of thermal transport comes down to micro/nanoscales (comparable to the MFP of energy carriers), the classical equations describing thermal transport cannot be applied directly for a few reasons. First, the thermal transport phenomenon is becoming noncontinuous at micro/nanoscales and it challenges the classical approaches based on the continuum assumptions. Second, the strong size effect by the nanoscale material or nanoscale inside structures significantly reduces the thermal conductivity, making the bulk counterpart's values incapable of being used anymore to describe the thermal transport at micro/nanoscales. Third, at micro/nanoscales, the thermal conductivity is not an intrinsic property of the material anymore. It depends on how it is defined and the physical process it gets involved. To address various challenges and issues in micro/nanoscale thermal transport, MD simulation has arisen as a powerful approach to provide insight into the fundamental physics in micro/nanoscale thermal transport and predict the material thermal behavior at micro/nanoscales under various optical and mechanical excitations. This section focuses on MD simulations to calculate the thermal conductivity of materials and study the nanoscale transport and phase change in laser-assisted SPM-based surface nanostructuring. It is intended to give readers the first (not complete) impressions on how MD simulation is used in studying thermal transport.

1.2.1 Equilibrium MD Prediction of Thermal Conductivity

The basis of MD simulation is to solve the Newtonian equation to obtain the position, force, and velocity of each atom in the system. Each atom has the following movement equation:

1.4

where mi and ri are the mass and position, respectively, of atom i and N is the total number of atoms in the system. Fij is the interaction force between atoms i and j. For many materials of face-centered crystal (FCC) structure, the Lennard–Jones (LJ) 12-6 potential is a good choice to describe the atomic interaction (2):

1.5

1.6

One of the widely used algorithms in MD simulation for integration is the Verlet algorithm, in which the half-step leap-frog scheme is usually used. This algorithm can be expressed as (2)

1.7a

1.7b

1.7c

1.7d

where vi is the velocity of atom i and Δt is the time step, which should be chosen to be much smaller than the phonon relaxation time. Another popular algorithm is the Gear predictor-corrector algorithm. More details about MD simulation can be found in an excellent book by Allen and Tildesley (7). At the beginning of MD simulation, a material with desired geometry is first constructed with specified initial velocities and boundary conditions. Then the system is run to reach an equilibrium state of desired temperature. During this equilibrium calculation, the velocities of atoms are adjusted each time step targeting the desired system temperature. It is always a good idea to run the system without velocities adjustment after the equilibrium calculation is carried out. This is intended to relax the disturbance introduced to the system by velocity adjustment during temperature scaling. After the system reaches equilibrium, the velocity distribution of atoms should follow the Maxwell distribution. Also, the system should be able to maintain at the desired temperature for a long time run after the equilibrium calculation.

When the system becomes small, say submicro- or nanoscales, boundary scattering of the energy carriers become very important in comparison with the internal scattering, and the material's thermal conductivity is expected to reduce when compared with the bulk counterpart's value. Two methods are widely used to predict the thermal conductivity of materials using MD simulations: one uses equilibrium calculation, the Green–Kubo method, and the other one is the nonequilibrium method, which applies a heat flux to the material and studies the steady-state temperature gradient inside. The Green–Kubo expression of the thermal conductivity in direction m, which is related to the long-time autocorrelation function, is given as (7)

1.8

where V is the volume of the system under study. For a two-body potential system, the expression of heat flux qm is described as follows (2):

1.9

In fact, the heat flux in Equation .8 is derived from a general form. For more complicated potentials (three-body or multibody potentials), a generalized form is used to derive the heat flux as , where Ei is the energy (kinetic and potential) of atom i. For the three-body potential of silicon, Volz and Chen (8) have carried out the work in deriving the expression of qm and calculating the thermal conductivity.

Figure 1.7a shows how the heat flux autocorrelation function relaxes with time for an argon crystal at 30 K. The autocorrelation function becomes close to zero when the time reaches 3 ps for the z direction and 6 ps for the x and y directions. This indicates that phonon transport in the z direction experiences stronger scattering and relaxation than in the x and y directions. The first part of the curves looks exponential, while there are some vibrations in the long-time part that takes a long time to eliminate. It needs to be pointed out that the autocorrelation function 〈q(0)q(t) 〉 shown in Figure 1.7a is averaged over a large time span of 2 ns, which is intended to smooth out the oscillation in the curve. After this long time calculation, the oscillation of the curve is weak and has negligible effect on the integration of 〈q(0)q(t) 〉 over time. Figure 1.7b shows how the thermal conductivity value changes with the integration time. It is evident that a long time calculation and integration is needed to obtain a relatively stable thermal conductivity. Such convergence of the thermal conductivity against the integration time is needed in the Green–Kubo method to ensure that the reported thermal conductivity represents the real one of the system. In thermal conductivity calculation using the Green–Kubo method, the long tail of the heat flux autocorrelation function sometimes has large oscillations with time. This imposes considerable uncertainty in the integral time cutoff and thermal conductivity calculation. Also, it could take a very long time calculation to suppress the oscillation and make it reach a value close to zero. These characteristics of the Green–Kubo technique should be kept in mind during MD simulation in order to obtain physically reasonable thermal conductivity. From the autocorrelation function relaxation shown in Figure 1.7a, the average relaxation time of phonons can be directly calculated as follows:

1.10

Figure 1.7 (a) Variation of the autocorrelation function for heat flux in the three directions (z for thickness direction and x and y for in-plane directions) for an argon film at 30 K. The film thickness is 4.28 nm. (b) Variation of the integral of the autocorrelation function against the computational time for the film (2).

Equation .10 simply represents the integration of the heat flux autocorrelation function over time. The average relaxation time of phonons can also be obtained using Equation .1. The specific heat of the material can be obtained using MD simulation through the following formula (2):

1.11

1.12

where

1.13

1.2.2 Nonequilibrium MD Study of Thermal Transport

Figure 1.8 Schematic to show how the computational domain is designed in NEMD simulation to calculate the thermal conductivity.

Figure 1.9 Configuration of the sample (a) Ge with embedded Si nanowires and (b) how the computational domain is designed to calculate its thermal conductivity (11). In this domain design, one heating region and one cooling region are used instead of the two cooling regions shown in Figure 1.8. The periodic boundary conditions used in the simulation make it similar to that in Figure 1.8.

Figure 1.10 shows the temperature distribution inside the material and the temperature jump across the Si/Ge interface.

It is observed from Figure 1.10 that the temperature distribution with Si and Ge is quite linear, and a temperature drop arises at their interface. Such a temperature drop is a direct consequence of the interface thermal resistance. The result shown in Figure 1.10 is based on NEMD simulations with 0.5-fs time step. Before heat flux is applied, the system is run for 200,000 steps to build an equilibrium state. After the temperature gradient is established, another 300,000-step calculation is conducted for temperature average and further system stabilizing.

In MD simulations, the temperature can be easily calculated from the time average kinetic energy of atoms in the sample section within the simulation time using the energy equipartition theorem:

1.14

where 〈 〉 denotes averaging over the total simulation time and kB the Boltzmann constant. Equation .14 is only valid at temperatures much higher than the Debye temperature. When the system temperature is lower than the Debye temperature, the quantum definition of the temperature in the Debye model can be used, which could be calculated from the following equation:

1.15

1.2.3 MD Study of Thermal Transport Constrained by Time

In addition to studying the thermal conductivity of micro/nanoscale materials, MD simulation has been used extensively for studying the dynamic thermal transport and phase change at micro/nanoscales in ultrafast laser–material interaction and nanomanufacturing using laser-assisted techniques. This is a very broad area, and the discussion provided below is only to give a very brief introduction about how the laser–material interaction is modeled.

In laser–material interaction simulation, periodical boundary conditions are usually used on the surfaces in the x and y directions and free boundary conditions on the surfaces in the z direction. This is usually intended to study the fundamental mechanisms in laser–material interaction. The target itself has free spaces above and below it, allowing atom movement in this direction on laser irradiation. The laser beam energy is absorbed exponentially in the target and expressed as

1.16

1.17

where δz is the thickness of layer 1 in the z direction. This laser beam absorption is achieved by adjusting the velocity of atoms in layer 1 with a factor χ:

1.18a

1.18b

Figure 1.11 Schematic to show a general computational domain in MD simulation of laser–material interaction at micro/nanoscales.

Figure 1.12 shows typical pictures of the material behavior in laser–material interaction. At 60 ps, it is evident that the thermal expansion of the material is replaced by intense phase explosion indicated by the strong nonuniform atomic distribution in space. Plots at 60 through 200 ps are characterized by large clusters separating from each other to form individual particles. As a direct consequence of the surface tension force, the initial irregular nanoparticles gradually change to spheres, and this process is demonstrated in plots of 200 through 400 ps. In MD simulation, numerous physical phenomena can be revealed in detail, such as phase explosion, mechanical wave propagation and dissipation in the target (12), and extremely fast ablated plume movement. In the past, significant work has been done by researchers on MD simulation of material behavior under intensive laser irradiation. Most work is for a target placed in a vacuum, while in practice, an ambient gas is present during laser–material interaction, which usually leads a very strong shock wave into the ambient gas. Such a shock wave will strongly change the plume behavior and the material structure evolution. Recent work by Dr. Xinwei Wang (13–18) and his group has pioneered research on studying shock waves in laser–material interaction using MD simulations and revealed unprecedented detail about the internal shock wave structure, such as mass, velocity, temperature, and pressure distributions within a nanoscale domain constrained by the shock wave.

Figure 1.12 Snapshots of atomic configurations of an argon solid (at 50 K initially) irradiated by a picosecond laser pulse. The pulse energy is 1.2 J/m2 and pulse width is 5 ps full-width at half maximum (FWHM) centered at 10 ps. The laser optical absorption depth takes 2.5 nm (12).

Figure 1.13 shows the shockwave development and propagation in laser–material interaction based on MD simulations. At 0.5 ns, a denser region in red color is already visible, which represents the expansion front of the shock wave (marked with arrows). The applied laser energy forces the target material to evaporate because its energy intensity exceeds the material ablation threshold, leading to a strong shock wave composed of compressed adjacent gas above the target. In the initial stages, the ejected plume immediately exerts forward, being induced by the high pressure mainly from intense phase explosion and expands into the background gas until the end of laser pulse. When the high energy plume propagates through the background gas, the interrelation between solid and gas becomes more significant. More mass of the ambient gas is being entrained in the shock wave front. Meanwhile, the ejected plume is being restrained due to increasing repulsive effect from the ambient gas. This restraint prevents the plume from developing freely in space. Consequently, thermalization of the plume occurs because slowing of the plume velocity converts its kinetic energy into thermal energy. With the time evolving, the coexisting length between the plume and the background gas increases because of the relative movement between the plume and the ambient gas. A very interesting phenomenon observed in Figure 1.13a is that starting from 3 ns, the expansion of the plume in space is significantly slowed down. Moreover, some of the particles/clusters in the plume start to move down toward the target surface, although the shock wave front continues to propagate out. From 3 to 5 ns, it is also observed that some clusters/particles stop moving out. Instead, they float and mix with the ambient gas.

Figure 1.13 A comparison of snapshots for the dynamics of shock wave formation and evolution for picosecond laser pulse energy of 3 J/m2. The ambient gas pressure is 0.22 MPa, and three different laser absorption depths are used: 5, 10, and 15 nm. Dark dots: target material atoms; gray dots: ambient gas atoms. The target is argon initially at 50 K, and the ambient gas has the same atomic mass of argon, but only with repulsive force between atoms (19).

Figure 1.14 shows the internal structure of the shock wave, including the atomic configuration, and velocity distribution within a nanometer-thick domain. The velocity (green) within the solid part represents the stress wave propagation in it. This stress wave is formed by the recoil pressure applied by the material ablation. It is surprising to see that at the plume–ambient gas interface, the velocities of the target and ambient gas atoms are not continuous. This reflects the momentum exchange at the interface, which is driven by the velocity difference between them. Also, inside the shock wave (the compressed ambient gas), the atom velocity is not continuous. The atoms at the front of the shock wave have the highest velocity and those inside move much slower. It is evident that MD simulation proves to be an extremely powerful tool in studying the micro/nanoscale thermal phenomena in laser–material interaction and could provide unprecedented details of the involved physics.

1.3 Boltzmann Transportation Equation for Thermal Transport Study

Besides MD simulations, the BTE has been used widely to study the energy carriers' (mostly phonons) transport behavior and the constraint of the material size/structure on thermal transport. In general, the BTE is written as (4)

1.19

where f is the distribution of phonons as a function of position , time t, frequency ω, and velocity represents the streaming of phonons with their own velocities; and is the change in the phonon distribution under the effect of external force. The right-hand side of Equation .19 deals with all scattering factors, including phonon–phonon, phonon–defect, and phonon–electron collisions. It is important to note that the BTE is a pure transport equation, and besides phonons' transport, it has been used very widely in fluid mechanics. For fluid movement, the scattering term on the right-hand side of Equation .19 basically is for the viscous force in the Navier–Stokes equation. One broad way to solve Equation .19 is the lattice Boltzmann method (LBM), which proves to be a very successful method for studying fluid behavior. However, to predict the phonon transport behavior using the BTE, many properties about the phonons are needed, such as their dispersion relation and scattering behavior (mirror or diffuse scattering) at the boundary/grain interface. Therefore, accuracy of the results by solving the BTE is somehow limited by the knowledge about the material structure and phonon behavior. On the other hand, the BTE still works as a good method for parametric studies of the size and structure effect on phonon behavior and thermal transport at micro/nanoscales. The example discussed below is a simplified LBM to solve the BTE and study the thickness effect on thermal conductivity in the thickness direction of thin films.

In nonmetallic solid materials, energy transport is dominated by the propagation of lattice vibrations, which are quantized as phonons. The amount of energy carried by a phonon with a frequency ω is ω, where is Plank's constant divided by 2π. Within solids, phonons travel at the speed of sound and undergo scattering when encountering other phonons, imperfections, impurities, or boundaries. These scattering events redistribute the energy among phonons and tend to restore the local thermal equilibrium. It is these processes that result in thermal resistance in heat conduction. In phonon–phonon collisions, normal processes do not contribute to thermal resistivity because they cannot alter the phonon momentum. Instead, only umklapp processes between phonons are responsible for thermal resistance.

To linearize Equation .19, the scattering term on the right-hand side is often expressed with the single-relaxation-time approximation:

1.20

1.21

where D(ω) is the density of states and the subscript p stands for a particular polarization type. Incorporating Equation .21 into 1.20 and taking the summation over polarizations, a simplified BTE with respect to total energy distribution is obtained as

1.22

where τ is regarded as an averaged thermal relaxation time independent of frequencies and polarizations. τ can be calculated as l/v, where l can be obtained from Equation .1, which encompasses all the internal scattering effects, provided the material is homogeneous. Although Equation .22 is an approximation that describes the average behavior of phonon transport, it suffices to capture the essential physics of interest.

In the LBM, an artificial lattice is attached to the space and the BTE is discretized by laying all particles at grid intersections and allowing them to move only along the linkages of nodes. This finite set of movements can well represent the real particle motion if the lattice pattern is properly chosen according to the nature of the transport process in question. In the example discussed here, the ballistic aspect of phonon transport is implemented by boundary conditions. Therefore, the internal processes can still be viewed as diffusive phenomena and characterized by an orthogonal lattice. Here, we introduce the discrete form of velocity, , and express the energy distribution as , where i denotes a set of directions along the grid sides. It follows that

1.23

1.24

Figure 1.15 Schematic of a three-dimensional lattice point arrangement in the LBM. The “lattice” here is different from that in crystal structure. It is only meant to define virtual points in space to conduct the calculation in the LBM.

Because there is only one single speed, the expected equilibrium distribution in each direction is equal and given as follows:

1.25

The net heat flux along each coordinate can be readily obtained as

1.26

From a conventional point of view, temperature is related to energy density by

1.27

In the study of thermal transport, the change in temperature is more meaningful than its absolute value; therefore, the differential form of Equation .27 is used in this discussion:

1.28

where ΔT is assumed to be small enough to keep ρcp constant.

Consider the heat transfer across a film shown in Figure 1.16. The top and bottom boundaries are fixed at temperatures TH and TL, respectively, such that the phonons at the two surfaces are always in thermal equilibrium and their energy densities are constant, termed uH and uL. Periodic boundary conditions are imposed on the boundaries in the x and y directions. Provided that the lattice size is L03, Equation .23 is hereby nondimensionalized as

1.29

where , and . Notice that in this case is a unit vector along the Cartesian coordinates, the final discretized equation in the Lagragian form can be achieved by selecting both the time and space step as unity:

1.30

where is the normalized node position. According to the relation described by Equation .28, energy density and temperature are equivalent in the dimensionless form

1.31

Figure 1.16 The film structure simulated using the LBM in the discussion.

Therefore, the lower and upper boundaries can be specified as

1.32

and the energy distribution in each direction is identical and equal to their equilibrium values. Additionally, Equation .31 is used to compute the temperature distribution after the LBM simulation is accomplished. On the basis of Equation .26, the dimensionless heat flux at a certain point is defined as follows:

1.33

The computational domain in Figure 1.16 is partitioned with the square lattice shown in Figure 1.15. To initialize the simulation, the distribution of the whole domain is set in equilibrium. The energy densities at each time step evolve through two sequential steps. The first step is collision, which occurs when phonons collide with each other and possibly change their distribution according to the scattering rule designated by (e0 − e)/τ. The second step is streaming, where phonons move to the nearest node in the direction of their velocity. This process represents the diffusion effect in Equation .23. After all sites have been updated by the above procedure, the expected equilibrium distribution at the new time step is determined from Equation .25. This evolution is repeated stepwise until a specified criterion is achieved. It is evident that the LBM can be used for solving both transient and steady-state problems. As mentioned earlier, phonons will approach thermal equilibrium immediately after reaching the upper or lower boundaries because the local temperatures are fixed. Likewise, when phonons originate from these two surfaces, they carry the same amount of energy as in the boundaries. From a physical viewpoint, this condition is similar to blackbody absorption and emission of radiation. In other words, the transmissivity of phonons is unity at the interfaces; hence, the thermal boundary resistance is inherently excluded in our model. On the other hand, attributed to the periodic boundary conditions in the x and y directions, phonons traveling out of one border will reenter the region from the opposite side. This ensures that the phonon transport in the x and y directions of the film is unconfined and has no directional preference. As a result, the net heat fluxes in the x and y